Application of Polarization Imaging Techniques in Brain Tumor Detection
偏振成像技术在脑肿瘤检测中的应用

论文ppt：https://blog.csdn.net/yyyyang666/article/details/128536637
论文原文：https://blog.csdn.net/yyyyang666/article/details/128516882
校内课程介绍：外院、光电联手打造学术英语协同教学新模式 (usst.edu.cn)

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本文链接：https://blog.csdn.net/yyyyang666/article/details/128516882

摘要： 癌症是公众健康的一大风险，以胶质瘤为代表的脑肿瘤在我国是目前最难被攻克的癌症之一。当前的诊断技术无法准确的识别肿瘤边界，给脑肿瘤的手术切除带来了很大挑战。大部分生物体组织在可见光波段为高散介质，由于其微观结构存在差异，会改变光的偏振状态，测量散射光的偏振度可以获得丰富的散射体结构信息，在脑肿瘤检测的应用上有很大的潜力。本文介绍了偏振呈现技术在脑肿瘤检测的应用：首先介绍了偏振成像的基本原理，接着给出了Lu-Chipman极化分解成像和偏振椭圆参数成像两种成像方式及其在脑肿瘤检测中的应用，最后给出了偏振成像技术在肿瘤检测中的未来发展前景。目前，脑肿瘤的发病机制尚不十分明确，脑肿瘤的检测仍是一个很有前途的研究方向。

关键词： 偏振成像脑肿瘤穆勒偏振测量法偏振椭圆参数

Abstract: Cancer is a risk for public health, and brain tumor represented by glioma is one of the most challenging cancers to overcome in China. The current diagnostic techniques can not accurately identify the tumor boundary, which brings a great challenge to the surgical resection of brain tumors. Most biological tissues are highly scattered medium in the visible band, which will change the polarization state of light because of the microstructure. So, measuring the polarization of scattered light can obtain abundant structure information, which could be used in brain tumor detection. This paper introduces the application of polarization imaging techniques in brain tumor detection. Firstly, we introduce the basic principle of polarization imaging, and then we give two imaging methods of Lu-Chipman polarization decomposition imaging and polarization ellipse parameter imaging and their application in brain tumor detection. Finally, we give a future development prospect of polarization imaging technology in tumor detection. At present, the pathogenic mechanism of brain tumor is still not precise. Polarization imaging techniques in brain tumor detection are still a promising future research direction.
Keywords: Polarization imaging, Brain tumor, Muller polarimetry, polarization ellipse parameters
　

Completion time: 2022-01

Content

1. Introduction
2. The basic principle of polarization imaging
- 2.1 The polarization of light and its characterization method
- 2.2 Stokes vector-Mueller matrix
- - 2.2.1 Measurement of Mueller Matrix
  - 2.2.2 Decomposition of Mueller matrix
3. Application of polarization imaging in brain tumors detection
- 3.1 Lu-Chipman polar decomposition imaging
- 3.2 PEPs imaging
- - 3.2.1 Method of polarimetric measurements and analyze
  - 3.2.2 Application on brain tissue
4. Conclusion
References

