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Analysis and reduction of noise-induced depolarization in catheter based polarization sensitive optical coherence tomography

Qingrui Li, Yin Yu, Zhenyang Ding, Fengyu Zhu, Yuanyao Li, Kuiyuan Tao, Peidong Hua, Tianduo Lai, Hao Kuang, and Tiegen Liu

Open Access Open Access

Abstract

In catheter based polarization sensitive optical coherence tomography (PS-OCT), a optical fiber with a rapid rotation in the catheter can cause low signal-to-noise ratio (SNR), polarization state instability, phase change of PS-OCT signals and then heavy noise-induced depolarization, which has a strong impact on the phase retardation measurement of the sample. In this paper, we analyze the noise-induced depolarization and find that the effect of depolarization can be reduced by polar decomposition after incoherent averaging in the Mueller matrix averaging (MMA) method. Namely, MMA can reduce impact of noise on phase retardation mapping. We present a Monte Carlo method based on PS-OCT to numerically describe noise-induced depolarization effect and contrast phase retardation imaging results by MMA and Jones matrix averaging (JMA) methods. The peak signal to noise ratio (PSNR) of simulated images processed by MMA is higher than about 8.9 dB than that processed by JMA. We also implement experiments of multiple biological tissues using the catheter based PS-OCT system. From the simulation and experimental results, we find the polarization contrasts processed by the MMA are better than those by JMA, especially at areas with high depolarization, because the MMA can reduce effect of noise-induced depolarization on the phase retardation measurement.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Catheter based polarization sensitive optical coherence tomography (PS-OCT) is a powerful technique to measure polarization properties of the vessel wall and offers characterization of coronary atherosclerotic lesions beyond the cross-sectional image of arterial microstructure available to conventional intensity based OCT [1]. A notable challenge in the catheter based PS-OCT is how to reduce polarization properties variations owing to a rapid rotating optical fiber in the catheter. The polarization determination method is need to measure polarization information of the sample with immunity of polarization properties variation of optical fiber based PS-OCT systems. Several methods determining the phase retardation of samples are widely used in the catheter based PS-OCT including Stokes vectors rotation angle method [2–5], differential Mueller matrices method [6–8], single input polarization state method [9–11]. A polarization symmetry constraint method was used to recover optic axis orientation in the catheter based PS-OCT [12]. Here, the matrix similarity method is a widely used and accepted polarization determination method for PS-OCT. For example, Jones matrix diagonalization (JMD) method performs eigenvalue decomposition of Jones matrix to acquire phase retardation of the sample based on matrix similarity [13–23]. Wang et al. [19] and Li et al. [21] also proposed catheter based PS-OCT systems using the JMD method. In addition, several de-noise methods have been attempted for JMD [20,22,24,25]. We previously proposed a similar Mueller matrix (SMM) method for polarization determination in the catheter based PS-OCT [26]. The experimental results verify that the SMM method can provide a better imaging quality of phase retardation for biological tissue. However, the advancements of the SMM method compared with the JMD method are not theoretically illustrated in [26].

Depolarization is defined by the randomization of the incident polarization state by processes associated with scattering, diattenuation and retardation, which can vary in space, time and/or wavelength. Depolarization is the result of incoherent superposition of pure, fully polarized states, leading to a mixed state [27,28]. The random change of polarization states is not only dependent on the tissue property, but is also significantly impacted by non-tissue factors, such as the signal-to-noise ratio (SNR), namely noise-induced depolarization [29]. In the catheter based PS-OCT, light travels through an optical fiber with a rapid rotation in the catheter, which causes low SNR, polarization states instability, phase change of PS-OCT signals. These factors will produce heavy noise-induced depolarization, which has a strong impact on the phase retardation measurement of the sample. Jones matrix averaging does not output any depolarization effect. In contrast, Mueller matrix averaging results in partially depolarizing Mueller matrix. Because of inherent nature of interferometric and phase-resolved detection in our PS-OCT, only Jones matrices can be acquired directly [30,31]. Mueller matrices for SMM method are converted from Jones matrices, namely Mueller-Jones matrix, in which process does not alter any of the information. Whereas, when a spatial kernel averaging is applied to these measured Mueller matrices, the noise on these measured Mueller matrices after averaging will be reduced. SMM with real field averaging is called as Mueller matrix averaging (MMA) method. More importantly, measured Mueller matrices after spatially averaging carry depolarization, which can be removed by the polar decomposition. JMD also can be applied by a complex field averaging to reduce noise [21,32]. JMD with complex field averaging is called as Jones matrix averaging (JMA) method. Since Jones matrix can treat only coherent phenomenon, JMA does not output any depolarization effect. Yamanari et al. developed a better birefringence imaging quality using an averaged 4 × 4 covariance matrix that is converted from the measured Jones matrix and a Cloude-Pottier decomposition than JMA [33]. Compared with the method presented by Yamanari et al., the presented MMA also has a similar processing including Mueller matrix averaging and decomposition. Compared with the JMA method, MMA method has a potential to reduce influence of noise-induced depolarization on the phase retardation measurement. Namely, MMA has a potential to reduce impact of noise on phase retardation mapping.

In this paper, we analyze the noise-induced depolarization and find that noise-induced depolarization of the catheter based PS-OCT can be reduced by Mueller matrix polar decomposition after incoherent averaging in the MMA method. The degree of polarization uniformity (DOPU) is a widely-used parameter to evaluate depolarization in the PS-OCT with only two orthogonal polarization channels [34,35]. We analyze the relationship between DOPU and noise and verify that noise can induce serious depolarization. We also analyze why non-depolarization Mueller matrices after spatially averaging can carry depolarization. We present a Monte Carlo method based on PS-OCT to numerically describe noise-induced depolarization effect and contrast phase retardation imaging results by MMA and JMA methods. Based on simulation results, the polarization contrasts processed by MMA is better than those by JMA, because MMA can reduce effect of noise-induced depolarization. The peak signal to noise ratio (PSNR) of simulated images processed by the MMA method is higher than about 8.9 dB than that processed by the JMA method. We also compare experimental results of multiple biological tissues including ex vivo biological tissues including chicken breast muscle, human nails and porcine cardiac blood vessel processed by JMA and MMA methods using the catheter based PS-OCT system. We find that the polarization contrast of MMA is much clearer than that of JMA method at areas with low DOPU, which further verifies the MMA method can reduce effect of noise-induced depolarization on the phase retardation measurement.

