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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
Data Literacy
•
•
•
•
Data Analytics
NIST Big Data
Retention
Jobs
BIO 2010:
Specific Strategies
A strong interdisciplinary curriculum
that 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
How do students interact with the
mathematical model underlying the biology?
?
Black box:
Hide the model
y=
axb
Glass box:
Study the model
y = axb
No box:
Build the model!
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/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
Transfer In
Amount in
Compartment
Transfer Out
• Balance Law:
Rate of Change = Rate In – Rate Out
Pharmacokinetics Model
• One-Compartment Model
– Bloodstream
• Two-Compartment Model
– Bloodstream and Gastrointestinal Tract
• Track relative concentrations
• Dimensionless model
One-Compartment Model
Injection
f (t)
Blood
b( t )
Metabolism
α
db
= f (t) - a b(t)
dt
• 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.
Two-Compartment Model
Decay in GI
tract and
absorption
into blood
Ingestion
GI Tract
f (t)
g (t )
Metabolism
Blood
β
b( t )
α
Create the Model
Decay in GI
tract and
absorption
into blood
Ingestion
GI Tract
f (t)
g (t )
Metabolism
Blood
β
b( t )
α
Time remaining: 10:00
Create the Model
Decay in GI
tract and
absorption
into blood
Ingestion
GI Tract
f (t)
g (t )
Metabolism
Blood
β
b( t )
Time remaining: 5:00
α
Create the Model
Decay in GI
tract and
absorption
into blood
Ingestion
GI Tract
f (t)
g (t )
Metabolism
Blood
β
b( t )
Time remaining: 4:00
α
Create the Model
Decay in GI
tract and
absorption
into blood
Ingestion
GI Tract
f (t)
g (t )
Metabolism
Blood
β
b( t )
Time remaining: 3:00
α
Create the Model
Decay in GI
tract and
absorption
into blood
Ingestion
GI Tract
f (t)
g (t )
Metabolism
Blood
β
b( t )
Time remaining: 2:00
α
Create the Model
Decay in GI
tract and
absorption
into blood
Ingestion
GI Tract
f (t)
g (t )
Metabolism
Blood
β
b( t )
Time remaining: 1:00
α
Create the Model
Decay in GI
tract and
absorption
into blood
Ingestion
GI Tract
f (t)
g (t )
Metabolism
Blood
β
b( t )
Time’s Up!
α
Two-Compartment Model
dg
= f (t) - b g(t)
dt
db
= b g(t) - a b(t)
dt
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
Model description
and assumptions
Dosing Function
Sliders to
change
dosage function
dynamically.
Drug Concentrations
versus Time
Blood concentration
versus GI concentration
Time is still independent variable.
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
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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.
2.
3.
4.
5.
6.
7.
8.
Aygestin® Product Insert, Duramed Pharm., Inc., Pomona, NY, July 2007
Drugs@FDA, http://www.accessdata.fda.gov/scripts/cder/drugsatfda/
Neuhauser, C. (2004) Calculus for Biology and Medicine. 2 ed. Pearson
Education.
Physician Information for Depo-Provera Pharmacia & Upjohn Company,
Kalamazoo, MI, Feb. 2004
Robeva, R., et al. (2008) An Invitation to Biomathematics. Elsevier.
Spitznagel, E. (Fall 1992) Two-Compartment Pharmacokinetic Models CODE-E. Harvey Mudd College, Claremont, CA.
Strogatz, S.H. (1994) Nonlinear Dynamics and Chaos with Applications to
Physics, Biology, Chemistry, and Engineering. Addison-Wesley.
Yeargers, E.K., Shonkwiler, R.W., and Herod, J.V. (1996) An Introduction to
the Mathematics of Biology. Birkhäuser.