1. Introduction

Cancer is one of the leading causes of premature death and burden public health, and brain tumors have been a hot research direction in recent years. For the brain tumors patient, Surgery is the critical treatment step. Although some well-delineated tumors can be removed as a whole, most gliomas, which are infiltrative, are hard to identify the precise borders of the tumor cells only removed piece by piece. So, subtotal resection increases the risk of recurrence; if healthy tissue is removed due to the complete resection, it may irreversibly damage the brain, resulting in irreversible disability and a decline in quality of life. Even though some methods using widely for determining tumors, such as preoperative medical imaging diagnosis and intraoperative microdissection, it is still hard to accurately distinguish the boundaries of tumors^[1].
　
Optical imaging method has become a widely used tool in biomedical and many other research fields because of its non-invasive, low damage, subcellular resolution, and other characteristics. Light has four basic properties: intensity, wavelength, phase, and polarization. Most biological tissues are highly scattering media in the visible band, which makes light often undergoes multiple scattering when it propagates in the tissue because of the microstructure, making the state of polarization change^[2]. Therefore, measuring the polarization of the scattered light can get the abundant structure information of the scatterer.
　
So, polarization imaging has the prospect of solving these problems in brain tumor detection. The white matter in the mammalian brain is mainly composed of nerve fiber bundles that arrange closely and exhibit a certain degree of optical anisotropy. On the other hand, the tumor destroys the original structure, erases the optical anisotropy in the lesion area of the brain. Therefore, using polarization imaging can identify the boundary of the tumor. It can also determine nerve fibers’ structure and sketch the orientations of nerve fibers inside and outside the plane of human brain tissue sections^[3].
　
Difference polarization (DP) and degree of polarization (DOP) are the earliest polarization imaging methods used in clinical diagnosis. In the 1990s, Anderson et al. first applied the DP imaging method to detect skin cancer, eliminate the influence of surface reflected light on the image, and improve superficial tissue imaging contrast. Then, Jacques et al. first tried to use the DOP imaging method, and the results show that the DOP parameter can distinguish the boundary between skin canceration and normal tissue. However, the complex internal structure of biological tissue affects the polarization measurement parameters, including DOP. Recently, the Muller matrix has attracted much attention as a characterization method that can fully reflect the optical polarization properties of media and has been initially used to detect cancerous tissues. Since then, Muller matrix decomposition has been proposed to solve the problems that the physical meaning of Muller matrix elements is not clear enough, and it is hard to extract information^[2]. However, biological tissue is not a pure retarder or polarization. Therefore, recently, Jain et al. has considered that the Muller matrix decomposition method does not necessarily understand the polarization behavior; on the contrary, in most cases, it only transfers it to the interpretation of the extracted amount, we should develop some new method to better solve the problem^[4].
　
This article summarizes the principle of polarization scattering imaging and its latest application in diagnosing brain tumors, showing the future development prospect of polarization imaging technology in cancer detection.
　
　
　

2. The basic principle of polarization imaging

2.1 The polarization of light and its characterization method

The polarization of light comes from the electromagnetic shear wave characteristics: when the electric field vibration of light shows a specific law along a particular direction or its change, it is polarized light. In contrast, when the electric field vibration is distributed randomly in all directions, it is natural light^[2].
　
Three commonly used methods characterize the polarization and its changes: Jones vector-Jones matrix, Poincare sphere, and Stokes vector-Mueller matrix. Stokes vector uses the intensity component of light and does not contain phase information, and can quantitatively describe partially polarized light and non-polarized light. Stokes vector is also the mainstream method to describe the polarization characteristics of tissue samples in biomedical research^[2].

2.2 Stokes vector-Mueller matrix

Stokes vector uses four-dimensional column vectors to represent four different light intensity components:
S=[IQUV](1)\bf S= \left[ \begin{matrix} I\\ Q\\ U\\ V \end{matrix} \right] \tag{1} S=IQUV(1)
where I is the total power, Q and U are linear polarization directions, and V is the circularly polarized component.

When interacting with a sample, the transformation of the Stokes vector of the incident light is described by a 4×4 real matrix M, called the Mueller matrix (MM), which reflects all the optical polarization properties of the medium^[2,4]
[I′Q′U′V′]=[M11M12M13M14M21M22M23M24M31M32M33M34M41M42M43M44][IQUV](2)\left[ \begin{matrix} I'\\ Q'\\ U'\\ V' \end{matrix} \right]= \left[ \begin{matrix} M_{11}& M_{12} & M_{13} & M_{14}\\ M_{21}& M_{22} & M_{23} & M_{24}\\ M_{31}& M_{32} & M_{33} & M_{34}\\ M_{41}& M_{42} & M_{43} & M_{44} \end{matrix} \right] \left[ \begin{matrix} I\\ Q\\ U\\ V \end{matrix} \right] \tag{2} I′Q′U′V′=M11M21M31M41M12M22M32M42M13M23M33M43M14M24M34M44IQUV(2)
However, the 16 elements of the Muller matrix are affected by multiple structures simultaneously, and the physical meaning of the representative is not clear. Therefore, choosing appropriate methods to clarify the polarization parameters corresponding to the structure and other physical parameters^[2].

