Lead in the BodyLead in the BodyUsing Euler’s Method to Using Euler’s Method to

Explore Compartment ModelsExplore Compartment Models

Maria HernandezMaria HernandezNCSSMNCSSM

TCM January 2007TCM January 2007hernandez@ncssm.eduhernandez@ncssm.edu

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

BonesCompartment C

C(t)

TissueCompartment B

B(t)

BloodCompartment A

A(t)

Compartment D – External Environment

Lead Inputfood, air, water

urine hair,nails,sweat

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

0500

1000150020002500

0 100 200 300 400

time(days)

Blood

Tissue

Bones

• 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

-500

500

1500

2500

0 200 400 600 800 1000

time(days)

Blood

Tissue

Bones

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

0

500

1000

1500

2000

2500

0 200 400 600 800 1000

time(days)

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 UniversityJohn Michel, Marietta Collegehttp://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