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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_6000_Библиотеки_им_академика_М_И_Перельмана.pdf
Pharmacokinetics
149
3. Since there exists a good number of tissue organs present in a human body, it becomes quite nec-
essary and important that the tissue volume should be determined, and hence, its ensuing drug
concentration (mg mL
–1
or mcg mL
–1
) specifi ed appropriately.
4. However, most of the valid information which is required essentially for proper description of a
physiologic pharmacokinetic model experimentally is diffi cult to achieve.
Note
Contrary to the serious limitations, the physiologic pharmacokinetic model fails miserably to give
a much better insight of how the ensuing physiologic factors would cause a defi nite change in the
drug-distribution profi le from a particular animal species to another.
Figure 3.9 illustrates explicitly a physiologic pharmacokinetic model (or Flow Model) where the various
compartments are duly arranged in a series in a descriptive fl ow diagram.
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Figure 3.9 The schematic representation of a physiologic pharmacokinetic model of drug perfusion.
Essentials of Biopharmaceutics and Pharmacokinetics
150
Description: The various aspects in Figure 3.9 may be described as below:
1. The IV administration of drug carries it to the lungs, and from there right into the arterial blood.
2. From the arterial blood, the drug passes into
∑ heart,
∑ gut and liver—in a sequential manner—and gets away via the hepatic elimination (K
m
),
∑ kidney and gets away through the urinary excretion (K
e
),
∑ venous blood after passing via the highly perfused tissues (HPT) and
∑ venous blood after entering through the poorly perfused tissues (PPT).
3. Thus, one may explicitly observe the ‘drug’ administered through IV passing across the venous
blood to the arterial blood by means of an array of physiologic pharmacokinetic modes.
Three Major Differences There are in all three major differences with regard to the physiologic phar-
macokinetic models, namely:
1. Devoid of Data-Fitment in Perfusion Model: The perfusion model is devoid of any sort of
data-fi tment demand. Consequently, the necessary drug concentrations in different tissues are pre-
dicted precisely by such critical requirements:
∑ organ tissue-size,
∑ ensuing blood-fl ow and
∑ drug–tissue–blood ratios
24
estimated experimentally.
2. Observed Variance in Blood-fl ow/Tissue-size/Drug–Tissue–Blood Ratios: One may
apparently observe a signifi cant variance in blood-fl ow, tissue-size and drug–tissue–blood ratios
neces sarily caused by virtue of certain pathophysiologic status.
25
Therefore, it is almost mandatory to
take into consideration the overall effects of these variants upon the drug distribution specifi cally in
physiologic pharmacokinetic models.
3. Possibility of Extrapolation: The most vital aspect of all physiologic pharmacokinetic models
may be applicable to many species and certain critical drug-human data may be extrapolated easily
and conveniently.
Following are two typical examples:
∑ A good number of drugs, viz. Digoxin, Lidocaine, Methotrexate, Thiopental, can be described by
the aid of the perfusion models exclusively.
∑ Tissue levels of quite a few of these drugs may not be predicted as accurately as can be accom-
plished with the compartment models.
Examples
Importantly, the real signifi cance of a physiologic–pharmacokinetic model of drug perfusion is solely
based upon the ensuing potential application of this particular model in the viable and accurate predic-
tion of the so-called ‘human pharmacokinetics’ from the accomplished ‘animal data’. However, one may
achieve the following critical information in humans and other species, such as:
24
That is, partition of the drug between the tissue and blood.
25
Pathophysiologic Status: It refers to the physiology of disordered function.
Pharmacokinetics
151
∑ mass of different body organs or tissues,
∑ degree of protein-binding,
∑ capacity of ‘drug metabolism’ and
∑ blood fl ow.
Interestingly, the aforesaid anatomical and physiological parameters may be employed explicitly to
∑ predict the ensuing effects of drug substances in humans and
∑ their consequent effects upon the animal models in all such instances whereby the ‘human experi-
mentation’ becomes rather diffi cult to accomplish.
Merits of Physiologic–Pharmacokinetic Models The physiologic–pharmacokinetic model essen-
tially has certain glaring merits, namely:
1. The mathematical treatment is mostly of a straight-forward nature.
2. Obviously, here the ‘fi tment of data’ is hardly necessary. Besides, one may gainfully predict the
actual drug concentrations duly present in various regions of the body based upon such factors as
∑ tissue or organ volume,
∑ rate of perfusion and
∑ tissue to plasma ‘partition coeffi cient’ estimated experimentally.
3. It does provide a precise description of the prevailing drug-concentration time profi le present duly
in any specifi c organ or tissue and, therefore, it may provide a much better picture of the ensuing
drug-distribution features in a human body.
4. It is quite possible to record the infl uence of an altogether changed pathology or physiology upon
the disposition of drug, which may be predicted both easily and conveniently based upon the
changes taking place in different pharmacokinetic parameters.
