## 2 Methods  
### The Influence of Inaccurate Sound Speed Fields on PAI  
Assuming that biological tissue is a homogeneous medium, the basic equation of PAI can be expressed as follows [17]:  
\[\
abla^{2}p(\mathbf{r}_{0},t)-\frac{1}{v^{2}}\frac{\partial^{2}p(\mathbf{r}_{0},t)}{ \partial t^{2}}=-\frac{\beta}{C}A(\mathbf{r})\frac{\partial I(t)}{\partial t}. \tag{1}\]  
In Equation (1), \(p(\mathbf{r}_{0},\ t)\) is the photoacoustic pressure received by the ultrasonic transducer at position \(\mathbf{r}_{0}\) at time \(t\). \(v\) is the homogeneous sound speed. \(\beta\) is the coefficient of thermal expansion. \(C\) is the specific heat capacity. \(A(\mathbf{r})\) represents the light absorption coefficient distribution function. \(I(t)\) represents the time distribution function of the incident laser. Due to the short duration of the laser pulse, \(I(t)\) can be replaced by an excitation pulse \(\delta(t)\) approximation [17]. Using the Green function to solve \(p(\mathbf{r}_{0},\ t)\), the general relationship between photoacoustic pressure and the light absorption coefficient is obtained as follows [17]:  
\[p(\mathbf{r}_{0},t)=\frac{\beta}{4\pi C}\frac{\mathrm{d}}{\mathrm{d}t}\int\frac{ A(\mathbf{r})}{|\mathbf{r}_{0}-\mathbf{r}|}\delta(t-\frac{|\mathbf{r}_{0}-\mathbf{r}|}{v})\mathrm{d}\mathbf{r}. \tag{2}\]  
The temporal integral function of \(p(\mathbf{r}_{0},\ t)\) is introduced as follows:  
\[G(\mathbf{r}_{0},t)=\frac{4\pi C}{\beta}t\int_{0}^{t}p(\mathbf{r}_{0},t)\mathrm{d}t. \tag{3}\]  
Equation (2) is brought into Equation (3). The constant terms of the two formulas cancel, and the time derivative term \(\mathrm{d}/\mathrm{d}t\) in Equation (2) cancels the time integral term \(\int_{0}^{t}\mathrm{d}t\) in Equation (3), so Equation (4) is obtained, as follows:  
\[G(\mathbf{r}_{0},t)=\int A(\mathbf{r})\frac{1}{v}\delta(t-\frac{|\mathbf{r}_{0}-\mathbf{r}|}{ v})\mathrm{d}\mathbf{r}, \tag{4}\]  
where \(G(\mathbf{r}_{0},t)\) represents the spherical Radon transform of the \(A(\mathbf{r})\); therefore, the photoacoustic image can be obtained by inverting the spherical Radon transform [18]. However, Equation (4) is only suitable for PAI under homogeneous sound speed fields. When the presence of an inhomogeneous sound speed field is taken into account in PAI, the speed field causes photoacoustic signals to refract during propagation [19]. The sound speed at any position in this sound speed field can be set to \(v(\mathbf{r}^{\prime\prime})\). The TOF of the photoacoustic signal is obtained by calculating the line integral of the reciprocal of the sound speed along the approximate propagation path as follows:  
\[t_{f}(\mathbf{r},\mathbf{r}_{0})=\int_{l}\frac{1}{v(\mathbf{r}^{\prime\prime})}\mathrm{d}l,\ l=|\mathbf{r}_{0}-\mathbf{r}|, \tag{5}\]  
where \(l\) is the distance from the sound source position \(\mathbf{r}\) to the ultrasonic transducer \(\mathbf{r}_{0}\). Equation (5) is brought into Equation (2). The relationship between the photoacousticpressure \(p_{inh}(\mathbf{r}_{0},\ t)\) obtained under the inhomogeneous sound speed field and \(A(\mathbf{r})\) can be expressed as follows:  
\[p_{inh}(\mathbf{r}_{0},t)=\frac{\beta}{4\pi\mathbb{C}}\frac{\mathrm{d}}{\mathrm{d}t }\int\frac{A(\mathbf{r})}{|\mathbf{r}_{0}-\mathbf{r}|}\delta(t-t_{f}(\mathbf{r},\mathbf{r}_{0})) \mathrm{d}\mathbf{r}. \tag{6}\]  
Equations (5) and (6) are brought into Equation (3) as follows:  
\[G_{inh}(\mathbf{r}_{0},t)=\int A(\mathbf{r})\frac{t_{f}(\mathbf{r},\mathbf{r}_{0})}{|\mathbf{r}_{0 }-\mathbf{r}|}\delta(t-t_{f}(\mathbf{r},\mathbf{r}_{0}))\mathrm{d}\mathbf{r}, \tag{7}\]  
where \(G_{inh}(\mathbf{r}_{0},\ t)\) is called the generalized Radon transformation of the \(A(\mathbf{r})\) in the inhomogeneous sound speed field. The image reconstruction process can be achieved by performing the inverse Radon transform on Equation (7) [18]. In the process of image reconstruction, the TOF is related to the propagation path of photoacoustic signal and depends on the sound speed distribution of biological tissue.