Pharmacokinetics
Syllabus:
3.1 Significance
of plasma drug concentration measurement.
3.2 Compartment
model – Definition and scope.
3.3 Pharmacokinetics
of drug absorption – Zero-order and first order absorption rate constant using
Wagner-Nelson method.
3.4 Volume
of distribution and distribution coefficient.
3.5 Compartment
kinetics – One compartment and two compartment models. Determination of
pharmacokinetic parameters from plasma and urine data after drug administration
by intravascular and oral route.
3.6 Curve
fitting (method of residuals), regression procedures.
3.7 Clearance
concept. Mechanism of renal clearance. Clearance ratio. Determination of renal
clearance.
3.8 Extraction
ration, hepatic clearance, biliary excretion, extra-hepatic circulation.
3.9 Non-linear
pharmacokinetics with special reference to one compartment model after I.V.
drug administration. Michaelis-Menten equation. Determination of non-linearity
(saturation mechanism).
Pharmacokinetic is defined as the kinetics of drug
absorption, distribution, metabolism and excretion (ADME) and their
relationship with the pharmacologic, therapeutic or toxicologic response in
human and animals.
3.1 Significance of plasma drug concentration measurement.
N.B. Measurement of drug
concentration in blood, serum, or plasma is the most direct method to assess
the pharmacokinetics of the drug in the body.
Whole blood contains cellular
elements (red blood corpuscles, white blood corpuscles and platelets) and
various proteins (albumin, globulin, prothrombin and fibrinogen, etc.).
To obtain plasma, the whole blood preserved with some anticoagulant (e.g.
citrate or heparin) is centrifuged and the supernatant liquid is collected as
plasma.
To obtain serum the whole blood is allowed to clot, then centrifuged and the
serum is collected from the supernatant.
3.2 Compartment model – Definition and scope.
A model is a hypothesis using mathematical terms to
describe quantitative relationships. Various mathematical models can be devised
to simulate the rate processes of drug absorption, distribution, and
elimination. This mathematical models make possible the development of
equations to describe drug concentrations in the body as a function of time.
In compartmental modeling the body is considered as composed
of several compartments that communicate reversibly with each other.
N.B. If one considers every organ, tissue or body fluid that can
get equilibrated with the drug as compartments, then infinite number of
compartments can exist in the body and the mathematical description of such a
model will be too complex to be handled. Hence, tissues which are approximately
similar in their drug distribution characteristics are pooled to form a
kinetically homogeneous hypothetical compartment. Such a compartment is not a
real physiologic or anatomic region but a fictitious or virtual one.
The kinetics of most drugs can be described by a
hypothetical model consisting of one, two or at the most three functional
compartments arranged either in series or parallel to each other.
** It is also assumed that the rate of drug movement between
compartments (i.e. entry and exit) follows first-order kinetics.
Depending upon whether the compartments are arranged
parallel or in a series, compartment models are divided into two categories - mammillary model and caternary model.
Mammillary model
·
This model is the most common compartment model
used in pharmacokinetics. It consists of one or more peripheral
compartments connected to the central
compartment in a manner similar to connection of satellites to a planet.
·
The central
compartment (or compartment -I) comprises of plasma and highly perfused tissues such as lungs, liver, kidneys,
etc. which rapidly equilibrate with the drug.
·
The peripheral
compartments or tissue compartments
(denoted by numbers 2,3, etc.) are those with low vascularity and poor perfusion. Distribution of drugs to these
compartments is through blood.
·
Movement of the drug between compartments is
defined by characteristics first-order
rate constants denoted by letter K. The subscript indicates the direction of
drug movement. Thus K12 refers to drug movement from compartment 1
to 2 and reverse for K21.
Model-1
One-compartment open model, intravenous
administration
I K10
Model-2
One-compartment open model, extra-vascular
administration
K01 I K10 
Model-3 Two-compartment open model, intravenous administration
K12
1 2
K21
K10
Model-4 Two-compartment open model, extravascular
administration
K12
K01 1 2
K21
K10
Model-5 Three-compartment open model, intravenous
administration
2
K12 K21 
K13
1 3
K31
K10
Model-6 Three-compartment open model, extravascular
administration
2 K12 K21 K13
K01 1 3 K31
K10
For intravenous
administration
Number of compartments
|
Rate constants
|
Number of rate
constants
|
1
2
3
|
K10
K10, K12,
K21
K10, K12,
K21, K13, K31
|
1
3
5
|
For extarvascular
administration
Number of
compartments
|
Rate constants
|
Number of rate
constants
|
1
2
3
|
K10, K01
K10, K01,
K12, K21
K10, K01,
K12, K21, K13, K31
|
2
4
6
|
So the number of rate constants that will appear in a
particular compartment model is given by R, where
For intravenous administration, R = 2n - 1
For extravascular administration, R = 2n where, n = the number of compartments.
Advantages of
compartmental modeling
1.
It gives a visual representation of various rate
processes involve in drug disposition.
2.
It show how many rate constants are necessary to
describe these processes.
3.
