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Pharmacokinetics
169
By applying common logarithms on either sides of the above equation
a E max a max
E
log =
2.303 2.303
K Kt Kt
K
−−
+
or
()
a
max
aE E
2.303
log
K
t
KK K
=
−
(bb)
As it has been expatiated earlier that ‘t
max
’ is solely dependent upon both the rate constants K
a
and K
E
,
the observed maximum concentration of the drug substance in plasma concentration takes place at t
max
predominantly. Therefore, by substituting ‘t
max
’ for ‘t’ in Equation (ay) we obtain:
()
E max a m ax
a0
max
da E
Kt Kt
KFX
Cee
VK K
−−
⎡⎤
=−
⎣⎦
−
(bc)
Nevertheless, one may judiciously obtain a rather much simpler expression based on Equation (bc)
above, by affecting the substitution
a
max
t
K
e
−
equivalent to
max
E
a
E
KTt
K
e
K
−
as per Equation (ba):
()
max E max
a0 a
max
da E E
Kt K t
KFX K
Cee
VK K K
−−
⎡⎤
∴= −
⎢⎥
−
⎣⎦
(bd)
()
Emax
a0 a
da E E
1
Kt
KKFX e
VK K K
−
⎛⎞
⋅
=−
⎜⎟
−
⎝⎠
()
()
Emax
aE
a0
da E a
Kt
KK
KFX
e
VK K K
−
−
=⋅⋅
−
or
E
p
0
max
d
Kt
FX
Ce
V
−
=⋅
(be)
Conclusion: In conclusion, it may be observed that the estimated C
max
immediately after the fi rst-order
input represents prominently a defi nite function of the resulting fraction of the dose (of the drug) enter-
ing the human body systematically,
∑ apparent ‘volume of distribution (V
d
)’,
∑ apparent ‘fi rst-order absorption’ and
∑ determined ‘elimination rate constants (K
E
)’
4.3.4. Influence of K
a
and K
E
upon C
max
, t
max
and AUC
The various articulated infl uence caused specifi cally due to the changes in K
a
at constant K
E
, and that
of K
E
at constant K
a
upon the three cardinal components C
max
, t
max
and AUC of a drug substance when
administered via the extravascular (EV) route is recorded explicitly in Table 3.8.
Essentials of Biopharmaceutics and Pharmacokinetics
170
5. Volume of Distribution and Distribution Coefficient
First and foremost, let us understand the meaning of certain terminologies, such as distribution and
distribution coeffi cient.
Distribution: It refers to the partitioning of a drug to the several locations or compartments in the
body or another heterogeneous system.
Distribution Coeffi cient: The distribution coeffi cient designates the ratio of the solubility (or
concentration) of a substance in an organic immiscible system to the solubility (or concentration) of the
same substance in water when observed in the same system under the specifi ed conditions at equilibrium.
5.1. Volume of Distribution (V
d
)
The volume of distribution (V
d
) is a biopharmaceutical terminology which specifi cally expresses the per-
ceived volume of the body wherein a certain quantum of an unchanged drug (A
b
) gets properly dis-
tributed depending on the ensuing concentration (C
b
) of the drug present in the circulating blood.
Importantly, V
d
= A
b
/C
b
designates a characteristic value for a particular drug-substance. Besides, ‘V
d
’,
represents an altogether abstract volume which may be easily calculated from the ratio of the quantum
of drug present duly in a human body to its actual concentration in plasma only when the partitioning
has been stabilized adequately (i.e., at steady-state parameters achieved).
In other words, the volume of distribution or perhaps more logically ‘apparent volume of distribution’
of a drug is neither literally nor actually a ‘volume’ and, therefore, it must not be regarded as a specifi c
physiologic volume/space present very much within the body. Thus, truly speaking, the apparent volume
of distribution vividly designates the apparent volume into which a drug substance gets evenly distrib-
uted in the human body at equilibrium. Hence, in a broader perspective (apparent), volume of distribu-
tion, V
d
, represents the volume of plasma
41
at a drug concentration (C), needed essentially to account for
the total quantum of drug present in the body (X).
