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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 availability Half-life Absorption Clearance Volume of Distribution Permea bility Efflux Aqueous Solubility Metabolic Stability Renal Excretion Biliary Excretion CNS Penetration Protein 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

1. One compartment Drug in

Hydrodynamic analogy

Drug in Drug out 2. Two compartment Drug in central tissue Drug out ____________________________ 3. Three compartment Drug in Tissue 1 central Tissue 2 Drug out Tissue 1 Drug in central Tissue 2 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 = - k el A where k el = k e + k m rearrange to : dA/A = - k el dt A Integrate:

∫

A 0 dA/A = - k el

∫

t t 0 dt Gives: ln A

|

A A 0 = - k el t

|

t 0 t or ln A – ln A 0 = - k el . t – t 0 This yields the familiar exponential or logarithmic expressions C 0 A = A 0 e – Kel t C = C 0 e – Kel t log C = log C 0 – k el . t /2.3

log C - K el /2.3

K el = 2.3/t . log C/C 0 time

I. Biological half-life (T

1/2

)

Consider again the rearranged expression dA/A = - k el dt Integrate between limits A and A/2

∫

A/2 A dA/A = - k el

∫

t/2 t 0 dt Gives: ln A – ln (A/2) = k el t 1/2 ln 2 = k el t 1/2 = 0.693

Therefore: t 1/2 = 0.693 / k el

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 t 0 to t ∞ i) Linear trapezoidal method: AUC t1  t2 = Area of a trapezoid t1  t2 = (t 2 – t 1 ). (C 2 + C 1 )/2 ii) Log trapezoidal method: AUC t 1  t 2 = (t 2 – t 1 ). (C 2 + C 1 )/ln(C 2 /C 1 ) iii) Lagrange method: cubic polynomial equation iv) Spline method: piecewise polynomials for curve-fitting

2000 1500 1000 500 0 0 5 10 15 20 25 time (hour) 30 35 40 45 50 Observed Predicted

Linear and/or Log trapezoidal method 2000 1500 1000 500 Observed Predicted 0 0 5 10 15 20 25 30 35 40 45 time (hour)

T1, T2, T3, T4, T5, T6 T7

50 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 (Cl

)

Clearance is a measure of the ability of the body to eliminate a drug from the blood circulation.

Cl s = elimination rate of drug from body/drug concentration in plasma Cl s = - [dA(t)/dt]/Cp(t) Integrate between limits 0 and ∞ Cl s =

∫

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: Cl s = 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 T 1/2 = 0.693 x V/Cl Therefore: V = T 1/2 X Cl / 0.693

Other terms: Vd Vz V β Vss Note: V is a concept, not a real value

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

170 0.5

122 1.0

74 1.5

45 2.0

28 2.5

17 3.0

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, 3 rd edition, p33

220 200 180 160 140 120 100 80 60 40 20 0 0.0

1000 0.5

1.0

1.5

time (hour) 2.0

2.5

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

Disposition kinetics:

• single iv administration • repeated blood sampling • plasma concentration-time profile Elimination

AUC

(inf)

k el

Plasma Half-life: Plasma Clearance: T 1/2 = ln 2 k el CL = Dose AUC Volume of Distribution at steady-state: V dss = CL k el T 1/2 = 0.693 x Vd/CL

Time

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.

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 .

Log Cp(t) Distribution Elimination phase phase Log A 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 -80 ºC until analysis

SUMMARY OF RESULTS: Plasma concentration in ng/ml:

SUMMARY OF RESULTS: Plasma concentration in ng/ml : animal # animal wt (kg) dose volume(ml) rat 1

