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Population PKPD

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Title: Population PKPD


1
Introduction to non-linear mixed models in
Pharmakokinetics
Ziad Taib Biostatistics, AStrazeneca
Vålådalen, March 2009
2
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.

3
Typical data
One curve per patient
Concentration
Time
4
Proposed model
where
5
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

6
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

7
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).

8
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.

9
Distribution
PK
PK
What happens to the drug in the body?
10
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.

11
Plasma concentration curve
12
Pharmacodynamics
  • Study of effect of the drug on the body
  • PD parameters Emax and EC50

13
Analysis of PK/PD data
Two types of approaches
14
PK/PD
15
PK non-compartmental approach
16
Pharmacokinetic models
Various types of models
17
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

18
One compartment model
  • General model
  • From outside depo (tablet)
  • IV
  • Infusion

C(t) , V
Ve
Vin
ka
ke
19
Half life
Plasma concentration
a phase (distribution)
C
C/2
Half life
Time
T½
0
20
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)
21
Tmax and Cmax in general
Cmax peak of plasma concentration
Distribution
absorption
Elimination
Tmax
Time
22
Area Under the Curve (AUC)
23
Slope -k/2.3
Time
Regular coordinates
Log coordinates
24
Calculation of Volume of Distribution
25
Drug clearance refers to the volume of plasma
fluid that is cleared of drug per unit time.
D, VD
26
ELIMINATION RATE CONSTANT
27
The general Nonlinear Mixed Effects Model
Pharmacokineticists use the term population
model when the model involves random effects.
28
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.
29
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
30
An example
One PCC per patients
Concentration
Time
31
Step 1 Write a PK (PK/PD) model
A statistical model
Mean model functional relationship
Variance model Assumptions on the residuals
32
Step 1 Write a deterministic (mean) model to
describe the individual kinetics
33
One compartment model with constant intravenous
infusion rate
34
Step 1 Write a deterministic (mean) model to
describe the individual kinetics
35
Step 1 Write a deterministic (mean) model to
describe the individual kinetics
36
Step 1 Write a model (variance) to describe the
magnitude of departure to the kinetics
Residual
Time
37
Step 1 Write a model (variance) to describe the
magnitude of departure to the kinetics
Residual
Time
38
Step 1 Describe the shape of departure to the
kinetics
Residual
Time
39
Step 1 Write an "individual" model
jth concentration measured on the ith patient
jth sample time of the ith patient
40
Step 2 Describe variation between individual
parameters
Distribution of clearances
Population of patients
41
Step 2 Our view through a sample of patients
Sample of patients
Sample of clearances
42
Step 2 Two main approaches
Sample of clearances
43
Step 2 Two main approaches
Sample of clearances
Semi-parametric approach (e.g. kernel estimate)
44
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
45
Step 2 Two main approaches
Sample of clearances
46
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,)

47
Step 2 Parametric approach
A simple model
48
Step 2 Population parameters
49
Step 2 Population parameters
Mean parameters
Variance parameters measure inter-individual
variability
50
Step 2 Parametric approach
A model including covariates
51
Step 2 A model including covariates
X2i
Age
X1i
BMI
52
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

53
1. Naive pooled data a single patient
Does not allow to estimate inter-individual
variation.
Concentration
Time
54
2. Two stages method stage 1Within individual
variability
Concentration
Time
55
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
56
3. The Maximum Likelihood Estimator
Let
57
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)
58
3.1 Laplacian expansion based methods
First Order (FO) (Beal, Sheiner 1982)
NONMEM Linearisation about 0
59
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
60
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
61
Gaussian quadratures
Approximation of the integrals by discrete sums
62
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

63
4. Simulations methods
Simulated Pseudo Maximum Likelihood (SPML)
Minimize
simulated variance
64
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
65
Model check Graphical analysis
Variance reduction
Predicted concentrations
Observed concentrations
66
Graphical analysis
Time
The PK model is inappropriate
The PK model seems good
67
Graphical analysis
Age
Age
BW
BW
Variance model seems good
Variance model not appropriate
68
Graphical analysis
Normality should be questioned
add other covariables or try semi-parametric model
Normality acceptable
69
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.

70
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

71
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
72
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
73
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