## Detecting axial heterogeneity of birefringence in layered turbid media using polarized light imaging |

Biomedical Optics Express, Vol. 3, Issue 12, pp. 3250-3263 (2012)

http://dx.doi.org/10.1364/BOE.3.003250

Acrobat PDF (4120 KB)

### Abstract

The structural anisotropy of biological tissues can be quantified using polarized light imaging in terms of birefringence; however, birefringence varies axially in anisotropic layered tissues. This may present ambiguity in result interpretation for techniques whose birefringence results are averaged over the sampling volume. To explore this issue, we extended the polarization sensitive Monte Carlo code to model bi-layered turbid media with varying uniaxial birefringence in the two layers. Our findings demonstrate that the asymmetry degree (ASD) between the off-diagonal Mueller matrix elements of heterogeneously birefringent samples is higher than the homogenously birefringent (uniaxial) samples with the same effective retardance (magnitude and orientation). We experimentally verified the validity of ASD as a birefringence heterogeneity measure by performing polarized light measurements of bi-layered elastic and scattering polyacrylamide phantoms.

© 2012 OSA

## 1. Introduction

1. M. F. G. Wood, N. Ghosh, M. A. Wallenburg, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Polarization birefringence measurements for characterizing the myocardium, including healthy, infarcted, and stem-cell-regenerated tissues,” J. Biomed. Opt. **15**(4), 047009 (2010). [CrossRef] [PubMed]

4. M. F. G. Wood, N. Ghosh, E. H. Moriyama, B. C. Wilson, and I. A. Vitkin, “Proof-of-principle demonstration of a Mueller matrix decomposition method for polarized light tissue characterization *in vivo*,” J. Biomed. Opt. **14**(1), 014029 (2009). [CrossRef] [PubMed]

5. T. Courtney, M. S. Sacks, J. Stankus, J. Guan, and W. R. Wagner, “Design and analysis of tissue engineering scaffolds that mimic soft tissue mechanical anisotropy,” Biomaterials **27**(19), 3631–3638 (2006). [PubMed]

3. S. Alali, K. Aitken, A. Shröder, D. Bagli, and I. A. Vitkin, “Optical assessment of tissue anisotropy in *ex vivo* distended rat bladders,” J. Biomed. Opt. **17**(8), 086010 (2012). [CrossRef]

4. M. F. G. Wood, N. Ghosh, E. H. Moriyama, B. C. Wilson, and I. A. Vitkin, “Proof-of-principle demonstration of a Mueller matrix decomposition method for polarized light tissue characterization *in vivo*,” J. Biomed. Opt. **14**(1), 014029 (2009). [CrossRef] [PubMed]

6. S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, and J. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express **14**(1), 190–202 (2006). [CrossRef] [PubMed]

10. N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Influence of the order of the constituent basis matrices on the Mueller matrix decomposition-derived polarization parameters in complex turbid media such as biological tissues,” Opt. Commun. **283**(6), 1200–1208 (2010). [CrossRef]

11. T. W. Gilbert, S. Wognum, E. M. Joyce, D. O. Freytes, M. S. Sacks, and S. F. Badylak, “Collagen fiber alignment and biaxial mechanical behavior of porcine urinary bladder derived extracellular matrix,” Biomaterials **29**(36), 4775–4782 (2008). [CrossRef] [PubMed]

12. K. D. Costa, E. J. Lee, and J. W. Holmes, “Creating alignment and anisotropy in engineered heart tissue: role of boundary conditions in a model three-dimensional culture system,” Tissue Eng. **9**(4), 567–577 (2003). [CrossRef] [PubMed]

13. J. F. de Boer, T. E. Milner, M. J. C. van Gemert, and J. S. Nelson, “Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography,” Opt. Lett. **22**(12), 934–936 (1997). [CrossRef] [PubMed]

15. E. Götzinger, M. Pircher, B. Baumann, C. Ahlers, W. Geitzenauer, U. Schmidt-Erfurth, and C. K. Hitzenberger, “Three-dimensional polarization sensitive OCT imaging and interactive display of the human retina,” Opt. Express **17**(5), 4151–4165 (2009). [CrossRef] [PubMed]

16. S. Brasselet, “Polarization-resolved nonlinear microscopy: application to structural molecular and biological imaging,” Adv. Opt. Photonics **3**(3), 205–271 (2011). [CrossRef]

17. S. Alali, M. Ahmad, A. Kim, N. Vurgun, M. F. G. Wood, and I. A. Vitkin, “Quantitative correlation between light depolarization and transport albedo of various porcine tissues,” J. Biomed. Opt. **17**(4), 045004 (2012). [CrossRef] [PubMed]

