## 3 Results and Discussions
### Most Sensitive Spectral Band  
We first tried to find out the most sensitive spectral band which can lead to accurate evaluation of average adipocyte size. Based on features of our data set, we predicted the spectral band of 12.5 to 16.5 MHz as the most sensitive. PA signals are wide-band signals and can reflect histological microfeatures of adipocyte tissues. The average adipocyte size of our data set was mainly within the range of 3600 to 6400 \(\upmu\)m\({}^{2}\). As depicted in Figure 2c, an adipocyte is shaped like a polygon. If we consider the adipocyte as a square, the relationship between its diagonal length \(l\) and area (i.e., size) \(S\) can be expressed as the equation \(S=l^{2}/2\). Thus, the average diagonal length of our data set mainly ranged from 84.85 \(\upmu\)m to 113.14 \(\upmu\)m. The corresponding time \(t\) it takes for sound to pass through adipocyte along diagonal can be computed, using the equation \(t=l/c_{0}\), where \(c_{0}\) is the speed of sound considered as 1450 m/s (a typical value in human adipose tissue [28]). Here, we used diagonal to estimate \(t\) because it is the longest path for sound to pass through, which varies when size changes. Therefore, the most sensitive frequency of our data set is considered to be  
\[f=\frac{1}{t}=\frac{c_{0}}{d} \tag{5}\]  
which ranged from 12.82 MHz to 17.09 MHz. Therefore, we first tested our deep learning method on the spectral band of 12.5 to 16.5 MHz. We picked out the normalized magnitude of spectral band of 12.5 to 16.5 MHz as the input of the deep network; the corresponding average adipocyte size acquired from standard histology of the tissue worked as the output. The input had 80 elements, each correspondent to the magnitude over a 0.05-MHz step in the 4-MHz spectral band. To initialize and train the deep network, a MATLAB (2017, MathWorks, Natick, MA, USA) toolbox called Neural Network Toolbox was employed. When training, the Levenberg-Marquardt method [29] was selected as the optimization algorithm. As described in Section 2, the deep network had several layers and each layer was a full connected layer shown in Figure 1. The output of each layer works as the input of the next layer. Different numbers of layers can make up different networks which may show various performance. We tried on networks with different layer numbers to find the optimal layer number. For each layer number, the element number of the output of the first hidden layer was set to about half of the input element number to reduce the number of parameters in the network and for rest layers, the element number of the output decreased gradually. The network was retrained for a total of 10 times for each layer number. For each time, data augment, described in Section 3.2, was performed and the minimum, maximum, and average values of the 10 MREs of the test set were computed, the result of which is shown in Figure 4.  
The depth (number of layers) of the deep network largely influences the performance of the network, as illustrated in Figure 4. Initially when the layer number increases, the network becomes deeper and a more complicated model is established to process the sample data, as a result of which,the MRE of the test set becomes smaller and the deep network is more stable as demonstrated by the smaller deviation range. However, additional layers mean more parameters of the deep network and the longer training time. In addition, problems such as vanishing gradient problem of tan-sigmoid function, overfitting, and the stronger influence of the noise on the network may appear when the network becomes too deep [30]. Therefore, when the layer number is larger than seven, further increasing the layer number leads to larger MRE and larger deviation range reflecting unstable performance. As a result, a 7-layer network turned out to be the best choice.