## Monte Carlo model of the depolarization of backscattered linearly polarized light in the sub-diffusion regime |

Optics Express, Vol. 22, Issue 5, pp. 5325-5340 (2014)

http://dx.doi.org/10.1364/OE.22.005325

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### Abstract

We present a predictive model of the depolarization ratio of backscattered linearly polarized light from spatially continuous refractive index media that is applicable to the sub-diffusion regime of light scattering. Using Monte Carlo simulations, we derived a simple relationship between the depolarization ratio and both the sample optical properties and illumination-collection geometry. Our model was validated on tissue simulating phantoms and found to be in good agreement. We further show the utility of this model by demonstrating its use for measuring the depolarization length from biological tissue *in vivo*. We expect our results to aid in the interpretation of the depolarization ratio from sub-diffusive reflectance measurements.

© 2014 Optical Society of America

## 1. Introduction

1. N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt. **16**(11), 110801 (2011). [CrossRef] [PubMed]

*g*= 0). The validity of the expression was further extended across the range of anisotropy values by the work of Rojas-Ochoa et al [3

3. L. F. Rojas-Ochoa, D. Lacoste, R. Lenke, P. Schurtenberger, and F. Scheffold, “Depolarization of backscattered linearly polarized light,” J. Opt. Soc. Am. A **21**(9), 1799–1804 (2004). [CrossRef] [PubMed]

4. M. Xu and R. R. Alfano, “Light depolarization by tissue and phantoms,” Proc. SPIE **60840**, 60840T (2006). [CrossRef]

5. 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]

*in vivo*polarization-gated fiber-optic probe measurements from human esophageal tissue and demonstrate that the depolarization ratio, the transport mean free path, and the depolarization length are all altered with dysplasia. The work presented in this paper should aid in the use of the depolarization ratio of sub-diffusive reflectance to characterize the intrinsic optical properties of turbid media.

## 2. Materials and methods

### 2.1 Polarization-sensitive Monte Carlo simulations

### 2.2 Depolarization ratio model

*l*

_{t}and the characteristic length scale of depolarization

*l*

_{p}. These properties in turn depend on the properties of the refractive index correlation function as summarized by

*dn*

^{2},

*l*

_{c}, and

*m*. Explicit relationships between the index correlation parameters and scattering properties of the medium under the Whittle-Matérn model are given by Rogers et al. [5

5. 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]

5. 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]

*kl*

_{c}= 10, an increase in

*m*from 1.6 to 1.9 is expected to decrease

*l*

_{t}(decrease

*d*due to increased scattering) according to Fig. 5 of [5

**34**(12), 1891–1893 (2009). [CrossRef] [PubMed]

*l*

_{p}/

*l*

_{t}is decreased (decrease

*d*) according to Eqs. (2)–(4). The decrease in the depolarization length combines with the decrease in

*l*

_{t}such that

*d*is decreased with increasing

*m.*We performed MC simulations as described in Section 2.1 and calculated

*d*from each simulation according to Eq. (1). We then fit Eq. (5) to the MC data using a least squares approach to determine the coefficients

*f*

_{1},

*f*

_{2}, and

*f*

_{3}.

*l*

_{p}is calculated for a given sample. For example, Mie theory can be used in the case of discrete spherical particles to calculate the amplitude scattering matrix used in Eqs. (2)–(4) while Whittle-Matérn theory is used for continuous media. The second important consequence of Eq. (5) is that it implicitly predicts that samples with matched bulk optical properties (

*l*

_{t}) but different phase functions (and hence

*l*

_{p}’s) will have different depolarization properties. This is in agreement with experimental evidence showing that tissue and phantoms depolarize light differently and that the size distribution of scatterers affects light depolarization [20

20. M. Ahmad, S. Alali, A. Kim, M. F. Wood, M. Ikram, and I. A. Vitkin, “Do different turbid media with matched bulk optical properties also exhibit similar polarization properties?” Biomed. Opt. Express **2**(12), 3248–3258 (2011). [CrossRef] [PubMed]

22. V. Sankaran, M. J. Everett, D. J. Maitland, and J. T. Walsh Jr., “Comparison of polarized-light propagation in biological tissue and phantoms,” Opt. Lett. **24**(15), 1044–1046 (1999). [CrossRef] [PubMed]

### 2.3 Probe instrumentation

15. 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]

*R*) of the illumination/collection areas on the sample surface for the current probe is 475 μm and the angle (θ

_{c}) between illumination and collection beams is ~14°.

