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Abstract
Polarization imaging is a powerful tool, which can be applied in biomedical diagnosis and many research fields. Here, we propose a new application of the indices of polarimetric purity (IPPs) composed of P1, P2, P3, to describe the glucose concentrations (GC) changes in the scattering system. The results suggest that P1 of the IPPs is a better indicator to GC in the solution than the degree of polarization (DoP) for the forward scattering detection. Meanwhile, the fitting relation among radius of scattering particle, GCs and P1 parameter has also been calculated, in which the error of inversion is no more than 4.73%. In the backscattering detection, the fitted frequency statistical histogram of the IPPs is used to measure the GCs, and their modes can represent changing trend of GCs.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As one of the intrinsic properties of light, in recent years, the polarization state of the light has attracted a lot of attentions due to its great application potentials in communications [1–3], navigation [4,5] and detection [6–8]. In general, the absorption and scattering of light by complex dispersion media lead to the loss of target information and serious degradation of image quality. How to remove the haze and underwater environment on the target acquisition and image quality, accurately obtain the target information and improve the image clarity, is a hot topic of current researches [9–17]. Polarization, as one of the fundamental physical properties of light, can carry important target information, and improve imaging quality. Due to the powerful ability of describing the interaction between the polarization and materials, Mueller matrix (MM) polarimetry has attracted considerable attentions in recent years because of its numerous practical applications in various branches of science and technology [18–20]. In order to restore the information with large noise transmitted in a long distance or high scattering media, and effectively increase the image contrast and sharpness, polarization retrieve (PR) method based on MM of scattering system has been proposed to detect the target in the complex background, which can greatly improve the imaging performances [21–24]. The 4*4 MM represents the transfer function of the interaction between the optical system and polarized light and contains complete information about the optical properties of the medium [18–20]. Therefore, MM polarimetry provides the possibility to quantitatively describe the polarization characteristics of samples, which contain abundant morphological, structural and functional information. However, the media environment is complex and contains multiple polarization effects, and three basic medium polarization properties of attenuation, retardance and depolarization [19], which will contribute to the MM elements in a complex interrelated way, masking the underlying optical indicators [25]. Therefore, MM decomposition method [26–28] is very important to the real applications, and recently, Indices of Polarimetric Purity (IPPs) method [29,30] has been proposed to solve the problems of unclear physical meaning of MM elements and difficulty in extracting information.
High GC in blood make the infection difficult to cure, causing many diseases. The treatment of diabetes with higher or lower GCs, may also directly cause the patients died. Thus, a tremendous need exists for a noninvasive glucose monitor for diabetics to increase the monitoring frequency and better guiding treatments. A variety of optical methods have been proposed for monitoring blood glucose, including optical coherence tomography [31], near-IR spectroscopy [32], fluorescence techniques [33], photoacoustic techniques [34] and so on. Although some of these techniques have achieved considerable precision, until now, none of them has been approved for clinical use. Polarized light provides a possible way by utilizing changes in polarization caused by glucose concentrations (GC), which result from its optical activity, and induced changes in the scattering properties due to the rise in Refractive Index (RI) with additional glucose [35]. Mehrubeoglu et al. measured the cross-polarization patterns of glucose in the turbid medium [36]. Cote and coworkers [37,38], Chou et al. [39,40] and Pu et al. [41] used highly sensitive polarimetric technology to identify small rotations in polarized light associated with physiological glucose levels. Ablitt et al. using Monte Carlo (MC) simulation studied the effect of a turbid medium containing small chiral spherical particles on the scattering of polarized light [42]. Wang [43] developed an MC method for extracting the MM elements of birefringent turbid media containing glucose using either a single-scattering model or a double-scattering model. Wood et al. [44] developed an MC model for polarized light propagation in birefringent, optically active, multiply scattering media. The problem of combining the effects of birefringence and optical activity in scattering media is resolved through use of the Jones N-matrix formalism, from which the MM for the combined effect can be obtained for being applied to photons as they propagate between scattering events. Phan and Lo [45] used a Stokes-Mueller matrix polarimetry system consisting of a polarization scanning generator and a high-accuracy Stokes polarimeter to detect low-concentration glucose samples in aqueous solutions with and without scattering effects, respectively. With a poor signal/noise ratio, detection of the small rotations of the polarization plane caused by glucose in physiological levels is quite difficult.
