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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5849_Библиотеки_им_академика_М_И_Перельмана
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Of all PK concepts, clearance is one of the most closely associated with the
physiology of the body (volume of distribution also has physiological significance).
By recognizing that the maximum clearance by an organ will be limited by the blood
flow to the organ, a direct comparison is made of the estimated clearance (using
Equation 5.16) and the blood flow through the major clearing organs of the body as in
Appendix 5.A.5. If the calculated clearance value approaches the hepatic and renal
blood flows, the clearance of the compound is considered high.
The liver clears drugs by metabolism and by excretion of the unchanged drug into
the bile so
CL
hepatic
¼ CL
metabolic
þ CL
biliary
ð5:19Þ
The hepatic clearance is dependent on (a) the hepatic blood flow and (b) the
extraction efficiency (or ratio) of the liver such that
CL
hepatic
¼ QH E
H
ð5:20Þ
where Q
H
is the hepatic blood flow and EHis the extraction efficiency. The EHis
dependent on the inherent ability of the liver to clear a drug substance or the intrinsic
clearance (CL
intrinsic
) of the liver for the particular drug such that
E
H
¼
fub CL
intrinsic
QHþ fub CL
intrinsic
ð5:21Þ
where fubis the fraction of the drug unbound in blood (this is one of the few times that
the unbound concentration in blood rather than plasma is important).
From Equation 5.21, the maximum value of the extraction efficiency (a fraction)
is 1. When E
H
approaches unity, the term fub CL
intrinsic
QHand hence, the hepatic
clearance in Equation 5.20 approaches the hepatic blood flow Q
H
. This means that
the impact of CL
intrinsic
is negligible. Chemically manipulating such compounds may
not result in affecting the hepatic clearance. Usually, an E
H
value > 0.7 for a drug is
considered a high extraction ratio.
Similarly, if Q
H
fub CL
intrinsic
, then EHdepends on QH, fub, and CL
intrinsic
and
when substituted in Equation 5.20, the CL
hepatic
depends on fuband CL
intrinsic
. Here the
hepatic clearance can be manipulated by changing the intrinsic clearance of the drug.
Typically an E
H
< 0.3 is considered a low extraction efficiency for the drug.
The E
H
range between 0.3 and 0.7 is considered as intermediate. Very few drugs
fall into this intermediate range and simpl e substitutions in Equation 5.21 after
assuming fu
b
¼ 1 show that only drugs with intrinsic clearance values in the range
between 0.43 Q
H
to 2.3 QHwill be in the intermediate range.
The fu
b
must be the fraction unbound in blood and can be substituted for the
fraction unbound in plasma provided that the blood-to-plasma ratio (C
blood/Cplasma
)
is 1.
Notice that no suggestion is made to manipulate the fu
b
since trying to design a
drug with a certain fu
b
or plasma protein binding is a futile exercise.
218 PHARMACOKINETICS FOR MEDICINAL CHEMISTS
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Such individual organ clearance can be estimated for other organs such as kidneys
in a similar manner as shown for the liver.
Estimating CL
intrinsic
The intrinsic clearance is estimated by in vitro experiments
where the drug is incubated with hepatic (or renal) microsomes or hepatocytes and the
degradation of the drug is studied at the initial conditions of the Michaelis–Menten
kinetic equation
Rate of metabolism ¼
V
max
C
K
m
þ C
ð5:22Þ
where V
max
is the maximum rate of the metabolism (also called the velocity) and
K
m
is the concentration “C” of the metabolite where the rate is1/2V
max
.
At low concentrations where K
m
C, the impact of C in the denominator is
ignored resulting in
Rate of metabolism ¼
V
max
C
K
m
ð5:23Þ
which is a first-order equation in concentration. The ratio V
max/Km
is called the
intrinsic CL. Most discovery organizations use this feature to determine the intrinsic
CL of their chemical series via high-throughput assays (usually 2–4 points) at low
concentrations of the drug relative to its K
m
.