2. Principle

2.1 Noise-induced depolarization

We firstly analyze the relationship between DOPU and noise. An additive noise model is usually used to analyze measured Jones matrices and $J(z)$ can be expressed as [32]:

$$J(z) = \left[ {\begin{array}{*{20}{c}} {{H_1}}&{{H_2}}\\ {{V_1}}&{{V_2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{H_{01}} + \Delta {H_1}}&{{H_{02}} + \Delta {H_2}}\\ {{V_{01}} + \Delta {V_1}}&{{V_{02}} + \Delta {V_2}} \end{array}} \right],$$
where H1,2 are the electric fields of horizontal polarized channel containing an original term ${H_{01,2}}$ and a noise term ΔH1,2, V 1,2 is the electric field of vertical polarized channel containing an original term ${V_{01,2}}$ and a noise term ΔV1,2. This PS-OCT system needs two different input states of polarization (SOPs). Here the subscript 1, 2 represent SOP1 and SOP2. Here we analyze any one of the two SOPs and assume that H and V of SOP 1 or 2 can be expressed as:
$$\left[ {\begin{array}{*{20}{c}} H\\ V \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{H_0} + \Delta H}\\ {{V_0} + \Delta V} \end{array}} \right],$$
where ΔH and ΔV are complex-valued and follow the zero-mean Gaussian distribution with the variance of $\sigma _H^2$ and $\sigma _V^2$ respectively [15,20,29]. $SN{R_H}$ and $SN{R_V}$ represent the SNR at the two channels H and V respectively.
$$SN{R_H} = \frac{{{{|H |}^2}}}{{\sigma _H^2}},\;\;\;\;\;SN{R_V}\textrm{ = }\frac{{{{|V |}^2}}}{{\sigma _V^2}}.$$

Stokes vectors of SOP1 or SOP2 S converted from [H, V], which can be expressed as:

$${\textbf S} = \left[ {\begin{array}{*{20}{c}} {{S_0}}\\ {{S_1}}\\ {{S_2}}\\ {{S_3}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {HH\mathrm{\ast } + VV\mathrm{\ast }}\\ {HH\mathrm{\ast{-}\ }VV\mathrm{\ast }}\\ {HV\mathrm{\ast{+}\ }VH\mathrm{\ast }}\\ {j \cdot (HV\mathrm{\ast{-}\ }VH\mathrm{\ast )}} \end{array}} \right],$$
where ${S_0}$, ${S_1}$, ${S_2}$, ${S_3}$ are the four elements of S, also functions of H and V. $\mathrm{\ast }$ represents the conjugate operation. DOPU is the degree of polarization (DOP) within an averaging window or kernel of the spatial domain. The average Stokes vector $\overline {\textbf S}$ in a n-pixel kernel can be
$$\overline {\textbf S} = \frac{1}{n}\sum\limits_{i = 1}^n {{{\textbf S}_i}} ,$$
where $\overline {\textbf S}$ represents the averaging operation for four elements of S. DOPU is calculated based on Eq. (2) to (5), which can be expressed as
$$DOPU = \frac{{\sqrt {{{\overline {{S_1}} }^2} + {{\overline {{S_2}} }^2} + {{\overline {{S_3}} }^2}} }}{{\overline {{S_0}} }} = \frac{{\sqrt {{{(\overline {{{|H |}^2}} \textrm{ + }\overline {{{|V |}^2}} )}^2} - \textrm{4}K} }}{{\overline {{{|H |}^2}} \textrm{ + }\overline {{{|V |}^2}} }},$$
where
$$K\textrm{ = }\sum\limits_{{q_1} = 0}^2 {\sum\limits_{{q_2} = 0}^2 {\sum\limits_{{q_3} = 0}^2 {{\mu _{{q_1},{q_2},{q_3}}}{{\overline {SN{R_V}^{ - 1}} }^{{q_1}}}} } {{\overline {SN{R_V}^{ - 1}} }^{{q_2}}}{{\overline {SN{R_H}^{ - 1}SN{R_V}^{ - 1}} }^{{q_3}}}} .$$

Here ${\mu _{{q_1},{q_2},{q_3}}}$ are constants as functions of ${H_0}$ and ${V_0}$. q1, q2, q3 are 0, 1, 2. Based on Eq. (7), we find that DOPU is as a function of K, and the value of K is depended on the randomness of H and V during averaging. When and only when all [H, V] within the kernel are equal to each other, K = 0 and DOPU = 1. Based Eq. (1), the noise ΔH and ΔV contribute to a Gaussian distribution, finally causes a decreasing on DOPU. To intuitively observe the relationship between DOPU and noise, we apply a numerical evaluation to Eq. (6) shown in Fig. 1. Here we set ${H_0}$ and ${V_0}$ equal to 1. We calculate the DOPU at different the SNR’s reciprocal of H and V channel namely $\overline {SNR_H^{ - 1}}$ and $\overline {SNR_V^{ - 1}}$. Here $\overline {SNR_H^{ - 1}} \in [\textrm{0},\textrm{10}]$, $\overline {SN{R_V}^{\textrm{ - 1}}} \in [\textrm{0},\textrm{10}]$. In Fig. 1, DOPU decreases from 1 along with $\sigma _H^2$ and $\sigma _V^2$ rising, namely $\overline {SNR_H^{ - 1}}$ and $\overline {SNR_V^{ - 1}}$ rising. Figure 1 verifies that noise can induce serious depolarization and we will discuss how to eliminate depolarization.

 figure: Fig. 1.

Fig. 1. Numerical evaluation of relationship between DOPU and the SNR’s reciprocal of H and V channel namely $\overline {SNR_H^{ - 1}}$ and $\overline {SNR_V^{ - 1}}$ based on Eq. (6). Here $\overline {SNR_H^{ - 1}} \in [\textrm{0},\textrm{10 }]$, $\overline {SN{R_V}^{\textrm{ - 1}}} \in [\textrm{0},\textrm{10}]$.