In this paper, we have investigated the effect of optical activity on the scattering medium through the MC simulation, and use the parameters from Indices of Polarimetric Purity (IPPs) as the indicator to monitor GC changes in the solution in real time. Here, the GC effects on degree of polarization (DoP) and IPPs during forward scattering and back scattering processes have been studied fully in simulations. The results show that P1 in the IPPs is sensitive to the GC changes, and P1 could be used to detect and indicate the GCs in the solution. In the forward scattering detection, the larger the particle size, the more obvious the effect. In the backscattering detection, IPPs cannot indicate the GC changes in the medium. And the histogram of frequency distribution has been used to display the GC change. Meanwhile, we have also investigated the effect of the medium containing chiral material on the variations of the scattering effect, which show that chiral materials do not affect the scattering effect. We have demonstrated that the P1 is a significant indicator which can provide a feasible way for noninvasive GC measurements in clinical diagnostic applications.
2. Theoretical background
2.1 Improved MC method
Starting from the theory of atmospheric radiation transmission, researchers have proposed a variety of theoretical transmission models for different research purposes. The radiation transmission theory is a propagation theory which studies the scattering and absorption of the electromagnetic radiation in non-uniform and random medium. On the research of radiation transmission in the scattering medium, the discrete ordinate method [46], the accumulation-multiplication method [47] have been proposed. Simulation algorithms to realize the efficient transmission of light waves in media are constantly studied, in which the MC method has been used to simulate physical processes through statistical probability methods for decades. The MC method is able to model physical processes accurately by generating a large number of repeated computations that accumulate statistically to a meaningful and representative output value. Due to the wide applicability and computing advantages, and the rapid development of computers, MC algorithm is widely used to study the transmission characteristics of polarization information in turbid media [48–50]. In 1973, Kattawar first applied MC algorithm to the characteristics of the polarization transmission [48]. In 1977, Carter used MC algorithm to solve the polarization radiation transmission equation [49]. A so-called rejection method has been proposed to solve the conditional probability problem, which can effectively judge the photon collision transmission in MC algorithm. In 1979, Wending used MC algorithm to deal with the problem of atmospheric radiation transmission. After then, Wilson and Adam first used MC algorithm to study polarization information characteristics in biological tissue environment [50]. Subsequently, the application of polarization MC algorithm in biomedicine began to be widely developed.
Here, we simulated the transmitting process of 106 photons into the scattering medium, during which the photons would interact with the scattering particles and experience incessant scattering until they reached to forward and backward detector. The collision process of photons and scattered particles is simplified to simulate the interaction between light and media. Then, the single scattering distance of the photon in the scattering medium can be determined by the following formula:
where is a random number between 0 and 1, and ${u_e}$ is the extinction coefficient of the transmission or reflection media, related to the incident wavelength, size, type of the scattering particles, and RI of the surrounding medium, which can be calculated from Mueller's theory. When a photon is traveling in the medium, it will interact with the scattering particles (colliding) due to the scattering effect, and the light’s polarization states defined by Stokes vector will also change. At the same time, each state change will produce a new meridional plane, and the scattering occurs in the scattering plane. Therefore, to solve the new exiting Stokes vector, we need to rotate to the corresponding plane, and the expression is: where is the single-scattering MM of the scattering particles, $R(\gamma )$ and $R(\beta )$ are the rotation angles of two rotations between the meridional plane and the scattering plane, which are completed by the rotation matrix R expressed as:The MM that represents the contribution of glucose to a certain propagation path can be considered the matrix for circular birefringence $R(\alpha )$:
The backscattered Stokes vector (Sbs) of the total radiance received by the detector on the upper surface of medium can be expressed as:
2.2 IPPs of material media
In the Stokes-Mueller formalism, the Stokes vectors contain the intensity and the polarization states of incident and outgoing light, and the MM can fully describe the properties of the scattering medium. According to the concept of parallel decomposition of Stokes vectors, passive linear optical systems have been investigated, which can be used as a parallel combination of several pure elements [52]. On the other hand, a Hermitian (covariance) positive semi-definite matrix H, which can be extracted from the elements of the MM, gives the statistical information of the scattering medium such as entropy and depolarization index [53–55]. The relationship between the Hermitian matrix H and its corresponding Mueller matrix MM can be expressed as follows:
The eigenvalue spectrum of the H can be written in the following form [29],
Using this size arrangement, three peculiar quantities can be obtained, which can provide complete information of the polarimetric purity of the MM. These quantities are called the Indices of Polarimetric Purity (IPPs) [29]. The IPPs are obtained from the eigenvalues of H such that [29]
