## A predictive model of backscattering at subdiffusion length scales |

Biomedical Optics Express, Vol. 1, Issue 3, pp. 1034-1046 (2010)

http://dx.doi.org/10.1364/BOE.1.001034

Acrobat PDF (1227 KB)

### Abstract

We provide a methodology for accurately predicting elastic backscattering radial distributions from random media with two simple empirical models. We apply these models to predict the backscattering based on two classes of scattering phase functions: the Henyey-Greenstein phase function and a generalized two parameter phase function that is derived from the Whittle-Matérn correlation function. We demonstrate that the model has excellent agreement over all length scales and has less than 1% error for backscattering at subdiffusion length scales for tissue-relevant optical properties. The presented model is the first available approach for accurately predicting backscattering at length scales significantly smaller than the transport mean free path.

© 2010 OSA

## 1. Introduction

1. T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. **19**(4), 879–888 (1992). [CrossRef] [PubMed]

*r*) is much greater than the transport mean free path (

*ls**). Diffusion approximations are not accurate at small distances of light transport (e.g. source-detector separation),

*r*, because they do not take the shape of the phase function into account. While at diffusion length scales (

*r/ls**>>1), light transport is primarily governed by the value of the transport mean free path (

*ls**), at sub-diffusion distances (

*r/ls**<1), the shape of the phase function may significantly affect the radial reflectance distribution. Foster and others have shown that the accuracy can be improved by accounting for higher order moments of the phase function in the P3 approximation [2

2. E. L. Hull and T. H. Foster, “Steady-state reflectance spectroscopy in the P-3 approximation,” J. Opt. Soc. Am. A **18**(3), 584–599 (2001). [CrossRef]

3. I. Seo, C. K. Hayakawa, and V. Venugopalan, “Radiative transport in the delta-P1 approximation for semi-infinite turbid media,” Med. Phys. **35**(2), 681–693 (2008). [CrossRef] [PubMed]

*ls**[2

2. E. L. Hull and T. H. Foster, “Steady-state reflectance spectroscopy in the P-3 approximation,” J. Opt. Soc. Am. A **18**(3), 584–599 (2001). [CrossRef]

*ls**. For example, cancer detection often requires the isolation of a signal from superficial tissue such as the epithelium or mucosa. In many tissues, the thickness of the epithelium is much smaller than

*ls**therefore requiring a source detector separation much smaller than

*ls**. For this reason, several groups have developed fiber probes that sample small source-detector separations [4

4. M. C. Skala, G. M. Palmer, C. F. Zhu, Q. Liu, K. M. Vrotsos, C. L. Marshek-Stone, A. Gendron-Fitzpatrick, and N. Ramanujam, “Investigation of fiber-optic probe designs for optical spectroscopic diagnosis of epithelial pre-cancers,” Lasers Surg. Med. **34**(1), 25–38 (2004). [CrossRef] [PubMed]

6. R. Reif, O. A’Amar, and I. J. Bigio, “Analytical model of light reflectance for extraction of the optical properties in small volumes of turbid media,” Appl. Opt. **46**(29), 7317–7328 (2007). [CrossRef] [PubMed]

7. Y. L. Kim, Y. Liu, V. M. Turzhitsky, H. K. Roy, R. K. Wali, and V. Backman, “Coherent backscattering spectroscopy,” Opt. Lett. **29**(16), 1906–1908 (2004). [CrossRef] [PubMed]

8. V. M. Turzhitsky, A. J. Gomes, Y. L. Kim, Y. Liu, A. Kromine, J. D. Rogers, M. Jameel, H. K. Roy, and V. Backman, “Measuring mucosal blood supply in vivo with a polarization-gating probe,” Appl. Opt. **47**(32), 6046–6057 (2008). [CrossRef] [PubMed]

9. J. C. Ramella-Roman, S. A. Prahl, and S. L. Jacques, “Three Monte Carlo programs of polarized light transport into scattering media: part I,” Opt. Express **13**(12), 4420–4438 (2005). [CrossRef] [PubMed]

10. L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Mcml - Monte-Carlo Modeling of Light Transport in Multilayered Tissues,” Comput Meth. Prog Biol. **47**(2), 131–146 (1995). [CrossRef]

