# Quantitative Reconstruction of Absorption Coefficients for Photoacoustic Tomography
## 2 Methods  
According to the photoacoustic effect, when laser pulses are delivered into biological tissues, optical absorbers in tissue absorb photons and generate ultrasound pressure waves which can be detected by ultrasound transducers around the tissue and then analyzed to reconstruct images [1; 2; 6]. The measured initial pressure distribution [15; 16; 17], \(P_{0}\) is proportional to the absorption energy distribution, \(A\), and the measured initial PA signal may be written as  
\[P_{0}=\hat{\Gamma}A, \tag{1}\]  
where \(\hat{\Gamma}\) is the PA efficiency, which usually can be regarded as a constant value. In addition, the absorption energy distribution can be written as  
\[A=\mu_{a}\Phi, \tag{2}\]  
where \(\mu_{a}\) is the optical absorption coefficient and \(\Phi\) is the light fluence. Therefore, the initial acoustic pressure distribution may be written as the product of three quantities:  
\[P_{0}=\hat{\Gamma}\mu_{a}\Phi, \tag{3}\]  
Since it is obvious that the measured photoacoustic signal is derived from the interaction between the light fluence and absorption coefficient, the quantitative PA measurements of the optical absorption coefficient requires an accurate estimation of optical fluence [7; 15; 16; 17]. In this paper, we address this issue for fluence estimation using the 3D Monte Carlo simulation, which is often regarded as the \"gold standard\" for simulating light transport in turbid media and provides a reliable and accurate method for calculating light transport. For this method, we use the mcvyz.c program, a computer simulation of light transport in a heterogenous tissues with varying absorption and scattering properties developed by [PERSON] et al. [7], and the Monte Carlo toolkit can be downloaded freely from [[https://omlc.org/software/mc/mcvyz/index.html](https://omlc.org/software/mc/mcvyz/index.html)]([https://omlc.org/software/mc/mcvyz/index.html](https://omlc.org/software/mc/mcvyz/index.html)). This paper introduces a novel method for quantitative reconstruction of the distribution of optical absorption coefficients in tissue. In this method, we implemented an iterative algorithm for recovering absorption coefficients from optical absorbed energy maps based on a 3D Monte Carlo simulation of light transport and integrated with fluence compensation to obtain the initialization parameters. At each iteration, we calculate the deviation between the detected and the computed absorbed energy distribution. By minimizing the deviation in the absorbed energy, the recovered values converge to the true absorption distribution, the flow-chart of the proposed method is shown in Figure 1. After obtaining the absorbed energy deposition, initialize the Monte Carlo simulation model. At this point, parameters such as the illumination geometry, the scattering distribution and the anisotropy factor are used as prior knowledge. It is necessary to combine the pure background for fluence compensation, and to obtain more structural priors to make the quantitative results more accurate. Then based on \(\mu_{a}=A/\Phi\), obtain the initial absorption coefficient \(\mu_{a}(0)\) and start iterating the loop. Calculate the deviation between the detected absorbed energy distribution \(A(0)\) and the computed absorbed energy distribution \(A(k)\) at each iteration. If the deviation exceeds the stop criterion \(\epsilon\), calculate the increment of the new absorption estimate \(\Delta\mu_{a}\), and update the new absorption estimate value. The iterative calculation through the loop until the error in the absorbed energy is sufficiently small or exceeds the set maximum number of times. By minimizing the deviation in the absorbed energy, output the recovered values of absorption distribution. In addition, we used three evaluation indicators to quantitatively evaluate the performance of the algorithm: structural similarity (SSIM), root mean square error (RMSE), and peak signal-to-noise ratio (PSNR) [40; 41; 42].