Introduction

Cancer pathology studies show that skin cancer is one of the most prevalent cancers in the world, and its incidence, treatment costs and associated health burden have been steadily increasing1. The precancerous skin characteristics at both surface and depth may change, so that the monitoring and analysis of lesions must be carried out at the same time, the changes at the surface project artifacts or introduce bias into the detection of deeper regions, which makes the identification of the characteristics of precancerous lesions complicated and difficult. Conventional diagnostic methods rely on the subjective judgment of doctors, which may not provide enough sensitivity for lesion diagnosis at early stage, leading to the easily missed precancerous lesion, and the diagnosis is mostly in the middle and late stage, which increases the physical and mental pressure of patients and may miss the best stage of treatment2. The combination of optical coherence tomography and polarization detection method, as a high-resolution, non-invasive and multidimensional characteristic imaging method to identify the depth of precancerous lesion, has garnered great attention for disease risk assessment, early diagnosis and precise treatment3,4. At present, many research results have shown that the contrast of dichroism, birefringence and depolarization can reveal the change of microstructure and morphology of biological tissues. The existence of birefringence characteristics is related to the anisotropy of structures such as fiber orientation and collagen, which are important components of biological tissues. Dichroism parameters and images have gradually become an important method to measure the physical state of tissues such as thickness and density. The obvious inhomogeneity and different roughness of biological tissues make the tissue structure depolarizes the polarized light. Therefore, polarization detection is an efficient imaging method that can provide multi-dimensional contrast information for precancerous lesion with fibrous changes and changes in physical properties. PS-OCT is a typical optical system that can obtain depth-resolved polarization images. Compared with traditional polarization detection methods, its obvious advantage is that it can distinguish the polarization characteristic information of the tissue in the effective imaging depth, rather than only obtaining the polarization information of the tissue surface, such as polarization microscopic imaging methods. In addition, two-dimensional or three-dimensional scanning can be realized according to the needs of the complexity of microstructure within the imaging depth of sample, and its polarization information can be restored with maximum non-destructive and high resolution.

Mueller matrix algorithm and Jones matrix method are usually used for characterizing polarization properties of biological tissues. However, only Mueller matrix algorithm can be used when the tissue exhibits depolarization5. At present, most studies analyze the polarization properties of the experimental Mueller matrix by decomposing it into several components with clear physical significance and simple matrix form, and some studies calculate the polarization characteristics using several elements of the experimentally measured Mueller matrix6,7. Polarization-sensitive optical coherence tomography is a technology that combines optical coherence tomography with polarization measurement methods to obtain two-dimensional or three-dimensional depth-resolved images of sample intensity and polarization characteristic parameters by using the low coherence characteristics of light source8. In terms of acquiring depth-resolved images of Mueller matrix of samples based on PS-OCT system, the research group of Wang and Yasuno carried out earlier research and achieved certain achievements9,10. In 1999, Wang et al. detected the Mueller matrix of sample by building a PS-OCT system containing a single detector, and proposed for the first time a method to reconstruct 16 element images of the Mueller matrix by detecting the intensity of polarization components in different directions8. Yasuno et al. measured the image of Jones matrix elements of the pig esophagus, and obtained its image of Mueller matrix elements through the transformation relationship, from which the layered organizational structure was initially observed9. After that, the group used the transformation method to obtain an image of the elements of the Mueller matrix of the bovine tendon11. At present, the method of obtaining polarization images based on Mueller matrix method, combined with specific incident and outgoing polarization state properties (described by Stokes vector or Poincare ball method) and structural characteristics of tissues to be measured is also constantly developing12.

However, the decomposition method has complex processes and limited application conditions. The method of direct element calculation only involves a small number of elements in the matrix, thus the accuracy needs to be verified by a large number of samples. At present, studies have focused on demonstrating the application potential of the element characterization method of polarization parameters based on Mueller matrix through the analysis of a large number of pathological samples. Ma Hui et al. used Mueller matrix to represent samples or systems with polarization characteristics. By exploring the decomposition, symmetry and transformation properties of Mueller matrix, combined with Monte Carlo (MC) simulation, machine learning, statistical analysis7,13,14,15 and a large number of biological and pathological tissue experiments, A method of directly calculating the polarization parameters from the elements of Mueller matrix is proposed and verified many times. A large number of experimental results provide an important reference for pathological diagnosis.

Therefore, a method of analyzing polarization properties using elements of Mueller matrix with concise algorithm, high imaging contrast and more accurate results is the focus of biomedical research. In recent years, targeted exploration to define polarization parameters with clear mechanisms as indicators of specific organizations has proved to be an effective way to improve contrast3,16. Nevertheless, only the contrast improvement of images with a few parameters, such as depolarization parameters, is reported, and there is a lack of effective methods to improve the contrast of images with multiple polarization parameters at the same time, resulting in the differences of information reflected among the multi-polarization parameter images and affect the accuracy of sample structure analysis. In vitro Basal cell carcinoma was detected using the scattering intensity and birefringence characteristics measured by PS-OCT, which are usually low in tumors due to collagen destruction, with sensitivity and specificity values exceeding 95%17. It was found that higher surface roughness was associated with lower degree of polarization (DOP), and melanoma was successfully distinguished from other types of skin lesions by virtue of its higher DOP than benign nevus18. Therefore, using multiple polarization parameters is beneficial for improving the diagnosis, discrimination and visualization contrast of skin lesions.

