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Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 9, Iss. 5 — Apr. 29, 2014
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Monte Carlo model of the depolarization of backscattered linearly polarized light in the sub-diffusion regime

Andrew J. Gomes, Herbert C. Wolfsen, Michael B. Wallace, Frances K. Cayer, and Vadim Backman  »View Author Affiliations


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

Polarized light scattering methods have found numerous biomedical applications ranging from early cancer detection to glucose sensing owing to their ability to both suppress multiple scattering and obtain intrinsic tissue polarization metrics [1

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

]. Relating the depolarization of light to sample properties is thus an important area of research. Strides in this area have been made in the case of the transmission and reflection of diffuse polarized light. Beginning with Akermans et al [2

2. E. Akkermans, P. E. Wolf, R. Maynard, and G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. France 49, 77–98 (1988).

], an expression for the depolarization ratio was derived for the case of completely isotropic scattering (anisotropic factor 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]

]. Further refinements led to the understanding of diffuse polarized transmission through a slab of fractal continuous media [4

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

].

In this paper, we empirically develop a simple analytical model that relates the reflectance depolarization ratio to the sample transport mean free path and the depolarization length scale, as well as to aspects of the illumination and collection geometry such as the collection angle and size of the illumination/collection area. Our model was first created using polarization-sensitive Monte Carlo simulations that were validated against experimental phantoms. An important aspect of these simulations is that they employed a phase function based on the Whittle-Matérn model [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]

] of light scattering from media with spatially continuous refractive index fluctuations such as those found in biological tissue. We tested our model on a tissue simulating Intralipid phantom and found that our model could accurately measure the depolarization length of the phantom. We further applied our model to 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

The intrinsic properties of the sample that compose Eq. (5) are the transport mean free path lt and the characteristic length scale of depolarization lp. These properties in turn depend on the properties of the refractive index correlation function as summarized by dn2, lc, 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]

]. To determine how the changes in the correlation function parameters influence the terms in Eq. (5) and the resulting light depolarization two things need to be calculated: 1.) The effect of the index correlation on the scattering parameters which can be determined from [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]

] and 2.) The effect of the index correlation function on the depolarization length scale which can be determined from Eqs. (2)(4). As an example, for a fixed klc = 10, an increase in m from 1.6 to 1.9 is expected to decrease lt (decrease d due to increased scattering) according to Fig. 5 of [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]

] while lp/lt is decreased (decrease d) according to Eqs. (2)(4). The decrease in the depolarization length combines with the decrease in lt 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 f1, f2, and f3.

There are two important aspects of Eq. (5). The first is that it is applicable to systems having continuous refractive index profiles and also potentially applicable to systems having discrete refractive index profiles. The difference will involve in how the theoretical value lp 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 (lt) but different phase functions (and hence lp’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

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

Depolarization ratio measurements from phantoms and human esophageal tissue were taken with a polarization-gated spectroscopy probe that has been characterized and described in detail previously [15

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]

]. In brief, the probe consists of three 200 μm diameter fibers (NA = 0.22): an illumination fiber and two collection fibers. Linear polarizer placement allows the simultaneous collection of co-polarized (I ) and cross-polarized (I) reflected light relative to the incident linearly polarized beam. A gradient refractive index (GRIN) lens collimates the incident light (half-angle divergence of 3 degrees) and focuses light backscattered from the sample onto the collections fibers. A white light-emitting diode (WT&T) is used for illumination while fiber optic spectrometers (Ocean Optics) processes light from the collection fibers as a function of wavelength. The radius (R) of the illumination/collection areas on the sample surface for the current probe is 475 μm and the angle (θc) between illumination and collection beams is ~14°.

2.4 Experimental phantoms

We carried out several experiments on microsphere phantoms with the polarization-gated probe and compared the depolarization ratios observed with the probe with results from Mie based MC simulations of the microsphere phantom and probe geometry. The purpose of these experiments was to determine if the MC simulations could accurately predict the depolarization ratio observed by the probe. We prepared phantoms consisting of polystyrene microspheres (Thermo-Scientific) of different sizes and concentrations in deionized water. The optical properties of the prepared phantoms were calculated using Mie theory and the phantom properties are summarized in Table 2

Table 2. Comparison of probe measurements of the depolarization ratio (d) from microsphere phantoms with Monte Carlo measurements of d from microsphere phantoms

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. A series of 20 individual spectral measurements were taken with the fiber-optic probe immersed in the microsphere solution. The solution was illuminated with light from a white light-emitting diode (WT&T). Reflected light was collected by two Ocean Optics fiber-optic spectrometers. These measurements were background subtracted and normalized to a diffuse reflectance standard. The signal intensity at a wavelength of 589 nm was used to calculate the depolarization from each microsphere solution. These results were then directly compared with Mie-based MC simulations of the exact phantom optical properties and probe geometry. From these experiments, we calculated a correction coefficient to apply to Eq. (5) that takes into account incomplete overlap of illumination-collection areas of the probe as well as the below 100% contrast polarizers used by the probe. The correction coefficient is described in Section 3.4.

