# A Review of Image Reconstruction Algorithms for Diffuse Optical Tomography
## 3 Forward Process
### Diffusion Approximation  
In DA, light propagation in the TD measurement is described by the following PDE [23]:  
\[\bigg{\{}\frac{1}{c}\frac{\partial}{\partial t}+\mu_{a}(\mathbf{r})-\
abla \cdot D(\mathbf{r})\
abla\bigg{\}}\Phi(\mathbf{r},t)=q_{0}(\mathbf{r},t), \tag{14}\]  
where \(D=\)1/\(\{3(\mu_{a}+\mu_{s}\cdot)\}\) is the diffusion coefficient, \(\Phi(\mathbf{r},t)=\int_{4\pi}l(\mathbf{r},\mathbf{s},t)d\Omega\) the fluence rate (photon density), and \(q_{0}(\mathbf{r},t)\) an isotropic light source. For the CW measurement, the first term in Equation (14) is removed. Equation (14) is the Fourier transform for the FD measurement. To compute light propagation with the PDE, the Robin boundary condition, which is often used in image reconstruction, is as follows:  
\[\frac{1}{2A}\Phi(\mathbf{r},t)=-\mathbf{n}\cdot D(\mathbf{r})\
abla\Phi( \mathbf{r},t), \tag{15}\]  
where \(A\) is a parameter that depends on the internal reflection ratio, and \(\mathbf{n}\) is the outward normal vector. Early studies by [PERSON] and [PERSON] [28] established the standard nonlinear image reconstruction scheme for DOT. The computation of the PDE is typically performed using the FEM. The Jacobian matrix \(J\) appearing in Equation (3) is obtained using the adjoint method based on the reciprocity theorem [70; 71]. In the TD measurement, the perturbation \(\eta\) in \(\Phi\) with \(\mu_{s}{}^{\prime}\) and \(\mu_{a}\) at the measured position \(\mathbf{r}_{d}\) associated with small changes \(\
u\) and \(\alpha\) in \(\mu_{s}{}^{\prime}\) and \(\mu_{a}\), respectively, is calculated as  
\[\eta(\mathbf{r}_{d},t)=-\int_{\Omega}\int_{-\infty}^{\infty}\
u(\xi)\
abla \Psi(\mathbf{r}_{d},\xi,t,\tau)\cdot\
abla\Phi(\xi,\tau)+\alpha(\xi)\Psi( \mathbf{r}_{d},\xi,t,\tau)\Phi(\xi,\tau)d\tau d\xi, \tag{16}\]  
where \(\Psi\) is [PERSON]'s function of the adjoint PDE, which calculates the light propagation from \(\mathbf{r}_{d}\) with position \(\xi\) and time \(\tau\). From Equation (16), the entries of \(J\) with respect to \(\mu_{s}{}^{\prime}\) are computed with the inner product of the gradients of \(\Psi\) and \(\Phi\), discretized with the FEM, whereas those with respect to \(\mu_{a}\) are computed with the product of \(\Psi\) and \(\Phi\) on the right-hand side of Equation (16). Various types of measurement exist for image reconstruction. [PERSON] et al. compared reconstructed images from the measured intensity (CW measurement), mean time-of-flight (first moment), variance (second moment), skew (third moment), and full-time-resolved DTOF [72]. The normalized DTOF provides a better image in terms of spatial resolution and quantification in 2D numerical experiments using DA, although the image reconstruction from full-time-resolved measurements required a 120-fold longer computational time than that from the mean time-of-flight and variance. DA image reconstruction from Laplace-transformed DTOF using a modified generalized pulse spectrum technique (mGPST) has also been proposed [73]. DA is obtained from the \(P_{1}\) approximation with the assumption that the time derivative of the net flux vector, \(\mathbf{J}=\int_{4\pi}\mathbf{s}\,\mathbf{l}(\mathbf{r},\mathbf{s},t)d\Omega\), equals zero, and the light source is isotropic. The DA is valid while \(\mu_{a}<<\mu_{s}{}^{\prime}\). It is not valid to use the PDE for the nonscattering void region. Moreover, DA requires spatiotemporal conditions in which the directions of photons are sufficiently randomized by many scattering events and the light intensity changes very slowly [23; 74]. Although the limitation of DA requiring the abovementioned assumptions and conditions, which often seem difficult to be fulfilled, has been debated, image reconstruction employing PDE has been successfully applied in in vivo and clinical DOT studies [75; 76; 77; 78; 79]. Several studies on image reconstruction based on DA with various regularizations and inversion schemes have been reported because the computational burden of the PDE is reasonable.