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Abstract
Polarization imaging can quantitatively probe the characteristic microstructural features of biological tissues non-invasively. In biomedical tissues, layered structures are common. Superposition of two simple layers can result in a complex Mueller matrix, and multi-color backscattering polarimetry can help to probe layered structures. In this work, multi-color backscattering Mueller matrix images are measured for living nude mice skins. Preliminary analysis of anisotropy parameter A and linear polarizance parameter b show signs of a layered structure in the skin. For more detailed examinations on polarization features of layered samples, we generate Mueller matrices by experimenting with two-layered thick tissues and concentrically aligned silk submerged in milk. Then we use supervised machine learning to identify polarization parameters that are sensitive to layered structure and guide the synthesis of more parameters. Monte Carlo simulation is also adopted to explore the relationship between parameters and microstructures of media. We conclude that multi-color backscattering polarimetry combined with supervised machine learning can be applied to probe the characteristic microstructure in layered living tissue samples.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Polarization imaging is a non-invasive optical method sensitive to microstructural characteristics of biological tissues [1–7]. It can quantitatively probe the characteristic microstructural features of biological specimens [8–11]. Layered structures are common to biological specimens [12]. However, superposition of layers with simple optical characteristics can result in complex Mueller matrices [13,14]. Distinguishing features of such layered structures can help to characterize microstructures of complex tissues.
Backscattering polarimetry has been applied to detect layer information of biological specimens. Images of degree of polarization have potential to identify skin cancer margins and guide surgical excision of skin cancer [15]. Gray-level co-occurrence matrices and texture features of Mueller matrix polar decomposition (MMPD) parameters can distinguish skin located in different anatomical sites [16]. These two researches can probe features of biological specimens at certain depth. Since incident light of different wavelengths can penetrate different depths in tissues [17], multi-color polarization imaging can be applied to probe layered structures. Diagonal elements of multi-color Mueller matrix (MM) have been used to probe the layers of ex vivo human colon [18]. Polarization parameters with clear physics meanings [19] may serve as even better indexes to probe optical properties of layered samples.
Machine learning methods have been applied to Mueller matrix imaging for various tasks [20–23]. In this work, multi-color backscattering polarimetry and supervised machine learning (SML) are combined to probe microstructures of layered structures. Using a backscattering MM imaging system, polarization features of living nude mice skins under five different wavelengths are measured. Anisotropy parameter A [24] and linear polarizance parameter b [24] are functions of the MM elements, A represents anisotropy degree of the fibrous structures, and b is closely related to linear polarizance of the scattering system. Experimental results of A and b clearly show different behaviors between the shorter and longer wavelengths, indicating layered structures in nude mice skins. H&E-stained slide of nude mice skin confirms the layered structure of the sample. Two-layer artificial thick tissues and a two-layer system (MS) with adjustable properties of individual layers are also constructed. Results of artificial layered samples and MS show that polarization parameters can distinguish distinctive features between layers.
Besides A, b, many other polarization parameters are ranked by SML classifiers to identify their sensitivity to the layered structure. We find out that parameters with high classification accuracy are related to anisotropy and polarizance, which indicates anisotropy and polarizance are the main differences between the two layers. Several new layered parameters based on the parameters with high classification are proposed, and some of which are even more sensitive to layered structures, such as Dtb, DtΔ and Dtk. These parameters are synthesis of polarization parameters that are sensitive to anisotropy (t1 [24]) and polarizance (b, 1-Δ [25], kc [26]). Experimental results of artificial thick tissues and MS show again that polarization parameter Dtb, DtΔ and Dtk are sensitive to the layered structures.
Monte Carlo (MC) simulations are also deployed to provide further insights on the parameters. Using the sphere-cylinder birefringence model (SCBM), which consists of spherical and cylindrical scatters buried in birefringent interstitial medium [27–29], Monte Carlo simulations show the change of polarization parameters as microstructures vary continuously. Results show that DtΔ has better performance than the others, which is applicable to a wider range and easy to calculate. DtΔ is also plotted against wavelength on nude mice skins, artificial thick tissues, MS, and MC simulations. It demonstrates promising performance in all these samples.
Therefore, the combination of multi-color backscattering polarimetry and SML can help us find new polarization parameters for probing layered structures. This method can be applied to probe microstructures in living layered biological samples, and to distinguish features between layers.