3. S. Alali, K. Aitken, A. Shröder, D. Bagli, and I. A. Vitkin, “Optical assessment of tissue anisotropy in *ex vivo* distended rat bladders,” J. Biomed. Opt. **17**(8), 086010 (2012). [CrossRef]

17. S. Alali, M. Ahmad, A. Kim, N. Vurgun, M. F. G. Wood, and I. A. Vitkin, “Quantitative correlation between light depolarization and transport albedo of various porcine tissues,” J. Biomed. Opt. **17**(4), 045004 (2012). [CrossRef] [PubMed]

18. X. Guo, M. F. G. Wood, and I. A. Vitkin, “A Monte Carlo study of penetration depth and sampling volume in of polarized light in turbid media,” Opt. Commun. **281**(3), 380–387 (2008). [CrossRef]

_{R}, diattenuator M

_{D}and depolarizer M

_{Δ}[19

19. S. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A **13**(5), 1106–1113 (1996). [CrossRef]

_{R}, the retardance magnitude δ can be derived as a measure of tissue anisotropy [1

1. M. F. G. Wood, N. Ghosh, M. A. Wallenburg, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Polarization birefringence measurements for characterizing the myocardium, including healthy, infarcted, and stem-cell-regenerated tissues,” J. Biomed. Opt. **15**(4), 047009 (2010). [CrossRef] [PubMed]

4. M. F. G. Wood, N. Ghosh, E. H. Moriyama, B. C. Wilson, and I. A. Vitkin, “Proof-of-principle demonstration of a Mueller matrix decomposition method for polarized light tissue characterization *in vivo*,” J. Biomed. Opt. **14**(1), 014029 (2009). [CrossRef] [PubMed]

19. S. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A **13**(5), 1106–1113 (1996). [CrossRef]

_{R}can be further decomposed to a linear part and a circular part [20,21]. From the linear part M

_{LR}, the retardance orientation (slow axis) can be obtained as [2

2. M. A. Wallenburg, M. Pop, M. F. G. Wood, N. Ghosh, G. A. Wright, and I. A. Vitkin, “Comparison of optical polarimetry and diffusion tensor MR imaging for assessing myocardial anisotropy,” J. Innovative Opt. Health Sci. **03**(02), 109–121 (2010). [CrossRef]

3. S. Alali, K. Aitken, A. Shröder, D. Bagli, and I. A. Vitkin, “Optical assessment of tissue anisotropy in *ex vivo* distended rat bladders,” J. Biomed. Opt. **17**(8), 086010 (2012). [CrossRef]

*θ*can be regarded as the anisotropy direction if the turbid media is homogenously birefringent (in this paper we assume positive birefringence and thus

*θ*is the extraordinary axis as well [20]). Another parameter derived from polar decomposition is the retardance ellipticity which may give an indication of birefringence (in)homogeneity [3

*ex vivo* distended rat bladders,” J. Biomed. Opt. **17**(8), 086010 (2012). [CrossRef]

27. X. Cheng and X. Wang, “Numerical study of the characterization of forward scattering Mueller matrix patterns of turbid media with triple forward scattering assumption,” Optik (Stuttg.) **121**(10), 872–875 (2010). [CrossRef]

11. T. W. Gilbert, S. Wognum, E. M. Joyce, D. O. Freytes, M. S. Sacks, and S. F. Badylak, “Collagen fiber alignment and biaxial mechanical behavior of porcine urinary bladder derived extracellular matrix,” Biomaterials **29**(36), 4775–4782 (2008). [CrossRef] [PubMed]

27. X. Cheng and X. Wang, “Numerical study of the characterization of forward scattering Mueller matrix patterns of turbid media with triple forward scattering assumption,” Optik (Stuttg.) **121**(10), 872–875 (2010). [CrossRef]

## 2. Mueller matrix of turbid homogenous (uniaxial) birefringent media from polarization sensitive Monte Carlo simulations

28. M. F. G. Wood, X. Guo, and I. A. Vitkin, “Polarized light propagation in multiply scattering media exhibiting both linear birefringence and optical activity: Monte Carlo model and experimental methodology,” J. Biomed. Opt. **12**(1), 014029 (2007). [CrossRef] [PubMed]

29. N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Polarimetry in turbid, birefringent, optically active media: a Monte Carlo study of Mueller matrix decomposition in the backscattering geometry,” J. Appl. Phys. **105**(10), 102023 (2009). [CrossRef]

_{o}, extraordinary refractive index n

_{e}= n

_{o}+ Δn (with Δn being the birefringence magnitude) and extraordinary axis

*θ*relative to the x axis as shown in Fig. 1a ). Let us denote the forward transmission Mueller matrix at the detection facet by M(x, y), as depicted in Fig. 1b).