*r/ls**[2

2. E. L. Hull and T. H. Foster, “Steady-state reflectance spectroscopy in the P-3 approximation,” J. Opt. Soc. Am. A **18**(3), 584–599 (2001). [CrossRef]

11. F. Bevilacqua and C. Depeursinge, “Monte Carlo study of diffuse reflectance at source-detector separations close to one transport mean free path,” J. Opt. Soc. Am. A **16**(12), 2935–2945 (1999). [CrossRef]

*ls**. The approach involves the construction of a simple model that predicts an infinitely narrow normally incident illumination beam response, termed p(r), from a turbid scattering medium. The response p(r) is a fundamental property of the turbid medium and is the objective for predictive modeling of most diffusion approximation models and Monte Carlo methods. If p(r) is known, both the effects of a finite source and a numerical aperture can be modeled. We consider two types of phase functions in order to construct the models: the commonly used Henyey-Greenstein phase function, and a more general two parameter phase function which encompasses the Henyey-Greenstein phase function and uses parameters that quantify the sample refractive index correlation function. As an experimental example, we measure the reflectance distribution from a tissue phantom composed of a mixture of polystyrene microspheres using Low-coherence Enhanced Backscattering and compare the measured distribution at small length scales to the newly developed model.

## 1. Monte Carlo Simulation of Reflectance

*r~ls**), accurate modeling of small length scales (

*r*<<

*ls**) requires a more general choice of phase function. For this purpose, we will follow a recently developed model that is based on the Whittle-Matérn correlation function [12

12. C. J. R. Sheppard, “Fractal model of light scattering in biological tissue and cells,” Opt. Lett. **32**(2), 142–144 (2007). [CrossRef] [PubMed]

13. J. D. Rogers, I. R. Capoğlu, and V. Backman, “Nonscalar elastic light scattering from continuous random media in the Born approximation,” Opt. Lett. **34**(12), 1891–1893 (2009). [CrossRef] [PubMed]

14. I. R. Çapoğlu, J. D. Rogers, A. Taflove, and V. Backman, “Accuracy of the Born approximation in calculating the scattering coefficient of biological continuous random media,” Opt. Lett. **34**(17), 2679–2681 (2009). [CrossRef] [PubMed]

16. P. Guttorp and T. Gneiting, “Studies in the history of probability and statistics XLIX On the Matern correlation family,” Biometrika **93**(4), 989–995 (2006). [CrossRef]

*Δn*is the variance of the refractive index fluctuations,

^{2}*l*is the correlation length, and

_{c}*m*is a parameter that determines the form of the function. The function

*K*denotes the modified Bessel function of the second kind of order

_{m-3/2}*m-3/2*. When

*m*< 1.5,

*B*is a power law, thus corresponding to a mass fractal medium with mass fractal dimension

_{n}(r)*D*= 2m. 1.5<

_{mf}*m*<2 corresponds to a stretched exponential function,

*m*= 2 corresponds to an exponential function, and as

*m*becomes much larger than 2,

*B*approaches a Gaussian function. The correlation length

_{n}(r)*l*has different physical meaning depending on the type of the correlation function. For

_{c}*m*= 2,

*m*< 1.5,

*l*represents the upper length scale at which the correlation function loses its fractal behavior. The differential scattering cross section can be derived by applying the Born approximation to the Whittle-Matérn correlation function [13

_{c}13. J. D. Rogers, I. R. Capoğlu, and V. Backman, “Nonscalar elastic light scattering from continuous random media in the Born approximation,” Opt. Lett. **34**(12), 1891–1893 (2009). [CrossRef] [PubMed]

*k = 2π/λ*and the normalization is such that

*m*and

*klc*without any change to the normalization:

*m*= 1.5. In this special case the correlation function is that of the space filling random field and the phase function simplifies to the commonly used form known as the Henyey-Greenstein phase function. The parameter

*ĝ*then becomes the average cos(θ), also known as the anisotropy factor

*g*. For other values of

*m*,

*g*is given by taking the forward moment:

