PK-PD analysis and modelling

Transcription

1 PK-PD analysis and modelling Paul M. Tulkens Cellular and Molecular Pharmacology & Center for Clinical Pharmacy Louvain Drug Research Institute Université catholique de Louvain, Brussels, Belgium a With the support of Wallonie-Bruxelles-International 1

2 Why modelling? to move from mere description to underlying phenomena nature can often be better explained in terms of equations than mere description this has been essential in physics (think about gravity law, radioactive decay, study of electromagnetic field and optics, up to the equivalence of mass and energy ) to allow predictions over and beyond what is immediately accessible by the experience to generate rules that can be applied widely

3 In vitro studies

4 Response to an antimicrobial an example with ceftobiprole and S. aureus (one strain) log1 CFU Effect-over-time concentration (multiples of MIC) time (h)

5 Response to an antimicrobial an example with ceftobiprole and S. aureus (2 strains) log1 CFU Effect-over-time (2 strains) concentration (multiples of MIC) This where the phenomenon happens time (h)

6 Response to an antimicrobial: the model an example with ceftobiprole and S. aureus (2 strains) 3 MSSA #1 MSSA #2 2 log cfu (24 h - h) log 1 multiples of MIC measured at ph 7.4

7 Response to an antimicrobial: the model an example with ceftobiprole and S. aureus (multiple strains) MIC E min 3 2 MSSA HA-MRSA CA-MRSA log cfu (24 h - h) 1 C S -1 E max log 1 multiples of MIC measured at ph 7.4

8 Analyses Sigmoidal dose-response: also called "4-parameters logistic equation", i.e. bottom (E min ) Top (Emax) EC 5 Hill slope Sigmoid dose-response 1. E max response EC 5 Hill slope = Equation for Prism Equation:Sigmoidal dose-response Y=Bottom + (Top-Bottom)/(1+1^((LogEC 5 -X))) E min concentration (log 1 ) ;X is the logarithm of concentration. Y is the response ;Y starts at Bottom and goes to Top with a sigmoid shape

9 Analyses Equation Equation:Sigmoidal dose-response Y=Bottom + (Top-Bottom)/(1+1^((LogEC 5 -X))) ;X is the logarithm of concentration. Y is the response ;Y starts at Bottom and goes to Top with a sigmoid shape

10 Type of functions Do not forget to use the appropriate axes! how would you fit those data

11 Type of functions This would be a good model

12 Run statistics

13 Run tests

14 Two examples

15 Impact of MIC on the response of intracellular bacteria to moxifloxacin MIC (mg/l) NRS192 SA1 NRS384 NRS386 SA69 KKH II-7924 HMC 551 SA481 after normalization for MIC log 1 extracellular concentration (mg/l) log 1 extracellular concentration (x MIC) Lemaire et al. Journal of Antimicrobial Chemotherapy (211) 66:596-67

16 Colistin and inoculum effect high inoculum ( in vivo low inoculum ( in vitro testing) colistin concentration (mg/l) The extent and rate of killing of P. aeruginosa by colistin were markedly decreased at high CFUo compared to those at low CFUo. Bulita et al. Antimicrob. Agents Chemother. (21) 54:

17 In search of models with Prism

18 In search of models (including your own)

19 In search of models (including your own)

20 And here you are azithromycin clarithromycin telithromycin log cfu from time (48 h) MIC,.1-.3 mg/l Cs, 3 mg/l C max.5 mg/l Log concentration (mg/l) log cfu from time (48 h) MIC,.8 mg/l Cs,.6 mg/l C max 1 mg/l Log concentration (mg/l) log cfu from time (48 h) MIC,.8 mg/l Cs,.4 mg/l C max 1 mg/l Log concentration (mg/l) ciprofloxacin moxifloxacin finafloxacin log cfu from time (48 h) MIC,.1 mg/l Cs,.2 mg/l C max 4 mg/l Log concentration (mg/l) log cfu from time (48 h) MIC,.1 mg/l Cs,.1 mg/l C max 4 mg/l Log concentration (mg/l) log cfu from time (48 h) MIC,.1 mg/l Cs,.5 mg/l C max 12 mg/l Log concentration (mg/l)

21 In vivo pharmacokinetics

22 What is PK analysis and modeling? Noncompartmental analysis Noncompartmental PK analysis examines total drug exposure and looks for function(s) fitting the change of concentration over time without reference to where the drug may distribute. Analysis is simple and does not imply anything concerning the actual fate of the drug. The results are purely descriptive and non-predictive unless the function selected is linked to physical phenomena (e.g. 1 st order kinetics). 22

23 What is PK analysis and modeling? Compartmental analysis Describes and predicts the concentration-time curve based on the movements of the drug between compartments (kinetic or physiological model) Once the model is indentified, it can be used to predict the concentration at any time. The model may be (very) difficult to develop The simplest PK compartmental model is the onecompartmental PK model with IV bolus administration and first-order elimination. The most complex PK models rely on the use of physiological information to ease development and validation. 23

