Systems and methods for determining plasma pharmacokinetics from high temporal resolution measurements of interstitial fluid drug concentrations

Abstract

Methods and systems that use high-frequency molecule or ion concentration measurements collected in subcutaneous or intradermal interstitial fluid to estimate a plasma concentration time course are described. Several embodiments can estimate the plasma concentration time courses from measurements of interstitial fluid taken at two or more sites. This ability to estimate clinically important plasma concentrations or concentration time courses using minimally invasive, subcutaneous or intradermal sensor placements can improve the precision and accuracy of the diagnosis, treatment, and monitoring of many diseases.

Description

SYSTEMS AND METHODS FOR DETERMINING PLASMA PHARMACOKINETICS FROM HIGH TEMPORAL RESOLUTION MEASUREMENTS OF INTERSTITIALFLUID DRUG CONCENTRATIONSCROSS-REFERENCE TO RELATED APPLICATIONS

[0001] The current application claims the priority to U.S. Provisional Patent Application No. 63 / 617,691 entitled “Systems and Methods For Estimating Plasma Pharmacokinetics From High Temporal Resolution Measurements Of Interstitial Fluid Drug Concentrations” filed January 4, 2024. The disclosure of U.S. Provisional Patent Application No. 63 / 617,691 is hereby incorporated by reference in its entirety for all purposes.GOVERNMENT SPONSORED RESEARCH

[0002] This invention was made with government support under EB022015 awarded by the National Institutes of Health. The government has certain rights in the invention.FIELD OF THE INVENTION

[0003] The current disclosure is directed to systems and methods for determining plasma pharmacokinetics; and more particularly to systems and methods for determining plasma pharmacokinetics from measurements of interstitial fluid drug concentrations.BACKGROUND

[0004] Therapeutic drug monitoring (TDM) and the monitoring of specific metabolites, biomarkers, and ions often provides lifesaving advantages in many therapies. However, current monitoring approaches for such molecules and ions may require blood draws and laboratory analysis. The cumbersome processes can limit the use of such measurements in clinical practice. In addition, because of the cumbersome processes involved, such measurements typically rely on only one or two measurements per day. Such sparse sampling reduces the precision with which trends can be determined, likely also reducing clinical effectiveness. Improved methods to monitor the plasma concentration orconcentration time courses of drugs, metabolites, biomarkers, and ions may thus be needed to improve clinical practice.BRIEF SUMMARY

[0005] Systems and methods for determining the plasma concentration time courses of drugs, metabolites, biomarkers and ions from measurements of interstitial fluid drug concentrations are described.

[0006] Some embodiments include a method for determining a plasma concentration time course of a molecule or ion, comprising: collecting a plurality of concentration time courses of the molecule or ion from at least two interstitial fluid (ISF) sites, wherein each of the at least two ISF sites has a non-identical distribution rate constant describing a rate of transport between the plasma and the at least two ISF sites; and estimating the plasma concentration time course by fitting the plurality of concentration time courses to a plurality of mathematical relationships; wherein the molecule or ion is eliminated from the body via the plasma instead of the at least two ISF sites; and wherein the at least two ISF sites communicate diffusively with the plasma.

[0007] In some embodiments, a structure of a plasma pharmacokinetics of the molecule or ion is known.

[0008] In some embodiments, the structure of the plasma pharmacokinetic comprises a first order plasma pharmacokinetic order, a second order plasma pharmacokinetic order, and a third order plasma pharmacokinetic order.

[0009] In some embodiments, a structure of a plasma pharmacokinetics is unknown, wherein at least one of the plurality of mathematical relationships selected from the group consisting of: the Bayesian Information Criterion; and the Akaike Information Criterion (AIC) is used to determine a preferred structure of the plasma pharmacokinetics.

[0010] In some embodiments, a plasma pharmacokinetic order is unknown, and the method further comprises estimating the distribution rate constant for transport between the plasma and the at least two ISF sites.

[0011] Some embodiments further comprise applying a smoothing time interval to the plurality of concentration time courses.

[0012] In some embodiments, an interval length with a lowest AIC among a plurality of time interval length options is chosen as a suitable smoothing time interval length.

[0013] In some embodiments, the plasma pharmacokinetic order is configured to be a first, second or third order plasma pharmacokinetic order or an enterohepatic reabsorption model.

[0014] Some embodiments further comprise using a plurality of electrochemical aptamer-based (EAB) sensors to collect the plurality of concentration time courses of the molecule or ion.

[0015] In some embodiments, each of the plurality of EAB sensors comprises: a working electrode, wherein the working electrode comprises a recognition element modified with a redox molecule; wherein the recognition element specifically binds to the molecule or ion and is configured to undergo a conformational change upon binding with the molecule or ion; wherein a current readout from the EAB sensor reflects a concentration of the molecule or ion.

[0016] In some embodiments, the recognition element is a nucleic acid aptamer or a protein.

[0017] In some embodiments, the working electrode is selected from the group consisting of: a wire, a rod, a planar electrode, a needle, and a microneedle.

[0018] In some embodiments, the molecule or ion is a drug is selected from the group consisting of: a small molecule drug molecule, an amino acid-based drug molecule, a protein-based drug molecule, a nucleic-acid-based drug molecule, an ion drug, and a monatomic ion drug.

[0019] In some embodiments, the distribution rate constant for transport for each of the at least two ISF sites differs by at least 10%.

[0020] Some embodiments include a method for determining a plasma concentration time course of a molecule or ion, comprising: collecting a plurality of concentration time courses of a molecule or ion from at least two interstitial fluid (ISF) sites, wherein each ofthe at least two ISF sites has a non-identical distribution rate constant describing a rate of transport between the plasma and each of the at least two ISF sites; and estimating the plasma concentration time course by fitting the plurality of concentration time courses to a plurality of mathematical relationships; wherein the molecule is eliminated from the body via the plasma instead of the at least two ISF sites; and wherein the at least two ISF sites communicate diffusively with the plasma.

[0021] In some embodiments, the molecule or ion is selected from the group consisting of: a small molecule drug, a macromolecular drug, an amino acid-based drug, a protein-based drug, a nucleic-acid-based drug, a small molecule metabolite, a hormone, a protein, an enzyme, an antibody, a biomarker, an ion, and a monatomic ion.

[0022] In some embodiments, a structure of a plasma pharmacokinetics of the molecule is known.

[0023] In some embodiments, a structure of a plasma chemical kinetics of the molecule or ion is unknown, wherein at least one of the plurality of mathematical relationships selected from the group consisting of: the Bayesian Information Criterion; and the Akaike Information Criterion (AIC) is used to determine a preferred structure of the plasma chemical kinetics.

[0024] In some embodiments, a plasma pharmacokinetic order is unknown, and the method further comprises estimating the distribution rate constant for transport between the plasma and the at least two ISF sites.

[0025] Some embodiments further comprise applying a smoothing time interval to the plurality of concentration time courses.

[0026] In some embodiments, an interval length with a lowest AIC among a plurality of time interval length options is chosen as a suitable smoothing time interval length.

[0027] In some embodiments, the plasma pharmacokinetic order is configured to be first or second or third order pharmacokinetics or an enterohepatic reabsorption model.

[0028] Some embodiments further comprise using a plurality of electrochemical, aptamer-based (EAB) sensors to collect the plurality of concentration time courses of the molecule.

[0029] In some embodiments, each of the plurality of EAB sensors comprises: a working electrode, wherein the working electrode comprises a recognition element modified with a redox molecule; wherein the recognition element specifically binds to the molecule or ion and is configured to undergo a conformational change upon binding with the molecule or ion; wherein a current readout from the EAB sensor reflects a concentration of the molecule or ion.

[0030] In some embodiments, the recognition element is a nucleic acid aptamer or a protein.

