A METHOD AND DEVICE FOR OPTICAL QUANTIFICATION OF OXYGEN PARTIAL PRESSURE IN BIOLOGICAL TISSUES

Abstract

The disclosure relates to methods and devices for monitoring the concentration of a substance, preferably oxygen, in a cell or tissue, e.g., in cells of the human skin. In particular, it provides a method for determining the concentration of a quencher, such as oxygen and/or the concentration of a probe, e.g., a heme precursor such as protoporphyrin IX (PpIX), wherein the probe is capable of exhibiting luminescence (delayed fluorescence (DF) or phosphorescence) and or transient triplet absorption, preferably, deDF, in a living cell. The method comprises steps of exciting the probe, measuring the lifetime of the luminescence exhibited by said probe, herein, in the presence of the quencher, the lifetime is shortened as compared to the lifetime in the absence of the quencher, and correlating said lifetime with said concentration. The disclosed method leads to more precise results than conventional methods, because of adaptations based on the understanding of the influence of the concentration of the probe and its excitation fluence rate (intensity) on the analysis. For example, the simultaneous time-resolved detection of the probe excimer and monomer DF allows estimation of the probe concentration and compensation of the probe self-quenching effect in the quencher concentration calculation, increasing the measurement precision. Taking into account second order triplet interactions also permits the interpretation of non-exponential decays and further improvement of the quencher and probe concentration estimation. Disclosed methods rely, e.g., on measurement at different emission wavelengths and application of an adaptive Stern-Volmer relationship, the decay central fitting method and/or a mixed orders approach. Said method can be applied, e.g., for bedside monitoring of patients. Also disclosed is the use of the PpIX precursor 5-aminolevulinic acid (5-ALA), or derivatives thereof, in this method, and a device suitable therefor.

Claims

1. A method for determining the concentration of a quencher, and/or the concentration of a probe capable of exhibiting a triplet-state based luminescence or transient triplet absorption in a cell in a tissue or organ, comprising steps of a) exciting the probe by irradiation with light having a wavelength and a fluence rate suitable for excitation of the probe, b) measuring the temporal evolution of the triplet-state based luminescence or transient triplet absorption of the probe at at least one emission or absorption wavelength, wherein, in the presence of the quencher, the triplet-state luminescence or transient triplet absorption decays more quickly compared to the decay in the absence of the quencher, and c) correlating said temporal evolution with said concentration(s), wherein second order reactions involving two excited triplet states of the probe are considered.

2. The method of claim 1, wherein the temporal evolution of triplet-state based luminescence measured in step b) is delayed fluorescence or phosphorescence.

3. The method of claim 1, wherein the concentration of excited triplet states of the probe is considered for determining the concentration of the quencher or of the probe, wherein, the concentration of the probe is determined, and, then, the concentration of the quencher is determined.

4. The method of claim 2, wherein the emission of the delayed fluorescence is measured simultaneously at at least two wavelengths, in the same sample volume, wherein these two wavelengths are suitable for distinguishing delayed fluorescence of probe monomers and of probe excimers.

5. The method of claim 4, wherein the adaptive Stern-Volmer relationship is used for determining the concentration of the quencher, wherein the adaptive Stern-Volmer relationship i) calculates the concentration of the probe based on the ratio of the probe excimer DF energy or initial intensity to the probe monomer DF energy or initial intensity ii) calculates the delayed fluorescence lifetime in absence of quencher at the probe concentration calculated in step i), and iii) calculates the quencher concentration by feeding the DF lifetime in absence of oxygen calculated at step ii) into the Stern-Volmer relationship, thus adapting the Stern-Volmer relationship to the probe concentration.

6. The method of claim 1, wherein the fluence rate (intensity) or fluence of the light for excitation is used to determine the relative contribution of first and second order triplet deactivation and/or luminescence processes wherein, the probe is excited by irradiation with light having at least two, different fluence rates or fluences.

7. The method of claim 1, wherein the decay central fitting method is used for correlating said temporal evolution with said concentration of quencher, wherein the decay central fitting method rejects the initial non-exponential part of the decay and, optionally, rejects the tail of the decay dominated by noise, thus selecting the central part of the decay of delayed fluorescence intensity over time.

8. The method of claim 1, wherein the concentration of the quencher is determined on the basis of the assumption that the excited triplet state of the probe deactivates through a mix of first and second order reactions and/or on the assumption that the delayed fluorescence is caused by a mix of first and second order reactions.

9. The method of claim 1, wherein the probe is protoporphyrin IX (PpIX) and the quencher is oxygen, and, before step a), a precursor of PpIX, is administered to the cell or has been administered to the cell.

10. The method of claim 1, wherein the cell is part of a tissue or organ.

11. A method for assessment of mitochondrial function in a sample comprising a tissue or organ, comprising restricting or ceasing the supply of oxygen to said sample and carrying out the method of claim 1 with said sample, wherein the probe is PpIX protoporphyrin IX (PpIX) and the quencher is oxygen.

12. A method for assessment of the status of a patient selected from the group comprising a sepsis patient, a critically ill patient, a patient undergoing a tumor treatment (e.g., phototherapy or photodynamic therapy), a patient undergoing surgery, a patient suffering from a neurodegenerative condition, or a decubitus patient, or for selecting an organ potentially suitable for transplantation to a patient, comprising carrying out the method of claim 1, wherein the cell is a cell of said patient or organ.

13. (canceled)

14. A device suitable for carrying out the method of claim 1, comprising an excitation light source arranged to illuminate a sample volume, and a time and intensity-resolved light detector arranged to detect luminescence, such as fluorescence or phosphorescence from the sample volume, or triplet absorption by the sample volume, and a control unit configured to obtain measurements of luminescence, from or transient triplet absorption by the sample volume, wherein said device comprises i) means for determining the delayed fluorescence lifetimes and intensities of a probe simultaneously at two emission wavelengths suitable for distinguishing delayed fluorescence of probe monomers and of probe excimers, wherein, preferably, the probe is a heme precursor delayed fluorescence is determined, and the first wavelength is in the range of 615-645 nm, and the second wavelength is in the range of 646 nm-700 nm; and/or ii) means for exciting the probe with at least two different excitation fluence rates or fluence; and/or iii) means for exciting the probe with at least two, different excitation wavelengths.

15. The device of claim 14, further comprising a pressure pad for applying local pressure on tissue containing arterioles, veins and/or capillary bed that supply oxygen to the sample volume, and/or a processing unit capable of processing the obtained measurements and configured to apply the adaptive Stern-Volmer relationship and/or the decay central fitting method and/or the mixed orders fitting method.

16. (canceled)

17. The method of claim 1, wherein the quencher is oxygen, and wherein the probe is a heme precursor.

18. The method of claim 17, wherein the heme precursor is protoporphyrin IX (PpIX).

19. The method of claim 2, wherein the temporal evolution of triplet-state based luminescence measured in step b) is temporal evolution of delayed fluorescence.

20. The method of claim 4, wherein the probe is a heme precursor and the first wavelength is in the range of 615-645 nm, optionally, about 630 nm and the second wavelength is in the range of 646 nm-700 nm, optionally, about 670 nm.

21. The method of claim 5, wherein the probe is a heme precursor and the ratio E.sub.670/E.sub.630 or I.sub.0.sup.670/I.sub.0.sup.630 or another parameter allowing to distinguish the contribution of excimer and monomer states to the luminescence is used in step i).

22. The method of claim 9, wherein the precursor of PpIX is 5-aminolevulinic acid.

Description

[0242] FIG. 1 Simplified Jablonski diagram of delayed fluorescence, with the name of the corresponding processes. Dotted line: non-radiative, continuous line: radiative. This diagram characterizes the excited states of molecules: on the y-axis, the energy of the outer orbit electrons is represented, and on the x-axis the configuration of the spin these electrons. Singlet state: antiparallel spins, spin multiplicity 1. Triplet state: parallel spins, spin multiplicity 3.

