Pharmacokinetic Models

Outline of Lecture
 Definition for Pharmacokinetic model
 Classification of Pharmacokinetic model
 One compartmental model IV bolus
 One compartmental model IV infusion
 One compartmental model extravascular
 Physiological Model
 Tissue Dosimeter

Intended Learning outcome
ILOS
Upon successful completion of this course student will be able to
define and classify different pharmacokinetic models
Understand the differences between the pharmacokinetic models and
know the advantages and limitations of each model.
Calculate important pharmacokinetic parameters
Will be able to solve case studies based on pharmacokinetic data
Estimate the values of different pharmacokinetic parameters from
plasma drug concentration and urinary excretion data.
Understand physiological model and its application in clinical study.

Learning resources
• M. Gibaldi and D. Perrier, Pharmacokinetics, Latest Edition,
Marcel Dekker, Inc., NY, USA.
• L. Shargel and A.B.C. Yu, Applied Biopharmaceutics and
Pharmacokinetics, Latest edition, Appleton & Lange, Stanford,
CT, USA.
• W.A. Ritschel and G.L. Kearns, Handbook of Basic
Pharmacokinetics including Clinical Applications, Latest
edition, American Pharmaceutical Association, Washington,
DC, USA.
• A.R. Gennaro, ed., Remington. The Science and Practice of
Pharmacy. Latest edition, Mack Publishing Co., Easton, PA,
USA.

Define: pharmacokinetic models provides concise means of
expressing mathematically or quantitatively, the time course
of drug throughout the body
Use of Pharmacokinetic Model
1. To know the concentration of drug in various body fluids
2. To get best dosage regime for each patient
3. Evaluate the toxicity with certain drugs
4. Correlation of Plasma drug concentration with
Pharmacological response
5. To evaluate the bioequivalence between different
formulation of the same drug
6. Estimate the possible drug or metabolite accumulation
7. Determine the influence of disease state on drug ADME

No Pharmacokinetic parameter Abbreviation Units example
Area under the curve AUC µg x hr/mL
Total body clearance ClT Litres/time
Renal clearance ClR Litres/time
Hepatic clearance ClH Litres/time
Apparent volume of distribution VD Litres
Vol. of distribution at steady state VSS Litres
Peak plasma drug concentration CMAX mg/L
Plasma drug concentration CP mg/L
Initial drug concn at t0 from graph Cp
o mg/L
Steady-state drug concentration Css mg/L

Time for peak drug concentration TMAX Hr
Dose DO mg
Loading dose DL mg
Maintenance dose DM mg
Amount of drug in the body DB Mg
First order rate constant for drug absorption Ka 1/hr
Zero order rate constant for drug absorption KO mg/hr
First order rate constant for drug elimination K 1/hr
Elimination half-life t½ hr
Fraction of drug absorbed F Ranges from 0-1
Fraction of drug excreted Fe Ranges from 0-1
Rate of drug infusion R mg/hr

Pharmacokinetic models are divided into 3 types
Compartment
Modeling
Non-Compartment
Analysis
Physiologic Modeling

Pharmacokinetic Modelling
Compartment
Models
Non-Compartment
Models
Physiologic
Models
Caternary
Model
One compt
Mamillary
Model
Three compt
Two compt
i v bolus
i v infusion
Extravascular
i v bolus
Extravascular
AUC, MRT, MAT, Cl, VSS
i v bolus
Extravascular

1. The body can be represented as a series of
compartments that communicate reversibly
to each other.
2. A compartment is not a real physiologic or
anatomic region but is considered as a tissue
or group of tissues that have a similar blood
flow and drug affinity.
3. Within each compartment, the drug is
uniformly distributed.
4. Drugs move dynamically in and out of
compartments.
Compartment models

5. Kinetics of most drugs can be described by a
hypothetical model consisting of one, two or
maximum three compartmental model
6. In compartmental model always central
compartment will be (compt No 1, which consist of
plasma and highly perfused tissues like lungs, liver, kidney etc)
entry & exit of drug will take place from central
compartment

7. The peripheral compartment will be
(compartment No 2 or 3, are those tissues
which have less blood flow)
Compartment # 2 or 3

8. Rate constants are used to represent the overall rate
processes of drug entry into and exit from the
compartment.
9. The model is an open system since drug is eliminated
from the system.
10. At any time, the amount of drug in the body is simply
the sum of drug present in the compartments.

