Method for estimating the concentration of gas released by a medium.

Abstract

A method for estimating a content of a gas of interest in a medium, using a measuring device, intended to be placed against the medium, the device extending between a contact face, intended to be applied against the medium and a distal end, the device comprising a side wall, extending between the contact face and the distal end, the device comprising: at the contact face, at least one inlet opening, configured to collect the gas of interest emitted by the medium, the inlet opening being made through the contact face; a measuring chamber, comprising a gas sensor, the gas sensor being configured to measure a concentration of gas of interest flowing through the measuring chamber;a collection chamber (30), connected to the measuring chamber, and delimited by the side wall, the collection chamber comprising at least one lateral opening (34), arranged through the side face, so as to admit ambient air into the collection chamber;

Description

Title of the invention: Method for estimating a concentration of gas released by a medium. Technical field

[0001] The technical field of the invention is the measurement of a gas released by a medium using a compact device. The compact device is applied against the medium. The medium can be the skin of a living being. The method then makes it possible to estimate a content of a gas released by the medium. The gas can in particular be carbon dioxide, for capnometry applications. It is then a question of estimating the content of carbon dioxide dissolved in the blood. The medium can be a liquid medium, for example water, or slurry. The gas can be carbon dioxide or methane. The medium can also be a plant medium, in which case the device can be used to study the respiration of the medium. PREVIOUS ART

[0002] Some respiratory diseases affect the gas exchange between blood and exhaled air. Blood contains dissolved gases, including oxygen and carbon dioxide (CO2), whose respective partial pressures reflect the gas exchange occurring in the lungs and organs.

[0003] To assess the concentration of dissolved CO2 in the blood, a blood sample can be taken. This is an invasive method, which can be painful and difficult to apply, particularly in neonatology. In addition, it can only be applied occasionally. Despite these drawbacks, its reliability has been validated by the medical profession and it constitutes a reference method. Another method consists of estimating the CO2 content of the blood non-invasively, by measuring the partial pressure of CO2 diffusing through the tissues, and in particular the skin. This method is referred to as transcutaneous capnometry.

[0004] In the blood, carbon dioxide is dissolved according to a molecular species (CO2), and according to two ionic species: carbonate ions CO32 and bicarbonates HCO3. The balance between these species depends on the pH of the blood, because the carbonate and bicarbonate ions are in equilibrium with the hydrogen ions H+. Thus, the concentration of carbonate or bicarbonate ions influences the pH of the blood. An increase in the concentration of dissolved CO2 (hypercapnia), results in an increase in the quantity of carbonate and bicarbonate ions, and, by equilibrium, of hydrogen ions, which causes a decrease in the pH of the blood, or acidosis. An increase in the concentration of CO2 can occur when the elimination of CO2 by the respiratory tract laboratory tests are insufficient, for example in the case of chronic obstructive pulmonary disease (COPD) or infectious diseases affecting the lungs, an example being infection due to COVID 19.

[0005] Conversely, a decrease in the concentration of dissolved CO2 (hypocapnia) lowers the concentration of hydrogen ions, which leads to an increase in pH, or alkalosis. Hypocapnia can be caused, for example, by hyperventilation associated with an increase in respiratory rate. The occurrence of acidosis or alkalosis can have consequences on metabolism. Thus, the concentration of CO2 in the blood is an important vital parameter, which should be monitored regularly for certain at-risk patients.

[0006] Monitoring CO2 concentration can also be used for patients in intensive care, or for newborns placed in incubators. This can also find applications in monitoring efforts, as physical activity promotes the production of carbon dioxide.

[0007] This type of analysis based on transcutaneous measurements was introduced in the 1980s. Transcutaneous measurements allow for continuous monitoring, for example to monitor the immediate effects of therapeutic treatment influencing the concentration of CO2 in the blood. It can also be used to determine the times at which more precise quantification, by blood sampling, is preferable. It is thus understood that invasive and non-invasive methods can be complementary: one is precise and punctual, while the other can be implemented continuously for longitudinal monitoring.

[0008] A compact device for measuring transcutaneous CO2 has already been described in WO2020 / 249466. It is a non-invasive measuring device, worn by a user, to estimate a concentration of a gas of interest emitted transcutaneously, the gas of interest being able for example to be carbon dioxide. The circulation of the gas in the device allows collection of the gas of interest. The gas of interest propagates through the device by convection, under the effect of a heat source. The device comprises a contact face, intended to be applied against the user. The device also comprises a measuring chamber, comprising a sensor of the gas of interest, typically CO2. For the increase in temperature to result in convection, it is preferable for the measuring chamber to be arranged above the contact face.

[0009] The inventors propose an improvement to the device described in WO2020 / 249466, aimed at improving certain performances, in particular the time response performances. The inventors also propose a method for estimating, using the device, the concentration of an analyte in the medium facing which the device is placed. Statement of the invention

[0010] An object of the invention is a method for estimating a content of a gas of interest in a medium, using a measuring device, intended to be placed against the medium, the measuring device extending between a contact face, intended to be applied against the medium and a distal end, the measuring device comprising a side wall, extending between the contact face and the distal end, the measuring device comprising: - at the contact face, at least one intake opening, configured to collect the gas of interest emitted by the medium, the intake opening being made through the contact face; - a measuring chamber, comprising a gas sensor, the gas sensor being configured to measure a concentration of gas of interest flowing through the measuring chamber; - a collection chamber, connected to the measuring chamber, and delimited by an opening on the side wall, the collection chamber comprising at least one side opening, arranged through the side face, or on the upper wall of the collection chamber so as to admit ambient air into the collection chamber;

[0011] the device being such that: - the measuring chamber is arranged between the contact face and the collection chamber; - the device comprises a drive means, configured to drive the air from the collection chamber towards an evacuation opening, the air drive inducing a transport of the gas of interest from the contact face towards the collection chamber, through the measuring chamber: - the process comprising: - a) measurement of the concentration of gas of interest in the measuring chamber; - b) using a processing unit, modeling of gas transport of interest, between the medium and the collection chamber, the modeling including consideration of the diffusion of the gas from the medium to the collection chamber through the contact face and the device, as well as convection of the gas in the collection cell resulting from the entrainment produced by the entrainment means; - c) from the measurement resulting from step a), and taking into account the model resulting from step b), estimation of the content of the gas of interest in the medium.

[0012] According to one embodiment, - the device is spatially discretized according to a spatial mesh, defining mesh points between the contact face and the collection chamber; - the model is a discretized spatio-temporal model, so as to estimate a gas content of interest at different points of the mesh, and at different times.

[0013] The method may be such that step b) comprises: - modeling of the diffusion of the gas of interest through the measuring chamber; - modeling of the diffusion and convection of the gas of interest in the collection chamber.

[0014] Step b) includes modeling the transport of the gas of interest in the medium.

[0015] Preferably, step c) implements a recursive estimator. It may be a linear recursive estimator. The linear recursive estimator may be a Kalman filter.

[0016] According to one application, the medium is the skin of a user, the skin extending above a blood vessel. Step b) may then comprise: i. from the concentration of gas of interest in the measuring chamber, resulting from step a), estimation of a transcutaneous concentration of gas of interest; ii. from the transcutaneous concentration of gas of interest resulting from sub-step (i), estimation of a concentration or partial pressure of gas of interest dissolved in the blood of the user.

[0017] The method may include modeling the transport of the gas of interest emitted by the medium through the device.

[0018] The medium to be analyzed can be a solid or liquid medium.

[0019] The drive means may be an air propellant or an air aspirator.

[0020] The modeling can be based on a two-dimensional model, the medium, the chamber measuring and the collection chamber being discretized according to different mesh points distributed along an axis parallel to the contact face and an axis perpendicular to the contact face.

[0021] The invention will be better understood upon reading the description of the exemplary embodiments presented in the remainder of the description, in conjunction with the figures listed below. FIGURES

[0022] [Fig.1A] shows a first embodiment of a device according to the invention.

[0023] [Fig.lB] shows schematically the flows inside the device.

[0024] [Fig.2A] shows schematically the main components of a gas sensor, arranged in the measuring chamber of the device. The view is taken in a plane perpendicular to a transverse axis.

[0025] [Fig.2B] shows schematically the main components of a gas sensor, based on a absorption of infrared light, and placed in the measuring chamber of the device. The view is taken in a plane parallel to the transverse axis.

[0026] [Fig.3] shows a diagram of a breakdown of the measuring device and the medium to be analyzed into compartments.

[0027] [Fig.4] schematizes a discretization of each compartment represented on the [Fig.3].

[0028] [Fig.5] illustrates the main steps of implementing a method according to the invention.

[0029] Figures 6A, 6B and 6C show an estimation of a CO2 concentration at different sampling points when the concentration in the blood follows a stepwise variation.

[0030] [Fig.6D] shows an estimate of the CO2 concentration measured by the gas sensor of the device, as well as an application of white noise to said concentration. The application of white noise

[0031] Figures 7A, 7B, 7C, 7D show an estimation of a CO2 concentration at different sampling points when the concentration measured by the sensor follows the noisy concentration simulated in [Fig.6D].

[0032] [Fig.7E] is a comparison between the concentration in the blood in the slot, which represents reality, and the concentration in the blood estimated by the recursive estimator.

[0033] [Fig.8] schematizes a discretization of each compartment according to two dimensions.

[0034] [Fig.9] illustrates an interface that separates two adjacent compartments. PRESENTATION OF SPECIAL METHODS OF IMPLEMENTATION

[0035] Figures 1A and 1B are general views of an example of a device 1 according to the invention. The device 1 is intended to be placed in contact with a medium that is to be analyzed. In the example described, the medium is the skin S of a user, human or animal. The device comprises a main body 2 as well as a fixing element 3, the latter being, in this example, a bracelet. Alternatively, the main body can be placed in contact with the lobe of an ear, or a finger. The support can be a clamp or be integrated into a hearing aid. More generally, the support is configured to keep the main body in contact with the medium to be analyzed.