_{k}(x,y) will be proportional to the product of Mueller matrices M

_{j}of consecutive scattering events in each trajectory k:

*r*,

*ξ*,

*ϕ*) denotes the polar coordinate system and

*N*is the number of scattering events in the trajectory

_{j}*k*. For notational simplicity, we will write

*M*(

_{j}*r*,

*ξ*,

*ϕ*) as

*M*. The symmetry properties of M

_{j}_{j}, the Mueller matrix between two scattering events, will influence the symmetry properties of M

_{k}the Mueller matrix of that particular path, and thus the final sample Mueller matrix M(x,y). M

_{j}itself can be represented as a product of Mueller matrices which account for different optical effects acting on photons. Upon each scattering, the photon’s reference frame is changed to the scattering plane (rotation of ζ

_{j}degrees), and the single scattering Mueller matrix is applied [28

28. M. F. G. Wood, X. Guo, and I. A. Vitkin, “Polarized light propagation in multiply scattering media exhibiting both linear birefringence and optical activity: Monte Carlo model and experimental methodology,” J. Biomed. Opt. **12**(1), 014029 (2007). [CrossRef] [PubMed]

32. 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]

_{j}degrees, for the parallel polarization of the photon to be parallel with the direction of maximum refractive index (which is the projection of orientation

*θ*in the plane perpendicular to the propagation direction), as described in detail in Wood et al. [28

28. M. F. G. Wood, X. Guo, and I. A. Vitkin, “Polarized light propagation in multiply scattering media exhibiting both linear birefringence and optical activity: Monte Carlo model and experimental methodology,” J. Biomed. Opt. **12**(1), 014029 (2007). [CrossRef] [PubMed]

_{j}in the global coordinate (lab’s reference) frame can be written as

_{j}= π.Δn(ϕ

_{j}).dz/λ is half the retardance over the short pathlength dz between the two scattering events and Δn(ϕ) is the difference between the refractive indices seen by the photon calculated as [28

**12**(1), 014029 (2007). [CrossRef] [PubMed]

_{j}is the angle between the photon propagation direction after the scattering event j and the extraordinary axis. The total retardance δ of the medium is the accumulation of the retardances 2g

_{j}over the whole pathlength along the trajectory k. R (β

_{j}) and R (ζ

_{j}) are the rotation matrices which rotate the photon reference frames and can be written as (with α representing β

_{j}or ζ

_{j}) [20]

_{s}(ψ) can be the single scattering Mueller matrix for a spherical scatterer, with ψ

_{j}being the angle of the scattering. It can be written as

*a*,

*b*,

*c*, and

*d*elements are calculated from Mie theory [23]. Thus knowing the form of the rotation, retardance and scattering matrices as per Eqs. (8)–(10), we can find the symmetric properties of the Mueller matrix M

_{j.}Following the approach presented by Rakovic et al. [24

24. M. J. Raković, G. W. Kattawar, M. B. Mehruűbeoğlu, B. D. Cameron, L. V. Wang, S. Rastegar, and G. L. Coté, “Light backscattering polarization patterns from turbid media: theory and experiment,” Appl. Opt. **38**(15), 3399–3408 (1999). [CrossRef] [PubMed]

^{2}= I being the identity matrix. With direct calculation, it can be shown that

^{t}is the transpose of M. From matrix product properties transpose(A × B) = transpose(B) × transpose(A), then plugging Eq. (12) into Eq. (6) results in

*θ*in the old coordinate to

*θ*in the new coordinate and switching the input/output plane as shown in Fig. 1.c relative to Fig. 1.a). Both these configurations result in the same forward Mueller matrix M(x,y). Hence, in a homogenously birefringent medium, the reverse order of the events in each trajectory is the same as the regular order and Eq. (15) can be rewritten as

_{k}along each trajectory k and dictates its general form. The general form of M

_{k}that satisfies Eq. (16) is

## 3. Mueller matrix of turbid heterogeneous (bi-layered) birefringent media from polarization sensitive Monte Carlo simulations

**12**(1), 014029 (2007). [CrossRef] [PubMed]

33. D. Côté and I. A. Vitkin, “Balanced detection for low-noise precision polarimetric measurements of optically active, multiply scattering tissue phantoms,” J. Biomed. Opt. **9**(1), 213–220 (2004). [CrossRef] [PubMed]

35. A. Surowiec, P. N. Shrivastava, M. Astrahan, and Z. Petrovich, “Utilization of a multilayer polyacrylamide phantom for evaluation of hyperthermia applicators,” Int. J. Hyperthermia **8**(6), 795–807 (1992). [CrossRef] [PubMed]

17. S. Alali, M. Ahmad, A. Kim, N. Vurgun, M. F. G. Wood, and I. A. Vitkin, “Quantitative correlation between light depolarization and transport albedo of various porcine tissues,” J. Biomed. Opt. **17**(4), 045004 (2012). [CrossRef] [PubMed]

