Use of polar decomposition for the diagnosis of oral precancer

The Mueller matrix describes all the polarizing properties of a sample and, therefore, the optical differ- ences between noncancerous and precancerous tissue that may be present within the matrix elements. A high-speed polarimetry system that generates 16 (cid:1) 4 (cid:1) 4 (cid:2) full Mueller matrices to characterize tissues is presented. Feature extraction is done on the Mueller matrix elements resulting in depolarization and retardance images by polar decomposition. These are used to detect and classify early oral cancers and precancerous changes in epithelium such as dysplasia. These images are compared with orthogonal polarization images and analyzed in an attempt to identity useful factors for the differentiation between cancerous lesions and their benign counterparts. Our results indicate that polarimetry has potential as a method for the in vivo early detection and diagnosis of oral premalignancy. © 2007 Optical Society of America OCIS codes: 230.0230, 170.3890, 230.5440, 290.7050.


Introduction
An estimated 1,444,920 new cases of cancer will be diagnosed in the United States in the year 2007, and an estimated 559,650 Americans will die from cancer. 1 Cancer (190.1 per 100,000) was the second leading cause of death in the United States in the year 2003, exceeded only by heart disease (231.6 per 100,000). 1 More than 85% of cancers originate in the epithelium, and epithelial cancers are preceded by a curable precancerous stage. As such, early detection is paramount for the successful treatment of this disease. If detected at precancerous stage, 95% of the cases have a complete recovery. 2 However, many forms of precancerous changes are difficult to detect using conventional techniques, which require histological examination of biopsies obtained from visible lesions or random surveillance biopsies.
Biological tissues are optically inhomogeneous, birefringent, and absorbing media. 3 Precancerous lesions are characterized by increased nuclear size and nuclear and͞or cytoplasmic ratio. The scattering from the epithelial layer of tissue can provide information on nuclear morphology. 4 The rate of depolarization of incident polarized light depends on the morphological and optical features such as the density, size, shape, and refractive index in the tissue. 5 Cancerous tissues depolarize light less than their surrounding tissues do, whereas noncancerous tissues have the same depolarizing properties as the surrounding tissue. 6 This effect is caused by changes in the subsurface structures of cancerous tissues that prevent light from penetrating the tissue as deeply as it would in normal tissue. These facts support an optical approach to noninvasive cancer detection.
In the field of optical imaging, there are a number of subsurface imaging techniques such as optical coherence tomography, 7 linear and nonlinear emission and harmonic generation imaging, 8,9 and confocal microscopy. 10 These techniques provide high resolution images of subsurface structures with the drawback that the imaging depth is small (i.e., 1 to 2 mm or less). To provide larger imaging depths, other techniques need to be explored. The imaging depth into the tissue depends on the scattering and absorption characteristic of the tissue. Demos and Alfano 11 showed that the spectral polarization differences imaging techniques (SPDI) was an imaging tool that can provide subsurface imaging at larger depth, and deep surface imaging in tissue (1.5 cm beneath the surface) was demonstrated by using both spectral and polarization discrimination of the backscattered photons. 12 In 1976, Bickel et al. 13 advanced polarimetry into the field of biomedical sensing when they described a technique that measured the polarization effects of the scattered light from bacterial suspensions to yield useful information to characterize the sample. In the ensuing decades, several groups have shown that a considerable amount of information, such as the average particle size, 14 photon path length, 15 and particle shape 16 -18 can be obtained from polarization sensitive measurements of the sample under investigation. 19 In addition, others have shown that polarization-based imaging measurements can provide enhanced visualization of superficial structures 20 -22 to allow for subsurface imaging. 23 The aforementioned applications of polarimetry for biomedical imaging involve the use of Mueller-Stokes calculus to mathematically depict how a biological sample affects the polarization vector of an incident light beam, determined by either backscattered 14,15,24,25 or transmitted 26 -28 light intensities from the sample. The experimental Mueller matrix of a sample contains information on retardance, diattenuation, and depolarization, which is not readily apparent in the original images. These sample discriminating parameters can be extracted using polar decomposition. The decomposition of Mueller matrices into a combination of product matrices has been addressed by many authors. 29 -32 Lu and Chipman 29 were able to decompose the Mueller matrix into three factors, namely, diattenuation, retardance, and depolarization. Several groups have developed a Mueller matrix imaging polarimeter capable of collecting the full Mueller matrix of tissue in vivo. 3,33 They have shown that by matrix decomposition the retardance and depolarization parameters show potential for distinguishing between varieties of dermatological conditions including malignant moles. A system similar to that presented here has been previously reported in a publication in 2002. 34 However, that system did not have enough resolution to allow analysis of celluar and nuclear morphologic features of tissue sections. To the best of our knowledge, no research has been performed to correlate the observed changes in the Mueller matrix to the changes in the tissue structure that occur as the tissue becomes precancerous. The main innovations in this current system are the histologic-quality images of cell morphology obtained by using objective lens and the better acquisition time to obtain the Mueller matrices. This system is capable of collecting the Mueller matrix, and the time taken to do so would be much less than 15 s. The basis for the optical detection of a precancerous stage, such as dysplasia, is the exploitation of pathology related changes in the optical properties of the tissue. 35 We hypothesize that it is through the observation of these morphologic and biochemical changes that precancerous tissue can be distinguished from noncancerous tissue. These images are compared with the orthogonal polarization image and analyzed in an attempt to determine specific diagnostic criteria for the identification of cancerous le-sions from their benign counterparts. Our results indicate that polarimetry has potential as a method for the in vivo early detection and diagnosis of oral premalignancy.

