Using Drugs 2CoPE: Pharmacokinetics in Excel Gretchen A. Koch-Noble Department of Mathematics and Computer Science Goucher College BioQUEST Summer Workshop June 16 & 17, 2012

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BIO 2010: Specific Strategies A strong interdisciplinary curriculum that includes physical science, information technology, and math. Early opportunities for independent research. Meaningful laboratory experiences.

Unpacking “ESTEEM” Excel: ubiquitous, easy, flexible, non-intimidating Exploratory: multifaceted, open-ended questions: students choose their own approach Experiential: see how math techniques enable more precise models, more rigorous tests

Three Boxes Black box: Hide the model ? y = ax b Glass box: Study the model y = ax b No box: Build the model! How do students interact with the mathematical model underlying the biology?

Pharmacokinetics “The study of the process by which a drug is absorbed, distributed, and metabolized.” ( What kinds of things do we want to consider when building a model? – Absorbed, distributed, metabolized? – How do drugs get into body? Ten minute breakout session in pairs!

Think, Pair, Share “The study of the process by which a drug is absorbed, distributed, and metabolized.” ( What kinds of things do we want to consider when building a model? – Absorbed, distributed, metabolized? – How do drugs get into body? Time remaining: 10:00

Think, Pair, Share “The study of the process by which a drug is absorbed, distributed, and metabolized.” ( What kinds of things do we want to consider when building a model? – Absorbed, distributed, metabolized? – How do drugs get into body? Time remaining: 5:00

Think, Pair, Share “The study of the process by which a drug is absorbed, distributed, and metabolized.” ( What kinds of things do we want to consider when building a model? – Absorbed, distributed, metabolized? – How do drugs get into body? Time remaining: 4:00

Think, Pair, Share “The study of the process by which a drug is absorbed, distributed, and metabolized.” ( What kinds of things do we want to consider when building a model? – Absorbed, distributed, metabolized? – How do drugs get into body? Time remaining: 3:00

Think, Pair, Share “The study of the process by which a drug is absorbed, distributed, and metabolized.” ( What kinds of things do we want to consider when building a model? – Absorbed, distributed, metabolized? – How do drugs get into body? Time remaining: 2:00

Think, Pair, Share “The study of the process by which a drug is absorbed, distributed, and metabolized.” ( What kinds of things do we want to consider when building a model? – Absorbed, distributed, metabolized? – How do drugs get into body? Time remaining: 1:00

Think, Pair, Share “The study of the process by which a drug is absorbed, distributed, and metabolized.” ( What kinds of things do we want to consider when building a model? – Absorbed, distributed, metabolized? – How do drugs get into body? Time’s Up!

Think, Pair, Share Results Effect of exercise on the concentration of the drug in the body; stomach -> portal vein -> liver -> bloodstream; if the rate of blood flow in the liver increases/decreases, how does that effect the concentration in the bloodstream? Function that describes the rate of absorption: variables to include: rate of blood flow, sex, height, weight, condition of individual, chemical nature of drug; functions for delivery, absorption, and elimination Interaction of drug with foods, other drugs; effects of the byproducts of metabolism; amount lost versus amount used; age Administration of drug – intravenous, patch, swallowed, sublingual, inhaled, injected, suppositories Dose of drug Liquid or solid form of drug Systems model – what has to be done to get to the free drug that has effect on target organ; threshold level of free drug Feedback loops – molecules in the plasma – timing of attachment on and off; absorption by tissue and sequestration; feedback loop for metabolism – excretion versus toxicity

Think, Pair, Share Results – Group 1 Rate – how quickly metabolized, linear, exponential, how ingested, how excreted – linear, exponential Rate of delivery, method of delivery – different models of distribution and absorption? Which organs to use? Global controls for research. Wide margins of error with different organs absorbing vs intravenous. Compartments inside organism where drug can accumulate versus target Type of drug – water soluble, lipid soluble, etc. Age, underlying disease, disease affecting metabolization sites Mechanism of action of drug – competive inhibitor, cofactors, etc.

