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Pharmacokinetics
159
Nevertheless, Equation (aa) elaborates explicitly the duration of time of the drug-concentration in
plasma (C) soon after the oral route of administration. It has been amply demonstrated and proven that for
majority of drug substances that are usually given in readily available dosage forms, one may observe the
ensuing absorption rate constant [K
a
] being appreciably greater than the elimination rate constant [K
E
].
Important Observations: In a situation, when the fi rst-order absorption rate constant (K
a
) turns out
to be appreciably larger in comparison to the corresponding elimination rate constant (K
E
)
29
the second
exponential
a
()
t
K
e
−
in Equation (aa) shall usually tend to approach ‘zero’ rather more swiftly than the fi rst
exponential
E
()
t
K
e
−
.
Furthermore, at larger values of time ‘t’, the Equation (aa) shall get reduced to the following expression:
a0
da E
()
t
K
KFX
Ce
VK K
−
=⋅
−
(ab)
Ultimately, Equation (ab) expatiates the critical elimination phase
30
of the following plasma drug con-
centration–time curve as depicted in Figure 3.13
Equation (ab) may be expressed in the form of common logarithms as follows:
a0
da E
log log
( ) 2.303
t
KFX K
C
VK K
⎡⎤
=−
⎢⎥
−
⎣⎦
(ac)
Note
Figure 3.12 illustrates a typical plasma concentration–time profi le of a drug substance adminis-
tered duly via the extravascular route.
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Figure 3.13 Logarithm of plasma drug concentration–time curve for a single oral dose of a drug substance.
29
That is, a ‘good rule’ suggests that it must be at least fi vefolds larger
aE
(or 5 ).KK=
30
That is, the particular time when the ensuing ‘absorption’ no longer comes into play.
Essentials of Biopharmaceutics and Pharmacokinetics
160
Obviously, the ‘slope’ of the terminal end of the linear segment of the plasma drug concentration–
time graph which is almost equal to K
t
/2.303, whereas the intercept is equivalent to log [K
a
FX
0
/V
d
(K
a
– K
E
)].
Therefore, one may easily obtain the following two values from Figure 3.13 as given below:
∑ elimination rate constant (K
E
) from the slope of the terminal linear segment and
∑ elimination of the plasma half-life (t
1/2
) of the drug substance as given by 0.693/K.
4.3. The Loo–Riegelman and Wagner–Nelson Equations
In fact, both Loo–Riegelman and Wagner–Nelson equations do help in calculating the actual percentage
of a drug substance, i.e., amount remaining to be absorbed (ARA) at any time. Importantly, the result-
ing percentage is exclusively based upon the critical absorbable fraction of the dose particularly in the
instance of an incomplete absorption (Notari et al., 1972).
31
Nevertheless, a suitable plot (which could be either zero-order or fi rst-order) duly obtained from
these data allows the logical calculation of the respective absorption rate constants, K
a
or K
0
.
4.3.1. Loo–Riegelman Equation
The usual effective application of the Loo–Riegelman equation to a drug-substance is described explic-
itly by these two distinct aspects, namely:
∑ two-compartment open model and
∑ absorption via a fi rst-order rate process.
Vaughan and Dennis (1980)
32
and Wagner (1983)
33
have demonstrated adequately that the Loo–
Riegelman equation, though based upon the model of Equation (ac) below as given in Figure 3.4 yet is
reasonably applicable even the model of Equation (ii) or Equation (iii) is in effect.
Based on the above facts, the Loo–Riegelman equation is found to be model independent as long as
the ‘input’ is confi ned to Compartment 1 exclusively.
(i)
(ii)
(iii)
Figure 3.13 clearly depicts a plot of the ‘log’ of plasma drug concentration vs. time (t) which gives
rise to a typical ‘biexponential curve’ at the very initial stage; and eventually tends to be ‘linear’ in
the course of the ensuing elimination phase (as duly described by Equation (ac) above).
Explanation
31
Notari RE et al.: J. Pharm. Sci., 61: 135, 1972.
32
Vaughan DP and Dennis MJ: J. Pharmacokinet. Biopharm., 8: 83, 1980.
33
Wagner JG: Fundamentals of Clinical Pharmacokinetics, 2nd edn., Hamilton, IL: Drug Intelligence, 1983, p. 196.
Pharmacokinetics
161
We may use the following equation:
b
10
C
AUC(0 )
nn
n
tnt
t
A
CK tT
V
⎛⎞
=+ −+
⎜⎟
⎝⎠
(ad)
where
bC
AV
is the total amount of drug absorbed,
V
C
is the volume of the central compartment,
C and T are the concentrations of drug in plasma and tissue and
n
t
T
is the time of absorption.
