Introduction to Pharmacokinetic Modelling Rationale

Michael Weiss

Martin Luther University

Halle-Wittenberg

michael.weiss@medizin.uni-halle.de

“Everything is a poison... the dose differentiates a poison from a remedy.”

Philippus von Hohenheim, known as Paracelsus (1493-1541)

Archives internationales de pharmacodynamie et de therapie

1937

This means that the principles governing

plasma time-concentrations are not only

capable of a mathematical adaptation,

but are expected to undergo an advancement

through the application of the

powerful resources of mathematics.

Where the word pharmacokinetics first appeared 1953

Question 1

Question 2

The maximum plasma concentration immediatly (1 min)

after a bolus dose of 1 mg digoxin is > 200 ng/ml.

Is this dose toxic in view of the therapeutic window

(target concentration) of 0.5 – 1 ng/ml (toxicity > 2 ng/ml) ?

Question 3

When will be more than 90% of this dose be elimitated?

CL = 0.2 l/min , Vss = 600 l

Dose Effect Pharmacokinetics Pharmacodynamics

Dosis

Effect

Time

Renal failure

Dosis

Effect

Time

Receptor-

Upregulation

PK/PD

Pharmacodynamics Pharmacokinetics

PK/PD

Pharmacokinetic(PK)-Pharmacodynamic(PD) systems analysis

Pharmacometrics

Physiologically based (mechanistic) modeling

Alterations in diseased states

Disease progression

Translational PK/PD modeling

Receptor binding & signal transduction

Variability

Sparse data

Clinical trial simulation

Pharmacokinetic system

Parameter estimation Model structure

Model

Data

Modeling

methodology

Modeling

purpose

Cobelli and Carson, 2005

Input Output

Disturbance

System Dose C(t)

Systems Approach

1

2

Black Box

Pragmatic Validity

Heuristic

Validity

Behavioural model Structural model

Linear system theory

Neural network

empirical physiological/mechanistic

Prediction

+Explanation

Oral Drug Dose

Dissolution

Absorption

Metabolismn

Distribution

Excretion

Effect Site (Receptors)

Pharmacological Effect

Ph

arm

aco

dyn

am

ics

Ph

arm

aco

kin

eti

cs

Ph

arm

acy

Elimination

sc, im, nasal, pulmonary, transdermal

Structure of pharmacokinetic system

Pharmacokinetics =

Transport across membranes

Transport with flowing blood

Perfusion or barrier limited Severely permeability limited

Tissue

Blood Blood

Tissue Lymph

Small Molecules Large Molecules

Passive Transport

P-Glycoprotein Pump (MDR1)

intestine

blood-brain barrier

kidney

liver

testis

cancer cells (MDR1)

etc.

Active Transport

BLOOD

BRAIN

BARRIER

KIDNEY

OATPs

MDR1 MRP2

OATs

OCTs

LIVER

GUT

BILE

OATPs MRP3

MRP2

MDR1

MRP1

OATPs

OCT1

OATPs

OATs MDRs

MRP2

BSEP MRP3

MDR1

OATs

Heart-Lung

Transporter in PK

OATP: Organic AnionTransporting Polypeptide

OAT: Organic Anion Transporter

OCT: Organic Cation Transporter

MDR: Multidrug Resistance protein

MRP: Multidrug Resistance-associated Protein

BSEP: Bile Salt Export

Metabolism

drug metabolite

enzymes

Diffusion

C1 C2

)]()([)(

211 tCtCSPdt

tdA

permeability surface

Active Transport

rate

C1

1

1max1 )(

CK

CV

dt

tdA

M

)()(

11 tCCLdt

tdAperm

10-1.0 100.0 101.0 102.04 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7

100

101

6

78

2

3

4

5

6

78

2

3

Time (h)

Pla

sma

Dig

oxin

Conce

ntr

atio

n (

ng/m

l)

?

Fit:

3-compartment model

3-exponential function

Initial phase

< 5 min

Terminal phase

t →∞

First Principles Based Modeling

?

First Principles

Convective transport

Q

Convective dispersion Vascular mixing

Permeation (Capillary uptake)

Diffusion (Extravascular)

Binding

Not well-mixed!

