Ligand-binding protein assay for estimation of free hormone concentrations

Abstract

Disclosed are methods of deriving the amount of a free ligand or lipophilic hormone in a sample based on a series of measurements that allow for the amount of free ligand or lipophilic hormone to be derived.

Description

Attorney Docket No.37759.0576P1 LIGAND-BINDING PROTEIN ASSAY FOR ESTIMATION OF FREE HORMONE CONCENTRATIONS CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Provisional Application No.63 / 568,016, filed March 21, 2024. BACKGROUND

[0002] Lipophilic hormones, such as steroid, vitamin D metabolites, and thyroid hormones, listed in Table 1 below, typically circulate in plasma in association with high- affinity serum transport or binding proteins (BPs). Clinically important examples of ligand- BP interactions that govern the transport of hormones (ligands) in plasma include (i) corticosteroids (and progesterone) to corticosteroid binding globulin (CBG), (ii) testosterone and other sex steroids to sex hormone binding globulin (SHBG), (iii) vitamin D metabolites to vitamin D binding protein (DBP), and (iv) thyroid hormones to thyroid binding globulin (TBG). These binding interactions are characterized by specificity and high affinity. The high affinity of BPs for their cognate ligand(s) is illustrated in Table 1 by the high equilibrium association (affinity) constants (Ka), which are expressed in units of 1 / concentration(M-1). The magnitude of these affinities (for free ligand) may be more intuitively appreciated by the equilibrium dissociation constant (Kd), which are expressed in units of ligand concentration and are in the nmol / L range. Not shown in Table 1 is the concentration of cognate binding proteins. Except for DBP, which is present in concentrations far greater than those of vitamin D metabolite ligands, CBG, SHBG, and TBG are present in limiting concentrations. As such, these ligand-BP interactions are characterized by saturable binding kinetics and under certain circumstances, concentrations of free ligand(s) may exceed the concentration of cognate BP. Even though several different ligands can bind a given BP with high affinity, these ligand-BP interactions are highly specific across ligand classes. For example, thyroid hormone binds CBG, SHBG, and DBP with negligible affinity. Table 1 Hormone BP Ka % Albumin Ka % TTR Ka % % Bound HAS Bound TTR Bound Free -Attorney Docket No.37759.0576P1 T4 TBG 1 x 70 HSA 1.5 x 5 TTR 2 x 20 0.03 1010106108T3 TBG 1 x 75 HSA 2 x 20 TTR 1 x 5 0.3 um

[0003] In addition to specific, high-affinity binding to their cognate serum transport protein(s), lipophilic hormones also bind in relatively non-specific fashion and with far lower affinities to albumin. As shown in Table 1, the affinity constant for ligand-albumin bindings is orders of magnitude less than specific BPs. However, because the concentration of albumin is so high in plasma, the albumin-bound fraction of ligand is substantial, typically several- fold (in the case of cortisol) and up to 20-fold (in the case of testosterone) higher than the free ligand concentration.

[0004] According to the free hormone hypothesis, only the free hormone, but not protein- bound hormone, is able to diffuse across the cell membrane. Thus, clinicians are primarily interested in the free rather than protein-bound fraction of these lipophilic hormones. Using the example of cortisol, it is not at present clinically feasible to measure free cortisol concentrations at the intracellular site of biological action. Interstitial cortisol concentrations have been experimentally measured, but only in a highly specialized research setting. Therefore, measurements of concentrations of free cortisol (in plasma or serum) represent the next best approximation of intracellular cortisol concentrations. Moreover, sampling of plasma or serum by venipuncture is safe, convenient, and well-validated. However, serum concentrations of free cortisol are rarely obtained experimentally. Instead, clinicians and to a large extent investigators rely on measurements of total cortisol concentrations. These provide a useful approximation of free cortisol concentration, but one that is subject to perturbations due to extraneous factors. These extraneous factors include variation in concentrations and affinities of cortisol binding globulin (CBG) and albumin, temperature, as well as competition by other compounds that also bind CBG or albumin with significant affinity.

[0005] In summary, there is no scientific rationale for preferring measurements of totalAttorney Docket No.37759.0576P1 rather than free cortisol concentrations in clinical practice. Rather, clinicians make do with measurements of total rather than free cortisol due to ease, tradition, and feasibility. By contrast, measurement of free cortisol by equilibrium dialysis or ultracentrifugation is cumbersome, expensive, and not well suited to rapid turnaround in a clinical laboratory setting. Thus, measurements of free cortisol concentration require send-out to specialty laboratories, with added expense and delay. In the case of cortisol, the delay in obtaining free cortisol concentrations in clinical samples obviates the use of such data in the acute care, such as assessment of adrenocortical function in critically ill patients. There is therefore a need in the art for improved methods for measuring free hormone concentrations, such as cortisol concentrations, in a clinically meaningful and useful manner. SUMMARY

[0006] The disclosed methods allow for the determination of the amount of free ligand or lipophilic hormone in a sample based on a series of measurements that provide data allowing for the derivation of the free amount of ligand or lipophilic hormone in the sample. The method is suitable for a variety applications, including diagnostic methods for diagnosing a disease or disorder, as well as detection methods for detecting a variety of ligands, lipophilic hormones, and binding partners for the ligands and lipophilic hormones.

[0007] In some aspects, the methods disclosed herein can permit calculation of total binding protein concentration as an alternative to direct measurement of binding protein, In some aspects, the methods disclosed herein can be suitable for a variety of applications, including (i) diagnostic methods for diagnosing a disease or disorder, (ii) monitoring treatment effects of hormone replacement, (iii) determination of free and total concentrations of specific binding protein, (iv estimation of the concentration of free albumin, (v) derivation of pharmacokinetic parameters to predict dynamic changes in total and free hormone concentrations in vivo, and (vi) development of pharmacodynamic measures of time-varying tissue exposure to free hormone concentrations in vivo.

[0008] In one aspect, the method involves determining the concentration of free cortisol in a sample. This method comprises obtaining or having a sample obtained, the sample comprising total cortisol, which is the sum of the concentration free cortisol and cortisol bound to corticosteroid-binding globulin (CBG) and albumin; dividing the sample into multiple sample portions;Attorney Docket No.37759.0576P1 performing at least one assay on a first sample portion to determine the concentration in the first sample portion of (i) total CBG, (ii) total albumin, and (iii) total cortisol; contacting a second sample portion with an effective amount of a labeled ligand, the labeled ligand having an equilibrium association constant of at least 20 M-1for CBG and albumin; and determining the concentration in the second sample portion of labeled ligand bound to CBG; performing a series of steps on a third sample portion, the series of steps comprising, in sequence: contacting the third sample portion with an effective amount of a first unlabeled ligand, the first unlabeled ligand having (i) an equilibrium association constant for albumin which is higher than the labeled ligand’s equilibrium association constant for albumin, and (ii) an equilibrium association constant for CBG which is less than 0.1 M-1; contacting the third sample portion with an effective amount of the labeled ligand; and determining the concentration in the third sample portion of labeled ligand bound to CBG; performing a series of steps on a fourth sample portion, the series of steps comprising, in sequence: contacting the fourth sample portion with an effective amount of albumin; contacting the fourth sample portion with an effective amount of the labeled ligand; and determining the concentration in the fourth sample portion of labeled ligand bound to CBG; and deriving, based on the concentrations determined in steps (c)-(f), the concentration of free cortisol in the sample.

[0009] In a further aspect, the method is more general, applicable to a variety of free ligands or lipophilic hormones in a sample. This method for example comprises: obtaining or having a sample obtained, the sample comprising total ligand or lipophilic hormone, which is the sum of the concentration of free ligand or lipophilic hormone and ligand or lipophilic hormone bound to a ligand or lipophilic hormone binding partner, albumin, or both; dividing the sample into multiple sample portions; performing at least one assay on a first sample portion to determine the concentration in the first sample portion of (i) total ligand or lipophilic hormone binding partner, (ii) optionally, total albumin, and (iii) ligand or lipophilic hormone; contacting a second sample portion with an effective amount of a labeled ligand or lipophilic hormone, the labeled ligand or lipophilic hormone having an equilibriumAttorney Docket No.37759.0576P1 association constant of at least 20 M-1for the ligand or lipophilic hormone binding partner, albumin, or both; and determining the concentration in the second sample portion of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner; performing a series of steps on a third sample portion, the series of steps comprising, in sequence: contacting the third sample portion with an effective amount of a first unlabeled ligand or lipophilic hormone, the first unlabeled ligand or lipophilic hormone having (i) an equilibrium association constant for albumin which is higher than the labeled ligand or lipophilic hormone’s equilibrium association constant for albumin, and (ii) an equilibrium association constant for the lipophilic hormone binding partner which is less than 0.1 M-1; contacting the third sample portion with an effective amount of the labeled ligand or lipophilic hormone; and determining the concentration in the third sample portion of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner; and deriving, based on the concentrations determined in steps (c)-(e), the concentration of free ligand or lipophilic hormone in the sample.

[0010] Also described are methods of diagnosing a disease or disorder, the method comprising performing one of the disclosed methods, and diagnosing the disease or disorder based on the concentration of free ligand or lipophilic hormone in the sample.

[0011] Additional advantages of the disclosed methods will be set forth in part in the description which follows, and in part will be understood from the description, or may be learned by practice of the disclosed method. The advantages of the disclosed methods will be realized and attained by means of the elements and combinations particularly pointed out in the appended claims. It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the invention as claimed. DETAILED DESCRIPTION

[0012] The disclosed methods may be understood more readily by reference to the following detailed description of particular embodiments and the Example included therein and following description.

[0013] It is to be understood that the disclosed methods are not limited to specific synthetic methods, specific analytical techniques, or to particular reagents unless otherwise specified, and, as such, may vary. It is also to be understood that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to beAttorney Docket No.37759.0576P1 limiting.

[0014] Disclosed are materials, compositions, and components that can be used for, can be used in conjunction with, can be used in preparation for, or are products of the disclosed methods and compositions. These and other materials are disclosed herein, and it is understood that when combinations, subsets, interactions, groups, etc. of these materials are disclosed that while specific reference of each various individual and collective combinations and permutation of these compounds may not be explicitly disclosed, each is specifically contemplated and described herein. Thus, if a class of molecules A, B, and C are disclosed as well as a class of molecules D, E, and F and an example of a combination molecule, A-D is disclosed, then even if each is not individually recited, each is individually and collectively contemplated. Thus, in this example, each of the combinations A-E, A-F, B-D, B-E, B-F, C- D, C-E, and C-F are specifically contemplated and should be considered disclosed from disclosure of A, B, and C; D, E, and F; and the example combination A-D. Likewise, any subset or combination of these is also specifically contemplated and disclosed. Thus, for example, the sub-group of A-E, B-F, and C-E are specifically contemplated and should be considered disclosed from disclosure of A, B, and C; D, E, and F; and the example combination A-D. This concept applies to all aspects of this application including, but not limited to, steps in methods of making and using the disclosed compositions. Thus, if there are a variety of additional steps that can be performed it is understood that each of these additional steps can be performed with any specific embodiment or combination of embodiments of the disclosed methods, and that each such combination is specifically contemplated and should be considered disclosed.

[0015] Headings are provided for convenience only and are not to be construed to limit the invention in any manner. Embodiments illustrated under any heading or in any portion of the disclosure may be combined with embodiments illustrated under the same or any other heading or other portion of the disclosure. A. Definitions

[0016] It is understood that the disclosed methods and compositions are not limited to the particular methodology, protocols, and reagents described as these may vary. It is also to be understood that the terminology used herein is for the purpose of describing particular embodiments only, and is not intended to limit the scope of the present invention which will be limited only by the appended claims.

[0017] It must be noted that as used herein and in the appended claims, the singular formsAttorney Docket No.37759.0576P1 “a,” “an,” and “the” include plural reference unless the context clearly dictates otherwise. Thus, for example, reference to “an assay” includes a plurality of such assays, reference to “the assay” is a reference to one or more assays and equivalents thereof known to those skilled in the art, and so forth.

[0018] The term “subject” refers to the target of administration, e.g., an animal. Thus, the subject of the disclosed methods can be a vertebrate, such as a mammal. For example, the subject can be a human. The term does not denote a particular age or sex. “Subject” can be used interchangeably with “individual” or “patient.” For example, the subject of administration can mean the recipient of the alternating electrical field. For example, the subject of administration can be a subject with cancer, e.g., ovarian cancer or lung cancer.

[0019] “Optional” or “optionally” means that the subsequently described event, circumstance, or material may or may not occur or be present, and that the description includes instances where the event, circumstance, or material occurs or is present and instances where it does not occur or is not present.

[0020] Ranges may be expressed herein as from “about” one particular value, and / or to “about” another particular value. When such a range is expressed, also specifically contemplated and considered disclosed is the range from the one particular value and / or to the other particular value unless the context specifically indicates otherwise. Similarly, when values are expressed as approximations, by use of the antecedent “about,” it will be understood that the particular value forms another, specifically contemplated embodiment that should be considered disclosed unless the context specifically indicates otherwise. It will be further understood that the endpoints of each of the ranges are significant both in relation to the other endpoint, and independently of the other endpoint unless the context specifically indicates otherwise. Finally, it should be understood that all of the individual values and sub- ranges of values contained within an explicitly disclosed range are also specifically contemplated and should be considered disclosed unless the context specifically indicates otherwise. The foregoing applies regardless of whether in particular cases some or all of these embodiments are explicitly disclosed.

[0021] Unless defined otherwise, all technical and scientific terms used herein have the same meanings as commonly understood by one of skill in the art to which the disclosed method and compositions belong. Although any methods and materials similar or equivalent to those described herein can be used in the practice or testing of the present method and compositions, the particularly useful methods, devices, and materials are as described.Attorney Docket No.37759.0576P1 Publications cited herein and the material for which they are cited are hereby specifically incorporated by reference. Nothing herein is to be construed as an admission that the present invention is not entitled to antedate such disclosure by virtue of prior invention. No admission is made that any reference constitutes prior art. The discussion of references states what their authors assert, and applicants reserve the right to challenge the accuracy and pertinence of the cited documents. It will be clearly understood that, although a number of publications are referred to herein, such reference does not constitute an admission that any of these documents forms part of the common general knowledge in the art.

[0022] Throughout the description and claims of this specification, the word “comprise” and variations of the word, such as “comprising” and “comprises,” means “including but not limited to,” and is not intended to exclude, for example, other additives, components, integers or steps. In particular, in methods stated as comprising one or more steps or operations it is specifically contemplated that each step comprises what is listed (unless that step includes a limiting term such as “consisting of”), meaning that each step is not intended to exclude, for example, other additives, components, integers or steps that are not listed in the step. B. Determination of Free Ligand or Binding Protein (BP) Concentrations

[0023] The general purpose of the disclosed procedures is to obtain rapid and accurate estimates of free ligand and / or BP concentrations in a clinical sample. The disclosed procedures and measurements adjunctive to usual measurement of total (protein-bound plus free ligand) and albumin, will facilitate accurate estimation of free, BP-bound, and albumin- bound ligand. These procedures also provide a useful measure of BP concentration in the serum sample. Ligand-BPs of interest include but are not limited to (i) CBG-cortisol, (ii) CBG-progesterone, (iii) SHBG-testosterone, (iv) DBP-25OH vitamin D and DBP-1,25-OH vitamin D (calcitriol). Ligand-BPs of interest include, but are not limited to: (i) CBG-cortisol, (ii) CBG-progesterone, (iii) CBG-aldosterone, (iv) SHBG-testosterone, (v) SHBG-estradiol, (vi) SHBG-dihydrotestosterone (vii) DBP-25OH vitamin D and (viii) DBP-1,25-OH vitamin D (calcitriol).

[0024] Measurement of free hormone concentration (e.g. free cortisol, free testosterone, free 25-OH vitamin D, and the like) provides added value relative to the current practice of measuring total hormone concentration (e.g. total cortisol, total testosterone, total 250H vitamin D). Therefore, the objective of the present analysis is to establish mathematical principles and methodology for rapid and convenient procedures that, in combination with routine clinical measurements of total cortisol (or other ligand / lipophilic hormone) andAttorney Docket No.37759.0576P1 albumin (or other binding partner) in serum, provide a reliable estimation of free cortisol (or other ligand / lipophilic hormone) and total CBG concentrations (or binding partner concentrations).

[0025] The estimation of free ligand / lipophilic hormone concentration should be accurate and available to assist the clinician in real-time clinical management decisions. That is, the goal is to provide clinicians clinical measurements of free as well as total free ligand / lipophilic hormone concentration in the same time frame and using the same clinical laboratory platform in which the latter (total ligand / lipophilic hormone) measurements are typically available. The general methodology may be categorized as a competitive ligand- transport-protein binding assay. As described below, in an aspect, the test involves addition of a labeled ligand, such as horseradish peroxidase (HRP) conjugated cortisol, designated E, to the clinical sample, followed by a separation technique to assay the concentration of binding partner (e.g., CBG)-bound E. By this means, the concentration of free ligand / lipophilic hormone (F) and binding partner in the original sample (before addition of E) may be determined.

[0026] Table 2 shows non-limiting examples of lectin-affinity and antibody-affinity methods for separating transport protein-bound hormone to determine free hormone concentrations, as well as clinical applications. Table 2 Lectin-affinity assay Transport protein Analyte estimated Clinical application(s) al d m yAttorney Docket No.37759.0576P1 SHBG Free testosterone Dx of male hypogonadism, management of testosterone re lacement in ,clinical utility include without limitation adrenocorticosteroids, sex steroids, sex hormones, vitamin D metabolites, thyroid hormones, progesterone, and the like. In the case of thyroid hormones, a convenient and reasonably accurate analog immunometric method has been developed to directly measure free T4 and free T3. However, there is no corresponding rapid, simple, and accurate method for determination of the free hormone concentration for other ligands of interest (cortisol, progesterone, testosterone, estradiol, 25-OH vitamin D, 1,25-OH vitamin D, etc.). Thus, there is a clinical need for methods for determination of the free hormone concentrations of these ligands that can be performed routinely in the clinical laboratory setting.

[0028] Although the Example below concerns the adrenal corticosteroid hormone, cortisol, the mathematical principles and experimental approach are generalizable to other lipophilic hormones, including, but not limited to the examples summarized in Table 2. While one objective is determination of free hormone concentration, the disclosed methodology also provides a meaningful estimation of the concentration of the cognate transport protein concentrations (CBG in the case of competitive cortisol-binding assay). There are a variety of clinical situations in which determination of transport protein concentrations may provide additional, independent clinical information. For example, serum concentrations of CBG are an independent predictor of mortality in septic shock. The disclosed methodology directlyAttorney Docket No.37759.0576P1 measures the concentration of hormone-bound binding globulin. The computational advantages afforded by the equations developed herein represent a much more complete and useful formulation of equilibrium relationships that define the concentration binding globulin and the free hormone concentration compared to the empiric qualities of the historic T3RU assay.

[0029] Both CBG and SHBG are glycoproteins that contain N-linked oligosaccharides. The glycosylation of CBG and SHBG results in high affinity binding of these transport proteins to the plant lectin, concanavalin A (ConA). The disclosed assays for free corticosteroids and free sex steroids (and concentrations of their binding globulins, CBG and SHBG, respectively) both take advantage of this ConA binding activity of CBG and SHBG as a separation technique. In the case of vitamin D metabolites, 25-OH and 1,25-OH vitamin D (see Table 1), DBP is also glycosylated, but the oligosaccharide attachments to the DBP protein do not support binding to ConA. Therefore, an alternative approach to separation of DBP-bound ligand, using antibody-coated rather than ConA-coated microtiter plates, can be used for DBP capture.

[0030] There are several elements of the disclosed methodology that support translation to the clinical laboratory setting. These include without limitation (i) simplicity of assay reagents, such as HRP-labeled hormone without need for proprietary, monoclonal antibodies and (ii) development of the assay in a microtiter plates platform used in most clinical laboratories. In addition, the fact that heat denaturation irreversibly abolishes steroid binding activity of CBG, SHBG, and DBP without affecting corresponding albumin-binding activity provides a simple and cost-effective control for specific transport-protein binding. C. Hormone and Ligand Binding to CBG 1. Cortisol

[0031] The stoichiometry of CBG-cortisol binding is well understood. CBG circulates as a monomer, and one molecule of CBG binds a single molecule of cortisol. Sites for N-linked glycosylation and their respective effects on CBG-cortisol binding affinities have been defined experimentally. As well, the crystal structure for cortisol- and progesterone bound CBG has been studied. In addition, kinetics and temperature sensitivity of CBG-cortisol binding interactions, as well as genetic variants affecting CBG-cortisol binding affinity, have been previously evaluated. At 37°C, the equilibrium dissociation constant (KD) for CBG- cortisol binding has been reported to be in the range of 13-33 nmol / L. CBG is expressed in liver, which is the primary source of circulating CBG. CBG is also expressed in a variety ofAttorney Docket No.37759.0576P1 extra-hepatic sites, including adrenal gland and hypothalamus, where it may act to modulate local corticosteroid signaling. CBG has a half-life of 4-5 days in vivo. CBG concentrations demonstrate diurnal variation, with peak levels observed in afternoon. However, these variations in CBG concentration appear to have minimal influence on total cortisol concentrations. 2. Other Ligands

[0032] A variety of ligands other than cortisol are also conditionally present in human plasma. Depending on their concentration and affinity, they may represent a significant source of competition for cortisol binding to CBG. For example, these include progesterone, cortisone, 11-deoxycortisol, 21-deoxycortisol, prednisolone, and other compounds. Cortisol and corticosterone both bind CBG with high affinity, with a relative binding affinity of 1,000 and an equilibrium association constant of 76,000,000 / M (which corresponds to equilibrium dissociation constant (KD) of 13.2 nmol / L). Table 3 shows examples of additional other steroids and their affinities for CBG. Table 3 Steroid CBG RBA (x1000) K(106M-1)

[0033] The significant binding affinity of several steroids for CBG are shown in Table 3. For example, both cortisol and corticosterone bind CBG with high affinity, as do other corticosteroids in the adrenocortical steroidogenic pathway, such as deoxycorticosterone andAttorney Docket No.37759.0576P1 deoxycortisol. Progesterone also binds CBG with relatively high affinity. Cortisone is one of the biologically inactive metabolites of cortisol and binds CBG with significant affinity. Whether other metabolites of cortisol also compete for CBG binding is not fully understood. Not shown in the above table is prednisolone, which also binds CBG with relatively high affinity. Depending on their concentration in plasma, all these steroids have the potential to compete with cortisol (as well as labeled CBG-binding ligand (E)) for binding to CBG.

[0034] In addition to intact and fully glycosylated CBG that binds cortisol with high affinity and circulates in healthy human subjects at concentrations in the range of 400-800 nmol / L, the presence of a structurally distinct CBG molecule has been identified that in some studies, may be distinguished by differential recognition of monoclonal antibodies. It has been suggested that the relatively low abundance CBG molecule recognized by 12G2 but not reactive-center loop (RCL)-specific loop (G12V) monoclonal antibodies, represents elastase- cleaved CBG. Based on that inference, CBG moiety may exist as low-affinity (la) CBG, since elastase-cleaved CBG has roughly 10-fold lower cortisol binding affinity than intact, high- affinity (ha) CBG. Dynamic and equilibrium models suggest that cortisol may include both laCBG and haCBG.

[0035] However, elastase-cleaved CBG is not present is the systemic circulation of healthy or critically ill human subjects. While CBG cleaved by neutrophil elastase may contribute to higher free cortisol concentrations as well as higher rates of tissue delivery of cortisol at sites of inflammation, it appears that elastase-cleaved CBG is rapidly desialylated and cleared by the asialo-glycoprotein receptor at sites of inflammation before reentry to the systemic circulation. It has also been demonstrated that differential recognition of CBG in human plasma by 12G2 and G12V monoclonal antibodies is related to differential patterns of glycosylation. Specifically, an N-glycan at N347 of human CBG limits 9G12 antibody recognition of the epitope, while 12G2 antibody reactivity is unaffected. Moreover, qualitative differences in N-glycosylation at N238 negatively affect steroid-binding activity of CBG. Thus, while it remains theoretically possible that there is heterogeneity of cortisol- binding affinities of CBG in human plasma, the percent of ‘low-affinity’ CBG and its KD for cortisol binding remain incompletely characterized at present. In any case, genome wide association studies establish that genetic variation in the SERPINA6 (CBG gene) locus contributes to variability in CBG concentrations and cortisol-binding affinities; this point is further reinforced by characterization of patients with SERPINA6 mutations that interfere with expression and cortisol-binding affinity in patients with CBG deficiency.Attorney Docket No.37759.0576P1 D. Methods 1. Methods of determining the concentration of free cortisol in a sample

[0036] Disclosed herein are methods of determining the concentration of free cortisol in a sample, the method comprising: (a) obtaining or having a sample obtained, the sample comprising total cortisol, which is the sum of the concentration free cortisol and cortisol bound to corticosteroid-binding globulin (CBG) and albumin; (b) dividing the sample into multiple sample portions; (c) performing at least one assay on a first sample portion to determine the concentration in the first sample portion of (i) total CBG, (ii) total albumin, and (iii) total cortisol; (d) contacting a second sample portion with an effective amount of a labeled ligand, the labeled ligand having an equilibrium association constant of at least 20 M-1for CBG and albumin; and determining the concentration in the second sample portion of labeled ligand bound to CBG; (e) performing a series of steps on a third sample portion, the series of steps comprising, in sequence: (1) contacting the third sample portion with an effective amount of a first unlabeled ligand, the first unlabeled ligand having (i) an equilibrium association constant for albumin which is higher than the labeled ligand’s equilibrium association constant for albumin, and (ii) an equilibrium association constant for CBG which is less than 0.1 M-1; (2) contacting the third sample portion with an effective amount of the labeled ligand; and (3) determining the concentration in the third sample portion of labeled ligand bound to CBG; (f) performing a series of steps on a fourth sample portion, the series of steps comprising, in sequence: (1) contacting the fourth sample portion with an effective amount of albumin; (2) contacting the fourth sample portion with an effective amount of the labeled ligand; and (3) determining the concentration in the fourth sample portion of labeled ligand bound to CBG; and (g) deriving, based on the concentrations determined in steps (c)-(f), the concentration of free cortisol in the sample. In some aspects, the labeled cortisol is a known / predetermined concentration of cortisol.

[0037] In some aspects, the labeled ligand is directly or indirectly labeled. In some aspects, the labeled ligand comprises labeled cortisol, corticosterone, cortisone, deoxycorticosterone, deoxycortisol, progesterone, or prednisolone. In some aspects, step (d) further comprises heating the second sample portion after contacting the second sample portion with an effective amount of the labeled ligand and prior to determining the concentration of labeled ligand bound to CBG. Heating can be optional and can be used to determine the amount of CBG bound to labeled cortisol on the solid support and this can be used to measure non-specific binding of cortisol to CBG.Attorney Docket No.37759.0576P1

[0038] In some aspects, the methods can further comprise adding an excess of unlabeled cortisol prior to contacting the second sample portion with a ligand immobilized on a solid support. As such, the method can be used to determine amount of CBG bound to labeled cortisol on the solid support and can be used to measure non-specific binding of cortisol to CBG.

[0039] In some aspects, the methods can further comprise adding a ligand that competes with cortisol and labeled cortisol for binding to CBG. In some aspects, a known, low concentration of a ligand (i.e. progesterone) that competes with cortisol for CBG (lower, but substantial affinity) can be added before immobilizing and the determine amount of CBG bound to labeled cortisol on the solid support and can be used to measure specific binding of cortisol to CBG.

[0040] In some aspects, step (d) further comprises contacting the second sample portion with an effective amount of unlabeled cortisol prior to determining the concentration of labeled ligand bound to CBG. In some aspects, step (d), (e), or (f), determining the concentration of labeled ligand bound to CBG comprises performing a lectin-affinity assay on the second, third, or fourth sample portion using a plant lectin which is immobilized on a solid support. In some aspects step (d), (e), or (f), determining the concentration of labeled ligand bound to CBG comprises performing a lectin-affinity assay on the second, third, or fourth sample portion using a plant lectin which is immobilized on a solid support and the plant lectin is Concanavilin-A. In some aspects, step (d), (e), or (f) further comprises adding an effective amount of a second unlabeled ligand prior to determining the concentration of first labeled ligand bound to CBG, the second unlabeled ligand having a equilibrium association constant for CBG which is lower than cortisol’s equilibrium association constant for CBG. In some aspects, step (d), (e), or (f) further comprises adding an effective amount of a second unlabeled ligand prior to determining the concentration of first labeled ligand bound to CBG, the second unlabeled ligand having a equilibrium association constant for CBG which is lower than cortisol’s equilibrium association constant for CBG and the second unlabeled ligand is cortisone, deoxycorticosterone, progesterone, or any combination thereof. In some aspects, step (d), (e), or (f) further comprises adding an effective amount of a second unlabeled ligand prior to determining the concentration of first labeled ligand bound to CBG, the second unlabeled ligand having a equilibrium association constant for CBG which is lower than cortisol’s equilibrium association constant for CBG and the second unlabeled ligand is cortisone, deoxycorticosterone, progesterone, or any combination thereof and theAttorney Docket No.37759.0576P1 second unlabeled ligand is added at a known concentration. In some aspects, the first unlabeled ligand is thyroxine (T4).

