Case studies for CT systems - Continuous and Discrete Time Signals and Systems - page 384

Number of antigens At any given time, the total number of antigens present
in the human body depends on three factors: (i) external antigens entering
the human body from outside; (ii) reproduced antigens produced within the
human body by the already existing antigens; and (iii) destroyed antigens that
are eradicated by the antibodies. The net change in the number of antigens is
modeled by the following equation:

d a
d t

= α a ( t ) − η b ( t ) + g ( t ) , (8.49)

where α denotes the reproduction rate at which the antigens are multiplying
within the human body and η is the destruction rate at which the antigens are
being destroyed by the antibodies.

Number of lymphocytes Assuming that the number of lymphocytes is pro-
portional to the number of antigens, the number of lymphocytes present within
the human body is given by

l ( t ) = β a ( t ) , (8.50)

where β is the proportionality constant relating the number of lymphocytes to
the number of antigens. The value of β generally depends on many factors,
including the health of the patient and external stimuli. In general, β varies
with time in a non-linear fashion. For simplicity, however, we can assume that
β is a constant.

Number of plasma cells The change in the number of plasma cells is pro-
portional to the number of lymphocytes l ( t ). Typically, there is a delay of τ
seconds between the instant that the antigens are detected and the instant that
the plasma cells are generated. Therefore, the number of plasma cells depends
on l ( t − τ ), where the proportionality constant is assumed to be unity. Also, a
large portion of plasma cells die due to aging. The number of plasma cells at
any time t can therefore be expressed as follows:

d p
d t

= l ( t − τ ) − γ p ( t ) , (8.51)

where γ denotes the rate at which the plasma cells die due to aging.

Number of antibodies The number of antibodies depends on three factors:
(i) new antibodies being generated by the human body (the rate of generation
of the new antibodies is proportional to the number of plasma cells in the human
body); (ii) destroyed antibodies lost to the antigens (the rate of destruction σ
of such antibodies is proportional to the number of existing antigens); and
(iii) dead antibodies lost to aging. We assume that the antibodies die at the rate
of λ because of aging. Combining the three factors, the number of antibodies