Gretchen A. Koch Goucher College BioQUEST Summer 2007 Using Drugs 2CoPE? Gretchen A. Koch Goucher College BioQUEST Summer 2007

Pharmacokinetics Study of how drugs move through the body (1) Enter via ingestion, injection, absorption (1) Exit via excretion, metabolism (1) Possible Lesson: Map the pathway of several kinds of drugs Use in biochemistry or anatomy and physiology class Different levels of explanation – how do different organs metabolize drugs? What chemical processes occur?

Model Bloodstream and Gastrointestinal tract Two-Compartment Model Bloodstream and Gastrointestinal tract Track relative concentrations Dimensionless model Units depend on the initial amounts of drug in body (i.e. 0 units) Time is given in hours as half-life is calculated as hours. Model is designed to not rely on units so it can be applied to many situations. Referred to as a “unit dosage” Rates are based on half-life of drugs in each compartment. The half-life does not depend on the units assigned to the measurement. Possible Lesson: Explore an exponential decay model and half-lives. Difficulty with half-life not depending on units

Decay in GI tract and absorption into blood A little mathematics… GI Tract Blood Ingestion Decay in GI tract and absorption into blood Metabolism f(t) is a “pulse” function that gives the dosage profile. (2) The half-life rates are given by a and b. g(t) and b(t) give the concentration of drug in the GI tract and bloodstream, respectively.

Pulse Function f(t) depends on many different factors like buffers, the manufacturer, etc. (2) Gives how often the drug is taken and how long it takes to dissolve.

Mathematical Equations Rate of change = Rate In – Rate Out The rate of change in the concentration of the drug in the GI tract is equal to the amount being ingested minus the concentration that is decaying. The rate of change in the concentration of the drug in the blood is equal to the concentration that is decaying from the GI tract minus the concentration decaying in the blood.

Possible Mathematical Lessons How do we get from the half-life to the rate of decay with no numbers? What other models use a similar differential equation system? How does having a non-constant dosage function affect the analytical solutions? What are the analytical solutions? How do we solve this system of equations numerically?

2CoPE Module Dynamic module where student chooses: Half-life of drugs in GI tract and bloodstream Parameters for the pulse function What is the unit dosage (think number of pills) taken? How often is the drug taken? How long does it take for the drug to dissolve? Models the metabolism of the drug over 6 days.

Solving the equations for Sliders to change dosage function dynamically. User entry of drug half-lives Model description and assumptions Solving the equations for model – all automatic based on user input.

drug concentration changes in GI tract with respect to time Graph showing how drug concentration changes in GI tract with respect to time in blood with respect to time Four different views of solutions to the model Blood and GI drug concentrations as a function of time on same graph Blood concentration versus GI concentration Time is still independent variable.

Topics to Explore Using 2CoPE How long does it take for the concentration of the drug in the blood to reach a steady state? The steady state can be thought of as even oscillations with no additional growth in concentration. For example, one must take many allergy medications for several days before having any consistent effect; this can be attributed to achieving the steady state in the blood. What effect does the half-life of the drug in either the GI tract or blood have on reaching a steady state? What about the dosing function? What about drugs like Lithium? (1) Achieving a steady state is difficult. When does the concentration become toxic? How easy is it to perturb the system so the concentration in the blood gets knocked out of the steady state? Alcohol metabolism?

References Spitznagel, E. (Fall 1992) Two-Compartment Pharmacokinetic Models C-ODE-E. Harvey Mudd College, Claremont, CA. Yeargers, E.K., Shonkwiler, R.W., and Herod, J.V. (1996) An Introduction to the Mathematics of Biology. Birkhäuser.