[0243] FIG. 2 Evolution of the DF lifetime τ as a function of time on healthy volunteers' sternum. The DF was collected on the whole [600; 750] nm range. There is a distinct minimum around 4-5 h

[0244] FIG. 3 Absorption (black, left curve and axis) and emission (grey, right curve and axis) spectrum of PpIX. PpIX can be excited at 405 (Soret band), 506, 515 or 630 nm (Q-bands) depending on the excitation penetration and excited states concentration desired. PpIX fluorescence is mainly detectable around 630 nm.

[0245] FIG. 4 One option for an improved intracellular oxygen measurement device. This version could include a variable excitation laser, which would allow the quantification of first and second-order processes. The excitation diameter would be large compared to the penetration length (about 100 μm), to minimize variations in the excitation fluence rate. It would also include bandpass filters at 630 and 670 nm, to distinguish between monomer and excimer DF. Finally, it would include a new pO.sub.2 calculation algorithm: preferably the adaptive Stern-Volmer method, but the Central Decay Fitting and Mixed-orders methods are also possible.

[0246] FIG. 5 Evolution of τ.sub.0 with [PpIX]. We observe a decrease, following the equation

[00022]$\frac{1}{{\tau }_0}=36.8\left[\mathrm{PpIX}\right]+2.74\times {10}^3,$

τ.sub.0 in s and [PpIX] in μM. We have identified a Stern-Volmer like self-quenching of PpIX of constant k.sub.q.sup.PPIX=36.8 μM.sup.−1s.sup.−1.

[0247] FIG. 6 Evolution of the DF spectrum with PpIX concentration, in absence of oxygen. A broad, featureless, red shifted (compared to the 630 nm peak) band develops at 670 nm.

[0248] FIG. 7 Updated Jablonski diagram taking excimers into account. Excimers form according to two pathways: 2T.sub.1.fwdarw.D* and s.sub.0+s.sub.1.fwdarw.D*; and subsequently desexcite at 670 nm. Their lifetime is comparable to that of S.sub.1, as the DF at 630 and 670 nm are detected simultaneously.

[0249] FIG. 8 Compared evolution of

[00023]$\frac{E_{\mathrm{DF}}^{670}}{E_{\mathrm{DF}}^{630}}$

and T.sub.630 over time, on one specific patient. The anticorrelation is clear, strongly suggesting the presence of excimers in vivo. This shows that the local PpIX concentration induces a strong noise-like variation of the lifetime which is not related to physiology. Averaging such graphs for different volunteers is not relevant here: as the variation of the energy ratio/τ.sub.630 depends on the local PpIX concentration, it strongly depends on the probe positioning, and is relatively variable between patients. The variation of

[00024]$\frac{E_{\mathrm{DF}}^{670}}{E_{\mathrm{DF}}^{630}}$

reflects more local rpm inhomogeneities than average PpIX concentration trends, contrary to FIG. 2 which includes the whole [600;750] nm window).

[0250] FIG. 9 Confirmation of the anticorrelation of τ.sub.630 with

[00025]$\frac{E_{\mathrm{DF}}^{670}}{E_{\mathrm{DF}}^{630}}$

on 4 other volunteers. The anticorrelation is visible in every graph. The data presented on the bottom left hand corner was obtained while pressing the skin (pO.sub.2=0). Continuous line; DF lifetime 625 nm, Dashed line

[00026]$\frac{E_{\mathrm{DF}}^{670}}{E_{\mathrm{DF}}^{630}}.$

[0251] FIG. 10 Compared evolution of τ.sub.670 and τ.sub.630 over time, as the skin is compressed to reach pO.sub.2˜0. The anticorrelation is visible, confirming the existence of two competitive fluorescence process at 670 and 630 nm, thus further proving the existence of excimers. The repeatability of a lifetime measurement in our conditions is around 20%.

[0252] FIG. 11 Compared evolution of τ.sub.670 and τ.sub.630 over time in the standard uncompressed case. The lifetimes are clearly correlated, as they are defined by the amount of oxygen quenching. Observations are coherent with our new model.

[0253] FIG. 12 Splitting of a DF signal (semilogscale) according to different kinetic behaviour: non mono-exponential initially (not linear on the graph), followed by a linear, [T.sub.1]o independent monoexponential decay, and finally noise

[0254] FIG. 13 Evolution of the DF lifetime fitted by the Origin algorithm as a function of the fitting range lower boundary t.sub.1. Clearly, the quadratic region is visible as the lifetime increases sharply before 200 μs, and differs between datatsets. Then the mono-exponential region appears with a plateau between 200 μs and 350 μs. The plateau region is not only relatively flat, but also close between datasets, which were measured at different times in the day: it is free of the influence of differences in PpIX concentration. Lastly, after 350 μs, erratic values due to noise are visible.

[0255] FIG. 14 DF lifetime at 630 nm (uncompressed measurement, 8 h ALA) fitted on the reduced interval [275; 600] μs, in accordance with the DCF method. Although the curve is less erratic than on FIG. 2, it is not as constant as expected.

[0256] FIG. 15A Relationship between the DF energy ratio at 670 and 630 nm

[00027]$\frac{E_{\mathrm{DF}}^{670}}{E_{\mathrm{DF}}^{630}}$

and the PpIX concentration, as measured in vitro. The curve is linear:

[00028]$\frac{E_{\mathrm{DF}}^{670}}{E_{\mathrm{DF}}^{630}}=0.09+2.5\times {10}^{-3}\left[\mathrm{PpIX}\right],$

[PpIX] in μM, r.sup.2=0.94. This linearity is characteristic of excimer DF.

[0257] FIG. 15B Excimer to monomer initial DF intensity ratio as a function of PpIX concentration, and associated linear fit at concentrations above 80 μM.

[0258] FIG. 16 Comparison between the pO.sub.2 as calculated by the standard Stern-Volmer equation (uncorrected pO.sub.2 and the adaptative SV equation (corrected pO.sub.2), on the sternum of a volunteer. The correction reduces the pO.sub.2 spread from [10;43] mmHg to [18;30] mmHg. Parameters: k.sub.q.sup.PpIX and τ.sub.0.sup.0.fwdarw.PpIX self-quenching constant and lifetime in absence of PpIX, as defined by

[00029]$\frac{1}{{\tau }_0^{630}}=\frac{1}{{\tau }_0^0}+k_q^{\mathrm{PpIX}}\left[\mathrm{PpIX}\right]$

(defined in FIG. 5); a, b.fwdarw.linear link between energy ratio and PpIX concentration:

[00030]$\frac{E_{\mathrm{DF}}^{670}}{E_{\mathrm{DF}}^{630}}=a+b\left[\mathrm{PpIX}\right]$

(defined in FIG. 15).

[0259] FIG. 17 Influence of fitting an incorrect model of PpIX DF (mono-exponential decay +Standard Stern-Volmer) on the pO.sub.2 measured in an oxygen consumption experiment. The two curves are very similar in their outlook: an improper understanding of the DF of PpIX produces values that are only slightly different from reality (real pO.sub.2: curve starts at 2 mmHg).