[0031] In some embodiments, the working electrode is selected from the group consisting of: a wire, a rod, a planar electrode, a needle, and a microneedle.

[0032] In some embodiments, the distribution rate constant for transport for each of the at least two ISF sites differs by at least 10%.

[0033] Additional embodiments and features are set forth in part in the description that follows, and in part will become apparent to those skilled in the art upon examination of the specification or may be learned by the practice of the disclosed subject matter. A further understanding of the nature and advantages of the present disclosure may be realized by reference to the remaining portions of the specification and the drawings, which form part of this disclosure.BRIEF DESCRIPTION OF THE DRAWINGS

[0034] The description will be more fully understood with reference to the following figures, which are presented as example embodiments of the invention and should not be construed as a complete recitation of the scope of the invention, wherein:

[0035] Figure 1A illustrates the concentration of a drug over time in accordance with an embodiment. Figure 1 A shows under the assumption that the drug transport between ISF and the plasma is governed by the equation (1 )(1 ), drug concentrations in the ISF are a reasonably good approximate of plasma drug concentrations as long as the rate constant for distribution between the plasma and the ISF, kD, is at least few-fold higher than the fastest rate of change in the plasma concentrations, here the elimination ratefrom the blood, kE. As long as the rate constant for the distribution, kD, is sufficiently high, some clinically important constants describing drug pharmacokinetics in plasma can be inferred directly by observing the ISF pharmacokinetics of the same drug.

[0036] Figure 1 B illustrates that, when the distribution rate constant is 10-times higher than the elimination rate constant, the half-life (after the peak concentration is achieved) seen in the ISF is about 15% longer than that seen in the plasma in accordance with an embodiment.

[0037] Figure 1 C illustrates the peak concentration in ISF, C^axreaches about 80% of when the two rate constants differ by the same factor of ten in accordance with an embodiment. The ratios considered in Figure 1A are highlighted in 1 B and 1 C.

[0038] Figures 2A and 2B show under a two-compartment model, observations in the secondary compartment (here ISF) do not provide enough information to unambiguously predict the concentrations in the central compartment (here plasma) in accordance with an embodiment. Figure 2A shows the model includes drug delivery into the plasma followed by transport into the ISF compartment with a distribution rate constant, kDand elimination from the plasma via first-order kinetics with an elimination rate constant, kE. Figure 2B shows under this model, any observed ISF concentration time profile could arise due to either of two non-equivalent plasma concentration time profiles.

[0039] Figure 3A shows due to a mathematical redundancy, concentration time course observations collected at a single site in the ISF are unable to uniquely define the plasma concentration time course that is generating them, thus leading to infinitely many mathematically equivalent estimates of plasma concentration-time profiles in accordance with an embodiment.

[0040] Figure 3B shows under the assumptions employed here, the ambiguity can be overcome by employing observations from a pair of non-redundant ISF measurement sites differing in the rate constants that describes molecular or ionic transport between the plasma and these sites in accordance with an embodiment.

[0041] Figure 3C shows such a pair of measurements can be used to derive a unique estimate of the plasma concentration time profile in accordance with an embodiment.

[0042] Figures 4A through 4C show that with prior knowledge of the appropriate pharmacokinetic model order, a non-redundant pair of ISF measurements can be used to estimate plasma concentration time courses with high precision in accordance with an embodiment. If the true compartmental pharmacokinetic model orders are known (e.g., of orders 1 , 2, or 3, respectively on figures 4A, 4B, 4C) in plasma, then the plasma timecourses with high precision by fitting to the non-redundant noisy ISF measurements can be estimated. Here the true plasma profiles are shown by solid black lines, and the simultaneously estimated plasma and ISF profiles by dashed green lines.

[0043] Figures 5A through 5C show that Bayesian Information Criterion (BIC) can be used to select the plasma pharmacokinetic model most likely to explain an observed, non- redundant pair of ISF concentration profiles, assuming the true plasma concentration time courses follow one of several possible models in accordance with an embodiment. Figures 5A through 5C show the “true” plasma pharmacokinetics follow a second order compartmental model (equation (5), with the noise-free true concentration profile in plasma is given by the black solid line and the fits are presented under the assumption of plasma pharmacokinetics being order one (Figure 5A), two (Figure 5B) and three (Figure 5C). The minimal BIC is seen for a second order model, thus recovering the correct model order.

[0044] Figures 6A through 6D show measurements collected at two non-redundant ISF sites are sufficient to accurately estimate the plasma concentration-time profile even in the absence of foreknowledge of even the structure (e.g., knowledge that it is a compartmental model of some order) of the plasma pharmacokinetics in accordance with an embodiment. To demonstrate this “Minimum Assumptions Fit” approach, two scenarios are employed: one in which the plasma kinetics follow a single-compartment model (Figure 6A, Figure 6B), and another in which the plasma concentration exhibits the effect of enterohepatic reabsorption (EHR), a pharmacokinetic mechanism that can generate non-monotonic plasma concentration time courses (Figure 6C, Figure 6D). These two plasma profiles can be used to generate corresponding ISF time courses to which simulated noise is added at typical experimental levels (signal-to-noise ratio = 30).(Figure 6A, Figure 6C). Fitting the resulting noisy ISF profiles (dots) returns the original plasma time course (solid black lines) to a clinically relevant degree of accuracy. The fits are shown by blue dots with 95% confidence intervals shown as solid blue lines. The estimated interpolated values in-between freely estimated values are shown by the green dashed lines (Figure 6B, Figure 6D).

[0045] Figures 7A through 7D show that, even when the distribution rate constants for the two measurements sites approach one another somewhat closely, the plasma concentration time course can still be estimated to a clinically relevant degree of accuracy in accordance with an embodiment. Two simulated plasma concentration time courses are employed: one when the plasma kinetics obeys a first-order exponential decay (Figure 7A, Figure 7B), and another when the plasma concentration exhibits the effect of enterohepatic recirculation, a mechanism that generates particularly complex, nonmonotonic pharmacokinetic behavior (Figure 7C, Figure 7D). Using these two simulated plasma time courses, pairs of ISF time courses were generated using pairs of distribution rate constants that differ by as much as a factor of 2 or as little as 10%. Simulated noise was then added after typical experimental noise (signal-to-noise ratio is about 30) to generate synthetic “experimental” pairs of ISF profiles (Figure 7A, Figure 7C). Fitting these returns the original plasma concentration time course (solid lines) to a clinically valuable but diminishing degree of accuracy when the two distribution rate constants approach within 10% of one another. The fits are shown by dots with 95% confidence intervals shown as solid lines matching the ratio between diffusion rates. The estimated interpolated values in-between freely estimated values are shown by the dashed lines again with corresponding colors (Figure 7B, Figure 7D).DETAILED DESCRIPTION

[0046] Therapeutic drug monitoring (TDM), where the individualization of dosages to maintain plasma drug concentrations within specified parameters, is an important component of many pharmacological therapies. Similarly, many therapeutic interventions derive value from measurements of plasma metabolite, biomarker, or ion concentrations.However, current approaches for measuring the concentrations of such molecules and ions in the plasma require blood draws and benchtop laboratory analysis. The cumbersome nature of these processes can limit their adoption. In addition, due to the cumbersome processes involved, these molecules and ions are typically only measured one or two times per day. The sparse sampling can significantly reduce the ability to monitor clinically important trends, likely also reducing the clinical value of such measurements.