In this work, a method considering the depolarization mechanism is presented for enhancing the polarization contrast of ex-vivo mouse skin using PS-OCT. Expressions of multiple polarization parameters based on two methods of the light propagation model and the exploration of the one unique form of canonical decomposition are first classified and combined according to the polarization characteristics. Then, combined expressions of depolarizing parameters and non-depolarizing parameters are combined and averaged respectively. Finally, depth-resolved images of these parameters of ex-vivo mouse skin are calculated using PS-OCT, the combined non-depolarizing images are integrated into the combined depolarizing image, and the combined contrast image is formed to demonstrate the clinical potential of the enhanced contrast method in skin disease diagnosis.

Method

There are 14 polarization parameters derived in the two methods of the light propagation model and the exploration of the one unique form of canonical decomposition19,20. Note that, the propagation model is based on a theoretical model in which a thin scattering layer within the medium is represented by a depolarization Mueller matrix and the effects of medium on the forward-pass and the backward-pass propagation of light through it are represented by two non-depolarizing matrices. The explicit expressions of the polarization properties are then derived for the formulas of elements of the macroscopic Mueller matrix. The other method based on the exploration of the one unique form of canonical decomposition allows directly obtaining the polarization properties from measured macroscopic Mueller matrices, in which an interpretation method of the canonical decomposition is developed by using the relationships derived with the form of one possible product in polar decomposition with the specific physical meanings of component matrices expressed in four different conditions. All these formulations are used and introduced here to demonstrate how the proposed parameters are obtained. After the classification and the combination according to the polarization characteristics. Six parameters can be obtained including: combined depolarization \({\Delta _{1 - \operatorname{int} }}\), and combined degrees of linear, circular parameter, \({e_{1 - \operatorname{int} }}\), combined diattenuation \({D_{1 - \operatorname{int} }}\), and combined diattenuation axis \(\alpha {\varepsilon _{d1 - \operatorname{int} }}\), and combined retardance \({R_{1 - \operatorname{int} }}\), and combined retarder fast axis \(\alpha {\varepsilon _{r1 - \operatorname{int} }}\)21:

$${D_{1 - \operatorname{int} }}=\frac{1}{2}\left[ {\sqrt {M_{{12}}^{2}+M_{{13}}^{2}+M_{{14}}^{2}} +\left| {\frac{{{M_{12}}\sqrt {M_{{42}}^{2}+M_{{34}}^{2}} }}{{{M_{34}}}}} \right|} \right]$$
(1)
$$\alpha {\varepsilon _{d1 - \operatorname{int} }}=\frac{1}{4}\left[ {\arctan (\frac{{M_{{13}}^{{}}}}{{M_{{12}}^{{}}}})+\arctan (\frac{{M_{{14}}^{{}}sin(arctan(\frac{{M_{{13}}^{{}}}}{{M_{{12}}^{{}}}}))}}{{M_{{13}}^{{}}}})} \right]$$
(2)
$${R_{1 - \operatorname{int} }}=\left\{ \begin{gathered} {M_{33}},{M_{44}}>\left| {{M_{34}}} \right|,\left| {{M_{43}}} \right|: \hfill \\ \frac{1}{2}\left[ {co{s^{ - 1}}\left( {\frac{{1 - {M_{21}}/{M_{12}} - {M_{31}}/{M_{13}} - {M_{41}}/{M_{14}}}}{2} - 1} \right)+\frac{{\sqrt {M_{{42}}^{2}+M_{{34}}^{2}} }}{{{M_{22}}+{M_{33}}}}} \right]{\text{ }} \hfill \\ {M_{21}},{M_{31}},{M_{41}} \approx 0,{M_{12}},{M_{13}},{M_{14}} \approx 0,{M_{33}},{M_{44}}>\left| {{M_{34}}} \right|,\left| {{M_{43}}} \right|: \hfill \\ \frac{1}{2}\left[ {co{s^{ - 1}}\left( {\frac{1}{{\sqrt {1+M_{{21}}^{2}+M_{{31}}^{2}+M_{{41}}^{2}} }}} \right){\text{+}}\frac{{\sqrt {M_{{42}}^{2}+M_{{34}}^{2}} }}{{{M_{22}}+{M_{33}}}}} \right] \hfill \\ {M_{33}},{M_{44}}<\left| {{M_{34}}} \right|,\left| {{M_{43}}} \right|: \hfill \\ \frac{1}{2}\left[ {co{s^{ - 1}}\left( {\frac{{{M_{33}}/\left| {{M_{34}}} \right|+{M_{44}}/\left| {{M_{43}}} \right|}}{2}} \right)+\frac{{\sqrt {M_{{42}}^{2}+M_{{34}}^{2}} }}{{{M_{22}}+{M_{33}}}}} \right] \hfill \\ \end{gathered} \right.$$
(3)
$$\alpha {\varepsilon _{r1 - \operatorname{int} }}=\left\{ \begin{gathered} {M_{33}},{M_{44}}>\left| {{M_{34}}} \right|,\left| {{M_{43}}} \right|: \hfill \\ \frac{1}{2}\left[ {\frac{1}{4}\left[ {\arctan \left( {\frac{{{M_{34}}/\left| {{M_{44}}} \right|+{M_{43}}/\left| {{M_{33}}} \right|}}{{{M_{24}}/\left| {{M_{44}}} \right|+{M_{42}}/\left| {{M_{22}}} \right|}}} \right)+\arctan \left( {\frac{{{M_{24}}/\left| {{M_{44}}} \right|+{M_{42}}/\left| {{M_{22}}} \right|}}{{{M_{23}}/\left| {{M_{33}}} \right|+{M_{32}}/\left| {{M_{22}}} \right|}}sin(2{\alpha _{Jr}})} \right)} \right] - \frac{{{M_{24}}}}{{{M_{34}}}} } \right] \hfill \\ {M_{33}},{M_{44}}<\left| {{M_{34}}} \right|,\left| {{M_{43}}} \right|: \hfill \\ \frac{1}{2}\left[ {\frac{1}{2}\left[ {\sqrt {\frac{{\left( {co{s^2}\left( {\frac{{{R_J}}}{2}} \right) - \frac{{{M_{22}}/\left| {{M_{22}}} \right|+{M_{33}}/\left| {{M_{34}}} \right|}}{2}} \right)}}{{si{n^2}\left( {\frac{{{R_J}}}{2}} \right)}}} +\sqrt {\frac{{\left( {co{s^2}\left( {\frac{{{R_J}}}{2}} \right) - \frac{{{M_{22}}/\left| {{M_{22}}} \right|+{M_{44}}/\left| {{M_{43}}} \right|}}{2}} \right)}}{{\left( {co{s^2}(2{e_{Jr}})si{n^2}\left( {\frac{{{R_J}}}{2}} \right)} \right)}}} } \right] - \frac{{{M_{24}}}}{{{M_{34}}}}} \right] \hfill \\ \end{gathered} \right.$$
(4)
$${\Delta _{1 - \operatorname{int} }}=\left\{ \begin{gathered} \frac{1}{2}\left[ {2 - \frac{{{M_{22}}+\left| {{M_{33}}} \right|+\left| {{M_{44}}} \right|}}{3} - \sqrt 2 \sqrt {M_{{42}}^{2}+M_{{34}}^{2}+M_{{44}}^{2}} } \right]\left( {{M_{33}},{M_{44}}>\left| {{M_{34}}} \right|,\left| {{M_{43}}} \right|} \right) \hfill \\ \frac{1}{2}\left[ {2 - \frac{{{M_{22}}+\left| {{M_{34}}} \right|+\left| {{M_{43}}} \right|}}{3} - \sqrt 2 \sqrt {M_{{42}}^{2}+M_{{34}}^{2}+M_{{44}}^{2}} } \right]\left( {{M_{33}},{M_{44}}<\left| {{M_{34}}} \right|,\left| {{M_{43}}} \right|} \right) \hfill \\ \end{gathered} \right.$$
(5)
$${e_{1 - \operatorname{int} }}=\left\{ \begin{gathered} \frac{1}{3}\left[ {{M_{22}}+\left| {{M_{33}}} \right|+\left| {{M_{44}}} \right| } \right]({M_{33}},{M_{44}}>\left| {{M_{34}}} \right|,\left| {{M_{43}}} \right|) \hfill \\ \frac{1}{3}\left[ {{M_{22}}+\left| {{M_{34}}} \right|+\left| {{M_{43}}} \right| } \right]({M_{33}},{M_{44}}<\left| {{M_{34}}} \right|,\left| {{M_{43}}} \right|) \hfill \\ \end{gathered} \right.$$
(6)

where \({M_{mn}}(m,n=1,2,3,4)\) represents the element at the \({m^{th}}\) row and the \({n^{th}}\) column of the measured Mueller matrix.

Then, the expressions of depolarizing parameters \({\Delta _{1 - \operatorname{int} }}\) and \({e_{1 - \operatorname{int} }}\) need to be combined to ensure that the components characterizing depolarization are all included. Therefore, the combined depolarization parameter can be more sensitive to the small changes of the depolarization of the tissue:

$$\Delta - {e_{com}}=\left\{ \begin{gathered} \frac{1}{4}\left[ {2 - \frac{{{M_{22}}+\left| {{M_{33}}} \right|+\left| {{M_{44}}} \right|}}{3} - \sqrt 2 \sqrt {M_{{42}}^{2}+M_{{34}}^{2}+M_{{44}}^{2}} } \right]+\frac{1}{6}\left[ {{M_{22}}+\left| {{M_{33}}} \right|+\left| {{M_{44}}} \right| } \right]({M_{33}},{M_{44}}>\left| {{M_{34}}} \right|,\left| {{M_{43}}} \right|) \hfill \\ \frac{1}{4}\left[ {2 - \frac{{{M_{22}}+\left| {{M_{34}}} \right|+\left| {{M_{43}}} \right|}}{3} - \sqrt 2 \sqrt {M_{{42}}^{2}+M_{{34}}^{2}+M_{{44}}^{2}} } \right]+\frac{1}{6}\left[ {{M_{22}}+\left| {{M_{34}}} \right|+\left| {{M_{43}}} \right| } \right]({M_{33}},{M_{44}}<\left| {{M_{34}}} \right|,\left| {{M_{43}}} \right|) \hfill \\ \end{gathered} \right.$$
(7)

