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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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collected) in each interval is computed and all these amounts are added to help
estimate the renal clearance as
CL
ren
¼
X
U
t
AUC
ð5:1Þ
where SU
t
is the cumulative amount of unchanged drug excreted in the urine from all
collections and the AUC is the plasma area under curve (AUC) for the duration of the
collection.
It is important that times for AUC calculation should match the time duration for
which the urine was collected. If one uses the AUC
1
value in the denominator of
Equation 5.1, then the numerator is the value of U
1
, whi ch is the complete excretion
of the drug through this pathway (Figure 5.4).
Glomerulus: Filtration
Proximal tubule:
reabsorption
Tubular Secretion
Distal tubule: Tubular
Loop of Henle
Common collecting tubule
Figure 5.3 Kidney diagram of excretory pathways (www.abbysenior.com/biology/kidney_
system.htm).
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
30
25
20
15
10
5
0
TIME
CUMULATIVE AMOUNT EXCRETED IN URINE
Uinfinity
Figure 5.4 The cumulative amount of drug excreted in the urine reaching an asymptote
indicating that all drug has been excreted through this pathway.
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The fraction of the dose that is excreted through the urinary pathway is shown in
Equation 5.2. It is important to collect urine for at least 4–6 half-lives of plasma
disposition. The collected urine data will yield a more accurate estimate of renal
clearance. It is also important to recognize the relationship, where
U
1
Dose
¼
CL
ren
CL
tot
¼
k
ren
k
e
ð5:2Þ
where k
ren
is the first-order rate constant for renal elimination (from plasma).
A semimechanistic understanding ([15], pp.169–178) of the value of the unbound
renal clearance CL
ren
/fu ( fu ¼ fraction unbound to plasma proteins) is made by
comparing it to the glomerular filtration rate (GFR) in an animal [16].
If the CL
renal
¼ fu GFR, then it is assumed that the only renal excretory
mechanism for the drug is filtration (GF), which is considered a first-order (diffusion-controlled) process or it is filtration with equal amounts of TS and TR such that
the latter cancels out.
If the CL
renal
> fu GFR, then it is assumed that the renal excretory mechanism
involves GF as well as TS and perhaps a small amount of TR.
If the CL
renal
< fu GFR, then the excretory mechanism involves GF with a large
TR but perhaps an insignificant TS.
5.2.4.2 Biliary Excretion The biliary process in mammals is responsible for
aiding in the saponification, digestion, and absorption of fats. The bile is formed
in the canaliculi of the liver and accumulated into the gall bladder (except in rodents
who do not have a gall bladder and continuously secrete bile into the intestine).
The bile is concentrate d in the gall bladder and eventually secreted into the intestine.
The biliary excretory system is also controlled by a plethora of transporters similar to
the tubules of the kidneys. Some of the major transporters are P-gp, MRP1, and MRP2
(Figure 5.5). These transporters maintain bile physiology and are responsible for
excreting toxic chemicals [17]. Many drugs are substrates and are excreted through
these transporters. There have been no examples of any uptake transporters that
reabsorb drugs back from the bile. It has been observed that typically drugs excreted
through the biliary pathway have a molecular weight > 350 Da.
The biliary data is treated very similar to the urine data. Bile is collected in intervals
and the cumulative amounts excreted are determined. The complete biliary excretion
profile is as shown in Figure 5.6.
Using a similar approach to the data analysis, a biliary clearance value can be
calculated from the bile data as
B
1
Dose
¼
CL
bil
CL
tot
¼
k
bil
k
e
ð5:3Þ
where k
bil
is similar to k
ren
in Equation 5.2
Noteworthy, the biliary clearance is a part of the hepatic clearance and cannot
exceed it (Equation 5.31).
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For all drugs, it is important to have multiple pathways of clearance so that if any
one pathway is immobilized due to saturati on or a DDI, the other pathways can take
over and clear out the drug, thereby reducing the potential for an adverse event to
occur. Although the potential for urinary or biliary excretion is difficult to engineer in
a drug, the discovery team should consider advancing compounds with multiple
pathways of clearance preferably over those that do not possess such properties.
