Calculation of gas biomarker concentrations in tissue using boundary inverse problems.

Abstract

A system and method for determining a level of an analyte parameter, comprising: causing a photon source to send photons to a probe sensitive to an analyte, exciting the probe to emit light; receiving optical data from a photodetector based on the light emitted from the probe; determining a difference between a first level of the analyte parameter at a first location adjacent to a first surface of the probe and an intermediate level of the analyte parameter at an intermediate location between the first location and a second location adjacent to a second surface of the probe; and determining a second level of the analyte parameter adjacent to the second surface of the probe based on the difference between the first level of the analyte parameter and the intermediate level of the analyte parameter.

Description

[Technical Field]

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS This application is based on and claims the benefit of and priority to U.S. Patent Application Nos. 63 / 429,074 (filed November 30, 2022) and 63 / 490,982 (filed March 17, 2023), the entire contents of which are incorporated herein by reference.

[0002] <Statement regarding federally sponsored research> This invention was made with government support under Grant No. FA9550-17-1-0277 awarded by the Air Force Office of Scientific Research and Grant No. HU0001-17-2-0009 awarded by the Henry M. Jackson Foundation for the Advancement of Military Medicine. [Background technology]

[0003] The present invention relates to a sensor system and method for detecting analyte concentrations, and more particularly to a transcutaneous gas concentration monitor that provides accurate tissue measurements of gases such as O2.

[0004] Quantitative assessment of gas (e.g., O2, CO2) concentrations in biological tissues is important in diagnosing and treating various conditions, particularly in situations where traditional clinical tools are inadequate, such as when blood flow is absent or blood pools [Reference 1]. Transcutaneous oxygen monitor (TCOM) devices can be used to directly and noninvasively measure tissue-available gases. Measurement of transcutaneous oxygenation (transcutaneous pO2 or tcpO2) can be a valuable indicator in burns, tourniquet application, and can be used to detect the onset of ischemia, particularly in diabetic ulcers [References 2-5]. However, currently available transcutaneous measurement devices are costly, bulky, and labor-intensive to operate, which pose significant obstacles to the practical application of this technology. Furthermore, these bulky devices require relatively pristine conditions to provide reliable measurements, such as requiring the patient to remain motionless or requiring long equilibration times, and in the case of TCOM, the skin must be heated to temperatures above 40°C.

[0005] Among the issues mentioned above, the key limitations of transcutaneous gas measurements are the question of whether the measurements reflect the true partial pressure (e.g., pO2) of the tissue at the skin surface [Ref. 6] and the need for a long equilibration period to obtain a stable reading. These issues arise because the devices are placed on the skin rather than within it, and the measured values ​​may lag behind the skin values ​​due to the time required for oxygen to diffuse into the device headspace and equilibrate therein. When such devices are applied, an exponential decline from atmospheric pressure is typically observed (for pO2, this decline occurs from approximately 160 mmHg to values ​​between 60 and 100 mmHg). The rate and amplitude of the decay depend on the volume of air trapped between the detector and the skin, the diffusion of gas through the various materials used, the skin's gas consumption, and other factors [Refs. 7, 8]. When a gas-impermeable material is used as a barrier to atmospheric gases [Ref. 9], it is assumed that the resulting concentration readings will correspond to the actual tissue values, given sufficient time. However, in applications where the sensor must be worn for extended periods, such as in post-operative situations

[10] , the sensor should ideally utilize a semi-breathable material to allow the skin to "breathe" (e.g., to allow some diffusion of moisture, oxygen, etc.), meaning that the resulting equilibrium value will be intermediate between the gas concentration in the atmosphere and that on the skin. It is also anticipated that some applications will need to avoid direct contact between the skin and the gas sensing material by adding a semi-permeable barrier layer, for example, to avoid the use of potentially harmful sensing materials

[11] , which would further alter the difference between the measured value and the true tissue value.

[0006] Therefore, what is needed is a transcutaneous gas pressure monitoring device that systematically produces true tissue tension readings while reducing the need for long equilibration times despite the use of a barrier or intermediate layer between the tissue and the sensor. Summary of the Invention

[0007] To address these limitations, the present disclosure provides systems and methods that address these limitations so that measured gas concentrations in transcutaneous gas pressure monitors can be converted to true gas pressure values ​​while using materials that can be used for long-term wear.

[0008] In one aspect, the present disclosure provides a sensor system comprising: a probe sensitive to an analyte, the probe having a body having a first surface and an opposing second surface; a photon source configured to send photons to the probe, the probe emitting light upon receiving photons from the photon source; a photodetector configured to detect light emitted from the probe; and a controller in communication with the photon source and the photodetector. The controller may be configured to execute a program stored therein to (i) cause the photon source to send photons to the probe and excite the probe to emit light when the probe receives the photons; (ii) receive optical data from the photodetector based on an interaction between the light emitted from the probe and the photodetector; (iii) determine, based on the optical data from the photodetector, a difference between a first level of the parameter of the analyte at a first position adjacent to the first surface of the probe and an intermediate level of the parameter of the analyte at an intermediate position between the first position and a second position adjacent to the second surface of the probe; and (iv) determine a second level of the parameter of the analyte adjacent to the second surface of the probe based on the difference between the first level of the parameter of the analyte and the intermediate level of the parameter of the analyte.

[0009] In one embodiment of the sensor system, the controller executes a program stored in the controller to determine the second level of the analyte adjacent the second surface of the probe by solving a mathematical diffusion problem involving the first level of the analyte and the intermediate level of the analyte.

[0010] In one embodiment of the sensor system, the controller executes a program stored in the controller to determine the second level of the analyte adjacent the second surface of the probe by solving an inverse mathematical boundary condition problem involving the first level of the analyte and the intermediate level of the analyte. In one embodiment of the sensor system, the inverse mathematical boundary condition problem includes a thickness-dependent diffusivity variable. In one embodiment of the sensor system, the inverse mathematical boundary condition problem includes a Dirichlet boundary condition.

[0011] In yet another aspect, the present disclosure provides a sensor system comprising: a probe sensitive to an analyte, the probe having a body having a first surface and an opposing second surface; a photon source configured to send photons to the probe, the probe emitting light upon receiving photons from the photon source; a photodetector configured to detect light emitted from the probe; and a controller in communication with the photon source and the photodetector. The controller may be configured to execute a program stored therein to (i) cause the photon source to send photons to the probe and excite the probe to emit light when the probe receives the photons; (ii) receive optical data from the photodetector based on an interaction between the light emitted from the probe and the photodetector; (iii) determine, based on the optical data from the photodetector, a gradient of the parameter of the analyte extending from a first position adjacent to the first surface of the probe to an intermediate position between a second position adjacent to the second surface of the probe and the first position; and (iv) calculate, based on the gradient, a level of the parameter of the analyte adjacent to the second surface of the probe.

[0012] In yet another aspect, the present disclosure provides a method for determining a level of a parameter of an analyte, the method including the steps of: (a) disposing a probe near a surface for monitoring the analyte; (b) disposing a photon source to deliver photons to the probe to excite the probe; (c) disposing a photodetector to receive light emitted by the probe when emitted by the photon source; (d) causing the photon source to send photons to the probe, which excites the probe to emit light when the probe receives the photons; and (e) collecting optical data from the photodetector based on an interaction between the light emitted from the probe and the photodetector. The method may include (f) determining, based on the optical data from the photodetector, a difference between a first level of the parameter of the analyte at a first location adjacent the first surface of the probe and an intermediate level of the parameter of the analyte at an intermediate location between the first location and a second location adjacent the second surface of the probe; and (g) determining a second level of the parameter of the analyte adjacent the second surface of the probe based on the difference between the first level of the parameter of the analyte and the intermediate level of the parameter of the analyte. In one embodiment of the method, step (g) includes determining the second level of the analyte adjacent the second surface of the probe by solving a mathematical diffusion problem involving the first level of the analyte and the intermediate level of the analyte. In one embodiment of the method, step (g) includes determining the second level of the analyte adjacent the second surface of the probe by solving a mathematical boundary condition inverse problem involving the first level of the analyte and the intermediate level of the analyte. The mathematical boundary condition inverse problem may include a thickness-dependent diffusivity variable. The mathematical boundary condition inverse problem may include Dirichlet boundary conditions.

[0013] In yet another aspect, the present disclosure provides a method for determining a level of an analyte parameter, the method including: (a) disposing a probe near a surface for monitoring an analyte; (b) disposing a photon source to deliver photons to the probe to excite the probe; (c) disposing a photodetector to receive light emitted by the probe when emitted by the photon source; (d) causing the photon source to send photons to the probe and excite the probe to emit light when the probe receives the photons; (e) collecting optical data from the photodetector based on an interaction between the light emitted from the probe and the photodetector; (f) determining, based on the optical data from the photodetector, a gradient of the analyte parameter from a first location adjacent the first surface of the probe to an intermediate location between a second location adjacent the second surface of the probe and the first location; and (g) calculating a level of the analyte parameter adjacent the second surface of the probe based on the gradient.

[0014] In yet another aspect, the present disclosure provides a computer system having instructions stored on a non-transitory computer-readable medium to cause at least one processor on a computer to: (i) cause a photon source to send photons to a probe sensitive to an analyte and excite the probe to emit light when the probe receives the photons, the probe comprising a body having a first surface and an opposing second surface; (ii) receive optical data from the photodetector based on an interaction between the light emitted from the probe and a photodetector; (iii) determine, based on the optical data from the photodetector, a difference between a first level of a parameter of the analyte at a first location adjacent the first surface of the probe and an intermediate level of the parameter of the analyte at a location intermediate between the first location and a second location adjacent the second surface of the probe; and (iv) determine, based on the difference between the first level of the parameter of the analyte and the intermediate level of the parameter of the analyte, a second level of the parameter of the analyte adjacent the second surface of the probe. In one embodiment of the computer system, the at least one processor determines the second level of the analyte adjacent to the second surface of the probe by solving a mathematical diffusion problem involving the first level of the analyte and the intermediate level of the analyte. In one embodiment of the computer system, the at least one processor determines the second level of the analyte adjacent to the second surface of the probe by solving an inverse mathematical boundary condition problem involving the first level of the analyte and the intermediate level of the analyte. The inverse mathematical boundary condition problem can include a thickness-dependent diffusivity variable. The inverse mathematical boundary condition problem can include a Dirichlet boundary condition.