By the convexity property of MM, any scattering medium MM can be decomposed into four components, whose relative weights are determined by the magnitude of the eigenvalues of the H. Meanwhile, the IPPs are restricted by the following conditions:
The following quadratic relation between the polarization purity and the three indices of purity (P1, P2 and P3) can be obtained as [29]
3. Effect of optical rotation on polarization measurement
It is well known that the presence of asymmetric chiral molecules like glucose leads to rotation of the plane of linearly polarized light about the axis of propagation. Since optical rotation measurement may be an attractive non-invasive method for detecting glucose levels in human tissues, some research has been devoted to measure the polarization characteristics of scattered light in chiral turbid media [56–58]. Biological tissue is a very complex system, in which light scattering is produced by vast kinds of macromolecules, organelles, small water droplets and other particles. To simplify the simulation process, we can use the sphere scattering model to simulate these tissues. The model is an infinite plate medium, in which the simulated parameters of the scatterers include size, RI, scattering coefficients. The parameters of the medium around the scatterer include RI, absorptivity, optical activity. Based on previous knowledge, single-layer models can be used to simulate specific biological tissues [59]. Meanwhile, we define that the forward scattering photons are scattered and collected in the forward detector when they are far away from the incident plane, and the backward scattering photons are scattered backward and collected in the backward detector. The system model is shown in Fig. 1. We focus on the effect of GC changes in the surrounding medium to the resulted polarization states.
3.1 Effect of optical rotation on the MM
Based on Mie theory, MC simulation has been used to simulate the effective back scattering MM of a scattering medium containing spherical scatters and glucose. We added the optical activity module to the previous MC program [60]. Firstly, the wavelength of incident light is 589 nm, and scattering particles with a radius of 0.5 µm, the RI (${n_s}$) of polystyrene spherical particles is 1.57, but the RI (${n_0}$) of the medium is 1.33. In the randomly scattering media, absorption coefficient (${u_a}$), scattering coefficient (${u_s}$) and GC (c) are set as 0.01 cm−1, 1 cm−1 and 300 g/dl respectively. We use a 200*200 pixels’ detector, to receive the scattered photons. Because of the rotation in polarized light caused by glucose, the Ref. [61] shows that some intensity patterns of the matrix elements have obvious rotation about the incident point, which is different from the matrix elements without glucose. The rotation of matrix element also happen in our simulation results, and are basically consistent with the results in Ref. [61], which demonstrates the correctness of our simulation system. In Fig. 2, with the other optical parameters unchanged, we exhibit sixteen MM elements of the turbid medium without glucose (Fig. 2(a)) compared with the MM elements of the turbid medium with glucose (Fig. 2(b)). The rotation material has little effect on matrix elements M11 which represents the reflectivity of non-polarized light and M44 which represents the reflectivity of circularly polarized light, because the rotation material does not change the parts of non-polarized light and circularly polarized light. But matrix elements M22, M23, M23 and M24 have obvious rotation angles and are greatly affected by optical rotation material. Therefore, these MM elements can be used to detect whether the system contains chiral material. As a chiral material, glucose has preferential handedness because it has an asymmetric molecular structure. Thus, glucose interacts differently with left and right circularly polarized light, causing the linearly polarized plane to rotate [62].
Fig. 2. Intensity results comparison of resulted back scattering MMs from turbid media without glucose (a), with glucose (b)
3.2 Scattering effect on depolarization in optical rotation media
The simulation results show that the improved MC algorithm can accurately simulate the effect of optical rotation on polarization detection. Meanwhile, we have also investigated the effect of optical rotation on its own scattering effect. Here, we choose to detect the forward scattering photons, in which the scattering particle radius and the range of scattering coefficient for the system are set as 0.05 µm and 1∼3 cm−1 respectively while the thickness of medium is 1 cm, and the other parameters are consistent with the above simulations. As shown in Fig. 3, both DoP of emergent light and IPPs of scattering system decreases with increasing scattering effect. This is consistent with the Mie scattering theory, and the greater the scattering effect, the higher the collision probability between photon and particle, so there will be serious depolarization effect. Meanwhile, as depicted in the Fig. 3(b), the changing trend of P1 is still greater than the DoP, which means that P1 is more sensitive to the changing scattering coefficients. P1 is the relative measure of the difference between the weights of the two more significant pure components of the system. The solution (system) with different scattering coefficients can be represented by different MM, which can be mapped into different pure systems according to IPPs decomposition. This property magnifies the representing differences of MM for different systems. Thus, P1 is more sensitive to slight change of system. These results also suggest that chiral material does not affect the main scattering effect of the turbid medium. The existing asymmetric chiral molecules, such as glucose, would result in the rotating plane of linearly polarized light around the propagation axis [63].
