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Pharmacokinetics
179
3. Water-Loading: The water-loading must be carried out by taking 400 mL of water soon after the
overnight fasting so as to
∑ cause diuresis and
∑ enable suffi cient collection of required ‘urine samples’.
4. The underlying procedure needs to be adopted rigidly:
(a) Before the actual administration of a drug, the urinary bladder should be emptied completely
as far as possible, i.e., up to a duration of 60 min from water-loading.
(b) The ‘urine sample’ obtained in (a) above is to be taken as the ‘blank sample’.
(c) ‘Drug’ is now administered with 200 mL of water.
(d) Ingestion of water may be followed by 200 mL at every hour on the hour, for the next 4 hr.
5. Healthy volunteers should be adequately instructed to empty their urinary bladders as completely
as possible while collecting the respective ‘urine samples’.
6. More ‘frequent sampling of urine’ must be carried out so as to accomplish a reasonably ‘good curve’.
7. Precise time (duration) and volume (mL) of ‘urine sample’ excreted must be noted carefully in the
course of the sampling procedure.
8. Care should be taken that an ‘individual collection period profi le’ must not exceed one biological
half-life (t
1/2
) of the ‘drug substance’ under investigation. It should be ideally much less.
9. Usually the urine samples should always be collected at least up to seven biological half-lives (t
1/2
)
so as to ensure complete collection (> 99.5%) with respect to the excreted drug.
10. Ultimately, any change incurred in
∑ pH of the urine and
∑ volume of urine collected
may alter directly or indirectly the ‘urinary excretion rate’.
Table 3.10 records the ‘urinary excretion data’ immediately followed by an IV bolus of 100 mg of a
drug substance that essentially consist of two separate criteria, viz. observations and treatment of data.
Thus, the various aspects of observations include
∑ number of samples collected,
∑ time (hr) of urine collection (t),
∑ volume of urine sample collected (mL) and
∑ concentration (mcg mL
−1
) of unchanged drug in urine sample.
Likewise, the various components of the treatment of data include
∑ urine collection interval (dt or
tΔ
),
∑ mid-point of urine collection (t
*
),
∑ quantum of drug excreted in time interval ΔX
u
,
∑ excretion rate (mg hr
−1
) (or dX
u
/dt),
∑ cumulative quantum of drug excreted (mg)
u
t
X
and
∑ quantum of drug remains to be excreted
()
uu
t
XX
α
−
.
Note
The various data duly calculated and recorded in Table 3.10 are processed in a systematic manner
to arrive at an array of further valuable information.
Estimation of Elimination Rate Constant (K
E
) from Urinary Excretion Data In fact, the fi rst-
order elimination (and/or excretion) rate constant (K
E
) may be easily computed from the urinary excre-
tion data by either of the following two methodologies, namely:
Essentials of Biopharmaceutics and Pharmacokinetics
180
∑ excretion rate method and
∑ sigma minus method,
which shall now be discussed briefl y in the following sections.
Excretion Rate Method It refers to the actual rate of urinary drug excretion, dx/dt, which is propor-
tional to the quantum of drug ‘X’ present in the human body, and may be expressed as:
u
e
d
d
X
KX
t
=
(bo)
where K
e
is the fi rst-order urinary excretion rate constant.
Hence, according to the fi rst-order pharmacokinetics, i.e.,
E
0
t
K
XXe
−
=⋅
(bp)
where K
E
is the elimination rate constant.
Substituting Equation (bp) in Equation (bo), we may have
E
u
e0
d
d
t
K
X
KXe
t
⋅
−
=⋅⋅
(bq)
where X
0
is the dose of drug administered (via IV bolus).
Transforming Equation (bq) into the corresponding ‘log’ form, we have:
uE
e0
d
log log
d 2.303
t
XK
KX
t
=⋅−
(br)
Table 3.10 Urinary excretion date immediately followed by IV bolus of 100 mg of a ‘drug substance’
Observations Treatment of data
No. of
samples
Time
of urine
collec-
tion t
(hr)
Volume
of urine
sample
collected
(mL)
Concen-
tration of
unchanged
drug in
urine
(mcg
mL
−1
)
Urine
collection
interval
dt or Δt
Midpoint
of urine
collection
t*
Quan-
tum of
drug
excreted
in time
interval
(ΔX
u
)
Excre-
tion rate
(mg hr
−1
)
dX
u
/dt)
Cumu-
lative
quantum
of drug
excreted
(mg)
(X
u
t
)
Quantum of
drug remains
to be excreted
t
uu
XX
a
⎡⎤
−
⎣⎦
00–– ––––067.5
1 0–2 150 250 2 1 35 17.5 35 32.5
2 2–4 160 100 2 3 15 7.5 50 17.5
3 4–6 100 80 2 5 7.4 3.7 58.7 8.8
46–821020274261.2 6.3
5 8–12 320 10 4 10 3.2 0.8 64.3 3.2
6 12–14 610 04 12 18 2.3 0.2 67.5
↓
X
u
α
–
Pharmacokinetics
181
Obviously, the Equation (br) clearly states that the respective semi-log plot of the rate of excretion vs.
