Using Drugs 2CoPE: Pharmacokinetics in Excel

Gretchen A. Koch-NobleDepartment of Mathematics and Computer Science

Goucher CollegeBioQUEST Summer Workshop

June 16 & 17, 2012

Data Literacy

• Data Analytics• NIST Big Data• Retention• Jobs

BIO 2010:Specific Strategies

A strong interdisciplinary curriculumthat includes physical science, information technology, and math.

Early opportunities for independent research.

Meaningful laboratory experiences.

http://bioquest.org

http://bioquest.org/esteem

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 = axb

Glass box:Study the model

y = axb

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.”

(www.thefreedictionary.com)• 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.”

(www.thefreedictionary.com)• 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.”

(www.thefreedictionary.com)• 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.”

(www.thefreedictionary.com)• 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.”

(www.thefreedictionary.com)• 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.”

(www.thefreedictionary.com)• 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.”

(www.thefreedictionary.com)• 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.”

(www.thefreedictionary.com)• 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/combination)

• Injection (vein/muscle/subcutaneous)

• 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 InAmount 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.

Injection Metabolism

f (t)

Blood

α

Two-Compartment Model

Ingestion

Decay in GI tract

and absorption into blood

MetabolismGI Tract

( )g tBlood

( )b tf (t) αβ

Create the Model

Time remaining: 10:00

Ingestion

Decay in GI tract

and absorption into blood

MetabolismGI Tract

( )g tBlood

( )b tf (t) αβ

Create the Model

Time remaining: 5:00

Ingestion

Decay in GI tract

and absorption into blood

MetabolismGI Tract

( )g tBlood

( )b tf (t) αβ

Create the Model

Time remaining: 4:00

Ingestion

Decay in GI tract

and absorption into blood

MetabolismGI Tract

( )g tBlood

( )b tf (t) αβ

Create the Model

Time remaining: 3:00

Ingestion

Decay in GI tract

and absorption into blood

MetabolismGI Tract

( )g tBlood

( )b tf (t) αβ

Create the Model

Time remaining: 2:00

Ingestion

Decay in GI tract

and absorption into blood

MetabolismGI Tract

( )g tBlood

( )b tf (t) αβ

Create the Model

Time remaining: 1:00

Ingestion

Decay in GI tract

and absorption into blood

MetabolismGI Tract

( )g tBlood

( )b tf (t) αβ

Create the Model

Time’s Up!

Ingestion

Decay in GI tract

and absorption into blood

MetabolismGI Tract

( )g tBlood

( )b tf (t) αβ

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 functiondynamically.

User entry of drug

half-lives

Model descriptionand assumptions

Blood concentration versus GI concentration

Time is still independent variable.

Dosing FunctionDrug 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

References1. Aygestin® Product Insert, Duramed Pharm., Inc., Pomona, NY,

July 20072. Drugs@FDA,

http://www.accessdata.fda.gov/scripts/cder/drugsatfda/ 3. Neuhauser, C. (2004) Calculus for Biology and Medicine. 2 ed.

Pearson Education.4. Physician Information for Depo-Provera Pharmacia & Upjohn

Company, Kalamazoo, MI, Feb. 20045. 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.