2. Method and materials
2.1 Experimental setup and samples
The experimental setup is a typical backscattering MM imaging system based on dual rotating retarder method [30–33]. Figure 1(a) shows the schematic of the setup. The light source consists of five 0.1W light-emitting diodes (LED) (Daheng Optic, China) at 475 nm, 495 nm, 525 nm, 585 nm, and 625 nm wavelengths. And the Full Width at Half Maximum (FWHM) of the LEDs are 20 nm, 20 nm, 35 nm, 15 nm, and 15 nm respectively. It is specially designed to change the wavelength of the incident lights through simple manual adjustment. Then the lights go through a polarization states generator (PSG), which consists of a lens (L1, Daheng Optic, China), a polarizer (P1, extinction ratio>1000:1, Daheng Optic, China) and a quarter-wave plate (R1, Wuhan Union Optic, China) fixed on a rotation stage (Thorlabs, PRM1Z8, USA). The lens collimates the diverging light from the LED, the linear polarizer and the rotatable quarter-wave plate are used to control the polarization states of the lights. The diameter of the circular illumination area is about 0.02 m, and irradiance level on the sample surface is about 4.5W/m2. The backscattered photons from the sample pass through a polarization states analyzer (PSA), which consists of the same type of polarizer (P2), quarter-wave plate (R2) and rotation stage as in the PSG and are collected by a lens (L2). The photons are recorded by a CCD camera (QImaging 32–0122 A, 12 bits, Canada). There is an oblique angle between the illumination and detection directions to avoid surface reflections by the samples. During the measurements, the polarizers (P1, P2) are fixed in the horizontal positions and the two quarter-wave plates (R1, R2) rotate at a fixed rate of 1:5 [30]. For different combinations of the PSG and PSA, 30 polarization component images are recorded corresponding to different polarization states of the incident and scattered lights. Then the MM of the sample is calculated from the 30 intensity images using the standard algorithm [33]. The experimentally obtained Mueller matrices are calibrated via an instrumentation matrix which is measured by measuring the Mueller matrices of the standard samples such as air, polarizers and waveplates [30,31,33]. Maximum errors among the matrix elements of air are less than 1%. The setup resolution is higher than 62.5μm.
The living samples are eight 11-week-old nude mice divided into two groups. One group of the nude mice are normal (NM-1) and the other group are injected with cells under their upper epidermis (NM-2), as shown in Fig. 1(b) and Fig. 1(c). The red squares mark approximately the region of interest (ROI) on the back of nude mice. The mice are sourced from Guangdong Medical Lab Animal Centre, Guangzhou, Guangdong, China. The nude mice are injected with chloral hydrate (10%) to make sure they remain stationary during the measurement. The use of the nude mice in the experiments is approved by the Administrative Committee on Animal Research of the Graduate School at Shenzhen, Tsinghua University. All the operations are performed in accordance with the applicable guidelines and regulations. Figure 1(j) is microscopic image of H&E-stained slide of normal nude mice skin.
Fig. 1. Schematic of experimental setup for multi-color backscattering MM measurements, nude mice specimens and other samples. (a) Schematic of experimental setup. Light Source: light-emitting diodes of different wavelengths; L1, L2: lens; P1, P2: polarizer; R1, R2: quarter-wave plate. The oblique incident angle θ is approximately 18° to avoid the surface reflection from the sample. The diameter of the illumination area is approximately 2 cm. (b) Normal nude mice (NM-1). (c) Nude mice with cells mass injected under their upper epidermis (NM-2). Red squares show the region of interest (ROI). (d) Porcine liver. (e) Porcine fat. (f) Chicken heart muscle. (g) Beef tendon. (h) Adjustable layered sample, concentrically aligned silk immerged in isotropic scattering media. (i) Concentrically aligned silk fiber disk. (j) Microscopic image of H&E-stained slide of normal. nude mice skin.
To verify that polarization parameters can probe layered structures, two-layered artificial tissue samples are composed as phantoms. Tissues of low anisotropy: porcine liver Fig. 1(d), porcine fat Fig. 1(e), and tissues of high anisotropy: chicken heart muscle Fig. 1(f), beef tendon Fig. 1(g), are selected. These tissues are randomly combined as artificial layered samples (TS1, TS2, TS3, TS4), whose upper layers have low anisotropy and lower layers have high anisotropy [34]. TS1 and TS2 are composed of porcine liver as upper layer and chicken heart as lower layer. The only difference between TS1 and TS2 is the thickness of the upper layer. The thickness of the upper layer of TS1 is approximately 0.1 mm, and of TS2 is approximately 0.6 mm. Thicknesses of both chicken heart layers are approximately 8 mm. TS3 and TS4 are composed of porcine fat as upper layer and beef tendon as lower layer. The thickness of the upper layer of TS3 is approximately 0.1 mm, and of TS4 is approximately 0.6 mm. Thicknesses of both beef tendon layers are approximately 8 mm.