With the consideration of the depolarization mechanism, the enhanced contrast method can be obtained through combining each expression of these non-depolarizing parameters in Eqs. (1–4) and the expression of \(\Delta - {e_{com}}\):

$${D_{\Delta e - com}}=\left\{ \begin{gathered} \frac{1}{8}\left[ {2 - \frac{{{M_{22}}+\left| {{M_{33}}} \right|+\left| {{M_{44}}} \right|}}{3} - \sqrt 2 \sqrt {M_{{42}}^{2}+M_{{34}}^{2}+M_{{44}}^{2}} } \right]+\frac{1}{{12}}\left[ {{M_{22}}+\left| {{M_{33}}} \right|+\left| {{M_{44}}} \right| } \right] \hfill \\ +\frac{1}{4}\left[ {\sqrt {M_{{12}}^{2}+M_{{13}}^{2}+M_{{14}}^{2}} +\left| {\frac{{{M_{12}}\sqrt {M_{{42}}^{2}+M_{{34}}^{2}} }}{{{M_{34}}}}} \right|} \right]({M_{33}},{M_{44}}>\left| {{M_{34}}} \right|,\left| {{M_{43}}} \right|) \hfill \\ \frac{1}{8}\left[ {2 - \frac{{{M_{22}}+\left| {{M_{34}}} \right|+\left| {{M_{43}}} \right|}}{3} - \sqrt 2 \sqrt {M_{{42}}^{2}+M_{{34}}^{2}+M_{{44}}^{2}} } \right]+\frac{1}{{12}}\left[ {{M_{22}}+\left| {{M_{34}}} \right|+\left| {{M_{43}}} \right| } \right] \hfill \\ +\frac{1}{4}\left[ {\sqrt {M_{{12}}^{2}+M_{{13}}^{2}+M_{{14}}^{2}} +\left| {\frac{{{M_{12}}\sqrt {M_{{42}}^{2}+M_{{34}}^{2}} }}{{{M_{34}}}}} \right|} \right]({M_{33}},{M_{44}}<\left| {{M_{34}}} \right|,\left| {{M_{43}}} \right|) \hfill \\ \end{gathered} \right.$$
(8)
$$\alpha {\varepsilon _{d\Delta e - com}}=\left\{ \begin{gathered} \frac{1}{8}\left[ {2 - \frac{{{M_{22}}+\left| {{M_{33}}} \right|+\left| {{M_{44}}} \right|}}{3} - \sqrt 2 \sqrt {M_{{42}}^{2}+M_{{34}}^{2}+M_{{44}}^{2}} } \right]+\frac{1}{{12}}\left[ {{M_{22}}+\left| {{M_{33}}} \right|+\left| {{M_{44}}} \right| } \right] \hfill \\ +\frac{1}{8}\left[ {\arctan \left( {\frac{{M_{{13}}^{{}}}}{{M_{{12}}^{{}}}}} \right)+\arctan \left( {\frac{{M_{{14}}^{{}}sin\left( {arctan\left( {\frac{{M_{{13}}^{{}}}}{{M_{{12}}^{{}}}}} \right)} \right)}}{{M_{{13}}^{{}}}}} \right)} \right]({M_{33}},{M_{44}}>\left| {{M_{34}}} \right|,\left| {{M_{43}}} \right|) \hfill \\ \frac{1}{8}\left[ {2 - \frac{{{M_{22}}+\left| {{M_{34}}} \right|+\left| {{M_{43}}} \right|}}{3} - \sqrt 2 \sqrt {M_{{42}}^{2}+M_{{34}}^{2}+M_{{44}}^{2}} } \right]+\frac{1}{{12}}\left[ {{M_{22}}+\left| {{M_{34}}} \right|+\left| {{M_{43}}} \right| } \right] \hfill \\ +\frac{1}{8}\left[ {\arctan \left( {\frac{{M_{{13}}^{{}}}}{{M_{{12}}^{{}}}}} \right)+\arctan \left( {\frac{{M_{{14}}^{{}}sin\left( {arctan\left( {\frac{{M_{{13}}^{{}}}}{{M_{{12}}^{{}}}}} \right)} \right)}}{{M_{{13}}^{{}}}}} \right)} \right]({M_{33}},{M_{44}}<\left| {{M_{34}}} \right|,\left| {{M_{43}}} \right|) \hfill \\ \end{gathered} \right.$$
(9)
$${R_{\Delta e - com}}=\left\{ \begin{gathered} {M_{33}},{M_{44}}>\left| {{M_{34}}} \right|,\left| {{M_{43}}} \right|: \hfill \\ \frac{1}{4}\left[ {co{s^{-1}}(\frac{{1-{M_{21}}/{M_{12}}-{M_{31}}/{M_{13}}-{M_{41}}/{M_{14}}}}{2}-1)+\frac{{\sqrt {M_{{42}}^{2}+M_{{34}}^{2}} }}{{{M_{22}}+{M_{33}}}}} \right] \hfill \\ {\text{+}}\frac{1}{8}\left[ {2 - \frac{{{M_{22}}+\left| {{M_{33}}} \right|+\left| {{M_{44}}} \right|}}{3} - \sqrt 2 \sqrt {M_{{42}}^{2}+M_{{34}}^{2}+M_{{44}}^{2}} } \right]+\frac{1}{{12}}\left[ {{M_{22}}+\left| {{M_{33}}} \right|+\left| {{M_{44}}} \right| } \right]{\text{ }} \hfill \\ {M_{21}},{M_{31}},{M_{41}} \approx 0,{M_{12}},{M_{13}},{M_{14}} \approx 0,{M_{33}},{M_{44}}>\left| {{M_{34}}} \right|,\left| {{M_{43}}} \right|: \hfill \\ \frac{1}{4}\left[ {co{s^{-1}}(\frac{1}{{\sqrt {1+M_{{21}}^{2}+M_{{31}}^{2}+M_{{41}}^{2}} }}){\text{ +}}\frac{{\sqrt {M_{{42}}^{2}+M_{{34}}^{2}} }}{{{M_{22}}+{M_{33}}}}} \right] \hfill \\ +\frac{1}{8}\left[ {2 - \frac{{{M_{22}}+\left| {{M_{33}}} \right|+\left| {{M_{44}}} \right|}}{3} - \sqrt 2 \sqrt {M_{{42}}^{2}+M_{{34}}^{2}+M_{{44}}^{2}} } \right]+\frac{1}{{12}}\left[ {{M_{22}}+\left| {{M_{33}}} \right|+\left| {{M_{44}}} \right| } \right] \hfill \\ {M_{33}},{M_{44}}<\left| {{M_{34}}} \right|,\left| {{M_{43}}} \right|: \hfill \\ \frac{1}{4}\left[ {co{s^{-1}}(\frac{{{M_{33}}/\left| {{M_{34}}} \right|+{M_{44}}/\left| {{M_{43}}} \right|}}{2})+\frac{{\sqrt {M_{{42}}^{2}+M_{{34}}^{2}} }}{{{M_{22}}+{M_{33}}}}} \right] \hfill \\ +\frac{1}{8}\left[ {2 - \frac{{{M_{22}}+\left| {{M_{34}}} \right|+\left| {{M_{43}}} \right|}}{3} - \sqrt 2 \sqrt {M_{{42}}^{2}+M_{{34}}^{2}+M_{{44}}^{2}} } \right]+\frac{1}{{12}}\left[ {{M_{22}}+\left| {{M_{34}}} \right|+\left| {{M_{43}}} \right| } \right] \hfill \\ \end{gathered} \right.$$
(10)
$$\alpha {\varepsilon _{r\Delta e - com}}=\left\{ \begin{gathered} {M_{33}},{M_{44}}>\left| {{M_{34}}} \right|,\left| {{M_{43}}} \right|: \hfill \\ \frac{1}{4}\left[ {\frac{1}{4}\left[ {\arctan \left( {\frac{{{M_{34}}/\left| {{M_{44}}} \right|+{M_{43}}/\left| {{M_{33}}} \right|}}{{{M_{24}}/\left| {{M_{44}}} \right|+{M_{42}}/\left| {{M_{22}}} \right|}}} \right)+\arctan \left( {\frac{{{M_{24}}/\left| {{M_{44}}} \right|+{M_{42}}/\left| {{M_{22}}} \right|}}{{{M_{23}}/\left| {{M_{33}}} \right|+{M_{32}}/\left| {{M_{22}}} \right|}}sin(2{\alpha _{Jr}})} \right)} \right] - \frac{{{M_{24}}}}{{{M_{34}}}} } \right] \hfill \\ +\frac{1}{8}\left[ {2 - \frac{{{M_{22}}+\left| {{M_{33}}} \right|+\left| {{M_{44}}} \right|}}{3} - \sqrt 2 \sqrt {M_{{42}}^{2}+M_{{34}}^{2}+M_{{44}}^{2}} } \right]+\frac{1}{6}\left[ {{M_{22}}+\left| {{M_{33}}} \right|+\left| {{M_{44}}} \right| } \right] \hfill \\ {M_{33}},{M_{44}}<\left| {{M_{34}}} \right|,\left| {{M_{43}}} \right|: \hfill \\ \frac{1}{4}\left[ {\frac{1}{2}\left[ {\sqrt {\frac{{\left( {co{s^2}\left( {\frac{{{R_J}}}{2}} \right) - \frac{{{M_{22}}/\left| {{M_{22}}} \right|+{M_{33}}/\left| {{M_{34}}} \right|}}{2}} \right)}}{{si{n^2}\left( {\frac{{{R_J}}}{2}} \right)}}} +\sqrt {\frac{{\left( {co{s^2}\left( {\frac{{{R_J}}}{2}} \right) - \frac{{{M_{22}}/\left| {{M_{22}}} \right|+{M_{44}}/\left| {{M_{43}}} \right|}}{2}} \right)}}{{\left( {co{s^2}(2{e_{Jr}})si{n^2}\left( {\frac{{{R_J}}}{2}} \right)} \right)}}} } \right] - \frac{{{M_{24}}}}{{{M_{34}}}}} \right] \hfill \\ +\frac{1}{8}\left[ {2 - \frac{{{M_{22}}+\left| {{M_{34}}} \right|+\left| {{M_{43}}} \right|}}{3} - \sqrt 2 \sqrt {M_{{42}}^{2}+M_{{34}}^{2}+M_{{44}}^{2}} } \right]+\frac{1}{6}\left[ {{M_{22}}+\left| {{M_{34}}} \right|+\left| {{M_{43}}} \right| } \right] \hfill \\ \end{gathered} \right.$$
(11)

