Lecture 4 Non-Linear and Generalized Mixed Effects Models

1. Ziad Taib Biostatistics, AZ MV, CTH Mars 2009 Lecture 4Non-Linear and Generalized Mixed Effects Models 1 Date

2. Part IIIntroduction to non-linear mixed models in Pharmakokinetics

3. Typical data One curve per patient Concentration Time

4. Common situation (bio)sciences: • A continuous responseevolves over time (or other condition) within individuals from a population of interest • Scientific interest focuses on featuresormechanisms that underlie individual time trajectoriesof the response and how these vary across the population. • A theoretical or empirical modelfor such individual profiles, typically non-linear in the parameters that may be interpreted as representing such features or mechanisms, is available. • Repeated measurementsover time are available on each individual in a sample drawn from the population • Inferenceon the scientific questions of interest is to be made in the context of the model and its parameters

5. Non linear mixed effects models Nonlinear mixed effects models: or hierarchical non-linear models • A formal statistical framework for this situation • A “hot” methodological research area in the early 1990s • Now widely accepted as a suitable approach to inference, with applications routinely reported and commercial software available • Many recent extensions, innovations • Have many applications: growth curves, pharmacokinetics, dose-response etc

6. PHARMACOKINETICS • A drugs can administered in many different ways: orally, by i.v. infusion, by inhalation, using a plaster etc. • Pharmacokinetics is the study of the rate processes that are responsible for the time course of the level of the drug (or any other exogenous compound in the body such as alcohol, toxins etc).

7. PHARMACOKINETICS • Pharmacokineticsis about what happens to the drug in the body.Itinvolves the kinetics of drug absorption, distribution, and elimination i.e. metabolism and excretion (adme). The description of drug distribution and elimination is often termed drug disposition. • One way to model these processes is to view the body as a system with a number of compartments through which the drug is distributed at certain rates. This flow can be described using constant rates in the cases of absorbtion and elimination.

8. Plasma concentration curves (PCC) • The concentration of a drug in the plasma reflects many of its properties. A PCC gives a hint as to how the ADME processes interact. If we draw a PCC in a logarithmic scale after an i.v. dose, we expect to get a straight line since we assume the concentration of the drug in plasma to decrease exponentially. This is first order- or linear kinetics. The elimination rate is then proportional to the concentration in plasma. This model is approximately true for most drugs.

9. Concentration Time Plasma concentration curve

10. Pharmacokinetic models Various types of models

11. k i.v. D, VD One-compartment model with rapid intravenous administration: The pharmacokinetics parameters • Half life • Distribution volume • AUC • Tmax and Cmax • D: Dose • VD: Volume • k: Elimination rate • Cl: Clearance

12. General model Tablet IV C(t) , V Ve Vin ka ke One compartment model

13. Typical example in kinetics A typical kinetics experiment is performed on a number, m, of groups of h patients. Individuals in different groups receive the same formulation of an active principle, and different groups receive different formulations. The formulations are given by IV route at time t=0. The dose, D, is the same for all formulations. For all formulations, the plasma concentration is measured at certain sampling times.

14. Random or fixed ? The formulation Fixed Fixed Dose Thesamplingtimes Fixed Analytical error Departure to kinetic model Theconcentrations Random Thepatients Random Population kinetics Classical kinetics Fixed

15. An example One PCC per patients Concentration Time

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

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

18. One compartment model with constant intravenous infusion rate

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

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

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

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

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

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

25. 0 0.1 0.2 0.3 0.4 Clearance Step 2 : Describe variation between individual parameters Distribution of clearances Population of patients

26. Step 2 : Our view through a sample of patients Sample of patients Sample of clearances

27. Semi-parametric approach Step 2 : Two main approaches:parametric and semi-parametric Sample of clearances

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

29. Step 2 : Semi-parametric approach • Does require a large sample size to provide results • Difficult to implement • Is implemented on “commercial” PK software Bias?

30. 0 0.1 0.2 0.3 0.4 Parametric approach Step 2 : Two main approaches Sample of clearances

31. 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 software (NONMEM, S+, SAS,…)

32. Step 2 : Parametric approach A simple model :

33. ln V Mean parameters ln Cl Step 2 : Population parameters Variance parameters : measure inter-individual variability

34. Step 2 : Parametric approach A model including covariates

35. 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

36. 1. Naive pooled data : a single patient Naïve Pooled Datacombines all the data as if they came from a single reference individual and fit into a model using classical fitting procedures. It is simple, but can not investigate fixed effect sources of variability, distinguish between variability within and between individuals.

37. The naïve approach does not allow to estimate inter- individual variation. Concentration Time

38. 2. Two stages method: stage 1Within individual variability Concentration . . . Time

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

40. 3. The Maximum Likelihood Estimator Let

41. 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) are used.

42. 3.1 Laplacian expansion based methods First Order (FO) (Beal, Sheiner 1982) NONMEM Linearisation about 0

43. 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

44. Gaussian quadratures Approximation of the integrals by discrete sums

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

46. 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

47. Model check: Graphical analysis Variance reduction Predicted concentrations Observed concentrations

48. Graphical analysis Time The PK model is inappropriate The PK model seems good

49. under gaussian assumption Graphical analysis Normality should be questioned add other covariates or try semi-parametric model Normality acceptable

50. The Theophylline example • An alkaloid derived from tea or produced synthetically; it is a smooth muscle relaxant used chiefly for its bronchodilator effect in the treatment of chronic obstructive pulmonary emphysema, bronchial asthma, chronic bronchitis and bronchospastic distress. It also has myocardial stimulant, coronary vasodilator, diuretic and respiratory center stimulant effects. http://www.tau.ac.il/cc/pages/docs/sas8/stat/chap46/sect38.htm