[0015] In yet another aspect, the present disclosure provides a computer system having instructions stored on a non-transitory computer-readable medium to cause at least one processor on a computer to: (i) cause a photon source to send photons to a probe sensitive to an analyte and excite the probe to emit light when the probe receives the photons, the probe having a body having a first surface and an opposing second surface; (ii) receive optical data from the photodetector based on an interaction between the light emitted from the probe and a photodetector; (iii) determine, based on the optical data from the photodetector, a gradient of a parameter of the analyte extending from a first location adjacent the first surface of the probe to an intermediate location between a second location adjacent the second surface of the probe and the first location; and (iv) calculate, based on the gradient, a level of the parameter of the analyte adjacent the second surface of the probe.

[0016] One advantage of the disclosed system and method is that the system and method can take into account gas concentration gradients in the transcutaneous pressure monitor when calculating the true gas concentration at the skin surface and output a report of the true tissue gas concentration.

[0017] Another advantage of the disclosed systems and methods is that they dramatically reduce equilibration time from approximately 30-60 minutes for conventional devices to just a few minutes. Commercially available devices and methods typically require an equilibration period of approximately 30-60 minutes, which can be prohibitive for measurements needed in clinical settings, particularly emergency medical applications where information on tissue gas concentrations (e.g., pO2) is needed within seconds to minutes.

[0018] The sensor systems and methods disclosed herein have numerous applications, non-limiting examples of which include (i) software that can be adapted to any transcutaneous sensor for obtaining skin surface readings, (ii) algorithms that can be included as part of the signal processing, and (iii) calibration tools that can be used to determine material properties for the software and algorithms. These products can be used in many areas of analyte measurement, including commercially available transcutaneous sensors that measure true skin O2 and CO2 based on wearable technology developed by, for example, the Wellman Center for Photomedicine at Massachusetts General Hospital in Boston, Massachusetts, USA.

[0019] Beyond gas sensing, the disclosed systems and methods can also be used in other biomedical applications, including sensing of dissolved compounds and liquids, including metabolites and drugs in tissues. Despite differences between gas and liquid diffusion, the diffusion processes are similar, making the applications described herein equally useful. Other applications include materials science and characterization, where dynamic diffusion processes exist. These applications include the development, testing, and operational monitoring of batteries, including lithium-ion and liquid-phase batteries, as well as the manufacturing of materials containing polymers, carbon fiber materials, and composites, as well as multilayer materials. For example, carbon fiber materials must be manufactured in the absence of oxygen, otherwise the resulting material will be highly brittle. Gas sensors, such as the disclosed systems, can detect true material gas concentrations, which can be a valuable aid in these critical manufacturing steps.

[0020] These and other features, aspects, and advantages of various embodiments of the present disclosure will become better understood with reference to the following description, appended claims, and accompanying drawings. [Brief explanation of the drawings]

[0021] [Figure 1A]FIG. 1 illustrates a wearable transcutaneous oxygenation monitor system of the present disclosure, which uses a numerical model based on the inverse boundary problem of a parabolic equation with Dirichlet boundary conditions to obtain accurate tissue pO2 measurements without long equilibration times. [Figure 1B] Panel (a) shows the layer structure of the oxygen sensing film, which is a clinical application for use with a wearable oxygen sensor. Panel (b) is a schematic of the oxygen gradient obtained through the different layers of the oxygen sensing film, showing the unknown true tissue pO2 (RBC = ???). [Figure 2a] This figure shows a test of the boundary inverse problem algorithm using a sample problem, which uses Neumann boundary conditions. The right boundary condition (RBC) and left boundary condition (LBC) are initially known, and a direct problem for intermediate values ​​of x = x* is calculated using a direct diffusion model. The boundary inverse problem is then used to calculate the RBC (RBCIBP) using u(x*,t) close to the initial known value. [Figure 2b] This figure shows a test of the boundary inverse problem algorithm using a sample problem, which uses Dirichlet boundary conditions. The right and left boundary conditions (RBC and LBC) are initially known, and a direct problem for intermediate values ​​of x = x* is calculated using a direct diffusion model. The boundary inverse problem is then used to calculate the RBC (RBCIBP) using u(x*,t) close to the initial known value. [Figure 3a] Since φPPMA>>dPPMA, the film structure is expressed in cylindrical coordinates to reduce the above diffusion problem to one dimension. [Figure 3b] FIG. 1 shows a system for performing control experiments. [Figure 3c] 10 shows the fitting of the boundary inverse algorithm to the measured right boundary using the TCOM and known left boundary, with the intermediate u(x*,t)=pO2 where x* and the diffusion coefficient of the film layer are found. The inset in the figure shows the underestimation of the change in pO2 of the TCOM raw reading during the "occlusion" period. [Figure 3d]The residuals of the fitting are shown, indicating that the difference between the fitting and the signal is small. [Figure 4a] Figure 10 shows that the direct diffusion model with Dirichlet conditions successfully reconstructs mean values ​​from wearable pO2 readings that are in good agreement with the experimental data, using fitting constants from the inverse model fitting and a constant left boundary of 160 mmHg and a measured right boundary (RBCIBP). [Figure 4b] FIG. 1 shows the oxygen profile (left axis) at different time points during the measurement and the diffusion coefficient profile (right axis) found from the fitting. [Figure 5a-b] FIG. 1 shows the fitting of the boundary inverse problem (panel a) and residuals (panel b) of measurements made on four additional films. [Figure 5c-d] Figure 1 shows the fitting of the boundary inverse problem (panel c) and residuals (panel d) of measurements made on four additional films. [Figure 5e-f] Figure 10 shows the fitting of the boundary inverse problem (panel e) and residuals (panel f) of measurements made on four additional films. [Figure 5g-h] Figure 10 shows the fitting of the boundary inverse problem (panel g) and residuals (panel h) of measurements made on four additional films. [Figure 6] Panel (a) shows the application of the boundary inverse algorithm to TCOM clinical data, where blood flow to the lower leg is restricted for 10 minutes (highlighted), and the recalculated pO2 is close to the tissue pO2. Panel (b) shows the relative change in pO2 during occlusion, with the true tissue pO2 showing a large and clear change compared to the original TCOM reading. DETAILED DESCRIPTION OF THE INVENTION

[0022] In the following detailed description, like parts in the various figures are designated by like reference numerals. None of the figures are drawn to scale. When dimensions are indicated in the text or figures, such dimensions are merely numerical examples that may be used in implementing one or more specific examples and are not intended to limit the scope of the invention disclosed herein. Some embodiments of the present disclosure are described herein with reference to the accompanying drawings. Reading this description together with the drawings will make it clear to those skilled in the art how the embodiments of the present disclosure may be implemented. The figures are for illustrative purposes and do not attempt to show structural details of the embodiments in more detail than is necessary for a fundamental understanding of the teachings of the present disclosure.

[0023] Before describing the present invention in detail, it should be understood that the present invention is not limited to the specific embodiments described herein. Furthermore, the terminology used herein is used solely for the purpose of describing specific embodiments and is not intended to limit the present invention. The scope of the present invention is limited only by the claims. As used herein, the singular forms "a," "an," and "the" include plural embodiments unless the context clearly dictates otherwise.

[0024] While the invention disclosed herein has been described with respect to particular embodiments and examples, the invention is not limited thereto, but rather is intended to encompass within the scope of the appended claims numerous other embodiments, examples, uses, modifications, and derivatives of the embodiments, examples, and uses.

[0025] It will be apparent to those skilled in the art that many further modifications beyond those already described herein can be made without departing from the spirit of the invention. In interpreting this disclosure, all terms should be interpreted in the broadest manner consistent with the surrounding context. The terms "comprising," "including," or "having" should be understood to describe elements, components, or steps in a non-exclusive manner, so that such elements, components, or steps can be combined with other elements, components, or steps not expressly mentioned. Embodiments described as "comprising," "including," or "having" a particular element are also intended to mean "consisting essentially of" or "consisting of" that element, unless the context clearly indicates otherwise.

[0026] In non-limiting embodiments, the present invention discloses mathematical and calibration methods for obtaining estimates of gas biomarker concentrations in tissue from noninvasive measurements at the skin surface. The methods include embodiments for obtaining oxygen concentrations in tissue, and while the specific embodiment of oxygen as the biomarker of interest is disclosed, the methods of the present invention can also be applied to other gaseous compounds originating from the skin, such as carbon dioxide, nitric oxide, or volatile organic compounds. The methods of the present invention can also be applied to any scenario requiring sensing of an analyte in a material, but where the sensor is external to the material itself or separated from the material by a semi-permeable barrier. Thus, this approach has applications in biomedicine, but also potentially in materials characterization, production, and manufacturing.

[0027] Referring to FIG. 1B, a non-limiting example embodiment of a wearable device 100 for transcutaneous oxygen monitoring of the present invention is shown. The device 100 includes a photoluminescent oxygen-sensitive probe having a film 105 including a polymer matrix and porphyrin molecules. A photon source is configured to transmit photons at a specific wavelength to the probe. A photodetector is configured to detect light emitted from the probe when the photon source transmits photons to the probe. Illumination from the photon source can be modulated (e.g., square wave, sine wave, triangular wave, sawtooth wave, etc.) or constant. The photodetector can be a PIN photodiode. The device 100 includes electronics 110 including a controller in electrical communication with the photon source and the photodetector. The controller stores and is configured to execute a program for calculating the oxygen level adjacent to the probe from the electrical signal received from the photodetector. The device 100 includes a highly breathable white scattering layer W and a transparent, semi-permeable transparent membrane SP that partially shields the skin from atmospheric oxygen. Because device 100 can directly measure pO2, it does not require perfusion or the presence of a pre-existing blood vessel. Optical device 100 can require little setup time, and device 100 readings can be virtually instantaneous. Optical device 100 eliminates the need for bedside calibration or heating, eliminates the need for pre-existing blood flow, and minimizes equilibration time before a simple readout can provide a presentable result, thereby providing a solution to problems that plague clinical oxygen sensing overall.