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
30
25
20
15
10
5
0
TIME
CUMULATIVE AMOUNT EXCRETED IN BIL
Binfinity
Figure 5.6 Biliary excretion profile.
b
Sinusoid
Hepatocyte
Bile ductule
Periductular capilary plexus
Liver transporters
MRP3
Diffusion
FAB P
(fatty
acids)
(bile salts)
(hydro-
philic
anions)
(hydrophobic
anions and
cations)
(type 1
cations)
(type 2
cations)
NTCP OAT
OATP OCTP OCTP
MRP1
MRP2
(bile salts)
(phospholipids)
(cations)
(bile salts)
MDP3
BSEP
MDR1
MRP6
Sinusoid
HA H
2CO2
HCO
3
–
HCO3–
CO
2
A
–
A
–
A
–
A
–
A
–
A
–
HA
A
–
HAHAH
+
A
–
A
–
H
+
Bile
canaliculus
Lateral
membrane
Figure 5.5 The biliary excretion apparatus in the liver.
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5.3 THE MATHEMATICS OF PHARMACOKINETICS
The rate of decline of the amount (or concentration) of a drug in the body depends
on the amount present at that time. This kinetic first-order process is depicted by a
differential equation of the form
dC
dt
¼ k C ð5:4Þ
where dC/dt is the rate of change of concentration, which is proportiona l to the firstorder of C. The negative sign indicates the declining concentration. The proportionality constant k is a first-order rate constant and is an indication of the rate of the
decline.
Separating the variables in Equation 5.4
dC
C
¼ k t ð5:5Þ
Integrating throughout provides
Ln Cj
t
0
¼ k t ð5:6Þ
which simplifies to
C ¼ C
0
e
kt
ð5:7Þ
where C
0
is the concentration at time ¼ 0. Transforming both sides of Equation 5.7 by
taking the natural log (log to the base “e”) obtains
Ln C ¼ Ln C
0
k t ð5:8Þ
It is for this reason that an exponential equation of the form in Equation 5.7 when
plotted on a rectilinear graph gives a curved profile, while the same equation plotted
on a semilog paper is a straight line as in Equation 5.8.
Note the semilog plots seen in pharmacokinetic figures are based on the log to the
base 10 (abbreviated simply as Log). This form of log transformation is only meant
for visualization purposes and no calculations are usually undertaken in this format.
However, it is easy to transform the natural log “Ln” to the base log base 10 “Log” as
Ln (X) ¼ 2.303 Log (X).
As long as the pharmacokinetics remains linear (i.e., the rate of change of
concentration is proportional to the concentration), the equations in PK are very
symmetrical and all PK data can be explained by a sum of exponentials depending
on the number of kinetic phases seen in the semilog plots of concentration versus
time. Just as the plasma or blood data (when plotted) reflect the route of administration and type of dosing, these equations reflect the curves and hence the route of
THE MATHEMATICS OF PHARMACOKINETICS 211
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administration and type of dosing (Table 5.2). Further, based on the number of
exponentials used to describe the plasma data, a number of distinct body spaces
(arbitrarily based on blood perfusion concepts—to be discussed later) are assigned to
the disposition of the drug.
5.3.1 Compartmental Versus Noncompartmental Analysis
In a discovery setting and particularly at the early stages of lead identification, PK data
analysis using simple noncompartmental methods is adequate to help in estimating
most of the basic PK parameters. Although, commercially available software
packages readily guide through the noncompartmental data analysis process, these
analyses can be done in Microsoft Excel or in some cases on a hand-held calculator.
Many of the LIMS (laboratory information management systems) used to capture
bioanalytical data have built-in PK parameters estimation capability and typicall y use
the noncompartmental methods of estimation.
If PK data is required to understand mechanistic aspects (how many distinct body
compartments, movement of drug from each compartment, rates of metabolism,
sequential metabolism, multiple excretory pathways, etc.) of the disposition of the
drug, then compartmental sums of exponentials modeling is a must. Some of the basic
models can be managed using Microsoft Excel or a hand-held calculator, but it is
advisable to do the calculations and curve fitting on commercially available software
platforms, many of which are tailored for PK data analysis.
A comparison of noncompartmental and compartmental method of PK data
analysis is shown in Table 5.3.
5.4 DRUG ADMINISTRATION AND PK OBSERVATIONS
Once the drug has been administered, the blood, urine, bile, feces, and other tissues
(depending on the objectives of the study) are sampled and submitted for chemical or
biochemical analysis using sensitive, reproducible, and rugged analytical methods.
