# Photoacoustic Imaging with a Finite-Size Circular Integrating Detector
## 2 Methods  
### Generation and Propagation of Photoacoustic Waves  
A laser pulse irradiates biological tissue, where the absorbers absorb light energy, triggering instantaneous thermal expansion. This results in a pressure increase proportional to the absorbed light energy, generating pressure waves. The general equation for the generation and propagation of PA waves is given by [29]:  
\[\frac{\partial^{2}p(\mathbf{r},t)}{\partial t^{2}}-c^{2}\Delta p(\mathbf{r},t)=p _{0}(\mathbf{r})\cdot\frac{d}{dt}\delta(t) \tag{1}\]  
where \(p(\mathbf{r},t)\) is the pressure at position \(\mathbf{r}\) and time \(t\): it describes the pressure signal that is propagating through the medium (such as biological tissue). \(p_{0}(\mathbf{r})\) represents the initial pressure distribution due to thermal expansion, \(c\) is the speed of sound in the tissue, and \(\delta(t)\) is the Dirac delta function representing the instantaneous excitation.  
Equation (1) corresponds to an initial value problem for the homogeneous wave equation:  
\[\frac{\partial^{2}}{\partial t^{2}}p(\mathbf{r},t)-c^{2}\Delta p(\mathbf{r},t)=0 \tag{2}\]  
with the initial conditions:  
\[p(\mathbf{r},0)=p_{0}(\mathbf{r});\;\frac{\partial}{\partial t}p(\mathbf{r},t )_{t=0}=0 \tag{3}\]  
In PAI, the main goal is to reconstruct the spatial distribution of the sample's internal energy absorption, which is proportional to the initial pressure distribution \(p_{0}(\mathbf{r})\). To achieve this, the initial pressure distribution \(p_{0}(\mathbf{r})\) needs to be recovered from the pressure signals \(p(\mathbf{r_{d}},t)\) measured by the detector D on the sample surface S (as shown in Figure 1).  
The pressure signal \(p(\mathbf{r},t)\) at position \(\mathbf{r}\) and time \(t\) can be obtained as follows:  
\[p(\mathbf{r},t)=\frac{1}{4\pi c}\iint_{|\mathbf{r}-\mathbf{r}^{\prime}|=ct} \frac{p_{0}(\mathbf{r}_{0})}{|\mathbf{r}-\mathbf{r}^{\prime}|}dS(\mathbf{r}^ {\prime}) \tag{4}\]  
In this equation, \(\mathbf{r}^{\prime}\) represents points on a spherical surface centered around \(\mathbf{r}\) with a radius equal to \(ct\).  
The research in this study is based on the circular integrating detector, and its structure is shown in Figure 2. The finite size of the detector causes it to perform a weighted integral of the acoustic field signal \(p(\mathbf{r},t)\) within its sensing region. The recorded signal \(p_{\omega}(\mathbf{r_{d}},t)\) is essentially the result of the convolution between the SIR \(\omega_{\text{det}}(\mathbf{r},t)\) and the acoustic field signal \(p(\mathbf{r_{s}},t)\). The SIR describes the spatial averaging effect of the detector on the received acoustic signal. The mathematical expression for the recorded signal of a finite-size detector can be written as follows [30]:  
\[p_{\omega}(\mathbf{r_{d}},t)=\int\limits_{\text{S}}\omega_{\text{det}}(| \mathbf{r_{d}}-\mathbf{r_{s}}|,t)\cdot p(\mathbf{r_{s}},t)dS(\mathbf{r_{s}}) \tag{5}\]  
Figure 1: Illustration of photoacoustic tomography, the detector D, positioned at \(\mathbf{r_{d}}\), scans the tissue, which occupies a volume R, along a closed surface S. The initial pressure within the tissue is denoted as \(p_{0}(\mathbf{r})\).  
Here, \(\mathbf{r_{d}}\) indicates the center of the detector, while \(\mathbf{r_{s}}\) represents positions on the detector's surface. S is the sensing area of the detector. When we apply reconstruction algorithms designed for ideal point detectors, such as time reversal or backprojection, to \(p_{\omega}(\mathbf{r_{d}},t)\), the reconstructed image will exhibit blurring.