Page 201 - Bio Engineering Approaches to Cancer Diagnosis and Treatment

200 CHAPTER 8 Ultrasound applications in cancer therapy

8.3.3 Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation
The KZK equation includes diffraction, absorption, and nonlinear effects have been
the most widely used model. The KZK equation accounting for energy losses was
first published in 1971 [64]. The KZK equation is less accurate in the near field and
at a position off the main axis due to be a parabolic approximation of the Westervelt
equation. For a focused transducer, the KZK equation is in theory valid for waves
traveling within 15–16 degrees of the nominal axis of the beam (typically is the
z-axis) [63]. The parabolic form of the wave equation can be solved with efficient
numerical techniques, which is the main advantage of the KZK equation. The KZK
equation is an equation of evolution type and has the first-order derivative with
respect to the propagation main axis.
The KZK equation for thermoviscous and homogeneous material is as follows [63]:
3
2
2
∂ p c δ ∂ p β ∂ p 2
2
∂ p∂z ∂t′−c 2 ∇∇2p−δ 2c ∂ p∂t′ −β 2 − 0 ∇ p − 0 3 3 − 0 3 2 = 0 (8.14)
3 3
2
3
⊥
0
0
0
0
zt
2
2 2
ρ c ∂ p ∂t′ =0 ∂∂ ′ 2 c 2 0 ∂ ′ t 2 ρ c ∂ ′ t
3
00
0 0
∂ 2 ∂ 2
2
2
2
2
∇∇2=∂ ∂x +∂ ∂y 2 where t′ is the retarded time (t′ = t − z/c ), ∇= ∂x 2 + ∂y 2 is the transverse Laplacian.
⊥
0
8.3.4 Kuznetsov equation
The Kuznetsov equation is more accurate than the Westervelt equation. For nondirec-
tional beams, it describes as two forms as follow [64]:
Base on velocity potential (φ):
∂ 2 φ ∂  1  4  2 β −1  φ ∂  2 
+

2
2
2 2
∂ φ∂t −c ∇ φ= ∂∂t 1ρ µB+ 43 2 − c 0 2 ∇ 2 φ =  ρ   µ + µ ∇ 2 φ (∇ φ) + 2     (8.15)

B
0
t
0
2
2
µ ∇ φ+ ∇φ + β−1 c ∂φ∂t ∂t ∂   0 3 c 0 ∂  t  
2
2
0
where φ, µ, and µ are the velocity potential, the coefficient of shear viscosity, the
B
coefficient of bulk viscosity, respectively.
Base on acoustic pressure:
2
2
3
1 ∂ P δ ∂ p β ∂ p 2  2 1 ∂ 
2
2
2
2
2
2
4 3
∇ p− 1c ∂ P∂t +δ c ∂ p∂t +β ∇ p − 2 2 + 0 4 3 + 0 4 2 +∇ + 2 ∂   L = 0 (8.15)
3

2

0 0
0
2
2
2
2
4 2 2
2
00
ρ c ∂ p ∂t +∇ + 1c ∂ ∂t L=0 c 0 ∂t c 0 ∂t ρ c ∂t c 0 t
0 0 0 0
where L is the Lagrangian density of acoustical energy that for plane progressive
waves is zero [63].
8.4 Ultrasonic-activated drug delivery
For the past two decades, ultrasonic-activated drug delivery has a major role in can-
cer therapy. Developed technology helps scientist to improve the delivery of drugs
and genes to target tissues while reducing systemic dose and toxicity. In the above
section, the physics of US is discussed. In this section, a brief introduction of drug
carriers for ultrasound drug delivery and the progress and challenges of ultrasonic-
activated drug delivery with special focus on cancer therapy are summarized.