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  • Vol. 4, Iss. 4 — Apr. 1, 2009
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Mueller decomposition images for cervical tissue: Potential for discriminating normal and dysplastic states

Prashant Shukla and Asima Pradhan  »View Author Affiliations


Optics Express, Vol. 17, Issue 3, pp. 1600-1609 (2009)
http://dx.doi.org/10.1364/OE.17.001600


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Abstract

We report the potential of Mueller decomposition images to discriminate the normal against the dysplastic (precancerous) states in cervical tissue. It is observed that in the epithelium region, depolarization power is sensitive to morphological changes during progression from normal to dysplastic state while retardance and diattenuation do not show any significant change. These morphological changes have been correlated with the microscopic images of the tissues. In contrast, it is the retardance which reveals the morphological changes around the stromal region. Additionally, we have evaluated the arithmetic mean of depolarization power and retardance from their respective decomposed images and have shown that this parameter has a potential to discriminate normal tissues against dysplastic ones.

© 2009 Optical Society of America

1. Introduction

Precancerous stages (dysplasia) in cervical tissue are generally estimated by noting changes in the epithelium region. The grade of dysplasia is decided by the thickness up to which structural changes occur in the epithelium region [1

1. R. S. Cotran, V. Kumar, and S. L. Robbins, Robbins Pathologic Basis of Disease (W.B. Saunders Company, 1989), Chap. 24.

]. However, it has been reported in the literature that there is a distortion in the collagen fibers during the development of dysplastic stage in stroma [2

2. D. Arifler, I. Pavlova, A. Gillenwater, and R. R. Kortum, “Light Scattering from Collagen Fiber Networks: Micro-Optical Properties of Normal and Neoplastic Stroma,” J. Biophys. 92, 3260 – 3274 (2007). [CrossRef]

]. Such changes that occur in the stroma are generally not used for histopathological analysis. Light scattering is known to reveal subtle structural changes in cells [3

3. V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, S. Shapshay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. van Dam, and M. S. Feld, “Detection of preinvasive cancer cells,” Nature 406, 35 – 36 (2000). [CrossRef] [PubMed]

]. Polarized light scattering in the form of Mueller matrix describes completely the optical properties of any scattering medium like tissue, polystyrene micro-spheres etc.. In the recent past, substantial theoretical [4–10

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

] and experimental [11–17

11. A. H. Hielscher, A. A. Eick, J. R. Mourant, D. Shen, J. P. Freyer, and I. J. Bigio,“Diffuse backscattering Mueller matrices of highly scattering media,” Opt. Express 1, 441–453 (1997). [CrossRef] [PubMed]

] studies based on the Mueller matrix have been done. The 16 elements of Mueller matrix generated by recording images of any scattering medium for various combinations of polarizer and analyzer can provide information such as the size and refractive index of the scattering medium [16

16. J. Dillet, C. Baravian, F. Caton, and A. Parker, “Size determination by use of two-dimensional Mueller matrices backscattered by optically thick random media,” Appl. Opt. 45, 4669 – 4677 (2006). [CrossRef] [PubMed]

]. In addition to these Mueller matrix elements, diattenuation, retardance and depolarization power can also reveal more information about the structure and morphology of highly scattering media such as biological tissues. These parameters (diattenuation, retardance and depolarization power) for any scattering medium can be extracted by polar decomposition of Mueller matrix given by Lu et. al. [4

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

]. Liu et. al. have used polar decomposition algorithm for rat skin and melanoma phantoms [14

14. G. L. Liu, Y. Li, and B. D. Cameron, “Polarization Based Optical imaging and processing techniques with application to the Cancer Diagnostics,” Proc. SPIE 4617, 208 – 219(2002). [CrossRef]

]. They have found that considerable information on the morphological structures of any scattering medium can be extracted from Mueller decomposition images in comparison to standard imaging techniques. Using 3×3 Mueller decomposition technique, Swami et. al. have reported high retardance value shown by collagen fibers extracted from eggshell membrane [17

17. M. K. Swami, S. Manhas, P. Buddhiwant, N. Ghosh, A. Uppal, and P.K. Gupta, “Polar decomposition of 3×3 Mueller matrix: a tool for quantitative tissue polarimetry,” Opt. Express 14, 9324 – 9337 (2006). [CrossRef] [PubMed]

]. Recently Anastasiadou et. al. have used DOP polarimetric technique for detection of cervical cancer and compared their results with classical colposcopy. They concluded that DOP technique shows changes from normal to cancer states and can be used for diagnosis purposes [18

