Pharmacokinetic Modeling Methods and their Integration with Pharmacodynamics

Cory Langston, DVM, PhD, Diplomate ACVCP [email protected]

Drug effect relationships Dose

Response

Response

Response

•

dose

dose loglog concentration High Lowvariability variability

Drug effect relationships •

Dose

•

Dose

Response

pharmacokinetics

Concentration

pharmacodynamics

Response

Pharmacokinetics – Pharmacodynamics (PK – PD) •

If a concentration can be easily measured (blood) and this concentration directly correlates with an effect, then the ability to predict concentrations becomes of therapeutic benefit.

Pharmacokinetics •

Pharmacokinetics (toxicokinetics) is a mathematical description of drug (toxin) disposition in the body. A complete model will address: → absorption

(A) → distribution (D) → metabolism (M) → excretion (E)

Types of Pharmacokinetic Modeling

•

Data-Based compartmental models (Classical; Compartmental pharmacokinetics) Physiologically-Based pharmacokinetic models Population-Based pharmacokinetic models

•

Pharmacokinetic-Pharmacodynamic models

•

•

Data-Based Compartmental Pharmacokinetics • • •

•

Classic kinetics Views the body as a series of compartments Those compartments have a mathematical volume in which the drug is distributed. Transfer of drug to and from compartments is described by rate constants.

Graphical representation Cartesian

semilog

11

10

10 9 8 amount (mg)

7 6 5

zero-order

4 3 2 1

1

0 0

1

2

3

4

5

6

7

8

9

10

0

11

2

4

6

8

10

tim e (h o u rs )

time (hours)

10

11 10 9 8 7 6

first-order

5 4 3 2 1 0

1 0

1

2

3

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5

6

hours

7

8

9

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11

0

1

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hours

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Mixed-order process d

Zero-order

Transitional

mc g/ml

1 0 0 .0 0

First-order 1 0 .0 0

1 .0 0 0

5

10 ho urs

15

Semi-log plot of first-order one-compartment model dose

Slope (λ) is a proportion/time i.e., /hr or hr-1

10

1 C = C0 • e-λt

Vc k10=λ

1 0

1

2

3

4

5

6

hours

7

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Two compartment model • Some drug plasma concentration-time profiles are biphasic.

10

m cg/m l

Distribution phase Elimination phase

1 0

1

2

3

4

5

6

hours

7

8

9

10

11

Method of residuals (feathering or curve stripping) 100

C = (C1 • e-λ1t ) + (Cz • e-λzt)

y-intercept = Cz Slope = λz

mcg/ml

10

1

0.1 0

5

10 hr

15

20

Method of residuals (feathering or curve stripping) 100.0

mcg/ml

C1

C = (C1 • e-λ1t ) + (Cz • e-λzt)

10.0

1.0

λ1

0.1 0

5

10 hr

15

20

Microconstants • •

k12, k21, k10 etc. are “microconstants To calculate microconstants: → 1st step: k21 = C1•λz + Cz • λ1 C1 + C z → 2nd

step: k10 = λ1 • λz

k21 → 3rd step: k12 = λ1 + λz - k21 - k10

Two-compartment model dose

k12

1 Vc k10

k21

2 Vp

Three-compartment model Equation

•

10

= (C1 • e-λ1t ) + (C2 • e-λ2t) + (Cz • e-λzt)

mcg/ml

→C

1 0

1

2

3

4

5

6 days hours

7

8

9

10

11

Three-compartment model dose

k13

3

k31

k12

1

k21

k10

2

Three-compartment model •

Compartment #1 → central

compartment → blood, extracellular fluid, highly perfused tissues •

Compartment #2 → less

•

perfused tissues; e.g., muscle

Compartment #3 → deep

compartment → poorly perfused tissue; e.g., fat, bone

Physiologic-based pharmacokinetics •

•

Compartments correspond to anatomical spaces so that physiologic interactions can be incorporated into the model. Allows extrapolation outside the range of data to deal with altered physiology (disease states).

Physiologic pharmacokinetics •

Model may incorporate these factors → anatomic Ê

organ volume

→ physiologic

blood flow Ê chemical reactions Ê

→ transport Ê

membrane permeabilities

→ thermodynamic Ê

protein or tissue binding

Physiologic approach to clearance Q • Ca

Elimination organ

Eliminated drug • • •

CL = Q(ER) ER = (Ca - Cv) / Ca CL = Q [ (Ca - Cv) / Ca ]

Q • Cv

Rate of exit from the rumen Dose theophylline

Dose Cr-EDTA

4

6 Vr

k40

Vcr k60

k14 k41

Computer programs •

Most programs can be used, but WINNONLIN and Advanced Continuous Simulation Language (ACSL) are commonly employed software programs

Population pharmacokinetics Reference: JVPT 21(3), 167-189, 1998.