2.2.1 Measurement of Mueller Matrix

The Muller polarimeter is a measuring instrument for determining the MM of the sample.
The Muller polarimeter’s operation describes the following: the polarization state generator (PSG) generates a set of input 4×1 Stokes vector Si\bf S_iSi and the sample converts it into a set of output Stokes vector M⋅Si\bf M \cdot S_iM⋅Si. Then, the polarization state analyzer (PSA), which provides the original intensity signal Bij\bf B_{ij}Bij by projecting each vector M⋅Si\bf M \cdot S_iM⋅Si onto the PSA ground state to analyze these output Stokes vectors. The simple matrix equation can summarize this schema B=A⋅M⋅W\bf B=A\cdot M\cdot WB=A⋅M⋅W, where the modulation matrix W\bf WW, which characterizes the PSG, is formed by the Si\bf S_iSi vectors in columns, whereas the Si\bf S_iSi are the line vectors of the analysis matrix A\bf AA characterizing the PSA^[3].

Fig. 1. A typical Muller polarimeter structure schematics: wide-field imaging Mueller polarimeter operating at multiple wavelengths in the visible wavelength range. Figure from the support material of Ref [3].

2.2.2 Decomposition of Mueller matrix

To further describe the physical meaning of Muller matrix elements, the scientists propose various calculation methods of MM in recent years, such as Muller matrix decomposition and transformation. Muller matrix decomposition converts Muller matrix into parameters with clear physical meaning by mathematical operation. Muller matrix polarization decomposition (MMPD), which Lu and Chipman first proposed in 1996, is still widely used^[2]. This method can generate high contrast images and assist in diagnostics.
Lu-Chipman polar decomposition proved that MM could be described as the product of the MMs of three essential optical elements, which are diattenuator MD\bf M_DMD, retarder MR\bf M_RMR, and polarization depolarizer M∆\bf M_∆M∆^[4]:

M=MD⋅MR⋅M∆(3)\bf M=M_D \cdot M_R \cdot M_∆ \tag{3} M=MD⋅MR⋅M∆(3)

MD=Tu[1DTDmD],MR=Tu[10T0mR],M∆=Tu[10TPm∆],(4){\bf M_D}=T_u \left[ \begin{matrix} \bf 1 & \bf D^T\\ \bf D & \bf m_D \end{matrix} \right], {\bf M_R}=T_u \left[ \begin{matrix} \bf 1 & \bf 0^T\\ \bf 0 & \bf m_R \end{matrix} \right], {\bf M_∆}=T_u \left[ \begin{matrix} \bf 1 & \bf 0^T\\ \bf P & \bf m_∆ \end{matrix} \right], \tag{4} MD=Tu[1DDTmD],MR=Tu[100TmR],M∆=Tu[1P0Tm∆],(4)
where TuT_uTu is the transmittance or reflectivity of unpolarized light, D is a 3×1 vector of diattenuation, P is a 3×1 vector of polarizing. mD\bf m_DmD, mR\bf m_RmR and m∆\bf m_∆m∆ are the 3×3 real matrix.
Then we can calculate the total depolarization^[4] as:
∆=1−∣tr(M∆−1)∣3,0≤∆≤1.(5)∆=1-\frac{|tr(M_∆-1)|}{3},0≤∆≤1. \tag{5} ∆=1−3∣tr(M∆−1)∣,0≤∆≤1.(5)

The combined effect of linear and circular birefringence^[4] is

RLC=cos−1(tr(MR)2−1),(6)R_{LC}=cos^{-1} (\frac { tr(\bf M_R )}{2}-1), \tag{6} RLC=cos−1(2tr(MR)−1),(6)
where MR\bf M_RMR can be describe as a combination of a linear retarder with linear retardance RRR and the azimuth of optical axis φφφ , and a MM of a circular retarder with optical rotation angle ψψψ^[4]. They can calculate as:
R=cos−1((MR(2,2)+MR(3,3))2+(MR(3,2)+MR(2,3))2−1),(7)R=cos^{-1} (\sqrt{{(M_R (2,2)+M_R (3,3))}^2+{(M_R (3,2)+M_R (2,3))}^2 }-1), \tag{7} R=cos−1((MR(2,2)+MR(3,3))2+(MR(3,2)+MR(2,3))2−1),(7)
φ=0.5tan−1(MR(2,4)MR(4,3)),(8)φ={0.5tan}^{-1} (\frac{M_R (2,4)}{M_R (4,3) }), \tag{8} φ=0.5tan−1(MR(4,3)MR(2,4)),(8)
ψ=tan−1(MR(3,2)−MR(2,3)MR(2,2)−MR(3,3)).(9)ψ=tan^{-1} (\frac{M_R (3,2)-M_R (2,3)}{M_R (2,2)-M_R (3,3) }). \tag{9} ψ=tan−1(MR(2,2)−MR(3,3)MR(3,2)−MR(2,3)).(9)
　
　
　

