Page 196 - Computational Modeling in Biomedical Engineering and Medical Physics

Magnetic drug targeting 185

Figure 6.9 Flow structure interaction in the absence of a magnetic field. Deformations are amplified
by a factor of 122. (A) No flow, no deformation. (B) No magnetic field, maximum flow rate, maximum
deformation, 0.5 μm. (C) Magnetic field, maximum flow rate, maximum deformation, 0.52 μm.

et al., 2011; Dobre, 2012) outlines the displacements of the biological tissues, vessel
wall and embedding tissue, assuming their structural hyperelastic behavior.
Apparently the deformation is related to the blood flow, therefore in what follows it
will be neglected and the vessels will be treated as rigid. The optimization of the mag-
net—its position, size, shape, magnetic properties, and its field spectrum—is proposed next.

The constructal optimization of the magnetic field source
The magnetic field sources in MDT are PMs and electromagnets, and particular attention
is allotted to their design subjected to several optimization criteria (Preis et al., 1991; Hoke
et al. (2008); Schenck, 2000). Recently a constructal optimization (Chapter 2: Constructal
Law Criteria in Morphing Shape and Structure of Systems With Internal Flows) of a PM
was suggested (Morega et al., 2018; S˘ andoiu, 2019), with the aim to enhance the targeting
effect of the magnetic field—“shape with a purpose.” Starting from the standard, uni-
formly magnetized parallelepiped bar, different PM configurations may be envisaged. Of
these we single out the optimization path where the PM bar is split into several smaller,
identical, parallelepiped blocks, with the final aim to optimally cover a specific ROI. The
block width, SW (in split direction, aligned with the hemodynamic flow) and the spacing
between the blocks (the gap size, GS) are the design optimization parameters. The total
volume of the magnetic material is invariant. Moreover, only the inside vessel volume—a
MAF—has magnetic properties.
Instead of Eq. (6.8) we use the magnetic scalar potential V m , H 52 rV m ,which yields

rU μ μ rV m 1 B rem 5 0;
0 r ð6:10Þ
where μ 5 fμ ; μ ; 1g for the PM, MAF, and elsewhere, respectively. The boundary
r r;mag r;ff
condition, magnetic insulation, becomes @V m =@n 5 0where n is the outward normal to
the boundary.