*m*, and varying values of

*g*, while Fig. 1(b) shows examples of phase functions with the same value of

*g*and varying values of

*m*. The parameter

*g*influences the width of the phase function while

*m*influences the shape of the phase function independently of the width. There are two cases in the generalized phase function which are removable discontinuities:

*m*= 1 and

*g*= 0. We can evaluate the phase function for these cases by employing L’Hospital’s rule:

*g*becomes:

9. J. C. Ramella-Roman, S. A. Prahl, and S. L. Jacques, “Three Monte Carlo programs of polarized light transport into scattering media: part I,” Opt. Express **13**(12), 4420–4438 (2005). [CrossRef] [PubMed]

17. J. C. Ramella-Roman, S. A. Prahl, and S. L. Jacques, “Three Monte Carlo programs of polarized light transport into scattering media: part II,” Opt. Express **13**(25), 10392–10405 (2005). [CrossRef] [PubMed]

*m*= 1.5 (Henyey-Greenstein case) with existing codes that implement the Henyey-Greenstein phase function [10

10. L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Mcml - Monte-Carlo Modeling of Light Transport in Multilayered Tissues,” Comput Meth. Prog Biol. **47**(2), 131–146 (1995). [CrossRef]

*g*varying from 0 to 0.98 and values of

*m*varying from 1.01 to 1.9 were obtained for the backscattering direction (0-10°). We found that the variations of backscattering probability distributions were small within the 10° angular collection range when the backscattering probability distribution was stored as a function of the position of the final scattering event. Therefore, all reflectance distributions were stored as a function of the position of the last scattering event.

*ls**was maintained at 100μm with a scattering slab thickness of 1cm, resulting in a scattering medium that approaches semi-infinite, with less than 2% of the intensity transmitting through the entire thickness of the slab. The infinitely narrow illumination beam was oriented orthogonally to the scattering medium. The boundary at the interface of the scattering medium was assumed to be index-matched and absorption was not present. The scattering angle,

*θ*, in the Monte Carlo simulation was chosen by expressing the probability of a selected angle as a function of the random variable

*ξ*:where

*ξ*is uniformly distributed between 0 and 1 [10

10. L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Mcml - Monte-Carlo Modeling of Light Transport in Multilayered Tissues,” Comput Meth. Prog Biol. **47**(2), 131–146 (1995). [CrossRef]

*ψ*= 2π

*ξ*. The remaining Monte Carlo simulation elements were identical to previously developed methodology for light propagation in turbid media [10

**47**(2), 131–146 (1995). [CrossRef]

## 2. Model of Reflectance: Henyey-Greenstein Phase Function

*r*and

*θ*being the polar coordinates in a plane perpendicular to the illumination beam. In a Monte Carlo simulation,

*P(r)*is the obtained reflectance distribution that is collected with azimuthally integrated radial storage (

*θ*is the azimuth angle). Figure 2(a) shows example

*P(r/ls*)*curves obtained from Monte Carlo simulations using the Henyey-Greenstein phase function for four different values of

*g*. All of the length scales in the Monte Carlo simulation are determined by

*ls*, the mean free path. Additionally, it is known that the determining length scale in the diffusion regime is

*ls**. Therefore, the axes in Fig. 2 are normalized with respect to

*ls**in order to be scalable for any value of

*ls*as well as observe the convergence of the results in the diffusion regime. All of the curves can be translated into units of

*P(r)*by multiplying the abscissa axis by

*ls**and dividing the ordinate axis by

*ls**. Another words,

*P(r) = P(ls*·s)/ls**. The division by

*ls**is required due to the change of variable in the normalization integral (

*ds = dr/ls**) such that

*P(r/ls*)*at subdiffusion length scales by subtracting

*P(r/ls*)*curves for isotropic scattering from

*P(r/ls*)*for non-isotropic cases (i.e.

*g*> 0). In Fig. 2(b), three difference curves are plotted for

*g*values of 0.9, 0.8, and 0.7. Note that the integral of each difference curve is 0 because the integral of

*P(r/ls*)*is always 1. The curves in Fig. 2(b) have very similar shapes but varying amplitudes. When each of these curves is rescaled by a constant that depends on

*g*, they closely overlap [Fig. 2(c)].