[0036] The device is intended to estimate a concentration of a gas of interest emanating from the skin of a user. By gas of interest is meant a gas for which one wishes to determine a concentration in a living human or animal body, and more particularly in the blood. In the example described below, in a non-limiting manner, the gas of interest is carbon dioxide, for which one seeks to estimate a content in the blood of the user. According to other possibilities, the gas of interest may be, in a non-limiting manner limiting, oxygen, ethyl alcohol, carbon monoxide, methane, nitric oxide, acetone or isoprene, hydrogen, certain drugs or volatile substances.

[0037] The main body 2 comprises a contact face 10, intended to be affixed to the skin S. The contact face 10 is substantially planar, in the sense that it extends parallel to a plane XY, certain portions being able to be inclined relative to the plane XY. The main body also comprises a distal end 4, opposite the contact face 10. The contact surface 10 and the distal end 4 are connected to each other by a lateral face 5, extending around a central axis A, parallel to a transverse axis Z, the latter being perpendicular to the plane XY.

[0038] The contact face 10 may be formed by a CO2-permeable membrane or a plate comprising inlet openings 12 allowing CO2 to pass through. The contact face may comprise a hydrophobic membrane, so as to prevent the diffusion of water vapor through the device 1. The contact face may be heated, which makes it possible to increase the blood flow rate. This improves collection.

[0039] According to one possibility, the device also comprises a heating element, making it possible to bring the contact face 10, delimiting the measuring chamber 20, to a temperature above 37°C, and preferably between 40°C and 50°C, and preferably between 40°C and 45°C, for example 42°C. The heating element is for example a resistor arranged on the contact face, producing heating by the Joule effect. A local and moderate increase in temperature, in the vicinity of the skin, in fact promotes an increase in blood flow by dilation of the blood capillaries, which increases the diffusion of a transcutaneous gas of interest, through the skin.

[0040] The contact face 10 opens onto a measuring chamber 20. The function of the measuring chamber is to estimate a CO2 concentration of the gas mixture circulating in the main body 2, parallel, or substantially parallel, to the central axis A. For this purpose, the measuring chamber 20 comprises a gas sensor 23. Several types of sensors can be used for this purpose, for example optical sensors or electrochemical sensors, the latter being able in particular to be based on metal oxides (MOX sensors), or photoacoustic sensors. The inventors considered that it was preferable to use an optical sensor, and more precisely an infrared sensor. Such a sensor does not require any particular maintenance, and is particularly compact, as well as inexpensive. In addition, such a sensor is very specific for characterizing chemical bonds.It is suitable for the detection of small molecules, for example carbon dioxide.

[0041] The gas sensor is an NDIR (Non Dispersive Infra Red) type sensor. This type of sensor comprises an infrared radiation source 24, generally emitting in a spectral band between 1 pm and 20 pm. It also comprises at at least one measuring photodetector 25, sensitive to infrared radiation. The principle is based on the attenuation, by the analyzed gas, of the infrared radiation, emitted by the source. The infrared source 24 and the measuring photodetector 25 form the gas sensor 23. The measuring photodetector 25 is for example a thermopile. A filter 26 placed in front of the thermopile determines the wavelength of the infrared radiation measured by the thermopile. Different configurations of the gas sensor are described below.

[0042] The gas sensor 23 is configured so that the transcutaneous gas of interest, in this case CO2, propagates between the light source and the photodetector, parallel to the central axis A, or substantially parallel to the central axis A. By substantially parallel, we mean parallel while admitting an angular tolerance of less than 30° or ± 20° relative to the parallel.

[0043] The infrared source emits light propagating perpendicular to the transverse axis Z. Preferably, the light emitted by the source is a beam in the form of a light sheet perpendicular to the transverse axis Z. Along the transverse axis Z, the thickness of the beam is preferably less than 1 cm, and preferably less than 5 mm. Preferably, the light sheet extends over at least 50%, or even at least 80% of the cross-section of the contact face 10, the latter being between a few cm2 and 25 or 30 cm2. By cross-section, we mean a surface perpendicular to the transverse axis Z (or to the central axis A). The small thickness of the beam makes it possible to reduce the response time. The high surface area of the cross-section of the beam makes it possible to increase the quantity of gas sampled and therefore the sensitivity of the measurement.

[0044] In order to promote a flow along the transverse axis Z, through the measuring chamber 20 the contact face comprises a multitude of intake openings 12, distributed according to a cross-section of a few cm2, for example between 5 and 25 cm2. Alternatively, the contact face is a porous membrane, permeable to CO2. The measuring chamber 20 opens onto a collection chamber 30. The interface between the measuring chamber 20 and the collection chamber 30 can be formed by a wide collection opening, or by a plurality of collection openings 22, arranged in a collection plate 21 and distributed according to a high surface cross-section, in a manner similar to the intake openings 12. Each collection opening 22 opens into a collection chamber 30.

[0045] The collection chamber 30 comprises lateral openings 34, arranged in the lateral face 5 of the device 1. The lateral openings are configured to allow an admission of ambient air inside the device 1. The ambient air is the air located outside the device. The flow of ambient air flowing through the lateral openings 34 is directed towards the central axis A. Each lateral opening 34 is preferably oriented perpendicular to the central axis A, so that the ambient air enters the device 1, through each lateral opening 34, in a direction perpendicular, or substantially perpendicular, to the central axis A.

[0046] The admission of ambient air is obtained by a pump 41, arranged to apply a depression in the collection chamber 30 relative to the ambient pressure. The pump 41 may be located in an exhaust duct 40, adjacent to the collection chamber 30. In addition to the admission of ambient air into the collection chamber, the pump allows the flow of the gas of interest through the measurement chamber 10 to the collection chamber 20, through the collection openings 22. Preferably, the flow rate is adjusted so that the flow of the gas of interest through the measurement chamber 20 is laminar. The air flow rate in the collection chamber may be for example between 0.1 and 10 mL / min, depending on the volume of the chamber.

[0047] In the collection chamber 30, the ambient air, admitted through the lateral openings 34, mixes with the gas of interest, admitted through the collection openings 22. The evacuation conduit 40 is also put under vacuum by the pump 41, so that the air, mixed with the gas of interest, is evacuated into the evacuation conduit, up to an evacuation opening 42, forming the distal end 4 of the device. In the example shown in FIGS. 1A and 1B, the air propagates through the evacuation opening 42 around the central axis A. According to another possibility, the air propagates through the evacuation opening 42 along an evacuation axis intersecting, and in particular perpendicular, to the central axis A.

[0048] In the example shown, the lateral openings are arranged on two opposite faces 5i, 52 of the side wall. This promotes symmetry of the air flow in the device. However, such a symmetrical arrangement of the lateral openings is not necessary. The lateral openings 34 may be provided on only one face of the side wall, the exhaust duct then being arranged on an opposite face of the side wall.

[0049] Each side opening can be associated with a CO2 filter, for example a filter comprising lime, so as to trap the CO2 present in the ambient air.

[0050] The device comprises a processing unit 50. The processing unit 50 comprises calculation means, for example a microprocessor or a microcontroller, embedded on the device, being connected by a wired or wireless link to a mobile phone or to a PC-type calculator. The processing unit 50 is also configured to implement the method for estimating the dissolved CO2 content of the user's blood, from the CO2 concentration resulting from the gas sensor 23.

[0051] The general principles governing the estimation of the dissolved CO2 content in the blood from the CO2 concentration detected in the gas sensor have been described in WO2020 / 249466.

[0052] The method comprises a calculation of the concentration of carbon dioxide in the measuring chamber from the measurements made by the two thermopiles using models describing the attenuation of infrared radiation at the two wavelengths associated with each of the thermopiles (Beer Lambert model, quadratic linear model). These models are described in WO2020 / 249466.

[0053] [Fig.lB] shows schematically the fluid currents formed inside the device 1. The transcutaneous CO2 is admitted into the device through the inlet openings 12 made in the contact face 10, and opening into the measuring chamber 20 (see arrow Fi). The CO2 propagates through the measuring chamber by diffusion, parallel to the central axis, the latter being parallel to the transverse axis Z. (see arrows F2). The ambient air is admitted through the lateral openings 34 made in the side wall (see arrows F3). The gas mixture comprising the ambient air and the transcutaneous CO2 is produced in the collection chamber 30. This mixture propagates towards the discharge opening 42 (see arrows F4) to exit the body 2 of the device (see arrows F5)

[0054] An important aspect of the device is the use of the pump 41, which makes it possible to form a convection current of air at the level of the collection chamber 30, to evacuate the transcutaneous air / gas mixture of interest. This also makes it possible to accelerate the circulation of air in the collection chamber. This results in a pumping effect accelerating the transport of CO2 from the blood to the air, through the skin and the measuring chamber. This results in a reduction in the response time of the device.

[0055] The measuring chamber 20 preferably comprises a temperature sensor as well as a humidity sensor and a pressure sensor.

[0056] The temperature sensor is used to convert the measured concentration into pressure according to the ideal gas law. Using the ideal gas law, we deduce the partial pressure of carbon dioxide in the measuring chamber at the temperature temperature T”™:

[0057] p^ = (1)

[0058] Where C'co is expressed in (mol / m3), R is the ideal gas constant (0.0623637 m3 .mmHg.K '.mol *), in K and P^co^ in mmHg.

[0059] The atmospheric pressure sensor P^^ is used to calculate the relative pressure p1^ rel expressed in ppm (parts per million):

[0060] nmes rel Pco, । rp> / "CO, - TrôF-W Z 'air \ /

[0061] Where and P™™ are expressed in the same unit, here in mmHg.

