Lead in the
Body
Part 3: Ingesting Lead
In a study published in
1973 (see title page for reference), Rabinowitz
and colleagues measured over an extended period of time the lead levels
in bones, blood, and tissue of a healthy male volunteer living in Los Angeles.
Their measurements produced the following transfer coefficients for movement
of lead between various parts of the body and for excretion from the body.
Note that, relatively speaking, lead is somewhat slow to enter the bones
and very slow to leave them.
Lead Transfer Coefficients (Rabinowitz, et al.)
Units: days-1
| a21 = 0.011 |
a12 = 0.012 |
from blood to tissue and back |
| a31 = 0.0039 |
a13 = 0.000035 |
from blood to bone and back |
| a01 = 0.021 |
a02 = 0.016 |
excretion from blood and tissue |
The Rabinowitz study also
showed that the average rate of ingestion of lead in Los Angeles over the
period studied was 49.3 micrograms per day.
In your worksheet, set the
values of the transfer coefficients aij and the ambient
lead level L to those given above. In Part 2 you computed an equilibrium
solution xe of the model system
x' = Ax + b.
The equilibrium solution
is a particular solution of the nonhomogeneous system. Thus, to
find the general solution. we need to find the general solution
xh of the corresponding homogeneous system x'
= Ax, and then add xh and xe.
- Evaluate xe
with the given numbers. Interpret the components of xe
in terms of what would be expected to happen to an individual who lived
in Los Angeles for a long time (before clean-up of lead in the atmosphere).
- Use eigenvalues and eigenvectors
of A to construct the general solution xh of the
homogeneous system x' = Ax.
- Construct the general solution
of x' = Ax + b.
- Solve the initial value
problem
x' = Ax + b, x(0) = 0. That is, find the vector function
that describes the state of lead in the volunteer's body if there were
no lead in his body when he moved to Los Angeles.
- Define the individual component
functions x1, x2, x3
that give lead levels in the blood, tissue, and bones of the volunteer.
Plot these functions over a span of 400 days. How do the levels
at the end of that time compare with the levels in the equilibrium solution?
About how long would it take to reach equilibrium levels? ("They are
never reached" is not an adequate answer -- that's true only in theory.)