*P(r/ls*)*that depends on just two simulation results:

*P(r/ls*)*for

*g*= 0 and

*P(r/ls*)*for a particular

*g*>0. While any value of

*g*>0 can be used, we use

*g*= 0.9 in the following analysis for convenience (this results in accurate prediction within the range of tissue anisotropy):where

*c(g)*is an empirical model for the coefficients that multiply the difference term. The values of the constants

*a*and

*b*are approximately 1.244 and 2.338 respectively. The shortened notation

*P*represents

_{g}*P(r/ls*)*for a given value of

*g*(e.g.

*P*for

_{0.9}= P(r/ls*)*g*= 0.9). The values of

*c(g)*were determined by fitting Monte Carlo results for a particular

*g*to the expression for

*P*in Eq. (7). The values of

_{g}*c(g)*and the empirical model for

*c(g)*are plotted in Fig. 3(a) . We can understand the difference between

*P*and

_{0.9}*P*as the alteration in the backscattering due to anisotropy. As

_{0}*g*increases, the anisotropy contribution increases in amplitude but retains a very similar radial shape. This allows for a predictability of

*P(r)*for any value of

*g*with only two reference

*P(r)*distributions. Fig. 3(b) shows a comparison of the Monte Carlo simulations and the model based on the difference relationship. Fig. 3(c) further illustrates the details of the model fit at small values of

*r/ls**. Note that the fits for

*g*= 0 and

*g*= 0.9 are not shown because the model and the Monte Carlo result are identical for those two cases (

*c*= 1 when

*g*= 0.9 and

*c*= 0 when

*g*= 0). The model has excellent agreement for values of

*g*that are close to 0.9, but begins to deviate slightly at

*g*= 0.7. As

*r/ls**becomes large, all of the curves converge and the backscattering can be predicted with an isotropic scattering model of equivalent

*ls**.

*P(r/ls*)*, we used each value in r/ls* as an input variable. Instead of mean centering, we subtracted the

*P(r/ls*)*curve for

*g*= 0. The effect of subtracting the isotropic

*P(r/ls*)*is similar to that of mean-centering, but results in a more predictable model that is independent of the particular reflectance distributions used in the PCA analysis. We then obtained a series of principle components and found that when the first three components are used,

*P(r,ls*)*can be predicted more accurately than the single-component difference model described above. The first three principle components (PC1-PC3) predicted 99.966%, 0.027%, 0.002% of the variance in the data, respectively. Based on this model,

*P(r/ls*)*can be predicted according to:where

*c*,

_{1}*c*, and

_{2}*c*are the weights of the principle components. We utilized a polynomial equation to fit the weights with the order of the polynomial chosen such that the R

_{3}^{2}coefficient is greater than 0.99. Fig. 4(a) shows a comparison of the principle component model with Monte Carlo simulation results for three tissue-relevant values of

*g*. The

*r/ls**axis is in log scale, showing that the model is in excellent agreement with the Monte Carlo simulations for the entire simulated range of 0.001<

*r/ls**<10. Fig. 4(b) shows the same comparison in linear scale for the subdiffusion range of

*r/ls**< 1, again, showing excellent agreement. Fig. 4(c) shows the distributions of the three principle components that were used in the model. Note that the contribution of each successive component decreases, with higher components being noisier. Fig. 4(d) is a plot of the weights of the three components along with the polynomial fits that are used for the predictive model.

## 3. Model of Reflectance: Whittle-Matérn Phase Function

*P(r/ls*)*distributions for varying values of

*g*and

*m*. In Fig. 5(a) ,

*P(r/ls*)*curves for four values of

*m*are shown with a constant anisotropy factor of

*g*= 0.9. The isotropic component is subtracted from these curves in Fig. 5(b). From Fig. 5(b), it is apparent that a simple scaling in amplitude cannot account for the difference between these curves. There is an

*m*-dependent alteration in the shape of the non-diffuse component of the curves. However, for a given value of

*m*, changes in

*g*only alter the amplitude of the non-diffuse component [Fig. 5(c)]. Therefore, it is clear that another component needs to be introduced that can account for the alterations in the shape of

*P(r/ls*)*due to varying

*m*. We can extend the difference model developed for the Henyey-Greenstein phase function (

*m*= 1.5) discussed in the previous section by defining a second difference component that is calculated by subtracting the isotropic probability from

*P(r/ls*)*for

*g*= 0.9 and a particular

*m*. In our analysis, we chose

*m*= 1.01. This value of

*m*was chosen because the shape of

*P(r/ls*)*becomes dramatically altered as

*m*approaches 1. The shapes of the two difference components are compared in Fig. 5(d).