EXAMPLES

[0260] In Vitro Discovery of PpIX Excimers

[0261] Measurement & Methods

[0262] The inventors worked with a solution of Protoporphyrin IX (PpIX, Fluka, >99.5%) diluted in Dimethylformamide (DMF, Sigma Aldrich D4551, >99%). Concentration ranged from 1 to 400 μM. The solution was placed in a 1 cm wide quartz cuvette. Delayed fluorescence curves were acquired on a calibrated Horiba Fluorolog-3 spectrofluorometer, operated in front face mode. The cuvette was illuminated by a pulsed Xenon lamp (pulse duration FWHM: 1 μs, 10000 pulses per curve), filtered using a monochromator. The sample was excited at 405 nm, with a FWHM of 5 nm. The emission at different wavelengths was collected with a FWHM of 3 nm. The temperature of the setup was maintained at 32° C. (or piloted when temperature was the variable) using a Pelletier Carry PCB-150 thermoregulator. The pO.sub.2 was brought to zero by bubbling N.sub.2 in the cuvette at a flow rate of 3 L/min, using a Brick™ gas mixer. The zone of bubbling was not illuminated by the excitation beam, and the samples were constantly stirred during measurements. The excitation intensity was modified by placing neutral density filters in the excitation beam. Exponential decays were fitted using Origin V8 (OriginLab, Northampton, Mass.): in the expression of a mono-exponential decay

[00031]$\left(t\right)=I_0\exp \left(-\frac{t}{\tau }\right)+y_0,$

the initial tensity l.sub.0, the DF lifetime τ, and the offset y.sub.0 were determined using the built-in Levenberg-Mar-quardt algorithm. The delayed fluorescence signals were collected and fitted from 0,13 to 11 ms.

[0263] Results

[0264] The first and simplest experiment was to quantify the DF lifetime in absence of oxygen as a function of the concentration of PpIX. In theory,

[00032]${\tau }_0=\frac{1}{k_E+k_{\mathrm{IC}}}$

should be independent of the concentration of PpIX ([PpIX]). The curve is presented on FIG. 5. Contrary to our ex-pectations, the inventors however observed a self-quenching phenomenon: the higher the concentration of PpIX, the smaller the lifetime: as the solution gets crowded, PpIX molecules stay a shorter time in the triplet state T.sub.1. This result is in complete disagree-ment with the established model of PpIX DF. While, so far, dependency on the concentration of PpIX in cells has not yet been observed or postulated, the inventors considered it could explain the three incoherencies they have identified earlier: [0265] The observed variations of T.sub.0 in different studies could simply be that the parametriza-tion (reducing the pO.sub.2 to 0) was done at different concentrations of PpIX, thus there were different self-quenching levels. This is plausible, as the intracellular concentration of PpIX is hard to control in vivo. [0266] The decrease of τ, measured in the skin up to 4-5 h, could be due to the increasing intracellular concentration of PpIX, due to the transformation of ALA in the skin into PpIX, causing self-quenching, and thus reducing the lifetime. Later, when PpIX starts to be catabolized, its concentration decreases and τ increases again. [0267] Finally, on patients, when the ALA application was done the evening before surgery, [PpIX] reaches its maximum before the first measurement, and is steadily decreasing during the day: the device misinterprets the increase of τ for a decrease of pO.sub.2, while in reality, it could simply be a reduction of the self-quenching effect. Combining the two quenching relationships (eq. 2 and FIG. 5), we obtain the following link between pO.sub.2 and [PpIX] variations:

[0268] The rate of PpIX catabolization in human cells:

[00033]$\frac{\partial \left[\mathrm{PpIX}\right]}{\partial t}=\frac{k_q^{O_2}}{k_q^{\mathrm{PpIX}}}\frac{\partial {pO}_2}{\partial t}$

[0269] was estimated by inventors as on average, −226 μM/h.

[0270] Conversely, by looking at the decrease of DF lifetime between 2 and 4-5 hours of PpIX application, the inventors could extract the average DF lifetime variation:

[00034]$\frac{\partial \tau }{\partial t}\in \left[-10;-5\right]\mu s.h^{-1}$

[0271] which gives a PpIX concentration rate of:

[00035]$\begin{array}{cc}\frac{\partial \left[\mathrm{PpIX}\right]}{\partial t}=-\frac{1}{k_q^{\mathrm{PpIX}}{\stackrel{\_}{\tau }}^2}\frac{\partial \tau }{\partial t}\in \left[85;170\right]\mu M.h^{-1}& \left(4\right)\end{array}$

[0272] These rates are surprisingly high, suggesting that the concentration in cells could easily be above mM. Interestingly, other methods of oxygen sensing using porphyrin nanoparti-cles have reported unexplained quenching effects, which the inventors postulate to be based on the same mechanism (Kubat et al., 2017. ACS Applied Materials and Interfaces 9: 36229-36238).

[0273] A second question is the molecular origin of the self-quenching. By measuring the DF spectrum of different concentrations of PpIX solutions, it appeared that a peak at 670 nm was gradually developing as concentration increased, as presented on FIG. 6. The change of spectrum shows that another desexcitation process than S.sub.1 —>S.sub.0 takes place at high concentration. This added to similar observations made in the past on pyrene (Parker et al., 1968; Birks et al., 1966. Spectrochimica Acta 22: 323-331) allowed to con-clude to the existence of dimers of excited T.sub.1 molecules, so called excimers.

[0274] It appears that, similarly to other aromatic molecules, PpIX excimers can be formed either by the encounter of two molecules in the first excited triplet state T.sub.1, or by the reaction of an excited singlet S.sub.1 with a triplet T.sub.1. The excited triplet state concentration can influence the relative contribution of different modes of its decay and hence affect the lifetime of the excited triplet state. For high excited triplet state concentrations, such as those expected after active cells have generated PpIX after administration of 5-ALA, it can be important to consider the excited triplet state concentration or parameters that serve as a proxy thereof as described in this application in order to accurately determine quencher (oxygen) concentrations from observed triplet state lifetimes. The initial triplet state concentration is determined by the excitation radiant exposure per pulse H times the molar extinction coefficient at excitation wavelength ε. At low excited triplet state concentration (HE[PpIX]<6,0 μJ. cm.sup.−3, E-type, thermal DF is dominant, and as the triplet concentration increases (HE[PpIX]>12 μJ. cm.sup.−3), P-type DF and excimer DF contribute predominantly.

[0275] The formation and desexcitation of excimers is presented on an updated Jablonski diagram (FIG. 7). While these in vitro studies do not allow for the conclusion that the same mechanism applies in cells, the inventors set out to find out if this was the case, and if it could account for the inconsistencies encountered. The question was whether excimers also form in vivo: the environment around PpIX molecules is so different from the homo-geneous, freely diffusing in vitro environment, that their existence has to be proven before any conclusions can be drawn.

[0276] Assessement of the Existence of PpIX Excimers In Vivo and Contribution of P-Type Delayed Fluorescence

[0277] Materials & Methods

[0278] “Dumbo” Setup

[0279] Plasters of 8 mg of ALA (Alacare 8 mg medicated plaster, Photonamic GmbH, Wedel) were applied on the skin of 20 healthy volunteers, in a mediothoracic position, on the sternum. The 2×2 cm.sup.2 plasters contain a suspension of 5-ALA HCl crystals suspended in a polymer matrix. The delayed fluorescence of PpIX was readily detectable from 3 to >15 h of ALA application, with a maximum intensity around 8 h. The outer layer of dead skin cells (stratum corneum) was removed using an abrasion pad, following the standard microdermabrasion method (Mik et al., 2013). Every hour, from 3 to 8 h of ALA application, the DF in the skin was quantified using a dedicated optical setup. The skin was excited by a series of 3 to 10 27 ns pulses at 515 nm (light doses 300 μJ/cm.sup.2. Energy per pulse: 15 μJ according to Innolas Service Test sheet Inno P1166, 18/02/2016. Laser Innolas Mod3 Yb-Yag, passive cooling, frequency doubling, passive Q-switch, fiber coupling (SMA)) via a frontal light diffuser providing a circular spot directly in contact with the skin (illuminated surface: 5 mm.sup.2). 1 ms separated each pulse, so that the system would have time to desexcite fully between each pulse.