Next, we took spectral measurements with the polarization-gated probe on a 20% Intralipid (Sigma-Aldrich) solution. The purpose of this experiment was to test if our depolarization model in Eq. (5) could be utilized to predict the depolarization length of the sample. Equation (5) was derived from MC simulations using a Whittle-Matérn model of light scattering from continuous refractive index media. The refractive index variations from Intralipid cannot be considered continuous. However, Eq. (5) suggests that the depolarization ratio will only depend on lt and lp. 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]

]. We determined the lt 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

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]

]. From Eqs. (2) and (3), lp depends on the elements of the amplitude scattering matrix and ls. The parameter ls can be determined from the lt 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]

]. The elements of the amplitude scattering matrix for 20% Intralipid can be computed by superimposing the Mie theory results for each particle size in the Intralipid particle size distribution which has also been provided in [24

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

]. The measured d value from the probe and the lt values of the phantom were used to calculate experimentally measured lp values using Eq. (5) which were then compared with the theoretical lp values for 20% Intralipid.

2.5 Measurement of depolarization length with polarization-gating

From Eq. (5), the depolarization length lp can be extracted if both d and lt can be measured. The polarization-gated probe directly measures d while lt 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 lt [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]

]. The physical basis for this observation is that since it takes several scattering events to depolarize the incident light, the cross-polarized signal originates from multiple scattering even when the illumination-collection radius is small. Using this fact, we have previously shown that spectral measurement of the cross-polarized intensity can be used to measure both lt 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]

]. In brief, the reflectance intensity of the cross-polarized signal can be written as
I(λ)=[cf1(m,θc)(Raλ2m4)f2(m,θc)]exp(μa(λ)L¯(λ)),
(6)
where c is a calibration constant determined experimentally on a phantom with known optical properties, f1 and f2 are functions of m and θc, a is a fitting a parameter, μa is the absorption coefficient, and L¯is the mean effective path length whose functional dependence on optical properties and illumination-collection geometry has been given previously [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]

]. In the Whittle-Matérn model, the quantity aλ2m4 determines the value of 1/lt(λ) given that klc>>1which is satisfied for most biological tissue [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]

]. By fitting Eq. (6) to the collected spectral intensity of the cross-polarized signal using a and m as free parameters, both m and lt can be measured.

2.6 Human studies

The study was conducted at Mayo Clinic Florida in, Jacksonville FL and approved by the Institutional Review Board. Data were collected between 2010 and 2012 from a total of 63 patients undergoing upper GI endoscopy. Barrett’s esophagus (BE) surveillance was the primary reason for endoscopy though some patients were evaluated for anemia, dyspepsia, or heartburn. After informed consent, patients underwent upper endoscopy with conscious sedation using midazolam, fentanyl and meperidine or propofol. After the initial diagnostic evaluation but prior to any tissue biopsy sampling, approximately 10 readings, each taking 15 ms, were taken with a polarization-gated fiber optic probe passed down the endoscope accessory channel. The probe was placed in gentle contact with the gut mucosa surface in BE regions within the distal esophageal mucosa. If a dysplastic nodule was seen, probe measurements were also taken from the nodule itself. The co-polarized and cross-polarized reflectance intensities were analyzed at a wavelength of 630 nm to mitigate the effect of hemoglobin absorption. The depolarization ratios of tissue sites from the same region were averaged together to form a single 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

