Добавил:
Sekretar
kiopkiopkiop18@yandex.ru
t.me/Prokururor I Вовсе не секретарь, но почту проверяю
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5722_Библиотеки_им_академика_М_И_Перельмана.pdf
Pharmacokinetics
199
Therefore, we may have
[]
[]
[]
t
max 3
C
E
EES
VK
=
or
[]
[]
t
max 3
C3
E
E
VK
VK
=
or
[]
[]
t
max
C
E
E
V
V
=
Now, on the basis of the above derivations, Equation (ci) may be written as
[]
[]
m
max
S
S
K
V
V
+
=
or
[]
[]
max
m
S
S
V
VK
=
+
(cl)
or
[]
[]
max
m
S
S
V
V
K
×
=
+
(cm)
Thus, Equation (cm) designates the Michaelis–Menten Equation.
7.2.2. Graphical Representation of Michaelis–Menten Equation
The ‘non-linear pharmacokinetics’ essentially involves as a necessary consequence ‘V’ as the rate of change
of drug concentration
()
dd,Ct−
and ‘S’ as drug concentration (C).
Figure 3.21 illustrates the graphical representation of Michaelis–Menten equation.
Figure 3.21 clearly reveals that types of ‘reaction kinetics’ as follows:
∑ First-Order Kinetics [Marked ‘X’]: It may be observed that at the initial stages, where low dose levels
prevail, the change in drug concentration (–dC/dt) is found to be linear with respect to change in
total concentration.
∑ Mixed-Order Kinetics [Marked ‘Y’]: In this instance one may observe an increase in concentration,
(–dC/dt), that is seen to be neither ‘linear’ nor ‘independent’ vis-à-vis the ensuing alteration in the
total concentration.
∑ Zero-Order Kinetics [Marked ‘Z’]: Ultimately, when the total concentration remains to be very
high (i.e., high-dose levels), (–dC/dt) is found to be absolutely independent of the prevailing total
concentration of the drug substance.
Essentials of Biopharmaceutics and Pharmacokinetics
200
Let us take into consideration the following three typical situations:
Situation 1: When K
m
= C:
Here, Equation (cc) becomes
max
d
=
d2
VC
C
tC
×
−
×
or
max
d
d2
V
C
t
−=
(cn)
In other words, the Michaelis–Menten constant, K
m
, is equivalent to the drug concentration at which
∑ rate of process or
∑ rate of reaction
is almost equal to one-half of the ‘theoretical maximum rate’ of the reaction.
Situation 2: When K
m
> C:
It shows evidently that K
m
is reasonably greater than C. Therefore, for the sole purpose from a practi-
cal angle, we may have the following expression:
mm
KCK+
If that is so, Equation (cc) will be rendered to
max
m
d
d
VC
C
tK
×
−=
(co)
However, Equation (co) is very much akin to the fi rst-order elimination of the drug substance, wherein,
the fi rst-order rate constant K
E
= V
max
/K
m
.
Observations: There are three important observations from the above statement of facts:
∑ Elimination: The critical elimination of an array of drugs invariably follow the fi rst-order reaction
kinetics, which suggests explicitly that ‘the plasma-drug concentration prevailing at the site of reac-
tion remains lower than K
m
’.
=
<
G&GW
;
.
P
9
PD[
9
PD[
&RQFHQWUDWLRQ&
;
)LUVWRUGHU5DWH
>/RZGRVH/HYHOV@
=
=HURRUGHU5DWH
>+LJKGRVH/HYHOV@
<
0L[HGRUGHU5DWH
>,QWHUPHGLDWH'RVH
/HYHOV@
Figure 3.21 A graphical plot showing the Michaelis–Menten equation.
Pharmacokinetics
201
∑ Exceptions: The aforesaid phenomena do have certain glaring exceptions, e.g., ethanol, salicylates
and phenytoin.
∑ Deviations: There are certain prominent deviations duly observed in certain typical case of drug
toxicities, which could eventually result from the ‘abnormally high concentration of drug’ in the body.
Situation 3: When K
m
< C:
In other words, the drug concentration is found to be suffi ciently greater than K
m
. Therefore, for
practical purposes, we may have
mm
KCK+
Thus, Equation (cc) becomes
max
d
d
C
V
t
−=
(cp)
Based on Equation (cp), one may conclude that the ‘rate of reaction’ usually comes into play at a con-
stant rate V
max
, and the former is found to be absolutely independent of the drug concentration. Hence,
it is more or less identical to the ‘zero-order kinetics.’
Ethanol and its respective metabolism process.
Examples
7.3. Drug Elimination by Capacity-Limited Process
Let us look into the typical instance when an IV administered drug in a subject gets adequately elimi-
nated by means of the capacity-limited process.
