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CHAPTER 3
PHARMACOKINETIC
MODELS
1
PHARMACOKINETIC
MODELING
A Model is a hypothesis using mathematical
terms to describe quantitative relationships
MODELING REQUIRES:
Thorough knowledge of anatomy and
physiology
Understanding 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 time
HOW?
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
Oral
Administration
G.I.
Tract
Intravenous
Injection
Circulatory
System
Intramuscular
Injection
Subcutaneous
Injection
Tissues
Excretion
ADME
Metabolic
Sites
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Complexity of PK model will vary
with:
1- Route of administration
2- 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
Percent of Dose
Metabolism
Blood
RET
Muscle
Lung
Adipose
Time
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COMPARTMENTAL MODELS
The body is represented by a series of
compartments that communicate reversibly
with each other.
k12
1
1
k21
2
2
3
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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 MODELS
Is the most common compartmental model used in
PK. The model consists of one or more
compartments connected to a central compartment
ka
1
kel
k12
1
2
k21
1
2
3
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Intravenous and Extravascular
Administration
IV, IM, SC
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Intravenous and extravascular
Route of Administration
Difference in plasma conc-time curve
Cp
Cp
Time
Time
Intravenous
Administration
Extravascular
Administration
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One Compartment Open Model
Intravenous Administration
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One Compartment Open Model
Intravenous Administration
The one compartment model offers the simplest
way to describe the process of drug distribution
and elimination in the body.
i.v.
Input
Blood
(Vd)
kel
Output
When the drug is administered i.v. in a single rapid
injection, the process of absorption is bypassed
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One Compartment Open Model
Intravenous Administration
The 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 EQUATIONS
The rate of change of drug plasma conc over
time is equal to:
dC p
dt
 kel C p
This expression shows that the rate of
elimination of drug from the body is a firstorder process and depends on kel
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INTEGRATED EQUATIONS

dC p
dt
  kel C p
Cp = Cp0e-k
el
t
ln Cp = ln Cp0  kelt
DB = Dose . e-k
el
t
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 DB0
Therefore,
t1/ 2
0.693

kel
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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.
DB
 kel t
e
Dose
The fraction of the dose lost after a time t
can be then calculated from:
1 e
 k el t
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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 liter
Also, 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).
Dose D
Vd 


Cp
C

B

p
Where Cp is the intercept obtained by plotting
Cp vs. time on a semilog paper.
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For One Compartment Model with IV
Administration:
o
Cp

D
Dose
B
Vd 

C p
C p
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Calculation of Vd from the AUC
Since,
Since,
Then,
dDB/dt = -kelD = -kelVdCp
dDB = -kelVdCpdt
 dDB = -kelVd  Cpdt
 Cpdt = AUC
AUC = Dose / kelVd
Dose
Vd 
kel [ AUC]
Model
Independent
Method
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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:
3.5Kg
100  5% of body weight
70 Kg
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.33
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.
Cl  Vd kel
AUC

0
Dose

Cl
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INTEGRATED EQUATIONS
0  kel t
p
Cp  C e
ln C p  ln C  kel t
0
p
t1 / 2
DB  Dose e
ln DB  ln Dose  k el t
0.693

k el
Cl p  Vd  k el
Dose
AUC 
Cl p
 kel t
Dose
Vd 
0
Cp
AUC 
C
0
p
k el
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ESTIMATION OF PK PARAMETERS
A plot of Cp vs. time
Cpo
kel
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