Transcript Slide 1

Pharmacokinetics –
a practical application of calculus
November 4, 2009
Elena Ho
Protein Bioanalytics / Pharmacokinetics
Protein Therapeutics
1
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Dosing Regimen
How much ?
How often ?
Oral bio-
Half-life
availability
Permeability
Efflux
Aqueous
Solubility
Volume
of Distribution
Clearance
Absorption
Metabolic
Stability
Renal
Excretion
Biliary
Excretion
CNS
Protein
Penetration Binding
Tissue
Binding
Typical Study Tools
Barriers between Dose and Target
Kerns & Li 2003, DDT 8:316-323
Compartment models
A compartment is an entity which can be described by a
definite volume and a concentration (of drug)
Concentration
Dose
V
Dose (mg) = C (ug/ml) x V (ml)
V = Dose/Concentration
One compartment model: the drug enters the body, distributes instantly
between blood and other body fluid or tissues.
Model
Hydrodynamic analogy
Drug in
1. One compartment
Drug in
Drug out
Drug out
__________________________
2. Two compartment
Drug in
central
tissue
Drug out
____________________________
3. Three compartment
Tissue 1
Drug in
Tissue 1
central
Drug in
central
Tissue 2
Tissue 2
Drug out
Drug recycle
Drug out
The human body is a multimillion compartment
model considering drug concentration in
different organelles, cells, or tissues
We have access to only two types of body fluid –
blood and urine
We will begin with the simplest model
Single dose, IV, one compartment : dose of drug introduced rapidly and completely and
quickly distributes into its homogenous volume of distribution. Drug is then eliminated by metabolism and excretion.
Then : dA/dt = - kel A
where kel = ke + km
rearrange to : dA/A = - kel dt
Integrate:
Gives:
A
A0 dA/A = - kel
∫
|
A
ln A A0= - kel t
∫
t
t0 dt
t
|t
0
or ln A – ln A0 = - kel . t – t0
This yields the familiar exponential or logarithmic expressions
C0
A = A0 e – Kel t
C = C0 e
– Kel t
- Kel/2.3
log C
log C = log C0 – kel . t /2.3
Kel = 2.3/t . log C/C0
time
I. Biological half-life (T1/2)
Consider again the rearranged expression
dA/A = - kel dt
Integrate between limits A and A/2
t/2
A/2
∫A
dA/A = - kel
Gives:
∫t
0
dt
ln A – ln (A/2) = kel t1/2
ln 2 = kel t1/2 = 0.693
Therefore:
t1/2 = 0.693 / kel
II. Area Under the Curve (AUC)
The integral of drug blood level over time from zero to infinity
and a measure of quantity of drug absorbed in the body
Area = A o  ∞
Sum of all concentration from t0 to t∞
i)
Linear trapezoidal method: AUC t1t2 = Area of a trapezoid t1t2
= (t2 – t1). (C2+ C1)/2
ii) Log trapezoidal method: AUC t1t2 = (t2 – t1). (C2+ C1)/ln(C2/C1)
iii) Lagrange method: cubic polynomial equation
iv) Spline method: piecewise polynomials for curve-fitting
2000
1500
1000
Observ ed
Predicted
500
0
0
5
10
15
20
25
30
time (hour)
35
40
45
50
Linear and/or Log trapezoidal method
2000
1500
1000
Observ ed
Predicted
500
0
0
5
10
15
20
25
30
35
40
45
50
time (hour)
T1, T2, T3,
T4,
T5,
T6
T7
Advantages: Easy to use. Reliable for slow declining or ascending curves
Disadvantages: error-prone for data points with a wide interval; over or under
estimate the true AUC; log 0 is not defined; not good for multiexponential curve
III. Clearance (Cls)
Clearance is a measure of the ability of the body to eliminate a drug
from the blood circulation.
Cls = elimination rate of drug from body/drug concentration in
plasma
Cls = - [dA(t)/dt]/Cp(t)
Integrate between limits 0 and ∞
∞
Cls =
∫t
0
∞
[dA(t)/dt] dt /
∫t
0
Cp(t) dt = A (0) - A(∞)/AUC 0-∞, iv
Gives: A (0) - A(∞) = total amount of drug (or total dose)
AUC 0-∞, iv = total AUC
Therefore:
Cls = Dose/AUC
IV. Volume of distribution (V)
The apparent volume of distribution of a drug can be viewed as the
total amount of drug present in the entire body and the drug concentration
in plasma at any given time
V (t) = A(t)/Cp(t)
Vc = Dose iv / Cp(0)
Cp(0) is the concentration at time 0, and can be
backextrapolated from the slope
T1/2 = 0.693 x V/Cl
Therefore:
V = T1/2 X Cl / 0.693
Note: V is a concept, not a real value
Other terms:
Vd
Vz
Vβ
Vss
The following table is the plasma concentrations of cocaine as a function of
Time after i.v. administration of 33 mg cocaine hydrochloride to a subject.