### 2.4 Experimental phantoms

*l*

_{t}and

*l*

_{p}. These values can also be calculated for discrete systems [4

4. M. Xu and R. R. Alfano, “Light depolarization by tissue and phantoms,” Proc. SPIE **60840**, 60840T (2006). [CrossRef]

*l*

_{t}spectrum of the Intralipid phantom using integrating sphere measurements coupled with the adding-doubling algorithm [23

23. S. A. Prahl, M. J. van Gemert, and A. J. Welch, “Determining the optical properties of turbid media by using the adding-doubling method,” Appl. Opt. **32**(4), 559–568 (1993). [CrossRef] [PubMed]

25. A. J. Gomes and V. Backman, “Analytical light reflectance models for overlapping illumination and collection area geometries,” Appl. Opt. **51**(33), 8013–8021 (2012). [PubMed]

*l*

_{p}depends on the elements of the amplitude scattering matrix and

*l*

_{s}. The parameter

*l*

_{s}can be determined from the

*l*

_{t}measurements using the anisotropy factor (

*g*) of 20% Intralipid which has been previously measured [24

24. R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express **16**(8), 5907–5925 (2008). [CrossRef] [PubMed]

24. R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express **16**(8), 5907–5925 (2008). [CrossRef] [PubMed]

*d*value from the probe and the

*l*

_{t}values of the phantom were used to calculate experimentally measured

*l*

_{p}values using Eq. (5) which were then compared with the theoretical

*l*

_{p}values for 20% Intralipid.

### 2.5 Measurement of depolarization length with polarization-gating

*l*

_{p}can be extracted if both

*d*and

*l*

_{t}can be measured. The polarization-gated probe directly measures

*d*while

*l*

_{t}can be quantified with the cross-polarized reflectance signal. The cross-polarized reflectance signal is advantageous because its signal intensity has no added relationship to the phase function than what is incorporated into

*l*

_{t}[6

6. A. J. Gomes, S. Ruderman, M. DelaCruz, R. K. Wali, H. K. Roy, and V. Backman, “In vivo measurement of the shape of the tissue-refractive-index correlation function and its application to detection of colorectal field carcinogenesis,” J. Biomed. Opt. **17**(4), 047005 (2012). [CrossRef] [PubMed]

*l*

_{t}and

*m*using a least-squares fitting method [25

25. A. J. Gomes and V. Backman, “Analytical light reflectance models for overlapping illumination and collection area geometries,” Appl. Opt. **51**(33), 8013–8021 (2012). [PubMed]

*c*is a calibration constant determined experimentally on a phantom with known optical properties,

*f*

_{1}and

*f*

_{2}are functions of

*m*and θ

_{c},

*a*is a fitting a parameter,

25. A. J. Gomes and V. Backman, “Analytical light reflectance models for overlapping illumination and collection area geometries,” Appl. Opt. **51**(33), 8013–8021 (2012). [PubMed]

**34**(12), 1891–1893 (2009). [CrossRef] [PubMed]

*a*and

*m*as free parameters, both

*m*and

*l*

_{t}can be measured.

### 2.6 Human studies

*d*for that region and patient. Of the 63 patients analyzed, 6 had dysplastic nodules from which probe measurements were taken.

## 3. Results

### 3.1 Dependence of depolarization ratio on sample optical properties

### 3.2 Dependence of depolarization ratio on illumination-collection geometry

*R*) on the sample surface and the collection angle θ

_{c}relative to the exact backscattering direction. Due to the scaling property of Monte Carlo, the

*d*can be plotted versus the dimensionless parameter

*R*/

*l*

_{s}as was done in Fig. 2(b). Thus, the effect of

*R*on

*d*for a fixed

*l*

_{s}can also be inferred from Fig. 2(b). Figure 2(b) shows that

*d*decreases as

*R*increases. This can be understood by studying the radial probability distribution (

*P*(

*r*)) of light exiting the sample surface. Previous studies have shown that

*R*→ 0 and approaches 0 as

*R*→ ∞ [18

18. Y. Liu, Y. Kim, X. Li, and V. Backman, “Investigation of depth selectivity of polarization gating for tissue characterization,” Opt. Express **13**(2), 601–611 (2005). [CrossRef] [PubMed]

_{c}on

*d*in Fig. 3. The parameter

*d*decreases as θ

_{c}increases, a finding in accordance with the observation that large angle scattering events are more depolarizing than small angle scattering events [19

19. X. Guo, M. F. G. Wood, N. Ghosh, and I. A. Vitkin, “Depolarization of light in turbid media: a scattering event resolved Monte Carlo study,” Appl. Opt. **49**(2), 153–162 (2010). [CrossRef] [PubMed]

