Pharmacodynamic models

1

Dose – response relation : PK and PD stages

Administered drug ABSORPTION

Bacteria Insects Parasites

Plasma Concentrations DISTRIBUTION Biophase Concentrations

Interactions Pharmacological Targets

ELIMINATION PHARMACOKINETICS

Cellular Action

Functional Therapeutic Response PHARMACODYNAMICS 2

Population Dose-Response : Variability

Many Resistant Individuals Majority of Individuals Average Effect Sensitive Individuals Minimal Effect Few Mild Response to SAME dose Maximal Effect Extreme 3

Variability of pharmacodynamic origin

Digoxin in Human: Therapeutic and adverse effects

5

Pharmacokinetics / Pharmacodynamics

 Quantification of drug disposition processes 

To link the quantity of administered drug with plasma and tissular concentrations



Objective: to determine the external (administered) doses that produce a given exposure

 Quantification of drugs effects 

To link intensity of the effect with drug concentration



Objective: to determine the range of drug concentrations (drug exposure) associated with a desired effect 6

Effect Endpoints

Graded

• Continuous scale (dose  effect) • Measured in a single biologic unit • Relates dose to intensity of effect

Quantal

• All-or-none pharmacologic effect • Population studies • Relates dose to frequency of effect

7



Relation between concentration and the intensity of an effect



Direct effects models

 Indirect effects models  Relation between concentration and probability of occurrence of an effect  Fixed-effect model

8

Direct effect models

Models describing relations between intensity of an effect and drug concentrations at the site of action Can be used in

in vivo

PK/PD modelling when it exists a direct and immediate link between plasma concentrations and effect



Emax model

 Simplifications of the Emax model :  Linear model  Log-linear model  A useful extension of the Emax model : 

Sigmoïd-Emax model 9

Effect /response concentration 10

Effect /response concentration 11

Effect /response EFFICACY Emax Emax / 2 EC 50 E = E max . C EC 50 + C POTENCY concentration 12

Emax model

E max . C E = EC 50 + C

 Relation described by two parameters 

E max



EC 50

: intrinsic activity,

EFFICACY

: conc. Associated with half-maximal effect

POTENCY

 Empirical justifications  The most simple mathematical description of the occurrence of a maximum  Theoretical justifications  Ligand-receptor interaction

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Drug-Receptor Interactions

Drug Drug Receptor Complex Ligand-binding domain Effector domain Receptor k 1 k 2

 Complex   Receptor K D    max  Drug   Drug 

Effect (K D = k 2 /k 1 ) 14

Consequences of amplification phenomenon

100 % Effect Binding to the receptor EC 50 < K D 50 % EC 50 K D Log[conc.] 15

Consequences of amplification phenomenon

Range of therapeutic concentrations : 100 % Effect

-

No enzyme saturation

-

Linear kinetics Binding to enzyme 50 % EC 50 K D Log[conc.] 16

Emax model

 Graphical representations

concentrations Log [concentrations] 17

Emax model

 Theoretical basis [L] + [R] [RL] Effect  relations

K D / EC 50

 Graphical representation  conc. in arithmetic scale :  conc. in logarithmic scale :

hyperbola sigmoïd

 Comparison of drugs in term of potency efficacy and

18

Emax model

 Efficacy and potency Effect

E max , B Less potent, more efficacious B E max , A More potent, less efficacious A EC 50 , A EC 50 , B

Log (concentrations)

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Emax-inhibition

 Inhibition of an effect :  Emax-inhibition  Fractional Emax-inhibition E = E 0 I max . C IC 50 + C E = E 0 .

