Page 200 - Bio Engineering Approaches to Cancer Diagnosis and Treatment

8.3 Governing equations 199

The vector field particle velocity is similar to wave equation:
1 ∂ u 2 2 2 2
2
∇ u = (8.9) ∇ u=1c ∂ u∂t
2
c 2 ∂t 2
The sound wave equation in one dimension is as follow:
2
2
∂ P − 1 ∂ P = 0 (8.10) ∂ P∂x −1c ∂ P∂t =0
2
2
2
2 2
∂x 2 c 2 ∂t 2
where c and P are the speed of sound and the acoustic pressure (the local deviation
from the ambient pressure), respectively.
8.3.2 Westervelt equation
The Westervelt equation is a very popular and high accuracy model for ultrasound
modeling. The generalized Westervelt equation is as follow [59, 60]

 1  1 ∂ P δ ∂ p β ∂ p 2
3
2
2
ρ∇⋅  ∇p  − 2 2 + 4 3 + 4 2 = 0 (8.11) ρ∇⋅1ρ ∇p− 1c ∂ P∂t +δc ∂ p∂t +
2
2
4 3
3
2


ρ
c
∂t
c
ρc
∂t
∂t
4 2 2
2
βρc ∂ p ∂t =0
where p, c, δ, β, and ρ are the sound pressure, the speed of sound, the sound dif-
fusivity, the coefficient of nonlinearity and the ambient density, respectively. In Eq.
(1), the first term takes diffraction into account. The third term accounts for attenu-
ation. The last term introduces the quadratic nonlinearity. Several simplified ver-
sions can be derived from Eq. (1). By using constants for all acoustic parameters, the
Westervelt equation for homogeneous media can be recovered [61].
1 ∂ P δ ∂ p β ∂ p 2
2
2
3
2
4 3
2
2
3
∇ p − + 0 + 0 = 0 (8.12) ∇ p− 1c ∂ P∂t +δ c ∂ p∂t +β
2
2
c 0 2 ∂t 2 c 0 4 ∂t 3 ρ c 4 ∂t 2 0 4 2 2 0 0
2
ρ c ∂ p ∂t =0
00
0 0 0
where c , ρ , δ , and β are the acoustic parameters for the background medium.
0
0
0
0
The linear acoustic wave equation can be derived by setting β to 0, so that
0
3
2
1 ∂ P δ ∂ p
2
3
4 3
∇ p − + 0 = 0 (8.13) ∇ p− 1c ∂ P∂t +δ c ∂ p∂t =0
2
2
2
2
c 0 2 ∂t 2 c 0 4 ∂t 3 0 0 0
Eqs (2) and (3) are useful for studies on characterizing acoustic fields of trans-
ducers and for approximately estimating the acoustic field in biological tissue [62].
Eq. (8.11) is used when heterogeneous medium such as skull [59]. On the other
hand, nonlinear Eqs (8.11) and (8.12) are used for lithotripsy, histotripsy, and tissue
harmonic imaging [63], when high pressure of ultrasound is present. Researchers
reported that using the linear acoustic approximation could underestimate the tem-
perature elevation in tissue [62].