[0280] The fluorescence 10 ns) and delayed fluorescence (5 to 500 μs) were collected by a fiber next to the light diffuser, and subsequently sent to a Photomultiplier (PMT) (Hamamatsu H11526-20-NF) and amplifier (Hamamatsu C6438-01). The PMT was gated to reject the prompt fluorescence. The decays were recorded by a data acquisition board (DAQ NI USB 6259, National Instruments) and processing unit. The emission light was filtered using a band-pass filter in the [600; 700] nm region, to reject excitation light and keep only the DF of PpIX. For certain measurements, the DF signal was additionally filtered to extract specific spectral bands, using bandpass filters at 625±12 nm, 670±12 nm or 700±12 nm. OD outside of bandpass: >4. When needed, the excitation intensity was varied by placing neutral density filters (OD E [0.15; 0.9]) in the excitation beam. In this work, DF energy E.sub.DF is understood as the total energy of the DF: if I(t) is the DF intensity as a function of time as measured by the data acquisition, then the intensity of the DF is E.sub.DF=∫.sub.0.sup.+∞I.sub.DF(t) dt. When changing the excitation intensity, the initial concentration of T.sub.1 states is linearly changed, as all processes leading to its formation are described by linear Einstein equation (absorption and Intersystem Crossing), and the illumination zone is constant in size. In short, [T.sub.1].sub.0∝l.sub.ex. As a consequence, when [T.sub.1].sub.0 changes, I.sub.DF is likely to vary linearly if the DF mechanism involves one T.sub.1 molecule, or quadratically, if it involves two T.sub.1 molecules. Unless otherwise mentioned, the DF intensity was manually fitted to a decreasing mono-exponential I.sub.DF(t)=.sub.0e.sup.−t/τ+y.sub.0 using the built-in fitting algorithm of the Origin V8 software.

[0281] Results

[0282] Contrary to the in vitro situation, PpIX concentration is not known in vivo, which makes the analysis more complex: we are unable to compare the DF at different concentrations, we have to track other proofs of the existence of excimers. The inventors chose to inves-tigate the competition process that occurs between the 630 and 670 nm luminescence pathways.

[0283] Evolution of E.sub.670/E.sub.630 compared to τ.sub.630

[0284] To assess the existence of excimers, the inventors looked at the DF energy at 670 nm. But the monomer form also has a slight DF signal at this wavelength (FIG. 3). We therefore compute the ratio of energy at 670 and 630 nm, E.sub.670/E.sub.630.

[0285] The inventors could have chosen to study the ratio of initial intensities, because it would not depend on the quenching by oxygen, which can be slightly varying between the measurement at 670 and 630 nm. But, in reality, the initial intensity is highly dependent on the second-order mechanisms described rigorously below, which do not follow the Stern Volmer equation. The inventors thus concluded that the DF energy (time integral of the intensity) is more robust for such an analysis.

[0286] If excimers did not exist, the ratio of energy at 670 and 630 nm would be constant over time. Showing its variations could be enough a proof. But to make it more compelling, the inventors compared the possible variations of this ratio with the DF lifetime at 630 nm τ.sub.630: if excimers exist, these two variables should be anti-correlated: When the PpIX concentration is high, τ.sub.630 small. At the same time, excimers form and produce DF at 670 nm, which causes E.sub.670/E.sub.630 to be high. Conversely, when the PpIX concentration is reduced, τ.sub.630 is longer, and no excimers form so that E.sub.670/E.sub.630 is small.

[0287] The result is shown in FIG. 8. Averaging the variations for a large number of patients would erase the erratic changes due to the local PpIX inhomogeneities and would only reflect the average bulk PpIX concentration (thus repeating FIG. 2). To detect the effect of local excimer formation, one has to compare τ.sub.630 and E.sub.670/E.sub.630 for each patient.

[0288] Nevertheless, to prove that this phenomenon was common in our measurements, we add the corresponding graphs of 4 other volunteers on FIG. 9. The inventors try to confirm their finding by comparing the DF lifetime at 630 and 670 nm in two different conditions. First, they apply a standard method of compression of the probe against the skin for one minute before measuring. As documented, the pressure blocks the microcirculation, and soon the quasi-totality of the intracellular oxygen is consumed, bringing the pO.sub.2 close to 0. At this point, the quenching by oxygen is removed, and the competition between the monomer and excimer becomes clearly visible. In such case, τ.sub.630 and τ.sub.670 should, once again, be anticorrelated over time.

[0289] The result is presented in FIG. 10. In the case where the microcirculation was not compressed, and pO.sub.2 is around physiologic values, a different behaviour is found: since the quenching by oxygen is very strong, it defines the lifetime, and the competition between monomer and excimer form becomes secondary. The inventors found τ.sub.630 and τ.sub.670 to be correlated, as they are defined by the oxygen quenching. FIG. 15A or/and FIG. 15B show the results.

[0290] These measurements all speak in favour of the existence of excimers in vivo. The inconsistencies detailed above could potentially have found an answer. However, the fact that excimer formation, a second-order process, plays a significant role in vivo in the tested conditions, opens an unforeseen question: is it sure that P-type DF, a second order DF process (2T.sub.1.fwdarw.S.sub.1+S.sub.0), does not occur? Could one bimolecular process appear and another one, starting with the same reactants, be negligible?

[0291] Contribution of P-type DF

[0292] The easiest method to discriminate between E and P-type DF is to study the dependence of DF energy on excitation intensity: in the case of a monomolecular process (E-type), this dependency should be linear, while in the case of a bimolecular one (P-type), it should be quadratic. Indeed, the initial concentration of triplet state [T.sub.1].sub.0 is proportional to the excitation intensity, as the illuminated volume is constant. By fitting a second order polynome

E.sub.DF=aI.sub.ex.sup.2bI.sub.ex+c

[0293] one can compare the linear contribution bI.sub.ex to the quadratic one aI.sup.2.sub.ex. An example of such curve is presented in FIG. 11. This type of measurement was repeated several times to confirm its conclusions, in various conditions (compressed or not, different times of ALA application). In addition to the Dumbo setup (described above), the same experiment was conducted on the “Opotek” setup, described in Harms et al., 2013. The radiant exposure in the Dumbo setup is 300 μJ.cm.sup.−2 per pulse, which is comparable to the Opotek (10 to 100 μJ.cm.sup.−2 per pulse). The averaged quadratic contributions are presented in table 1.

TABLE-US-00001 TABLE 1 P-type contribution in the total DF signal, at different ALA application time and pO.sub.2 levels, on two distinct setups Time of ALA application Dumbo Dumbo − pO.sub.2 = 0 Opotek 3 h / 30% / 5 h 70% 40% 40% 7 h 70% 40% 40%

[0294] Several conclusions can be drawn from these experiments: [0295] In standard, uncompressed measurements, P-type DF contributes between 40% to 70% to the total DF signal. [0296] The absence of oxygen significantly reduces the contribution of P-type DF. The increase of the contribution of E-type DF can be understood as a kinetic competition between first order processes, E-type DF and oxygen quenching: T.sub.1.fwdarw.S.sub.1 versus T.sub.1+.sup.3 O.sub.2.fwdarw.S.sub.0+.sup.1 O.sub.2. [0297] Consequently, it is preferable not to neglect the contribution of P-type DF in the analysis of DF signals, as has been done so far. In the next section, the inventors present a precise model to take different DF processes into account.