### 3.3 Monte Carlo model for the depolarization ratio

*R*and θ

_{c}. An example fit is shown in Fig. 4(a) for θ

_{c}= 0-18°. The individual black data points are MC outcomes while the shaded surface is the best fit to the data points (

*r*

^{2}> 0.99). The functional relationship between [

*f*

_{1}

*f*

_{2}

*f*

_{3}] and θ

_{c}was found to be linear of the form

*x*

_{1}+

*x*

_{2}θ

_{c}and these values are shown in Table 1. Using the values of these parameters, we computed

*d*generated by Eq. (5) (

*d*

_{Model}) and compared it with

*d*from the actual MC calculations (

*d*

_{MC}). In Fig. 4(b),

*d*

_{MC}is plotted versus

*d*

_{Model}with the result showing a strong linear correlation (R

^{2}> 0.99). The percentage error between

*d*

_{MC}and

*d*

_{Model}is shown in Fig. 4(c) as a function of

*d*

_{MC}. The mean percentage error was 8 percent across the entire range of

*d*

_{MC}while it was 3 percent in the more biologically relevant regime of

*d*

_{MC}> 0.5.

### 3.4 Experimental verification of the depolarization ratio model

*d*from microsphere phantoms with the polarization-gated probe and compared it to

*d*calculated from Mie-based MC simulations of the corresponding phantom. The purpose of this was two-fold. The first objective was to demonstrate that our MC simulations were predictive of the results observed experimentally with the probe. The second objective was to quantify how the non-ideal polarization behavior of a realistic probe would cause deviations from Eq. (5). Equation (5) was derived under the assumptions of ideal polarizers and completely overlapping illumination-collection areas. In our probe, we have observed that the polarizer and GRIN lens contrast is on the order of 94% while there is a small 30 µm center-to-center separation between the illumination-collection areas. To examine these effects, we conducted two Mie-based MC simulations: one assuming ideal polarization and complete overlap and the other simulating polarizers with 94% contrast and 30 µm center-to-center separations. The experimental comparison with these two MC simulations is summarized in Table 2. Each row corresponds to a separate microsphere phantom and the columns give the microsphere diameter of the phantom, the

*g*and

*d*measured experimentally with the probe, and the

*d*’s calculated using ideal and non-ideal MC simulations. Table 2 demonstrates that there is an appreciable difference between the

*d*computed by the ideal and non-ideal MC simulations. The probe measurement of

*d*closely matches the non-ideal MC simulation with an average percent error of <5%. This demonstrates that the MC model of the probe geometry accurately predicts the depolarization observed experimentally by the probe.

*d value*and the ideal MC

*d value*was ~12%. These results suggest that an error term needs to be introduced into Eq. (5) to account for the non-ideal behavior of realistic fiber-optic probes. To determine this error term we calculated the

*d*from Whittle-Matérn based MC simulations for ideal and non-ideal cases across a wide range of optical properties described in Section 2.1. The probe geometry was held constant with

*R*= 475 µm and θ

_{c}= 0-18°. The result of this comparison is shown in Fig. 5 where it seen that there is a consistent proportional relationship (r

^{2}> 0.99) between the

*d*calculated from the ideal and non-ideal simulations. For this specific probe geometry, Eq. (5) can then be modified asFor probes with other

*R*, θ

_{c}, or center-to-center separation the scaling factor may be different than 0.89. The scaling factor should be calculated independently for each probe design.

*l*

_{p}using both the measured depolarization ratio from the probe and the

*l*

_{t}values of the phantom. The

*l*

_{t}values can be determined using an integrating sphere measurement coupled with the adding-doubling algorithm or by analysis of the cross-polarized reflectance signal as described in Section 2.5. In Fig. 6 we present the

*l*

_{p}/

*l*

_{t}spectrum of the Intralipid sample. The

*l*

_{t}spectrum was measured with an integrating sphere and the theoretical

*l*

_{p}spectrum of the Intralipid was calculated from Eqs. (2)–(4) as well as the known particle size distribution and

*g*spectrum of Intralipid as described in Section 2.4. Using the known

*l*

_{t}spectrum and the measured depolarization ratio as a function of wavelength, we computed an experimentally measured

*l*

_{p}at each wavelength. These experimentally measured

*l*

_{p}values are compared with the theoretical

*l*

_{p}values in Fig. 7(a). There is a strong linear correlation between the experimentally derived and theoretical

*l*

_{p}values with a linear correlation coefficient greater than 0.99. We further quantified the percent error between the measured and theoretical

*l*

_{p}values as illustrated in Fig. 7(b). The percent error is defined as

*l*

_{p}values are generally lower than the theoretical

*l*

_{p}by an average of 7%. The error magnitude decreases for higher

*l*

_{p}which is consistent with Fig. 4(c) showing a lower error for higher d (and hence higher

*l*

_{p}). Sources of discrepancy include noise in the polarization-gated measurement, possible error in the measurement of

*l*

_{t}, as well as uncertainty regarding the precise value of the scaling factor in Eq. (7). Despite these sources of error, Fig. 7 demonstrates good correlation between the measured and theoretical

*l*

_{p}values.

