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Pharmacokinetics
139
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Figure 3.3 Various mammillary compartment models. (The pharmacokinetic rate constant is basically
designated as ‘K
a
’, i.e., the fi rst-order absorption rate constant and the fi rst-order elimination rate constant
as ‘K
E
’.)
Essentials of Biopharmaceutics and Pharmacokinetics
140
In usual practice, the exact number of rate constants that may eventually appear in a specifi c compart-
ment model is designated by ‘R’ and, therefore, we may have:
for IV administration: R = 2n – 1 and
for EV administration: R = 2n
where n is the number of compartments.
In general, looking closely at these different models, it is rather important to ascertain whether one
may have an exact and precise access to the ‘drug concentration data’ obtained directly from each com-
partment separately. In Figure 3.3, the data of Models 3 and 4 pertaining to Compartment 2 may not be
obtained so easily by virtue of the fact that the respective tissues do not usually:
∑ get sampled frequently and
∑ comprise a homogeneous concentration of ‘drug’.
Let us assume a typical situation whereby the exact quantum of drug substance being duly absorbed and
eliminated per unit time is accomplished adequately by sampling Compartment 1, one may estimate suit-
ably using various mathematical models the exact amount of the ‘drug’ present in the tissue compartments
(or peripheral compartments).
Note
With the help of suitable ‘mathematical equations’ elaborating these typical models, it is a lot
easier to determine and arrive at the different pharmacokinetic parameters being employed.
Multi-Compartment Properties It has been duly observed that by either
∑ an IV administration or
∑ a bolus
18
administration,
quite a few drug substances do acquire enough distribution in vivo rather slowly in order that a ‘sig-
nifi cant portion of the intended dose’ gets eliminated adequately before accomplishing the so-called
‘distribution equilibrium’.
19
Thus, we may have two different types of ‘plots’ of a drug having multi-compartment properties, such as:
∑ semi-logarithmic plot of plasma-drug concentration vs. time after IV administration of drug and
∑ semi-logarithmic plot of plasma-drug concentration vs. time after extravascular administration of drug.
The aforesaid two aspects shall now be treated separately along with their respective plots and appropri-
ate explanations in the following sections:
Semi-logarithmic Plot of Plasma-Drug Concentration vs. Time After IV Administration of Drug: It is,
however, pertinent to state here that in a specifi c instance when the drug is administered via IV, usually
more than one exponential term may be needed critically to differentiate and characterize the percep-
tive changes in the ensuing plasma-drug concentration with respect to time. In fact, the number of the
18
Bolus: It refers to the rounded mass of a pharmaceutical preparation ready to swallow or such a mass passing
through the GI tract.
19
Distribution Equilibrium: The equilibrium achieved due to the partitioning of drug to the various locations
or compartments in the body or another heteroge system.
Pharmacokinetics
141
prevailing exponential terms does signify the actual number of compartments involved. Therefore, for a
two compartment model one may have
∑ a central compartment and
∑ a peripheral compartment,
as illustrated explicitly in Figure 3.4.
∑ COMPARTMENT IN SERIES
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K
a
is the pharmacokinetic rate constant at input,
K
e
is the pharmacokinetic rate constant at output,
V
p
is the volume of ‘plasma’,
V
C
is the volume of ‘intracellular fl uid’,
C
1
is the concentration in Compartment 1,
C
2
is the concentration in Compartment 2,
D = Drug used,
D
1
= Quantum of drug in Compartment 1,
D
2
= Quantum of drug in Compartment 2.
∑ COMPARTMENT IN PARALLEL
Drugs Adopting Two-Compartment Model: There are a good number of drugs that predomi-
nantly adopt the two-compartment model. Amazingly, vancomycin,
20
a glycopeptide antibiotic, represents
a classical example, which follows specifi cally a two-compartment model, having a distribution phase
extending for 1–2 hr.
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Figure 3.4 Diagrammatic illustrations of an open two-compartment pharmacokinetic model.
20
Vancomycin: It is obtained from Amycolatopsis orientalis originally from the culture of an Indonesian soil sample
and later on from Indian soil. In 1958, it was duly introduced as an antibiotic against certain gram +ve cocci, viz.
streptococci, staphylococci and pneumococci.
Essentials of Biopharmaceutics and Pharmacokinetics
142
Several other equally important drugs that critically follow either a two or even more compartment
model are as stated below:
∑ amphetamine
∑ chlordiazeperoxide
∑ digoxin
∑ epinephrine
∑ gentamycin
∑ lidocaine
∑ methicilline
∑ oxacillin
∑ sulphisoxazole
∑ theophylline
∑ warfarin.
