Page 209 - Bio Engineering Approaches to Cancer Diagnosis and Treatment

208 CHAPTER 8 Ultrasound applications in cancer therapy

∂ρ ∂t=−∇⋅ρ ν→ ∂ ρ 1 =−∇⋅ ( ρν ) (8.17)
1
1
0
∂t 01

ν ∂
v
ρ ∂ν→ ∂t=−∇p +∇⋅µ ∇v→ +∇v→ T+∇µ ρ 1 =−∇p + ∇⋅ µ ( (∇+ ∇v T ) ) + ∇ µ ( (∇ ⋅v ) ) (8.18)
0 1 1 0 1 1 0 ∂t 1 0 1 1 b 1
b∇⋅v→
1
p =c ρ p 1 = c 2 ρ 1 (8.19)
2
1 1

∇⋅ρ ν→ +ρ ν→ =0 ∇⋅ ( ρν + ρν ) = 0 (8.20)
0
11
2
0 2 1 1

ν ∂ ( ( T ) )
ρ ∂ν→ ∂t+ρ ν→ ⋅∇ν→ =−∇p +∇⋅µ ρ 1 1 + ρν ( 1 ⋅∇ ν ) 1 = −∇ p 2 +∇ ⋅ µ ∇ ν 2 +∇ ν 2 +∇ λ ( (∇⋅ ν ) ) (8.21)
0
0
2
0
0
1
2
1
1
1
∇ν→ +∇ν→ T+∇λ ∇⋅ν→ ∂t
0 2 2 0 2
2
2
p =c ρ +12∂c ∂ρρ12 p 2 = c 2 ρ + 1 2 ∂c ρ 2 ρ 1 2 (8.22)
2
2
2
∂
The relations between wave and bio-heat equations in tissue are as below:
∂T
2
2
k
ρtcpt∂Tt∂t−kt∇ Tt=Qaco+Qbio ρ c pt ∂t t −∇ T t = Q aco + Q bio (8.23)
t
t
Qbio While blood perfusion effect Q bio is simply calculated by using Eq. (8.23), thermal
Qaco source of HIFU Q aco should be calculated by solving nonlinear Westervelt equation.
Qbio=ρ cpwbT −T Q bio = ρ c p ω (T 2 − T 0 ) (8.24)
0
b
0 2 0
2 α  ∂p  2
t
2
Qaco=2αabsw ⋅ρt⋅cst∂pt∂t 2 Q aco = ω ⋅ abs   ∂   (8.25)
2
ρ ⋅cs
t t t
3
2
2
3
3
2
2
∇ pt−1cst2∂ pt∂t +δcst4∂ pt∂t +β ∇ p t − 1 2 ∂ p 2 t + δ ∂ p 3 t + β 4 ∂ p 2 t = 0 (8.26)
2
2
4
4 2
2
ρtcst ∂ pt∂t =0 cs t ∂t cs t ∂t ρ cs t ∂t
t
To solve these equations physical properties of blood and tissue are set as
Table 8.5.
Satisfying continuity equation at vessel wall, velocity for second-order perturba-
tions should not be zero.
Boundary conditions for zero and second-order perturbations as well as for first-
order perturbations are listed in Tables 8.6 and 8.7, respectively.
As known, noninvasive and painless ultrasonic energy transmission into a tissues
is widely used for thermal therapy. HIFU beams do not affect tissues in propagation
path. So it is used as a practical method of hyperthermia, which is a major source