# 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  
To analyze the influence of the water temperature in the UBT scanner tank on the refraction phenomenon of the ultrasound wave that penetrates the cross-section of the breast immersed in this water, two different curvilinear layers of water/skin/breast fat/breast glandular tissue were created, shown in Figure 8.  
Figure 7: View of the prototype of the hybrid UBT scanner developed by DRAMINSKI S.A., the ultrasonic ring array, and elementary transducers. The adoption of these two models is based on the assumption that shortly after the breast is immersed in vivo in water at a temperature different from the body temperature (36.6 \({}^{\circ}\)C), a value of about 36.6 \({}^{\circ}\)C is briefly still maintained in the subcutaneous fat layer, which is, on average, 5 mm thick. After keeping the breast submerged for a longer time, a temperature gradient should develop in the subcutaneous fat layer. This is assumed to be a linear gradient from the temperature value in the surrounding water to the value in the glandular tissue of the breast located at the fat layer. To simplify the calculations, the average, constant value of the ultrasound speed in the skin (1642 m/s) with assumed thickness of 2 mm was set. The fat layer was divided into five sublayers with linearly varying temperature (Figure 8). It has been verified that at this fat layer thickness, the inclusion of more sub-layers has no significant effect on the refraction phenomenon in the developed model. Calculations were performed for four different water temperatures: 20, 25, 30, 35 \({}^{\circ}\)C and both models--without and with gradient. Ultrasound velocity conditions in water for these temperatures were taken as: 1482.4 m/s, 1496.7 m/s, 1509.2 m/s, and 1519.8 m/s (Figure 6). The ultrasound velocity in the subcutaneous fat layer at body temperature was taken as 1470 m/s (Figure 6), and in the glandular tissue of the breast, as 1515 m/s, since most refractive errors in ultrasound transmission tomography images were observed for dense breasts. Ultrasound velocities in the gradient layers of breast adipose tissue were determined from the temperature coefficient (Figure 6) in the model. For example, for a water temperature of 25 \({}^{\circ}\)C, the temperatures in the five individual layers of adipose tissue are, respectively: 26.9, 28.9, 30.8, 32.7, and 34.7 \({}^{\circ}\)C and the corresponding ultrasound velocities in these layers: 1500, 1494, 1488, 1482, and 1476 m/s. The calculations were performed for beam rays that ranged from 5\({}^{\circ}\) to 37\({}^{\circ}\) (the angle from the \(Y\)-axis to the beam ray), which ranged from 7.2\({}^{\circ}\) to 60.4\({}^{\circ}\) after recalculation to incidence angles from water to the curvilinear skin surface. Furthermore, for each water temperature (20, 25, 30, 35 \({}^{\circ}\)C) and both models, calculations were also made for the maximum possible values of the beam ray angle, still allowing this ray to pass through all layers (water/skin/fat/breast/skin/water). These critical angles of beam rays are 62\({}^{\circ}\), 63\({}^{\circ}\), 64\({}^{\circ}\), and 64.8\({}^{\circ}\), respectively, after recalculation into incidence angles from water to the curvilinear skin surface, regardless of the model (Figures 9 and 10). The angle of incidence \(\alpha_{i}\) is equal to or greater than the first critical angle determined by the following relation, and the longitudinal wave disappears in the medium \(i+1\):  
\[\sin\alpha_{critical_{i}}=\frac{c_{L_{i}}}{c_{L_{i+1}}}\, \tag{2}\]  
Figure 8: Layered computational models of the water/skin/breast fat/breast glandular tissue media: (**a**) assuming body temperature in the breast fat layer; (**b**) assuming a linear temperature gradient changing from water temperature to body temperature in the breast fat layer. (where \(c_{L_{i}}\)--ultrasound velocity in the medium \(i\), \(c_{L_{i+1}}\)--ultrasound velocity in medium \(i+1\)).