Method to determine the absorption coefficient in turbid media

Abstract

The invention relates to a method to determine the wavelength dependent absorption coefficient of a turbid medium using overlapping illumination-detection areas comprising the steps of a) retrieving a calibration spectrum (CA) from a reference measurement using a reference sample; b) carrying out a measurement on an actual sample for determining the absolute reflection spectrum (R.sub.abs) using a raw spectrum measured on the sample (S.sub.medium) and the calibration spectrum (C.sub.); C) using the absolute reflection spectrum (R.sub.abs) for determining the wavelength dependent absorption coefficient by minimizing the difference between the measured absolute reflection spectrum (R.sub.abs) and a model function (R.sub.abs.sup.model). wherein the model function (R.sub.abs.sup.model) is modelled using a predetermined equation based on prior knowledge of the combination of a dependence of the effective photon path length (L.sub.PF) on a scattering phase function (PF); a dependence of the absolute reflectance in the absence of absorption (R.sub.abs.sup.0) on scattering phase function (PF). The invention further relates to a system and a computer program product for determining the wavelength dependent absorption coefficient of a turbid medium.

Claims

1. A method to determine the wavelength dependent absorption coefficient of an actual sample of a turbid medium without prior knowledge of an actual scattering coefficient using overlapping illumination and detection areas, wherein measurements are carried out using a single optical fiber for delivering a light beam towards the actual sample for illumination and for collecting a reflected light beam from the actual sample for detection, wherein a light spot of the delivered light beam used for illumination overlaps a light spot of the reflected light beam used for detection, the method comprising the steps of: a. retrieving a calibration spectrum from a reference measurement using a reference sample; b. measuring the actual sample using the single optical fiber to deliver a light beam towards the actual sample to determine an absolute reflection spectrum using a raw spectrum measured on the actual sample and the calibration spectrum; c. using the absolute reflection spectrum to determine the wavelength dependent absorption coefficient by minimizing the difference between the measured absolute reflection spectrum and a model function, wherein d. the model function is modelled using a pre-determined equation based on prior knowledge of the combination of: i. a dependence of an effective photon path length on a scattering phase function; ii. a dependence of the absolute reflectance in the absence of absorption on the scattering phase function; and iii. an assumed estimate for the scattering coefficient, wherein the effective photon path length and the absolute reflectance in the absence of absorption and are both a function of the assumed estimate for the scattering coefficient, wherein an influence of a mis-estimation of the scattering coefficient on the value of the absolute reflectance in the absence of absorption is at least partially compensated by the influence of the mis-estimation of the scattering coefficient on the value of the effective photon path length such that the wavelength dependent absorption coefficient of the actual sample of the turbid medium is determined without prior knowledge of the actual scattering coefficient of the actual sample.

2. The method according to claim 1, wherein the calibration spectrum is retrieved by a measurement performed with the single fiber in contact with the calibration sample for retrieving an absolute device calibration spectrum.

3. The method according to claim 2, where for the calibration measurement a scattering reference sample is used having a scattering coefficient such that .sub.s d.sub.fiber>10, wherein .sub.s is the calibration sample's reduced scattering coefficient and d.sub.fiber is the single fiber's diameter.

4. The method according to claim 1, wherein light used for the measurements is monochromatic or continuous spectrum light.

5. The method according to claim 1, wherein light used for measurements comprises a continuous spectrum of wavelengths.

6. The method according to claim 1, wherein the turbid medium is tissue.

7. The method according to claim 1, wherein the absorption coefficient is related to the concentration and/or packaging of absorbing molecules selected from the group consisting of hemoglobin, bilirubin, beta-carotene, melanin, cytochrome, glucose, lipid and water.

8. The method according to claim 1, wherein for the assumed scattering phase function a gamma value between =1.6 and =1.8 is used, where gamma is related to the first and second moments, g1 and g2, respectively, of the scattering phase function according to =(1g2)/(1g1).

9. The method according to claim 1, where for the calibration measurement a reference sample of known phase function and scattering coefficient is used.

10. The method according to claim 1, wherein the model function is described by the following equation:
R.sub.abs.sup.model=R.sub.abs.sup.0.Math.exp(.sub.a.Math.L.sub.SF.sup.model); wherein R.sub.abs.sup.model is the model function; .sub.a is the wavelength dependent absorption coefficient; R.sub.abs.sup.0 is the absolute reflectance in the absence of absorption; and L.sub.SF.sup.model is the effective photon path length.

