Chapter 23

Pharmacodynamic Models and Physiologically‐Based Pharmacokinetic Models

Spring 2012

PHAR 7633 Basic Pharmacokinetics 

Student Objectives for This Chapter

• To understand the development and use of physiologically based pharmacokinetic (PBPK) models 

• To understand the different types of concentration ‐effect relationships 

• To understand the mathematical relationships involved with direct reversible pharmacological effect kinetics 

2

Goals of Modeling

• Codify current facts

• Testing competing hypotheses

• Predicting system response under new conditions

• Estimating inaccessible system variables

F. Eugene Yates (1973) 3

Pharmacokinetics (PK)

4

Important are:F = BioavailabilityCL = ClearanceV = Volume of Distribution

IV

PO

Cp

t

Description of the time course and factors affecting the handling of drugs by the body.

Specific models are needed to make predictions of kinetic behavior of drugs at any dosages

• Compartmental Models (e.g., 1 CM, 2 CM) • Physiologic Models (e.g., PBPK)

Physiologically Based Pharmacokinetic Models (PBPK)

5Physiologically‐Based PK Model

PBPK model segregates the body into separate compartments representing organs interconnected by the blood circulation.

PBPK Model Requirements:• Experimental data 

• Physiological Parameters(Tissue size, blood flow)

• Model construction

• Program for numerical solution of differential equations

QL

QHR

QM

QBM

QH

QG

QK

QO

IV

PO

Model of Simple Tissue Distribution

6

Assumptions: Rapid equilibrium between tissue & venous blood

)C(CQdt

dAvTaT

T

AT = Amount in tissueQT = Tissue blood flowCa =  Conc. arterial bloodCvT = Conc. tissue venous bloodKP =  Tissue/plasma partition coefficient

Initial uptake (amount/time) = QT x Ca

CaCvT

VT

Capillary

QTQT

ArteryVein

CvT

CT

Tissue

vTPT CKC KP

Fick’s Law of Perfusion

)K

C(CQ

dt

dA

P

TaT

T

Drug Distribution within Tissue – Permeability Issues

7

Capillary 

Tissue cells

Artery

FiltrationReabsorption

Vein

Interstitial fluidOsmosis

Fick’s Law of Diffusion

)C(CPSdt

dA21

T

AT = Amount in tissue

PS  = Permeability/Surfacearea constant

C1, C2 = Concentration

Initial uptake (amount/time) = PSxC1

Size, nature, permeability, binding and transporters can determine tissue uptake

Tissue Subcompartments

CaCv

Interstitial 

Intracellular

Capillary

PS

PBPK Model Example: Thiopental 

8

Figure 23.3.4 Some Results for Thiopental Bischoff and Dedrick 1968(http://www.boomer.org/c/p4/c23/c2303.html )

PBPK Models: Pros vs. Cons

Advantages:– The prediction of plasma (blood) and tissue PK of drug candidates prior to in vivo experiments

– Support a better mechanistic understanding of PK properties from tissue kinetic data predicted, facilitating a more rational design during clinical candidate selection. 

– Extrapolation across species, routes of administration, duration and timing of drug input and dose levels.

Limitations:– Considerable experimental and computational effort– Difficulty in obtaining realistic parameters

9

Pharmacodynamics (PD)

Description of the time ‐course and factors controllingdrug effects on the body.

Important are:

Emax

EC50

keo, kin , 

= Capacity constant

= Sensitivity constant

= Various time constantsfor specific models

kin kin kin

S(t)C(t)

S(t-TP) S(t-TP-TR)

P R

Serum rHuEP

OConc., IU/L

Reticulocytes, %

10000

1000

100

100 5 10 15

765432

10 5 10 15 20

Time, day

SCDoses

EPO stimulatesproduction of RBC.10

Basic Tenets of Pharmacodynamics

Capacity-Limitation Turnover and HomeostasisE

ffec

t, %

Concentration

γγ50

γmax

CEC

CEE

Hill Function

The Law of Mass Action( D + R DR ) and smallquantity of targets leads to capacity-limitations in most responses.

Production LossBiologicalFactor

(R)

Rkkdt

dRlossproduction

Both diseases and therapeuticagents often interfere with thehomeostasis in the body resulting from the natural turnover of biological substances or functions.

