Page 22 - Computational Modeling in Biomedical Engineering and Medical Physics

8 Computational Modeling in Biomedical Engineering and Medical Physics

is key, if the model is to provide for a realistic image of the underlying physics. In par-
ticular these laws of Physics provide the divergence and curl sources of the vector
fields.
As already stated, this first step in building the physical model goes hand in hand with
a correct definition of the system and its boundary. An initial sketch with the pencil on
the paper is successively morphed into the physical system, the physics and the couplings
that describe the internal and external interactions are stated and possibly rescaled in space
and time, such that the mathematical model, which then builds upon, may produce solu-
tions that comply with and represent within acceptable accuracy limits the reality. At
some point, eventually, the resulting mathematical model may require reconsidering the
physical model for reasons of affordable, convenient complexity, efficiency, accuracy, and
predictive validity. Each of the models presented in this work went through such process,
but usually the final, resulting physical mathematical numerical simulation results are
presented.

1.5 Mathematical models
Complete and independent, coherent, and noncontradictory system
of laws
Completeness, independency, and coherency of the noncontradictory system of laws
that present a physical problem should be the concern when the physical model, the
physical laws, is stated and the mathematical model [the partial differential equation
(PDEs)] is abstracted. Completeness means that all relevant laws are included.
Independence is needed to ascertain that the equations are independent, in the sense that
they may not be obtained from each other. Coherency requires the usage of a system of
units containing a set of fundamental or base units from which all other units in the
system are derived (Coherence, 2018). Noncontradictory means the equations are not
adversarial.
A qualitative analysis of the system of equations that represents the physical model
and perhaps its dynamic properties (Michel and Wang, 1995) is then advisable. This
may be performed starting with the fundamental theorem of the vector fields
(Annex 1) and the material properties that relate the quantities through constitutive
laws. An example may be the dynamic EMF problem in immobile, nonlinear, homo-
geneous, and isotropic media. Maxwell laws, used to formulate the physical model,
include the magnetic flux law (a scalar equation) and the electromagnetic induction
(Faraday) law (a vector equation). It may be easily verified that the magnetic flux law
can be deduced from Faraday’s law (the divergence of a curl is identically zero) up to
a time-independent additive constant (Purcell, 1984). Therefore the magnetic flux law
could be discarded form the system, without any difficulty—mathematically there are
as many equations left as the number of unknown are. However, if the working