## χ^{2} analysis for estimating the accuracy of optical properties derived from time resolved diffuse-reflectance

Optics Express, Vol. 17, Issue 22, pp. 20521-20537 (2009)

http://dx.doi.org/10.1364/OE.17.020521

Acrobat PDF (481 KB)

### Abstract

Weighted residuals and the reduced χ^{2} (χ_{R}^{2}) value are investigated with regard to their relevance for assessing optical property estimates using the diffusion equation for time-resolved measurements in turbid media. It is shown and explained, for all photon counting experiments including lifetime estimation, why χ_{R}^{2} increases linearly with the number of photons when there is a model bias. Only when a sufficient number of photons has been acquired, χ_{R}^{2} is a pertinent value for assessing the accuracy of μ_{a} and μ_{s}' estimates. It was concluded that χ_{R}^{2} is of particular interest for cases of small interfiber separation, low-level scattering, strong absorption and incorrect measurement of instrument response function. It was also found that χ_{R}^{2} is less pertinent for judging μ_{a} in case of air boundary effects.

© 2009 OSA

## 1. Introduction

1. V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. **23**(3), 313–320 (2005). [CrossRef] [PubMed]

3. L. Spinelli, A. Torricelli, A. Pifferi, P. Taroni, G. M. Danesini, and R. Cubeddu, “Bulk optical properties and tissue components in the female breast from multiwavelength time-resolved optical mammography,” J. Biomed. Opt. **9**(6), 1137–1142 (2004). [CrossRef] [PubMed]

4. B. C. Wilson and M. S. Patterson, “The physics, biophysics and technology of photodynamic therapy,” Phys. Med. Biol. **53**(9), R61–R109 (2008). [CrossRef] [PubMed]

5. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. **50**(4), R1–R43 (2005). [CrossRef] [PubMed]

6. R. Choe, A. Corlu, K. Lee, T. Durduran, S. D. Konecky, M. Grosicka-Koptyra, S. R. Arridge, B. J. Czerniecki, D. L. Fraker, A. DeMichele, B. Chance, M. A. Rosen, and A. G. Yodh, “Diffuse optical tomography of breast cancer during neoadjuvant chemotherapy: a case study with comparison to MRI,” Med. Phys. **32**(4), 1128–1139 (2005). [CrossRef] [PubMed]

7. A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express **15**(11), 6696–6716 (2007). [CrossRef] [PubMed]

8. A. Koenig, L. Hervé, V. Josserand, M. Berger, J. Boutet, A. Da Silva, J. M. Dinten, P. Peltié, J. L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. **13**(1), 011008 (2008). [CrossRef] [PubMed]

*μ*and absorption coefficient

_{s}’*μ*. Knowledge of these optical properties is necessary for the above applications. As examples, it has been shown that errors on background optical properties can result in misestimation of other optical properties or the position of an absorbing inclusion [9

_{a}9. X. F. Cheng and D. A. Boas, “Systematic diffuse optical image errors resulting from uncertainty in the background optical properties,” Opt. Express **4**(8), 299–307 (1999). [CrossRef] [PubMed]

10. V. Chernomordik, D. Hattery, I. Gannot, and A. H. Gandjbakhche, “Inverse method 3-D reconstruction of localized in vivo fluorescence - Application to Sjogren syndrome,” IEEE J. Sel. Top. Quant. **5**(4), 930–935 (1999). [CrossRef]

*μ*and

_{a}*μ*estimation [11

_{s}’11. J. Swartling, J. S. Dam, and S. Andersson-Engels, “Comparison of spatially and temporally resolved diffuse-reflectance measurement systems for determination of biomedical optical properties,” Appl. Opt. **42**(22), 4612–4620 (2003). [CrossRef] [PubMed]

12. B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, and R. Boretsky, “Comparison of Time-Resolved and -Unresolved Measurements of Deoxyhemoglobin in Brain,” Proc. Natl. Acad. Sci. U.S.A. **85**(14), 4971–4975 (1988). [CrossRef] [PubMed]

15. M. S. Patterson, B. Chance, and B. C. Wilson, “Time Resolved Reflectance and Transmittance for the Noninvasive Measurement of Tissue Optical-Properties,” Appl. Opt. **28**(12), 2331–2336 (1989). [CrossRef] [PubMed]

16. B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, and G. Holtom, “Time-resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscle,” Anal. Biochem. **174**(2), 698–707 (1988). [CrossRef] [PubMed]

