Page 24 - Computational Modeling in Biomedical Engineering and Medical Physics

10 Computational Modeling in Biomedical Engineering and Medical Physics

which means that the total flux across the boundary is zero—actually it equals the total
internal scalar source). However, even if this condition is verified, the solution is still
not unique: it can be proven that it is unique but to the limit of an additive arbitrary
constant (Mocanu, 1981).
The initial conditions define the initial state of the system, and its initial interactions
with the environment. As with the boundary conditions, their knowledge is required to
predict the state of system at any time later on, during the process. The initial conditions
have to be set inside the system and on its boundary, for the unknown, primitive func-
tion, and its derivatives, up to the highest but one order of the time derivative in the
PDE. In fact, with respect to time, the PDE poses a Cauchy problem.
Whereas time and space are treated as variables in Mathematics, in Physics their
significance is quite different. Therefore the mathematical modeling techniques used
to solve boundary values and initial physical problems ought to consider this aspect.

Initial values problems

A class of only initial conditions problems exits and these are described by ordinary
differential equations (ODEs). The electrical circuit problems are such examples
(Mocanu, 1979). Conveniently the electrical circuits are treated as complex systems
constructed by interconnecting dipolar, elementary, ideal, passive circuit elements,
each representing and underlying electromagnetic process, on one hand, and active
circuit elements, that is, the power sources, on the other hand. Thus the resistance
presents the electrical conduction (heat source by Joule effect); the inductance synthe-
sizes the electromagnetic induction (magnetic energy); the capacitance electrical mod-
els the electrical charge conservation (electrical energy). The active elements are
voltage and current sources. In this approach, the terminals of the dipolar elements are
their boundaries. The physics inside these idealized systems are present through specific
working conditions (equations) between electrical currents and voltages at the term-
inals. These equations may be algebraic, for resistances, and differential time derivatives
or integral time integrals for the inductances and capacitances (reactive elements).
Inductances and capacitances are then susceptible of initial state conditions, voltages
and currents, respectively, which actually represent internal electric or magnetic fluxes,
respectively. Their connectivity, or the topological assembly of these circuit elements,
closes the definition of the external interactions between the circuit elements.
Topology provides the big picture of the macro system, the electrical network, which
connects ideal elements that may be in internal and external disequilibrium. This dis-
equilibrium manifests itself through conjugated fluxes (terminal electrical currents) and
gradients (terminal voltages). Kirchhoff’s laws are used to present them through inte-
gral differential, algebraic systems of ODEs. To this adds the initial conditions—as
many as reactive elements are. The ODEs system may be reduced to a single, higher