{'fraction_non_alphanumeric': 0.033094578344748264, 'fraction_numerical': 0.014638992000836513, 'mean_word_length': 5.013203395158755, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Depth estimation of tumor invasion in early gastric cancer using scattering of circularly polarized light: Monte Carlo Simulation study

Quantitative depth estimation of tumor invasion in early gastric cancer by scattering of circularly polarized light is computationally investigated using the Monte Carlo method. Using the optical parameters of the human stomach wall and its carcinoma, the intensity and circular polarization of light scattered from pseudo-healthy and cancerous tissues were calculated over a wide spectral range. Large differences in the circular polarization with opposite signs, together with the large intensity, are obtained at wavelengths 600 nm and 950 nm. At these two wavelengths, the sampling depth of the biological tissues can be modulated by tuning the detection angle. In bi-layered pseudo-tissues with a cancerous layer on a healthy layer and vice versa, the degree of circular polarization of scattered light shows systematic changes depending on the thickness and depth of the cancerous layer, which indicates the feasibility of in vivo quantitative estimation of cancer progression in early gastric cancer.


Introduction
Recent developments in diagnostics and treatments are one of the major causes for the gradual reduction in mortality rate due to gastric cancer in the past few decades [1].
Nevertheless, gastric cancer remains a serious health issue, especially in East Asian countries.
Above all, for Japanese males, the age-standardized incidence rate of stomach cancer per 100,000 is extremely high, which is probably because of the high infection rate of chronic Helicobacter pylori and large intake of salted foods.
Early detection of cancer is the most crucial strategy for increasing the chance of effective treatment and reducing mortality due to cancer. Accurate diagnostic results regarding the tissue types, locations, distributions, and progress degrees of the cancer are required for the appropriate treatment. Gastric cancers have been classified according to the TNM staging system, where the three key parameters are T (size or direct extent of the primary tumor), N (degree of spread to regional lymph nodes), and M (presence of distant metastasis). Early-stage cancers have been identified almost only with T stage classification ( Figure 1) [2][3][4]. The cancers categorized in the Tis and T1 stages rarely spread to a lymph node. It can be removed without sacrificing the entire organ by using endoscopic surgical procedures, such as endoscopic submucosal dissection (ESD) [5][6][7]. Approximately 20 to 30 % of cancers in T2 stages spread to lymph nodes, which have been treated by surgical treatment with lymph node dissections. Therefore, accurate tumor invasion depth in gastric cancer is of primary importance in determining the therapeutic approach. Recently, imageenhanced endoscopy (IEE), represented by narrow-band imaging (NBI) [8], which is used for the diagnosis of tissue types and degree of spread in cancer has drastically improved the quantitative and qualitative diagnosis of gastrointestinal cancers. Magnifying endoscopy with NBI (ME-NBI) has been used for determining the invasion depth by observing the intrapapillary capillary loops for esophageal tumors and pit patterns for colon tumors [9].
For these tumors, ME-NBI has obtained a high diagnosis rate. In contrast, for the depth estimation of tumor invasion for gastric cancers, magnifying endoscopy with indigo carmine dye contrast or endoscopic ultrasonography (EUS) has been used. The correlation between the invasion depth and abnormal blood vessels or mesh patterns has been reported in recent years [10,11]. However, only a few cases were analyzed for these studies. Methods to determine the invasion depth in gastric cancer have not yet been established, necessitating a 3 breakthrough from a different technological viewpoint.
One of the techniques is the optical biological observation using light polarization [12].
Scattering of incident linear polarized light (LPL) provides structural information of biological tissues. This is carried out by analyzing the degree of birefringence and depolarization from the supplying images that reflect the anisotropy of the tissues [13][14][15][16][17].