[0047] In vivo molecular sensing using electrochemical, aptamer-based (EAB) sensors can provide seconds-resolved drug concentration measurements in situ in the living body can provide a method in order to improve the convenience and accuracy of in vivo molecular monitoring (See, e.g., Arroyo-Curras, N., et al., (2017). Proceedings of the National Academy of Sciences, 114(4), 645-650; Chamorro-Garcia, A., et al., (2022). ACS Sensors, acssensors.2c01894; Dauphin-Ducharme, P., et al., (2019). ACS Sensors, 4(10), 2832-2837; Lin, S., et al., (2022). Science Advances, 8(38), eabq4539; Wu, Y., et al., (2022). Analytical Chemistry, 94(23), 8335-8345; the disclosures of which are incorporated by reference.)

[0048] Although the sub-second-resolved, EAB-derived molecular measurements may be collected in plasma using intravenous sensors (Arroyo-Curras, N., et al., (2018). ACS Sensors, 3(2), 360-366, Idili, A., et al., (2019). Chemical Science, 10(35), 8164- 8170; Idili, A., et al., Analytical Chemistry, 93(8), 4023-4032; Vieira, P. A., et al., (2019). Frontiers in Molecular Biosciences, 6; the disclosures of which are incorporated by reference.), the placements required for such measurements may be more invasive than ideal for day-to-day clinical use. In contrast, real-time drug measurements performed in the interstitial fluid (ISF) of the subcutaneous or intradermal space would be preferable due to safety and convenience considerations, as sensors can access the ISF with far less invasive placements than those required for plasma measurements (Lin, S., et al., (2022). Science Advances, 8(38), eabq4539; Wu, Y., et al., (2022). Analytical Chemistry, 94(23), 8335-8345; the disclosures of which are incorporated by reference.) In order to apply ISF-derived concentration data to patient care and make effective and accurateclinical diagnostics, however, methods that can convert such measurement results into clinically accurate estimates of the plasma concentrations are needed.

[0049] Many embodiments provide methods to accurately determine plasma concentration-time profiles by using highly time resolved measurements of concentrations in the ISF. Several embodiments can determine plasma concentration time courses with clinically acceptable accuracy using realistically noisy measurements performed in the ISF, under the assumptions that the molecule or ion in question is eliminated from the body from the plasma, and not from the ISF, and that the ISF communicates only with and (i.e., without active transport) passively with the plasma. Provided that the time scale for distribution into the ISF is several times more rapid than the time scale over which plasma concentrations meaningfully change, ISF time courses closely parallel plasma time courses in accordance with some embodiments. For molecules or ions for which this condition does not hold, plasma concentration time courses can still be accurately estimated from measurements collected at a pair of ISF sites in accordance with certain embodiments, provided the latter differ even slightly in the time scales with which the molecule or ion diffuses to them. Under the assumptions employed, some embodiments can estimate full plasma kinetic time profiles without a priori knowledge of the structure of the plasma time course. The resulting ability to estimate plasma pharmacokinetics from minimally invasive, subcutaneous or intradermal sensor placements has the potential to improve the precision and reach of therapeutic drug monitoring, the safety and efficacy of drug delivery, and the clinical value of metabolite, biomarker, and ion concentration measurements. Methods for estimating plasma concentration-time profiles can be used for (but not limited to) therapeutic drug monitoring, drug development, and / or pharmacological studies, the diagnosis, treatment, and monitoring of metabolic and endocrinological disorders, the diagnosis, treatment, and monitoring of inflammatory diseases, and the diagnosis, treatment, and monitoring of electrolyte imbalances.

[0050] Many embodiments can determine and / or estimate the plasma concentrationtime profiles for various types of molecules including (but not limited to) small moleculedrugs, macromolecular drugs, amino acid-based drugs, protein-based drugs, nuclei acidbased drugs, metabolites, biomarkers including (but not limited to) hormones, proteins indicative of disease, and ions, including but not limited to sodium, potassium, calcium, magnesium, chloride. The drugs, metabolites, and biomarkers can have a variety of sizes and / or molecular weights.

[0051] In many embodiments, the EAB sensors can be used for measuring molecules (such as drug molecules) concentrations at various ISF sites. EAB sensors can perform seconds and / or sub-seconds resolved measurements of multiple molecules and / or ions in situ in the living body of an animal or a non-animal human. For in vivo applications, the EAB sensors may include an aptamer-coated microneedle or wire as a working electrode. The microneedle or wire may be inserted through the surface of the skin such that the aptamer-coated portion contacts a biological fluid of the subcutaneous tissues. Alternatively, the EAB sensors may be placed in another bodily compartment, such as a vein. The EAB sensors can also include a counter electrode and a reference electrode. These electrodes can also be in the form of a microneedle, or a wire similarly inserted under the skin.

[0052] The EAB sensor electrodes may remain in situ for minutes, hours or even days and over that period provide clinically valuable information on the amount of analyte in the bodily fluid. Reasonable extrapolation to estimate amounts of analyte in the general circulation may be made. In this way, an EAB sensor can provide clinically relevant information on the amount of an exogenous analyte (such as a drug) or an endogenous analyte (such as a hormone) in the subject. The information may be used in the diagnosis, treatment, and / or monitoring of disease.

[0053] The working electrode of an EAB sensor may have at least one associated counter electrode and at least one associated reference electrode. Each working electrode may have a dedicated counter electrode, however in some embodiments the counter electrode is shared amongst some or all the assembled working electrodes. Each working electrode may have a dedicated reference electrode, however in someembodiments the reference electrode is shared amongst some or all the assembled working electrodes.

[0054] In many embodiments, an EAB sensor may be voltametric, chronoamperometric, or impedimetric. In a voltametric sensor, a potential waveform is applied to the sensor interface, and the resulting current response is recorded. In chronometric approaches, a step potential is applied, and the resulting time-evolving current response is recorded. In impedimetric sensing, a sinusoidal potential waveform is applied, and the resulting sinusoidal current response is recorded.

[0055] In some embodiments, EAB sensors are of the voltametric type, with a recognition element being bound to the working electrode. Gold can be used as the probe surface for the working electrode. The recognition element can be a nucleic acid or an aptamer. The recognition element has an associated redox-active species which acts as a reporter. The redox reporter can be (but not limited to) methylene blue. Upon target (e.g., drug) binding, the recognition element undergoes a conformational change, bringing the redox reporter more proximal to the working electrode surface. This increase in proximity increases electron transfer from the redox reporter to the electrode. The increase in speed of electron transfer contributes to a change in Faradaic current that is detected by a potentiostat. EAB sensors can be incorporated into a circuit having a reference electrode. The reference electrode is the site of a known chemical reaction that has a known redox potential. For example, a reference electrode based on the silversilver chloride (Ag / AgCI) redox pair has a fixed and known potential forming the point against which the redox potential of the working electrode is measured. Also typically included in the circuit is a counter electrode which functions as a cathode or an anode to the working electrode. Because current does not pass through the reference electrode (due to an impedance of the potentiostat), any current generated is attributed to the working and counter electrodes. Current is measured as a function of potential of the interrogating electrode versus the reference electrode. The difference in potential produces the current in the circuit thereby generating an output signal. The signalquantifies target binding depending on electron transfer that is ideally stoichiometrically proportional to target binding.

[0056] In some embodiments, the working electrode or any other electrode may be a wire, a needle, a microneedle, an electrode array, a microneedle array, which contact the ISF of a subject. Microneedles and / or microneedle arrays are preferred for transdermal applications where piercing of the skin is necessary to contact the ISF.

[0057] Electrodes in accordance with many embodiments can be fabricated in a range of various shapes and geometries, although their specific geometry for transdermal applications be optimized to breach the stratum corneum for reliable skin penetration. In some embodiments, the apparatus may be configured to be urged into the skin of a subject to facilitate the electrodes breaching the stratum corneum and to penetrate through the skin layers. For non-human applications, the stratum corneum may be replaced by an analogous, or even a non-analogous layer on the surface of the subject.