[0041] In some aspects, the methods disclosed herein involve determining the concentration of free cortisol in a sample. In some aspects, the methods disclosed herein comprise obtaining or having a sample obtained, the sample comprising total cortisol, which is the sum of the concentration free cortisol and cortisol bound to corticosteroid-binding globulin (CBG) and albumin; dividing the sample into multiple sample portions; performing at least one assay on a first sample portion to determine the concentration in the first sample portion of (i) total CBG, (ii) total albumin, and (iii) total cortisol; contacting a second sample portion with an effective amount of a labeled ligand, which could be labeled cortisol (F*) or, alternatively, a labeled other than cortisol ligand (E*) that has an equilibrium association constant of at least 1x107M-1for the specific BP (CBG); separation of CBG-bound F* using lectin-affinity matrix; using appropriate standards and controls, determining the equilibrium concentration of CBG-bound F* [F*C] in the sample; performing a series of equilibrium perturbation steps on a third-sixth sample portion. The assay signal (equilibrium concentration of F*C) can be obtained by a lectin-affinity method in the second sample, and in subsequent samples, each perturbation of equilibrium will result in a measurable change in [F*C] relative to the second sample. In this aspect, the four perturbations can be: (i) contacting the third sample portion with an effective amount of (free) CBG, which will increase [F*C] relative to the second sample (+^[F*C]), (ii) contacting the fourth sample with an effective amount of (free), unlabeled ligand (P), having equilibrium association constant of at least 1x107M-1for the specific BP (CBG), that competes for CBG-F* binding. Addition of P in a sample after the second sample (e.g., a fourth sample) will decrease [F*C] (relative to the second sample) in dose-dependent fashion (-^[F*C]), (iii) contacting the fifth test tube with an effective amount of free albumin. Addition of free albumin will decrease [F*C] (relative to the second sample) in dose-dependent fashion (-^[F*C]); and (iv) contacting the sixth sample with unlabeled ligand Q having Ka for CBG ^ 0 and Ka for albumin ^ 1x106M-1albumin. Addition of Q in a sixth sample for instance will increase [F*C] relative to test tube #2 (+^[F*C]) in dose-dependent fashion. At pharmacologic concentrations of Q, the concentration of free albumin is decreased, resulting in lower concentrations of albumin-bound F and albumin-bound F*.Attorney Docket No.37759.0576P1 2. Methods of determining the concentration of free ligand or lipophilic hormone in a sample

[0042] Disclosed herein are methods of determining the concentration of free ligand or lipophilic hormone in a sample, the method comprising: (a) obtaining or having a sample obtained, the sample comprising total ligand or lipophilic hormone, which is the sum of the concentration of free ligand or lipophilic hormone and ligand or lipophilic hormone bound to a ligand or lipophilic hormone binding partner, albumin, or both; (b) dividing the sample into multiple sample portions; (c) performing at least one assay on a first sample portion to determine the concentration in the first sample portion of (i) total ligand or lipophilic hormone binding partner, (ii) optionally, total albumin, and (iii) ligand or lipophilic hormone; (d) contacting a second sample portion with an effective amount of a labeled ligand or lipophilic hormone, the labeled ligand or lipophilic hormone having an equilibrium association constant of at least 20 M-1for the ligand or lipophilic hormone binding partner, albumin, or both; and determining the concentration in the second sample portion of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner; (e) performing a series of steps on a third sample portion, the series of steps comprising, in sequence: (1) contacting the third sample portion with an effective amount of a first unlabeled ligand or lipophilic hormone, the first unlabeled ligand or lipophilic hormone having (i) an equilibrium association constant for albumin which is higher than the labeled ligand or lipophilic hormone’s equilibrium association constant for albumin, and (ii) an equilibrium association constant for the lipophilic hormone binding partner which is less than 0.1 M-1; (2) contacting the third sample portion with an effective amount of the labeled ligand or lipophilic hormone; and (3) determining the concentration in the third sample portion of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner; and (f) deriving, based on the concentrations determined in steps (c)-(e), the concentration of free ligand or lipophilic hormone in the sample. In some aspects, the disclosed methods can further comprise performing a series of steps on a fourth sample portion, the series of steps comprising, in sequence: (a) contacting the fourth sample portion with an effective amount of albumin; (b) contacting the fourth sample portion with an effective amount of the labeled ligand or lipophilic hormone; and (c) determining the concentration in the fourth sample portion of labeled ligand or liphophilic hormone bound to the ligand or lipophilic hormone binding partner. In some aspects, the disclosed methods can further comprise performing a series of steps on a fourth sample portion, the series of stepsAttorney Docket No.37759.0576P1 comprising, in sequence: (a) contacting the fourth sample portion with an effective amount of albumin; (b) contacting the fourth sample portion with an effective amount of the labeled ligand or lipophilic hormone; and (c) determining the concentration in the fourth sample portion of labeled ligand or liphophilic hormone bound to the ligand or lipophilic hormone binding partner, wherein determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner comprises performing a lectin-affinity assay on the fourth sample portion using a plant lectin which is immobilized on a solid support. In some aspects, the plant lectin is Concanavilin-A. In some aspects, the labeled ligand or lipophilic hormone is directly or indirectly labeled. In some aspects, the labeled ligand or lipophilic hormone comprises labeled cortisol, corticosterone, cortisone, deoxycorticosterone, deoxycortisol, progesterone, or prednisolone. In some aspects, step (d) further comprises heating the second sample portion after contacting the second sample portion with an effective amount of the labeled ligand or lipophilic hormone and prior to determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner.

[0043] In some aspects, step (d) further comprises contacting the second sample portion with an effective amount of unlabeled ligand or lipophilic hormone prior to determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner.

[0044] In some aspects, step (d) or (e), determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner comprises performing a lectin-affinity assay on the second or third sample portion using a plant lectin which is immobilized on a solid support. In some aspects, the plant lectin is Concanavilin-A.

[0045] In some aspects, step (d) or (e) further comprises adding an effective amount of a second unlabeled ligand or lipophilic hormone prior to determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner, the second unlabeled ligand having a equilibrium association constant for the ligand or lipophilic hormone binding partner which is lower than the ligand or lipophilic hormone’s equilibrium association constant for CBG. In some aspects, the second unlabeled ligand or lipophilic hormone is cortisone, deoxycorticosterone, progesterone, or any combination thereof. In some aspects, the second unlabeled ligand or lipophilic hormone is added at a known concentration. In some aspects, the first unlabeled ligand is thyroxine (T4). In some aspects, the ligand or lipophilic hormone binding partner is a serum transport or bindingAttorney Docket No.37759.0576P1 protein. In some aspects, the ligand or lipophilic hormone is a corticosteroid, sex steroid, vitamin D, or a thyroid hormone. In some aspects, the ligand or lipophilic hormone is cortisol and the serum transport or binding protein is CBG. In some aspects, the ligand or lipophilic hormone is progesterone and the serum transport or binding protein is CBG. In some aspects, the ligand or lipophilic hormone is testosterone and the serum transport or binding protein is sex hormone binding globulin. In some aspects, the ligand or lipophilic hormone is a vitamin D metabolite and the serum transport or binding protein is vitamin D binding protein. 3. Methods of diagnosing a disease or disorder

[0046] Disclosed herein are methods of diagnosing a disease or disorder, the method comprising: (a) obtaining or having a sample obtained, the sample comprising total ligand or lipophilic hormone, which is the sum of the concentration of free ligand or lipophilic hormone and ligand or lipophilic hormone bound to a ligand or lipophilic hormone binding partner, albumin, or both; (b) dividing the sample into multiple sample portions; (c) performing at least one assay on a first sample portion to determine the concentration in the first sample portion of (i) total ligand or lipophilic hormone binding partner, (ii) optionally, total albumin, and (iii) ligand or lipophilic hormone; (d) contacting a second sample portion with an effective amount of a labeled ligand or lipophilic hormone, the labeled ligand or lipophilic hormone having an equilibrium association constant of at least 20 M-1for the ligand or lipophilic hormone binding partner, albumin, or both; and determining the concentration in the second sample portion of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner; (e) performing a series of steps on a third sample portion, the series of steps comprising, in sequence: (1) contacting the third sample portion with an effective amount of a first unlabeled ligand or lipophilic hormone, the first unlabeled ligand or lipophilic hormone having (i) an equilibrium association constant for albumin which is higher than the labeled ligand or lipophilic hormone’s equilibrium association constant for albumin, and (ii) an equilibrium association constant for the lipophilic hormone binding partner which is less than 0.1 M-1; (2) contacting the third sample portion with an effective amount of the labeled ligand or lipophilic hormone; and (3) determining the concentration in the third sample portion of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner; and (f) deriving, based on the concentrations determined in steps (c)-(e), the concentration of free ligand or lipophilic hormone in the sample and diagnosing the disease or disorder based on the concentration of free ligand or lipophilic hormone in the sample. In some aspects, the disclosed methods canAttorney Docket No.37759.0576P1 further comprise performing a series of steps on a fourth sample portion, the series of steps comprising, in sequence: (a) contacting the fourth sample portion with an effective amount of albumin; (b) contacting the fourth sample portion with an effective amount of the labeled ligand or lipophilic hormone; and (c) determining the concentration in the fourth sample portion of labeled ligand or liphophilic hormone bound to the ligand or lipophilic hormone binding partner. In some aspects, the disclosed methods can further comprise performing a series of steps on a fourth sample portion, the series of steps comprising, in sequence: (a) contacting the fourth sample portion with an effective amount of albumin; (b) contacting the fourth sample portion with an effective amount of the labeled ligand or lipophilic hormone; and (c) determining the concentration in the fourth sample portion of labeled ligand or liphophilic hormone bound to the ligand or lipophilic hormone binding partner, wherein determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner comprises performing a lectin-affinity assay on the fourth sample portion using a plant lectin which is immobilized on a solid support. In some aspects, the plant lectin is Concanavilin-A. In some aspects, the labeled ligand or lipophilic hormone is directly or indirectly labeled. In some aspects, the labeled ligand or lipophilic hormone comprises labeled cortisol, corticosterone, cortisone, deoxycorticosterone, deoxycortisol, progesterone, or prednisolone. In some aspects, step (d) further comprises heating the second sample portion after contacting the second sample portion with an effective amount of the labeled ligand or lipophilic hormone and prior to determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner.

[0047] In some aspects, step (d) further comprises contacting the second sample portion with an effective amount of unlabeled ligand or lipophilic hormone prior to determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner.

[0048] In some aspects, step (d) or (e), determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner comprises performing a lectin-affinity assay on the second or third sample portion using a plant lectin which is immobilized on a solid support. In some aspects, the plant lectin is Concanavilin-A.

[0049] In some aspects, step (d) or (e) further comprises adding an effective amount of a second unlabeled ligand or lipophilic hormone prior to determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner, the second unlabeled ligand having a equilibrium association constant for the ligandAttorney Docket No.37759.0576P1 or lipophilic hormone binding partner which is lower than the ligand or lipophilic hormone’s equilibrium association constant for CBG. In some aspects, the second unlabeled ligand or lipophilic hormone is cortisone, deoxycorticosterone, progesterone, or any combination thereof. In some aspects, the second unlabeled ligand or lipophilic hormone is added at a known concentration. In some aspects, the first unlabeled ligand is thyroxine (T4). In some aspects, the ligand or lipophilic hormone binding partner is a serum transport or binding protein. In some aspects, the ligand or lipophilic hormone is a corticosteroid, sex steroid, vitamin D, or a thyroid hormone. In some aspects, the ligand or lipophilic hormone is cortisol and the serum transport or binding protein is CBG. In some aspects, the ligand or lipophilic hormone is progesterone and the serum transport or binding protein is CBG. In some aspects, the ligand or lipophilic hormone is testosterone and the serum transport or binding protein is sex hormone binding globulin. In some aspects, the ligand or lipophilic hormone is a vitamin D metabolite and the serum transport or binding protein is vitamin D binding protein. E. Analytical Modeling 1. Coolen’s Equation

[0050] As described herein, Coolen’s equation can be used in any of the methods disclosed herein. By way of illustration, but not in limiting to the particular embodiment, Coolen’s equation is described here as applied to measuring cortisol. A quadratic equation was developed (the Coolen’s equation) to estimate free cortisol using measured concentrations of CBG (XTotCBG) and total cortisol (XTotF). In this analysis, it was assumed both albumin concentration (40 g / L = 580,000 nmol / L) and albumin-cortisol binding affinity (with equilibrium dissociation constant [KA] for albumin-cortisol binding of 330,000 nmol / L). A cubic equation as well as Coolens’ quadratic equation has also been developed. The quadratic formulation of the solution for free cortisol has some advantages relative to Coolens’ insofar as it preserves familiar elements of the quadratic polynomial (see Equation C below). (The quadratic polynomial is obscured in Coolens’ equation owing to their need to divide through by the term ‘2a‘).

[0051] A cubic equilibrium solution was derived for computing free cortisol using mass action equations for the following plasma compartments: (i) free cortisol (F), (ii) CBG-bound cortisol (FC), (iii) unbound CBG (C), (iv) albumin-bound cortisol (FA), and (v) unbound albumin (A) : FC ^ F+C and FA ^ F+A. Two equilibrium equationsequations result in five equations,Attorney Docket No.37759.0576P1 which may be combined to yield a cubic equation for free cortisol. In these equations, F, C, FC, and FA are defined as above, TotC represents total CBG concentration, and TotA represents total albumin concentration. Eq .1) [ F ][ C ]^KC,[FC],,Eq .5) TotF ^ F ^ FC ^ FA .

[0052] There are six concentrations and five equations; the objectives are to solve for TotF in terms of F and then solve the inverse equation for F in terms of TotF. First, using Equation 4 to eliminate A in Equation 2, FA in terms of F can be obtained: Eq .6) FA^F*TotA.^With a similar formula for FC, a firstEq . 7) TotF ^ F ^ F ^ TotC / ^ KC^ F ^^ .For a secondto obtain F as the solution of the cubic equation: Eq. A ) F 3 ^ pF 2 ^ qF ^ r ^ 0 , where^ ^ ^ ^ ^, , A standard formula forAttorney Docket No.37759.0576P1 Eq . B ) F_ cubic ^p ^2 ^ a 3 ^ cos ^^3 ^ ^ ,3 ^^2^

[0053] Coolens equilibrium equationsby assuming that FA / F the physiological range of F. This can be seen in Equation 6, where the value of F is always much smaller than the equilibrium dissociation constant for albumin-cortisol binding (KA). Assuming a simple stoichiometry for cortisol binding to albumin and for F up to 1000 nM, it can be computed that the error in assuming the multiplier N is constant in F is less than 0.25%. To derive Coolens’ equation, note that the multiplier of F in the Equation 6) for FA is Eq .8) N ^ TotA^ TotA.^ Rewriting Equation 7),Eq.9) TotF ^ F (1 ^ N ) ^ F ^ TotC / ^ KC ^ F ^ .Clearing the denominator provides a quadratic equation in F, which is Coolens equation: Eq.C) aF2^ bF ^ c ^ 0, whereb^(1^N)KC^TotC^TotF,c^^TotF^KC, and whose solution is the quadractic formula.In summary, using the Coolens equation, F can be calculated given N and KCand measured TotC and TotF. Coolens equation is often used assuming N= 1.74.

[0054] Coolens’ N also figures importantly into a solution algorithm for free cortisol. This proposition takes advantage of the simplifying assumption of Coolen’s equation, namely that KA>> F. As a result, the binding reaction of albumin and cortisol, at least at physiologicAttorney Docket No.37759.0576P1 concentrations of free cortisol, never approaches a level at which saturable binding kinetics might obtain. This assumption allows for FA / F to be treated as a constant, which is also equal to TotA / KA(see Eq 8, above and also section III-G, below).

[0055] There is a body of experimental evidence suggesting that the reaction involving reversible binding of albumin and cortisol in human plasma at 37°C is complex, involving multiple binding sites, possibilities of competitive binding ligands, and allosteric interactions. However, none of these complexities challenge the assumption, based on KA>>F, that N may be treated effectively as a constant. That is, it is reasonable to infer that in human serum, albumin-bound cortisol (XFA) increases proportionately to free cortisol concentration (XF). As to what the exact value of that ratio of proportionately between XFand XFAis human plasma (37°C) and whether it can safely be assumed to be the same from one person to the next (or one clinical sample test tube to the next) is an issue that pertains more directly to the numerical values applied in equations of Coolens and Heyns. Thus, Coolens’ equation assumes the following values: (i) albumin concentration is 40 g / L, (ii) simple (1:1) stoichiometry of albumin-cortisol binding equilibrium dissociation constant (KD) for albumin-cortisol binding (KA) of 330,000 nmol / L, (iii) a uniform binding affinity (equilibrium dissociation constant KD) for CBG-cortisol (Kc) of 33 nmol / L, and (iv) absence of any compounds in plasma that might compete for binding of cortisol to CBG and / or albumin. Note that since N equals XTotA / KA = XFA / XF, the Coolens’ formula also assumes N=1.74, where (N+1)*XF= [XFA+ XF] = bioavailable cortisol. A lower affinity of albumin- cortisol binding (N=1.3) has been suggested, whereas a higher affinity for albumin-cortisol (N=3.0) was obtained in ligand-binding studies using purified albumin solutions.

[0056] ^1-acid glycoprotein (AAG), also known as orosomucoid, circulates in human plasma at a concentration of approximately 24 ^M and binds cortisol with an affinity (KD=62 ^M) intermediate to CBG and albumin. In a matrix formulation of mass action and mass conservation equations for competitive ligand-protein binding described below, the identification of cortisol binding proteins in addition to CBG and albumin may be represented as an additional column in the matrix. Note that each additional column adds one additional order of complexity to the coupled polynomial equations. By contrast, treatment of albumin- cortisol binding as a constant (N) reduces the order of complexity by one.

[0057] Differential equations for and solutions to the competitive binding of aldosterone (E) cortisol (F) in a dynamic model have been evaluated in vivo, which replicated previously reported experimental observations. The development of the model and associated equationsAttorney Docket No.37759.0576P1 provided the theoretical background for the in vivo (dynamic model) introduction to the disclosed competitive ligand-transport protein binding assay.

[0058] The term “bioavailable testosterone” originated in the field of andrology, since SHBG-bound testosterone can be separated by ammonium sulfate precipitation. Thus, bioavailable testosterone (sum of free testosterone XT and albumin bound testosterone (XTA) can also be derived by subtraction of SHBG-bound testosterone (XTS) from the total testosterone concentration (XTotT). The disclosed methodology provides an assessment of CBG-bound cortisol (XC). If XTotF= XC+ XA+ XF, one can obtain bioavailable cortisol (sum of albumin-bound and free cortisol) by subtracting XC from XTotF. The term bioavailable is used to represent [albumin-bound plus free ligand] without implying the physiological significance or biological activities associated with bioavailable ligand. In the case of cortisol, for example, the term bioavailable cortisol is used as a convenient term to represent the sum of albumin-bound and free cortisol.

[0059] The clinical laboratory problem of determining free cortisol based on measured total cortisol and (total) albumin concentrations addressed using the methodologies disclosed herein can be parsed into two distinct operations: (i) determination of bioavailable cortisol (sum of free and albumin-bound cortisol) and (ii) determination of N in order to quantify XF and XFAfractions of bioavailable cortisol.

[0060] Gold-standard methods for measurement of free cortisol involve (i) pre-analytic separation phase and (ii) analytic procedures. Separation methods typically involve equilibrium dialysis (ED), ultrafiltration (UF), or gel filtration (GF). Analytic methods typically involve cortisol immunoassay, which may be variably affected by cross-reactivity with other adrenocorticosteroid compounds. Liquid chromatography with tandem mass spectrometry (LC-MS / MS) is a preferred and more specific method for measurement of cortisol. However, even LC-MS methods are potentially subject to artifact and inaccuracy. Historically, indirect methods were used to estimate free ligand fraction. This approach utilized addition of labeled ligand, determination of percentage of free labeled T, and calculation of free ligand concentration by multiplying the percentage of free labeled ligand by total ligand concentration. However, several sources of error in this approach, including impurities in the labeled ligand, have been identified, and recent comparisons suggest that the ED with direct measurement of ligand in the dialysate is the contemporary gold standard method for measurement of free ligands such as free cortisol, free testosterone, and 25-OH vitamin D. Regardless of analytic measurement procedures, pre-analytic separation methodsAttorney Docket No.37759.0576P1 such as ED and UF are labor-intensive and not routinely available in the clinical laboratory setting. 2. Quadratic, Cubic, and Quartic Formulae for Estimating Free Cortisol Concentrations

[0061] Three different polynomial equations for calculation of free cortisol, based on measured concentrations of total cortisol (XTotF) and total CBG (XTotCBG) have been suggested. A quadratic formula, Coolens’ equation, is described in detail above. The Coolens’ formula assumes a specified concentration of albumin (using a median population value of ^580,000 nmol / L) as well as a specific binding affinity of albumin for cortisol. The cubic equation is more complex insofar as it uses measured rather than assumed concentrations of albumin. In addition, the cubic solution accommodates individual or group- specific variation in albumin-cortisol binding affinity.

[0062] A variation on the cubic solution is a quartic solution, which includes the additional presence of a low-affinity CBG moiety, originally thought to represent elastase- cleaved CBG. While the more complex quartic solution can achieve superior fits to experimental data relative to quadratic and cubic formulas, it does so at the expense of additional degrees of freedom, including the relative concentration of ‘low-affinity’ CBG. In addition, the assumption that ‘elastase-cleaved CBG’ has a 10-fold lower affinity compared to full-length CBG is open to question in view of demonstration that elastase-cleaved CBG is not detectable at measurable concentrations in human plasma.

[0063] In all these considerations of the binding affinity of cortisol for CBG, albumin, and perhaps other serum proteins as well, it should be recognized that the presence of other (unmeasured) ligands that compete with cortisol for binding to CBG and / or albumin would be identified by alterations in apparent cortisol-binding affinity for respective binding proteins. Examples of iterative solutions that account for competition of ligands for cognate binding proteins using competitive binding principles and coupled polynomial equations have been proposed for both testosterone-SHBG and cortisol-CBG interactions. However, these concepts of competition of various ligands for binding proteins has not been routinely incorporated into formulae for estimation of free ligand concentrations. Limitations of all the proposed formulas for estimation of free cortisol are assumptions that cortisol binding affinities are known and homogeneous. Neither of these assumptions are fully supported by experimental data.

[0064] In the formulae described above, measured levels of XTotCBG and XTotF are used asAttorney Docket No.37759.0576P1 input variables to calculation of free cortisol. It follows that error or bias in measurement of these values will lead to corresponding error or bias in the determination of free cortisol. Thus, the accuracy of estimated free cortisol depends on accuracy of primary measurements of total concentrations of cortisol, CBG, and, in the case of cubic and quartic solutions, albumin in the clinical sample of serum or plasma. The concern for accuracy of primary measurements of total concentrations is highlighted by different reference ranges for comparable populations obtained using different methods for CBG assay and lack of alignment between CBG concentrations measured using two distinct monoclonal antibodies, compared to those measured by commercial RIA using a single, polyclonal antibody to CBG.

[0065] Bias in the determination of total cortisol concentrations in serum or plasma is also well established. This is true for immunoassays, where cross-reactivity (binding of antibody to compounds other than cortisol) have been well-documented. Depending on the design of the assay, this cross-reactivity can result in under- or over-estimation of actual serum cortisol concentrations. An additional factor affecting the reliability of immunometric methods for measurement of total cortisol concerns the chemical methods used to dissociate CBG-bound cortisol; these methods are often proprietary to commercial assays and may result in underestimation of total cortisol when CBG concentrations / affinity are increased.

[0066] While LC-MS / MS methods provide more specific measurements of serum cortisol, they too are subject to various sources of error, as exemplified by the co-elution of cortisol metabolites having similar molecular mass and mass spectrometric fragmentation patterns with cortisol. For example, the presence of co-eluting metabolites such as 20-^ and 20-^-dihydrocortisone results in overestimation of serum cortisol concentrations unless they can be separated from cortisol by careful adjustment of chromatographic conditions. The use of2H-labeled internal standards has reduced the coefficient of variation in cortisol measurements using LC-MS / MS, but differences in recovery rates remains a potential source of variation in cortisol measurements that tends to increase the CV of duplicate measurements.Attorney Docket No.37759.0576P1 EXAMPLES

[0067] The following examples further illustrate this disclosure. The scope of the disclosure and claims is not limited by the scope of the following examples. A. Dynamic In Vivo Model for Cortisol (F) and Labeled Cortisol

[0068] The competition of ligands (e.g. labeled and unlabeled cortisol) for a shared serum binding protein (e.g. CBG), was originally developed for the in vivo setting and was made clear by application of the following simultaneous differential equations. These differential equations are analogous to the 4-compartment, diffusion model equations previously reported in U.S. Patent Application Number 17 / 993,282, incorporated by reference in its entirety for its teaching of the diffusion model equations.

[0069] The eight simultaneous differential equations are taken to describe the relationships between time varying concentrations of four cortisol compartments, free, CBG-bound and albumin-bound, extra-vascular written as XF , X FC , X FA , ^^ி^ respectively, and offour comparable compartments for labeled cortisol (E) compartments, i.e. free, CBG-bound,albumin-bound, and extravascular E (designated XE , X EC , X EA , and ^^ா^.

[0070] Differential equations for endogenous cortisol (F) and labeled ligand (E) are said to be coupled systems of equations, since terms used for F in equations 1a-d also appear in the differential equations for E in equations 1e-h include the competition between cortisol (F) and labeled cortisol (E). That is, equations 1a-h are coupled since both E and F compete for binding to CBG.

[0071] It was assumed that all of the rate constants of E are equal to the corresponding rates for F; that is, labeled cortisol behaves the same as the unlabeled cortisol in the body. In a subsequent model, this requirement can be relaxed such that the labeled hormone behaves in the same way as the unlabeled hormone. This latter permutation will facilitate use of other labeled ligands, such as prednisolone or aldosterone, that bind CBG (and albumin) with different affinity than cortisol. By the same token, the label proposed in experimental methods, cortisol conjugated with the enzyme horseradish peroxidase (HRP), likely has a lower affinity for CBG binding compared to unlabeled (endogenous) cortisol.Attorney Docket No.37759.0576P1^^ி,^^ி^ ,^^ி^= concentrations of free, CBG bound, and albumin bound cortisol = concentrations of free, CBG bound, and albumin bound labeled cortisol ^^ି^^ ,^^ି^^ = cortisol unbinding rate factors for CBG and albumin (“off” rates) = cortisol binding rate factors for CBG and albumin (“on” rates) ^̃^^^=^^^^^^^^ ,  ^^ℎ^^^^^^ ^^^^ ൌ ^^^^^^^^ െ ^^ி^ െ ^^ா^ ZF= cortisol secretion rate ZE = labeled cortisol secretion rate ^= free cortisol elimination rate constant*For diagram and equations, F in reference to cortisol is used, C in reference to CBG, A in reference to albumin, and E in reference to labeled cortisol. The single letter designation for labeled cortisol is not intended to refer to corticosterone, which is sometimes referred to as compound E.Attorney Docket No.37759.0576P1

[0072] Concentrations E and F compete for binding to CBG and albumin as indicated in the mass action equations for free [C] and [A]. This implies that the time courses of XF(t) and XE(t) in the dynamic solution are modified by this competition. Since albumin concentrations greatly exceed cortisol under physiologic conditions, cortisol-albumin binding reactions are generally non-saturable. The issues of E and F binding to albumin is discussed below. Although albumin-cortisol binding is treated as saturable in the 4-compartment model, the vast molar excess of albumin to cortisol allows for the simplification of the relationship using concepts developed in quadratic and cubic (mass action) formulations of cortisol-CBG binding mass action such that a linear relationship between free cortisol (F) and albumin- bound cortisol (FA) obtains and, if albumin concentration is measured experimentally, the concentration of albumin-bound cortisol can be reasonably approximated as a constant (N in Coolens’ equation) that can be simply obtained by multiplying N by the plasma free cortisol concentration (bioavailable cortisol [XFA + XF] = (N+1)*XF.

[0073] With this simplification of cortisol-albumin binding and at physiologic cortisol concentrations, the competition reactions of practical interest concern (i) competition of E and F CBG binding and (ii) additional competition of other (measured or unmeasured) compounds (other than E and F) that may have significant affinity for CBG and circulate at concentrations that result in significant CBG binding activity, and (iii) the existence of other (measured or unmeasured) serum proteins that significantly bind E and F at measurable concentrations. For example, the existence of alternate species of CBG in serum that bind cortisol with lower affinity than full-length, fully glycosylated CBG (having KD in the range of 13-33 nmol / L) has been proposed. 1. Interrupted Steady State

[0074] Following a period of continuous cortisol (F), it can be assumed that steady state conditions of unchanging cortisol concentrations are achieved. Applying the four- compartment model (Eq 1a-d above), a steady state concentration of free cortisol (denoted as FSS) obtains. While the continuous infusion of F continues, a second infusion of E is then initiated. Concentrations of E gradually increase and, like F, reach a new steady state. The continuous infusion of E is referred to as a disruption or interruption of steady state. Since E and F both compete for CBG binding, the concentration of CBG-bound cortisol is lower in the disrupted steady state condition compared to the original steady state condition. 2. Equilibrium in a Plasma Sample in a Test Tube

[0075] The same discussion applies to a test tube containing a sample of bloodAttorney Docket No.37759.0576P1 maintained at body temperature. When a blood sample is obtained by venipuncture and placed in a test tube maintained a body temperature (37°C), a cortisol equilibrium obtains in milliseconds and is maintained because there is no input (ZF=0) and there is no elimination of F (α=0) from the test tube. Thus, equilibrium in the test tube can be viewed as analogous to the steady state condition in vivo. The forward solutions of the dynamic (in vivo) model require that rate constants (on- and off-rates for cortisol-CBG and cortisol-albumin binding reactions, respectively), be known and applied. By contrast, equilibrium solutions in the test tube (and also the in vivo model at steady state) require equilibrium association (Ka) (or dissociation (Kd)) constants (rather than on- and off-rate constants). 3. Labeled Ligand and Original Sample

[0076] In the original sample (e.g., test tube #1), TotF is distributed to F0, FC’ and FA’. The goal is to estimate F0 without measuring it; TotF is measured. In test tube #2 (an aliquot of the original sample), a known amount of labeled cortisol (E) is added so that a known concentration (E0 = TotE) is added. (These calculations involve known volumes.) A new equilibrium is formed quickly (100s of msecs). TotE = E0competes for free and F-bound CBG forming EC, which events predictably result in lower FC and increased F and FA. The notation in test tube #2 will be the same as in Eqs 1a-h.

[0077] With E0 labeled and distinguishable, this total mass of labeled cortisol TotE = E0 is redistributed to E, EC, and EA in test tube #2, though the distribution may be unknown. This result can be described in terms of FS < F + E< FS + E0. FS^E0 = F + E, which is less than FS + E0 because some of the free cortisol is bound to CBG in equilibrium and is no longer free. ^ represents the combination of Fs and E0, which is dynamically defined but in steady state will represent the effect on Fs by E0. 4. Relative Equilibrium Concentrations of E and F In Vitro

[0078] In the test tube #2 experiment above, the additional concentration TotE = E0 is redistributed to E, EC, and EA in the same proportions as the new distribution of TotF into F, FC, and FA. This notion that E, EC, and EA are proportional to the new distribution of F, FC, and FA was known to investigators infusing labeled cortisol in vivo in the past, as it informs the approach of multiplying steady state concentrations of labeled and unlabeled cortisol concentrations by the infusion rate of labeled cortisol.

