Page 175 - Bio Engineering Approaches to Cancer Diagnosis and Treatment

7.6 Application of external magnet on cancerous solid tumors 173

Finally, the concentration equation for blood in capillary is obtained by [37]:
2 

∂ C =−∇⋅ − ∇ + 0.5∀ core µ χ MNP ∇ H ( )
C Cv +
t ∂  D Blood C 6πµ r 0 1+ χ MNP  (7.38)
  MDC 3   ∂C∂t=−∇⋅−DBlood∇C+Cv→+C0.5∀co
The last term in right hand of Eq. (7.38) is mass transfer due to magnetic force re6πµrMDCµ χMNP1+χMNP3 ∇H 2
0
(influence of external magnet). As shown this term is function of   ∀ MNP   . By insert- ∀MNPrMDC
MNP 
ing ∀ MNP from Eq. (7.33) the term    ∀ MDC   is as follows:  r MDC  ∀ M N P
∀MNPrMDC
r
9 )
−
∀ MNP = 0.7 (∀ MDC − ∀ shell ) = 2.8 π r ( MDC − e 5 ( ) 3 (7.39)
r MDC r MDC 3 r MDC ∀MNPrMDC= 0.7 ∀MDC− ∀shellr-
MDC= 2.8 π rMDC−5e−9 3 rMDC
3
This means that this term becomes greater as MDCs radius (r MDC ) increases.
Capillary wall (endothelium) is modeled as a porous media. The concentration
equation as follows [37]:
∂ C =−∇⋅ [− ∇ + G − EC ()
t ∂ D Endo C Cv MDC ] + ε (7.40) ∂C∂t=−∇⋅−DEndo∇C+Cv→MDC+Gε−EC
Also, it is assumed that the fluid is motionless in endothelium layer and the tissue
where the maximum velocity is very small (0.016 µm/s for 1 cm radius tumor sur-
rounded with normal tissue) [37].
Endothelium diffusion coefficient (D Endo ) is given by [1]:

D endo = D ×   ε  ×× J (7.41)
S
2 
∞
 λ g  Dendo=D∞×ελg2×S×J
where D is diffusion coefficient of particle in unbounded fluid.
∞
The gaps between endothelial cells are filled with plasma. The diffusion coef-
ficient of particle in unbounded fluid, D , is given by Brownian diffusion coefficient
∞
of particles in the plasma as follow:
B
D plasma = kT (7.42)
r
6πµ plasma MDC Dplasma=kBT6πµplasmarMDC
µ plasma is plasma viscosity and it is equal to1.24 mPa s [38].
Also S (steric coefficient), J (hydrodynamic coefficient) and ε (the porosity of
endothelium layer) are calculated as follow [37]:
1 α)
S ( − 2 α = r MNP (7.43)
=
2
r core S=1−α α=rMNPrcore
J ( 1 2.1044α + 2.089α − 0.948α ) (7.44)
−
=
3
5
J=1−2.1044α+2.089α −0.948α 5
3
Intercellulargap in Endotheiumlayer
ε = (7.45)
AveragesizeofEndotheium cell ε=Intercellular gap in Endotheium lay-
The concentration equation for tumor tissue, a porous media, is: er Average size of Endotheium cell
∂ C =−∇× − ∇ +  1  s × ∀ MNPe µ χ MNP ∇ 2 

C C ε
2 
t ∂  D Tissue   λ g  J 12πµ r 0 1+ χ MNP H ( )  (7.46)
  plasma MDC 3   ∂C∂t=−∇×−DTissue∇C+C ε1λg2 s ×J∀MNPe12πµplasmarMDCµ χMNP1+χMNP3 ∇H 2
0