For further study on layered structures, a layered adjustable sample MS consisting of two layers of different anisotropy is designed as shown in Fig. 1(h). The upper layer is fresh milk as the nearly isotropic scattering media with variable scattering coefficient and thickness, and the lower layer is concentrically winded silk fibers as the anisotropic fibrous structures. The silk fibers are sealed between two coverslips to prevent its optical properties from being affected by the surrounding liquid shown as Fig. 1(i). Both scattering coefficient and thickness of the isotropic layer can be changed.
Samples are summarized in the following table 1.
Table 1. Samples used in this study
2.2 Mueller matrix transformation (MMT) parameters and Mueller matrix polar decomposition (MMPD) parameters
MM can characterize the comprehensive polarization properties of the medium. However, individual MM elements often fail to reveal explicit relationship with specific microstructural features. Polarization parameters derived from MM are proposed to help characterizing the sample’s microstructural properties [19]. MMT parameters and MMPD parameters are adopted to analyze the experimental data quantitatively in this work.
MMT parameters are functions of the MM elements. Parameter A and b are often used in our previous works [11,24,34,35]. A [24] represents anisotropy degree of the fibrous structures, and b [24] represents linear polarizance of the scattering system, which is related to the size and shape of scatterers in the system. Besides anisotropy parameter A and linear polarizance parameter b, there are also many other polarization parameters available to characterize the microstructural features of the samples. Parameter t1[24], PL, DL [36], qL, and rL [26] are related to anisotropic properties from different origins. Parameter α1 [35], αP, αD, αq, and αr[26] are related to azimuth angles of these anisotropy, while α1 has smaller periods but better signal to noise ratio [35]. Parameter kC [26] is related to polarizance or depolarization. Parameters PC, DC are related to circular polarization and circular diattenuation [36]. A closer examination revealed that A consists of contributions from parameter t1 and parameter b [37].
MMPD parameters are proposed by Lu and Chipman [38]. MM can be decomposed to obtain parameters with clear physics meanings, i.e., depolarization (Δ), linear retardance (δ), retardance (R), diattenuation (D), orientation of fast axis (θ) and optical rotation (φ).
Expressions of the parameters are shown in Table 2.
Table 2. Polarization parametersa
2.3 Supervised machine learning (SML) algorithms
SML is used in this work to evaluate all the polarization parameters mentioned above in pixel level. In order to reduce the impact of a single algorithm on the results, six mature and basic SML algorithms are adopted [39], i.e., decision tree (DT, split criterion is Gini's diversity index), linear discriminant analysis (LDA), logistic regression (LR), naive Bayesian model (NBM, kernel is Gaussian), support vector machine (SVM, kernel function is linear) and k-nearest neighbors (KNN, distance is Euclidean), to estimate capabilities of parameters probing layered structures. Values of parameters from ROI at different incident wavelengths are fed into the classifiers as features of the instances. We label the instances by whether the incident light can penetrate into the lower layer and detect information of lower layer (yes or no). The way to differentiate penetration characteristic of incident lights will be introduced in the following sections.
To validate the result of SML, 10-fold cross-validation method is used [40]. Data is randomly divided into 10 sets, with one set is selected as test set and the other 9 sets are combined as the training data, so that the test instances do not overlap with the instances used in the training set. This procedure is repeated for each of the 10 sets, and the average of the classification accuracies (number of correctly classified instances/total number of instances) is the final result. The classification accuracies of the parameters show their capabilities to differentiate superficial layer and two layered structures.
2.4 Monte Carlo simulations
The basic idea of Monte Carlo simulation method is to construct a probability model or random process, which corresponds to the physical process to be solved. A large number of random process calculations are performed by generating random numbers in line with the distribution of variables in reality. The expected value of the random variable is obtained by statistical average. Monte Carlo simulations for the propagation of polarized photons is to simulate the scattering process of light in a turbid medium, which can track the trajectories and polarization states of the photons as they propagate through the media [41].