Time (h)
Conc (ug/l)
0.16
0.5
1.0
1.5
2.0
2.5
3.0
170
122
74
45
28
17
10
Molecular weight of cocaine Hydrochloride = 340 g/mole; MW of cocaine = 303 g/mole
Prepare a semilog plot of plasma concentration versus time
Calculate the half-life
Calculate the total clearance
Given the body weight of the subject is 75 kg, calculate the volume of distribution
in L/kg
Source: adapted from Chow, M.J. et el (Clin. Pharm Ther 38:318-324, 1985) &
Tozer, T “Clinical Pharmacokinetics, 3rd edition, p33
220
200
180
160
140
120
100
80
60
40
20
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
time (hour)
1000
100
10
0.0
0.5
1.0
1.5
time (hour)
2.0
2.5
3.0
Kel = 0.9922
T1/2 = 0.7 hr
AUC = 194 h* ug/L
Cl = 145 L/h
V = 146L or 1.9L/kg
In vivo Pharmacokinetics in Rodents
Distribution
Plasma Concentration
Disposition kinetics:
• single iv administration
• repeated blood sampling
• plasma concentration-time profile
Plasma Half-life:
T1/2 =
ln 2
kel
Plasma Clearance:
CL =
Dose
AUC
Elimination
AUC(inf)
Volume of
Distribution at
steady-state:
Vdss =
CL
kel
The clearance of compounds is evaluated in
relation to the liver blood flow which is
20, 60, and 90 mL/min/kg in human, rat, and mouse,
respectively.
kel
Time
T1/2 = 0.693 x Vd/CL
The volume of distribution should exceed that of
total body water, i.e. 0.6-0.7 L/kg which indicates that
the compound distributes freely into tissues.
Distribution
phase
Elimination
phase
Log A
Log Cp(t)
Slope = - β/2.303
Log B
Slope = - α/2.303
A 2-compartment model = biexponential equation
Cp(t) = A*e –α*t + B*e – β*t
At time 0, Cp(0) = A + B
Vc = Dose/Cp(0) or Dose/(A + B)
Example of a pharmacokinetic study:
single dose IV in the rat
Study design
Animal : Sprague-Dawley male rat, approximately 10 weeks old weighing
~250 g each (n=4)
Compound : BAY xxxxxx supplied by ABC, lot # AaBbCc
Dosing : each animal will receive a dose equivalent to 0.7 mg/kg.
Time points: pre dose, 5 min, 30 min, 1, 2, 4, 7, 24, 28, 31, 48 hours post dose
Blood sample : collect 225 ul of blood in 25 ul of 5% Na Citrate at each time
point. Centrifuge blood at 5000 g for 5 minutes. Separate the plasma and
keep at -80ºC until analysis
SUMMARY OF RESULTS: Plasma concentration in ng/ml:
SUMMARY OF RESULTS: Plasma concentration in ng/ml:
animal #
rat 1
rat 2
rat 3
rat 4
Mean
animal wt (kg)
0.293
0.310
0.306
0.292
0.300
dose
volume(ml)
0.30
0.31
0.31
0.30
0.305
predose
<LLOQ
<LLOQ
<LLOQ
<LLOQ
0.083
6570.0
8115.7
12997.3
3724.8
0.5
5455.0
5630.4
9316.9
1
3935.3
4809.3
2
3873.1
4
SEM
%CV
7851.9
2241.5
49.4
6165.9
6642.1
1044.2
27.2
8203.2
4525.1
5368.2
1111.2
35.9
3932.2
6542.8
3457.6
4451.4
814.2
31.7
1952.7
2050.6
2758.8
2092.2
2213.6
212.6
16.6
7
1126.0
1385.8
1298.6
1305.4
1279.0
63.2
8.6
24
175.8
208.2
217.9
177.8
194.9
12.3
10.9
28
132.2
146.0
148.5
133.8
140.1
4.8
5.9
31
125.8
174.3
153.4
129.0
145.6
13.1
15.6
48
40.4
43.5
53.4
40.0
44.3
3.6
14.1
time (hr)
LLOQ = 7.81 ng/ml
Retain = 625 ug/ml (target: 700 ug/ml)
Pharmacokinetic parameters:
Animal No.
rat 1
rat 2
rat 3
rat 4
Mgeo
Sdgeo
Mari
Sdari
CV
Dose
[mg/kg]
0.700
0.700
n.c.
