# Analysis of the Refraction Effect in Ultrasound Breast Tomography
## 2 Materials and Methods
### Acoustic Parameters of Water, Fat, and Glandular Tissue of the Breast  
In Chapter 2.2, breast tissue was considered homogeneous. In reality, there is a fat layer under the breast skin where the ultrasound velocity is much lower than the average velocity in the glandular tissue. Thus, Figure 5 illustratively analyzes three scenarios for the passage of four ultrasound beam rays in divergent scanning geometry from the transmitter to four different receivers of a multi-element ultrasonic ring array through a breast (with a subcutaneous fat layer, and without considering the thin skin layer at this stage) submerged in water. These are illustrative drawings obtained using the computational model described in Chapter 3.1. All three scenarios assume passage of the rays at angles of \(0^{\circ}\), \(15^{\circ}\), \(45^{\circ}\), and \(52^{\circ}\), respectively, to the beam propagation direction. The first and second ray pass successively through the water/fat/glandular tissue/fat/water layers. The angle of the third ray was chosen so that, in the absence of refraction, it passes through the water/fat/water layers just at the edge of the glandular tissue. The fourth and final ray from the right side of the beam passes through water only, just at the edge of the breast fat tissue. The three scenarios illustrate the refraction pattern of the ultrasound beam rays (black lines) for 3 different cases defining the mutual relationships of ultrasound velocities in water (1), fat (2), and glandular breast tissue (3), respectively: \(c_{1}<c_{2}<c_{3}\); \(c_{1}=c_{2}<c_{3}\); \(c_{1}>c_{2}\) and \(c_{2}<c_{3}\), with only the adjusted velocity value \(c_{1}\) in water (changing its temperature). The white lines illustrate the ray trajectories after the transition without taking refraction into account. As can be seen, the subcutaneous fat layer can lead to a significant deviation of the ultrasonic beam path from a rectilinear transition due to  
Figure 4: Artefacts in the example of reconstruction of ultrasound velocity distribution image in the cross-section of an ostrich egg measured with the prototype of the hybrid UBT scanner. the refraction phenomenon. Thus, this chapter analyzes the acoustic parameters of water, fat, and glandular breast tissue, which can affect the refraction phenomenon of ultrasonic beam rays as well as the acoustic parameters of the skin, a thin layer of which is above the fat layer. The most important acoustic parameter affecting the refraction of ultrasound wave beam rays is its propagation velocity in the medium, especially its changes as a result of inhomogeneity or layering of the tested medium. The weakening of the energy of the ultrasound wave after its passage through the medium is also important, as it depends mainly on the ultrasound attenuation coefficient and the wave transmission coefficient (Figure 1b) through the layers of different acoustic impedances (the product of the ultrasound velocity and the density of the medium). The ultrasound velocity is temperature dependent and increases with increasing temperature in most soft tissues and in water. In fat, the ultrasound velocity decreases with increasing temperature. The average value of the propagation velocity of the longitudinal ultrasound wave in soft tissues is assumed to be \(c\) = 1540 m/s (density \(\rho\) = 1058 kg/m\({}^{3}\), acoustic impedance \(Z\) = 1.62\(\cdot\)10\({}^{6}\) kg/(m\({}^{2}\cdot\)s)), and the mean value of the attenuation coefficient of the amplitude of the ultrasound wave is \(\alpha\) = 0.5 (dB/cm)/MHz \(\approx\) 5.8 (Np/m)/MHz [40]. With some approximation, it can be assumed that in the frequency range of 1-6 MHz, the ultrasound attenuation coefficient \(\alpha/f\) in soft tissues is constant. Increasing the water or fat content in the tissue generally decreases the ultrasound velocity value.