[0062] The humidity sensor is used to calculate the concentration of water vapor in the 10

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[0074] measuring chamber. An important radiation attenuation factor concerns water vapor. Taking humidity into account requires converting the relative humidity value RHmes provided by the humidity sensor into water vapor concentration C^'o-Relative humidity corresponds to the partial pressure of water in the measuring chamber, noted on the saturation vapor pressure at the tem temperature of the measuring chamber. HR mes ^ —^^-.100% 3 The H2O concentration in the measuring chamber can be obtained according to the expression: Ph2o-^RT -Ch^oRT (41 Combining the two previous equations, we obtain: rmes - M Ch i°~ 100%^ 161 v; Where VPH7O( T1^ ) represents the values of saturated water vapor pressure. As a function of temperature, these values can be obtained from tables, or from an analytical expression, for example the Tetens equation VP H2O ( T™*) = 0.61078 (6) The concentration can be used to estimate the CO2 concentration in the measuring chamber, using the Beer-Lambert expression ] = -^03(42)1^02+ [U / 2o(4i ) -kapiA)} + [Aà-Ui ) -4^(22)] (7) Where Ceo, and Ch^o are the molar concentrations, &coXA) ' ^co,(42), kn^o{^\ ), kn^oi ) the attenuation coefficients of carbon dioxide and water vapor, 2] ), AairÇ^ ) the attenuations of air at wavelengths 42. Thus, taking into account the H2O concentration makes it possible to improve the precision with which the CO2 concentration is measured, as discussed in patent application WO2020 / 249466. [Fig.2B] shows the measuring chamber in a plane Pxz parallel to the transverse axis Z. The measuring chamber may extend between two grids 21 and 21' permeable to the gas of interest. The grid 21 forms the interface wall with the collection chamber 30. It comprises collection openings, allowing the flow of the gas of interest from the measuring chamber to the collection chamber. Preferably, each grid is reflective in the emission spectral band of the infrared source. The measuring chamber may comprise a reflective grid 21', similar to the grid 21, at the contact face 10. The presence of each reflective grid makes it possible to increase the quantity of light propagating through the gas of interest, towards each channel of the photodetector. It also makes it possible to limit the thickness of the light sheet, along the transverse axis.

[0075] The distance d between the grid 21 and the contact face, or between the grids 21 and 21', is for example between 3 mm and 8 mm.

[0076] The measurements resulting from the sensor may be noisy, in which case low-pass filtering may be applied. Low-pass filtering may be carried out, for example, by a sliding average filter whose length is adapted to the level of the measurement noise.

[0077] From C^o2' 'a Partial pressure of CO2 ?i) is deduced by: j-jnes / ^my ladies

[0078] From the atmospheric pressure sensor of the measuring chamber, the relative pressure is calculated by: 100791 = (9) *tjir

[0080] Where is the atmospheric pressure in the measuring chamber.

[0081] From the CO2 concentration resulting from the measuring sensor, the carbon dioxide concentration in the blood can be estimated by applying a model of CO2 transport between the blood and the measuring chamber. A one-way transport model is described below, modeling CO2 transport by diffusion and by convection. First model: 1D one-dimensional model

[0082] Figure 3 shows a diagram of an example of a device placed on the user's skin S. The blood occupies a blood compartment B, between the coordinates zm and zmü blood, and its thickness is A 7Cmg. The blood compartment B is covered by skin S. The skin S extends between the coordinates zln Skin and zouï and its thickness is \ ?skin Ztn skin — ^out blood _ The skin cs( covered by the device. In the device of the prior art, the device comprises a superposition of the collection chamber and the measurement chamber. The collection chamber extends between the coordinates zm co) and and its thickness is Az^. The measurement chamber extends between the coordinates zm mei and and its thickness is A Z1**. zP col — z°'a• zm mes — zpUt cal_ $c|on the invention, the device comprises a superposition of the measuring chamber 20, of the collection chamber 30.

[0083] Diffusion is characterized by a diffusion coefficient, which depends on the medium crossed, in this case blood, skin and air.

[0084] In the 1D modeling, the transport of carbon dioxide is described only along the central axis A corresponding to the Z axis. The blood flows, in the blood compartment, according to a convection current of average speed us"ng. In the skin, the CO2 migrates only by diffusion. In the device, in the measuring cell the CO2 migrates only by diffusion and in the collection cell CO2 migrates by diffusion and convection. Convection results from the air circulation induced by the pump in the collection chamber. Air convection in the collection chamber results in diffusion through the contact face and through the measuring chamber. It is characterized by an average speed W01 along the Z axis.

[0085] Based on a propagation of the gas of interest in a Z direction perpendicular to the surface of the skin (one-dimensional model), we can write in any homogeneous medium (_i-e of the same Henry constant): [00861 = (w)

[0087] where: - is the propagation speed of CO2 at a coordinate along the Z axis; - C(t, z) is the concentration of CO2 at time t and at coordinate z; - DL) is the diffusion coefficient of CO2 according to the direction of propagation Z, at the z coordinate. - R(z, t) corresponds to the exit of CO2 in the blood at the skin level (sink term) as well as to the arrival of CO2 through the side walls in the collection chamber (source term). R(z, t) is therefore non-zero only at the blood / skin interface and the measuring chamber / collection chamber interface inside the device.

[0088] Considering a homogeneous medium: J. / VA = Q À JL) = Q

[0089] Equation (10) becomes

[0090] dazj) ri dc(zJ) , LA

[0091] In expression (11), the term corresponds to the dynamics due to the dz convection. The term corresponds to the dynamics due to diffusion. Dà.; Expression (11) corresponds to a convection-diffusion equation.

[0092] Expression (11) models the transport of the gas of interest along the Z axis inside each compartment previously described (measuring chamber, collection chamber).

[0093] The model also takes into account an observation equation:

[0094] C(zme,? t) = y (12) where 2' corresponds to the measured concentration Ccq in the measuring cell and Lwev is the spatial coordinate along the Z axis of the point where the measurement is taken.

[0095] The boundary conditions are: t 0096 '(12)

[0097] where zbiood o corresponds to the coordinate, along Z, of the entrance to the blood compartment. And

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[0113] (13) where corresponds to the coordinate, along Z, of the distal end of the device. The conditions at the interfaces are now described. In the following equations, the indices i and j designate two successive environments respectively upstream and downstream of the interface, the upstream and downstream being taken into account in the direction of net propagation of CO2, i.e. oriented from the blood compartment towards the collection chamber. Considering that zt\j represents the spatial coordinate along the Z axis of the interface between medium i and medium j, j < j is the gas flux leaving medium i and j is the gas flux entering medium j, per unit area. For a gas transparent interface we have: ¢ / , f) = Jj(z (15) with : 0 r) (16) And In expressions (16) and (17), u.(z ^(z^ t) and Uj(zi\j)C(zi}j, t) cor respond to convection, while £).^ .jvC(z ] ' 0 and 0 matches to the broadcast. It is considered that at the interfaces between blood and skin, or between skin and measuring chamber, or between the measuring chamber and the collection chamber, the speed is zero on either side of the interface: uj _ Uj^z — q . Indeed, the Convection only occurs in the collection chamber. In the skin and in the measuring chamber, CO2 propagates only by diffusion. By combining (16) and (17), due to the continuity of the flow, and by introducing a transparency factor, we then obtain: ùCdï,., / ) ÙC / j,. / ) / \ rn • 'I'! — T) |J ■ I 1RI f transparent i\ j g- I 1 Where rtranspa i\j corresponds to the transparency factor between compartment i and compartment i]. The transparency factor corresponds to a ratio between the open surface of the interface, allowing the gas of interest to pass through, on the total surface of the interface. On both sides of the interface, the pressure is identical: (19)

[0114] Gold 101151 (20)

[0116] Where T ; is the temperature of medium i, and T i — Tj

[0117] C^r) H s- H, L dA 4 'i\J" 1 / L /

[0118] The symbol — means “equality that serves as a definition.”

[0119] Taking into account the flow of blood in the blood compartment

[0120] In blood compartment B: ,

[0121] ^zblood 3 = _ (22) - A x is the width of the input window - Cmt is the concentration of CO2 leaving the blood compartment under the effect of blood flow; - is the flow rate of blood in a direction perpendicular to the Z axis. R(zbl°od 3 corresponds to a well term.

[0122] uMng =--S-- 23

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[0126] g'""" is the blood flow rate and '^ is the exit area on the left and right of the blood compartment. This is a cross-section of the blood vessels. Taking into account the admission of air into the collection chamber. In the collection chamber, the air intake results in a source term corresponding to the carbon dioxide present in the ambient air (24) - A x corresponds to the width of the device - liÿ'1 is the air inlet current _ çm coi au- csl |the concentration of air entering the collection chamber; - R(z,co1, t) corresponds to a source term. 0”' / \ 25

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[0130] is the air flow rate and co! is the inlet area along the side walls of the collection cell. Discretization of the model. The device and the medium on which it is applied have been discretized The convection-diffusion equation (11) can be spatially discretized as

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[0144] shown in [Fig.4]. A spatial mesh was created, generating 17 sampling points equally distributed in each compartment, between the blood compartment B and the collection chamber. Each compartment has 5 sampling points, 2 of which are placed at a lower or upper interface. At an interface between a downstream compartment and an upstream compartment, the concentration leaving the downstream compartment is referenced by an “out” superscript while the concentration entering the upstream compartment is referenced by an “in” superscript. The concentration inside the blood compartment is discretized according to 3 sampling points: fblood,! ■ fblood,2 ■ fblood3 {'COI ' L CO2 ' ^CO2 The concentration inside the skin compartment is discretized according to 3 sampling points: ^skin,ï • fskin.2 • fskinS {'CO2 ' ^002 ' CCO2 The concentration inside the measuring chamber is discretized according to 3 sampling points: fno.' • f mes?. ■ f mes,3 ^CO2 ' ^002 ' {'COI The concentration inside and at the outlet of the collection chamber is discretized according to 4 sampling points: col,} ■ f col2 ■ f col3 ■ foui col LCO2 ' ^002 ' ^CO2 ' {'COI At the interfaces between compartments, the concentrations are We define CO2 concentrations at the interfaces: At the blood / skin interface: _ c'^n; jjblood cl j^skm correspond to the Henry constants of blood and skin. At the skin / measuring cell interface __ H("' corresponds to the Henry constant of air. At the measuring cell / collection cell interface fuck my_ end col {'COI “ ^002 The input concentration is In the direct problem formulation, end blood CS( known. When solving the inverse problem, this is who is ^002 4 1 ^002 4 research. The spatial distance 8z between each point of the mesh of the same compartment is identical. Subsequently, the distance between each point of the mesh is noted ôz- , the index i designating the medium: i = 1 for blood, i = 2 for skin, i = 3 for the chamber of

[0145]

[0146]

[0147]

[0148]

[0149]

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[0155] measurement and i = 4 for the collection chamber. Given the spatial discretization carried out, the convection diffusion equation (expression 11) becomes: (30) JL (5¾)- JL 2(5¾)- jv ' 7 In this example, r — t + ôt; we use an implicit integration scheme. We define a vector z, of dimension (17, 1), which includes all the mesh points. z = (3D 7 0=1 The total number of samples is N, such that ( 32 ) where Ni is the number of samples within compartment i. » T 1 Where A Zi is the height of the midpoint i and the distance between two successive points in the midpoint i is noted ôz^. The position following z in the column, at which each midpoint begins, is defined by ^b i. This position can be calculated in general by the following relations:

[0156] z0, for i = 1 (32') Zdeb i— , V'-1 to ■ o Zq + Lj=x^Zf for 1^2 Direct model

[0157] A vector c is defined with respect to the concentrations evaluated in different sampling points defined by the vector z.