A proper explanation of Equation (ad) and its application is obtained through a typical example.
A ‘drug substance’ exhibits a bi exponential disposition soon after an IV injection. The
calculated values for the two-compartment model rate constants are found to be K
12
= 0.30;
K
21
= 0.40 and K
10
= 0.25 (per hr). The same drug was given orally, and the data thus obtained
are provided in Table 3.6. The aforesaid information shall be utilized to calculate the two
unknown, quantities given in Equation (ad), namely; AUC(0 – t
n
); and
b
C
A
V
⎛⎞
⎝⎠
34
.
Table 3.6 Observed plasma levels of a drug substance soon after the oral administration of 490 mg
S. No. Time (hr) Concentration (mg%) S. No. Time (hr) Concentration (mg%)
1 0.5 3.2 7 4.0 4.8
2 1.0 4.8 8 5.0 4.1
3 1.5 5.5 9 7.0 3.1
4 2.0 5.7 10 9.0 2.2
5 2.5 5.7 11 11.0 1.8
6 3.0 5.4 12 13.0 1.4
Solution The results are duly calculated in the following three steps in a sequential manner and
subsequently recorded in Table 3.7.
Example
Step 1: It essentially involves the calculation of the tissue concentrations as a function of time. The
following equation has been employed:
()
21Dt 21Dt
12
12
11
21
1
1DCDt
2
KK
tn tn tn
K
TTe C e K
K
−−
−−
=+− +
(ae)
34
Adapted from: Notari RE: Biopharmaceutics and Clinical Pharmacokinetics, 4th edn., New York: Marcel Dekker Inc., 1987.
Essentials of Biopharmaceutics and Pharmacokinetics
162
Table 3.7 Answers duly recorded from stepwise calculations for the Loo–Riegelman equation
Step 1 Step 2 Step 3
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
t
n
Δt
C
t
n
ΔC
T
t
n
Area t
n
– 1
to t
n
Area t
0
to t
n
K
10
×
col. 7
A
b
/V
C
= cols.
3 + 5 + 8
Percent
A
b
/V
C
100% –
col. 10
0.5 0.5 3.2 3.2 0.240 0.80 0.80 0.20 3.64 26.8 73.2
1.0 0.5 4.8 1.6 0.752 2.00 2.80 0.70 6.25 46.0 54.0
1.5 0.5 5.5 0.7 1.32 2.58 5.38 1.35 8.17 60.1 39.9
2.0 0.5 5.7 0.2 1.84 2.80 8.18 2.04 9.58 70.4 29.6
2.5 0.5 5.7 0.0 2.28 2.85 11.03 2.76 10.7 78.7 21.3
3.0 0.5 5.4 –0.3 2.62 2.78 13.81 3.45 11.5 84.6 15.4
4.0 1.0 4.8 –0.6 3.00 5.10 18.91 4.73 12.5 91.9 8.1
5.0 1.0 4.1 –0.7 3.09 4.45 23.36 5.84 13.0 95.6 4.4
7.0 2.0 3.1 –1.0 2.78 7.20 30.56 7.65 13.5 99.3 0.7
9.0 2.0 2.2 –0.9 2.26 5.30 35.86 8.96 13.4 98.5 1.5
11.0 2.0 1.8 –0.4 1.80 4.00 39.86 9.96 13.6 100.0 0.0
13.0 2.0 1.4 –0.4 1.43 3.20 43.06 10.8 13.6 100.0 0.0
35
Because it relates to the time of the previous data point.
Now, Equation (ae) shall be solved appropriately for each data point. In the above example, let us con-
sider the fi rst-set of points, i.e., 0.5 hr at concentration 3.2 mg%. Hence, we may have these data:
1
0.5; 3.5 and 0
n
tC t
−
Δ= Δ = =
35
In this manner, the concentrations of a drug in plasma and tissue at time (t
n
– 1
) obtained are also ‘zero’,
as there is absolutely ‘no drug’ present in the body at time ‘zero’. Thus, the fi rst entry in Table 3.7 under
the Step 1 is carefully calculated from the following expression:
T
0.5
= 0 + 0 + ½(0.3) (3.2) (0.5)
(af )
= 0.24
and likewise, the second entry from the following equation:
()
()
0.20 0.2
1.0
3.2 0.3 1
0.24 1 0.3 1.6 0.5
0.4 2
Te e
−−
×
=+− +⎡ ⎤
⎣⎦
= 0.751 (ag)
and so on so forth. All the entries of
n
t
T
is recorded in Table 3.7.