Models of process PK

Models of data Statistics

Structural model Intra- and

inter-individual variability

Probability distribution of the model

parameters in the target population

(population approach)

Compartments (differential equations)

„Model independent“ (numerical integration)

Subsystems (Laplace transformation)

NONMEM

ADAPT 5

MONOLIX, etc

Covariates

Poor data perfect model poor result

Poor model perfect data poor result

Design of PK experiments

Model building

Identifiability

Feasibility

Pharmacokinetic System and Experimental Designs

C(tsurgery) In vivo

Isolated perfused organ

Compartmental

Destructive sampling

PBPK

Systemic Transit Time Density

Pulmonary Transit Time Density

Cardiac output

Brain

Heart

Kidney

Testes

Fat

Gut

Bone

Vei

ns

Art

erie

s

Lung

Pancreas

Spleen

Skin

Liver

Muscle

Structure of the body

Routes of drug administration

oral (enteral)

intramuscular

Organs of drug elimination

intra

venous

Dose Dor , Div etc.

Brain

Heart

Kidney

Testes

Fat

Gut

Bone

Vei

ns

Art

erie

s

Lung

Pancreas

Spleen

Skin

Liver

Muscle

What can we measure ?

Renal excretion (?) AeR(t)

Venous blood (plasma) concentration

C(t)

Modelling of PK transport in terms of mass (amount) !

(“Transport or elimination of concentration” is nonsense.)

Dilemma: we measure concentration.

C(t) A(t) ?

)()(

tCCLdt

tdAe

Basic Equation

Rate of drug elimination = Clearance x Plasma concentration

(1)

dttCCLAdt

tdAe

e )()(

00

AUCCLDiv

Note: ive DA )( (nothing remains in the body)

Well-mixed plasma

compartment !

“ model independent “ or noncompartmental analysis)

Estimation of Clearance (single dose)

AUC

DCL iv

AUC

C(t)

t

Single dose

Div dttCAUC

0

)(

!

Intravenous dose

Area Under the Curve

Estimation of Clearance (infusion)

ssCCLSteady state after continuous

i.v. infusion DR

Output (elimination rate) = Input (dose rate, infusion rate)

t

C(t)

Css

DR

ssC

DRCL

Elimination rate

Dose rate

Nonlinear Pharmacokinetics

Dose

AUC linear nonlinear

Saturable Metabolism

RH

uHM

uH

HCK

CVR

,

,max

Michaelis-Menten equation

uHC , uHM CK

VCL

,

maxint

MK

VCL max

int

MuH KC ,

linear (dose independent)

kinetics

C(t)

saturation

exponential

high bolus dose

Compartmental Models

C(tsurgery) In vivo

Isolated perfused organ

Compartmental

Destructive sampling

PBPK

Systemic Transit Time Density

Pulmonary Transit Time Density

Cardiac output

2-Compartment Model

)()()(

)()()()(

1122212

2211121101

txktxkdt

tdx

txktxktxkdt

tdx

x1(0) = Div

1

1 )()(

V

txtC

Div k10

1

2

k21 k12

peripheral

compartm.

sampling (central)

compartm.

2

21

1

12

1

10V

CLk

V

CLk

V

CLk dd reparameterisation

)1( 2112121 kkVVVVss

10 VV

Distribution Kinetics

t ~ 2-10 h

Terminal half-life

zt ,2/1

Div CL

1

2

CL12

3

CL13

3-Compartment Model

V1

Fit excellent for Civ(t) of most drugs-useful as empirical model

V1 : no clear meaning in terms of initial

distribution

CL12, CL21 : no meaning in terms of

underlying distribution

processes

Estimation and interpretation of

steady-state parameters (CL, Vss) is

straightforward:

Vss , CL model independent

Physiological Based Pharmacokinetic Modelling

C(tsurgery) In vivo

Isolated perfused organ

Compartmental

Destructive

sampling

PBPK

Systemic Transit Time Density

Pulmonary Transit Time Density

Cardiac output

Brain

Heart

Kidney

Testes

Fat

Gut

Carcass

Vei

ns

Art

erie

s

Lung

Pancreas

Spleen

Skin

Liver

Muscle

Physiological Based Pharmacokinetic Modelling (PBPK)