In this work, Monte Carlo (MC) simulations based on the sphere-cylinder birefringence model (SCBM) [27–29] is adopted to understand the relationship between the polarization parameters and the properties of microstructures. SCBM contains scatters and birefringent interstitial medium. Spherical scatters and infinitely long cylindrical scatters are used to approximate the microstructures in tissues, such as cell nucleus and organelles, collagens and muscle fibers. Properties of the layered structures and illumination conditions can be easily changed in MC simulations, such as the diameters and scattering coefficients of the scatterers, refractive indices and the birefringence of the scatters and the medium, the wavelengths and incident angle of the incident light.
3. Results
3.1 Parameters A and B probing layered structures
Using the multi-color backscattering MM imaging system, MM of living nude mice skins at five different wavelengths are measured. Anisotropy parameter A and polarizance parameters b derived from MM are plotted as functions of the incident wavelengths on living nude mice skin, shown as Fig. 2. The horizontal axis is wavelengths of illuminating lights, and the vertical axis is value of the parameters. The mean values of parameters are calculated from all samples from each group, and the error bars represent a distribution of mean values from measurements on individual mice from each group.
Fig. 2. Anisotropy parameter A (a) and linear polarizance parameter b (b) of NM-1 and NM-2 at different illuminating wavelengths. The mean value of parameters is calculated from all samples from each group, and the error bars represent a distribution of mean values from measurements on individual mice from each group.
In Fig. 2(a), value of A increases with illuminating wavelength of NM-1, while value of A of NM-2 stays almost the same when illuminating wavelength is shorter than 525 nm. In Fig. 2(b), the value of b decreases with illuminating wavelength of NM-1, while there is a stable stage of NM-2 when illuminating wavelength is shorter than 525 nm. This may because there is a mass of cell under upper epidermis of NM-2. Here come the assumptions: 1. layered structures of nude mice skin leads to the different behaviors of the parameters. 2. the upper layer of NM-2 is thicker than NM-1, or the scattering coefficient of NM-2 is bigger than NM-1. Figure 1(j) is another supporting evidence shows that nude mice skin has layered structures. The upper layer of skin is epidermis, which is almost isotropic [42]. And below the epidermis is dermis, which contains collagen and elastic fibers, having higher anisotropy than epidermis [43,44].
Experimental results in Fig. 2(a) and Fig. 2(b) can be interpreted as: in Fig. 2(a), NM-1 contains two layers of different anisotropies. The lower layer has higher anisotropy than the upper layer. All the lights used in the experiments penetrates into the lower layer, so there is no inflection of the spectral curves. When the illuminating wavelength of the light is longer and the light penetrates deeper, characteristics of the lower layer contribute more to the value of the parameter. The upper layer of NM-2 is thicker, or the scattering coefficient is bigger than NM-1. When the illuminating wavelength is shorter than 525 nm, incident light cannot penetrate to the lower layer in NM-2, resulting in the value of A staying almost the same. Phenomenon of parameter b varying with wavelength in Fig. 2(b) also can be interpreted by nude mice skin containing two layers and the upper layer of NM-2 is thicker, or the scattering coefficient is bigger than NM-1. Depolarization of these two layers is different and the upper layer induces less depolarization than the lower layer.
Figure 2 shows that anisotropy and polarizance are different between the two layers of nude mice skin. To verify that polarization parameter can probe layered structures, artificial two-layer biological samples are composed as phantoms. Tissues with distinctive structural properties are selected based on our previous work [34], then combined randomly as TS1, TS2, TS3 and TS4. Detail of the samples has been demonstrated in Sec. 2.1. The most apparent difference between the two layers of phantoms is anisotropy. Anisotropy parameter A varying with wavelength of TS1 and TS2 are shown in the Fig. 3(a), and anisotropy parameter A varying with wavelength of TS3 and TS4 are shown in the Fig. 3(b).
Fig. 3. Anisotropy parameter A of manual layered structures at different illuminating wavelengths. (a) The upper layer is porcine liver and the lower layer is chicken heart. Upper layer of TS2 is thicker than TS1. (b) The upper layer is porcine fat and the lower layer is beef tendon. Upper layer of TS4 is thicker than TS3. The mean values are calculated from five rounds of experiments, and the error bars represent a distribution of mean values from measurements on each tissue sample.