[0058] In several embodiments, the plasma concentration time courses of various molecules can be accurately estimated from measurements at two or more ISF sites. The ISF measurement sites can be a certain distance away from each other. The ISF measurement sites can be at different depths in the skin. Each ISF measurement site in accordance with some embodiments has different rate constant for target molecule distribution (or non-identical target molecule or ion concentration changes) such that the ISF measurement sites produce non-redundant or non-identical concentration time courses.

[0059] The determination and / or estimation of plasma concentration-time profiles in accordance with several embodiments may employ three assumptions. The first is that the molecule or ion in the test may not be eliminated from the body via or produced in the tissue compartment that is being observed. Given that observations are made in the subcutaneous or intradermal ISF, this assumption holds for most drugs and many metabolites and biomarkers, which, for example, are cleared hepatically or renally. The second assumption is that the ISF is a terminal compartment (or is in rapid equilibrium with intracellular fluids, rendering intracellular and intercellular fluid a single compartment)that communicates only with the plasma and only via passive diffusion (or mathematically equivalently, the bulk flow of fluid into the tissues from the capillaries and out of tissue via the lymph system), an assumption that is commonly ascribed to, for example, glucose. The third assumption is that the rate constants describing molecular transport between the ISF and the plasma are constant (continuous glucose monitors may have a “lock out period,” presumably to allow the tissues to adjust to the insertion of the device). Under these assumptions, transport between the plasma and the ISF is described by,where and CP(t) are the time-dependent concentrations in the ISF and plasma, respectively, and kDis the rate constant for the target molecule’s distribution between the two compartments.

[0060] When equation (1 ) holds and when the timescale associated with distribution between the plasma and the ISF (~ 1 / kD) is significantly shorter than the timescale of the rapid and clinically relevant changes in plasma concentration, several embodiments provide a method for estimating plasma pharmacokinetics, CP(t), using concentration measurements performed in the ISF, C / SF(t). As this gap in timescales increases, the concentration time course in the ISF approaches the plasma concentration time course. Several embodiments provide a quantitative description of a drug that exhibits first-order elimination kinetics, (2)where kEis the first order elimination rate constant and the Cmaxis the maximum plasma concentration seen after, for example, an injection. Under this scenario, if kDis 10-fold greater than kE, the plasma and ISF pharmacokinetics track one another rather closely (Fig. 1A). In Fig. 1 A through Fig. 1 C, 101 represents the curve kD= 10 / cF. 102 represents the curve kD= 3kE. 103 represents the curve kD= kE. 104 represents the curve kD= ^kE. 105 represents the curve 106 represents plasma. Specifically, in thisscenario the half-life observed in ISF is about 15% longer than the half-life seen in the plasma (Fig. 1 B). Similarly, the maximum concentration reached in the ISF, C^Fax, is within about 20% of the plasma maximum,despite this parameter being much more sensitive to the ratio of the distribution to elimination timescales (Fig. 1 C). Due to the low molecular weight of glucose (and thus its rapid diffusion into the tissues) and the relatively small magnitude and slow timescale with which its plasma concentrations fluctuate (e.g., the concentration rise and fall after eating is typically less than 2-fold and occurs over many minutes), this scenario is a reasonable approximation for the relationship between its plasma and ISF concentrations, and thus continuous glucose monitors employ this assumption in their analysis. As kDapproaches kE, however, the accuracy of simply assuming that the plasma time course equals the ISF time course degrades. For example, when kDis about 3-fold faster than kEthe half-life in ISF is about 50% longer than that in plasma and C™xfalls approximately 40% short of C^sxza. When kDequals kE, the halflife in ISF stretches to about 240% that of the plasma and CIFFfalls to about 1 / 3 of the^-.plasma max

[0061] Under the assumptions as discussed above, measurements performed in the ISF can also provide clinically important constraints on plasma pharmacokinetics in scenarios in which the timescales of the most rapid changes in plasma concentration are similar to the timescales of distribution into the ISF. Specifically, while ISF concentration time courses might not clearly define C^sxia, as long as equation (1 ) holds, ISF-derived measurements can be used to determine plasma exposure (as the “area under the curve”;This is because, under the assumptions, AUCptasmaasymptotically approaches the area under the curve observed in the ISF (AUClsp), and thus the latter can be used to estimate the former to any desired precision degree if one collects data for sufficiently long (see Theorem 1 in the Examples section). However, while this result provides a means of estimating A UCpiasmafrom AUCISF, it does not provide a means of estimating the underlying plasma concentration time profile, which may have a greater clinical value.

[0062] Several embodiments provide that under the simplifying approximations outlined above, a molecule’s or ion’s plasma concentration time course cannot be uniquely determined from measurement of its concentration time course at a single site in the ISF. To understand the observation, consider a drug that obeys a two-compartment model where the drug pharmacokinetics in the plasma is dominated by a constant elimination rate and the drug can only enter to and exit from the ISF through passive means from and to the plasma. Under these circumstances the drug’s pharmacokinetics can be described by:where kEis the rate constant for elimination from the plasma, kDis the rate constant for transport between the plasma and the ISF, VPis the apparent distribution volume of the plasma, and CP(t) and C / SF(t) are the concentration time courses in plasma and ISF, respectively. For this case, there are always two plasma time courses (differing in kEand kD) that could produce any given ISF time course (see Fig. 2A, Fig. 2B, and Proposition 2 in the Example section), such that time courses collected at a single site in the ISF will produce ambiguous estimates of the plasma time course. And the situation for models that include more than two compartments may be worse. For example, there are a number of plasma time courses consistent with any given ISF time course under a three- compartmental model (see Fig. 3A through Fig. 3C and Proposition 3).

[0063] Provided that the assumptions hold (i.e., that equation (1 ) holds), a solution to resolve the ambiguous estimates of plasma pharmacokinetics may include: the use of measurements performed at two, non-redundant sites in the ISF for which kDdiffers by some degree. This results in the equations,where Q(t) and C2(t) stand for the molecule’s or ion’s concentration time courses observed in the first and the second measurement sites, / q and fc2are the distribution coefficients for transport between the plasma and each of these two sites, and CP(t) is the plasma concentration time course.

[0064] The plasma concentration time course can be uniquely estimated from dual ISF time courses in the simplifying case of having some a priori knowledge of the structure of the concentration plasma time course. Specifically, assuming that the plasma time course after, for example, an injection can be described as the sum of a specific, known number, n, of exponential decays,where Atand kE. are the relative magnitude and the elimination rate for the ithexponential decay. The number of exponential decay profiles appear in the pharmacokinetic order (see Methodology for details), for example the equation (2) is of pharmacokinetic order 1 (i.e. , a one-compartment model). The distribution coefficients can then be estimated along with parameters that define the plasma concentration profile given the noisy ISF measurements and the intravenous drug injection profile. To demonstrate this, the systems can be simulated in which (a) the plasma pharmacokinetics are of pharmacokinetic order 1 , 2 or 3; (b) the signal-to-noise ratio is 30; and the rate constants for distribution into the tissues at the two observation sites differ by some factor. The first analysis assumes that the rate constants for distribution to the two ISF sites are ki = 0.15 min-1and k2 = 0.075 min-1; i.e., they differ by a factor of 2. Under these circumstances, the plasma concentration time course can be estimated from the paired ISF measurements uniquely and with great precision (Fig. 4A through Fig. 4C).

