# Analysis of the Refraction Effect in Ultrasound Breast Tomography
## 3 Results
### Analysis of the Effect of Water Temperature on the Refraction of the Beam Path
This means that in the angular range of 0-37\({}^{\circ}\) we can find the optimum value of the water temperature (between 30 \({}^{\circ}\)C and 35 \({}^{\circ}\)C) due to the refraction minimization. For beam ray angles close to the critical incidence angles at the breast cross-section boundary (see the second row in Figures 9 and 10), the refraction phenomenon successively increases intensity with increasing water temperature from 20 \({}^{\circ}\)C to 35 \({}^{\circ}\)C. Moreover, in this case, the refraction angles especially at the water/skin and fat/skin boundaries increase significantly, and deviation of the ray path from straightness are large. This is also associated with an elongation of the transition paths through these layers, resulting in a weakening of the ultrasonic signal. In order to optimize the value of the water temperature for minimizing the refraction phenomenon, the parameter \(\Delta_{opt}\) was chosen to characterize the difference in the angle of deviation from the \(Y\)-axis of the beam ray incident from the water to the breast border (skin layer) and the ray exiting the breast cross-section (skin layer) into the water:  
\[\Delta_{opt}=\alpha_{IN}-\alpha_{OUT}\left[{}^{\circ}\right] \tag{4}\]  
where \(\alpha_{IN}\)--angle of deviation from the \(Y\)-axis of the beam ray incident from the water on the boundary of the breast skin layer, \(\alpha_{OUT}\)--angle of deviation from the \(Y\)-axis of the beam ray exiting the breast skin layer into the water. The optimal value of \(\Delta_{opt}=0\) means that the beam ray incident from the water on the breast skin has the same directional coefficient as the beam ray that exits from this section into the water. Figure 11 shows the results of the calculation of the parameter \(\Delta_{opt}\) as a function of the angle \(\alpha_{IN}\), for different values of the water temperature and both models (without and with a temperature gradient in the fat layer). On the auxiliary \(X\)-axis, the values of incidence angles of the beam rays at the water/skin boundary determined from \(\alpha_{IN}\) angles are given. Figure 11: Results of the calculation of the parameter \(\Delta_{opt}\) as a function of the angle \(\alpha_{IN}\) for different values of the water temperature and both models: without (**a**) and with temperature gradient (**b**) in the fat layer. Due to the negligible differences in the refraction of the ultrasound beam rays in the cross-section of the breast with uniform and stratified fat (cf. Figures 9 and 10), the differences are also small for the parameter \(\Delta_{opt}(\alpha_{IN})\) (Figure 11). In the plot, the angle \(\alpha_{IN}\approx 32^{\circ}\) (corresponding to the angle of incidence at the water/skin boundary \(\sim\)50\({}^{\circ}\)) is characteristic, for which the course of \(\Delta_{opt}(\alpha_{IN})\) changes its character and starts to increase or decrease more intensively depending on the temperature. Then \(\Delta_{opt}\) starts to decrease from an angle \(\alpha_{IN}\approx 35^{\circ}\) (i.e., incidence angle of 55.9\({}^{\circ}\)) at each water temperature, until finally \(\Delta_{opt}\) decreases sharply for angles \(\alpha_{IN}>37^{\circ}\) (i.e., incidence angles larger than about 60\({}^{\circ}\)). Positive but close to zero values of \(\Delta_{opt}\) can be observed for \(\alpha_{IN}\leq 35^{\circ}\) for a water temperature of 30 \({}^{\circ}\)C. Figure 12 shows the results of the optimization of the water temperature obtained with the developed models for the parameter \(\Delta_{opt}\). The optimization was performed by selecting the water temperature values separately for each angle \(\alpha_{IN}\) of the beam ray (blue line and scale of the main \(Y\)-axis), so that the coefficient \(\Delta_{opt}\) tends to zero (orange line and scale of the auxiliary \(Y\)-axis). The water temperature in the optimization procedure was adjusted with a resolution of 0.1\({}^{\circ}\) because it is difficult to stabilize it more precisely in the real measurement system.