[0028] In device 100, the photoluminescent oxygen-sensitive probe can include a polymer matrix. In some embodiments, the polymer matrix includes a hydrophobic polymer. In some embodiments, the polymer matrix includes a polymer selected from the group consisting of acrylate polymers, methacrylate polymers, and blends and copolymers thereof. Non-limiting examples of acrylate polymers include poly(acrylic acid), poly(methyl acrylate), poly(ethyl acrylate), poly(propyl acrylate), and poly(butyl acrylate). Non-limiting examples of methacrylate polymers include poly(methyl methacrylate), poly(ethyl methacrylate), poly(propyl methacrylate), poly(butyl methacrylate), and poly(hydroxyethyl methacrylate). In some embodiments, the polymer matrix includes a polymer selected from the group consisting of alkyl methacrylate polymers. In some embodiments, the polymer matrix includes poly(propyl methacrylate) (PPMA). In some embodiments, the polymer matrix includes a polymer selected from the group consisting of siloxane polymers and blends and copolymers thereof. In one non-limiting example, the polymer comprises polydimethylsiloxane (PDMS). For example, silicones such as PDMS can have extremely high gas permeability, which can allow for rapid readings of tissue oxygen dynamics.

[0029] As a non-limiting example, the porphyrin molecule of the photoluminescent oxygen-sensitive probe can be a metalloporphyrin capable of emitting red phosphorescence when excited by blue light, and the intensity and lifetime of the phosphorescence can be inversely proportional to pO2. The oxygen-sensitive probe can also incorporate a reference sensor in the form of a green-emitting dye to serve as a reference standard for accurate pO2 measurements. In another non-limiting example, the concentration of CO2 can be detected using a fluorophore that senses the presence of CO2 bound to the material.

[0030] In some embodiments, porphyrin-based oxygen sensing molecules embedded in a polymer matrix are designed to provide exceptional sensitivity and accuracy for measuring tissue oxygenation. Porphyrin-based oxygen sensing molecules can be formed through a modular synthetic route that allows for tuning both the oxygen sensitivity range of the oxygen sensing molecule and its compatibility with the matrix material in which it is embedded. The matrix material can be further configured to tune the oxygen sensitivity range of the oxygen sensing molecule. In one embodiment, the film changes include changes in phosphorescence. For example, oxygen sensing molecules can be specifically designed to feature bright red phosphorescent emissions, providing a visual response to changes in oxygenation levels that are visible under ambient light. These properties simplify the collection and interpretation of oxygen-dependent emissions, allowing analysis to be performed using simple and inexpensive equipment.

[0031] In one embodiment, the oxygen sensing molecule comprises a phosphorescent meso-unsubstituted porphyrin having the formula (I): [ka] wherein M is a metal, each R is independently an atom or group of atoms, and at least one R is -OR 1 and R 1 is an atom or group of atoms.

[0032] In the porphyrin of formula (I), R 1 can be selected from the group consisting of hydrogen, substituted or unsubstituted alkyl, substituted or unsubstituted alkylcarbonyl, substituted or unsubstituted alkenyl, substituted or unsubstituted alkynyl, substituted or unsubstituted cycloalkyl, substituted or unsubstituted heterocycloalkyl, substituted or unsubstituted aryl, heteroaryl, halo, cyano, and nitro. In one example of a porphyrin of formula (I), R 1 is hydrogen. In another example of a porphyrin of formula (I), R 1 is alkynyl, for example 2-propynyl (propargyl). In yet another example of a porphyrin of formula (I), R1 is alkylcarbonyl, such as 2,2-dimethylpropanoyl (also called trimethylacetyl or pivaloyl). In the porphyrin of formula (I), multiple R's are -OR 1 and optionally, all R's can be -OR'd 1 It can be said that:

[0033] In one example of a porphyrin of formula (I), R 1 contains a triazolyl group. The triazolyl group can be attached to O through an alkyl chain. In one example of a porphyrin of formula (I), R 1 contains an alkyl glutamate group. 1 can be terminated with a pair of alkyl glutamate groups. In another example of a porphyrin of formula (I), R 1 contains a triazolyl group, and R 1 is a terminal pair of ethyl glutamate groups, and all R are -OR 1 In one example of a porphyrin of formula (I), the metal is platinum or palladium.

[0034] The porphyrin of formula (I) can be an oxygen-sensing phosphor with an emission intensity that depends on the partial pressure of oxygen. In one example of the porphyrin of formula (I), the porphyrin can be excited when irradiated with a first wavelength in the range of 350 to 600 nanometers and then emit phosphorescence at a second wavelength in the range of 600 to 700 nanometers. The first wavelength can be 532 nanometers and the second wavelength can be 644 nanometers. The first wavelength can also be 546 nanometers and the second wavelength can be 674 nanometers.

[0035] In another embodiment, the oxygen sensing molecule comprises a phosphorescent meso-unsubstituted porphyrin having the formula (II): [ka] wherein M is a metal, each R is independently an atom or group of atoms, and at least one R is -OR 1and R 1 is an atom or group of atoms.

[0036] In the porphyrin of formula (II), R 1 can be selected from the group consisting of hydrogen, substituted or unsubstituted alkyl, substituted or unsubstituted alkylcarbonyl, substituted or unsubstituted alkenyl, substituted or unsubstituted alkynyl, substituted or unsubstituted cycloalkyl, substituted or unsubstituted heterocycloalkyl, substituted or unsubstituted aryl, heteroaryl, halo, cyano, and nitro. In one example of a porphyrin of formula (II), R 1 is hydrogen. In another example of a porphyrin of formula (II), R 1 is alkynyl, for example 2-propynyl (propargyl). In yet another example of a porphyrin of formula (II), R 1 is alkylcarbonyl, such as 2,2-dimethylpropanoyl (also called trimethylacetyl or pivaloyl). In the porphyrin of formula (II), multiple R's are -OR 1 Optionally, all R's can be -OR 1 It can also be done as follows.

[0037] In one example of a porphyrin of formula (II), R 1 contains a triazolyl group. The triazolyl group can be attached to O through an alkyl chain. In one example of a porphyrin of formula (II), R 1 contains an alkyl glutamate group. 1 can be terminated with a pair of alkyl glutamate groups. In another example of a porphyrin of formula (II), R 1 contains a triazolyl group, and R 1 is a terminal pair of ethyl glutamate groups, and all R are -OR 1 In one example of a porphyrin of formula (II), the metal is platinum or palladium.

[0038] Porphyrins of formula (II) can be oxygen-sensing phosphors with emission intensities that depend on oxygen partial pressure. In one example of a porphyrin of formula (II), the porphyrin can be excited when irradiated with a first wavelength in the range of 350-650 nanometers and subsequently emit phosphorescence at a second wavelength in the range of 700-800 nanometers. The first wavelength can be 594 nanometers and the second wavelength can be 740 nanometers. The first wavelength can be 605 nanometers and the second wavelength can be 770 nanometers. The first wavelength can be 600-615 nanometers and the second wavelength can be 760-800 nanometers. In one non-limiting embodiment of the device 100, the photon source can be a light-emitting diode emitting light at a wavelength of 405 nanometers. Photons from the photon source can be filtered using a long-pass filter.

[0039] The controller can be a microcontroller or a system-on-chip and can include non-transitory memory capable of storing an executable program on the controller. In some embodiments, the controller can store an analyte calculation program that calculates an analyte (e.g., acid) level adjacent to the probe from one or more electrical signals received from the photodetector. The controller can also include an output, which can be a wire bundle. The output can be connected to an external interface that can be used to display, store, and / or analyze results of the executable program of the device 100. The device 100 can include a display in electrical communication with the controller, which can execute a program stored therein to display the oxygen level on the display and / or store values ​​of the oxygen level at multiple points in time over a period of time. In other embodiments, the controller can be configured with a wireless output, which can provide wireless communication. Non-limiting examples of wireless communication that can be incorporated include Wi-Fi, Bluetooth, near-field communication, cellular networks, radio frequency, etc. The controller may include an on-board power source, such as a battery, that can provide power to the controller so as to power the photon source, the photodetector, and the controller. In other embodiments, the controller may include an external power source, such as an electrical connection to grid power, so as to provide power to the photon source, the photodetector, and the controller.

[0040] In the device 100, the highly breathable white scattering layer W can comprise a silicone film containing a pigment. The pigment can be white. The scattering layer W can backscatter phosphorescent emissions to a photodetector. The scattering layer W also serves as a light shield, preventing external illumination from affecting the measurement and providing a reading that is independent of tissue or skin color. In the device 10, a semi-permeable transparent membrane SP partially shields the tissue from atmospheric oxygen and can keep room air out of the probe and scattering layer W, allowing the material to equilibrate to the tissue's pO2. The device 100 for transcutaneous O2 sensing is biocompatible because only the multilayer film contacts the skin. The light-scattering layer W prevents the probe from direct contact with the tissue.

[0041] With the components of device 100 described in detail above, the method of operation of optical device 100 can be understood. Device 100 is particularly useful for detecting oxygen concentrations, but can also be used for detecting concentrations of other analytes. In one non-limiting example method of operation of optical device 100, a photon source is activated by a controller to send photons of a particular wavelength to the probe. A photodetector detects light emitted from the probe when the photon source sends photons to the probe. The controller executes a program executed on the controller to calculate the level of an analyte parameter adjacent the probe based on the light emitted from the probe detected by the photodetector.

[0042] The controller can execute a program executed on the controller to calculate the level of the analyte parameter adjacent to the probe based on a normalization factor, algorithm, and / or calibration determined by sending photons of the wavelength to the probe to account for changes in brightness of the polymer matrix. The controller can execute a program executed on the controller to calculate the level of the analyte parameter adjacent to the probe based on a luminescence ratio (e.g., phosphorescence ratio) that provides an index, the index being proportional to the level of the analyte parameter adjacent to the probe and normalized using the normalization factor. The controller can execute a program executed on the controller to calculate the level of the analyte parameter adjacent to the probe based on a calibration algorithm such that the device can output a report of the level of the analyte parameter adjacent to the probe via wired or wireless communication. The calibration algorithm can use different calibration parameters depending on whether the derivative of the luminescence ratio (e.g., phosphorescence ratio) is positive or negative.

[0043] In one aspect, the invention provides a sensor system comprising: a probe sensitive to an analyte, the probe having a body having a first surface and an opposing second surface; a photon source configured to send photons to the probe, the probe emitting light upon receiving photons from the photon source; a photodetector configured to detect light emitted from the probe; and a controller in communication with the photon source and the photodetector. The controller may be configured to execute a program stored therein to (i) cause the photon source to send photons to the probe and excite the probe to emit light when the probe receives the photons; (ii) receive optical data from the photodetector based on an interaction between the light emitted from the probe and the photodetector; (iii) determine, based on the optical data from the photodetector, a difference between a first level of the parameter of the analyte at a first position adjacent to the first surface of the probe and an intermediate level of the parameter of the analyte at an intermediate position between the first position and a second position adjacent to the second surface of the probe; and (iv) determine a second level of the parameter of the analyte adjacent to the second surface of the probe based on the difference between the first level of the parameter of the analyte and the intermediate level of the parameter of the analyte.

