Time-varying covariates

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Covariate-parameter relationships are usually defined via the Monolix GUI, and are available once the covariate columns of the dataset have been tagged as either CATEGORICAL COVARIATE or CONTINUOUS COVARIATE. However, these columns type can only be used for covariates which are constant over time. Time-varying covariates need to be handled in a different way. Two solutions are proposed below.

Case study introduction

We are considering a simple TMDD model where the drug is eliminated via a linear elimination as well as a non-linear elimination. There is a simple compartment and the drug is given via an iv infusion. The drug concentration is measured over several months.

We would would to assess the impact of WT on V and Cl, as well as the effect of comedications COMED on Cl. Both WT and COMED are varying over time.

The effect of time-varying weight (WT) on the volume V and clearance Cl follows a usual power-law relationship. COMED is a categorical covariate with can have value 0 (no comedication), 1 (concomitant administration of drug 1) or 2 (concomittant administration of drug 2). In this example, we assume that concomittant administration of another drug leads to a higher clearance due to drug-drug interactions.

The model we want to implement is:

Time-varying covariates as regressor

Time-varying covariates can be tagged as REGRESSOR in the dataset and the parameter-covariate relationship implemented in the structural model directly.

Tagging of dataset columns

To make the WT and COMED covariates available as a variable is the model file, the first step is to tag them as REGRESSOR column-type when loading the data set (instead of using the CONTINUOUS COVARIATE or CATEGORICAL COVARIATE column-type). Note that regressors needs to be numerical values. So categorical covariates such as COMED need to be appear in numeric format, e.g with value 0, 1 and 2.

 SCREENSHOT

Structural model

In the model file, the WT and COMED covariates are passed as input argument and designated as being regressors. The volume without covariate effect VnoCov is given as usual in the input, and similarly for the clearance ClnoCov. In addition, as the covariate effect is defined in the structural model, the covariate effect parameters beta_V, beta_Cl, are also passed as input parameters.

[LONGITUDINAL]
input = {..., VnoCov, ClnoCov, beta_V, beta_Cl, WT}
WT = {use=regressor}

If several regressors are used, be careful that, if the column headers and regressor names in the model do not match, the regressors are mapped by order with the data set columns tagged as REGRESSOR.

Note that the parameter VnoCov in the input corresponds to the individual parameter with IIV (random effects) but no covariate effect yet, so . The relationship between the volume V and the weight WT is defined in the EQUATION: or PK: block to create a new individual parameter:

For categorical covariates,

PK:
depot(target=A)

EQUATION:
V = VnoCov * (WT/70)^beta_V  ; VnoCov and ClnoCov already contains the random effects
Cl= ClnoCov * (WT/70)^beta_Cl

k = Cl/V
ddt_A = - k * A - Vm*A/(Km + A/V)
Cc = A/V

The parameters V and Cl are then used is the rest of the code.

The individual parameters which will be displayed in the result section after running the project will correspond to input of the structural model, i.e VnoCov and ClnoCov. In order to output the individual parameter V and Cl (which includes the covariates effect), they can be added to the table statement in the output:

OUTPUT:
output = {Cc}
table = {V, Cl}

Statistical model

The definition of the parameters in the GUI deserves special attention. Indeed the parameters beta_V and beta_Cl characterize the covariate effect and are the same for all individuals: their inter-individual variability must be removed by unselecting the random effects. To allow their value to go to either positive or negative values, their distribution is set to normal.

On the opposite, the parameters VnoCov and ClnoCov keeps their inter-individual variability and usual log-normal distribution. 

Results

Model comparison and automated seach

Using occasions

Comparison of the two methods