Biomedical Engineering Reference
In-Depth Information
making it more complex. By adding complexity, we add additional parameters, which must
be estimated based on the data collected using parameter estimation techniques.
Given a system described by a group of compartments, some exchange of solute (i.e., a
radioactive tracer, a molecule like glucose or insulin, a gas like oxygen or carbon dioxide)
is expected between compartments by diffusion. Compartmental analysis predicts the
quantity or concentration of solutes under consideration in each compartment as a function
of time using conservation of mass—that is, accumulation equals input minus output. The
model may be linear, nonlinear, continuous, or discrete, and may even have time-varying
or stochastic parameters. If the model is continuous and linear, then the change in solute
concentration is described as a sum of exponential and sinusoidal terms.
The following assumptions are made when describing the transfer of a solute by diffu-
sion between any two compartments:
1.
The volume of each compartment remains constant.
2.
Any solute
entering a compartment is instantaneously mixed throughout the entire
compartment.
3.
The rate of loss of a solute from a compartment is proportional to the amount of solute in
the compartment times the transfer rate,
q
K
, given by
Kq
. The transfer rate typically has
units of liters per minute.
If two solutes are being tracked in a system, the overall model can be described using two
parallel models. For instance, if each solute flows in and out of the plasma, each model can
have its own plasma compartment separate from the other. It follows that we can track
n
solutes, with each solute having its own model, all separately sharing the plasma. This
concept follows with additional tissues and blood vessels.
From a modeling perspective, identifying compartments and the number of compart-
ments to describe a system is a difficult step. Acquiring measurement data for model facili-
tation is another difficult step because some compartments are inaccessible. Both of these
steps are beyond the scope of this topic; interested readers can examine topics listed at
the end of this chapter for more information.
7.4.1 Inputs to a Compartmental System
The inputs to a compartmental system are discussed following.
Bolus Injection
A bolus injection is an immediate injection of a solute into a compartment. It is assumed
that the injected solute instantaneously mixes with the solution in the compartment. Math-
ematically, a bolus is approximated as either a change in initial conditions or as an impulse
function, d(
t
).
Constant Continuous Infusion
A constant continuous infusion input is delivered by an infusion pump or an intrave-
nous drip into a compartment. It is assumed that the injected solute instantaneously mixes
with the solution in the compartment. Mathematically, a constant continuous infusion input
is approximated as a unit step function,
u(t).