Results of artificial two-layer tissue samples are similar to that in nude mice skin, which demonstrate that when the upper layer of the sample is thick enough, incident lights of shorter wavelength cannot penetrate into the lower layer, value of parameter stays stable. Thus, multi-color backscattering polarimetric is a powerful instrument to detect layered structures quantitatively.
For further study on layered structures, an adjustable two-layer sample MS is designed, which consists of concentrically aligned silk as the lower layer and isotropic scattering media as the upper layer. Both scattering coefficient and thickness of the isotropic layer can be adjusted easily by changing the concentration or thickness. The most apparent difference between the two layers is anisotropy. Series experiments have been taken on MS and the behavior of parameter A is observed to probe microstructures of the sample. Figure 4(a) shows how the average values of A varies with wavelengths for different concentrations of isotropic layer, and the thickness of the upper layer is fixed as 2.0 cm. The A-λ curves vary almost linearly with wavelengths when the concentration is 0.10%. And when scattering coefficient of the upper layer is high enough so that the photons cannot reach the lower layer, platforms appear at the short wavelength ends. The width of the platform of constant A increases with increasing scattering coefficient of the upper layer. Figure 4(b) shows how the average values of A varies with wavelengths when the concentration of the upper layer is 0.50% but vary its thickness from 0.5 cm to 2.0 cm. When the thickness of the upper layer is 0.5 cm or 1.0 cm, A varies almost linear with wavelength. When the thickness increases, parameter A stays unchanged at short wavelengths when the photons cannot reach the lower layer, and increase at longer wavelengths once photons reach the lower layer.
Fig. 4. Anisotropy parameter A of MS at different illuminating wavelengths. (a) Concentration of the dilute milk varies from 0.10% to 0.30%, and the thickness of milk is 2.0 cm. (b) Thickness of dilute milk varies from 1.0 cm to 2.0 cm, and the concentration of milk is 0.50%.
From the experimental results of adjustable two-layer sample, here comes a conclusion that the width of platform of constant A appearing at the short wavelength ends has a positive correlation with the thickness or scattering coefficient of the upper layer.
3.2 Other parameters capable of probing layered structures
Other parameters besides A and b are fed into the classifiers mentioned in Sec. 2.3 to figure out their ability of probing layered structures. Based on the result of the classification accuracy of SML, new parameters are synthesis by parameters with high ranking.
Results of NM-2 in Fig. 2 show a sharp turn at approximately 525 nm in both parameters, indicating a crossing of the layer boundary around this wavelength. Value of polarization parameters of illuminating wavelength at 475 nm and 495 nm are labeled as one group whose incident light only penetrates into the upper layer, and data of illuminating wavelength at 585 nm and 625 nm can be labeled as the other group whose incident light penetrates into the lower layer. The instances with different labels are equal in quantity, so there is no bias. Polarization parameters mentioned above are calculated from Mueller matrices from NM-2. The size of ROI is $150 \times 150$ pixels. To reduce noise, we take average values of parameters in $10 \times 10$ pixels as one set of data. There are 900 sets of data with different illuminating wavelength in total.
In the same manner, MM elements of NM-2 are also fed into the SML classifiers mentioned above. Results of best 10 elements are shown in Fig. 5(a). Element m34 is on the right edge of the MM, parameter rL and ${\alpha _r}$ are relevant to right edge. Elements m22 and m33 are on the diagonal line, parameter A, b, t1 and ${\alpha _1}$ are relevant to m22 and m33. Elements m44 and m14 are on the corners, m44 is kC and m14 is DC. Polarization parameters are fed into the SML classifiers. Results of best 10 parameters are shown in Fig. 5(b). We can see that the top five parameters are better than the best element. Classification accuracy of A is above 90%. Classification accuracies of rL, Δ, b and kC are above 80%. Results in Fig. 5(a) and Fig. 5(b) show the consistence between the MM elements and polarization parameters, and the superiority of parameters when probing layered structures.
Fig. 5. Classification accuracy of MM elements and parameters. Dots of different shapes or colors represent accuracy of different classifiers. Bar column represent average accuracy of all the classifiers. (a) MM elements to differentiate nude mice data. (b) Existing parameters to differentiate nude mice data. (c) New polarization parameters to differentiate nude mice data. (d) New polarization parameters to differentiate MS experiment data. (e) New polarization parameters to differentiate TS2 data. (f) New polarization parameters to differentiate TS4 data.