The combined polarization parameters under the depolarization mechanism are \({D_{\Delta e - com}}\), \(\alpha {\varepsilon _{d\Delta e - com}}\), \({R_{\Delta e - com}}\) and \(\alpha {\varepsilon _{r\Delta e - com}}\). It can be found that, on the one hand, each expression contains multiple Mueller matrix elements, indicating the complementarity of advantages of the two initial approaches and reduction of the inaccuracy of either method. Note that, instead of replacing non-depolarized components with depolarized components to improve accuracy, each newly obtained polarization parameter expression contains multiple Mueller matrix elements instead of several Mueller matrix elements. As the number of matrix elements involved in the characterization of polarization parameters increases, and the images of 16 elements of Muller matrix all reflect information related to the micro-structure of tissues, the accuracy of the proposed expressions involving multiple parameters is thus improved. On the other hand, according to the results reported in the literature, the high sensitivity of the depolarization properties to image the skin tissue may help to improve the contrast of the non-depolarizing feature in the whole effective imaging depth around 1 mm using PS-OCT, so as to improve the imaging contrast of multiple non-depolarization parameters at the same time. Compared to the widely used dimensionality reduction techniques, such as principal component analysis, the most essential advantage is that our method is derived by considering the principle of optical propagation and the mechanism of polarization characteristics, which can not only specifically enhance the contrast of polarized images and explain the microstructure in the images through the polarization mechanism.

Results and discussions

To demonstrate the application of the proposed 4 combined polarization parameters under the depolarization mechanism, images of ex-vivo skin of mouse were obtained and calculated by a Mueller polarization-sensitive spectral-domain optical coherence tomography system set up in our Lab as shown in19. Figure 1 shows schematic of the PS-OCT system. Figure 2 shows the experimental results of ex-vivo skin at two different stages.

Fig. 1
figure 1

Schematic of the PS-OCT system. SLD: Super-luminescent diode; SMF: Single-mode fiber; PC-FFB: Polarization Controller-Free-Space Fiber Bench; RQW: Rotated quarter-wave plate; RP: Rotated plate; FC: 50:50 Fiber coupler; PC1-3: Polarization controllers; FH1-3: Fiber holders; CL1-3: Collimated lens; GM: Galvo mirror; FL: Focusing lens; S:Sample; M: Mirror; TG: Transmission grating; AFL: Achromatic focusing lens; PBS: Polarization beam splitter; CCD 1–2: Line-scan CCD cameras.

Full size image

Figure 1 shows the PS-OCT system, imaging principle, and depth-resolved polarization imaging process. The yellow boxes highlight the system imaging setup, sample preparation, data acquisition and analysis process. Under the condition of 4 different incident polarized light, the PS-OCT system is firstly used to obtain the information of 16 elements of Mueller matrix per pixel within the effective imaging depth of sample, then the corresponding polarization images based on the proposed parameter method are calculated, and the region of interest is determined to complete the image analysis. In the imaging process, the system adopts a SLD light source with a central wavelength of 840 nm. The light emitted through a 50:50 fiber coupler, which enters the sample arm and the reference arm respectively. The backscattered light of sample interferes with the light from the reference arm at the fiber coupler and to be detected through the collimating lens, grating, converging lens and polarization beam splitter by two linear array CCDS. The CCD signal is collected by the data acquisition card (DAQ) and sent to the computer, and the program is run through the Matlab to complete the image reconstruction of the sample using the expressions proposed. In the process of data analysis and processing, it is first necessary to construct the Stokes vector of the corresponding outgoing light according to the horizontal and vertical components of the interference light:

$${S_{out}}=\left[ {\begin{array}{*{20}{c}} {{S_1}} \\ {{S_2}} \\ {{S_3}} \\ {{S_4}} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {{H_{(z)}}H_{{(z)}}^{*}+{V_{(z)}}V_{{(z)}}^{*}} \\ {{H_{(z)}}H_{{(z)}}^{*} - {V_{(z)}}V_{{(z)}}^{*}} \\ {{H_{(z)}}V_{{(z)}}^{*}+H_{{(z)}}^{*}{V_{(z)}}} \\ {\operatorname{i} ({H_{(z)}}V_{{(z)}}^{*} - H_{{(z)}}^{*}{V_{(z)}})} \end{array}} \right]$$
(12)