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2.2 Analysis and reduction of depolarization in spatially averaged SMM method

The JMD method is a widely used and accepted polarization determination method, which performs eigenvalue decomposition of Jones matrix to acquire phase retardation of the sample based on matrix similarity [13–22]. However, Jones matrix is non-depolarizing and only contains birefringent retardation and diattenuation, so depolarization cannot be extracted from any Jones matrix. In the catheter based PS-OCT, optical fiber with a rapid rotation in the catheter can cause heavy noise-induced depolarization, which has a strong impact on the phase retardation measurement of the sample. Learning the idea of the JMD method, we proposed a SMM method for polarization determination in the catheter based PS-OCT [26]. Since only two orthogonal polarization channels are used to acquire data in the presented catheter based PS-OCT, acquired Jones matrices need to be converted to Mueller matrices for SMM method, in which process does not alter any of the information. Whereas, if we apply a spatial kernel averaging to measured Mueller matrices namely SMM is converted to MMA, measured Mueller matrices after spatially averaging carry depolarization. Depolarization can be removed by the polar decomposition in MMA method. We will analyze why Mueller matrices after spatial averaging can carry depolarization and how to reduce depolarization by MMA method in detail.

In the MMA method, we can transform measured Jones matrix $J(z)$ in Eq. (1) into Mueller-Jones matrix $M(z)$ by this conversion formula as [35]:

$$M(z) = U[J(z) \otimes J{(z)^ \ast }]{U^{ - 1}},$$
where U is Pauli matrix and ${\otimes}$ is Kronecker product sign. Based on Eq. (8), since the matrix transformation does not add any information, all Mueller-Jones matrices $M(z)$ are non-depolarizing. Whereas, when we apply a spatial kernel averaging to measured Mueller matrices and these non-depolarization Mueller matrices add up in a spatial kernel, the noise on these measured Mueller matrices after averaging will be reduced. More importantly depolarization would be generated during the accumulation process [36]. In the detail proof, we operate an averaging operation among all pixels in the kernel and the measured Mueller matrix $\overline {M(z)}$ after averaging can be expressed as:
$$\overline {M(z)} = \frac{1}{n}({A + B + C + \ldots ..} ),$$
where A, B, C ……are n different non-depolarizing normalized Mueller-Jones matrices corresponding to n pixels in the kernel. Fry and Kattawar [37] proved that for every non-depolarizing Mueller matrix M we can write as:
$$\textrm{tr}({{M^T}M} )= 4m_{00}^2,$$
where $m_{00}^{}$ is the element of at the first row and first column of M. $tr({\ast} )$ is the matrix trace operation. According to Eq. (9) and (10), we can write:
$$\textrm{tr}({{{\overline {M(z)} }^T}\overline {M(z)} } )\textrm{ = }\frac{1}{{{n^2}}}\textrm{tr}({{A^T}A + {B^T}B + C{C^T} + \ldots + {A^T}B + {B^T}A + \ldots } ),$$
$${m_{00}} = \frac{1}{n}({{a_{00}} + {b_{00}} + {c_{00}} + \ldots } ),$$
where ${a_{00}}$, ${b_{00}}$, ${c_{00}}$,…are the elements at the first row and first column of A, B, C…respectively. When and only when A = B = C=…, the average matrix $\overline {M(z)}$ satisfies the condition of Eq. (10) and $\overline {M(z)}$ is a non-depolarizing Mueller matrix [38], else $\overline {M(z)}$ becomes a depolarization-including matrix. Due to the existing of noise and tissue depolarization, these matrices A, B, C…in the kernel would not be equal to each other and $\overline {M(z)}$ is a depolarizing Mueller matrix.

We will analyze how to reduce depolarization by the MMA method. The SMM method has been described previously [26]. Here the MMA is averaged SMM. We just introduce them briefly. To reduce polarization properties variations owing to a rapidly rotating optical fiber in the catheter, the reference point need to be chosen as the front surface of samples or the outside surface of the catheter probe’s sheath. We assume the $\overline {M({z_{ref}})}$ is the reference measured Mueller matrix after spatial kernel averaging, which is also converted from Jones matrix $J({z_{ref}})$. The calibrated Mueller matrix characterizing the combined system optical fibers and sample can be obtained as:

$$M({z_{ref}},z) = \overline {M(z)} {\overline {M({z_{ref}})} ^{ - 1}},$$

To reduce the impact of depolarization on the phase retardation measurement of the sample, a Lu-Chipman polar decomposition is applied to a matrix M and then it can be separated three components as depolarization part $M_{}^\Delta (z,{z_{ref}})$, birefringence part $M_{}^R(z,{z_{ref}})$ and diattenuation part $M_{}^D(z,{z_{ref}})$[39–41], which can be expressed as:

$$M(z,{z_{ref}}) \Rightarrow {M^\Delta }(z,{z_{ref}}){M^R}(z,{z_{ref}}){M^D}(z,{z_{ref}}).$$

Based on the matrix similarity, the phase retardation of the sample r(z) can be measured by this relation, which can be expressed as [26]:

$$r(z) = \arccos [tr({M^R}({z_{ref}},z))/2 - 1].$$

Based on Eq. (15), $M_{}^R(z,{z_{ref}})$ is immunity for depolarization, so the impact of noise-induced depolarization on the phase retardation measurement of the sample is reduced.

3. Monte Carlo simulation

3.1 Monte Carlo simulation modeling

We utilize Monte Carlo simulation to quantitatively compare JMA and MMA, because we can set material parameters such as birefringence, depolarization of simulated sample and SNR. Namely, we can compare measured phase retardation with true values using MMA and JMA. In the experiments, these true parameters of biological tissues are difficult to acquire accurately. In addition, Monte Carlo simulation method is a widely accepted for tissue optics simulation [42–45] and PS-OCT research [22,46,47]. We construct Monte Carlo simulation modeling and process for a PS-OCT as shown Fig. 2. Through tracing the propagations of the virtual photons in the simulated birefringence material, we observe the SOP changing for each photon and record in the simulated orthogonal polarization signals. Before the Monte Carlo simulation, we construct the multi-scattering model of the photon under Jones matrix in birefringence medium as a preparation. Inspired by Wang’s modeling [42], the preparation of the presented simulation is designed as a combination of two parts, the initialization of photons, and the specification for the simulated birefringence material. First, the photons are profiled with total launching number, the input electric field vectors. After that we tag each photon with a wavenumber k according to an expected light source spectrum distribution G(k). Second, the propagation parameters of the photon should be initialized before launching, including the propagation direction vector [ux, uy, uz], the position vector [x, y, z], the step length of the photon trace s, the scattering angle θ, the azimuth angle ϕ, the scattering weight ω [42] , etc. At each scattering, the change of [ux, uy, uz] and [x, y, z] are quantified according to basic geometric transformations as functions of θ, ϕ, and s, which can be expressed as [43]:

$$\begin{array}{*{20}{c}} {{u_x}^{\prime} = {{(1 - {u_z}^2)}^{ - {\textstyle{1 \over 2}}}}\sin \theta ({u_x}{u_y}\cos \phi - {u_y}\sin \phi ) + {u_x}\cos \theta }\\ {{u_y}^{\prime} = {{(1 - {u_z}^2)}^{ - {\textstyle{1 \over 2}}}}\sin \theta ({u_x}{u_z}\cos \phi - {u_x}\sin \phi ) + {u_y}\cos \theta }\\ {{u_z}^{\prime} = {{(1 - {u_z}^2)}^{ - {\textstyle{1 \over 2}}}}\sin \theta \cos \phi ({u_y}{u_z}\cos \phi - {u_x}\sin \phi ) + {u_z}\cos \theta ,} \end{array}$$
$$\begin{array}{*{20}{c}} {x^{\prime} = x + {u_x} \cdot s}\\ {y^{\prime} = y + {u_y} \cdot s}\\ {z^{\prime} = z + {u_z} \cdot s,} \end{array}$$
where x’, y’, z’ and ux’, uy’, uz’ are entries of new position vector and new direction vector after a scattering, respectively. s is the step length of the photon packet. Hence, all the propagation parameters will change on each scattering and are recorded before the next scattering, until the last scattering of the last photon are recorded. Therefore, the calculation principles of θ, ϕ, and s are important. The scattering happens as a result of the optic-material interaction [44]. Besides the size of birefringence material need to be specified, optical parameters of birefringence material also should be adequately specified, which includes scattering coefficient μs, the absorption coefficient μa, and the anisotropy coefficient g. After launching, for each time the photon scatters, the new propagation parameters are calculated from both the propagation parameters at the last time and the material parameters of the nearby voxel, namely the area around the scattering point. Here ϕ is the azimuth angle which counts the rotation angle of the electric field coordinates axis after one scattering, ϕ as a random number has a uniform distribution from 0 to 180°. θ as a random number obeys probability density function Henyey-Greenstein (H-G) phase function ${f_{HG}}(\theta )$ determined by g, which can be expressed as [48]:
$${f_{HG}}(\theta ) = \frac{{1 - {g^2}}}{{2[1 + {g^2} - 2g\cos (\theta )]}}.$$
s obeys a material-relative probability density function f(s) determined by μs and μa, which can be expressed as [46]:
$$f(s) = ({\mu _s} + {\mu _a}){e^{ - ({\mu _s} + {\mu _a})s}}.$$

 figure: Fig. 2.

Fig. 2. Flow chart of Monte Carlo simulation for PS-OCT

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When the preparation is finished, the Monte Carlo simulation can be started by launching photons. The scattering is a random process, hence the photon has the possibility to propagate out of the simulated material or be severely absorbed in an endless path. This phenomenon that the photon packet carries few energy to backscattering signals is called as the photon death. The scattering weight ω is reduced after the photon scattering occurs each time, which can be expressed as:

$${\omega _{n + 1}} = {\mu _s}{({\mu _s} + {\mu _a})^{ - 1}}{\omega _n}. $$
where ωn and ωn+1 represent the scattering weight before and after the nth scattering, respectively. The simulation program can detect the situation of the photon death by judging whether ω is lower than the weight threshold of the photon alive ωth or the current position of the photon is out of the boundary. If so, a new photon should be launched. If not, we continue to trace the path of the photon to construct a transitive Jones matrix relationship as a function of propagation parameters, which can be expressed as [48,49]:
$${\left( {\begin{array}{*{20}{c}} h\\ v \end{array}} \right)_{n + 1}} = \left( {\begin{array}{*{20}{c}} {\cos \phi }&{\sin \phi }\\ { - \sin \phi }&{\cos \phi } \end{array}} \right)\left( {\begin{array}{*{20}{c}} 1&0\\ 0&{{e^{ - i\delta }}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\cos \phi }&{ - \sin \phi }\\ {\sin \phi }&{\cos \phi } \end{array}} \right){\left( {\begin{array}{*{20}{c}} h\\ v \end{array}} \right)_n},$$
where h and v are the horizontal and vertical optical fields of the photon. δ is birefringence retardation, which be expressed as [50]:
$$\delta = ks[{n_s}{n_f}\sqrt {{{({n_s}\cos \alpha )}^2} + {{({n_f}\sin \alpha )}^2}} - {n_f}],$$
where ns and nf are refractive index along the slow axis and the fast axis of the birefringent material, respectively. α is the angle between the propagation direction of the photon and the slow axis of the birefringent material. During the propagation of the photons, L and ω are updated along with each scattering. We assume that H(k) and V(k) represent horizontal and vertical collected total photon as signals at k-domain, which can be expressed as [47]:
$$\left( {\begin{array}{*{20}{c}} {H(k)}\\ {V(k)} \end{array}} \right) = \sum\limits_{{k_i} = k - \Delta k/2}^{k + \Delta k/2} {\sqrt {G({k_i})} \sum\limits_{n = 0}^{{N_k}} {{{\left\{ {\sqrt {L\omega } {e^{j{k_i}z}}\left( {\begin{array}{*{20}{c}} h\\ v \end{array}} \right)} \right\}}_n}} } ,$$
where Δk is the width of the wavenumber interval for PS-OCT. k is the center wavenumber for each Δk. ΔK is the total wavenumber range of the light source. Here ΔK =Δk·l, where l is the number of the wavenumber interval corresponding to the pixel number for each A-scan. In each Δk, every photon has each wavenumber ki generated randomly. The total number of photon for simulation is M. According to G(k) distribution and M, we set a proportional number of the photons for each Δk. Nk represents the total scattering times of the photon with k. L is the back-scattering likelihood [51]. The subscript n stands for the scattering time of the photon. H(k) and V(k) are operated interference with the reference light and then converted to H(z) and V(z) in the spatial domain by a Fourier transform to complete the polarization imaging.