It enables the pharmacokineticist to write differential
equations for each of the rate processes in order to describe
drug-concentration changes in each compartment.
4.
It is useful in predicting drug concentration-time
profile in both normal physiological and i pathological conditions.
5.
It is important in the development of dosage regimens.
Disadvantages of
compartmental modeling:
1.
The compartments and parameters bear no relationship
with the physiologic functions or the anatomic structure of the species;
several assumptions have to be made to facilitate data interpretation.
2.
Extensive efforts are required in the development of an
exact model that predicts and describes correctly the ADME of a certain drug.
3.
The model is based on curve fitting of plasma
concentration with complex multi-exponential mathematical equations.
4.
The model may vary within a study population.
5.
The approach can be applied only to a specific rug
under study.
6.
The drug behavior within the body may fit different
compartmental models depending on the route of administration.
Difficulties generally arise when using models to interpret
the differences between results from human and animal experiments.
Catenary model
In this model, the compartments are joined to one another in
a series like compartments of a train. This is not observable physiologically /
anatomically as the various organs are directly linked to the blood
compartment. Hence this model is rarely used.
K01 K12 K23
1 2 3
K21 K32
Physiologic Model
(Perfusion rate limited model)
The rate of appearance of a drug in a
tissue depends on two processes:
(i)
perfusion of
blood into that tissue and
(ii)
permeation of
drug from blood capillaries to the tissue fluid
In case of highly membrane permeable drugs e.g. low
molecular weight, poorly ionized and highly lipophilic drugs the permeation
step is much faster than the perfusion step. Hence the process is called
perfusion rate limited (because
perfusion step is the slowest step and the over all rate can be controlled by
controlling this step only).
Drug concentrations in various tissues are predicted by
organ tissue size, blood flow, and experimentally determined drug tissue-blood
ratios (i.e. partition of drug between the tissue and blood).
Because there are many tissue organs in the body, each
tissue volume must be obtained and its drug concentration is described.
Unfortunately, it is difficult to obtain data from different tissues
experimentally (the
animal or subject should be sacrificed).
3.5 Compartmental kinetics
3.5.1 One compartment open model
·
The one-compartment open model is the simplest
model which depicts the body as a single, kinetically homogeneous
unit that has no barriers to the movement of drug and final distribution equilibrium
between the drug in plasma and other body fluids is attained instantaneously and
maintained at all the time.
·
This model thus applies only to those drugs that
distribute rapidly though out the body.
·
The concentration of drug in plasma represents
the drug concentration in all body tissues.
·
The term “open” indicates that the input (availability)
and output (elimination) are unidirectional and that the drug can be eliminated
from the body.
Fig1. Representation of one-compartment open model showing
input and output process.
3.5.2 Two compartment open model
In this model, the drug distributes into two compartments.
The central compartment represents
the blood and highly perfused tissues. The drug distributes rapidly and
uniformly in the central compartment. The second compartment known as peripheral compartment or tissue compartment, contains tissues in
which drug equilibrates slowly. Drug transfer between the two compartments is
assumed to take place by first-order processes.
There are several possible two compartment open models:
Model A
Drug elimination takes place from central compartment (k10).
This model is generally found true for all drugs because the major drug
eliminating organs are kidney and liver which are highly perfused tissues.
Model B
Drug elimination takes place from peripheral compartment (k20).
Model C
Drug elimination takes place both from central (k10)
and peripheral compartments (k20).
3.5.3 Determination of pharmacokinetic parameters from plasma concentration data
3.5.3.1 Intravenous bolus administration
When a drug that distributes rapidly in the body is given in
the form of a rapid intravenous injection (i.e IV bolus dose), it takes about 2
to 3 minutes for complete circulation and therefore the rate of absorption is
neglected in calculations. The model can be depicted as follows:
Elimination Rate Constant (kE)
Change in amount of drug in the body 
where X = amount of drug in the body remained to be
eliminated at time, t.
= rate of input - rate of output
= 0 - kEX ----------(i) where kE is the first order
elimination rate constant
This kE includes both the rate constants of
metabolism (km) (or biotransformation) and excretion (ke)
of the drug.
Integrating eqn (i) 
ln
(X/X0) = - kE t ----------------(ii)
Since X = C Vd where Vd is apparent volume of
distribution and
C =
concentration of the drug in plasma at time t.
 |
Replacing X in equation (i)
ln
(C/C0) = - kE
t
or
C/C0 = e - kE t
or C = C0 e - kE t
--------------(iii)
Taking logarithm of eqn (iii)
log
C = log C0 - (kE / 2.303) t
Constant kE can be calculated from the slope of
fig.2
Slope = - (kE /2.303)
\ kE = -
2.303 x SlopeElimination half life (or Biological half life)
It is defined as the time taken for the amount of drug in
the body as well as plasma concentration to decline by one half its initial
value.
i.e. at time t =
0 C = C0.
and at time t = t1/2 
Substituting the values of t and C in the logarithmic form
of eqn. (iii) yields:
log (C0/2)
= log C0 -
(kE / 2.303) t1/2.
or, 
or, 
It is expressed by 
Apparent volume of distribution (Vd)
·
The apparent volume of distribution is a
parameter of the one-compartment open model because the volume of distribution
governs the plasma concentration of the drug after a given dose.