12. B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, and R. Boretsky, “Comparison of Time-Resolved and -Unresolved Measurements of Deoxyhemoglobin in Brain,” Proc. Natl. Acad. Sci. U.S.A. **85**(14), 4971–4975 (1988). [CrossRef] [PubMed]

17. T. Svensson, J. Swartling, P. Taroni, A. Torricelli, P. Lindblom, C. Ingvar, and S. Andersson-Engels, “Characterization of normal breast tissue heterogeneity using time-resolved near-infrared spectroscopy,” Phys. Med. Biol. **50**(11), 2559–2571 (2005). [CrossRef] [PubMed]

18. R. Cubeddu, C. D’Andrea, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “Effects of the menstrual cycle on the red and near-infrared optical properties of the human breast,” Photochem. Photobiol. **72**(3), 383–391 (2000). [PubMed]

19. T. Svensson, S. Andersson-Engels, M. Einarsdóttír, and K. Svanberg, “In vivo optical characterization of human prostate tissue using near-infrared time-resolved spectroscopy,” J. Biomed. Opt. **12**(1), 014022 (2007). [CrossRef] [PubMed]

*μ*and

_{a}*μ*, by fitting the data with a photon propagation model. The algorithm seeks to minimize a χ

_{s}’^{2}error function between the temporal profile and the model. An important issue, which has been discussed for many years, is then the choice of an appropriate model, and fast enough to compute for estimation algorithm [20

20. K. M. Yoo, F. Liu, and R. R. Alfano, “When Does the Diffusion Approximation Fail to Describe Photon Transport in Random Media?” Phys. Rev. Lett. **64**(22), 2647–2650 (1990). [CrossRef] [PubMed]

21. R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “Experimental test of theoretical models for time-resolved reflectance,” Med. Phys. **23**(9), 1625–1633 (1996). [CrossRef] [PubMed]

*μ*has to be large compared to

_{s}’*μ*, the source-to-detector distance has to be large enough and early arriving photons fail to fit the model [22

_{a}22. R. Cubeddu, M. Musolino, A. Pifferi, P. Taroni, and G. Valentini, “Time-Resolved Reflectance - a Systematic Study for Application to the Optical Characterization of Tissues,” IEEE J. Quantum Electron. **30**(10), 2421–2430 (1994). [CrossRef]

23. A. Laidevant, A. da Silva, M. Berger, and J. M. Dinten, “Effects of the surface boundary on the determination of the optical properties of a turbid medium with time-resolved reflectance,” Appl. Opt. **45**(19), 4756–4764 (2006). [CrossRef] [PubMed]

*μ*and

_{a}*μ*estimates are inaccurate. Many works have thus been performed to assess accuracy of

_{s}’*μ*and

_{a}*μ*estimates when using diffusion approximation and TRS measurements [21

_{s}’21. R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “Experimental test of theoretical models for time-resolved reflectance,” Med. Phys. **23**(9), 1625–1633 (1996). [CrossRef] [PubMed]

*μ*>0.5cm

_{a}^{−1}and

*μ*<5 cm

_{s}’^{−1}), errors in both parameters can exceed 30% [24

24. E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. Biomed. Opt. **13**(4), 041304 (2008). [CrossRef] [PubMed]

26. A. Pifferi, A. Torricelli, P. Taroni, D. Comelli, A. Bassi, and R. Cubeddu, “Fully automated time domain spectrometer for the absorption and scattering characterization of diffusive media,” Rev. Sci. Instrum. **78**(5), 053103 (2007). [CrossRef] [PubMed]

27. B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. **10**(6), 824–830 (1983). [CrossRef] [PubMed]

29. S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte Carlo Modeling of Light Propagation in Highly Scattering Tissue .1. Model Predictions and Comparison with Diffusion-Theory,” IEEE Trans. Biomed. Eng. **36**(12), 1162–1168 (1989). [CrossRef] [PubMed]

*et al.*[24

24. E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. Biomed. Opt. **13**(4), 041304 (2008). [CrossRef] [PubMed]

25. E. Alerstam, S. Andersson-Engels, and T. Svensson, “Improved accuracy in time-resolved diffuse reflectance spectroscopy,” Opt. Express **16**(14), 10440–10454 (2008). [CrossRef] [PubMed]