^{−1}[17

**17**(4), 045004 (2012). [CrossRef] [PubMed]

**12**(1), 014029 (2007). [CrossRef] [PubMed]

_{eff}, and orientation

*θ*

_{eff}[1

1. M. F. G. Wood, N. Ghosh, M. A. Wallenburg, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Polarization birefringence measurements for characterizing the myocardium, including healthy, infarcted, and stem-cell-regenerated tissues,” J. Biomed. Opt. **15**(4), 047009 (2010). [CrossRef] [PubMed]

*in vivo*,” J. Biomed. Opt. **14**(1), 014029 (2009). [CrossRef] [PubMed]

7. N. Ghosh, M. F. Wood, and I. A. Vitkin, “Mueller matrix decomposition for extraction of individual polarization parameters from complex turbid media exhibiting multiple scattering, optical activity, and linear birefringence,” J. Biomed. Opt. **13**(4), 044036 (2008). [CrossRef] [PubMed]

10. N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Influence of the order of the constituent basis matrices on the Mueller matrix decomposition-derived polarization parameters in complex turbid media such as biological tissues,” Opt. Commun. **283**(6), 1200–1208 (2010). [CrossRef]

19. S. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A **13**(5), 1106–1113 (1996). [CrossRef]

_{eff}and

*θ*

_{eff}were simulated in the new PolMC code. Each of the layers in the homogenous samples has the same anisotropy direction

*θ*

_{eff}, and half of the effective retardance value δ

_{eff}/2. These simulations will thus highlight the differences in Mueller matrices of homogenous and heterogeneous samples as a function of birefringence orientation difference in the two layers. We are expecting the homogenous birefringent sample to show higher degree of symmetry between their off-diagonal Mueller matrix elements (as per Eq. (17)) compared to axially heterogeneous birefringent samples. To quantify the symmetries in equivalent homogenous (EH) and heterogeneous samples, we define the asymmetry degrees (ASD) based on Eq. (17), as the sum of normalized differences (or sums if there is a sign change) between the off-diagonal elements - excluding those of the first column and row because of their lower values as mentioned before—as follows:

_{ij}is defined as the off-diagonal Mueller matrix element M

_{ij}normalized by M

_{11,}x

_{i}and y

_{j}are the spatial position and N is the number of the pixels in each dimension of the element’s image. Normalization of m

_{ij}with respect to its maximum in the defining equations above will help quantify the difference in the spatial profile (patterns) in the image, rather than the differences in the magnitude of the off-diagonal elements. The ASD metric will give an indication of the sample’s axial heterogeneity in comparison to its EH counterpart.

*homogenous*samples with different δ

_{eff}and

*θ*

_{eff}will have different ASD values. For example, if the values of the elements in the Mueller matrix of a pure retarder with retardance δ

_{eff}and orientation

*θ*

_{eff}are large, then the symmetry in Eq. (16) will be more manifest, yielding lower ASD [21]. Otherwise, owing to relatively small magnitude of the off-diagonal elements of a pure scatterer, the spatial symmetry in the images will be less and the total ASDs will be higher. Thus, heterogeneity can be best gauged by comparing the unknown sample’s ASD to its equivalent homogenous ASD (ASD

_{EH}). Moreover, depending on the values of δ

_{eff}and

*θ*

_{eff}, different axial heterogeneities will yield different deviations from the corresponding ASD

_{EH}.

## 4. Simulation results

### 4.1. Modeling heterogeneous anisotropic samples and their equivalent homogenous (EH) counterparts

_{EH}For this initial study, we chose to examine the effects of direction change only; the possible parameter space to explore is simply too large (orientation and magnitude values and their changes, varying layer thicknesses, varying optical properties in the layers, etc), so we start with simple, and biologically relevant, case of depth-dependent anisotropy axis change. The modeled samples I, II and III have the same constant birefringence magnitude and orientation in the first layer, but a different orientation relative to the first one (equal to 30°, 60° and 90°) in the second layer. Table 1 shows the birefringence magnitude and orientation of the two layers in sample I, II and III. Also shown are the δ

_{eff}and

*θ*

_{eff}values derived from polar decomposition of the PolMC-generated Mueller matrix. δ

_{eff}and

*θ*

_{eff}listed in Table 1 are the mean value of the images (δ

_{eff}(x,y) and

*θ*

_{eff}(x,y)) obtained from polar decomposition.

