An introduction to population kinetics Didier Concordet NATIONAL VETERINARY SCHOOL Toulouse

Preliminaries Definitions : Random variable Fixed variable Distribution

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.

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 t 1 = 2 mn, t 2 = 10mn, t 3 = 30 mn, t 4 = 1h, t 5 =2 h, t 6 = 4 h.

Random or fixed ? The formulation Dose The sampling times The concentrations The dogs Fixed Random Fixed Random Analytical error Departure to kinetic model Population kinetics Classical kinetics

Distribution ? The distribution of a random variable is defined by the probability of occurrence of the all the values it takes. Clearance Concentrations at t=2 mn

An example 30 horses Time Concentration

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

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

residual

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

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

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

Step 1 :Write an "individual" model j th concentration measured on the i th animal j th sample time of the i th animal residual CV Gaussian residual with unit variance

Step 2 : Describe variation between individual parameters Distribution of clearances Population of horses Clearance

Step 2 : Our view through a sample of animals Sample of horsesSample of clearances

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

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

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

Step 2 : Two main approaches Sample of clearances Parametric approach

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

Step 2 : Parametric approach A simple model :

ln Cl ln V Step 2 : Population parameters

Mean parameters Variance parameters : measure inter-individual variability

Step 2 : Parametric approach A model including covariables

BW Age Age i BW i Step 2 : A model including covariables

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

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

Two stages method: stage 1 Concentration Time

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

The Maximum Likelihood Estimator Let

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, ) cannot be computed exactly Several approximations of l(y, )

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

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

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

Gaussian quadratures Approximation of the integrals by discrete sums

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

Properties Naive pooled dataNeverEasy to useDoes not provide consistent estimate Two stagesRich data/ Does not require Overestimation of initial estimatesa specific softwarevariance components FOInitial estimate quick computation Gives quickly a result Does not provide consistent estimate FOCE/NLMERich data/ smallGive quickly a result. Biased estimates when intra individual available on specific sparse data and/or variance softwares large intra Gaussian Alwaysconsistent andThe computation is long quadratureefficient estimates when P is large provided P is large SMPLAlwaysconsistent estimates The computation is long when K is large Criterion When Advantages Drawbacks

Step 4 : Graphical analysis Observed concentrations Predicted concentrations Variance reduction

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

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

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

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

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