11

Biological Turnover Rates of Structures or Functions

Electrical Signals (msec)

Chemical Signals (min)

Mediators, Electrolytes (min)

Hormones (hr)

mRNA (hr)

Proteins / Enzymes (hr)

Cells (days)

Tissues (mo)

Organs (year)

Person ( .8 century)

Fast

Slow

BIOMARKERS

CLINICALEFFECTS

12

Components of PK/PD Models

Jusko et al., JPB 23: 5, 1995

Review article:Mager DE, Wyska E, Jusko, WJ, Diversity of Mechanism‐Based Pharmacodynamic Models, Drug Met. Disp. 31: 510 (2003).

R

ResponseDisposition

KineticsBiophase

DistributionBiosensor

ProcessBiosignal

FluxTrans-

duction

Pharmacokinetics Pharmacodynamics

Drug

keoCP Ce

Biosignal

kin

kout

H

H

13

Types of Drug Effects

Reversible• Direct

- Rapid- Slow

• Indirect- Synthesis, secretion- Cell trafficking- Enzyme induction

Irreversible• Chemotherapy• Enzyme inactivation

14

Direct Reversible Effects

Clark’s Occupancy Theory: A. J. Clark (1933) proposed the first model to account for the quantitative behavior of a receptor-mediated process.

R + L RL Effectkon

koff

R = Receptor, L = Ligand, RTOT = Total ReceptorKD = Equilibrium Dissociation Constant (KD = kon/koff) 15

LK

LRRL

D

TOT

L

RLor

Effect

The “Law of Mass Action”

Hill Function:Emax = CapacityEC50 = Sensitivity

Hill AV , The possible effect of the aggregation of molecules of haemoglobinon its dissociation curve, J. Physiol. (London) 40: iv (1910).

= Hill Factorγγ

50

γmax

CEC

CEE

100

80

60

40

20

00 20 40 60 80

Eff

ect,

% 0.5

1

Concentration

= 2

16

Properties of Hill Function

17

γγ

γmax

CEC

CEE

50

When C << EC50

E = S · CS = Emax / EC50

When C = EC50

E = ½Emax

When C >> EC50

E = EmaxC

Eff

ect

Emax

EC50

Emax/2

LINEAR MODEL

CSEE 0

Relationship between change in QTc intervals and N - acetylprocainamide plasmaconcentrations in several patients. From Piergies et al, CPT 42: 107 (1987).

50max /ECES

18

Red

ucti

on in

Pre

mat

ure

Ven

tric

ular

Con

trac

tion

Fre

quen

cy (

%)

Tocainide plasma concentration (g/ml)

Slopes of E vs. log C Plots are Often Linear

Intercept =

γ2

ECln 50

Slope = m =4

γEmax

19

Simple Direct Effect:  Inhibition of Cyclooxygenase Enzyme Activity in Blood by Dexamethasone

Zamacona et al, 2001

TIME, hr

0 8 16 24 32

COX  A

CTIVITY (% of Baselin

e)

0

20

40

60

80

100

120

DEX

AMETHASO

NE CONC, n

M

0.1

1

10

100

DEXAMETHASONE CONC, nM

0

20

40

60

80

100

120

0 0.1 1 10 100 1000

IC50 = 1.42 nM

CIC

CIEE

50

max0 1

20

PK/PD Expectations For Simple Direct Effects

tk0 eCC

CEC

CEE

50

max

Effect

Concentration

Time, hr

Profiles based on  k = 0.4, Emax = 100, EC50 = 100. 21

PK PD

0

20

40

60

80

100

0.1 1 10 100 1000 10000

PK/PD Expectations For Simple Direct Effects, con’t

Effect

Concentration

Time, hr

22

CEC

CEE

50

max

0

20

40

60

80

100

0.1 1 10 100 1000 10000

Time, hr

Effect

Concentration

PK PD

“No hysteresis”

Kinetics of Pharmacologic EffectsGerhard Levy,  Clin. Pharmacol. Ther. 7:362 (1966)

tmkoEE logC

t

PK

2.3

k-

log C

E

Pharmacology

m

t

Eo

PD

mk

E

NB:  Half‐life applies in PK but not PD!

23

Slope = m =4

γEmax

Decline of Effect:  Derivation

eAlogmE

meE

Alog

Kinetics:

2.3kt

AlogAlog

Combine:

2.3kt

meE

meE

t2.3km

EE

k = elimination constant

A = Amount of drug

m = Slope

e = Intercept

E° = Peak effect

24

Pharmacodynamic Models Producing Delayed Responses

• Direct Effect:  Active Metabolite 

• Direct Effect:  Biophase Distribution

• Direct Effect:  Slow Receptor kon/koff

• Antibody‐Ligand Interaction

• Indirect Response:  Inhibition of kin

• Indirect Response:  Stimulation of kout

• Indirect Response:  Inactivation

• Indirect Response:  Cell Life‐Span Loss

• Irreversible Effect:  Regeneration 

• Transduction Process

PK

log C

25

TimeResponse

Time

PDResponse

Conc.