### 3.5 Application of the depolarization model to biological tissue

*in vivo*as an illustrative example of applying the model. Polarization-gated probe measurements were taken from regions of the esophageal mucosa with and without dysplasia. We found that the depolarization ratio was significantly altered in dysplastic sites as shown in Fig. 8(a). The depolarization ratio increases from a mean value of 0.56 in non-dysplastic regions to a mean value of 0.7 in dysplastic ones with a statistically significant p-value of 0.005. Figure 8(b) illustrates that this elevation of

*d*with dysplasia is driven by both an increase in

*l*

_{t}(from 0.08 to 0.12 cm, p-value = 0.02) and

*l*

_{p}(from 0.12 to 0.19 cm, p-value = 0.02). The value of

*l*

_{t}was determined from the cross-polarized signal as described in Section 2.5, while

*l*

_{p}was determined from Eq. (7). Our observations of an increase in

*l*

_{t}with dysplasia in Barrett’s esophagus are consistent with previous studies using diffuse reflectance spectroscopy [26

26. I. Georgakoudi, B. C. Jacobson, J. Van Dam, V. Backman, M. B. Wallace, M. G. Müller, Q. Zhang, K. Badizadegan, D. Sun, G. A. Thomas, L. T. Perelman, and M. S. Feld, “Fluorescence, reflectance, and light-scattering spectroscopy for evaluating dysplasia in patients with Barrett’s esophagus,” Gastroenterology **120**(7), 1620–1629 (2001). [CrossRef] [PubMed]

*m*and

*kl*

_{c}which are the parameters that govern the shape of the Whittle-Matérn phase function [11

11. V. Turzhitsky, A. Radosevich, J. D. Rogers, A. Taflove, and V. Backman, “A predictive model of backscattering at subdiffusion length scales,” Biomed. Opt. Express **1**(3), 1034–1046 (2010). [CrossRef] [PubMed]

*g*) where

*g*is average cosine of the scattering angle. The value

*g*can be expressed in terms of

*m*and

*kl*

_{c}such that

*m*and

*kl*

_{c}[5

**34**(12), 1891–1893 (2009). [CrossRef] [PubMed]

*m*from the cross-polarized reflectance signal (from Eq. (6)), the value of

*kl*

_{c}can be determined from the measurement

*g*is very sensitive to the precise magnitude of

*m*= 1.6, a shift of 10% in

*g*from 0.8 to 0.95. Thus very precise and accurate measurements of

*l*

_{p},

*l*

_{t}, and

*m*measurements, accuracy of the depolarization ratio model, as well as tissue inhomogeneity and variability, we do not believe that we can obtain a robust measurement of

*g*from each tissue site at this time. In an attempt to minimize these sources of error, we took the average values of all the depolarization ratios and m values measured from each tissue site in our dataset. These average ± standard deviation values were m = 1.7 ± 0.09 and d = 0.57 ± 0.06. Using these values in conjunction with of the value of

*l*

_{t}for esophagus, the parameter

*l*

_{t}for esophagus at ~ 630 nm varies in the literature from ~ 0.06-0.13 cm with our own measured value of 0.08 cm falling in between this range [26

26. I. Georgakoudi, B. C. Jacobson, J. Van Dam, V. Backman, M. B. Wallace, M. G. Müller, Q. Zhang, K. Badizadegan, D. Sun, G. A. Thomas, L. T. Perelman, and M. S. Feld, “Fluorescence, reflectance, and light-scattering spectroscopy for evaluating dysplasia in patients with Barrett’s esophagus,” Gastroenterology **120**(7), 1620–1629 (2001). [CrossRef] [PubMed]

28. C. Holmer, K. S. Lehmann, J. Wanken, C. Reissfelder, A. Roggan, G. Mueller, H. J. Buhr, and J. P. Ritz, “Optical properties of adenocarcinoma and squamous cell carcinoma of the gastroesophageal junction,” J. Biomed. Opt. **12**(1), 014025 (2007). [CrossRef] [PubMed]

28. C. Holmer, K. S. Lehmann, J. Wanken, C. Reissfelder, A. Roggan, G. Mueller, H. J. Buhr, and J. P. Ritz, “Optical properties of adenocarcinoma and squamous cell carcinoma of the gastroesophageal junction,” J. Biomed. Opt. **12**(1), 014025 (2007). [CrossRef] [PubMed]

*g*values obtainable from the

*l*

_{t}range suggests that the ratio must be estimated precisely.