[0079] In the test tube experiment, if the additional concentration E0 were unlabeled, then E0is just incremental free cortisol that can be rewritten as ΔF. Total cortisol becomes TotF + ΔF which is redistributed to F, FC and FA. Though increasing TotF to TotF + ΔF implies F,Attorney Docket No.37759.0576P1 FC and FA are increased, they have the same percentage distribution as the E, EC, and EA above. The assumption that the addition of free cortisol E0 (labeled or unlabeled) to the test tube only binds to free CBG (and to free albumin) and leaves the original FC and FA unchanged is not correct. The problem is that the assumption does not account for the competition between E and F (for example to CBG).

[0080] These considerations can provide for an algebraic estimate of the original free cortisol (F), which is simpler and quicker to do than contemporary methods for measurement of free cortisol concentration, which typically involve equilibrium dialysis or ultracentrifugation. This method also algebraically obtains the concentration of total CBG (if binding affinity and stoichiometry of CBG-cortisol binding are known). B. Estimation of Albumin-Cortisol Binding from Purified Albumin Solution Data

[0081] The experiment involves addition of labeled cortisol to solutions of varying concentrations of purified albumin. Thus, the test tube consists of TotA and TotE, which at equilibrium also gives a free E and a free A. In this experiment, E represents labeled cortisol (3H-cortisol), which is assumed to bind albumin with an affinity equal to that of endogenous (unlabeled) cortisol (F). Purified Albumin Solution XA tE

[0082] There is nocompetition for combining of E and A with each other. A method was developed for representation of the equilibrium ligand-protein binding equations in a matrix format, as described in below. In this format, the experiment may be formulated as a (1x1) matrix, in which K is the equilibrium association (affinity) constant for albumin-cortisol binding and the concentration of albumin-bound cortisol is determined by K and respective concentrations of free cortisol and (free) albumin according to law of mass action and as illustrated below. For cortisol binding in a purified solution of albumin, this 1x1 matrix represents 1 ligand (cortisol) and 1 binding protein (albumin). Here, the labeled ligand, e.g.3H-cortisol (E), cells are measured, and other cells are known experimentally.

[0083] The objective is to estimate the equilibrium association (affinity) constant (Ka = 1 / Kd) for albumin-cortisol binding) from this series of experiments using the known values of XTotA and the measured values of the percent free cortisol (R) as given in Table 4. Table 4Attorney Docket No.37759.0576P1 TotA R 1 / R 0.1 0.84 1.19 1 071 141 7 4 5 Conservation of mass laws give:

[0084] Equation 1) R = 1 / (1 +KA ) becomes 1’ ) 1 / R = 1 +KA TotA A graphical estimate of K is obtained by plotting 1 / R versus TotA and fitting the regression line where the slope is the estimate K. Data are shown in Table 4 above, where R represents the free 3H-cortisol concentration expressed as a fraction of total 3H-cortisol [free 3H- cortisol] / [albumin-bound 3H-cortisol+ free 3H-cortisol].

[0085] The slope is KA=0.0771 per g / L. The conversion to units nmol / L gives 0.0771 / (15047) = 5.124 x 106 per nmol / L and in terms of KD = 1 / 5.124 =195,200 nmol / L. At median concentrations of albumin in human plasma (40 g / L = 580,000 nmol / L), this corresponds to an N of 3.0, which is substantially higher than N=1.74 used in Coolens’ equation.

[0086] The concentration of albumin-bound cortisol (XFA) was higher than predicted at the lowest concentration of albumin (0.1 g / L) examined. To address the problem that the y- intercept is lower than predicted in the 1 / R vs. XTotA graph above, an alternative solution algorithm using a non-linear function of R can be used. Applying Feldman equations without Coolens’ approximation, Equation 1) can be written: ^ൌ 1 ^ ^^ ^^ ൌ^ಲ்^௧^ ^1 ^after substituting for A from eq 2). SoAttorney Docket No.37759.0576P1 1’’’) ^ ோെ ^^ ∗ ^^ ൌ 1 െ ^^ ^ ^^^ ^^^^^^^^, where B = ^^^ TotFplotting f(R) = ^ ோെ ^^ ∗ ^^ versusestimate ^^^. The parameter B can be varied until the linear fit is good and / or the y-intercept is 1-B.units as in the above example, this formulation yields a slightly higher affinity of albumin-cortisol binding, corresponding to KA of around 176,000 nmol / L and, for XTotA of 580,000 nmol corresponds to N of 3.3.

[0088] An alternative consideration is that the stoichiometry of albumin-cortisol binding is not 1:1. For example, if it is assumed that cortisol-albumin binding was n:1, where multiple cortisol molecules may bind to a single albumin molecule. In that case, the representation of the Feldman equations in matrix format still applies. The order of binding cannot be determined by the number of cortisol binding sites on albumin; it could still be 1:1, just one at a time, concatenated (represented mathematically as a composition of saturable binding functions), or involving allosteric interactions. The composition of two or more Michaelis- Menten-like functions is a Michaelis-Menten-like function. Therefore, Coolens’ approximation still holds because n:1 binding still has only one protein.

[0089] The Feldman equations are: 1) ^^ ൌ ்^௧ி ^ା ^ಲ^ ி^షభand ^^ ൌ ^ವ ்^௧^ = To ^^ವା ி^tA if ^^ ≪ ^^^ and^^ℎ^^ ^^^^^^^^^^^^^^^^ ^^ ൌ ^^ ^^^^^^^^ andN=^^^ TotAof the Feldman equations in matrix format still applies as follows: XF + KXFn XA = XTotF, factoring F out yields XF (1+KXFn-1XA) = XTotF

[0090] An alternative, sequential approach to the various binding sites on the albumin molecule yields similar results but with a reduced n.

[0091] This model can be represented as a 2x1 matrix, treated radiolabeled cortisol (XE) and unlabeled cortisol (XF) as two ligands having the same albumin binding affinity (KA) and where XA represents ‘free’ albumin. Purified Albumin Solution XAAttorney Docket No.37759.0576P1 XE KAXEXA(XEA) XTotE XF KAXFXA(XFA) XTotF

[0092] In this scnstant in the test tube, though the distribution between free- and albumin-bound E can vary (XE+ XEA= XTotE) and the concentrations of XTotA is also experimentally constant for the test tube. Increasing the concentration of XTotFin the test tube to high levels (55,000 nmol / L) had no effect on the concentration of albumin-bound E (XEA). These observations support the assumption that (i) KA>>XFand (ii) XA>> XF, such that the concentration of XAis effectively constant and, consequently, the concentration of albumin-bound cortisol (XFA = KAXFXA) varies in proportion to XF. These conclusions are consistent with the experimental observation in that addition of cortisol to the test tube, even at supraphysiologic concentrations of XTotF (55,000 nmol / L) do not substantially lower the concentration of free albumin (XA). This may be contrasted with the saturable kinetics of the CBG-cortisol binding reaction, in which the concentration of XF may equal or exceed KC during physiologic conditions and the concentration ‘free CBG’ (XC) varies substantially in relation to XF.

[0093] The above matrix may be viewed as a standardized binding assay for ligand-BP interactions in which3H-cortisol (E) is used as label and unlabeled cortisol (F) is treated as the competitor for a purified solution of albumin, which is present at high concentrations and demonstrates significant and measurable cortisol binding affinity. Assuming Coolens’ assumptions apply to both E and F in the 2x1 matrix (with same N), XTotE = XE + NE and XTotF= XF+ NF. Increasing TotF increases F in a proportional manner (XF= XTotF / (N+1)), however the labeled total (XTotE) is unchanged and XE = XTotE / (N+1) is unchanged. Thus, the split between XEand XEAis unchanged, and the albumin bound labeled cortisol (XEA) is constant.

[0094] The principle of Coolens’ approximation where KA>>F is reasonable, and there is good experimental evidence that albumin-cortisol binding kinetics are non-saturable at physiological concentrations of serum albumin. In other words, at any physiologic concentration of free cortisol (F), a linear relationship between albumin concentration and albumin-bound cortisol concentration (XFA) obtains. The slope of this linear relationship, with XF on abscissa and XFA on ordinate, is dependent on the constant (N), which in turn depends on albumin-cortisol binding properties summarized in a simplified stochiometricAttorney Docket No.37759.0576P1 model by albumin concentration (XTotA) and the affinity of the albumin-cortisol binding reaction. Where the equilibrium dissociation constant (KD) for albumin-cortisol binding (KA) is used to summarize the affinity of the reactants, N = XTotA / KA.

[0095] At physiological concentrations of purified albumin solutions (40 g / L) and at 37C°, N=3.0 is realistic. This conclusion is also supported by studies where plasma was heated to 60°C for 60 min, which irreversibly denatures cortisol-binding activity of CBG, an N=3.0 was also observed. These experimental observations suggest KA = 195,200 nmol / L may provide a realistic, initial estimate of N based on measured concentrations of total albumin (XTotA).

[0096] Although the possibility that other compounds present in human plasma might compete with cortisol for albumin binding or binding interactions cannot be excluded, the fact that addition of unlabeled cortisol to relatively high concentrations (55,000 nmol / L) in vitro had no effect on3H-cortisol binding to albumin (or heat-treated human plasma) supports the view that albumin-cortisol binding during physiologic conditions may be reasonably considered to be non-saturable. However, the existence of other substances, such as free fatty acids, that may circulate at sufficiently high concentrations (and bind albumin with sufficiently high affinity) to competitively decrease albumin-cortisol binding cannot be ruled out. Moreover, while the studies demonstrating a ratio of XFAto XFof approximately 3.0 in representative samples of heat-treated human serum are re-assuring in their similarity to results obtained in purified albumin solutions, they do not exclude the possibility of heat- labile compounds that competitively inhibit albumin-cortisol binding in vivo.

[0097] The equations may be modified to accommodate that possibility of multiple (n) binding sites, such that each molecule of albumin is capable of binding multiple cortisol molecules. One approach to this scenario treats albumin-cortisol binding as 1:1 stoichiometry, but this approach tends to overestimate cortisol binding affinity (lower KA) and underestimates the concentration of cortisol needed to saturate albumin binding sites. The presence of multiple binding sites for albumin still accommodates the use of a constant (N) to reflect the proportionate relationship between free cortisol (XF) and albumin-bound cortisol (XFA). C. Estimation of Albumin-Cortisol Binding Constant (N) and Biological Variation in Albumin-Cortisol Binding in Non-Denatured Human Serum

[0098] As to what value for albumin-cortisol binding constant (N) is and how it may be estimated in a plasma sample is a matter of less certainty. One concern with Coolens’Attorney Docket No.37759.0576P1 equation is the assumption of albumin concentration (~40 g / L = 585,000 nmol / L). Previous work supports the use of measured rather than assumed albumin concentrations. This is reasonable since there is substantial normal variation in albumin concentrations and the measurement of albumin in clinical samples is simple, accurate, and inexpensive. As to what KA for albumin-cortisol binding should be used when applying Coolens’ ‘N’, there are major variations reported in the literature, ranging from KAof 80,000 to 810,000.

[0099] To further explore this issue, post-hoc analysis of data that includes serial measures of XTotFand XF, as well as XTotCBG, was evaluated in subjects administered oral and iv hydrocortisone. Useful elements of the work are (i) development of Feldman’s coupled mass action and conservation of mass equations in a matrix format and (ii) introduction of the idea of interruption of matrix to solve the inverse matrix problem. The analysis used total and free cortisol concentration data. One finding in this analysis relevant to the above discussion concerning the albumin-cortisol binding constant (N) in human sera that has not been subject to heat-denaturation of CBG was the observation that N varied substantially by individual and condition.

[0100] If the equilibrium of free cortisol concentration (^^ி) binding to free CBG (^^^) and free albumin ^^^^^ as well as a known competitor (^^^) that also binds to CBG and albumin*, the equilibrium can be represented in matrix format as: ^^^ ^^^The matrix of affinities ൫^^൯ is 2x2. In the above matrix format, the bounded concentrations ^^ி^, ^^ி^, ^^^^, and ^^^^have been replaced by mass action formula (assuming 1:1 stochiometric binding). The four total concentrations give rise to four non-linear equations; and assuming total concentrations and all affinities ^^^^are known; there are four free concentration unknowns. Thus, based on conservation of mass and mass action formula, these four equations are: ^^்^௧ி ൌ ^^ி^1 ^ ^^^^ ^^^ ^ ^^^ଶ ^^^^^^்^௧^ ൌ ^^^^1 ^ ^^^^ ^^^ ^ ^^^ଶ ^^^^^^்^௧^ ൌ ^^^^1 ^ ^^^^ ^^ி ^ ^^ଶ^ ^^^^Attorney Docket No.37759.0576P1

[0101] In the conventional (forward) solution of the matrix, the total concentrations are measured and the respective ligand binding affinities are known. The solution of the above coupled equations is a non-linear problem. The iterative nature of this problem can be seen by writing each equation above as the free ligand concentration equal to the total concentration divided by the corresponding quantity in the parentheses (i.e. cross dividing by rhs). These equations were reported to be: It is possiblerespectively. This is a generalization to (nxm) matrix of affinities ൫^^^^൯. The ratios for ^^^into the ratios for ^^^can be substituted, giving the ^^^as of ^^^. This isconvenient for iterative solution; though continuity gives the if the functions for ^^^. and ^^^to be iterated as a system.

[0102] It is possible to clear the fractions in the equation(s) ^^^as a function of ^^^to obtain the solution as the root of resulting polynomial. The degree of the resulting polynomial is m+1 which depends on the number of binding proteins represented by the ^^^. The 1x1, 1x2, and 1x3 matrices lead to the quadratic polynomial, the cubic polynomial, or the quartic polynomial. Several of these formulations have also been adapted to include the possibility of additional ligands represented by 2x1, 2x2, and 2x3 matrices, as illustrated also by the model that includes a competitive inhibitor of CBG-cortisol binding. These polynomials have closed-form algebraic formula; however, larger systems (m>3) lead to quintic (or higher degree) polynomials, which are not tractable to closed-form solution (Abel’s impossibility theorem). Application of Coolens’ assumption, namely that XFA / XF (N) is a constant at physiological concentrations of XF(and the corollary premise that N=XTotA / KA, reduces the degree of the polynomial equations by 1. Thus, adoption of Coolens’ N means that the degree of the polynomial is equal to the number of columns in the above matrix.

[0103] The inverse problem introduces an addition problem: the 2x2 matrix has 4 unknowns and has 4 equations; however, the 2x3 or 3x2 matrix has 6 unknowns but only 5 equations. This difficulty can be remedied by more observations, e.g., by having multiple assays per subject. This approach relies on the assumption that cortisol concentrations are time-varying; that is, multiple assays would not be useful under steady state conditions. An additional assumption in this analysis is that the concentrations of binding proteins are time- invariant during the sampling period.Attorney Docket No.37759.0576P1

[0104] The equations can be interrupted for the solution when one of marginal totals is unknown (either in the far-right column and bottom row, in the present case ^^்^௧^) but (in the present case) free cortisol concentration (^^ி) is known. ^^்^௧^can be considered to be a parameter of the system. The equation for ^^்^௧^above is still useful in the iterative solution for TotP; however, more assays per subject are needed and the known total concentrations must vary across the different assay time points. The data consists of 14 or 15 time points per subject which vary dynamically (non-steady state condition) due to experimental intervention (a bolus of hydrocortisone at time zero).

[0105] To estimate parameters in the inverse problem, a programmable algorithm is needed; iterations will be a useful part of this algorithm. Non-linear regression provides least sum of squared errors estimators for ^^^^and now ^^்^௧^given an adequate number of observations. This outer loop gives a functional estimate of the predicted free cortisol, so that the sum of squared errors (measured ^^ி– predicted ^^ி) are minimized. The functional evaluation is also a non-linear problem and could require iteration as well. This is the inner loop providing the equilibrium solution (discussed above) that provides assurance that the predicted values and the total values are in equilibrium and thus coherent. This development is generalized to n ligands and m binding proteins (nxm) matrix K of affinities). The iterative equations can be replaced by the roots of polynomials. These examples can be generalized to include additional ligands by obtaining roots of coupled polynomial equations. However, this approach is limited by the fact that in general there is no closed form solution for quintic polynomial equations. By contrast, the disclosed approach using iteration is robust and versatile, as it may be applied over a broad range of conditions involving multiple binding proteins and / or ligands, including all those mentioned above. The use of this notation of XCand XAin the matrix below differs from the use of these terms in previous reports; in previous reports XC and XA were used to denote CBG-bound and albumin-bound cortisol, respectively. D. Distribution of Cortisol in Human Serum in Vitro: Role of Competitive Ligand- Protein Interactions in Women on Oral Contraceptives

[0106] The relationship between serum concentrations of total (XTotF) and free (XF) cortisol (test tube equilibrium at 37°C) is influenced by many factors. These include concentrations and affinities of serum binding proteins (BP), such as CBG and albumin (A). Among women taking oral contraceptives (OC), XF was found to be higher than predicted using ligand-BP association parameters developed in healthy volunteers (HV). WeAttorney Docket No.37759.0576P1 hypothesized that ligand(s) (XP) competing for XFbinding to CBG may contribute to altered relationships between XTotF and XF observed in OC relative to HV.

[0107] Feldman’s system of iterative, non-linear equilibrium equations were developed in (nxm) matrix format. Data included measured XTotF and XF at each time point (0-480 min) after 20 mg hydrocortisone. Groups were stratified by (i) mode of administration (iv vs. po) and (ii) condition (HV, n=13 vs. OC, n=12). The matrix was interrupted to obtain iterative equilibrium solutions for parameters of interest using (1x2) minimal model (MM) and (2x2) ligand competition model (LCM). Solutions were optimized by minimizing least squares differences between measured and model predicted XF. Albumin-bound cortisol was parameterized as a constant ratio of XF(N) to simplify the matrix problem. Comparisons and interactions within the 2x2x2 design (delivery, group, and model) were analyzed by ANOVA.

[0108] In OC, LCM provided significantly better fit to measured XF relative to MM (13 vs.19% error, P=0.003). MM solutions for CBG-cortisol affinities were significantly higher in HV (KD=26.6) vs. OC (KD=71.9 nmol / L), but LCM yielded similar affinities in both groups (KD= 23.9 in HV and 16.1 nmol / L in OC, interaction P<0.001). LCM solutions for concentrations and CBG-binding affinities of XP were significantly increased in OCP vs. HV (both P<0.001). LCM solutions for HV yielded higher N for po (3.1±1.2) vs. iv (1.4±0.4) (P=0.007). Model solutions depend on the precision and accuracy of measurements made in clinical samples. Although the competitor (XP) is treated as a single compound in LCM, it may represent the combined effects (e.g., harmonic means) of multiple ligands.

[0109] Matrix notation was useful to development of non-linear, iterative equations for competitive ligand-BP interactions, which were applicable to both forward and inverse solutions of the matrix problems. LCM provided a better fit to experimental data for OC compared to MM, providing (indirect) evidence that competing ligand(s) (XP) influence the distribution of cortisol in serum samples obtained from women on OC, (iii) inter- and intra- subject variability for N was observed for both models, suggesting that assignment of a single population value for cortisol-albumin binding constant may be unrealistic.

[0110] In the conventional application of Feldman-like equations, the challenge is to solve for the concentration of free ligand(s) when all the totals (of both ligands and BPs) are known and all the equilibrium association constants (K, L / nmol), which are equal to the inverse of the cognate BP-ligand equilibrium dissociation constant (KD, nmol / L), are known. As to the method for solving these coupled equations to estimate the free ligandAttorney Docket No.37759.0576P1 concentration, suffice to say that there are several different approaches. These solution methods may require iteration, which in a programming language involves a loop (perhaps a do loop).

[0111] Considering the simple 1x2 matrix, without the additional complexity of additional ligand(s) that compete with cortisol for CBG binding. In this 1x2 matrix, the unknown quantities to be estimated can be determined. Of the variables to be estimated, XFis of primary interest. It is possible but not necessary to estimate XC and XA during estimation of XFby iteration, but in any case, these values are of (possible) secondary interest and generally go unreported. 1 ^^^^^^^^ ^^^^^^ ^^^^^ ^^ ^^^ ிThe estimation of the albumn constant (K12) may be re- parameterized as a constant (NA), as discussed above (III-D and III-E); this re- parameterization is especially convenient in the data of Perogamvros et al, since XTotAwas not measured in those experiments. However, under usual conditions XTotA is measured and the solution for NAobtained by solving for K12, as shown in the matrix above.

[0112] Re-writing the above matrix using Coolens-like assumption (N), the following matrix can be obtained. The use of Coolens’ N simplifies the complexity of the matrix problem; that is, the degree of the coupled polynomial equations is reduced by 1. (Coolens’ approximation replaces two potential parameters with one parameter (e.g. replaces K for cortisol-albumin binding, K12 in above example, and XTotA with N), and at the same time reduces the degree (complexity) of the polynomial solution by one.

[0113] Another problem is also a simple 1x2 matrix and is like the above one insofar as the concentrations of both BPs (XTotC and XTotA) and total ligand (XTotF) are measured. In this problem, XFis also measured, and these measured concentrations (blue) are then used to estimate solutions for the parameter(s) of interest (shaded), i.e. equilibrium affinity constants for CBG-cortisol and albumin-cortisol binding reactions, respectively. Thus, the total BPs, total ligand, and free ligand concentrations are known, and it is possible to solve for the unknown K’s. This involves a least squares solution, which itself involves iteration, and thus can be viewed an outer loop. In this problem, the function evaluation may also require iteration.Attorney Docket No.37759.0576P1 2 ^^^^^^^^ி^^^^^^ி^^^^^^ଶ^^ி^^்^௧ி

[0114] An additional prd some K’s are unknown—so it is a mixed problem. In this case, effective XTotPand KPCare unknown, so it is needed to solve for these as well as the free ligands. This requires the right design in terms of data provided, concentration-time series. The third problem is a more comple problem by virtue of the existence of a putative ligand, which competes with cortisol for CBG-binding and yields a a 2x2 matrix. Parameters to estimate include NA, XTotP, KPC, KFC. The competitor P binds to CBG with an affinity that is significant but lower than cortisol (higher Kd), but binds non- specifically to albumin with similar affinity as cortisol. In this example: 3^^^ ^^^^^ ^^ ^^ ^^ ^^ ^^ ^^ி^

[0115] Although P is fou a e as a s g e co pou , can be recognized that it may represent a summation function (harmonic mean) of multiple discrete ligands having different KD’s and concentrations. The use of weighted harmonic means using linearized functions, analogous to Lineweaver-Burke plots for combining enzyme reactions, is a related investigation to be covered elsewhere.

[0116] The term interruption is used for a mixed type of unknowns between the three types of quantitative parameters, i.e. (i) total concentrations (of BPs and ligands), (ii) free concentrations (of BPs and ligands), and (iii) affinity constants (K’s); or, interruption also occurs concentration(s) of BP-ligand complex in certain position(s) of the matrix is measured. In the first problem, only the free ligand concentration(s) are unknown. In the second problem, only the affinity constant(s) are unknown. However, in the third problem, some (but not all) total ligand concentrations and some (but not all) of the affinity constants (K’s) are unknown. In this context, the matrix is interrupted when more than one type of unknown needs to be estimated.

[0117] For sake of completeness and illustration, an equilibrium model can be considered wherein the existence of an elastase cleaved CBG molecules of known (reduced) affinity forAttorney Docket No.37759.0576P1 cortisol. This is the so-called quartic method for estimation of free cortisol, which becomes a 1x3 matrix having measured concentrations for some but not all BPs and ligands, and either assumed or solved values for affinities and the fractional concentrations of XTotC1and XTotC2, respectively. By comparing measured vs. model-predicted concentrations of XF, solutions for these parameters of interest may be obtained by iterative procedures that minimize residual errors (least squares solutions). 4^^^^ ^^^ଶ ^^^^^ி^^^^^^ி^^^^ଶ^^^ଶ^^ி^^^ଶ^^^ଷ^^ி^^்^௧ி

[0118] In this mTotC1 + XTotC2 = XTotC (measured) is known, where C1 may be considered usual high=affinity (ha) CBG KD^12.5 െ 33 nmol / L and parameterized as p*TotC, and C2 is the putative elastase-cleaved,low-affinity (la) CBG having ^10-fold lower affinity for cortisol binding (e.g., KD^ 125- 330 nmol / L) and at a fractional concentration (relative to total CBG) of XTotC2 = (1-p)*XTotC. Also, treating K11as 10-fold greater than K12reduces the number of parameters to be estimated. The fourth problem may also be considered to be a form of matrix interruption, in the sense that both affinity constants (K11, K12, and K13) as well as fractional distribution of ha and la CBG are to be estimated.

[0119] The model illustrated in the fourth problem has been developed in the endocrine literature. Interest in this model has been based on preliminary but indirect evidence of significant concentrations of elastase-cleaved CBG in human plasma. However, the more recent understanding that elastase-cleaved CBG does not circulate in plasma at significant, measurable concentrations and recognition that the differential recognition of CBG epitopes by various monoclonal antibodies is due to differences in N-linked glycosylation that do not appear to impact CBG-cortisol binding affinities leave several areas of uncertainty for Problem #4 in its present constitution. In particular, there is limited basis to assume that value of K12 is 10% that of K11. E. Equilibrium Solutions Constrained Where Binding Affinities of Ligands E and F for Cognate Binding Protein(s) are Equal

[0120] It is instructive to use the equations in 1a-h) to investigate equilibrium conditions. The steady state condition for F is based on dynamic processes in 1a) into secretion- elimination that equilibrates over several hours and the equilibrium is based on free cortisolAttorney Docket No.37759.0576P1 binding to 1b) CBG and 1c) albumin, both of which equilibrate over milliseconds. This binding equilibrium takes place under conditions that do not involve secretion or elimination; it is of interest that this condition obtains when a blood sample is drawn into a test tube and maintained at the body temperature. Thus, every time series of sampled cortisol satisfies this binding equilibrium and is only slightly different than the in vivo values. To examine this important binding equilibrium, note that two of the simultaneous non-linear differential equations for cortisol binding become 0ൌ ^̃^^^^^ி^^^^    െ ^^ି^^ ^^ி^^^^^ ^̃^^^. These equations simply of CBG-bound and albumin-bound cortisolequations 2 a and 2 b below found in many chemistry textbooks. ^^^^^^^^ ^^^^^^^^ ^^^^^^^^^^^^^^^^ ^^^ ^ ^2^^^ ൌ ^^^ , ^^ℎ^^^^^^ ^^^^^^^^^^^^^^^^^^ ൌି^  ^^^^^^^ ^^^^^Similarly, (equations 1f and 1g set equal to zero) become ^2^^^^^^^^^^^ ൌ^^^^, ^^^^^^ ^^. These four protein-bindingalso referred to as mass action equations. There are four mass conservation equations 3(a-d); the first two are already mentioned in the “where clause” of equation 1), and the last two (3c and 3d) are mass conservation equations not explicitly mentioned in the differential equations. ^3^^^ ^^^^^^^^ ൌ ^^ ^ ^^^^ ^ ^^^^^3^^^ ^^^^^^^^ ൌ ^^ ^ ^^^^ ^ ^^^^^3^^^ ^^^^^^^^ ൌ ^^ ^ ^^^^ ^ ^^^^ ^3^^^ ^^^^^^^^ ൌ ^^ ^ ^^^^ ^ ^^^^.

[0121] The two formulae in the Eq.1) where clause are equivalent to equations 3a) andAttorney Docket No.37759.0576P1 3b) and these can be rewritten as C = TotC - FC – EC and A = TotA – FA – EA. These formulae can be substituted for concentration C in mass action equations 2a-b) and for concentrations of A in mass action equations 2c-d)

[0122] Since cortisol F and labeled cortisol E are linked in Equation 1) and 3), the following equilibrium relationship of F and E can be obtained by equating [C] in equations 2a) and 2d). ^^^^ ^^^^ ^4^ൌ . ^^ ^^ Similarly, equating [A] in equations 2b) and ^^5^^^^ ^^^^ ൌ . ^^ ^^ Note that under conditions the FA / F ^ N is constant, as in Coolens, EA / E is also constant and approximately equal to N. So, equation 3b) becomes 3b’) TotA = A + N*(F+E).Now [C] in 2a) is C=^^^^ ൌ ^ವ^^ி^^ி^^^^^^ ^^^^^^ ൌ^ி^^ி ^^ ^^^^4^.^^^^ ൌ ^^^ ி^ ^ ^^ ி^^ி^^ ^ி ^^ ^^^^ TotC = ி^ ி {^^^^ ^ ^^ ^ ^^^ , so that equation 3a) becomes^7^ ^^ ^^ ^^^^^^^^ ^^^^^^^^ ൌ ൌൌ ^^ , ^^^^^^^^^^^^^^ ^ ^^^ ^^^ ^ ^^ ≫ ^^ ^ ^^.^ ^ ^^ ^ ^^^^^Similar formulae for EC and EA can be used; by symmetry we have^8^ ^^^^ ൌ ா ்^௧^^ಶ^ be obtained by substituting 6) and 7) into equation 3c) to obtain:^ ^^^ ^^^^^^^^ ^^ ^^^^^^^^  ^^^^^^^^ 10 ^^^^^^^^ ^^ ^ ^^^ ^^ ∗ ^1 ^ ^^ ^ ^^Attorney Docket No.37759.0576P1^ ^^^ ^^^^^^^^ ^^ ^^^^^^^^  ^^^^^^^^ 11 ^^^^^^^^ ൌ ^^ ^ ^^ ^^^ ^^^^ ^ ^^ ^ ^^ ൌ ^^ ∗ ^1 ^ ^^ ^^ ^^^ ^^ ^^^^^ ^ ^^ ^ ^^the following12^ ்^௧ா ்^௧ி ா ൌ ி , 13^ %E ൌ %F, where %E ൌ 100*E / TotE, etc. and 14^ ா^ ൌ ா^ ா ்^௧ா ி^ ி^ ൌ ி ൌ ்^௧ி Comparing்^௧ி்^௧ா ி in 10) to ா ^^^^ 11) shows equation 12) is true in equilibrium. The reciprocal of expressions in 12) shows 13). Equations 4) and 5) and cross multiplicationin 12) shows 14).

[0125] For the test tube #2, TotE = the original E0. So measuring TotF, means it is possible to compute the values of ratios in the lemmas above. Eq.14) implies E, EC, EA are proportional to F, FC, FA with the same constant of proportionality. The above lemma and proofs work if F and E share the same KD for relevant mass action binding reactions (to CBG and albumin) and if there only two relevant binding proteins in serum (CBG and albumin). The issues of using a labeled steroid that has different KD than cortisol and variation of compounds that compete with F for binding to CBG are discussed separately. The ratios in Eq 14) are unaffected by the additional of competitor(s) (for which there is a proof). If the ratio TotE / TotF is known, then all the ratios in 14) are known. The addition of competitor(s) increases F and %F. The addition of a competitor to the system does not change Lemma 1, even though it will decrease both free CBG and free albumin. The factor reflecting these decreases are the same in the formulas for TotE and TotF and cancel in ratios.