[0298] Influence of Second Order Reactions on DF Processes: The Kinetics of Non-MonoexPonential Decays

[0299] Evolution of the Triplet State: The Depopulation Equation

[0300] Reminder on Kinetics

[0301] Considering the reaction

[00036]$A\stackrel{k}{.fwdarw.}B$

we define the reaction rate constant:

[00037]$\begin{array}{cc}\xi \left(t\right)=-\frac{\partial A}{\partial t}=\frac{\partial B}{\partial t}& \left(5\right)\end{array}$

[0302] under the hypothesis of the law of mass action (reaction rate proportional to the product of the concentration of all reactants at the power of their stoichiometric coefficients), the reaction rate is

r=k[A]  (6)

[0303] with k the reaction rate constant, which, as a first approximation, depends minimally on concentrations of reactants. In our case of excited states dynamics, the situation is iden-tical: reaction rates are also proportional to the reactants concentration, following the linear Einstein equations. When a reactant appears with a coefficient 1, in the equation, its concentration can be computed by solving the linear differential equation:

[00038]$\begin{array}{cc}\frac{\partial A}{\partial t}=-k\left[A\right]& \left(7\right)\end{array}$

[0304] The result is an exponential function:

[A](t)=[A].sub.0e.sup.−kt=[A].sub.0e.sup.−t/τ  (8)

[0305] which allows the introduction of a characteristic time τ=1/k, which does not depend on [A]. It is called the lifetime of A. To be completely clear, the population evolves as a mono-exponential if and only if it is only depopulated by first-order reactions. A mono-exponential kinetics is equivalent to a first-order depopulation. But as soon as two or more coefficients appears in front of A, we are in the non-linear case:

[00039]$\begin{array}{cc}2A\stackrel{k^{\prime }}{.fwdarw.}B& \left(9\right)\end{array}$

[0306] This case follows the kinetics:

[00040]$\begin{array}{cc}\frac{\partial A}{\partial t}=-2{k^{\prime }\left[A\right]}^2& \left(10\right)\end{array}$

[0307] where the concentrations evolution is harder to determine, and is not mono-exponential. When a reactant appears with a coefficient one, we will indifferently designate the process as linear, monomolecular or first-order. Conversely, when the stoichiometric coefficient is 2, we use the terms non linear, bimolecular or second-order.

[0308] The Traditional Stern-Volmer Model

[0309] In the traditional model, the assumption is that only E-type DF, Internal Conversion and oxygen quenching contribute to the triplet state desexcitation:

[00041]$\begin{array}{cc}\frac{\partial \left[T_1\right]}{\partial t}=-\left(k_E+k_{\mathrm{IC}}+k_q^{O_2}{pO}_2\right)\left[T_1\right]& \left(11\right)\end{array}$

[0310] Taking Second-Order Desexcitation in Account

[0311] The inventors have shown above that the contribution of P-type DF was not negligible. They have also identified a potentially strong effect of excimer formation, which is also bimolecular. Based on this changed technical understanding of the processes involved, they include the quadratic desexcitation terms in the depopulation equation, which thus loses its linear nature (see definition of reaction rates on FIG. 7:

[00042]$\begin{array}{cc}\frac{\partial \left[T_1\right]}{\partial t}=-\left(k_E+k_{\mathrm{IC}}+k_q^{O_2}{pO}_2\right)\left[T_1\right]-2{\left(k_P+k^d\right)\left[T_1\right]}^2& \left(14\right)\end{array}$

[0312] This solves into a more complex function. As it involves both first and second-order reactions, it is designated the mixed-orders depopulation function:

[00043]$\begin{array}{cc}{\left[T_1\right]}_{\mathrm{mixed}}\left(t\right)=\frac{k_E+k_{\mathrm{IC}}+k_q^{O_2}{pO}_2}{\begin{array}{c}\left(\frac{k_E+k_{\mathrm{IC}}+k_q^{O_2}{pO}_2}{{\left[T_1\right]}_0}+2\left(k_P+k^d\right)\right)e^{\left(k_E+k_{\mathrm{IC}}+k_q^{O_2}p{O}_2\right)t}-\\ 2\left(k_P+k^d\right)\end{array}}& \left(15\right)\end{array}$

[0313] In this expression, one can spot the exponential term

[00044]$e^{-\left(k_E+k_{\mathrm{IC}}+k_q^{O_2}p{O}_2\right)t},$

perturbed by the second-order constants k.sub.p+k.sup.d. We can study approximate forms of the function at short and long times:

[0314] Mixed-Orders Depopulation at Short Times

[00045]$\begin{array}{cc}\mathrm{When}t.fwdarw.0& \left(16\right)\end{array}$${\left[T_1\right]}_{\mathrm{mixed}}{\left(t\right)}_0^{~}\frac{{\left[T_1\right]}_0}{1+\left(k_E+k_{\mathrm{IC}}+k_q^{O_2}{pO}_2+2{\left(k_P+k^d\right)\left[T_1\right]}_0\right)t}$

which yields, [T.sub.1](0)=[T.sub.1].sub.0, but also

[00046]$\frac{\partial \left[T_1\right]}{\partial t}\left(t=0\right)=-{\left(k_E+k_{\mathrm{IC}}+k_q^{O_2}{pO}_2+2{\left(k_P+k^d\right)\left[T_1\right]}_0\right)\left[T_1\right]}_0$

[0315] And if we assume that [T.sub.1].sub.0 is big enough, this even reduces to

[00047]$\begin{array}{cc}{\left[T_1\right]}_{\mathrm{mixed}}{\left(t\right)}_0^{~}\frac{{\left[T_1\right]}_0}{1+2{\left(k_P+k^d\right)\left[T_1\right]}_{0}t}\mathrm{and}& \left(17\right)\end{array}$$\frac{\partial \left[T_1\right]}{\partial t}\left(t=0\right)=-2{\left(k_P+k^d\right)\left[T_1\right]}_0^2$

[0316] Thus, the beginning of the depopulation is completely controlled by the second-order processes, and its slope reflects the reaction rate constants k.sub.p and k.sup.d and the initially excited population [T.sub.1].sub.0.

[0317] Mixed-Orders Depopulation at Long Times

[0318] When t.fwdarw.+∞

[00048]$\begin{array}{cc}{\left[T_1\right]}_{\mathrm{mixed}}{\left(t\right)}_{+\infty }^{~}\frac{{\left(k_E+k_{\mathrm{IC}}+k_q^{O_2}{pO}_2\right)\left[T_1\right]}_0}{k_E+k_{\mathrm{IC}}+k_q^{O_2}{pO}_2+2{\left(k_P+k^d\right)\left[T_1\right]}_0}{e}^{-\left(k_E+k_{\mathrm{IC}}+k_q^{O_2}{\mathrm{pO}}_2\right)t}& \left(18\right)\end{array}$

a mono-exponential decay of lifetime follows. exactly like the Stern-Volmer theory:

[00049]${\tau }_{\infty }=\frac{1}{k_E+k_{\mathrm{IC}}+k_q^{O_2}p{O}_2}$

[0319] This shows that the later part of the desexcitation follows a Stern-Volmer quenching by oxygen.

[0320] However, the DF signal I.sub.DF(t) is only an indirect image of the triplet population [T.sub.1](t): to draw conclusions on how to analyse the decays, the inventors investigated this link.