Compartment 1:
Represents the plasma or central compartment
Compartment 2:
Represents the tissue compartment
Compartment 3:
Represents an additional deep tissue compartment
1
1
1
2
2
3
ka kel
k12
k21

The parameters needed to describe the model are:
- The volume of each compartment
- The rate constants coming into and out of each
compartment
e.g. 3 parameters are needed to describe model 1:
- the volume of the compartment
- the absorption rate constant ka
- the elimination rate constant kel
The drug is usually sampled from compartment 1.
1
ka kel

Drug sampling from compartment 2 or 3 is difficult to obtain
because tissues are not easily sampled.
If the amount of drug absorbed and eliminated per unit time is
obtained by sampling compartment 1, then the amount of drug
contained in the tissue compartment 2 can be estimated
mathematically.

Advantages of compartmental model
 It gives a visual representation of various rate processes
 It shows how many rate constant are required to describe
the processes
 It is important in the development of dosage regime
Limitations of compartmental model
 Drug given by IV route may behave according to single
compartment model but the same drug given by oral route
may show 2 compartment behavior.
 Hard work is require to develop a model to correctly
describe ADME of drug
 Complex mathematical equation based on plasma drug
concentration
 The model may vary within a study population group
 This method can be applied to only a specific drugs

Rapid IV Injection (i.v. bolus)
 The entire dose of the drug enters the
general circulation immediately.
One Compartment Open Model
The drug usually distributes via the circulation to all tissues
in the body.
Due to rapid drug equilibrium between blood and tissue,
drug elimination occurs as if the dose is all dissolved in a
tank of uniform fluid from which the drug is eliminated.

 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 levels.
 The one-compartment model offers the simplest way
to describe the process of drug distribution and
elimination in the body.
 It can be represented schematically as:

Blood
(Vd)
IV
Input
kel
Parameters:
1. The volume in which the drug is assumed to be uniformly
distributed is termed the apparent volume of distribution
(Vd)
2. Elimination rate constant (kel) which determines the
elimination rate of the drug over time.

Mathematically
This behavior can be expressed mathematically as :

The decrease in the plasma drug
concentration in the body can be
expressed as follows:
dCp
dt
= - kel Cp
This equation shows that the rate of drug elimination in the
body is a first elimination process, depending on:
kel: the elimination rate constant
Cp: plasma drug concentration at time t
The –ve sign is an indication for a decrease in concentration.
1. Elimination Rate Constant (kel)

Integration of the equation gives the following:
Cp
º : plasma drug concentration at t = 0
The same equation can be expressed as:
DB Dose . e-kelt
=
logCp + logCp
º
- kelt
2.303
=
Cp Cp
º e-kelt
=
ln Cp ln Cp
º - kelt
=
DB: drug remaining in the body after time t
Remember:
Straight line equation: y = ax + b a= slope; b= intercept

Plotting the drug concentration versus
time we can detect a curve showing a
continuous decrease in concentration
as the drug is eliminated.

We can obtain a straight line by:
1. plotting the log conc. versus time.
slope : -kel /2.303 intercept: logCp
º
2. plotting the ln conc. versus time.
slope : -kel intercept: lnCp
º
3. plotting the conc. versus time in a
semilog paper
Time
log
conc
slope : -kel /2.303 intercept: Cp
º

2. Elimination half life (t1/2)
The elimination half life is defined as the time required
for 50% of the drug to be eliminated from the body.
When the elimination process follows first order
kinetics, the elimination half life of the drug is constant
and is related to the elimination rate constant (kel).
The half life can be determined from the following
equation:
t1/2
0.693
kel
=

Half-Life
First order
 C = Co e - k
el
t
If C/Co = 0.50 for half of the
original amount
 0.50 = e – k
el
t
Simplify by applying integration
 ln 0.50 = -kel t ½
 -0.693 = -kel t ½
 t 1/2 = 0.693 / kel
Zero order
Graph method or
equation method
t1/2 = (C0 / 2) ÷Kel
Where C0 is initial dose

3. Apparent Volume of distribution (Vd)
Because the value of the volume of distribution does not
have a true physiologic meaning in terms of an anatomic
space, then the term apparent volume of distribution is
used.
It is a measure of:
- the extent of distribution and
- the affinity of drug to various tissues in the body.
For each drug Vd is constant except in certain pathologic
conditions
The Vd relates the concentration of the drug on plasma
Cp to the amount of drug in body (DB)
DB = Vd Cp