Note
These ‘pharmacokinetic parameters’ predominantly are related to the actual anatomic and
physiologic values.
5. Logical and judicious correlation of data in many animal models is evidently possible. Even certain
drug substances may be carefully extrapolated to humans.
Important Note: The one and only major demerit of such comprehensive models is retaining
meaningful experimental data that certainly proves to be a Herculean and exhaustive task.
3.2.3. Non-Compartmental Pharmacokinetics
Due to the well-know fact there exist numerous disadvantages along with several diffi culties invariably
encountered with the ‘classical compartment models’, gave birth to many ‘newer approaches’ to carry out an
elaborated study confi ned to the time-course of drug substances in a human body. It is usually termed
as the non-compartmental pharmacokinetic model.
However, it is also known as the ‘independent-model method’. Besides, it may be applicable to any avail-
able compartment model only in such conditions where
∑ the drug substances and
∑ their respective metabolites
strictly adhere to the ‘linear kinetics’ exclusively.
Essentials of Biopharmaceutics and Pharmacokinetics
152
Statistical Moments Theory The non-compartmental pharmacokinetic models normally employed
for the calculation of absorption distribution, and elimination parameters are solely based upon the
statistical moments theory.
It essentially involves the meticulous collection of various experimental data obtained duly from
the single-administration of a ‘drug’. Now, considering the time-course of a drug concentration
duly present in plasma representing specifi cally the ‘statistical distribution curve’, we may have the
following expression:
UAMC
MRT
AUC
=
(r)
where
MRT
26
is the mean residence time,
AUC is the area under ‘zero-moment curve’ and
AUMC is the area under ‘fi rst-moment curve’.
Importantly, one may obtain AUMC easily from a plot:
∑ product of plasma-drug concentration (at time ‘t’, i.e., ‘C
t
’) and
∑ time ‘t’ (from 0 → α),
as illustrated in Figure 3.10
Hence, AUMC may be expressed mathematically by the following expression:
0
UAMC d
t
Ct
α
=
∫
(s)
In the same manner, AUC may be arrived at from a plot of the ensuing plasma-drug concentration vs.
time duly extending from 0 → α. Therefore, AUC can be expressed mathematically by the following
expression:
0
AUC dCt
α
=
∫
(t)
26
MRT: It may be defi ned as ‘the average amount of time consumed by a drug in the body before being elimi-
nated’. Evidently, it refers to the statistical moment analogy of plasma half-life
1/2
()t
. In true sense, MRT desig-
nates the actual time taken for ~63.2% of the IV bolus dose to be eliminated. Obviously, the MRT values shall
always be greater in a situation when the drug is being administered via a route other than the IV bolus.
The various vital aspects of Figure 3.10 may be expatiated as follows:
Both area under the ‘fi rst-moment curve’ (AUMC) and area under the ‘zero-moment curve’
may be calculated based on the respective graphs (or plots) duly obtained by the ‘trapezoidal rule’.
In other words, the entire curve is duly divided into a series of vertical lines (see Figure 3.10) into
an array of trapezoids. Subsequently, the specifi c area of each trapezoid is calculated individually,
and then they are added up together fi nally.
Explanation
Pharmacokinetics
153
Important Applications The important applications of the non-compartmental pharmacokinetic model
are concerned with the determination of various pharmacokinetic aspects, namely:
∑ bioavailability,
∑ clearance,
∑ volume of distribution (V
d
),
∑ plasma half-life (t
1/2
),
∑ rate of absorption and
∑ ‘fi rst-order’ absorption rate of the drug substance.
Merits of Non-Compartmental Pharmacokinetics These essentially comprise the following:
1. The pharmacokinetic parameters may be derived quite easily by using simple algebraic equations.
2. Importantly, one may make use of similar mathematical treatment to practically any sort of
∑ drug substance or
∑ metabolite
only if they rigidly follow the ‘fi rst-order’ kinetics.
3. The necessity of an elaborated description of the drug disposition characteristic feature is ruled
out completely.
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Figure 3.10 Diagrammatic representations of AUMC and AUC plots.
* Trapezoidal Rule: It refers to the method for estimating integrals (i.e., areas under absorption curves) by adding the areas of discrete
trapezoids. It is frequently used for estimating several other pharmacokinetic integrals to determine either quantity and/or extent of
occurrence of a process.
Essentials of Biopharmaceutics and Pharmacokinetics
154
3.2.4. Non-Linear Pharmacokinetics
The pharmacokinetic models discussed so far have been duly constructed on the assumption that the
ensuing pharmacokinetic pathways are invariably described by these two vital factors:
∑ linear differential equations and
∑ ‘fi rst-order’ rate constants.