18. M. Anastasiadou, A. De Martino, D. Clement, F. Liège, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Stat. Sol. (C) 5, 1423–1426 426 (2008). [CrossRef]

]. In a complex turbid medium like biological tissue, many polarization effects occur simultaneously (the most common polarimetry effects are depolarization, linear birefringence and optical activity). Thus the Stokes parameter-based measure of degree of polarization represents the value of degree of polarization resulting from several ‘lumped, polarization effects. In contrast, the Mueller matrix decomposition approach enables one to extract, quantify and interpret the individual intrinsic polarimetry characteristics of tissue. Each of these, individual polarization parameters, holds promise as a useful biological metric, as is also apparent from the results of our studies. In the present work the polar decomposition algorithm on Mueller matrix [4

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

] has been used to study its potential to discriminate the normal against the precancerous stage (dysplastic state) of human cervical tissues. Significant differences are observed in the decomposition images of normal and dysplastic tissues of the same patient. Changes are observed in the stroma of the cervical tissue through the retardance parameter while epithelial changes are noticed through depolarization power.

2. Theory

2.1 Polar decomposition of the Mueller Matrices

Let M be a 4×4 Mueller matrix as given below

M=(m00m01m02m03m10m11m12m13m20m21m22m23m30m31m32m33)
(1)

This Mueller matrix M can be decomposed into three elementary matrices representing a depolarizer (M), a retarder (MR) and a diattenuator (MD) [4

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

]. The decomposition of Mueller matrix depends upon the order in which the diattenuator, depolarizer and retarder matrices are multiplied. Based on the order of these matrices, six possible decompositions can be performed. Among these, the process in which the diattenuator matrix comes ahead of the retardance and the depolarization matrix [M=MMRMD] always leads to a physically realizable Mueller matrix [5

5. J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices,” Opt. Lett. 29, 2234 – 2236 (2004). [CrossRef] [PubMed]

]. The decomposition process discussed in this paper is based on this approach. Therefore, any Mueller matrix M can be mathematically decomposed as

M=MΔMRMD
(2)

where the depolarizing matrix M accounts for the depolarizing effects of the medium, the retarder matrix MR describes the effects of linear birefringence and optical activity, and the diattenuator matrix MD includes the effects of linear and circular dichroism.

We can calculate directly diattenuation D from the Mueller matrix M as

D=(m012+m022+m033)1/2m00
(3)

From the Mueller matrix M, first we can construct a diattenuator Mueller matrix by taking diattenuation vector D⃗ as

D=1m00(m01m02m03)
(4)

Thus the first row of M gives the diattenuation vector. Then from this diattenuation vector the diattenuator Mueller matrix can be constructed as

MD=(1DTDmD)
(5)

Here mD is defined as

mD=1D2I+(11D2)D̂D̂T
(6)

where I is 3×3 identity matrix and D̂(=DD)denotes the unit vector along D⃗.

Further a Mueller matrix M′ is defined based on M, as

M=MMD1
(7)

This M′ contains only retardance and depolarization and no diattenuation. M′ can be further decomposed as a retarder followed by a depolarizer

MΔMR=(10PΔmΔ)(100mR)
=(10PΔmΔmR)(10PΔm|)
=M
(8)

Here m′ is a 3×3 sub-matrix of M′. Equations (7) and (8) lead to

PΔ=PmD1D2
(9)
m=mΔmR
(10)

where polarizance vector P⃗ can be expressed in terms of Mueller matrix elements as follows

P=1m00(m10m20m30)
(11)

Let λ 1, λ 2 and λ 3 be the eigen values of m′(m′)T. m has eigen values √λ 1, √λ 2 and √λ 3. m can be obtained by

mΔ=±[m(m)T+(λ1λ2+λ2λ3+λ3λ1)1/2I]1
×[{(λ1)1/2+(λ2)1/2+(λ3)1/2}m(m)T+(λ1λ2λ3)1/2I]
(12)

If determinant of m′ is negative then minus sign is applied. So M can be determined by Eqs. (9) and (12). Once M is determined then we can evaluate depolarization power ∆ as

Δ=1tr(MΔ)13
(13)

Now MR can be obtained by

MR=MΔ1M
(14)

From the retardance Mueller matrix, the retardance can be obtained by

R=cos1[tr(MR)21]
(15)

Finally we obtain three parameters as:

D=1m00[m012+m022+m032]1/2
(16)
Δ=1tr(MΔ)13
(17)
R=cos1[tr(MR)21]
(18)

A program was written in MATLAB (Math work) to decompose the Mueller matrices in terms of the above parameters for each pixel of the illuminated region of CCD. Microscopic images of these slides were also recorded to see the morphological changes histopathology. A black mark was put on the tissue slide before recording the microscopic images so as to ensure the illumination of the same spot for optical studies.

