Title: Population PKPD
1Introduction to non-linear mixed models in
Pharmakokinetics
Ziad Taib Biostatistics, AStrazeneca
Vålådalen, March 2009
2Some 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.
3Typical data
One curve per patient
Concentration
Time
4Proposed model
where
5Common 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
6Non 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
7PHARMACOKINETICS
- 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).
8PHARMACOKINETICS
- 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.
9Distribution
PK
PK
What happens to the drug in the body?
10Plasma 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.
11Plasma concentration curve
12Pharmacodynamics
- Study of effect of the drug on the body
- PD parameters Emax and EC50
13Analysis of PK/PD data
Two types of approaches
14PK/PD
15PK non-compartmental approach
16Pharmacokinetic models
Various types of models
17One-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
18One compartment model
- From outside depo (tablet)
- IV
- Infusion
C(t) , V
Ve
Vin
ka
ke
19Half life
Plasma concentration
a phase (distribution)
C
C/2
Half life
Time
T½
0
20Distribution 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)
21Tmax and Cmax in general
Cmax peak of plasma concentration
Distribution
absorption
Elimination
Tmax
Time
22Area Under the Curve (AUC)
23Slope -k/2.3
Time
Regular coordinates
Log coordinates
24Calculation of Volume of Distribution
25Drug clearance refers to the volume of plasma
fluid that is cleared of drug per unit time.
D, VD
26ELIMINATION RATE CONSTANT
27The general Nonlinear Mixed Effects Model
Pharmacokineticists use the term population
model when the model involves random effects.
28Example 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.
29Random 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
30An example
One PCC per patients
Concentration
Time
31Step 1 Write a PK (PK/PD) model
A statistical model
Mean model functional relationship
Variance model Assumptions on the residuals
32Step 1 Write a deterministic (mean) model to
describe the individual kinetics
33One compartment model with constant intravenous
infusion rate
34Step 1 Write a deterministic (mean) model to
describe the individual kinetics
35Step 1 Write a deterministic (mean) model to
describe the individual kinetics
36Step 1 Write a model (variance) to describe the
magnitude of departure to the kinetics
Residual
Time
37Step 1 Write a model (variance) to describe the
magnitude of departure to the kinetics
Residual
Time
38Step 1 Describe the shape of departure to the
kinetics
Residual
Time
39Step 1 Write an "individual" model
jth concentration measured on the ith patient
jth sample time of the ith patient
40Step 2 Describe variation between individual
parameters
Distribution of clearances
Population of patients
41Step 2 Our view through a sample of patients
Sample of patients
Sample of clearances
42Step 2 Two main approaches
Sample of clearances
43Step 2 Two main approaches
Sample of clearances
Semi-parametric approach (e.g. kernel estimate)
44Step 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
45Step 2 Two main approaches
Sample of clearances
46Step 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,)
47Step 2 Parametric approach
A simple model
48Step 2 Population parameters
49Step 2 Population parameters
Mean parameters
Variance parameters measure inter-individual
variability
50Step 2 Parametric approach
A model including covariates
51Step 2 A model including covariates
X2i
Age
X1i
BMI
52Step 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
531. Naive pooled data a single patient
Does not allow to estimate inter-individual
variation.
Concentration
Time
542. Two stages method stage 1Within individual
variability
Concentration
Time
55Two 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
563. The Maximum Likelihood Estimator
Let
57The 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)
583.1 Laplacian expansion based methods
First Order (FO) (Beal, Sheiner 1982)
NONMEM Linearisation about 0
59Laplacian 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
60Laplacian 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
61Gaussian quadratures
Approximation of the integrals by discrete sums
62NONMEM (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
634. Simulations methods
Simulated Pseudo Maximum Likelihood (SPML)
Minimize
simulated variance
64Properties
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
65Model check Graphical analysis
Variance reduction
Predicted concentrations
Observed concentrations
66Graphical analysis
Time
The PK model is inappropriate
The PK model seems good
67Graphical analysis
Age
Age
BW
BW
Variance model seems good
Variance model not appropriate
68Graphical analysis
Normality should be questioned
add other covariables or try semi-parametric model
Normality acceptable
69Extension 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.
70Advantages 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
71To 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
72What 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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