# Analysis of the Refraction Effect in Ultrasound Breast Tomography
## 3 Results
### Analysis of Beam Ray Incident Angles on a Cross-Section of a Breast in Water  
In the context of calculations and analyses performed in Chapter 3.1, it turns out that to be of interest to investigate the location and size of the area of the measured breast cross-section on which the ultrasonic wave beam rays coming out from the sending transducer of the ring array fall in water at specific angles (refraction/distortion ring). For this purpose, the simple geometric computational models shown in Figure 13 were developed. Figure 12: Results of optimization of the water temperature obtained with the developed models for the parameter \(\Delta_{opt}\): without (**a**) and with temperature gradient (**b**) in the fat layer. This model is an extension of the model described in Chapter 3.1. It additionally allows the angular steps of the rays to be set with respect to the placement of 1024 ultrasonic transducers in the ring array of a prototype of the UBT scanner. The breast cross-section was modeled as a circle with a given diameter \(D_{b}\), without taking into account additional layers. This cross-section was assumed to be centrally located in the ultrasound tomography ring array composed of 1024 uniformly spaced ultrasonic transducers (angular pitch = 360\({}^{\circ}\)/1024 \(\approx\) 0.3516\({}^{\circ}\)). The maximum diameter of the measurement area in water \(D_{p_{max}}\) when using 513 receiving transducers in one measurement projection is:  
\[D_{p_{max}}=\frac{D_{r}}{\sqrt{2}} \tag{5}\]  
where \(D_{r}\) is the diameter of the inner ring of the ultrasonic array. Using the developed models, it is possible to visualize the area of the largest refraction of the ultrasound wave in the cross-section of a breast immersed in water, inside the ring array. The internal diameter of the ring array \(D_{r}\) = 260 mm was assumed in the modeling. The maximum diameter of the measurement area in water when using 513 receiver transducers in one projection is \(D_{p_{max}}\) = 183.85 mm, in this case. This number of receiving transducers is matched to their horizontal directivity characteristics in a ring array arrangement. However, it should be noted that the reciprocal relationship between the inside diameter of the ring array and the diameter of the breast cross-section and the number of ultrasonic transducers are important in modeling. The angles of incidence of ultrasound beam rays in the cross-sectional area of the breast in the horizontal plane can approach 90\({}^{\circ}\) for the divergence angles of the beam that cover this cross-section (Figure 13), and the divergence angles decrease with the \(D_{b}/D_{r}\) ratio. In the vertical plane, because the beam is narrow and the array moves along the breast hanging in water, the planes of the breast sections examined are parallel to each other and critical incidence angles above 60-70\({}^{\circ}\) are obtained only for sections of the breast tip, near the nipple (Figure 13). However, the phenomenon of vertical ray refraction (even small) is much more dangerous than horizontally, because in this case the rays of a narrow beam, after refraction, may not reach the surface of the receiving transducer on the other side. Thus, it may be impossible to examine the breast tip over a fairly large range of cross-sections because of vertical refraction. The results of the model calculations of the area of the largest refraction of the ultrasound wave in the cross-section of a breast immersed in water, inside the ring array for three different diameters of these breast cross-sections (\(D_{b}\) = 180, 120, 40 mm) are presented in the horizontal plane in Figure 14 and in the vertical plane in Figure 15 (transducer No. 513 is the source). For the assumed value of the inner diameter of the ring array \(D_{r}\) = 260 mm  
Figure 13: Geometric computational models for investigating the area of greatest refraction of an ultrasound wave in a cross-section of a breast submerged in water, inside a ring array, successively in the horizontal and vertical planes.