11. The method according to claim 1, wherein the absolute reflectance in the absence of absorption is described by the following equation: $R_{\mathrm{abs}}^0={}_c\left(1+P_4{e}^{\left(-P_5{}_s^{}{d}_{\mathrm{fiber}}\right)}\right)\left[\frac{{\left({}_s^{}{d}_{\mathrm{fiber}}\right)}^{P_6}}{P_5+{\left({}_s^{}{d}_{\mathrm{fiber}}\right)}^{P_6}}\right],$ wherein R.sub.abs.sup.0 is the absolute reflectance in the absence of absorption; .sub.s is the assumed estimate for the scattering coefficient; .sub.c is an asymptotic value at a diffuse limit to a single fiber collection efficiency; d.sub.fiber is a diameter of a fiber which is used for measurements; and P.sub.4, P.sub.5, P.sub.6 are scattering phase function (PF) specific parameters.

12. The method according to claim 1, wherein the effective photon path length is described by the following equation: $L_{\mathrm{SFmodel}}=\frac{C_{\mathrm{PF}}{p}_1{d}_{\mathrm{fiber}}}{{\left({}_s^{}{d}_{\mathrm{fiber}}\right)}^{p_2}\left(p_3+{\left({}_a{d}_{\mathrm{fiber}}\right)}^{p_3}\right)},$ wherein L.sub.SF.sup.model is the effective photon path length; .sub.s is the assumed estimate for the scattering coefficient; d.sub.fiber is a diameter of a fiber which is used for measurements; C.sub.PF describes dependence of the effective photon path length on the scattering phase function; and P.sub.1, P.sub.2, P.sub.3 are empirically established constants.

13. A system for determining the wavelength dependent absorption coefficient of an actual sample of a turbid medium without prior knowledge of an scattering coefficient using overlapping illumination and detection areas comprising: 1) a light source adapted to generate a light beam for illumination of the actual sample; 2) a spectrometer for detecting a reflected light beam from the actual sample and carrying out spectrum analysis based on measurement data; 3) a bifurcated optical cable, a first end of which is connected to the light source, a second end of which is connected to the spectrometer, the first and second ends being connected to a probe formed by a single optical fiber for delivering the light beam from the light source towards the actual sample for illumination and for collecting the reflected light beam from the sample for detection, wherein a light spot of the delivered light beam used for illumination overlaps a light spot of the reflected light beam used for detection; and 4) a processor adapted for: a. retrieving a calibration spectrum from a reference measurement using a reference sample; b. retrieving results of a further measurement on the actual sample using the single optical fiber to deliver a light towards the actual sample to determine the absolute reflection spectrum using a raw spectrum measured on the actual sample and the calibration spectrum; c. using the absolute reflection spectrum for determining the wavelength dependent absorption coefficient by minimizing the difference between the measured absolute reflection spectrum and a model function, wherein for the model function is modelled using a pre-determined equation based on prior knowledge of the combination of: i. a dependence of the effective photon path length on a scattering phase function; ii. a dependence of the absolute reflectance in the absence of absorption on the scattering phase function; and iii. an assumed estimate for the scattering coefficient, wherein the effective photon path length and the absolute reflectance in the absence of absorption and are both a function of the assumed estimate for the scattering coefficient, wherein an influence of a mis-estimation of the scattering coefficient on the value of the absolute reflectance in the absence of absorption is at least partially compensated by the influence of the mis-estimation of the scattering coefficient on the value of the effective photon path length such that the wavelength dependent absorption coefficient of the actual sample of the turbid medium is determined without prior knowledge of the actual scattering coefficient of the actual sample.

14. The system according to claim 13, wherein the light source comprises a monochromatic or continuous spectrum light source.

15. The system according to claim 13, wherein the single fiber is provided in an interventional instrument.

16. The system according to claim 15, wherein the interventional instrument is a biopsy needle.

17. The system according to claim 13, where for the calibration measurement a scattering reference sample is used having .sub.s d.sub.fiber>10.

18. The system according to claim 13, where for the calibration measurement a reference sample of known phase function and scattering coefficient is used.

