Transcript Using Drugs 2CoPE?

Using Drugs
2CoPE?
Gretchen A. Koch
Goucher College
BioQUEST Summer 2007
Pharmacokinetics
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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
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Use in biochemistry or anatomy and
physiology class
Different levels of explanation – how do
different organs metabolize drugs? What
chemical processes occur?
Model
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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.
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Possible Lesson: Explore an exponential decay model and halflives.
 Difficulty with half-life not depending on units
A little mathematics…
Ingestion
GI Tract
f (t )
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g (t )
Decay in GI
tract and
absorption
into blood
b
Metabolism
Blood
b(t )
a
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
144.00
136.80
129.60
122.40
115.20
108.00
100.80
93.60
86.40
79.20
72.00
64.80
57.60
50.40
43.20
36.00
28.80
21.60
14.40
7.20
2.50
2.00
1.50
1.00
0.50
0.00
0.00
Unit Dosage
Dosage Pulse Function, f(t), versus Time
Time (hours)
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f(t) depends on many different factors like buffers,
the manufacturer, etc. (2)
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Gives how often the drug is taken and how long it
takes to dissolve.
Mathematical Equations
dg
 f (t )  b g (t )
dt
db
 b g ( t )  a b( t )
dt
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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
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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
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Dynamic module where student chooses:
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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?
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Models the metabolism of the drug over 6
days.
Model description
and assumptions
Sliders to
change
dosage function
dynamically.
Solving the equations for
model – all automatic
based on user input.
Graph showing how
drug concentration changes
in GI tract with respect to time
Graph showing how
drug concentration changes
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
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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
1.
Spitznagel, E. (Fall 1992) TwoCompartment 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.