5.4.1.3 Volume of Distribution Once a drug enters the blood stream, it distributes
to various parts and tissues of the body, to a degree dependent on its physicochemical
properties. A pseudoequilibrium is established (in reality a drug never experiences a
true equilibrium as clearance process begins as soon as the drug is in the systemic
circulation) and the concentration in plasma that is achieved is based on the amount
(dose) administered or absorbed and the extent of distribution.
The plasma concentration reflects the extent to which the dose was diluted (or
distributed). The factor that relates the amount of the drug to its concentration is
called the “volume of distribution” (V
d
). This extent of dilution could be due to
extensive distribution throughout the body or it could be due to binding to specific
tissues in the body or in som e unusual cases it could reflect the sequestration of the
drug to a single tissue. Assessing the true extent of distribution from the V
d
could be
misleading as many factors contribute to the magnitude of the estimate of V
d
. It is for
this reason that the term “apparent volume of distribution” is used to describe this
factor and hence
Apparent volume of distribution ¼
Amount ðdoseÞ
Concentration
ð5:24Þ
The volume of distribution does not reflect true physiological volumes but by
definition the value cannot be less than the blood (or plasma) volume of the animal.
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There is no upper limit to the volume of distribution, and it can exceed the total body
water of the animal.
Since the volume of distribution characterizes the relationship between the amount
of drug and its concentration, the value of the volume of distribution will change as
the shape of the pharmacokinetic profile changes. For illustration, a monoexponential
(Figure 5.10) and a biexponential PK (Figure 5.11) disposition curve (semilog
transformed) are considered.
In Figure 5.10, the entire curve needs a single volume term (V
d
) to describe the
relationship outlined by Equation 5.24.
In Figure 5.11, the plasma curve following an IV bolus is biphasic and each kinetic
phase is explained by a different volume term. This biphasic curve is explained by
C ¼ A e
at
þ B e
bt
ð5:25Þ
where A and B are the intercepts of the distribution and elimination phases of the drug,
respectively, and a and b are the corresponding distribution and elimination rate
constants, respectively. It is important to note that the terminal phase represents
elimination and the slope of the line defining this phase is “b” and the intercept on the
Y-axis is “B.” The preterminal phase, however, is related to distribution and
elimination. Once the elimination phase is deconvoluted (subtracted) from the
preterminal phase, the resultant curve represents the distribution phase and its slope
is “a” and intercept is “A.”
DETERMINING Vd
0.1000
1.0000
10.0000
100.0000
1211109876543210
TIME (hours)
CONCN (ug/mL)
IV
DOSE = 10 mg
C0 = 10 ug/mL
Vd = 1000 mL = 1 L
Figure 5.10 The one-compartment (monoexponential) model.
220
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Volume of Distribution of the Central Compartment (Vc) This term is estimated by
extrapolating the preterminal phase (also called the alpha phase) of the curve to
time ¼ 0, (Y-axis intercept), and can also be estimated by utilizing Equation 5.25 as
V
c
¼
Dose
A þ B
ð5:26Þ
The V
c
cannot be less than the blood volume because the drug, at a minimum, has to
equilibrate into the blood and will be diluted to that extent.
Volume of Distribution Extrapolated (V
extrap
) This term is obtained by dividing
the dose with “B,” the intercept on the ordinate axis that is extrapolated from the
terminal phase (Figure 5.11). This term is an overestimate of the volume of
distribution and does not have any scientific utility. It was sometimes calculated
mainly to inform “budding” pharmacokineticists of the perils of estimating erroneous
parameters.
Volume of Distribution of the Beta Phase (V
b
or V
darea
) This term explains the
relationship between amount and concentration in the terminal phase or the elimination phase of the disposition. It is estimated from the total body clearance as
V
b
¼
CL
tot
b
ð5:27Þ
where b is the slope of the terminal phase.
DETERMINING Vd
0.1
1
10
100
1000
30
25
20
15
10
5
0
TIME (hours)
CONCN (uM)
Vc
Vbeta or Varea
Vextrap
Figure 5.11 The two-compartment (biexponential) model.