[0044] In one embodiment of the sensor system, the controller executes a program stored in the controller to determine the second level of the analyte adjacent the second surface of the probe by solving a mathematical diffusion problem involving the first level of the analyte and the intermediate level of the analyte.

[0045] In one embodiment of the sensor system, the controller executes a program stored in the controller to determine the second level of the analyte adjacent the second surface of the probe by solving an inverse mathematical boundary condition problem involving the first level of the analyte and the intermediate level of the analyte. In one embodiment of the sensor system, the inverse mathematical boundary condition problem includes a thickness-dependent diffusivity variable. In one embodiment of the sensor system, the inverse mathematical boundary condition problem includes a Dirichlet boundary condition.

[0046] In yet another aspect, the invention provides a sensor system comprising: a probe sensitive to an analyte, the probe having a body having a first surface and an opposing second surface; a photon source configured to send photons to the probe, the probe emitting light upon receiving photons from the photon source; a photodetector configured to detect light emitted from the probe; and a controller in communication with the photon source and the photodetector. The controller may be configured to execute a program stored therein to (i) cause the photon source to send photons to the probe and excite the probe to emit light when the probe receives the photons; (ii) receive optical data from the photodetector based on an interaction between the light emitted from the probe and the photodetector; (iii) determine, based on the optical data from the photodetector, a gradient of the parameter of the analyte extending from a first position adjacent to the first surface of the probe to an intermediate position between a second position adjacent to the second surface of the probe and the first position; and (iv) calculate, based on the gradient, a level of the parameter of the analyte adjacent to the second surface of the probe.

[0047] In one embodiment of the sensor system, the parameter is partial pressure. In one embodiment of the sensor system, the analyte includes at least one of oxygen, carbon dioxide, nitric oxide, and volatile organic compounds. In one embodiment of the sensor system, the analyte includes oxygen.

[0048] In one embodiment of the sensor system, when a section of the sensor system is disposed on a portion of a third surface, the section of the sensor system is configured to define a zone adjacent the third surface, and the probe is in fluid communication with the zone. In one embodiment of the sensor system, the third surface includes the patient's skin, and the analyte includes transcutaneous oxygen.

[0049] In one embodiment of the sensor system, the probe includes a metalloporphyrin embedded in a layer of polymeric material. In one embodiment of the sensor system, an oxygen-permeable scattering layer is disposed between the layer of polymeric material and the third surface. The gas-permeable scattering layer increases the phosphorescent signal that reaches and is collected by the photodetector and serves as a light blocker.

[0050] In one embodiment of the sensor system, a semi-transparent layer is disposed between the layer of polymeric material and the photodetector, and in one embodiment of the sensor system, the semi-transparent layer is semi-permeable to oxygen diffusion therethrough.

[0051] In one embodiment of the sensor system, the third surface comprises the patient's skin, the analyte comprises transcutaneous oxygen, and the equilibration time of oxygen in the zone when the section of the sensor system is placed on the portion of the skin is less than 30 minutes, or the equilibration time of oxygen in the zone when the section of the sensor system is placed on the portion of the skin is less than 20 minutes, or the equilibration time of oxygen in the zone when the section of the sensor system is placed on the portion of the skin is less than 10 minutes.

[0052] In one embodiment of the sensor system, the third surface comprises the patient's skin and the analyte comprises a liquid-dissolved compound in tissue. In one embodiment of the sensor system, the liquid-dissolved compound is selected from the group consisting of a drug and a metabolite. In one embodiment of the sensor system, the third surface comprises a battery. In one embodiment of the sensor system, the third surface comprises a material selected from the group consisting of a polymer material, a carbon fiber material, a composite material, and a multilayer material.

[0053] In yet another aspect, the invention provides a method for determining a level of a parameter of an analyte, the method comprising the steps of: (a) disposing a probe near a surface for monitoring an analyte; (b) disposing a photon source to deliver photons to the probe to excite the probe; (c) disposing a photodetector to receive light emitted by the probe when emitted by the photon source; (d) causing the photon source to send photons to the probe, which excites the probe to emit light when the probe receives the photons; and (e) collecting optical data from the photodetector based on an interaction between the light emitted from the probe and the photodetector. The method may include (f) determining a difference between a first level of the parameter of the analyte at a first location adjacent the first surface of the probe and an intermediate level of the parameter of the analyte at an intermediate location between the first location and a second location adjacent the second surface of the probe based on the optical data from the photodetector; and (g) determining a second level of the parameter of the analyte adjacent the second surface of the probe based on the difference between the first level of the parameter of the analyte and the intermediate level of the parameter of the analyte. In one embodiment of the method, step (g) includes determining the second level of the analyte adjacent the second surface of the probe by solving a mathematical diffusion problem involving the first level of the analyte and the intermediate level of the analyte. The mathematical inverse boundary condition problem may include a thickness-dependent diffusivity variable. The mathematical inverse boundary condition problem may include a Dirichlet boundary condition.

[0054] In yet another aspect, the invention provides a method for determining a level of an analyte parameter, the method comprising: (a) disposing a probe near a surface for monitoring an analyte; (b) disposing a photon source to deliver photons to the probe to excite the probe; (c) disposing a photodetector to receive light emitted by the probe when emitted by the photon source; (d) causing the photon source to send photons to the probe and excite the probe to emit light when the probe receives the photons; (e) collecting optical data from the photodetector based on an interaction between the light emitted from the probe and the photodetector; (f) determining, based on the optical data from the photodetector, a gradient of the analyte parameter from a first location adjacent the first surface of the probe to an intermediate location between a second location adjacent the second surface of the probe and the first location; and (g) calculating a level of the analyte parameter adjacent the second surface of the probe based on the gradient.

[0055] In one embodiment of the method, the parameter is partial pressure. In one embodiment of the method, the analyte comprises at least one of oxygen, carbon dioxide, nitric oxide, and volatile organic compounds. In one embodiment of the method, the analyte comprises oxygen.

[0056] In one embodiment of the method, the method further includes disposing a section of the sensor system on a portion of a third surface to define a zone adjacent the third surface such that the probe is in fluid communication with the zone. In one embodiment of the method, the third surface includes the patient's skin and the analyte includes transcutaneous oxygen.

[0057] In one embodiment of the method, the probe comprises a metalloporphyrin embedded in a layer of polymeric material.

[0058] In one embodiment of the method, the method further comprises disposing an oxygen-permeable scattering layer between the layer of polymeric material and the third surface.

[0059] In one embodiment of the method, the method further includes disposing a semi-transparent layer between the layer of polymeric material and the photodetector, the semi-transparent layer being semi-permeable to oxygen diffusion therethrough.

[0060] In one embodiment of the method, the third surface comprises the patient's skin, the analyte comprises transcutaneous oxygen, and when the section of the sensor system is placed on the portion of the skin, the equilibration time of oxygen in the zone is less than 30 minutes, or the equilibration time of oxygen in the zone is less than 20 minutes, or the equilibration time of oxygen in the zone is less than 10 minutes.

[0061] In yet another aspect, the invention provides a computer system comprising instructions stored on a non-transitory computer-readable medium to cause at least one processor on a computer to: (i) cause a photon source to send photons to a probe sensitive to an analyte and excite the probe to emit light when the probe receives the photons, the probe comprising a body having a first surface and an opposing second surface; (ii) receive optical data from the photodetector based on an interaction between the light emitted from the probe and a photodetector; (iii) determine, based on the optical data from the photodetector, a difference between a first level of a parameter of the analyte at a first location adjacent the first surface of the probe and an intermediate level of the parameter of the analyte at a location intermediate between the first location and a second location adjacent the second surface of the probe; and (iv) determine a second level of the parameter of the analyte adjacent the second surface of the probe based on the difference between the first level of the parameter of the analyte and the intermediate level of the parameter of the analyte. In one embodiment of the computer system, the at least one processor determines the second level of the analyte adjacent to the second surface of the probe by solving a mathematical diffusion problem involving the first level of the analyte and the intermediate level of the analyte. In one embodiment of the computer system, the at least one processor determines the second level of the analyte adjacent to the second surface of the probe by solving an inverse mathematical boundary condition problem involving the first level of the analyte and the intermediate level of the analyte. The inverse mathematical boundary condition problem can include a thickness-dependent diffusivity variable. The inverse mathematical boundary condition problem can include a Dirichlet boundary condition.

[0062] In yet another aspect, the invention provides a computer system comprising instructions stored on a non-transitory computer-readable medium to cause at least one processor on a computer to: (i) cause a photon source to send photons to a probe sensitive to an analyte and excite the probe to emit light when the probe receives the photons, the probe having a body having a first surface and an opposing second surface; (ii) receive optical data from the photodetector based on an interaction between the light emitted from the probe and a photodetector; (iii) determine, based on the optical data from the photodetector, a gradient of a parameter of the analyte extending from a first location adjacent the first surface of the probe to an intermediate location between a second location adjacent the second surface of the probe and the first location; and (iv) calculate, based on the gradient, a level of the parameter of the analyte adjacent the second surface of the probe. In one embodiment of the computer system, the parameter is a partial pressure. In one embodiment of the computer system, the analytes include at least one of oxygen, carbon dioxide, nitric oxide, and volatile organic compounds. In one embodiment of the computer system, the analytes include oxygen.

[0063] Example The following examples are provided to illustrate and further explain certain embodiments and aspects of the present invention and should not be construed as limiting the scope of the invention. The descriptions in the following examples are offered without being bound by theory.

[0064] Overview of the Example A significant limitation of transcutaneous oxygen monitoring (TCOM) measurements is the long equilibration period required to obtain a stable reading that approximates the true partial pressure of oxygen (pO2, transcutaneous pO2, or tcpO2) in the tissue at the skin surface. Furthermore, the relationship between the obtained reading and the true tcpO2 is usually not discussed. When applying this TCOM device, an exponential decay from atmospheric pO2 (160 mmHg) to a value around 60–80 mmHg is typically observed. The rate and amplitude of the decay depend on the volume of air trapped between the detector and the skin and the diffusion of oxygen through the various materials used. Given sufficient time, it was believed that the obtained pO2 reading would match the actual tcpO2 value. However, when the device uses semi-breathable materials to allow the skin to breathe, the obtained equilibrium pO2 value is intermediate between atmospheric and skin pO2.

