Page 176 - Bio Engineering Approaches to Cancer Diagnosis and Treatment

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

D Tissue is the MNPs diffusion coefficient in the tissue and is obtained by:

D Tissue = D ×   ε  ×× J (7.47)
S
2 
∞
DTissue=D∞×ελg2×S×J  λ g 
Because the interstitial fluid is motionless the D is usually equal to D [1].
∞ plasma
The steric coefficient (S) and hydrodynamic coefficient (J) are obtained from [37]:
S = exp(− 0.84 k ) , k =   1+ r MDC   2 × φ (7.48)
1.09
S=exp−0.8  r fiber 
4k1.09, k=1+rMDCrfiber ×φ
2
J=e−α×φϑ J = e −× ϑ (7.49)
αφ
Φ is the fibers volume fraction. Collagen fibrils radius, r fibrils , is equal to 50 nm.
The value of r fibrils is between 15 and 100 nm [37]. Also, the value of the fiber volume
fraction of fibers, Φ is equal to 0.01 where the values for three different tumors are
Φ = 0.01 (LS174T), Φ = 0.03 (HSTS26T), and Φ = 0.045 (U87) [39].
The geometrical tortuosity (λ ) is expressed as a function of porosity [1]:
g
λg=ε−n λ = ε − n (7.50)
g
Tissue porosity, ε, varies between 0.06 and 0.6 for different tumors [1]. The value
of n has an upper and lower limit and is determined by:
+
+
2 2 Upperlimit n : = 0.23 0.3ε + ε 2 andLower limit : n = 0.23 ε 2 (7.51)
Upper limit:n=0.23+0.3ε+ε andLower limit:n=0.23+ε
For the safe simplicity an average value of n is utilized.
For initial condition dimensionless MDCs concentration (C) inside the vessel is equal
to 1 and no concentration is assumed in the tumor tissue and in the capillary wall (C = 0).
According to Fig. 7.8 the boundary conditions are as provided in Table 7.3.
Fig. 7.9 shows the distribution of dimensionless MDCs concentration in the tumor
tissue at 24 h time under influence of external magnet with different flux densities.
MDCs diameters are 50, 100, 166, 250 and 345 nm.
As shown, in the absence of external magnet (0 T), the small MDCs penetrate
deeper into the tumor, but their penetration into the tumor tissue, in general, is not

Table 7.3 Fig. 7.8 boundary conditions.
Boundary Conditions
Channel inlet Steady velocity (U = 0.2 mm/s [35])
in
−4
Steady MDCs inlet concentration (C = 10 )
0
Channel upper wall No-slip condition (u,v = 0)
MNPs can go through the wall
Channel lower wall No-slip condition (u,v = 0)
MNPs can’t go through the wall
Channel outlet Neumann boundary condition for both momentum and
concentration equations
Tissue upper, right and left wall MDCs cannot go through the wall