Simplifying pharmacokinetic covariance structures

It is widely known that certain pharmacokinetic parameters can be correlated across the population, such as clearance and volume.  This is particularly true in PK models after an oral dose, when post-absorption parameters are often expressed as CL/F, V/F, Q/F etc.

In theory, the random variation in the set of the pharmacokinetic parameters can be modelled using a full covariance matrix that contains elements that represent the covariance between each pair of parameters.  In NONMEM, this achieved by using the $OMEGA BLOCK()  syntax.  In WinBUGs, a Wishart prior can be specified for a multivariate normal distribution.

However, a full covariance matrix for a model with n fixed parameters requires n(n+1)/2 covariance parameters to be estimated.  Variance parameters are considerably hard to estimate accurately than fixed effects, and this may pose a problem in typical sized datasets.  At best, the covariance matrix will be imprecisely estimated.  Consider for example a 3 compartment model with 2 absorption parameters - 8 parameters in total means that a full covariance matrix would require 36 parameters!  

Rather than revert to a diagonal matrix that may miss key features of the data and hence produce poor simulations, it is sometimes possible to simplify the covariance matrix.  The idea is to retain the key covariances, but avoiding estimating those covariance that are not so critical.

Here is a checklist of things to consider:

1) After oral dosing, bioavailability (F) is often a key source of variability.  Even though a fixed effect for F cannot be estimated, it is nonetheless possible to assign a random effect to F, that will capture the covariance shared across all apparent parameters (those which are written as "parameter/F").  This random effect can often be specified as lognormal eg.

F=EXP(ETA(1))

CL=THETA(1)*EXP(ETA(2))/F

V=THETA(2)*EXP(ETA(3))/F etc.

In this context it is not necessary to constrain F to be between 0 and 1 by using a logistic transformation - doing so would require fixing F to an arbitrary value.  The remaining parameters will usually have much reduced covariance and can hopefully be handled using a diagonal matrix.

2)  Consider removing intercompartmental clearance (Q) from the block part of the covariance matrix.  If the model is correctly specified Q will usual represent predominantly passive processes, and should only be associated with small intersubject variability. 

3) Generally, if a variance component is small, losing a correlation will not be harmful for overall simulation.  Very small on-diagonal covariance components (<2%) can usually be deleted.  If you cannot get reliable estimation of a full or reduced covariance matrix, usually the key covariances are between clearance and the volume term affecting the dominant phases in the data.  If you fit a diagonal covariance model to the data, and then examine the posthoc parameters, important correlations should be evident - if you can't see a convincing correlation on these plots, it probably isn't reliably estimated, even if it's included.  

4) To check if your reduced model is adequate, simulate from your model and compare to the original dataset.