Step 2: This particular step takes care of the calculation of elimination as a function of time (t
n
). So
far, we have duly calculated the values for
n
t
T
. As we have proceed the data for
n
t
C
(i.e., concentration of
drug in plasma at time t
n
), there remains only one segment of Equation (ad) that need to be calculated,
Pharmacokinetics
163
which being [AUC(0 – t
n
)] designating the ‘area under the plasma time–concentration curve’ extending
from time ‘0’ to time ‘t
n
’.
Wagner (1983)
36
suggested emphatically that the much needed precision and accuracy may be
enhanced by incorporating the interpolated points.
37
Furthermore, by making use of the logarithmic
trapezoidal rule
38
during the terminal log linear segment,
39
and the critical extension of the aforesaid trap-
ezoidal rule right up the peak level.
Needless to add that in this ‘specifi c example’, it may be carried out very easily and conveniently by employ-
ing the ‘trapezoid rule’. Hence, one should be able to draw the ‘curve’ for C vs. t, and subsequently the respec-
tive ‘individual areas’ duly calculated for each trapezoid (or triangle) as elaborated by the various data points.
The answers thus obtained are provided in Table 3.7, column (6), which represent the areas of the dif-
ferent ‘trapezoids’. Therefore, it is absolutely necessary to sum up each and every area up to and including
t
n
so as to get the ‘total value’ of the AUC(0 – t
n
) in Equation (ad), as illustrated duly in Table 3.7, column
(7). At the end, each of these values is multiplied by the so-called ‘elimination constant K
10
= 0.25’ in
order to obtain the values in the respective column (8) of Table 3.7.
Step 3: Calculation of A
b
/V
C
in Equation (ad): The three major component segments of Equation
(ad) are now supposed to be calculated individually, e.g., columns 3, 4 and 8 in Table 3.7, which are ulti-
mately summed up to get at the respective values contained in column 9. Now, a critical examination
of the entries in column 9, as a function of time, shall reveal that A
b
/V
C
, seems to approach a ‘maximum
value’ of approximately 13.6. Consequently, the values of A
b
/V
C
thus accomplished are duly converted
to a percentage of the aforesaid ‘maximum value’, as per the following expression:
bb
CC
100
%
13.6
AA
VV
=
(ah)
and the accrued results are duly posted in column 10 of Table 3.7.
The percentage of ‘drug unabsorbed’, as a function of time, is represented in column 11, and this is cal-
culated by subtracting column 10 from 100%. Therefore, the fi rst-order plot of the percent unabsorbed
vs. time yields a value of 0.60 hr
−1
for K
a
.
Curve-Fitting Technique: The well-known method for the determination of K
a
, i.e., fi rst-order
absorption rate constant, designates a curve-fi tting technique. Importantly, this method is found to be most
suitable for such drug substances which essentially possess the following characteristic features:
∑ get absorbed rapidly,
∑ undergo complete absorption and
∑ rigidly follow the one-compartment kinetics even when administered via the IV route.
Nevertheless, in such instances when the particular drug substance undergoes the following three modes
of absorption, namely:
∑ gastrointestinal (GI) motility retardation,
∑ enzymatic degradation and
∑ exhibits typical ‘multi-component characteristic features’ second IV administration,
40
36
Wagner JG: J. Pharm. Sci., 72: 838, 1983.
37
That is, derived data from that which precedes and follows it.
38
Trapezoidal Rule: It is a method for estimating integrals (areas under absorption curves) by adding the
respective areas of discrete trapezoids.
39
Chiou Wl: J. Pharmacokinet. Biopharm., 6: 539, 1978.
40
That is found to be absolutely true for almost all drug substances.
Essentials of Biopharmaceutics and Pharmacokinetics
164
eventually one may arrive at the K
a
value (duly computed by the curve-fi tting technique), which proves to
be ‘incorrect’ even though the drugs were actually absorbed by the fi rst-order kinetics.
Obviously, under such circumstances, the value of K
a
so obtained probably at its best may provide a
reasonable estimate with regard to the fi rst-order disappearance of the specifi c drug substance from the
very GI tract instead of the fi rst-order appearance in the normal systemic circulation.
4.3.2. Wagner–Nelson Equation
The Wagner–Nelson equation serves as not only a better estimation of K
a
, but also a qualifi ed alternative
method to the ‘curve-fi tting technique’. Interestingly, the Wagner–Nelson method predominantly involves
∑ critical estimation of K
a
based upon the percent (%) unabsorbed–time plots and
∑ independent of the assumption of both zero-order and fi rst-order absorptions.
