Population PKPD

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Introduction to non-linear mixed models in
Pharmakokinetics
Ziad Taib Biostatistics, AStrazeneca
Vålådalen, March 2009
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Some references

• Davidian, M. and Giltinan, D.M. (1995). Nonlinear
  Models for Repeated Measurement Data. Chapman
  Hall/CRC Press.
• Davidian, M. and Giltinan, D.M. (2003). Nonlinear
  models for repeated measurement data An overview
  and update. Journal of Agricultural, Biological,
  and Environmental Statistics 8, 387419.
• Davidian, M. (2009). Non-linear mixed-effects
  models. In Longitudinal Data Analysis, G.
  Fitzmaurice, M. Davidian, G. Verbeke, and G.
  Molenberghs (eds). Chapman Hall/CRC Press, ch.
  5, 107141.
• (An outstanding overview ?) Pharmacokinetics and
  pharmaco- dynamics , by D.M. Giltinan, in
  Encyclopedia of Biostatistics, 2nd edition.

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Typical data
One curve per patient
Concentration
Time
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Proposed model
where
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Common situation in the biosciences

• A continuous response evolves over time (or other
  condition) within individuals from a population
  of interest
• Scientific interest focusses on features or
  mechanisms that underlie individual time
  trajectories of the response and how these vary
  across the population.
• A theoretical or empirical model for such
  individual profiles, typically non-linear in
  parameters that may be interpreted as
  representing such features or mechanisms, is
  available.
• Repeated measurements over time are available on
  each individual in a sample drawn from the
  population
• Inference on the scientific questions of interest
  is to be made in the context of the model and its
  parameters

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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
• Objective of this talk An updated review of the
  model and survey of recent pharmacokinetics
  applications

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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).

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PHARMACOKINETICS

• Pharmacokinetics is about what happens to the
  drug in the body. It involves 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.

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Distribution
PK
PK
What happens to the drug in the body?
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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.

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Plasma concentration curve
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Pharmacodynamics

• Study of effect of the drug on the body
• PD parameters Emax and EC50

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Analysis of PK/PD data
Two types of approaches
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PK/PD
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PK non-compartmental approach
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Pharmacokinetic models
Various types of models
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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

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One compartment model

• General model

• From outside depo (tablet)
• IV
• Infusion

C(t) , V
Ve
Vin
ka
ke
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Half life
Plasma concentration
a phase (distribution)
C
C/2
Half life
Time
T½
0
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Distribution volume

• Distribution volume When we administer a certain
  dose, d, of the drug, we obtain a certain
  concentration, C0 at time 0 assuming nothing is
  lost during distribution. Since concentration
  dose/volume, we have the simple equation (1)
  below where Vd, stands for distribution volume.

(1)
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Tmax and Cmax in general
Cmax peak of plasma concentration
Distribution
absorption
Elimination
Tmax
Time
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Area Under the Curve (AUC)
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Slope -k/2.3
Time
Regular coordinates
Log coordinates
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Calculation of Volume of Distribution
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Drug clearance refers to the volume of plasma
fluid that is cleared of drug per unit time.
D, VD
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ELIMINATION RATE CONSTANT
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The general Nonlinear Mixed Effects Model
Pharmacokineticists use the term population
model when the model involves random effects.
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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 formulatio.
The formulations are given by IV route at time
t0. The dose, D, is the same for all
formulations.
For all formulations, the plasma concentration is
measured at certain sampling times.
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Random or fixed ?

The formulation
Fixed
Fixed
Dose
The sampling times
Fixed
Analytical error Departure to kinetic model
The concentrations
Random
The patients
Random
Population kinetics
Classical kinetics
Fixed
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An example
One PCC per patients
Concentration
Time
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Step 1 Write a PK (PK/PD) model
A statistical model
Mean model functional relationship
Variance model Assumptions on the residuals
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Step 1 Write a deterministic (mean) model to
describe the individual kinetics
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One compartment model with constant intravenous
infusion rate
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Step 1 Write a deterministic (mean) model to
describe the individual kinetics
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Step 1 Write a deterministic (mean) model to
describe the individual kinetics
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Step 1 Write a model (variance) to describe the
magnitude of departure to the kinetics
Residual
Time
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Step 1 Write a model (variance) to describe the
magnitude of departure to the kinetics
Residual
Time
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Step 1 Describe the shape of departure to the
kinetics
Residual
Time
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Step 1 Write an "individual" model
jth concentration measured on the ith patient
jth sample time of the ith patient
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Step 2 Describe variation between individual
parameters
Distribution of clearances
Population of patients
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Step 2 Our view through a sample of patients
Sample of patients
Sample of clearances
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Step 2 Two main approaches
Sample of clearances
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Step 2 Two main approaches
Sample of clearances
Semi-parametric approach (e.g. kernel estimate)
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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
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Step 2 Two main approaches
Sample of clearances
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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,)

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Step 2 Parametric approach
A simple model
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Step 2 Population parameters
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Step 2 Population parameters
Mean parameters
Variance parameters measure inter-individual
variability
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Step 2 Parametric approach
A model including covariates
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Step 2 A model including covariates
X2i
Age
X1i
BMI
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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

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1. Naive pooled data a single patient
Does not allow to estimate inter-individual
variation.
Concentration
Time
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2. Two stages method stage 1Within individual
variability
Concentration
Time
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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 of W which tends to zero when the
number of observations per animal
increases Cannot be used with sparse data
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3. The Maximum Likelihood Estimator
Let
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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)
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3.1 Laplacian expansion based methods
First Order (FO) (Beal, Sheiner 1982)
NONMEM Linearisation about 0
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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
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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
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Gaussian quadratures
Approximation of the integrals by discrete sums
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NONMEM (Nonlinear Mixed-Effects Model)

• A FORTRAN 77 program
• Platform-independent
• Each time, users provide subroutine and data to
  be complied and run (needs FORTRAN Compilers)
• Mixed-effects
• Fixed effect mean population values and model
  parameters for covariates
• Random effect random variables (intersubject
  variability)
• Population model includes both effects,
  therefore, called Mixed-effects model

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4. Simulations methods
Simulated Pseudo Maximum Likelihood (SPML)
Minimize
simulated variance
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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
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Model check Graphical analysis
Variance reduction
Predicted concentrations
Observed concentrations
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Graphical analysis
Time
The PK model is inappropriate
The PK model seems good
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Graphical analysis
Age
Age
BW
BW
Variance model seems good
Variance model not appropriate
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Graphical analysis
Normality should be questioned
add other covariables or try semi-parametric model
Normality acceptable
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Extension Stochastic modelling

• Several approaches are possible in order to use
  stochastic models. An example is the use of
  Markov processes of different kinds. Another is
  the use of stochastic differential equations. As
  an example consider the simple i.v. case where
  the rate constant k is deterministically related
  to C.
• The use of Stochastic Differential Equations
  (SDEs) extend ODE The use of Stochastic
  Differential Equations (SDEs) extend ODE models
  by explicitly incorporating system noise, e.g.

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Advantages of use of SDEs

• Mean behaviour remains intact
• Naturally disentangles system error from
  measurement error
• Ability to account for structural
  misspecification
• Probabilistic framework for model determination
• Possibility of use of extended Kalman-filter for
  estimation of parameters

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