## 1 Introduction  
Obesity is a public health crisis which afflicts one third of the U.S. population [1] and is associated with a significant risk of metabolic diseases. Metabolic diseases come with abnormal chemical reactions in the body; diabetes mellitus (DM) is one of the most common metabolic diseases. The National Health and Nutrition Examination Survey (NHANES) reveals that diabetes is becoming increasingly prevalent and the corresponding weight classes are continually increasing, with nearly half of adult diabetics considered to be obese [2]. Prediction of obese patients at risk for developing metabolic disease is of central importance. Prior research demonstrates that adipose tissue phenotypic features,also known as tissue observable characteristics, such as adipocyte size, correlate with human metabolic disease [3; 4; 5]. That suggests the adipose size may serve as predictors of metabolic disease risk and a diagnostic and therapeutic target. Histology is the current gold standard for assessments of adipose size [3; 4; 5]. While accurate, histology requires adipose tissue biopsy, which involves surgery and microscope observation, which is invasive and labor- and cost-intensive. Emerging biomedical photoacoustic (PA) imaging technology is based upon the detection of laser-induced thermoacoustic signals from a biological sample which can then be used to produce images reflecting the optical absorption contrast in the sample [6; 7; 8]. Due to PA signal's broad-band feature, it can reflect sample microstructures of different scopes. Recently, various methods for analyzing the PA signals in the frequency domain have been developed by several research groups [9; 10; 11; 12], aiming to achieve quantitative evaluation of tissue microstructures. One of the methods termed as photoacoustic spectral analysis (PASA) [13; 14; 15; 16], borrowed the basic concept from ultrasound spectral analysis [17; 18; 19]. In PASA, by quantifying the key parameters, including slope, intercept, and mid-band fit, of the linear fit to the truncated signal power spectrum within the predetermined frequency range, histological microfeatures of the optically absorbing materials in target tissues were characterized. Later, by using ultrabroad-band ultrasonic detectors, the scope of PASA was further extended from the tissue level to the cellular level [16]. Using the three parameters of the linear fit (i.e., slope, intercept, and mid-band fit) to describe the main features of the power spectrum provides a practical way of addressing the stochastic nature of the tissue microstructure, and can lead to measurements that are not only quantitative but also robust. However, despite these advantages, PASA, relying on a simple linear fit to the power spectrum within the entire predetermined frequency range, utilizes only limited information of the signal power spectrum. Spectral analysis based on deep learning provides a potential solution to this problem. A network is established in deep learning which describes the relationships between input and output by parameters and functions. The process of training uses the data sets whose input and output are known to modify all the parameters in the network to achieve an optimal model under control of a learning rule [20]. Upon achieving optimal parameters, the network can be used to predict the corresponding output of data sets whose input is known. Varying layer numbers (numbers of processing steps) and layer sizes (numbers of parameters involved in each processing step) can provide different performance of data representation which results in varying accuracy of prediction [21]. Due to its excellent performance characteristics, such as high accuracy, deep learning methods in medical diagnosis and evaluation have been rapidly developed [22; 23; 24]. We studied the feasibility of measuring adipocyte size in human adipose tissue by analyzing PA spectrum with the help of deep learning in this research. A deep neural network was used to fit the relationship between the PA spectrum and an average adipocyte.