Thus, we may have the following expression:
d
X
V
C
=
(bf)
Table 3.8 Critical observed infl uence of K
a
and K
E
upon C
max
, t
max
and AUC
S. No.
Grossly affected
parameters
Observed infl uence when K
E
remains constant
Observed infl uence when K
a
remains constant
Lesser K
a
More K
a
Lesser K
E
More K
E
1 C
max
↓↑↑↓
2 t
max
Long Short Long Short
3 AUC No change No change
↑↓
(↑: shows an increase; and ↓: shows a decrease)
41
That is usually measured as plasma rather than blood.
Pharmacokinetics
171
or
Amount present in body
Apparent volume of distribution
Plasma drug concentration
=
(bg)
Advantages of V
d
: There are two major advantages of volume of distribution (V
d
), namely:
1. Determination of the plasma concentration in a situation when a known quantum of a drug sub-
stance is duly present in the body and
2. Estimation of the critically required ‘dose’ so as to attain a given (or targeted) plasma concentration
at time ‘t’, i.e., after a certain duration.
5.2. Presence of a Fraction of Drug Both Inside and Outside Plasma
From Equation (bf ), the amount of drug present in the body (X ) is given by the product of V
d
and C.
Hence, Amount of drug in plasma = V
p
· C
where
V
p
is the volume of plasma and
C is the plasma drug concentration.
Fraction of ‘drug’ inside body plasma is
p
d
VC
VC
⋅
=
⋅
or
p
d
V
V
=
(bh)
Likewise, fraction of ‘drug’ outside body plasma
()
dp
d
VV
V
−
=
(bi)
Remarks: Equation (bh) explains clearly that greater the volume of distribution (V
d
), the lesser would be the
fraction of drug present in the body plasma.
5.3. Determination of Plasma Volume
There are two chemical substances (or reagents), viz. Evan’s Blue and Indocyanine Green, both of which
are high-molecular-weight compounds, which possess a unique characteristic feature of remaining
essentially confi ned to the circulating plasma in the human body on being administered via the IV route.
1D
Â62
±
1+
&+
&+
+
1
+2
2+
11
1D
Â62
±
62
±
Â1D
62
±
Â1D
11
(YDQ¶V%OXH
Essentials of Biopharmaceutics and Pharmacokinetics
172
&+
&+
&+
2
6&+
+
&
1
1
,QGRF\DQLQH*UHHQ
±
&+
62
Â1D
±
These two reagents are invariably employed to determine the plasma volume and even the blood
volume accurately in a situation after the estimation of ‘hematocrit’.
42
Besides, the anions, viz. Cl
−
and
Br
−
, that readily and evenly get distributed throughout the extracellular fl uid but fail to cross over the
cell-membranes rapidly may be employed justifi ably to determine the following two values:
∑ extracellular water volume and
∑ extracellular fl uid volume.
5.4. Volume of Total Body Water (or Fluid)
It can be determined with reasonably good accuracy by making use of
∑ heavy water or deuterated water (D
2
O) or
∑ certain specifi c ‘lipid-soluble compounds’, e.g., ‘antipyrine’
that gets distributed swiftly in the entire ‘total body water/fl uid’.
Table 3.9 provides the approximate percentage or volumes for various body compartments.
5.5. Volume of Distribution (V
d
) vs. Protein/Tissue Binding of Drugs
It has been amply proved and established that the duly estimated ‘apparent volume of distribution’ for
each of the tracers, viz. plasma, extracellular fl uid, intercellular fl uid, vascular fl uid etc., as expatiated above
almost turns out to be very much similar to its ‘true volume of distribution’ based on the glaring fact that
Table 3.9 Components in body fl uid water
S. No.
Components in
body fl uid water
Percentage
(%) approx.
Volume (L) approx.
1 Plasma 5 3
2 Extracellular fl uid 20 12
3 Intercellular fl uid – 24
4 Vascular fl uid (or
blood)
–6
5 Total body water 70 42
42
Hematocrit: It refers to the percentage of erythrocytes in a specifi ed volume of whole blood. It designates the
ration of RBCs to plasma.
Pharmacokinetics
173
∑ plasma–protein binding and
∑ tissue–protein binding
are virtually negligible.