where \({H_{(z)}}\)and \({V_{(z)}}\)represent respectively the horizontal and vertical components of the interference signal, \({S_i}(i=1,2,3,4)\) denote the ith element of the vector, \({\text{*}}\)denotes the complex conjugate. Second, values of elements of \({{\mathbf{S}}_{out}}\)s corresponding to four incident polarization states are required to determine the Mueller matrix of the sample at each pixel:

$$\begin{gathered} \left\{ \begin{gathered} {{\mathbf{M}}_{11}}=({S_{{0^ \circ }}}(1)+{S_{{{90}^ \circ }}}(1))/2 \hfill \\ {{\mathbf{M}}_{21}}=({S_{{0^ \circ }}}(2)+{S_{{{90}^ \circ }}}(2))/2 \hfill \\ {{\mathbf{M}}_{31}}=({S_{{0^ \circ }}}(3)+{S_{{{90}^ \circ }}}(3))/2 \hfill \\ {{\mathbf{M}}_{41}}=({S_{{0^ \circ }}}(4)+{S_{{{90}^ \circ }}}(4))/2 \hfill \\ \end{gathered} \right.{\text{ }}\left\{ \begin{gathered} {{\mathbf{M}}_{12}}=({S_{{0^ \circ }}}(1) - {S_{{{90}^ \circ }}}(1))/2 \hfill \\ {{\mathbf{M}}_{22}}=({S_{{0^ \circ }}}(2) - {S_{{{90}^ \circ }}}(2))/2 \hfill \\ {{\mathbf{M}}_{32}}=({S_{{0^ \circ }}}(3) - {S_{{{90}^ \circ }}}(3))/2 \hfill \\ {{\mathbf{M}}_{42}}=({S_{{0^ \circ }}}(4) - {S_{{{90}^ \circ }}}(4))/2 \hfill \\ \end{gathered} \right. \hfill \\ \left\{ \begin{gathered} {{\mathbf{M}}_{13}}={S_{{{45}^ \circ }}}(1) - ({S_{{0^ \circ }}}(1)+{S_{{{90}^ \circ }}}(1))/2 \hfill \\ {{\mathbf{M}}_{23}}={S_{{{45}^ \circ }}}(2) - ({S_{{0^ \circ }}}(2)+{S_{{{90}^ \circ }}}(2))/2 \hfill \\ {{\mathbf{M}}_{33}}={S_{{{45}^ \circ }}}(3) - ({S_{{0^ \circ }}}(3)+{S_{{{90}^ \circ }}}(3))/2 \hfill \\ {{\mathbf{M}}_{43}}={S_{{{45}^ \circ }}}(4) - ({S_{{0^ \circ }}}(4)+{S_{{{90}^ \circ }}}(4))/2 \hfill \\ \end{gathered} \right.{\text{ }}\left\{ \begin{gathered} {{\mathbf{M}}_{14}}={S_Y}(1) - ({S_{{0^ \circ }}}(1)+{S_{{{90}^ \circ }}}(1))/2 \hfill \\ {{\mathbf{M}}_{24}}={S_Y}(2) - ({S_{{0^ \circ }}}(2)+{S_{{{90}^ \circ }}}(2))/2 \hfill \\ {{\mathbf{M}}_{34}}={S_Y}(3) - ({S_{{0^ \circ }}}(3)+{S_{{{90}^ \circ }}}(3))/2 \hfill \\ {{\mathbf{M}}_{44}}={S_Y}(4) - ({S_{{0^ \circ }}}(4)+{S_{{{90}^ \circ }}}(4))/2 \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered}$$
(13)

where\({M_{mn}}(m,n=1,2,3,4)\)represent the elements at the \({m^{th}}\) row and \({n^{th}}\) column of the Mueller matrix\({{\mathbf{M}}_{}}\), \({S_r}(i)(r={0^ \circ },{45^ \circ },{90^ \circ },Y;i=1,2,3,4)\)represent the ith element of \({{\mathbf{S}}_{out}}\)corresponding to the incident polarized light with the polarization state of r, Y represents the incident right-hand polarized light. At last, the 1st and the 2nd integrated parameters of mouse skin ex vivo can be calculated using these elements of Mueller matrix.

Fig. 2
figure 2

Experimental results of the ex-vivo healthy skin and skin with inflammation. Scale bar: 200 μm in both axial and lateral directions of experimental image. Each image is 2 mm(width)×1 mm(depth). The scale bar in the pathological image is about 500 μm.

Full size image

It can be seen from both the images of healthy skin and the inflamed skin that, images of combined polarization parameters under the depolarization mechanism:\({D_{\Delta e - com}}\), \(\alpha {\varepsilon _{d\Delta e - com}}\), \({R_{\Delta e - com}}\) and \(\alpha {\varepsilon _{r\Delta e - com}}\) show more clear structures compared to the images of simply combined parameters, where (1) images of \({D_{\Delta e - com}}\) of two stages show better contrast in the whole imaging depth rather than the high local contrast with the blurred remaining structural details, the display of different layers is consistent with the information of the image of simply combined parameter \({D_{1 - \operatorname{int} }}\), (2) images of \(\alpha {\varepsilon _{d\Delta e - com}}\) of two stages highlight the spot structures that can not be clearly distinguished in the image of \(\alpha {\varepsilon _{d1 - \operatorname{int} }}\), and may result from dehydration, (3) images of \(\alpha {\varepsilon _{r\Delta e - com}}\) of two stages enhance the contrast of skin surface so that the deeper structures can also be revealed better, and (4) images of \({R_{\Delta e - com}}\) of two stages show the higher contrast of the structural boundaries in deep tissues.

