Page 194 - Computational Modeling in Biomedical Engineering and Medical Physics

Magnetic drug targeting 183

The magnetic field produced by the PM is described through Ampere's law, the mag-
netic flux law (Chapter 1: Physical, Mathematical, and Numerical Modeling)
r 3 H 5 0; rUB 5 0 ð6:5Þ

and the material law B 5 B(H), which for linear magnetic media yields
μ 5 μ 1 1 χÞ 5 μ μ
0 ð 0 r
B 5 μ μ H 1 B rem ; for the permanent magnet;
0 r;mag ð6:1Þ

B 5 μ H 1 M ff HðÞ ; for the aggregate magnetic fluid;
0 ð6:2Þ
B 5 μ H; for arterial walls and embedding tissue: ð6:3Þ
0
In the above B is the magnetic flux density, H the magnetic field strength, and μ 0 the
magnetic permeability of free space. For the PM, μ r 5 μ r,mag and B rem is the remanent
magnetic flux density. For the MAF (superparamagnetic medium), M ff (H) 5 χH,with χ
magnetic susceptibility.
The magnetic vector potential, A (and the divergence free gauge condition)

B 5 r 3 A; rUA 5 0; ð6:7Þ
may be used to present the mathematical model

21 21
r 3 μ μ r 3 A 2 B rem 5 0: ð6:8Þ
r
0
“Infinite elements” are bordering the computational domain to provide for a
boundary that contains the magnetic field within a shorter distance from the magnet,
but conveniently sized for the hemodynamic flow and the structural interactions, and
where the magnetic field may be verified to be vanishingly small (magnetic insulation,
n 3 A 5 0). This approach has the advantage of a single and smaller computational
domain.
The MAF is assumed Newtonian. Its flow is pulsatile (arterial), incompressible and
laminar, described by Eqs. (6.1) and (6.2), with f 5 f mg magnetic body forces
(Chapter 1: Physical, Mathematical, and Numerical Modeling) (Rosensweig, 1997;
Morega et al., 2010)

ÞH:
0
f mg 5 μ MUrð ð6:9Þ
No-slip conditions are set for the walls and pressure conditions for the inlet and outlet:
2
2
2
2
p 1 5 13.300 N/m ; p 2 5 13.290 N/m ; p 3 5 13.040 N/m ; p 4 5 13.040 N/m , p i 5 1 1
21
K sin(t 1 3/2), with K a factor of the order O(10 )(Fig. 6.7)(Morega et al., 2011).