A. Stokes Vector and Mueller Matrix
In 1943, Hans Mueller developed a matrix that relates the Stokes vector of the light impinging on a sample to the Stokes vector leaving sample models. 36 Using the method with the input and output polarization states (Stokes vectors) known, the 4 ϫ 4 Mueller matrix can be used to describe the polarization properties of a sample. This relationship is shown below in Eq. (1), where M sample is the Mueller matrix of a sample and S out and S in are the output and input Stokes vectors respectively: The Stokes parameters S consist of four values. The first parameter, S 0 , is simply the overall intensity of the light. The second parameter, S 1 , is the tendency of the light to exhibit either a horizontal ͑S 1 Ͼ 0͒ or a vertical ͑S 1 Ͻ 0͒ polarization state. Similary, the third parameter, S 2 , is the tendency of the light to exhibit either a ϩ45°͑S 2 Ͼ 0͒ or a Ϫ45°͑ S 2 Ͻ 0͒ polarization state. Finally, the fourth parameter, S 3 , is the tendency of the light to exhibit either a right ͑S 3 Ͼ 0͒ or a left ͑S 3 Ͻ 0͒ polarization state. The value for a specific Stokes parameter will be zero if neither of the states for a given parameter is observed.
For simplicity, a Mueller matrix is normalized by dividing all the M ij elements by the M 00 element. The resulting normalized matrix has an M 00 element that is identically ϩ1 and the remaining elements range from Ϫ1 to ϩ1. For a given sample, a Mueller matrix M can be accomplished with 16 polarization images, as depicted in Table 1.
We calculate orthogonal polarization images ͑I OP ͒ by using 3 where I par is the image at light with the polarizer oriented to accept light parallel to the incident light, and I per is the image at light with the polarizer oriented to accept light perpendicular to the incident light.