Flow

Vascular volume

Partition coefficient K

Organ tissue volume VT

V1

V2

Q Q

V1

V2 = KVT

CLd CLd = fuPS

Brain

Heart

Kidney

Testes

Fat

Gut

Carcass

Vei

ns

Art

erie

s

Lung

Pancreas

Spleen

Skin

Liver

Muscle

)()()(

)()()()(

1

1

2

2

2

2

2

1

1

1

1

1

txV

CLtx

V

CL

dt

tdx

txV

CLtx

V

CLtx

V

Q

dt

tdx

dd

dd

System of Diffeq

e.g., noneliminating organ:

Flow

Vascular volume

Partition coefficient K

Organ tissue volume VT

V1

V2

Q Q

V1

V2 = KVT

CLd CLd = fuPS

PBPK

Simulation of Alfentanil Kinetics in Humans

Upscaling from rat data

Bjorkman, Wada, Stanski

Anesthesiology, 1998.

Human:

Tissue volumes (mass)

Vascular volumes

Blood flows

Rat:

Partition coefficients

Permeabilities (CLd)

Upscaling from Animal to Human

WaV Vss 75.0WaCL CLSince QL and GFR ~ W0.75

75.0WaBSA BSAAlternatively:

animalL

humanL

animalhumanQ

QCLCL

,

,

pb

pWaP

25.0WaMDRT MDRT

bWfunctionOrgan ~

Allometrie

Distributed Modelling

Transit Time Distribution

Advection-Dispersion Equation

(Microvascular Network)

x

Cv

x

CD

t

C

2

2

Dispersion

Coefficient

(geometrical

dispersion)

Blood Flow

Velocity

Normalized outflow concentration

Vascular marker

Transit time dispersion (RD2) Mean transit time (MTT)

f(t)

tMTTRD

MTTt

tRD

MTTtftC IG 2

2

32 2

)(exp

2)()(

t

t

Solution of Advection-Dispersion Equation: Inverse Gaussian distribution, density fIG(t) (Brownian passage time distribution)

Vascular Marker

Extent of distribution (VB= MTT×Flow) Rate of distribution

Subsystems (Transit Time Distribution)

C(tsurgery) In vivo

Isolated perfused organ

Compartmental

Destructive

sampling

PBPK

Systemic Transit Time Density

Pulmonary Transit Time Density

Cardiac output

Model Formulation in the Laplace Domain

Model Structure:

Compartments Differential Equations

Subsystems Transit Time Density Functions, fi(t)

(Impulse Response)

Limitation of using compartments as subsystems

exponential distributed transit times

Advantage of model building in Laplace domain

simple rules for connecting subsystems

Subsystems

Model building: Laplace Transformation

)(ˆ2 sf)(ˆ

1 sf

)(ˆ1 sf

)(ˆ2 sf

Q

Q

Q1

Q2

)(ˆ)(

)(ˆ)(ˆ)(ˆ

1

21

sfLtf

sfsfsf

)(ˆ)(

)(ˆ)1()(ˆ)(ˆ

1

1

21

sfLtf

QQq

sfqsfqsf

Numerical inverse Laplace transformation

Numerical Inverse Laplace Transformation

)}(ˆ{)( 1 sfLtf

ADAPT Schalla & Weiss , Eur J Pharm Sci,1999.

SCIENTIST 3.0

Implemented in nonlinear regression software:

FORTRAN implementation of Talbot's method

)}({)(ˆ tfLsf Model Equation in Laplace Domain

Recirculatory Model

C(tsurgery) In vivo

Isolated perfused organ

Compartmental

Destructive

sampling

PBPK

Systemic Transit Time Density

Pulmonary Transit Time Density

Cardiac output

Using Front-end Kinetics to Optimize

Target-controlled Drug Infusions

Initial Distribution (Front-End Kinetics) Thiopental

Determines the anesthetic induction dose!