3. Application of polarization imaging in brain tumors detection

3.1 Lu-Chipman polar decomposition imaging

Using Lu-Chipman polar decomposition, Muller matrix factorization imaging is widely used to detect skin cancer, oral cancer, gastric cancer, and other cancers^[2]. Recently, Schucht et al. applied this method to detect the brain tumor, one of the most challenging cancers, providing a new way to diagnose the brain tumor.
They used the Muller polarizer shown in Fig. 1 to measure fixed human brain tissue and image the grey-scale intensity, the total depolarization, the scalar linear retardance, and the azimuth of the optical axis calculated from MM. They measured in air and water with the light of wavelength 550nm, and both of those show the existence of fiber tracks on the image of the total depolarization and the scalar linear retardance. When measured in water, the brain’s contrast between gray matter and white matter is higher than in air because the impact of surface scattering on depolarization and scalar retardance is significantly diminished ^[3].
The gray matter has less depolarizing and scalar retardance than the white matter, as shown in Fig. 2(b, c). In Fig. 2(d), the optical axis azimuth diagram shows the orientation of the nerve fiber bundle. To verify the accuracy of this method, they selected an area where is between gray matter (also called the cortex) and white matter nerve fiber track, enlarged the image of the area, and compared it with the photo of the silver-stained histological section–the definitive way for brain fiber visualization, as shown in Fig. 3 and Fig. 4.

Fig. 2. Image of the fixed human brain specimen measured in water with the light wavelength 550nm. Figure from the Ref. [3].

Fig. 3. Image of the fixed human brain specimen. (a) image of azimuth of the optical axis and its enlarged image in an area between gray matter and white matter nerve fiber track. (b) photo of the silver-stained histological section for the whole specimen and the enlarged area corresponding in (a). A1-A4 are the zones in the white matter nerve fiber track, A5 is in the gray matter zone. Zones of B1-B5 correspond to A1-A5. Figure from the Ref. [3].

Fig. 4. (a) Circular histogram of the azimuth of optical axis for the zones A1-A5 in Fig. 3(a). (b) the enlarged image of zone B1-B5 in Fig. 3(b). C- gray matter, F-white matter nerve fiber track, white dashed line is the border of C and F. Figure from the Ref. [3].

The circular histogram of zone A1-A5 shows a convincing correlation with the nerve fiber direction in zone B1-B5 described by the silver-stained histological section photo, as shown in Fig. 4.

The results above also are proved by using fresh calf brain tissue^[3]. Further research proves that even in adverse measurement conditions, like complex surface topography and the presence of blood or saline, impact little for imaging^[4].

In conclusion, using Lu-Chipman polar decomposition imaging can clearly show that the nerve fiber track exists through the image of depolarization and scalar retardance. Besides, the map of the azimuth of optical axis visualized the orientation of the white matter nerve fiber track. Therefore, it provides a non-invasive, non-staining, fast, and repeatable method to identify the edge of tumor by detecting the orientation of the health white matter nerve fiber track in brain surgery^[3,4] .

However, Lu-Chipman’s polar decomposition assumes that the system is a non-commutative chain of diattenuator, retarder, and polarization depolarizer, which may not necessarily correspond to the complex of biological tissues structure^[5]. Instead of analyzing the measured MM, Jain et al. extract polarization ellipse parameters (PEPs) from the output Stokes vectors and produces a spatially resolved map of PEPs. It also has the potential to assist guide tumor resection by identifying nerve fiber tracks^[1,5,6] .

3.2 PEPs imaging

3.2.1 Method of polarimetric measurements and analyze

Fig. 5. (a) Structural diagram of the polarization imaging device. LC, L1 and L2 are lenses. LCR is liquid crystal retarders, and LP is linear polarizers. (b) Schematic diagram of polar coordinate setting. Define the probing beam as the origin of the polar coordinate, and each point can be defined in this coordinate system by (ρ,ϕ). S_in is expressed in ∑in and S_out is expressed in ∑out. Figure adapted from the Ref. [1].