3. Experimental methods and materials

The samples used in this study were pathologically characterized, stained vertical sections of human cervical tissues (containing both epithelial and stromal regions) for normal and dysplastic states on glass slides. The lateral dimensions of tissue sample were 4 mm × 6 mm, having thickness of 5 μm. These samples were illuminated with He-Ne laser (Melles Griot, 20mW, λ = 632.8 nm) having spot size of 1.1 mm. Mueller images of these tissue slides were recorded in the transmission mode on a CCD (Apogee 1E, USA) having resolution of 768 × 512 pixels. The incident beam was polarized using a linear polarizer (Glan Thompson having extinction coefficient 100000:1) placed between the sample and the source. To generate circularly polarized light, a quarter wave plate (05RP24-02, Newport) was introduced between the polarizer and the sample. The transmitted light from the sample was collected by a convex lens and after passing through subsequent polarization optics, was made to fall on the CCD. Altogether 49 images were recorded for each sample using various combinations of polarizer, analyzer and quarter wave plate to generate 16 elements of the Mueller matrix. Experiment has been done on 10 pairs (normal and dysplasia) of cervical tissues. The experiment was repeated three times for each sample to ensure reproducibility of experimental results.

4. Results and discussion

The experimental set-up was first calibrated by recording Mueller matrices for known optical elements such as linear polarizer and air. The Mueller matrices for horizontal polarizer and air are given below:

  1. For horizontal polarizer (1.00000.97910.00190.00950.96820.97620.01700.03080.04750.03550.00790.01810.03060.01970.01440.0162)
  2. For air (1.00000.01940.03510.02310.02010.97320.01910.02360.00400.02310.96320.03230.00870.00650.02650.9701)

Typical value of error in each element was found to lie between 1– 4%. After obtaining satisfactory results from these standard measurements, the set-up was used to record Mueller matrices for the samples investigated in this study. Figures 1 (a) and 1(b) show a typical Mueller image (M00) and the corresponding microscopic image for the dysplastic state of the cervical tissue respectively. The black point mark is visible in both images.

Fig. 1. (a) Mueller image M00 and (b) the microscopic image of a dysplastic cervical tissue for the entire illuminated region.
Fig. 2. Images of (a) depolarization power (b) diattenuation (c) retardance of a dysplastic cervical tissue section for the entire illuminated region.

Figure 2 represents the Mueller decomposition image in terms of depolarization power, diattenuation and retardance for dysplastic cervical tissue section. It may be observed that the black mark is reflected well in the depolarization power and the diattenuation images as compared to the retardance image. This is expected since depolarization power and diattenuation both reflect scattering and absorbance changes whereas retardance reflects the birefringence and hence the black mark should not show up in the retardance image.

To correlate the alteration in morphology precisely at different regions of the normal and dysplastic tissues, Mueller decomposition images were investigated separately for the epithelium and stromal sides of tissue. Figure 3 shows the 2-D Mueller decomposition images in terms of depolarization power for the epithelium region. The top of the image corresponds to the outer region of the epithelium while the bottom corresponds to its lowest side (i.e. basal layer which can be seen in the microscopic image also).

Fig. 3. Typical depolarization power images of (a) normal and (b) dysplastic state in the epithelium region of the cervical tissue section.

In the depolarization power images significant changes were observed in the epithelium region of the dysplasia tissue as compared to the normal one. The value of depolarization power in this region is large over the entire range of pixels along the vertical direction. The large value of depolarization power in the dysplastic tissue indicates an increase in the value of scattering coefficient. The increased value of scattering coefficient is equivalent to reduced scattering mean free path which implies a higher density of scatterers. This is confirmed in the microscopic image of the dysplastic tissue in Fig. (4), which shows the growth of cell density in the epithelium region starting from the basal layer. A layered optical contrast is noticed in the dysplastic state in contrast to an almost uniform density in the normal state. This variation in the color contrast reflects the variation in the growth of cell density from basal to the outer region of epithelium, during progression of dysplasia. It is observed in all the samples consistently and may be a useful parameter for grading the various dysplastic states. The number of samples studied here are not enough to confirm this. However, this gradation in density is a definite parameter to discriminate dysplasia in general, from normal state with 100% sensitivity in 10 samples.