[0158] is a vector of dimension (17, 1). It is obtained by concatenation of the terms of the vectors ¢.5( / ) with i between 1 and 4.

[0159] (34) c0 = £ (Zdeb 2 3" k' ' ' ' 2 C^deb?> + k-Ô^\ ...,A3 + <%)' k= (N4+ 1).

[0160]

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[0175] ii C(Zdeb 1 + ^1'^1) C(Zdeh2 + 1' ^¾) t(Zdeb 2 + ^2-^2) c(w^) ^( Zdeb3 + ^^ C(Zdeb4+ ^4) C(^È4+(^4+1)-¾) In this example, we assume that CO2 is measured at the outlet of the measuring chamber: y = C ( z deb 3 + N3 ôz3 ) Netat groups the points at the upper edge and strictly inside the middles, ignoring the 3 points which correspond to the blood / skin, skin / mes, mes / col interfaces. Netat = N-3 In our case, we have: 7V1 = N2 = N3 — N4 = 3 so N = 16 and Netat =13 We define a vector c with respect to the concentrations evaluated at the different state points. We define an augmented state vector C of dimension (Netat + 1) by adding to the vector c the concentration Cm blood at the entrance to the blood medium: Expression (30) can be written in matrix form: c(t + dt) -c(t) - ôt ■ F ■ c(r) (36) c(t) etc(t + ôt:) are vectors of dimension (13, 1). Each term of these vectors corresponds to a concentration at a measurement point represented in figure 4, respectively at time f and at time t + ôt F is a passage matrix, of dimension (13,13), such that F — Fconv+Fdif + Fint + Fpun (37) Or: Fcom groups together the discretizations linked to convection: Fcom, of dimension [Netab Netat] (Netat =13 in this example) is calculated in the following way on the terms of rank (%, q + 1) ( 38 )

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[0188] And on the terms of rank (^, q -1): f — JL. (39) J 2ôz{ Apart from the points of rank (q,q+l) and (q, q-1), the terms of Fconv have a zero value. F(nf groups the discretizations linked to diffusion; Fint groups together discretizations related to interfaces and edges; and F contains the term well (i.e. the exit of blood at the skin level into the bloodstream). uî corresponds to the propagation speed in the z direction for compartment i and ÔZj is the distance between two adjacent sampling points of the same compartment i F(jif of dimension [Netat, Netat] (Netat =13 in this example) is calculated in the following way by having: • on the main diagonal (row and column of the same rank ¢): above the main diagonal <M+1 ( / below the main diagonal Apart from the points of rank (q,q+l) and (q, q-1), (q,q), the terms of F^j have a zero value. The Finh matrix describing the conditions at the interfaces between two adjacent compartments, of dimension [Netat, Nefat] (Netat =13 in this example), is calculated as follows: All terms in the Fint matrix are null, except: - The terms [1,1] and [1,2], which correspond to the lower edge, to the coordinate ^1 = ^+1 The terms [(zdebi + Ni ' 5zi, zdebi + Ni ' Szi),(zdebi + N] • ôzb zdeb2 + ôz2)] and [(zdeb2 + $Z2, Zdebl + ' §Zi),( Zdefc>2 + ÔZ2, Zdeb2 + 5z2)] which correspond respond to the first interface; The terms [(zdeb2 + N2 ■ ôz2, zdeb2 + N2 • 02^),( zdeb2 + N2 • ôz2, zdeb3 + ôz3 )] and [(zdeb3 + ôz3, zdeb2 + N2 ■ ôz2),( zdeb3+ ôz3, zdeb3 + ôz3)] which correspond to the second interface; The terms [(zdeb3 + N3 • ôz3, zdeb3 + N3 • ôz3), (zdeb3 + N3 • ôz3, zdeb4 + ôz4 )] and [( zdeb4+ ôz4, zdeb3 + N3 ■ ôz3), (zdeb4 + ôz4, zdeb4+ ôz4)] which correspond respond to the third interface; The terms (zdeM + ( N4 + 1 ) • 5¾ z.deM + AQ • ôz4), ( zdeM + ( N4 + 1 ) ■ ôz4, zdeM + ( N4 + 1 ) • 5z4) are equal to correspond to the upper edge.

[0189] The terms of the matrix different from those previously explained are equal to 0.

[0190] Each non-zero term fqq of the matrix Fint is such that:

[0191] . iA t — —-—r 43 2{ôz^ \ /

[0192] with [' / - <?] = [z^dNdôzv zdM + Nl■fej, [zdeb2 + N2■ 5z.2, zdeb2 + N2'8z2], [^3+^3-0¾ +

[0193] Each non-zero term f of the matrix Fint is such that:

[0194] f .^...^(44^ ' q^+2 “ 'Uôz.j r4;

[0195] with [4 7 + 2] = [z^ + iV] -ôz^ zdeb2 + &2], [ zdeh2 + N2 • 8z2, ^43+^23], zdeb4 + ôz4]

[0196]

[0197]

[0198]

[0199] And " H - 'iranspa i[j “

[0200] 77j and Hj are the respective Henry coefficients of the media i and j and r troupa i\j is the transparency factor between media i and j

[0201] The matrix Fpun, describing the conditions at the interfaces between two adjacent compartments, of dimension [A^ / af, Af^J (Netat = 13 in this example), is calculated as follows. The only non-zero term is the term located at position (3, 2) corresponding to the interface between the blood and the skin

[0202]

[0203]

[0204] The term corresponds to the convection exit from the blood compartment The air inlet into the collection cell is taken into account in the matrix of command described below. 20 Application of a recursive estimator.

[0205] In this example, a recursive estimator is implemented to estimate the CO2 concentration in the blood, as a function of time. This is a linear recursive estimator, of the Kalman filter type.

[0206] The estimator is based on the spatio-temporal state space model previously described, to which an observation equation is added, which translates the relationship between each measurement and the states.

[0207] In a manner known with this type of estimator, the concentration, at an instant, is calculated recursively, from an estimate of said concentration, resulting from a previous iteration, and from a measurement carried out at instant t.

[0208] Starting from equation (36):

[0209] C(z,t+ôt)-C(z,t)=dt-FC(z,t+ôt);

[0210] We can write:

[0211] / (I-Ôt-F)C(zJ + ôt)=C(z,t) 37 S

[0212] Subsequently, the following notations are used: - Q>1: state vector at time k predicted at time k - 1 previous: dimension (^eta?+l) i - ^kji: state vector at time k calculated at time k; dimension (Ne[at+1 ); - control vector, of dimension (Q, 1). In this example, Q = 1. - Q+u: state vector at time k + 1 predicted at time k; dimension (Netat +D; - Pk,ki: covariance matrix of the error at time k predicted at the previous time k - 1: dimension (N elal+1, Netat+1); - ; the covariance matrix of the error at time k calculated at time k; dimension (A^îtrf+1, A^tof+1); - Pk+\.k: covariance matrix of the error at time k + 1 predicted at time k; dimension (Netat+1, A / ^a?+l); - Kk; Kalman gain matrix calculated at time k; dimension (A^wire, M) - Vk: measurement observed at time k; dimension (M). In this example, the measurement is a scalar: M = 1. -H; observation matrix: dimension (A^etof+1,M); -G; order matrix: dimension (Netat+1, Q); - Wk: modeling noise: dimension (A^tf+l); - ; observation noise: dimension (M); It is assumed that the noises observation and modeling are white and independent; Rk: covariance matrix of the observation noise at time k. Dimension

[0213]

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[0233] (M, M); Qk: covariance matrix of the system noise at time k. Dimension ( Netat+1, state+1 ); F: transition matrix increased by dimensions (Netat + 1, Netat + 1). W[ k ] ~N(0, (5^ is the noise that integrates the errors related to the model and v[&]~N(0, cr?) describes the noise on the values provided by the sensor. Their covariance matrices are defined: Q,. k * n 0, k = n E[wT] = Rk, k # n 0, k = n (38) N(0, cr^) means a normal distribution with zero mean and variance cr&. means is distributed according to E denotes the expectation operator w[£] and v[ Æ ] are assumed to be mutually independent. E[^]=0 (39) The method includes an initialization phase, during which an initial state vector and a covariance matrix of the error at the initial time Pk=w are used. The correction step is defined by the following equations: The Kalman gain is estimated by Kk = PkkAH [HP^H +Rk} (40] The Kalman gain reflects the weights (i.e. the confidence) given to the noisy measurement and its estimate Hckk.^ Estimate of the concentration ^kk at time k; Qj - + Kk(yk~HxkJc-l) (41) Estimation of the covariance matrix of the estimation error Pkk: Pu = ( / - KiHiPtk <I-Kl,H) + <42> The estimation of Ckjc allows us to know Cin biood, the latter being the rank 1 term of the vector ^kj<. From Cin biood, we can estimate The prediction step is defined by the following equations: Prediction of concentration at time k+1:

[0234]

[0235]

[0236]

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[0238]

[0239]

[0240]

[0241]

[0242]

[0243] Prediction of the error covariance matrix Pk+ijc P^X'PuX'^Q^ The variable that we are trying to estimate, Cin bjood, is located in the state vector Q at position (rank 1 term). It is linked to the following state variable, ChloodA, by the terms of the transport equation. The state equation associated with it is: C t „ [ k + 1 ] = <fCm k ] + U’ [ k + 1 ] (46) Where V is a parameter characterizing our prior on the signal Cin blgod . If our prior is that the input signal is stationary, we will choose (p = 1 Expression (37) translates to: ^-+1 = ^ + ^5 + ^+1 (38) Or Q is the augmented state vector defined in (41). ck = state vector increased includes the concentration in each compartment as well as the concentration in the blood. We define for an implicit time integration scheme  = IF-ôt (39) Where the augmented transition matrix F is F[(N+l)^N+l)]is the matrix defined by: [J7]