[0065] In some applications, direct contact between the skin and the O2 sensing material must be avoided, and the addition of a semi-permeable layer further alters the difference between the measured pO2 and the true tcpO2. Therefore, it would be valuable to significantly reduce or completely eliminate the latency of a TCOM device, and for the device to consistently produce true tissue oxygenation readings even as device environmental conditions (temperature, humidity, movement, etc.) change.

[0066] This example introduces a toolset for accurately measuring skin oxygenation by fully utilizing wearable transcutaneous oxygenation monitor (TCOM) technology. We describe a numerical model and experimental characterization techniques that enable the extraction of accurate tissue oxygenation measurements. The numerical model is based on a boundary inverse problem of a parabolic equation with Dirichlet boundary conditions. To validate the model and characterize oxygen diffusion through the oxygen-sensing material, we designed a series of control / calibration experiments modeled after the device's clinical application, using oxygenation values ​​in the physiological range expected in healthy tissue. Our results demonstrate that it is possible to obtain accurate tissue pO2 measurements with a compact, wearable device without requiring long equilibration times.

[0067] Introduction As noted above, even though TCOM devices use a barrier or intermediate layer between the tissue and the sensor, it would be valuable to systematically generate true tissue oxygenation readings while reducing the need for long equilibration times. Here, we report the development of a mathematical toolset to address these limitations so that device-measured oxygen concentrations can be converted to true oxygenation values ​​while using materials that can be used for long-term wear applications. We describe and test a numerical model, along with experimental characterization, to easily calculate pO2 values, creating a practical approach for future applications.

[0068] Experimental and Numerical Methods Wearable Transcutaneous Oxygen Monitor A recently developed wearable TCOM device

[12] detects molecular oxygen through changes in the phosphorescence lifetime of a laboratory-synthesized, ultrabright metalloporphyrin

[13] , which exhibits oxygen-induced phosphorescence quenching. The change in the lifetime τ of the fluorophore depends on oxygen according to the Stern-Volmer relationship

[14] . τ0 / τ=1+K SV ×pO2(1) where τ is the phosphorescence lifetime of the probe in the absence of oxygen, τ is the phosphorescence lifetime of the probe in the presence of oxygen, and K SVis the Stern-Volmer quenching constant, which depends on the lifetime of the probe and its environment (polymer matrix, solvent, etc.), and pO2 is the partial pressure of oxygen. This fluorophore can be easily embedded into polymer-based films, resulting in ultrathin, breathable films that exhibit high-intensity emission throughout the physiological pO2 range and are unaffected by changes in relative humidity [15, 16]. This device is truly wearable (no external readout electronics or cable connections are required), lightweight, completely self-contained, and measures tissue oxygenation without the use of heating elements, making it an improvement over existing technologies. Porphyrins have been successfully implemented in several different applications with multiple form factors, including clinically tested applyable bandages

[10] , wound-healing hydrogel dressings

[17] , skin-mountable devices [12, 18], intramuscular needles

[19] , and subcutaneous microneedles

[20] .

[0069] In the TCOM wearable device 100 shown in Figures 1A and 1B, the oxygen sensing film 105 is composed of a medical-grade semi-permeable transparent membrane SP (Bioclusive, McKesson) that partially shields the skin from atmospheric oxygen; a thin poly(propyl methacrylate) layer embedded with metalloporphyrins forming the probe PPMA; and a highly breathable white scattering layer W (Figure 1B, panel a) that serves as a light shield and increases the phosphorescent signal collected and reaches the photodetector. This O2 sensing film 105 design allows for a short settling or equilibration time by providing an airtight seal to the skin and minimizes the volume of void space or trapped air between the device and the skin. Furthermore, the use of the medical-grade semi-permeable layer SP allows some moisture and oxygen to diffuse away from the skin, enabling the device 100 and materials to be worn for extended periods. The device 100 also includes an electronic circuit 110 with a controller that communicates with the O2 sensing film 105. The electronic circuit 110 can include photon source(s) and photodetector(s). The photon source can include one or more photon sources (e.g., one, two, three, four, etc.), and in some cases, the photon source can include a single photon source. Each photon source can be optically coupled to the probe PPMA and configured to emit a respective light toward the probe PPMA, which then emits light in response to absorbing the light. For example, the light can interact with a photoluminescent material in the probe PPMA to emit photoluminescent light. As a more specific example, the light can interact with a phosphorescent material in the probe PPMA to emit phosphorescence. The wearable device described in Reference 12 is attached to an O2 sensing film 105, excites the phosphorescence of porphyrins in the PPMA layer via two ultraviolet LEDs, and detects the phosphorescence via a small photodiode. A band 120 couples the electronic circuit 110 to a body part, such as the leg shown in FIG. 1A or the arm shown in FIG. 1B.

[0070] Boundary inverse problems for parabolic equations with Dirichlet boundary conditions A key detail of this example is that the measurement of oxygen partial pressure is obtained from porphyrins embedded in a PPMA layer that is sandwiched between two other layers and does not directly contact the patient's skin. Looking at panel b of Figure 1B, the film and device provide a pO2 reading, which is the intermediate concentration between the atmosphere on one side of the O2 sensing film 105 and the tissue tcpO2 on the other. This scenario corresponds to a diffusion problem where pO2 is known at one boundary (air on the left—e.g., RBC = 160 mmHg) and one midpoint (pO2 at the center of the film—e.g., the TCOM reading), and we are interested in the pO2 at the other boundary (right, in this example, tissue, e.g., RBC = ???). This particular type of mathematical problem is known as a boundary condition inverse problem. In this example, we develop and demonstrate a one-dimensional boundary condition inverse problem solution for estimating true tissue oxygenation from readings obtained by the oxygen sensing film and TCOM wearable technology, along with a control experiment.

[0071] Since the objective in this example is to find the concentration u, or pO2, value at one boundary, it is necessary to solve the boundary inverse problem of a parabolic equation with thickness-dependent diffusivity D(x).

number

[0072] midpoint x * The additional conditions in u(x * ,t)=φ(t),0 <t≦T (4) , which is the pO2 measurement obtained using the wearable device, pO2 TCOM Corresponds to.

[0073] Finally, the initial conditions are set as follows: u(x,0)=g(x),0 <x<l (5) This typically corresponds to an initial pO2 value of atmospheric oxygenation for any x, i.e., g(x) = 160 mmHg. This initial condition corresponds to the sensor material not yet attached to the skin surface.

[0074] Discrete Boundary Inverse Problem The method, described in detail in Reference 21, involves discretizing the problem described by Equations 2-5 using finite differences to include an x-dependent diffusivity, D. Here, we modify the above algorithm to use a Neumann left boundary condition instead of a Dirichlet boundary condition, since the problem investigated here requires specifying the concentration value rather than its derivative at the boundary condition.

[0075] The space and time coordinates x and t have the following nodes: ω h ={x i =ih,i=1,2,...,N x -1,h=l / N x} ω τ ={t j =jτ,j=1,2,···,N t -1,τ=l / N t}

[0076] Additional condition observation x * The node corresponding to x * =n * Defined as h. For brevity,

number

[0077] On the grid ωhτ = ωh × ωτ, ref. 21 employs an implicit difference scheme for the time approximation and a finite difference method for the boundary inverse problem. Then, for t ∈ ωτ, the discrete problem is expressed as

number

[0078] The left bound is modified from the method in Ref. 21, where the left Dirichlet bound is defined as

number

number

[0079] Algorithm for solving inverse problems To solve the above boundary inverse problem, at each time t j As discussed in Reference 21, we need to solve the following linear algebraic equation: i is decomposed into two grid functions (Reference 22): y i =v i +y n ·w i ,i=0,···,n (10) In this example, n=N x At each time point t j V in iThe value of can be determined by solving the following system of linear equations: The first case or first line below is the only modification to the work of ref. 21, since it depends on the left boundary condition.

number

[0080]

number

[0081] Each time point t j So, v i and w i Once the value of is known, the value of the function at the right boundary, y n Yes n* =φ and y n* =v n* +·y n* w n* It is calculated using and so it is as follows:

number

[0082] Finally, for the algorithm to be applicable, n ≠0 (or not too small), and this condition is met by C=τ / 2h 2 This is satisfied when the Courant stability criterion for a parabolic equation, defined as:

[0083] Testing the algorithm As a starting point, we first reproduced the boundary inverse problem reported in Reference 21, which uses the Neumann left boundary condition. Then, to test our algorithm, we solved the following sample problem given in Reference 21. First, we adopted an implicit difference scheme using GNU Octave [Reference 23] to solve the sample problem and implemented a direct initial-boundary value problem, which is

number

number

number

[0084] For the calculations, the following parameters were used: l = 1, N x =100,T=2,N t =100. Next, the additional condition u(x * ,t)=φ(t) to x * = 0.5. The left and right boundaries and φ(t) are plotted as solid lines in Figure 2, panel a.

[0085] Now we can use the direct model calculation, ie φ(t) as the left boundary and additional conditions, to calculate the right boundary using the boundary inverse problem algorithm of Ref. 21.

[0086] The calculated right boundary is plotted as scattered points in panel a of Figure 2, which is in agreement with the known function ρ(t) from the direct model and with the results of the reference article.

[0087] Now that this is complete, it is possible to test the above modified algorithm for solving the boundary inverse problem of a parabolic equation with left Dirichlet boundary conditions. This is because the left boundary u(0,t)=0 It was solved in the same way as above, except that

[0088] Here, the same parameters were reused: l=1, N x =100,T=2,N t =100,C=100,u(x * ,t)=φ(t),x * = 0.5. We solve this direct problem using the known left-right condition to obtain φ(t), which is plotted (solid line) in panel b of Figure 2.

[0089] Boundary inverse problems applied to transcutaneous oxygenation. Finally, the modified boundary inverse problem described above allows us to reconstruct the right boundary, plotted as scattered points in panel b of Figure 2, so that it closely matches the known function ρ(t) from the direct model, just as it does with the Neumann boundary condition.