3.3 Monte Carlo model for the depolarization ratio

Equation (5) was fit to the data generated by MC simulations for a variety of optical properties and for several R and θc. An example fit is shown in Fig. 4(a)
Fig. 4 (a) The depolarization ratio as a function of R/lt and R/lp from MC simulations (filled circles). The displayed surface is the best fit of Eq. (5) to the MC data for a θc of 14°. (b) The depolarization ratio from MC simulations (dMC) compared with the depolarization ratio as computed by the model in Eq. (5) (dModel). (c) The percent error between dMC and dModel as a function of dMC.
for θc = 0-18°. The individual black data points are MC outcomes while the shaded surface is the best fit to the data points (r2 > 0.99). The functional relationship between [f1 f2 f3] and θc was found to be linear of the form x1 + x2θc and these values are shown in Table 1. Using the values of these parameters, we computed d generated by Eq. (5) (dModel) and compared it with d from the actual MC calculations (dMC). In Fig. 4(b), dMC is plotted versus dModel with the result showing a strong linear correlation (R2 > 0.99). The percentage error between dMC and dModel is shown in Fig. 4(c) as a function of dMC. The mean percentage error was 8 percent across the entire range of dMC while it was 3 percent in the more biologically relevant regime of dMC > 0.5.

3.4 Experimental verification of the depolarization ratio model

We measured 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 μs'evaluated at λ = 589 nm, the 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.

The percent error between the probe experiment 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 θc = 0-18°. The result of this comparison is shown in Fig. 5
Fig. 5 Depolarization ratio (d) calculated from MC simulations incorporating ideal polarizers and complete illumination-collection area overlap (dideal) versus the depolarization ratio as calculated from MC simulations incorporating polarizers with 94% contrast and a 30 µm center-to-center separation between illumination and collection areas (dnon-ideal).
where it seen that there is a consistent proportional relationship (r2 > 0.99) between the d calculated from the ideal and non-ideal simulations. For this specific probe geometry, Eq. (5) can then be modified as
d=0.89(f1(θc)+f2(θc)Rlt+f3(θc)Rlp)1.
(7)
For probes with other R, θc, 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.

3.5 Application of the depolarization model to biological tissue

Encoded in the ratio lp/lt is information about the scattering phase function. This is apparent from Eqs. (2)(4) that show lp/lsdepends on the variables m and klc 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]

]. Furthermore, we arrive at lp/ltby multiplying lp/lsby (1-g) where g is average cosine of the scattering angle. The value g can be expressed in terms of m and klc such that lp/ltis entirely a function of m and klc [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]

]. Since we measure m from the cross-polarized reflectance signal (from Eq. (6)), the value of klc can be determined from the measurementlp/lt. This would allow us to reconstruct the phase function shape and determine the anisotropy factor from each tissue site. However, the measured g is very sensitive to the precise magnitude oflp/lt. For example, for a fixed m = 1.6, a shift of 10% in lp/ltfrom 1.42 to 1.29 can alter the measured g from 0.8 to 0.95. Thus very precise and accurate measurements of lp/ltare required and due to inherent limitations on signal-to-noise ratio, accuracy of the lp, lt, 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 lt for esophagus, the parameter lp/lt may be determined. The value of lt 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

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]

]. The resulting range of lp/ltwould be from 1.19 – 1.67. These values correspond to klc = 1.26 – 30.9 and g = 0.58 – 0.99. It is encouraging that these values are within the expected range for the upper gastrointestinal tract [27

27. A. N. Bashkatov, E. A. Genina, V. I. Kochubey, A. A. Gavrilova, S. V. Kapralov, V. A. Grishaev, and V. V. Tuchin, “Optical properties of human stomach mucosa in the spectral range from 400 to 2000 nm - art. no. 673401,” Proc. Soc. Photo-Opt. Ins. 6734, 73401 (2007).

,28

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]

]. The lp/ltratio could potentially be used to measure the tissue anisotropy factor from each tissue site but the wide range of g values obtainable from the lt range suggests that the ratio must be estimated precisely.

In our depolarization model, for the sake of simplicity, we have not considered tissue inhomogeneity or other polarization effects such as birefringence. Our model is based on the assumption, common in the biomedical optics literature, that the scattering properties are distributed homogeneously within the medium [29

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

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]

]. Tissue, however, is a complex layered and inhomogeneous structure. The depolarization signal from our probe, for example, will sample both the epithelium and stroma of the esophageal tissue which are expected to have different optical properties. The optical properties we measure will have contributions from both layers. This can be mitigated by designing a probe whose penetration depth for the cross-polarized signal (which has the highest depth) does not exceed the depth of the top layer. This can be achieved by decreasing R and/or increasing θc [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]

]. Finally, tissue can be expected to have linear birefringence due to structural alignment of collagen fibers, lipid bilayers, muscle fibers, etc. To test whether this property could influence the depolarization ratio, we performed and compared two MC simulations: one where the sample had no birefringence and another where the sample possessed random birefringence (the case applicable to esophageal tissue where collagen fibers are oriented randomly) having a difference in ordinary and extraordinary refractive indices of 0.001 which is the maximum difference expected in tissue [33

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]

]. The method for the MC simulation with birefringence followed that given in [34

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]

]. We found that the depolarization ratio varied less than 4.2% between these two simulations indicating that the birefringence will have a minimal effect on the observed depolarization ratio. This is consistent with the finding in [34

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]

] that birefringence had minimal impact on the spatial reflectance profiles for linearly polarized light.