time plot shall give rise to a straight line having a slope –K
E
/2.303 as depicted clearly in Figure 3.17.
ORJ.
H
;
ORJ
6ORSH ±
7LPHW
.
(
>0LGSRLQW7LPHRI8ULQH&ROOHFWLRQ@
G;
X
GW
Figure 3.17 Semi-log plot of rate of excretion vs. midpoint time of urine collection duration (t*) for computing
the elimination rate constant (K
E
) after IV bolus administration of a drug.
Various aspects of Figure 3.17 are expatiated as follows:
1. It should be considered that the ‘slope’ duly obtained from a semi-log plot of an excretion
rate vs. time plot is invariably related to
∑ elimination rate constant K
E
and
∑ certainly not to excretion rate constant K
.
2. However, one may duly obtain the excretion rate constant (K
E
) from the intercept on the
Y-axis given by log K
e
X
0
.
3. Thus, both K
E
and K
e
values can be judiciously employed to compute the following two
determinations effectively:
∑ elimination half-life and
∑ non-renal elimination rate constant.
Explanation
Advantages of Excretion Rate Method: It has the following advantages, namely:
1. Drug substances with long inherent biological half-lives (t
1/2
), the urine samples should be col-
lected up to 3–4 half-lives only.
2. There is absolutely no necessity to collect ‘all urine samples’ because the subsequent collection of
‘any two consecutive urine samples’ usually gives rise to such valid points upon the excretion rate plot
from which one may easily construct a straight line (see Figure 3.17).
Sigma-Minus Method It is possible to estimate the exact ‘cumulative quantum’ of the unchanged drug
substance duly excreted in the urine sample up to anytime by integrating the following equation:
()
()
a
uea0
a
d
d
Kt
Kt
X KKFX
ee
tKK
−
−
=−
−
(bs)
Essentials of Biopharmaceutics and Pharmacokinetics
182
where
K
a
is the fi rst-order rate constant for drug absorption,
F is the fraction of the drug absorbed and
K is the fi rst-order rate constant for drug elimination.
Integrating Equation (bs) between the limits of ‘t’ ranging between ‘0’ and ‘t’, we have the following
expression:
()
a
ea 0
u
a
00
dd
tt
Kt
Kt
KKFX
Xeet
KK
−⋅
−
⎡⎤
=−
⎢⎥
−
⎣⎦
∫∫
or
()
a
0
Ea 0
uu
aaa
11
Kt
Kt
t
KKFX
ee
XX
KK KK K K
−⋅
−
⎡⎤
⎛⎞
−
−= +− +
⎢⎥
⎜⎟
−− −
⎝⎠
⎣⎦
(bt)
where X
u
0
is the cumulative quantum of drug substance duly excreted in urine at t = 0 equals ‘zero’.
Therefore, by simplifying Equation (bt), we may have
()
a
E0 Ea0
u
aa
Kt
Kt
t
KFX KKFX
ee
X
KKKKK
−⋅
−
⎛⎞
−
=+ +
⎜⎟
−
⎝⎠
(bu)
Plot of X
u
t
vs. Time (t): A carefully drawn plot of X
u
t
vs. t shall normally provide the so-called urinary
excretion curve as illustrated in Figure 3.18.
Thus, at a specifi c situation when the ‘total drug’ has ultimately been duly excreted from the body,
i.e., at t = α, Equation (bu) gets reduced to
e0
u
KFX
X
K
α
=
(bv)
where X
u
α
is the total amount of unchanged drug duly excreted in urine at an infi nite time (i.e., t
α
).
7LPH+UV
&XPXODWLYH$PRXQWRI
'UXJ([FUHWHGLQ8ULQH>;
X
W
@
Figure 3.18 A plot of cumulative amount of drug substance excreted in urine vs. time.