3.2 Monte Carlo simulation results

We use Monte Carlo method to numerically describe noise-induced depolarization and compare the phase retardation mapping by MMA and JMA. We set the optical parameters of Monte Carlo modeling of a biomaterial. Here we design the sample is uniform cube with 1mm×1mm×1mm. We set biomaterial tissues as simulation material and parameters for Monte Carlo simulation are discussed as below: The anisotropy g of biomaterial tissues varies from 0.9 to 0.99 [52], so we choose g = 0.9 for a typical physiological tissue. To make the retardation stripes more observable within 1mm-depth tissue, we set Δn = 0.008 which ranges from myosin fibrils 0.0015 to connective tissue 0.04 for common biomaterial [53]. Here ns =1.3000 nf = ns +Δn =1.3008. We aim to set the simulation in a turbid tissue-like scenario, hence the absorption coefficient μa and scattering coefficient μs are set according to low scattering and low absorption turbid tissue-like property [54]. Here μa = 0.001 cm−1, μs = 0.1 cm−1. The number of cubic voxels is 109. The light source has a Gaussian distribution G(k) with a center wavelength of 1310 nm. The spectrum range is 60 nm. The pixel number of A-scan is 1024, namely l = 1024. We set the number of photons M =80000. We initialize the photon propagation parameters including ωth = 0.0001, ω=1, L = 0.00001, x = 0, y = 0, z = 0, ux = 0, uy = 0, and uz = 1. Here we set 512 A-scans for each simulation imaging. The size of image is 512 × 512 pixels due to the symmetry of Fourier transform.

We firstly discuss the noise-induced depolarization based on Monte Carlo simulation’s results. Here we assume that SNR = SNRH = SNRV. The amplitude of signals and noise are set manually by SNR from 2.5 dB to 20 dB, which are added to H and V channels in k-domain. Here the added noise is used to represent the noise from the catheter based PS-OCT system. After the Monte Carlo PS-OCT simulation, signals are collected from the tissue and are used to calculate DOPU. The relationship between DOPU and SNR is shown in Fig. 3. We find that that DOPU is proportional to SNR, which verifies the noise-induced depolarization effect is heavy when SNR is low.

 figure: Fig. 3.

Fig. 3. Noise-induced DOPU under different SNR by Monte Carlo simulation.

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We contrast the phase retardation imaging results processed by MMA and JMA methods based on Monte Carlo simulation. The detail algorithm JMA method is shown in [18,21,32]. The averaging kernel is a sliding window with a size of 30 × 30 pixels. For complex field averaging in JMA and real field averaging in MMA, we operate averaging by the same sliding window on the whole image one pixel by one pixel. We show Monte Carlo simulation results of phase retardation images under different SNR in Fig. 4. The image size of Fig. 4 is 512 × 512 pixels. The length of A-scan is 1 mm and the length of B-scan is 1 mm. The noise and signals are set at the k-domain based on setting SNR. All setting SNR of each A-scan in a B-scan in an image of Fig. 4 are the same. The SNR from top to bottom are −3, 0, 3, 6 dB and no added noise, respectively. To a birefringent sample, the cumulative round-trip phase retardation varies from 0 to π along with the depth periodically, which generates parallel periodic straight stripes as polarization contrast shown Fig. 4(a) and 4(b). For each phase retardation image, the polarization contrast is deteriorated along with increasing of the imaging depth, which reason is that the SNR of backscattering signals is decreased along with increasing of the imaging depth. In Fig. 4(c), the DOPU is also decreased along with the increasing of the imaging depth. The reason is that depolarization is increased along with the depth due to more random scatterings or serpentine photons occurring with deeper depth. At the condition of the simulated data without added noise shown in Fig. 4(a5), (b5) and (c5). Since there is no noise floor and g is high, the phenomenon that polarization contrast and DOPU are depth-dependent is not obvious. We still observe relatively low DOPU stripes at the deep area caused by tissue depolarization shown in Fig. 4(c5), although the low DOPU area is not clear due to a consistent color bar in Fig. 4(c). Compared with images processed by JMA and MMA methods, the polarization contrasts at deeper locations processed by MMA is more obvious than those by JMA, especially in images with heavy noise shown in Fig. 4(a1) and 4(b1), 4(a2) and 4(b2). For phase retardation images with different SNR, the polarization contrast is deteriorated along with the decreasing of SNR. DOPU also has a positive correlation to SNR due to the noise-induced depolarization shown in Fig. 4(c). Similarly, the polarization contrasts processed by MMA is better than those by JMA in different SNR shown in Fig. 4(a) and 4(b). These results by Monte Carlo simulation above verifies that MMA can reduce effect of noise-induced depolarization on the phase retardation measurement.

 figure: Fig. 4.

Fig. 4. Phase retardation and DOPU imaging based on Monte Carlo simulation. (a1) to (a5) Phase retardation image processed by MMA with SNR of −3 dB to 6 dB and no added noise. (b1) to (b5) Phase retardation image processed by JMA with SNR of −3 dB to 6 dB and no added noise. (c1) to (c5) DOPU with SNR of −3 dB to 6 dB and no added noise.

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To quantitatively compare the JMA with MMA method, we compare phase retardation calculation results processed by these two methods with phase retardation without any noise namely true image. Here we use a peak signal to noise ratio (PSNR) to evaluate the data in Fig. 4 and PSNR can be expressed as:

$$PSNR = 10{\log _{10}}\frac{{MAX_I^2}}{{\frac{1}{{XZ}}\sum\limits_{i = 1}^X {\sum\limits_{j = 1}^Z {{{|{G(i,j) - F(i,j)} |}^2}} } }},$$
where X and Z are the length and width of the image, namely the A-scan number and the scanning pixel depth respectively. $MA{X_I}$ represents the theoretical maximum value of image pixels that is π. $G(i,j)$ and $F(i,j)$ are the pixel of the true image and images to be evaluated. We calculate PSNR under different SNR based on the data in Fig. 4 to evaluate the MMA and JMA methods as shown in Fig. 5. From Fig. 5, PSNR of the MMA method is higher than that of JMA method at each SNR. The averaged improvement of PSNR compared with MMA and JMA methods is about 8.9 dB.

 figure: Fig. 5.

