Page 269 - Computational Modeling in Biomedical Engineering and Medical Physics

258 Computational Modeling in Biomedical Engineering and Medical Physics

The work interaction of the EMF with the renal tissue, or the “resistive” heat
source (Joule heating), is readily available once the electric field is solved. Qualitatively
RF heating contributes to increasing the internal energy in the ROI, which results in
the local temperature increase to the desired level of hyperthermia. The local thermal
unbalance triggers the heat transfer, through diffusion and convection, to the neigh-
boring, cooler regions. The blood vessels, from capillaries to arteries and veins, convey
a significant part of the heat, and its accurate knowledge is important in adjusting the
right level of RF exposure.
The hemodynamic inside the kidney, a saturated porous medium, is slow (Re ,, 1);
therefore Stokes Brinkman momentum conservation is recommended. In the larger vessels
2
(arterial and venal trees), the flow is faster Re B O(10 ), and Navier Stokes form of
momentum equation may be utilized. Moreover theflowtimescale (B seconds) is smaller
than the heat transfer time scale (B minutes), which yields the stationary forms of these
momentums equations

T
ρ uUrÞuŠ 52 pI 1 μ ru 1 ruÞ ; ð8:5Þ
ð
ð ½
!

rU 2pI 1 μ 1 ru 1 ruÞ T 2 μκ 21 1 β u jj 1 Q br 2 pI 1 μ 1 ru 1 ruÞ T u 5 0: ð8:6Þ
ð
ð
F
ε p ε 2 ε p
p
The mass conservation law (incompressible flow)
ρrUu 5 Q br ; ð8:7Þ
adds to the mathematical model. In the above equations u is the velocity, p the
T
pressure, ρ the mass density, μ the dynamic viscosity, ( ) the transposition opera-
tor, I the unity matrix, κ the porosity, ε p the permeability, Q br is a mass source,
and β F is a Forchheimer term (drag coefficient). The larger vessels and the porous
medium flows are connected through boundary conditions that refer to pressure
and velocity. Moreover at the vascular level of the kidney, the rheology of the
blood is presented through a power law, where the dynamic viscosity is
η 5 mU_ γ n21 , Table 8.3 (Morega et al., 2010, 2020; Shibeshi and Collins, 2005).
The boundary conditions of the hemodynamic model are presented in Fig. 8.3
(central image).
In the general heat transfer (GHT) model, the energy equation is (Chapter 1:
Physical, Mathematical, and Numerical Modeling)

@T
ρC 1 uUrÞT 5 r krTÞ 1 _q: ð8:8Þ
ð
ð
@t
The basal metabolic rate, much smaller than the RFA heat source, is neglected
here. The upstream temperature of the arterial blood is 37 C. Convection heat