Equation (cc) is as
max
m
d
d
VC
C
tKC
×
−=
+
Rearranging Equation (ca), we have
()
max m
d
d
C
Vt CK
C
×=− +
(cq)
Integrating Equation (cq), we get
V
max
.t + I = –C – K
m
. InC (cr)
where I is an integration constant.
Now, at time (t) = 0, C = C
0
and hence, we have
I = C
0
– K
m
. InC (cs)
Substituting Equation (cs) in Equation (cr), we get
()
0
0
m
max max
In
CC
C
K
t
VVC
−
⎛⎞
=+
⎜⎟
⎝⎠
(ct)
Essentials of Biopharmaceutics and Pharmacokinetics
202
Changing to the ‘common logarithm’ we may obtain:
()
0
max
0
mm
log log
2.303 2.303
CC
V
CC
KK
−
=+ −
(cu)
However, from Equation (cu) one may get a plot between log plasma drug concentration vs. time (t), which
would give rise to a curve having its terminal linear segment with a slope equivalent to V
max
/2.303K
m
as illustrated in Figure 3.22.
7LPH+UV
ORJ3ODVPD'UXJ&RQFHQWUDWLRQ
ORJ&
′
ORJ&
6WDUWLQJ3RLQWRI
)LUVWRUGHU.LQHWLFV
7HUPLQDO/LQHDU
6HJPHQW
6ORSH ±
9
PD[
 .
P
Figure 3.22 A plot between plasma-drug concentration vs. time (hr): Drug being administered via IV route
with capacity-limited process.
Note
In the drug elimination phenomenon by the capacity-limits process, one may conveniently
calculate the two kinetic parameters, V
max
and K
m
, by various laid down techniques.
7.4. Important Recognized Plot Variants in Non-Linear Pharmacokinetics
There are, in fact, four important plot variants with regard to the non-linear pharmacokinetics, namely:
∑ Lineweaver–Burk plot (or doubled-reciprocal plot),
∑ Hanes–Woolf plot,
∑ Woolf–Augustinsson–Hofstee plot and
∑ Direct linear plot,
which will be treated individually in the following sections.
7.4.1. Lineweaver–Burk Plot (or Doubled-Reciprocal Plot)
The Lineweaver–Burk plot refers to the straight-line plot obtained when the reciprocal of the velocity
of an enzyme reaction is plotted against the reciprocal of the substrate concentration. In other words, it
Pharmacokinetics
203
designates the straight line duly obtained from the reciprocal of the Michaelis–Menten equation and is also
known as the ‘Doubled-Reciprocal Plot’.
In the course of an intensive prolonged research, it becomes almost necessary and important to esti-
mate precisely the rate of change of plasma-drug concentration with time. Therefore, in order to accom-
plish this critical objective, the ‘blood samples’ are normally withdrawn at different pre-determined time
intervals and ultimately, the data is plotted graphically. Thus, one of this most abundantly utilized plot is
termed as the Lineweaver–Burk plot. It is a plot of two entities:
m
11
dd
vs.
Ct C
(where C
m
represents the plasma-drug concentration at the midpoint of the ensuing ‘sampling profi le’).
Figure 3.23 depicts the graphical representation of the Lineweaver–Burk plot.
Let us take the reciprocal of Equation (cc):
m
max m max
11
dd .
K
Ct V C V
=+
(cv)
where C
m
is the plasma-drug concentration at the midpoint of the sampling interval.
Now, a plot of the reciprocal of dC/dt vs. reciprocal of C
m
gives rise to a straight line (Figure 3.23)
having the following two components:
Y-Intercept =
max
1
V
and
Slope =
m
max
K
V
Variants in Lineweaver–Burk Plot: There are three prominent variants in the Lineweaver–Burk
plot which shall be treated separately in the following sections:
1. Identifi cation of Competitive Enzyme Inhibition: Figure 3.24(a) shows a plot between 1/V and
1/(D), wherein the two extrapolated lines from the point of intersection, i.e., each for the inhibitor
and control diverge from each other.
G&
GW
6ORSH
9
PD[
.
P
9
PD[
FP
Figure 3.23 Graphical representation of Lineweaver–Burk plot (or doubled-reciprocal plot).
Essentials of Biopharmaceutics and Pharmacokinetics
204
2. Identifi cation of Non-competitive Enzyme Inhibition: Figure 3.24(b) depicts a plot between
1/V and 1/(D), wherein the extrapolated lines from the points of interception on the Y-axis (i.e.,
1/V) viz. inhibitor and control, diverge from each other.
3. Identifi cation of Uncompetitive Enzyme Inhibition: Figure 3.24(c) illustrates a plot between
1/V and 1/(D), wherein the two extrapolated lines from two different point on the X-axis(i.e., 1/CD)
have two points of intersection on the Y-axis (i.e., 1/V) that are almost parallel to each other.