4. Conclusion

The methods outlined in this paper provide a model to predict the depolarization of backscattered linearly polarized light for different sample optical properties as well as different illumination-collection geometries. The model can be applied to probes and systems that collect sub-diffusive photons. Our model was validated on tissue simulating phantoms and we applied our depolarization model to calculate the depolarization length from biological tissue 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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M. Xu and R. R. Alfano, “Fractal mechanisms of light scattering in biological tissue and cells,” Opt. Lett. 30(22), 3051–3053 (2005). [CrossRef] [PubMed]

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]

12.

M. Hunter, V. Backman, G. Popescu, M. Kalashnikov, C. W. Boone, A. Wax, V. Gopal, K. Badizadegan, G. D. Stoner, and M. S. Feld, “Tissue self-affinity and polarized light scattering in the born approximation: a new model for precancer detection,” Phys. Rev. Lett. 97(13), 138102 (2006). [CrossRef] [PubMed]

13.

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

14.

O. Nadiarnykh, R. B. LaComb, M. A. Brewer, and P. J. Campagnola, “Alterations of the extracellular matrix in ovarian cancer studied by Second Harmonic Generation imaging microscopy,” BMC Cancer 10(1), 94 (2010). [CrossRef] [PubMed]

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]

16.

D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: Influence of the size parameter,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 49(2), 1767–1770 (1994). [CrossRef] [PubMed]

17.

M. Xu and R. R. Alfano, “Random walk of polarized light in turbid media,” Phys. Rev. Lett. 95(21), 213901 (2005). [CrossRef] [PubMed]

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]

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]

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]

21.

N. Ghosh, H. Patel, and P. Gupta, “Depolarization of light in tissue phantoms - effect of a distribution in the size of scatterers,” Opt. Express 11(18), 2198–2205 (2003). [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]

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]

24.

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

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]

27.

A. N. Bashkatov, E. A. Genina, V. I. Kochubey, A. A. Gavrilova, S. V. Kapralov, V. A. Grishaev, and V. V. Tuchin, “Optical properties of human stomach mucosa in the spectral range from 400 to 2000 nm - art. no. 673401,” Proc. Soc. Photo-Opt. Ins. 6734, 73401 (2007).

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]

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]

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 18(6), 5580–5594 (2010). [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]

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]

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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References