[0126] With 11) TotE = ^^ ∗ ^1 ^ ^^ ^  ்^௧^^^^ = E(1+N) + EC, but EC is measured in the Test, and %E= E / TotE by definition. With 13) %F=%E and F = %F *TotF. With E and F computed and with Eq 11):^^^^ ൌ ா ்^௧^^ವ^ାிାா, TotC can be computed.

[0127] For the test tube #2 experiment and from an experimental procedure (see details below) by which the signal EC can be measured, algebraically computing the concentrations of total CBG and then subsequently computing algebraically the concentration of free cortisol (F) is possible. This theorem is applicable to conditions where the affinities of E and F for both CBG and albumin binding are equal.Attorney Docket No.37759.0576P1

[0128] In original sample (test tube #1), TotF is distributed in equilibrium to free cortisol (F0) and cortisol bound to CBG (FC’) and albumin (FA’). The goal is to estimate F0 without measuring it; total cortisol (TotF) and total albumin (TotA) are measured. In test tube #2 (an aliquot of the original sample), a known mass of labeled free cortisol (QE) is added so that a known concentration E0 is added to tube #2. (These calculations involve known volumes.) A new equilibrium is quickly established (100s of msecs). E0competes for CBG forming CBG- bound E (EC). Now the concentration of labeled EC in test tube #2 is measured experimentally.

[0129] The following computations are needed to compute total CBG concentration (XTotCBG) and then free cortisol concentrations (XF) based on the experimental assay measuring the concentration of CBG-bound E (EC). Concentrations of total cortisol and total albumin are measured in test tube #1 and unchanged in test tube #2.(15) ^^ ൌ ாబ ି^େ^ା^^^^^^^ %E= E / TotE , ^^ℎ^^^^^^ ^^^ ^^^^ ^^ℎ^^ ^^^^^^^^^^^^^^^^, ^^^^^^^^^^ ^^^^^^^^^^^^^^ free E,^^^^^^^^ ^^ ൌ ^^^^^ ^^^^^^^^^^^^^^^^ ^^^^^^ ^^^^^^^^,^^^^^^^^,^^^^^^ ^^ℎ^^ ^^^^^^^^^^^^ ^^^^ ^^^^^^ ^^^^^^^^^^^^^^^^, ^^^^16^ F ൌ TotF ∗ %EAlso,^17^ ^^^^^^^^ ൌ ா^ா∗ ^^^^^ ^ ^^ ^ ^^^ can be estimated.Eq 16) isoriginal F0in test tube #1,Finally, ^18^ ^^^^^^^^ ൌ ^^ ்^௧^ ^∗ ^1 ^ ^^ ^^ವ^ାிబ^. So≈ 33 nmol / L and computing TotC from test tube #2, F0 can be solved by Coolens quadratic formula. The ability to experimentally measure EC simplifies expressions and makes the problem solvable.

[0130] Here if there are an unknown endogenous cortisol infusion (secretion) rate (ZF) and a known rate of infusion of labeled cortisol (ZE) that runs to steady state, the unknown endogeneous infusion ^^^^^^^^ ^^ிcan be determined by measuring free F and free E in steady state and computing as follows.

[0131] In Steady State in the 4- compartment model constant, and for continuous infusions ^^ி^^^^^^ ^^ா, the following proportionality holds. ^ಷ ൌ ி^ orி^ ^ಶா^^^ி ൌா^ ^^ா.Attorney Docket No.37759.0576P1 This can also be done using்^௧ிೄ்^௧ா by equation 14): ೄ19) ^^^^^^^^^^^^ ிൌ ^^^^^^^^^^^^ா.

[0132] The steady state equations with E and F occur independently and can be divided to give ^^ி ൌி^^ ா^^ ^^ா. So the above arguments can be viewed as proofing the steady state in E and F are not affected by the competition between E and F and showing that free cortisol ratio Fs / Es is equal to the total cortisol ratio TotFs / TotEs. Equation 1a) becomes ^^ி^^^^ ൌ^^ ∗ ^^ி^^^^ and for constant, continuous infusion, then ^^ி^^^^ ^^^^^^^^^^^^^^ ^^ ^^^^^^^^^^^^^^^^ ൌ ^^^^say and 1h) becomes ^^ா^^^^ ൌ ^^ ∗ ^^^^ . This theorem shows how if one measuresTotFS / TotESat steady state then the unknown constant secretion rate ZF(t) can be computed knowing the constant infusion rate ZE(t) by ZF = ^்^௧ிೄ^*ZE. ்^௧ாೄon equilibrium results as well as steady state. For this restricteda relationship between R (XF / XTofT) and the concentration of CBG-bound E (XEC) was observed in simulation.

[0133] There may be a variety of circumstances where it may be preferable to use another ligand than labeled cortisol to perform the above calculations. For example, ligands such as prednisolone, aldosterone, and HRP-labeled cortisol bind CBG but with different affinity than cortisol. Depending on the situation, it may be preferable from experimental considerations to use labeled ligand having either high or lower affinity than the endogenous compound of interest.

[0134] Then the same argument yields protein binding equations 2) unchanged except there are two different KD (K1 and K2 as above), but the same mass conservation equations 3) and equating free CBG, in 2a) and 2d) obtained are:Attorney Docket No.37759.0576P110)-14) are changed. However, the arguments remain tractable. That is, it is not required that the labeled ligand E have the same binding affinities as the endogenous ligand F). Eq 15) remains unchanged with the clarification that the Kd for E binding to albumin (NE is analogous to Coolens’ constant, i.e. albumin concentration divided by the dissociation constant for albumin-E binding) may be distinct from NF, the Coolens’ constant for albumin-cortisol binding (albumin concentration divided by the dissociation constant for albumin-cortisol binding).

[0136] For the test tube #2 experiment described above with different CBG affinity for E and from the procedure with a coated well that binds CBG from which the signal EC can be measured, F can be computed:in both test tubes (though unknown until eq.17)).

[0137] Coolens’ assumption (N represents the constant ratio of albumin-bound cortisol to free cortisol) for computing F0:in test tube #1:Attorney Docket No.37759.0576P1 ^18′^ ^^^^^^^^ ൌ ^^^ ∗ ^1 ^ ^^ி ^ ்^௧^ ^భାிబ^. for CBG-cortisol binding ≈ 33 nmol / L, andby Coolens’ quadratic formula. As noted concentrations of albumin were measured to estimate albumin-bound cortisol concentration rather than assuming a fixed albumin concentration using measured albumin concentration. Coolens’ approximation also assumes that there are no other ligands competing for CBG binding in the test tube.

[0139] The use of matrix notation was introduced above. The equilibrium association constants for ligand-BP binding (affinities), as shown in Table 1, can be represented by a matrix K = (Kij). The mass action equations (6 equations) in matrix format are ^^^^^^^^ ^^^^^^ଶ^^^^^^^^^^^^^^ଶ^^^^^ଶ20) ^^^^ ^^^^ ^^^ ^^ ^ ^ ൌ ൭^^ ^^ ^^^^^ ^^ ^^ ^where ^^^^^^^൧ represent concentrations , and where the entries on the right hand side (r.h.s.) are products of concentrations ^^^^^ ^^^^^^ ^^^^൧. We delete the brackets when this will not lead to ambiguity. Next, the mass conservation equations are ^^^^^^^^^ ^^^ ^ ^^^^^^^^ ^ ^^^^^^ଶ^^^^^^^^^^ ^ ^^ ^ ^^^ ^^^^ ^ ^^^ ^^ ^ ^^^^.Q1Q2123Attorney Docket No.37759.0576P1

[0140] Given [PQ], K , TotP and TotQ solutions for P and Q can be obtained by numerical methods using eqs 20) and 21), which are summarized in Eq 22). Row sums (e.g. P1+ ^^^^^^^^^^+ ^^^ଶ^^^^^ଶ= TotP1) give rise to the first three rows of Eq 23) and column sums give rise to the last two rows of Eq 23).23) TotP1 = P^^1 ^ ^^^^^^^ ^ ^^ଶ^^^ଶ ^ ^^ଷ^^^ଷ^ ,Tot^^ଶ ൌ Pଶ^1 ^ ^^^ଶ^^^ ^ ^^ଶଶ^^ଶ ^ ^^ଷଶ^^ଷ^ , ^^^^^^TotP3 = Pଷ^1 ^ ^^^ଷ^^^ ^ ^^ଶଷ^^ଶ ^ ^^ଷଷ^^ଷ^TotQ1 = Q^^1 ^ ^^^^^^^ ^ ^^ଶ^^^ଶ ^ ^^ଷ^^^ଷ^ andNow there are 5 equations in 5 unknowns P (in row sums) and Q (in column sums). Feldman proposes solving for P’s as a function the Q’s and then solving for the Q’s as a function of the P’s, which proves Feldman’s iterative equations introduced above and repeated here. ்^௧^ 22a) ^^^்^௧ொೖ^=^^ା ∑ೕ^ ொ^ where 22b) ^^^ൌ^ ∑ೕ ^ೖ ೖ ା ^^ೖ^^, Pଶ ^^^^^^ Pଷ ; in fact Q=g(P)represents a singularity in the original 5 equations. If these 5 equations were linear then they would need to be linearly independent for there to be a unique solution and linear Q=g(P) implies the matrix of coefficients would not be full rank and not linearly independent. Nonlinear equations have similar problems. Thus, substituting for Q using Q=g(P) is more than helpful; it is necessary to find an algebraic solution. Clearing fractions results in a polynomial equation, which in this example would be a cubic polynomial equation.

[0141] The complexity of the model can be explained in terms of the equivalent problem of coupled polynomial equations. Feldman’s non-linear iterative equations for P’s involve ratios of Q’s; these ratios can be cleared algebraically to give polynomials expressed only in terms of P. The first feature of complexity is related to the order of the polynomials, which depends on the number of Q’s (or number of columns) (plus 1). Thus, having a greater number of unique ligand-binding proteins in the test tube (columns in Feldman matrix) results in higher degree polynomials. For example, considering cortisol binding to CBG and albumin represents, with separate columns for CBG and albumin, represents a 3rdorder (cubic) polynomial. However, the order of the polynomial can be reduced by simplifications, such as Coolens’ approximation for cortisol-albumin binding, which reduces the problem to a 2ndorder (quadratic) polynomial. The second feature of complexity comes from the numberAttorney Docket No.37759.0576P1 of rows (of P’s), because these are coupled in the sense that the polynomial for Pimay contain Pj, where j ≠ i. Since we are unable at present to solve such a system of equations, we will use an iterative solutions approach given by Feldman’s non-linear iterations. This does point out the need to determine whether the solutions exist and are unique. For example, the roots of a polynomial may be negative numbers or complex numbers, both of which are extraneous, as we have previously described for the cubic solution and others have demonstrated for quartic solutions.

[0142] Though the Feldman equilibrium equations treated iteratively have been used to obtain equilibrium solutions of free ligand concentrations (P’s), it would be helpful to demonstrate existence and uniqueness of these solutions more generally. If P^^is the iterations of Eq 22a) and Q^^^be the iterations of Eq 22b). The limits limPఔ ൌ ^^ and limQఔൌ ఔ→^ ఔ→^ ^^ exist. Writing Feldman’s iterative equations as the functions ఔ = f (Q ) and Q ^^^g(PఔP^ ^ ି^), the composition is Pఔ= f∘g(Pఔି^) and obtainedP = f∘g(P), since f, g and ^^ ∘g are continuous functions. This last equation can be solved for Pas a system of coupled polynomial equations. Coupled refers to the fact the coefficients of the polynomial in Pi depend on the other Pk, k ≠ i .

[0143] If the limits P and Q do not exist, then we note that each Pi satisfies the inequalities, 0 ^ Pi ^ TotPi, which is a closed (compact) box and the Heine-Borel theorem guarantees at least one limit point P in this box and a subsequence of Pఔ’s converging to this P. The same argument as above guarantees that this P satisfies the system of coupled polynomials. In fact, for every limit point this is true. The existence and uniqueness of solutions of Feldman’s iterative equations does not require that we obtain analytic solutions, i.e., closed form formula. In fact, analytic solutions are readily available for the roots of polynomials of degree 4 or less, but it has been shown that the general 5thdegree polynomials do not have analytic solutions. Most think solutions should be obtained by iterative (or perhaps differential equation) numerical methods. Fortunately, showing existence and uniqueness of solutions does not require analytic solutions.

[0144] A relatively simple presentation of the system of coupled polynomials is obtained by eliminating fractions where possible in Feldman’s equations (22a and 22b). So,^^ 1 ^ ∑^^^ ^^்^௧ொ = ^ೕ^ ^ ^ୀ^ ^^ ^^ ^^^^^^^^and ^^^ൌ ^ . Substituting for ^^^in the first equation givesAttorney Docket No.37759.0576P124) ^^^ ^1 ^ ∑^ ^ୀ^ ^^்^௧ொೕ^^ ^ା ∑^^ ^ൠ = ^^^^^^^^^. which expresses the iteration in terms of P ೖసభೖೕ ೖalone.Define ^ =∏^ఔୀ^ ^1 ^ ∑^^ୀ^ ^^^ఔ^^^ ^ ൌ ∏^ ఔୀ^ ^^^^ ^ఔ^^^ ^ 1 ^ ∑^ஷ^ ^^^ఔ^^^ ^ ൌ ∏ఔୀ^^^^^ఔ^^^ ^^^ ^ ^^ 1 ^^^^ ^^^ ^ 1 ^^^^^^ ^^^^ ^^^^^^^^^^^^^^^^ ^^^^^^ℎ ^^^^^^^^^^^^^^ ^^^^ ^^^ ^^ഌ;^^So,25) ^^^ ^^^ ∑^ ^ୀ^ ^^^^ ^^^^^^^^^ ∏ఔஷ^ ^^^^ఔ^^^ ^ ^^^ఔ^^ = ^^^^^^^^^^^ ^ is a polynomial.a Equation 25) for Pi as a (m+1) degree polynomial in each Pi can be written as ^^^ା^^^^ା^^ ^ ^^^^^ ^^ ^ ⋯^ ^^^^^^^^ C = 0, where ^^ ^ ^^^^^^^^^^^^ ^ ^ ^^^^^^ ^^^ା^ ൌ ∏^^^ ^

[0145] Without loss in generality, it can be supposed ^^^ା^ ^ 0. It will become clear why^^^ା^ ൌ 0 is not a difficulty in this proof. Descartes’ Rule of Signs for polynomials says thenumber of positive roots is related to the number sign changes of the coefficients of the polynomial. For (m+1) degree polynomial 25), there is at least one sign change regardless of whether coefficients ^^ଶ⋯^^^ା^are positive or negative or zero. This implies there is leastone positive solution for Pi. For m replaced by m+1 and for ^^^^ା^ ^ 0 25) becomes a m+2degree polynomial. Note that ^^^^ା^ ൌ 0 leaves the degree of the polynomial and the patternof sign changes unaltered at m+1, which justifies the assumption that, without loss in generality, the leading coefficient ^^^ା^can be taken to be positive.

[0146] A scenerio was considered in which there is only one binding proteins Q and m=1. The 1x1 matrix format for equilibrium is: Q PNow K TotP and TotQ are known and conservation of mass gives two equations TotP =P(1+KQ) where TotQ=Q(1+KP). Substituting for Q,Attorney Docket No.37759.0576P1 TotP = P (1+ KTotQ / (1+KP)) = P (1+ TotQ / (Kd + P)). Clearing the fraction results in a quadratic polynomial: AP2+BP +C =0 where A= 1, B= Kd +TotQ -TotP and C= -Kd TotP and A is positive, C is negative and B could be either or 0. This implies there is exactly one sign change, so Descartes’ Rule of Signs implies there exists exactly one positive root.

[0147] For the nx1 matrix format for equilibrium, Q P1K1P1Q TotP1i n^^^^^^^^^^ ൌ ^^^^ ^ 1 ^ ^^^^ ^^^ and^=1 ^ ∑^ ^ୀ^ ^^^^^^ ൌ 1 ^ ^^^^^^ ^ ∑^ ^ஷ^ ^^^^^^, and clearingthe fraction, results in^^^^^^^^^^ ^1 ^ ^^^^^^ ^ ∑^ ^ஷ^ ^^^^^^ ^ ൌ ^^^^ ^ ^1 ^ ^^^^^^ ^ ∑^ஷ^ ^^^^^^ ^ ^ ^^^^ ^^^^^^^^^B=൫1 ^ ∑^^^ ^^ ൯ ^ ^^^^ ^^^^^^^^ െ ^^^^ ^^^^^^^^ and C= - ^^^^^^^^ ^1 ^ ∑^ ^ஷ^ ^ ^ ^ ^ ^ஷ^ ^^^^^^^ and the ^and Bbe either or 0,Rule of Signs implies there exists exactly one positive root.

[0148] When there is exactly one sign change, there exists exactly one positive root as the unique limit is true for the iterations for each Pi; Therefore, there exists one vector limit P. This shows the limit of iterations=P exists and is unique.

[0149] Rewriting Equation 25) using ^ = ∏^ఔୀ^ ^^^^ఔ^^^ ^ ^^^ఔ^ = ^^^^^^^^ ^Attorney Docket No.37759.0576P1 This second factor is a product of linear factors and thus is an (m-1) degree polynomialwhose roots are all negative ^^ ൌ^^ഌ^ െ^^ഌ for all ^^ ് ^^. The first factor is a quadraticpolynomial with A positive, C and one sign change so that it has exactly onepositive root (and one negative root). 1. Degree of Characteristic Polynomial

[0150] The degree (m+1) of the polynomial (see Section III-F) has been related to the number of binding proteins (Qj); however, we can reverse the roles of ligands and BPs and obtain (n+1) degree polynomial. This points out a slight inaccuracy in the presentation above; the truth relates to the number of Kii>0, which is related to r equals rank of the matrix K (r=rank(K)). Since the rank of K must be less than or equal to min (m,n) and often is less, the number of positive Kii’s may be less than (m+1) and (n+1). The factoring of the limiting polynomial into one quadratic factor and multiple linear factors in 25) provides a partial answer to the question. The factors of the limiting polynomial for Pi involve the Ki’s in the ith row of K. For the product of linear factors ∏_(ν≠i)▒{K_iν P_i+C_iν } , if K_iν = 0; then this term becomes a positive constant, the negative root is removed, and the degree of the polynomial is reduced by one. This is true for every K_iν=0 in the ith row of K. Now if Kii = 0, then A= Kii is zero and B becomes B = C_ii+ ∑_(j=1)^m▒〖K_ij TotQ_j〗_ , which is positive. Thus, the quadratic factor (AP_i^2 +BP_i +C ) becomes linear with one positive root and the negative root has been removed and the degree of the polynomial is reduced by 1. The result is that there is always one positive root and a reduced number of (extraneous) negative roots, and the degree of characteristic polynomial in P_(i ) equals 1 + number of positive Ki in the ith row of K.

[0151] For another view of the degree of the characteristic polynomial (in a different space), a matrix version of the Feldman equations can be used and the singular value decomposition of the matrix K. See Appendix 3 in Dorin RI, Qualls CR. Distribution of cortisol in human plasma in vitro: Equilibrium solutions for free cortisolusing equations of mass conservation and mass action.2024. In: Cortisol: Between Phsyiology and Pathology [Internet]. London, UK: IntechOpen.

[0152] . This decomposition says every nxm matrix can be factored as K= Q_(1 )∑〖 Q 〗_2^T where〖nxn matrix Q〗_(1 ) and mxm matrix〖 Q〗_(2 )are each orthonormal (〖 Q〗_1^T Q_(1 )=I_n and〖 Q〗_2^T〖 Q〗_(2 )=I_m) and where there are r ( =Attorney Docket No.37759.0576P1 rank(K)) positive singular values μ_i which form the diagonal of ∑ such that ∑▒〖= (■(diag(μ)&0@0&0))〗. Another way of saying this is K= i=1rμi p_i qiT where p_i ia a left singular nx1 vector corresponding to μ_i (column of Q_(1 )) and q_i is a right singular mx1 vector corresponding to μ_i (column of Q_(2 )).

[0153] ^^⊙ ^^ ൌ ^^^^^ ∙ ^^^^൧. Hadamard products are commutative:^^⊙ ^^ ൌ ^^ ⊙ ^^ ;where as the usual matrix product of square matrices is not generally commutative. TheHadamard (Schur) product of two vectors of the same size is similarly defined as ^^ ⊙ ^^ ൌ^^^^ ∙ ^^^൧. The Schur product of a vector and a matrix follows quickly. The Schur product ofvectors can be realized in the matrix algebra using square diagonal matrices like diag (v) whose diagonal is made up of the elements of the vector v and off diagonal entries are allzero: ^^ ⊙ ^^ ൌ ^^^^^^^^^^^^ ^^ ൌ ^^^^^^^^^^^^ ^^ . Also, diag(v) 1 ൌ ^^ , where 1 is a vector withentries all equal to 1. Conversely, diag (v) can be realized using a Schur product:^^^^^^^^ ^^^^ ൌ ^^ ⊙ ^^ = ^^ ⊙ ^^ , where ^^ is the usual matrix identity. The important part of theFeldman equations involves diag(P) K diag(Q) =^^^^^^^^^^^^ .

[0154] The pre-multiplication by diag(P) can be represented by ^^⊙ ^^, but the post-multiplication by diag(Q) operates on the columns of K and is not easily represented by the Schur product ⊙ since the candidate expression of ^^⊙^^ operates on rows by commutativity and not on columns as required. Attempts at two other expressions give Kdiag(Q) = K (Q⊙ ^^^. This requires a distributive law for computations, which is said not toexist; or alternatively K diag(Q) = ^Q ⊙^^்^், which is also difficult to use in computations.

[0155] Feldman equations 22a) and 22b) with the nxm matrix K, nx1 vector ^^ ൌ ^^^^^ ,and mx1 vector ^^ ൌ ^^^^൧ can be written asAttorney Docket No.37759.0576P1 This last matrix equation is comparable to Feldman’s system of polynomial equations expressed in matrix form. The term T1 = ^^^^^^^^^^^^ ^^ ^^^^^^^^^^^^ ^^்^^^^^^^^^^^^ is a quadraticform. Now substitute ^^ ൌ ∑^ ^ୀ^ ^^^ ^^^^^்^ ^^^^^^ ^^் ൌ ∑^ ^ୀ^ ^^^ ^^^^^^்into this term of the matrix equation in order to write the product of sums as a double sum does not seem to advance toward ait is also a quadratic matrix equation in X= diag (P) of the form. 26) XAX + BX + CX = D, where ^^ ൌ ^^ ^^^^^^^^^^^^ ^^் which is nxn, symmetric, nonnegative definite matrix ofrank r = rank (K) < min(n,m)., B = 0 , a nxn matrix of zeros C = - (I + ^^ ^^^^^^^^^^^^^^^^^^1^^, D = - diag (TotP), where the coefficient C requires some modification to be (nxn) matrix.

[0156] Since the coefficients still depend on Q, which requires iterative solution, Eq 26) will not provide an analytic solution for computing a numerical answer. It might show that the solution P exists and is unique as the only positive root of equation 26).22a) P + (^^ ⊙ ^^^^^ ൌ ^^^^^^^^^^^ ^^^^^^^^ ^^^^^^^^ ^^^solutions could be addressedby using an exhaustive set of initial values (for either P’s or Q’s). Solutions that are consistent across a broad array of initial conditions are more likely to represent unique solutions. These considerations also acknowledge the potential application of iterations and domains of strange attractions. 2. Three Methods for Obtaining Equilibrium Solutions

[0158] A third method can be accomplished using differential equations, as suggested by the development of dynamic solutions. This same approach using differential equations can also be applied to in vitro conditions of the test tube, for which dynamical factors such as appearance, elimination, and diffusion are no longer required.

[0159] The 4-compartment cortisol model differential equations conserve mass over time (summed across the four compartments) and the equilibrium is obtained in less than one second. Steady state in vivo takes a much longer period of time. For example, the steady state that obtains as an asymptotic function of cortisol concentrations in both vascular andAttorney Docket No.37759.0576P1 extravascular distribution volumes during continuous cortisol infusion takes 6-12 h in vivo. By contrast, the equilibrium related to binding of free ligands to BPs in the plasma volume in vivo or in the test tube ex vivo, takes place on a much shorter time scale than steady state related to in vivo elimination and diffusion.

[0160] Most blood tests are measurements of equilibrium concentrations in a test tube. In the test tube, the blood can be considered as being in a vascular space and there is no diffusion of cortisol to interstitial / intracellular space. There is no continuous elimination of cortisol in the test tube, and no appearance / infusion of cortisol in the test tube. Elimination, appearance (infusion), and diffusion were the processes that are balanced over the longer time-scale in steady state.

[0161] In the test tube, the 4-compartment model of cortisol is reduced to 3 compartments (without the extravascular compartment) and without the terms for appearance, elimination, and diffusion. 1^ ௗ^ಷௗ௧ൌ ^^^ି^ ∗ ^^^ி^ െ ^^^ ^X்^௧^^ீ െ ^^ி^^ ∗ ^^ி^ + ^^^ି^ ∗ ^^^ி^ െ ^^^ ^^^்^௧^ െ ^^ி^^ ∗Where ^^^^ భ^ ൌ^ ^^ష^భand ^^^^^^^^^ ^^ି^are on- and off-rates for cortisol-CBG binding, respectively.nd ^^^ ൌ ^ಲ Aభಲand ^^^^^^^^^^ ^^ି^^ are on and off rates for cortisol-albumin binding,By setting the derivative of Eq 2 to zero, which gives: [FC] = ^^^[C][F], which is a mass action equation. Here [C] = [TotCBG]-[FC] is due to conservation of mass law. Similarly, setting the derivative in Eq 3) to zero results in a mass action equation [FA] = ^^^[A][F], where free albumin concentration [A] = [TotA]-[FA]. To understand Eq 1) about free cortisol concentration (XF), add Eq 1, 2, and 3 together. On the left-hand side: ௗ^ಷ+ ௗ^ಷ^+ௗ^ಷಲ= ௗ^^^^ಷ, since by conservation of mass XTotF = XF + XFC + XFA.Attorney Docket No.37759.0576P1 On the right-hand side, there is 0; that is, derivative of ௗ^^^^ಷௗ௧ = 0, which implies XTotFis equal to a constant.

[0162] Thus the constant = TotF(0) at time = 0 and all times thereafter. So this implies that the mass of total cortisol (summed across the three compartments) is constant across all time and, assuming that the volume remains constant, the concentration (XTotF) is also constant across all time.

[0163] Now the connection to the iterative equations coupling mass action and mass conservation follows as described in Feldman et al. For example, [TotF] = [F]+ ^^^[C][F] + ^^^[A][F] = [F](1+ ^^^[C] + ^^^[A]), or [F] = ^்^௧ி^ ^ା ^^^େ^ ା ^ಲ^^^ , which is a Feldman first equation.

[0164] Feldman’s be developed similarly. Insummary, this means of Eq 1-3) above will provide solutions that also satisfy Feldman’s equilibrium equations. Thus, the proof of existence and uniqueness also applies to the limiting differential equation solution.

[0165] The requirements of no infusion, no elimination, and no diffusion of cortisol in the test tube represents an in vitro vascular compartment. These requirements are equivalent to the total cortisol (XTotF) being constant. For example, if there were diffusion, XTotFwould not be constant for all time. This is true in steady state (both in vivo and in vitro), which is typically a limit (in time), i.e. asymptote. Eq 1) is needed to represent these requirements in the system of differential equations; in fact, Eq 1) enforces these requirements.

[0166] To summarize the impact of this matrix point of view, there are 3 elements represented in Feldman’s matrix equation: Affinity matrix (K) representing ligand-protein binding equilibrium association (affinity) constants (typically known from experimental studies reported in the literature, see for example Table 1); Total concentrations of both binding protein(s) and ligand(s) (typically measured); Concentration of free ligands and free binding proteins (here typically computed using Feldman’s iterative equations or the roots of coupled polynomial equations or by differential equations). Estimation of these elements may give rise to different types of problems:

[0167] Direct solution of free ligand concentrations, herein called ‘inner loop’ solutions, i.e. computation of free ligand concentrations using known / measured values of totals and cognate binding affinities. This represents the traditional application of Feldman equations, by which measured total concentrations of ligand(s) and binding protein(s) are applied in conjunction with known equilibrium association parameters to derive free ligandAttorney Docket No.37759.0576P1 concentrations, typically using algebraic solutions for coupled polynomial equations or iterative solutions. Given k’s and totals, compute free.

[0168] Inverse problem of estimating selected binding affinities, which requires data and non-linear regression, herein called ‘outer loop’ solutions. Note that the outer loop for least squares solutions may require iteration to obtain ligand solutions (as an inner loop) at each iteration of the outer loop, as illustrated in the example shown in Section III-G. This use of iterative procedures for function evaluation is distinct from the more typical problem where an algebraic function evaluation is used in the inner loop. Given totals and free, compute selected K’s.

[0169] Estimation of free ligand concentrations without experimental measurement of high-affinity binding protein concentration(s) by interruption of the equilibrium matrix. This approach requires an assay procedure for measuring concentrations of protein-bound ligand(s) in select cells (in the matrix). We have used the term of interruption of the equilibrium matrix to describe such procedures. In one example, shown in Section III-G, the equilibrium matrix was interrupted by measurement of free cortisol concentration (XF) as well as total (XTotF). In the proposed competitive ligand-binding assay, the matrix is interrupted by measurement of CBG-bound ligand. In this situation, the (labeled) ligand need not have affinities for CBG and albumin binding identical to that of the endogenous ligand of interest, as described in Section V. Interruption of the matrix by measurement of concentrations of BP-bound ligand would permit the estimation of concentrations of free ligand (and BP) without the need to directly measure concentration(s) of the high-affinity transport protein. Given most totals and concentrations of selected protein-bound ligand, compute the missing totals and / or free ligand concentrations. F. Solutions for Ligand-Protein Binding Matrix and Interrupted Ligand-Protein Binding Matrix

[0170] Hormone disposition modeling can often be recognized as manipulations in compartmental models having mathematical constructs. The compartments may be physical in nature, such as rational blood flow and organs and interstitial volumes, or different species of the hormone such as free cortisol, CBG-bound cortisol, both in the vascular system. Test beds discussed here focus on circulation in human physiology but could be applied to mammalian models or other species. The main concept of a hormone test bed is that a realistic, true to physiology, hormone disposition model can be constructed without regard of whether parameters can be estimated or not. Such a test bed model can be used to estimateAttorney Docket No.37759.0576P1 realistic data and this data can be used to develop a minimum disposition model where the parameters are estimable.