[0321] From Triplet Concentration Evolution to DF Signals

[0322] The DF intensity produced by E-type DF is worth:

I.sub.DF.sup.E(t)=θ.sub.S.sub.1k.sub.E[T.sub.1](t)  (19)

with

[00050]${\theta }_{S_1}=\frac{k_f^s}{k_f^s+k_{\mathrm{nr}}^s+k_{\mathrm{ISC}}+k_S^d\left[S_0\right]}$

the fluorescence quantum yield of the S.sub.1 state. Conversely the

[0323] DF intensity produced by P-type DF is:

I.sub.DF.sup.P(t)=θ.sub.S.sub.1k.sub.p[T.sub.1].sup.2(t)  (20)

and finally the excimer DF intensity:

I.sub.DF.sup.excimer(t)=θ.sub.D.Math.(k.sub.T.sup.d[T.sub.1].sup.2(t)+[S.sub.1](t)[S.sub.0])  (21)

with

[00051]$\left[S_1\right]=\frac{{\theta }_{S_1}}{k_f^s}\left(k_E\left[T_1\right]+{k_P\left[T_1\right]}^2\right)$

with

[00052]${\theta }_{D^{*}}=\frac{k_f^d}{k_f^d+k_{\mathrm{nr}}^d}$

the quantum yield of the excimer state. The expected DF intensity can be deduced by injecting the expression of [T1] of equation (15) into equations (19) to (21).

[0324] Pure Second-Order Desexcitation and DF

[0325] The inventors have further explored the result of a process where second-order reactions are so much stronger than first order ones that the depopulation equation becomes:

[00053]$\begin{array}{cc}\frac{\partial \left[T_1\right]}{\partial t}\approx -2{\left(k_P+k^d\right)\left[T_1\right]}^2& \left(22\right)\end{array}$

[0326] In that case, the solution is

[00054]$\begin{array}{cc}\left[T_1\right]\left(t\right)=\frac{{\left[T_1\right]}_0}{1+2{\left(k_P+k^d\right)\left[T_1\right]}_{0}t}& \left(23\right)\end{array}$

and finally, it is logical to consider that only P-type will produce DF:

[00055]$\begin{array}{cc}{I}_{\mathrm{DF}}^P\left(t\right)={\theta }_{S_1}{k_P\left(\frac{{\left[T_1\right]}_0}{1+2{\left(k_P+k^d\right)\left[T_1\right]}_{0}t}\right)}^2& \left(24\right)\end{array}$

[0327] To assess the existence of P-type or excimer DF in vivo, the inventors compared the quality of a mono-exponential fit versus a mixed-orders decay fit on the DF signals collected. This is done in the next section.

[0328] Results

[0329] The inventors assessed the nature of the DF process in vivo, at standard excitation intensity. The central hypothesis is that the depopulation equation is of mixed orders, and the DF is caused by a mix of E-type and P-type. Thus, the inventors' proposition for the most exact fit is:

I.sub.DF.sup.630(t)=αI.sub.DF.sup.E+βI.sub.DF.sup.p=αθ.sub.S.sub.1k.sub.E[T.sub.1](t)+βθ.sub.S.sub.1k.sub.p[T.sub.1].sup.2(t)  (25)

with [T.sub.1](t) from equation (15). Unfortunately, this function requires sophisticated fitting methods.

[0330] To allow for fitting with more standard type fitting methods, e.g., as implemented in Origin, the inventors compared the fitting quality of 4 distinct functions: Mono-exponential (1.sup.st order depopulation and E-type DF), Mixed-orders+E-type DF, mixed-orders+P-type DF, and 2.sup.nd order+P-type DF (equation (24)). In essence, the inventors are assessing the nature of two reactions: [0331] The reaction that will depopulate the triplet, whose kinetics will define the depopulation equation. The inventors call it the depopulation pathway. This reaction can be of the first order (Internal Conversion, quenching by oxygen), or of the second order (P-type RISC). [0332] The reaction that will produce the delayed fluorescence. The inventors call it the measurement pathway. In our case, it can be E-type DF or P-type DF.

[0333] The combination of these two reactions defines the expected kinetics of the DF. The different possibilities and associated kinetics are summarized in Table 2.

TABLE-US-00002 TABLE 2 Row: measurement pathway Column: depopulation pathway E-type DF P-type DF 1.sup.st order (ex: IC) S.sub.DF(t) = θ.sub.S.sub.1k.sub.E[T.sub.1].sub.0e.sup.−at S.sub.DF(t) = θ.sub.S.sub.1k.sub.P[T.sub.1].sub.0.sup.2e.sup.−2at [00056]$\frac{\partial \left[T_1\right]}{\partial t}=-a\left[T_1\right]$ 2.sup.nd order (ex: P-type) [00057]$\frac{\partial \left[T_1\right]}{\partial t}=-{b\left[T_1\right]}^2$ [00058]$S_{DF}\left(t\right)={\theta }_{S_1}{k}_E\frac{{\left[T_1\right]}_0}{1+{b\left[T_1\right]}_{0}t}$ [00059]$S_{DF}\left(t\right)={\theta }_{S_1}{k_P\left(\frac{{\left[T_1\right]}_0}{1+{b\left[T_1\right]}_{0}t}\right)}^2$ Mixed-orders S.sub.DF(t) = θ.sub.S.sub.1k.sub.E[T.sub.1].sub.mixed(t) S.sub.DF(t) = θ.sub.S.sub.1k.sub.P[T.sub.1].sub.mixed.sup.2 [00060]$\frac{\partial \left[T_1\right]}{\partial t}=-a\left[T_1\right]-{b\left[T_1\right]}^2$ Delayed fluorescence kinetics for different depopulation and measurement pathways. Notation: a = k.sub.E + k.sub.IC + k.sub.q.sup.O.sup.2pO.sub.2, b = 2(k.sub.P + k.sup.d) and [T].sub.mixed: expression from equation 15.1st order depopulation and P-type DF is unlikely to happen in reality, because 1st order is associated to low concentration of excited states, while P-type corresponds to high concentrations. The same remark goes for 2nd order depopulation and E-type DF.

[0334] To determine which reactions are the measurement and depopulation pathways, the inventors consider standard 630 nm DF signals obtained on volunteers' sternum after 3 to 7 hours of ALA application. The inventors compared the statistical quality of different fits in Table 3.

[0335] These measurements further confirm that P-type and excimer DF are so strong that they significantly change the shape of the decays. Moreover, the great success of second-order+P-type fitting function (introduced above) suggests that most triplet state molecules desexcite via P-type RISC and DF. This does not mean that oxygen quenching is negligible: as described in equation 18, it controls the slope of the decay at longer times. Even if it has a moderate influence on the whole decay (and thus the global shape mainly depends on second-order processes), it has a strong influence on the lifetime that one can extract from the signal. It should also be considered that multi-parameters non-linear fittings are challenging: it is possible that, mixed-orders+P-type DF should the best fit.

[0336] Conclusion

[0337] In this section, the inventors have shown the necessity to take second order processes into account in order to correctly interpret the DF kinetics, and have exhibited the appropriate decay function. This allows for the development of methods that take these effects

TABLE-US-00003 TABLE 3 Fitting parameters and precision of different fitting functions. Notations defined in table 2. The 2.sup.nd order + P-type DF function is the most precise. Different methods indicate the second-order rate constant, b = 2(k.sub.P + k.sup.d) = 10.sup.8 M.sup.−1s.sup.−1, and the initial concentration of excited triplet states [T.sub.1].sub.0 ∈ [28; 51] μM. This high value is another element which suggests that intracellular PpIX concentration could be higher than previously thought. Fitting function r.sup.2 a [s.sup.−1] b [M.sup.−1s.sup.−1] [T.sub.1].sub.0 [M] Mono-exp 0.913 1.15 × 10.sup.4  / / Mixed- 0.936 .sup. 8.46 × 10.sup.−10 1.14 × 10.sup.9 2.82 × 10.sup.−5 orders + E-type Mixed- 0.933 2.73 × 10.sup.−8 2.04 × 10.sup.8 4.79 × 10.sup.−5 orders + P-type 2.sup.nd 0.996 /  2.5 × 10.sup.8  5.1 × 10.sup.−5 order + P-type
into account and measure the pO.sub.2 more precisely. The correction can be done at different levels of accuracy and practical feasibility, described below.