However, in a turbid medium like biological tissues, LPL is readily lost by multiple scattering, therefore, it provides poor information from deep regions of tissues. In contrast, circularly polarized light (CPL) has comparatively more endurance against multiple scattering than LPL [18,19]; its polarization survives even after a large number of scattering events in biological tissues with a thickness of several millimeters. Utilizing this characteristic of CPL, Meglinski et al. [20] pioneered the application of CPL for optical cancer detection by mapping the polarization-dependent optical properties of tissues on the Poincaré sphere. They verified the use of the Stokes vector of backscattered light from tissues for non-invasive optical tissue biopsy. Kunnen et al. [21] reported that when CPL impinges on a human lung tissue, the polarization states of the scattered light show clear differences between normal and tumor tissues. These findings were interpreted as follow: the CPL beams impinged on a biological tissue are scattered multiple times by cell nuclei and gradually depolarized. In the Mie regime, where the scatterers (cell nuclei) are larger than the wavelength, the depolarization sensitively depends on the size of the scatterers [22]. Therefore, the resultant degree of circular polarization (DOCP) of the scattered light provides structural characteristics, such as the size, anisotropy, distribution, and density of the cell nucleus. In cancerous tissue, the cell nuclei are enlarged due to the abnormal growth of cancer, which can be detected by the difference in the DOCP of the normal and tumor tissues. Following these reports, the polarimetry technique for tissue observation using the polarization of light, including CPL, has been extensively studied [23][24][25][26][27][28][29][30]. Recently, the polarimetry technique has been applied for grading colon cancer [31], Alzheimer's disease [32], and early-stage breast cancer [33].
Following these studies, we also have investigated the scattering phenomena of incident CPL in biotissues for cancer detection. Before then, we have studied CPL-emitting diodes, also called spin-LEDs, and we have achieved fully polarized CPL emission at room temperature [34], electrical switching of CPL helicity at high speed [36] [39]. The line-scanning experiments along a region incorporating normal and cancerous parts show steep changes in the DOCP value depending on the state of the tissue, which indicates the feasibility of this technique in identifying the carcinoma concealed in healthy tissues. Meanwhile, we conducted theoretical and computational analyses using Monte Carlo (MC) simulation methods for the scattering process of CPL with cell nuclei in pseudo-tissues [40]. The single scattering of CPL against particles with diameters corresponding to cell nuclei in healthy and cancerous tissues was investigated first. Then, we introduced them into multiple scattering systems in pseudohealthy and cancerous tissues to clarify the contribution of optical and structural conditions to the resultant polarization of scattered light. These studies revealed three major pointsthe resultant DOCP values of the scattered light showed obvious differences between pseudo-healthy and cancerous tissues irrespective of the scattering angles, the intensity of scattered light obtained from healthy and cancerous tissues are approximately the same, and the scattering volume (depth) can be controlled by changing the scattering angle. The third deduction suggests that the depth profile of the tissue can be obtained by analyzing the scattering-angle dependences in the DOCP of the scattered CPL.
In this study, we computationally investigated the quantitative measurement of tumor invasion depth in layered structures that consists of cancerous and healthy tissues. In the preliminary stage, the wavelengths were optimized for detecting cancer and estimating distributions in terms of the polarimetric response for biological tissues and the intensity of scattered light. Subsequently, the changes in the DOCP values were analyzed using the optimized wavelengths for various structure of biological tissues; a cancerous layer lying on the surface [41], and a cancerous layer hiding under the healthy tissue.

