Page 281 - Computational Modeling in Biomedical Engineering and Medical Physics

270 Computational Modeling in Biomedical Engineering and Medical Physics

computational domain might be described by scattering, or by low reflecting bound-
ary conditions. The area of penetration of each antenna into the domain could be
described by a port boundary condition (coaxial in this case), introducing in that
way the source of electromagnetic waves.
2. The heat transfer problem is physically described by the balance of energy in the inter-
ventional region. When it applies to homogeneous tissue irrigated by a fine capil-
lary blood network, the bioheat equation (Pennes, 1948) is the common
mathematical representation, with its term that depends on the local vascularization
(Chapter 1: Physical, Mathematical, and Numerical Modeling)

@T
2
ρC 5 kr T 2 ρ C b ω b T 2 T b Þ 1 Q met 1 Q electric ; ð8:11aÞ
ð
b
@t
while forced convection heat transfer in large blood vessels is described by

@T 2
ρC 1 uUrÞT 5 kr T 1 Q met 1 Q electric : ð8:11bÞ
ð
@t
In both heat transfer equations the metabolic energy intake and consumption,
3
Q met , are commonly neglected, while the external heat source, Q electric (W/m ), is the
so-called resistive heat (i.e., the absorbed power density); it is generated by the EMF
in the exposed tissue and results by processing the solution of the EMF problem

1

Q emf 5 Re σ 1 jωεð ÞEUE : ð8:12Þ
2
It is important to notice that Eqs. (8.11) refer here to the harmonic regime
hence the electric and magnetic field strengths are introduced in their complex
representation; commercial electromagnetic analysis software commonly operates
with the peak values of harmonic quantities.
The interventional region, extended up to adiabatic boundaries is the computa-
tional domain for the thermal problem.
3. The hemodynamic problem is defined inside the blood vessels; it is representative of
the dynamics of blood flow (pulsating, incompressible, laminar) through a large
vessel, sufficiently close to the interventional region, to influence the heat transfer.
The mathematical form is described by the Navier Stokes equation, which states
for the momentum balance, associated with the mass conservation law; blood is
assumed to be a Newtonian fluid for the particular conditions of this study.

@u
T

ρ 1 ðuUrÞu 5 r 2 pI 1 ηðrUu 1 ðrUuÞ Þ and rUu 5 0: ð8:13Þ
@t
The velocity is set to zero at the walls of the blood vessels.