Therefore, \({D_{\Delta e - com}}\) can be used as an effective tool to improve and smooth the contrast throughout the imaging depth. \(\alpha {\varepsilon _{d\Delta e - com}}\) can be used as an indicator of skin dehydration. \(\alpha {\varepsilon _{r\Delta e - com}}\) is found to be sensitive to the changes of skin surface micro-structure. \({R_{\Delta e - com}}\) can be used as a recognizer for structural boundaries in deep skin tissues. The proposed combined polarization parameters under the depolarization mechanism show the potential for identifying the structures of skin tissue ex-vivo.

Here, the increase in contrast of polarization parameters should not be measured by parameter values, it should be determined by whether some micro-structure can be identified. By comparing and analyzing the two tissues with the blue arrows and boxes in the parameter graphs of depolarization characteristics considered in Fig. 2 with those polarization characteristics of the corresponding positions, it can be concluded that the parameter graphs with depolarization characteristics considered have a better ability to distinguish organizational structure details.

In addition, combined with the analysis of the tissue structure chart, it can be seen that for healthy tissues, the epidermis and internal pores can be observed within the imaging depth of the PS-OCT system. According to the scale, the thickness of the epidermal layer is consistent with the organization chart. Note that, the imaging depth of inflamed tissue was deeper than that of resting normal skin, This is because the adipose layer of healthy tissue is thicker and has exceeded the effective imaging depth of the system, resulting in the absorption of a large amount of incident light, while the adipose layer of inflammatory tissue is thinner, so that the muscle layer under it can also be revealed within the range of effective depth. The above conclusions can be used to prove the performance and accuracy of the proposed method.

It should be pointed out that the pores in the epidermal layer of skin can be clearly distinguished in the \({R_{\Delta e - com}}\) image considering depolarization instead of in the retardance image. In addition, the boundary of the layered feature can be roughly identified in the image of \({R_{\Delta e - com}}\), while it can be clearly revealed in the initial retardance image. In a word, the information of pores and the boundary can be revealed from the image of \({R_{\Delta e - com}}\), and only the boundary can be found in the initial retardance image. Therefore, the derived parameter considering depolarization can be the more comprehensive parameter. The boundaries in the retardance images show better contrast. However, it may not be the potential suppresion of the birefrignence sensitivity in \({R_{\Delta e - com}}\), more tissues of different kinds of mouse skin need to be imaged to illustrate the extent to which this parameter is affected by skin type and complete the explaination.

Note that, dehydration tissue can be regarded as a kind of thermal degeneration. Experimental results show that phase-based polarization contrast is more sensitive to thermal degeneration of biological tissues than amplitude-based contrast22. The combination of amplitude-based contrast with phase-based polarization contrast and the orientation-based contrast provides more comprehensive information about biological tissues. They imaged an ex vivo skin sample—from a rat belly—containing a burn lesion, where the calculated intensity image, the retardation image, the diattenuation image and the histological image are shown. The burn region cannot be identified in the intensity image, but it can be clearly seen with marked contrast in the retardation and diattenuation images as verified by the polarization histological image. In our work, \(\alpha {\varepsilon _{d\Delta e - com}}\) is the parameter combined by the azimuth and ellipticity characterizing the diattenuation axis. Therefore, it can be used to reveal the degeneration of skin tissue. The dot-like shape stricture is similar to the results in the literature that may have formed as a result of dehydration and detachment from surrounding tissue.

Conclusions

In conclusion, a method considering depolarization mechanism for enhancing the polarization contrast of ex-vivo mouse skin using PS-OCT is presented. 6 combined parameters obtained by the classification and combination of 14 polarization parameters derived in the two methods of the light propagation model and the exploration of the unique form of canonical decomposition are firstly introduced. By combining the expressions of depolarizing parameters \({\Delta _{1 - \operatorname{int} }}\) and \({e_{1 - \operatorname{int} }}\) to get the parameter\(\Delta - {e_{com}}\), the enhanced contrast method is obtained with the consideration of depolarization mechanism, through combining each expression of these non-depolarizing parameters given and the expression of \(\Delta - {e_{com}}\). To demonstrate the capability of the proposed method, depth-resolved images of theses parameters of ex vivo skin of mouse are reconstructed using PS-OCT. Our results show that these parameters considering the depolarization yields better polarization contrast compared to those simply combined parameters to the same area of skin tissue. Moreover, these proposed parameters are proved to be an effective tool for enhancing the local contrast and identify the local structures, providing a compelling contrast mechanism to investigate the micro-structure in ex-vivo mouse skin and overcome some limitations of the existing polarization parameters.