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CHAPTER 3

PHARMACOKINETIC MODELS

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PHARMACOKINETIC MODELING

A Model is a hypothesis using mathematical terms to describe quantitative relationships

MODELING REQUIRES:Thorough knowledge of anatomy and

physiologyUnderstanding the concepts and limitations

of mathematical models. Assumptions are made for simplicity

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OUTCOME

The development of equations to describe drug concentrations in the body as a function of timeHOW?

By fitting the model to the experimental data known as variables.

A PK function relates an independent variable to a dependent variable.

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FATE OF DRUG IN THE BODY

ADME

Oral

Administration

G.I.

Tract

Circulatory

System

Intravenous

Injection

TissuesMetabolic

Sites

Intramuscular

Injection

Subcutaneous

Injection

Exc

reti

on

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Complexity of PK model will vary with:

1- Route of administration2- Extent and duration of distribution into various body fluids and tissues.3- The processes of elimination.4- Intended application of the PK model.

We Always Choose the SIMPLEST Model

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Types of PK Models

1- Physiologic (Perfusion) Models

2- Compartmental Models

3- Mammillary Models

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PHYSILOGIC PK MODELS

Models are based on known physiologic

and anatomic data.

Blood flow is responsible for distributing

drug to various parts of the body.

Each tissue volume must be obtained and

its drug conc described.

Predict realistic tissue drug conc

Applied only to animal species and human

data can be extrapolated.

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PHYSILOGIC PK MODELS

Can study how physiologic factors may change drug distribution from one animal species to another

No data fitting is required Drug conc in the various tissues are predicted

by organ tissue size, blood flow, and experimentally determined drug tissue-blood ratios.

Pathophysiologic conditions can affect distribution.

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Physiological Model Simulation

Perfusion Model Simulation of Lidocaine IV Infusion in Man

Blood

Metabolism

RET

Muscle

AdiposeLung

Time

Per

cen

t o

f D

ose

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COMPARTMENTAL MODELS

The body is represented by a series of compartments that communicate reversibly with each other.

1 2 3

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k12

k21

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COMPARTMENTAL MODELS A compartment is not a real physiologic or

anatomic region, but it is a tissue or group

of tissues having similar blood flow and drug

affinity. Within each compartment the drug is considered

to be uniformly distributed. Drug move in and out of compartments Compartmental models are based on linear

differential equations. Rate constants are used to describe drug entry

into and out from the compartment.

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COMPARTMENTAL MODELS

The model is an open system since drug is

eliminated from the system. The amount of drug in the body is the sum

of drug present in the compartments. Extrapolation from animal data is not

possible because the volume is not a true

volume but is a mathematical concept. Parameters are kinetically determined from

the data.

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MAMMILLARY MODELSIs the most common compartmental model used in PK. The model consists of one or more

compartments connected to a central compartment

2 1 3

1ka kel

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k12

k21

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Intravenous and Extravascular AdministrationIV, IM, SC

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Intravenous and extravascular Route of

AdministrationDifference in plasma conc-time curve

Intravenous

Administration

Extravascular

Administration

Time

Cp

Time

Cp

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One Compartment Open Model Intravenous

Administration

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One Compartment Open Model Intravenous

AdministrationThe one compartment model offers the simplest way to describe the process of drug distribution and elimination in the body.

When the drug is administered i.v. in a single rapid injection, the process of absorption is bypassed

Blood (Vd)

i.v. Input

kel

Output

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One Compartment Open Model Intravenous

AdministrationThe one-compartment model does not predict actual drug levels in the tissues, but does imply that changes in the plasma levels of a drug will result in proportional changes in tissue drug levels.

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FIRST-ORDER KINETICS The rate of elimination for most drugs is a

first-order process.

kel is a first-order rate constant with a unit

of inverse time such as hr-1.