The instrument used in measurement and imaging is shown in Fig. 5(a), nearly the same as before shown in Fig. 1. To better describe the incident and reflection position of light on the material, they established the coordinate system as shown in Fig. 5(b). So the reflection position of light at (ρ,ϕ)(ρ,ϕ)(ρ,ϕ)can be expressed by a Stokes vector^[5]:
S⃗out(ρ,ϕ)=(Iout(ρ,ϕ),Qout(ρ,ϕ),Uout(ρ,ϕ),Vout(ρ,ϕ))T,(10)\vec{\bf S}_{out} (ρ,ϕ)={(I_{out} (ρ,ϕ),Q_{out} (ρ,ϕ),U_{out} (ρ,ϕ),V_{out} (ρ,ϕ))}^T, \tag{10} Sout(ρ,ϕ)=(Iout(ρ,ϕ),Qout(ρ,ϕ),Uout(ρ,ϕ),Vout(ρ,ϕ))T,(10)
and it can be calculated in any incident light state S⃗in\vec{\bf S}_{in}Sin with MM^[5] by
S⃗out(ρ,ϕ)=M(ρ,ϕ)S⃗in.(11)\vec{\bf S}_{out}(ρ,ϕ)=M(ρ,ϕ) \vec{\bf S}_{in}. \tag{11} Sout(ρ,ϕ)=M(ρ,ϕ)Sin.(11)

Then, they extract the PEPs from the output Stokes vectors, which can help comprehend the polarization altering properties and produce a map of PEPs in the polar coordinate system (ρ,ϕ)(ρ,ϕ)(ρ,ϕ)^[6].

Ellipticity ϵϵϵ in the PEPs^[5] can calculate as:
ϵ(ρ,ϕ)=Mmin(ρ,ϕ)Mmax(ρ,ϕ),(12)ϵ(ρ,ϕ)=\frac{M_{min} (ρ,ϕ)}{M_{max} (ρ,ϕ) }, \tag{12} ϵ(ρ,ϕ)=Mmax(ρ,ϕ)Mmin(ρ,ϕ),(12)
which Mmin(ρ,ϕ)M_{min} (ρ,ϕ)Mmin(ρ,ϕ) and Mmax(ρ,ϕ)M_{max} (ρ,ϕ)Mmax(ρ,ϕ) can be calculated as:
Mmax2(ρ,ϕ)=12Qout2(ρ,ϕ)+Uout2(ρ,ϕ)+Vout2(ρ,ϕ)+12Qout2(ρ,ϕ)+Uout2(ρ,ϕ),(13)M^2_{max} (ρ,ϕ)=\frac12 \sqrt {Q^2_{out} (ρ,ϕ)+U^2_{out} (ρ,ϕ)+V^2_{out} (ρ,ϕ) }+\frac12 \sqrt {Q^2_{out} (ρ,ϕ)+U^2_{out} (ρ,ϕ) } , \tag{13} Mmax2(ρ,ϕ)=21Qout2(ρ,ϕ)+Uout2(ρ,ϕ)+Vout2(ρ,ϕ)+21Qout2(ρ,ϕ)+Uout2(ρ,ϕ),(13)
Mmin2(ρ,ϕ)=12Qout2(ρ,ϕ)+Uout2(ρ,ϕ)+Vout2(ρ,ϕ)−12Qout2(ρ,ϕ)+Uout2(ρ,ϕ).(14)M^2_{min} (ρ,ϕ)=\frac12 \sqrt {Q^2_{out} (ρ,ϕ)+U^2_{out} (ρ,ϕ)+V^2_{out} (ρ,ϕ) }-\frac12 \sqrt {Q^2_{out} (ρ,ϕ)+U^2_{out} (ρ,ϕ) } . \tag{14} Mmin2(ρ,ϕ)=21Qout2(ρ,ϕ)+Uout2(ρ,ϕ)+Vout2(ρ,ϕ)−21Qout2(ρ,ϕ)+Uout2(ρ,ϕ).(14)

Helicity h in the PEPs^[5] can calculate as:
h(ρ,ϕ)=sgn(Vout(ρ,ϕ)).(15)h(ρ,ϕ)={\rm sgn}(V_{out} (ρ,ϕ)). \tag{15} h(ρ,ϕ)=sgn(Vout(ρ,ϕ)).(15)