_{eff}and

*θ*

_{eff}from these birefringently heterogeneous samples, the next task was to generate Mueller matrices for EH samples (homogenously birefringent samples) which exhibit the same δ

_{eff}and

*θ*

_{eff}. Obviously, the birefringence orientation in both layers of an EH sample should be equal to

*θ*

_{eff}. But choosing the appropriate birefringence Δn value is not trivial, since the samples are turbid and the overall pathlength is not known before simulations. So we ran several PolMC simulations, with birefringence orientation

*θ*

_{eff}and different birefringence magnitudes Δn. After some trial and error, the parameter values for the three EH samples (to match the heterogeneous samples I-III) were selected, as shown in Table 2 (compare its last two columns with those in Table 1). We now have the PolMC-generated Mueller matrices for the three heterogeneous samples and their EH counterparts, and can proceed with ASD analysis (Eqs. (15)–(17)).

### 4-2. Symmetry analysis of the Mueller matrix images

_{11}) of the sample I-III and their respective EHs respectively. As seen, the homogenous samples follow the symmetry patterns predicted in Eq. (17); however, as mentioned previously the symmetries are different among EHs. To quantify the symmetry of these Mueller matrix elements, we calculated ASD numbers from Eqs. (18)–(19) (Table 3 ).

_{eff}and

*θ*

_{eff}in the homogenous sample EH II causes a ΔASD

_{EH}of ~15%. Given this uncertainty in ASD

_{EH}, for a sample with effective values of δ

_{eff}and

*θ*

_{eff}to be classified as heterogeneous, its ASD should obey the condition: (ASD

_{Sample}- |ΔASD

_{EH}|) /ASD

_{EH}> 1. This condition will ensure that the ASD difference between the sample and its EH arises solely from heterogeneity. For example, the large difference between the ASDs in sample II and EH II, is due to heterogeneity and not from homogenous variations in δ

_{eff}and

*θ*

_{eff}. This condition holds for samples I-III verifying their heterogeneity.

_{sample}/ASD

_{EH}are different for each case and cannot be regarded as a measure of higher or lower heterogeneity. For example, ASD

_{sample}/ASD

_{EH}of sample II is about 8 times that of sample III, while the heterogeneity (in this case change of anisotropy axis) is larger in sample III.

_{eff}and

*θ*

_{eff}are prepared. Generating such a table requires high computational power to run the PolMC code many times.

## 5. Experimental validation with turbid birefringent phantoms

**12**(1), 014029 (2007). [CrossRef] [PubMed]

35. A. Surowiec, P. N. Shrivastava, M. Astrahan, and Z. Petrovich, “Utilization of a multilayer polyacrylamide phantom for evaluation of hyperthermia applicators,” Int. J. Hyperthermia **8**(6), 795–807 (1992). [CrossRef] [PubMed]

^{−5}per 1 mm of stretch [28

**12**(1), 014029 (2007). [CrossRef] [PubMed]

35. A. Surowiec, P. N. Shrivastava, M. Astrahan, and Z. Petrovich, “Utilization of a multilayer polyacrylamide phantom for evaluation of hyperthermia applicators,” Int. J. Hyperthermia **8**(6), 795–807 (1992). [CrossRef] [PubMed]

^{−1}, by adding polystyrene beads of 1.2 μm size and of refractive index of 1.59 using Mie theory [22,36

36. S. Prahl, “Mie Scattering Calculator,” http://omlc.ogi.edu/calc/mie_calc.html.

**17**(4), 045004 (2012). [CrossRef] [PubMed]

_{eff}) and rotating their orientation (till it is equal to

*θ*

_{eff}). The properties of each slab and the effective retardance and orientation of the two layers together, found from polar decomposition, are listed in Table 4. As seen, the heterogeneous sample and its EH have a difference of about 20% in the experimental values of δ

_{eff}and

*θ*

_{eff}. This is the closest we could achieve with this phantom system. Our simulation results indicate that 20% homogenous change in δ

_{eff}and

*θ*

_{eff}will result in ΔASD

_{EH}< 25%.

_{Sample}- ΔASD

_{EH}) /ASD

_{EH}> 1, and is consistent with MC. Hence, experiments and MC simulation agree in the trends and both give farther credence to the proposed Mueller matrix asymmetry formalism for detecting and quantifying axial birefringence heterogeneity.

