Biomedical Engineering Reference
In-Depth Information
It is interesting to note that all deterministic models include some measurement error.
The measurement error introduces a probabilistic element into the deterministic model,
so it might be considered stochastic. However, in this chapter, models are deterministic
if their principle features lead to definitive predictions. On the other hand, models are
stochastic if their principle features depend on probabilistic elements. This chapter is
primarily concerned with deterministic models.
13.1.2 Solutions
There are two types of solutions available to the modeler. A closed form solution exists
for models that can be solved by analytic techniques, such as solving a differential equation
using the classical technique or by using Laplace transforms. For example, given the follow-
ing differential equation
‥
x
þ
4
x
þ
3
x
¼
9
with initial conditions
x
ð
0
Þ¼
0 and
x
ð
0
Þ¼
1, the solution is found as
u
ð
t
Þ
x
ð
t
Þ¼
e
3
t
e
t
þ
4
3
A numerical or simulation solution exists for models that have no closed form solution.
Consider the following function:
33
2
33
1
t
7
Z
20
2
e
p
x
¼
dt
2p
20
This function (the area under a Gaussian curve) has no closed form solution and must be
solved using an approximation technique, such as the trapezoidal rule for integration. Most
nonlinear differential equations do not have an exact solution and must be solved via an
iterative method or simulation package such as SIMULINK. This was the situation in
Chapter 12 when the Hodgkin-Huxley model was solved.
Inverse Solutions
Engineers often design and build systems to a predetermined specification. They often
use a model to predict how the system will behave because a model is efficient and
economical. The model that is built is called a plant and consists of parameters that
completely describe the system: the characteristic equation. The engineer selects the pa-
rameters of the plant to achieve a certain set of specifications such as rise time, settling time,
or peak overshoot time.
In contrast, biomedical engineers involved with physiological modeling do not build the
physiological system but only observe the behavior of the system—the input and output of
the system—and then characterize it with a model. Characterizing the model as illustrated
in Figure 13.1 involves identifying the form or structure of the model, collecting data, and
then using the data to estimate the parameters of the model. The goal of physiological
modeling is not to design a system but to identify the components (or parameters) of the
system. Most often, data needed for building the model are not the data that can be