[0171] The dynamics of compartmental models are represented by differential equations, typically one differential equation for each compartment. In forward-solved differential equations, values of parameters and initial conditions are specified in advance and then the dynamic disposition of each compartment is computed by numerical methods appropriate to solving a system of differential equations. This is suitable for use as a test bed model. The inverse problem consists of fitting experimentally derived data with a model of differential equations by adjusting the parameters. This is suitable for evaluation of a minimal model. The test bed is useful in the development of a minimal model by providing realistic simulated data for testing variations in the minimal model using the inverse (backwards solutions) for the parameters of the candidate minimal model. A basic methodology is the comparison of two candidate minimal models in terms of the accuracy of parameter estimates (bias, SD, root mean squared errors) computed with respect to the test bed model using simulated data (that simulated the data): this accuracy is compared between the two candidate minimum models.

[0172] However, it is known that there are algebraic solutions up to 4 degrees (quartic equations), which generally do not exist for degree 5 (quintic equations). It may be possible to reduce the degrees of the polynomials, as for example in Coolens’ approximation that turns a cubic equations (3rddegree polynomial) into a quadratic equation (2nd degree polynomial). The reason the Coolens’ simplification was feasible is because the albumin-cortisol binding reaction is considered to be non-saturable. That is because the KD for albumin-cortisol binding is very high relative to physiologic concentrations of free cortisol. A similar approach to simplifying sources of complexity may also be applicable to analogous problems or situations in the Feldman matrix.

[0173] Using 1x2 matrix model of Feldman equations with Coolen’s approximations, FA=NFA*F and EA=NEA*F, iterating to equilibrium solutions of F0 in test tube#1 for multiple cases of TotC and TotF and separately equilibrium solutions of F, E, and EC in test tube#2 for these same cases of TotC and TotF as well as for varying added concentrations TotE (experimentally, amounts of E and volumes are used to calculate the added concentration TotE). This is a test bed for computing the calibration curve of F0 vs EC. Assuming TotF and TotA (or N) are known in test tube#1 and input TotE and measured EC are known in test tube#2. but in the assay, F0 and TotC are unknown. Equations 15’-18’) of the assay are used to estimate E, then F in test tube#2, then TotC and then F0. In this testbed, these estimatesAttorney Docket No.37759.0576P1 match the input TotC and F0 with relative errors of 10-4(0.01%) or less. This assay can be described by an interrupted Feldman 2x2 matrix, where the measurement of EC instead of TotC interrupts the need to iterate in order to know equilibrium solutions. An algebraic solution is obtained. The solutions are accurate across a wide range of total cortisol concentrations, including those encountered during physiological tests for Cushing’s syndrome (e.g. DEX suppression test), adrenal insufficiency (cosyntropin stimulation test), and hypercortisolism associated with critical illness, septic shock, and Covid-19 pneumonia. Obs TotC TotF TotE assay F0 F_assay F0_assay 1 300 50 33 300.007 4.140 4.365 4.140Attorney Docket No.37759.0576P1 300 800 33 300.030 139.427 141.171 139.360 300 800 66 300.031 139.427 142.892 139.360Attorney Docket No.37759.0576P1 67 600 100 528 600.048 5.060 9.576 5.060 68 600 150 33 600.017 8.059 8.390 8.059

[0174] If ththen two interruptions EC1 and EC2 are needed. Consideration of a distinct species of CBG having lower affinity that usual would add an additional column to the Feldman matrix. Consideration of additional ligands competing with cortisol for CBG binding would add an additional row to the Feldman matrix.

[0175] The above formulation also addresses the situation where there is a significant concentration of unlabeled E in the serum sample. 12’)்^௧ா்^௧ா∗ ்^௧ிாൌா∗ൌி , %E = 100*E / TotE, etc. and14’) ி^ൌி^ൌிൌ்^௧ி In this formulation, the equations in Lemma 1 still obtain and are still tractable to solution so long as the initial mass (and concentration) of unlabeled E is known.

[0176] The above formulation also addresses the situation where there is an unknown concentration of a compound that also binds to CBG with significant affinity. Where there is a significant concentration of progesterone (P4) in the sample, which concentration may not be recognized or accounted for in the analysis, should be considered. As noted above, CBG has a significant affinity for P4, albeit reduced relative to CBG-cortisol binding. The presence of compounds having significant CBG binding affinity that would compete with F and E for CBG binding. If unaccounted for, the presence of additional competitive ligand (e.g. P4, prednisolone or aldosterone) would lead to underestimation of CBG concentration and also underestimation of free cortisol concentration. However, if the concentration of ‘unknown’ CBG binding ligands or their affinity for CBG is sufficiently low relative to cortisol, then the extent of bias related to unknown, unmeasured compounds may be negligible.

[0177] There are a number of situations in which CBG may have a different binding affinity for cortisol than usual. For example, there are genetic mutations of CBG that result in decreased cortisol binding affinity (increased KD), and also genetic variation in the CBG gene affecting its concentration and cortisol binding affinity. In addition, there are alterations in CBG glycosylation that can affect immunoreactivity and cortisol binding activity of CBG in conditions such as sepsis and inflammation. In such cases, discordance between immunoreactive CBG concentration, measured by ELISA for example, and cortisol bindingAttorney Docket No.37759.0576P1 capacity is expected. Indeed, variation in binding affinity of the serum transport protein has confounded accurate calculation of free hormone concentrations using mass action equations. In this regard, the functional nature of the proposed assay if an important advantage. Since the disclosed assay only detects CBG by virtue of its cortisol binding activity, CBG proteins having decreased binding affinity (increased KD) would at a lower concentration compared to immunometric assay. Thus, the CBG concentration measured by the lectin-affinity method is treated as an effective CBG concentration. This is useful, since discordance between cortisol- binding activity and immunoreactive CBG has been observed for certain anti-CBG antibodies. G. Methods for Measurement of CBG-Bound Labeled Ligand (EC) in Clinical Samples

[0178] Two components of the experimental method involve (i) development of conditions for specific and high-affinity binding of labeled ligand (E) and CBG and (ii) separation of CBG-bound E (EC) from albumin-bound and free E. Once conditions for obtaining a colorimetric signal specific for EC using appropriate positive and negative controls are established experimentally, reference standards will be generated for development of a standard curve that can be used to quantify concentration of EC in test tube #2. 1. Labeled Ligands In Vitro

[0179] Initial conditions examined CBG binding activities of two alternative labeled cortisol ligands that have been previously validated in immunoassays (but not in CBG- binding assay), namely (i) cortisol conjugated with the enzyme horseradish peroxidase (HRP) and (ii) cortisol conjugated with biotin. Binding reactions are performed at 37°C, at which temperature the reaction is expected to rapidly achieve equilibrium following addition of labeled cortisol (E) to test tube #2. The presence of conjugate on the cortisol molecule may reduce affinity for CBG binding relative to unlabeled cortisol. Crystal structures of CBG- bound cortisol and CBG-bound progesterone suggest that hydroxyl groups associated with C- 11 and C-17 form hydrogen bonds within the CBG steroid binding pocket and are critical for high-affinity CBG-cortisol binding. Therefore, initial evaluation of options for HRP- or biotin-labeled ligands take advantage of reagents that avoid disruption of at these sites.

[0180] For example, commercially available HRP-labeled cortisol is conjugated on the 4- position of the cortisol A ring. Alternatively, a 3-(o-carboxymethyl)oxime conjugate has been developed for cortisol immunoassays. Use of HRP-labeled cortisol provides a direct signal for ES detection following addition of substrate (tetramethylbenzidine (TMB). Use of biotin-Attorney Docket No.37759.0576P1 labeled cortisol would require secondary addition of streptavidin-conjugated HRP (SA-HRP). High biotin concentrations in clinical samples obtained in patients who had recently ingested exogenous biotin would affect assay results, as has been recognized for many other biotin- streptavidin based immunoassays. Although the preference is to use labeled ligand, other approaches may be considered, including a sandwich-type assay by which CBG is ‘captured’ by adsorption to the ConA affinity matrix and EC recognized by (labeled) x-cortisol antibody.

[0181] Initial conditions and time course for ConA binding can be established using recombinant, human CBG protein. Recombinant human CBG expressed in Sf9 insect cells is glycosylated and retains steroid-binding activity. Binding reactions are performed in 350 µL volume using commercially prepared, ConA-coated microtiter plates. As appropriate, further characterization of E-C binding by traditional ligand-protein methods, such as Scatchard analysis, may also be used. 2. Separation of CBG-Bound E (EC) from Albumin-Bound and Free E

[0182] In initial studies the binding reactions for CBG and E can be performed in a separate reaction tube and then transferred to the ConA coated plate. ConA binding buffer consists of 50mM Tris, 0.5 M NaCl, 1 mM CaCl2, 1 mM MgCl2, and 1 mM MnCl2 at pH 7.4. Among healthy l adult males, the concentration of CBG is in the range of 400-650 nmol / L, and cortisol (XTotF) is measured in the range of 50-800 nmol / L. Therefore, we will attempt to vary concentrations of these reactants in a similar range of concentrations. An alternative separation technique uses antibody to CBG to localize CBG to the microtiter plate surface.

[0183] CBG binding to ConA may be nearly quantitative. However, quantitative binding of CBG is not a requirement of the proposed lectin-affinity assay. Rather, the objective of the assay is to ascertain the concentration of EC using a standard curve. Nonetheless, the efficiency of ConA binding of EC will be assessed experimentally. There are many other glycoproteins that demonstrate N-linked glycosylation and, in theory, might compete with EC for binding to immobilized ConA. Assessing saturable binding of CBG to the ConA matrix can be done by examining the effect of additional glycoproteins containing N-linked oligosaccharides and ConA binding activity but devoid of cortisol binding activity (e.g. recombinant SHBG). The contents of test tube #2, having reached equilibrium after incubation at 37°C, is transferred to ConA-coated multiplate.

[0184] After incubation of the contents of test tube #2 with adsorption of EC to the ConA matrix and subsequent wash cycles, to which For HRP-labeled cortisol, the following washAttorney Docket No.37759.0576P1 cycles, 150 µL of HRP substrate, TMB (3,3',5,5'-tetramethylbenzidine), is added and microplates are incubated for 15 min on plate shaker (200 rpm). Cleavage of TMB by HRP to yield colorimetric signal is time and temperature-dependent; these variables will be examined experimentally. The HRP enzymatic reaction is concluded by addition of stop solution (2N hydrochloric acid). After this step microplate can be read at 450 nm for blue color detection on the Biotek Cytation5 station. The optimization of and number of wash cycles will be evaluated experimentally. Various blocking agents are used in the preparation of ConA microtiter plates, including albumin. Use of positive and negative controls can be used to assess the potential impact of non-specific binding of E to albumin or other blocking agents used in the microtiter plates (i.e. non-specific matrix binding). with two negative controls, which will also be run in duplicate.

[0185] Conditions and times for optimization of the assay procedure can be developed using recombinant CBG. Negative controls include (i) paired samples subjected to heat denaturation at 60°C for 60 min, which irreversibly denatures CBG but does not interfere with albumin-cortisol binding and (ii) paired samples to which an excess of unlabeled cortisol (20,000 nmol / L) has been added.

[0186] A standard plot using either lots of heat-inactivated human serum to which known mass / concentration of recombinant CBG has been added or, alternatively, use of standardized lots human serum having known (measured) concentrations of CBG can be developed. Initial conditions for the standard curves can be developed by varying the concentration of E. Once those conditions are optimized, the usual standard curve can involve addition of known concentrations of unlabeled cortisol, which are expected to diminish the colorimetric signal in dose-dependent fashion. As above, paired negative controls to ascertain specific binding include heat-activated sample. Potential utility of measuring concentration of E in the supernatant (unbound fraction) and expression of the relationship by bound / free ratios (B / F) using Scatchard type plot can also be evaluated.

[0187] To validate the competitive ligand binding assay, comparing estimated values for CBG and free cortisol concentration with those measured by traditional experimental methods (immunoassay for CBG, ultracentrifugation for free cortisol). CBG concentrations measured by immunoassay can be compared to those computed using the lectin-affinity assay in a subset of serum samples. Results can be compared by Spearman correlation to capture non-linearities and Bland-Altman plots. Comparison of measured and computed free cortisol concentrations can also be assessed. Measured concentrations can be compared to freeAttorney Docket No.37759.0576P1 cortisol concentrations computed by the lectin-affinity method using Eq 15’-18’ and compared by correlation and Bland-Altman plots as above. A sub-analysis can also be performed to assess the role of competitive binding by endogenous ligands other than cortisol by adding known concentrations of competing steroids (e.g. synthetic cortisone, progesterone, or prednisolone). Studies can also address microheterogeneity in glycosylation affecting lectin binding.

[0188] There may be additional information regarding the concentration of XF in test tube 1 that can be gleaned, with little additional effort or expense, by determination of the concentration of EC in additional test tubes to which known quantities (and concentrations) of E, F, albumin, or recombinant CBG have been added.

[0189] The presence of unmeasured compounds that significantly compete with E and F for CBG binding could be (incorrectly) interpreted as a reduced apparent affinity for CBG- cortisol binding (reduced Ka, increased Kd). Allowing for that simplification, in clinical samples where concentrations of [EC] are low, it is possible to use in experimental methods that might usefully distinguish relative contributions of decreased concentration of CBG vs. decreased CBG-cortisol binding affinity. The former situation is associated with decreased maximal binding capacity (Bmax), whereas the latter situation is associated with ‘normal’ Bmax.

[0190] Using a test bed approach, as described above, addition of CBG-binding ligands at physiologic concentrations results in disruption of EC binding leading to a concentration- dependent decrease in [EC]. Therefore, in situations where it may be clinically useful to distinguish between low CBG concentrations vs. low CBG-cortisol binding affinity, additional test tubes in which conditions that systematically disrupt the non-linear Feldman equilibrium may be usefully applied (^ [EC]) as a function of exogenous progesterone added to the test tube. 3. Addition of Ligands that Selectively Disrupt CBG-Cortisol vs. Albumin-Cortisol Binding Reactions

[0191] As an example of differential affinities, in testbed solutions, [EC] is significantly decreased at relatively modest concentrations (0-200 nmol / L) of added progesterone. However, these concentrations have negligible impact on albumin-cortisol binding. Therefore, any ^ [EC] observed following addition of nanomolar concentrations of progesterone can be attributed to its effects on CBG-ligand binding. Much higher concentrations of progesterone, in the range of 600-1200 ^mol / L, would be required toAttorney Docket No.37759.0576P1 decrease the concentration of albumin-bound cortisol (and albumin-bound E).

[0192] As an example of differential specificities, the contrast between specific binding of high-affinity transport proteins (compared to relatively non-specific binding of albumin) to isolate the effects of added ligand on albumin-cortisol binding can be used. T4 and cortisol share and compete for binding to the albumin molecule. The affinity constant for CBG-T4 binding is effectively zero. Large quantities of T4 may be added to test tubes #3, #4, etc. to achieve test tube concentrations in the range of 600-1200 ^mol / L. These high concentrations of T4 lower [EC] by disrupting albumin-cortisol binding; the resulting increase in free cortisol concentration in test tubes #3, #4, etc. then compete for CBG binding. Examining ^ [EC] as a function of T4 concentrations provides additional insight into the apparent Ka for albumin-cortisol binding. This amount of cortisol displaced from albumin may be more precisely understood by using antecedent studies with unlabeled progesterone (or cortisol) as a sort of calibration curve. By measuring albumin concentration and calculating the apparent sample-specific affinity constant for albumin-cortisol binding, a more realistic value for N may be ascertained. The term ‘effective’ binding affinity is used to acknowledge that apparent Kd may be a simplified representation of (multiple) competitive binding reactions involving other hormones, free fatty acids, etc. that non-specifically interact with circulating albumin molecules in the plasma volume.

[0193] Dilution of serum / plasma in ligand-binding buffer can present several difficulties. So-called matrix effects in this case refer to pleiotropic anomalies in experimental results related to idiosyncratic effects of serum that indirectly influence apparent ligand-protein binding characteristics. These matrix effects may be related to diverse factors, including presence of proteins or other compounds that inhibit or promote lignad-protein binding reaction(s) in competitive or non-competitive fashion.

[0194] As highlighted in Table 1 above, there are several differences between testosterone / SHBG and cortisol / CBG that impact the application of the proposed methodology to estimation of free testosterone (or free estradiol). These include (i) lower concentrations of SHBG (~-10-60 nmol / L) relative to CBG (~600 nmol / L), (ii) higher affinity of albumin-testosterone binding, and (iii) potential for allosteric interactions affecting SHBG- testosterone binding kinetics. Consequent to these differences, concentration of SHBG- and albumin-bound testosterone concentrations in the normal, healthy adult male population are roughly equal, and the concentration of free testosterone as a percent of total testosterone (XTotT) (~2%) is substantially lower than percent free cortisol (~4%).Attorney Docket No.37759.0576P1

[0195] The Endocrine Society Clinical Practice Guideline for Testosterone Therapy in Men with Hypogonadism recommends measurement of free testosterone in conditions associated with increased or decreased SHBG levels, as summarized in Table 2. With regard to measurement of free testosterone, the guideline states: “In men who have conditions that alter SHBG or whose initial total testosterone concentrations are at or near the lower limit of the normal range, clinicians should determine free testosterone concentrations either directly from equilibrium dialysis assays or by calculations that use total testosterone, SHBG, and albumin concentrations. Clinicians should not use direct analog-based free testosterone immunoassays, as they are inaccurate.” However, equilibrium dialysis methods are cumbersome and expensive, while calculations of free testosterone using measured concentrations of total testosterone, SHBG, and albumin have several limitations.

[0196] The human SHBG gene is located on chromosome 17 and expressed primarily in the liver. Human SHBG has MW ≈90kDa and circulates as a homodimer; consequently, each SHBG dimer binds two molecules of testosterone, with potential for cooperative binding interactions. Two biantennary N-linked oligosaccharides and one O-linked oligosaccharide are attached to each monomer. The N-glycosylation consensus site positioned closest to the C terminus of SHBG (Asn367-Gly-Thr in human SHBG) is highly conserved in all known mammalian sequences, which suggests that glycosylation of this site in particular is functionally important. In addition to testosterone, SHBG binds dihydrotestosterone (DHT) and estradiol (E2) with relatively high affinity.

[0197] Several genetic variations of SHBG gene have been identified and linked to serum concentrations of both SHBG and testosterone in genome wide association studies. In addition to its role in transport of steroid hormones, SHBG has been linked to biological activities of sex steroids and other conditions. The binding activity of SHBG, like CBG and DBP, can be irreversibly denatured with heating to 60°C for 60 min, which process does not affect testosterone-albumin or cortisol-albumin binding activities. Thus, as in the CBG- cortisol assay described above, heat-denaturation of clinical samples serves as a useful, simple, and inexpensive negative control in the ligand-transport protein binding assay. The presence of multiple sites for N-linked glycosylation should support binding to the concanavalin matrix using the proposed lectin-affinity separation method. There is good experimental evidence that hormone-bound SHBG binds ConA with the same avidity as free SHBG; steroid-binding does not interfere with lectin binding of SHBG. There does appear to be genetic variation in the number of glycosylation sites of human SHBG, which could resultAttorney Docket No.37759.0576P1 in differential affinities for the lectin matrix. The well-characterized variations in SHBG glycosylation and genetic variation in SHBG binding affinities for testosterone likely contribute to the unsatisfactory performance of free testosterone estimates calculated on the basis of measured concentrations of total testosterone (XTotT) and SHBG (XTotSHBG). Similar limitations associated with calculation of free cortisol and vitamin D metabolites using measured concentrations of CBG or DBP, respectively, have been similarly observed.

[0198] As in the CBG-cortisol assay, labeled ligand (E), such as HRP- or biotin- conjugated testosterone, need not have the same affinity as testosterone for SHBG (or albumin) binding. The lower concentration of SHBG in human plasma could pose limitations in sensitivity of detection of EC in the free testosterone assay. The flexibility of the proposed assay to use ligands of different transport protein binding affinities than the endogenous hormone of interest are again noted to be potentially advantageous insofar as DHT has greater than 2-fold higher affinity for SHBG than testosterone. Use of HRP- or biotin- labeeled DHT may therefore prove useful in obtaining a reliable signal for the lectin-bound EC complex. Recombinant human SHBG expressed in Sf9 insect cells is glycosylated and retains steroid-binding activity and can be used as a positive control and to establish initial experimental conditions.

[0199] Also shown in Table 1, albumin binds testosterone with ~10-fold higher affinity compared to cortisol However, the KD for albumin-testosterone binding still greatly exceeds the concentration of free testosterone and Coolen’s simplification assumption of non- saturable binding, as discussed above, is still applicable. As in the case of cortisol, we would advocate for application individualized N values based on measured rather than assumed concentrations of serum albumin. Also, similarly to cortisol, we believe that additional experimental work confirming the assumptions of Coolens’ N and evaluating the influence of other albumin-binding substances in plasma are warranted. There is an extensive literature evaluating the clinical value of bioavailable testosterone measurements. Bioavailable testosterone, i.e. the sum of albumin-bound and free testosterone, may be measured in the laboratory ammonium sulfate precipitation of SHBG-bound testosterone. While the theory that ‘weakly bound’ albumin-testosterone can dissociate and cross the endothelial cell membrane during capillary transit has its limitations, the separation of SHBG-bound from [albumin-bound plus free testosterone] is similar in concept to the proposed ligand-transport protein assay. Moreover, if the KD for albumin-testosterone binding is well-defined and serum albumin is appropriately measured, the concentration of free testosterone can beAttorney Docket No.37759.0576P1 reasonably estimated from the bioavailable testosterone concentration. However, ammonium sulfate precipitation is not routinely performed in the clinical laboratory setting, and the technique typically involves use of a tracer, such as3H-testosterone. Ascertainment of free testosterone can be done using multi-well plate format without requirement for specialized procedures or reagents would provide simpler, less expensive, and more reliable estimate of free testosterone concentrations in clinical samples. 4. Adaptation of Binding Assay in Human Plasma

[0200] The rationale for determination of free (rather than total) concentrations of vitamin D metabolites of interest, such as 25-OH vitamin D (25OHD), 1,25-OH vitamin D (calcitriol), and 24,25-(OH)2-vitamin D (24,25-OHD) (see Table 2), has been previously studied. There are several distinctive features of DBP binding to its associated ligands that directly impact the application of the proposed ligand-transport protein binding assay. For example, one notable feature is that concentrations of DBP in human plasma (~6 ^mol / L = 6,000 nmol / L) greatly exceeds those of CBG (400-750 nmol / L) and SHBG (10-60 nmol / L). As well, the concentration of DBP in human plasma greatly exceeds the concentrations of its cognate binding ligands (free 25OHD, free calcitriol, and free 24,25-OHD). As a result, the concentration of free vitamin D metabolites as a percent of total concentrations (the sum of DBP-bound, albumin-bound, and free vitamin D metabolites) is remarkably reduced (<0.5% for calcitriol and <0.1% for 25OHD). The substantial excess of DBP relative to free ligand means that, unlike the situations for SHBG / testosterone and CBG / cortisol, the binding reactions for vitamin D metabolites and DBP are, for practical purposes, non-saturable. Indeed, simplifications such as Coolens’ N could be considered for the estimation of plasma / serum concentrations of free 25OHD, calcitriol, and 24,25OHD were it not for (i) difficulty in measuring DBP and (ii) remarkable variability in DBP affinity for its cognate ligands. DBP is highly polymorphic, as suggested by its original characterization as group- specific component (GC). This heterogeneity appears to be driven in part by genetic factors, as genome-wide association studies for 25OHD concentrations in US women have established significant linkage with both DBP (chromosome 4) and CYP2R1, a cytochrome P450 enzyme that hydroxylates vitamin D to form 25OHD (chromosome 11).

[0201] A second notable feature is the absence of N-glycosylation of DBP. Although DBP is a glycoprotein characterized by the presence of O-linked glycosylation sites, the absence of N-glycosylation sites demonstrates that, in contrast to CBG and SHBG, DBP would not be a suitable candidate for lectin-affinity separation. Therefore, other methods,Attorney Docket No.37759.0576P1 such as antibody-coated microtiter plates using antibodies that bind DBP, would be useful for the separation of ligand-bound DBP. For such ligand-binding assays for DBP, HRP-labeled or biotinylated 25-OH (or calcitriol) could serve as labeled ligand (E).

[0202] As noted in the above discussion of CBG / cortisol and SHBG / testosterone, the flexibility to use ligands having different binding affinity to DBP than the endogenous ligand of interest may be advantageous. For example, a higher affinity ligand (HRP-DHT) may be advantageous for the free testosterone assay in order to adapt to the relatively low plasma concentrations of SHBG. Conversely, a labeled ligand having lower affinity for DBP, such as HRP-labeled calcitriol, may be useful to adapt to the relatively high plasma concentrations of DBP and could provide better separation of EC values than HRP-labeled 25OHD. Similar principles likely inform the successful application of labeled T3, which has a lower affinity for TBG than T4 (see Table 1), in the T3RU assay. H. References

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[0332] 130. Van Baelen, H., et al., Genetic variation of human sex hormone- binding globulin: evidence for a worldwide bi-allelic gene. J Clin Endocrinol Metab, 1992. 75(1): p.135-9.

[0333] 131. Jayaraj, A., et al., Allosterically Coupled Multisite Binding of Testosterone to Human Serum Albumin. Endocrinology, 2021.162(2).

[0334] 132. Shea, J.L., P.Y. Wong, and Y. Chen, Free testosterone: clinical utility and important analytical aspects of measurement. Adv Clin Chem, 2014.63: p.59-84.

[0335] 133. Schwartz, J.B., et al., Determination of Free 25(OH)D Concentrations and Their Relationships to Total 25(OH)D in Multiple Clinical Populations. J Clin Endocrinol Metab, 2018.103(9): p.3278-3288.

[0336] 134. Schwartz, J.B., et al., A comparison of measured and calculated free 25(OH) vitamin D levels in clinical populations. J Clin Endocrinol Metab, 2014.99(5): p. 1631-7.

[0337] 135. Molenaar, N., A.B. Groeneveld, and M.F. de Jong, Three calculationsAttorney Docket No.37759.0576P1 of free cortisol versus measured values in the critically ill. Clin Biochem, 2015.48(16-17): p. 1053-8.

[0338] 136. Hogeveen, K.N., et al., Human sex hormone-binding globulin variants associated with hyperandrogenism and ovarian dysfunction. J Clin Invest, 2002.109(7): p. 973-81.

[0339] 137. Bocchinfuso, W.P., S. Warmels-Rodenhiser, and G.L. Hammond, Structure / function analyses of human sex hormone-binding globulin by site-directed mutagenesis. FEBS Lett, 1992.301(2): p.227-30.

[0340] 138. Bikle, D.D., S. Malmstroem, and J. Schwartz, Current Controversies: Are Free Vitamin Metabolite Levels a More Accurate Assessment of Vitamin D Status than Total Levels? Endocrinol Metab Clin North Am, 2017.46(4): p.901-918.

[0341] 139. Bikle, D.D. and J. Schwartz, Vitamin D Binding Protein, Total and Free Vitamin D Levels in Different Physiological and Pathophysiological Conditions. Front Endocrinol (Lausanne), 2019.10: p.317.

[0342] 140. Schwartz, J.B., et al., Variability in free 25(OH) vitamin D levels in clinical populations. J Steroid Biochem Mol Biol, 2014.144 Pt A: p.156-8.

[0343] 141. Bouillon, R., et al., Vitamin D Binding Protein: A Historic Overview. Front Endocrinol (Lausanne), 2019.10: p.910.

[0344] 142. O'Brien, K.M., et al., Genome-Wide Association Study of Serum 25- Hydroxyvitamin D in US Women. Front Genet, 2018.9: p.67.

[0345] Dorin RI, Qualls CR. Distribution of cortisol in human plasma in vitro: Equilibrium solutions for free cortisolusing equations of mass conservation and mass action. 2024. In: Cortisol: Between Phsyiology and Pathology [Internet]. London, UK: IntechOpen.