30. T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J. Biophoton. **1**(3), 200–203 (2008). [CrossRef]

^{2}criterion together with the weighted residuals between model and data. This fitting method has proven efficiency for different time-resolved photon counting applications, notably for TCSPC lifetime measurements for choosing the right number of parameters in multi-exponential decay fluorescence. Fitting values are correct inasmuch that data follow a certain set of assumptions. According to [31], these assumptions are:

- (i) data uncertainty is the dependent variable (y-axis), meaning in this case that time uncertainty (x-axis) is negligible,
- (ii) this uncertainty has a Gaussian distribution, centered around the model value,
- (iii) there is no systematic error whether in time or in intensity,
- (iv) data points correspond to independent observations,
- (v) the number of data points T is greater than the number of estimated parameters p,
- (vi) the model is correct (incorrect models yield incorrect fitted parameters).

^{2}value (

*μ*and

_{a}*μ*estimation in TRS. As proof of the principle of using

_{s}’## 2. Materials and methods

### 2.1 Experimental system

*μ*=180+/−20cm

_{s}’^{−1}. The absorption property of the ink stock solution is

*μ*(ink)=5.35+/−0.15cm

_{a}^{−1}, determined with an absorption UV-Visible spectrophotometer (Cary 300 Scan, Varian). Note that this is only the ink contribution to absorption as a water tank is used in the second line of the spectrometer. This measurement is used to cross-check the μa estimate by the time-resolved measurements using the following formula:

*μ*(water,775nm)=0.027cm

_{a}^{−1}and depends only slightly on temperature and purity [32].

^{−1}for absorption, and 1 to 15 cm

^{−1}for isotropic scattering. Values of 0.1 cm

^{−1}for absorption, and 10 cm

^{−1}for isotropic scattering are considered as reference values. Therefore, strong absorption is defined by values greater than 0.1 cm

^{−1}and weak scattering by values smaller than 10 cm

^{−1}.

### 2.2 Monte Carlo simulations

33. S. Prahl, “Monte Carlo Simulations,” http://omlc.ogi.edu/software/mc/.

28. L. H. Wang, S. L. Jacques, and L. Q. Zheng, “MCML-Monte Carlo Modeling of Light Transport in Multi-layered Tissues,” Comput. Methods Programs Biomed. **47**(2), 131–146 (1995). [CrossRef] [PubMed]

_{max}=7.6ns. It means that a photon can generate multiple detection events implying entangled photon events [24

24. E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. Biomed. Opt. **13**(4), 041304 (2008). [CrossRef] [PubMed]

*μ*=14.2cm

_{s}’^{−1}, source-detector distance of

*r*=1.7cm, and time channel resolution of 3ps, it leads to correlation between typically five adjacent time channels (data not shown), the two first neighbors having a correlation coefficient of 0.5. This seems not to be a problem in this configuration as the number of fitting channels T is generally a few hundreds, large enough compared to 5. However, this point would require a more exhaustive study. In the case of strong absorption (less fitting channels) discussed in subsection 3.6, the effects of these correlations start to be visible.

*μ*into account, data are further multiplied by

_{a}*c*is the speed of light in the medium. We call such simulation scalable MC, or MCs (even if the space is not scaled as in [24

**13**(4), 041304 (2008). [CrossRef] [PubMed]

*a*stands for absorption) have also been performed. After each scattering event, MCa has classically the probability (1-albedo) to kill a photon, where albedo=(μ

_{s}/(μ

_{s}+μ

_{a})). However, for a question of computation time, MCa simulations were used only when the statistic issue is at stake.

### 2.3 Model

22. R. Cubeddu, M. Musolino, A. Pifferi, P. Taroni, and G. Valentini, “Time-Resolved Reflectance - a Systematic Study for Application to the Optical Characterization of Tissues,” IEEE J. Quantum Electron. **30**(10), 2421–2430 (1994). [CrossRef]

23. A. Laidevant, A. da Silva, M. Berger, and J. M. Dinten, “Effects of the surface boundary on the determination of the optical properties of a turbid medium with time-resolved reflectance,” Appl. Opt. **45**(19), 4756–4764 (2006). [CrossRef] [PubMed]

15. M. S. Patterson, B. Chance, and B. C. Wilson, “Time Resolved Reflectance and Transmittance for the Noninvasive Measurement of Tissue Optical-Properties,” Appl. Opt. **28**(12), 2331–2336 (1989). [CrossRef] [PubMed]