[0244] G-

[0245]

[0246]

[0247]

[0248]

[0249] Where g — 2^ for the term defined by the height along the z axis And w is the vector containing the noise on the states. It has the dimension {Netat + l)xl (yyin blood \ W / For an implicit time integration scheme, the discrete equation of state

[0250]

[0251]

[0252]

[0253]

[0254]

[0255]

[0256]

[0257]

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[0259]

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[0264]

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[0269]

[0270] is written: Âc[U 1] = + GCincolair[k + 1] +h{£+ 1] wÎW hiood suj( a |oj normal N(0, cr2) Where the matrix A of dimension ( Netat + 1 ) X ( Netat 4-1 ) is defined by: To = IF-ôl Where I is the identity matrix of dimension (Nerat + 1) x (Netat + 1) The observation equation for the augmented system is written: yk=^ck+Vk (40) Or: • is the noisy observation from the real concentration value C^. asS • v[£]~N(0, 0¾ describes the observation noise on the values provided by the sensor • h is the augmented observation vector defined by: In our case, since the observation is defined by a single value, the augmented observation matrix H reduces to an observation vector h. Or: H^h The system of equations is At èbl = ^ + GC1^ M + %+1 The Kalman algorithm recursively chains a step of correction of the state vector at an instant k (or 4) and prediction of the state vector From Cin hiood, the CO2 content of the blood can be Blond pin in blood (47) COy — _ p»ang es(; Ostwald solubility coefficient of carbon dioxide in blood. [Fig.5] shows schematically the main steps of implementing a method according to the invention. Step 100: Place the measuring device against the analyzed medium and activate the pump.

[0271] Step 110: Initialization: During this step, an initial state vector Q-1,0 and an error covariance matrix at the initial time Pk=to are defined. The measurement time k is then incremented: k = 2.

[0272] Step 120: measurement of the CO2 concentration in the measuring chamber. This gives the measured quantity Jj,

[0273] Step 130: From the initial state vector ck=W, or from the state vector estimated during a previous iteration Q>l, and from the covariance matrix of the error at the initial instant ^=10, or resulting from a previous iteration Pkjc-i, determination of the Kalman gain at instant k (expression 41) and estimation of the state vector (expression 42).

[0274] The estimation of allows us to know Citl bi„mb the latter being the term of rank Aa*i of the vector: this corresponds to step 135. From Cin biood, we can estimate pinhlood (expression 47): step 136.

[0275] Step 140: Estimation of the covariance matrix Pkx of the estimation error (expression 43).

[0276] Step 150: Prediction of the state vector at time k+1.(expression 44) and prediction of the error covariance matrix at the following time Pk+\x

[0277] Step 160: Incrementing the measurement time

[0278] Step 170: repeat steps 110 to 160 or exit the algorithm

[0279] The iterative steps 110 to 170 are repeated until the algorithm exits. The output of the algorithm can be decided by the user, or correspond to a predetermined number of iterations.

[0280] The method previously described was implemented taking into account the following parameters: - length (along the X axis) blood compartment: A x™8 = A ,v =5 cm; - width (along the Y axis) blood compartment: A y^8 = A y = 2 cm; - blood compartment height (along the Z axis): A = 0.3 cm; - surface area of the blood compartment in contact with the skin: — 10 cm2; - volume of the blood compartment: ysa,ig = 3 cm3; - Henry's coefficient of carbon dioxide in the blood: uiang — 0.54 - Diffusion coefficient of carbon dioxide in the blood (water coefficient): D™8 = 2.2 105 cm2.s 1 - Initial CO2 concentration in the blood: = 1.0990 pmokcm3 - Blood flow: Qsang — 1.83 103 ml / s (cm3s ') which corresponds to the following speed: • Transverse speed (along x): = 0.0031 cm.s 1 • Axial velocity (along z): U^8 = 1.83 10 4 cm.s 1 skin compartment length (along the X axis): A = A x = 5 cm; skin compartment width (along the Y axis): A yf™" — A y = 2 cm; skin compartment height (along the Z axis): A £Skin = 16 10 4 cm; skin surface in contact with the device: ^eaa — 10 cm2; Henry's coefficient of carbon dioxide in the skin: — 1.6 Diffusion coefficient of carbon dioxide in the skin: jji,water _ 1 107 cm2 .s1 measuring chamber temperature: Tmes — 315.15 K; length of the measuring chamber (along the X axis): A xnes = A x = 5 cm; width of the measuring chamber (along the Y axis): A = A y = 2 cm; height of the measuring chamber (along the Z axis): Az!nes= 0.25 cm; Henry coefficient of carbon dioxide in the measuring chamber: Hmes = H air 1 Diffusion coefficient of carbon dioxide in the measuring chamber: Dmes = Dalr = 0.18 cm2, s 1 CO2 concentration in the incoming air, assuming there is a filter; Cair = 0 Convective air flow in the collection chamber: Qar — 1.67 102 ml / s • Transverse speed (along the x axis): — 33,103 cm.s 1 • Axial speed (along the z axis): = 1.67.103 cm.s 1 surface area of the contact face between the measuring chamber and the collection chamber: Ames = Ac°l =10 cm2; collection chamber temperature: Tco1 = 315.15 K; length of the collection chamber (along the X axis): A x'^1 = A x = 5 cm; width of the collection chamber (along the Y axis): A yco! = A y = 2 cm; height of the collection chamber (along the Z axis): A zco! — 0.25 cm; Henry's coefficient of carbon dioxide in the collection chamber: ^col_ _ | Diffusion coefficient of carbon dioxide in the collection chamber: = [y™' = 0.18 cm2, s 1 Transparency factor between skin and the measuring cell: skin|mes= 1; Transparency factor between the measuring cell and the collection cell: r meslcol 1 Time sampling step: 5t = Measurement noise variance: 106 mol / m3 Model noise variance: 108 mol / m3 Coefficient of the autoregressive model: (p = 0

[0281] A direct model was first implemented, making it possible to estimate the concentration at each sampling point from a known concentration in the blood. The known concentration in the blood follows a niche. Figures 6A, 6B, 6C describe the evolution of the concentration as a function of time respectively in the blood compartment, in the skin, and in the collection chamber. [Fig.6D] shows the evolution of the concentration at the gas sensor, in the measurement chamber (curve a). Curve b of [Fig.6D] shows the concentration at the gas sensor of curve a, to which Gaussian noise has been added to simulate noisy measurements. In each of Figures 6A to 6D, the x-axis corresponds to time and the y-axis corresponds to the estimated CO2 concentration.

[0282] From the simulated noisy measurements (curve b of [Fig.6D]), the recursive estimator previously described was implemented. The state vector estimated at each measurement time makes it possible to obtain an estimate of the concentration at different sampling points. Figures 7A, 7B, 7C, 7D show the evolution of the CO2 concentration respectively at the level of the collection chamber, the measurement chamber, the skin compartment and the blood. [Fig.7E] shows the actual blood concentration (curve a) and the blood concentration resulting from the estimate.

[0283] In each of Figures 7A to 7E, the x-axis corresponds to time and the y-axis corresponds to the estimated CO2 concentration. Second model: 2D two-dimensional model

[0284] We describe in the following a model according to which the evacuation in the collection cell is done transversely (perpendicular to the z axis), as shown in [Fig.8]. The chosen geometry fixes the calculation of the velocity vector field for the propagation of the carrier air in the collection cell, the convection diffusion equations define the propagation of carbon dioxide. The air entry is done through one or both side faces.

[0285] Blood propagation is modeled by circulation along an X axis, perpendicular to the Z axis.

[0286] Each of the compartments defined previously is also discretized along the x axis, in the same way as along z, namely with 3 concentration values internal to the compartments and two concentration values on the edges.

[0287] Figure 8 represents the division into compartments. In this example, the position index in X varies from 0 to 4 as indicated at the bottom of the figure. The position index along the z axis varies from 0 to 16 with the edges and from 1 to 15 for the internal points, as indicated on the right of the figure. The index notation kx or kz is used depending on the axis studied.

[0288] {0,1,2,3,4)

[0289]

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[0310] k z e {0.1, 2.3, 4.5, 6.7, 8.9, 10,H, 12,13,14,15,16} For example, for the “Skin” compartment, along the X axis the kx index varies from 0 to 4 for the points including the edges and from 1 to 3 for the internal points; along the Z axis the Damage index from 4 to 8, with the interfaces and from 5 to 7 for the internal points. In a homogeneous medium, equation (11) becomes: dC(x.Zj} r.(<XC(x> Z,t) . T rrrzMLTrn kx .......--5----- - -u x .......5-- + D—+2D-AUT0NUM at A dv ( dz2 dx2 ) The boundary conditions are: At the lower edge - level U equation (13) becomes =0; e {0.1, 2.34} (2D- AUTONUM ) At the upper edge - level A» / ~ equation (13) becomes = Qç {0,1,2,3,4} (2D-AUTONUM ) Along the left side, at coordinate U • • At the left lateral edge of the blood cell =Q. EU {0} (2D- AUTONUM ) C( x^ Z k , 0 = C in k z e {1,2,3} (2D- AUTONUM ) • Along the left side of the skin compartment and the measuring chamber: = q ^e{4, 12} (2D-AUTONUM ) • At the left side edge of the collection cell = Q; k z e {16} (2D- AUTONUM ) C(x i>Zk ,t) = {13,14,15} (2D-AUTONUM) • Along the right side, according to the U coordinate; {0,1,16} (2D- AUTONUM ) Conditions at interfaces Notation: the index i will be used as a reference to the medium before the interface, the index j for the medium after the interface. The coordinate at the interface is indicated by the index i\J- Continuity of flow, equation (18) becomes r,^, imD. V = O; 5 / ' (2D- AUTONUM ) Continuity of pressure equation (21) becomes = AJU jg C zl Ix, ;iU (2D-AUTONUM) 17 ilj * ], / f ' '

