# Ultrasound-Guided Detection and Segmentation of Photoacoustic Signals from Bone Tissue In Vivo
## 2 Materials and Methods
### 2D Simulations of PA Signal Generation and Propagation  
In the PA mode, the laser illuminated the bone and was absorbed by the bone tissue; it was converted into heat and thermal expansion to generate the PA pressure. In response to a heat source, \(H(\mathbf{r},t)\), the pressure, \(p(\mathbf{r},t)\) at position \(\mathbf{r}\) and time \(t\) in an acoustically homogeneous medium obeys the following wave equation (ignoring the thermal diffusion and kinematic viscosity) [21; 35]:  
\[\
abla^{2}p(\mathbf{r},t)-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}p(\bm {r},t)=-\frac{\beta}{C_{p}}\frac{\partial}{\partial t}H(\mathbf{r},t), \tag{2}\]  
where \(H(\mathbf{r},t)\) is a heating function defined as the thermal energy converted at spatial position \(\mathbf{r}\) and time \(t\) by the electromagnetic (EM) radiation per unit volume per unit time. \(\beta\) is the isobaric volume expansion coefficient in K\({}^{-1}\), and \(C_{p}\) is the isobaric specific heat in J (K kg)\({}^{-1}\), and \(c\) is the acoustic speed. Under both thermal and stress confinement conditions, the heating time can be treated as a delta function, i.e.,  
\[H(\mathbf{r},t)\approx\mu_{a}(\mathbf{r})\Phi(\mathbf{r})\delta(t), \tag{3}\]  
and the initial pressure \(p_{\mathbf{0}}\) of the absorber at location \(\mathbf{r}\) after absorbing the light energy can be calculated by  
\[p_{\mathbf{0}}(\mathbf{r})=\frac{c^{2}\beta}{C_{p}}\mu_{a}(\mathbf{r})\Phi(\mathbf{r}). \tag{4}\]  
Here, \(\mu_{a}\) and \(\Phi\) are the absorption coefficient and the optical radiation fluence rate, respectively. We assume that biological tissue can be regarded as \"macro-homogeneous\" [36]. The \(\Phi(\mathbf{r})\) can be expressed as [36; 37]  
\[\Phi(\mathbf{r})=\frac{1}{4\pi D|\mathbf{r}-\mathbf{r}^{\prime}|}\exp\left(-\mu_{eff}|\bm {r}-\mathbf{r}^{\prime}|\right) \tag{5}\]  
where \(\mu_{eff}=\sqrt{3\mu_{a}(\mu_{a}+\mu_{s}^{\prime})}\) is the effective attenuation coefficient, \(\mu_{a}\) is the optical absorption coefficient, \(\mu_{s}^{\prime}\) is the reduced scattering coefficient, parameter \(D\) is the diffusion coefficient given as \(D=1/(3\mu_{s}^{\prime})\), and \(\mathbf{r}^{\prime}\) is the spatial position of the source.  
In the simulation, the optical radiation fluence \(\Phi\) in bone was simulated by using the NIRFAST-MATLAB 9.1 toolbox (Dartmouth College, University of Birmingham and Kitware Inc., Hanover, NH, USA) based on the finite element method (FEM) [38; 39]. The \(\mu_{a}\) and \(\mu_{s}^{\prime}\) were set as 0.08 cm\({}^{-1}\) and 14.00 cm\({}^{-1}\) for bone tissue [14], and 0.12 cm\({}^{-1}\) and 14.8 cm\({}^{-1}\) for soft tissue [40] in this study. In the simulation of PA signal generation and propagation, since none of \(\beta\) and \(C_{p}\) affected the profile of the PA signal waveform, they were set at 1 in the simulation. In addition, the Gaussian-modulated sine pulse defined in Equation (1) was used as the PA pressure source for each light absorber in the simulation. Then, the excitation condition of the PA source \(p_{c}(\mathbf{r_{0}},t)\) as a function of time can be expressed as  
\[p_{c}(\mathbf{r_{0}},t)=\cdot\mu_{a}(\mathbf{r_{0}})\cdot\Phi(\mathbf{r_{0}})A\cdot exp( \frac{-\left(t-\frac{t}{2}\right)^{2}}{s^{2}})sin(2\pi ft) \tag{6}\]  
The simulation of PA propagation in the bone is the same as that of US propagation in the bone based on the FDTD method [33; 34]. The signal propagation lasted for 100 us in the simulation, which allowed it to travel through the whole bone and beyond the specified domain.