[0090] Boundary inverse problems applied to transcutaneous oxygenation. The above example deals with a constant diffusion coefficient for all x. To fully describe the multilayer system of the oxygen sensing film, one needs to use a three-dimensional model with a diffusivity that varies with depth, i.e., D(z). However, as we will explain below, in our case, the use of a conservative finite difference scheme for the 1D diffusion equation [Ref. 24] is sufficient.

[0091] As shown in panel a of Figure 3, the diameter of the PPMA acid-sensing disk is

number

number

[12] , thereby providing a pO2 reading. The semi-transparent film SP is much larger than the PPMA disk (see the photograph of the film in panel b of Figure 3), and its thickness is d SP The white scattering layer W had a thickness of d W = 45 μm, covering the entire PPMA disk.

[0092] Only the central part of the PPMA disk was probed to measure pO2, and the film thickness was much smaller than the film width.

number

[0093] The diffusion equation in three dimensions is

number

number

number

number

[0094] It should be noted that considering molecular emission to occur at a specific depth in the PPMA layer is a simplification. In a wearable sensor, this measurement reflects the average oxygenation throughout the volume of the PPMA layer, excited by the LED and sampled by the photodiode. Finally, we use the measured layer thicknesses to propose a smooth (differentiable) D(x) function to obtain the thickness-dependent diffusion coefficient to account for each of the different layers in the O2 sensing film. To create a smooth (differentiable) and sharp transition in diffusivity to reflect the changes between each layer, we use the following function:

number

[0095] Results and Discussion Controlled experiments and modeling We designed an experiment to solve the boundary inverse problem by estimating the unknown diffusion coefficients of the three-layer SP, PPMA, and W layers. In this experiment, the pO2 at the left boundary (air), the intermediate value (TCOM reading), and the pO2 at the right boundary (oxygenated with the film attached) were known. To perform this experiment, shown in panel b of Figure 3, we designed and 3D printed a calibration block made of rubbery photopolymer using a Formlabs Form 3B printer. The block contained an inner chamber that could be connected to a tube and filled with a controlled gas mixture. An additional opening allowed for sampling of the pO2 of the gas mixture using a commercially available oxygen sensor (Presens). A 1 cm diameter area was designed on the surface of the block, with multiple small holes drilled into it, and an oxygen-sensing film was placed across this area. The bottom of the film was exposed to the pO2 of the inner chamber.

[0096] As shown in panel b of Figure 3, a film is applied to the calibration block, which simulates clinical use, and then a wearable device is attached to the film to obtain a pO2 reading or pO2TCOM of the PPMA layer. EXP Probing.

[0097] The pO2 of this gas mixture was varied by changing the mixture of nitrogen and air supplied to an automatic gas mixer

[25] , which was programmed to simulate the clinical outcome from the following steps: (a) The device probes atmospheric pO2; (b) Place the device on the subject's skin and allow it to equilibrate for 20-30 minutes; (c) Restriction of blood flow via a pressure cuff for approximately 10 minutes, resulting in decreased oxygenation; (d) After removing the cuff and restoring blood flow, an increase in pO2 is observed (reactive hyperemia), which then settles to the initial value before cuffing. (e) The device is removed from the subject's arm and exposed to the atmosphere.

[0098] pO2 TCOM and RBCs EXP Both are shown in panel c of Figure 3. Five different films were tested in this system, the first of which is shown in panel c of Figure 3. The inset shows how the TCOM raw readings underestimated the change in pO2 compared to the right boundary measurements during periods simulating blood flow restriction or occlusion.

[0099] The data were then fitted using an inverse diffusion equation model with a thickness-dependent diffusion coefficient defined by the piecewise function described above. We used the following parameters for all the above boundary inverse calculations: l = 145 μm, N x =1000,T=78.5min,N t = 200. This gives a Courant stability criterion C = 9.341. The algorithm results in a quality comparable to that of C = 93.41, but the calculation is significantly faster. Known left boundary and intermediate reading pO2 TCOM A nonlinear least squares algorithm was used to calculate the reconstructed right boundary using the experimentally measured values ​​(RBC EXP ) This model matches x * The values ​​of the three layers D SP , D PPMA , D W Using four fitting coefficients for the diffusivity values ​​of . The fitting results are shown in Figure 3, panel c, and the reconstructed right boundary is in good agreement with the experimental value. The residual plot in Figure 3, panel d, provides further evidence of the goodness of the fit. The fitting and residual results for the other three remaining films are shown in Figure 5.

[0100] The fitting coefficient values ​​obtained for all films are shown in Table 1, where film number 1 corresponds to the results shown in Figures 3 and 4. [Table 1]

[0101] The most consistent value among these is x * The average value is 80.7±0.6 μm, which is near the center of the PPMA film (77.5 μm). For the semi-transparent layer SP, the coefficient D SP However, the mean and standard deviation were not calculated because the upper limit set for the fitting algorithm was reached for three of the films. The values ​​for films 1 and 2 are within the experimentally observed range of 10 at room temperature. -8 ~10 -6 cm 2 / s [References 26, 27]. For PPMA, the relative error is large, (1.05±0.22)×10 -7 cm 2 / s, but the average value is within a known range (2 -8 ) x 10 -8 cm 2 / s [References 28-30]. A highly breathable white scattering layer W composed of dimethylsiloxane and TiO2 powder [References 12, 19] has an average diffusivity of (1.33±0.44) × 10 -6 cm 2 / s, which is consistent with the experimental value [Reference 31], but the relative error is large, (1.33±0.44)×10 -6 cm 2 / s. We also consider an exponential decay function (f(t) = A1 + A2 e) for the initial decline from atmospheric pO2 to its equilibrium value. -t / τ ) and the average decay time for all five films was τ = 2.25 ± 0.08 min for the TCOM data and τ = 2.25 ± 0.08 min for the RBC data. EXP The reading was τ = 1.7 ± 0.1 min, indicating a fast equilibration time for our film.

[0102] As a precaution, we have already solved the inverse problem, as in the example considered in Figure 2, and we will check whether solving this direct problem will yield values ​​that are consistent with experiment. We feed fitting coefficients into a direct model with thickness-dependent diffusivity, with known left and right boundaries, x *The oxygen concentration calculated by the experimental results, pO2 TCOM This can be seen in panel a of Figure 4. The diffusivity profile or D(x) is shown in panel b of Figure 4 along with the O2 concentration gradient through the layer.

[0103] Applying algorithms to clinical data Now, the above boundary inverse problem can be adjusted to fit the individual diffusion coefficient of the O2 sensing film and used to calculate the "true" skin pO2 using TCOM oxygenation readings. Figure 6 shows the results of applying the algorithm to clinical study measurements. The measurements were carried out according to the following procedure. (a) The device probes atmospheric pO2; (b) The device is placed on the subject's lower leg (calf) and allowed to equilibrate for approximately 40 minutes; (c) restricting blood flow in the thigh via a pressure cuff for 10 minutes, resulting in reduced oxygenation; (d) Reactive hyperemia is produced by removing the cuff, resulting in an increased increase in pO2, followed by a return of pO2 to the initial value.

[0104] As shown in panel a of Figure 6, IBP pO2 readings equilibrated from ambient pO2 to tissue levels within 30 minutes after application. Fitting an exponential function to both time series resulted in exponential decay times of approximately 13 minutes for both TCOM and IBP readings (13.56 minutes for TCOM readings and 13.43 minutes for IBP readings). Differences relative to the values ​​obtained above for the film on the calibration block may be due to air pockets trapped in the creases when the adhesive film was applied to the skin or due to the response of the tissue itself.

[0105] The resulting equilibrium pO2 value was approximately 50 mmHg, within the range expected for healthy tissue [6]. Furthermore, panel b of Figure 6 plots the relative change in oxygenation of the TCOM and IBP readings during pressure cuff application (the pO2 value at the time the pressure cuff was applied was subtracted from the measurement). The graph shows that the pO2 changes due to increases and decreases in blood flow are sharper and larger in amplitude than the raw readings reported by the TCOM device, and also more accurately estimate the actual decrease in tissue oxygenation.

[0106] conclusion The above results show that by characterizing the oxygen gradient through each layer of the O2 sensing film, it is possible to obtain numerical pO2 values ​​that are closer to tissue values ​​and quantitatively reflect changes in oxygenation at the skin surface, thereby enabling highly accurate measurement of oxygenation when blood flow changes.

[0107] In this example, a methodology (numerical and experimental) was designed to characterize the diffusion of oxygen through multilayer films and materials. It is interesting to note that we observed that in the same experimental setup, the results obtained from different films were qualitatively similar but showed slight quantitative differences. For example, the equilibrium pO2 values ​​measured by the apparatus were found to be different for each individual film. This explains the difference in some of the fitting parameters in Table 1 (e.g., D SP) is manifested in inconsistent values ​​between films. Because the films were prepared manually, these differences may be due to human factors. For example, differences in air pockets trapped between the film layers may have occurred. Also, the adhesive ring support used to attach the measurement device to the film may have resulted in different exposures of the left boundary to ambient air. The reported pO2 differences between films may be due to experimental factors, such as incomplete application of the film to the chamber. The approach developed here can be used to investigate the effects of film construction and improve the consistency of the sensor film material. For example, the difference in exposure of the left boundary can be addressed by adding holes to the sensor head to ensure direct contact between ambient air and the top of the film.

[0108] The apparent higher diffusivity of PPMA measured here compared to literature data could be due to changes in pore size upon embedding of the metalloporphyrin

[30] or to cracking of the glassy PPMA during layer stacking manipulation. However, because we are proposing a model of the system under study, we did not determine the diffusion coefficients. These values ​​can and should be determined experimentally.

[0109] We found parameter values ​​for dx and dt that ensure stable calculations (Courant stability criterion C>1) while also achieving fast and accurate calculations. The algorithm can be incorporated into the device firmware to calculate tissue pO2 in real time after each pO2 sample, since the sampling times used can range from a few seconds to minutes. Real-time implementation of this algorithm can also dramatically reduce the settling or equilibration time required for the device to report skin pO2 values.

[0110] We investigated algorithm improvements to more accurately predict tissue oxygenation. For example, this algorithm uses pO2 TCOM The signal is at a certain depth (x in the model) * This can be attributed to phosphorescence throughout the PPMA layer rather than in the PPMA layer itself, or to the change in the diffusivity of the material as it changes with temperature. The model can also be extended by adding a simulation of oxygen consumption and transport in the skin

[32] , to remove the effect of the low permeability of the epidermis and obtain values ​​closer to those of the subcutaneous tissue.