Importantly, for an IV injection of a drug showing glaringly two-compartment models, one would
invariably come across two distinct phases of defi nite alteration in the ‘plasma-drug concentration’ with
regard to ‘time’. Thus, we may have in the respective
Central Compartment: Firstly, a marked reduction in the drug concentration shall be rather fast and
rapid (i.e., the distribution phase); and secondly, it would be sluggish (i.e., the
elimination phase or post-distributive phase).
Peripheral Compartment: Firstly, there would be a sharp increase in the drug concentration which
will be followed by a peak (i.e., the distributive phase); and secondly, it will
distinctly exhibit an apparent decline in the ensuing drug concentration
(i.e., the elimination phase or post-distributive phase).
Obviously, the distribution of a drug substance usually taking place between the central compartment
and the peripheral compartment is being governed by the fi rst-order kinetics.
Figure 3.5 depicts the plot between the log plasma drug concentration of a drug substance versus the time
profi le after the IV administration which clearly exhibits a two-compartment model. Hence, it will give rise
to two phases of change occurring in the plasma drug concentration with regard to time profi le.
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Figure 3.5 Plasma drug concentration vs. time profi le after the IV administration of a drug.
Pharmacokinetics
143
Nevertheless, the observed distribution of drug taking place between the central and peripheral com-
partments invariably is guided by the fi rst-order kinetics. Therefore, K
12
and K
21
(as seen in Figure 3.4)
do represent the fi rst-order rate constants.
For Central Compartment—The observed change in the drug concentration may be expressed as:
C
21 p 12 C E C
d
d
C
KC KC KC
t
=⋅−⋅−⋅
(d)
C
C
p
ewher
d d is the change in drug concentration in central compartment,
is the drug concentration in central compartment and
is the drug concentration in peripheral compartment.
Ct
C
C
Remembering that
X = Vd.C
where
X is the amount of drug in the body at time ‘t’ after IV injection,
V
d
is the volume of distribution
21
and
C is the amount of drug in the body.
Therefore, one may now express Equation (d) as follows
21 p
C12CEC
pCC
d
d
X
KX
CKXK
tV V V
⋅
⋅⋅
=−−
(e)
where
X
C
is the amount of ‘drug’ present in the central compartment,
X
p
is the amount of ‘drug’ present in the peripheral compartment,
V
C
is the volume of distribution of the central compartment and
V
p
is the volume of distribution of the peripheral compartment.
For Peripheral Compartment—The observed rate of change of the drug concentration may be duly
expressed as follows:
p
12 C 21 p
d
d
C
C
KC K
t
=⋅−⋅
(f )
However, Equation (f ) can also be written in terms of the actual amount of drug present:
p21p
12 C
Cp
d
=
d
XCK
KX
tV V
⋅
⋅
−
(g)
21
Volume of Distribution: It is referred to as an apparent biopharmaceutical terms expressing the perceived vol-
ume (V
d
) of the body in which a specifi c amount of the unchanged drug (A
b
) is duly distributed based upon the
ensuing concentration (C
b
) of the drug in the blood. Thus, V
d
= A
b
/C
b
, a characteristic value for a specifi c drug,
i.e., an ‘abstract volume’ which is calculated from the ratio of the amount of drug in the body to its concentration
in plasma once partitioning has been stabilized adequately (at steady-state conditions).
Essentials of Biopharmaceutics and Pharmacokinetics
144
Therefore, one may virtually estimate the ensuing drug concentration present in both central as well as
peripheral compartments at any point in time by integrating Equations (e) and (g):
..
0
21 21
C
C
tt
K
X
K
Cee
V
−α −β
⎡⎤
⎧⎫⎧⎫
−α −β
=⋅+⋅
⎨⎬⎨⎬
⎢⎥
β−α α−β
⎩⎭⎩⎭
⎣⎦
(h)
..
0
12 12
p
p
tt
e
X
KK
Ce
V
−α −β
⎡⎤
⎧⎫ ⎧⎫
=⋅+⋅
⎨⎬ ⎨⎬
⎢⎥
β−α α−β
⎩⎭ ⎩⎭
⎣⎦
(i)
where
C
C
is the drug concentration in the central compartment,
C
p
is the drug concentration in the peripheral compartment,
X
0
is the dose of the drug in IV administration,
α is the fi rst-order kinetic constant for rapid distributive phase and
β is the fi rst-order kinetic constant for slow elimination phase.