(

1 C IC 50 + C

)

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Simplifications of the Emax model



Linear model



Log-linear model 21

Linear model

E = S.C + E 0  Effect is linearly related to concentrations  Parameters of the model (S, E 0 ) are estimated by linear regression

22

Linear model

Effect /response Emax Emax / 2 EC 50 conc 23

Linear model

 Examples :

in vivo

E = S.C + E 0 plasma concentrations of … … digoxin and systolic function … quinidine and duration of Q-T interval … verapamil and duration of P-R interval … pilocarpine and salivary flow

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Log-linear model

E = S.logC + b  Developed with

in vitro

pharmacology  Graphical characteristic of log transformation  Wide concentration ranges : concentrations “zoom” on the small  « Linearization » of the portion of the curve from 20% to 80% of maximal effect : linear regression to estimate the slope  Problem : maximal effect is not estimated

25

Log-linear model

Effect /response Emax Emax / 2 EC 50 Log conc 26

Log-linear model

E = S.logC + E 0  Examples :

in vivo

plasma concentrations of … … propranolol and reduction of exercise-induced tachycardia

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Extension of Emax model



Sigmoïd Emax model 28

Sigmoïd Emax model

Sensitivity of the concentration-effect relation

Effect E 80 E 20

E = E max . C n EC 50 n + C n

29 Log[conc.]

Sigmoïd Emax model

 E max . C n Empirical model E = EC 50 n + C n  when conc.-effect relation cannot be not fitted with Emax  the third parameter provides hyperbola « flexibility » around the  Influence of n the shape of the relation  n = 1 : classical Emax  n < 1 : upper before EC 50 , lower after EC 50  n > 1 : lower before EC 50 , upper after EC 50

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Sigmoïd Emax model

 Empirical model  Introduced by Archibald Hill to describe the cooperative binding of oxygen to haemoglobin : « Hill coefficient »  Theoretical basis : receptor occupancy  Examples :

in vivo

plasma concentrations  n < 1 : Conc.-effect relation very flat propranolol  n > 5 : all-or-none response tocaidine /NSAID 

n = « SENSITIVITY » of the conc-effet

.

relation 31

Sigmoïd Emax model

Sensitivity : influence of the pharmacodynamic endpoint

Effect NSAID E 80

 

COX inhibition Quantification of lameness (force plate) Surrogate endpoint

versus

Clinical endpoint 32 Log[conc.]

Sensitivity of the concentration-effect relation  Impact on selectivity and safety

Therapeutic index TD 50 ED 50 TD 1 ED 99 Safety factor 33

Extension of Emax model



Sigmoïd Emax model



Sigmoïd Emax inhibition 34

Sigmoid Emax-inhibition

Y



E

0 

E

max

n EC

50  

X n X n Y



D

 1 

A



D X C B

100 90 80 70 60 50 40 30 20 10 0 1 10 Melatonine (ng/mL) 100 Observed Predicted

35



Relation between concentration and the intensity of an effect

 Direct effects models 

Indirect effects models

 Relation between concentration and probability of occurrence of an effect  Fixed-effect model

36

Indirect effect models

Kin Kout Response (R) dR dt + = Kin - Kout*R + Decrease of the response Increase of the response 37

 Relation between concentration and the intensity of an effect  Direct effects models  Indirect effects models 

Relation between concentration and probability of occurrence of an effect



Fixed-effect model 38

Fixed-effect model

 The link between a concentration and the probability of occurrence of a defined effect  Concept of threshold concentration  The threshold concentration is different from a subject to another one : it is a random variable , characterized by a distribution in the population  We can association concentrations with a of occurrence of the effect probability  Example : adverse effects of digoxin

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Fixed-effect model

Histogram 120 100 80 60 40 20 C 10% C 50%

 Variability of pharmacodynamic origin  Determination of the therapeutic window

100 % 80 % 60 % 40 % 20 % 40

Sensitivity of the concentration-effect relation  Impact on selectivity and safety

Sensitivity of the relation = variability of the response in the population 41

Fixed-effect model : the logistic regression

 Transformation of the probability of the response P   Logit  Ln  1  P P    ;     Assumption: the Logit is linearly linked to the explicative variable Logit  θ 1  θ 2 .X

 Reciprocal of the Logit equation :

P

 1  1

e



Logit

P  1  e  1  θ 1  θ 2 .X



42