Transcript Slide 1

Pharmacokinetics –
a practical application of calculus
April 6, 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
Interesting facts about a human body
Absorbing surface area of skin: 1.73 m2
 Absorbing surface area of the lung: 70 m2
 Absorbing surface area of GI tract: ~200m2
(1/2 basketball court)
 Small intestine is ~2” around, 22’ long
 Total length of capillaries is ~ 37,000 miles

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
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
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
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
kel
Time
T1/2 = 0.693 x Vd/CL
The clearance of compounds is evaluated in
The volume of distribution should exceed that of
relation to the liver blood flow which is
total body water, i.e. 0.6-0.7 L/kg which indicates that
60 and 90 mL/min/kg in rat and mouse, respectively. the compound distributes freely into tissues.
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%
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 AABBCC. Dissolve 0.7 mg in 10 ul of
DMSO, bring it up to 1 mL with PBS.
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 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:
animal #
animal wt (kg)
dose volume(ml)
rat 1
rat 2
rat 3
rat 4
0.278
0.28
0.296
0.30
0.295
0.30
0.29
0.29
<LLOQ
<LLOQ
<LLOQ
<LLOQ
2259.1
1045.4
754.5
1888.1
977.5
639.0
2044.2
1005.3
678.9
2162.7
1095.3
838.2
519.2
271.2
251.7
444.9
238.6
196.5
513.4
273.3
177.2
415.5
254.1
209.8
30.8
25.5
18.8
30.7
20.2
18.0
35.6
21.5
16.4
36.6
23.0
16.8
time (hr)
predose
0.083
0.5
1
2
4
7
24
28
31
LLOQ = 15.6 ng/ml
Retain = 866 ug/ml
Rat plasma concentration was determined using ELISA immunoassay method:
MW:
2/3 rule
fu [%]:
Animal No.
rat 1
rat 2
rat 3
rat 4
Mgeo
SDgeo
LOG
SDgeo
Mari
orig. dose
SDari
CV
orig. dose
orig.
orig.dose
dose
[mg/kg]
0.7
0.7
0.7
0.7
norm.
norm.dose
dose
[mg/kg]
1
1
1
1
[#]
[h]
time point
time
1
0
2258
1960
2093
2150
2113
1.06
0.0256
2115
124
5.87
2
0.083
2259
1888
2044
2163
2084
1.08
0.0336
2089
160
7.66
3
0.5
1045
978
1005
1095
1030
1.05
0.0214
1031
51.2
4.96
4
1
754
639
679
838
724
1.13
0.0519
728
87.9
12.1
5
2
519
445
513
416
471
1.12
0.0475
473
51.2
10.8
6
4
271
239
273
254
259
1.07
0.0276
259
16.3
6.27
7
7
252
196
177
210
207
1.16
0.0641
209
31.6
15.1
8
24
30.8
30.7
35.6
36.6
33.3
1.10
0.0405
33.4
3.12
9.32
9
28
25.5
20.2
21.5
23.0
22.5
1.11
0.0434
22.5
2.28
10.1
10
31
18.8
18.0
16.4
16.8
17.5
1.06
0.0270
17.5
1.09
6.24
[µg/L]
concentrations
plasma conc. of BAY 877030
Xxxxxx(Prep.No.= WANG1010-1-1) after iv bolus
administration of 0.7mg/kg to male Sprague Dawley Rat, (n=4 of 4)
10000
rat 1
rat 2
rat 3
rat 4
conc. [µg/L]
1000
100
10
0
5
10
15
20
25
30
time [h]
Pharmacokinetic parameters:
Animal No.
rat 1
rat 2
rat 3
rat 4
Mgeo
Sdgeo
Mari
Sdari
CV
Dose
[mg/kg]
0.700
0.700
0.700
0.700
0.700
1.00
0.700
0.00
0.00
AUC
[µg·h/L]
5603
4819
5047
5299
5184
1.07
5192
337
6.49
AUCnorm
[kg·h/L]
8.00
6.88
7.21
7.57
7.41
1.07
7.42
0.481
6.49
%AUC(t last -∞)
[%]
3.01
3.31
3.19
3.19
3.17
1.04
3.17
0.122
3.84
CLplasma
[L/h/kg]
0.125
0.145
0.139
0.132
0.135
1.07
0.135
0.00873
6.45
Vss
[L/kg]
0.945
1.12
1.05
1.02
1.03
1.07
1.04
0.0736
7.11
MRT
[h]
7.56
7.73
7.58
7.73
7.65
1.01
7.65
0.0916
1.20
t1/2
[h]
6.63
6.85
6.79
6.81
6.77
1.01
6.77
0.0974
1.44
t1/2,a
[h]
0.588
0.592
0.583
0.681
0.610
1.08
0.611
0.0469
7.67
This compound represents a 2-compartment model.
Com1: BAY 877030 = HKB11-R338A-HG3 (HTI AHIX-5041 was
used as capture antibody in the assay)
Summary
Elimination T1/2 = 6.8 hours
Total plasma clearance = 135 ml/h/kg
Vss = 1.03 L/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
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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