# Implementation of a Breast Phantom with Acoustic Properties for Ultrasonic Thermometry
## 4 Results
### Results of Signal Processing  
To calculate the SOSs for the individual phantoms of each medium, the mean delay (\(\Delta t\)) was first determined across three experimental repetitions using Equation (7).  
\[\Delta t_{mean}=\frac{\Delta t_{Phantom\ 1}+\Delta t_{Phantom\ 2}+\Delta t_{Phantom\ 3}}{3} \tag{7}\]  
A propagation distance of 5 mm was considered. The reference ToF at 32 \({}^{\circ}\)C was determined for each phantom, and by adding the average \(\Delta\)t value for each temperature, the corresponding ToF was obtained at each temperature for each medium. In the two- and three-layer phantoms, the same procedure was followed for each layer taking the corresponding average \(\Delta\)t values. SOS computations were conducted within a controlled temperature range of 32 to 36 \({}^{\circ}\)C, in increments of 0.5 \({}^{\circ}\)C.  
Once the velocities for single-, two- and three-layer phantoms were obtained, these were averaged for each kind of mimicked tissue. The corresponding mean and standard deviation values for each mimicked tissue are reported in Table 2. Additionally, the mean coefficients of variation equal to 0.83, 1.48, and 1.10 were obtained for the fat, gland, and tumor phantoms, respectively.  
Figure 9: Graph showing the signals acquired from the tumor case phantom (fat, gland, and tumor) within the range of interest, along with the filtered echo windows. (**a**) Raw signal acquisition of tri-layer phantom in the tumoral case, (**b**) filtered signal of echo 1, (**c**) filtered signal of echo 2, and (**d**) filtered signal of echo 3.  
Finally, a linear regression was performed both for the SOS as a function of temperature and for the temperature as a function of the SOS, using the thermocouple temperatures as a reference. The linear regressions are shown in Figure 10, where the 95% confidence intervals are also included.  
Figure 10a presents the results of the linear regression of temperature as a function of velocity for fat within the range of 32 \({}^{\circ}\)C to 36 \({}^{\circ}\)C. The corresponding mathematical expressions are provided in Equation (8) for sound velocity as a function of temperature and in Equation (9) for temperature as a function of velocity, both specifically for fat and within the same temperature range.  
\[C(T)=(-3.8061)T+1583.2\qquad[\mathrm{m/s}] \tag{8}\]  
\[T(C)=(-0.2616)C+414.2752\quad[^{\circ}\mathrm{C}] \tag{9}\]  
Figure 10b presents the results of the linear regression of temperature as a function of velocity for the glandular phantom within the range of 32 \({}^{\circ}\)C to 36 \({}^{\circ}\)C. The mathematical expressions for sound velocity as a function of temperature are provided in Equation (10), while those for temperature as a function of sound velocity are given in Equation (11). Both of these are specifically for the glandular phantom.  
\[C(T)=(0.6204)T+1515.0\qquad[\mathrm{m/s}] \tag{10}\]  
\[T(C)=(1.5870)C-2403.8\qquad[^{\circ}\mathrm{C}] \tag{11}\]  
Figure 10c presents the results of the linear regression of temperature as a function of velocity for the tumor phantom within the range of 32 \({}^{\circ}\)C to 36 \({}^{\circ}\)C. The mathematical  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Temperature (\({}^{\circ}\)C)**} & \multicolumn{2}{c}{**Fat**} & \multicolumn{2}{c}{**Gland**} & \multicolumn{2}{c}{**Tumor**} \\ \cline{2-7}  & **Mean** & **STD** & **Mean** & **STD** & **Mean** & **STD** \\ \hline
32.0 & 1461.55 & 14.78 & 1534.74 & 23.22 & 1572.58 & 21.48 \\
32.5 & 1460.02 & 14.22 & 1535.24 & 22.78 & 1573.27 & 20.63 \\
33.0 & 1457.34 & 14.07 & 1535.68 & 22  
expressions for sound velocity as a function of temperature are provided in Equation (12), while those for temperature as a function of sound velocity are given in Equation (13). Both of these correspond specifically to the tumor phantom.  
\[C(T)=(1.5743)T+1522.3\qquad[\mathrm{m/s}] \tag{12}\]  
\[T(C)=(0.6251)C-951.0423\ \ [^{\circ}\mathrm{C}] \tag{13}\]