Experimental methods
The polarization state of light is expressed by the Stokes vector , given by the equation In this study, we used the polarization-light MC algorithm developed by Ramella-Raman et al. [43], also known as "meridian plane MC algorithm", to investigate the intensity, polarization, and passage distribution of scattered light. In the polarization-light MC algorithm, light beams are traced using absorption and scattering accompanied by depolarization in a medium. A light beam propagates for a random length associated with the mean free pass in the medium, ( + ) −1 , where and are the absorption and scattering coefficients of the medium, respectively. The propagation length ∆ is determined by a random number (0 < ≤ 1), ∆ = − ln( ) ( + ) ⁄ . After a traveling in the medium for a distance ∆ , the light beam runs into a scatterer and is subjected to a scattering event. In each scattering process, a random direction and a particular axis are chosen by the rejection method depending on the single scattering phase function in the Mie scattering process [44]. Accordingly, the polarization state after the scattering event is rewritten by the Stokes vector with respect to the meridian planes, which is determined by the new direction and axis. Simultaneously, a light beam is absorbed at a certain ratio, All Stokes parameters are multiplied by to denote a decrease in intensity due to absorption. Subsequently, the light beam travels in a new direction. After this series of processes, the beam going outwards the medium are collectively detected, and the beam whose intensity is less than a certain value falls out as 6 they are fully absorbed. Because of the refraction at the interface to the air, the light beams having an angle less than the critical angle, sin −1 (1.00 1.33 ⁄ ) ≈ 48.75° , can emit an outward air and can be detected, whereas the beams with angles larger than the critical angle are totally reflected and returned to the scattering process loops. These totally reflected light beams scarcely arrive at the surface again, emit outward, and are detected, therefore they are ignored in this study. By repeating these processes from injection to outgoing or full absorption, we analyzed the detected beam for every angle of emergence, hereafter called "detection angle".
Polarized MC simulations were carried out for pseudo-biological tissues with a bi-layered structure: a cancerous layer on a healthy layer and vice versa. The cancerous and healthy tissues are the aqueous dispersions of the particles with 5.9 and 11.0 m in diameters, respectively. These diameters correspond to the typical nuclear size [21,45] and the average values we experimentally measured in biological specimens [39]. The refractive indices of the particle and matrix were chosen as 1.59 and 1.33, respectively. The first layered structure, in which a cancerous layer with various thicknesses t is located on a healthy layer, corresponds to cancer that is progressing from the surface to the interior of the tissues. In contrast, the second layered structure, where a cancerous layer lies deep with a depth , corresponds to a buried cancer without exposure to the surface.

Optical parameters
The optical parameters, scattering and absorption coefficients, were obtained using semiempirical formulae and experimental measurements obtained from the stomach wall and its carcinoma.
The approximate scattering coefficient ′ as a function of the wavelength is given by Eq. (1) [46,47].
In this equation, and are constants specified by the tissue type. For a stomach wall, = 792 (mm −1 ) and = 0.97 (no units) [48].
In this equation, is the oxidation degree of hemoglobin, = HbO 2 (HbO 2 + Hb) ⁄ , and and are heuristic scaling factors adjusted to match the absorption data currently available for each tissue. In ref. [48], the values of these parameters for a normal stomach wall were = 0.7 , = 0.01 , and = 0.8 . The absorption spectra of these three constituents, ,HbO 2 ( ), ,Hb ( ), and ,W ( ), were reported and summarized by Prahl et al. [49].
The obtained spectra of the optical parameters were adjusted by the experimental values for stomach wall tissue and stomach tumor tissue. The experimental data were obtained by an integrating sphere and semiconductor lasers of wavelengths 532, 820, and 914 nm. Figure   2 shows the spectra of the optical parameters of healthy stomach wall (red) and gastric cancer Therefore, we adopted the semiempirical spectra of the optical parameters as those for healthy tissues in this study. The difference in scattering coefficients between the healthy stomach wall and tumor tissue was negligible. As for the absorption coefficient, each experimental value for tumor tissue at three wavelengths was smaller (−2.7 ~− 3.4 %) than the values for normal tissues. The average rate of decrease was approximately 3.0 %. Therefore, the spectra obtained by multiplying 0.97 to the spectra obtained by Eqn. (2) were employed as ( ) spectra for the tumor. These values were then introduced into the algorithms of Monte Carlo simulations to investigate the possibility of cancer detection using the proposed method.

