Lag-time with NONMEM and nlmixr2
By Matt Fidler and the nlmixr2 Development Team in nlmixr2
November 10, 2022
This is more of a methodology post, pointing out how things are done
in nlmixr2 and how it likely doesn’t match what is done in NONMEM
(and at least one reason why a drop-in replacement of rxode2 by
another tool like PKPDsim, mrgsolve, or deSolve is not an easy
project).
For the impatient, adding focei lag time (and other dose-based
events) have improved in stability for this release of nlmixr2.
A little background about focei
Gradients and Sensitivities
One of the differences between focei and saem is that focei
needs gradients to determine the best solution for the individual and
for the population best fit. (For the population fit you may not need
the gradient depending on the outer optimization method, for example
outerOpt="bobyqa" does not need gradients).
For nlmixr2 (and likely for NONMEM, based on some discussions with
others) the gradients for each individual are calculated based on the
forward sensitivity equations.
These forward sensitivity equations expand the ODE systems with extra
compartments. Each extra compartment will give the gradient of the
original compartment with respect to one of the between subject
variability (or etas). This means that the ODE system is expanded by
the number of ODEs in the original system multiplied by the number
etas in the system (hence the stability of focei is dependent on the
number of etas and ODEs in the original problem). The forward
sensitivities also has a way to handle changes in initial conditions.
As an aside, this needed sensitivity expansion is one (of a few)
reasons why nlmixr2 and rxode2 are bound so tightly together (and
other tools cannot be easily used or adapted).
Where sensitivities fall short
What the method cannot handle is parameter-based changes in dosing.
This includes lag-time, bio-availability, as well as modeled duration
of infusion or modeled rate of infusion in nlmixr2 models.
These are types of gradients are undetermined by pure forward
sensitivities. The last few releases of nlmixr2 took the first
observation and tried to figure out the best step-size to calculate
the gradients for the whole individual based on the Gill, Murray,
Saunders, and Wright (1) method.
New method for handling dose-based sensitivities
Now in nlmixr2, we use the method proposed in Shi et al (2)
(https://doi.org/10.48550/arXiv.2110.06380). This method is extended
so that instead of optimizing the gradient of one rate, we take the
harmonic mean of all the calculated rates for each observation and
optimize the best step size for that value (which is the same method
we use for the Hessian calculation in the generalized log-likelihood).
In our testing of lag-times, this seems to perform better than the previous method.
Because this method was only proposed last year, I don’t believe that
NONMEM uses this method. This is one of the many things that are
different between NONMEM focei and nlmixr2 focei.
Overall conclusions
In my estimation, this should show:
Dose-related parameters perform better when they have
etasthan the did before the log-likelihood releaserxode2integration is more than simply a solver you can replace with another solver.NONMEMandnlmixr2, while producing similar objective functions forfocei, have different internal choices and it should not be considered exactly identical. (ienlmixr2is not a R version ofNONMEM)