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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

3212

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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

33

21

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

2

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323

2

2321

2

11

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