A and rL represent anisotropy. (1-Δ) represents total polarizance, b and kC represent linear and circular polarizance respectively. The results suggest that anisotropy and polarizance are the key differentiating characteristics between two layers of the nude mice skin. The best parameter A is derived from the anisotropy parameter t1 normalized by polarizance parameter b [37], which contains polarizance information besides anisotropy. Inspired by the calculation of A, we attempt to propose some new polarization parameters containing both anisotropy information and polarizance information. New parameters are derived from anisotropy parameter normalized by polarizance parameter, shown as followings,
Figure 5(c) shows the classification accuracy of new parameter Dtb is higher than A, and parameters Dtb, DtΔ and Dtk are higher than rL in nude mice skin.
While the concentration of milk in MS is 0.50% and the thickness of milk is 2.0 cm, almost the same phenomenon as NM-2 in Fig. 1(a) of A can be observed. Data of this set of MS experiments are fed into the classifiers as procedure of NM-2. Anisotropy parameter A is the best polarization parameter to probe layered structures in MS. New parameters in MS experiments are calculated, and are fed into the classifiers. Figure 5(d) shows that classification accuracy of parameters Dtk, Dtb, DtΔ are almost the same with that of A.
Dtb, DtΔ, Dtk can probe layered structures in nude mice skin, and still perform well in MS. The results indicate that the three parameters can probe layered structures, especially when anisotropy and polarizance are different between layers. New parameters of TS2 and TS4 are also fed into the classifiers. The results are shown in Fig. 5(e) and Fig. 5(d). Dtb, DtΔ and Dtk still work well in these samples.
Besides classification accuracy between data of short wavelengths and long wavelengths, classification accuracy between NM-1 and NM-2 at different wavelengths is also calculated. The results show that the best classification examples correspond to wavelength 525 nm. When wavelength is shorter than 525 nm, NM-1 contains information of the upper layer and the lower layer, and NM-2 contains information of upper layer only. When the wavelength is longer, the information of lower layer of NM-1 contributes more, and the classification accuracy between NM-1 and NM-2 is higher. When wavelength is longer than 525 nm, NM-1 and NM-2 both contains information of the upper layer and the lower layer. When the wavelength is longer, the information of upper layer of NM-2 contributes less, and classification accuracy between NM-1 and NM-2 is lower.
3.3 Validation of new parameters
Now that new parameters have been proposed via SML. In this part, we adopted MC simulations to analyze these parameters.
Microstructures are changed in the SCBM and the value of Dtb, DtΔ and Dtk are plotted, while value of A is plotted for reference. Figure 6(a) shows that A, Dtb, DtΔ and Dtk increase almost linearly with diameter of sphere scatterers. We do linear polynomial least square fitting [45] on these parameters separately, coefficient of determination R2 (R2∈[0,1], and R2=1 means entire fitting) of A, Dtb, DtΔ and Dtk are 0.950, 0.986, 0.997 and 0.990. R2 of DtΔ is larger than the other three parameters, which means DtΔ is easier to be predicted, making it more suitable for probing microstructures. Figure 3(b) shows that A, Dtb and Dtk are not stable when diameter of cylinder scatterers changes, only DtΔ is stable and monotone decreasing when the diameter of cylinder scatterers is bigger than 0.3 μm. Figure 5(c) shows that A is unchanged with σ when cylinder scatterers are well aligned. We do linear polynomial fitting using least square approximant on Dtb, DtΔ and Dtk separately, the R2 are 0.951, 0.964 and 0.938. DtΔ is more suitable for probing microstructures. Distribution of cylinder scatterers follows Gaussian distribution in SCBM. σ is standard deviation of variance of the distribution. Figure 5(d) shows that A remains unchanged when scattering coefficient of cylinder scatterers changes, Dtb is irregular, while DtΔ and Dtk is monotone increasing. R2 of DtΔ and Dtk are 0.842 and 0.861.
Fig. 6. A, Dtb, DtΔ and Dtk vary with change of microstructures. (a) parameters vary with diameter of sphere scatterers. (b) parameters vary with diameter of cylinder scatterers. (c) parameters vary with σ of cylinder scatterers. (d) parameters vary with scattering coefficient of cylinder scatterers.
Results are concluded in the following table 3.