7.4.2. Hanes–Woolf Plot
It has been observed that many a times in the doubled-reciprocal plot, the plotted ‘points’ appear to be
too close to one another. Hence, to circumvent this sort of problem, the Hanes–Woolf plot is employed.
It is solely based on the following equation:
mmm
max max
dd
CKC
Ct V V
⎡⎤
=+
⎢⎥
⎣⎦
(cw)
9
'
,QKLELWRU
&RQWURO
Figure 3.24(a) Lineweaver–Burk plot to identify competitive enzyme inhibition.
'
9
,QKLELWRU
&RQWURO
Figure 3.24(b) Lineweaver–Burk plot to identify non-competitive enzyme inhibition.
Pharmacokinetics
205
Equation (cw) is duly formed by the rearrangement of Equation (cv). Hence, based on Equation (cw), the
observed plot of
m
d
d
C
C
t
vs. C
m
shall turn out to be a straight line having the following two components:
Slope =
max
1
V
and
Y-Intercept =
m
max
K
V
7.4.3. Woolf–Augustinsson–Hofstee Plot
In a further attempt to overcome the appearance of doubled-reciprocal plot and the closeness of ‘points’,
the Woolf–Augustinsson–Hofstee plot came into being. It is exclusively based on the following equation:
()
m
max
m
dd.
d
d
CtK
C
V
tC
=−
(cx)
Now, based on Equation (cx) the graphical plot of
m
ddd
d
CCt
vs.
tC
will appear to be straight line
having the following two components:
Slope = –K
m
and
Intercept = V
max
7.4.4. Direct Linear Plot
The direct linear plot comes into being when two altogether different dosing rates, viz. DR
1
and DR
2
,
invariably give rise to two corresponding steady states designated as C
SS
1
and C
SS
2
.
Figure 3.25 clearly illustrates a ‘direct linear plot’ designating steady-state concentrations when the
drug substance is duly administered at two different rates.
In Figure 3.25, the points between C
SS
1
and D
R
1
and C
SS
2
and D
R
2
are carefully joined to obtain
straight lines. Now, the point of intersection of these two straight lines is duly extrapolated along the
Y-axis (i.e., the DR-axis) to obtain V
max
and likewise along the X-axis (i.e., the C
SS-axis
) to get K
m
.
'
9
,QKLELWRU
&RQWURO
Figure 3.24(c) Lineweaver–Burk plot to identify uncompetitive enzyme inhibition.
Essentials of Biopharmaceutics and Pharmacokinetics
206
8. Pharmacokinetic Profiles of Some Specific Examples
from Various Classes of Drugs
Perhaps it would be worthwhile to examine the reported pharmacokinetic profi les of at least one rep-
resentative typical example belonging to the following class of drugs with particular reference to the
following aspects:
∑ chemical structure, name, uses;
∑ pharmacokinetic profi les in animal models;
∑ experimental procedure;
∑ analytical protocol;
∑ results and discussion;
∑ summary;
Following are the selected ‘Classes of Drugs’ that have been discussed briefl y by citing the example
of a specifi c example:
∑ Antihypertensives, e.g., Captopril
∑ Antiarrhythmics, e.g., Procainamide Hydrochloride
∑ Antianginal drugs, e.g., Nitroglycerine
∑ Antineoplastic agents, e.g., Fluorouracil
∑ Antibiotics, e.g., Piperacillin Sodium
The aforesaid drugs have been treated individually in the following sections.
'5D[LV
9
PD[
'5
'5
&66
&66
&66
.
P
Figure 3.25 A direct linear plot showing steady-state concentrations (C
SS
) when drug is administered at distinctly
two different rates.
Pharmacokinetics
207
8.1. Antihypertensives
Captopril (Singhvi et al., 1981)
49
+2
2
&22+
+
1
&+
I = [(2S)-3-Mercapto-2-methyl-l-oxopropyl]–L-proline [· = sites labeled with
14
C]
Brand Name: Capoten
®
[Squibb].
Captopril is a potent, orally active, specifi c inhibitor of the enzyme that catalyzes the conversion of angi-
otensin I to angiotensin II. It has been shown clinically to be an effective antihypertensive agent.
Pharmacokinetics of Captopril in Dogs and Monkeys: Captopril is a chemically unstable thiol
compound in biological samples and undergoes rapid auto-oxidation to form the disulfi de dimer of the
parent compound and some other products. To minimize such processes, captopril was converted to its
N-ethylmaleimide derivative in blood, urine and bile during or immediately after sample collection.
Experimental Procedure: Clinical Protocol–[
14
C] Captopril was universally labelled with carbon
14 in the proline moiety. A non-ionic surfactant (Triton X-14), a tissue solubilizer (Soluene-350) and
N-ethylmaleimide were used. Other chemicals were reagent grade. Three male adult purebred beagles
(dogs) and three male adult rhesus monkeys were fasted overnight prior to the study, hydrated orally
with water, anaesthesized with intravenous pentobarbital sodium, intubated with an endotracheal cath-
eter and maintained under anaesthesia by constant intravenous infusion of pentobarbital sodium.
