## 2 The Methods of PAI and LUI
### Forward and Inverse Problems of PAT and LUT--Reconstruction Algorithms  
#### 2.2.1 Algorithms for PAT  
A forward problem for PAT, under the assumption of absorption of short laser pulse and acoustically homogeneous lossless medium, is formulated as a wave equation with initial pressure source:  
\[\frac{\partial^{2}p}{\partial t^{2}}-c^{2}\
abla^{2}p=p_{0}(\mathbf{r})\frac{d \delta(t)}{dt}, \tag{1}\]  
where \(p(\mathbf{r},t)\) is the acoustic pressure field, \(c\) is the speed of sound in the medium, \(p_{0}(\mathbf{r})\) is the initial pressure distribution in the light absorbing material, and \(\delta(t)\) is the Dirac delta function. Note, that \(p_{0}(\mathbf{r})=\Gamma(\mathbf{r})H(\mathbf{r})\), where \(\Gamma(\mathbf{r})=\beta c^{2}/c_{p}\) is the Gruneisen parameter, \(H(\mathbf{r})\) is the amount of optical energy, absorbed in the medium per unit volume [70], \(\beta=-\rho^{-1}(\partial\rho/\partial T)_{p}\) is the volume thermal expansion coefficient, \(\rho\) and \(T\) are the density and temperature of the medium, \(c_{p}\) is the specific heat capacity at constant pressure. Solution of Equation (1) is given by  
\[p(\mathbf{r},t)=\frac{1}{4\pi c}\frac{\partial}{\partial t}\int_{|\mathbf{r}-\mathbf{r}^{ \prime}|=ct}\frac{p_{0}(\mathbf{r}^{\prime})}{|\mathbf{r}-\mathbf{r}^{\prime}|}dS^{\prime} \big{(}\mathbf{r}^{\prime},t\big{)}, \tag{2}\]  
where the integral is taken over a surface of a sphere in 3D, and \(dS^{\prime}\) is the surface element. The inversion of Expression (2) to obtain \(p_{0}(\mathbf{r})\), using known dependences of pressure on time \(p(\mathbf{r}_{m},t)\) in a set of points \(\mathbf{r}_{m}\), \(m=1,\ldots,N\), where \(N\) is the number of sensors on the detection surface, is the basis for the majority of the algorithms in PAT. All reconstruction algorithms can be divided into two groups. The first group considers the PA source (the initial pressure distribution) to be continuous, and it includes time-reversal, Fourier-domain, and back-projection algorithms. The second group, which is represented by the model-based algorithms, considers the PA source to be inherently discrete (typically a sum of a finite number of functions with limited support). As there are good reviews on algorithms for PAT (e.g., [72]), they are reviewed here briefly. _Time reversal algorithm._ Time reversal is based on the invariance of the wave equation to the substitution \(t\to-t\). Let us denote sufficiently large time by \(t_{0}\), so that acoustic pressure in medium volume \(V\) decays to \(p(\mathbf{r},t_{0})=0\). Let us define the reversed pressure wave as  
Figure 1: Specific temporal shape (**a**) and spectrum (**b**) of laser-induced photoacoustic (PA) pulse. Reproduced with permission from [41], Springer Nature, 2009. \[p_{tr}(\mathbf{r},t)=p(\mathbf{r},2t_{0}-t),\ t_{0}\leq t\leq 2t_{0}. \tag{3}\]  
The inhomogeneous wave equation (Equation (1)) is equivalent to the homogeneous wave equation with initial conditions \(p(\mathbf{r},0)=p_{0}(\mathbf{r})\) and \((\partial p/\partial t)(\mathbf{r},0)=0\). The homogeneous wave equation is invariant under transform \(t\to-t\), then \(p_{tr}(\mathbf{r},t)\) is a solution of homogeneous wave equation with initial conditions \(p_{tr}(\mathbf{r},t_{0})=p(\mathbf{r},t_{0})=0\) and \((\partial p_{tr}/\partial t)(\mathbf{r},t_{0})=-\ (\partial p/\partial t)(\mathbf{r},t_{0})=0\). According to Equation (3), time reversal gives the initial distribution of acoustic pressure \(p_{0}(\mathbf{r})\) at time \(t=2t_{0}\). Measured values are known on the boundary of medium volume \(\partial V\)the detection surface \(S_{0}\). Hence, time reversal assumes the solution of system:  
\[ \begin{cases}\frac{\partial^{2}p_{tr}}{\partial t^{2}}-c_{0}^{2}\Delta p_{tr} =0&(\mathbf{r},t)\in V\times[t_{0},2t_{0}]\\ p_{tr}(\mathbf{r},t_{0})=\frac{\partial p_{tr}}{\partial t}(\mathbf{r},t_{0})=0&\mathbf{r }\in V\\ p_{tr}(\mathbf{r},t)=p(\mathbf{r},2t_{0}-t)&(\mathbf{r},t)\in\partial V\times[t_{0},2t_{0} ]\end{cases} .