[0346] Features and advantages of this disclosure are apparent from the detailed specification, and the claims cover all such features and advantages. Numerous variations will occur to those skilled in the art, and any variations equivalent to those described in this disclosure fall within the scope of this disclosure. Those skilled in the art will appreciate that the conception upon which this disclosure is based may be used as a basis for designing other compositions and methods for carrying out the several purposes of this disclosure. As a result, the claims should not be considered as limited by the description or examples.Attorney Docket No.37759.0576P1 CLAIMS What is claimed is: 1. A method of determining the concentration of free cortisol in a sample, the method comprising: a) obtaining or having a sample obtained, the sample comprising total cortisol, which is the sum of the concentration free cortisol and cortisol bound to corticosteroid-binding globulin (CBG) and albumin; b) dividing the sample into multiple sample portions; c) performing at least one assay on a first sample portion to determine the concentration in the first sample portion of (i) total CBG, (ii) total albumin, and (iii) total cortisol; d) contacting a second sample portion with an effective amount of a labeled ligand, the labeled ligand having an equilibrium association constant of at least 20 M-1for CBG and albumin; and determining the concentration in the second sample portion of labeled ligand bound to CBG; e) performing a series of steps on a third sample portion, the series of steps comprising, in sequence: i) contacting the third sample portion with an effective amount of a first unlabeled ligand, the first unlabeled ligand having (i) an equilibrium association constant for albumin which is higher than the labeled ligand’s equilibrium association constant for albumin, and (ii) an equilibrium association constant for CBG which is less than 0.1 M-1; ii) contacting the third sample portion with an effective amount of the labeled ligand; and iii) determining the concentration in the third sample portion of labeled ligand bound to CBG; f) performing a series of steps on a fourth sample portion, the series of steps comprising, in sequence: i) contacting the fourth sample portion with an effective amount of albumin;ii) contacting the fourth sample portion with an effective amount of the labeled ligand; and iii) determining the concentration in the fourth sample portion of labeled ligand bound to CBG; and g) deriving, based on the concentrations determined in steps (c)-(f), the concentration of free cortisol in the sample. The method of claim 1, wherein the labeled ligand is directly or indirectly labeled. The method of claim 1, wherein the labeled ligand comprises labeled cortisol, corticosterone, cortisone, deoxycorticosterone, deoxycortisol, progesterone, or prednisolone. The method of claim 1, wherein step (d) further comprises heating the second sample portion after contacting the second sample portion with an effective amount of the labeled ligand and prior to determining the concentration of labeled ligand bound to CBG. The method of claim 1, wherein step (d) further comprises contacting the second sample portion with an effective amount of unlabeled cortisol prior to determining the concentration of labeled ligand bound to CBG. The method of claim 1, wherein in step (d), (e), or (f), determining the concentration of labeled ligand bound to CBG comprises performing a lectin-affinity assay on the second, third, or fourth sample portion using a plant lectin which is immobilized on a solid support. The method of claim 6, wherein the plant lectin is Concanavilin-A. The method of claim 1, wherein step (d), (e), or (f) further comprises adding an effective amount of a second unlabeled ligand prior to determining the concentration of first labeled ligand bound to CBG, the second unlabeled ligand having a equilibrium association constant for CBG which is lower than cortisol’s equilibrium association constant for CBG.The method of claim 8. wherein the second unlabeled ligand is cortisone, deoxycorticosterone, progesterone, or any combination thereof. The method of claim 8, wherein the second unlabeled ligand is added at a known concentration. The method of claim 1, wherein the first unlabeled ligand is thyroxine (T4). A method of determining the concentration of free ligand or lipophilic hormone in a sample, the method comprising: a) obtaining or having a sample obtained, the sample comprising total ligand or lipophilic hormone, which is the sum of the concentration of free ligand or lipophilic hormone and ligand or lipophilic hormone bound to a ligand or lipophilic hormone binding partner, albumin, or both; b) dividing the sample into multiple sample portions; c) performing at least one assay on a first sample portion to determine the concentration in the first sample portion of (i) total ligand or lipophilic hormone binding partner, (ii) optionally, total albumin, and (iii) ligand or lipophilic hormone; d) contacting a second sample portion with an effective amount of a labeled ligand or lipophilic hormone, the labeled ligand or lipophilic hormone having an equilibrium association constant of at least 20 M'1for the ligand or lipophilic hormone binding partner, albumin, or both; and determining the concentration in the second sample portion of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner; e) performing a series of steps on a third sample portion, the series of steps comprising, in sequence: i) contacting the third sample portion with an effective amount of a first unlabeled ligand or lipophilic hormone, the first unlabeled ligand or lipophilic hormone having (i) an equilibrium association constant for albumin which is higher than the labeled ligand or lipophilic hormone’s equilibrium association constant for albumin, and (ii) anequilibrium association constant for the lipophilic hormone binding partner which is less than 0. 1 M’1; ii) contacting the third sample portion with an effective amount of the labeled ligand or lipophilic hormone; and iii) determining the concentration in the third sample portion of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner; and f) deriving, based on the concentrations determined in steps (c)-(e), the concentration of free ligand or lipophilic hormone in the sample. The method of claim 12, further comprising performing a series of steps on a fourth sample portion, the series of steps comprising, in sequence: a) contacting the fourth sample portion with an effective amount of albumin; b) contacting the fourth sample portion with an effective amount of the labeled ligand or lipophilic hormone; and c) determining the concentration in the fourth sample portion of labeled ligand or liphophilic hormone bound to the ligand or lipophilic hormone binding partner. The method of claim 13, wherein determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner comprises performing a lectin-affinity assay on the fourth sample portion using a plant lectin which is immobilized on a solid support. The method of claim 14, wherein the plant lectin is Concanavilin-A. The method of claim 12, wherein the labeled ligand or lipophilic hormone is directly or indirectly labeled. The method of claim 12, wherein the labeled ligand or lipophilic hormone comprises labeled cortisol, corticosterone, cortisone, deoxycorticosterone, deoxycortisol, progesterone, or prednisolone.The method of claim 12, wherein step (d) further comprises heating the second sample portion after contacting the second sample portion with an effective amount of the labeled ligand or lipophilic hormone and prior to determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner. The method of claim 12, wherein step (d) further comprises contacting the second sample portion with an effective amount of unlabeled ligand or lipophilic hormone prior to determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner. The method of claim 12, wherein in step (d) or (e), determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner comprises performing a lectin-affmity assay on the second or third sample portion using a plant lectin which is immobilized on a solid support. The method of claim 20, wherein the plant lectin is Concanavilin-A. The method of claim 12, wherein step (d) or (e) further comprises adding an effective amount of a second unlabeled ligand or lipophilic hormone prior to determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner, the second unlabeled ligand having a equilibrium association constant for the ligand or lipophilic hormone binding partner which is lower than the ligand or lipophilic hormone’s equilibrium association constant for CBG. The method of claim 22, wherein the second unlabeled ligand or lipophilic hormone is cortisone, deoxycorticosterone, progesterone, or any combination thereof. The method of claim 22, wherein the second unlabeled ligand or lipophilic hormone is added at a known concentration. The method of claim 12, wherein the first unlabeled ligand is thyroxine (T4). The method of claim 12, wherein the ligand or lipophilic hormone binding partner is a serum transport or binding protein.The method of claim 12, wherein the ligand or lipophilic hormone is a corticosteroid, sex steroid, vitamin D, or a thyroid hormone. The method of claim 26, wherein the ligand or lipophilic hormone is cortisol and the serum transport or binding protein is CBG. The method of claim 26, wherein the ligand or lipophilic hormone is progesterone and the serum transport or binding protein is CBG. The method of claim 26, wherein the ligand or lipophilic hormone is testosterone and the serum transport or binding protein is sex hormone binding globulin. The method of claim 26, wherein the ligand or lipophilic hormone is a vitamin D metabolite and the serum transport or binding protein is vitamin D binding protein. A method of diagnosing a disease or disorder, the method comprising performing the method of claim 12, and diagnosing the disease or disorder based on the concentration of free ligand or lipophilic hormone in the sample.LIGAND-BINDING PROTEIN ASSAY FOR ESTIMATION OF FREE HORMONE CONCENTRATIONSCROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Provisional Application No. 63 / 568,016, filed March 21, 2024.BACKGROUND

[0002] Lipophilic hormones, such as steroid, vitamin D metabolites, and thyroid hormones, listed in Table 1 below, ty pically circulate in plasma in association with high- affinity' serum transport or binding proteins (BPs). Clinically important examples of ligand- BP interactions that govern the transport of hormones (ligands) in plasma include (i) corticosteroids (and progesterone) to corticosteroid binding globulin (CBG), (ii) testosterone and other sex steroids to sex hormone binding globulin (SHBG), (iii) vitamin D metabolites to vitamin D binding protein (DBP), and (iv) thyroid hormones to thyroid binding globulin (TBG). These binding interactions are characterized by specificity and high affinity. The high affinity of BPs for their cognate ligand(s) is illustrated in Table 1 by the high equilibrium association (affinity) constants (Ka). which are expressed in units of l / concentration(M-1). The magnitude of these affinities (for free ligand) may be more intuitively appreciated by the equilibrium dissociation constant (Kd), which are expressed in units of ligand concentration and are in the nmol / L range. Not shown in Table 1 is the concentration of cognate binding proteins. Except for DBP, which is present in concentrations far greater than those of vitamin D metabolite ligands, CBG, SHBG, and TBG are present in limiting concentrations. As such, these ligand-BP interactions are characterized by saturable binding kinetics and under certain circumstances, concentrations of free ligand(s) may exceed the concentration of cognate BP. Even though several different ligands can bind a given BP with high affinity, these ligand-BP interactions are highly specific across ligand classes. For example, thyroid hormone binds CBG, SHBG, and DBP with negligible affinity.Table 11BP = binding protein; CBG = cortisol-binding globulin: DBF = vitamin D-binding protein: HAS = human serum albumin: Ka = association constant; SHBG = sex hormonne-binding globulin; T3 = triiodothyronine; T4 = thyroxine; TBG = thyroid-binding globulin; TTR = transthyretin

[0003] In addition to specific, high-affinity binding to their cognate serum transport protein(s), lipophilic hormones also bind in relatively non-specific fashion and with far lower affinities to albumin. As shown in Table 1, the affinity' constant for ligand-albumin bindings is orders of magnitude less than specific BPs. How ever, because the concentration of albumin is so high in plasma, the albumin-bound fraction of ligand is substantial, typically several- fold (in the case of cortisol) and up to 20-fold (in the case of testosterone) higher than the free ligand concentration.

[0004] According to the free hormone hypothesis, only the free hormone, but not proteinbound hormone, is able to diffuse across the cell membrane. Thus, clinicians are primarily interested in the free rather than protein-bound fraction of these lipophilic hormones. Using the example of cortisol, it is not at present clinically feasible to measure free cortisol concentrations at the intracellular site of biological action. Interstitial cortisol concentrations have been experimentally measured, but only in a highly specialized research setting. Therefore, measurements of concentrations of free cortisol (in plasma or serum) represent the next best approximation of intracellular cortisol concentrations. Moreover, sampling of plasma or serum by venipuncture is safe, convenient, and w ell-validated. However, serum concentrations of free cortisol are rarely obtained experimentally. Instead, clinicians and to a large extent investigators rely on measurements of total cortisol concentrations. These provide a useful approximation of free cortisol concentration, but one that is subject to perturbations due to extraneous factors. These extraneous factors include variation in concentrations and affinities of cortisol binding globulin (CBG) and albumin, temperature, as well as competition by other compounds that also bind CBG or albumin with significant affinity.

[0005] In summary, there is no scientific rationale for preferring measurements of total2rather than free cortisol concentrations in clinical practice. Rather, clinicians make do with measurements of total rather than free cortisol due to ease, tradition, and feasibility. By contrast, measurement of free cortisol by equilibrium dialysis or ultracentrifugation is cumbersome, expensive, and not well suited to rapid turnaround in a clinical laboratory' setting. Thus, measurements of free cortisol concentration require send-out to specialty laboratories, with added expense and delay. In the case of cortisol, the delay in obtaining free cortisol concentrations in clinical samples obviates the use of such data in the acute care, such as assessment of adrenocortical function in critically ill patients. There is therefore a need in the art for improved methods for measuring free hormone concentrations, such as cortisol concentrations, in a clinically meaningful and useful manner.SUMMARY

[0006] The disclosed methods allow for the determination of the amount of free ligand or lipophilic hormone in a sample based on a series of measurements that provide data allowing for the derivation of the free amount of ligand or lipophilic hormone in the sample. The method is suitable for a variety applications, including diagnostic methods for diagnosing a disease or disorder, as well as detection methods for detecting a variety of ligands, lipophilic hormones, and binding partners for the ligands and lipophilic hormones.

[0007] In some aspects, the methods disclosed herein can permit calculation of total binding protein concentration as an alternative to direct measurement of binding protein, In some aspects, the methods disclosed herein can be suitable for a variety of applications, including (i) diagnostic methods for diagnosing a disease or disorder, (ii) monitoring treatment effects of hormone replacement, (iii) determination of free and total concentrations of specific binding protein, (iv estimation of the concentration of free albumin, (v) derivation of pharmacokinetic parameters to predict dynamic changes in total and free hormone concentrations in vivo, and (vi) development of pharmacodynamic measures of time-varying tissue exposure to free hormone concentrations in vivo.

[0008] In one aspect, the method involves determining the concentration of free cortisol in a sample. This method comprises obtaining or having a sample obtained, the sample comprising total cortisol, which is the sum of the concentration free cortisol and cortisol bound to corticosteroid-binding globulin (CBG) and albumin; dividing the sample into multiple sample portions;performing at least one assay on a first sample portion to determine the concentration in the first sample portion of (i) total CBG, (ii) total albumin, and (iii) total cortisol; contacting a second sample portion with an effective amount of a labeled ligand, the labeled ligand having an equilibrium association constant of at least 20 M1for CBG and albumin; and determining the concentration in the second sample portion of labeled ligand bound to CBG; performing a series of steps on a third sample portion, the series of steps comprising, in sequence; contacting the third sample portion with an effective amount of a first unlabeled ligand, the first unlabeled ligand having (i) an equilibrium association constant for albumin which is higher than the labeled ligand’s equilibrium association constant for albumin, and (ii) an equilibrium association constant for CBG which is less than 0. 1 M’1; contacting the third sample portion with an effective amount of the labeled ligand; and determining the concentration in the third sample portion of labeled ligand bound to CBG; performing a series of steps on a fourth sample portion, the series of steps comprising, in sequence: contacting the fourth sample portion with an effective amount of albumin; contacting the fourth sample portion with an effective amount of the labeled ligand; and determining the concentration in the fourth sample portion of labeled ligand bound to CBG; and deriving, based on the concentrations determined in steps (c)-(f), the concentration of free cortisol in the sample.

[0009] In a further aspect, the method is more general, applicable to a variety of free ligands or lipophilic hormones in a sample. This method for example comprises: obtaining or having a sample obtained, the sample comprising total ligand or lipophilic hormone, which is the sum of the concentration of free ligand or lipophilic hormone and ligand or lipophilic hormone bound to a ligand or lipophilic hormone binding partner, albumin, or both; dividing the sample into multiple sample portions; performing at least one assay on a first sample portion to determine the concentration in the first sample portion of (i) total ligand or lipophilic hormone binding partner, (ii) optionally, total albumin, and (iii) ligand or lipophilic hormone; contacting a second sample portion with an effective amount of a labeled ligand or lipophilic hormone, the labeled ligand or lipophilic hormone having an equilibrium4association constant of at least 20 M1for the ligand or lipophilic hormone binding partner, albumin, or both; and determining the concentration in the second sample portion of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner; performing a series of steps on a third sample portion, the series of steps comprising, in sequence: contacting the third sample portion with an effective amount of a first unlabeled ligand or lipophilic hormone, the first unlabeled ligand or lipophilic hormone having (i) an equilibrium association constant for albumin which is higher than the labeled ligand or lipophilic hormone’s equilibrium association constant for albumin, and (ii) an equilibrium association constant for the lipophilic hormone binding partner which is less than 0.1 M’1; contacting the third sample portion with an effective amount of the labeled ligand or lipophilic hormone; and determining the concentration in the third sample portion of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner; and deriving, based on the concentrations determined in steps (c)-(e), the concentration of free ligand or lipophilic hormone in the sample.

[0010] Also described are methods of diagnosing a disease or disorder, the method comprising performing one of the disclosed methods, and diagnosing the disease or disorder based on the concentration of free ligand or lipophilic hormone in the sample.

[0011] Additional advantages of the disclosed methods will be set forth in part in the description which follows, and in part will be understood from the description, or may be learned by practice of the disclosed method. The advantages of the disclosed methods will be realized and attained by means of the elements and combinations particularly pointed out in the appended claims. It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the invention as claimed.DETAILED DESCRIPTION

[0012] The disclosed methods may be understood more readily by reference to the following detailed description of particular embodiments and the Example included therein and following description.

[0013] It is to be understood that the disclosed methods are not limited to specific synthetic methods, specific analytical techniques, or to particular reagents unless otherwise specified, and, as such, may vary. It is also to be understood that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be5limiting.

[0014] Disclosed are materials, compositions, and components that can be used for, can be used in conjunction with, can be used in preparation for, or are products of the disclosed methods and compositions. These and other materials are disclosed herein, and it is understood that when combinations, subsets, interactions, groups, etc. of these materials are disclosed that while specific reference of each various individual and collective combinations and permutation of these compounds may not be explicitly disclosed, each is specifically contemplated and described herein. Thus, if a class of molecules A, B, and C are disclosed as well as a class of molecules D, E, and F and an example of a combination molecule, A-D is disclosed, then even if each is not individually recited, each is individually and collectively contemplated. Thus, in this example, each of the combinations A-E, A-F, B-D, B-E, B-F, C- D, C-E, and C-F are specifically contemplated and should be considered disclosed from disclosure of A, B, and C; D, E, and F; and the example combination A-D. Likewise, any subset or combination of these is also specifically contemplated and disclosed. Thus, for example, the sub-group of A-E, B-F, and C-E are specifically contemplated and should be considered disclosed from disclosure of A, B, and C; D, E, and F; and the example combination A-D. This concept applies to all aspects of this application including, but not limited to, steps in methods of making and using the disclosed compositions. Thus, if there are a variety of additional steps that can be performed it is understood that each of these additional steps can be performed with any specific embodiment or combination of embodiments of the disclosed methods, and that each such combination is specifically contemplated and should be considered disclosed.

[0015] Headings are provided for convenience only and are not to be construed to limit the invention in any manner. Embodiments illustrated under any heading or in any portion of the disclosure may be combined with embodiments illustrated under the same or any other heading or other portion of the disclosure.A. Definitions

[0016] It is understood that the disclosed methods and compositions are not limited to the particular methodology, protocols, and reagents described as these may vary. It is also to be understood that the terminology used herein is for the purpose of describing particular embodiments only, and is not intended to limit the scope of the present invention which will be limited only by the appended claims.

[0017] It must be noted that as used herein and in the appended claims, the singular forms6“a,” “an,” and “the” include plural reference unless the context clearly dictates otherwise. Thus, for example, reference to “an assay” includes a plurality of such assays, reference to “the assay” is a reference to one or more assays and equivalents thereof known to those skilled in the art, and so forth.

[0018] The term “subject” refers to the target of administration, e.g., an animal. Thus, the subject of the disclosed methods can be a vertebrate, such as a mammal. For example, the subject can be a human. The term does not denote a particular age or sex. “Subject” can be used interchangeably with “individual” or “patient.” For example, the subject of administration can mean the recipient of the alternating electrical field. For example, the subject of administration can be a subject with cancer, e.g., ovarian cancer or lung cancer.

[0019] “Optional” or “optionally” means that the subsequently described event, circumstance, or material may or may not occur or be present, and that the description includes instances where the event, circumstance, or material occurs or is present and instances where it does not occur or is not present.

[0020] Ranges may be expressed herein as from “about” one particular value, and / or to “about” another particular value. When such a range is expressed, also specifically contemplated and considered disclosed is the range from the one particular value and / or to the other particular value unless the context specifically indicates otherwise. Similarly, when values are expressed as approximations, by use of the antecedent “about,” it will be understood that the particular value forms another, specifically contemplated embodiment that should be considered disclosed unless the context specifically indicates otherwise. It will be further understood that the endpoints of each of the ranges are significant both in relation to the other endpoint, and independently of the other endpoint unless the context specifically indicates otherwise. Finally, it should be understood that all of the individual values and subranges of values contained within an explicitly disclosed range are also specifically contemplated and should be considered disclosed unless the context specifically indicates otherwise. The foregoing applies regardless of w hether in particular cases some or all of these embodiments are explicitly disclosed.

[0021] Unless defined otherwise, all technical and scientific terms used herein have the same meanings as commonly understood by one of skill in the art to which the disclosed method and compositions belong. Although any methods and materials similar or equivalent to those described herein can be used in the practice or testing of the present method and compositions, the particularly useful methods, devices, and materials are as described.7Publications cited herein and the material for which they are cited are hereby specifically incorporated by reference. Nothing herein is to be construed as an admission that the present invention is not entitled to antedate such disclosure by virtue of prior invention. No admission is made that any reference constitutes prior art. The discussion of references states what their authors assert, and applicants reserve the right to challenge the accuracy and pertinence of the cited documents. It will be clearly understood that, although a number of publications are referred to herein, such reference does not constitute an admission that any of these documents forms part of the common general knowledge in the art.

[0022] Throughout the description and claims of this specification, the word “comprise” and variations of the word, such as “comprising” and “comprises,” means “including but not limited to,” and is not intended to exclude, for example, other additives, components, integers or steps. In particular, in methods stated as comprising one or more steps or operations it is specifically contemplated that each step comprises what is listed (unless that step includes a limiting term such as “consisting of’), meaning that each step is not intended to exclude, for example, other additives, components, integers or steps that are not listed in the step.B. Determination of Free Ligand or Binding Protein (BP) Concentrations

[0023] The general purpose of the disclosed procedures is to obtain rapid and accurate estimates of free ligand and / or BP concentrations in a clinical sample. The disclosed procedures and measurements adjunctive to usual measurement of total (protein-bound plus free ligand) and albumin, will facilitate accurate estimation of free, BP-bound, and albuminbound ligand. These procedures also provide a useful measure of BP concentration in the serum sample. Ligand-BPs of interest include but are not limited to (i) CBG-cortisol, (ii) CBG-progesterone, (iii) SHBG-testosterone, (iv) DBP-25OH vitamin D and DBP-1,25-OH vitamin D (calcitriol). Ligand-BPs of interest include, but are not limited to: (i) CBG-cortisol, (ii) CBG-progesterone. (iii) CBG-aldosterone. (iv) SHBG-testosterone, (v) SHBG-estradiol, (vi) SHBG-dihydrotestosterone (vii) DBP-25OH vitamin D and (viii) DBP-1 ,25-OH vitamin D (calcitriol).

[0024] Measurement of free hormone concentration (e.g. free cortisol, free testosterone, free 25 -OH vitamin D, and the like) provides added value relative to the current practice of measuring total hormone concentration (e.g. total cortisol, total testosterone, total 250H vitamin D). Therefore, the objective of the present analysis is to establish mathematical principles and methodology' for rapid and convenient procedures that, in combination with routine clinical measurements of total cortisol (or other ligand / lipophilic hormone) and8albumin (or other binding partner) in serum, provide a reliable estimation of free cortisol (or other ligand / lipophilic hormone) and total CBG concentrations (or binding partner concentrations).

[0025] The estimation of free ligand / lipophilic hormone concentration should be accurate and available to assist the clinician in real-time clinical management decisions. That is, the goal is to provide clinicians clinical measurements of free as well as total free ligand / lipophilic hormone concentration in the same time frame and using the same clinical laboratory platform in which the latter (total ligand / lipophilic hormone) measurements are typically available. The general methodology may be categorized as a competitive ligand- transport-protein binding assay. As described below, in an aspect, the test involves addition of a labeled ligand, such as horseradish peroxidase (HRP) conjugated cortisol, designated E, to the clinical sample, followed by a separation technique to assay the concentration of binding partner (e.g., CBG)-bound E. By this means, the concentration of free ligand / lipophilic hormone (F) and binding partner in the original sample (before addition of E) may be determined.

[0026] Table 2 shows non-limiting examples of lectin-affinity and antibody-affinity methods for separating transport protein-bound hormone to determine free hormone concentrations, as well as clinical applications.Table 2

[0027] The lipophilic hormones for which free hormone concentrations would have clinical utility include without limitation adrenocorticosteroids, sex steroids, sex hormones, vitamin D metabolites, thyroid hormones, progesterone, and the like. In the case of thyroid hormones, a convenient and reasonably accurate analog immunometric method has been developed to directly measure free T4 and free T3. However, there is no corresponding rapid, simple, and accurate method for determination of the free hormone concentration for other ligands of interest (cortisol, progesterone, testosterone, estradiol, 25-OH vitamin D. 1,25-OH vitamin D, etc.). Thus, there is a clinical need for methods for determination of the free hormone concentrations of these ligands that can be performed routinely in the clinical laboratory setting.

[0028] Although the Example below concerns the adrenal corticosteroid hormone, cortisol, the mathematical principles and experimental approach are generalizable to other lipophilic hormones, including, but not limited to the examples summarized in Table 2. While one objective is determination of free hormone concentration, the disclosed methodology also provides a meaningful estimation of the concentration of the cognate transport protein concentrations (CBG in the case of competitive cortisol-binding assay). There are a variety of clinical situations in which determination of transport protein concentrations may provide additional, independent clinical information. For example, serum concentrations of CBG are an independent predictor of mortality in septic shock. The disclosed methodology directlymeasures the concentration of hormone-bound binding globulin. The computational advantages afforded by the equations developed herein represent a much more complete and useful formulation of equilibrium relationships that define the concentration binding globulin and the free hormone concentration compared to the empiric qualities of the historic T3RU assay.

[0029] Both CBG and SHBG are glycoproteins that contain N-linked oligosaccharides. The glycosylation of CBG and SHBG results in high affinity binding of these transport proteins to the plant lectin, concanavalin A (ConA). The disclosed assays for free corticosteroids and free sex steroids (and concentrations of their binding globulins, CBG and SHBG, respectively) both take advantage of this ConA binding activity of CBG and SHBG as a separation technique. In the case of vitamin D metabolites, 25-OH and 1,25-OH vitamin D (see Table 1), DBP is also glycosylated, but the oligosaccharide attachments to the DBP protein do not support binding to ConA. Therefore, an alternative approach to separation of DBP-bound ligand, using antibody-coated rather than ConA-coated microtiter plates, can be used for DBP capture.

[0030] There are several elements of the disclosed methodology that support translation to the clinical laboratory7setting. These include without limitation (i) simplicity7of assay reagents, such as HRP-labeled hormone without need for proprietary, monoclonal antibodies and (ii) development of the assay in a microtiter plates platform used in most clinical laboratories. In addition, the fact that heat denaturation irreversibly abolishes steroid binding activity of CBG, SHBG, and DBP without affecting corresponding albumin-binding activity7provides a simple and cost-effective control for specific transport-protein binding.C. Hormone and Ligand Binding to CBG1. Cortisol

[0031] The stoichiometry7of CBG-cortisol binding is well understood. CBG circulates as a monomer, and one molecule of CBG binds a single molecule of cortisol. Sites for N-linked glycosylation and their respective effects on CBG-cortisol binding affinities have been defined experimentally. As well, the crystal structure for cortisol- and progesterone bound CBG has been studied. In addition, kinetics and temperature sensitivity of CBG-cortisol binding interactions, as well as genetic variants affecting CBG-cortisol binding affinity, have been previously evaluated. At 37°C, the equilibrium dissociation constant (KD) for CBG- cortisol binding has been reported to be in the range of 13-33 nmol / L. CBG is expressed in liver, which is the primary source of circulating CBG. CBG is also expressed in a variety of11extra-hepatic sites, including adrenal gland and hypothalamus, where it may act to modulate local corticosteroid signaling. CBG has a half-life of 4-5 days in vivo. CBG concentrations demonstrate diurnal variation, with peak levels observed in afternoon. However, these variations in CBG concentration appear to have minimal influence on total cortisol concentrations.2. Other Ligands

[0032] A variety of ligands other than cortisol are also conditionally present in human plasma. Depending on their concentration and affinity7, they may represent a significant source of competition for cortisol binding to CBG. For example, these include progesterone, cortisone. 11 -deoxy cortisol, 21 -deoxy cortisol, prednisolone, and other compounds. Cortisol and corticosterone both bind CBG with high affinity, with a relative binding affinity of 1,000 and an equilibrium association constant of 76, 000, 000 / M (which corresponds to equilibrium dissociation constant (KD) of 13.2 nmol / L). Table 3 shows examples of additional other steroids and their affinities for CBG.Table 3

[0033] The significant binding affinity of several steroids for CBG are shown in Table 3. For example, both cortisol and corticosterone bind CBG with high affinity, as do other corticosteroids in the adrenocortical steroidogenic pathway, such as deoxycorticosterone anddeoxy cortisol. Progesterone also binds CBG with relatively high affinity. Cortisone is one of the biologically inactive metabolites of cortisol and binds CBG with significant affinity. Whether other metabolites of cortisol also compete for CBG binding is not fully understood. Not shown in the above table is prednisolone, which also binds CBG with relatively high affinity. Depending on their concentration in plasma, all these steroids have the potential to compete with cortisol (as well as labeled CBG-binding ligand (E)) for binding to CBG.

[0034] In addition to intact and fully glycosylated CBG that binds cortisol with high affinity and circulates in healthy human subjects at concentrations in the range of 400-800 nmol / L, the presence of a structurally distinct CBG molecule has been identified that in some studies, may be distinguished by differential recognition of monoclonal antibodies. It has been suggested that the relatively low abundance CBG molecule recognized by 12G2 but not reactive-center loop (RCL)-specific loop (G12V) monoclonal antibodies, represents elastase- cleaved CBG. Based on that inference, CBG moiety may exist as low-affinity (la) CBG, since elastase-cleaved CBG has roughly 10-fold lower cortisol binding affinity than intact, high- affinity (ha) CBG. Dynamic and equilibrium models suggest that cortisol may include both laCBG and haCBG.

[0035] However, elastase-cleaved CBG is not present is the systemic circulation of healthy or critically ill human subjects. While CBG cleaved by neutrophil elastase may contribute to higher free cortisol concentrations as well as higher rates of tissue delivery of cortisol at sites of inflammation, it appears that elastase-cleaved CBG is rapidly desialylated and cleared by the asialo-gly coprotein receptor at sites of inflammation before reentry to the systemic circulation. It has also been demonstrated that differential recognition of CBG in human plasma by 12G2 and G12V monoclonal antibodies is related to differential patterns of glycosylation. Specifically, an N-glycan at N347 of human CBG limits 9G12 antibody recognition of the epitope, while 12G2 antibody reactivity is unaffected. Moreover, qualitative differences in N-glycosylation at N238 negatively affect steroid-binding activity of CBG. Thus, while it remains theoretically possible that there is heterogeneity of cortisol- binding affinities of CBG in human plasma, the percent of Tow-affinity’ CBG and its KD for cortisol binding remain incompletely characterized at present. In any case, genome wide association studies establish that genetic variation in the SERPINA6 (CBG gene) locus contributes to variability in CBG concentrations and cortisol-binding affinities; this point is further reinforced by characterization of patients with SERPINA6 mutations that interfere with expression and cortisol-binding affinity in patients with CBG deficiency.13D. Methods1. Methods of determining the concentration of free cortisol in a sample

[0036] Disclosed herein are methods of determining the concentration of free cortisol in a sample, the method comprising: (a) obtaining or having a sample obtained, the sample comprising total cortisol, which is the sum of the concentration free cortisol and cortisol bound to corticosteroid-binding globulin (CBG) and albumin; (b) dividing the sample into multiple sample portions; (c) performing at least one assay on a first sample portion to determine the concentration in the first sample portion of (i) total CBG, (ii) total albumin, and (iii) total cortisol; (d) contacting a second sample portion with an effective amount of a labeled ligand, the labeled ligand having an equilibrium association constant of at least 20 M’1for CBG and albumin; and determining the concentration in the second sample portion of labeled ligand bound to CBG; (e) performing a series of steps on a third sample portion, the series of steps comprising, in sequence: (1) contacting the third sample portion with an effective amount of a first unlabeled ligand, the first unlabeled ligand having (i) an equilibrium association constant for albumin which is higher than the labeled ligand’s equilibrium association constant for albumin, and (ii) an equilibrium association constant for CBG which is less than 0. 1 M’1; (2) contacting the third sample portion with an effective amount of the labeled ligand; and (3) determining the concentration in the third sample portion of labeled ligand bound to CBG; (f) performing a series of steps on a fourth sample portion, the series of steps comprising, in sequence: (1) contacting the fourth sample portion with an effective amount of albumin; (2) contacting the fourth sample portion with an effective amount of the labeled ligand; and (3) determining the concentration in the fourth sample portion of labeled ligand bound to CBG; and (g) deriving, based on the concentrations determined in steps (c)-(T). the concentration of free cortisol in the sample. In some aspects, the labeled cortisol is a known / predetermined concentration of cortisol.