0.700
0.700
1.00
0.700
0.00
0.00
AUC
[µg·h/L]
31044
35403
n.c.
31989
32760
1.07
32812
2293
6.99
AUCnorm
[kg·h/L]
44.3
50.6
n.c.
45.7
46.8
1.07
46.9
3.28
6.99
%AUC(t last -∞)
[%]
2.16
1.96
n.c.
2.04
2.05
1.05
2.05
0.0968
4.72
CLplasma
[mL/h/kg]
22.5
19.8
n.c.
21.9
21.4
1.07
21.4
1.45
6.77
Vss
[L/kg]
0.204
0.183
n.c.
0.201
0.195
1.06
0.196
0.0114
5.85
MRT
[h]
9.04
9.23
n.c.
9.17
9.14
1.01
9.15
0.0988
1.08
t1/2
[h]
11.3
10.6
n.c.
11.1
11.0
1.03
11.0
0.372
3.38
t1/2,a
[h]
1.55
0.790
n.c.
0.573
0.889
1.66
0.972
0.515
53.0
Com1: BAY 861789 = BeneFIX lot #D26525 // Com2: rat 3
was eliminated due to dosing line was used in sampling
Summary
This compound represents a 2-compartment model.
Elimination T1/2 = 11.0 hours
Total plasma clearance = 21.4 mL/h/kg
Vss = 195 mL/kg
Remark
This profile suggests a slow clearance compound with a moderate elimination
half life. The volume of distribution at steady state is high, suggesting the
compound distribution is beyond the plasma volume compartment
Absorption
GI Tract
Stomach:
Dissolution
Stability at pH 1
Intestines: Dissolution
Stability at pH 3-8
Permeability
Metabolic stability
Compound properties controlling absorption:
• size
MW
• aqueous solubility
Sw
• lipophilicity
logP
• polarity
PSA
• ionization
pKa
• ...
Absorption
Deriving Models of the Gastrointestinal Tract
Oral bioavailability: Barriers and In vitro Models
Fraction of
dose absorbed:
FA%
Gut Wall
Liver
Oral
bioavailabilit
F%
Oral Absorption
limited by:
Stability
Solubility
Permeability
ATP-dependent Efflux
Drug Metabolism
Hepatocellular Uptake,
Drug Metabolism and
Biliary Excretion
In Vitro Models:
Gastric and
Intestinal
Juice
Phys.-Chem. Descr.
Caco-2
Liver Microsomes
Hepatocytes
S9 mix, Cytosol
Intest. Microsomes
In vivo Pharmacokinetics in Rodents
Oral kinetics:
Cmax
• single
Max. plasma conc. and
Time of max. pl. conc.
Tmax
Distribution
Plasma Concentration
po administration
• repeated blood sampling
• plasma concentration-time profile
Absorption
Elimination
kel
AUC(inf)
Time
Tmax
Oral Bioavailability:
Cmax
F=
There is no possibility to extrapolate the bioavailability in
rodents to that in man. The sources of its limitation are often
more important than the actual value as this information may
allow to study the corresponding mechanism using human in
vitro systems and to extrapolate the expected bioavailability .
AUC po / Dpo
AUC iv / D iv
x 100%
Extravascular application: oral, subcutaneous, inhalation…etc
conc
Time
Sustain release: oral, dermal patches
conc
Time
IV infusion
Conc
Time
Plasma conc. -Time profile after an oral dose of 10 ug/kg in rabbits
0.07
0.06
0.05
0.04
animal 3
0.03
animal 4
animal 8
0.02
0.01
0.00
0
1
2
3
4
time (hour)
5
6
7
8
Simulation of plasma conc. vs. time profile after multiple doses
Rabbit IV 81.7 IU/kg dose daily for 14 days
1600
1400
1200
1000
800
600
400
200
0
0
50
100
150
200
250
Time (hour)
300
350
400
450
500
Modeling of increasing doses
.
.
821581 (CCR5)
40
35
30
25
20
15
10
5
0
0
0.25
0.5
Time (hour)
0.75
1
Predicting Human PK
• Direct Scaling of in vitro rate of metabolism to the CL in vivo
in vitro
CL
c
t
in vivo
CL
• physiologically based
• metabolic CL only
•  first-pass effect
•  oral bioavailability
• Allometric Scaling of human PK based on animal data in vivo
CL
Vdss
• empirical
• total CL and Vss
• requires mech. to be scalable
•  t1/2
body weight
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