•

•

Traditional kinetic studies usually conducted in small number of healthy individuals How to account for disease effects and differences in population → therapeutic

drug monitoring → physiologic kinetics → population kinetics

Population pharmacokinetics •

•

Compartmental and physiologic kinetics derive their information from extensive sampling of a small number of animals, usually in good health. Population kinetics derive their information from limited sampling of a much larger number of animals, often representing the target population (diseased animals).

Population pharmacokinetics •

Population kinetics identify surrogate parameters (age, body weight, common clinical test results) as covariates that relate the physiologic factors altering the underlying pharmacokinetic model. → e.g.,

•

age → % body water → volume of distribution Cr clearance → GFR → CL

“In other words, one must determine the sources of pharmacokinetic variability in a patient population as well as the magnitude of that variability, in order to design dosage regimens that account for individual patient characteristics.”

Pharmacostatistical Model

Pharmacokinetic model (Fixed Effects) Structural Model Ci = D / Vd • e- CL / Vd • t •Fixed effects Êdose Êtime •Fixed-effect parameters Êclearance Êvolume of distribution

Regression Model Clavg = θ1 + (θ2 • CRCL) Vdavg = θ3 +(θ4 • Age) •Fixed effects Êage Êcreatinine clearance •Fixed-effect parameters Êθ1, θ2, ... θn

Statistical Model (Random Effects) Statistical Model •Intraindividual random effects ÊCij = Ci + εij •Intraind. random-effect parameter Êσ2 (variance of ε) •Interindividual random effects ÊCLj = CLavg + ηCLj ÊVdj = Vdavg + ηVdj •Interind. random-effect parameters Êω2CL (variance of ηCL) Êω2Vd (variance of ηVd)

Types of true population pharmacokinetic methods •

Parametric → assumes

a normal or log-normal distribution → usually simpler; computer program NONMEM •

Nonparametric → does

not require a normal distribution and can identify deviations such as bimodal or skewed distributions → computes a “joint probability density function’, which measures the variance of two parameters and how they are related → Computer programs NPEM, NPML, and NPAG (part of USC-PACK)

Pharmacokinetic model only

Population model Includes age, CrCL, and body wt as covariates

Population pharmacokinetic methods •

• •

More representative of the population to which the drug is targeted. Requires less rigid experimental design. Less extensive sampling per subject creating less patient stress.

‘True’ population pharmacokinetic methods •

•

Characterizes random effects including both inter- and intraindividual (residual) variability of the estimated parameters. Allows not only for the prediction of the effect of clinical features on kinetic parameters, but the degree of confidence in those predictions.

PK – Pharmacodynamic models When hysteresis occurs in the time-concentration profile versus the time-effect profile, a PK-PD model should be developed. •

All figures from: “Riviere, J. E. Comparative pharmacokinetics : principles, techniques, and applications; Ames, : Iowa State University press, 1999.

PK-Pharmacodynamic models

Effector compartment

PK-Pharmacodynamic models

Reflects barriers between the central compartment and the receptors and is a function of anatomical location, blood perfusion, and tissue permeabilities.

Hill equation often used to describe concentration-effect relationship γ determines slope of sigmoid C-E curve; related to drugreceptor binding ratio

PK-PD model of meperidine in goats

Validation of meperidine model

PB-PK-PD models •

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Some models have been created for toxicology risk assessment using ACSL. Example: “A Physiologically Based Pharmacokinetic and Pharmacodynamic Model of Paraoxon in Rainbow Trout. Toxicology & Applied Pharm 145, 1997, 192-201”

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Used “Continuous System Modeling Program III (CSMP III) software

PB-PK model

Water CLD

CLu

Gill Cv

V E

E Ca

Brain

QBr

A

Heart

QH

R

Liver

QL

T

I Muscle

N Kidney

QM • 0.6

QM

E R

QK

Y

PD model of cholinesterase inactivation RO AChE

+ Paraoxon

KAChE

[PO + AChE]

KD

+ RO

CaE

KCaE

KD AChE-1 = (KAChE/Ri) [paraoxon] + KD/Ro

[PO + CaE]

Parameters Used in the Model • • • • • • • •

Paraoxon conc. AChE conc. CaE conc. Tissue/plasma partition coeff. Brain AChE synthesis rate Brain AChE degradation rate Blood flow to each tissue Tissue volume

• • • • •

AChE bimolecular rate constant CaE bimolecular rate constant Hepatic clearance Water uptake clearance Water depuration clearance

Model-predicted AChE activity after water exposure to 75 ng/ml paraoxon

• exp. data point …. predicted conc; model w/o CaE  predicted conc; model w/ CaE

Table 3 Sensitivity of Brain AChE Inhibition to Changes in the PBPK-PD Model Parameters Parameter [AChE] [CaE] Q R Ro KD CLh KAChE KCaE

AChE conc Carboxylesterase conc. Blood flow Tissue/plasma Partition coeff. Brain AChE synthesis rate Brain AChE degradation rate Hepatic clearance AChE bimolecular rate constant CaE bimolecular rate constant

Percent change in brain AChE inhibition when model parameter is increased 10% +1.0 +1.8 -0.8