Fig. 5. PSNR of images processed by JMA and MMA method under different SNRs based on Monte Carlo simulation results in Fig. 4.

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4. Experiment and discussion

We compare experimental results of multiple biological tissues processed by the JMA and MMA method using our catheter based PS-OCT system. The experimental setup of the catheter based PS-OCT system has been described previously shown in Fig. 6 [26]. Our experimental setup is to use the PMF based depth multiplexing method. The light source is a swept laser (HSL-20-100-M, Santec, Inc.). The sweep rate, sweep range and starting wavelength of this swept laser source are 100 kHz, 80 nm and 1270 nm, respectively. The average output power of this swept laser source is 20 mW. An auxiliary Mach-Zehnder interferometer is built into the swept laser, which provides an external clock (k-clock) to trigger the data acquisition and sample the main interference signals at equidistant wavenumber points. The delay length in air of this auxiliary interferometer is 20 mm, then the corresponding imaging depth is 5 mm in air in this PS-OCT system. The main interference in this PS-OCT system is a modified fiber-based Mach-Zehnder structure, which applies a 99/1 coupler to split the light into the sample and reference arms. An 18.5 m PMF in the sample arm as a passive polarization delay is used to generate a 2.5 mm depth separation in air between two orthogonal input SOPs. A corresponding 18.5 m SMF in the reference arm is used to match the length of the PMF in the sample arm. Two circulators in the sample and reference arms guide light to the catheter probe and reflected variable optical delay line (RVDL), respectively and then guide the reflected light to the 50/50 coupler. The inference between light in the reference and sample arm in the 50/50 coupler. Two fiber based polarization beam splitters (PBS) and balanced photo detectors (BPD) are implemented to achieve the polarization diversity detection. A 500 M sample/s 12-bit waveform digitizer card (ATS9350, Alazar Technologies Inc.) is used to data acquisition. Polarization controller (PC) 1 is used to balance the probe light between these two input SOPs in the sample arm and PC 2 to PC 4 are used to balance the referenced light between the polarization diversity channels and calibrate two PBSs. We assemble the optical rotary junction and standard clinical catheter (C7 Dragonfly catheter, St. Jude Medical Inc.) to realize a circumferential scanning. The outer diameter of the catheter probe is about 0.9 mm. The fiber in the catheter probe is single-mode fiber. In the PS-OCT system, the spatial resolution of A-scan namely the axial resolution is about 10 μm. The effective A-scan contains about 200 points in the k domain and then is interpolated to 3000 pixels in the z domain by N points Fourier transform with a zero padding that is corresponding to the imaging depth of 2 mm. The rotational speed is about 2000 rotations/min. Since the sweep rate of TLS is 100 kHz, the system completes 100 k A-scan per second. The catheter rotating one cycle contains 3000 A-lines, namely a rotation of the catheter as B-scan contains 3000 pixels. The sensitivity of this system is about 90 dB. We apply averaging on the data in polar coordinates. The size of image is 3000 × 3000 pixels.

 figure: Fig. 6.

Fig. 6. Schematic of the catheter based PS-OCT system. BPD is balanced photo detector; PC is polarization controller; PBS is polarization beam splitter; DAQ is data acquisition card; RVDL is reflected variable optical delay line. PMF is polarization maintaining fiber marked as red line. SMF is single mode fiber marked as black line.

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We widely select several biological tissues including ex vivo biological tissues including chicken breast muscle, human nails and porcine cardiac blood vessel for polarization imaging. To biological tissues with a strong birefringence such as muscle and nails, their cumulative round-trip phase retardation changes from 0 to π periodically, generating circular periodic stripes as polarization contrast in a 2D circumferential scan image. Colors of stripes change from blue to green to yellow periodically. We show images of cumulative round-trip phase retardation and DOPU in Fig. 7–11. When we compare JMA and MMA, the same kernel and data are applied to be processed by these two methods. Different size sliding averaging kernel is applied for a complex field averaging in JMA and a real field averaging in MMA shown in Fig. 7. From Fig. 7, when kernel size is 15 × 15 pixels, these stripes caused by phase retardation are not very clear due to noise shown in Fig. 7(b1) to (d1). When kernel size is 45 × 45 pixels, these stripes caused by phase retardation are burring, because details of phase retardation changes are missing shown in Fig. 7(b2) to (d2). When kernel size is 30 × 30 pixels, these stripes caused by phase retardation are clear. Therefore, we choose the kernel size of 30 × 30 pixels to process other biological tissues shown in Fig. 8–11.

 figure: Fig. 7.

Fig. 7. Comparison of chicken muscle phase retardation images by JMA and MMA methods and DOPU images by different kernel sizes of 15 ×15 pixels, 30 ×30 pixels and 45 ×45 pixels. (a) Structure images in grayscale. (b1)-(b3) DOPU images. (c1)-(c3) Cumulative round-trip phase retardation images using the JMA method. (d1)-(d3) Cumulative round-trip phase retardation images using the MMA method. Compared with the results with different kernel size, the results of 30 ×30 pixels are optimal. These areas with low DOPU pointed by black arrows are corresponding to these areas with MMA having a higher polarization contrast than JMA.

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 figure: Fig. 8.

Fig. 8. Comparison of phase retardation images by JMA and MMA methods and DOPU images by chicken breast muscle. (a) Structure images in grayscale. (b) Cumulative round-trip phase retardation images using the JMA method. (c) Cumulative round-trip phase retardation images using the MMA method (d) DOPU images. The area with low DOPU pointed by black arrows is corresponding to the area with MMA having a higher polarization contrast than JMA.

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 figure: Fig. 9.

Fig. 9. Comparison of phase retardation images by JMA and MMA methods and DOPU images by chicken breast muscle. (a) Structure images in grayscale. (b) Cumulative round-trip phase retardation images using the JMA method. (c) Cumulative round-trip phase retardation images using the MMA method (d) DOPU images. These areas with low DOPU pointed by black arrows are corresponding to the areas with MMA having a higher polarization contrast than JMA.

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 figure: Fig. 10.

Fig. 10. Comparison of phase retardation images by JMA and MMA methods and DOPU images by ex vivo human finger nails. (a) Structure images in grayscale. (b) Cumulative round-trip phase retardation images using the JMA method. (c) Cumulative round-trip phase retardation images using the MMA method (d) DOPU images. The area with low DOPU pointed by black arrows is corresponding to the area with MMA having a higher polarization contrast than JMA.

