Download Pharmacodynamic models

Pharmacodynamic models
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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
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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
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Variability of pharmacodynamic origin
Digoxin in Human: Therapeutic and adverse effects
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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
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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
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 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
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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
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Effect /response
concentration
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Effect /response
concentration
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EFFICACY
Effect /response
Emax
Emax . C
E=
EC50 + C
Emax / 2
EC50
concentration
POTENCY
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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
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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)
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Consequences of amplification phenomenon
Effect
Binding to the receptor
100 %
EC50 < KD
50 %
EC50
KD
Log[conc.]
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Consequences of amplification phenomenon
Range of therapeutic concentrations :
Effect
100 %
- No enzyme saturation
- Linear kinetics
Binding to enzyme
50 %
EC50
KD
Log[conc.]
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Emax model

Graphical representations
concentrations
Log [concentrations]
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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
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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)
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Emax-inhibition

Inhibition of an effect :
 Emax-inhibition
 Fractional
Emax-inhibition
E = E0 E = E0.(1 -
Imax . C
IC50 + C
C
)
IC50 + C
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Simplifications of the Emax model
Linear model
 Log-linear model

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Linear model
E = S.C + E0

Effect is linearly related to concentrations
 Parameters
of the model (S, E0) are estimated by linear
regression
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Linear model
Effect /response
Emax
Emax / 2
EC50
conc
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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
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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
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Log-linear model
Effect /response
Emax
Emax / 2
EC50
Log conc
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Log-linear model
E = S.logC + E0

Examples : in vivo plasma concentrations of …
… propranolol and reduction of
exercise-induced tachycardia
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Extension of Emax model

Sigmoïd Emax model
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Sigmoïd Emax model
Sensitivity of the concentration-effect relation
Effect
E80
Emax . C n
E=
EC50n + C n
E20
Log[conc.]
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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
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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
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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.]
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Sensitivity of the concentration-effect relation

Impact on selectivity and safety
Therapeutic index
TD50
ED50
TD1
ED99
Safety factor
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Extension of Emax model
Sigmoïd Emax model
 Sigmoïd Emax inhibition

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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
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B
30
20
10
0
1
10
100
1000
Melatonine (ng/mL)
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 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
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Indirect effect models
Kin
Kout
Response
(R)
Increase
of the response
dR
dt
Decrease of the response
+
-
= Kin - Kout*R
-
+
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 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
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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
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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
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Sensitivity of the concentration-effect relation

Impact on selectivity and safety
Sensitivity of the relation
=
variability of the response in the population
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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 
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