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Drugs ‘R’ Us: Using Drugs
2CoPE?
Gretchen A. Koch
Goucher College
BioQUEST Summer 2009
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
 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
A little mathematics…
Ingestion
GI Tract
f (t )
Decay in GI
tract and
absorption
into blood
g (t )

Metabolism
Blood
b( t )
 f(t) is a “pulse” function that gives the dosage profile. (2)
 The half-life rates are given by α and .
 g(t) and b(t) give the concentration of drug in the GI tract and
bloodstream, respectively.

Pulse/Dosing 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
dg
 f (t )   g (t )
dt
db
  g ( t )   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 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?
 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 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?
One Compartment Model
Ingestion
Metabolism
Blood
f (t )
b( t )

 f(t) is a “pulse” function that gives the dosage profile. (2)
 The half-life rate is given by α.
 b(t) gives the concentration of drug in the bloodstream.
Mathematical Equations
db
 f (t)   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 ingested minus the concentration that is
being metabolized.
References
1.
Spitznagel, E. (Fall 1992) Two-Compartment
Pharmacokinetic Models C-ODE-E. Harvey Mudd College,
Claremont, CA.
2.
Yeargers, E.K., Shonkwiler, R.W., and Herod, J.V. (1996) An
Introduction to the Mathematics of Biology. Birkhäuser.
Acknowledgements
Tony Weisstein
John Jungck
Raina Robeva
Michael Garman, Sean Lonsdale, David Ludgin, Kelly
Moran, and Katrina Ramirez-Meyer