Table 3. Parameters probing microstructures (monotonous for Yes, non-monotonous for No)
DtΔ is monotonous and can be used to probe samples quantitatively. Results of MC simulation show that DtΔ is better than A, Dtb, and Dtk. Dtb and A are derived from anisotropy parameter t1 and linear polarizance parameter b. Dtk is derived from anisotropy parameter t1 and circular polarizance parameter kC. And DtΔ is derived from t1 and total polarizance parameter (1-Δ). Biological samples contain rich and complex microstructural information. DtΔ can probe microstructures in more general situations than A, Dtb, and Dtk, for containing information of both linear and circular polarizance. When the sample is layered, DtΔ is more capable of probing difference between layers in multi-color backscattering polarimetry. We can conclude that DtΔ is a better polarization parameter for probing layered structures. DtΔ can be obtained from elements of MM as shown in the following,
Values of DtΔ are plotted against wavelength of different samples, and shown in the following figure. Figure 7(a) shows average values of DtΔ varying with wavelengths of nude mice skins. Figure 7(b) shows average values of DtΔ of two-layered SCBM in MC simulations. The upper layer contains sphere scatterers, and the lower layer contains cylinder scatterers. Figure 7(c) shows average values of DtΔ of TS1 and TS2. Figure 7(d) shows average values of DtΔ of TS3 and TS4. Figure 7(e) shows average values of DtΔ of MS, concentration of the diluted milk varies from 0.10% to 0.30%. Figure 7(f) shows average values of DtΔ of MS, thickness of diluted milk varies from 1.0 cm to 2.0 cm. The results show that DtΔ has ability to probe microstructures in layered structures quantitatively.
Fig. 7. Parameter DtΔ at different illuminating wavelengths of different samples. (a) nude mice skins. (b) MC simulations. The upper layer contains sphere scatterers, and the lower layer contains cylinder scatterers. (c) TS1 and TS2. (d) TS3 and TS4. (e) MS, concentration of the diluted milk varies from 0.10% to 0.30%. (f) MS, thickness of diluted milk varies from 1.0 cm to 2.0 cm.
4. Conclusions
Polarization parameters of multi-color backscattering Mueller matrix (MM) have potential to probe optical properties of layered samples. Anisotropy parameter A and linear polarizance parameter b in multi-color backscattering polarimetry show apparently different behaviors at the shorter and longer wavelengths on living nude mice, indicating layered structures in nude mice skins. The experimental data shows that the lower layer of nude mice skin has higher anisotropy and induces more depolarization than the upper layer. Tissues of different anisotropy are combined as artificial layered biological phantoms, and the results confirmed that polarization parameters can probe layered structures. For further study, a two-layer adjustable sample MS is designed, which consists of concentrically aligned silk and isotropic diluted milk. Scattering coefficient and thickness of diluted milk can be changed easily. Series of experiments are carried out and the results show that parameter A can probe the thickness and the scattering coefficient of the upper layer of MS quantitatively. Results of living nude mice skin and other artificial layered samples show that polarization parameters varying with illuminating wavelengths have potential to quantitatively characterize microstructures of layered sample.
To find more parameters capable of probing layered structures besides A and b, parameters are evaluated by supervised machine learning (SML). Results of MM elements and polarization parameters of nude mice skin experiments demonstrate that polarization parameters are better indexes to probe layered structures than MM elements. Anisotropy and polarizance are the most prominent differentiating characteristics between the two layers of nude mice skin. Based on this study, some new parameters relevant to anisotropy and polarizance are proposed. Three of these new parameters, DtΔ, Dtb and Dtk, are capable of probing nude mice skin and artificial layered samples.
MC simulations are introduced to evaluate the parameters A, DtΔ, Dtb and Dtk. Results of MC simulations show that DtΔ is a better parameter to probe layered structures in more general situations since it varies monotonously when microstructures change continuously. Besides, DtΔ contains information of anisotropy, linear polarizance and circular polarizance, which contains more information than other parameters. To validate the function of DtΔ, its values are plotted against wavelength of different samples. Results show that DtΔ in multi-color backscattering polarimetry can probe layered structures, which makes extracting information from MM more effectively.
In this work, we combine multi-color backscattering polarimetry with SML, and propose an approach to find new polarization parameters which are capable of probing layered structures. This technique paves the way for probing microstructures in layered living tissue samples.
Funding
National Natural Science Foundation of China (11974206, 61527826); Shenzhen Fundamental Research Program (JCYJ20170412170814624).
Acknowledgments
The authors are also thankful to Pengcheng Li and Ruoyu Meng from the Tsinghua Shenzhen International Graduate School for the helpful discussions.
Disclosures
The authors declare no conflict of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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