Steady-state concentrations of unchanged captopril in the blood were determined. Constant intra-
venous infusion doses were maintained for 4 hr in three anaesthetized dogs and three anaesthetized
monkeys. [
14
C] captopril was infused, using a constant infusion pump after which 5% manitol was given,
and the animal was kept under anaesthesia for an additional 8 hr.
Urine and blood samples were collected every 30 min during [
14
C] captopril infusion, for 3 hr after
the end of infusion, and then every 1 hr for the next 5 hr. A blood sample was withdrawn at the mid-
point of each collection. All samples were tested with N-ethylmaleimide and analyzed for total radioac-
tivity and unchanged captopril.
Analytical Protocol—Blood and urine samples were analyzed using a thin-layer radiochromatographic
assay for captopril. This method is quite specifi c for unchanged captopril and has a limit of detection of
10 ng mL
−1
.
Results and Discussion: The pharmacokinetic profi le was evaluated exclusively on data obtained
for unchanged captopril in blood, urine and bile during the steady-state and the post-infusion phase.
A semi-logarithmic curve was obtained from the plot of concentrations of unchanged captopril in
blood versus time. The bi-exponential decline of blood captopril concentrations indicates that the drug
exhibits characteristics of a two-compartment open model, with elimination occurring from the central
compartment.
The results of the various pharmacokinetic parameters were averaged for all three animals. The vol-
ume of distribution of captopril was higher in the monkey (3.61 kg
−1
) than in the dog (2.51 kg
−1
); the
49
Singhvi SM et al.: J. Pham. Sci., 70(10): 1108–1111, 1981.
Essentials of Biopharmaceutics and Pharmacokinetics
208
volume of the central compartment (V
1
) was about the same (0.51 kg
−1
) for both species. The values for
distribution rate constants (K
12
and K
21
) were comparable for both species. However, the overall elimina-
tion rate constant (K
CL
) in the monkey (2.6 hr
−1
) was about twice that found in the dog (1.3 hr
−1
) due
to higher renal clearance in the monkey.
The body and renal clearance values in mL kg hr
−1
of captopril in the monkey 1135 (CL
T
), 944 (CL
R
)
were much greater than the corresponding values for the dog 605 (CL
T
), 341 (CL
R
), primarily because
the net tubular secretion of captopril was three times higher in the monkey than in the dog.
The terminal half-life (t
1/2
β) value was slightly higher in the dog than in the monkey, 2.8 and 2.2 hr,
respectively.
Summary: The disposition of captopril in dog and monkeys following intravenous infusion of the
drug was studied to gain a good understanding of captopril pharmacokinetics and to provide informa-
tion for properly designing pharmacokinetic studies in humans. Captopril appeared to exhibit charac-
teristics of a two-compartment open model in both the dog and monkey. The difference between the
total body clearance and renal clearance of captopril in both species was primarily due to the metabolic
clearance of captopril. The large values of volume of distribution obtained indicated extensive distribu-
tion of captopril throughout the body.
8.2. Antiarrythmics
Procainamide Hydrochloride: [BP, BPC] (Kamath et al., 1981)
50
2
1
+
1
+
1
&+
&+
 +&O
4-Amino-N-(2-diethylaminoethyl) benzamide hydrochloride.
Brand names: Procan
®
(Parke-Davis); Pronestyl Hydrochloride
®
(Squibb).
Procainamide hydrochloride is an effective antiarrythmic drug.
Pharmacokinetics of Procainamide and N-Acetylprocainamide in Rats:
14
C-labelled procainamide
and
14
C-labelled N-acetylprocainamide (the major metabolite of procainamide in humans) as hydrochloride
salts were diluted with unlabelled drugs.
Clinical Protocol: Eight male rats were used as subjects. Each animal received procainamide (75 mg kg
−1
)
and N-acetylprocainamide (86 mg kg
−1
) intravenously according to a two-way cross-over design. Four
of the eight rats received procainamide fi rst, while the other four received N-acetylprocainamide. The
day before the drug was administered, a cannula was inverted into the jugular vein of each rat under
light ether anaesthesia.
The drug solutions were administered intravenously through the jugular cannula after an overnight
fast. Water was freely available at all times. Serial blood samples (0.4 mL each) were withdrawn at dif-
ferent time intervals after drug administration. After each withdrawal, 0.4–0.5 mL of normal saline
containing 1% heparin solution was infused to replace the lost volume. Plasma was separated and frozen
50
Kamath BL et al.: J. Pharm. Sci., 70: 667–669, 1981.