Salient Features: The various salient features are as follows:
1. A majority of drug substances are appreciably bound to
∑ vascular components,
∑ extravascular components or
∑ both components.
2. The drug substances that are bound signifi cantly to the respective ‘plasma proteins’ do exhibit
‘apparent volumes of distribution’ which are indeed lesser than the ‘true volume of distribution’.
Besides, ‘V
d
’ of this kind of drug substances actually ranges between the two components, namely
blood volume and total body water volume, i.e., 6–42 L.
Warfarin has V
d
~7 L.
Example
3. Such drug substances which have a tendency to get bound intimately and extensively to the EV
tissues do exhibit the ‘apparent volume of distributions’ that appear to be larger in comparison to
their actual distribution space.
Amitriptyline has V
d
~ 1,400 L, which shows explicitly that the quantum of drug present in plasma
happens to be very meagre and small vis-à-vis the quantum present in the EV compartments,
thereby suggesting broadly that the tissue concentrations of drug are defi nitely very high.
Example
Note
In general, the ‘apparent volume of distribution’ (V
d
) implies that it is the most prominent char-
acteristic feature of a drug and hence, may vary from 0.04 to more than 20 L kg
−1
.
6. Organ-Specific Clearance
Clearance may be explained as a phenomenon that removes the chemical compound (drug ) completely
by the kidneys or liver specifi cally from a requisite volume of blood
43
per unit of time.
Total Clearance: In general, the ‘total clearance’ of a drug substance from the body almost invariably
encompasses more than one organ of elimination. The human body, the God-gifted anatomy, is created
in a manner such that the ensuing observed clearance from the composite clearing organs critically takes
place in parallel and, therefore, proves to be of an ‘additive nature’.
43
Blood: It refers to the fl uid tissue that circulates via the heart, arteries, veins and capillaries, carrying oxygen and
nutrients throughout the body.
Essentials of Biopharmaceutics and Pharmacokinetics
174
6.1. Concept of Organ-Specific Clearance
It is indeed possible and a lot easier to have the concept of organ-specifi c clearance duly visualized with
the help of Figure 3.16.
In a situation when a drug substances gets duly eliminated exclusively either by renal and hepatic
elimination, the ‘total clearance’, designed as CL
T
, of the drug substance may be expressed as
CL
T
= CL
R
+ CL
H
(bj)
where
CL
R
is the renal clearance and
CL
H
is the hepatic clearance.
Example
5HPRYDO
RI
'UXJ
0HPEUDQH
12
3
Figure 3.16 Illustration of organ-specifi c clearance whereby introduction of the drug into Compartment 1,
from which drug gets distributed into Compartment 2. The solution being circulated at a constant rate ‘q’ via
Compartment 3 wherein drug undergoes critical removal from a fi xed volume per unit time.
Figure 3.16 may be expatiated with the aid of the following sequential steps:
1. The ‘drug substance’ is carefully introduced as a single dose into Compartment 1 (in beaker),
and from which it further gets partitioned into Compartment 2 (in beaker).
2. The ‘drug solution’ is now circulated almost at a constant rate ‘Q’ via Compartment 3, wherein
the effective removal of the ‘drug substance’ right from the solution designates elimination.
Note: At this stage, the aforesaid ‘extraction of the drug substance’ may or may not be complete at all.
3. Importantly, the extent to which the effective extraction takes place is critically refl ected by
the actual difference observed between the following two components:
∑ C
in
, i.e., concentration of the solution gaming entry into the ‘extracting compartment’ and
∑ C
out
, i.e., concentration of the solution leaving the ‘extracting compartment’.
4. In a situation when the drug gets extracted almost completely, one would expect C
out
= 0
and when none is extracted, one may have C
out
= C
in
.
Explanation
Pharmacokinetics
175
5. Ultimately, the prevailing limits for the critical concentration that leave Compartment 3 are
as follows:
0 ≤ C
out
≤ C
in
Alternatively, the three terminologies, viz. clearance, systemic clearance and normalized sys-
temic clearance, may be explained as following:
Clearance [CL] is usually calculated soon after the IV administration (i.e., 100% bioavailability)
and may be expressed as
IV
LC
AUC
D
=
where
D
IV
is the amount of drug administered via IV route and
AUC is the area under the curve.
