# Implementation of a Breast Phantom with Acoustic Properties for Ultrasonic Thermometry
## 3 Methodology
### Methods for Determining the Speed of Sound in Phantoms
#### 3.3.2 Bilayer SOS Characterization  
For the characterization of the phantom composed of two layers simulating healthy breast tissue, the procedure was adjusted to account for the delay of each layer. Two echoes were obtained: one from the (a) Fat/Gland interface and another from the (b) Gland/Hard Wall interface. In a phantom made up of different types of tissues, each echo ToF represents the time taken for sound to traverse all the layers until the interface (accumulated ToF). Since the echoes from deeper layers accumulate delays from the previous ones, it is essential to eliminate these accumulated delays subtracting delays from previous layers.  
The time delays, \(\Delta\)t, were calculated for the first layer by correlating the window associated with the (a) Fat/Gland interface at 32 \({}^{\circ}\)C with the windows corresponding to other temperatures at the same interface (a) Fat/Gland, ranging from 32.5 \({}^{\circ}\)C to 36 \({}^{\circ}\)C in 0.5 \({}^{\circ}\)C increments. The correlations provided the lag in samples relative to the reference temperature (32 \({}^{\circ}\)C), which was then converted to time delays (\(\Delta\)t) by dividing the shifted  
Figure 6: Graph illustrating the behavior of cross-correlation: (**a**) representation of echoes acquired in a fatty medium at two different temperatures; (**b**) result of the correlation of echoes associated with the fatty medium; (**c**) representation of echoes acquired in a glandular medium at two different temperatures; (**d**) result of the correlation of echoes associated with the glandular medium.  
samples by the sampling frequency. Using the same approach, the procedure was applied to the (b) Gland/Hard Wall interface.  
The ToF to the first echo (a) Fat/Gland interface at an initial reference temperature is represented as \(t_{01}\) in Figure 7, while the ToF to the second echo at the same reference temperature is represented as \(t_{02}\).  
In this context, when internal temperature increases uniformly inside the phantom body, \(t_{11}\) and \(t_{12}\) correspond to the new ToF's to the echoes of the (a) Fat/Gland interface and (b) Gland/Hard Wall interface, respectively, measured at the new temperature. These time measurements, which are typically on the order of microseconds, allow us to calculate the differences (\(\Delta t\)) in ToF due to temperature changes for each echo using Equations (2) and (3).  
These differences can represent time delays (+) if the echo moves away from the origin (zero seconds) or time advances (--) if the echo moves closer to it.  
For the second echo associated with the (b) Gland/Hard Wall interface, the cumulative effect of the delay (or advance) of echo for the (a) Fat/Gland interface must be subtracted from the delay of the echo of the (b) Gland/Hard Wall interface (Equation (4)) to isolate the effect of gland medium from the fat medium. This approach allows us to relate the observed delay or advance directly to changes in ultrasonic velocity in the gland medium.  
\[\Delta t_{11}=t_{01}-t_{11} \tag{2}\]  
\[\Delta t_{12}=t_{02}-t_{12} \tag{3}\]  
\[\Delta{t^{\prime}}_{12}=\Delta t_{12}-\Delta t_{11} \tag{4}\]