35. J. C. J. Paasschens, “Solution of the time-dependent Boltzmann equation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **56**(1), 1135–1141 (1997). [CrossRef]

36. M. Bassani, F. Martelli, G. Zaccanti, and D. Contini, “Independence of the diffusion coefficient from absorption: experimental and numerical evidence,” Opt. Lett. **22**(12), 853–855 (1997). [CrossRef] [PubMed]

*r*is the distance between a virtual isotropic source and the detector fiber. The virtual isotropic source is classically assumed to be l*=1/

*μ*beneath the fiber source [15

_{s}’15. M. S. Patterson, B. Chance, and B. C. Wilson, “Time Resolved Reflectance and Transmittance for the Noninvasive Measurement of Tissue Optical-Properties,” Appl. Opt. **28**(12), 2331–2336 (1989). [CrossRef] [PubMed]

### 2.4 Fitting procedure

*μ*and

_{a}*μ*, from the temporal profile recorded, containing numerous data points. The Nonlinear Least Squares method is used for the estimation; more precisely, a Nelder-Mead simplex algorithm drives minimization of a reduced

_{s}’*χ*error function which measures the deviation with respect to the above model. The

_{R}^{2}*χ*error value is expressed as:where T (typically a few hundred) is the number of temporal channels i used in the fitting procedure,

_{R}^{2}*p*is the number of estimated parameters (

*p*=2 here,

*μ*and

_{a}*μ*), m

_{s}’_{i}and s

_{i}represent the model and measured signal per temporal channel

*t*.

_{i}*W*stands for weighted residual:

_{Ri}*W*) should oscillate around zero and each term of the sum contributes to one on average, and so the reduced

_{Ri}*χ*can be expected to be 1.

^{2}*et al.*[21

21. R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “Experimental test of theoretical models for time-resolved reflectance,” Med. Phys. **23**(9), 1625–1633 (1996). [CrossRef] [PubMed]

_{1}corresponding to 95% of the maximum on the rising edge (because the first arriving photons do not follow the diffusion equation) and t

_{2}corresponding to 5% of the maximum on the trailing edge (noise limitation), except where otherwise stated.

*s*is modified by the absorption weighting

_{i}_{a}– but surprisingly, the error due to the omitted correction is very small (just a few percents).

### 2.5 Goodness of fit

*χ*, together with a non-random pattern of weighted residual, is indicative of a bad fit, thus providing poor confidence in estimated parameters. Assuming both the model is correct and there are 200 independent measurements (in this case time points), there are less than five chances in 100 of finding a

_{R}^{2}*χ*value above 1.17 [31]. Such probabilities can be found using

_{R}^{2}*χ*distribution which tends towards normal distribution for high degrees of liberty (in practice, a normal distribution is used for T>200). Table 1 gives a few probabilities for usual situations in TCSPC when all the above assumptions are verified, in particular model validity. Moreover, properties of normal distribution imply that less than 5% of the absolute weighted residual points should be above 2, and less than 1% above 3.

^{2}*χ*and behavior of weighted residuals is not proof that the fit is good and that the estimated parameters are correct. This study aims at giving an insight into what can be expected from these criteria to estimate the optical parameters,

_{R}^{2}*μ*and

_{a}*μ*.

_{s}’## 3. Results

### 3.1 χ_{R}^{2} and number of photons

_{i}is greater for early times, because the first arriving photons do not follow the diffusion model as they do not undergo sufficient scattering events. Both m

_{i}and b

_{i}depend on the experimental conditions, such as optical properties of the medium, and distance between fibers r. For a particular time channel

*χ*components then become:

_{R}^{2}_{1}and t

_{2}, and after background noise is subtracted. More precisely, the signal, the model and the bias also increase linearly with

*N*, with the standard deviation of the signal increasing with

^{7}, and a corrected

### 3.2 Influence of the distance between fibers

*r*, distance between source and detector fibers. For each of these temporal profiles, a fitting procedure is performed providing

*μ*and

_{a}*μ*estimates as well as the

_{s}’*r*. Both

*μ*and

_{a}*μ*are overestimated when

_{s}’*r*is too short (respectively 30% and 45% error for

*r*=0.5cm), before reaching a plateau for

*μ*=7.9 cm

_{s}’^{−1}and

*μ*=0.106 cm

_{a}^{−1}, considered as “the true values”, the latter being checked by the absorption spectroscopy method. Measurements at the two first distances lead to a high value of

*r*=0.5cm), and non-random weighted residuals, which warn experimenters about the problem. For