24 What is PK analysis and modeling? Compartmental analysis The simplest PK compartmental model is the onecompartmental PK model with IV bolus administration and first-order kinetic elimination This can be developed with simple software accessible to lay users such as Prism (with some sophistication sometimes) More complex PK models rely on the use of physiological information to ease development and validation. This requires "high capacity" software that is often impossible to use without serious introduction 24

25 Simple compartmental models

26 Integrating (calculus)

27 From model to data and finding "best parameters" with a computer (curve fitting) choose (or enter) your equation enter your data enter initial parameter values (best estimate; optional but useful) the computer will then compare equation-based curve to actual data modify parameters by successive iterations until a "best" fit is obtained the limit is the number of iterations numerical integration

28 From data to model with a computer (no calculus)

29 Example of monocopartmental analysis (*) Exponential-decay (1 compartment) 15 concentratoin (mg/l) theoretical curve C o = 142 mg/l - 2 g - 7 kg - V d =.2 L/kg ke =.185 (t1/2 = 2h) plateau = equation: C = C.e -kt time (h) * this analysis and the following ones concern ceftazidime IV 29

30 Fitting to ideal population data (*) Ceftazidime: ideal patients concentration (mg/l) 15 SPAN K PLATEAU HalfLife 125 Std. Error SPAN K 1 PLATEAU 95% Confidence Intervals SPAN K 75 PLATEAU HalfLife Goodness of Fit Degrees of Freedom 5 R² to to to to * data from a few volunteers time (h) 3

31 Ideal population: tests for 95 % CI Ceftazidime: ideal patients concentration (mg/l) 15 SPAN K PLATEAU HalfLife 125 Std. Error SPAN K 1 PLATEAU 95% Confidence Intervals SPAN K 75 PLATEAU HalfLife Goodness of Fit Degrees of Freedom 5 R² to to to to time (h) 31

32 Ideal population: residuals ideal-valuesnonlin fit of ideal-valuesdata Table-1:Residuals time (h) why are they much larger here? 32

33 Real population (*) ceftazidime: real population 15 concentration (mg/l) n = 27 patients C o = 98 mg/l (2 g kg) ke =.1865 (t 1/2 = h) plateau = * data from several patients time (h) 33

34 Real population: 95 % CI concentration (mg/l) ceftazidime: real population One phase exponential decay Best-fit values SPAN K PLATEAU HalfLife Std. Error SPAN K PLATEAU 95% Confidence Intervals SPAN K PLATEAU HalfLife Goodness of Fit Degrees of Freedom R² Absolute Sum of Squares Sy.x to to to time (h) 34

35 Real population: residuals residuals Real population: residuals time (h) residuals Ideal population: Residuals time (h) 35

36 More complex models: accumulation / decay concentration (mg/l) Bateman function (applied to ceftazidime) theoretical curve D = 2 g / 7 kg (28.57 mg/kg) - V d =.2 L/kg ka = 8 ke =.3465 (t1/2 = 2h) plateau = equation: C = D/Vd x ka/(ka-ke) [e -ket e- kat ] time (h) 36

37 In search of more complex models with Prism

38 Accumulation / decay with Prism (*) Ceftazidime with Bateman function 1 real data concentration (mg/l) R 2 = time (h) equation: C = D/Vd x ka/(ka-ke) [e -ket e- kat ] 38

39 Examples d'analyse monocompartimentale (*) concentration (mg/l) Ceftazidime with Bateman time (h) accumulation-decay Best-fit values D VD KA KE E Std. Error D VD KA KE E 95% Confidence Intervals D VD KA KE E Goodness of Fit Degrees of Freedom R² equation: C = D/Vd x ka/(ka-ke) [e -ket e- kat ] 3.545e e e to 7.89e to to 4.245e to to 1.152e Prism has a problem here! 39

40 When the data become really too complex 4

41 The Mixed non-lin approaches Different softwares, but alll working by numerical integration based on pre-defined models

42 The Monlolix project

43 The Monlolix project

44 The Monlolix software

45 Temocillin project (full) Concentration (mg/l) ID: 1 ID: 2 ID: 3 ID: 4 ID: 5 ID: 6 ID: 7 ID: 8 ID: 9 ID: 1 ID: Time (h)

46 Projet temocillin (simplified)

47 Outputs: individual curves

48 Outputs: spaghetti plot (*) 25 Total number of subjects: 4 Average number of doses per subject: 1 Total/Average/Min/Max numbers of observations: observations time * not noodles!

49 Outputs: population curves

50 Outputs: population DV Percentile 9 : Obs. Percentile 5 : Obs. Percentile 1 : Obs. Percentile 9 : 9% CI Percentile 5 : 9% CI Percentile 1 : 9% CI Outliers Outlier (area) TIME

51 Outputs: observations vs. predictions Using the population parameters Using the individual parameters 2 2 observations 15 1 observations predictions predictions

52 Outputs: residuals PWRES 2 IWRES 2 NPDE time time time PWRES IWRES NPDE predictions predictions predictions