Systemic clearance relates to a single oral dose administration of a drug substance (with < 100%
bioavailability) and is usually based upon AUC ranging with in time 0 → t (i.e., AUC
0–t
) Hence, the
resulting trapezoid calculation is invariably given by the following expression:
0
CL
AUC
t
FD
−
⋅
=
where
F is the fraction of the drug (dose) absorbed and
D is the dose of the drug administered.
Normalized systemic clearance refers to the body weight of a patient and may be suitably calculated by
the help of the following expression:
Normalized
CL
CL
Body weight (kg)
=
6.2. Renal Clearance
Renal clearance may be defi ned as ‘removal by the kidney of a solute (or other substance) from a specifi c
volume of blood per unit of time’.
In other words, renal clearance refers to the ratio of the product of urine concentration of the solute
and the rate of urine fl ow to the plasma concentration of the solute.
Fundamental Basis: A survey of literature reveals that the physiologists carried out an exhaustive
study with regard to the renal clearance of both:
∑ endogenous chemical entities and
∑ exogenous chemical substances
much before the recognition of the application(s) of the clearance concepts in pharmacokinetics. In fact,
the wonderful revelations proclaimed by the renal physiologists, almost stretched over a few decades,
paved the way for a better understanding pertaining to the clearance of drug in pharmacokinetics.
Essentials of Biopharmaceutics and Pharmacokinetics
176
It is important to state here that there exists three most vital primary phenomena, namely:
∑ fi ltration,
∑ secretion and
∑ reabsorption.
Indeed metabolism designates actually a minor on-going phenomena. Interestingly, each of the aforesaid
major phenomena is affected predominantly by several known decisive factors.
Thus, the rate of fi ltration (RF) by the kidney for a drug substance is usually given by the following
expression:
U
R
GFR
ate of filteration (RF)
C
=
(bk)
where
GFR is the glomerular fi ltration rate,
C
U
is the free-drug concentration.
Now, the observed clearance due to fi ltration is given by the ‘quotient of the rate of fi ltration at a given
concentration’ and, therefore, the renal clearance, CL
R
, of a particular drug substance caused due to fi ltra-
tion is given by the expression:
CL
R
= (GFR) ( f
u
) (bl)
where f
u
is the free-fraction of drug.
From Equation (bl) one may obtain the renal clearance of a drug substance which gets eliminated only
due to fi ltration and can be determined provided the following two components are known:
∑ glomerular fi ltration rate (GFR) and
∑ free-fraction of the drug substance.
6.2.1. Two Substances Used to Determine GFR
Following two substances are commonly used to determine the GFR precisely, such as:
1. Creatinine, i.e., an endogenous by product of muscle metabolism and
2. Inulin, i.e., a polysaccharide substance.
Importantly, it has been observed that both creatinine and inulin are essentially 100% eliminated in the
urine due to fi ltration via the kidneys and, therefore, the respective CL
T
values of these two aforesaid
constituents may be used judiciously as the reasonable acceptable estimates for the GFR.
6.2.2. Active Tubular Secretion (ATS)
The active tubular secretion (ATS) serves as another primary process closely involved in the renal elimi-
nation of drugs. In fact, there are quite a few active-transport systems prevailing in the proximal renal
tubule which are solely responsible of the critical excretion of the drugs from the circulating blood into
the urine. Interestingly, one may come across a multiplicity of systems for the ATS of both cations and
anions. There should always be a net tubular secretion together with the clearance by fi ltration in a situ-
ation when the prevailing renal clearance (CL
R
) comes out to be greater than:
Pharmacokinetics
177
∑ product of GFR and
∑ free fraction ( f
u
).
Hence, the renal clearance due to ATS (CL
ATS
) may be expressed by the following equation:
()
()
()
RP u u,s,int
ATS
RP u u,s ,int
C
CL
L
CL
Qf
Qf
=
+
(bm)
where
Q
RP
is the effective renal plasma fl ow and
CL
u,s,int
is the unbound intrinsic secretary clearance.
