# Implementation of a Breast Phantom with Acoustic Properties for Ultrasonic Thermometry
## 2 Theoretical Framework
### Method for Determining Temperature in Phantoms  
Most of the temperature estimation ultrasonic methods are based on the thermal dependence of the ToF of ultrasound echoes, which considers two different physical phenomena: the local change in the SOS due to temperature variations and the thermal expansion of the propagation medium. Along an A-line (which, in ultrasound, refers to a one-dimensional representation of the echo along the beam's propagation direction), both effects produce echo time shifts. These can be estimated and have been shown to be related to local temperature changes in the propagation medium. Since these effects are generally small, a linear approach is used for their analysis.  
This approach enables temperature estimation along the ultrasound propagation axis. The temperature change \(\Delta T(z)\) can be determined by measuring the axial displacements of the echo as a function of depth \((z)\), and it is influenced by the initial SOS, \(c_{0}\), in the medium before heating, as well as by the thermal properties of the tissue, represented by the thermal expansion coefficient, \(\alpha\), and the coefficient \(\beta=1/c_{0}\cdot\partial c/\partial T\), which describes how the SOS varies with temperature. This relationship allows for temperature estimation using Equation (1):  
\[\Delta T(z)=\frac{c_{0}}{2(\alpha-\beta)}\times\frac{\partial t(z)}{\partial z} \tag{1}\]  
This method assumes a linear variation in the SOS with temperature, facilitating its application across different tissue types, although the values of \(\alpha\) and \(\beta\) may vary depending on fat content and other tissue characteristics [17; 24; 27; 43; 50; 70].  
To calculate the time delays associated with SOS variations, the classical cross-correlation method was used, where the signal acquired at a reference temperature was correlated with the signal corresponding to a higher temperature. In this way, the number of samples shifted relative to the reference signal was obtained, which was then converted into a relative time shift.  
The cross-correlation method employed in this study was calibrated under controlled experimental conditions using sunflower oil. The ultrasonic signals were acquired with a 1 MHz transducer at a sampling frequency of 1 GHz. Prior to processing, the signals were windowed to isolate the region of interest, and then the bandpass filtered between 0.5 and 1.5 MHz to retain the relevant frequency components. Cross-correlation was applied directly to the filtered signals to estimate the time delays induced by temperature variations. Calibration was performed over a temperature range from 19.2 \({}^{\circ}\)C to 30.2 \({}^{\circ}\)C to evaluate the method's sensitivity to thermal changes. Sunflower oil provides a thermally stable medium, making it suitable for initial calibration.  
A linear regression of ultrasound velocity as a function of temperature yielded a negative slope of -3.10 m/s\({}^{\circ}\)C\({}^{-1}\) and an intercept of approximately 1423.79 m/s at 19.2 \({}^{\circ}\)C, indicating a moderate decrease in SOS with increasing temperature. These results are consistent with previous reports, where the temperature coefficient of sound speed (dC/dT) for sunflower oil has been found to be -3.2 m/s\({}^{\circ}\)C\({}^{-1}\) over temperature ranges of 26.85-69.85 \({}^{\circ}\)C (with an intercept of 1387.6 m/s at 44.85 \({}^{\circ}\)C) and 20-40 \({}^{\circ}\)C to -3.4 m/s\({}^{\circ}\)C\({}^{-1}\) with an intercept of 1470.4 m/s at 20 \({}^{\circ}\)C, respectively [71; 72; 73].