Nonlinear Mixed-Effects Modeling (population …fcrauste/teaching/EUR...Nonlinear Mixed-Effects...

Digital Public Health Program Modeling Life Science - Dynamical Processes

Nonlinear Mixed-Effects Modeling (population approach)

Fabien Crauste Chargé de Recherches CNRS Institut de Mathématiques de Bordeaux UMR 5251 Université de Bordeaux

1. Motivation

2. Linear mixed-effects models

3. Nonlinear mixed-effects models

4. Identifiability

Motivation

MotivationsMain objective: reproducing data using a mathematical model

Math ModelData


Math ModelData

What data?

?

?


Math ModelData

What data?

?

?

What model?

?

?


Math ModelData

What data?

?

?

What model?

?

?

How to compare them?

Collinson-Pautz et al. (2016) Crauste et al. (2015)C

D8+

T c

ell c

ount

s(lo

g sc

ale)

Days post-infection0 5 10 15 20 25 30 35 40 45 50

101

102

103

104

105

106

Time (days)

CD

8 T

cell

counts

Lm

Flu

VV

MotivationsLongitudinal data: mean +/- standard deviation

2 examples: Tumor growth (left) and immune cell counts (right), in various conditions ; mean and standard deviations

MotivationsLongitudinal data: individual dynamics

CD

8+ T

cel

ls(lo

g sc

ale)

Days after infection

Number of CD8+ T cells in Balbc mice

mean +/- s.d. and individual dynamics

Time (hours) after injection

War

farin

Con

cent

ratio

n (m

g/L)

Concentration of warfarin in 32 healthy patients

(1.5 mg/kg for each patient)

Bazzoli et al. (2014)

From Lavielle and Bleakley (2015)

1. Viral load of Hepatitis C (4 patients)


From Lavielle and Bleakley (2015)

2. Weight curves (4 rats)

1. Viral load of Hepatitis C (4 patients)


MotivationsHow to account for inter-individual variability?

Option 1: Use a model to characterize each individual’s behavior-> not enough data to estimate all parameter values -> identifiability issue -> impossibility to characterize individuals’ behaviors


Option 1: Use a model to characterize each individual’s behavior-> not enough data to estimate all parameter values -> identifiability issue -> impossibility to characterize individuals’ behaviors

Option 2: Use a Population Approach and Mixed-Effects Models-> consider individuals belong to the same population -> use statistical model -> validated in PKPD works


(Non)Linear (mixed-effects) models

(Non)Linear (mixed-effects) modelsLinear models

Assume a linear relationship between observations and explanatory variables

Methodology (simple): - pick up explanatory variables - estimate parameters using least-square method




But: a linear relationship is not always appropriate





d = 1

d = 3 d = 2





d = 1

d = 3 d = 2

Question: is weight a polynomial function of time?

Instead of assuming a linear relationship between observations and explanatory variables

(Non)Linear (mixed-effects) modelsNonlinear models

assume a nonlinear relationship between observations and explanatory variables:

where f is a structural model, defined by a set (vector) of parameter values and an explanatory variable .

Instead of assuming a linear relationship between observations and explanatory variables

(Non)Linear (mixed-effects) modelsNonlinear models

assume a nonlinear relationship between observations and explanatory variables:

where f is a structural model, defined by a set (vector) of parameter values and an explanatory variable .

In the case of rats weight, one can for instance assume

Assume a linear relationship between observations and explanatory variables for each individual:

(Non)Linear (mixed-effects) modelsLinear mixed-effects models

with



with

fixed effect



with

fixed effect random effect



with

fixed effect random effect

Random effects and residuals are assumed to be random variables, with mean = 0 and a variance to be estimated. Assuming they are normally distributed, then:

Then one needs relevant and appropriate methods to estimate parameter values… (I’ll come back to that)

Nonlinear mixed-effects models

- observed data of individual i

- explanatory variables

- vector of parameters, with

- the structural model, here a function of explanatory variables and a vector of parameter values

- the error model,

Nonlinear mixed-effects modelsStructural and error models

From the previous considerations, we can then assume

where

[constant]

[proportional]

[combined]

Nonlinear mixed-effects modelsStructural and error models: comments

Comments


Comments

- observed data of individual i-> the number of observations may vary from one individual to another


Comments


- explanatory variables-> time is often an explanatory variable-> solutions of a dynamical system may also be explanatory variables


Comments



- vector of parameters, with -> the law of random effects has to be chosen-> it may depend on one’s knowledge or on classical laws


Comments



- vector of parameters, with -> the law of random effects has to be chosen-> it may depend on one’s knowledge or on classical laws

- the error model,

-> often, a constant error is used, -> then

How does the drug circulate in the body?