In a broader perspective, these models need not be true and is quite important to state here that these are
the typical instances wherein both the non-ideal and non-linear behavioural pattern may take place effectively.
Evidently, the determination of a host of pharmacokinetic parameters, for instance,
∑ absorption rate constant,
∑ elimination rate constant and
∑ plasma clearance factor,
are entirely based upon the assumption that all these conditionalities (i.e., parameters) actually fails to
bring about either a signifi cant alteration on the administration of a drug at different dose levels or in a
situation when the multiple doses of a drug are duly administered.
Note
The only major disadvantage of this technique being the scanty provision of absolute restricted
information pertaining to the critical plasma-drug concentration–time profi le.
It has been duly observed that either the increased doses of certain drugs or the chronic medi-
cation may ultimately give rise to the apparent deviations with respect to the ensuing linear-
pharmacokinetic profi le as shown with single-low doses of the same drug.
Example
Furthermore, the non-linear pharmacokinetic behavioural pattern is usually termed as the ‘dose-
dependent pharmacokinetics’. It has been discussed more elaborately under Section 3(7).
Time-Dependent Pharmacokinetics When the actual time of the administration of a drug sub-
stance particularly infl uences the pharmacokinetic parameters of the drug, the overall phenomenon is
known as the time-dependent pharmacokinetics.
However, it is pertinent to state at this point in time that the aforesaid non-linear pharma-
cokinetics may be the result of a typical pathologic alteration with regard to the respective drug
absorption–distribution–elimination.
The non-linear pharmacokinetics shall be treated in details, under Section 3(7).
4. Pharmacokinetics of Drug Absorption
The ‘orders of processes’ related to the pharmacokinetics of drug absorption could be ‘zero order’, ‘fi rst order’
or ‘second order’. In true sense, the ‘order’ solely depends upon the number of variables that ultimately
determine the ‘rate of reaction’. However in the domain of pharmacokinetics, the zero order and fi rst order
do play an extremely important role and, therefore, shall be discussed in the following sections.
Pharmacokinetics
155
4.1. Zero-Order Absorption Rate Constant
The zero-order absorption is prominently characterized by a constant rate of absorption. Importantly, it
is found to be absolutely independent of
∑ actual amount remaining to be absorbed (ARA) and
∑ regular ARA vs. t plot appears to be linear with respect to ‘slope’ that is equivalent to the rate of
absorption, whereas, the ‘semi-log plot’ is duly shown by an increasing-gradient with time (t), as
illustrated in Figure 3.11.
Contrarily, the fi rst-order absorption rate constant determining phenomenon is markedly distinguished
by means of a visible decline in the constant rate of absorption with ARA.
27
Thus, the actual ‘Cartesian
Plot’
28
appears as a curvilinear type, while the ‘Semi-log Plot’ comes out as a straight line having the
desired absorption rate constant designated duly by its ‘slope’ (Figure 3.11).
It has been established beyond any reasonable doubt that soon after the extravascular administration
(EV) of a drug, the observed rate of change in the actual quantum of the same in the body, dx/dt, shows
the prevailing difference between the rate of input (i.e., absorption) given by dx
EV
/dt, and the rate of
input [i.e., elimination (E)] given by dx
E
/dt. Thus, we may have
EV
E
d
d
d
ddd
x
x
x
ttt
=−
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Figure 3.11 An explicit distinction between zero-order and fi rst-order absorption phenomena (a) cartesian plot
and (b) semi-log plot.
27
That is, the rate of absorption is solely dependent upon ARA.
28
Also known as a ‘Regular Plot’.
Essentials of Biopharmaceutics and Pharmacokinetics
156
4.1.1. Plasma Drug Level–Time Curve for a Drug Administered in Single Oral Dose
In a real perspective, it may be understood thoughtfully that the critical rate of an on-going phenom-
enon exclusively depends upon the actual quantum of the drug duly available for such a phenomenon
to accomplish at given time and, therefore, is equivalent to the ultimate product of
∑ rate constant and
∑ quantum of ‘drug’ at a given time.
Obviously, one may conclude generously that the ‘rate of a phenomenon’ undergoes a defi nite change
with respect to time because the ensuing quantum of ‘drug substance’ alters with time.
Figure 3.12 records evidently the plasma drug level–time curve duly accomplished following a ‘single-
oral dose of a drug substance’. Thus, the resulting ‘curve’ may be further subdivided into different zones
genuinely based upon the following two major factors:
∑ relative magnitudes of rate of absorption and
∑ rate of elimination at different time points.
The following four different phases of the plasma drug level–time curve obtained from a single
oral dose of a drug substance are as given below:
∑ absorption phase,
∑ plateau phase,
∑ post-absorption phase and
∑ elimination phase.
Explanation
as shown by A, B, C and D in Figure 3.12 which will now be treated briefl y in the following sections.
