Page 38 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

MODELING OF BIOMEDICAL SYSTEMS 15

The number of engineers dying (before retirement) per unit time is proportional to the number of
engineers at that time:
M(t) = k E(t) (1.32)
4
The demand for engineers at any given time is proportional to the number of jobs available at that
time (J(t)) and is inversely proportional to the number of engineers available at that time:

D(t) = kJ(t)/E(t) (1.33)
The number of jobs available depends on various factors such as government spending for R&D
projects, economic growth, sales of medical products, number of hospitals, etc. Let us assume in this
case (biomedical engineering) that the number of jobs is directly proportional to the sales of medical
products (p), directly proportional to government spending for health care R&D (e), and directly pro-
portional to the number of new medical product company startups (i):
J(t) = (k e + k c + k i + k + kp ) (1.34)
6 7 8 9
Although we assumed that the number of jobs at the present time is dependent on e(t), c(t), h(t),
i(t), and p(t), in reality the number of jobs at present may depend on previous values of these para-
meters, or on the history of these parameters.
Let us now analyze the demand history. This history depends on the memory function. Let us
assume that the effect of demand existing at a time decays exponentially (exponentially decaying
memory). The net effect of demands from time = 0 to t can be expressed as
H (t) = I τ = t {D(τ) exp[−k (t − τ)]}dτ (1.35)
1 τ = 0 10
The number of students entering the engineering school per unit time is

S(t) = k H (t) (1.36)
11 1
Immigration rate can similarly be expressed as
I(t) = k H (t) (1.37)
12 2
where
H (t) = I τ = t {D(τ)exp[−k (t − τ)]}dτ (1.38)
2 τ = 0 13
H and H are called hereditary functions. Instead of an exponential decay of memory, we could have
1
2
a sinusoidal or some other functional form of memory decay, depending on the physical situation.
dE/dt = k k H (t − 4) + k H (t) − (k + k + k )E(t) (1.39)
1 10 1 11 2 2 3 4
In this analysis, making various assumptions, we have formulated a lumped parameter determin-
istic model to predict the number of engineers (biomedical) present in the United States at any given
time. If we want to know the geographical distribution, we can take two approaches. We can divide
the entire United States into a number of compartments (e.g., northeast, east, west, etc.) and study
the intercompartmental diffusion. Alternatively, we can make E a continuous variable in space and
time I (x, y, t) and account for spatial diffusion.

1.4.2 Modeling the Cell-Mediated Immunity in Homograft Rejection
In cell-mediated immunity, lymphocytes in the tissue become sensitized to the target (graft) cells and
travel to the regional lymph nodes where they initiate an immunological response by increasing the
production of immunocompetent lymphocytes. The newly produced lymphocytes are then transported