Warfarin injections in 32 patients

Nonlinear mixed-effects modelsAn example in PharmacoKinetics-PharmacoDynamics (PK-PD)

Time (hours) after injectionW

arfa

rin C

once

ntra

tion

(mg/

L)

How does it affect the patient?

PCA

(%)

Time (hours) after injectionBazzoli et al. (2014)

Consider a simple model (1 compartment)

Drug Main Compartment

dose F

absorption ka clearance kevolume V

Warfarin injections in 32 patients PK model

Concentration: , where



Consider a classical model: direct response, with inhibitory effect

Effect(Y )PCA synthesis

Rindegradation

kout

function(C(t))

Warfarin injections in 32 patients PD model

Effect of the drug on PCA



Warfarin injections in 32 patients PK-PD model

Effect of the drug on PCA:

Drug concentration:

2 quantities, 9 parameters, and non-identifiability (see for instance F and V)




The observation model is




2 types of data (PK and PD)





structural models





explanatory variable (time)





individual-dependent parameters





One parameter fixed within the population





combined error model





constant error model




Nonlinear mixed-effects modelsCovariates

Covariates (age, weight, group…) allow to account for specificities in the population that could explain, at least partially, the data:

Covariates are a priori individual-dependent.




Example: weight curves of rats





males





males females

Nonlinear mixed-effects modelsCovariates on the PK-PD example



covariate on all parameters?


Categorical covariates (age, weight, group…) allow to account for specificities in the population that could explain, at least partially, the data:


covariate on all parameters?(Wald test)

log(individual weights)

log(

k a)

log(

V)

log(

k e)

gender



Categorical covariates (age, weight, group…) allow to account for specificities in the population that could explain, at least partially, the data:


covariate on all parameters?Wald test



Nonlinear mixed-effects modelsEstimating parameter values

This approach aims at modeling the behavior of a population of individuals, hence the objective is to characterize the population behavior. This is done by estimating population parameters, using data of individual dynamics.

Statistical methods used to estimate parameter values are based on specific probability distributions: we denote by the set of population parameters (fixed effects, error parameters…) and by the likelihood of observations y with respect to




Frequentist approach: maximize the likelihood of observations with respect to parameters




Frequentist approach: maximize the likelihood of observations with respect to parameters

Bayesian approach: maximize the conditional probability of parameters with respect to observations

Nonlinear mixed-effects modelsEstimating parameter values: likelihood maximization

- Model linearization: use an order 1 development of f, compute the integral, and obtain a ‘linearized’ likelihood

- Numerical integration: use a numerical integration of the integral, computationally very expensive

- Stochastic integration: use Monte-Carlo simulations instead of numerical integration of the integral, computationally very expensive

The first question is: how to compute the likelihood?

Nonlinear mixed-effects modelsEstimating parameter values: likelihood maximization

The second question is: how to estimate population parameters?

- Standard minimization algorithms: like the Newton-Ralphson algorithm…

- Specific algorithms: SAEM (stochastic approximation expectation maximization), a modified version of the EM algorithm, is adapted to NMEM, it:

๏ first computes the conditional expectancy of the likelihood, using a stochastic approximation,

๏ then maximizes this quantity with respect to the parameters.

Once population parameters have been estimated, individual parameters are estimated by computing the conditional distribution of

Nonlinear mixed-effects modelsEstimating parameter values: individual parameters

In the frequentist approach, one can focus on the maximum a posteriori (MAP) estimator, given by

and the conditional mean of is the mean of the conditional distribution of

Consequence: the more information the better for the estimation, so individuals with few information may tend to select the population parameter as an individual parameter…

Nonlinear mixed-effects modelsEstimating parameter values: individual parameters & shrinkage

We made the assumption

and when estimating parameter values we obtained values that should follow this law ; hence, we can compute the






PKPD(warfarin)example

75% 10%




Are probability laws correct?

Are data informative

enough?


PKPD(warfarin)example

75% 10%

Nonlinear mixed-effects modelsValidating model & assumptions


The whole method is based on a set of assumptions/hypotheses:


The whole method is based on a set of assumptions/hypotheses: The whole method is based on a set of assumptions/hypotheses:

- Structural model: which formalism? which interactions? how many parameters?






- Parameters: fixed effects? random effects? which probability law for random effects?










- Error model: constant? proportional? combined? something else?















- Covariates: meaningful? meaningless?















- Covariates: meaningful? meaningless?