Fig. 4. Microscopic images of epithelial layer of (a) normal and (b) dysplastic state.

On the other hand, diattenuation and retardance images (not shown) were found to be inconsistent to variations in normal and dysplastic tissues for the epithelium region. It thus appears that morphological and structural changes of the epithelium region are better reflected in terms of depolarization as compared to diattenuation and retardance in the Mueller decomposition images.

Fig. 5. Typical retardance images of (a) normal and (b) dysplastic state in the stromal region of the cervical tissue section.

In contrast, it is the retardance image of the stromal region in Fig. (5), which shows consistent changes from the normal to the dysplastic state. The value of retardance is found to be lower for the dysplastic state as compared to the normal counterpart. This reduction in the value of retardance implies a decrease in the birefringence in the stromal region. Birefringence arises due to fibrous structure of collagen present in the stromal region of cervical tissue. Decrease in birefringence indicates deformation of the regular molecular binding structure and even damage of the helix type molecules of collagen as reported in the literature [2

2. D. Arifler, I. Pavlova, A. Gillenwater, and R. R. Kortum, “Light Scattering from Collagen Fiber Networks: Micro-Optical Properties of Normal and Neoplastic Stroma,” J. Biophys. 92, 3260 – 3274 (2007). [CrossRef]

]. The diattenuation and depolarization images, however, do not show any consistent changes in the stromal region. Therefore, morphological and structural changes of the stromal region are less pronounced in terms of depolarization and diattenuation as compared to retardance in the Mueller decomposition images.

The arithmetic mean of retardance, diattenuation and depolarization power have also been calculated from their respective decomposed images to see whether the changes in these mean values can discriminate among normal and dysplastic tissues. This arithmetic mean has been calculated by summing over the values of retardance, diattenuation and depolarization power of each and every pixel of their corresponding decomposed images and then dividing the sum by the total number of pixels. The fact that the retardance in the stromal region decreases from normal to dysplastic tissues while the reverse happens in case of depolarization power in the epithelium region, has been confirmed by this additional parameter.

Figures 6(a) and 6(b) show the mean values of depolarization power of epithelium region and that of retardance of stromal region respectively. It is clear from the figure that the mean values of depolarization power increase from normal to dysplastic tissues. It is found that the mean value of depolarization power for normal tissues is generally less than 0.3 while for dysplastic tissues, it is greater than 0.3 from the ten samples studied. Therefore, one can observe that there is a clear demarcation for the mean values of depolarization power between normal and dysplastic tissues which can serve as a benchmark for differentiating normal against dysplastic tissues. However, for the 10 samples studied, such a distinct demarcation is not apparent in the mean values of retardance of the stromal region as seen in Fig. 6(b). Nevertheless, we observe that its value decreases from normal to dysplastic tissues significantly for each patient, which is consistent with what we observed with the values of retardance for stromal region in its corresponding decomposed images shown in Fig. 6(b). We have found that the mean values of diattenuation did not show any consistent changes from normal to dysplastic tissue, which is again consistent with what we observed in its corresponding decomposed images.

Fig. 6. (a) Mean value of depolarization power for the epithelium region and (b) mean value of retardance for the stromal region.

5. Conclusion

In conclusion, our study shows that Mueller decomposition images have a potential to reveal the structural and morphological changes in the dysplasia state of human cervical tissue efficiently. We observe that in the epithelium region, depolarization power is sensitive to morphological changes during progression from normal to dysplastic state while diattenuation and retardance do not show any significant change. However, it is the retardance which reveals the morphological changes around the stromal region. Changes in epithelium region are conventionally used for diagnosis of dysplasia while changes in the stromal section are generally not mentioned in the normal histopathology. However, with our observation of stromal changes through retardance images, one can strengthen the diagnostic technique. Additionally, we have defined a parameter in terms of mean values of depolarization power, diattenuation and retardance calculated from their respective decomposed images. We have shown that the mean values of depolarization power in epithelium region and retardance in stromal region show significant changes from normal to dysplastic tissue while diattenuation does not show a consistent change. A consistent variation in the density, noticed in the epithelial layer of dysplastic tissues may also be a useful parameter to determine various stages of dysplasia. Further study on a larger number of samples is currently being pursued to confirm this. We are of the opinion that this parameter which appears to show a cut-off in the depolarization power values can be taken as a benchmark for discriminating normal tissues against dysplastic one after performing a careful statistical analysis on a large number of samples.