[0111] References 1. Vegfors, M.; Lindberg, LG; Lennmarken, C., “The influence of changes in blood flow on the accuracy of pulse oximetry in humans”, Acta Anaesthesiol. Scand. 1992, 36, 346-349. 2. Li, Z.; Navarro-Alvarez, N.; Keeley, EJ; Nowell, NH; Goncalves, BM; Huang, CA; Evans, CL, “Non-invasive monitoring of skin inflammation using an oxygen-sensing paint-on bandage”, Biomedical optics express 2017, 8, 4640-4651. 3. McMillan, TE; Gardner, WT; Schmidt, AH; Johnstone, AJ, “Diagnosing acute compartment syndrome-where have we got to?”, Int. Orthop. 2019, 43, 2429-2435. 4. McPhail, L.R.; Cooper, L.T.; Hodge, D.O.; Cabanela, M.E.; Rooke, T.W., “Transcutaneous partial pressure of oxygen after surgical wounds”, Vasc. Med. 2004, 9, 125-127. 5. Moosa, H.H.; Makaroun, M.S.; Peitzman, A.B.; Steed, D.L.; Webster, M.W., “TcPO2 values in limb ischemia: Effects of blood flow and arterial oxygen tension”, Journal of Surgical Research 1986, 40, 482-487. 6. Baumgartl, H.; Ehrly, A.; Saeger-Lorenz, K.; Lubbers, D., “Clinical Oxygen Pressure Measurement: Tissue Oxygen Pressure and Transcutaneous Oxygen Pressure; Springer, 1987; pp 121-128. 7. Holst, G. A.; Koster, T.; Voges, E.; Lubbers, D.W., “FLOX-an oxygen-flux-measuring system using a phase-modulation method to evaluate the oxygen-dependent fluorescence lifetime”, Sensors and Actuators: B. Chemical 1995, 29, 231-239. 8. Stucker, M.; Struk, A.; Altmeyer, P.; Herde, M.; Baumgartl, H.; Lubbers, D.W., “The cutaneous uptake of atmospheric oxygen contributes significantly to the oxygen supply of human dermis and epidermis”, Journal of Physiology 2002, 538, 985-994. 9. Babilas, P.; Lamby, P.; Prantl, L.; Schreml, S.; Jung, E. M.; Liebsch, G.; Wolfbeis, O.S.; Landthaler, M.; Szeimies, R.-M.; Abels, C., “Transcutaneous pO2 imaging during tourniquet-induced forearm ischemia using planar optical oxygen sensors”, Skin Research and Technology 2008, 14, 304-311. 10. Marks, H.; Bucknor, A.; Roussakis, E.; Nowell, N.; Kamali, P.; Cascales, J. P.; Kazei, D.; Lin, S.J.; Evans, C.L., “A paintable phosphorescent bandage for postoperative tissue oxygen assessment in DIEP flap reconstruction”, Science Advances 2020, 6, No. eabd1061. 11. Alford, R.; Simpson, H. M.; Duberman, J.; Hill, G.C.; Ogawa, M.; Regino, C.; Kobayashi, H.; Choyke, P.L., “Toxicity of organic fluorophores used in molecular imaging: literature review”, Molecular imaging 2009, 8, 6. 12. Cascales, J.P.; Roussakis, E.; Witthauer, L.; Goss, A.; Li, X.; Chen, Y.; Marks, H.L.; Evans, C.L., “Wearable device for remote monitoring of transcutaneous tissue oxygenation”, Biomed. Opt. Express 2020, 11, 6989-7002. 13. Roussakis, E.; Li, Z.; Nowell, N.H.; Nichols, A.J.; Evans, C.L., “Bright,“clickable” porphyrins for the visualization of oxygenation under ambient light”, Angew. Chem., Int. Ed. 2015, 54, 14728-14731. 14. Stern, O.; Volmer, M. Uber die abklingzeit der fluoreszenz”, Phys. Z. 1919, 20, 183-188. 15. Roussakis, E.; Cascales, J.P.; Marks, H.L.; Li, X.; Grinstaff, M.; Evans, C.L., “Humidity-Insensitive Tissue Oxygen Tension Sensing for Wearable Devices”, Photochem. Photobiol. 2020, 96, 373-379. 16. Li, X.; Roussakis, E.; Cascales, J.P.; Marks, H.L.; Witthauer, L.; Evers, M.; Manstein, D.; Evans, C.L., “Optimization of bright, highly flexible, and humidity insensitive porphyrin-based oxygen-sensing materials”, Journal of Materials Chemistry C 2021, 9, 7555-7567. 17. Marks, H. L.; Cook, K.; Roussakis, E.; Cascales, J.P.; Korunes-Miller, J.T.; Grinstaff, M.W.; Evans, C.L., “Quantitative Luminescence Photography of a Swellable Hydrogel Dressing with a Traffic-Light Response to Oxygen (Adv. Healthcare Mater. 10 / 2022)”, Adv. Healthcare Mater. 2022, 11, 2270048. 18. Cascales, J.P.; Greenfield, D.A.; Roussakis, E.; Witthauer, L.; Li, X.; Goss, A.; Evans, C.L., “Wireless wearable sensor paired with machine learning for the quantification of tissue oxygenation”, IEEE Internet of Things Journal 2021, 8, 17557-17567. 19. Witthauer, L.; Cascales, J.P.; Roussakis, E.; Li, X.; Goss, A.; Chen, Y.; Evans, C.L., “Portable oxygen-sensing device for the improved assessment of compartment syndrome and other hypoxiarelated conditions”, ACS sensors 2021, 6, 43-53. 20. Muller, M.; Cascales, J.P.; Marks, H.L.; Wang-Evers, M.; Manstein, D.; Evans, C.L., “Phosphorescent Microneedle Array for the Measurement of Oxygen Partial Pressure in Tissue”, ACS Sensors 2022, 7, 3440-3449. 21. Vasilev, V.; Vasilyeva, M.; Kardashevsky, A., “The numerical olution of the boundary inverse problem for a parabolic equation”, AIP Conf. Proc. 2016, 100010. 22. Vabishchevich, P.; Vasilev, V., “Computational determination of the lowest order coefficient in a parabolic equation”, Doklady Mathematics 2014, 89, 179-181. 23. Eaton, J. W.; Bateman, D.; Hauberg, S.; Wehbring, R. GNU Octave version 3.8.1 manual: a high-level interactive language for numerical computations; CreateSpace Independent Publishing Platform, 2014; ISBN 1441413006. 24. Lee, J.; Jeong, M.S., Stability of Finite Difference Schemes on the Diffusion Equation with Discontinuous Coefficients; Massachusetts Institute of Technology, 2017. 25. Cascales, J.P.; Li, X.; Roussakis, E.; Evans, C.L., “ A Patient-Ready Wearable Transcutaneous CO2 Sensor”, Biosensors 2022, 12, 333. 26. Wang, Y.; Gupta, M.; Schiraldi, D.A., “Oxygen permeability in thermoplastic polyurethanes”, J. Polym. Sci., Part B: Polym. Phys. 2012, 50, 681-693. 27. Coelho, I.; Pires, R.F.; Goncalves, S.B.; Bonifacio, V.D.; Faria, M., “Gas Permeability and Mechanical Properties of Polyurethane-Based Membranes for Blood Oxygenators”, Membranes 2022, 12, 826. 28. Klinger, M.; Tolbod, L.P.; Gothelf, K.V.; Ogilby, P.R., “Effect of polymer cross-links on oxygen diffusion in glassy PMMA films”, ACS Appl. Mater. Interfaces 2009, 1, 661-667. 29. Poulsen, L.; Ogilby, P.R., “Oxygen diffusion in glassy polymer films: effects of other gases and changes in pressure”, J. Phys. Chem. A 2000, 104, 2573-2580. 30. Hu, C.-C.; Liu, T.-C.; Lee, K.-R.; Ruaan, R.-C.; Lai, J.-Y., “Zeolite-filled PMMA composite membranes: influence of coupling agent addition on gas separation properties”, Desalination 2006, 193, 14-24. 31. Merkel, T.; Bondar, V.; Nagai, K.; Freeman, B.; Pinnau, I, “Gas sorption, diffusion, and permeation in poly (dimethylsiloxane)”, J. Polym. Sci., Part B: Polym. Phys. 2000, 38, 415-434. 32. Wang, W., “Oxygen partial pressure in outer layers of skin: simulation using three-dimensional multilayered models”, Microcirculation 2005, 12, 195-207.

[0112] The citation of any publication or reference shall not be construed as an admission that it is prior art with respect to the present invention.

[0113] Thus, the present invention provides a sensor system and method for detecting analyte concentrations, and more specifically, a transcutaneous oxygen pressure monitor that provides accurate tissue measurements without requiring long equilibration times. Here, we describe the development of a system and method to fully utilize our previously developed TCOM wearable technology, for example, by applying it to sensors based on O2 quenching of phosphorescence. However, these methods can be applied to transcutaneous gas sensing in general. We present experimental characterization and a numerical model for extracting true tissue oxygenation. The model was tested with clinical trial data. Results demonstrate that it is possible to obtain true tissue pO2 without waiting for long equilibration periods, which is currently a major limitation of the technology's applicability.

[0114] In light of the principles and example embodiments described and illustrated herein, it is recognized that example embodiments can be modified in arrangement and detail without departing from such principles. Additionally, while the foregoing discussion has focused on specific embodiments, other configurations are contemplated. In particular, even when phrases such as "in one embodiment," "in another embodiment," "in other embodiments," and "in some embodiments" are used herein, these phrases are intended to generally refer to possible embodiments and are not intended to limit the invention to specific embodiment configurations. Terms used herein may refer to the same or different embodiments that can be combined with other embodiments. In principle, any embodiment referenced herein can be freely combined with one or more other embodiments referenced herein, and any number of features of different embodiments can be combined with each other.

[0115] Although the present invention has been described in detail with reference to specific embodiments, those skilled in the art will understand that the present invention can be used in embodiments other than those described, which are presented for purposes of illustration and not limitation. Accordingly, the scope of the appended claims should not be limited to the description of the embodiments contained herein.