[0037] In some aspects, the labeled ligand is directly or indirectly labeled. In some aspects, the labeled ligand comprises labeled cortisol, corticosterone, cortisone, deoxycorticosterone, deoxy cortisol, progesterone, or prednisolone. In some aspects, step (d) further comprises heating the second sample portion after contacting the second sample portion with an effective amount of the labeled ligand and prior to determining the concentration of labeled ligand bound to CBG. Heating can be optional and can be used to determine the amount of CBG bound to labeled cortisol on the solid support and this can be used to measure non-specific binding of cortisol to CBG.14

[0038] In some aspects, the methods can further comprise adding an excess of unlabeled cortisol prior to contacting the second sample portion with a ligand immobilized on a solid support. As such, the method can be used to determine amount of CBG bound to labeled cortisol on the solid support and can be used to measure non-specific binding of cortisol to CBG.

[0039] In some aspects, the methods can further comprise adding a ligand that competes with cortisol and labeled cortisol for binding to CBG. In some aspects, a known, low concentration of a ligand (i.e. progesterone) that competes with cortisol for CBG (lower, but substantial affinity) can be added before immobilizing and the determine amount of CBG bound to labeled cortisol on the solid support and can be used to measure specific binding of cortisol to CBG.

[0040] In some aspects, step (d) further comprises contacting the second sample portion with an effective amount of unlabeled cortisol prior to determining the concentration of labeled ligand bound to CBG. In some aspects, step (d), (e), or (I), determining the concentration of labeled ligand bound to CBG comprises performing a lectin-affinity assay on the second, third, or fourth sample portion using a plant lectin which is immobilized on a solid support. In some aspects step (d), (e), or (I), determining the concentration of labeled ligand bound to CBG comprises performing a lectin-affinity assay on the second, third, or fourth sample portion using a plant lectin which is immobilized on a solid support and the plant lectin is Concanavilin-A. In some aspects, step (d), (e), or (f) further comprises adding an effective amount of a second unlabeled ligand prior to determining the concentration of first labeled ligand bound to CBG, the second unlabeled ligand having a equilibrium association constant for CBG which is lower than cortisol’s equilibrium association constant for CBG. In some aspects, step (d), (e), or (I) further comprises adding an effective amount of a second unlabeled ligand prior to determining the concentration of first labeled ligand bound to CBG, the second unlabeled ligand having a equilibrium association constant for CBG which is lower than cortisol’s equilibrium association constant for CBG and the second unlabeled ligand is cortisone, deoxycorticosterone, progesterone, or any combination thereof. In some aspects, step (d), (e), or (I) further comprises adding an effective amount of a second unlabeled ligand prior to determining the concentration of first labeled ligand bound to CBG, the second unlabeled ligand having a equilibrium association constant for CBG which is lower than cortisol’s equilibrium association constant for CBG and the second unlabeled ligand is cortisone, deoxycorticosterone, progesterone, or any combination thereof and the15second unlabeled ligand is added at a known concentration. In some aspects, the first unlabeled ligand is thyroxine (T4).

[0041] In some aspects, the methods disclosed herein involve determining the concentration of free cortisol in a sample. In some aspects, the methods disclosed herein comprise obtaining or having a sample obtained, the sample comprising total cortisol, which is the sum of the concentration free cortisol and cortisol bound to corticosteroid-binding globulin (CBG) and albumin; dividing the sample into multiple sample portions; performing at least one assay on a first sample portion to determine the concentration in the first sample portion of (i) total CBG, (ii) total albumin, and (iii) total cortisol; contacting a second sample portion with an effective amount of a labeled ligand, which could be labeled cortisol (F*) or, alternatively, a labeled other than cortisol ligand (E*) that has an equilibrium association constant of at least 1x107M'1for the specific BP (CBG); separation of CBG-bound F* using lectin-affinity matrix; using appropriate standards and controls, determining the equilibrium concentration of CBG-bound F* [F*C] in the sample; performing a series of equilibrium perturbation steps on a third-sixth sample portion. The assay signal (equilibrium concentration of F*C) can be obtained by a lectin-affinity method in the second sample, and in subsequent samples, each perturbation of equilibrium will result in a measurable change in [F*C] relative to the second sample. In this aspect, the four perturbations can be; (i) contacting the third sample portion with an effective amount of (free) CBG, which will increase [F*C] relative to the second sample (+A[F*C]), (ii) contacting the fourth sample with an effective amount of (free), unlabeled ligand (P), having equilibrium association constant of at least 1x107M'1for the specific BP (CBG), that competes for CBG-F* binding. Addition of P in a sample after the second sample (e.g., a fourth sample) will decrease [F*C] (relative to the second sample) in dose-dependent fashion (-A[F*C]), (iii) contacting the fifth test tube with an effective amount of free albumin. Addition of free albumin will decrease [F*C] (relative to the second sample) in dose-dependent fashion (-A[F*C]); and (iv) contacting the sixth sample with unlabeled ligand Q having Kafor CBG ® 0 and Kafor albumin ~ 1x106M'1albumin. Addition of Q in a sixth sample for instance will increase [F*CJ relative to test tube #2 (+A[F*CJ) in dose-dependent fashion. At pharmacologic concentrations of Q, the concentration of free albumin is decreased, resulting in lower concentrations of albumin-bound F and albumin-bound F*.162. Methods of determining the concentration of free ligand or lipophilic hormone in a sample

[0042] Disclosed herein are methods of determining the concentration of free ligand or lipophilic hormone in a sample, the method comprising: (a) obtaining or having a sample obtained, the sample comprising total ligand or lipophilic hormone, which is the sum of the concentration of free ligand or lipophilic hormone and ligand or lipophilic hormone bound to a ligand or lipophilic hormone binding partner, albumin, or both; (b) dividing the sample into multiple sample portions; (c) performing at least one assay on a first sample portion to determine the concentration in the first sample portion of (i) total ligand or lipophilic hormone binding partner, (ii) optionally, total albumin, and (iii) ligand or lipophilic hormone; (d) contacting a second sample portion with an effective amount of a labeled ligand or lipophilic hormone, the labeled ligand or lipophilic hormone having an equilibrium association constant of at least 20 M1for the ligand or lipophilic hormone binding partner, albumin, or both; and determining the concentration in the second sample portion of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner; (e) performing a series of steps on a third sample portion, the series of steps comprising, in sequence: (1) contacting the third sample portion with an effective amount of a first unlabeled ligand or lipophilic hormone, the first unlabeled ligand or lipophilic hormone having (i) an equilibrium association constant for albumin which is higher than the labeled ligand or lipophilic hormone’s equilibrium association constant for albumin, and (ii) an equilibrium association constant for the lipophilic hormone binding partner which is less than 0. 1 M’1; (2) contacting the third sample portion with an effective amount of the labeled ligand or lipophilic hormone; and (3) determining the concentration in the third sample portion of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner; and (f) deriving, based on the concentrations determined in steps (c)-(e), the concentration of free ligand or lipophilic hormone in the sample. In some aspects, the disclosed methods can further comprise performing a series of steps on a fourth sample portion, the series of steps comprising, in sequence: (a) contacting the fourth sample portion with an effective amount of albumin; (b) contacting the fourth sample portion with an effective amount of the labeled ligand or lipophilic hormone; and (c) determining the concentration in the fourth sample portion of labeled ligand or liphophilic hormone bound to the ligand or lipophilic hormone binding partner. In some aspects, the disclosed methods can further comprise performing a series of steps on a fourth sample portion, the series of steps17comprising, in sequence: (a) contacting the fourth sample portion with an effective amount of albumin; (b) contacting the fourth sample portion with an effective amount of the labeled ligand or lipophilic hormone; and (c) determining the concentration in the fourth sample portion of labeled ligand or liphophilic hormone bound to the ligand or lipophilic hormone binding partner, wherein determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner comprises performing a lectin-affinity assay on the fourth sample portion using a plant lectin which is immobilized on a solid support. In some aspects, the plant lectin is Concanavilin-A. In some aspects, the labeled ligand or lipophilic hormone is directly or indirectly labeled. In some aspects, the labeled ligand or lipophilic hormone comprises labeled cortisol, corticosterone, cortisone, deoxycorticosterone, deoxy cortisol, progesterone, or prednisolone. In some aspects, step (d) further comprises heating the second sample portion after contacting the second sample portion with an effective amount of the labeled ligand or lipophilic hormone and prior to determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner.

[0043] In some aspects, step (d) further comprises contacting the second sample portion with an effective amount of unlabeled ligand or lipophilic hormone prior to determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner.

[0044] In some aspects, step (d) or (e), determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner comprises performing a lectin-affinity assay on the second or third sample portion using a plant lectin which is immobilized on a solid support. In some aspects, the plant lectin is Concanavilin-A.

[0045] In some aspects, step (d) or (e) further comprises adding an effective amount of a second unlabeled ligand or lipophilic hormone prior to determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner, the second unlabeled ligand having a equilibrium association constant for the ligand or lipophilic hormone binding partner which is lower than the ligand or lipophilic hormone's equilibrium association constant for CBG. In some aspects, the second unlabeled ligand or lipophilic hormone is cortisone, deoxycorticosterone, progesterone, or any combination thereof. In some aspects, the second unlabeled ligand or lipophilic hormone is added at a known concentration. In some aspects, the first unlabeled ligand is thyroxine (T4). In some aspects, the ligand or lipophilic hormone binding partner is a serum transport or binding18protein. In some aspects, the ligand or lipophilic hormone is a corticosteroid, sex steroid, vitamin D, or a thyroid hormone. In some aspects, the ligand or lipophilic hormone is cortisol and the serum transport or binding protein is CBG. In some aspects, the ligand or lipophilic hormone is progesterone and the serum transport or binding protein is CBG. In some aspects, the ligand or lipophilic hormone is testosterone and the serum transport or binding protein is sex hormone binding globulin. In some aspects, the ligand or lipophilic hormone is a vitamin D metabolite and the serum transport or binding protein is vitamin D binding protein.3. Methods of diagnosing a disease or disorder

[0046] Disclosed herein are methods of diagnosing a disease or disorder, the method comprising: (a) obtaining or having a sample obtained, the sample comprising total ligand or lipophilic hormone, which is the sum of the concentration of free ligand or lipophilic hormone and ligand or lipophilic hormone bound to a ligand or lipophilic hormone binding partner, albumin, or both; (b) dividing the sample into multiple sample portions; (c) performing at least one assay on a first sample portion to determine the concentration in the first sample portion of (i) total ligand or lipophilic hormone binding partner, (ii) optionally, total albumin, and (iii) ligand or lipophilic hormone; (d) contacting a second sample portion with an effective amount of a labeled ligand or lipophilic hormone, the labeled ligand or lipophilic hormone having an equilibrium association constant of at least 20 M1for the ligand or lipophilic hormone binding partner, albumin, or both; and determining the concentration in the second sample portion of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner; (e) performing a series of steps on a third sample portion, the series of steps comprising, in sequence: (1) contacting the third sample portion with an effective amount of a first unlabeled ligand or lipophilic hormone, the first unlabeled ligand or lipophilic hormone having (i) an equilibrium association constant for albumin which is higher than the labeled ligand or lipophilic hormone’s equilibrium association constant for albumin, and (ii) an equilibrium association constant for the lipophilic hormone binding partner which is less than 0.1 M’1; (2) contacting the third sample portion with an effective amount of the labeled ligand or lipophilic hormone; and (3) determining the concentration in the third sample portion of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner; and (f) deriving, based on the concentrations determined in steps (c)-(e), the concentration of free ligand or lipophilic hormone in the sample and diagnosing the disease or disorder based on the concentration of free ligand or lipophilic hormone in the sample. In some aspects, the disclosed methods can19further comprise performing a series of steps on a fourth sample portion, the series of steps comprising, in sequence: (a) contacting the fourth sample portion with an effective amount of albumin; (b) contacting the fourth sample portion with an effective amount of the labeled ligand or lipophilic hormone; and (c) determining the concentration in the fourth sample portion of labeled ligand or liphophilic hormone bound to the ligand or lipophilic hormone binding partner. In some aspects, the disclosed methods can further comprise performing a series of steps on a fourth sample portion, the series of steps comprising, in sequence: (a) contacting the fourth sample portion with an effective amount of albumin; (b) contacting the fourth sample portion with an effective amount of the labeled ligand or lipophilic hormone; and (c) determining the concentration in the fourth sample portion of labeled ligand or liphophilic hormone bound to the ligand or lipophilic hormone binding partner, wherein determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner comprises performing a lectin-affinity assay on the fourth sample portion using a plant lectin which is immobilized on a solid support. In some aspects, the plant lectin is Concanavilin-A. In some aspects, the labeled ligand or lipophilic hormone is directly or indirectly labeled. In some aspects, the labeled ligand or lipophilic hormone comprises labeled cortisol, corticosterone, cortisone, deoxycorticosterone, deoxycortisol, progesterone, or prednisolone. In some aspects, step (d) further comprises heating the second sample portion after contacting the second sample portion with an effective amount of the labeled ligand or lipophilic hormone and prior to determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner.

[0047] In some aspects, step (d) further comprises contacting the second sample portion with an effective amount of unlabeled ligand or lipophilic hormone prior to determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner.

[0048] In some aspects, step (d) or (e), determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner comprises performing a lectin-affinity assay on the second or third sample portion using a plant lectin which is immobilized on a solid support. In some aspects, the plant lectin is Concanavilin-A.

[0049] In some aspects, step (d) or (e) further comprises adding an effective amount of a second unlabeled ligand or lipophilic hormone prior to determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner, the second unlabeled ligand having a equilibrium association constant for the ligand20or lipophilic hormone binding partner which is lower than the ligand or lipophilic hormone’s equilibrium association constant for CBG. In some aspects, the second unlabeled ligand or lipophilic hormone is cortisone, deoxycorticosterone, progesterone, or any combination thereof. In some aspects, the second unlabeled ligand or lipophilic hormone is added at a known concentration. In some aspects, the first unlabeled ligand is thyroxine (T4). In some aspects, the ligand or lipophilic hormone binding partner is a serum transport or binding protein. In some aspects, the ligand or lipophilic hormone is a corticosteroid, sex steroid, vitamin D, or a thyroid hormone. In some aspects, the ligand or lipophilic hormone is cortisol and the serum transport or binding protein is CBG. In some aspects, the ligand or lipophilic hormone is progesterone and the serum transport or binding protein is CBG. In some aspects, the ligand or lipophilic hormone is testosterone and the serum transport or binding protein is sex hormone binding globulin. In some aspects, the ligand or lipophilic hormone is a vitamin D metabolite and the serum transport or binding protein is vitamin D binding protein.E. Analytical Modeling1. Coolen’s Equation

[0050] As described herein, Coolen’s equation can be used in any of the methods disclosed herein. By way of illustration, but not in limiting to the particular embodiment, Coolen’s equation is described here as applied to measuring cortisol. A quadratic equation was developed (the Coolen’s equation) to estimate free cortisol using measured concentrations of CBG (XTOICBG) and total cortisol (XTOIF). In this analysis, it was assumed both albumin concentration (40 g / L = 580,000 nmol / L) and albumin-cortisol binding affinity (with equilibrium dissociation constant [KA] for albumin-cortisol binding of 330,000 nmol / L). A cubic equation as well as Coolens' quadratic equation has also been developed. The quadratic formulation of the solution for free cortisol has some advantages relative to Coolens’ insofar as it preserves familiar elements of the quadratic polynomial (see Equation C below). (The quadratic polynomial is obscured in Coolens’ equation owing to their need to divide through by the term ‘2a‘).

[0051] A cubic equilibrium solution was derived for computing free cortisol using mass action equations for the following plasma compartments: (i) free cortisol (F), (ii) CBG-bound cortisol (FC), (iii) unbound CBG (C), (iv) albumin-bound cortisol (FA), and (v) unbound albumin (A) :Two equilibrium equations and three conservation of mass equations result in five equations.21which may be combined to yield a cubic equation for free cortisol. In these equations, F, C, FC, and FA are defined as above, TotC represents total CBG concentration, and TotA represents total albumin concentration.Eq. 4) A + FA = TotA ,Eq. 5) TotF = F + FC + FA .

[0052] There are six concentrations and five equations; the objectives are to solve for TotF in terms of F and then solve the inverse equation for F in terms of TotF. First, using Equation 4 to eliminate A in Equation 2, FA in terms of F can be obtained:Eq. 6)With a similar formula for FC, a first objective can be obtained:For a second objective, the denominators of Equation 6 can be cleared to obtain F as the solution of the cubic equation:A standard formula for the solution of a cubic equation is:

[0053] Coolens quadratic equation can be derived from the above equilibrium equations by assuming that FA / F (=N in Coolens formula) is constant across the physiological range of F. This can be seen in Equation 6, where the value of F is always much smaller than the equilibrium dissociation constant for albumin-cortisol binding (KA). Assuming a simple stoichiometry for cortisol binding to albumin and for F up to 1000 nM, it can be computed that the error in assuming the multiplier N is constant in F is less than 0.25%. To derive Coolens’ equation, note that the multiplier of F in the Equation 6) for FA isRewriting Equation 7), provides:Eq.9) TotF=F(l+N)+F*TotC / (Kc+F).Clearing the denominator provides a quadratic equation in F, which is Coolens equation:a =l+ N, b=(l+N)KC+TotC-TotF, c=-TotF*KQ , and whose solution is the quadractic formula.In summan . using the Coolens equation, F can be calculated given N and Kc and measuredTotC and TotF. Coolens equation is often used assuming N= 1.74.

[0054] Coolens’ N also figures importantly into a solution algorithm for free cortisol.This proposition takes advantage of the simplifying assumption of Coolen’s equation, namely that KA » F. As a result, the binding reaction of albumin and cortisol, at least at physiologic23concentrations of free cortisol, never approaches a level at which saturable binding kinetics might obtain. This assumption allows for FA / F to be treated as a constant, which is also equal to TotA / KA (see Eq 8, above and also section III-G, below).

[0055] There is a body of experimental evidence suggesting that the reaction involving reversible binding of albumin and cortisol in human plasma at 37°C is complex, involving multiple binding sites, possibilities of competitive binding ligands, and allosteric interactions. However, none of these complexities challenge the assumption, based on KA»F, that N may be treated effectively as a constant. That is, it is reasonable to infer that in human serum, albumin-bound cortisol (XFA) increases proportionately to free cortisol concentration (XF). AS to what the exact value of that ratio of proportionately between XF and XFA is human plasma (37°C) and whether it can safely be assumed to be the same from one person to the next (or one clinical sample test tube to the next) is an issue that pertains more directly to the numerical values applied in equations of Coolens and Heyns. Thus, Coolens’ equation assumes the following values: (i) albumin concentration is 40 g / L, (ii) simple (1: 1) stoichiometry of albumin-cortisol binding equilibrium dissociation constant (KD) for albumin-cortisol binding (KA) of 330,000 nmol / L, (iii) a uniform binding affinity (equilibrium dissociation constant KD) for CBG-cortisol (Kc) of 33 nmol / L, and (iv) absence of any compounds in plasma that might compete for binding of cortisol to CBG and / or albumin. Note that since N equals XTOIA / K = XFA / XF. the Coolens’ formula also assumes N=1.74, where (N+1)*XF = [XFA + XF] = bioavailable cortisol. A lower affinity of albumin- cortisol binding (N=1.3) has been suggested, whereas a higher affinity for albumin-cortisol (N=3.0) was obtained in ligand-binding studies using purified albumin solutions.

[0056] ai-acid glycoprotein (AAG), also known as orosomucoid. circulates in human plasma at a concentration of approximately 24 pM and binds cortisol with an affinity (KD =62 pM) intermediate to CBG and albumin. In a matrix formulation of mass action and mass conservation equations for competitive ligand-protein binding described below', the identification of cortisol binding proteins in addition to CBG and albumin may be represented as an additional column in the matrix. Note that each additional column adds one additional order of complexity to the coupled polynomial equations. By contrast, treatment of albumin- cortisol binding as a constant (N) reduces the order of complexity by one.

[0057] Differential equations for and solutions to the competitive binding of aldosterone (E) cortisol (F) in a dynamic model have been evaluated in vivo, which replicated previously reported experimental observations. The development of the model and associated equations24provided the theoretical background for the in vivo (dynamic model) introduction to the disclosed competitive ligand-transport protein binding assay.

[0058] The term “bioavailable testosterone” originated in the field of andrology. since SHBG-bound testosterone can be separated by ammonium sulfate precipitation. Thus, bioavailable testosterone (sum of free testosterone XT and albumin bound testosterone (XTA) can also be derived by subtraction of SHBG-bound testosterone (XTS) from the total testosterone concentration (Xiccr). The disclosed methodology provides an assessment of CBG-bound cortisol (Xc). If XiotF = Xc + XA + XF, one can obtain bioavailable cortisol (sum of albumin-bound and free cortisol) by subtracting Xc from XiotF. The term bioavailable is used to represent [albumin-bound plus free ligand] without implying the physiological significance or biological activities associated with bioavailable ligand. In the case of cortisol, for example, the term bioavailable cortisol is used as a convenient term to represent the sum of albumin-bound and free cortisol.

[0059] The clinical laboratory problem of determining free cortisol based on measured total cortisol and (total) albumin concentrations addressed using the methodologies disclosed herein can be parsed into two distinct operations: (i) determination of bioavailable cortisol (sum of free and albumin-bound cortisol) and (ii) determination ofN in order to quantify XF and XFA fractions of bioavailable cortisol.

[0060] Gold-standard methods for measurement of free cortisol involve (i) pre-analytic separation phase and (ii) analytic procedures. Separation methods typically involve equilibrium dialysis (ED), ultrafiltration (UF), or gel filtration (GF). Analytic methods typically involve cortisol immunoassay, which may be variably affected by cross-reactivity with other adrenocorticosteroid compounds. Liquid chromatography with tandem mass spectrometry (LC-MS / MS) is a preferred and more specific method for measurement of cortisol. However, even LC-MS methods are potentially subject to artifact and inaccuracy. Historically, indirect methods were used to estimate free ligand fraction. This approach utilized addition of labeled ligand, determination of percentage of free labeled T, and calculation of free ligand concentration by multiplying the percentage of free labeled ligand by total ligand concentration. However, several sources of error in this approach, including impurities in the labeled ligand, have been identified, and recent comparisons suggest that the ED with direct measurement of ligand in the dialysate is the contemporary' gold standard method for measurement of free ligands such as free cortisol, free testosterone, and 25-OH vitamin D. Regardless of analytic measurement procedures, pre-analytic separation methods25such as ED and UF are labor-intensive and not routinely available in the clinical laboratory setting.2. Quadratic, Cubic, and Quartic Formulae for Estimating Free Cortisol Concentrations

[0061] Three different poly nomial equations for calculation of free cortisol, based on measured concentrations of total cortisol (XTOIF) and total CBG (XTOICBG) have been suggested. A quadratic formula, Coolens’ equation, is described in detail above. The Coolens’ formula assumes a specified concentration of albumin (using a median population value of =580,000 nmol / L) as well as a specific binding affinity of albumin for cortisol. The cubic equation is more complex insofar as it uses measured rather than assumed concentrations of albumin. In addition, the cubic solution accommodates individual or groupspecific variation in albumin-cortisol binding affinity.

[0062] A variation on the cubic solution is a quartic solution, which includes the additional presence of a low-affinity CBG moiety, originally thought to represent elastase- cleaved CBG. While the more complex quartic solution can achieve superior fits to experimental data relative to quadratic and cubic formulas, it does so at the expense of additional degrees of freedom, including the relative concentration of Tow-affinity' CBG. In addition, the assumption that ‘elastase-cleaved CBG' has a 10-fold lower affinity compared to full-length CBG is open to question in view of demonstration that elastase-cleaved CBG is not detectable at measurable concentrations in human plasma.

[0063] In all these considerations of the binding affinity of cortisol for CBG, albumin, and perhaps other serum proteins as well, it should be recognized that the presence of other (unmeasured) ligands that compete with cortisol for binding to CBG and / or albumin would be identified by alterations in apparent cortisol-binding affinity for respective binding proteins. Examples of iterative solutions that account for competition of ligands for cognate binding proteins using competitive binding principles and coupled polynomial equations have been proposed for both testosterone-SHBG and cortisol-CBG interactions. However, these concepts of competition of various ligands for binding proteins has not been routinely incorporated into formulae for estimation of free ligand concentrations. Limitations of all the proposed formulas for estimation of free cortisol are assumptions that cortisol binding affinities are known and homogeneous. Neither of these assumptions are fully supported by experimental data.

[0064] In the formulae described above, measured levels of XTOICBG and XTOIF are used as26input variables to calculation of free cortisol. It follows that error or bias in measurement of these values will lead to corresponding error or bias in the determination of free cortisol. Thus, the accuracy of estimated free cortisol depends on accuracy of primary measurements of total concentrations of cortisol, CBG, and, in the case of cubic and quartic solutions, albumin in the clinical sample of serum or plasma. The concern for accuracy of primary measurements of total concentrations is highlighted by different reference ranges for comparable populations obtained using different methods for CBG assay and lack of alignment between CBG concentrations measured using two distinct monoclonal antibodies, compared to those measured by commercial RIA using a single, polyclonal antibody to CBG.

[0065] Bias in the determination of total cortisol concentrations in serum or plasma is also well established. This is true for immunoassays, where cross-reactivity (binding of antibody to compounds other than cortisol) have been well-documented. Depending on the design of the assay, this cross-reactivity7can result in under- or over-estimation of actual serum cortisol concentrations. An additional factor affecting the reliability of immunometric methods for measurement of total cortisol concerns the chemical methods used to dissociate CBG-bound cortisol; these methods are often proprietary to commercial assays and may result in underestimation of total cortisol when CBG concentrations / affinity are increased.

[0066] While LC-MS / MS methods provide more specific measurements of serum cortisol, they too are subject to various sources of error, as exemplified by the co-elution of cortisol metabolites having similar molecular mass and mass spectrometric fragmentation patterns with cortisol. For example, the presence of co-eluting metabolites such as 20-a and 20-p-dihydrocortisone results in overestimation of serum cortisol concentrations unless they can be separated from cortisol by careful adjustment of chromatographic conditions. The use of2H-labeled internal standards has reduced the coefficient of variation in cortisol measurements using LC-MS / MS, but differences in recovery7rates remains a potential source of variation in cortisol measurements that tends to increase the CV of duplicate measurements.T1EXAMPLES

[0067] The following examples further illustrate this disclosure. The scope of the disclosure and claims is not limited by the scope of the following examples.A. Dynamic Jn Vivo Model for Cortisol (F) and Labeled Cortisol

[0068] The competition of ligands (e.g. labeled and unlabeled cortisol) for a shared serum binding protein (e.g. CBG), was originally developed for the in vivo setting and was made clear by application of the following simultaneous differential equations. These differential equations are analogous to the 4-compartment, diffusion model equations previously reported in U.S. Patent Application Number 17 / 993,282, incorporated by reference in its entirety for its teaching of the diffusion model equations.

[0069] The eight simultaneous differential equations are taken to describe the relationships between time varying concentrations of four cortisol compartments, free. CBG- bound and albumin-bound, extra-vascular written as X . ,XFC,XFA, XFerespectively, and of four comparable compartments for labeled cortisol (E) compartments, i.e. free, CBG-bound, albumin-bound, and extravascular E (designated XE,XEC,XEA, and XEe.

[0070] Differential equations for endogenous cortisol (F) and labeled ligand (E) are said to be coupled systems of equations, since terms used for F in equations 1 a-d also appear in the differential equations for E in equations le-h include the competition between cortisol (F) and labeled cortisol (E). That is, equations 1 a-h are coupled since both E and F compete for binding to CBG.

[0071] It was assumed that all of the rate constants of E are equal to the corresponding rates for F; that is, labeled cortisol behaves the same as the unlabeled cortisol in the body. In a subsequent model, this requirement can be relaxed such that the labeled hormone behaves in the same way as the unlabeled hormone. This latter permutation will facilitate use of other labeled ligands, such as prednisolone or aldosterone, that bind CBG (and albumin) with different affinity than cortisol. By the same token, the label proposed in experimental methods, cortisol conjugated with the enzyme horseradish peroxidase (HRP), likely has a lower affinity for CBG binding compared to unlabeled (endogenous) cortisol.(where , XFA= concentrations of free, CBG bound, and albumin bound cortisolX£, XEC, XEA= concentrations of free, CBG bound, and albumin bound labeled cortisol= cortisol unbinding rate factors for CBG and albumin (‘'off’ rates)Ki >Ki ~ cortisol binding rate factors for CBG and albumin (“on” rates)ZF = cortisol secretion rateZE = labeled cortisol secretion rateCl = free cortisol elimination rate constant*For diagram and equations, F in reference to cortisol is used, C in reference to CBG, A in reference to albumin, and E in reference to labeled cortisol. The single letter designation for labeled cortisol is not intended to refer to corticosterone, which is sometimes referred to as compound E.29

[0072] Concentrations E and F compete for binding to CBG and albumin as indicated in the mass action equations for free [C] and [A], This implies that the time courses of Xr(t) and XE(t) in the dynamic solution are modified by this competition. Since albumin concentrations greatly exceed cortisol under physiologic conditions, cortisol-albumin binding reactions are generally non-saturable. The issues of E and F binding to albumin is discussed below. Although albumin-cortisol binding is treated as saturable in the 4-compartment model, the vast molar excess of albumin to cortisol allows for the simplification of the relationship using concepts developed in quadratic and cubic (mass action) formulations of cortisol-CBG binding mass action such that a linear relationship between free cortisol (F) and albuminbound cortisol (FA) obtains and, if albumin concentration is measured experimentally, the concentration of albumin-bound cortisol can be reasonably approximated as a constant (N in Coolens’ equation) that can be simply obtained by multiplying N by the plasma free cortisol concentration (bioavailable cortisol [XFA + XF] = (N+1)*XF.