*P(r/ls*)*can then be predicted according to a two-component model:

*c*and

_{1}*c*vary smoothly and continuously with

_{2}*g*and

*m*. These coefficients can be fit to a variety of functions, depending on the desired simplicity and accuracy of the model. We fit these coefficients to a third order polynomial in two dimensions described by Eq. (10).where

*x = ln(m)*and

*y = 1/(1-g)*. The constants,

*a – j*, are supplied in Table 1 . These constants were optimized to obtain a minimized error for

*g*≥0.6.

*c*,

_{1}*c*, and

_{2}*c*each vary as a function of

_{3}*g*and

*m*in the generalized model. The variation of these coefficients is also smooth and continuous and can be fit to a polynomial equation based model such as the one in Eq. (10). Fig. 6(a) shows a comparison of the difference model (PΔ model) for the Whittle-Matérn phase function with Monte Carlo results for varying values of

*m*and a constant

*g*of 0.9. The agreement is excellent, although the error slightly increases for larger values of

*m*. The agreement is improved for the PCA based model, shown in Fig. 6(b). The error is quantified for the entire range of

*g*and

*m*for the PΔ and PCA models in Fig. 6(c) and Fig. 6(d), respectively. Although the equations were optimized for

*g*≥0.6, the average error for

*r/ls**between 0 and 1 is less than 2% for the entire range of

*g*and

*m*values that were evaluated. The error was less than 1% for all values of

*m*and biologically relevant anisotropy factors (

*g*≥0.6). The PCA model had less error than the PΔ model for this biologically relevant anisotropy range.

## 4. Experimental Measurement of P(r) from Tissue Phantom

*P(r)*for

*r<ls**. The experimental system used for the measurements has been described elsewhere and validated with

*P(r)*measurements and simulations of mono-dispersed polystyrene microspheres in water [20

20. V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy, and V. Backman, “Characterization of Light Transport in Scattering Media at Subdiffusion Length Scales with Low-Coherence Enhanced Backscattering,” IEEE J. Sel. Top. Quantum Electron. **16**(3), 619–626 (2010). [CrossRef]

*m*of 1.6 and

*g*of 0.8 is compared to the resulting phase function obtained from the microsphere mixture in Fig. 8(a) . In principle, the agreement may further be improved by using a larger variety of sphere sizes. We then obtained measurements for the unpolarized

*P(r)*by measuring the unpolarized LEBS signal and dividing the Fourier transform of the signal by the coherence function [20

20. V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy, and V. Backman, “Characterization of Light Transport in Scattering Media at Subdiffusion Length Scales with Low-Coherence Enhanced Backscattering,” IEEE J. Sel. Top. Quantum Electron. **16**(3), 619–626 (2010). [CrossRef]

21. E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. **56**(14), 1471–1474 (1986). [CrossRef] [PubMed]

20. V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy, and V. Backman, “Characterization of Light Transport in Scattering Media at Subdiffusion Length Scales with Low-Coherence Enhanced Backscattering,” IEEE J. Sel. Top. Quantum Electron. **16**(3), 619–626 (2010). [CrossRef]

*P(r)*from mono-disperse microsphere suspensions with varying sizes and comparing to Monte Carlo simulations that track polarization and utilized the Mie phase function. The experimentally measured

*P(r)*and the

*P(r)*obtained from the PCA model were in excellent agreement for

*r*<<

*ls**[Fig. 8(b)], with an average error of 7.1% for the range shown. It is interesting to note that the noise level from the experimental measurement in Fig. 8(b) varies with the radial position. The LEBS method measures the two-dimensional quantity

*p(x,y)*via a two-dimensional Fourier transform. The higher noise level at small radii is a result of the decreased number of pixel elements that contribute to the summation over the polar angle. As

*r*increases, the noise level decreases until the coherence function begins to approach small values, at which point the noise level quickly rises due to the division of two numbers that have values near 0.

