Page 37 - Computational Modeling in Biomedical Engineering and Medical Physics

Physical, mathematical, and numerical modeling 23

Methods and Chapter 6: Magnetic Drug Targeting). Changes in the electrical conduc-
tivity of the aortic blood during a cardiac cycle are marked by a switch in orientation
and shape of the red blood cells (RBC) due to the hemodynamic flow, which is the
reason for change in the electrical conductivity: during the diastole when RBCs are
randomly distributed the electrical resistance is high (low conductance) whereas during
the systolic period the RBCs are aligned streamwise, and change their shape to favor
flow, which leads to low electrical resistance (high conductance) (Hoetink et al., 2004;
Gaw et al., 2008; Visser, 1992). Thus blood electrical conductivity is
1 2 H
σ b 5 σ pl ; ð1:30Þ
1 1 C 2 1ÞH
ð
where σ b and σ pl [S/m] are the electric conductivities of blood and plasma, H repre-
sents the hematocrit, and C is defined as nondimensional geometric factor for the
RBC. In the round tube theory, C is a function of the tube radius, the geometry of
the RBC approximated as prolate ellipsoid, the local the shear rate and an empirical
time constant for cell orientation—cells changing from random to aligned orientation;
Gaw et al., 2008. More details are found in Chapter 5: Bioimpedance Methods.
In electrophysiology and bioelectromagnetism, the electrically conductive medium extends
continuously; it is three-dimensional, and referred to as a volume conductor (Malmivuo and
Plonsey, 1995). Capacitance is distributed too because capacitive effects are related to cellular
membranes, which extend continuously throughout a three-dimensional region. It may be
noted that usually the electrical conductivity is seen as a constant, subject to the regular vari-
ability due to the different tissues and individuals (Gabriel and Gabriel, 1996; Valvano, 2010),
and less concern is given to its hemodynamic flow dependence. However, for modeling elec-
trical fields in anatomic regions with arterial flows it should be ascertained whether the con-
stancy of electrical conductivity is acceptable. Thus for instance, a group of impedance
methods aimed at the evaluation of the hemodynamic flow parameters, the Impedance
Cardiography (Miller and Horvath, 1978) and the Electrical Cardiometry (Osypka, 2009)
(Chapter 5: Bioimpedance Methods) are based on, sense, and use it.

Rheological properties of blood
The rheology of blood is of concern in the arterial hemodynamic flow numerical simula-
tion, which implements momentum equation, in either Navier Stokes, for “clear” fluid,
or Brinkmann, Forchheimer, and Darcy, in porous media formulations, for which specific
models of fluid are presumed (Chapter 5: Bioimpedance methods, and Chapter 8:
Hyperthermia and ablation (Thermotherapy methods). Depending on the group of blood
vessels the hemodynamic analysis may require different rheological models for blood.
Fig. 1.3 renders, qualitatively, the pressure levels for the types of flow in relation to the
specific vascular segment of concern (Feijóo, 2000; Morega et al., 2010).