Biomedical Engineering Reference
In-Depth Information
FIGURE 12.5
Two-compartment model,
separated by a semipermeable membrane of
thickness
x. This type of model can be used
to analyze the solute transport through the
nephron. To increase the complexity of the
model, multiple compartments can be in-
cluded separated by different semipermeable
membranes.
Δ
Compartment 1
Compartment 2
V
1
V
2
c
1
c
2
Δ
x
dc
1
dt
1
DA
Δ
c
1
V
1
1
ð
V
2
Þ
DA
Δ
c
0
V
2
ð
12
:
4
Þ
5
x
V
1
V
2
x
which is a first-order linear ordinary differential equation.
Equation 12.4
can be solved in
a variety of ways, depending on the conditions of the problem. For instance, if
V
1
is equal
to
V
2
, it will significantly simplify
Equation 12.4
. The use of Laplace Transforms to solve
the differential equation may prove to be easy, or other mathematical methods can be
used. Regardless of the solution method, it will be typical to see that the solute concentra-
tion change with respect to time in one compartment will be described by an exponential
decay, while the solute concentration in the other compartment will be described by an
exponential rise (for a two-compartment system).
To use compartmental analysis to model the transfer of solutes between two compart-
ments, the following assumptions are useful. First, assume that the volume of each com-
partment remains the same with time. This makes the time derivative only a function of
concentration and not of volume (
Equation 12.4
). In fact, the equations derived previously
make this assumption. Also, we make the assumption that the solute is homogeneously
mixed throughout the compartment immediately after it crosses the semipermeable mem-
brane. The only factors that affect the transfer of the solute across the membrane are
accounted for within the diffusion coefficient, the cross-sectional area of the membrane,
the membrane thickness, and the concentration gradient across the membrane.
Example
Consider the reabsorption/metabolism of glucose within the nephron as modeled by a two-
compartment system. Assume that the input of glucose into the nephron is constant and defined
by K
1
. The output of glucose from the nephron can be described by the metabolism of glucose
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