Wavelength optimization for cancer detection
The intensity and polarization of light scattered from pseudo-healthy and cancerous tissues were calculated using the meridian plane MC algorithm. The optical geometry for the calculations is shown in Figure 3(a). The CPL beams whose polarization is 3 = +1 are irradiated into pseudo-biological tissues with an incident angle = 1°. The light beams that reflected directly from the surface and the light beams that underwent a few scattering events did not contain important information on the tissue. To exclude these light beams from consideration, the detection region is laid 1 mm from the point of incidence with a width of 1 mm, as shown using the thick red line on the pseudo tissue horizontally 1 to 2 mm in Figure   3  The total intensity shows large values within the range of 500 nm to 1300 nm, which is because of the small . To assess the ability to detect and estimate cancerous tissue depending on the wavelength, the figure of merit is defined as = Δ ���� × , and its spectral profile is shown in Figure 4(c). In the spectral range less than 500 nm and longer 9 than 1400 nm, significant performances in Δ ���� are lost owing to the weak intensities.
Meanwhile, in the spectral region from green to near infrared, inherits Δ ���� behavior, with a negative large peak at around 600 nm, a monotonic increase with large vibrations, and a large positive peak at around 950 nm. The spectral dependence of clearly indicates that 600 and 950 nm are the appropriate wavelengths for cancer detection, which has a significant difference in polarization with the opposite signs. The largest derived from the largest Δ ���� is also obtained at 1050 nm. However, there is a larger fluctuation at around 1050 nm than at 950 nm, due to which this excluded for the optimization. In conventional pulse oximeters, two wavelengths-665 and 880 nm-are used to evaluate the degree of oxygen saturation in the blood. Light at 665 nm has a larger absorbance for hemoglobin than oxyhemoglobin, whereas light at 880 nm has the opposite absorption characteristics [49]. The oxygen saturation is not evaluated by absolute values of absorption but by the ratios between two wavelengths in the pulse oximeters, which ensures the measurement accuracy. Similarly, the accuracy of cancer detection is expected to be enhanced by using the two wavelengths, 600 and 950 nm, which have opposite polarization tendencies for cancerous and healthy tissues.
In this study, we analyzed the reason for the opposite sign in Δ ���� between the two wavelengths by reviewing a single scattering behavior.  shown in Figure 4 cannot be applied to a transparent configuration that is employed, for example, in a fingertip pulse oximeter, because the forward scatterings are used dominantly.

Depth estimation of tumor invasion
The measurements of the depth profiles were verified using the two optimized wavelengths. An example of the distribution of simulated light beam paths is shown in the area of a biological tissue of Figure 3(a), which is obtained under the conditions that the detection angle is = 35 ± 5° , the wavelength = 950 nm, the medium comprises spheres of diameter = 5.9 μm, (a pseudo-healthy tissue), and a photon number of the incident CPL is 500,000. The simulated light beam paths under the other condition ( and ) are shown in Figure S1 to S4 in the Supporting Information. The distribution of light beams drastically varied with : changing from −90° to +90° while maintaining = 1°.
When is close to zero (vertical incidence), the detected light beams contain many light beams that dive deeper. In contrast, light beams scattered in a shallow volume tend to exit from the surface with a large . The averaged maximum depth of the detected light beams is defined here as , which can be paraphrased as a sampling depth.

Conclusions
We the tip of an endoscope, which we proposed in Ref. [40]. However, the tip of the endoscope is crowded because of the recent multi-functionalization of endoscopes. Alternatively, we propose to attach the endoscopic probe to the sidewall of an endoscope, which we obtained from the intravascular optical coherence tomography [50,51]. The smaller spatial restrictions will enable the longer semi-latus rectum of a parabolic mirror to detect light with a detection angle within a large angular range. To estimate the depth of invasion tumors in gastric cancer using the CPL scattering technique, experimental demonstrations using cancerous biological specimens at various stages of early gastric cancer are required.
Moreover, further developments are needed in the CPL emitter and detector based on spin-LEDs used for this technique.

Ethics approval
Ethics approval is not required to carry out this work.

AUTHOR CONTRIBUTIONS
N. N. was involved in conceptualization, investigation, calculations, writing-original draft preparation, writing-review and editing, project management. T. K. prepared the biological tissues and was involved in review and editing from the biological viewpoint.       Figure S5) and the depth of cancer ( Figure S6). Figure 7 (b) and (d) correspond and dependences of DOCP values shown in Figure S5(b)(d) and S6(b)(d), respectively.

FIGURE S5
The detection angle dependence of the calculated (a)(c) intensity and (b)(d)DOCP values for (a)(b) = 600 nm and (c)(d) 950 nm, respectively. The colors of the plot represents the thickness of cancer .