“Counter clockwise Hysteresis”

Modeling a Biophase

Furchgott R (1955)Segre, Il Pharmaco (1968)Sheiner et. al., CPT (1979)

Pancuronium PK/PD – Muscle Relaxation

eC50ECeCmaxE

E

eCpCeokdt

edC

26

Paralysis by Mechan

ical 

Twitch Response, %

Pancuronium CP or CE (g/ml)

CPCe

keo = 0.1

0.3

0.5

2.0

keo = 0.1

2.0

EFFECT

CONCEN

TRATION

Role of keo in Determining Simple Drug Effects

PK/PD Model:CP Ce

keo

kele50

emax

CEC

CEE

TIME

Simulations of Plasma Drug Concentrations (Cp, dashed line) for monoexponential kinetics(Co = 1000, kel = 0.4), Effect Site Concentrations (Ce, solid lines) for the indicated values of keo, and Effect profiles for the Emax model with Emax = 100 and EC50 = 100.

TIME

0.3

Basic Indirect Response Models

(Dayneka et al., JPB 21: 457 1993).

Response(Mediator)

(R)

Production Removal

kin kout

Drugs can alter the production (kin) or dissipation (kout) process normally controlling endogenous levels of R.  Drugs can inhibit (    ) or stimulate (    ) any of these processes.

28

Irreversible Drug Effects: Antibiotics on Cell Killing

(Jusko, JPS 60: 892 1971)(Zhi et al, JPB 16: 1988)

RCkRkdtdR

s

Cell Growth

ks

k • C

29

PK PD

Effects of Various Intraperitoneal Doses of Piperacillin on Killing and Growth Kinetics of Pseudomonas Aeruginosa in Neutropenic Mice

Cell Killing

Factors Affecting PK/PD: Covariates

Physiological

AGE:  Young/Old

Sex

Pregnancy

Race

Genetics

Stress

Disease

Hepatic

Renal

Thyroid

Cystic Fibrosis

Obesity

Drug Interactions

Inhibitors

Inducers

Joint Mechanisms

Opposing Mechanisms

30

Pharmacokinetics and Pharmacodynamics of d ‐ Tubocurarine in Infants, Children, and Adults

Fisher DM et al, Anesthesiology 57: 203 (1982).

Time (min)

ml∙min

‐1∙m

‐2

dTC Clearance (Cl)

0.6

0.3

0.0

EC50 g/ml

Neonates      Infants     Children     Adults

Neonates      Infants     Children     Adults

Infants show both reducedclearance (related to GFR)and greater sensitivity to dTC.

31

Influence of extreme obesity on the body disposition and neuromuscular blocking effect of atracurium

Varin F, et al, CPT 48: 18 (1990).

ATR

ACURIUM, n

g/m

l, PLA

SMA

TIME (min)

NEU

ROMUSCULA

R BLO

CKADE (%

)

ATRACURIUM EFFECT COMPARTMENT CONCENTRATION (ng/ml)

PK Parameters:

Vss, L/kgTBW CL, ml/min/kgTBW

Normal 0.141 6.6Obese 0.067 3.5

PD Parameters: (Emax = 100)

t½keo, min EC50, ng/ml

8.3 3127.4 470 32

PK/PD and Pharmacogenetics

Albuterol Concentration (ng/m

l)

% Chan

ge in

 FEV

1Time (h) Time (h)

The 2 ‐ adrenergic receptor gene polymorphism was assessedby PCR and found to be a major determinant of bronchodilatorresponse to albuterol.  J.J. Lima et al, CPT 65: 519 (1999).

S(t)

2 phenotypes

33

Pharmacodynamic Caveats

Measurements should be sensitive, gradual, reproducible, objective, and meaningful.

Studies should include baseline and span 2 or 3 dose levels with effects from 0 to Emax.

Measure major intermediary steps.

Models should recognize mechanism(s) of drug action.

Covariates are important!

34

Modeling Dynamic Effects

PHARMACOKINETICS

BIOPHASE

MECHANISM

MODEL TYPE

TURNOVER

TRANSDUCTION

ADAPTATION

PATHOPHYSIOLOGY

DISEASE PROGRESSION

COMPLEXITIES

Requires Integration of Diverse Components35