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 figure: Fig. 11.

Fig. 11. Comparison of phase retardation images by JMA and MMA method and DOPU images by ex vivo porcine cardiac blood vessel. (a) Structure images in grayscale. (b) Cumulative round-trip phase retardation images using the JMA method. (c) Cumulative round-trip phase retardation images using the MMA method (d) DOPU images. The area with low DOPU pointed by black arrows is corresponding to the area with MMA having a higher polarization contrast than JMA.

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We compare the phase retardation results from the same raw data using the JMA method shown in Fig. 7(c) to 11(c) with using the MMA method shown in Fig. 7(d) to 11 (d). The circular periodic stripes in phase retardation images by these two methods are clear. Whereas, the images processed by the MMA method have a higher polarization contrast than those by the JMA method shown in Fig. 7 to 11 pointed by black arrows. We also show DOPU of these biological tissues in Fig. 7(b) to 11 (b). We point out these areas with low DOPU by black arrows. We find that these areas with low DOPU are corresponding to these areas with MMA having a higher polarization contrast than JMA, which verifies that depolarization effect on phase retardation measurement can be reduced by Mueller matrix polar decomposition after incoherent averaging in the MMA method. Here biological tissues such as muscle, human nails and porcine cardiac blood vessel do not contain very strong tissue depolarization, so the decreasing of DOPU is caused by noise, namely noise-induced depolarization. There are not obvious periodic stripes in the phase retardation images of porcine cardiac blood vessel shown in Fig. 11(c) and (d) because porcine cardiac blood vessel has not very strong birefringence. Noise-induced depolarization will cause more phase retardation change processed by the JMA method shown in Fig. 11(b) and (c). Whereas, these noise-induced phase retardation change are reduced processed by the MMA method shown in Fig. 11(d), especially in the area with low DOPU pointed by black arrows shown in Fig. 11(b).

In Fig. 7(b) to 11(b), DOPU cannot correctly represent the tissue depolarization. Since tissue depolarization is defined by the variation of polarization states of backscattered light from the local space in a tissue, which is not only dependent on the tissue property, but also is significantly impacted by non-tissue factors, such as the signal-to-noise ratio (SNR), namely noise-induced depolarization [29,33,55]. In this algorithm, we cannot distinguish tissue depolarization and noise-induced depolarization. We only reduce the impact of depolarization on the phase retardation measurement of the sample. Tissue depolarization is linked to tissue heterogeneity and can reveal clinically relevant features, especially in plaque characterization detection [56]. In the future, we will consider to apply noise corrected DOPU [29], depolarization index [57], differential depolarization index [55], depolarization metric based on Müller matrix polar decomposition [28], entropy [33] and so on to extract tissue depolarization.

Since true phase retardation data of biological tissues is difficult to be acquired, we are not able to use PSNR to evaluate the experimental data. We only qualitatively compare experimental results of multiple biological tissues processed by the JMA and MMA method using our catheter based PS-OCT system. In the future, we will implement tissue-like birefringence phantoms [58] to quantitatively compare JMA and MMA based on the experimental system, because we can acquire true phase retardation data of tissue-like birefringence phantoms. In the simulation, we choose zero-mean Gaussian distribution as noise distribution. We plan to do a further research based on more noise distribution including Rice distribution for catheter-based PS-OCT [59,60].

5. Conclusion

We analyze noise-induced depolarization and find that depolarization can be reduced by polar decomposition after incoherent averaging in the MMA method based on the catheter based PS-OCT. We present a Monte Carlo method based on PS-OCT to numerically describe noise-induced depolarization effect and contrast phase retardation imaging results by MMA and JMA methods. PSNR of simulated images processed by MMA is higher than about 8.9 dB than that processed by JMA. We also implement experiments of multiple biological tissues using the catheter based PS-OCT system. From the simulation and experimental results, we find the polarization contrasts processed by MMA is better than those by JMA, especially at areas with high depolarization, because MMA can reduce effect of noise-induced depolarization on the phase retardation measurement. The presented algorithm and analysis are independent from the hardware, which are suitable any PS-OCT system with two input polarization states. In the future, we will distinguish tissue depolarization and noise-induced depolarization. In addition, we also explore the local birefringence measurement based on the MMA method.

Funding

National Natural Science Foundation of China (61975147, 61635008, 61735011, 61505138); Key Technologies Research and Development Program (2019YFC0120701); Special Technical Support Project of China Market Supervision and Administration (2021YJ027).

Disclosures

The authors declare no conflicts 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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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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Figures (11)
Fig. 1.
Fig. 1. Numerical evaluation of relationship between DOPU and the SNR’s reciprocal of H and V channel namely $\overline {SNR_H^{ - 1}}$ and $\overline {SNR_V^{ - 1}}$ based on Eq. (6). Here $\overline {SNR_H^{ - 1}} \in [\textrm{0},\textrm{10 }]$, $\overline {SN{R_V}^{\textrm{ - 1}}} \in [\textrm{0},\textrm{10}]$.

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Fig. 2.
Fig. 2. Flow chart of Monte Carlo simulation for PS-OCT

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Fig. 3.
Fig. 3. Noise-induced DOPU under different SNR by Monte Carlo simulation.

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Fig. 4.
Fig. 4. Phase retardation and DOPU imaging based on Monte Carlo simulation. (a1) to (a5) Phase retardation image processed by MMA with SNR of −3 dB to 6 dB and no added noise. (b1) to (b5) Phase retardation image processed by JMA with SNR of −3 dB to 6 dB and no added noise. (c1) to (c5) DOPU with SNR of −3 dB to 6 dB and no added noise.

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Fig. 5.
Fig. 5. PSNR of images processed by JMA and MMA method under different SNRs based on Monte Carlo simulation results in Fig. 4.

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Fig. 6.
Fig. 6. Schematic of the catheter based PS-OCT system. BPD is balanced photo detector; PC is polarization controller; PBS is polarization beam splitter; DAQ is data acquisition card; RVDL is reflected variable optical delay line. PMF is polarization maintaining fiber marked as red line. SMF is single mode fiber marked as black line.