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Volume of Distribution at Steady State (Vss) When the elimination of a drug is
very rapid as compared to its distribution, the drug does not get an opportunity to
distribute to a pseudo equilibrium. Consequently, using Equations 5.26 and 5.27 to
estimate the volumes of distribution will result in inflated numbers, rather than the
true volume of distribution. In such cases and as applied more generally, the
volume of distribution measured at a steady state would give a more accurate value.
If one was to infuse a drug to a steady state and measure the volume of distribution,
the value would be a true representation of the distribution of the drug. It can be
derived that
V
ss
¼ Vc1 þ
k
12
k
21
¼ MRT CL
tot
ð5:28Þ
where V
c
is as described in Equation 5.25. The ratio of k12to k21is the ratio of the
intercompartmental rate constants in a two-compartment model. MRT is the mean
residence time (Section 5.4.1.4) and CL
tot
is the total body clearance.
As in the case of clearance, the volume of distribution value can be compared to the
volumes of the various body spaces in an animal. Although this comparison does not
have absolute physiological significance, it nevertheless gives a sense of the extent of
distribution that the drug undergoes. Table 5.9 (Appendix 5.A.3) compares the various
body space volumes in a variety of laboratory animal species.
The volume of distribution is a difficult variable to control via chemical manipulations of the structure as opposed to clearance, which can be manipulated by
changing chemical structure.
5.4.1.4 Mean Residence Time Mean residence time is defined as the time for
63.2% of the administered drug molecules (dose) to be eliminated from the body.
Statistical Moments The concept of residence times has been adopted into the PK
arena from its extensive use in chemical engineering where it is used to describe
flow data.
Every time a drug is dosed, a large number of drug molecules are introduced into
the body (system). If the body is considered to be a stochastic space (drug molecules
do not interact with each other) and these drug molecules distribute in this space,
then each molecule spends a finite amount of time residing in the body before it is
eliminated. If the residence times of each molecule in the body were plotted, a
normal statistical distribution would be obtained. The probability of finding a
molecule in this distribution can be considered to be a probability density
function [18].
Simple statistical evaluations to this probability density function and estimating
parameters that are equivalent to the mean, standard deviation, and so on, of a normal
distribution can be applied (Table 5.4). A mathematical description of such probability density functions are called “moments.”
222 PHARMACOKINETICS FOR MEDICINAL CHEMISTS
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TABLE 5.4 Similarities of Statistical Moments and Normal Statistics
Moment (M) Description Normal Distribution Probability Density
Function
0 Number N AUC
1 Mean
X ¼
X
X
i
N
MRT
2 Variance s
2
¼
X
ðX
i
XÞ
2
N
VRT
3 Skewness g ¼
X
ðX
i
XÞ3=N
ðs
2
Þ
3=2
SRT
Hence, the plasma (or blood) PK curve can be considered a statistical distribution
and the descriptors for this distribution can be calculated as
M
r
¼
ð
1
0
tr CðtÞdtr¼ 0; 1; 2; 3; ...; n ð5:29Þ
and for r ¼ 0,
M
0
¼
ð
1
0
t0 CðtÞdt ¼ AUC ð5:30Þ
and for r ¼ 1,
M
1
¼
ð
1
0
t1 CðtÞdt ¼ AUMC ð5:31Þ
and for r ¼ 3,
M
2
¼
ð
1
0
t2 CðtÞdt ¼ VRT ð5:32Þ
These parameters can also be estimated by the trapezoidal rule and typically for
most PK anal yses, the zero and the first moments are estimated by the trapezoidal rule
where AUC is estimated as in Equations 5.29 and 5.30 and
M
1
¼ AUMC ¼
X
Cntnþ C
n 1tn 1
2
ðt
2
t1Þð5:33Þ
DRUG ADMINISTRATION AND PK OBSERVATIONS 223
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AUMC1¼ AUMC
last
þ
C
lasttlast
k
e
þ
C
last
k
2
e
ð5:34Þ
where AUMC is the area under of the first moment curve and terms C
last
, t
last
, and
k
e
are as defined before (Equations 5.29 and 5.30).