Fig. 3. Cumulative IPPs and polarization purity of medium and DoP of emitted light, for forward scattered, as a function of the scattering coefficient.
3.3 Characterizing GC with IPPs from forward scattering
We found that changes in the polarization of emitted light caused by glucose can be observed in the MM patterns of the turbid medium. The MM patterns of backscattering and forward scattering changes with the GC changes in the turbid medium, which enables the non-invasive monitoring of the GC in the highly scattering tissues. Here, we set up homogeneous mono-dispersion system, in which the size of scattering particles in the medium is 0.05 µm, 0.125 µm and 0.5 µm respectively, and the medium depth (thickness) and the GC are set as L=1 cm and 0–400 g/dl respectively. We use MC simulation to track how the incident polarized light varies with GC changes in the turbid medium with scattering particles of different sizes. The IPPs of the corresponding scattering system and DoP of the emitted light under different GCs can be obtained. From Eq. (11), we can calculate and obtain polarization purity (${P_\Delta }$) as shown in Fig. 4, from which we can see that the polarization purity (${P_\Delta }$) is not sensitive to the GC changes. According to the simulation data, we can conclude that the P1 parameter in the IPPs can well describe the GC change in the turbid medium. In other words, GC changes can also result in the changes of relative difference between pure components of the system.
As shown in Fig. 5, in medium with three-size scattered particles, The P1 has an overall downward tendency with the increased CG. The scattering particles in the medium are 0.05 µm, 0.125 µm and 0.5um respectively, and DoP of the emitted light decreases slightly with increasing GCs (Fig. 5(a)). However, when the scattering particle size is increased to 0.5 µm, the DoP has not changed with the increasing GCs in the medium, and remains small (Fig. 5(b)). However, the changing trends of P1 with the increasing GCs in three turbid media systems are obviously greater than those of DoPs (Fig. 5). At the same time, the larger the particle size for the forward-scattering environments, the more obvious the changing trend of P1. This suggests that for the forward-scattering environments, the parameter of P1 can well monitor GC changes in solution.
Fig. 5. Cumulative P1 of medium and DoP of emitted light, for forward scattered, as a function of the GC
In order to describe the GCs in solution quantitatively, we hope to obtain the functional relations of the GCs in the scattering medium with the P1 value. However, for scattering particles with different radius, the relation between P1 parameter and GCs should be also different. From the data tracing points in Fig. 5, it can be fitted by polynomials. According to the curve of GCs and P1 parameters in Fig. 5, we use the MATLAB platform to obtain the fitting function of each curve. Then several polynomials of P1 parameter about concentration are obtained under different scattering particle sizes. Finally, the function relation between each constant term and the radius of the scattering particles can be obtained by fitting.
According to the curve in Fig. 5, if the scattering particle size is fixed, it is assumed that the P1 parameter has a cubic relation with the GCs. It is represented by the following formula:
where $a,b,c,d$ are constants, x is GC.After fitting the three curves in Fig. 5, the cubed function can be expressed as follows:
Then the relation between the radius and the four constants in the cubic polynomial is also fitted. By fitting the relation between the four constants in trinomial and the radius of scattering particle, we find that the relation between them and the radius is quadratic. The specific relationship can be expressed as follows:
In order to verify the accuracy of the above fitting results, we have respectively selected some GCs in the range of 0–100 g/dl, 100–300 g/dl, 300–400 g/dl and 400–500 g/dl for the scattering system with the particle radius of 0.05 µm. The P1 values of scattering system for forward detection can be obtained by MC simulation. Through multiple verifications, the inversion error values on each interval were obtained and recorded in Table 1.
Table 1. Inversion of GCs and the relative error
As can be seen from Table 1, the errors between the presetting GCs and the retrieving GCs are different with the changes of the presetting GCs, and the bigger error occurs in the GCs’ ranging from 0 to 100 g/dl, but the error is still acceptable. Through verification, the function values generated from the fitting of the cubic polynomial basically coincide with the original simulation data (presetting GCs), and the error does not exceed 5% (the biggest error is 4.73%). Using this inversion method, the GCs in the media can be retrieved according to the value measured by polarimetric technology, which can provide an effective scheme for quantifying the blood glucose level of diabetic patients.