Figure 3.12 Different phases of a plasma level–time curve for a drug administered via a single oral dose.
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Pharmacokinetics
157
Absorption Phase (A) Immediately after the oral administration of a particular drug substance, almost the
entire dose of the same is duly available for commencing the absorption phase at zero time. Thus, the quan-
tum of the drug substance prevailing critically at the absorption site starts declining as a function of time (t).
Therefore, one may obviously observe during the absorption phase pertaining to a plasma level–time curve,
the attained rate of drug absorption appears to be markedly greater vis-à-vis the rate of drug elimination.
Thus, we may have
ae
dd
dd
XX
tt
>
(u)
Plateau Phase (B) It refers to the absorption of drug substance into the systemic circulation rather
slowly, which enhances the plasma level of the drug substance till such time the rate of absorption remains
obviously greater in comparison to the rate of elimination. In this manner, the slow and steady enhance-
ment of the evening drug-level in plasma helps to increase the rate of elimination of the drug substance,
dX
e
/dt, as a function of time. It obviously suggests that the rate of absorption of a drug substance decreases
with time (t) whereas, the resulting rate of elimination of the same shall almost render to be ‘equal’.
Importantly, the critical ‘time’ at which the aforesaid ‘equality’ is accomplished is usually termed as the
‘time of peak-drug concentration (t
p
)’ in plasma; and hence, the attained concentration of the drug sub-
stance is invariably called as the ‘maximum plasma drug concentration (C
max
)’, that may be duly accom-
plished soon after a single oral dose of the drug is administered. Evidently, one may observe that at ‘t
p
’:
Rate of absorption = Rate of elimination
i.e.,
ae
dd
dd
XX
tt
=
(v)
Post-Absorption Phase (C) Soon after the completion of the ‘plateau phase (B)’, the observed rate of
elimination of a drug substance appears to be faster vis-à-vis rate of absorption, as designated explicitly
by the distinct ‘post-absorption phase (C)’ (see Figure 3.12). Interestingly, in the course of this particu-
lar phase, one may observe predominantly that there exists a defi nite decline in the evening ‘drug levels’
because the on-going absorption of the drug substance remains still in active progress.
Thus, we have the following expression:
ae
dd
dd
XX
tt
<
(w)
Elimination Phase (D) It represents vividly a typical situation when the drug substance strategically
located at the absorption site gets depleted overwhelmingly and, therefore, the rate of drug absorption
critically attains the ‘zero level’, i.e., dX
a
/dt = 0. Thus, the elimination phase [D] of the plasma level–time
curve eventually represents the overall elimination of the drug substance from the body exclusively, and
that happens to be a fi rst-order phenomenon. It follows from the above stated facts that the actual rate
of change observed in the quantum of the drug substance present duly in the human body specifi cally
in the aforesaid ‘elimination phase’ is frequently expressed by the fi rst-order phenomenon, as given below:
e
d
d
X
KX
t
=−
(x)
Essentials of Biopharmaceutics and Pharmacokinetics
158
where
K is the overall elimination rate constant and
X is the quantum of drug substance present in the human body at any time.
4.2. First-Order Absorption Rate Constant
The First-order absorption rate constant usually designates the oral-administration of a drug substance, i.e.,
the critical presence of the unchanged drug substance in blood/plasma.
Therefore, for a specifi c drug substance which gains entry into the body by means of a fi rst-order
absorption phenomenon fi nally gets distributed throughout the human body via a one-compartment
kinetics. Consequently, it is duly eliminated by the help of fi rst-order phenomena and hence, the said
model may be expressed as follows:
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where
X
0
is the oral dose of a drug substance administered,
X is the amount of drug present in blood and body tissues at any time (t) (i.e., equivalent to the prod-
uct of the volume of distribution (V
d
) of the drug and the plasma drug concentration),
K
a
is the fi rst-order rate constant of the drug absorption,
K
E
is the fi rst-order rate constant of the elimination phenomena.
We have seen earlier that
dX/dt = Rate of absorption – Rate of elimination
and the differential form of the above equation is given by the following expression:
aa E
d
d
X
KX K X
t
=−
(y)
where
K
a
is the fi rst-order absorption rate constant and
X
a
is the amount of drug substance present at the ‘absorption site remaining to be absorbed (ARA)’.
Integrating Equation (y) we may have
Ea
a0
aE
()
tt
KK
KFX
Xee
KK
−−
⎡⎤
=−
⎣⎦
−
(z)
where F is the fraction of the drug substance being absorbed systemically soon after an extravascular
administration.
Now, transforming Equation (z) in terms of the concentrations, we may obtain the following
expression:
Ea
a0
da E
()
tt
KK
FKX
Cee
VK K
−−
⎡⎤
=−
⎣⎦
−
(aa)
where V
d
is the volume of distribution (i.e., X = V
d
× C).