Page 216 - Computational Modeling in Biomedical Engineering and Medical Physics

Magnetic drug targeting 205

intense, MD transfer through the endothelial membrane and tissue are described
by, respectively,
@
c M 1 v mg Ur c M 5 rU D M rc M Þ;
@t ð ð6:30Þ

@
c T 1 v mg Ur c T 5 rU D T rc T Þ;
@t ð ð6:31Þ

where c M and c T denote the species (MNPs) in the membrane and tissue, respectively,
D M and D T are the diffusivities of MNPs through the membrane and tissue, respec-
tively. The recommended magnetic velocity, v mg ,is (Grief and Richardson, 2005;
Nacev et al., 2011)

a μ χ
2
2
v r 5 krH ; k 5 0 ;
9η 1 1 χ ð6:32Þ
3
where k is the magnetic drift coefficient, which depends on: a—MNP radius,
χ—MNPs susceptibility, and η the dynamic viscosity of blood.
In fact, the suggested approach to calculate the magnetic velocity, v r , that is used as
a macroscopic quantity, is a homogenization technique that raises the physical model
scale from the MNP nanometric level to a macroscopic, continuous, magnetizable
medium, superparamagnetic like, which is characterized by an apparent susceptibility,
χ app , that depends on the (constant) viscosity of a Newtonian fluid—the coefficient k
in Eq. (6.32), which is proportional with the susceptivity of the MNP and the
dynamic viscosity of the solute fluid. In view of Eq. (6.32), for constant χ (same mate-
rial for the MNP), η - 0 yields k - N, consequently less viscous the fluid is the
higher the magnetic velocity is.
The bidimensional model in Fig. 6.28 refers to the membrane of a blood vessel
and the surrounding tissue: a longitudinal section through a cylindrical blood vessel
(R1) with a diameter d 5 5 mm. The vessel is bounded by an endothelial layer
(R2) of thickness g 5 0.5 mm, and embedded into the target tissue (R5), 50 mm
thick. The PM (R4, NdFeB, B rem 5 1T, of size 10mm 3 30 mm) is outside the
target tissue, and its magnetization is set to favor the transendothelial transfer of
MNPs. MNPs may cross through the wall with a diffusion coefficient D M .The
magnetic field problem is closed by free space (R3). The membrane face inside the
3
vessel has C b 5 4mol/m , and all other parts of the species transfer boundary are
2
2
impermeable. The properties are D M 5 1.5 3 10 212 m /s, D T 5 1.2 3 10 214 m /s,
23 2 217 4 2
a 5 2.5 3 10 m , k 5 5 3 10 m /(A s), and χ 5 0.003 (Swabb et al., 1974;
Nacev et al., 2011; Soltani and Chen, 2013; Winner et al., 2016; Zhan et al., 2019).