Biomedical Engineering Reference
In-Depth Information
Encapsulated Pill or Hypodermic Needle Injection
An encapsulated pill or hypodermic needle injection provides a constant continuous
input over a period of time. It is assumed that the diffused or injected solute instanta-
neously mixes with the solution in the compartment. Mathematically, an encapsulated pill
or hypodermic needle injection is approximated by a pulse function,
u
(
t
)
u
(
t
t
1
), where
t
1
is the duration of the pulse.
Depending on the type of input, the solute moves through the body via the circulatory
system. We assume instantaneous mixing as the solute enters the system by action of the
heart. In some situations, the solute enters the body through the digestive system and then
through the plasma. The solute diffuses out of the circulatory system into the other com-
partments of the body. In general, the elimination of a solute from the body occurs through
the kidneys; intestines; lungs, skin, and sweat; biotransformation (converted to another
form) in the liver and other organs; and metabolized in tissues.
7.5 ONE-COMPARTMENT MODELING
The simplest compartment model consists of only one compartment. A one-compartment
model is shown in Figure 7.8, where a box is used to define the compartment, and the flow
of solute is defined by arrows. The input to the compartment can be any of those described
in Section 7.4.1 or other types of functions. The output transfer rate,
K
10
, depicts the flow of
solute from compartment 1 to the environment space, 0. The convention used in writing the
transfer rate,
K
ij
, describes the flow of solute leaving compartment
i,
and entering compartment
0. While only one output is shown in Figure 7.8, there can
be multiple outputs to different spaces, such as the urine, liver, and so on, and multiple inputs.
As we will see, all of the outputs in this case can be combined into a single output by summing
the transfer rates into a single transfer rate if convenient. There can be more than one input to
the compartment, and if so, each input can be solved for separately using superposition, with
zero initial conditions, and the natural response to the initial conditions.
To analyze the system in Figure 7.8, we use conservation of mass to write the differential
equation describing the rate of change of the quantity of solute in the compartment, given
as accumulation
K
ij
j
. All transfer rates are given by
¼
input - output, where
Accumulation
¼
q
1
Input
¼
f
ð
t
Þ
Output
¼
K
10
q
1
Input
K
10
q
1
FIGURE 7.8
A one-compartment model. Assume the volume of the compartment is
V
1
. The input is
f
(
t
).
