Transcript Lead in the Body

Lead in the Body
Using Euler’s Method to
Explore Compartment Models
Maria Hernandez
NCSSM
TCM January 2007
[email protected]
Some Facts About Lead
• Lead is one of the oldest chemical toxins.
• Reports of lead poisoning date to ancient Greece and
high levels of lead have been found in ancient Egyptian
mummies.
• Lead is the most widely used metal after iron. About 2.5
million tons of lead are produced yearly in the world.
Most of this lead goes into battery production and most
of the remainder is used in cable covering, plumbing,
ammunition and in the manufacture of compounds, used
as fuel additives.
• Lead compounds have been widely used as
pigments in paints, although recently their use
has been drastically curtailed to reduce the
health hazards. Most commonly it is found in
lead paint for protecting metal surfaces.
• Occupations that may expose workers to lead
include painters, smelters, firearms instruction,
automotive repair, brass or copper foundries,
and bridge, tunnel and elevated highway
construction.
Effects of Lead on the Body
• Once lead has entered the human body it
causes adverse effects. It combines with
red blood cells causing anemia. Lead also
settles in bones taking the place of
calcium. It can be found in the liver and
kidneys.
• Lead affects the nervous system, including
the brain.
Our Compartment Model
Lead Input
food, air, water
Bones
Compartment C
C(t)
Blood
Compartment A
A(t)
Tissue
Compartment B
B(t)
urine
hair,
nails,
sweat
Compartment D – External Environment
What We Plan To Do Today:
• Write coupled differential equations.
• Write Euler’s Method equations.
• Use technology to generate values for lead in
the blood, tissue and bones - Calculator,
MathCad, Spreadsheet
• Graph lead levels for blood, tissue and bones.
• We let A(t), B(t) and C(t) be the amount of lead in each
compartment at time t, as shown in the diagram.
• 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 Kab. 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 Kba, etc.
Let
dA/dt = rate of change of level of lead in the blood,
dB/dt = rate of change of level of lead in the tissue,
dC/dt = rate of change of level of lead 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.
The measurements produced the following transfer
coefficients for movement of lead between various parts
of the body and for excretion from the body. The daily
exposure is 49.3 micrograms per day.
Lead Transfer Coefficients (Rabinowitz, et al.)
Units: days-1
Kab = 0.0111
Kba = 0.0124
from blood to tissue and back
Kac = 0.0039
Kca = 0.000035
from blood to bone and back
Kad = 0.021
Kbd = 0.016
excretion from blood and tissue
Write the DEs
and
the Euler’s Method Equations
Solution
Lead in Body
2500
2000
1500
1000
500
0
Blood
Tissue
Bones
0
100
200
300
time(days)
400
• Explore Equilibrium Levels for Blood
Lead Level and Tissue Lead Level.
• Numerically or Analytically (Solving a
System of Linear Equations).
Note: using analytic methods we can find
equilibrium for bone lead level.
Part II.
Explore new scenario – The effect of
placing the volunteer in a lead-free
environment after 400 days of
exposure.
More Calculator or Spreadsheet
Lead after no threat
2500
Blood
Tissue
Bones
1500
500
-500 0
200
400
600
time(days)
800
1000
Part III.
Explore new scenario – The effect of
placing the volunteer in a lead-free
environment and administering an
anti-lead drug.
After No Threat and Drug
2500
2000
1500
1000
500
0
0
200
400
600
time(days)
800
1000
Information on
Blood Lead Level Tolerance
• Some sources suggest a tolerance of 25
micrograms per deciliter of blood for adults
= 250 micrograms per liter of blood.
• We have approximately 5 liters of blood in
our bodies.
• Tolerance =1250 micrograms
What would make this a better lab?
• More information on how chelating drugs
lower the BLL – Blood Lead Level.
• Change the model to reflect that
information.
References
• The Connected Curriculum Project, 1998 – 2000
David Smith, Duke University with the assistance of
Jer-Chin Chuang, Furman University
John Michel, Marietta College
http://www.math.duke.edu/education/ccp/materials/diffeq/bodylead/
• “Differential Equations A Modeling Approach”, Borrelli and
Coleman, John Wiley & Sons, 1996
• Article on web – “Lead” , September 2002
www.minerals.csiro.au/safety/lead.htm
• Article on wed – “Dangers of Lead Still Linger” U.S. Food and
Drug Administration, Dixie Farley, 1998
www.cfsna.fda.gov/~dms/fdalead.html