[0065] While a priori knowledge of the order of the plasma pharmacokinetics is employed in the above study (Fig. 4A through Fig. 4C), the ability to estimate plasmaconcentration time courses from time-resolved measurements at two ISF sites in accordance with various embodiments does not require such foreknowledge. One way to achieve accurate plasma estimates without foreknowledge of their structure is to perform fits to a set of candidate pharmacokinetic orders, and then use a selection criterion, such as Bayesian Information Criterion (BIC), to determine the preferred model. BIC punishes model complexity while rewarding the fit quality, which results in selecting the least complex model that still explains the model well. To illustrate this approach, some embodiments generate realistically noisy ISF data driven by second order plasma kinetics and then fit the resulting ISF data assuming first, second, or third order plasma kinetics. The BIC values associated with the three models, 1169, 481 , and 492 for first, second, and third order, correctly identify the second order model (lowest BIC) as the correct model of the plasma pharmacokinetics (Fig. 5A through Fig. 5C).

[0066] While the above-described estimation approach does not require knowing the correct pharmacokinetic order of the plasma pharmacokinetics, it does require that the correct model is one of a set of possibilities to be explored. Several embodiments provide that it is possible to use a pair of non-redundant ISF time courses to accurately estimate the plasma time course that is driving them without making any assumptions about the specific model that describes the latter (see Theorem 2). This renders it possible to estimate plasma concentration time courses from noisy ISF measurements even when the plasma pharmacokinetics are unknown or do not follow any particular model structure. The approach is referred as the “Minimum Assumptions Fit.” In addition to estimating CP(t), the rate constants,and fc2are also estimated, associated with the measurements, ^(t) and C2(t), collected at sites 1 and 2. A difficulty might be that because they are related to the first derivatives of Cj(t) and C2(t), the estimates of k±and k2are sensitive to noise. To circumvent this, some embodiments use a second order polynomial spline approach to “smooth” the noisy concentration measurements. Fitting a second order polynomial to every sampling interval would overfit the observations, rendering estimating rate constants and CP(t) impossible. To fix this, an appropriate time interval has to be selected over which the observed data can be smoothed by fitting thespline, which can be performed using approaches such as the Akaike Information Criterion (AIC). This constrains us to estimating only a few points of the plasma concentration time course, with concentration values between these points being obtained by interpolation. Given the few-second resolution of subcutaneous EAB sensor measurements, such interpolation is not expected to reduce the accuracy of the estimated plasma concentration time course in any clinically meaningful way.

[0067] To explore the effectiveness of the Minimum Assumptions Fit approach, some embodiments employ two different examples of plasma drug pharmacokinetics. One is a drug exhibiting the same second order plasma pharmacokinetics (2PK) as the previous example, and a second is a drug following an enterohepatic reabsorption (EHR) model (Fig. 6A through Fig. 6D). The latter model is particularly challenging, as this class of pharmacokinetics produces significantly non-monotonic plasma time courses. The simulations use distribution rate constants of k = 0.15 min’1and / c2= 0.075 min’1and noise at a signal-to-noise ratio of 30.

[0068] Minimum Assumptions Fit approach can accurately estimate plasma concentration time courses under clinically plausible conditions without any prior knowledge of the structure of the plasma pharmacokinetics. To achieve this, the procedure selected a smoothing interval of about 4 min and 5 min for 2PK and EHR cases, respectively, and thus estimates the plasma concentration with these time resolutions. The resulting plasma concentration estimates define curves that match the true plasma pharmacokinetics quite well. For example, the approach’s estimates of Cmax(plasma), which, because it reflects a single, fleeting time point, is one of the more difficult parameters to estimate, are about 53±2 pM and 49±3 pM (confidence intervals are standard deviations derived analytically from solving the optimization problems) versus the “true” values of about 57.3 pM and 49.8 pM for the 2PK and EHR cases, respectively (Fig. 6A through Fig. 6C).

[0069] Many embodiments provide that the ability to accurately estimate plasma concentration time course depends on the assumption that the measurement sites are non-redundant, i.e., that the distribution rates / and k2of each site differ significantly.This raises the question of how strongly the accuracy of the estimated plasma concentration time courses depends on this difference. To answer this, the Minimum Assumptions Fit procedure can be run for data generated using pairs of distribution rate constants varying in relative magnitude, while keeping the signal-to-noise ratio of the ISF measurements fixed at the experimentally observed value of 30. As two distribution rates approach one another, the estimation of the plasma concentration time course becomes more uncertain (Fig. 7A through Fig. 7D). Nevertheless, the results can retain clinically useful accuracy even when the two distribution rate constants are quite similar. Even when the distribution rates differ by about 10%, for example, the estimated Cmnvare about 57±12 pM and 41 ±18 pM (versus the true values of about 57.3 and 49.8 pM), respectively, for the 2PK and EHR examples, respectively.EXEMPLARY EMBODIMENTS

[0070] The following examples are put forth so as to provide those of ordinary skill in the art with a complete disclosure and description of how to make and use the present invention, and are not intended to limit the scope of what the inventors regard as their invention nor are they intended to represent that the experiments below are all or the only experiments performed. Efforts have been made to ensure accuracy with respect to numbers used (e.g., amounts, temperature) but some experimental errors and deviations should be accounted for.Example 1 : Known Pharmacokinetic Order

[0071] Several embodiments model the distribution of a molecule or ion (such as a drug) between the plasma and the interstitial fluid by first order dynamics and assume measurements from two sites are as in (4). Some embodiments aim to estimate the plasma concentration CP(t) based on observations of concentrations Ct(t) and C2t) at two sites in the ISF.

[0072] Several embodiments start by studying the case when the structure of the plasma pharmacokinetics is known. Assuming that the pharmacokinetics in plasma followa sum of exponential decays produced by a generic stable linear time-invariant system. This structure is defined as follows,

[0073] Definition 1. A compartment is said to exhibit nthorder pharmacokinetics if the concentration profile after an injection that terminated at or before time t = 0 can be expressed as a sum of n exponential decay terms of the form, all t > 0, (6)where kE. for i = 1,2, ..., n are distinct elimination rates and Atare relative weights of each process.

[0074] Remark 1. Equation (6) represents the response to a particular injection to the compartment, taking place before time t. However, if assuming that the underlying dynamics follow a linear time invariant (LTI) compartmental model, meaning the pharmacokinetics is governed by constant diffusion rate constants such as equation (3)(3), then the response defines the input-output relation from a generic injection (viewed as the input) to the drug concentration (viewed as the output). This observation can be stated as follows.

[0075] Proposition 1. Assume that the pharmacokinetic response is generated by a linear time invariant (LTI) compartmental system and that the input injection remains constant over time intervals of length T. Then the input time series u(0), u(T), ...,u(NT) and the output time series x(Sl), x(T), ..., x(NT) for a compartmental model exhibiting nthorder pharmacokinetic satisfy the following relationship:where the input is assumed to remain constant between the sampling instances, and x(t) stand for a general response of the compartmental system to a generic input (not necessarily zero after time t = 0).

[0076] Assume that the underlying pharmacokinetic order in the sense of Definition 1 of the plasma concentration profile is known and, therefore, there is a difference equation of the form of in (7) in Proposition 1 that defines the relation between the concentration in plasma at a given time and the details of the injection history. Since the focus is to study the values of CP(t) at sampling times t = 0, T, ..., NT, it is convenient to convert the differential equation (4) into a difference equation. To do that, assume CP(t) remains constant, which is justified when the sampling time, t, is much smaller than the time constants associated with the pharmacokinetic dynamics. This results in the difference equations,Further, assume that the measurements of the ISF concentration profiles are corrupted by zero-mean Gaussian noises with variancescr^, respectively. Denote the two ISF noisy measurements by y^ / cT) and y2( / cT) for k = 0,1,This allows to formulate the log-likelihood of the observed measurements for given parameter values.

[0077] This formulation along with the system equations given in equation (8) allows to cast the problem of estimating the parameter values 0 = ({a;, $i-1}”=i,k1,k2cyl,ol)' as well as the plasma concentration CP(t) for t = 0, T, ..., NT as the following maximumlikelihood optimization max 0A(t),c2(t),cP(t) 7(0)where the maximization is taken over a set defined by the equality constraints (8), and the inequality constraints overlisted in, as well as positivity constraints on the diffusion rates klrk2, and the noise variances cri, cr2.