In one-compartment model with IV injection,
the Vd is calculated from the following equation:
Vd
Dose
Cp
°
= =
DB
°
Cp
°
kel[AUC]
Dose
= ∞
o
Vd
• The apparent volume of distribution may be determined by
knowing the dose, elimination rate constant and the AUC.
NONCOMPARTMENTAL METHOD

First:
Last: AUCt(last)- = Cp last/ kel
AUC0- = AUC0-t(first) + AUCt(last)-
Assessment of AUC
The AUC can be
determined by the
trapezoidal method

4. Clearance
 Clearance is a measure of drug elimination from the
blood.
 Drug elimination from the body is due to both,
metabolism through the liver and drug excretion
through the kidneys.
 Clearance is defined as:
the volume of blood that is cleared per unit time.
 Clearance for a first-order elimination process is
constant regardless of the drug concentration because
clearance is expressed in volume per unit time rather
than amount of drug per unit time.

Clearance can be also calculated from the dose and AUC:
[AUC]
Dose
Cl = ∞
o
or
From the equation it is obvious that there is no need
to know the half-life or the volume of distribution to
calculate the clearance.
Cl = kel Vd

logCp + logCp
º
- kelt
2.303
=
Cp Cp
º e-kelt
=
ln Cp ln Cp
º - kelt
=
t1/2
0.693
kel
=
Vd
Dose
Cp
°
=
k [AUC]
Dose
= ∞
o
Vd
Cl = kel Vd
[AUC]
Dose
Cl = ∞
o
kel
Cl
=
Vd

Estimation of Pharmacokinetic Parameters:
After an IV bolus injection and blood sampling, plasma
concentration is plotted versus time.
The plot indicates that the values are dropping progressively
i.e. first-order
In order to obtain a straight line the data is plotted semi-
logarithmically.
kel : the elimination rate constant is the –ve slope of the line
t½ : 0.693/kel
Cl : Dose/AUC (AUC: estimated using the trapezoidal rule)
Vd : Cl/kel

Case study
The following data for decomposition of two drugs, A
and B are given in the table below :
1Brahmankar DM
Time (hr) Drug A (mg) Drug B (mg)
0.0
0.5 379 181.2
1.0 358 164.0
1.5 337 148.6
2.0 316 134.6
3.0 274 110.4
4.0 232 90.6
6.0 148 61.0
8.0 64 41.0

1) Determine (by plotting or otherwise) the rate of process for
both drugs.
2) What is the rate constant for both drugs?
3) What were the original amounts of drug before elimination?
4) What is their half-life?
5) If the original quantities of drug taken were 800 mg for A and
400 mg for B then what will be their new half-lives?
6) Write equations for the straight line that best fits the
experimental data for both drugs.

1. Drug A = Zero, and Drug B = first order
2. Drug A Kel = 42 mg/hr from slope equation and Drug
B Kel = 0.198mg/hr by using integration equation
3. Initial dose of Drug A = 400mg and Drug B = 200mg
from extra-plotation of graph to up to y-axis.
4. Drug A t1/2= 4.76hrs from graph direct and Drug B
t1/2 = 3.5hrs by 0.693/Kel equation
5. Drug A double dose t1/2 = 9.52 and Drug B double
dose t1/2 = will remain unchanged i:e 3.5hrs
6. Straight line equation for drug A zero order C = Co –
Kelt (64 = 400 – 42*t) and Drug B first order Log C=
Log C0 – Kel t/2.303 (log 41= log 200 –
0.198*t/2.303)

IV Infusion
 A single iv dose may rapidly produce a desired
therapeutic concentration, but this route is
unsuitable when a constant plasma or tissue drug
concentrations and effect is desired for a certain
period of time.
 To maintain a constant plasma drug concentration,
drug must be administered at a constant rate.
One Compartment Open Model

 This can be obtained with high degree of precision
by infusing drug intravenously via either a drip or a
pump usually done in hospital settings.

Because no drug was present in the body at zero time,
drug level rises from zero drug concentration and
gradually becomes constant when a plateau or steady-
state drug concentration is reached.