0.293

0.30

time (hr) predose

0.083

0.5

1 2 4 7 24 28 31 48 LLOQ = 7.81 ng/ml

5455.0

3935.3

3873.1

1952.7

1126.0

175.8

132.2

125.8

40.4

rat 2

0.310

0.31

5630.4

4809.3

3932.2

2050.6

1385.8

208.2

146.0

174.3

43.5

rat 3

0.306

0.31

9316.9

8203.2

6542.8

2758.8

1298.6

217.9

148.5

153.4

53.4

rat 4

0.292

0.30

Mean 0.300

0.305

SEM

6165.9

4525.1

3457.6

7851.9

6642.1

5368.2

4451.4

2241.5

1044.2

1111.2

814.2

2092.2

1305.4

177.8

133.8

129.0

2213.6

1279.0

194.9

140.1

145.6

212.6

63.2

12.3

4.8

13.1

40.0

44.3

Retain = 625 ug/ml (target: 700 ug/ml) 3.6

%CV

49.4

27.2

35.9

31.7

16.6

8.6

10.9

5.9

15.6

14.1

Pharmacokinetic parameters: Animal No.

Dose AUC AUC norm %AUC(t last -∞) CL plasma V ss MRT t 1/2 t 1/2,a [mg/kg] [µg·h/L] [kg·h/L] [%] [mL/h/kg] [L/kg] [h] [h] [h] rat 1 0.700

31044 44.3

2.16

22.5

0.204

9.04

11.3

1.55

rat 2 0.700

35403 50.6

1.96

19.8

0.183

9.23

10.6

0.790

rat 3 n.c.

n.c.

Com1: BAY 861789 = BeneFIX lot #D26525 // Com2: rat 3 was eliminated due to dosing line was used in sampling rat 4 0.700

31989 45.7

2.04

21.9

0.201

9.17

11.1

0.573

Mgeo 0.700

32760 46.8

2.05

21.4

0.195

9.14

11.0

0.889

Sdgeo 1.00

1.07

1.05

1.07

1.06

1.01

1.03

1.66

Mari 0.700

32812 46.9

2.05

21.4

0.196

9.15

11.0

0.972

Sdari 0.00

2293 3.28

0.0968

1.45

0.0114

0.0988

0.372

0.515

CV 0.00

6.99

4.72

6.77

5.85

1.08

3.38

53.0

Elimination T 1/2 = 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

GI Tract Absorption

Stomach: Dissolution Stability at pH 1 Intestines: Dissolution Stability at pH 3-8 Permeability Metabolic stability

Compound properties controlling absorption:

• size • aqueous solubility • lipophilicity • polarity • ionization • ...

MW Sw logP PSA pKa

Deriving Models of the Gastrointestinal Tract

Absorption

Oral bioavailability: Barriers and In vitro Models Fraction of dose absorbed: FA% Gut Wall Liver Oral bioavailability: F%

Oral Absorption limited by:

Stability Solubility

In Vitro Models:

Gastric and Intestinal Juice Permeability ATP-dependent Efflux Drug Metabolism Phys.-Chem. Descr.

Caco-2 Intest. Microsomes Hepatocellular Uptake, Drug Metabolism and Biliary Excretion Liver Microsomes Hepatocytes S9 mix, Cytosol

In vivo Pharmacokinetics in Rodents

Oral kinetics:

C max

• single po administration • repeated blood sampling • plasma concentration-time profile Absorption Distribution Elimination

AUC

(inf)

Max. plasma conc. and Time of max. pl. conc.

T max C max 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 .

T max Oral Bioavailability:

F = AUC po / D po AUC iv / D iv x 100%

Time el

conc Extravascular application: oral, subcutaneous, inhalation…etc Time

conc Sustain release: oral, dermal patches Time

Conc IV infusion Time

Plasma conc. -Time profile after an oral dose of 10 ug/kg in rabbits 0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.00

0 1 2 3 4 time (hour) 5 animal 3 animal 4 animal 8 6 7 8

Simulation of plasma conc. vs. time profile after multiple doses 1600 1400 1200 1000 800 600 400 200 0 0 Rabbit IV 81.7 IU/kg dose daily for 14 days 50 100 150 200 250 Time (hour) 300 350 400 450 500

40 35 30 25 20 15 10 5 0 0 Modeling of increasing doses .

. 821581 (CCR5) 0.25

0.5

Time (hour) 0.75

Predicting Human PK

•

Direct Scaling

of in vitro rate of metabolism to the CL in vivo

c t in vitro CL 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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