[0078] Remark 2. Estimating noise variances can be interpreted as choosing optimal weights on the measurements from different compartments so that the estimation avoids a potential bias toward one dataset over the other.

[0079] Following this, maximum likelihood estimation problem (10) can be solved using nonlinear programming tools for an assumed number of exponential processes n. Once these estimated parameter values are ready, assign a measure of fitness to the assumed pharmacokinetic order n, by computing the BIC.where np= 2n + 4 is the number of parameters estimated,are the estimates of the model parameters, as introduced in (4), (8), (9), for an assumed pharmacokinetic order n in plasma. This measure allows to pick the most reasonable pharmacokinetic order that explains the observed measurementsC2(t) (Fig. 5A through Fig. 5C).Example 2: Minimum Assumptions Fit

[0080] Several embodiments estimate CP(t) given the measurements of Q(t) and C2(t) while making as few assumptions as possible regarding the structure of the plasma pharmacokinetics. The dual ISF concentration time courses (4) can be rearranged to have a familiar regression structure with the plasma concentrations CP(t) as part of the regression vector.where Q, G are vectors of values C^Ct), C2(t), and CP(t) over the available time instants, dQ and dC2are vectors of valuesrespectively, 0 is a vector of zerosand / is the identity matrix of appropriate sizes.

[0081] Equation (10) permits the estimation of the diffusion coefficients / q, / c2and the plasma concentration values CP(t) that appear in the right-hand side vector, based on the left-hand side vector and the matrix at the center. However, to form this matrix, some embodiments compute derivatives of concentration time courses at the ISF measurement sites. To accomplish this, some embodiments use smoothing-spline approach to estimate the derivatives of Cx(t) and C2(t). Specifically, fit second order polynomials to the data as follows:for Tj < t < Tj+1, j = 0,1, ... , NSand i = 1,2, where [7},7}+1) are smoothing intervals over which the data will be smoothed by a polynomial fit,are the coefficients for such polynomials and Nsis the number of such intervals. The smoothing intervals can be of all equal length, corresponding to knot locations given meaning that eachsmoothing interval will be of lengthThen for a given interval length ns, define the cost function for the fit aswith1, 2 and j = 0, ... , NS(15) subject towhere 0 is the collection of all parameters that affects the overall cost, cr? is the measurement noise variance and 0; is the collection of Ns+ 1 sets of polynomial coefficients, aij. bij. ci j, for the Ithcompartment, pj = p - jnsis the difference between p and the largest integer that is smaller than p and divisible by ns, and Y^pT) is the noisy measurement from the Ithcompartment at time instant pT, and CP(jnsT) is the value of plasma concentration at time jnsT, i.e. , once every nsT periods. Here the smoothness constraints (16), (17), follow from the spline structure to make sure that is the values of concentration, or their derivatives do not suddenly change every spline period, i.e., every nsT time intervals. The dynamics constraints (18) allow to link two concentration time course and they follow from the assumption on the dynamics in (4) with the spline structure given at (15).Example 3: Proposition 1

[0082] Some embodiments establish the connection between (6) and (7). This connection is a known one in the control theory for an impulse response of the form and a corresponding difference equation of the transfer function for an LTI system. Here, the expression for a pharmacokinetic order of n = 1 can be derived. Suppose a system exhibiting first order pharmacokinetics in response to a unit injection at time 0. Specifically, suppose a concentration profile given by (19) in response to the unit input instance ( v20) '

[0083] The response of this system can be constructed against a more complicated input profile that has non-zero values after the initial time, t = 0. Any such input profile can be expressed as a combination of unit inputs such aswhere Uj(t) is the delayed unit input

[0084] Because the system is assumed to be time-invariant, the response of the system to the delayed unit input u^kT) is given by y^t) = AoekE^~l\ Furthermore, because the system is assumed to be linear, the response of the system at time t = k to the complicated input profile will be given byHence, ax= -e~kET, p0= A.Example 4: Theorem 1

[0085] Let CP(t) and C / SF(t) denote the concentration time courses in the plasma and the ISF, respectively, and define the respective areas under the curves (AUC) asIf AUCp is finite, then, under the passive diffusion assumption (1 ) with kD> 0 ,AUCp = AUCISP. (25)

[0086] By the definition of AUC]SP, there is

[0087] Furthermore, the diffusion dynamics given at (1 ) gives an exact description of CISP(t) in terms of the plasma concentration levels CP(t) by the variation of constants formulaHence, express AUCISFin terms of the plasma concentration levels CP(t).= AUCP

[0088] Let G(s, 0) be a parametrized transfer function with the parameter vector 0 G 0 where 0 is the set of all feasible parameter values. We call 0 uniquely identifiable ifExample 5: Theorem 2

[0089] Let CP(t) denote the concentration time course in the plasma. Assume that Cj(t) and C2(t) denote two different ISF concentration time courses resulting from CP(t) under the simple diffusion dynamics of (4) with diffusion rate constants krand k2, respectively. If assuming zero concentrations Cj(t) = C2(t) = CP(t) = 0 before time t = 0, then (i) krand CP(t) are not uniquely identifiable by using only QCt). (Similarly, k2and CP(t) are not uniquely identifiable by using only with C2(t)), and (ii) klfk2and CP(t) are uniquely identifiable by using both Ct(t) and C2(t) if / q #= k2.

[0090] First, prove the first statement for the case of / q and C / t), noting that the case of k2and C2(t) is identically proved. Let Q(s) and CP(s) be the Laplace transform of Cx(t)and CP(t), respectively. Then under the diffusion model (1 ) and zero initial concentrations, there is£Let kr* krbe a different rate constant. Then rearrange (30) by using this new rate constant.where t7(s) is the Laplace transform of the filtered version of CP(t). Then since theLaplace transform is injective, any pair s indistinguishable from the true{kp CpCt)}.

[0091] Next, prove the second statement. Let Cx(s), C2(s) and CP(s) be the Laplace transforms of Cx(t), C2(t) and Cp(t), respectively. Then under the passive diffusion model (1 ) and with k±#= k2, (32)2

[0092] Assume that there also exist diffusion rates and {fc such that(32)(32) also holds for them.Since equation (33) must hold for all s e (CThen there is (34)i t

[0093] Hence there is a unique pair of {k^ k^ Cptt)} corresponding to {^(0, ^(0}.Example 6: Proposition 2

[0094] The second order transport model with only 3 parameters given in (3) is not identifiable given the measurements of C / SF(t) and the input u(t), and kE=£ kD.

[0095] The defining features of (3) are:The pharmacokinetics in the plasma can be explained be a single elimination rate kE.Transfer in and out of the ISF is governed by principles of passive diffusion.The only route of injection is intravenous.Observe the concentration in the ISF.

[0096] Now the matrixis a similarity transformation for (3) that still satisfies our experimental assumptions 1-4. Using T of (35), there is the new system description,is the transformed plasma concentration time profile that is consistent with the same set of C / SF(t) and u(t). The equivalent system (36) also shows a way to transform pharmacokinetic parameters while preserving the observed data. kE— kDkE— kE(37)where kE, kD, V^ are the parameter values of the equivalent system.Example 7: Proposition 3

[0097] A third order compartmental system, given bywhere assume only Cj(t) can be measured and have access to u(t), cannot be uniquely identified and there are infinitely many concentration time profile candidates CP(t).