Note
It is worthwhile to state here that both α and β depend slowly upon K
12
, K
21
and K
E
.
It has been duly established that when a drug is administered via the IV route, it explicitly depicts a
plasma-drug concentration vs. time curve of the drug that essentially follows a two-compartment sys-
tem. Thus, it may be duly expressed by means of the ‘biexponential equation’ as given below:
tt
BCAe e
−α⋅ −β⋅
+=⋅ ⋅
(j)
where A and B represent the corresponding zero time intercepts.
It has been duly ascertained that Equation ( j) may be eventually resolved into its respective compo-
nents by adopting the ‘Method of Residuals’.
22
Figure 3.6 represents vividly the plot between the two components, viz. plasma-drug concentration
profi le vs. time after the IV administration of a drug.
22
Method of Residuals: It refers to the determination of absorption and elimination rate constants having fairly
comparable values.
It may be observed in Figure 3.6 that the initial decline duly caused by virtue of the ‘distribution
of drug’ appears to be much more rapid in comparison to the later phase i.e., elimination phase (or
post-distributive phase).
Explanation
In other words, α, i.e., the fi rst-order kinetic constant for the rapid distributive phase, happens to be
much larger than β, i.e., the corresponding fi rst-order constant for the corresponding slow elimination
phase. Therefore, one may critically take cognizance of the fact that
t
Ae
−α⋅
⋅
(i.e., the zero time intercepts
Pharmacokinetics
145
of the fi rst-order constant for rapid distribution phase) shall approach ‘zero’ rather more rapidly vis-à-vis
t
Be
−β⋅
⋅
(i.e., the zero time intercepts of the fi rst-order constant for slow elimination phase). If this is the
situation, Equation ( j) may become
t
CBe
−β⋅
=⋅
′
(k)
Thus, taking the logarithm we have
log log
2.303
t
CB
β⋅
=−
′
(l)
Besides, the ensuing semi-log plot of plasma-drug concentration vs. time shall evidently give rise to
a linear terminal phase having the ultimate slope equivalent to
2.303
t
s
β⋅
−
(or = 0.434β), that on being
subjected to extrapolation to ‘zero’ produces an intercept of log B.
Hence, the plasma half-life (t
1/2
) for the elimination phase comes out to be
12
0.693
t =
β
(m)
Further more, the subsequent subtraction of the concentration time values shall yield a series of residual
drug concentration time values (C
γ
) that may be given as:
CCCγ =−
′
(n)
where
C is the amount of ‘drug’ injected in the body and
C′ is the amount of plasma-drug concentration (after the distribution/elimination phases).
Now, based upon the Equations ( j) and (k) we have:
t
CAe
−α⋅
γ
=⋅
(o)
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Figure 3.6 Diagrammatic representation of plasma-drug concentration profi le vs. time after IV
administration of a drug.
Essentials of Biopharmaceutics and Pharmacokinetics
146
Again, taking the logarithm of Equation (o) we have
log log
2.303
t
CA
γ
α⋅
=−
(p)
Therefore, a semi-log plot of the residual-drug concentration (C
γ
) vs. time shall virtually give rise to
a ‘straight line’ having:
∑ slope =
2.303
α
−
and
∑ a zero time intercept upon Y-axis of log A.
At this point in time, it is absolutely necessary to point out that due to the method of residuals in
Equation ( j) all the parameters may be duly calculated with great case and fervour. Perhaps it would
certainly help to determine the various parameters of a two-compartment model, viz. K
12
, K
21
, V
d
.
Semi-logarithmic Plot of Plasma-Drug Concentration vs. Time After Extravascular
Administration of Drug: It has been duly observed that after the extravascular administration of a
‘drug’ particularly in a two-compartment model, one may come across three processes being involved
simultaneously, such as absorption–distribution–elimination. Importantly, the aforesaid fi rst-order
absorption phenomena in an extravascular administration may be expressed as follows:
0
p
tt
ka t
CAe Be C
−
−α⋅ −β⋅
⋅
=⋅ +⋅ −
(q)
where
A and B designate the corresponding zero time intercepts (as described earlier),
0
p
C
is the Y-axis inter-
cept of the absorption residual line, and theoretically equals to A + B.
Nevertheless, the above multi-exponential components may be suitably resolved by the help of the
‘method of residuals’ as already discussed earlier.
