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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.