19. A computer program product for determining the wavelength dependent absorption coefficient of an actual sample of a turbid medium without prior knowledge of an actual scattering coefficient using overlapping illumination and detection areas, wherein measurements are carried out using a single optical fiber for delivering a light beam towards the actual sample for illumination and for collecting a reflected light beam from the actual sample for detection, wherein a light spot of the delivered light beam used for illumination is overlapping a light spot of the reflected light beam used for detection, the computer program product comprising instructions for causing a processor to carry out the following steps: a. retrieving a calibration spectrum from a reference measurement using a reference sample; b. retrieving data of a measurement on the actual sample using the single optical fiber having dimensions between 10 m and 3 mm to deliver a light towards the actual sample to determine the absolute reflection spectrum using a raw spectrum measured on the actual sample and the calibration spectrum; c. using the absolute reflection spectrum for determining the wavelength dependent absorption coefficient by minimizing the difference between the measured absolute reflection spectrum and a model function, wherein d. the model function is modelled using a pre-determined equation based on prior knowledge of the combination of: i. a dependence of an effective photon path length on a scattering phase function; and ii. a dependence of the absolute reflectance in the absence of absorption on the scattering phase function; and iii. an assumed estimate for the scattering coefficient, wherein the effective photon path length and the absolute reflectance in the absence of absorption and are both a function of the assumed estimate for the scattering coefficient, wherein an influence of a mis-estimation of the scattering coefficient on the value of the absolute reflectance in the absence of absorption is at least partially compensated by the influence of the mis-estimation of the scattering coefficient on the value of the effective photon path length such that the wavelength dependent absorption coefficient of the actual sample of the turbid medium is determined without prior knowledge of the actual scattering coefficient.

Description

BRIEF DESCRIPTION OF THE FIGURES

(1) FIG. 1 presents in a schematic way an embodiment of a system which may be used for carrying out a calibration measurement.

(2) FIG. 2 presents a number of characteristic curves.

DETAILED DESCRIPTION

(3) FIG. 1 presents in a schematic way an embodiment of a system which may be used for carrying out a calibration measurement on a sample 8. For this purpose the system 10 comprises a probe 2, a bifurcated optical cable 4, one end of which is connected to a light source 6 and the other end of which is connected to a suitable spectrometer 7. Accordingly, the optical fiber 2 is used for delivering a light beam from the light source 6 to the sample and for collecting reflected light from the sample 8.

(4) It is further found to be advantageous to polish the probe 2 at an angle larger than arcsin(NA/n.sub.sample) with respect to a vertical line for minimizing specular reflections, where NA is the numerical aperture of the fiber and n.sub.medium is the refractive index of the sample.

(5) The reflectance of the sample is in case of a high scattering sample given by R.sub.sample=.sub.c(NA()).

(6) When the absorption coefficient of a turbid medium (tissue) is to be determined, the equation 1 has to be used in a Lambert-Beer equation, according to R.sub.abs=R.sub.abs.sup.0exp(.sub.aL.sub.SFmodel). In a general way, the equation 1 can be written as:

(7) $\begin{array}{cc}{L}_{\mathrm{SF}\mathrm{model}}=\frac{C_{\mathrm{PF}}{p}_1{d}_{\mathrm{fiber}}}{{\left({}_s^{}{d}_{\mathrm{fiber}}\right)}^{p2}\left(p_3+{\left({}_a{d}_{\mathrm{fiber}}\right)}^{p_3}\right)}& \left(5\right)\end{array}$

(8) As has been indicated earlier, in equation (5) PF and .sub.s of tissue are not known, which implies that C.sub.PF is not known and that specification of .sub.s from reflectance R.sub.abs.sup.0 also requires knowledge of PF for specifying the correct constants P.sub.4, P.sub.5, and P.sub.6 in equation 2.

(9) In accordance with the invention, .sub.s is estimated from reflectance R.sub.abs.sup.0 such that a potential mis-estimation of .sub.s is compensated by a corresponding mis-estimation of C.sub.PF.

(10) It is found that the ratio of C.sub.PF/(.sub.s).sup.p2 is approximately equal to its true value (within 7.5% for biological tissues), provided the C.sub.PF is properly linked to the phase function used to estimate .sub.s from R.sub.abs.sup.0 (i.e. C.sub.PF is linked to the values of P.sub.4, P.sub.5, and P.sub.6 in equation 2).

(11) It is found that as high angle scattering events become more likely, R.sub.abs.sup.0 increases because incident photons are more likely to be collected and the photon path length L.sub.SF decreases as those collected photons are likely to travel a shorter path.

(12) In FIG. 2, a number of characteristic curves is presented, wherein it is found that the curve for C.sub.PF=0.944, P.sub.4=1.55, P.sub.5=6.82 and P.sub.6=0.969 is an optimal curve for practicing the invention. FIG. 2 shows the relation between R.sub.abs.sup.0 and .sub.sd.sub.fiber for 3 exemplary embodiments of known samples having different known phase functions PF with different backscattering components. The C.sub.PF values and P.sub.4, P.sub.5, and P.sub.6 values for these phase functions PF are also indicated for each of these selected PF's. The graphs presented in FIG. 2 are calculated using equation (2) discussed with reference to the foregoing.