## 6. Conclusion

## References and links

1. | M. F. G. Wood, N. Ghosh, M. A. Wallenburg, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Polarization birefringence measurements for characterizing the myocardium, including healthy, infarcted, and stem-cell-regenerated tissues,” J. Biomed. Opt. |

2. | M. A. Wallenburg, M. Pop, M. F. G. Wood, N. Ghosh, G. A. Wright, and I. A. Vitkin, “Comparison of optical polarimetry and diffusion tensor MR imaging for assessing myocardial anisotropy,” J. Innovative Opt. Health Sci. |

3. | S. Alali, K. Aitken, A. Shröder, D. Bagli, and I. A. Vitkin, “Optical assessment of tissue anisotropy in |

4. | M. F. G. Wood, N. Ghosh, E. H. Moriyama, B. C. Wilson, and I. A. Vitkin, “Proof-of-principle demonstration of a Mueller matrix decomposition method for polarized light tissue characterization |

5. | T. Courtney, M. S. Sacks, J. Stankus, J. Guan, and W. R. Wagner, “Design and analysis of tissue engineering scaffolds that mimic soft tissue mechanical anisotropy,” Biomaterials |

6. | S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, and J. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express |

7. | N. Ghosh, M. F. Wood, and I. A. Vitkin, “Mueller matrix decomposition for extraction of individual polarization parameters from complex turbid media exhibiting multiple scattering, optical activity, and linear birefringence,” J. Biomed. Opt. |

8. | N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J Biophotonics |

9. | X. Li and G. Yao, “Mueller matrix decomposition of diffuse reflectance imaging in skeletal muscle,” Appl. Opt. |

10. | N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Influence of the order of the constituent basis matrices on the Mueller matrix decomposition-derived polarization parameters in complex turbid media such as biological tissues,” Opt. Commun. |

11. | T. W. Gilbert, S. Wognum, E. M. Joyce, D. O. Freytes, M. S. Sacks, and S. F. Badylak, “Collagen fiber alignment and biaxial mechanical behavior of porcine urinary bladder derived extracellular matrix,” Biomaterials |

12. | K. D. Costa, E. J. Lee, and J. W. Holmes, “Creating alignment and anisotropy in engineered heart tissue: role of boundary conditions in a model three-dimensional culture system,” Tissue Eng. |

13. | J. F. de Boer, T. E. Milner, M. J. C. van Gemert, and J. S. Nelson, “Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography,” Opt. Lett. |

14. | S. Jiao and L. V. Wang, “Two-dimensional depth-resolved Mueller matrix of biological tissue measured with double-beam polarization-sensitive optical coherence tomography,” Opt. Lett. |

15. | E. Götzinger, M. Pircher, B. Baumann, C. Ahlers, W. Geitzenauer, U. Schmidt-Erfurth, and C. K. Hitzenberger, “Three-dimensional polarization sensitive OCT imaging and interactive display of the human retina,” Opt. Express |

16. | S. Brasselet, “Polarization-resolved nonlinear microscopy: application to structural molecular and biological imaging,” Adv. Opt. Photonics |

17. | S. Alali, M. Ahmad, A. Kim, N. Vurgun, M. F. G. Wood, and I. A. Vitkin, “Quantitative correlation between light depolarization and transport albedo of various porcine tissues,” J. Biomed. Opt. |

18. | X. Guo, M. F. G. Wood, and I. A. Vitkin, “A Monte Carlo study of penetration depth and sampling volume in of polarized light in turbid media,” Opt. Commun. |

19. | S. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A |

20. | D. Goldstein, |

21. | R. A. Chipman, |

22. | H. C. V. de Hulst, |

23. | C. F. Bohren and D. R. Huffman, |

24. | M. J. Raković, G. W. Kattawar, M. B. Mehruűbeoğlu, B. D. Cameron, L. V. Wang, S. Rastegar, and G. L. Coté, “Light backscattering polarization patterns from turbid media: theory and experiment,” Appl. Opt. |

25. | A. J. Brown and Y. Xie, “Symmetry relations revealed in Mueller matrix hemispherical maps,” J. Quant. Spectrosc. Radiat. Transf. |

26. | C. Fallet, T. Novikova, M. Foldyna, S. Manhas, B. H. Ibrahim, and A. De Martino, “Overlay measurements by Mueller polarimetry in back focal plane,” J. Micro/Nanolithogr. MEMS MOEMS |

27. | X. Cheng and X. Wang, “Numerical study of the characterization of forward scattering Mueller matrix patterns of turbid media with triple forward scattering assumption,” Optik (Stuttg.) |

28. | M. F. G. Wood, X. Guo, and I. A. Vitkin, “Polarized light propagation in multiply scattering media exhibiting both linear birefringence and optical activity: Monte Carlo model and experimental methodology,” J. Biomed. Opt. |

29. | N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Polarimetry in turbid, birefringent, optically active media: a Monte Carlo study of Mueller matrix decomposition in the backscattering geometry,” J. Appl. Phys. |

30. | D. Côté and I. A. Vitkin, “Robust concentration determination of optically active molecules in turbid media with validated three-dimensional polarization sensitive Monte Carlo calculations,” Opt. Express |

31. | X. Wang and L. V. Wang, “Propagation of polarized light in birefringent turbid media: a Monte Carlo study,” J. Biomed. Opt. |