*r*=1.1cm and above,

*μ*has already reached the plateau and

_{a}*μ*is overestimated by 10% or less, and

_{s}’*r*=1.7cm when not specified. We attribute the high value of 1.5 at the plateau to unknown systematic errors (light leakage, slightly incorrect IRF) together with a high number of acquired photons which exhibits any little bias. Note that when the distance between fibers is increased, the collected intensity strongly decreases: as a result, the optical density before the PMT and the gain voltage applied to the PMT had to be changed. For each applied voltage, the IRF with the same gain was performed and used for deconvolution. The deviation due to the short detection distance can be explained by the fact that early arriving photons do not follow the diffusion behavior – these photons predominate because source and detector are close to each other.

*r*between fibers, especially in the case of

*μ*estimation.

_{a}### 3.3 Influence of the IRF used for deconvolution

37. A. Liebert, H. Wabnitz, D. Grosenick, and R. Macdonald, “Fiber dispersion in time domain measurements compromising the accuracy of determination of optical properties of strongly scattering media,” J. Biomed. Opt. **8**(3), 512–516 (2003). [CrossRef] [PubMed]

### 3.4 Weak scattering tissues or phantom

*μ*and

_{a}*μ*estimates can thus be rejected. Figure 5 was drawn up using a fit interval t

_{s}’_{2}corresponding to 0.5% of the maximum instead of 5% in order to show the slight decreasing trend of μ

_{a}due to dilution, which is expected. For t

_{2}taken at 5% of the maximum (data not shown), estimates are modified by less than 2% for

*μ*but are then not able to show the decrease, and less than 3% for

_{a}*μ*.

_{s}’### 3.5 Strong absorption tissues or phantom: experimental artifacts

**13**(4), 041304 (2008). [CrossRef] [PubMed]

*μ*and

_{a}*μ*[30

_{s}’30. T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J. Biophoton. **1**(3), 200–203 (2008). [CrossRef]

*μ*=0.1cm

_{a}^{−1}and

*μ*=15cm

_{s}’^{−1}, and further increasing ink concentration. Figure 6(a) shows how

*μ*and

_{a}*μ*are both underestimated for strongly absorbing media, which is in contradiction with previous results comparing experimental data, diffusion equation and MC simulations [24

_{s}’**13**(4), 041304 (2008). [CrossRef] [PubMed]

25. E. Alerstam, S. Andersson-Engels, and T. Svensson, “Improved accuracy in time-resolved diffuse reflectance spectroscopy,” Opt. Express **16**(14), 10440–10454 (2008). [CrossRef] [PubMed]

30. T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J. Biophoton. **1**(3), 200–203 (2008). [CrossRef]

*μ*=0.53cm

_{a}^{−1}(26% underestimation).

_{a}is explained by the tail which is higher than expected. We checked that this behavior cannot be explained by different IRF measurements, in particular with or without using a white paper to excite the different modes of the collection fiber [37

37. A. Liebert, H. Wabnitz, D. Grosenick, and R. Macdonald, “Fiber dispersion in time domain measurements compromising the accuracy of determination of optical properties of strongly scattering media,” J. Biomed. Opt. **8**(3), 512–516 (2003). [CrossRef] [PubMed]

19. T. Svensson, S. Andersson-Engels, M. Einarsdóttír, and K. Svanberg, “In vivo optical characterization of human prostate tissue using near-infrared time-resolved spectroscopy,” J. Biomed. Opt. **12**(1), 014022 (2007). [CrossRef] [PubMed]

38. A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, R. Cubeddu, H. Wabnitz, D. Grosenick, M. Möller, R. Macdonald, J. Swartling, T. Svensson, S. Andersson-Engels, R. L. van Veen, H. J. Sterenborg, J. M. Tualle, H. L. Nghiem, S. Avrillier, M. Whelan, and H. Stamm, “Performance assessment of photon migration instruments: the MEDPHOT protocol,” Appl. Opt. **44**(11), 2104–2114 (2005). [CrossRef] [PubMed]

### 3.6 Strong absorption: MC simulation

_{s}’ between 5 and 15 cm

^{−1}), strong absorption results in overestimation of both μ

_{a}and μ

_{s}’, together with elevation of the

_{a}=0.3cm

^{−1}and μ

_{s}’=5cm

^{−1}at 786nm [30

**1**(3), 200–203 (2008). [CrossRef]

_{a}and μ

_{s}’ compared to respectively 2% and 1% for the 95/5 fitting range). Figure 7(c) shows that weighted residuals are almost randomly distributed for a small number of photons but Fig. 7(d) clearly demonstrates the short time bias for a large number. Again, for a given model bias and a low number of photons, both the