29. A. Amelink, H. J. Sterenborg, M. P. 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]

31. R. Reif, M. S. Amorosino, K. W. Calabro, O. A’Amar, S. K. Singh, and I. J. Bigio, “Analysis of changes in reflectance measurements on biological tissues subjected to different probe pressures,” J. Biomed. Opt. **13**(1), 010502 (2008). [CrossRef] [PubMed]

*R*and/or increasing θ

_{c}[32

32. A. J. Gomes, V. Turzhitsky, S. Ruderman, and V. Backman, “Monte Carlo model of the penetration depth for polarization gating spectroscopy: influence of illumination-collection geometry and sample optical properties,” Appl. Opt. **51**(20), 4627–4637 (2012). [CrossRef] [PubMed]

33. M. J. Everett, K. Schoenenberger, B. W. Colston Jr, and L. B. Da Silva, “Birefringence characterization of biological tissue by use of optical coherence tomography,” Opt. Lett. **23**(3), 228–230 (1998). [CrossRef] [PubMed]

34. A. J. Radosevich, J. D. Rogers, I. R. Capoğlu, N. N. Mutyal, P. Pradhan, and V. Backman, “Open source software for electric field Monte Carlo simulation of coherent backscattering in biological media containing birefringence,” J. Biomed. Opt. **17**(11), 115001 (2012). [CrossRef] [PubMed]

34. A. J. Radosevich, J. D. Rogers, I. R. Capoğlu, N. N. Mutyal, P. Pradhan, and V. Backman, “Open source software for electric field Monte Carlo simulation of coherent backscattering in biological media containing birefringence,” J. Biomed. Opt. **17**(11), 115001 (2012). [CrossRef] [PubMed]

## 4. Conclusion

*in vivo*using a simple polarization-gated measurement. We expect that our depolarization model will help in the assessment of tissue optical properties using polarization-gating spectroscopy. Future work will further explore the effect of tissue inhomogeneity on the observed depolarization ratio as well as obtaining site-specific measurements of the anisotropy factor. The full range of applicability of the depolarization model to discrete systems is also of interest.

## References and links

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

30. | A. Kim, M. Roy, F. Dadani, and B. C. Wilson, “A fiber optic reflectance probe with multiple source-collector separations to increase the dynamic range of derived tissue optical absorption and scattering coefficients,” Opt. Express |

31. | R. Reif, M. S. Amorosino, K. W. Calabro, O. A’Amar, S. K. Singh, and I. J. Bigio, “Analysis of changes in reflectance measurements on biological tissues subjected to different probe pressures,” J. Biomed. Opt. |

32. | A. J. Gomes, V. Turzhitsky, S. Ruderman, and V. Backman, “Monte Carlo model of the penetration depth for polarization gating spectroscopy: influence of illumination-collection geometry and sample optical properties,” Appl. Opt. |

33. | M. J. Everett, K. Schoenenberger, B. W. Colston Jr, and L. B. Da Silva, “Birefringence characterization of biological tissue by use of optical coherence tomography,” Opt. Lett. |

34. | A. J. Radosevich, J. D. Rogers, I. R. Capoğlu, N. N. Mutyal, P. Pradhan, and V. Backman, “Open source software for electric field Monte Carlo simulation of coherent backscattering in biological media containing birefringence,” J. Biomed. Opt. |

**OCIS Codes**

(290.5855) Scattering : Scattering, polarization

(170.6935) Medical optics and biotechnology : Tissue characterization

**ToC Category:**

Scattering

**History**

Original Manuscript: September 12, 2012

Revised Manuscript: October 27, 2012

Manuscript Accepted: November 15, 2012

Published: February 28, 2014

**Virtual Issues**

Vol. 9, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Andrew J. Gomes, Herbert C. Wolfsen, Michael B. Wallace, Frances K. Cayer, and Vadim Backman, "Monte Carlo model of the depolarization of backscattered linearly polarized light in the sub-diffusion regime," Opt. Express **22**, 5325-5340 (2014)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-22-5-5325

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