Acknowledgments

The authors would like to acknowledge Dr. Asha Agrawal for providing the tissue slides and for fruitful discussions.

References and links:

1.

R. S. Cotran, V. Kumar, and S. L. Robbins, Robbins Pathologic Basis of Disease (W.B. Saunders Company, 1989), Chap. 24.

2.

D. Arifler, I. Pavlova, A. Gillenwater, and R. R. Kortum, “Light Scattering from Collagen Fiber Networks: Micro-Optical Properties of Normal and Neoplastic Stroma,” J. Biophys. 92, 3260 – 3274 (2007). [CrossRef]

3.

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, S. Shapshay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. van Dam, and M. S. Feld, “Detection of preinvasive cancer cells,” Nature 406, 35 – 36 (2000). [CrossRef] [PubMed]

4.

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

5.

J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices,” Opt. Lett. 29, 2234 – 2236 (2004). [CrossRef] [PubMed]

6.

B. D. Cameron, M. J. Rakovic, M. Mehrubeoglu, G. W. Kattawar, S. Rastegar, L.V. Wang, and G. L. Cote, “Measurement and calculation of the two-dimensional backscattering matrix of a turbid medium,” Opt. Lett. 23, 485 – 487 (1998). [CrossRef]

7.

B. Kaplan, G. ledanois, and B. Drevillon, “Mueller matrix of dense polystyrene latex sphere suspension: measurements and monte Carlo simulation,” Appl. Opt. 40, 2769 – 2777 (2001). [CrossRef]

8.

S. Firdous and M. Ikram, “Characterization of turbid medium through diffusely backscattering polarized light with matrix calculus-II,” in Proceedings of IEEE Conference on International Networking and Communication (INCC2004) pp. 115 – 123.

9.

Y. Deng, Q. Lu, Q. Luo, and S. Zeng, “Third order scattering model for the diffuse backscattering intensity patterns of polarized light from a turbid medium,” Appl. Phy. Lett. 90, 153902 (1) –153902 (3) (2007). [CrossRef]

10.

Y. Deng, S. Zeng, Q. Lu, D. Zhu, and Q. Luo, “Characterization of back scattering muller matrix patterns of highly scattering media with triple scattering assumption,” Opt. Express 15, 9672 – 9680 (2007). [CrossRef] [PubMed]

11.

A. H. Hielscher, A. A. Eick, J. R. Mourant, D. Shen, J. P. Freyer, and I. J. Bigio,“Diffuse backscattering Mueller matrices of highly scattering media,” Opt. Express 1, 441–453 (1997). [CrossRef] [PubMed]

12.

A. H. Hielscher, J. R. Mourant, and I. J. Bigio, “Influence of particle size and concentration on the diffuse backscattering of polarized light from tissue phantoms and biological cell suspensions,” Appl. Opt. 36, 125–135 (1997). [CrossRef] [PubMed]

13.

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

14.

G. L. Liu, Y. Li, and B. D. Cameron, “Polarization Based Optical imaging and processing techniques with application to the Cancer Diagnostics,” Proc. SPIE 4617, 208 – 219(2002). [CrossRef]

15.

I. Berezhnyy and A. Dogariu, “Time resolved Mueller matrix imaging polarimetry,” Opt. Express 12, 4635 – 4649 (2004). [CrossRef] [PubMed]

16.

J. Dillet, C. Baravian, F. Caton, and A. Parker, “Size determination by use of two-dimensional Mueller matrices backscattered by optically thick random media,” Appl. Opt. 45, 4669 – 4677 (2006). [CrossRef] [PubMed]

17.

M. K. Swami, S. Manhas, P. Buddhiwant, N. Ghosh, A. Uppal, and P.K. Gupta, “Polar decomposition of 3×3 Mueller matrix: a tool for quantitative tissue polarimetry,” Opt. Express 14, 9324 – 9337 (2006). [CrossRef] [PubMed]

18.