Oral Administration of a Single Dose of a Drug Substance: It may be observed that soon after the
oral administration of a single dose of a drug at any material time, the actual quantum of drug being absorbed
right into the ensuing systemic circulation (X
0
), is given by the sum of quantum of drug present in the body
(A), and the quantum of drug eliminated from the body (A
un
) (i.e., amount unabsorbed).
Thus, we may have the following expression:
X
0
= A + A
un
(ai)
Therefore, the quantum of the drug absorbed (A) at any time (t) is equivalent to the sum of the quantum
of the drug present in the body (X), and also the precise quantum of the drug duly eliminated from the
body at any material time (X
E
). We have the following expression:
A = X + X
E
(aj)
Derivatization of Equation (aj) gives
E
d
dd
ddd
X
AX
tt t
=+
(ak)
As we know that X = V
d
× C, hence we may have the following version of Equation (ak):
d
dd
dd
XC
V
tt
=⋅
(al)
Besides,
E
d
,
d
X
X
K
t
=
but X = V
d
·C
Therefore, we have
E
d
d
d
C
X
KV
t
=⋅⋅
(am)
Putting the values of Equations (al) and (am) in Equation (ak) we have:
dd
dd
dd
AC
VKVC
tt
=⋅ +
(an)
Pharmacokinetics
165
or
dd
dd dAV CKVCt=⋅ +
(ao)
Now, integrating Equation (ao) between the limits t = 0 to t = t gives the following expression:
dd
00 0
dd d
tt t
AV CKV C t=+ ⋅
∫∫ ∫
or
()
0d10 d
0
d
t
t
AAVCC KVCt−= − + ⋅
∫
(ap)
Evidently, A
0
is equal to the quantum of a drug absorbed at t = 0 is also equivalent to ‘0’; besides, the
actual ensuing concentration of the drug in the human body at t = 0 also becomes equal to ‘0’. Hence,
Equation (ap) gets simplifi ed to the following expression:
d1 d
0
d
t
t
AVC KVCt=⋅+ ⋅
∫
(aq)
Equation (aq) may be suitably rearranged to
d
d
0
d
t
t
t
A
CKVCt
V
=+ ⋅
∫
(ar)
where
A
t
/V
d
is the amount of drug absorbed at time ‘t’ divided by the ‘volume of distribution’,
C
t
is the plasma (serum/blood) concentration at time ‘t’ and
0
d
t
Ct⋅
∫
is the area under the plasma concentration vs. time curve up to time ‘t’.
Again, integrating Equation (ao) but now between the limits t = 0 to t = α, and subsequently rearranging
the respective equation, we may obtain the following expression:
dd
00 0
dd dAV C KVCt
αα α
=+⋅
∫∫ ∫
or
()
d0d
0
dAVCC KVCt
α
αα
=−+ ⋅
∫
Now, we know that C
0
= 0 and C
α
= 0, hence the above expression may be represented as follows:
d
0
dAKVCt
α
α
=⋅
∫
(as)
Essentials of Biopharmaceutics and Pharmacokinetics
166
Equation (as) may be suitably rearranged as below:
d
0
d
A
KC t
V
α
α
=⋅
∫
(at)
where
A
α
/V
d
is the total quantum of drug absorbed from the dosage form up to an infi nite time (t = α)
divided by the ‘volume of distribution’, and
0
dCt
α
⋅
∫
is the area under the total plasma (serum/blood) concentration vs. time curve.
Fraction of the drug substance being critically absorbed at ‘any time’ is obtained realistically by dividing
Equation (ar) with Equation (at):
0
d
d
0
d
d
t
t
t
A
CKCt
V
A
KC t
V
α
α
⎛⎞
+⋅
⎜⎟
⎝⎠
=
⋅
∫
∫
(au)
or
0
0
d
d
t
t
t
t
CKCt
A
A
KC t
α
⎛⎞
+⋅
⎜⎟
⎝⎠
=
⋅
∫
∫
(av)
However, the ‘unabsorbed fraction’ at any time ‘t’ is usually given by the following expression:
0
0
d
11
d
t
t
t
CCK t
A
A
KC t
α
α
⎡⎤
⎛⎞
+⋅
⎢⎥
⎜⎟
⎡⎤
⎝⎠
⎢⎥
−=−
⎢⎥
⎢⎥
⎣⎦
⎢⎥
⋅
⎢⎥
⎣⎦
∫
∫
(aw)
or
0
0
d
100 1 100 1
d
t
t
t
CKCt
A
A
KC t
α
α
⎡⎤
⎛⎞
+⋅
⎢⎥
⎜⎟
⎡⎤
⎝⎠
⎢⎥
−= −
⎢⎥
⎢⎥
⎣⎦
⎢⎥
⋅
⎢⎥
⎣⎦
∫
∫
(ax)
Based on Equation (ax) one may easily calculate the actual percentage of the unabsorbed drug substance at
any time ‘t’. In this way, we may usually come across the following two commonly encountered situations:
Pharmacokinetics
167
1. Percent-Unabsorbed Drug vs. Time: It gives a straight-line thereby showing that the absorp-
tion phenomenon follows a zero-order kinetics (see Figure 3.14).