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Semi-log paper

Plotting the data

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INTEGRATED EQUATIONSThe rate of change of drug plasma conc over time is equal to:

This expression shows that the rate of elimination of drug from the body is a first-order process and depends on kel

pelp Ck

dt

dC

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INTEGRATED EQUATIONS

pelp Ck

dt

dC

Cp = Cp0e-kelt

ln Cp = ln Cp0 kelt

DB = Dose . e-kelt

ln DB = ln Dose kelt

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Elimination Half-Life (t1/2)

Is the time taken for the drug conc or the amount in the body to fall by one-half, such as Cp = ½ Cp0 or DB = ½ DB

0

Therefore,

elkt

693.02/1

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ESTIMATION OF half-life from graph

A plot of Cp vs. time

t1/2 = 3 hr

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Fraction of the Dose Remaining

The fraction of the dose remaining in the body (DB /Dose) varies with time.

The fraction of the dose lost after a time t can be then calculated from:

tkB eleDose

D

tkele 1

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Volume of Distribution (Vd)Is the volume in which the drug is dissolved in the body.Example: 1 gram of drug is dissolved in an unknown volume of water. Upon assay the conc was found to be 1mg/ml. What is the original volume of the solution?

V = Amount / Conc = 1/1= 1 literAlso, if the volume and the conc are known, then the original amount dissolved can be calculated

Amount = V X Conc= 1X1= 1 gram

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Apparent Vd

It is called apparent because it does not have any physiological meaning. Drugs that are highly lipid soluble, such as digoxin has a very high Vd (600 liters), drugs which are lipid insoluble remain in the blood and have a low Vd.

For digoxin, if that were a physiological space and I were all water, that would weigh about 1320 lb (599 kg).

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Apparent Vd

Vd is the ratio between the amount of drug in the body (dose given) and the concentration measured in blood or plasma.

Therefore, Vd is calculated from the equation:

Vd = DB / CP

where,

DB = amount of drug in the body

Cp = plasma drug concentration

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For One Compartment Model with IV Administration:

With rapid IV injection the dose is equal to the amount of drug in the body at zero time (DB).

Where Cp is the intercept obtained by plotting Cp vs. time on a semilog paper.

p

B

pd C

D

C

DoseV

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For One Compartment Model with IV Administration:

p

B

pd C

D

C

DoseV

Cpo

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Calculation of Vd from the AUC

Since, dDB/dt = -kelD = -kelVdCp

dDB = -kelVdCpdt

dDB = -kelVd Cpdt

Since, Cpdt = AUC

Then, AUC = Dose / kelVd

ModelIndependent

Method][AUCk

DoseV

eld

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Significance of Vd

Drugs can have Vd equal, smaller or greater than

the body mass

Drugs with small Vd are usually confined to the

central compartment or highly bound to plasma

proteins

Drugs with large Vd are usually confined in the

tissue

Vd can also be expressed as % of body mass and

compared to true anatomic volume

Vd is constant but can change due to pathological

conditions or with age

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Apparent Vd

Example: if the Vd is 3500 ml for a subject weighing 70 kg, the Vd expressed as percent

of body weight would be:

weightbodyofKg

Kg%5100

70

5.3

The larger the apparent Vd, the greater the amount of drug in the extravascular tissues. Note that the plasma represents about 4.5% of the body weight and total body water about 60% of body weight.

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CLEARANCE (Cl)Is the volume of blood that is cleared of drug per unit time (i.e. L/hr). Cl is a measure of drug elimination from the body without identifying the mechanism or process. Cl for a first-order elimination process is constant regardless of the drug conc.

eldkVCl Cl

DoseAUC

0

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INTEGRATED EQUATIONS

0p

d C

DoseV

elkt

693.02/1

pCl

DoseAUC

el

p

k

CAUC

0

eldp kVCl

tkpp

eleCC 0 tkB

eleDoseD

tkCC elpp 0lnln tkDoseD elB lnln

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ESTIMATION OF PK PARAMETERS

A plot of Cp vs. time

kel

Cpo