3.4 IPPs from backward scattering
The above simulation studies the behaviors of the forward scattering photons, and meanwhile we have also constructed a transmission model of optical-rotation MC to investigate the MM of backward scattering of medium in the homogeneous dispersion system. The system parameters set here are consistent with the above simulation model. Firstly, we can obtain the DoP of backscattering light from the media with three different scattering particles, as depicted in Fig. 6(a). We have found that the DoP of detected backward scattered light could not show the GC changes in the turbid media. Next, we simulate four different incident lights travels in these scattering systems, which are natural light, horizontally linearly polarized light, 45-degree linearly polarized light and right-circularly polarized light, respectively. The MMs representing the polarization characteristics of the scattering systems can be obtained by calculation, and then the corresponding parameters can be obtained according to the calculation formula of IPPs. As depicted in Fig. 6(b), the obtained three IPPs parameters (P1, P2, P3) and the polarization purity could not represent the GC changes in the medium (here, is not shown in the diagram)
Fig. 6. Cumulative IPPs of medium and DoP of emitted light, for backward scattered, as a function of the GC.
Based on the previous results, we infer that the parameter of P1 has some characteristics that can represent the GC changes in the turbid systems. So we tried to use mathematical tools to see if the spatial distribution of P1 could reflect the GC changes in the solution. Here, we chose the particle radius in the solution as 0.05 µm as an example. Figures 7(a-d) present four spatial distributions of parameter of P1 (200×200) for the turbid systems with different GCs. Figure 7(a) presents turbid systems without GCs.Fig. 7(b-d) present the resulted P1s with increasing glucose concentrations. The corresponding histograms N (P1) of the P1 distribution of the turbid medium with different GCs was counted on the MATLAB platform. Here, we divide the interval of P1 values into 30 equal parts, count the numbers on each interval separately, and the P1 values on each interval are replaced by the coordinates of the centerline position between the intervals. The histograms are fitted to the curve, and this is shown in the Figs. 7(e-h), and here N is the normalized frequency. Looking at the first row of images in Figs. 7, we find that the P1 parameter distributions of the two media with or without glucose is significant different, but in solutions with different GCs, the spatial distribution of P1 is indistinguishable. As shown in Fig. 7(e)-(h), histogram of frequency distribution of P1 parameter, have been found that the position of the maximum point gradually moves to the left with increasing GCs in the solution. We calculated the modes corresponding to the P1 parameters of these scattering systems, and record their specific values. As shown in Table 2, with increasing GCs in turbid media, the mode of P1 parameter is decreased. The mode which is accounting for most ratio in the detector can present the overall tendency of P1. The P1 is decreased with increasing GC, which is consistent with forward scattering.
Fig. 7. IPPs parameter spatial distribution P1 (200*200) and histograms of frequency distribution N of P1 (200*200) of four GCs in the turbid medium.
Table 2. The mode of statistical P1 parameter and the proportion of corresponding numbers
4. Conclusions
In this paper, based on the IPPs, we have investigated the effect of GC changes in turbid media on light depolarization. The GC influence on polarization parameters has been studied in the forward scattering and backward scattering detections. For the forward scattering, in the scattering systems composed of scattered particles of different sizes, the DoP of emitted light cannot represent the GC changes in the solution. However, P1 parameter of IPPs tends to decrease with increasing GCs. And the larger particle size of the turbid media, the more significant the change in P1 with GCs. Meanwhile, the fitting relation between radius of scattering particle and GCs and P1 parameter is calculated, and the error of inversion is no more than 4.73%. The P1 value cannot directly show the GC changes in the solution for the backscattering detection. However, using the frequency distribution histogram of P1, the mode of spatial distribution P1 is found to represent the GCs. The method we employ in this paper is the control variable method, in similar words, GC being the only variable. To be honest, it can’t respect the real human tissues preciously. However, the above results demonstrate that the IPPs are exactly more sensitive than other standard polarization indexes, revealing significant potential for noninvasive detecting of GC levels in human tissues. We will continually pursue the meaningful topic and establish more precious and intelligent models with experimentally support.
Funding
National Natural Science Foundation of China (11804073, 61775050, 61971177); Fundamental Research Funds for the Central Universities ((PA2019GDZC0098); Anhui Key Laboratory of Polarization Imaging Detection Technology (2018-KFJJ-02).
Disclosures
The authors declare no conflicts of interest.
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