Need for diagnosis tools

Nonlinear mixed-effects modelsValidating model & assumptions: Residuals

Estimated values versus Theoretical values

- k

- Other ‘shrinkage’ measure:

Individual Weighted Residuals (IWRES) should be normally distributed, with mean 0 and variance 1, so the should be close to 0.

If not, then this may indicate that data are scarce or that too many parameters are used

Nonlinear mixed-effects modelsValidating model & assumptions: Distribution errors


Normalized prediction distribution

errors (npde)

[normally distributed]

PK-PD(warfarin)example

npde

npde

time (hours) Predicted population

PK

PD


Nonlinear mixed-effects modelsValidating model & assumptions: Visual Predictive Check


Visual predictive check (VPC)

[percentiles, based on simulations]


War

farin

Con

cent

ratio

n (m

g/L)

PK

PCA

(%)


PD



Nonlinear mixed-effects modelsValidating model & assumptions: Observations vs Predictions


Observations versus

Predictions

PD

Population predictions Individual predictions

Obs

erve

d PC

A (%

)

PK

Obs

erve

d C

once

ntra

tion



Nonlinear mixed-effects modelsTo conclude: software

Software Algorithms for estimating parameter values

Monolix SAEM

R nlme, nlmixr, saemix (SAEM), lme (Laplacian approximation)

NONMEM several (Laplacian approximation, SAEM, first-order method…) [but PK modeling only]

SAS NLMIXED (adaptive gaussian quadrature, first-order method)

WinBugs Bayesian algorithm

Identifiability

IdentifiabiltyWhat is ‘identifiability’?

Main equation:

Initial condition equation:

Observation:

- given inputs - set of parameters (values to be determined) - solution at time t, with the parameters

,

IdentifiabiltyWhat is ‘identifiability’?

Main equation:

Initial condition equation:

Observation:

- given inputs - set of parameters (values to be determined) - solution at time t, with the parameters

,

Definition. Let be given. The model is identifiable in if for all parameters such that outputs are equal then The model is said to be identifiable if it is identifiable in every parameter

Methods

IdentifiabiltyTheoretical approaches

Linear models Nonlinear models

Laplace transform Taylor series expansion

Taylor series expansions (of the output) Algebro-Differential elimination

State isomorphism theorem

We will consider the case when the model is either an algebraic equation or a system of ODEs

IdentifiabiltyTheoretical approaches: an illustrative example

Hemodynamics model (C. Audebert’s PhD thesis)

Model



Model

Time (s)

Obs

erva

tion

(ml/s

)

Observation Qin



Model

Time (s)

Obs

erva

tion

(ml/s

)

Observation QinEstimate


Model

, with


using Laplace transform,

so

Hence parameters are identifiable if

IdentifiabiltyTheoretical approaches

Yet, most of the time it is not possible to prove identifiability

and data are not always ‘ideal’

IdentifiabiltyStatistical approaches

Raue et al. (2009)

‘Given a model that sufficiently describes the measured data, it is important to infer how well model parameters are

determined by the amount and quality of experimental data’ (Raue et al., 2009)


Raue et al. (2009)

Estimate parameters using likelihood maximizationParameter estimation


Raue et al. (2009)


Define confidence intervals of the estimate of a parameter

where is the Hessian matrix, and is the covariance matrix of

the parameter estimates

Confidence intervals


Raue et al. (2009)

One can also define likelihood-based confidence intervals, as a confidence region:


Define confidence intervals of the estimate of a parameter

where is the Hessian matrix, and is the covariance matrix of

the parameter estimates

Confidence intervals

IdentifiabiltyStructural identifiability

Raue et al. (2009)

Identifiability

Structural Identifiability

IdentifiabiltyStructural identifiability

Raue et al. (2009)

Identifiability

Structural Identifiability

identifiable structurally non-identifiable

IdentifiabiltyStatistical approaches: an example

Raue et al. (2009)


Raue et al. (2009)

Identifiable


Raue et al. (2009)


Raue et al. (2009)

Structurallynon-identifiable


Raue et al. (2009)


Raue et al. (2009)

????

IdentifiabiltyPractical identifiability

Raue et al. (2009)

identifiable practically non-identifiable

IdentifiabiltyConclusions

Identifiability.Some theoretical methods available, but most of them are limited. (There are also methods investigating ‘local’ identifiability)

Structural non-identifiability.The - structural - model is not well fitted to the data. The number of parameters must be reduced (e.g. using correlations).

Practical non-identifiability.Data being ‘non-ideal’, non-identifiability can come either from the model or from the data (amount, quality). Based on profile likelihood.

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