1. Lead in the Body Using Euler’s Method to Explore Compartment Models Maria Hernandez NCSSM TCM January 2007 hernandez@ncssm.edu

2. 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.

3. 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.

4. 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.

5. Our Compartment Model Bones Compartment C C(t) Tissue Compartment B B(t) Blood Compartment A A(t ) Compartment D – External Environment Lead Input food, air, water urine hair, nails, sweat

6. 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.

7. 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 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.

8. 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.

9. 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 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

10. Write the DEs and the Euler’s Method Equations

11. Solution

12. 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.

13. 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

15. Part III. Explore new scenario – The effect of placing the volunteer in a lead-free environment and administering an anti-lead drug.

17. 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

18. 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.

19. 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 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 www.cfsna.fda.gov/~dms/fdalead.html