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[0327] Expression of air speed. The notations used are: - Calculate the volume mass of the air - the molar mass of air in the standard proportions of its different components - Cair the molar volume concentration of air in the standard proportions of its different components - Pair the air pressure - increase air speed - y air the acceleration of the air - Tazi the air temperature - F j mass force field translating the effect of gravity on the gas. We have the relationship: Pair air^air According to the ideal gas law we have: n pp x air M air The momentum conservation equation leads to the Navier Stokes equation. Since the velocity is low, we can assume that the gas is incompressible and that the density remains constant. We also neglect the effect of gravity on the gas and the viscosity of air. The simplified equation becomes: Or: Where grad(uair) is the tensor: = ] The mass conservation equation is written (see equation 14 in ): Either in steady state: div(pa.uair^ If we assume that Pair is constant in the chamber of the device, we obtain as a simplified equation: div(uair) =0 Knowing the air flow rate at the inlet of the collection cell sets the value of uair at the inlet of the device. Solving this equation allows the velocity field inside the device to be calculated. In the case of propagation in a single direction, for example X, the equation becomes :

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[0349] 2“^ _ „ dx “U The air velocity field is constant throughout the device and equal to the velocity vector at the device input. We denote (X], x?) the spatial coordinates (x, z) and li^r) 'cs coordinates of the air velocity vector u«ir = ( u- : Or again: E2 3¾¾ _ t=} dx, “ And : V-12 duiMir j 7=1.2 In the steady state, assuming negligible effect of gravity on the gas ( / ^ = 0), this equation becomes: 2 of^, _ 1 dP^, <^Fair - even d Navier Stokes equations for a perfect viscous incompressible fluid in 2D: We denote (x1? X-,) the spatial coordinates (x, z) and u^r) the coordinates of the velocity vector uZjair): dîv(uajr) =0 Or again: L2 _ dx,- ~ And : — +L^d — j- 1.2 Where v is the kinetic viscosity of the fluid (unit: m2^1). In the steady state, assuming negligible the effect of gravity on the gas ( Fj — 0 ), this equation becomes: y2 _ i 3rd part y2 dx: ^1=1 dx? 7=1.2 In the case of the 2D model with transverse evacuation, we propose to use a simplified model of the velocity field assuming that it is uniform throughout the chamber, and parallel to the x axis. The air inlet is from one side only and equation (25) becomes: (2D-AUT0NUM )

[0350] Where Qair is the air flow rate and A™ coi is the inlet area along the side wall of the collection cell 5, A^ut co! is the outlet area along the side wall of the collection cell 5. We further assume that A"ut cot — A™

[0351] We assume that the axial velocity is zero.

[0352] z^ = 0 (2D-AUTONUM ) Expression of blood speed.

[0353] a™ sans is the left entry surface into the blood compartment, a™1 blood is the right exit surface from the blood compartment. This is to model a section of the blood vessels. We further assume that = A“ scmg- We assume that the entry is made through all the sub-compartments of the blood compartment, that is to say for U e {1,2, 3}. [03 5 4] = (2D-AUTONUM)

[0355] We assume that the axial velocity is zero:

[0356] = 0 (2D-AUTONUM )

[0357] The convection velocity in the measuring cell is assumed to be zero:

[0358] = 0 (2D- AUTONUM )

[0359] ^ = 0 (2D-AUTONUM )

[0360] The convection velocity in the skin is assumed to be zero:

[0361] 11^ = 0 (2D-AUTONUM )

[0362] u?water = 0 (2D-AUTONUM)

[0363] In summary, over the entire 2D model, the discretized convection velocity field on the sampling points are written:

[0364] ' 0 ; kz = 0 ; U{1,2,3) X. y “

[0365] 0 ; kz e {4, ..., 12} (2D- AUTONUM ) ; UE {13,14,15} , 0 ; U=16 0 ; ^ = 0 0 ; U e {1,2, 3 ) uz(x, zk„ t) = 0 ; k.^{4, ...,12) (2D-AUTONUM ) 0 ; kz(= {13,14,15) ,0 ; U=16 Discretization of the 2D model

[0366] For the discretization of the continuous 2D model, the approach remains the same as for the 1D model previously described. From the continuous equations we establish the discrete equations using the centered finite difference method. We formulate the problem in a vector / matrix form in order to use the Kalman filter for the estimation of the hidden states of the system, and for the estimation of the variable of interest sant>.

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[0379] The spatial mesh is performed along both X and Z axes as shown in [Fig.8]. As in 1D, we define the distance ÔZj between two successive points for each medium / according to z: Where is the total number of sample points along the z axis for medium i (N^ — 5 in our case). The position following z from the start of the middle i in the column is denoted '-deb i. It is defined by the expression z0 for i = 1 ^' debl Zo + (Ç=iAzy) for you {2,3,4} Where ^-o is the Z-axis dimension of the lower edge of the blood compartment and A Zj is the height of the middle j. We proceed in the same way along the X axis: We know the assumed equal width of each middle: A x{ = A x We choose the number of discretization points assumed to be the same for each medium along the X axis, including 2 internal points and two points on the edges, i.e. Nx points in total (Nx =5 in our case). The distance 5x between two successive points for each medium i along x is: — Nx-1 The concentrations located in the vector co are associated with the spatial sampling points corresponding to the different heights, then to the different abscissas, and finally to the different environments. The components of the vector co are thus defined in the following manner, by first varying the index kx corresponding to a scan along the X axis by adding the two edges, then by varying the index kz corresponding to a scan along the Z axis by adding the edges and the interfaces, then by varying the index i of the middle:

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[0386] >C(x Q +k»ôx, z^ + kx.ôz^ k x =Q, …,N X -U k z = 0, ...,(^-1)-C^ + k^z^^ k x = 0, …,^−1; k z = 1, …, (<2-l) k x = 0, …,N X -V, k z =L …, (2V-3-1) ‘C(x n +k x .ôx,zk x = 0, …, N x -1; kr= t (Na−1) > C ( An + 0.5xz + 0− 5z . I \ 0 deb 1 1 / c(x0+CV −lUxz + 0.&J \ u A ' ' deb\ V c(x o +o.&- x \ should 1 ! * / C(x„+GV r -1)Sx. z M t +(N a c{x a +0.Sx.z M2 +l.Sz2) ^0 = dx^+IN -Úôx,z +(N^-ï).5z9} C^ + û&.^+L&J C^+W,-1). Sx, z M} + (N, r 1). Please,) C(xo +O. Sx, ^,4+1.¾) C ( x,,+(N Y -1). Sx, z,, + (N.., -1). Sz4) N defines the total number of sample points considered for the spatial discretization of the transport model. These samples include points strictly inside the media, points at the upper and lower edges, points at the lateral edges and points corresponding to the three interfaces blood / skin, skin / measuring chamber, measuring chamber / collection chamber N = N X .N Z Where Nz is the total number of discrete samples in z , ^=1+ state groups the points at the edges (lateral, upper and lower) and strictly inside the midpoints in z, not taking into account the 3x(Nx) points which correspond

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[0410] at the blood / skin, skin / blood, blood / neck interfaces. In our case, we have: Nx = 5; N-] = N= N= N = 5 so Nz = 17; N = 85 and Netat = 70. Once this vector eo is defined, the resolution of the system follows the same procedure as that described for the resolution of the 1D system. Considering the spatial discretization carried out, the 2D-1 convection diffusion equation within a homogeneous medium becomes: C( x, z, t) is the concentration at point z which follows the concentration C(x Z-ÔZ., t) defined at point z - ôz. and which precedes the concentration C ( X, z + &+, t ) defined at point z + <5<+ if we follow the (arbitrary) orientation of the z axis. r indicates the time reference. For equal sampling steps Ô<+ = ÔZ. — ÔZj, within the same medium i homogeneous the equation becomes: (2M2) Or again: Discretization at the edges We obtain the following equivalences for the boundary equations: C( ) -C ( xfyôx,~j ) = 0C(x0-Ôx, z, t) — C(xQ + Ôx, z, t) ^Q^C(x4-ôx,z,t) ~C(x4+dx, z,t) = o C(x, z0- Ôz, t) = C(x, z0+ôz,t) = 0 - C(x, z 16 -^ t) = C(x, z i6 +oz. t) Substituting into the discrete convection-diffusion equation: Special case of the lower left corner} = [ W 4 + [ W5 ]C(-V& r) + [-p- jC(.¾ q, t) + jc(.Xo z„ + &p r) (2D-23) Special case of the lower edge (x^, 70)lk E [1,2, 3,4} (1,2,3,4) (2D-24) Special case of the lower right corner ( (2D-25) Special case of the upper left corner) 3 F -2L\ 1 ~ ~ r) + [T^p ^16'r) + [ pjT)2 pr) + [TJj1(2D-^6) Special case of the upper edge. 7); k E {1,2, 3,4}

[0411] ^27)

[0412] Special case of the upper right corner ( )

[0413] £S2£B!i«£&ad = [^]C(^^^ (2D-28)

[0414] Special case of the left edge; fc,e{0,4, ..., 12,16}

[0415] £152^^2^=(^^^)+(^^+^0+^(.^^^ ■■■■ i2|d

[0416] Special case of the right edge x4, z^, G {0, .... 16}

[0417] 15^22^^4^^(^,0 + (^^^0+^(¾^^ ^e(a ■■-' 16! ^3°) Discretization at interfaces

[0418] According to the same principle as for the 1D model, we introduce on the interface points and 4(3 and z3|4 the corresponding fictitious concentration variables: C112, 64(3 and 6'3(4, cf. figure 11. Cji? and et will be eliminated from the system by substitution.

[0419] In [Fig.9], we illustrate how to discretize around an interface that separates two physical compartments.

[0420] The notations zt?"f 1 and zm J correspond to kz= 4, 8 or 12 and "in" indicates an entry into a medium and "out" an exit from the medium.