2. Logarithm of Percent-Unabsorbed Drug vs. Time: It also yields a straight line which shows
that the absorption phenomenon is said to follow fi rst-order kinetics (see Figure 3.15).
Remarks: In short, it may be added that the Wagner–Nelson equation particularly helps one to have a clear
vision, concept and understanding with respect to the critical ‘absorption kinetics’ without making any
sort of assumptions whatsoever. Importantly, the methodology appears to be very much useful and effi ca-
cious for enabling an in-depth investigative study for deciphering the actual in-built mechanism for the
specifi c release profi le of drug substances from the dosage forms (or drug products) in vivo. Besides, the
inherent absorption phenomenon itself invariably accounts for the fi rst-order kinetics, whereas, the sub-
sequent on-going dissolution of capsules, tablets, more emphatically the meticulously designed sustained
release drug products, should usually be elaborated by relatively more complex kinetic mechanisms.
The Wagner–Nelson method essentially needs the collection of ‘blood samples’ soon after a single-oral
dose at regular intervals of time till such time the entire quantum of the drug substance gets duly eliminated
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.
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Figure 3.14 Plot between percent of unabsorbed drug vs. time for a single oral dose of a drug exhibiting
a zero-order absorption phenomenon.
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Figure 3.15 Plot between percent of unabsorbed drug (ARA) vs. time ‘t’ showing the fi rst-order absorption
process (i.e., according to Wagner–Nelson method).
Essentials of Biopharmaceutics and Pharmacokinetics
168
from the body. Thus, K
E
is usually derived from ‘log C’ vs. ‘ t’ plot, and a semi-log plot of percent unabsorbed
(viz. percent ARA) vs. ‘t’ gives a straight line whose slope is –K
a
/2.303 (see Figure 3.15). In case, a
regular plot of the same happens to be a straight line, one may conclude that the absorption process is
of the zero-order kinetics.
The major disadvantage of the Wagner–Nelson method is its being applicable exclusively to
such drug substances having one-compartment characteristic features. The real problem gets trig-
gered in a particular situation when a drug which obeys the one-compartment model after an
extravascular administration typically exhibits the characteristic features of a multi-component
model after an IV administration.
4.3.3. Estimation of C
max
and t
max
The maximum plasma concentration of a drug substance, C
max
, as observed after the administration of a
single-oral dose of the same usually takes place at the plateau of the ensuing plasma concentration time
curve. Thus, the actual time essentially required to attain the maximum concentration is termed as the
peak time, t
max
. Furthermore, the time actually required to approach C
max
is found to be quite independ-
ent of the ‘dose of the drug’. Instead, it is solely dependent upon the rate constants for the absorption (K
a
)
and also the elimination (K
E
).
However, it has been observed that at the maximum level of concentration of the drug, invariably
known also as the ‘peak concentration’, the rate of drug absorption appears to be almost equal to the rate
of the drug elimination, which ultimately renders the ensuing rate of concentration change profi le, dc/
dt, to become equivalent to ‘zero’.
Now, the rate of concentration change observed may be accomplished by differentiating the follow-
ing equation:
()
a
a0
ddaE
nt
KK
KFX
X
Cee
VVKK
−−
⎡⎤
== −
⎣⎦
−
(ay)
and the above equation on being differentiated with respect to time gives the following expression:
() ()
Ea
2
a0 a 0
da E da E
d
d
tt
KK
KK FX K FX
c
ee
tVKK VKK
−−
=− +
−−
(az)
Since, the plasma drug concentration reaches C
max
at time t
max
, then ultimately dc/dt becomes ‘zero’ and,
therefore, we may have
() ()
max a max
2
Ea 0 a 0
da E da E
Kt K t
KKFX K FX
ee
VK K VK K
−−
=
−−
and the above expression may be easily simplifi ed to
max a max
Ea
Kt K t
Ke Ke
−−
=
(ba)
or
max
amax
a
E
Kt
Kt
K
e
K
e
−
−
=