Importantly, for such drugs which undergo both ‘fi ltration’ and ‘active tubular secretion (ATS)’, the
actual renal clearance (CL
R
) is given by the simple sum total of:
∑ clearance due to ‘fi ltration’ and
∑ clearance due to ‘secretion’,
which may be given as
CL
R
= (GFR) ( f
u
) + CL
ATS
(bn)
6.2.3. Kinetics of Urinary Excretion
In a broader perspective, it is possible to obtain useful information(s) based upon the urinary data per-
taining to the elimination kinetics of a drug substance, particularly in the absence of available plasma
level time data. Therefore, the kinetics of urinary excretion has several glaring advantages in the analysis
of an established pharmacokinetic system. These essentially comprise the following:
1. It serves as a useful technique to determine precise concentration of drugs in plasma in the absence
of a sensitive accurate procedure.
2. The present methodology is defi nitely a non-invasive procedure and hence, proves to be more
patient (subject) friendly and improved patient compliance.
3. It is a lot convenient to collect ‘urine samples’ at regular intervals in comparison to drawing blood
samples periodically.
4. For estimating the ‘urine-drug concentrations’ one may require a less sensitive method vis-à-vis the
‘plasma-drug concentrations’.
Note
In case, the urine-drug concentrations are found to be in a much lower range, assaying of ‘larger
sample volumes’ need to be carried out.
5. These generated data may help to determine in the effective computation of such vital factors:
∑ fi rst-order elimination,
∑ excretion and absorption rate constants and
∑ drug-fraction excreted unchanged.
Essentials of Biopharmaceutics and Pharmacokinetics
178
6. Besides, the fi rst-order metabolism, and/or extrarenal excretion rate constant may also be duly
calculated accordingly from the difference:
(K
E
– K
C
) = K
m
where
K
E
is the fi rst-order constant of the elimination phenomenon, (or overall elimination rate constant).
K
C
is a constant for the drug concentration and
K
m
is the rate constant for ‘hepatic elimination’ (see Figure 3.8).
7. Both absolute and relative bioavailability can be measured directly and this is quite possible with-
out the necessity of the fi tment of data into a known mathematical model.
8. In association with the plasma level-time data, one may also make use of the urinary excretory data
(or pharmacokinetics of urinary excretion) in order to estimate precisely the renal clearance of the
unchanged drug as per the following expression:
R
Total quantum of ‘drug’ unchanged
CL
Area under the plasma-level-time curve
=
Computation of Volume of Distribution (V
d
) and Renal Clearance (CL
R
) from Pharmacokinetics
of Urinary Excretion Data It has been observed that if V
d
is known, one may calculate the following
two values easily:
∑ total systemic clearance and
∑ non-renal clearance.
In actual practice, it may not be possible to compute the V
d
and CL
R
from the urine data exclusively.
Nevertheless, the estimated renal clearance (CL
R
) from pharmacokinetics of urinary excretion data may
not be an ‘accurate equivalent substitute’ for the plasma concentration-level data. Therefore, the data
obtained could be utilized judiciously as a more or less representation of a ‘rough estimate of the phar-
macokinetic parameters’.
At this point in time, two typical situations may arise, namely:
∑ Dosage form: showing an extremely slow and sluggish drug-release pattern and
∑ Drug product: exhibiting a very prolonged biological half-life (t
1/2
).
Consequently, the ultimate low-level of urinary drug concentration may be too dilute for enabling a
precise determination of the ‘drug’ released.
Importantly, in the second instance, viz. for such drug substances having a long biological half-life
(t
1/2
), one is required to collect urine output for several days at a stretch in order to determine the total
drug excreted.
Factors for Generating Valid Renal Clearance (CL
R
) from Pharmacokinetics (or Urinary Excretion
Data Profi le) Following are the 10 most important and critical factors that are invariably required for
generating valid renal clearance (CL
R
) based on the pharmacokinetic evaluation studies:
1. Excretion of an appreciable quantum of drug in an unchanged form in the urine up to a level of
at least 10%.
2. The selected ‘analytical procedure’ for the assay of unchanged drug should be highly specifi c, and also
it must not indulge in any sort of interference with the ensuing metabolites.