[0421] We obtain for the discretization of equations 2D-10 and 2D-11:

[0422] . 0,(¾. ,-^4 -6,-(¾^ — Cj(^zz. iy+Ôz^tÿ-Cj^Xa^^ (2D-31)

[0423] Replacing the concentration determined by Henry's law (2D-32)

[0424] ^transpa ï\j 'j 4 0 / ] f 0 $<■ ( i'-j Iraaspa i\ji / | y " j

[0425] We make the following notation:

[0426] C —ôt (2D-33)

[0427] Equation (2D-31) becomes

[0428] ' ' C7î,-(r,z / h,f)=C,|4^ t)=Ci[j + i\j^Ci(x.ziU-ôzl,t) □ □ (2D-34)

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[0447] In this equation, we recognize the two terms Cy^xz । • + ÔZ / j < 7 ( f 7 -Ôz- t ) as being respectively the values of Cj ( x, z'" À t ) and If we place ourselves at the point just before the interface ( 7(x ' t) = C ï " / ) ' 'a concentration at the previous point ( x, zoul i-SZf, t ) is internal to the medium, the concentration at the next point is placed on the interface is Q(x, zout' + ôz t) which is equal to q. route i The expression for the concentration Ct (x, ' + dz# t ) thus determined is integrated in the discrete convection-diffusion equation (2D-22) in order to calculate the dynamics in the point Or again: Or: (2D-36) (2D-37) If we are located at the point just after the interface (J ,^nj — (Z7 q. , the concentration at the next point (J. j zw j q. ^7^ / j is internal to the medium, the concentration at the previous point is located on the interface Cj x, zin - ôZj, t ) cst c^x, z .h, r) which is equal to q(< t): (2D-38) The expression for the concentration c( z'" 7 - ôz^ t ) is integrated into the discrete convection-diffusion equation (2D-22) in order to calculate the dynamics in the point zOT j 4337)+-^^) +) + 137^)+^33)+ + ^^+)+-3^43(+^0 + / -,,, - CO-» Or again: Or: (2D-41.) The preceding equations can be written in matrix form by defining a spatial differential operator matrix that represents an operator with elements that are the terms in brackets of these equations:

[0448] c0(x, z, t + ôt)-€0(x, z, t) = Ôt-F0-c0(x, z,t)

[0449] This spatial differential operator matrix defines the temporal discretization:

[0450] If T = 1 we use an explicit Euler time integration scheme:

[0451] cQ(x, z, t + ôt) -c0(x, z, t) = ôt-FQ-c0(x, z, t)

[0452] Or again:

[0453] c0(x,z, t + 5t) = (I + ôt-F0) •CqÇx.z, t)

[0454] If t — 1 + ôt we use an implicit Euler time integration scheme:

[0455] c0(x, z, t + Ôt) -c0(x, z, t) = ôt-F0-c0(x, z, t + ôt)

[0456] Or again:

[0457] (I-ôt-F0) 'C0(x, z, t+ôt) =Cq(x,zA)

[0458] The spatial differential operator matrices Fq are expressed as follows by isolating the terms linked respectively to convection, diffusion, and conditions at the interfaces and edges:

[0459] Fq — Fq conv + Fq + Fq jni

[0460] Fo com groups together the terms of the equation linked to convection,

[0461] Fq f groups together the terms of the equation linked to diffusion;

[0462] Fq il]t groups together the terms of the equation linked to the interfaces and the edges;

[0463] The discretization points are scanned in the x direction and then in the z direction to construct the matrices. We denote kF the indices of the elements of this matrix.

[0464] If the position ( x, z ) has the index ( kF, kF = G {0, N - 1} ) in these matrices according to this scan in x then in z, the following positions will have the following indices: (x-ôx, z) index (kF- 1) (x, z) index (kF) (x + ÔX,z) index (^F+l) (x, z-, - 5zt ) index ( kF- Nx) ( x, z; + ôz, ) index ( kF + Nx )

[0465] The positions of the edges are defined by the points (x, z) having in these matrices the following indices (kF): lower edge index kF - (kx); kx - G {0, (Nx- 1)} top edge index kF = ( kx + ( Nz - 1 ) N v ) ; kx = G {0,(TVx - 1)} left edge index = (0 + kz = g{1,_Æz-2) right edge index kF = ( (N x -1) + kzNx ) ; kz = G { L _ Nz - 2}

[0466] The positions of the interfaces are defined by the points (x, z) having in these matrices the following indices ( kF ): first interface = + (^-1)W j; = G {0,.., (A^ second interface k F =(k x + + ); k x = e {0,„(Nx-1)) \ \ V- / r / third interface kF = lkx+ + - l)jv ykx = e {(U <Wx-l)} These matrices are sparse, we denote by fk k , the matrix element at position kF,

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[0486] Fo conv is constructed as follows: 7 kF, VI 2ôx p _ ux the other terms of Fq conv take a zero value. Fo dif is constructed as follows: = _2D ——t kpkr (ôx)* D, kpkrl (gxy Sun kF,k^i (gxy (5< l). the other terms of Fq take a zero value. Fo int is constructed as follows: For samples below the interface: fi _ ) \ F f interface x k F =(k x + (N x -2).n2. k,= e(ft,(Mr-l)} k F =(k x + Vl + .V ,-2kv ); k x = e {0, _ (Vx-1)} k F = (to a + (^-1 + ^-1+^ -2)^ W.= e {0,., (2Vx-l)} +• „ u fix+lD ; J kpjcrl~ 2(&J2 _ −2D, −21), . 12 7 kF,kF~ (Sx f + +rtranspa i\jÀ i\jôZi ,, -uxÔx+^-Di ■^kpjip+l 2(ox)2

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[0503] 2D, + _ i J kFJcF+2Nx - A >\Right P _ j D.î J krN^kp+Nx ~ THE OWNER For samples above the / 4 interface, _ , - KFifteen^ k F =(k x + (MJjvJ; k x = and {0,_(A^%-1)} k F = (N z i - 1+AT W ); k x = and {0,., (Nx−1)} p— I kx “f 2V[ −1 + ^Vj,2 u-fix+lDj kpkF-\ 2(&v)2 -2D, 2D, 2 A1 '• _ JI (Ær)2 {BzjŸ ÔZ} -u.Ax+lDj 2(5x)2 r I transparent i\jA 2Di Where Ç and À i^j and À yp were defined above (equations 2D-33 and 2D-37 and 2D-41). For the edges: -2D. -2D ___ __________1. 1 __________i — °'° I.5.M (,¾ f 2Di (ôx)1 1 W r _ -22¾ J (NA)JV,(Nrl)W “ (¾¾2 (-5¾)2 • ~ । 4-4 (¾2 (¾)2 f 2pi 7 43 (&)2 f = 2P' 74"V (& J3 1 „ _ -2D4 -2D4 {5xÿ + (ô. y f _ 2D, 4 (¾]).^.)¾])^ +1 - i8vŸ 2D, J (NA).NjNp2H'y~ (q-^ k F=(IA k.^e{K,(Nx-^} j? __ 2 4+(3⁄4] )^,,3+(3⁄4].).^. - 2D, J 4+(^1)^4+(^-2),^. y

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[0511] . c. JUx+ZD^ J kFJcrl~ 2(&)2 f _ −2Dl , −ZD, j , −x+2Di JkpkF+i. Z(bottom)2 f .=-^- J kFkr+Dx k F = (k x + (N z - l)JVj; k x = and {1,„ (Nx−2)} ' „ use+2D4 hFJcrl~ 2(Ox)2 f - ; ' 2D 4 JkFijir. ( Sxy ~ - / / :t5x+2D4 ' 2\ ox)2 F and J kFkrNx~ k F = (0+ ^JVj; k z = g {1, _ 2Vz − 2} y −2D; −2D, T — __________L. |. __________L-. ' kfkp (Sx)2 (Bz,)2 f = 2D ' kFkF+\ (Qx(2 f =-^ J kpJcrNx (d?.y f - D ' J kFkp+Nx (s?-)2 k F = ( (N x -1) + kJV x ) ; k z = g {1,.. N z - 2} f _ -ZD, -ZD, (Sx)2 + (szy f _ 2°, J kFjc.ri (oxy f - D > J kFkFNx j2 f =-^- 2 kF,kF+Nx (Bz-)2 The other terms of Fq int take a zero value. Restricted direct problem

[0512] To describe the direct transport model we only use the terms located at the strict interior of the z-centers, ignoring the 3x(Nx) points which correspond to the blood / skin, skin / mes, mes / collar interfaces and the points at the edges

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[0528] (lateral, upper and lower). We described above how to eliminate from the equations the points located at the exact level of the interfaces by points before or after the interface. The points of the matrix FQba corresponding to the interfaces are eliminated from the matrices. We define a new matrix F of dimension (A^tJtoZ) in which the interface points have been deleted. Let the points of indices kF be defined above. F — Fconv + Fdif 4- Fdlt We define a control matrix G which includes the input values known. G is a two-column matrix of size (Ne1at, 2) of the form (case of transverse evacuation): *c=(0+ k= e (1,2 3) fc G =(o + (n^ + N^ + N + k z =e {1,2,3} 0 for all other terms Or ~ Sx™* 9 is the vector containing the exogenous inputs into the system. These are the initial concentration introduced into the system in the liquid phase San8 on the one hand, and the concentration of CO2 in the ambient air Cm c°lAlr on the other hand. ,in Blood çdn Col Air The concentration data are generated using a signal model following the recursion relationship: Where A is the transition matrix obtained from the implicit time discretization etc - the Netat-dimensional state vector A = IF-Ôt In the case of transverse counter-current evacuation, the air intake is through the opposite side, the matrix G is expressed as follows: ^ot Or gk^ kG=(0 + kzNxy kz= e{1,2,3} k G = ( 0 + ( N?,, + N z2 +N _ 3 - 6 + k z ) JV X +(N x -1) j ; E = e {1,2,3} 0 for all other tarnished ôx™"* The Kalman algorithm allows us to linearly estimate a variable of interest, here çln ScmR, by seeking to minimize the mean square error between the actual value and estimated value. It is based on the formulation of a state-space model by associating a state evolution equation and an observation equation which expresses the link between measurements and states.

[0529] c|^+l] =Fc[^]+G£[M1]+w[^+1] (2D-42)

[0530] yÇk^ = Hc(kt) + (2D-43)

[0531] kt represents the time discretization index.