TABLE 5.2 Sum of Exponentials Equations Used to Model Typical PK Data
Route/Type Equation Exponentials Compartments
IV/Bolus C ¼ C
0
e
kt
11
IV/Bolus C ¼ A e
at
þ B e
bt
22
IV/Bolus C ¼ A e
at
þ B e
bt
þ C e
g t
33
IV/Infusion C ¼ C
0
e
kt
þ Css1 e
kt
11
EV/Bolus C ¼ I e
ket
e
kat
21
EV/Bolus C ¼ A e
at
þ B e
bt
D e
kat
32
Note: (1) The number of exponentials (or compartments) is based on the number of distinct parameters
associated with the exponent. (2) The negative sign before the exponent indicates absorption or increasing
concentrations from an infusion.
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The actual time of data (not the nominal protocol specified times) and sample
collection is recorded and this time is used for PK calculations. The resultant data
is analyzed and interpreted.
Depending on the matrix, each PK data set is treated differently and subsequently
plotted to examine the profile depending on the route of administration and the dose.
Blood/Plasma Data Analysis The blood/plasma concentrations are plotted
against time on a rectilinear graph and examined for inconsistencies. The data
is subsequently plotted on a semilogarithmic graph and checked to identify the
number of distinct kinetic phases. Based on this observation, an initial estimate
is made on the type of PK model that may be applied to analyze the data.
Excreta Data Analysis Excreta (urine, bile, feces)—The amounts excreted during
a collection interval (0–4, 4–8, etc.) are estimated and the cumulative amounts
up to the interval are calculated. The cumulative amounts are plotted against
time to establish the extent of excretion (fraction of the dose) and estimate
clearance parameters through the particular excretion pathway.
Tissue Data Analysis The concentration of the drug in the tissue expressed as
concentration of the drug per gram of tissue or per milliliter (if the density is
estimated) of tissue matrix is plotted similar to the blood and plasma data. The
tissue data can be overlaid on the same graph as the plasma data to assess the
kinetics of the drug in the tissue.
5.4.1 Analysis of Intravenous PK Data
The plasma (blood) data following an IV bolus dose if administered correctly always
shows a declining profile as time increases (Figure 5.7).
The following parameters can be derived from the plasma (blood) PK profile
shown above including the area under the curve, clearance, mean residence time
(MRT), and half-life (t
1/2
) of the compound.
TABLE 5.3 Comparison of Noncompartmental and Compartmental Method of PK
Data Analysis
Noncompartmental Compartmental
Uses terminal phase data and AUC to derive
all other PK parameters
Uses the entire data to obtain the PK
parameters
Quick, easy, and high throughput. Ideal for
screening/binning compounds
Relatively difficult. Ideal for late-stage
compound development
Requires basic knowledge of math/statistics Requires a more detailed understanding
of math/statistics
Not as detailed and provides no mechanistic
interpretation of the data
Very robust and can provide mechanistic
detail of drug disposition
Not helpful in simulations or for the
treatment of PK–PD data analysis
Valuable for future predictions, simulations,
and treatment of PK–PD data
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5.4.1.1 Area under the Curve The area under the plasma (blood) concentration
versus time curve is a scalar quantity that measures drug exposure. This PK parameter
cannot be compared across drugs but for a given drug the AUC values for different
doses can be compared. The AUC parameter is useful in toxicology, biopharmaceutics, and pharmacokinetics.
AUC in Toxicology Since AUC is a measure of drug exposure, the AUC values
plotted against dose (Figure 5.21) can be evaluated to assess dose linearity. Dose
linearity is when AUC is proportional to dose (solid line).
Thus if the AUC values increase more than proportionally to increasing dose
(dashed line), then a clearance pathway has been saturated resulting in a systemic
nonlinearity. If the AUC values plateau as dose increases (dotted line) then either
absorption is saturated or nonlinear protein binding occurred (Figure 5.21).
AUC in Biopharmaceutics The AUC value can be used for a direct comparison of
different formulations of the same drug. In bioequivalency testing, the AUC values
following administration of a generic product must be within 80–120% of the value of
ONE COMPARTMENT BODY MODEL
0
0.2
0.4
0.6
0.8
1
1.2
302520
15
105
0
TIME
CONCN. (mass/volume)
ONE COMPARTMENT BODY MODEL
0.01
0.1
1
30
25
20151050
TIME
CONCN. (mass/volume)
Figure 5.7 Rectilinear (top) and semilogarithmic (bottom) IV PK.
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the innovator product administered to the same individuals. In a discovery setting, the
AUC values can be used to compare the performance of different formulations or the
impact of salt forms, or crystal structure of the same drug administered at the same dose.