The Grand Model Dosing – How much and how often – How taken Transdermal Inhaled Swallowed (liquid/solid/combinatio n) Injection (vein/muscle/subcutaneo us) Absorption – Chemical changes – Fat cells Distribution – Freely circulating/carried – Movement from bloodstream to affected area – How long Elimination – Metabolism in liver – Excretion via kidneys – Special case: Radioactive decay Excess via kidneys Rest via half-life decay

Goals for the Model Determine how much medication is required to achieve the desired effect. – Minimum effective concentration – Minimum toxic concentration – Therapeutic window How long does it take a drug to reach a steady state in the body? – Consistently in therapeutic window

Key Ideas for the Model Every drug has a half-life that must be published by the manufacturer. The model will combine discrete and continuous processes. – Discrete: Dosing – Continuous: Distribution, Absorption, and Elimination Keep it simple to start.

Compartment Models Look at the transfer of materials from one compartment to another Balance Law: Rate of Change = Rate In – Rate Out Transfer In Amount in Compartment Transfer Out

Pharmacokinetics Model One-Compartment Model – Bloodstream Two-Compartment Model – Bloodstream and Gastrointestinal Tract Track relative concentrations Dimensionless model

One-Compartment Model Rate of Change = Rate In – Rate Out The rate of change in the concentration of the drug in the blood is equal to the amount being injected minus the concentration that is metabolized. InjectionMetabolism f (t) Blood α

Two-Compartment Model Ingestion Decay in GI tract and absorption into blood Metabolism GI Tract Blood f (t) α β

Create the Model Time remaining: 10:00 Ingestion Decay in GI tract and absorption into blood Metabolism GI Tract Blood f (t) α β

Create the Model Time remaining: 5:00 Ingestion Decay in GI tract and absorption into blood Metabolism GI Tract Blood f (t) α β

Create the Model Time remaining: 4:00 Ingestion Decay in GI tract and absorption into blood Metabolism GI Tract Blood f (t) α β

Create the Model Time remaining: 3:00 Ingestion Decay in GI tract and absorption into blood Metabolism GI Tract Blood f (t) α β

Create the Model Time remaining: 2:00 Ingestion Decay in GI tract and absorption into blood Metabolism GI Tract Blood f (t) α β

Create the Model Time remaining: 1:00 Ingestion Decay in GI tract and absorption into blood Metabolism GI Tract Blood f (t) α β

Create the Model Time’s Up! Ingestion Decay in GI tract and absorption into blood Metabolism GI Tract Blood f (t) α β

Two-Compartment Model 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 that is decaying in the blood.

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

Using Drugs 2CoPE Module Dynamic module where user 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? Single dose Missed doses

Sliders to change dosage function dynamically. Sliders to change dosage function dynamically. User entry of drug half-lives User entry of drug half-lives Model description and assumptions Model description and assumptions Blood concentration versus GI concentration Time is still independent variable. Blood concentration versus GI concentration Time is still independent variable. Dosing Function Drug Concentrations versus Time Drug Concentrations versus Time

Topics to be Explored Using 2CoPE How long does it take for the concentration of the drug in the blood to reach a steady state? 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 that have a narrow therapeutic range? Caffeine metabolism? Other drugs?

Example Laboratory Exercise Birth control and hormones – How many doses can one skip before you get pregnant? – Must it correspond to the hormonal cycle? Aygestin – Half-life: 8.51 hours – Standard dose: 5 mg Depo-Provera – Half-life: 50 days – Standard dose: 150 mg/mL Units!!

Conclusion Built mathematical models of Pharmacokinetics Many topics to explore Useful in both a research and classroom setting Explore on your own!

Acknowledgements PEER – UTK Organizers John Jungck, Beloit College Anton Weisstein, Truman State University Ethel Stanley, Beloit College Sam Donovan, University of Pittsburgh Claudia Neuhauser, U. Minn. – Rochester Marsha Timmerman, LaSalle University Tracey Schatteman, Lincoln Land Community College BioQUEST Curriculum Consortium Goucher College

References 1.Aygestin® Product Insert, Duramed Pharm., Inc., Pomona, NY, July Neuhauser, C. (2004) Calculus for Biology and Medicine. 2 ed. Pearson Education. 4.Physician Information for Depo-Provera Pharmacia & Upjohn Company, Kalamazoo, MI, Feb Robeva, R., et al. (2008) An Invitation to Biomathematics. Elsevier. 6.Spitznagel, E. (Fall 1992) Two-Compartment Pharmacokinetic Models C- ODE-E. Harvey Mudd College, Claremont, CA. 7.Strogatz, S.H. (1994) Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry, and Engineering. Addison-Wesley. 8.Yeargers, E.K., Shonkwiler, R.W., and Herod, J.V. (1996) An Introduction to the Mathematics of Biology. Birkhäuser.