### 3.7 Boundary effects

*μ*and

_{a}*μ*[23

_{s}’23. A. Laidevant, A. da Silva, M. Berger, and J. M. Dinten, “Effects of the surface boundary on the determination of the optical properties of a turbid medium with time-resolved reflectance,” Appl. Opt. **45**(19), 4756–4764 (2006). [CrossRef] [PubMed]

*μ*parameter. This implies that the curve tail is steeper, so that the time measured to obtain the peak intensity, t

_{a}_{max}, is underestimated. And given that

*μ*increases with t

_{s}’_{max}[15

**28**(12), 2331–2336 (1989). [CrossRef] [PubMed]

39. L. Leonardi and D. H. Burns, “Quantitative measurements in scattering media: Photon time-of-flight analysis with analytical descriptors,” Appl. Spectrosc. **53**(6), 628–636 (1999). [CrossRef]

*μ*is also underestimated. These deviations from an infinite medium as a function of depth are shown in Fig. 8 ; the plateau of an infinite medium is almost reached at z = 2cm. Except for the first two depth values, where the

_{s}’*μ*is reached without any clear indication on

_{a}*μ*). The distance to all boundaries has to be chosen precisely by experimenters in order to avoid any such misestimation which cannot be highlighted by

_{s}’_{a}and μ

_{s}’ estimates are correct anymore. It is noticed that early photons are more affected by the large holder. Consequently, Fig. 9(a) shows weighted residuals and

### 3.8 Experimental uncertainties

*r*is completely compensated by the diffusion coefficient D, leading to an incorrect estimate of

*μ*, but the same estimate of

_{s}’*μ*and the same residuals between model and data: there is no clue given to experimenters of this potential error. Due to the squared dependence of

_{a}*r*in the exponential, a 5% error in

*r*will be responsible for a 10% error in

*μ*.

_{s}’*μ*and 2% in

_{s}’*μ*. As the

_{a}*μ*estimation is directly linked with t

_{s}’_{max}, it is normal that

*μ*is more affected than

_{s}’*μ*by a time shift. On the other hand, for abnormally big time shifts (such as 100ps), a high

_{a}## 4. Conclusion and summary

_{a}and μ

_{s}’ estimation cannot be assessed.

## Acknowledgments

## References and links

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8. | A. Koenig, L. Hervé, V. Josserand, M. Berger, J. Boutet, A. Da Silva, J. M. Dinten, P. Peltié, J. L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. |

9. | X. F. Cheng and D. A. Boas, “Systematic diffuse optical image errors resulting from uncertainty in the background optical properties,” Opt. Express |

10. | V. Chernomordik, D. Hattery, I. Gannot, and A. H. Gandjbakhche, “Inverse method 3-D reconstruction of localized in vivo fluorescence - Application to Sjogren syndrome,” IEEE J. Sel. Top. Quant. |

11. | J. Swartling, J. S. Dam, and S. Andersson-Engels, “Comparison of spatially and temporally resolved diffuse-reflectance measurement systems for determination of biomedical optical properties,” Appl. Opt. |

12. | B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, and R. Boretsky, “Comparison of Time-Resolved and -Unresolved Measurements of Deoxyhemoglobin in Brain,” Proc. Natl. Acad. Sci. U.S.A. |

13. | D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, and J. Wyatt, “Estimation of Optical Pathlength through Tissue from Direct Time of Flight Measurement,” Phys. Med. Biol. |

14. | S. L. Jacques, “Time-resolved reflectance spectroscopy in turbid tissues,” IEEE Trans. Biomed. Eng. |

15. | M. S. Patterson, B. Chance, and B. C. Wilson, “Time Resolved Reflectance and Transmittance for the Noninvasive Measurement of Tissue Optical-Properties,” Appl. Opt. |

16. | B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, and G. Holtom, “Time-resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscle,” Anal. Biochem. |

17. | T. Svensson, J. Swartling, P. Taroni, A. Torricelli, P. Lindblom, C. Ingvar, and S. Andersson-Engels, “Characterization of normal breast tissue heterogeneity using time-resolved near-infrared spectroscopy,” Phys. Med. Biol. |