(13) For an unknown sample, such as tissue, utilization of equations (1) and (2) to calculate the photon path length L.sub.SF requires an assumption about the phase function PF, which is also unknown.

(14) It is found that it is particularly suitable to assume that the phase function PF is characterized by C.sub.PF=0.944, P.sub.4=1.55, P.sub.5=6.82 and P.sub.6=0.969 (see solid line, curve 1 in FIG. 2). However, it is also possible to implement the equation using the other curves given in FIG. 2, or any other alternative curves which may be produced using equation 2 or using Monte Carlo simulations applied to a known sample. However, it is found advantageous to select a curve whose reflectance properties are close to the reflectance properties which may be expected from a sample under investigation, such as tissue.

(15) Utilization of the P.sub.4, P.sub.5 and P.sub.6, discussed with reference to the foregoing regarding an assumed phase function PF in equation (2) corresponding to curve number 1 in FIG. 2, would yield an estimated value of .sub.s (Est) for an unknown sample based on a measured reflectance R.sub.abs.sup.0(Real).

(16) The following effect has been found when analyzing equations (1), (2) and the graphs given in FIG. 2. Should the true sample phase function PF of an unknown sample be characterized by a higher backscattering component than the assumed PF (e.g. the true combination of parameters corresponding to true curve 2 in FIG. 2 is C.sub.PF=0.86, P.sub.4=1.24, P.sub.5=4.47 and P.sub.6=0.82), then the .sub.s would be overestimated from R.sub.abs.sup.0(Real) since .sub.s (Est)>.sub.s(Real), see corresponding notations in FIG. 2.

(17) However, the initially assumed C.sub.PF (0.944), corresponding to the assumed sample curve 1 is larger than the true C.sub.PF (0.86), corresponding to the true sample curve 2. Accordingly, an over-estimation of C.sub.PF compensates for the effect of over-estimation of .sub.s on L.sub.SF in Eq. (1).

(18) Next, if the true phase function PF has in fact a smaller backscattering component than the value assumed for the phase function PF (e.g. the true sample PF corresponds to curve 3 in FIG. 2, instead of the assumed curve 2), then the resulting .sub.s (Est) obtained by using the assumed R.sub.abs.sup.0 curve 2 of FIG. 2 would be underestimated. Accordingly, also for this situation an effect of compensation takes placei.e. an under-estimation of C.sub.PF in Eq. (1), since in this case the assumed C.sub.PF (0.944) was smaller than the true C.sub.PF (1.0).

(19) Preferably, for the assumed scattering phase function PF.sup.assumed a gamma value between =1.6 and 1.8 is used, where gamma is related to the first and second moments (g.sub.1 and g.sub.2, respectively) of the scattering phase function according to =(1g.sub.2)/(1g.sub.1).

(20) The inter-related, compensating effects of mis-estimation of C.sub.PF and .sub.s through assumption of an estimated phase function PF can be further analysed by evaluation of the ratios C.sub.PF.sup.est/C.sub.PF.sup.real and (.sub.s(Est)/.sub.s(Real)).sup.0.18.

(21) It is found that these two metrics either both are smaller than unity or both are greater than unity, indicating a compensating effect on estimates of L.sub.SF. Moreover, the magnitudes of these effects are also very similar: C.sub.PF.sup.est/C.sub.PF.sup.real ranges from 0.9 to 1.12 in biological tissues, while (.sub.s(Est)/.sub.s(Real)).sup.0.18 ranges from 0.85 to 1.25 in case equation (2) is used to calculate .sub.s(Est) from R.sub.abs.sup.0.

(22) The inset (I) in FIG. 2 shows a histogram plot of the ratio of these 2 metrics (defined as the compensation factor). It will be appreciated that perfect compensation of the effect of mis-estimation of C.sub.PF and .sub.s on path length would yield a compensation factor of 1.0. The histogram clearly shows a narrow distribution centred around 1.0, with 76% of the data within 5% of this value, and 99% of the data within 10% of this value.

(23) It will be appreciated that while specific embodiments of the invention have been described above, the invention may be practiced otherwise than as described. For example, for specific turbid media different constants in the equations may be used. However, the method for determining the appropriate constants will lie within the ordinary skill of the person skilled in the art, when reducing the invention into practice.