Orientation ψ in the PEPs^[5] can calculate as:
tan⁡[2ψ(ρ,ϕ)]=Uout(ρ,ϕ)Qout(ρ,ϕ),−π2≤ψ(ρ,ϕ)≤π2.(16)tan⁡[2ψ(ρ,ϕ)]=\frac{U_{out}(ρ,ϕ)}{Q_{out} (ρ,ϕ) } , -\fracπ2≤ψ(ρ,ϕ)≤\fracπ2. \tag{16} tan⁡[2ψ(ρ,ϕ)]=Qout(ρ,ϕ)Uout(ρ,ϕ),−2π≤ψ(ρ,ϕ)≤2π.(16)

And the degree of polarization Π in the PEPs^[5] can calculate as:
Π(ρ,ϕ)=Qout2(ρ,ϕ)+Uout2(ρ,ϕ)+Vout2(ρ,ϕ)Iout(ρ,ϕ).(17)Π(ρ,ϕ)=\frac{\sqrt {Q^2_{out} (ρ,ϕ)+U^2_{out} (ρ,ϕ)+V^2_{out} (ρ,ϕ) }}{I_{out}(ρ,ϕ) }. \tag{17} Π(ρ,ϕ)=Iout(ρ,ϕ)Qout2(ρ,ϕ)+Uout2(ρ,ϕ)+Vout2(ρ,ϕ).(17)

After that, Jain et al. used PEPs and added the degree of linear polarization Π_L, a valuable indicator with the ellipticity, to analyze the polarization on the anisotropic nanofibers^[5]. The degree of linear polarization can be calculated as:
ΠL(ρ,ϕ)=Qout2(ρ,ϕ)+Uout2(ρ,ϕ)Iout(ρ,ϕ).(18)Π_L (ρ,ϕ)=\frac{\sqrt {Q^2_{out} (ρ,ϕ)+U^2_{out} (ρ,ϕ)}}{I_{out}(ρ,ϕ) }. \tag{18} ΠL(ρ,ϕ)=Iout(ρ,ϕ)Qout2(ρ,ϕ)+Uout2(ρ,ϕ).(18)

They find that the degree of linear polarization is a valuable indicator for inferring the degree of alignment, which may also be valuable when assessing tissue pathology like brain tissue^[5].

3.2.2 Application on brain tissue

Jain et al. use a fixed human brain specimen and choose four regions, including white matter and gray matter, as the polarimetric imaging analysis areas. White matter consists of nerve fibers and has different structures, such as corpus callosum, external capsule, internal capsule, and gyri, making diverse nerves fiber structure in different parts^[1]. So they chose three regions in white matter and one in gray matter, as shown in Fig. 6. Region A and C are the typical gray matter area and white matter area, and region B and D are the other structs of white matter area.

Fig. 6. (a) Image of the fixed health human brain specimen in coronal cross section. (b) the enlarged image of region A-D mark in (a). The red points are the measurement points r_i. GM—Gray Matter, EC—External Capsule, IC—Internal Capsule, PCG—Pre-Central Gyrus, CC—Corpus Callosum. Figure from the Ref. [1].

As mentioned in the last section, the spatial distribution of the degree of linear polarization can most intuitively understand the arrangement degree of fiber samples. To quantify the differences between the images and simplify the analysis, they carried out a radical analysis around each rir_iri by calculating the average degree of polarization <ΠL(ϕ)>ρ<Π_L (ϕ)>_ρ<ΠL(ϕ)>ρ in different azimuthal angles ϕϕϕ ^[5]. Image of the spatial distribution of the degree of linear polarization (ΠL(ρ,ϕ)—image)(Π_L (ρ,ϕ)—image)(ΠL(ρ,ϕ)—image) in point r1r_1r1 of region A is shown in Fig. 7(a), and its radical analysis is shown in Fig. 7(b), which the value of ϕmaxϕ_{max}ϕmax when <ΠL(ϕ)>ρ<Π_L (ϕ)>_ρ<ΠL(ϕ)>ρ is maximum can figure out by the Gauss Fit. When change the orientations for the linear polarization S⃗in{\bf \vec{S}}_{in}Sin of the probing beam, we can get a new ϕmaxϕ_{max}ϕmax and the corresponding value of <ΠL(ϕ)>ρ<Π_L (ϕ)>_ρ<ΠL(ϕ)>ρ, then we can make a diagram between ϕmaxϕ_{max}ϕmax, maximum of<ΠL(ϕ)>ρ<Π_L (ϕ)>_ρ<ΠL(ϕ)>ρ and orientation of linear input states S⃗in{\bf \vec{S}}_{in}Sin , as shown in Fig. 7（c）(d). Different polarization state S⃗in{\bf \vec{S}}_{in}Sin of Jain et al.'s choice^[6] was shown in Table 1.