Avram & Krejcie, Anesthesiology, 2003

3-fold higher VC

Brain

Heart

Kidney

Testes

Fat

Gut

Carcass

Vei

ns

Art

erie

s

Lung

Pancrea

s

Spleen

Skin

Liver

Muscle

Lumping Dose

C(t)

Cardiac output

Pulmonary

Systemic

Dose C(t)

Minimal Circulatory Model

Hepatic Clearance

Cin Cin

Cout

Ae,B

HHH EQCL

QH = 1500 mL/min

FH= 1- EH

FH: Hepatic (first pass) availability

Fraction escaping elimination by the liver

Absorption and Bioavailability

Gut Dor

FA

FH

HAFFF

Systemic circulation

Bioavailability F

= Fraction of Dor that

reaches the systemic

circulation

Subsystems: Isolated Perfused Organs

C(tsurgery) In vivo

Isolated perfused organ

Compartmental

Destructive

sampling

PBPK

Systemic Transit Time Density

Pulmonary Transit Time Density

Cardiac output

Input

Output

Hepatic Pharmacokinetics

Input Output

Cunningham & Van Horn, Alcohol Res & Health, 2003

Hepatic Clearance:

Sinusoidale uptake

Hepatocellular metabolism

Biliary excretion

Intravascular mixing (vascular marker)

Microcirculatory network

Cellular distribution

Microscopic volume element

vascular

tissue

phase

Vp

VT

Capillary flow

Intravascular Mixing + Cellular Distribution Stochastic model of transit time distribution

Weiss & Roberts, J Pharmacokin Biopharm, 1996

Advection-Dispersion Equation

Extravascular Space

Intravascular Space

Single Capillary

Well-mixed cellular

Slow binding Rapid diffusion

kin kout

kon

koff

kout kin

VC

kout kin

Cell

Vascular

Disse space

ke

ke

ke

Slow diffusion Rapid binding

eff

dD

L2

L

Diffusion time constant

Weiss et al., Br J Pharmacol, 2000

d

0 50 100 150

10-1

100

101

102

103

0 50 100 150

10-1

100

101

102

103

Slow binding

Well-mixed (cellular)

Time (s)

Weiss et al., Br J Pharmacol, 2000.

Hepatic Transt Time Density of Diclofenac Isolated perfused rat liver outflow curve

Slow Intracellular Carrier Diffusion

The free fraction is negligibly small

(hydrophobic and amphipatic molecules) soluble cytoplasmic binding proteins act as diffusing transport carriers (Dmob)

kin Diffusion, d

Flow

Luxon & Weisiger, Am J Physiol, 1993

Albumin (solid line) or [14C]sucrose (dotted line)

as the extracellular reference for [3H]palmitate

fits of the slow-diffusion model

Well-mixed model (dash-dotted line)

Slow-binding model (short dashed line)

Slow-diffusion/bound model (solid line)

Hung et al., Am J Physiol Gastrointest Liver, 2003

Multidrug resistance associated protein (MRP)

Organic cation transporter (OCT)

Organic anion transporter (OAT)

Organic anion transporting polypeptide (OATP)

sodium-taurocholate cotransporting polypeptide (NTCP)

P-glycoprotein (MDR1),

Breast cancer resistance protein (BCRP)

Bile salt export pump (BSEP)

Functional Characterization of Transporters

)(

max

tCK

Vk

M

in

kin

Flow

C(t)

Nonlinear system

1 2 3 5 7 8

4 6 vascular

Disse space

Vascular space

Blood flow

OATP2

Na

pu

mp

kon(C)

koff

kout ke

kin(C)

OATP2

Hepatocyte

CK

VCk

M

in

max)(

Digoxin Semi- distributed liver model

Weiss et al., Pharm Res, 2010

Brain

Heart

Kidney

Testes

Fat

Gut

Carcass

Vei

ns

Art

erie

s

Lung

Pancrea

s

Spleen

Skin

Liver

Muscle

Lumping

LLl Liver Gut

Rest Systemic

Systemic Transit Time Density

Pulmonary Transit Time Density

Cardiac

output

Distributed liver model

Uptake

Elimination

Weiss et al., Eur J Pharm Sci, 2011

ICG Kinetics in Dog

CLuptake

(ml/min)

ke (min-1)

CL

(ml/min)

Interplay between Hepatic Uptake and Excretion of ICG

Weiss et al., Eur J Pharm Sci, 2011

Question 4

Frequency of blood sampling proportional to the rate of change in blood

concentration?

0 20 40 60

100

2

3

456

2

3

4

Dig

oxin

co

nce

ntr

ation

(ng/m

l)

Time(h)

Statistics

formal

Modelling („Art“)

informal

intuitive

Picasso „The goat“

Design of Experiments

“Art is the lie that helps us see the truth, “ said Picasso, and the same can be said of

modelling.

