# Deep Learning of Diffuse Optical Tomography Based on Time-Domain Radiative Transfer Equation
## 2 Methodology  
### Model  
Throughout this paper, we work with a two-dimensional model for solving the RTE and performing image reconstruction. We consider a target tissue 4 cm \(\times\) 4 cm in size, which we divide into 28 domains to specify the positions of absorbers, as shown in Figure 1. We model the absorber as a round absorber with a diameter of 5 mm. This setup is based on an experiment using a phantom composed of polyurethane with titanium oxide (scatterer) and carbon black (absorber). In this paper, we present a two-dimensional model. We compare our RTE calculation and image reconstruction with results from the setup of the experiment, which is a three-dimensional cylinder model. The source of incident beam and detectors are located at the half of the cylinder in height. The details of the experiment and the 3D model will be separately reported. The optical properties of the phantom are characterized by  
\[\mu_{a}=0.21/\text{cm}\ \ \mu_{s}=22.45/\text{cm} \tag{1}\]where \(\mu_{a}\) and \(\mu_{s}\) are the absorption and scattering coefficient, respectively. Within this background material, an absorbing pole with the following coefficients  
\[\mu_{a}=0.64/\text{cm}\ \ \mu_{s}=22.63/\text{cm} \tag{2}\]  
is inserted. Scattering dominates absorption in the inserted pole as well as in the background. Absorption features appear in light emerging from the pole at the detectors, since the absorption coefficient is larger by a factor of three in the pole. This is a principle for detecting cancer positions in biological tissue. In this paper, we show the results for the cases in which the absorption coefficient of the absorber is 3 times larger than the surrounding tissue. However, we have also tested how smaller differences in absorption coefficient can be identified by deep learning. It is shown that our classification method can work even for cases in which the absorption coefficient is 1.5 times larger. The sources of the incident beams and detectors are located at eight points as shown in Figure 1, where the positions of the sources are labelled S1-S8 and positions of detectors are labelled D1-D8. We solve the following time-dependent radiative transfer equation in two-dimensional space to follow scattered and absorbed light,  
\[\frac{1}{c}\frac{\partial I}{\partial t}+\mathbf{n}\cdot\
abla I=-\mu I+\eta \tag{3}\]  
with  
\[\mu=\mu_{a}+\mu_{s}\eta=\mu_{s}\oint\phi(\mathbf{n},\mathbf{n}^{\prime})I(\mathbf{n}^{ \prime})d\Omega^{\prime}\,, \tag{4}\]  
where \(I\) is the specific intensity of light, \(\eta\) is the emissivity by scattered photons, and \(\phi\) is a phase function. The emissivity term represents radiative transfer from one direction \(\mathbf{n}^{\prime}\) to another direction \(\mathbf{n}\). We employ the Henyey-Greenstein function for \(\phi\), that is,  
\[\phi(\mathbf{n},\mathbf{n}^{\prime})=\frac{1}{2\pi}\frac{1-g^{2}}{1+g^{2}-2g(\mathbf{n} \cdot\mathbf{n}^{\prime})} \tag{5}\]  
Figure 1: The configuration of target tissue. The dimensions are 4 cm \(\times\) 4 cm. The tissue is divided into 28 domains, each of which can possess a round absorber with a diameter of 5 mm. Eight incident beam directions (labelled S1–S8) and eight detector positions (D1–D8) are shown. We divide the whole area into eight groups, each of which has four or two domains. where \(g\) is the scattering anisotropy parameter. The case for \(g=1\) or \(g=0\) means perfectly forward scattering and isotropic scattering, respectively. We take \(g=0.62\) for the simulations. This value is consistent with the phantom experiment. To solve Equation (3), we have developed a new solver, TRINITY (Time-dependent Radiation Transfer in Near-infrared Tomography)([PERSON] et al. [11]), which is based on our former steady-state RTE solver, ART (Authentic Radiative Transfer) [12]. In this method, the distribution of light rays is constructed independently from the the grids that possess source functions. The most important advantage is the simultaneous reduction of computational cost and numerical diffusion.