32. | 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 |

33. | D. Côté and I. A. Vitkin, “Balanced detection for low-noise precision polarimetric measurements of optically active, multiply scattering tissue phantoms,” J. Biomed. Opt. |

34. | K. C. Hadley and I. A. Vitkin, “Optical rotation and linear and circular depolarization rates in diffusively scattered light from chiral, racemic, and achiral turbid media,” J. Biomed. Opt. |

35. | A. Surowiec, P. N. Shrivastava, M. Astrahan, and Z. Petrovich, “Utilization of a multilayer polyacrylamide phantom for evaluation of hyperthermia applicators,” Int. J. Hyperthermia |

36. | S. Prahl, “Mie Scattering Calculator,” http://omlc.ogi.edu/calc/mie_calc.html. |

**OCIS Codes**

(170.3660) Medical optics and biotechnology : Light propagation in tissues

(260.1440) Physical optics : Birefringence

(260.5430) Physical optics : Polarization

(110.0113) Imaging systems : Imaging through turbid media

**ToC Category:**

Optics of Tissue and Turbid Media

**History**

Original Manuscript: September 11, 2012

Revised Manuscript: November 9, 2012

Manuscript Accepted: November 11, 2012

Published: November 14, 2012

**Citation**

Sanaz Alali, Yuting Wang, and I. Alex Vitkin, "Detecting axial heterogeneity of birefringence in layered turbid media using polarized light imaging," Biomed. Opt. Express **3**, 3250-3263 (2012)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-3-12-3250