  1. N. Ghosh, I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt. 16(11), 110801 (2011). [CrossRef] [PubMed]
  2. E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. France 49, 77–98 (1988).
  3. L. F. Rojas-Ochoa, D. Lacoste, R. Lenke, P. Schurtenberger, F. Scheffold, “Depolarization of backscattered linearly polarized light,” J. Opt. Soc. Am. A 21(9), 1799–1804 (2004). [CrossRef] [PubMed]
  4. M. Xu, R. R. Alfano, “Light depolarization by tissue and phantoms,” Proc. SPIE 60840, 60840T (2006). [CrossRef]
  5. J. D. Rogers, I. R. Capoğlu, V. Backman, “Nonscalar elastic light scattering from continuous random media in the Born approximation,” Opt. Lett. 34(12), 1891–1893 (2009). [CrossRef] [PubMed]
  6. A. J. Gomes, S. Ruderman, M. DelaCruz, R. K. Wali, H. K. Roy, 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]
  7. A. Radosevich, J. Rogers, V. Turzhitsky, N. Mutyal, J. Yi, H. Roy, V. Backman, “Polarized enhanced backscattering spectroscopy for characterization of biological tissues at subdiffusion length-scales,” IEEE J. Sel. Top. Quantum Electron. 18(4), 1313–1325 (2011).
  8. M. Moscoso, J. B. Keller, G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. Am. A 18(4), 948–960 (2001). [CrossRef] [PubMed]
  9. J. M. Schmitt, G. Kumar, “Turbulent nature of refractive-index variations in biological tissue,” Opt. Lett. 21(16), 1310–1312 (1996). [CrossRef] [PubMed]
  10. M. Xu, R. R. Alfano, “Fractal mechanisms of light scattering in biological tissue and cells,” Opt. Lett. 30(22), 3051–3053 (2005). [CrossRef] [PubMed]
  11. V. Turzhitsky, A. Radosevich, J. D. Rogers, A. Taflove, V. Backman, “A predictive model of backscattering at subdiffusion length scales,” Biomed. Opt. Express 1(3), 1034–1046 (2010). [CrossRef] [PubMed]
  12. M. Hunter, V. Backman, G. Popescu, M. Kalashnikov, C. W. Boone, A. Wax, V. Gopal, K. Badizadegan, G. D. Stoner, M. S. Feld, “Tissue self-affinity and polarized light scattering in the born approximation: a new model for precancer detection,” Phys. Rev. Lett. 97(13), 138102 (2006). [CrossRef] [PubMed]
  13. C. J. Sheppard, “Fractal model of light scattering in biological tissue and cells,” Opt. Lett. 32(2), 142–144 (2007). [CrossRef] [PubMed]
  14. O. Nadiarnykh, R. B. LaComb, M. A. Brewer, P. J. Campagnola, “Alterations of the extracellular matrix in ovarian cancer studied by Second Harmonic Generation imaging microscopy,” BMC Cancer 10(1), 94 (2010). [CrossRef] [PubMed]
  15. V. M. Turzhitsky, A. J. Gomes, Y. L. Kim, Y. Liu, A. Kromine, J. D. Rogers, M. Jameel, H. K. Roy, V. Backman, “Measuring mucosal blood supply in vivo with a polarization-gating probe,” Appl. Opt. 47(32), 6046–6057 (2008). [CrossRef] [PubMed]
  16. D. Bicout, C. Brosseau, A. S. Martinez, J. M. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: Influence of the size parameter,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 49(2), 1767–1770 (1994). [CrossRef] [PubMed]
  17. M. Xu, R. R. Alfano, “Random walk of polarized light in turbid media,” Phys. Rev. Lett. 95(21), 213901 (2005). [CrossRef] [PubMed]
  18. Y. Liu, Y. Kim, X. Li, V. Backman, “Investigation of depth selectivity of polarization gating for tissue characterization,” Opt. Express 13(2), 601–611 (2005). [CrossRef] [PubMed]
  19. X. Guo, M. F. G. Wood, N. Ghosh, 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]
  20. M. Ahmad, S. Alali, A. Kim, M. F. Wood, M. Ikram, 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]
  21. N. Ghosh, H. Patel, P. Gupta, “Depolarization of light in tissue phantoms - effect of a distribution in the size of scatterers,” Opt. Express 11(18), 2198–2205 (2003). [CrossRef] [PubMed]
  22. V. Sankaran, M. J. Everett, D. J. Maitland, J. T. Walsh., “Comparison of polarized-light propagation in biological tissue and phantoms,” Opt. Lett. 24(15), 1044–1046 (1999). [CrossRef] [PubMed]
  23. S. A. Prahl, M. J. van Gemert, 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]
  24. R. Michels, F. Foschum, A. Kienle, “Optical properties of fat emulsions,” Opt. Express 16(8), 5907–5925 (2008). [CrossRef] [PubMed]
  25. A. J. Gomes, V. Backman, “Analytical light reflectance models for overlapping illumination and collection area geometries,” Appl. Opt. 51(33), 8013–8021 (2012). [PubMed]
  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, 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]
  27. A. N. Bashkatov, E. A. Genina, V. I. Kochubey, A. A. Gavrilova, S. V. Kapralov, V. A. Grishaev, V. V. Tuchin, “Optical properties of human stomach mucosa in the spectral range from 400 to 2000 nm - art. no. 673401,” Proc. Soc. Photo-Opt. Ins. 6734, 73401 (2007).
  28. C. Holmer, K. S. Lehmann, J. Wanken, C. Reissfelder, A. Roggan, G. Mueller, H. J. Buhr, J. P. Ritz, “Optical properties of adenocarcinoma and squamous cell carcinoma of the gastroesophageal junction,” J. Biomed. Opt. 12(1), 014025 (2007). [CrossRef] [PubMed]
  29. A. Amelink, H. J. Sterenborg, M. P. Bard, 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]
  30. A. Kim, M. Roy, F. Dadani, 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 18(6), 5580–5594 (2010). [CrossRef] [PubMed]
  31. R. Reif, M. S. Amorosino, K. W. Calabro, O. A’Amar, S. K. Singh, 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]
  32. A. J. Gomes, V. Turzhitsky, S. Ruderman, 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, 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, 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]

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