Calculation of Mean Residence Time The central or the average tendency of this
residence distribution of drug molecules is called the mean residence time and is
defined as the time for 63.2% of the administered dose to be eliminated ([19], p. 409).
For an IV dose, the MRT is estimated as
MRT
IV
¼
AUMC
AUC
ð5:35Þ
The MRT is related to the half-life of the drug and
MRT
IV
¼
1
K
ð5:36Þ
where K ¼ CL
tot/Vss
¼ 0.693/t
1/2
or the elimination rate constant fol lowing IV dosing.
Sometimes for a drug with a multiexponential decline, it is more important to
estimate an effective half-life to understand the “meaningful” rate of elimination and
this is estimated as
t
eff
1=2
¼ 0:693 MRT
IV
ð5:37Þ
A similar approach can be taken for a PO (or EV) dosed system, except the drug is
administered into a separate compartment (GI tract, SC site, etc.) before it enters the
systemic circulation. In order to account for the time the drug spends in this dosed
compartment, the equatio n for PO or EV dosing is
MRT
PO
¼ MRTIVþ MAT ð5:38Þ
where MAT is the mean absorption time and is similar to considering the average
time the drug molecules spend in the absorption compartment waiting to be absorbed.
Similar to Equation 5.36, the MAT can be used to determine an absorption rate
constant (k
a
)
MAT ¼
1
k
a
ð5:39Þ
This value of k
a
is useful when a dose curve is to be predicted in humans following
extrapolations from preclinical data.
The MRT can also be used to estimate the volume of distribution at steady state
(V
ss
) as per Equation 5.28 where Vss¼ MRT CL
tot
.
224 PHARMACOKINETICS FOR MEDICINAL CHEMISTS
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5.4.1.5 Half-Life (t
1/2
) The time for the drug concentration to fall to half of its
(initial) maximum value is the half-life. This PK parameter is the most often used and
easiest to understa nd for the nonkineticist but is the most misinterpreted as well.
In a first-order process, the half-life is estimated as
t
1=2
¼
Ln 2
k
¼
0:693
k
ð5:40Þ
where k is any first-order rate constant.
The half-life is simple and easy to understand when describing a monoexponential
decline because a single half-life explains the entire disposition (Figure 5.12).
For a drug undergoing a multiexponential decline, there are many half-lives that
can be determined for each of the kinetic phases (Figure 5.13). The a-phase and
b-phase have different estimates. Which one is relevant? It could be argued that
the terminal phase represents the true elimination phase of the drug and the half-life
associated with that phase is more relevant. However, there are many instances where
DETERMINING HALF-LIFE
0.0001
0.0010
0.0100
0.1000
1.0000
10.0000
12
11
10
9
8
7
6
5
4
3
2
1
0
TIME (hours)
CONCN (uM)
IV
Figure 5.12 Determining half-life; concentration drop from 2 units to 1 unit is a half-life.
DETERMINING HALF-LIVES
0.1
1
10
100
1000
302520151050
TIME (hours)
CONCN (uM)
IV
t1/2 (_) = 1.1 hr
t1/2 (_) = 5.1 hr
Figure 5.13 A biphasic PK profile with different half-lives for each phase.
DRUG ADMINISTRATION AND PK OBSERVATIONS 225
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the terminal phase does not contribute to the overall disposition of the compound and
can be misleading in its interpretation (Figure 5.14).
In Figure 5.14, the drug has undergone three log orders of concentration change
by the time the terminal phase is realized. The terminal phase contributes to less than
10% of the total AUC of the disposition. If the half-life of the terminal phase is used
(7 h in this example) as a guide, then a once a day (QD) or twice a day (BID) profile
would suffice to achieve a steady state (it takes 6 t
1/2
to achieve a steady state). In
reality, either of these dosing regimens would not reach a steady state because the
kinetic phase with the 7 h half-life has an insignificant impact on the accumulation
to steady state. In fact the “meaningful” half-life is associated with the phase
immediately prior to the terminal phase (t
1/2
¼ 1.1 h in this example) and it would
be impractical to dose such a drug to a steady state unless one was using an
IV infusion.