[0338] Practical Consequences on Skin Mitochondrial Oxygen Measurement

[0339] Beside the new understanding of PpIX DF, the objective of the invention remains to improve the quality of cellular oxygen measurements, in particular, in patients. More specifically, the objective is to solve the inconsistencies described above. The inventors describe several methods in this section that solve this problem.

[0340] 1. The Decay Central Fitting (DCF) Method to Capture the Linear Quenching Processes

[0341] As detailed above, a mixed-order decay (equation (15)) can be temporally subdivided in three distinctive parts: [0342] The beginning of the decay is not mono-exponential, and governed by a mix of first and second order processes (equation (17)). Its expression involves [T.sub.1].sub.0, which means that it depends on excitation intensity and local chromophore concentration. One simple way to obtain a stable pO.sub.2 value thus is not to fit this part of the decay which is completely relative and changing. [0343] The central part of the decay is a mono-exponential whose inverse lifetime is directly proportional to the pO.sub.2 (equation (18)). Its lifetime does not depend on [T.sub.1].sub.0. Even if the amplitude depends on the local fluence rate and chromophore concentration, the lifetime (i.e. the slope of the decay) does not: this section of the decay may be fitted by a mono-exponential. [0344] The last part of the decay is dominated by noise. It is preferably neglected and not fitted.

[0345] This subdivision is illustrated in FIG. 12. This splitting shows that only the center of the decay should be fitted to a mono-exponential, and this is enough to precisely deduce the pO.sub.2, despite the second-order effects. The inventors call this approach the decay central fitting method. It is not an experimental optimization of the fitting quality: it reflects an improved conceptual understanding of the luminescence process.

[0346] The determination of t.sub.1, the time of transition between the initial non-exponential phase and the monoexponential phase, and t.sub.2, the time between the monoexponential phase and the noise-dominated section, works as follows: assuming that the DF intensity I(t) can be written as the sum of noise ΔI(t), centered around 0, and an average intensity Ī(t), which is typically the function obtained after fitting of the experimental signal, such that I(t)=Ī(t)+ΔI(t).

[0347] Defining the monoexponentiality indicator as

[00061]$M\left(t\right)={\left(1-\frac{{\stackrel{\_}{I}}^{″}\stackrel{\_}{I}}{{\left({\stackrel{\_}{I}}^{\prime }\right)}^2}\right)}^2$

[0348] With ′ denoting the derivation with respect to time. The thresholds are values such that: M(t.sub.1)=0.7=T.

[0349] Introducing the signal to noise ration SNR(t), the noise dominated section starts, for example, at the time value such that

[00062]$\mathrm{SNR}\left(t_2\right)=\frac{\stackrel{\_}{I}\left(t_2\right)+\Delta I\left(t_2\right)}{\stackrel{\_}{I}\left(t_2\right)}=2=T^{\prime }$

[0350] The inventors set the optimal values for the decay central fitting method to [t.sub.1; t.sub.2]=[275; 600]μs in the experimental conditions used (i.e., in the Dumbo setup) based on the criteria and considerations explained above.

[0351] A last point should be discussed: the lifetime indicated by the plateau (long 225 μs in this experiment on non-compressed skin),

is in reality not the triplet lifetime

[00063]${\tau }_T=\frac{1}{k_E+k_{\mathrm{IC}}+k_q^{O_2}{pO}_2}.$

When produced by E-type DF, the

[0352] DF intensity in the central range is indeed proportional to

[00064]$e^{-\frac{t}{\tau }},$

but when produced by P-type DF, which represents 40 to 70% of the signal, the decay lifetime is worth half of the triplet lifetime due to the quadratic dependence. Consequently, the observed lifetime is an average between τ.sub.T and τ.sub.T/2. The weights of this average cannot be simply measured by this method, and depends on the local PpIX concentration and fluence rate. The variable excitation fluence rate can be used to determine the relative contributions of E-type and P-type DF, and thus calculate T.sub.T.

[0353] To assess the efficiency of this method, the data of FIG. 2 was analysed on the reduced interval [275; 600] μs. The result is presented in FIG. 14.

[0354] The lifetime calculated is less erratic than before application of the DCF method, but some variability remains as τ varies from 100 to 350 μs. This could be due to the variable contribution of E and P-type giving a random fluctuation between τ.sub.T and τ.sub.T/2. It suggests that the DCF method can still be rendered more precise with a quantitative discrimination between mono and bimolecular processes.

[0355] Conclusion

[0356] The decay central fitting method constitutes a step-forward towards a stable, precise measurement of the pO.sub.2. It is easy to apply, as it only requires to place a 630 nm bandpass filter in the emission beam and to change the mono-exponential fitting range to the time between [t.sub.1; t.sub.2], e.g., [275; 600] μs.

[0357] As its precision is limited by the unknown E-type vs P-type DF ratio, it is preferred to simultaneously vary the excitation intensity, quantify the fraction of linear vs quadratic behaviour (as done in the section “contribution of P-type DF” above), and subsequently deduce a more precise Ty and pO.sub.2.

[0358] 2. Correction of PpIX Self-Quenching: The Adaptive Stern-Volmer Relationship

[0359] Another option, which would be conceptually less different from the standard model, but practically more complex, is to conserve the Stern-Volmer model, but to adapt the DF lifetime in absence of oxygen at 630 nm τ.sub.0.sup.630, in order to compensate for the variable self-quenching of PpIX. Keeping in mind the central results discovered by the inventors: first, that PpIX lifetime in absence of oxygen strongly depends on the local PpIX concentration, in a Stern-Volmer like quenching relationship, and second, that the ratio of DF energy at 670 and 630 nm is linearly linked to the local PpIX concentration. The method comprises:

[0360] measuring the DF signal at 630 nm and 670 nm simultaneously and in the same sample volume,

[0361] using the ratio of intensities I.sub.0.sup.670/I.sub.0.sup.630 and/or related parameters such as energies

[00065]$\frac{E_{670}}{E_{630}}$

or the inverse of such ratios or the ratio's of intensities at a certain time after the excitation pulse, or energies in a certain time period with respect to an excitation pulse (for example 1 μs or between 1 μs and 100 μs after the excitation pulse had ended or fallen back to below 1/10.000.sup.th of its peak) and/or with intensities or energies determined in a part of the spectrum including one of the wavelengths 670 nm or 630 nm, but not including wavelengths within more than 1 nm, preferably more than 5 nm, more preferably more than 15 nm, from the other and setting it in a relationship to an intensity or the energy observed in another part of the spectrum or in the entire spectrum, or from determining the phase shift with respect to a modulated excitation source of luminescence in such spectra, or the phase shift of relationships of spectra or the phase shift of the transient absorption of a probing wavelength to deduce the local concentration of PpIX,

[0362] deduce τ.sub.0.sup.630 out of [PpIX] according to the Stern-Volmer quenching relationship established earlier.

[0363] injecting τ.sub.0.sup.630 and τ.sup.630 into the adaptative Stern-Volmer relationship, and finally calculating the corrected pO.sub.2:

[00066]$\begin{array}{cc}{pO}_2^{\mathrm{corrected}}=\frac{1}{k_q^{O_2}}\left(\frac{1}{{\tau }_{630}}-\frac{1}{{\tau }_0^{630}}\right)& \left(26\right)\end{array}$

[0364] As shown in FIG. 22, it has been proven in vitro that the energies (time integrals if the decays) at 670 nm and 630 nm were linearly correlated to the concentration of PpIX.

[0365] An application of this method is provided in FIG. 16. The graph shows a much more constant value for the concentration of oxygen, and thus, a significantly better assessment of the status of the subject.