Figure 3.7 depicts the semi-logarithmic plot of the plasma-drug concentration vs. time after the
extravascular administration of drug.
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Figure 3.7 Semi-logarithmic plot of the plasma-drug concentration vs. time for a two-compartment model:
After extravascular administration of a drug.
Pharmacokinetics
147
Caternary Model In the caternary model, the various compartments are specifi cally joined to one
another in a row very much similar to the compartments of a train, as shown in Figure 3.8.
It essentially has the following three important features:
∑ Quite contrarily, the mammalian model comprises either one or more compartments around a cen-
tral compartment very much akin to satellites.
∑ The caternary model can be noticeable neither anatomically or physiologically since the different
organs are linked almost directly to the blood compartments.
∑ Because most of the so-called ‘functional organs of the body’ are associated directly to the plasma,
caternary model fi nds its remote usage in a human body.
Merits and Demerits of Compartment Modelling Approach: The various merits and demerits
of the compartment modelling approach have been enumerated in Table 3.5.
3.2.2. Flow Models (or Physiologic–Pharmacokinetic Models)
In general, these models are also known as
∑ blood-fl ow rate-limited models and
∑ perfusion rate-limited models.
As the name suggests, the physiologic pharmacokinetic models are duly based upon the known values of
both physiologic and anatomic data and, therefore, signifi es a rather more realistic and true picture of the
drug disposition in different tissues and organs in a human body.
The various steps essentially comprise the following:
1. The pioneer step being the actual subtraction of the ‘line of elimination’ (see Figure 3.7 the
‘solid line’ with slope Z ) from the curve.
2. It will eventually result into the formation of a two-compartment residual plot.
3. Importantly, the aforesaid two-compartment residual plot bears essentially a critical post
absorptive linear segment (see in Figure 3.7 the slope Y shown by
—⊗⊗
).
4. In case, the residuals are subtracted carefully from this plot, one may obtain a plot having the
slope X as depicted by ‘. . .’ in Figure 3.7.
5. The two observed intercepts, namely ‘A’ and ‘B’, remain the same as described earlier.
6.
0
p
C
designates the Y-axis intercept of the respective absorption residual line, and, therefore,
equals to A + B theoretically.
Explanation
.
.
.
.
Figure 3.8 The caternary model using three compartments.
Essentials of Biopharmaceutics and Pharmacokinetics
148
In fact, the said models do describe kinetically the data with critical consideration of the fact that the
blood fl ow is exclusively responsible in the proper distribution of a drug substance to various segments of
a human body. Besides, the actual uptake of a drug into various organs is estimated ultimately due to the
obvious binding of a drug in these tissues.
Salient Features Following are same of the salient features, such as:
1. The actual number of compartments that are involved in these models solely depends upon the
disposition properties of the individual drug substance.
2. Bones
23
which have absolutely no penetration of drug are totally excluded from such models.
Table 3.5 Merits and demerits of compartment modelling
S. No. Merits of compartment modelling S. No. Demerits of compartment modelling
1 Provides an explicit visual representation of
rate process variants intimately involved in
drug disposition.
2 Determines the precise number of rate
constants that are absolutely necessary to
describe these process vividly.
3 The pharmacokinetic experts are duly
empowered to derive the various ‘differen-
tial equations’ pertaining to each of the rate
phenomena so as to elaborate drug
concentration changes occurring in each
compartment.
4 Extremely important aspect with regard to the
development of the dosage regimens.
5 Perceptively useful in predicting the
relationship between the drug concentration
and time profi le in
• normal physiologic and
• pathologic conditions.
23
Bones: They refer to the hard, rigid form of connective tissue constituting most of the skeleton of vertebrates,
composed chiefl y of calcium salts.
1 There exists no relationship between
• physiologic functions and
• anatomic structure of species.
Hence, there is a dire need to have several
assumptions so as to facilitate the data
interpretation.
2 Exhaustive efforts are needed to develop a
befi tting ‘model’ which would predict and
designate suitably the ADME of a typical
‘drug’.
3 Compartment model is solely based upon
curve fi tting of plasma concentration to
the resulting complex multi-exponential
mathematic equations.
4 Model may invariably exhibit variance very
much within a study population.
5 Applicability only confi ned to a particular
drug under investigation.
6 Drug profi le within the body may fi t into
various compartmental models pertaining
to the route of administration.
7 Complexities may arise when
comparing results using both human
and animal experiments.