Biomedical Engineering Reference
In-Depth Information
increase in blood volume, the opposite response will occur, causing the permeability of
the collecting ducts toward water to decrease.
12.5 COMPARTMENTAL ANALYSIS FOR URINE FORMATION
This section will describe one of the mathematical techniques that can be used to
describe the movement of solutes through the nephron. When analyzing the transfer of
substrates within the body, it is typically convenient to describe the regions that are
involved in the transfer as compartments. For instance, the tubule lumen, the tubule epi-
thelial cells, and the peritubular lumen can all be individual compartments used to
describe solute movement. With this analysis method, it is typical to assume that the
boundary has a uniform permeability (although it can change with time, but including
this would only increase the complexity of the solution), and that the compartments have
some known volume as a function of time. In general, compartmental analysis problems
are solved using mass conservation laws.
We will begin our analysis by solving for the transfer of a substance between two com-
partments separated by a thin permeable membrane. To solve for the change in solute con-
centration with time across a thin membrane, Fick's law of diffusion can be used:
dc
dt
52
DA
V
dc
dx
ð
12
:
1
Þ
In
Equation 12.1
,
c
is the concentration of the solute,
D
is the diffusion coefficient,
A
is
the surface area of the membrane,
V
is the compartment volume (constant), and
dx
is the
thickness of the membrane. The simplest system is composed of two compartments
(
Figure 12.5
), and each quantity of interest would be denoted with a subscript to declare
which compartment is being referenced.
Equation 12.1
can be simplified to
dc
1
dt
52
DA
V
1
c
1
c
2
2
ð
:
Þ
12
2
x
Δ
assuming uniformity across the semipermeable membrane and that we are referencing
compartment 1. If at time zero, there is an initial concentration (
c
0
) of a solute in compart-
ment 1 (with no solute in compartment 2 at time zero), then mass conservation states that
V
1
c
1
V
2
c
2
V
1
c
0
ð
:
Þ
1
5
12
3
The multiplication of volume by a concentration is “quantity of solute” and that is what can
be measured and must be conserved within the two-compartment system. If
Equation 12.3
is solved for the concentration within compartment 2 and substituted into
Equation 12.2
,we
get
V
1
c
0
2
V
1
c
1
c
2
5
V
2
V
1
c
0
2
V
1
c
1
c
1
2
dc
1
dt
52
DA
V
1
DA
Δ
c
1
V
1
ð
1
V
2
Þ
2
V
1
c
0
V
2
52
x
x
V
1
V
2
Δ
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