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Some Deterministic Models in Mathematical Biology:Physiologically Based
Pharmacokinetic Models for Toxic Chemicals
Cammey E. ColeMeredith College
January 7, 2007
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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)
Models in Toxicology ResearchA 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 chemical in
the venous blood leaving the kidney.
QK CVKKidney
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Example of Compartment in PBPK Model
• CK is the concentration of chemical in the kidney at time t.
• CBl is the concentration of chemical in the blood at time t.
• CVK is the concentration of chemical 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
KBlKK
VCVCQ
dtdC −
=
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BenzeneAveloar Space
Lung Blood
Fat
Slowly Perfused
Rapidly Perfused
Liver
Stomach MuconicAcid
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)Other Tissues
Adipose
Kidney
Urine
Liver
B
l
o
o
d
Initial condition:IV exposure
Stomach
Initial condition: gavageexposure
Feces
Metabolite
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4-MI Female Rat Data (NTP TK)
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4-MI Female Rat Data (Chronic)
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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 Model
• x1, x2, and x3 represent amounts of the drug in the compartments.
• a, b, and c represent linear flow rate constants.
Stomach
Other Tissues
Liverx2
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.
=
=
=
dtdxdtdxdtdx
3
2
1
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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
3212
11
cxbxdtdx
cxbxaxdtdx
axdtdx
−=
+−=
−=
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Linear Model (continued)Let’s now write the system in matrix form.
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
3
2
1
3
2
1
0
00
xxx
cbcba
a
xxx
dtd
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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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[ ][ ][ ]
)(,,0))((
)()()(
))(()(0
00
2
2
cbacba
cbabccbbcabccba
cbcba
a
+−−=++−−=++−−=
−+++−−=
−−−−−−−=
−−−−
−−
λλλλ
λλλ
λλλλ
λλλλ
λλ
Finding Eigenvalues of ASet the determinant of equal to zero and solve for .IA λ− λ
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Finding Eigenvectors.0=λConsider
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
−=−
cbcba
aIA
0
000
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Finding Eigenvectors0=λ
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
bc0
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Finding Eigenvectors.a−=λConsider
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+−+−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−−−−
−−−=−
acbcaba
acbcaba
aaaIA
0
000
)(0)(
00)(
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Finding Eigenvectorsa−=λ
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−+
abaca
aacab2
2
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Finding Eigenvectors).( cb +−=λConsider
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ ++−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
++−++−
++−=++
bbcca
cba
cbcbccbba
cbaIcbA
0
00)(
)(0)(
00)()(
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Finding Eigenvectors)( cb +−=λ
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−110
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Linear Model (continued)Then, our general solution would be given by:
tcb
at
ek
eab
acaaacab
kbck
xxx
)(
3
2
2
21
3
2
1
110
0
+−
−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
−++
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
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Parameter Values and Initial Conditions
For our example, let a=3, b=4, and c=1, and usethe 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
===
xxx
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Linear Model (continued)Then, our particular solution would be given by:
with
tt ekekkxxx
5
3
3
21
3
2
1
110
1266
410
−−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
524,
32,
54
321 −=== kkk
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Graphical Results Link1 Link2
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Schematic of Nonlinear Model
x1, x2 , x3, and x4 represent amounts of the drug (or its metabolite).
Liverx2
Stomachx1
a
Other Tissuesx3
b c Metabolitex4
2
2
xKVx+
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Nonlinear Model Equations
2
24
323
2
2321
2
11
xKxV
dtdx
cxbxdtdx
xKxVcxbxax
dtdx
axdtdx
+=
−=
+−+−=
−=
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Nonlinear Model Link 1 Link2
a=0.2, b=0.4, c=0.1, V=0.3, K=4
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Exploration• What would happen if one of the parameter
values were doubled? halved?• What would happen if the initial conditions were
changed to represent some background level present in the liver or other tissues?
We will now use Phaser to explore these questions.