18. | R. Cubeddu, C. D’Andrea, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “Effects of the menstrual cycle on the red and near-infrared optical properties of the human breast,” Photochem. Photobiol. |

19. | T. Svensson, S. Andersson-Engels, M. Einarsdóttír, and K. Svanberg, “In vivo optical characterization of human prostate tissue using near-infrared time-resolved spectroscopy,” J. Biomed. Opt. |

20. | K. M. Yoo, F. Liu, and R. R. Alfano, “When Does the Diffusion Approximation Fail to Describe Photon Transport in Random Media?” Phys. Rev. Lett. |

21. | R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “Experimental test of theoretical models for time-resolved reflectance,” Med. Phys. |

22. | R. Cubeddu, M. Musolino, A. Pifferi, P. Taroni, and G. Valentini, “Time-Resolved Reflectance - a Systematic Study for Application to the Optical Characterization of Tissues,” IEEE J. Quantum Electron. |

23. | A. Laidevant, A. da Silva, M. Berger, and J. M. Dinten, “Effects of the surface boundary on the determination of the optical properties of a turbid medium with time-resolved reflectance,” Appl. Opt. |

24. | E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. Biomed. Opt. |

25. | E. Alerstam, S. Andersson-Engels, and T. Svensson, “Improved accuracy in time-resolved diffuse reflectance spectroscopy,” Opt. Express |

26. | A. Pifferi, A. Torricelli, P. Taroni, D. Comelli, A. Bassi, and R. Cubeddu, “Fully automated time domain spectrometer for the absorption and scattering characterization of diffusive media,” Rev. Sci. Instrum. |

27. | B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. |

28. | L. H. Wang, S. L. Jacques, and L. Q. Zheng, “MCML-Monte Carlo Modeling of Light Transport in Multi-layered Tissues,” Comput. Methods Programs Biomed. |

29. | S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte Carlo Modeling of Light Propagation in Highly Scattering Tissue .1. Model Predictions and Comparison with Diffusion-Theory,” IEEE Trans. Biomed. Eng. |

30. | T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J. Biophoton. |

31. | J. R. Lakowicz, |

32. | H. Buiteveld, J. H. M. Hakvoort, and M. Donze, “Optical properties of pure water,” Ocean Optics XIIProc. SPIE |

33. | S. Prahl, “Monte Carlo Simulations,” http://omlc.ogi.edu/software/mc/. |

34. | S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” Dosimetry of Laser Radiation in Medicine and Biology-Proc. SPIE |

35. | J. C. J. Paasschens, “Solution of the time-dependent Boltzmann equation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

36. | M. Bassani, F. Martelli, G. Zaccanti, and D. Contini, “Independence of the diffusion coefficient from absorption: experimental and numerical evidence,” Opt. Lett. |

37. | A. Liebert, H. Wabnitz, D. Grosenick, and R. Macdonald, “Fiber dispersion in time domain measurements compromising the accuracy of determination of optical properties of strongly scattering media,” J. Biomed. Opt. |

38. | A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, R. Cubeddu, H. Wabnitz, D. Grosenick, M. Möller, R. Macdonald, J. Swartling, T. Svensson, S. Andersson-Engels, R. L. van Veen, H. J. Sterenborg, J. M. Tualle, H. L. Nghiem, S. Avrillier, M. Whelan, and H. Stamm, “Performance assessment of photon migration instruments: the MEDPHOT protocol,” Appl. Opt. |

39. | L. Leonardi and D. H. Burns, “Quantitative measurements in scattering media: Photon time-of-flight analysis with analytical descriptors,” Appl. Spectrosc. |

**OCIS Codes**

(290.7050) Scattering : Turbid media

(170.6935) Medical optics and biotechnology : Tissue characterization

**ToC Category:**

Scattering

**History**

Original Manuscript: June 12, 2009

Revised Manuscript: September 4, 2009

Manuscript Accepted: September 7, 2009

Published: October 23, 2009

**Virtual Issues**

Vol. 4, Iss. 12 *Virtual Journal for Biomedical Optics*

**Citation**

Laurent Guyon, Anabela da Silva, Anne Planat-Chrétien, Philippe Rizo, and Jean-Marc Dinten, "χ^{2} analysis for estimating the accuracy of optical properties derived from time resolved diffuse-reflectance," Opt. Express **17**, 20521-20537 (2009)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-17-22-20521

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