Lead in the Body – A Calculus Investigation Bones C(t) Tissue B(t ...

Lead in the Body – A Calculus Investigation
Adapted from Duke University’s Connected Curriculum Project and Differential Equations, A Modeling
Approach, Borrelli and Coleman, John Wiley and Sons Inc., 1996
Lead can be absorbed in the body by inhalation and ingestion. Once in your blood
stream, lead accumulates in the blood, soft tissues and bones. Lead is excreted from the
body primarily through the urinary system and through hair, nails, and sweat (see figure).
Lead Input
food, air, water
Bones
Compartment C
C(t)
k ca
k ac
Blood
Compartment A
A(t)
k
ab
k
ba
Tissue
Compartment B
B(t)
k urine
ad
k
bd
hair,
nails,
sweat
Compartment D – External Environment
We model the flow of lead into, through, and out of a body with separate compartments
for blood, bones, and other tissues, plus a compartment for the external environment.
We let A(t), B(t) and C(t) be the amount of lead in each compartment at time t, as
shown above. We assume that the rate of transfer from compartment A to compartment
B is proportional to the amount of lead in compartment A with proportionality constant
k
ab
. Similarly, we assume that the rate of transfer from compartment B to compartment
A is proportional to the amount of lead in compartment B with proportionality constant
k
ba
, etc. We are not concerned with the lead in compartment D because we are not
concerned with the quantity of lead outside the body. The body is not a closed system,
but including an extra compartment in the model creates a closed system. We assume
that exposure to lead in the environment results in ingestion of lead at a constant rate L,
measured in micrograms (ug) per day . This external input is called a “driving force”.
1. Let dA be the rate of change of the level of lead in the blood, dB be the rate of
dt
dt
change of the level of lead in the tissue and dC be the rate of change of the level of lead
dt
in the bones. Write a set of coupled differential equations representing the rates of
change for the amount of lead in the blood, bones and tissue compartments of the body.

2. In a study published in 1973 (see 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
k
ab
= 0.0111 k
ba
= 0.0124 from blood to tissue and back
k ac = 0.0039 k ca = 0.000035 from blood to bone and back
k
ad
= 0.021 k
bd
= 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.
Using the values given, write the Euler’s Method equations for the levels of lead in each
of the compartments. Graph the level of lead in the blood, tissue and bones for the
volunteer for his first 400 days. You may assume that A(0)=0, B(0) = 0, C(0) = 0, and
∆ t = 1. What do you notice about the levels of lead in each of the body’s
compartments
3. Now graph the level of lead in the blood, tissue and bones for 800 days. What do
you notice about the levels of lead in each of the body’s compartments
4. From your graphs in parts 2 and 3, we can see that the lead level in the blood and
tissue compartments reaches an equilibrium level. Assume that the lead level in the
bones reaches an equilibrium value. Using analytic methods, find the equilibrium levels
for all three compartments. Compare your analytic results to your numerical values for
the blood and tissue compartments.
Part II A Lead-Free Environment
Suppose now that the volunteer has been placed in a totally lead-free environment after
the 400 day exposure in Los Angeles. That is, the external driving force in the model is
now set to 0.
1. Modify your Euler’s Method equations to reflect this change in the model. Graph the
levels of lead in the blood, tissue and bones for the volunteer for his first 400 days and
his next 400 lead-free days on the same graph. What do you notice about the levels of
lead in each of the body’s compartments

2. Your plot should show that the lead level in the bones peaks at around 600 days.
What is that peak amount If the volunteer remains in a lead-free environment, how
long will it take for the lead level in his bones to reach half of its peak amount
Part III A Lead-Free Environment with Anti-lead Medication
Another way to remove lead from the bones is to administer an anti-lead medication that
increases the rate at which lead leaves the skeletal system. In particular, suppose that
the rate coefficient k ac = 0.000035 increases to k ac = 0.00035 with the anti-lead drug.
Suppose that after 400 days of exposure to lead, the volunteer is placed in a lead-free
environment and administered the anti-lead drug. Modify your Euler’s Method equations
to reflect this change in the model. Graph the level of lead in each compartment over
the total 800 days. Approximately when will the level of lead in the bones reach its peak
amount What is that peak amount How long will it take for the lead level in the bones
to reach half of its peak amount
Articles by Rabinowitz et al. in Science 182 (1973), 725-727, and by Batschelet et al. in Journal of
Mathematical Biology 8 (1979), 15-23.