# Analysis of the Refraction Effect in Ultrasound Breast Tomography
## 2 Materials and Methods
### Refraction in Ultrasound Transmission Tomography Scanning  
To illustrate refractive errors in the ultrasound transmission tomography scanning process of breast tissue, numerical calculations were performed for a homogeneous circular cylinder section simulating a coronal cross-section of a 150 mm diameter female breast with an assumed average ultrasound velocity for a premenopausal woman \(c_{prostst}=1510\) m/s (\(T=37\)\({}^{\circ}\)C) [40; 41], immersed in water at temperature \(T=25\)\({}^{\circ}\)C (\(c_{water}\approx 1496.7\) m/s) and \(T=37\)\({}^{\circ}\) C (\(c_{water}\approx 1523.7\) m/s), respectively. The distance between the transmitter and the receiver in water was set as \(l_{o}=300\) mm. The calculations assumed point sizes for each pair of ultrasonic transmitter and receiver. The method for determining the path of the beam rays from the transmitter to the receiver was based on an iterative search for the ultrasonic beam ray to reach the receiver fastest after bilateral refraction on a circular object of assumed ultrasound speed, immersed in water [22; 24; 42]. On the basis of the ray paths determined, the average values (projections) of the transit time were calculated, by which the distance \(l_{o}\) was divided, obtaining the ultrasound velocity projections in the parallel ray geometry, when the pair  
Figure 1: Dependence of: (**a**) angle of refraction of the ultrasonic wave at the water/breast interface from the angle of incidence; (**b**) coefficient of transmission of the ultrasonic wave \(D_{p}\) through a circular cross-section of a breast immersed in water. of transmitting and receiving transducers was moved stepwise along the cross-section of the cylinder. Paths of the refracted rays were determined with high accuracy using the equations of the circle and the equations of the rotated straight and tangential lines in the geometry resulting from the adopted scanning system. In this way, the spread (\(n=0.991\)) and focusing (\(n=1.009\)) of the transmitted ultrasound wave beam can be simulated in the breast cross-section with an index of refraction close to unity, respectively. For both cases, the ray paths of refracted wave beams and calculated ultrasound velocity projection values for the cylindrical cross-section with and without refraction are presented in Figures 2 and 3, respectively, in the parallel beam scanning geometry. Similar phenomena can be expected to occur at the soft tissue boundaries. For \(n<1\), there is always a wave transition from transmitter to receiver, and the blue areas in Figures 2 and 3 indicate the scanning range for which the transit time along refracted rays is less than that along straight lines through water. For transmission measurements, this means that a pulse (of small amplitude) will appear in this area before the pulse passes directly through the water because of refraction on the circular cross-section of the cylinder. Depending on the rise amplitude of this pulse, the signal-to-noise ratio, and the detection threshold of the transit time measurement method, it may or may not be subject to detection. The imaging error of the object and its size may therefore vary, causing an enlargement of the cylinder cross-section and/or a misrepresentation of the ultrasound velocity near its boundaries. For \(n>1\), there is always an area where transmission of the ultrasonic wave cannot be achieved along the path from the transmitter to the receiver. In Figures 2 and 3, this area is marked with red; blue is used to mark the area where the ultrasound wave is simultaneously transmitted along two different refracted rays, while the rays marked with black reach the path from the transmitter to the receiver the fastest and due to the smaller refraction angle, the pulse amplitude in this case is larger. Therefore, it can be assumed that the duplicated rays will not affect the measurement of the pulse transit time of the ultrasonic wave.