This suggests that the half-life can be a confusing parameter. Further, this
parameter is not an independent PK parameter. It in fact depends on two physiologically based PK parameters CL
tot
and Vd(Vssin a multiexponential decline) as
t
1=2
¼ 0:693
V
ss
CL
tot
ð5:41Þ
One can clearly appreciate the association between Equations 5.28, 5.37 and 5.40
in defining the relationships between half-life, effective half-life, and MRT.
In practice, it is important to compare the terminal half-life value with the MRT
estimate to check if the half-life being considered is meaningful (the MRT value and
the half-life value should be similar); otherwise it may be more appropriate to use the
effective half-life (Equation 5.37).
Further, it is difficult or next to impossible for chemists to conduct SAR around
half-life because it requires controlling both the volume of distribution and the
clearance. The latter can be manipulated by structural modifications of metabolic
“soft” spots on the molecule but the former (V
d
) is very difficult to control and predict.
IV
0.01
0.10
1.00
10.00
100.00
302520151050
TIME (hours)
CONCN (Xg/mL)
α - PHASE (5 min)
β - PHASE (1.1 hours )
γ - PHASE (6.65 hours)
CP = Ae
−αt
+ Be
−βt
+ Ce
−γt
Figure 5.14 IV profile showing a three-exponential decline with a rapid a- and b-phases and
a slower g-phase. The g-phase contributes to less than 10% of the total AUC.
226
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5.4.2 Analysis of Extravascular PK Data
A drug delivered into an extravascular space such as an oral, subcutaneous, intramuscular, inhalation, transdermal, etc. formulation, will result in a growth curve in the
plasma profile, reaching a maximum (C
max
) followed by a decline in plasma levels.
Drug delivered to any body compartment, where the drug has to traverse biomembranes before it can enter the blood/plasma circulation, will show a profile similar to
that shown in Figure 5.15.
The structure of the plasma or blood level curve following extravascular admin-
istration is characterized by C
max
, T
max
, absorption rate (for the absorption phase), and
the post absorption phase.
5.4.2.1 Maximum Concentration The maximum observed plasma (or blood)
level following EV dosing is the C
max
. It is obvious that since this is an observed value,
there should be adequate sampling postdose to be able to accurately identify and
estimate the C
max
of the drug. The C
max
of the drug is also dependent on the
formulation and a change in the formulation can cause a change in the C
max
.
In toxicokinetics, the C
max
value is used to establish “safety margins” in multiples
above some NOAEL (no observed adverse effect level) and hence it needs to be
determined accurately. In toxicokinetic studies, as the dose is increased, it is quite
possible that the absorption of the drug becomes saturated or dissolution-limited
(solubility is usually the major contributing factor to a slower dissolution). As the
overallrate of absorption slows, the plasma kinetics tends to reflect a “flip–flop” kinetic
pattern (see Section 5.4.2.5) and the profile acquires a flattened shape with a C
max
that is
shifted to a higher time value.Such a “right” shift in the C
max
along the time axis (higher
T
max
) is indicative of a saturable absorption or dissolution-limited absorption. It is
therefore important to interpret the C
max
value cautiously in a dose escalation protocol.
In bioequivalence studies, the FDA requires that the C
max
observed for the generic
product must be within 80–120% of the innovator compound.
5.4.2.2 Time to Maximum Concentration The time to obtain maximum plasma
concentration (C
max
)isT
max
and the challenges faced in determining the accuracy of
FEATURES OF AN EXTRAVASCULAR (PO) CURVE
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
10
9
8
7
6
5
4
3
2
1
0
TIME (hours)
CONCN (µM)
Cmax = 4.06 µM
Tmax = 1.3 hours
Figure 5.15 A typical plasma PK profile observed following EV administration.
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