## 2 Deep Neural Network with Fully Connected Layers  
The traditional PASA method only depends on a simple linear fit with the power spectrum. Therefore, it cannot accurately represent the relationship between PA spectrum and average adipocyte size which may be complex and difficult to be fit with an intelligible mathematical expression. Compared to PASA, deep learning methods can provide a much more complicated nonlinear model to fit the relationship and achieve higher accuracy for our study. More specifically, with each A-line PA signal from a tissue, the power spectrum was computed and normalized, and then analyzed using deep learning method to fully utilize the spectral information. With the method of deep learning, the relationship between the power spectrum and adipocyte size was learned by a deep network. The power spectrum and the corresponding adipocyte size worked as the input and output of the deep network, respectively, when training and testing. For an N-layer network, the element number of input and output of the network are I\({}_{0}\) and O\({}_{\text{N+1}}\). In our study, each element of the input was a power spectrum value of the PA signal while the output, with only one element, was the average adipocyte size. With the power spectrum of a PA signal fed into the first layer, all output elements of each layer functioned as the input of the next layer and the output of the last layer was the predicted average adipocyte size. In our study, N-1 hidden layers and one output layer (the last layer) were used to fit the nonlinear relationship between power spectrum and average adipocyte size. Each layer is a fully connected layer, which means all of the input elements are connected to each output element, as shown in Figure 1. All hidden layers and output layer did a linear transformation at first and then used different functions to map linear transformation result to output. The \(n\)-th layer can be defined as  
\[y_{j}=f\left(\sum_{i=0}^{I_{n}}w_{i,j}x_{i}+b_{j}\right),j=1,2,\ldots,O_{n}, \tag{1}\]  
where \(x\) is the input of the layer (\(I_{n}\) elements) and \(y\) is the output (\(O_{n}\) elements). The transfer function \(f(x)\), as shown in Figure 1b, in hidden layers was selected as tan-sigmoid function in our network, which is defined as  
\[f(x)=\frac{2}{1+e^{-2x}}-1,x\in(-\infty,+\infty), \tag{2}\]  
The tan-sigmoid function is continuous, smooth, and monotone, and increases quickly in the vicinity of \(x=0\) with a codomain of (\(-1\), \(1\)). Here, we chose a tan-sigmoid function as the transfer function because its features match with the human adipocyte size distribution, as their values should be continuous and bounded. For output layer, transfer function \(f(x)\) was selected as a linear function so that the output can match the adipocyte size appropriately. Through setting layer number N and the input and output element number for each layer, we establish a network and use all parameters (\(\mathbf{w}\) and \(\mathbf{b}\) in Equation (1)) to fit the relationship between input and output. When training, data set was divided into training set and test set. The training set was randomly picked out from the whole data set while the remains were defined as test set. All parameters (\(\mathbf{w}\) and \(\mathbf{b}\) in Equation (1)) in the network were optimized gradually through iteration to minimize the difference between the predicted average adipocyte size and the ground truth value from histology for all samples in the training set. When testing, the power spectrum of test set was fed into the network with the parameters obtained through training to predict the average adipocyte  
Figure 1: One layer of deep network. (**a**) Fully connected layer. All of the input elements are connected to each output element. (**b**) Obtaining \(j\)-th element of output from input elements. sizes.