Page 37 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

14 BIOMEDICAL SYSTEMS ANALYSIS

stress/strain histories. To illustrate the modeling of the effects of memory and time delay, let us con-
sider a model to predict the number of engineers in the United States. Then we will consider a model
of cell-mediated immunity which has similar delays and memory functions.

1.4.1 A Model to Predict the Number of Engineers in the United States
An easy-to-understand example of a deterministic model with time delay and memory is a model to
predict the number of biomedical engineers in the United States at any given time. Let us restrict our
analysis to a single discipline such as biomedical engineering. Let E be the number of engineers (bio-
medical) at any given time. The time rate of change of the number of engineers at any given time in
the United States can be expressed as
dE/dt = G + I − R − L − M (1.28)

where G represents the number of graduates entering the profession (graduating from an engineer-
ing program) per unit time, I represents the number of engineers immigrating into the United States
per unit time, R represents the number of engineers retiring per unit time, L represents the number
of engineers leaving the profession per unit time (e.g., leaving the profession to become doctors,
lawyers, managers, etc.), and M represents the number of engineers dying (before retirement) per unit
time.
In Eq. (1.28), we have lumped the entire United States into a single region (a well-stirred com-
partment) with homogeneous distribution. In addition, we have not made any discrimination with
regard to age, sex, or professional level. We have considered the entire pool as a well-stirred homo-
geneous compartment. In reality, there is a continuous distribution of ages. Even with this global
analysis with a lumped model, we could consider the age distribution with a series of compartments
with each compartment representing engineers within a particular age group. Moreover, we have
assumed that all engineering graduates enter the workforce. A percentage of them go to graduate
school and enter the workforce at a later time.
The number of graduates entering the profession is a function of the number of students entering
the engineering school 4 years before:

G(t) = k S(t − 4) (1.29)
1
where S(t) is the number of students entering the engineering school per unit time. The number of
students entering the engineering school depends on the demand for the engineering profession over
a period of years, that is, on the demand history.
The number of engineers immigrating into the United States per unit time depends on two factors:
demand history in the United States for engineers and the number of visas that can be issued per
unit time. Assuming that immigration visa policy is also dependent on demand history, we can
assume that I is dependent on demand history. Here we have assumed that immigrants from all for-
eign countries are lumped into a single compartment. In reality, each country should be placed in a
separate compartment and intercompartmental diffusion should be studied.
The number of engineers retiring per unit time is proportional to the number of engineers in the
profession at the time:

R(t) = k E(t) (1.30)
2
The number of engineers leaving the profession depends on various factors: the demand for the
engineering profession at that time and demand for various other professions at that time as well as
on several personal factors. For the purpose of this analysis, let us assume that the number of engi-
neers leaving the profession in a time interval is proportional to the number of individuals in the pro-
fession at that time:
L(t) = k E(t) (1.31)
3