Sort: Year | Journal | Reset

### References

- M. F. G. Wood, N. Ghosh, M. A. Wallenburg, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Polarization birefringence measurements for characterizing the myocardium, including healthy, infarcted, and stem-cell-regenerated tissues,” J. Biomed. Opt.15(4), 047009 (2010). [CrossRef] [PubMed]
- M. A. Wallenburg, M. Pop, M. F. G. Wood, N. Ghosh, G. A. Wright, and I. A. Vitkin, “Comparison of optical polarimetry and diffusion tensor MR imaging for assessing myocardial anisotropy,” J. Innovative Opt. Health Sci.03(02), 109–121 (2010). [CrossRef]
- S. Alali, K. Aitken, A. Shröder, D. Bagli, and I. A. Vitkin, “Optical assessment of tissue anisotropy in ex vivo distended rat bladders,” J. Biomed. Opt.17(8), 086010 (2012). [CrossRef]
- M. F. G. Wood, N. Ghosh, E. H. Moriyama, B. C. Wilson, and I. A. Vitkin, “Proof-of-principle demonstration of a Mueller matrix decomposition method for polarized light tissue characterization in vivo,” J. Biomed. Opt.14(1), 014029 (2009). [CrossRef] [PubMed]
- T. Courtney, M. S. Sacks, J. Stankus, J. Guan, and W. R. Wagner, “Design and analysis of tissue engineering scaffolds that mimic soft tissue mechanical anisotropy,” Biomaterials27(19), 3631–3638 (2006). [PubMed]
- S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, and J. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express14(1), 190–202 (2006). [CrossRef] [PubMed]
- N. Ghosh, M. F. Wood, and I. A. Vitkin, “Mueller matrix decomposition for extraction of individual polarization parameters from complex turbid media exhibiting multiple scattering, optical activity, and linear birefringence,” J. Biomed. Opt.13(4), 044036 (2008). [CrossRef] [PubMed]
- N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J Biophotonics2(3), 145–156 (2009). [CrossRef] [PubMed]
- X. Li and G. Yao, “Mueller matrix decomposition of diffuse reflectance imaging in skeletal muscle,” Appl. Opt.48(14), 2625–2631 (2009). [CrossRef] [PubMed]
- N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Influence of the order of the constituent basis matrices on the Mueller matrix decomposition-derived polarization parameters in complex turbid media such as biological tissues,” Opt. Commun.283(6), 1200–1208 (2010). [CrossRef]
- T. W. Gilbert, S. Wognum, E. M. Joyce, D. O. Freytes, M. S. Sacks, and S. F. Badylak, “Collagen fiber alignment and biaxial mechanical behavior of porcine urinary bladder derived extracellular matrix,” Biomaterials29(36), 4775–4782 (2008). [CrossRef] [PubMed]
- K. D. Costa, E. J. Lee, and J. W. Holmes, “Creating alignment and anisotropy in engineered heart tissue: role of boundary conditions in a model three-dimensional culture system,” Tissue Eng.9(4), 567–577 (2003). [CrossRef] [PubMed]
- J. F. de Boer, T. E. Milner, M. J. C. van Gemert, and J. S. Nelson, “Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography,” Opt. Lett.22(12), 934–936 (1997). [CrossRef] [PubMed]
- S. Jiao and L. V. Wang, “Two-dimensional depth-resolved Mueller matrix of biological tissue measured with double-beam polarization-sensitive optical coherence tomography,” Opt. Lett.27(2), 101–103 (2002). [CrossRef] [PubMed]
- E. Götzinger, M. Pircher, B. Baumann, C. Ahlers, W. Geitzenauer, U. Schmidt-Erfurth, and C. K. Hitzenberger, “Three-dimensional polarization sensitive OCT imaging and interactive display of the human retina,” Opt. Express17(5), 4151–4165 (2009). [CrossRef] [PubMed]
- S. Brasselet, “Polarization-resolved nonlinear microscopy: application to structural molecular and biological imaging,” Adv. Opt. Photonics3(3), 205–271 (2011). [CrossRef]
- S. Alali, M. Ahmad, A. Kim, N. Vurgun, M. F. G. Wood, and I. A. Vitkin, “Quantitative correlation between light depolarization and transport albedo of various porcine tissues,” J. Biomed. Opt.17(4), 045004 (2012). [CrossRef] [PubMed]
- X. Guo, M. F. G. Wood, and I. A. Vitkin, “A Monte Carlo study of penetration depth and sampling volume in of polarized light in turbid media,” Opt. Commun.281(3), 380–387 (2008). [CrossRef]
- S. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A13(5), 1106–1113 (1996). [CrossRef]
- D. Goldstein, Polarized Light, 2nd ed. (Marcel Dekker, New York, 2003).
- R. A. Chipman, Handbook of Optics, 2nd ed., M. Bass, ed. (McGraw-Hill, New York, 1994)
- H. C. V. de Hulst, Light Scattering by Small Particles (Dover, New York (1981)
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1998).
- M. J. Raković, G. W. Kattawar, M. B. Mehruűbeoğlu, B. D. Cameron, L. V. Wang, S. Rastegar, and G. L. Coté, “Light backscattering polarization patterns from turbid media: theory and experiment,” Appl. Opt.38(15), 3399–3408 (1999). [CrossRef] [PubMed]
- A. J. Brown and Y. Xie, “Symmetry relations revealed in Mueller matrix hemispherical maps,” J. Quant. Spectrosc. Radiat. Transf.113(8), 644–651 (2012). [CrossRef]
- C. Fallet, T. Novikova, M. Foldyna, S. Manhas, B. H. Ibrahim, and A. De Martino, “Overlay measurements by Mueller polarimetry in back focal plane,” J. Micro/Nanolithogr. MEMS MOEMS10(3), 033017 (2011).
- X. Cheng and X. Wang, “Numerical study of the characterization of forward scattering Mueller matrix patterns of turbid media with triple forward scattering assumption,” Optik (Stuttg.)121(10), 872–875 (2010). [CrossRef]
- M. F. G. Wood, X. Guo, and I. A. Vitkin, “Polarized light propagation in multiply scattering media exhibiting both linear birefringence and optical activity: Monte Carlo model and experimental methodology,” J. Biomed. Opt.12(1), 014029 (2007). [CrossRef] [PubMed]
- N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Polarimetry in turbid, birefringent, optically active media: a Monte Carlo study of Mueller matrix decomposition in the backscattering geometry,” J. Appl. Phys.105(10), 102023 (2009). [CrossRef]
- D. Côté and I. A. Vitkin, “Robust concentration determination of optically active molecules in turbid media with validated three-dimensional polarization sensitive Monte Carlo calculations,” Opt. Express13(1), 148–163 (2005). [CrossRef] [PubMed]
- X. Wang and L. V. Wang, “Propagation of polarized light in birefringent turbid media: a Monte Carlo study,” J. Biomed. Opt.7(3), 279–290 (2002). [CrossRef] [PubMed]
- 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. Express13(12), 4420–4438 (2005). [CrossRef] [PubMed]
- D. Côté and I. A. Vitkin, “Balanced detection for low-noise precision polarimetric measurements of optically active, multiply scattering tissue phantoms,” J. Biomed. Opt.9(1), 213–220 (2004). [CrossRef] [PubMed]
- K. C. Hadley and I. A. Vitkin, “Optical rotation and linear and circular depolarization rates in diffusively scattered light from chiral, racemic, and achiral turbid media,” J. Biomed. Opt.7(3), 291–299 (2002). [CrossRef] [PubMed]
- A. Surowiec, P. N. Shrivastava, M. Astrahan, and Z. Petrovich, “Utilization of a multilayer polyacrylamide phantom for evaluation of hyperthermia applicators,” Int. J. Hyperthermia8(6), 795–807 (1992). [CrossRef] [PubMed]
- S. Prahl, “Mie Scattering Calculator,” http://omlc.ogi.edu/calc/mie_calc.html .

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.