## 5. Conclusions

*g*>0.6 [Fig. 4(a) and Fig. 6(d)]. The results of the fits can potentially be improved by implementing more principle components, increasing the accuracy of the model fit to the coefficients, or fitting to a smaller range of optical properties. That said, the error for the model applied to biologically relevant optical properties (

*g≥*0.6 and 1<

*m*<2) was less than 1%. The same procedure that was presented here can be used for modeling other ranges of g and m in order to obtain improved accuracy.

*λ*>600nm. Therefore, the technique described here can be utilized to measure scattering properties in the non-absorbing wavelength regions. Absorption can then be characterized by understanding the path length distribution for varying optical properties and measuring the backscattering for varying wavelengths. From Fig. 7, we can conclude that absorption primarily alters the intensity of backscattering at larger radial distances and has a minimal effect at r <<

*l*. This is due to shorter path lengths at smaller radial distances resulting in less attenuation of the scattered rays (The Beer-Lambert law). In cases where absorption cannot be neglected, a traditional diffusion approximation model of absorption in order to quantify the backscattering contribution can be used. In this case, the isotropic scattering portion [

_{a}*P*from Eq. (7) to (9)] can be modeled with standard diffusion approximation equations for reflectance [18

_{g = 0}18. A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, and B. C. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. **35**(13), 2304–2314 (1996). [CrossRef]

19. T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. **19**(4), 879–888 (1992). [CrossRef] [PubMed]

*P(r)*, to a random medium with a tissue-relevant range of optical properties and without the need for performing a large number of Monte Carlo simulations. Only three simulations are required including a simulation for isotropic scattering and two simulations for anisotropic scattering (

*g*= 0.9 with

*m*of 1.5 and 1.01). A Henyey-Greenstein based

*P(r)*model is simpler in that it only requires two Monte Carlo simulations; however, it may not be as comprehensive of a model for tissue characterization. Finally, we presented a methodology for obtaining phantoms that have the potential to closely mimic optical properties of tissue, including the backscattering at small length-scales. The ability to predict the backscattering distribution at subdiffusion length scales holds promise for using techniques such as LEBS to measure optical properties of tissue (such as

*g*,

*m*and

*ls**) by measuring

*P(r)*. These results may also allow for faster, simpler and more accurate solutions to the inverse problem of measuring optical properties from tissue by providing an alternative for existing inverse Monte Carlo methods [5

5. A. Amelink, H. J. C. M. Sterenborg, M. P. L. Bard, and S. A. Burgers, “In vivo measurement of the local optical properties of tissue by use of differential path-length spectroscopy,” Opt. Lett. **29**(10), 1087–1089 (2004). [CrossRef] [PubMed]

6. R. Reif, O. A’Amar, and I. J. Bigio, “Analytical model of light reflectance for extraction of the optical properties in small volumes of turbid media,” Appl. Opt. **46**(29), 7317–7328 (2007). [CrossRef] [PubMed]

11. F. Bevilacqua and C. Depeursinge, “Monte Carlo study of diffuse reflectance at source-detector separations close to one transport mean free path,” J. Opt. Soc. Am. A **16**(12), 2935–2945 (1999). [CrossRef]

22. G. M. Palmer and N. Ramanujam, “Monte Carlo-based inverse model for calculating tissue optical properties. Part I: Theory and validation on synthetic phantoms,” Appl. Opt. **45**(5), 1062–1071 (2006). [CrossRef] [PubMed]

23. L. V. Wang, “Rapid modeling of diffuse reflectance of light in turbid slabs,” J. Opt. Soc. Am. A **15**(4), 936–944 (1998). [CrossRef] [PubMed]

*P(r)*measurement from the Whittle-Matérn phase function at

*r<ls**indicate that the presented models and experimental phantom will be useful for characterizing the optical properties of biological samples.

