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Pharmacodynamic models 1 Dose – response relation : PK and PD stages Administered drug Bacteria Insects Parasites ABSORPTION Plasma Concentrations Biophase Concentrations Interactions Pharmacological Targets DISTRIBUTION ELIMINATION PHARMACOKINETICS Cellular Action Functional Therapeutic Response PHARMACODYNAMICS 2 Population Dose-Response : Variability Number of Individuals Many Resistant Individuals Majority of Individuals Sensitive Individuals Average Effect Maximal Effect Minimal Effect Few Mild Response to SAME dose 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 EFFICACY Effect /response Emax Emax . C E= EC50 + C Emax / 2 EC50 concentration POTENCY 12 Emax model Relation described by two parameters Emax . C E= EC50 + C Emax : intrinsic activity, EFFICACY EC50 : 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 13 Drug-Receptor Interactions Drug Ligand-binding domain Effector domain Receptor Drug-Receptor Complex k1 k2 Effect Receptor max Drug Complex K D Drug (KD = k2/k1) 14 Consequences of amplification phenomenon Effect Binding to the receptor 100 % EC50 < KD 50 % EC50 KD Log[conc.] 15 Consequences of amplification phenomenon Range of therapeutic concentrations : Effect 100 % - No enzyme saturation - Linear kinetics Binding to enzyme 50 % EC50 KD Log[conc.] 16 Emax model Graphical representations concentrations Log [concentrations] 17 Emax model Theoretical basis [L] + [R] relations [RL] Effect KD / EC50 Graphical representation conc. in arithmetic scale : hyperbola conc. in logarithmic scale : sigmoïd Comparison of drugs in term of efficacy and potency 18 Emax model Efficacy and potency Less potent, more efficacious Effect Emax,B B More potent, less efficacious Emax,A A EC50,A EC50,B Log (concentrations) 19 Emax-inhibition Inhibition of an effect : Emax-inhibition Fractional Emax-inhibition E = E0 E = E0.(1 - Imax . C IC50 + C C ) IC50 + C 20 Simplifications of the Emax model Linear model Log-linear model 21 Linear model E = S.C + E0 Effect is linearly related to concentrations Parameters of the model (S, E0) are estimated by linear regression 22 Linear model Effect /response Emax Emax / 2 EC50 conc 23 Linear model E = S.C + E0 Examples : in vivo 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 24 Log-linear model E = S.logC + b Developed with in vitro pharmacology Graphical characteristic of log transformation concentration ranges : “zoom” on the small concentrations « Linearization » of the portion of the curve from 20% to 80% of maximal effect : linear regression to estimate the slope Wide Problem : maximal effect is not estimated 25 Log-linear model Effect /response Emax Emax / 2 EC50 Log conc 26 Log-linear model E = S.logC + E0 Examples : in vivo plasma concentrations of … … propranolol and reduction of exercise-induced tachycardia 27 Extension of Emax model Sigmoïd Emax model 28 Sigmoïd Emax model Sensitivity of the concentration-effect relation Effect E80 Emax . C n E= EC50n + C n E20 Log[conc.] 29 Sigmoïd Emax model Empirical model Emax . C n E= EC50n + C n when conc.-effect relation cannot be not fitted with Emax the third parameter provides « flexibility » around the hyperbola Influence of n the shape of the relation n = 1: classical Emax n < 1: upper before EC50 , lower after EC50 n > 1: lower before EC50 , upper after EC50 30 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 E80 COX inhibition Quantification of lameness (force plate) Surrogate endpoint versus Clinical endpoint Log[conc.] 32 Sensitivity of the concentration-effect relation Impact on selectivity and safety Therapeutic index TD50 ED50 TD1 ED99 Safety factor 33 Extension of Emax model Sigmoïd Emax model Sigmoïd Emax inhibition 34 Sigmoid Emax-inhibition E X Y E0 maxn EC50 X n 100 n 90 80 70 60 Y D 50 A D X 1 C Observed Predicted 40 B 30 20 10 0 1 10 100 1000 Melatonine (ng/mL) 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) Increase of the response dR dt Decrease of the response + - = Kin - Kout*R - + 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 probability of occurrence of the effect Example : adverse effects of digoxin 39 Fixed-effect model Histogram 120 100 100 % 80 80 % 60 60 % 40 40 % 20 20 % C10% C50% Variability of pharmacodynamic origin Determination of the therapeutic window 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 0 ; 1 P Logit P Ln 1 P - ; Assumption: the Logit is linearly linked to the explicative variable Logit P θ1 θ2 .X Reciprocal of the Logit equation : 1 P 1 e Logit P 1 1 e θ1 θ 2 .X 42