[0098] Let p e (0,1). Then the matrix (39)0 0 1 is a similarity transformation for the system description (38). This means to observe the same concentration time course in ISF, GCt), for the same infusion profile, u(t), even though the values of parameters and thereby the plasma concentration time course changes.Example 8: Proposition 4

[0099] Suppose that the response of a system exhibiting nt}lorder pharmacokinetics against an instantaneous injection is described by, all t > 0, (40)And the same system has a generic input-output relation given by,

[0100] Then if the Atand kE. are positive real numbers for every i = 1,then the values satisfy the following inequalities:For pharmacokinetic order n = 1,For pharmacokinetic order n = 2,For pharmacokinetic order n = 3,

[0101] List the relation between the elimination rates, kE., the relative weights, Aitand the coefficientsFor pharmacokinetic order n - 1,For pharmacokinetic order n = 2,For pharmacokinetic order n = 3,

[0102] Now with these relations, the maximum and minimum value restrictions can be derived since AitkEi> 0 for i = 1, ... , n and for n = 1,2,3. Since kE. > 0, 0 < e~kEtT< 1. For pharmacokinetic order n = 1,For pharmacokinetic order n = 2,For pharmacokinetic order n = 3,Furthermore, the e are roots of a polynomial with coefficientsor i =1, ... , n and n = 1,2,3. This provides another necessary condition for the pharmacokinetic order n = 2. SinceeEi and e distinct real roots for the polynomial, positivediscriminant can be0, for the polynomial in the case n = 2. Finally, since 0 < onditions on the combinations of pzfor i =and n = 1,2,3 may befound.For pharmacokinetic order n = 2,For pharmacokinetic order n = 3,Po + Pi > 0Pl + / ?2 <0

[0103] The condition onfor n = 3, can be reached in the same way as shown for n = 2. The condition on+ p2on the other hand is because if two numbers satisfy 0 < a, b < 1, then ab < a + b.EXAMPLES

[0104] Example 1 : A method for determining a plasma concentration time course of a molecule or ion, comprising: collecting a plurality of concentration time courses of the molecule or ion from at least two interstitial fluid (ISF) sites, wherein each of the at least two ISF sites has a non-identical distribution rate constant describing a rate of transport between the plasma and the at least two ISF sites; and estimating the plasma concentration time course by fitting the plurality of concentration time courses to a plurality of mathematical relationships; wherein the molecule or ion is eliminated from the body via the plasma instead of the at least two ISF sites; and wherein the at least two ISF sites communicate diffusively with the plasma.

[0105] Example 2: The method of example 1 , wherein a structure of a plasma pharmacokinetics of the molecule or ion is known.

[0106] Example 3: The method of example 1 or 2, wherein the structure of the plasma pharmacokinetic comprises a first order plasma pharmacokinetic order, a second order plasma pharmacokinetic order, and a third order plasma pharmacokinetic order.

[0107] Example 4: The method of example 1 , or 2, or 3, wherein a structure of a plasma pharmacokinetics is unknown, wherein at least one of the plurality of mathematical relationships selected from the group consisting of: the BayesianInformation Criterion; and the Akaike Information Criterion (AIC) is used to determine a preferred structure of the plasma pharmacokinetics.

[0108] Example 5: The method of any one of examples 1 to 4, wherein a plasma pharmacokinetic order is unknown, and the method further comprises estimating the distribution rate constant for transport between the plasma and the at least two ISF sites.

[0109] Example 6: The method of any one of examples 1 to 5, further comprising applying a smoothing time interval to the plurality of concentration time courses.

[0110] Example 7: The method of any one of examples 1 to 6, wherein an interval length with a lowest AIC among a plurality of time interval length options is chosen as a suitable smoothing time interval length.

[0111] Example 8: The method of any one of examples 1 to 7, wherein the plasma pharmacokinetic order is configured to be a first, second or third order plasma pharmacokinetic order or an enterohepatic reabsorption model.

[0112] Example 9: The method of any one of examples 1 to 8, further comprising using a plurality of electrochemical aptamer-based (EAB) sensors to collect the plurality of concentration time courses of the molecule or ion.

[0113] Example 10: The method of any one of examples 1 to 9, wherein each of the plurality of EAB sensors comprises: a working electrode, wherein the working electrode comprises a recognition element modified with a redox molecule; wherein the recognition element specifically binds to the molecule or ion and is configured to undergo a conformational change upon binding with the molecule or ion; wherein a current readout from the EAB sensor reflects a concentration of the molecule or ion.

[0114] Example 11 : The method of any one of examples 1 to 10, wherein the recognition element is a nucleic acid aptamer or a protein.

[0115] Example 12: The method of any one of examples 1 to 11 , wherein the working electrode is selected from the group consisting of: a wire, a rod, a planar electrode, a needle, and a microneedle.

[0116] Example 13: The method of any one of examples 1 to 12, wherein the molecule or ion is a drug is selected from the group consisting of: a small molecule drug molecule,an amino acid-based drug molecule, a protein-based drug molecule, a nucleic-acid-based drug molecule, an ion drug, and a monatomic ion drug.

[0117] Example 14: The method of any one of examples 1 to 13, wherein the distribution rate constant for transport for each of the at least two ISF sites differs by at least 10%.

[0118] Example 15: A method for determining a plasma concentration time course of a molecule or ion, comprising: collecting a plurality of concentration time courses of a molecule or ion from at least two interstitial fluid (ISF) sites, wherein each of the at least two ISF sites has a non-identical distribution rate constant describing a rate of transport between the plasma and each of the at least two ISF sites; and estimating the plasma concentration time course by fitting the plurality of concentration time courses to a plurality of mathematical relationships; wherein the molecule is eliminated from the body via the plasma instead of the at least two ISF sites; and wherein the at least two ISF sites communicate diffusively with the plasma.

[0119] Example 16: The method of example 15, wherein the molecule or ion is selected from the group consisting of: a small molecule drug, a macromolecular drug, an amino acid-based drug, a protein-based drug, a nucleic-acid-based drug, a small molecule metabolite, a hormone, a protein, an enzyme, an antibody, a biomarker, an ion, and a monatomic ion.

[0120] Example 17: The method of example 15 or 16, wherein a structure of a plasma pharmacokinetics of the molecule is known.

[0121] Example 18: The method of example 15, or 16, or 17, wherein a structure of a plasma chemical kinetics of the molecule or ion is unknown, wherein at least one of the plurality of mathematical relationships selected from the group consisting of: the BayesianInformation Criterion; and the Akaike Information Criterion (AIC) is used to determine a preferred structure of the plasma chemical kinetics.

[0122] Example 19: The method of any one of examples 15 to 18, wherein a plasma pharmacokinetic order is unknown, and the method further comprises estimating the distribution rate constant for transport between the plasma and the at least two ISF sites.

[0123] Example 20: The method of any one of examples 15 to 19, further comprising applying a smoothing time interval to the plurality of concentration time courses.

[0124] Example 21 : The method of any one of examples 15 to 20, wherein an interval length with a lowest AIC among a plurality of time interval length options is chosen as a suitable smoothing time interval length.

[0125] Example 22: The method of any one of examples 15 to 21 , wherein the plasma pharmacokinetic order is configured to be first or second or third order pharmacokinetics or an enterohepatic reabsorption model.

[0126] Example 23: The method of any one of examples 15 to 22, further comprising using a plurality of electrochemical, aptamer-based (EAB) sensors to collect the plurality of concentration time courses of the molecule.

[0127] Example 24: The method of any one of examples 15 to 23, wherein each of the plurality of EAB sensors comprises: a working electrode, wherein the working electrode comprises a recognition element modified with a redox molecule; wherein the recognition element specifically binds to the molecule or ion and is configured to undergo a conformational change upon binding with the molecule or ion; wherein a current readout from the EAB sensor reflects a concentration of the molecule or ion.

[0128] Example 25: The method of any one of examples 15 to 24, wherein the recognition element is a nucleic acid aptamer or a protein.

