MATH 164, Spring 2001 Due Date: Name(s):
Honors Project 2: Tracer Methods in Permeability
Potassium ions K+
flow into and out of red blood cells. Physiologists are concerned with understanding
this flow since it helps them understand the structure and behavior of red blood cells; hence it might help
them in combating blood diseases.
One approach to studying the flow of potassium ions into and out of red blood cells is to use the fact that
there is a radioactive isotope K42+
of potassium that that is indistinguishable from K+
except for the fact
that it is radioactive. Thus, if K42+
ions are introduced into the bloodstream, they can be sampled and
measured at subsequent times to provide information about the permeability of red blood cells.
To be more concrete, suppose C = C(t) represents the amount of potassium in corpuscles at time t,
and P = P(t) represents the amount of potassium in the plasma. If C and P measure the K42+
which is
introduced into the bloodstream at time t = 0 then C(0) = 0 and P(0) = P0, P0 being a known quantity.
Assuming that no potassium enters or leaves the bloodstream during the course of study amounts to assuming
that C(t) + P(t) = P0. (Since the half-life of K42+
is approximately 12.5 hr and this is significantly greater
than the time it takes to introduce K42+
into the bloodstream and to perform requisite sampling tests,
radioactive decay is not a significant source of error when it comes to the assumption that "no potassium
enters or leaves the bloodstream during the course of study".) If we model the change dP/dt of P as a linear
function of C (by K42+
ions moving from corpuscles to plasma) and P (by K42+
ions moving from plasma
to corpuscles)
dP
dt
= k1C - k2P where k1, k2 > 0
then we obtain a differential equation
dP
dt
= k1C - k2P = k1(P0 - P) - k2P = k1P0 - (k1 + k2)P where P(0) = P0 (1)
which can be solved for P.
Exercises:
1. a) Show that, under the above assumptions, the equilibrium level of K42+
in the plasma (the value of
P = P(t) for large t) is
P =
k1P0
k1 + k2
.
b) How could P be determined experimentally?
2. Show that the solution to Equation 1 is
P =
k1P0
k1 + k2
1 +
k2
k1
e-(k1+k2)t
.
It follows from Exercise 2 that the solution to our differential equation may be written
P = P 1 +
k2
k1
e-(k1+k2)t
or
P
P
- 1 =
k2
k1
e-(k1+k2)t
.
Plasma Corpuscles
By taking logarithms of both sides of this last equation, we find that
ln
P
P
- 1 = -(k1 + k2)t + ln
k2
k1
. (2)
Exercises:
3. How would you use Equation 2 to determine k1 and k2 from a data set {(ti, P(ti)), i = 1, . . . , N}?
4. Find P0, P, k1 and k2 given the following fictitious data set (in which we suppress units for simplicity):
t 0 1 2 3 4 5 6 7 8 9 10 11 12
P 1.60 1.37 1.27 1.22 1.19 1.18 1.18 1.18 1.17 1.17 1.17 1.17 1.17
5. Plot P(t) vs. C(t) (in a PC-coordinate plane) using k1 and k2 for the above data set (you may use
Maple if you wish), and describe in words the evolution of P(t) vs. C(t) for t > 0.
The model we discussed in this project is often referred to as a two compartment model since the passage of
K42+
ions occurs between two compartments: plasma and corpuscles. There are, of course, also three, four,
... compartment models, and these are of interest in the study of subjects such as epidemiology. A multiple
compartment model in which quantities change with time is often referred to as a dynamical system. One
goal of this project has been to illustrate that the study of dynamical systems has important applications in
both science and engineering.