Fig. 7. Result of point r_1 in region A, a gray matter area. (a) The Π_L (ρ,ϕ)—image in output coordinate system ∑out. The input polarization state (S_in=x) describes in the upper coordinate in red. The red circle is the size and position of the probing beam. (b) The radical analysis in the situation of (a). (c) The value of ϕ_max when <Π_L (ϕ)>_ρ is maximum with different orientations for the linear polarization S _in of the probing beam. The orientations were from π/2 to -π/2 in ∑out. (d) Peak value of <Π_L (ϕ)>_ρ at ϕ_max with different orientations for the linear polarization S_in of the probing beam. The orientations were from 0 to π in ∑in. Dotted line in (c)(d) represents the colloidal suspension, which is an isotropic medium. Figure from the Ref. [1]

Table 1. The polarization state S_in used to probe the sample. Table adapted from the Ref. [6]

Polarization state	Stokes vector
Linear	S⃗LX=(1+100)T{\bf \vec{S}}_{LX}=\left(\begin{matrix}1 & +1 & 0 & 0 \end{matrix}\right)^TSLX=(1+100)T
	S⃗LY=(1−100)T{\bf \vec{S}}_{LY}=\left(\begin{matrix}1 & -1 & 0 & 0 \end{matrix}\right)^TSLY=(1−100)T
Linear diagonal	S⃗L+=(10+10)T{\bf \vec{S}}_{L+}=\left(\begin{matrix}1 & 0 & +1 & 0 \end{matrix}\right)^TSL+=(10+10)T
	S⃗L−=(10−10)T{\bf \vec{S}}_{L-}=\left(\begin{matrix}1 & 0 & -1 & 0 \end{matrix}\right)^TSL−=(10−10)T

　
Similarly, for other points rir_iri, we can also get the relation between ϕmaxϕ_{max}ϕmax, maximum of <ΠL(ϕ)>ρ<Π_L (ϕ)>_ρ<ΠL(ϕ)>ρ and orientation of linear input states S⃗in{\bf \vec{S}}_{in}Sin. Compared with Fig. 8(a) and (c）, we can easily find the differences between gray matter and white matter. These results show gray matter has isotropy and the slope close to -1; the white matter has anisotropy, and the slope close to 0. It also proves the reproducibility of this method^[1]. Region B and D can also get the same result.

Fig. 8. Result of all points in region A and C. (a)(c) The value of ϕ_max when <Π_L (ϕ)>_ρ is maximum with different orientations for the linear polarization S_in of the probing beam. The orientations were from π/2 to -π/2 in ∑out. (b)(d) Peak value of <Π_L (ϕ)>_ρ at ϕ_max with different orientations for the linear polarization S_in of the probing beam. The orientations were from 0 to π in ∑in. Region A are the typical gray matter area, and region C is the typical white matter area. Figure from the Ref. [1].

To quantitatively evaluate the slope, Jain et al. calculate the derivative of each input state and take the median of its distribution, defining it as the slope parameter mmm. The result shows that mmm is close to -1 in all the gray matter, mostly between -0.5 and 0 in the white matter^[1]. Therefore, the value of mmm have strong correlation with tissue structure.

