Biomedical Engineering Reference
In-Depth Information
a mean step size of 5.6
10
3
s. The initial pressures,
required to start the solution, were computed by a linear
algebraic solution of a hemodynamic system where all
pressures are assumed constant.
machine epsilon (the smallest number that can be repre-
sented by the computer) will help the programmer un-
derstand the importance of robust program and system
design and controlling the propagation of error.
Lastly, this introduction includes a discussion of one of
the most important concepts in numerical analysis, the
role that a Taylor series approximation plays in mapping
continuous models to their discrete analogs and methods
for solving the discrete representation. The Taylor series
plays an important role in deriving numerical algorithms
and in characterizing the error introduced by the discrete
approximation and also in the error propagated by
performing a sequence of calculations.
2.1a.5 Overview of the text
The material presented in this text shows how to apply the
principlesandtechniquesofcomputingandnumerical
problem solving in a wide variety of problems that arise in
BME. The aimhere is to provide the studentwith a
working
knowledge
of Numerical Methods, i.e., to be able to read,
understand and useNumerical Methods to BME problems.
Aworking knowledge has as its emphasis the understanding
and application of the fundamentals. This approach pro-
vides the reader with exposure to a broad range of princi-
ples and techniques, but not theoretically rigorous
derivations of the methods; the mathematical foundations
are more appropriate for courses in Numerical Methods in
a mathematics or computer science curriculum.
Our second aim is to give the reader examples of how to
construct engineering models of biomedical systems. The
conservation laws' first
theme from this chapter is rein-
forced in the models presented in the examples. Thus, the
text is organized into four sections around the physical
principles: fundamentals, using models of steady-state
behavior, using models of finite time behavior and using
models of transient behavior. Throughout this text, exam-
ples froma varietyof BME specialties, including biomedical
instrumentation, imaging, bioinformatics, biomechanics,
and biomaterials, are used to reinforce the concepts.
2.1a.5.2 Part II: Steady-state behavior
(algebraic models)
Part II is an overview of techniques used to analyze sys-
tems that are in steady state and whose models are for-
mulated as algebraic equations that could be either linear
or nonlinear. A single equation that is explicit in the
unknown can easily be solved by methods from pre-
college algebra; if the equation is implicit in the un-
known, then root-finding techniques must be used. If the
model is a set of simultaneous equations then numerical
algebraic methods are used. Of course, the case of si-
multaneous, implicit equations is also treated. Each of
these techniques is presented in this part of the text.
2.1a.5.3 Part III: Dynamic behavior
(differential equations)
Part III is of greatest interest to the biomedical engineer:
modeling the transient behavior of dynamic systems and
solving for the output of such systems. This section of the
text includes methods for solving both ordinary and partial
differential equations using numericaltechniquesaswellas
Simulink.
Also in Part III, system behaviors during finite-time
intervals, modeled by integral equations, are considered.
In these cases, the solution of the model for an output
parameter requires numerical integration and differenti-
ation techniques. These methods are presented in this
section, along with methods for improving the accuracy of
the results. A recurring theme in this section is the trade-
off that must be made between accuracy of the solution
and the amount of computation that is performed.
2.1a.5.1 Part I: Fundamentals
The first part of the text is an introduction to the funda-
mental principles of numerical methods. As programming
is a necessary part of numerical methods, the examples,
problems and applications in the text are given in
MATLAB and basic terminology and principles of pro-
gram development in the MATLAB language are
reviewed. Since the reader may eventually implement
a numerical method using another programming language,
this section will help the reader relate implementation of
common concepts of computer science: block structured
design, data structures and analysis of algorithms. The
emphasis is on design and the tradeoffs that a programmer
makes when implementing an algorithm.
The introduction also includes a discussion of number
representation and the effect that number representation
has on the accuracy, precision and stability of the results of
the computation. It is especially important in biomedical
and healthcare applications that accuracy and stability be
preserved and the system is as robust as possible. Whether
or not MATLAB is used, keeping in mind concepts such as
2.1a.5.4 Part IV: Modeling tools
and applications
Part IV is an introduction to developing models of com-
plex systems and to tools and techniques for analyzing
complex behaviors. Examples of multicompartmental
