Page 174 - Bio Engineering Approaches to Cancer Diagnosis and Treatment

172 CHAPTER 7 Application of magnetic and electric fields for cancer therapy

The continuity and momentum equations for blood are as follows:
u ∂ + v ∂ = 0
∂u∂x+∂v∂y=0 ∂ x ∂ y (7.30)
 u ∂ u ∂  ∂ P  ∂ 2 u ∂ 2 u
ρ  u + v  =− + µ  + 2  + F x (7.31)
2
ρu∂u∂x+v∂u∂y=−∂P∂x+µ∂ u∂x +∂ u  ∂ x ∂ y ∂ x  ∂ x 2 ∂ y 
2
2
2
∂y +Fx  v ∂ v ∂  ∂  ∂ 2 ∂ 2 v
ρ  u + v  =− P + µ  v + 2  + F y (7.32)
2
2
2
ρu∂v∂x+v∂v∂y=−∂P∂y+µ∂ v∂x +∂ v  ∂ x ∂ y ∂ y  ∂ x 2 ∂ y 
2
∂y +Fy
The MDCs structure is depicted in Fig. 7.3. As shown the MDCs are core-shell
(spherical core with 5 nm biocompatible shell). The volume of MDC (∀ MDC ) is sum
of core sphere volume and coating volume. Total volume of MNPs (∀ MNP ) is 0.7
of core sphere volume. Therefore, total MNP‘s volume (∀ MNP ) of single MDC is
as follow:
∀MNP=0.7 ∀MDC− ∀shell ∀ MNP = 0.7 ( ∀ MDC − ∀ shell ) (7.33)
The magnetic body force inside the capillary is given by [1]:
χ C C
F = Fn 0.5 × ∀ MNP × µ 0 χ MNP ×∇ | H | × n p and n = ∀ MDC = C × ∀ 0 (7.34)
2
p=
p
1
F→=F→ np=0.5×∀MNP×µ χMNP1+χMNP3× 1+ MNP 3 MDC MDC
0
1
2
∇|H| ×npandnp=CMDC∀MDC=C×C ∀MDC
0
∀MNP where ∀ MNP is single MNP‘s volume and χ MNP is the magnetic susceptibility of the
MNPs and set equal to 3 [36]. Also C MDC , C , and C are volumetric concentrations of
0
MDCs in the blood, concentration of MDCs at inlet and dimensionless concentration
(C MDC /C ), respectively.
0
The concentration equation inside the capillary is as follows [1]:
∂ C +∇ C ( ⋅ v D ( ∇ C)
∂C∂t+∇⋅Cv→MDC=∇⋅Dblood∇C t ∂ MDC ) = ∇⋅ blood (7.35)

v→MNP where D blood , v MNP , and V relative are MNPs diffusion coefficient in blood, MNPs velocity
vector, and the MNPs relative velocity with respect to the blood flow, respectively.
Also G and E are generation and uptake terms, respectively. In this model, no genera-
tion and no uptake is considered. [1].

v
v→MNP=v→+v→relative v MNP =+ v relative (7.36)
B
D Blood = D + D S D = kT (7.37)
B
B
r
DBlood=DB+DSDB=kBT6πµbloodrMDC 6πµ blood MDC

F ∀ χ 2
v relative = 6πµ 1 = 12πµ MNP × µ 0 χ MNP ×∇ H incapillary
r
r
v→relative=F→ 6πµbloodrMDC= ∀MNP12πµ− blood MDC blood MDC 1+ MNP 3
1
2
bloodrMDC×µ χMNP1+χMNP3 ×∇H in capillary
0
where D is Brownian diffusion coefficient and is calculated from Einstein relation
B
[37] and D is scattering diffusion coefficient and it is equal to 3.5 × 10 −12 [37].
s