Sometimes the covariates affecting individual parameters change over the course of the study. For example, if an individual had heart failure for part of his observation period but not the rest, the metabolic clearance CL_met could change. Another example is when acute renal failure occurred to a patient, then the renal clearance CL_ren could change.

This time-dependency can be implemented in PREDPP recursively. Recall

$$ y_{ij} = f(x_{ij}, \phi_i) + \epsilon_{ij}. $$ $$ \phi_{i} = \mu(z_{i}, \theta) + \eta_{i}. $$ from population model, the covariate values at \(t_{j+1}\) determine \(\phi\) in time period \((t_j, t_{j+1}]\). That is, when NONMEM advances system states from time \(t_j\) to \(t_{j+1}\), it uses covariates \(z\) (and in general \(x\)) at time \(t_{j+1}\) to compute \(\phi\). Therefore in order to have \(\phi\) change appropriately with \(z\), on the data record for time point \(t_{j+1}\) we want to place a value of \(z\) corresponding to time period \((t_j, t_{j+1}]\). For example, if the $PK block is used to calculate clearance CL based on covariate eGFR, then CL is peice-wise constant between state times \(t_{j}\) and \(t_{j+1}\), and based on eGFR observation at \(t_{j+1}\). This is sometimes referred to as "Last Observation Carried Backward" (LOCB), as opposed to "Last Observation Carried Forward" (LOCF).

One challenge of the above approach is when the covariates in \(z\) are not measured during the time interval. In that case, one will have to use certain interpolation method.