Pharmacokinetics
183
Therefore, one may quite judiciously and skilfully develop an ‘appropriate equation’ which critically
describes the amount of the drug to be excreted vs. time based on Equations (bu) and (bv).
Further more, the total amount of an unchanged drug excreted in urine at an infi nite time (i.e., t
α
)
given by X
u
α
minus the amount of drug being excreted in time ‘t’ (i.e., X
u
t
) is represented by the follow-
ing equation:
()
()
a
u
uu a
a
Kt
tKt
X
XX Ke Ke
KK
a
α
−⋅
−
−= ⋅−⋅
−
(bw)
Pharmacokinetic Parameters of a Drug A plot of log X
u
t
vs. t is usually employed to calculate the ensuing
pharmacokinetic parameters of a drug, as described in Figure 3.19.
However, the residual line may be drawn by using the method of residuals, the slope of which is found
to be equivalent to –K
a
/2.303. Hence, in Equation (bv), if the values of F and X
0
are known, one may
calculate ‘K
e
’ quite easily.
Practice Problems Let us look into the following two representative practice problems:
Problem 1 Ceftizoxime was administered to an adult subject at a dose of 2 g via IV route, and was
duly advised to consume two glassfuls of water at the time of administration and again 2 hr later so as to
induce urination. Eventually, the ‘subject’ voided at two different intervals, viz. 2.5 and 7 hr. The urine
sample collected at 2.5 hr contained 1.44 g of ceftizoxime, and the one collected at 7 hr contained 490
mg. The observed serum concentration (mg L
−1
) as a function of time was found to be described by the
respective ‘bi-exponential equation’.
KU KU
tt
C
−−
−−
HH
(i) What would be the actual calculated renal clearance values for each of the two urinary excretion time
intervals?
Solution As we know that CL
R
= A
t
/AUC, where both A
t
and AUC designate specifi cally the stipu-
lated time duration.
Hence, at 2.5 hr: AUC (0 – 2.5 hr) = 141.7 mg L
−1
1
R
1440
CL 10.16 L hr
141.7
−
∴==
7LPH+UV
ORJ;
X
α
± ;
X
W
6ORSH ±
.
D
6ORSH ±
.
Figure 3.19 A plot between log (X
u
α
– X
u
t
) vs. time (hr).
Essentials of Biopharmaceutics and Pharmacokinetics
184
or
1
R
10.16 1,000
CL 169.3 mL m in
60
−
×
==
Likewise, at 7 hr: AUC (2.5 – 7 hr) = 48.5 mg L
−1
1
R
490
CL 10.1 L hr
48.5
−
∴==
or
1
R
10.1 1,000
CL 168.3 mL m in
60
−
×
==
(ii) What will be the corresponding ‘Total Body Clearance’ and ‘f
u
’ values?
Solution It is known that the total body clearance (CL
T
):
11
Total amount of ‘drug’ unchanged
CL
Area under the plasma-level time curve
2,000
10.16 L hr 169 mL m in
196.8
−−
=
== =
i.e.,
1
CL 169 mL m in
−
=
Since,
Ru
CL CL ; hence ‘ ’ 1f=
Problem 2 A ‘drug substance’ is administered via IM route to a healthy subject and the following two
data obtained (Table 3.11):
∑ plasma concentration of the drug substance (mg mL
−1
) and
∑ time (hr).
Calculate the absorption rate constant and order of the absorption process precisely from the above data.
Solution The Wagner–Nelson method (see 4.3.2 under Part III in this section) may be applied for the
assessment of the exact order of the ensuing absorption phenomenon. It also helps in the calculation
of the absorption rate constant.
In this particular instance, the following equation is employed:
d
0
d
t
t
t
K
A
CCt
V
=+ ⋅
∫
First of all a semi-log plot between plasma drug concentration vs. time must be prepared and, therefore, the
‘slope’ of the so-called ‘terminal linear segment of the plotted graph’ is found to be equivalent to
2.303
K
⎡⎤
−
⎢⎥
⎣⎦
:
Pharmacokinetics
185
i.e., Slope = 0.0651
K = 0.0651 × 2.303 = 0.15 hr
−1
The biological half-life (t
1/2
) is calculated as follows:
12
0.693
4.62 hrs
0.15
/
t ==
On the basis of the above two given values, e.g., plasma concentration of the drug substance (mg mL
−1
)
and time (hr), one may calculate the following determinations, such as
ddd
00
d; d; ; and
tt
tt
AAA
CtKCt
VVV
α
⋅⋅ −
∫∫
All these six determinants are arranged together in Table 3.12.
