# Analysis of the Refraction Effect in Ultrasound Breast Tomography
## 2 Materials and Methods
### Refraction in Ultrasound Transmission Tomography Scanning
As can be seen in Figures 2 and 3, the refraction phenomenon of ultrasound wave rays after passing through the cylinder cross-section is clearly visible, first of all, at its edges. In both cases, there are areas near the edges of the cross-section that will not be penetrated by the ultrasonic wave due to the refraction phenomenon, and these areas have a larger size for \(n>1\) than for \(n<1\) for the same absolute value of the difference \(1n-11\). In the case of wave beam spreading (Figures 2a and 3a), the area of overlap between the detection points of the rays passing after refraction through the cross-section of the cylinder and the rays passing directly through the water is only about 4.5 mm at its lateral edges (colored blue). It means that in transmission imaging, the diameter of the breast cross-section can be enlarged by a maximum of about 9 mm. In the case of wave focusing (Figures 2b and 3b), the transmission fading area (colored red) is approximately 7.2 mm at both lateral edges of the cylinder. This measurement situation is very unfavorable. The random noise pulse time, the start time, or the end time of the measurement window will then be determined when measuring the pulse transit time within a certain time window. Such measurements will introduce errors, especially for ultrasound tomography image reconstruction [42,43]. In order to minimize signal fading errors (Figures 2b and 3b), it is therefore necessary to develop algorithms that interpolate missing measurement values on the basis of neighboring values. For in vivo studies, it is also a good idea to set the temperature of water in which the breast is immersed in such a way that a refraction index close to unity can be obtained at the water/breast interface. Based on the results obtained from the calculations, it can be predicted that in the case of ultrasound transmission tomography measurements, the influence of refraction on the errors in the imaging of the structure of biological media is not significant for most soft tissues, which are characterized by the index of refraction \(0.9<n<1.1\). In the case of tissues, due to the refraction phenomenon, it is advantageous to use large \(l_{\text{o}}\) distances between the source and the detector in relation to the size of the cross-section of the examined object. For large distances, when the axis of the pair of sending and receiving transducers is outside the object, a wave ray passing directly through the water instead of a ray refracted in the object can be expected to reach the receiving transducer. For small \(l_{\text{o}}\) distances compared to the cross-sectional size of the object, the opposite situation is more likely, since transmission coefficients \(D_{p}(\alpha)\) at the water/tissue interface for most soft tissues are close to unity over a wide angular range (Figure 1b). In such a situation, the size of the structure under study in the ultrasound tomography image can be artificially slightly enlarged. A small \(l_{\text{o}}\) distance, on the other hand, is advantageous because of the improved resolution for detecting small changes in ultrasound velocity introduced by small inhomogeneities located in the path of the transmitted ultrasound wave. Experiments performed by the authors on tissue  
Figure 3: Calculated values of ultrasound velocity projection through a circular cross-section of a cylinder without and with refraction for refraction indices: (**a**) \(n=0.991\); (**b**) \(n=1.009\). The ultrasound velocity projection value \(c_{p}\) is determined as the ratio of the rectilinear path of the wave passage between the transmitter and receiver to the measured (simulated) transit time. phantoms and biological media have shown that for \(n<1\) (the ultrasound velocity in the object is higher than in its surroundings), as a result of intense refraction of passing rays near the edge of the water/phantom, a narrow ring of significantly overstated ultrasound velocity values appears in the reconstructed cross-sectional image of the phantom around its edges.