# Quantitative Reconstruction of Absorption Coefficients for Photoacoustic Tomography
## 3 Numerical Simulation  
We design a complex numerical simulation experiment to evaluate the proposed method, a 3D MC model of light transport was employed to simulate the measured absorbed energy density [7]. In this numerical simulation, we designed a 3D simulation of the hypothetical complex tissue model, as illustrated in Figure 2a. Figure 1: Flow chart for quantitative reconstruction the distribution of optical absorption coefficients. The 3D tissue model contains 5 targets with different absorption coefficients with a range of 0.45 cm\({}^{-1}\) to 7 cm\({}^{-1}\) embedded in a block tissue (1.8 \(\times\) 1.8 \(\times\) 1.8 cm\({}^{3}\)). Target 1 is a cube with a side length of 0.18 cm, the absorption coefficient is 2.0 cm\({}^{-1}\). Target 2 is a sphere (dia. 0.30 cm) with absorption coefficient \(\upmu_{\text{a}}\) of 6 cm\({}^{-1}\). Target 3 is a multi-layer nested structure, which is a large cube nested into four layers, and the absorption distribution of each layer is 1, 3, 5, 7 (cm\({}^{-1}\)). Target 4 is a little cube with a little ball inside, the \(\upmu_{\text{a}}\) of the inner-sphere is 4 cm\({}^{-1}\). Target 5 is a sphere (dia. 0.12 cm), the absorption coefficient is the same as for target 2. The simulation parameters are summarized in Table 1. The background's absorption coefficient was \(\upmu_{\text{a}}\) = 0.45 cm\({}^{-1}\), and its scattering coefficient was \(\upmu_{\text{s}}\) = 356.5 cm\({}^{-1}\). Figure 2b shows the measured absorption energy distribution on the cross section of z-x plane at y = 0. Obviously, the distribution of absorbed energy is affected by the depth-dependent attenuation of light fluence, which cannot provide sufficient tissue information, especially in the deep tissue. A uniformly distributed illumination was selected, irradiated from top to bottom, and the illumination diameter was 0.63 cm. The corresponding laser wavelength is 532 nm and the running time of simulation was 10 min. We set the stop criterion to 0.2 and the maximum number of iterations to 30. The z-x plane configuration for the simulation model is shown in Figure 3a, and Figure 3b is the fluence distribution of pure background tissue corresponding to the z-x section, which is calculated based on the Monte Carlo method according to the background tissue parameters. Then execute light fluence compensation to obtain initial absorption coefficient and implement the quantitative estimate of absorption coefficient iteratively. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Target Name** & **Target Structure** & **Target Size (x \(\times\) y \(\times\) z) (cm\({}^{3}\))** & \(\upmu\)**a (cm\({}^{-1}\))** & \(\upmu\)**s (cm\({}^{-1}\))** & **g** \\ \hline target1 & cube & (0.18 \(\times\) 0.18 \(\times\) 0.18) & 2.00 & 100.00 & 0.90 \\ target2 & sphere & dia. 0.30 & 6.00 & 100.00 & 0.90 \\ target3- layer1 & Cube -layer1 & (0.27 \(\times\) 0.54 \(\times\) 0.135) & 1.00 & 100.00 & 0.90 \\ target3- layer2 & Cube -layer2 & (0.27 \(\times\) 0.54 \(\times\) 0.135) & 3.00 & 100.00 & 0.90 \\ target3- layer3 & Cube -layer3 & (0.27 \(\times\) 0.54 \(\times\) 0.135) & 5.00 & 100.00 & 0.90 \\ target3- layer4 & Cube -layer4 & (0.27 \(\times\) 0.54 \(\times\) 0.135) & 7.00 & 100.00 & 0.90 \\ target4 & cube + sphere & (0.18 \(\times\) 0.18 \(\times\) 0.18), dia. 0.12 & 2.00,4.00 & 150.00 & 0.90 \\ target5 & small sphere & dia. 0.12 & 6.00 & 100.00 & 0.90 \\ background & cube & (1.80 \(\times\) 1.80 \(\times\) 1.80) & 0.45 & 356.00 & 0.90 \\ \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. Figure 2: **(a)** Complex numerical simulation 3D model, different objects are distinguished by different colors and shapes. **(b)** Measured absorption energy distribution, for the selected z-x section (y = 0 cm). The axes z and x (left and bottom) illustrate the spatial scale, the unit is cm, whereas the color scale (right) represents the absorbed energy density. Iterative recovery of the absorption coefficient results of three iterations (No. 1, 5 and 10) are shown in Figure 4 respectively.