AUC in Pharmacokinetics The AUC values can be used to determine other PK
parameters such as the total body clearance, fraction bioavailable, and mean residence
time.
Estimating AUC The easiest method of calculating the AUC is the linear trapezoidal rule. The trapezoidal rule is a general purpose method of graphically calculating the AUC (can be used for any type of curve) and is independent of the
pharmacokinetics associated with the curve.
The technique involves breaking down the curve into individual trapezoids
(Figure 5.8) and estimating the area under each trapezoid. Adding all the trapezoids,
then gives the area under the entire curve
One can further break down each trapezoid into a triangle (gray) and a rectangle
(black) and estimate the area of the triangle as 1/2 base height and the area of
the rectangle as length multiplied by width (Figure 5.9). Adding these two parts gives
the area of the trapezoid as follows.
Once all the trapezoids are summed, the area under the curve up to the last sampled
time point is obtained:
AUC
last
¼
X
Cnþ C
n 1
2
ðt
2
t1Þð5:9Þ
Since the plasma levels decline exponentially (approach but never get to zero;
as t !1, C ! 0), the area under the curve to infinity is a measure of the complete
exposure of the drug:
AUC
1
¼ AUC
last
þ
C
last
k
e
ð5:10Þ
Figure 5.8 A series of trapezoids making up the area under the curve.
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where C
last
is the last measured time point and keis the slope of the terminal phase of
the log transformed plasma concentration versus time profile.
If the plasma concentration versus time data has been fit to a sum of exponentials
then the AUC
1
is calculated mathematically using
AUC
1
¼
ð
1
0
C dt ¼
X
n
1
C
i 1
l
i
ð5:11Þ
and thus for a monoexponential decline (one-compartment model), the AUC
1
will be
AUC
1
¼
ð
1
0
C dt ¼
ð
1
0
C0 e
kt
dt ¼
C
0
k
e
ð5:12Þ
and for a biexponential decline (two-compartment model) the AUC
1
will be
AUC
1
¼
ð
1
0
C dt ¼
ð
1
0
ðA e
at
þ B e
bt
Þdt ¼
A
a
þ
B
b
ð5:13Þ
Similar expressions can be derived for every sum of exponential modeling
analysis, and most commercially available software will calculate the AUC value
by this method in compartmental analysis.
5.4.1.2 Clearance Every drug is a foreign substance and the body tends to remove
it once detected in circulation. In a first-order kineti c domain, this elimination is
proportional to the concentration of the drug. The clearance of the drug is the
proportionality constant between the rate of elimination and the concentration of the
Figure 5.9 An expanded view of the trapezoid to measure the AUC. Area of trapezoid ¼
1
/2(C1þ C2) (t2– t1).
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drug (usually measured in plasma, although it can refer to an individual tissue)
such that
dE
dt
/ C ð5:14Þ
where dE/dt is the rate of elimination and C is the plasma concentration. Solving
Equation 5.14 gives
dE
dt
¼ CL C ð5:15Þ
where CL is the proportionality constant called “clearance.” Solving for clearance
(separating the variables and integrating from zero to infinity) gives
CL ¼
Dose
AUC
1
ð5:16Þ
where dose is the dose that is syst emically available. The formula in Equation 5.16
is most often used for estimating clearance and although it does not help to define
the term, it is easy to note that CL is a volumetric term with units of volume/unit
time.
Clearance is defined as that volume of blood, which, when passing through an
organ per unit time, is completely cleared of drug. Thus, a CL value of 10 mL/min
would indicate that it takes 1 min to remove all traces of drug from 10 mL of blood.
Individual Organ Clearance Although drug clearance can take place from any
organ, the liver is the most significant organ for drug clearance and hence has been
studied extensively.
The liver is responsible for detoxifying the body of all foreign entities and
eliminating them from the body. Most drugs are metabolized by the liver through
the cytochrome P450 enzymes that biotransform the drug into more polar metabolites.
These polar metabolites are rendered ready for excretion and removed from the body.
Other organs that also play a significant role in clearing drugs are the kidneys, lungs,
blood, GI tract, skin, and so on.
The clearance estimated from Equation 5.16 is the total body clearance and is the
sum of all the individual clearance in the body so
CL
tot
¼ CL
hepatic
þ CL
renal
þ CL
other
ð5:17Þ
If the individual organ CL (renal CL as an example) is determined, one can use
Equation 5.18 to determine the fractional clearance of each organ as
f
renal
e
¼
CL
renal
CL
total
ð5:18Þ
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