B. Decomposition for Depolarizing Mueller Matrices
A practical method of categorizing Mueller matrices M is by describing the type of polarization element that is represented. The three polarization properties are retardance, diattenuation, and depolarization.
There are 16°of freedom in the Mueller matrix: one in intensity (transmittance, reflectance, scattering), three in retardance, three in diattenuation, and nine in depolarization. An object's Mueller matrix may contain any combination of these properties, which are extracted by using a polar decomposition process. This polar decomposition, explained in detail by Lu and Chipman, 37 Diattenuation D can be acquired directly through the algebraic calculation of the elements in the Muller matrix shown by The Mueller matrix for a diattenuator with a diattenuator vector D is then M can be written as in which D and P are the diattenuation and polarizance vectors of M. Define a Mueller matrix MЈ based upon M as Note that MЈ has no diattenuation. However, MЈ is not a pure retarder as in the case of a nondepolarizing element because MЈ contains both retardance and depolarization. MЈ is then further decomposed as a depolarizer followed by a retarder, i.e., Equations (9) and (10) lead to The four-by-four matrix decomposition reduces to a three-by-three matrix decomposition given Eq. (12). Actually this is equivalent to the polar decomposition of a three-by-three real matrix. Equation (12) leads to Let 1 , 2 , and 3 be eigenvalues of mЈ͑mЈ͒ T . From Eq. (14) it follows that m ⌬ has ͱ 1 , ͱ 2 , and ͱ 3 as eigenvalues, assuming that mЈ is nonsingular. Applying the Cayley-Hamilton theorem and Eq. (14), m ⌬ can be obtained by m ⌬ ϭ Ϯ ͕mЈ͑mЈ͒ T ϩ ͑ ͱ 1 2 ϩ ͱ 2 3 ϩ ͱ 3 1͒ I͖ Ϫ1 ϫ ͕͑ ͱ 1 ϩ ͱ 2 ϩ ͱ 3͒ mЈ ͑ mЈ ͒ T ϩ ͱ 1 2 3 I͖. (15) If the determinant of mЈ is negative then the minus sign is applied. Otherwise the plus sign is applied.
The notation is as follows: the first term represents the input polarization state, and the second term represents the output polarization state. The states are defined as: H for horizontal, V for vertical, P for ϩ45°, and R for right circular.
Then the submatrix m R of retardance matrix M R can be obtained by Its depolarization power denoted by ⌬ is given by Last, the retardance matrix M R can be known and the retardance can be acquired by

A. Experimental Setup
The system shown in Fig. 1

Tumor Induction
A total of nine cheek pouches in female golden Syrian hamsters (Mesocricetus auratus) were used for in vivo imaging. 38 Application of 0.5% DMBA (9, 10 dimethyl-1, 2-benzanthracene) in mineral oil three times per week produced mild to severe dysplasia after 3-6 weeks that progressed to squamous cell carcinoma at approximately 10 weeks. The right cheek pouch of each hamster was treated with DMBA carcinogen in mineral oil; the other cheek pouch was treated only with mineral oil. Previous studies have shown that the carcinogenesis process in one cheek pouch does not affect the other cheek. Therefore the treated cheek pouch does not affect the other cheek. 39,40 The histological features in this model have been shown to correspond closely with those of premalignancy and malignancy in human oral mucosa. 38

Imaging
Five hamsters were anesthetized using intraperitoneal 2:1 Ketamine HCL ͑100 mg͞ml͒: Xylazine ͑20 mg͞ml͒ at a dose of 0.75 cc͞kg. These animals were then wrapped in mylar with a cotton oversleeve to prevent hypothermia. The cheek pouches were everted, held in place with a tension clamp, kept moist by irrigation with normal saline, and imaged in vivo in near real time. In some animals, imaging at the same location was performed twice to provide an indication of image fidelity. Immediately after imaging, the hamsters were euthanized and the specimens excised for histologic preparation. Four specimens for in situ dysplasia and tumor imaging were obtained from four hamsters. The hamsters were euthanized and each cheek pouch was carefully dissected out and pinned onto a corkboard using 0.20 mm inset pins. The tissues were imaged immediately after positioning. The samples were then fixed and sent for histological processing.
The animals used in this study were housed and treated in accordance with animal research committee guidelines at the University of California, Irvine (approval number 1997-1972).