[0366] Conclusion

[0367] This method has shown very good results: on volunteers whose pO.sub.2 had no reason to vary abruptly, the adaptive Stern-Volmer relationship flattened the pO.sub.2 around a stable value, which is physiologically more realistic. The method was very robust and yielded good results even on noisy data.

[0368] In terms of practical modifications, the method requires the selection of 2 emission wavelengths or spectral bands at about 630 and about 670 nm, thus, e.g., bandpass filters may be installed. The algorithmic part remains simple, as the fitting function is a mono-exponential. The inventors present the adaptative Stern-Volmer relationship as a most satisfactory method, in terms of precision and ease of implementation.

[0369] 3. The Complete Method: Fitting Mixed-Orders Decays

[0370] As detailed above, when taking into account all different reactions, the DF intensity at 630 nm follows:

I.sub.630(t)=θ.sub.S.sub.1(k.sub.E[T.sub.1].sub.mixed+k.sub.P[T.sub.1].sub.mixed.sup.2)+y.sub.0  (27)

with

[00067]$\begin{array}{cc}{\left[T_1\right]}_{\mathrm{mixed}}\left(t\right)=& \left(28\right)\end{array}$$\frac{k_E+k_{\mathrm{IC}}+k_q^{O_2}{pO}_2}{\left(\frac{k_E+k_{\mathrm{IC}}+k_q^{O_2}{pO}_2}{{\left[T_1\right]}_0}+2\left(k_P+k^d\right)\right)e^{\left(k_E+k_{\mathrm{IC}}+k_q^{O_2}{\mathrm{pO}}_2\right)t}-2\left(k_P+k^d\right)}$

using notations introduced earlier.

[0371] The most precise method would be to fit this function on each decay. Once the different fixed parameters have been determined (O.sub.S, k.sub.E, k.sub.p,k.sub.1C, k.sub.q.sup.O.sup.2, k.sup.d), only two variables remain to be fitted: [T.sub.1].sub.0 and pO.sub.2.

[0372] To reinforce the result, it is also possible to fit the decay at 670 nm, which is worth

I.sub.DF.sup.670(t)=θ.sub.D.Math.(k.sub.T.sup.d[T.sub.1].sup.2(t)+k.sub.S.sup.d[S.sub.1](t)[S.sub.0])  (29)

with

[00068]$\left[S_1\right]=\frac{{\theta }_{S_1}}{k_f^s}\left(k_E\left[T_1\right]+{k_P\left[T_1\right]}^2\right)$

[0373] Also, once typical values of [T.sub.1].sub.0 and pO.sub.2 are known under the experimental conditions used, it is possible to start the optimization algorithm close to these values. A tailor-made algorithm can then be provided that reaches optimal parameters in a short time.

[0374] The application of this method in practice is complex: fitting of such a complex nonlinear function is a numerical challenge.

[0375] Exclusion of an In Vivo Contribution of Singlet Oxygen Feedback Delayed Fluorescence (SOFDF)

[0376] An additional mechanism of PpIX-DF has been recently described in artificial systems, e.g., by Vinklarek et al., 2017. It involves a double action of oxygen: a ground state triplet molecule collects the PpIX triplet energy, as in conventional quenching, but then this singlet oxygen excites another T.sub.1 molecule to S.sub.1, which subsequently produces SOFDF. The process can be summed up by the following reaction:

[00069]$\begin{array}{cc}\mathrm{SOFDF}:\{\begin{array}{c}{T}_1+{ }^3{O}_2.fwdarw.S_0+{ }^1{O}_2\\ T_1+{ }^1{O}_2.fwdarw.S_1+{ }^3{O}_2\\ S_1\stackrel{\mathrm{hv}}{.fwdarw.}{S}_0\end{array}& \left(30\right)\end{array}$

[0377] This process has been observed in vitro (Vinklarek et al., 2017; Scholz et al., 2013. Photochemical & Photobiological Sciences 12: 1873) and in isolated fibroblasts (Scholz et al., 2017. Photochemical & Photobiological Sciences 16: 1643-1653; Scholz et al., 2015. Photochemical & Photobiological Sciences 14: 700-713), with different photosensitizers, including PpIX. This type of DF is characterized by a rise-decay kinetics, described by

I.sub.SOFDF(t)=C(e.sup.−t/τ1−e.sup.−t/τ2)  (31)

with

[00070]$\tau =\frac{{\tau }_T}{1+{\tau }_T/{\tau }_{\Delta }}$

the decay time and τ.sub.2=τ.sub.T/2 the rise time (τ.sub.T: triplet state lifetime and τΔ: singlet oxygen lifetime).

[0378] Despite careful observations, the inventors were not able to identify the rise decay kinetics in vivo, suggesting that SOFDF did not significantly occur in human skin. This can be explained by two reasons: [0379] First and foremost, all SOFDF observations with cells in the state of the art were done on air saturated samples of isolated cells, where the pO.sub.2 must have been close to 160 mm Hg. In tissues such as human skin, the pO.sub.2 is closer to 50 mm Hg, i.e., at least three times smaller. Since SOFDF involves two oxygen molecules, it requires a high pO.sub.2 to be detectable. In the tissues and organs analysed in the context of the invention, the pO.sub.2 is too low to detect SOFDF. [0380] The spatial irradiance used by Vinklarek et al., 2017, was around 230 times more intense than what is used in the Dumbo setup (70 mJ.cm.sup.−2 vs 300 μJ.cm.sup.−2). This could explain a different behaviour.

[0381] Compatibility with Previous Measurements (Oxygen Consumption)

[0382] A natural question after the discovery of PpIX excimers is: how measurements and conclusions in the past have been performed, using an incomplete model of triplet-state based optical properties, such as delayed fluorescence of probes, e.g., PpIX? The case of Oxygen Consumption Measurements (OCM) is typical: as described in the literature described above, this method, based on the compression of the microcirculation to assess the rate of oxygen consumption by skin cells, has shown excellent results in recent years.

[0383] How can the results have been so coherent, if the fitting method was incorrect? To evaluate it, the inventors compared the pO.sub.2 consumption curve obtained 1) in the physiologic case, according to the models of oxygen consumption as provided by the present invention, and 2) if this physiologic pO.sub.2 was deduced using the standard Stern-Volmer method, while in reality the PpIX decay follows the mixed-orders+P-type function (equation (15)):

[0384] According to simple models of oxygen consumption by tissue, in case of microcirculation blockage by compression, the pO.sub.2 evolution can be simplified to:

pO.sub.2.sup.real(t)=1+e.sup.−t  (33)

where we arbitrarily set the time constant to 1. Assuming that the pO.sub.2 varies as pO.sub.2.sup.real, the DF signal that would be obtained if the DF process was a mixed-orders+P-type DF (equation (15)) was deduce, as suggested by table 3. This gives us a series of signals DF.sup.real(t) for each value of pO.sub.2.sup.real(t). Then mono-exponential decays were incorrectly fitted to the different DF.sup.real(t), thus obtaining an incorrect pO.sub.2.sup.incorrect(t) thanks to the standard Stern-Volmer relationship. Finally, pO.sub.2.sup.real was compared to pO.sub.2.sup.incorrect(t) to see how different the pO.sub.2 becomes when one uses an incomplete model of PpIX DF.

[0385] The result is presented in FIG. 17. This graph shows important information: even when using an improper Stern-Volmer model of PpIX DF, the pO.sub.2 obtained is not too different from the reality, for the simple reason that whatever the model, when the pO.sub.2 increases, the DF characteristic time decreases, and vice-versa.

[0386] As a conclusion, it is perfectly possible that a large number of studies were conducted based on an improper model of PpIX model, giving sometimes satisfactory results: it is only when looking in detail into the DF shape and lifetime variations that the inconsistencies can be detected, and then overcome by the method of the invention.