·
Due to rapid drug equilibration in between the
blood and tissues. The volume in which the drug is assumed to be uniformly
distributed is termed as the volume of
distribution.
·
The volume of distribution represents a volume
that must be considered in estimating the amount of drug in the body from the
concentration of drug found in the sampling compartment.
·
It is actually called the apparent volume of distribution because the value of the volume of
distribution does not have a true physiologic meaning in terms of an anatomic
space. It is only a hypothetical one.
Method-I
It is determined by administering it by rapid i.v. injection
and using the following equation:
 |
The C0 value is obtained by extrapolation of the
plot of log(plasma conc) vs time.
Method-II
Vd can be determined by another way if the AUC and the first order elimination rate constant, kE is known.
In the equation 
the X terms are
substituted by X = Vd x C,
the X terms are
substituted by X = Vd x C,
where C =
plasma concentration of unchanged drug in the body;
we get 
Integrating both sides upto infinite time
Since Xµ = 0
hence, 
Therefore, 
The calculation of Vd by means of the above
equation is model independent because
no pharmacokinetic model is considered while calculating and the AUC is
determined by the trapezoidal rule.
Significance of Volume of distribution
The apparent volume of distribution is not a true
physiologic volume. Most drug have an apparent volume of distribution smaller than, or equal to, the body mass.
·
From equation, 
it is evident that Vd
is dependent on C0. Drugs with a large Vd are more
concentrated in the extravascular tissues and less concentrated
intravascularly.
it is evident that Vd
is dependent on C0. Drugs with a large Vd are more
concentrated in the extravascular tissues and less concentrated
intravascularly.
Þ
When the drug is concentrated in the peripheral tissues the C0 is
small resulting in large Vd.
Þ
If a drug is highly
bound to plasma proteins or remains in the vascular region, then C0
will be higher; resulting in a smaller Vd.
·
Vd is a volume term that can be
expressed as a simple volume or in
terms of percent of body weight. In
expressing the Vd in terms of % body weight, a 1 L volume is assumed
to be equal to the weight of 1 kg. e.g. if the volume of distribution, Vd,
is 3500 mL for a subject weighing 70 kg, the Vd expressed as % body
weight would be:
If Vd is a large number
– i.e. > 100 % of body weight – then it may be assumed that the drug is
concentrated in certain tissue compartments. Thus the Vd is a useful
parameter in considering the relative amounts of drug in the vascular and in
the extravascular tissues.
·
Pharmacologists often attempt to conceptualize
the Vd as true physiologic or anatomic volume, however this
assumption is rarely correct.
·
Given the apparent volume for a particular drug,
the total amount of drug in the body at any time after administration of the
drug may be determined by measuring the plasma concentration according to the
following formula:
X
= Vd x C
·
For each drug the Vd is constant. in
certain physiologic cases, the apparent Vd for the drug may be
altered if the distribution of the drug is changed. For example in oedematous
conditions, the total body water and total extracellular water increase; this
is reflected in a large Vd. Similarly. changes in total body weight
and lean body mass (which normally occur with age) may also affect apparent Vd.
CLEARANCE
N.B. The body is considered as a system of organs perfused by plasma
and both body fluids. Drug elimination from the body is an ongoing process due
to both metabolism (i.e. biotransformation) and drug excretion through the
kidney and other routes. The mechanisms of drug elimination are complex, but
collectively drug elimination from the body may be quantitated using the
concept of drug clearance. The rate of elimination may be expressed in several
ways, each of which essentially describe the same process, but with different
levels of insight and application in pharmacokinetics.
Definition: Clearance is defined as the volume of plasma fluid that is cleared
of drug per unit time.
Drug Elimination
expressed as Amount Per Unit Time (i.e. Mass approach)
Unit: mg/min or mg/hr
Advantage:
1)
The expression of drug elimination from the body in
terms of mass per unit time is simple, absolute, and unambiguous.
2)
For a zero order elimination process, expressing the
rate of drug elimination as mass per unit time is convenient because the rate
is constant.
Disadvantage:
1.
For a first-order elimination, drug clearance expressed
as mass per unit time is not constant.
Drug Elimination
expressed as Volume Per Unit Time (i.e. Volume approach)
Unit: ml/min or litre/hr
Advantage:
For a first-order elimination, drug clearance expressed as
volume per unit time is constant. This approach of clearance is most common in
pharmacy.
N.B. The drug concentration in the body will gradually decline such
that the mass of drug removed over time
is not constant. The plasma volume in the healthy state is relatively constant
because water lost through the kidney is rapidly replaced with fluid absorbed
from the gastrointestinal tract.
Therefore 
The negative sign refers to the drug exiting from the body.
Questions:
Q1. What is the significance of measuring plasma level drug
concentrations?
Q2. What is the purpose of compartmental models?
Q3.