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Some Deterministic Models in Mathematical Biology: Physiologically Based
Pharmacokinetic Models for Toxic Chemicals
Cammey E. Cole
Meredith College
March 11, 2005
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Outline• Introduction to compartment models• Research examples• Linear model
– Analytics– Graphics
• Nonlinear model• Exploration
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Physiologically Based Pharmacokinetic (PBPK)
ModelsA physiologically based
pharmacokinetic (PBPK) model for the uptake and elimination of a chemical in rodents is developed to relate the amount of IV and orally administered chemical to the tissue doses of the chemical and its metabolite.
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Characteristics of PBPK Models
• Compartments are to represent the amount or concentration of the chemical in a particular tissue.
• Model incorporates known tissue volumes and blood flow rates; this allows us to use the same model across multiple species.
• Similar tissues are grouped together.• Compartments are assumed to be well-mixed.
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Example of Compartment in PBPK Model
• QK is the blood flow into the kidney.
• CVK is the concentration of drug in the venous blood leaving the kidney.
QK CVKKidney
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Example of Compartment in PBPK Model
• CK is the concentration of drug in the kidney at time t.
• CVK is the concentration of drug in the venous blood leaving the kidney at time t.
• QK is the blood flow into the kidney.
• VK is the volume of the kidney.
K
KKKK
V
CVCQ
dt
dC
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BenzeneAveloar Space
Lung Blood
Fat
Slowly Perfused
Rapidly Perfused
Liver
StomachMuconic
Acid
ExhaledInhaled
VenousBlood
CV
ArterialBlood
CA
SlowlyPerfused
Blood
Liver
Blood
Liver
Blood
Liver
BOBO
BO
BO
PH
PH
HQ
HQ
BZ
BZ
BZ
BZ
BZ
ConjugatesPH Catechol
THB
ConjugatesHQ
PMA
Benzene
Benzene Oxide Phenol Hydroquinone
RapidlyPerfusedFat
BO PH
SlowlyPerfused
PH
Fat
PH
RapidlyPerfused Fat
SlowlyPerfused
RapidlyPerfused
HQ HQ HQ
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Benzene Plot
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Benzene Plot
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4-Methylimidazole (4-MI)
Metabolism
Adipose
Other Tissues
Kidney
Urine
Liver
B
l
o
o
d
Initial condition:IV exposure
Other Tissues
Leaving System
Blood
Stomach
Initial condition: gavage exposure
Feces
4-MI
Metabolite
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4-MI Female Rat Data (NTP TK)
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Linear Model ExampleA drug or chemical enters the body via the stomach.
Where does it go?Assume we can think about the body as three
compartments:– Stomach (where drug enters)– Liver (where drug is metabolized)– All other tissues
Assume that once the drug leaves the stomach, it can not return to the stomach.
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Schematic of Linear ModelStomach
Other Tissues
Liver
• x1, x2 , and x3 represent amounts of the drug.• a, b, and c represent flow rates.
x2
x1
x3
a
b c
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Linear Model EquationsLet’s look at the change of amounts in each
compartment, assuming the mass balance principle is applied.
323
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cxbxdt
dx
cxbxaxdt
dx
axdt
dx
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Linear Model (continued)Let’s now write the system in matrix form.
3
2
1
3
2
1
0
00
x
x
x
cb
cba
a
x
x
x
dt
d
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Linear Model (continued)
• Find the eigenvectors and eigenvalues.
• Write general solution of the differential equation.
• Use initial conditions of the system to determine particular solution.
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Linear Model (continued)For our example, let a=3, b=4, and c=11. Then, our general
solution would be given by:
tt ekekk
x
x
x15
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3
2
1
1
1
0
1
2
3
4
11
0
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Initial ConditionsUsing the initial conditions of
we are representing the fact that the drug began in the stomach and there were no background levels of the drug in the system.
,0)0(
0)0(
9)0(
3
2
1
x
x
x
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Graphical Results
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Schematic of Nonlinear Model
x1, x2 , x3, and x4 represent amounts of the drug.
Stomach
Other Tissues
Liverx2
x1
x3
a
b c Metabolitex4
2
2
xK
Vx
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Nonlinear Model Equations
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323
2
2321
2
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xK
xV
dt
dx
cxbxdt
dx
xK
xVcxbxax
dt
dx
axdt
dx
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Nonlinear Model a=0.2, b=0.4, c=0.1, V=0.3, K=4
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Exploration
• What would happen if the parameters were changed?
• What would happen if the initial conditions were changed?
We will now use Phaser to explore these questions. Phaser files are on the website:
www.meredith.edu/math/faculty/cole/maaworkshop/pbpkmodels.htm