Claims

1. a probe sensitive to the analyte, the probe comprising a body having a first surface and an opposing second surface; a photon source configured to send photons to the probe; A sensor system comprising: the probe emits light when it receives photons from the photon source; The sensor system further comprises: a photodetector configured to detect light emitted from the probe; a controller in communication with the photon source and the photodetector; It is equipped with The controller executes a program stored in the controller, (i) causing the photon source to send photons to the probe, exciting the probe to emit light when the probe receives the photons; (ii) receiving optical data from the photodetector based on an interaction between light emitted from the probe and the photodetector; (iii) determining a difference between a first level of the parameter of the analyte at a first location adjacent to the first surface of the probe and an intermediate level of the parameter of the analyte at an intermediate location between the first location and a second location adjacent to the second surface of the probe based on the optical data from the photodetector; (iv) determining a second level of the parameter of the analyte adjacent the second surface of the probe based on a difference between the first level of the parameter of the analyte and the intermediate level of the parameter of the analyte. A sensor system characterized by being configured as follows.

2. the controller executes a program stored in the controller to determine the second level of the analyte adjacent the second surface of the probe by solving a mathematical diffusion problem involving the first level of the analyte and the intermediate level of the analyte; The sensor system of claim 1 .

3. the controller executes a program stored in the controller to determine the second level of the analyte adjacent the second surface of the probe by solving a mathematical boundary condition inverse problem involving the first level of the analyte and the intermediate level of the analyte; The sensor system of claim 1 .

4. the mathematical boundary condition inverse problem includes a thickness-dependent diffusivity variable; The sensor system of claim 3 .

5. the mathematical boundary condition inverse problem includes Dirichlet boundary conditions; The sensor system of claim 3 .

6. a probe sensitive to the analyte, the probe comprising a body having a first surface and an opposing second surface; a photon source configured to send photons to the probe; A sensor system comprising: the probe emits light when it receives photons from the photon source; The sensor system further comprises: a photodetector configured to detect light emitted from the probe; a controller in communication with the photon source and the photodetector; It is equipped with The controller executes a program stored in the controller, (i) causing the photon source to send photons to the probe, exciting the probe to emit light when the probe receives the photons; (ii) receiving optical data from the photodetector based on an interaction between light emitted from the probe and the photodetector; (iii) determining a gradient of the analyte parameter from a first location adjacent the first surface of the probe to an intermediate location between the first location and a second location adjacent the second surface of the probe based on the optical data from the photodetector; (iv) calculating a level of the parameter of the analyte adjacent the second surface of the probe based on the gradient. A sensor system characterized by being configured as follows.

7. The parameter is a partial pressure. The sensor system according to any one of claims 1 to 6.

8. The analyte includes at least one of oxygen, carbon dioxide, nitric oxide, and volatile organic compounds. The sensor system according to any one of claims 1 to 6.

9. the analyte comprises oxygen; The sensor system according to any one of claims 1 to 6.

10. when a section of the sensor system is disposed on a portion of a third surface, the section of the sensor system is configured to define a zone adjacent to the third surface; the probe is in fluid communication with the zone; The sensor system according to any one of claims 1 to 6.

11. the third surface comprises the patient's skin; the analyte comprises transcutaneous oxygen; The sensor system of claim 10.

12. the probe comprises a metalloporphyrin embedded in a layer of polymeric material; The sensor system of claim 10.

13. further comprising an oxygen-permeable scattering layer disposed between the layer of polymeric material and the third surface. The sensor system of claim 12.

14. the oxygen-permeable scattering layer increases the phosphorescent signal that reaches and is collected by the photodetector and serves as a light blocker; The sensor system of claim 13.

15. further comprising a semi-transparent layer disposed between the layer of polymeric material and the photodetector. The sensor system of claim 12.

16. the semi-permeable layer is semi-permeable to oxygen diffusion through the semi-permeable layer; The sensor system of claim 15.

17. the third surface comprises the patient's skin; the analyte comprises transcutaneous oxygen; the oxygen equilibration time in the zone when the section of the sensor system is placed on the portion of the skin is less than 30 minutes; The sensor system of claim 10.

18. the third surface comprises the patient's skin; The analyte comprises a liquid-dissolved compound in tissue; The sensor system of claim 10.

19. The liquid-dissolved compound is selected from the group consisting of drugs and metabolites.

20. The sensor system of claim 18.

20. the third surface includes a battery; The sensor system of claim 10.

21. the third surface comprises a material selected from the group consisting of a polymeric material, a carbon fiber material, a composite material, and a multi-layer material; The sensor system of claim 10.

22. 1. A method for determining a level of an analyte parameter, comprising: (a) placing a probe near a surface to monitor an analyte; (b) positioning a photon source to deliver photons to the probe to excite the probe; (c) positioning a photodetector to receive light emitted by the probe when emitted by the photon source; (d) causing the photon source to send photons to the probe, exciting the probe to emit light when the photon is received by the probe; (e) collecting optical data from the photodetector based on an interaction between light emitted from the probe and the photodetector; (f) determining a difference between a first level of the parameter of the analyte at a first location adjacent to the first surface of the probe and an intermediate level of the parameter of the analyte at an intermediate location between the first location and a second location adjacent to the second surface of the probe based on the optical data from the photodetector; (g) determining a second level of the parameter of the analyte adjacent the second surface of the probe based on a difference between the first level of the parameter of the analyte and the intermediate level of the parameter of the analyte; A method comprising:

23. step (g) comprising determining the second level of the analyte adjacent the second surface of the probe by solving a mathematical diffusion problem involving the first level of the analyte and the intermediate level of the analyte; 23. The method of claim 22.

24. step (g) comprising determining the second level of the analyte adjacent the second surface of the probe by solving a mathematical boundary condition inverse problem involving the first level of the analyte and the intermediate level of the analyte; 23. The method of claim 22.

25. the mathematical boundary condition inverse problem includes a thickness-dependent diffusivity variable; 25. The method of claim 24.

26. the mathematical boundary condition inverse problem includes Dirichlet boundary conditions; 25. The method of claim 24.

27. 1. A method for determining a level of an analyte parameter, comprising: (a) placing a probe near a surface to monitor an analyte; (b) positioning a photon source to deliver photons to the probe to excite the probe; (c) positioning a photodetector to receive light emitted by the probe when emitted by the photon source; (d) causing the photon source to send photons to the probe, exciting the probe to emit light when the photon is received by the probe; (e) collecting optical data from the photodetector based on an interaction between light emitted from the probe and the photodetector; (f) determining a gradient of the analyte parameter from a first location adjacent the first surface of the probe to an intermediate location between the first location and a second location adjacent the second surface of the probe based on the optical data from the photodetector; (g) calculating a level of the parameter of the analyte adjacent the second surface of the probe based on the gradient; A method comprising:

28. The parameter is a partial pressure.

28. The method of any one of claims 22 to 27.

29. The analyte includes at least one of oxygen, carbon dioxide, nitric oxide, and volatile organic compounds.

28. The method of any one of claims 22 to 27.

30. the analyte comprises oxygen; 28. The method of any one of claims 22 to 27.

31. and further comprising the step of: disposing a section of the sensor system on a portion of a third surface to define a zone adjacent the third surface such that the probe is in fluid communication with the zone.

28. The method of any one of claims 22 to 27.

32. the third surface comprises the patient's skin; the analyte comprises transcutaneous oxygen; 32. The method of claim 31.

33. the probe comprises a metalloporphyrin embedded in a layer of polymeric material; 32. The method of claim 31.

34. further comprising disposing an oxygen-permeable scattering layer between the layer of polymeric material and the third surface.

34. The method of claim 33.

35. further comprising disposing a semi-transparent layer between the layer of polymer material and the photodetector; the semi-permeable layer is semi-permeable to oxygen diffusion through the semi-permeable layer; 34. The method of claim 33.

36. the third surface comprises the patient's skin; the analyte comprises transcutaneous oxygen; the oxygen equilibration time in the zone when the section of the sensor system is placed on the portion of the skin is less than 30 minutes; 32. The method of claim 31.

37. The oxygen equilibration time in the zone is less than 20 minutes.

37. The method of claim 36.

38. The oxygen equilibration time in the zone is less than 10 minutes.

37. The method of claim 36.

39. At least one processor on the computer (i) causing a photon source to send photons to a probe sensitive to an analyte, exciting the probe to emit light when the probe receives the photons, the probe comprising a body having a first surface and an opposing second surface; (ii) receiving optical data from the photodetector based on an interaction between the light emitted from the probe and the photodetector; (iii) determining, based on the optical data from the photodetector, a difference between a first level of the parameter of the analyte at a first location adjacent to the first surface of the probe and an intermediate level of the parameter of the analyte at an intermediate location between the first location and a second location adjacent to the second surface of the probe; (iv) determining a second level of the parameter of the analyte adjacent the second surface of the probe based on a difference between the first level of the parameter of the analyte and the intermediate level of the parameter of the analyte.

11. A computer system comprising: a computer readable medium having stored thereon instructions for:

40. the at least one processor determines the second level of the analyte adjacent the second surface of the probe by solving a mathematical diffusion problem involving the first level of the analyte and the intermediate level of the analyte; 40. The computer system of claim 39.

41. the at least one processor determines the second level of the analyte adjacent the second surface of the probe by solving a mathematical boundary condition inverse problem involving the first level of the analyte and the intermediate level of the analyte.

40. The computer system of claim 39.

42. the mathematical boundary condition inverse problem includes a thickness-dependent diffusivity variable; 42. The computer system of claim 41.

43. the mathematical boundary condition inverse problem includes Dirichlet boundary conditions; 42. The computer system of claim 41.

44. At least one processor on the computer (i) causing a photon source to send photons to a probe sensitive to an analyte, exciting the probe to emit light when the probe receives the photons, the probe comprising a body having a first surface and an opposing second surface; (ii) receiving optical data from the photodetector based on an interaction between the light emitted from the probe and the photodetector; (iii) determining a gradient of the analyte parameter from a first location adjacent the first surface of the probe to an intermediate location between the first location and a second location adjacent the second surface of the probe based on the optical data from the photodetector; (iv) calculating a level of the parameter of the analyte adjacent the second surface of the probe based on the gradient.

11. A computer system comprising: a computer readable medium having stored thereon instructions for:

45. The parameter is a partial pressure.

45. A computer system according to any one of claims 39 to 44.

46. The analyte includes at least one of oxygen, carbon dioxide, nitric oxide, and volatile organic compounds.

45. A computer system according to any one of claims 39 to 44.

47. the analyte comprises oxygen; 45. A computer system according to any one of claims 39 to 44.

Images