On seeing a Picasso sculpture of a goat, we are amazed that his caricature seems

more goatlike than the real animal, and we gain a much stronger feeling for

“goatness”.

Modelling

Lee A. Segel, 1984

Similarly, a good mathematical model – though distorted and hence “wrong”

like any simplified representation of reality –

will reveal some essential components of a complex phenomenon.

Design of PK Experiments

‚intuitive‘

formal informal

mathematical

Experience

in PK/PD

System theory/Statistics Identifiability

Optimal sampling

Models and Modelling

Information from Experiments

Amount of information ~ log(1/p) p probability (expectation) of the result before the experiment is done

Model without model we have no expectations:

no basis for choosing what to observe, i.e., design of experiments;

an observation generates an infinite amount of information

Rescicno and Beck, 1987

A priori identifiability

Given the structure of the model and experimental design,

can the model parameters be estimated if the data are error free?

A posteriori identifiability

Given model and measured experimental data,

can the model parameters be estimated within a reasonable

degree of statistical precision?

Model misspecification Biased parameter estimates,

wrong conclusions

Model identifiability

Additional information (data):

e.g. on vascular mixing (MID),

Fit to low and high dose responses

(nonlinear systems)

Bayesian estimation, a priori information

Misspecified models can give very precise estimates of the wrong

answer. (Halloran et al., 1996)

A priori identifiability

Input location and duration

Output (sampling) location

Model appropriateness

A posteriori identifiability

Issues in Experimental Design

Number, range and spacing of sample times

Number of subjects

?

Choice of drug input function

Choice of route of administration

Choice of sampling site

Choice of sampling scheme

Population or individual analysis

• Model identification complexity reduction

• Model misspecification biased estimates

• Model validity modelling objectives

The art of asking the right

questions in mathematics

is more important than the art

of solving them.

Georg Cantor

1845-1918

References Avram, M. J. and T. C. Krejcie (2003). "Using front-end kinetics to optimize target-controlled

drug infusions." Anesthesiology 99(5): 1078-1086.

Bjorkman, S., R. D. Wada, et al. (1998). "Application of physiologic models to predict the

influence of changes in body composition and blood flows on the pharmacokinetics of

fentanyl and alfentanil in patients." Anesthesiology 88(3): 657-667.

Cobelli, C. and E. Carson (2008). Introduction to modeling in physiology and medicine, Academic

Press.

Cunningham, C. C. and C. G. Van Horn (2003). "Energy availability and alcohol-related liver

pathology." Alcohol Research and Health 27: 291-299.

Hung, D. Y., F. J. Burczynski, et al. (2003). "Fatty acid binding protein is a major determinant of

hepatic pharmacokinetics of palmitate and its metabolites." American Journal of

Physiology-Gastrointestinal and Liver Physiology 284(3): G423-G433.

Rescigno, A., J. S. Beck, et al. (1987). "The use and abuse of models." Journal of Pharmacokinetics

and Pharmacodynamics 15(3): 327-340.

Schalla, M. and M. Weiss (1999). "Pharmacokinetic curve fitting using numerical inverse Laplace

transformation." European journal of pharmaceutical sciences 7(4): 305-309.

Weiss, M., G. Hübner, et al. (1996). "Effects of cardiac output on disposition kinetics of sorbitol:

recirculatory modelling." British journal of clinical pharmacology 41(4): 261-268.

Weiss, M., T. C. Krejcie, et al. (2011). "A physiologically based model of hepatic ICG clearance:

Interplay between sinusoidal uptake and biliary excretion." European Journal of

Pharmaceutical Sciences 44(3): 359-365.

Weiss, M., P. Li, et al. (2010). "An improved nonlinear model describing the hepatic

pharmacokinetics of digoxin: evidence for two functionally different uptake systems and

saturable binding." Pharmaceutical research 27(9): 1999-2007.

Weiss, M. and M. S. Roberts (1996). "Tissue distribution kinetics as determinant of transit time

dispersion of drugs in organs: application of a stochastic model to the rat hindlimb." Journal

of pharmacokinetics and biopharmaceutics 24(2): 173-196.