[0073] With this simplification of cortisol-albumin binding and at physiologic cortisol concentrations, the competition reactions of practical interest concern (i) competition of E and F CBG binding and (ii) additional competition of other (measured or unmeasured) compounds (other than E and F) that may have significant affinity' for CBG and circulate at concentrations that result in significant CBG binding activity', and (iii) the existence of other (measured or unmeasured) serum proteins that significantly bind E and F at measurable concentrations. For example, the existence of alternate species of CBG in serum that bind cortisol with lower affinity' than full-length, fully glycosylated CBG (having KD in the range of 13-33 nmol / L) has been proposed.1. Interrupted Steady State

[0074] Following a period of continuous cortisol (F), it can be assumed that steady state conditions of unchanging cortisol concentrations are achieved. Applying the four- compartment model (Eq 1 a-d above), a steady state concentration of free cortisol (denoted as Fss) obtains. While the continuous infusion of F continues, a second infusion of E is then initiated. Concentrations of E gradually increase and, like F. reach a new steady state. The continuous infusion of E is referred to as a disruption or interruption of steady state. Since E and F both compete for CBG binding, the concentration of CBG-bound cortisol is lower in the disrupted steady state condition compared to the original steady state condition.2. Equilibrium in a Plasma Sample in a Test Tube

[0075] The same discussion applies to a test tube containing a sample of blood30maintained at body temperature. When a blood sample is obtained by venipuncture and placed in a test tube maintained a body temperature (37°C), a cortisol equilibrium obtains in milliseconds and is maintained because there is no input (ZF=0) and there is no elimination of F (a=0) from the test tube. Thus, equilibrium in the test tube can be viewed as analogous to the steady state condition in vivo. The forward solutions of the dynamic (in vivo) model require that rate constants (on- and off-rates for cortisol-CBG and cortisol-albumin binding reactions, respectively), be known and applied. By contrast, equilibrium solutions in the test tube (and also the in vivo model at steady state) require equilibrium association (Ka) (or dissociation (Ka)) constants (rather than on- and off-rate constants).3. Labeled Ligand and Original Sample

[0076] In the original sample (e g., test tube #1), TotF is distributed to Fo, FC’ and FA’. The goal is to estimate Fo without measuring it; TotF is measured. In test tube #2 (an aliquot of the original sample), a known amount of labeled cortisol (E) is added so that a known concentration (Eo = TotE) is added. (These calculations involve known volumes.) A new equilibrium is formed quickly (100s of msecs). TotE = Eo competes for free and F-bound CBG forming EC, which events predictably result in lower FC and increased F and FA. The notation in test tube #2 will be the same as in Eqs la-h.

[0077] With Eo labeled and distinguishable, this total mass of labeled cortisol TotE = Eo is redistributed to E, EC. and EA in test tube #2. though the distnbution may be unknown. This result can be described in terms of Fs < F + E< Fs + Eo. FsffiEo = F + E, which is less than Fs + Eo because some of the free cortisol is bound to CBG in equilibrium and is no longer free. ® represents the combination of Fs and Eo, which is dynamically defined but in steady state will represent the effect on Fs by Eo.4. Relative Equilibrium Concentrations of E and F In Vitro

[0078] In the test tube #2 experiment above, the additional concentration TotE = Eo is redistributed to E, EC, and EA in the same proportions as the new distribution of TotF into F, FC. and FA. This notion that E, EC, and EA are proportional to the new distribution of F, FC, and FA was known to investigators infusing labeled cortisol in vivo in the past, as it informs the approach of multiplying steady state concentrations of labeled and unlabeled cortisol concentrations by the infusion rate of labeled cortisol.

[0079] In the test tube experiment, if the additional concentration Eo were unlabeled, then Eo is just incremental free cortisol that can be rewritten as AF. Total cortisol becomes TotF + AF which is redistributed to F, FC and FA. Though increasing TotF to TotF + AF implies F,31FC and FA are increased, they have the same percentage distribution as the E, EC, and EA above. The assumption that the addition of free cortisol Eo (labeled or unlabeled) to the test tube only binds to free CBG (and to free albumin) and leaves the original FC and FA unchanged is not correct. The problem is that the assumption does not account for the competition between E and F (for example to CBG).

[0080] These considerations can provide for an algebraic estimate of the original free cortisol (F), which is simpler and quicker to do than contemporary' methods for measurement of free cortisol concentration, which ty pically involve equilibrium dialysis or ultracentrifugation. This method also algebraically obtains the concentration of total CBG (if binding affinity and stoichiometry of CBG-cortisol binding are known).B. Estimation of Albumin-Cortisol Binding from Purified Albumin Solution Data

[0081] The experiment involves addition of labeled cortisol to solutions of varying concentrations of purified albumin. Thus, the test tube consists of TotA and TotE, which at equilibrium also gives a free E and a free A. In this experiment, E represents labeled cortisol (3H-cortisol), which is assumed to bind albumin with an affinity equal to that of endogenous (unlabeled) cortisol (F).

[0082] There is no competition for combining of E and A with each other. A method was developed for representation of the equilibrium ligand-protein binding equations in a matrix format, as described in below; In this format, the experiment may be formulated as a (1x1) matrix, in which K is the equilibrium association (affinity) constant for albumin-cortisol binding and the concentration of albumin-bound cortisol is determined by K and respective concentrations of free cortisol and (free) albumin according to law' of mass action and as illustrated below. For cortisol binding in a purified solution of albumin, this 1x1 matrix represents 1 ligand (cortisol) and 1 binding protein (albumin). Here, the labeled ligand, e.g.3H-cortisol (E), cells are measured, and other cells are knowTi experimentally.

[0083] The objective is to estimate the equilibrium association (affinity) constant (Ka = 1 / Kd) for albumin-cortisol binding) from this series of experiments using the know n values of XTotA and the measured values of the percent free cortisol (R) as given in Table 4. Table 432Conservation of mass laws give:Rewriting 2 ) in terms of dissociation constant KD = 1 / Ka. we have A =KKpn1-f t Coolens’approximation assumes E« Kj> in which case A = TotA. Then = KA.4 = KATotA, SO that N = 7 and N = KATotA .

[0084] Equation 1) R = 1 / (1 +KA ) becomes T ) 1 / R = 1 +KA TotAA graphical estimate of K is obtained by plotting 1 / R versus TotA and fitting the regression line where the slope is the estimate K. Data are shown in Table 4 above, where R represents the free 3H-cortisol concentration expressed as a fraction of total 3H-cortisol [free 3H- cortisol] / [albumin-bound 3H-cortisol+ free 3H-cortisol],

[0085] The slope is KA=0.0771 per g / L. The conversion to units nmol / L gives 0.0771 / (15047) = 5.124 x 106 per nmol / L and in terms of K = 1 / 5.124 =195,200 nmol / L. At median concentrations of albumin in human plasma (40 g / L = 580,000 nmol / L), this corresponds to an N of 3.0, which is substantially higher than N=1.74 used in Coolens’ equation.

[0086] The concentration of albumin-bound cortisol (XFA) w as higher than predicted at the lowest concentration of albumin (0. 1 g / L) examined. To address the problem that the y- intercept is lower than predicted in the 1 / R vs. XTOIA graph above, an alternative solution algorithm using a non-linear function of R can be used. Applying Feldman equations w ithout Coolens’ approximation, Equation 1) can be written: tuting for A from eq 2). Soomes33

[0087] A graphical estimate of KAcan be obtained by plotting f(R) = - — B * R versusTotA and fitting the regression line where the slope is the estimate KA. The parameter B can be varied until the linear fit is good and / or the y-intercept is 1-B. Converting units as in the above example, this formulation yields a slightly higher affinity of albumin-cortisol binding, corresponding to KA of around 176,000 nmol / L and, for XTOIA of 580,000 nmol corresponds to N of 3.3.

[0088] An alternative consideration is that the stoichiometry of albumin-cortisol binding is not 1 : 1. For example, if it is assumed that cortisol-albumin binding was n: 1, where multiple cortisol molecules may bind to a single albumin molecule. In that case, the representation of the Feldman equations in matrix format still applies. The order of binding cannot be determined by the number of cortisol binding sites on albumin; it could still be 1 : 1. just one at a time, concatenated (represented mathematically as a composition of saturable binding functions), or involving allosteric interactions. The composition of two or more Michaelis- Menten-like functions is a Michaelis-Menten-like function. Therefore, Coolens’ approximation still holds because n: 1 binding still has only one protein.

[0089] The Feldman equations are: A = KKDd+ToFtnA= TotA if Fn« KDuand the constant N = KATotA and thus re N=K4 TotAIn this case, the representation of the Feldman equations in matrix format still applies as follows:XF + KXFn XA = XTotF, factoring F out yields XF (1+KXFn-lXA) = XTotF

[0090] An alternative, sequential approach to the various binding sites on the albumin molecule yields similar results but with a reduced n.

[0091] This model can be represented as a 2x1 matrix, treated radiolabeled cortisol (XE) and unlabeled cortisol (XF) as two ligands having the same albumin binding affinity (KA) and where XA represents ‘free’ albumin.34

[0092] In this scenario, the amount of labeled cortisol (XTOIE) is constant in the test tube, though the distribution between free- and albumin-bound E can vary (XE + XEA = XTOIE) and the concentrations of XTOIA is also experimentally constant for the test tube. Increasing the concentration of XiotF in the test tube to high levels (55,000 nmol / L) had no effect on the concentration of albumin-bound E (XEA). These observations support the assumption that (i) KA»XF and (ii) XA » XF, such that the concentration of XA is effectively constant and, consequently, the concentration of albumin-bound cortisol (XFA = KAXFXA) varies in proportion to XF. These conclusions are consistent with the experimental observation in that addition of cortisol to the test tube, even at supraphysiologic concentrations of XTotF (55,000 nmol / L) do not substantially lower the concentration of free albumin (XA). This may be contrasted with the saturable kinetics of the CBG-cortisol binding reaction, in which the concentration of XF may equal or exceed KC during physiologic conditions and the concentration ‘free CBG' (XC) varies substantially in relation to XF.

[0093] The above matrix may be viewed as a standardized binding assay for ligand-BP interactions in which3H-cortisol (E) is used as label and unlabeled cortisol (F) is treated as the competitor for a purified solution of albumin, which is present at high concentrations and demonstrates significant and measurable cortisol binding affinity. Assuming Coolens’ assumptions apply to both E and F in the 2x1 matrix (with same N). XTOIE = XE + NE and XiotF= XF + NF. Increasing TotF increases F in a proportional manner (XF = XTOIF / (N+1)), however the labeled total (XTOIE) is unchanged and XE = XTOIE / (N+1 ) is unchanged. Thus, the split between XE and XEA is unchanged, and the albumin bound labeled cortisol (XEA) is constant.

[0094] The principle of Coolens’ approximation where KA»F is reasonable, and there is good experimental evidence that albumin-cortisol binding kinetics are non-saturable at physiological concentrations of serum albumin. In other words, at any physiologic concentration of free cortisol (F), a linear relationship between albumin concentration and albumin-bound cortisol concentration (XFA) obtains. The slope of this linear relationship, with XF on abscissa and XFA on ordinate, is dependent on the constant (N), which in turn depends on albumin-cortisol binding properties summarized in a simplified stochiometricmodel by albumin concentration (XTOIA) and the affinity of the albumin-cortisol binding reaction. Where the equilibrium dissociation constant (KD) for albumin-cortisol binding (KA) is used to summarize the affinity of the reactants, N = XTOIA / KA.

[0095] At physiological concentrations of purified albumin solutions (40 g / L) and at 37C°, N=3.0 is realistic. This conclusion is also supported by studies where plasma was heated to 60°C for 60 min. which irreversibly denatures cortisol-binding activity of CBG, an N=3.0 was also observed. These experimental observations suggest KA = 195,200 nmol / L may provide a realistic, initial estimate of N based on measured concentrations of total albumin (XTOIA).

[0096] Although the possibility’ that other compounds present in human plasma might compete with cortisol for albumin binding or binding interactions cannot be excluded, the fact that addition of unlabeled cortisol to relatively high concentrations (55,000 nmol / L) in vitro had no effect on?H-cortisol binding to albumin (or heat-treated human plasma) supports the view that albumin-cortisol binding during physiologic conditions may be reasonably considered to be non-saturable. However, the existence of other substances, such as free fatty acids, that may circulate at sufficiently high concentrations (and bind albumin with sufficiently high affinity) to competitively decrease albumin-cortisol binding cannot be ruled out. Moreover, while the studies demonstrating a ratio of XFA to XF of approximately 3.0 in representative samples of heat-treated human serum are re-assuring in their similarity to results obtained in purified albumin solutions, they do not exclude the possibility of heat- labile compounds that competitively inhibit albumin-cortisol binding in vivo.

[0097] The equations may be modified to accommodate that possibility of multiple (n) binding sites, such that each molecule of albumin is capable of binding multiple cortisol molecules. One approach to this scenario treats albumin-cortisol binding as 1: 1 stoichiometry, but this approach tends to overestimate cortisol binding affinity (lower KA) and underestimates the concentration of cortisol needed to saturate albumin binding sites. The presence of multiple binding sites for albumin still accommodates the use of a constant (N) to reflect the proportionate relationship between free cortisol (XF) and albumin-bound cortisol (XFA).C. Estimation of Albumin-Cortisol Binding Constant (N) and Biological Variation in Albumin-Cortisol Binding in Non-Denatured Human Serum

[0098] As to what value for albumin-cortisol binding constant (N) is and how it may be estimated in a plasma sample is a matter of less certainty. One concern with Coolens'36equation is the assumption of albumin concentration (~40 g / L = 585,000 nmol / L). Previous work supports the use of measured rather than assumed albumin concentrations. This is reasonable since there is substantial normal variation in albumin concentrations and the measurement of albumin in clinical samples is simple, accurate, and inexpensive. As to what KA for albumin-cortisol binding should be used when applying Coolens’ ‘N’, there are major variations reported in the literature, ranging from KA of 80,000 to 810,000.

[0099] To further explore this issue, post-hoc analysis of data that includes serial measures of XTOIF and XF, as well as XiotcBG, was evaluated in subjects administered oral and iv hydrocortisone. Useful elements of the work are (i) development of Feldman’s coupled mass action and conservation of mass equations in a matrix format and (ii) introduction of the idea of interruption of matrix to solve the inverse matrix problem. The analysis used total and free cortisol concentration data. One finding in this analysis relevant to the above discussion concerning the albumin-cortisol binding constant (N) in human sera that has not been subject to heat-denaturation of CBG was the observation that N varied substantially by individual and condition.

[0100] If the equilibrium of free cortisol concentration (XF) binding to free CBG (Xc) and free albumin (2 ) as well as a known competitor (XP) that also binds to CBG and albumin*, the equilibrium can be represented in matrix format as:The matrix of affinities (K[y) is 2x2. In the above matrix format, the bounded concentrations XFC. XFA. XPC, and XPAhave been replaced by mass action formula (assuming 1 : 1 stochiometric binding). The four total concentrations give rise to four non-linear equations; and assuming total concentrations and all affinities Kj are known; there are four free concentration unknowns. Thus, based on conservation of mass and mass action formula, these four equations are:XrotF=^F (1 + Ku Xc+ K12XA~) XpotP=^p(l + K^ XC+ K12XA)

[0101] In the conventional (forward) solution of the matrix, the total concentrations are measured and the respective ligand binding affinities are known. The solution of the above coupled equations is a non-linear problem. The iterative nature of this problem can be seen by writing each equation above as the free ligand concentration equal to the total concentration divided by the corresponding quantity in the parentheses (i.e. cross dividing by rhs). These equations were reported to be:It is possible to use n Ptand m Qj for free concentrations of ligand and protein, respectively. This is a generalization to (nxm) matrix of affinities• The ratios for Qj into the ratios for Ptcan be substituted, giving the P, as continuous functions of Pj. This is convenient for iterative solution; though continuity gives the same solution if the functions for P(. and Qj to be iterated as a system.

[0102] It is possible to clear the fractions in the equation(s) Ptas a function of Ptto obtain the solution as the root of resulting polynomial. The degree of the resulting polynomial is m+1 which depends on the number of binding proteins represented by the Qj. The 1x1, 1x2, and 1x3 matrices lead to the quadratic polynomial, the cubic polynomial, or the quartic polynomial. Several of these formulations have also been adapted to include the possibility of additional ligands represented by 2x1, 2x2, and 2x3 matrices, as illustrated also by the model that includes a competitive inhibitor of CBG-cortisol binding. These polynomials have closed-form algebraic formula; however, larger systems (m>3) lead to quintic (or higher degree) polynomials, which are not tractable to closed-form solution (Abel's impossibility theorem). Application of Coolens’ assumption, namely that XF / XF (N) is a constant at physiological concentrations ofXp (and the corollary premise that N=XTOIA / KA, reduces the degree of the polynomial equations by 1. Thus, adoption of Coolens’ N means that the degree of the polynomial is equal to the number of columns in the above matrix.

[0103] The inverse problem introduces an addition problem: the 2x2 matrix has 4 unknowns and has 4 equations; however, the 2x3 or 3x2 matrix has 6 unknowns but only 5 equations. This difficulty can be remedied by more observations, e.g., by having multiple assays per subject. This approach relies on the assumption that cortisol concentrations are time-varying; that is, multiple assays would not be useful under steady state conditions. An additional assumption in this analysis is that the concentrations of binding proteins are timeinvariant during the sampling period.38

[0104] The equations can be interrupted for the solution when one of marginal totals is unknown (either in the far-right column and bottom row, in the present case XTotP) but (in the present case) free cortisol concentration (XF) is known. XTotPcan be considered to be a parameter of the system. The equation for XTotPabove is still useful in the iterative solution for TotP; however, more assays per subject are needed and the known total concentrations must vary across the different assay time points. The data consists of 14 or 15 time points per subject which vary dynamically (non-steady state condition) due to experimental intervention (a bolus of hydrocortisone at time zero).

[0105] To estimate parameters in the inverse problem, a programmable algorithm is needed; iterations will be a useful part of this algorithm. Non-linear regression provides least sum of squared errors estimators for KtJand now XTotPgiven an adequate number of observations. This outer loop gives a functional estimate of the predicted free cortisol, so that the sum of squared errors (measured XF- predicted XF) are minimized. The functional evaluation is also anon-linear problem and could require iteration as well. This is the inner loop providing the equilibrium solution (discussed above) that provides assurance that the predicted values and the total values are in equilibrium and thus coherent.This development is generalized to n ligands and m binding proteins (nxm) matrix K of affinities). The iterative equations can be replaced by the roots of polynomials. These examples can be generalized to include additional ligands by obtaining roots of coupled polynomial equations. How ever, this approach is limited by the fact that in general there is no closed form solution for quintic polynomial equations. By contrast, the disclosed approach using iteration is robust and versatile, as it may be applied over a broad range of conditions involving multiple binding proteins and / or ligands, including all those mentioned above. The use of this notation of Xc and XA in the matrix below differs from the use of these terms in previous reports; in previous reports Xc and XA were used to denote CBG-bound and albumin-bound cortisol, respectively.D. Distribution of Cortisol in Human Serum in Vitro: Role of Competitive Ligand- Protein Interactions in Women on Oral Contraceptives

[0106] The relationship betw een serum concentrations of total (XTOIF) and free (XF) cortisol (test tube equilibrium at 37°C) is influenced by many factors. These include concentrations and affinities of serum binding proteins (BP), such as CBG and albumin (A). Among women taking oral contraceptives (OC). XF was found to be higher than predicted using ligand-BP association parameters developed in healthy volunteers (HV). We39hypothesized that ligand(s) (Xp) competing for XF binding to CBG may contribute to altered relationships between XTOIF and XF observed in OC relative to HV.

[0107] Feldman’s system of iterative, non-linear equilibrium equations were developed in (nxm) matrix format. Data included measured XiotF and XF at each time point (0-480 min) after 20 mg hydrocortisone. Groups were stratified by (i) mode of administration (iv vs. po) and (ii) condition (HV. n=13 vs. OC. n=12). The matrix was interrupted to obtain iterative equilibrium solutions for parameters of interest using (1x2) minimal model (MM) and (2x2) ligand competition model (LCM). Solutions were optimized by minimizing least squares differences between measured and model predicted X . Albumin-bound cortisol was parameterized as a constant ratio of XF (N) to simplify the matrix problem. Comparisons and interactions within the 2x2x2 design (delivery, group, and model) were analyzed by ANOVA.

[0108] In OC, LCM provided significantly better fit to measured XF relative to MM (13 vs. 19% error, P=0.003). MM solutions for CBG-cortisol affinities were significantly higher in HV (KD=26.6) VS. OC (KD=7L9 nmol / L), but LCM yielded similar affinities in both groups (KD = 23.9 in HV and 16.1 nmol / L in OC, interaction P<0.001). LCM solutions for concentrations and CBG-binding affinities of Xp were significantly increased in OCP vs. HV (both P<0.001). LCM solutions for HV yielded higher N for po (3.1±1.2) vs. iv (1.4±0.4) (P=0.007). Model solutions depend on the precision and accuracy of measurements made in clinical samples. Although the competitor (Xp) is treated as a single compound in LCM, it may represent the combined effects (e.g., harmonic means) of multiple ligands.

[0109] Matrix notation was useful to development of non-linear, iterative equations for competitive ligand-BP interactions, which were applicable to both forward and inverse solutions of the matrix problems. LCM provided a better fit to experimental data for OC compared to MM, providing (indirect) evidence that competing ligand(s) (Xp) influence the distribution of cortisol in serum samples obtained from women on OC, (iii) inter- and intrasubject variability for N was observed for both models, suggesting that assignment of a single population value for cortisol-albumin binding constant may be unrealistic.

[0110] In the conventional application of Feldman-like equations, the challenge is to solve for the concentration of free ligand(s) when all the totals (of both ligands and BPs) are known and all the equilibrium association constants (K, L / nmol), which are equal to the inverse of the cognate BP-ligand equilibrium dissociation constant (KD. nmol / L), are known. As to the method for solving these coupled equations to estimate the free ligand40concentration, suffice to say that there are several different approaches. These solution methods may require iteration, which in a programming language involves a loop (perhaps a do loop).

[0111] Considering the simple 1x2 matrix, without the additional complexity of additional ligand(s) that compete with cortisol for CBG binding. In this 1x2 matrix, the unknown quantities to be estimated can be determined. Of the variables to be estimated, XF is of primary interest. It is possible but not necessary to estimate Xc and XA during estimation of XF by iteration, but in any case, these values are of (possible) secondary interest and generally go unreported.The estimation of the albumin-cortisol equilibrium association constant (K12) may be reparameterized as a constant (NA), as discussed above (III-D and III-E); this reparameterization is especially convenient in the data of Perogamvros et al, since XiotA was not measured in those experiments. However, under usual conditions XiotA is measured and the solution for NA obtained by solving for K12, as shown in the matrix above.

[0112] Re-writing the above matrix using Coolens-like assumption (N), the following matrix can be obtained. The use of Cool ens’ N simplifies the complexity of the matrix problem; that is, the degree of the coupled polynomial equations is re...

Claims

Attorney Docket No.37759.0576P1 ii) contacting the fourth sample portion with an effective amount of the labeled ligand; and iii) determining the concentration in the fourth sample portion of labeled ligand bound to CBG; and g) deriving, based on the concentrations determined in steps (c)-(f), the concentration of free cortisol in the sample.

2. The method of claim 1, wherein the labeled ligand is directly or indirectly labeled.

3. The method of claim 1, wherein the labeled ligand comprises labeled cortisol, corticosterone, cortisone, deoxycorticosterone, deoxycortisol, progesterone, or prednisolone.

4. The method of claim 1, wherein step (d) further comprises heating the second sample portion after contacting the second sample portion with an effective amount of the labeled ligand and prior to determining the concentration of labeled ligand bound to CBG.

5. The method of claim 1, wherein step (d) further comprises contacting the second sample portion with an effective amount of unlabeled cortisol prior to determining the concentration of labeled ligand bound to CBG.

6. The method of claim 1, wherein in step (d), (e), or (f), determining the concentration of labeled ligand bound to CBG comprises performing a lectin-affinity assay on the second, third, or fourth sample portion using a plant lectin which is immobilized on a solid support.

7. The method of claim 6, wherein the plant lectin is Concanavilin-A.

8. The method of claim 1, wherein step (d), (e), or (f) further comprises adding an effective amount of a second unlabeled ligand prior to determining the concentration of first labeled ligand bound to CBG, the second unlabeled ligand having a equilibrium association constant for CBG which is lower than cortisol’s equilibrium association constant for CBG. 86Attorney Docket No.37759.0576P1 9. The method of claim 8, wherein the second unlabeled ligand is cortisone, deoxycorticosterone, progesterone, or any combination thereof.

10. The method of claim 8, wherein the second unlabeled ligand is added at a known concentration.

11. The method of claim 1, wherein the first unlabeled ligand is thyroxine (T4).

12. A method of determining the concentration of free ligand or lipophilic hormone in a sample, the method comprising: a) obtaining or having a sample obtained, the sample comprising total ligand or lipophilic hormone, which is the sum of the concentration of free ligand or lipophilic hormone and ligand or lipophilic hormone bound to a ligand or lipophilic hormone binding partner, albumin, or both; b) dividing the sample into multiple sample portions; c) performing at least one assay on a first sample portion to determine the concentration in the first sample portion of (i) total ligand or lipophilic hormone binding partner, (ii) optionally, total albumin, and (iii) ligand or lipophilic hormone; d) contacting a second sample portion with an effective amount of a labeled ligand or lipophilic hormone, the labeled ligand or lipophilic hormone having an equilibrium association constant of at least 20 M-1for the ligand or lipophilic hormone binding partner, albumin, or both; and determining the concentration in the second sample portion of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner; e) performing a series of steps on a third sample portion, the series of steps comprising, in sequence: i) contacting the third sample portion with an effective amount of a first unlabeled ligand or lipophilic hormone, the first unlabeled ligand or lipophilic hormone having (i) an equilibrium association constant for albumin which is higher than the labeled ligand or lipophilic hormone’s equilibrium association constant for albumin, and (ii) an 87Attorney Docket No.37759.0576P1 equilibrium association constant for the lipophilic hormone binding partner which is less than 0.1 M-1; ii) contacting the third sample portion with an effective amount of the labeled ligand or lipophilic hormone; and iii) determining the concentration in the third sample portion of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner; and f) deriving, based on the concentrations determined in steps (c)-(e), the concentration of free ligand or lipophilic hormone in the sample.

13. The method of claim 12, further comprising performing a series of steps on a fourth sample portion, the series of steps comprising, in sequence: a) contacting the fourth sample portion with an effective amount of albumin; b) contacting the fourth sample portion with an effective amount of the labeled ligand or lipophilic hormone; and c) determining the concentration in the fourth sample portion of labeled ligand or liphophilic hormone bound to the ligand or lipophilic hormone binding partner.

14. The method of claim 13, wherein determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner comprises performing a lectin-affinity assay on the fourth sample portion using a plant lectin which is immobilized on a solid support.

15. The method of claim 14, wherein the plant lectin is Concanavilin-A.

16. The method of claim 12, wherein the labeled ligand or lipophilic hormone is directly or indirectly labeled.

17. The method of claim 12, wherein the labeled ligand or lipophilic hormone comprises labeled cortisol, corticosterone, cortisone, deoxycorticosterone, deoxycortisol, progesterone, or prednisolone. 88Attorney Docket No.37759.0576P1 18. The method of claim 12, wherein step (d) further comprises heating the second sample portion after contacting the second sample portion with an effective amount of the labeled ligand or lipophilic hormone and prior to determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner.

19. The method of claim 12, wherein step (d) further comprises contacting the second sample portion with an effective amount of unlabeled ligand or lipophilic hormone prior to determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner.

20. The method of claim 12, wherein in step (d) or (e), determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner comprises performing a lectin-affinity assay on the second or third sample portion using a plant lectin which is immobilized on a solid support.

21. The method of claim 20, wherein the plant lectin is Concanavilin-A.

22. The method of claim 12, wherein step (d) or (e) further comprises adding an effective amount of a second unlabeled ligand or lipophilic hormone prior to determining the concentration of labeled ligand or lipophilic hormone bound to the ligand or lipophilic hormone binding partner, the second unlabeled ligand having a equilibrium association constant for the ligand or lipophilic hormone binding partner which is lower than the ligand or lipophilic hormone’s equilibrium association constant for CBG.

23. The method of claim 22, wherein the second unlabeled ligand or lipophilic hormone is cortisone, deoxycorticosterone, progesterone, or any combination thereof.

24. The method of claim 22, wherein the second unlabeled ligand or lipophilic hormone is added at a known concentration.

25. The method of claim 12, wherein the first unlabeled ligand is thyroxine (T4).

26. The method of claim 12, wherein the ligand or lipophilic hormone binding partner is a serum transport or binding protein. 89Attorney Docket No.37759.0576P1 27. The method of claim 12, wherein the ligand or lipophilic hormone is a corticosteroid, sex steroid, vitamin D, or a thyroid hormone.

28. The method of claim 26, wherein the ligand or lipophilic hormone is cortisol and the serum transport or binding protein is CBG.

29. The method of claim 26, wherein the ligand or lipophilic hormone is progesterone and the serum transport or binding protein is CBG.

30. The method of claim 26, wherein the ligand or lipophilic hormone is testosterone and the serum transport or binding protein is sex hormone binding globulin.

31. The method of claim 26, wherein the ligand or lipophilic hormone is a vitamin D metabolite and the serum transport or binding protein is vitamin D binding protein.

32. A method of diagnosing a disease or disorder, the method comprising performing the method of claim 12, and diagnosing the disease or disorder based on the concentration of free ligand or lipophilic hormone in the sample. 9027. The method of claim 12, wherein the ligand or lipophilic hormone is a corticosteroid, sex steroid, vitamin D, or a thyroid hormone.

28. The method of claim 26, wherein the ligand or lipophilic hormone is cortisol and the serum transport or binding protein is CBG.

29. The method of claim 26, wherein the ligand or lipophilic hormone is progesterone and the serum transport or binding protein is CBG.

30. The method of claim 26, wherein the ligand or lipophilic hormone is testosterone and the serum transport or binding protein is sex hormone binding globulin.

31. The method of claim 26, wherein the ligand or lipophilic hormone is a vitamin D metabolite and the serum transport or binding protein is vitamin D binding protein.

32. A method of diagnosing a disease or disorder, the method comprising performing the method of claim 12, and diagnosing the disease or disorder based on the concentration of free ligand or lipophilic hormone in the sample.90

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