M. Anastasiadou, A. De Martino, D. Clement, F. Liège, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Stat. Sol. (C) 5, 1423–1426 426 (2008). [CrossRef]

OCIS Codes
(170.6930) Medical optics and biotechnology : Tissue
(290.0290) Scattering : Scattering
(290.7050) Scattering : Turbid media
(110.0113) Imaging systems : Imaging through turbid media
(290.5855) Scattering : Scattering, polarization

ToC Category:
Medical Optics and Biotechnology

History
Original Manuscript: November 21, 2008
Revised Manuscript: January 16, 2009
Manuscript Accepted: January 22, 2009
Published: January 27, 2009

Virtual Issues
Vol. 4, Iss. 4 Virtual Journal for Biomedical Optics

Citation
Prashant Shukla and Asima Pradhan, "Mueller decomposition images for cervical tissue: Potential for discriminating normal and dysplastic states," Opt. Express 17, 1600-1609 (2009)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-17-3-1600


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References

  1. R. S. Cotran, V. Kumar, and S. L. Robbins, Robbins Pathologic Basis of Disease (W. B. Saunders Company, 1989), Chap. 24.
  2. D. Arifler, I. Pavlova, A. Gillenwater, and R. R. Kortum, "Light Scattering from Collagen Fiber Networks: Micro-Optical Properties of Normal and Neoplastic Stroma," J. Biophys. 92, 3260 - 3274 (2007). [CrossRef]
  3. V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, S. Shapshay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. van Dam, and M. S. Feld, "Detection of preinvasive cancer cells," Nature 406, 35 - 36 (2000). [CrossRef] [PubMed]
  4. S.Y. Lu and R. A. Chipman, "Interpretation of Mueller matrices based on polar decomposition," J. Opt. Soc. Am. A 13, 1106 - 1113 (1996). [CrossRef]
  5. J. Morio and F. Goudail, "Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices," Opt. Lett. 29, 2234 - 2236 (2004). [CrossRef] [PubMed]
  6. B. D. Cameron, M. J. Rakovic, M. Mehrubeoglu, G. W. Kattawar, S. Rastegar, L. V. Wang, and G. L. Cote, "Measurement and calculation of the two-dimensional backscattering matrix of a turbid medium," Opt. Lett. 23, 485 - 487 (1998). [CrossRef]
  7. B. Kaplan, G. ledanois, and B. Drevillon, "Mueller matrix of dense polystyrene latex sphere suspension: measurements and monte Carlo simulation," Appl. Opt. 40, 2769 - 2777 (2001). [CrossRef]
  8. S. Firdous and M. Ikram, "Characterization of turbid medium through diffusely backscattering polarized light with matrix calculus-II," in Proceedings of IEEE Conference on International Networking and Communication (INCC 2004) pp. 115 - 123.
  9. Y. Deng, Q. Lu, Q. Luo, and S. Zeng, "Third order scattering model for the diffuse backscattering intensity patterns of polarized light from a turbid medium," Appl. Phys. Lett. 90, 153902 (1) -153902 (3) (2007). [CrossRef]
  10. Y. Deng, S. Zeng, Q. Lu, D. Zhu, and Q. Luo, "Characterization of back scattering muller matrix patterns of highly scattering media with triple scattering assumption," Opt. Express 15, 9672 - 9680 (2007). [CrossRef] [PubMed]
  11. A. H. Hielscher, A. A. Eick, J. R. Mourant, D. Shen, J. P. Freyer, and I. J. Bigio, "Diffuse backscattering Mueller matrices of highly scattering media," Opt. Express 1, 441- 453 (1997). [CrossRef] [PubMed]
  12. A. H. Hielscher, J. R. Mourant, and I. J. Bigio, "Influence of particle size and concentration on the diffuse backscattering of polarized light from tissue phantoms and biological cell suspensions," Appl. Opt. 36, 125 -135 (1997). [CrossRef] [PubMed]
  13. M. J. Rakovic, G. W. Kattawar, M. Mehrubeoglu, B. D. Cameron, L. V. Wang, S. Rastegar, and G. L Cote, "Light backscattering polarization patterns from turbid media: theory and experiment," Appl. Opt. 38, 3399 - 3408 (1999). [CrossRef]
  14. G. L. Liu, Y. Li, and B. D. Cameron, "Polarization Based Optical imaging and processing techniques with application to the Cancer Diagnostics," Proc. SPIE 4617, 208 - 219 (2002). [CrossRef]
  15. I. Berezhnyy and A. Dogariu, "Time resolved Mueller matrix imaging polarimetry," Opt. Express 12, 4635 - 4649 (2004). [CrossRef] [PubMed]
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