[0129] Example 26: The method of any one of examples 15 to 25, wherein the working electrode is selected from the group consisting of: a wire, a rod, a planar electrode, a needle, and a microneedle.

[0130] Example 27: The method of any one of examples 15 to 26, wherein the distribution rate constant for transport for each of the at least two ISF sites differs by at least 10%.DOCTRINE OF EQUIVALENTS

[0131] As can be inferred from the above discussion, the above-mentioned concepts can be implemented in a variety of arrangements in accordance with embodiments of the invention. Accordingly, although the present invention has been described in certain specific aspects, many additional modifications and variations would be apparent to those skilled in the art. It is therefore to be understood that the present invention may be practiced otherwise than specifically described. Thus, embodiments of the present invention should be considered in all respects as illustrative and not restrictive.

[0132] As used herein, the singular terms “a,” “an,” and “the” may include plural referents unless the context clearly dictates otherwise. Reference to an object in the singular is not intended to mean “one and only one” unless explicitly so stated, but rather “one or more.”

[0133] As used herein, the terms “approximately,” and “about” are used to describe and account for small variations. When used in conjunction with an event or circumstance, the terms can refer to instances in which the event or circumstance occurs precisely as well as instances in which the event or circumstance occurs to a close approximation. When used in conjunction with a numerical value, the terms can refer to a range of variation of less than or equal to ± 10% of that numerical value, such as less than or equal to ±5%, less than or equal to ±4%, less than or equal to ±3%, less than or equal to ±2%, less than or equal to ±1 %, less than or equal to ±0.5%, less than or equal to ±0.1 %, or less than or equal to ±0.05%.

[0134] Additionally, amounts, ratios, and other numerical values may sometimes be presented herein in a range format. It is to be understood that such range format is used for convenience and brevity and should be understood flexibly to include numerical values explicitly specified as limits of a range, but also to include all individual numerical valuesor sub-ranges encompassed within that range as if each numerical value and sub-range is explicitly specified. For example, a ratio in the range of about 1 to about 200 should be understood to include the explicitly recited limits of about 1 and about 200, but also to include individual ratios such as about 2, about 3, and about 4, and sub-ranges such as about 10 to about 50, about 20 to about 100, and so forth.

Claims

CLAIMS1. A method for determining a plasma concentration time course of a molecule or ion, comprising: collecting a plurality of concentration time courses of the molecule or ion from at least two interstitial fluid (ISF) sites, wherein each of the at least two ISF sites has a non-identical distribution rate constant describing a rate of transport between the plasma and the at least two ISF sites; and estimating the plasma concentration time course by fitting the plurality of concentration time courses to a plurality of mathematical relationships; wherein the molecule or ion is eliminated from the body via the plasma instead of the at least two ISF sites; and wherein the at least two ISF sites communicate diffusively with the plasma.

2. The method of claim 1 , wherein a structure of a plasma pharmacokinetics of the molecule or ion is known.

3. The method of claim 2, wherein the structure of the plasma pharmacokinetic comprises a first order plasma pharmacokinetic order, a second order plasma pharmacokinetic order, and a third order plasma pharmacokinetic order.

4. The method of claim 1 , wherein a structure of a plasma pharmacokinetics is unknown, wherein at least one of the plurality of mathematical relationships selected from the group consisting of: the Bayesian Information Criterion; and the Akaike Information Criterion (AIC) is used to determine a preferred structure of the plasma pharmacokinetics.

5. The method of claim 1 , wherein a plasma pharmacokinetic order is unknown, and the method further comprises estimating the distribution rate constant for transport between the plasma and the at least two ISF sites.

6. The method of claim 5, further comprising applying a smoothing time interval to the plurality of concentration time courses.

7. The method of claim 6, wherein an interval length with a lowest AIC among a plurality of time interval length options is chosen as a suitable smoothing time interval length.

8. The method of claim 7, wherein the plasma pharmacokinetic order is configured to be a first, second or third order plasma pharmacokinetic order or an enterohepatic reabsorption model.

9. The method of claim 1 , further comprising using a plurality of electrochemical aptamerbased (EAB) sensors to collect the plurality of concentration time courses of the molecule or ion.

10. The method of claim 9, wherein each of the plurality of EAB sensors comprises: a working electrode, wherein the working electrode comprises a recognition element modified with a redox molecule; wherein the recognition element specifically binds to the molecule or ion and is configured to undergo a conformational change upon binding with the molecule or ion; wherein a current readout from the EAB sensor reflects a concentration of the molecule or ion.11 . The method of claim 10, wherein the recognition element is a nucleic acid aptamer or a protein.

12. The method of claim 10, wherein the working electrode is selected from the group consisting of: a wire, a rod, a planar electrode, a needle, and a microneedle.

13. The method of claim 1 , wherein the molecule or ion is a drug is selected from the group consisting of: a small molecule drug molecule, an amino acid-based drug molecule, a protein-based drug molecule, a nucleic-acid-based drug molecule, an ion drug, and a monatomic ion drug.

14. The method of claim 1 , wherein the distribution rate constant for transport for each of the at least two ISF sites differs by at least 10%.

15. A method for determining a plasma concentration time course of a molecule or ion, comprising: collecting a plurality of concentration time courses of a molecule or ion from at least two interstitial fluid (ISF) sites, wherein each of the at least two ISF sites has a non-identical distribution rate constant describing a rate of transport between the plasma and each of the at least two ISF sites; and estimating the plasma concentration time course by fitting the plurality of concentration time courses to a plurality of mathematical relationships; wherein the molecule is eliminated from the body via the plasma instead of the at least two ISF sites; and wherein the at least two ISF sites communicate diffusively with the plasma.

16. The method of claim 15, wherein the molecule or ion is selected from the group consisting of: a small molecule drug, a macromolecular drug, an amino acid-based drug, a protein-based drug, a nucleic-acid-based drug, a small molecule metabolite, a hormone, a protein, an enzyme, an antibody, a biomarker, an ion, and a monatomic ion.

17. The method of claim 15, wherein a structure of a plasma pharmacokinetics of the molecule is known.

18. The method of claim 15, wherein a structure of a plasma chemical kinetics of the molecule or ion is unknown, wherein at least one of the plurality of mathematical relationships selected from the group consisting of: the Bayesian Information Criterion; and the Akaike Information Criterion (AIC) is used to determine a preferred structure of the plasma chemical kinetics.

19. The method of claim 15, wherein a plasma pharmacokinetic order is unknown, and the method further comprises estimating the distribution rate constant for transport between the plasma and the at least two ISF sites.

20. The method of claim 19, further comprising applying a smoothing time interval to the plurality of concentration time courses.21 . The method of claim 20, wherein an interval length with a lowest AIC among a plurality of time interval length options is chosen as a suitable smoothing time interval length.

22. The method of claim 21 , wherein the plasma pharmacokinetic order is configured to be first or second or third order pharmacokinetics or an enterohepatic reabsorption model.

23. The method of claim 15, further comprising using a plurality of electrochemical, aptamer-based (EAB) sensors to collect the plurality of concentration time courses of the molecule.

24. The method of claim 23, wherein each of the plurality of EAB sensors comprises: a working electrode, wherein the working electrode comprises a recognition element modified with a redox molecule; wherein the recognition element specifically binds to the molecule or ion and is configured to undergo a conformational change upon binding with the molecule or ion; wherein a current readout from the EAB sensor reflects a concentration of the molecule or ion.

25. The method of claim 24, wherein the recognition element is a nucleic acid aptamer or a protein.

26. The method of claim 24, wherein the working electrode is selected from the group consisting of: a wire, a rod, a planar electrode, a needle, and a microneedle.

27. The method of claim 15, wherein the distribution rate constant for transport for each of the at least two ISF sites differs by at least 10%.

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