## 2 Mathematical Model and Theory
### Speed of Sound Imaging  
SOS imaging is a non-invasive imaging modality that can show the acoustic property change in body tissue [34]. Previous studies have shown that the SOS in body tissue is sensitive to the temperature change [35]. In our study, the temperature control is necessary for the constant temperature hyperthermia system. Due to the available relationship between temperature and SOS (temperature and SOS vary in direct proportion) [36], the temperature monitoring in the medium is realized by the SOS imaging. There are basically three categories of SOS imaging methods. The first one is filtered back projection (FBP) reconstruction technique, which is also one of the main techniques in image reconstruction of clinical CT applications [37]. By employing the Radon transform, FBP technique can complete the image reconstruction in frequency domain, which avoids the direct solution of complex inverse problem including finding the mapping function that matches the measured projection data and the SOS distribution. In this way, the FBP reconstruction technique can provide fast computing speed with a reduction of computation complexity by using the Radon transform, which is the foundation of the FBP algorithm. A high-quality image can be produced when enough transmission data have been acquired, which is difficult to realize in practical application. Furthermore, FBP technique assumes that the propagation of acoustic waves travels along straight lines in body tissue [38]. The approximation causes the limitation of the precision in SOS reconstruction. The second one is bent-ray tracing method (BRTM) based on ray theory, which employs time of flight (TOF) of acoustic wave propagation paths to achieve the SOS distribution reconstruction [39]. In this method, the acoustic wave propagation paths can be estimated by using the principles of geometrical optics and these paths are not straight when traveling through different medium interfaces [40]. The precision of the TOF measurement is crucial to the accuracy of the SOS distribution estimation. In our application, the average length of acoustic wave propagation paths is around 120 and the average SOS is about 1580. Assuming that the sampling frequency of the ring ultrasound transducer array is 20, the average error of SOS is more than 1, which means that the deviation of temperature monitoring is not satisfied with the temperature fluctuation limitation. The third one is full-wave inversion (FWI) based on the theory of acoustic wave propagation, which has many applications in medical imaging and geophysics [41; 42; 43]. When applying the FWI method, the SOS image reconstruction is realized by estimating the SOS distribution using the acoustic wave equation and the measured data. The measured acoustic wave data can be acquired by the receiver ultrasound sensor elements and the estimated acoustic wave data can be acquired through pseudospectral k-space method. Given the gradient of a sum of squared norms between the measured data and the estimated data, a gradient descent-based optimization algorithm can be employed to update the SOS estimate at every iteration [44]. Considering high order diffraction and scattering, FWI method can provide the better image quality in robustness, resolution and accuracy compared with two previous methods that have an approximate hypothesis [45]. In our work, temperature monitoring in the soft tissue is realized by the FWI method, which depends on the optimization iteration algorithm to achieve the SOS distribution reconstruction. At the beginning of iterations, the initial assumed SOS values of the medium play an important role in calculation speed and accuracy of SOS image reconstruction. Through a correspondent B-mode ultrasound image, the boundary information and position information of human thigh tissues can be acquired.