Page 21 - Computational Modeling in Biomedical Engineering and Medical Physics

Physical, mathematical, and numerical modeling 7

where P mag [VA] is the magnetic power, P Hertz [VA] is the Hertzian power, P J is the
resistive (Joule) power [W], and ε is the complex permittivity. The bar below indi-
0

cates complex quantities, the upper symbol ( ) indicates complex conjugated quanti-
p ffiffiffiffiffiffiffiffi
ties, and j 5 2 1. The specific absorption rate (SAR) analysis in Chapter 8:
Hyperthermia and Ablation (Thermotherapy Methods), will single out the contribu-
tions and combined effect of the two power sources, resistive and dielectric, respec-
tively, in the EMF-induced local hyperthermia pending the EMF substance
interaction.

1.4 Multidisciplinary (multiphysics) problems
The problems presented throughout this book are multidisciplinary (multiphysics),
with multiple concurring and interacting “physics” with variable degrees of couplings:
hemodynamic activity monitoring (arterial hemodynamic flow, structural mechanics of
the arm, piezoelectric, and capacitive processes); magnetic drug targeting (hemody-
namic flow with magnetic field interactions, structural response of the blood vessels);
thermography (hemodynamic and heat flow); hyperthermia (RF and MW EMFs,
hemodynamic, heat flow); thermal ablation (hemodynamic flow, EMF, heat transfer);
EMF dosimetry (EMF, hemodynamic flow, heat transfer); and bioimpedance methods
(hemodynamic flow, EMF).
In general the steps to take in modeling multiphysics problems and medical engi-
neering problems make no exception to the rule, start with the definition of the sys-
tem and its boundary, its “fabric” (structure and composition), and the recognition
and description of the internal and external constraints and interactions to which this is
subjected as describable by the laws of Physics. Multiple phenomena of different
nature may occur, may interact, and may influence each other in a complex cau-
se effect web of conditionalities and dependencies.
More than often, such coupled problems of electromagnetism, heat and mass trans-
fer, structural mechanics, and others have different time and space scales, which may
raise concerns in formulating consistent, solvable mathematical models using finite
computational resources.
Mathematical modeling is used to find the underlying states of the evolution path
that the system pursues assuming the macroscopic, continuum media hypothesis for
the system and the physical phenomena to which this it subjected. The evolution
and interactions occur then contiguously in space and continuously time. The accom-
panying internal transformations and the internal and external interactions are pre-
sented through a set of physical laws that provide for a quantitative, deterministic
cause effect set of mathematical equations, differential, integral, algebraic, that make
the mathematical model. Which particular physics occur and what specific forms (dif-
ferential, integral, algebraic) may have the mathematical equations that describe them