At steady state, the rate of drug leaving the body is equal
to the rate of drug (infusion rate) entering the body.
Therefore, the rate of change of drug plasma concentration:
dCp / dt = 0
At steady state: Rate of drug input = Rate of drug output

The pharmacokinetics of a drug given by constant IV
infusion:
input process in which the drug is directly infused into the
systemic blood circulation is zero-order and
elimination of drug from the plasma is a first order
process.
Blood
(Vd)
IV Input
R
kel

The change in the plasma drug concentration at any
time (dC/dt) during the infusion =
the rate of input - the rate of output.
R : the infusion rate (zero - order)
kel : the elimination rate constant (first-order).
Integration of the above equation gives the following
expression:
Cp
R
Vd kel
= (1- e-kelt)
dCp
dt
= R - kel Cp

Whenever the infusion stops either at steady state or
before reaching steady state, the drug concentration
declines according to first-order kinetics with the slope
of the elimination curve equal to - kel

All infusions are stopped before true steady state is
reached, because in theory, an infinite time is needed
after the start of the IV infusion for the drug to reach
the steady-state drug concentration because drug
elimination is first order.
In clinical practice, a plasma drug concentration prior
to, but approaching, the theoretical steady state is
considered the steady-state plasma drug
concentration.

At infinite time, t =∞, e-kelt approaches zero
and the equation reduces to:
Cp
R
Vd kel
= (1- e-∞)
Css
R
Vd kel
= =
Cl
R

Time Needed to Reach Css
The time required to reach the steady-state drug
concentration in the plasma is dependent on the
elimination rate constant of the drug.
For most drugs, at therapeutic concentrations, drugs are
eliminated by a first-order process.
When infused at a constant rate R, the infusion rate will be
fixed while the rate of elimination steadily increases until,
steady state is reached because rate of elimination is first-
order which means it depends on the plasma concentration
which is also increasing until steady state is reached.
Rate of elimination = kelCp

After one half-life the plasma drug concentration is 50% the
steady state concentration (plateau) value, after 2 half-lives,
the plasma drug concentration is 75% of the steady state
value.
For practical purposes, however, the steady state may be
considered to be reached in 5 half lives.

For example, if the steady state of theophylline in a patient
is 4hrs half lives, then it must be infused for 20hrs before
steady state is reached.
Whereas it takes about 3 weeks of constant phenobarbital
infusion to reach steady state as it has a t1/2 of 100 hrs.

It has to be noticed that an increase in the infusion rate
will not shorten the time to reach the steady state drug
concentration.
If the drug is given at a more rapid infusion rate a higher
steady-state drug concentration will be obtained, but
the time to reach steady state is the same.

Css
R
Vd kel
= R Css Vd kel
= R = (10 mg/L) (10L) (0.2 hr-1)
= 20 mg/hr
Example
An antibiotic has a volume of
distribution of 10 L and a kel of 0.2 hr-1.
A steady-state plasma concentration of
10 mg/L is desired.
What is the infusion rate needed to
maintain this concentration?
R
2 R

Situations sometimes demand that the plateau be reached
rapidly.
How to solve this problem?
Solution:
At the start of an infusion, give a bolus dose equal to the
amount desired in the body at plateau.
Usually the bolus dose is a therapeutic dose, and the
infusion rate is adjusted to maintain the therapeutic level.
When the bolus dose and infusion rate are exactly matched
(the rate of infusion is exactly matched by the rate of
elimination), the amounts of drug in the body associated
with the 2 modes of administration are complementary; the
gain of one offsets the loss of the other.

Remember if a bolus dose of a certain drug is given at
the same time an IV infusion of the same drug is
infused, then the total drug concentration in the blood
is the sum of the 2 resulting concentrations.
The total concentration Cp at t hours after the IV dose
and the IV infusion is equal to Cp = C1 + C2

C1 Cp
º e-kelt
=
DB = Vd Cp
C2
R
Vd kel
= (1- e-kelt)
Don’t forget:
Bolus:
Infusion:

The loading dose DL or initial bolus dose of a
drug, is used to obtain stead- state
concentrations as rapidly as possible.
If a bolus dose of a certain drug is given and at
the same time an IV infusion of the same drug
is infused, then the total drug concentration in
the blood can be calculated as follows:
Cp = C1+ C2

 A large number of drugs Cp can be described by
one compartmental model with first-order
absorption and elimination
 After ev administration, the rate of change in the
amount of drug in the body dX/dt is the
difference between the rate of input (absorption)
dXev/ dt and rate of output (elimination) dXE/dt.
One Compartmental Open model
Extra-vascular administration

 
t
K
-
t
K
-
E
a
d
0
a a
E
e
-
e
)
K
-
(K
V
X
F
K
C 
dX/dt = Rate of absorption – Rate of elimination

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