Explanations of 3rd to 6th Columns in Table 3.12: These specifi ed columns may be further expatiated
as below:
3rd Column
⎡⎤
⋅
⎢⎥
⎣⎦
∫
t
Ct
0
d
: It clearly exhibits the area under the curve (C) vs. time curve duly calculated
in a sequential manner right from t = 0 to each steps of the time points by the help of the recognized
trapezoidal rule.
4th Column
⋅
∫
t
KC t
0
d
: It vividly shows that each of the preceding areas multiplied by K.
5th Column [A
t
/ V
d
]: It refers to the sum total of the values given in the 2nd and 4th columns as per
the following expression:
d
0
d
t
t
t
A
CKCt
V
=+ ⋅
∫
Table 3.11 Time vs. plasma drug concentration
Time (hr)
Plasma drug concentration (mg mL
−1
)
0.25 0.65
0.50 1.2
0.75 1.8
1.0 2.4
1.5 3.5
2.0 4.4
3.0 6.0
6.0 5.5
12.0 2.4
18.0 0.9
24.0 0.5
Essentials of Biopharmaceutics and Pharmacokinetics
186
However,
d
A
V
α
is the maximum value in the 5th column (10.00).
6th Column
⎡⎤
−
⎢⎥
⎣⎦
a
t
AA
VV
dd
: It evidently shows the ensuing residual existing between A
α
/V
d
and A
t
/V
d
(i.e., each sequential value in the 5th column.
Now, plot the graphic representation between the following pairs, namely:
∑ values in the 6th column vs. time and
∑ log of values in the 6th column vs. time.
Thus, two different situations may arise at this point in time:
Situation 1: When the absorption phenomenon obeys the fi rst-order kinetics, one may eventually
obtain the ‘semi log plot’ as a straight line having a slope of
a
.
2.303
K
−
Situation 2: When the absorption phenomenon critically follows a zero-order kinetics, the resulting
‘Cartesian plot’ shall invariably give rise to a straight line having the slope equivalent to the respective
zero-order rate constant.
Table 3.12 Values of six determinants present in plasma drug concentration
Time (hr)
Plasma drug
concentration
(mg mL
−1
)
0
d
t
Ct
⎡⎤
⋅
⎢⎥
⎣⎦
∫
0
d
t
KC t⋅
∫
G
t
A
V
dd
t
AA
VV
a
⎡⎤
−
⎢⎥
⎣⎦
0.25 0.65 0.1 0.02 0.67 0.38
0.50 1.2 0.3 0.05 1.25 0.75
0.75 1.8 0.7 0.11 1.91 8.09
1.0 2.4 1.2 0.18 2.58 7.52
1.5 3.5 2.6 0.39 3.89 6.21
2.0 4.4 4.5 0.68 5.08 5.02
3.0 6.0 9.7 1.45 7.45 2.54
.0 5.5 27.1 4.07 9.57 0.33
12.0 2.4 50.8 7.62 10.02 0.08
18.0 0.9 60.4 8.06 9.96 0.04
24.0 0.4 64.4 9.645 10.00
44
–
44
Where,
d
A
V
α
= 10.0, which is the actual amount of drug absorbed from the dosage from up to time (α) divided
by the volume of distribution (V
d
).
Note
Importantly, in this particular instance, one may obtain a straight line having the ‘Cartesian Plot’.
Therefore, the ultimate absorption phenomenon rigidly follows the zero-order kinetics.
Pharmacokinetics
187
Remarks: These essentially comprise the following:
1. The aforesaid example depicts clearly the usefulness of the Wagner–Nelson calculation for examining
critically the exact mechanism of release profi le of drugs from ‘dosage forms’ in vivo.
2. First-order phenomenon is usually observed due to the drug absorption from a solution, whereas,
any deviations from it evidently shows the release pattern of the dosage form.
6.3. Hepatic Clearance
Hepatic clearance is also known as the hepatic elimination of drugs.
Extraction Ratio (E): It refers to the ratio of the rate of drug elimination and the rate at which
the drug gains entry into the organ of elimination. The extraction ratio designates a measure of the effi -
ciency with which an organ of the human body actually eliminates a given drug.