## Acknowledgements

## References and links

1. | T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. |

2. | E. L. Hull and T. H. Foster, “Steady-state reflectance spectroscopy in the P-3 approximation,” J. Opt. Soc. Am. A |

3. | I. Seo, C. K. Hayakawa, and V. Venugopalan, “Radiative transport in the delta-P1 approximation for semi-infinite turbid media,” Med. Phys. |

4. | M. C. Skala, G. M. Palmer, C. F. Zhu, Q. Liu, K. M. Vrotsos, C. L. Marshek-Stone, A. Gendron-Fitzpatrick, and N. Ramanujam, “Investigation of fiber-optic probe designs for optical spectroscopic diagnosis of epithelial pre-cancers,” Lasers Surg. Med. |

5. | A. Amelink, H. J. C. M. Sterenborg, M. P. L. Bard, and S. A. Burgers, “In vivo measurement of the local optical properties of tissue by use of differential path-length spectroscopy,” Opt. Lett. |

6. | R. Reif, O. A’Amar, and I. J. Bigio, “Analytical model of light reflectance for extraction of the optical properties in small volumes of turbid media,” Appl. Opt. |

7. | Y. L. Kim, Y. Liu, V. M. Turzhitsky, H. K. Roy, R. K. Wali, and V. Backman, “Coherent backscattering spectroscopy,” Opt. Lett. |

8. | V. M. Turzhitsky, A. J. Gomes, Y. L. Kim, Y. Liu, A. Kromine, J. D. Rogers, M. Jameel, H. K. Roy, and V. Backman, “Measuring mucosal blood supply in vivo with a polarization-gating probe,” Appl. Opt. |

9. | J. C. Ramella-Roman, S. A. Prahl, and S. L. Jacques, “Three Monte Carlo programs of polarized light transport into scattering media: part I,” Opt. Express |

10. | L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Mcml - Monte-Carlo Modeling of Light Transport in Multilayered Tissues,” Comput Meth. Prog Biol. |

11. | F. Bevilacqua and C. Depeursinge, “Monte Carlo study of diffuse reflectance at source-detector separations close to one transport mean free path,” J. Opt. Soc. Am. A |

12. | C. J. R. Sheppard, “Fractal model of light scattering in biological tissue and cells,” Opt. Lett. |

13. | J. D. Rogers, I. R. Capoğlu, and V. Backman, “Nonscalar elastic light scattering from continuous random media in the Born approximation,” Opt. Lett. |

14. | I. R. Çapoğlu, J. D. Rogers, A. Taflove, and V. Backman, “Accuracy of the Born approximation in calculating the scattering coefficient of biological continuous random media,” Opt. Lett. |

15. | A. Ishimaru, |

16. | P. Guttorp and T. Gneiting, “Studies in the history of probability and statistics XLIX On the Matern correlation family,” Biometrika |

17. | J. C. Ramella-Roman, S. A. Prahl, and S. L. Jacques, “Three Monte Carlo programs of polarized light transport into scattering media: part II,” Opt. Express |

18. | A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, and B. C. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. |

19. | T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. |

20. | V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy, and V. Backman, “Characterization of Light Transport in Scattering Media at Subdiffusion Length Scales with Low-Coherence Enhanced Backscattering,” IEEE J. Sel. Top. Quantum Electron. |

21. | E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. |

22. | G. M. Palmer and N. Ramanujam, “Monte Carlo-based inverse model for calculating tissue optical properties. Part I: Theory and validation on synthetic phantoms,” Appl. Opt. |

23. | L. V. Wang, “Rapid modeling of diffuse reflectance of light in turbid slabs,” J. Opt. Soc. Am. A |

**OCIS Codes**

(170.7050) Medical optics and biotechnology : Turbid media

(290.1350) Scattering : Backscattering

(290.3200) Scattering : Inverse scattering

(170.6935) Medical optics and biotechnology : Tissue characterization

**ToC Category:**

Optics of Tissue and Turbid Media

**History**

Original Manuscript: July 12, 2010

Revised Manuscript: August 27, 2010

Manuscript Accepted: September 26, 2010

Published: September 30, 2010

**Citation**

Vladimir Turzhitsky, Andrew Radosevich, Jeremy D. Rogers, Allen Taflove, and Vadim Backman, "A predictive model of backscattering at subdiffusion length scales," Biomed. Opt. Express **1**, 1034-1046 (2010)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-1-3-1034

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### References

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