An introduction to population kinetics

1. NATIONAL VETERINARY SCHOOL Toulouse An introduction to population kinetics Didier Concordet

2. Preliminaries Definitions : Random variable Fixed variable Distribution

3. Random or fixed ? Definitions : A random variable is a variable whose value changes when the experiment is begun again. The value it takes is drawn from a distribution. A fixed variable is a variable whose value does not change when the experiment is begun again. The value it takes is chosen (directly or indirectly) by experimenter.

4. Example in kinetics A kinetics experiment is performed on two groups of 10 dogs. The first group of 10 dogs receives the formulation A of an active principle, the other group receives the formulation B. The two formulations are given by IV route at time t=0. The dose is the same for the two formulations D = 10mg/kg. For both formulations, the sampling times are t1 = 2 mn, t2= 10mn, t3= 30 mn, t4 = 1h, t5=2 h, t6 = 4 h.

5. Randomor fixed ? The formulation Fixed Fixed Dose Thesamplingtimes Fixed Analytical error Departure to kinetic model Theconcentrations Random Thedogs Random Population kinetics Classical kinetics Fixed

6. 7.8 8.0 8.2 8.4 0 0.1 0.2 0.3 0.4 Clearance Concentrations at t=2 mn Distribution ? The distribution of a random variable is defined by the probability of occurrence of the all the values it takes.

7. An example 30 horses Concentration Time

8. Step 1 : Write a PK (PK/PD) model A statistical model Mean model : functional relationship Variance model : Assumptions on the residuals

9. Step 1 : Write a deterministic (mean) model to describe the individual kinetics

10. Step 1 : Write a deterministic (mean) model to describe the individual kinetics

11. residual Step 1 : Write a deterministic (mean) model to describe the individual kinetics

12. Step 1 : Write a model (variance) to describe the magnitude of departure to the kinetics Residual Time

13. Step 1 : Write a model (variance) to describe the magnitude of departure to the kinetics Residual Time

14. Step 1 : Describe the shape of departure to the kinetics Residual Time

15. residual CV Gaussian residual with unit variance Step 1 :Write an "individual" model jth concentration measured on the ithanimal jth sample time of the ithanimal

16. 0 0.1 0.2 0.3 0.4 Clearance Step 2 : Describe variation between individual parameters Distribution of clearances Population of horses

17. Step 2 : Our view through a sample of animals Sample of horses Sample of clearances

18. Semi-parametric approach Step 2 : Two main approaches Sample of clearances

19. Step 2 : Two main approaches Sample of clearances Semi-parametric approach (e.g. kernel estimate)

20. Step 2 : Semi-parametric approach • Does require a large sample size to provide results • Difficult to implement • Is implemented on confidential pop PK softwares Does not lead to bias

21. 0 0.1 0.2 0.3 0.4 Parametric approach Step 2 : Two main approaches Sample of clearances

22. Step 2 : Parametric approach • Easier to understand • Does not require a large sample size to provide (good or poor) results • Easy to implement • Is implemented on the most popular pop PK softwares (NONMEM, S+, SAS,…) Can lead to severe bias when the pop PK is used as a simulation tool

23. Step 2 : Parametric approach A simple model :

24. ln V ln Cl Step 2 : Population parameters

25. Step 2 : Population parameters Mean parameters Variance parameters : measure inter-individual variability

26. Step 2 : Parametric approach A model including covariables

27. Agei Age BWi BW Step 2 : A model including covariables

28. Step 3 :Estimate the parameters of the current model Several methods with different properties • Naive pooled data • Two-stages • Likelihood approximations • Laplacian expansion based methods • Gaussian quadratures • Simulations methods

29. Naive pooled data : a single animal Does not allow to estimate inter-individual variation. Concentration Time

30. Two stages method: stage 1 Concentration Time

31. Two stages method : stage 2 Does not require a specific software Does not use information about the distribution Leads to an overestimation of W which tends to zero when the number of observations per animal increases Cannot be used with sparse data

32. The Maximum Likelihood Estimator Let

33. The Maximum Likelihood Estimator is the best estimator that can be obtained among the consistent estimators It is efficient (it has the smallest variance) Unfortunately, l(y,q) cannot be computed exactly Several approximations of l(y,q)

34. Laplacian expansion based methods First Order (FO) (Beal, Sheiner 1982) NONMEM Linearisation about 0

35. Laplacian expansion based methods First Order Conditional Estimation (FOCE) (Beal, Sheiner) NONMEM Non Linear Mixed Effects models (NLME) (Pinheiro, Bates)S+, SAS (Wolfinger) Linearisation about the current prediction of the individual parameter

36. Laplacian expansion based methods First Order Conditional Estimation (FOCE) (Beal, Sheiner) NONMEM Non Linear Mixed Effects models (NLME) (Pinheiro, Bates)S+, SAS (Wolfinger) Linearisation about the current prediction of the individual parameter

37. Gaussian quadratures Approximation of the integrals by discrete sums

38. Simulations methods Simulated Pseudo Maximum Likelihood (SPML) Minimize simulated variance

39. Properties Criterion When Advantages Drawbacks Naive pooled data Never Easy to use Does not provide consistent estimate Two stages Rich data/ Does not require Overestimation of initial estimates a specific software variance components FO Initial estimate quick computation Gives quickly a result Does not provide consistent estimate FOCE/NLME Rich data/ small Give quickly a result. Biased estimates when intra individual available on specific sparse data and/or variance softwares large intra Gaussian Always consistent and The computation is long quadrature efficient estimates when P is large provided P is large SMPL Always consistent estimates The computation is long when K is large

40. Step 4 : Graphical analysis Variance reduction Predicted concentrations Observed concentrations

41. Step 4 : Graphical analysis Time The PK model is inappropriate The PK model seems good

42. Step 4 : Graphical analysis Age Age BW BW Variance model seems good Variance model not appropriate

43. under gaussian assumption Step 4 : Graphical analysis Normality should be questioned add other covariables or try semi-parametric model Normality acceptable

44. No Simplify the model Yes No Yes To Summarise Write the PK model Write a first model for individual parameters without any covariable Interpret results Add covariables Are there variations between individuals parameters ? (inspection of W) Check (at least) graphically the model Is the model correct ?

45. What you should no longer believe Messy data can provide good results Population PK/PD is made to analyze sparse data No stringent assumption about the data is required Population PK/PD is too difficult for me