# 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
The calculations in the developed model are realized by determining angles of incidence and refraction of the ultrasonic beam rays on the individual layer boundaries using a generalized formula:  
\[\frac{\sin\alpha_{i}}{c_{L_{i}}}=\frac{\sin\beta_{i+1}}{c_{L_{i+1}}} \tag{3}\]  
(where \(\beta_{i+1}\)--refraction angle in the medium \(i+1\)). These angles and paths of rays in individual layers are calculated from geometric relations taking into account intersections of straight lines with circles constituting borders of individual layers. The developed model is universal and allows adjusting any values of the parameters (diameter of the breast section, distance of the ultrasound source from the breast section, thickness and ultrasound velocities in the layers, water temperature). Figure 9: Results of calculating the refraction of the incident ultrasonic wave beam rays from water on curvilinear breast layers without temperature gradient in fat (see the model in Figure 8a) for angles of beam rays in the range of respectively of 5–37\({}^{\circ}\) (first row), as well as for the maximum possible values of the angle of beam ray still allowing this ray to pass through all layers (second row), for temperatures of 20 \({}^{\circ}\)C, 25 \({}^{\circ}\)C, 30 \({}^{\circ}\)C, and 35 \({}^{\circ}\)C (rows). The results of the calculation of the refraction of the incident ultrasonic wave beam ray from water on both layer models (Figure 8) are visualized in Figures 9 and 10. The black diagonal dashed lines show the rectilinear course of ultrasonic beam rays passing through the breast without taking into account the refraction phenomenon at individual layer boundaries. The fragments of the outgoing ultrasonic rays, including refraction into the water after passing through the breast, are marked with a solid blue line. For incoming (black) and outgoing (blue) rays, the values of angles of deviation from the \(Y\)-axis are given. The short, dashed lines perpendicular to the curvilinear boundary of each layer illustrate the perpendiculars to the tangents at the intersections of the layer boundaries by the passing and refracted wave ray in succession. In this way, it is possible to visually assess the change in ray path due to passage through the layers compared to the rectilinear path (assumed in the image reconstruction) and the change in ray length in each layer for both models, different temperatures in the water, and different angles of incidence of the wave at the water/breast boundary. Figure 10: Results of calculating the refraction of the incident ultrasonic wave beam rays from water on curvilinear breast layers with temperature gradient in fat (see the model in Figure 8b) for angles of beam rays in the range of, respectively, 5–3\({}^{\circ}\) (first row), as well as for the maximum possible values of the angle of beam ray still allowing this ray to pass through all layers (second row), for temperatures of 20 \({}^{\circ}\)C, 25 \({}^{\circ}\)C, 30 \({}^{\circ}\)C, and 35 \({}^{\circ}\)C (rows). The differences between the results obtained for the model without and with the fat gradient appear to be insignificant even for the incidence angles close to the critical ones (compare Figures 9 and 10). When analyzing individual cases for beam ray angles in the range of 5\({}^{\circ}\) to 37\({}^{\circ}\) shown in Figures 9 and 10 (top rows), it can be observed that as the water temperature increases to 30 \({}^{\circ}\)C, the refraction phenomenon at individual layer boundaries proceeds with decreasing intensity. The minimization of refraction results in a correct (close to rectilinear) course of the ultrasonic wave beam ray through the layers. Furthermore, the refracted rays coming out of the breast section have larger \(Y\)-axis deviation angles compared to the incident rays. For a water temperature of 35 \({}^{\circ}\)C, the refraction phenomenon is still low, but the outgoing rays have smaller angles of deviation from the \(Y\)-axis compared to the incident rays.