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Fig. 7.
Fig. 7. Comparison of chicken muscle phase retardation images by JMA and MMA methods and DOPU images by different kernel sizes of 15 ×15 pixels, 30 ×30 pixels and 45 ×45 pixels. (a) Structure images in grayscale. (b1)-(b3) DOPU images. (c1)-(c3) Cumulative round-trip phase retardation images using the JMA method. (d1)-(d3) Cumulative round-trip phase retardation images using the MMA method. Compared with the results with different kernel size, the results of 30 ×30 pixels are optimal. These areas with low DOPU pointed by black arrows are corresponding to these areas with MMA having a higher polarization contrast than JMA.

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Fig. 8.
Fig. 8. Comparison of phase retardation images by JMA and MMA methods and DOPU images by chicken breast muscle. (a) Structure images in grayscale. (b) Cumulative round-trip phase retardation images using the JMA method. (c) Cumulative round-trip phase retardation images using the MMA method (d) DOPU images. The area with low DOPU pointed by black arrows is corresponding to the area with MMA having a higher polarization contrast than JMA.

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Fig. 9.
Fig. 9. Comparison of phase retardation images by JMA and MMA methods and DOPU images by chicken breast muscle. (a) Structure images in grayscale. (b) Cumulative round-trip phase retardation images using the JMA method. (c) Cumulative round-trip phase retardation images using the MMA method (d) DOPU images. These areas with low DOPU pointed by black arrows are corresponding to the areas with MMA having a higher polarization contrast than JMA.

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Fig. 10.
Fig. 10. Comparison of phase retardation images by JMA and MMA methods and DOPU images by ex vivo human finger nails. (a) Structure images in grayscale. (b) Cumulative round-trip phase retardation images using the JMA method. (c) Cumulative round-trip phase retardation images using the MMA method (d) DOPU images. The area with low DOPU pointed by black arrows is corresponding to the area with MMA having a higher polarization contrast than JMA.

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Fig. 11.
Fig. 11. Comparison of phase retardation images by JMA and MMA method and DOPU images by ex vivo porcine cardiac blood vessel. (a) Structure images in grayscale. (b) Cumulative round-trip phase retardation images using the JMA method. (c) Cumulative round-trip phase retardation images using the MMA method (d) DOPU images. The area with low DOPU pointed by black arrows is corresponding to the area with MMA having a higher polarization contrast than JMA.

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Equations (24)

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J ( z ) = [ H 1 H 2 V 1 V 2 ] = [ H 01 + Δ H 1 H 02 + Δ H 2 V 01 + Δ V 1 V 02 + Δ V 2 ] ,
[ H V ] = [ H 0 + Δ H V 0 + Δ V ] ,
S N R H = | H | 2 σ H 2 , S N R V  =  | V | 2 σ V 2 .
S = [ S 0 S 1 S 2 S 3 ] = [ H H ∗ + V V ∗ H H ∗ −   V V ∗ H V ∗ +   V H ∗ j ⋅ ( H V ∗ −   V H ∗ ) ] ,
S ¯ = 1 n ∑ i = 1 n S i ,
D O P U = S 1 ¯ 2 + S 2 ¯ 2 + S 3 ¯ 2 S 0 ¯ = ( | H | 2 ¯  +  | V | 2 ¯ ) 2 − 4 K | H | 2 ¯  +  | V | 2 ¯ ,
K  =  ∑ q 1 = 0 2 ∑ q 2 = 0 2 ∑ q 3 = 0 2 μ q 1 , q 2 , q 3 S N R V − 1 ¯ q 1 S N R V − 1 ¯ q 2 S N R H − 1 S N R V − 1 ¯ q 3 .
M ( z ) = U [ J ( z ) ⊗ J ( z ) ∗ ] U − 1 ,
M ( z ) ¯ = 1 n ( A + B + C + … . . ) ,
tr ( M T M ) = 4 m 00 2 ,
tr ( M ( z ) ¯ T M ( z ) ¯ )  =  1 n 2 tr ( A T A + B T B + C C T + … + A T B + B T A + … ) ,
m 00 = 1 n ( a 00 + b 00 + c 00 + … ) ,
M ( z r e f , z ) = M ( z ) ¯ M ( z r e f ) ¯ − 1 ,
M ( z , z r e f ) ⇒ M Δ ( z , z r e f ) M R ( z , z r e f ) M D ( z , z r e f ) .
r ( z ) = arccos ⁡ [ t r ( M R ( z r e f , z ) ) / 2 − 1 ] .
u x ′ = ( 1 − u z 2 ) − 1 2 sin ⁡ θ ( u x u y cos ⁡ ϕ − u y sin ⁡ ϕ ) + u x cos ⁡ θ u y ′ = ( 1 − u z 2 ) − 1 2 sin ⁡ θ ( u x u z cos ⁡ ϕ − u x sin ⁡ ϕ ) + u y cos ⁡ θ u z ′ = ( 1 − u z 2 ) − 1 2 sin ⁡ θ cos ⁡ ϕ ( u y u z cos ⁡ ϕ − u x sin ⁡ ϕ ) + u z cos ⁡ θ ,
x ′ = x + u x ⋅ s y ′ = y + u y ⋅ s z ′ = z + u z ⋅ s ,
f H G ( θ ) = 1 − g 2 2 [ 1 + g 2 − 2 g cos ⁡ ( θ ) ] .
f ( s ) = ( μ s + μ a ) e − ( μ s + μ a ) s .
ω n + 1 = μ s ( μ s + μ a ) − 1 ω n .
( h v ) n + 1 = ( cos ⁡ ϕ sin ⁡ ϕ − sin ⁡ ϕ cos ⁡ ϕ ) ( 1 0 0 e − i δ ) ( cos ⁡ ϕ − sin ⁡ ϕ sin ⁡ ϕ cos ⁡ ϕ ) ( h v ) n ,
δ = k s [ n s n f ( n s cos ⁡ α ) 2 + ( n f sin ⁡ α ) 2 − n f ] ,
( H ( k ) V ( k ) ) = ∑ k i = k − Δ k / 2 k + Δ k / 2 G ( k i ) ∑ n = 0 N k { L ω e j k i z ( h v ) } n ,
P S N R = 10 log 10 M A X I 2 1 X Z ∑ i = 1 X ∑ j = 1 Z | G ( i , j ) − F ( i , j ) | 2 ,
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