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Magnetic drug targeting 179

6.4 Magnetic drug mixing
The magnetic drug is injected into the blood stream where it is subjected to a mixing
process with the blood and conveyed to the ROI. The degree of mixing depends on
many factors (MD dilution, vessel caliber, blood flow rate, MD injection flow rate,
etc.) but it is important to find the distance from the injection point where this ini-
tially two-phase fluid, blood-MD, turns into a homogeneous, single-phase fluid flow.
From that point on, the blood and the MD are a homogeneous magnetizable aggre-
gate fluid (MAF) whose magnetic permeability accounts for the dilution of the MD
mixed with the blood.
A simpler, two-dimensional model may help to evidence the underlying features
of the two-phase flow produced by injecting the MD into a blood vessel, through a
catheter, in the absence of any external magnetic field. The level set method (Sussman
et al., 1998; Sethian, 1996; Sethian and Wiegmann, 2000) is used to solve this two-
phase flow problem that is expressed by the momentum balance and the continuity
laws (Chapter 1: Physical, Mathematical, and Numerical Modeling) (Dobre, 2012),
which for Newtonian fluids yield
@u

ρ T 1 f ;
@t 1 uUrÞu 5 r 2pI 1 η rUu 1 rUuÞ ð6:1Þ
ð
ð
rUu 5 0; ð6:2Þ

and the level set equation
@Φ rΦ
;
ð
ð
@t 1 uUrÞΦ 5 γrU εrΦ 2 Φ 1 2 ΦÞ j rΦj ð6:3Þ
where f 5 f ts stands for the surface tension forces, Φ is the level set function (nondimen-
sional, with values in the range 0 1), and γ and ε are stabilization parameters that
ensure the existence and uniqueness of the solution. The mass density, ρ, and the
kinematic viscosity, η, are provided by

ρ 5 ρ 1 ρ MD 2 ρ Φ; η 5 η 1 η MD 2 η Φ; ð6:4Þ

b
b
b
b
where ( ) b and ( ) MD denote the blood and the MD, respectively.
This method is useful in studying the dynamics of interfaces and geometric shapes,
making it possible to calculate the curvilinear surfaces in the Cartesian coordinate sys-
tem and does not require the parameterization of the computational domains. Also it
facilitates the analysis of the dynamics of the computational domain morphology,
when and where its deformation and, or its division into several subdomains occur.
On the other hand, there are applications of this method also in the field of image