The direction ϕmaxϕ_{max}ϕmax, the azimuthal angle where <ΠL(ϕ)>ρ<Π_L (ϕ)>_ρ<ΠL(ϕ)>ρ is maximum, is perpendicular to the fiber orientation, and the probing state S⃗max{\bf \vec{S}}_{max}Smax is roughly perpendicular to ϕmaxϕ_{max}ϕmax, so S⃗max{\bf \vec{S}}_{max}Smax can represent the average alignment direction^[1]. And its value can be estimated by the average orientation parameter ψψψ over ρρρ along ϕ=ϕmaxϕ=ϕ_{max}ϕ=ϕmax, which describe the orientation of the major axis of the ellipse in PEPs^[1]. So, by using mmm and the average orientation parameter ψψψ, the average alignment direction and degree of alignment of nerve fiber in the measurement point rir_iri can be encoded in ellipse and plotted onto the image^[1], as shown in Fig. 9. From the shape of the ellipses, we can infer the degree of the alignment of the tissue visually.

Fig. 9. The ellipse-encoding visualization of average alignment direction and degree of alignment for all the measurement points r_i, which is marked with red dots. The orientation of the major axis in the ellipse indicates the average alignment direction, and the ratio of the minor axis to the major axis represents the value of m. For high degree of alignment, the ellipse approaches a line; on the contrary, the ellipse becomes a circle. GM—Gray Matter, EC—External Capsule, IC—Internal Capsule, PCG—Pre-Central Gyrus, CC—Corpus Callosum. Figure from the Ref. [1].

In conclusion, using PEPs imaging, we can distinguish isotropy and anisotropy of the tissue by the value of mmm, which could be used to distinguish the boundaries of tumors accurately. We also can distinguish the degree of alignment inside white matter and the orientation of the nerve fiber by ellipse-encoding visualization, which could be used to guide the tumor resection^[1].

4. Conclusion

In China, brain tumor represented by glioma is one of the most challenging cancers to be cured. Accurately determining the boundary of tumors is very important to tumor resection^[1]. In this study, we give the basic principle of polarization imaging, introduce two latest polarization imaging and analyze methods applicated in brain tumor detection. Lu-Chipman polar decomposition imaging shows the result more visually through the image of depolarization, scalar retardance, and the azimuth of optical axis. However, this analysis method was founded in 1996, and this method has some disadvantages in the analysis of brain tissue with the further understanding of brain tissue these years. PEPs imaging extracts the parameters, which can reflect the polarization altering from the output Stokes vectors, realizing distinguishes isotropy and anisotropy of the tissue and the degree of alignment of the tissue, but the results are not visually enough.

There have been many new methods for polarization analysis and imaging in recent years. At present, the pathogenic mechanism of brain tumor is still not precise. Using polarization imaging techniques to detect brain tumor is still a promising research direction in the future.

References

A. Jain, L. Ulrich, M. Jaeger, P. Schucht, M. Frenz, and H. Gunhan Akarcay, “Backscattering polarimetric imaging of the human brain to determine the orientation and degree of alignment of nerve fiber bundles,” Biomed Opt Express 12, 4452-4466 (2021).

H. H. He, N. Zeng, R. Liao, and H. Ma, “Progresses of polarization imaging techniques and their applications in cancer detections,” Progress in Biochemistry and Biophysics 42, 419-433 (2015).

P. Schucht, H. R. Lee, H. M. Mezouar, E. Hewer, A. Raabe, M. Murek, I. Zubak, J. Goldberg, E. Kovari, A. Pierangelo, and T. Novikova, “Visualization of white matter fiber tracts of brain tissue sections with wide-field imaging mueller polarimetry,” IEEE Trans Med Imaging 39, 4376-4382 (2020).

O. Rodriguez-Nunez, P. Schucht, E. Hewer, T. Novikova, and A. Pierangelo, “Polarimetric visualization of healthy brain fiber tracts under adverse conditions: ex vivo studies,” Biomed Opt Express 12, 6674-6685 (2021).

A. Jain, A. K. Maurya, L. Ulrich, M. Jaeger, R. M. Rossi, A. Neels, P. Schucht, A. Dommann, M. Frenz, and H. G. Akarcay, “Polarimetric imaging in backscattering for the structural characterization of strongly scattering birefringent fibrous media,” Opt Express 28, 16673-16695 (2020).

M. Hornung, A. Jain, M. Frenz, and H. G. Akarcay, “Interpretation of backscattering polarimetric images recorded from multiply scattering systems: a study on colloidal suspensions,” Opt Express 27, 6210-6239 (2019).

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[原文] Application of Polarization Imaging Techniques in Brain Tumor Detection 偏振成像技术在脑肿瘤检测中的应用