C. Histology
After imaging the tissue was formalin fixed, photographed through a steromicroscope for gross visual documentation, and then processed for paraffin embedding. Several 6 m paraffin sections were cut and Hematoxylin and eosin (H&E) stained. The polarization images for each tissue sample were compared with the corresponding steromicrographs for orientation and identification of the imaged area. The histologic sections taken from the imaged areas were examined using standard light microscopy to determine the degree of pathology present.

Results
Hamster oral mucosa was imaged at all stages of carcinogenesis in the hamster cheek pouch model. Panel (a) in Fig. 2 presents an in situ microscopic image of hamster cheek pouch tissue with precancer. The dysplastic and normal regions are marked by red and blue dotted circles, respectively. The 2D polarization and 2D retardance images for both normal and cancerous regions are shown in Figs. 2(b) and 2(c), respectively. Figures 2(d) and 2(e) show the histology of normal and dysplastic regions, respectively. As theses figures show, depolarization and retardance images can be used to quantitatively identify the dysplastic region of tissue. These advances in depolarization and retardance imaging promise ultimately to permit earlier and more sensitive diagnosis and to improve the monitoring of disease progression, the identification of region margins, and the assessment of cancer response to therapy. Figures 3 and 4 depict the orthogonal polarization, depolarization, and retardance images in vivo for normal and precancerous tissues, respectively. These images were from three different hamster cheek pouches and the imaged fields were 2.25 mm ϫ 2.25 mm. Based on the polarization images, the Mueller matrix images were reconstructed, decomposed, and compared with orthogonal polarization images. Figures 3(a) and 4(a) are the orthogonal, polarization images ͑I OP ͒ for healthy and precancerous tissues, respectively, and no noticeable differences between them are apparent.
Depolarization describes the process of how polarized light can be converted into light with a lower degree of polarization. Depolarization can be the result of a disordered medium or scattering from the surface. One quantity that describes the depolarizing behavior of an element is the depolarization index (⌬). The depolarization index of a Mueller matrix can range from 1 for a perfect depolarizer to 0 for completely polarized light. Based on the data presented in Figs. 3(b) and 4(b), depolarization properties may be useful for differentiating between healthy and precancerous tissues, as precancerous tissue depolarizes light less than healthy tissue.
The greatest variation in retardance values was seen in the M 33 element of the Mueller matrix, which describes the circular retardance characteristic of a sample. Only the M 33 element of the Mueller matrix is presented along with the associated retardance image for healthy and precancerous tissues in Figs. 3(c) and 4(c). It indicates that retardance values change within the precancerous lesion, while no change is visible for the noncancerous tissues. These individual pixel value changes result in a lower average retardance value for precancerous tissues. These depolarization and retardance images are potentially useful not only for differentiating between samples but also for boundary identification.

Discussion and Conclusion
We have demonstrated the use of polarization for early detection and diagnosis of oral precancer. The fact that our system can be used for tissue characterization suggests its potential application as a noninvasive modality for the detection and monitoring of precancer and suspicious oral tissue lesions. A Mueller matrix is extremely powerful because it completely describes the polarization altering properties of a sample; however, a raw Mueller matrix is difficult to interpret and provides little physical insight. As such, polar decomposition of the Mueller matrix was performed to yield the more familiar quantities of retardance (birefringence), diattenuation (dichroism), and depolarization. It was shown that there is some measurable depolarization to differentiate between noncancerous and precancerous lesions. Therefore precancerous lesions depolarize light less than healthy tissue, and noncancerous lesions depolarize the light the same as the surrounding tissue. Finally, it indicates that the retardance value changes within the precancerous lesion, while no change is visible for the healthy lesion. These individual pixel value changes result in a lower average retardance value for the cancerous sample. These depolarization and retardance images are potentially useful not only for differentiating between samples but also for boundary identification.