Table 3.13 provides the hepatic and renal extraction ratios of certain important drug substances and
their respective metabolites.
Thus, the organ clearance of a drug may be defi ned as ‘the product of blood fl ow to the organ (Q),
and the respective extraction ratio (E)’.
Hence, the equation for the hepatic clearance (CL
H
) is expressed as:
CL
H
= Q
H
.E (bx)
where Q
H
is the hepatic blood fl ow.
Equation (bx) may suggest that a closer look at this ‘simple model’ meant for hepatic clearance (CL
H
)
would be directly proportional to the respective hepatic blood fl ow (Q
H
). However, this conclusion may not
prove to be absolutely correct since the extraction ratio (E) varies inversely with Q
H
. Based on the above
cited facts and observations, it may be inferred that for both qualitative and quantitative predictions of the
‘hepatic drug clearance’, it must require a defi nitely more complex model for hepatic clearance. In fact, one
may actually look for such a parameter which has got to be quite independent of any physiological factor(s).
The various types of ‘models’ have been duly proposed and tested to provide adequate mechanism for
the hepatic clearance of drug substances, which are as follows.
6.3.1. Venous Equilibrium Model of Hepatic Clearance
It has exhibited a signifi cant advantageous utility in the critical and precise prediction of
∑ pathophysiological and
∑ drug-induced
Table 3.13 Hepatic and renal extraction ratios (E) of certain important drug substances and their respective
metabolites
S. No. Mode of extraction
Calculated extraction ratio ( E)
Low Intermediate High
1 Hepatic
extraction
Diazepam, Phenobarbitone,
Phenytoin, Procaina-
mide, Theophylline
Aspirin, Codeine,
Nortriptyline,
Quinidine
Propranolol, Lidocaine,
Nitroglycerine, Morphine,
Isoprenaline
2 Renal extraction Digoxin, Furosemide,
Atenolol, Tetracycline
Penicillin’s (selected
ones) Procainamide,
Cimetidine
Penicillin’s (selected ones)
Hippuric Acid Sulphated
Drugs (Several drugs)
Glucuronides
Essentials of Biopharmaceutics and Pharmacokinetics
188
alterations in the overall hepatic clearance. Importantly, in the venous equilibrium model, the ultimate
hepatic extraction is usually given by the following expression:
ub u,int
H ub u,int
CL
QCL
f
E
f
=
(by)
where
f
ub
is the unbound fraction in blood and
CL
u, int
is the unbound intrinsic hepatic clearance.
In true sense, the unbound intrinsic hepatic clearance (CL
u, int
) evidently refl ects the inherent ability of
the liver to help in the removal of drug from blood in the absence of several other confounding factors,
viz. Q
H
and f
ub
. As it has already been demonstrated that the hepatic clearance is the product of Q
H
and
E, hence, we may have the following expression:
()
Hub u.int
H
H ub u.int
() CL
CL
QCL
Qf
f
⋅
=
⋅
(bz)
Equation (bz) designating a ‘model for hepatic clearance’ essentially gives rise to a reasonably powerful
tool for predicting such vital alterations in
∑ drug clearance and
∑ steady-state drug concentrations,
in typical conditions when some limiting parameters are dealt with. Following are two specifi c instances,
such as:
(a) when Q
H
Ⰷ f
ub
.CL
u.int
, CL
H
may be duly approximated by f
ub
.CL
u.int
(ca)
(b) when Q
H
Ⰶ f
ub
.CL
u.int
, CL
H
may be approximated by Q
H
. (cb)
Important Points These essentially comprise the following:
1. Drugs with a High f
ub
.CL
u.int
Values: These are believed to show an obvious perfusion rate-
limited elimination, i.e., their respective elimination rate shall be exclusively rate limited by
hepatic blood fl ow.
2. Drugs with a Low f
ub
.CL
u.int
Values: These are found to be absolutely independent of the perfusion rate.
Consequently, these limiting conditional ties imposed on drugs do permit us to put together
several drug substances into such classified categories which predominantly show ‘identical phar-
macokinetic parameters.’
Following are two typical examples:
Agents having an f
ub
.CL
u.int
< 0.2 L. min
−1
: These are usually classifi ed as ‘low intrinsic
clearance drug substances.’
Agents having an f
ub
.CL
u.int
> 5 L min
−1
: These are invariably defi ned as ‘high intrinsic clear-
ance drug substances’.
Examples