[0532] The vector w describes the model noise on the state evolution equation, with variance-covariance matrix Q.

[0533] Through the observation matrix H, we define the observable of the state vector c. In the specific case of our problem, we only have one observable for the state vector, all the other concentration variables being hidden, the matrix is reduced to a transposed vector.

[0534] H = ti

[0535] The vector r describes the observation noise of variance-covariance matrix R.

[0536] We need to augment the state vector with the desired variable Cjn xan? in the same way as in 1D.

[0537] The variable that we seek to estimate Cjn sang, is found in the vector Q at position -¾ It is linked to the following state variable Cv£,[ by the terms of the equation of transport. We assume that the signal can be modeled by a first-order autoregressive model defined by the following recurrence relation:

[0538] to 5 blood = <pCin sang + win sans (2D-44)

[0539] in discrete form:

[0540] ( l- <p) Citlsang[kt + 1] = ôt Cùlsang[kf] 1] (2D-45)

[0541] Where is a parameter of the model. It should be interpreted as a regularity parameter on the signal.

[0542] If (p = 0, the time derivative of the input signal is assumed to be zero. There is no variation.

[0543] Yes 0 the derivative of the input signal is assumed to be positive for a positive Cin blood concentration. This favors the increase of the signal, but this will penalize its decrease.

[0544] We define an augmented system by associating the physical model of the system described previously with this signal model.

[0545] [ c — P c 4- Gq + VV I . . (2D-46) 1 Çtn sanS _ sanü + win sang i dt '

[0546] The equation of state of the augmented system is written in the following form:

[0547] ±c = Fc + Gq + w (2D-47) 42

[0548] Discretization implicitly provides the following equation:

[0549] + +

[0550]

[0551]

[0552]

[0553] With : ',kF— 1 : N etaJ w[ / <] ~N(0, ô) is 'c noise which integrates the errors related to the dimension model. where N(0, g), is a normal vector distribution with zero mean for each component and variance-covariance matrix Q. the variance of the model noise describing the model-related errors common to each component of the noise vector W for each instant kt The variance-covariance matrix Q is Q —

[0554] G the control matrix now only applies to C'n co1 mr (deletion of the first column).

[0555] 0 '

[0556] kc= (0 + (^ + ^-2 + ^-6 + ^).^ kz = e {1,2, 3}

[0557] " kkc2~ ôx

[0558] As indicated above, in the case of transverse counter-current evacuation: [°559] ÆG=(o + (^ + ^,2 + ^^6 + ^-).^ + (^-1))-, Zq=g {1,2,3}

[0560] „ _ ^ka2~ ôx

[0561] Kahnan filtering algorithm. • ^ktJcri is the augmented state vector at time kt predicted at time kt -1 previous • CkiA, is the augmented state vector at time kt calculated at time kt • is the augmented state vector at time kt + 1 predicted at time kt • is the covariance matrix of the error at time kt predicted at time k{ - 1 previous • is the covariance matrix of the error at time kt calculated at time kt • Pkt+\kt is the covariance matrix of the error at time kt + 1 predicted at time kt • Kk( is the Kalman gain calculated at time kt • is the new measurement observed at time kt • A is the transition matrix applied to state Q(+i • G is the control matrix applied to the Q input of the augmented system • H is the augmented observation matrix • Q is the noise covariance matrix of the augmented restricted system • R is the covariance matrix of the observation noise

[0562] The problem to be solved is reduced to obtaining recurrent relations for the estimation of the state vector Ck^, as a function of the previous state vector Ckt-\kri and the new observation. Two steps are necessary to solve this problem:

[0563] #1 The step of correcting and updating the state vector and the co matrix variance from the observation equations and the state vector and covariance matrix estimated in the previous prediction step.

[0564] #2 The step of predicting the state vector and the covariance matrix at time next from the state vector and the covariance matrix at the previous time.

[0565] These two steps are preceded by an initialization:

[0566] #0 We assume that the state of the system c0.0 is known at the instant kt = 0 and the matrix designating the covariance of the error Pq,o.

[0567] The state vector at time kt = 0 is initialized by the zero vector : = 0 (of dimensions and the matrix designating the covariance of the error by the matrix identity (of NetatxN dimensions

[0568] The algorithm is stopped when no more new measurements are made.

[0569] We now present the computation loop of the Kalman filtering algorithm: Correction phase

[0570] We know the state vector at time kt predicted at time kt -1: and the covariance matrix of the error at time kt predicted at time kt - 1: Pkpkrl and n°us observes y^.

[0571] We want to estimate: ^k^,, Pk^

[0572] The Kalman gain Kkt is determined with the following formula where Pk^-i is calculated in equation 2D-44 or 2D-45:

[0573] Kk, = PtA.iHT(HPtlk,-iHT + «f I2D-48

[0574] The Kalman gain expresses the weights (confidence) given to the new noisy measurement yk or to the estimate of the state cktjkt-ï which is also subject to the different per-

[0575]

[0576]

[0577] external disturbances. The estimation of the current state (a posteriori state estimator) of the system is done by taking a linear combination between the estimation made at the previous instant kt- 1 (a priori state estimator) and the new recorded data. The Kalman gain intervenes to correct the estimation made at instant kt - 1 for the current instant kt according to the new recorded measurement yk. Knowing the expression for the Kalman gain K^, we can give the expression for estimating the state vector Cktkt at time kt as a function of the difference between the current measurement y^ and the estimated measurement Hc^ from the previous state estimate. This difference defines the prediction error which assesses the amount of new information brought by the current measurement. Ck^ = ]+Kkt (yk-Hckbkri) (2D - 49) The estimation of the covariance matrix of the estimation error at time kt from the covariance matrix Pk^-l estimated at time kt- 1 is done according to the following formula: 105781 Pk^IK^rw il-Kk,H)T + KtJŒl, (2D-50)

[0579] The estimation of Cktkt allows us to know Cin, the latter being the rank 1 term of the vector

[0580] From Cin hiood, we can estimate p^iood.

[0581] (2D.51) P

[0582] Where: _ csl |c Ostwald solubility coefficient of carbon dioxide in the blood. Prediction phase

[0583] We know the state vector and the covariance matrix of the estimation error Pk^ at time kt.

[0584] We wish to predict the new state of the system èkt+ikt. We place ourselves in the integration scheme corresponding to the implicit method. The prediction equation is as follows:

[0585] ÂQ, +1A = 8 Mj + GCly^ (2D-52)

[0586] We also want to predict the error covariance matrix Pkt+lkt: 105871 Pw, = + 0 (2D-53)

[0588] Using these formulas, we perform one-step prediction from the state equation. Only a past sample is retained in order to predict the future behavior of the system, through the state vector and the variance-covariance matrix.

[0589] Although described in connection with a transcutaneous CO2 measurement, the invention can be implemented in other applications, so as to measure a gas emitted by a medium, solid or liquid. The medium can in particular be: - a plant, for example a fruit, so as to follow the maturation process; - slurry, in order to monitor the emission of gases, for example methane; - a culture medium for biological organisms or microorganisms; - water, for example fresh water or sea water, for example for monitor gas concentration or regulate CO2 concentration;

[0590] a soil, in order to study its respiration.

Claims

1. Claims Method for estimating a content of a gas of interest in a medium, using a measuring device, intended to be placed against the medium, the measuring device extending between a contact face, intended to be applied against the medium and a distal end, the measuring device comprising a side wall, extending between the contact face and the distal end, the measuring device comprising: - at the contact face (10), at least one intake opening (12), configured to collect the gas of interest emitted by the medium, the intake opening being made through the contact face; - a measuring chamber (20), comprising a gas sensor (23), the gas sensor being configured to measure a concentration of gas of interest flowing through the measuring chamber; - a collection chamber (30), connected to the measuring chamber, and delimited by an opening on the side wall, the collection chamber comprising at least one lateral opening (34), arranged through the side face, or on the upper wall of the collection chamber so as to admit ambient air into the collection chamber; the device being such that: - the measuring chamber (20) is arranged between the contact face (10) and the collection chamber (30); - the device comprises a drive means (41), configured to drive the air from the collection chamber towards an evacuation opening (42), the air drive inducing a transport of the gas of interest from the contact face towards the collection chamber, through the measurement chamber: - the process comprising: - a) measurement of the concentration of gas of interest in the measuring chamber; - b) using a processing unit, modeling the transport of the gas of interest, between the medium and the collection chamber, the modeling including consideration of diffusion gas from the medium to the collection chamber through the contact face, as well as a convection of the gas in the collection cell resulting from the entrainment produced by the entrainment means; - c) from the measurement resulting from step a), and taking into account the model resulting from step b), estimation of the content of the gas of interest in the medium; the method being characterized in that: - the device is spatially discretized according to a spatial mesh, defining mesh points between the contact face and the collection chamber; - the model is a discretized spatio-temporal model, so as to estimate a content of gas of interest at different points of the mesh, and at different times;- the modeling is based on a two-dimensional model, the medium, the measuring chamber and the collection chamber being discretized according to different mesh points distributed along an axis parallel to the contact face and an axis perpendicular to the contact face.;

2. Method according to any one of the preceding claims, in which step b) comprises: - modeling the diffusion of the gas of interest through the measuring chamber; - modeling the diffusion and convection of the gas of interest in the collection chamber.

3. Method according to any one of the preceding claims, in which step b) comprises modeling the transport of the gas of interest in the medium.

4. A method according to any preceding claim, wherein step c) implements a recursive estimator.

5. The method of claim 4, wherein step c) implements a linear recursive estimator.

6. The method of claim 5, wherein the recursive estimator linear is a Kalman filter.

7. A method according to any one of the preceding claims, wherein the medium is the skin of a user, the skin extending over a blood vessel, wherein step b) comprises: i. from the concentration of gas of interest in the measuring chamber, resulting from step a), estimating a transcutaneous concentration of gas of interest; ii. from the concentration of transcutaneous gas of interest resulting from sub-step (i), estimating a concentration or partial pressure of gas of interest dissolved in the blood of the user.

8. Method according to any one of the preceding claims, comprising modeling the transport of the gas of interest emitted by the medium through the device.

9. A method according to any preceding claim, wherein the medium to be analyzed is a solid or liquid medium.

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