Background and Objectives

1
Transforming Parts of a Differential Equations
System to Difference Equations as a Method for
Run-time Savings in NONMEM VI
Klas JF Petersson, Lena E Friberg, Mats O Karlsson
Department of Pharmaceutical Biosciences, Uppsala
University, Uppsala, Sweden

• The mean absolute error for the fixed effects
  parameters in the faster models were between 1.5
  and 8.0 and between 2.2 and 19.8 for the
  variance parameters
• The difference in OFV compared to the original
  models ranged between -14.4 and 1.7 units. The
  small differences in OFV suggests similar
  prediction properties and this can also be seen
  in fig. 3.

Background and Objectives
Increased computing power gives us the
possibility of building more complex models that
more adequately describes the sometimes complex
mechanisms of diseases and drug effects. Even
with the modern computers of today these models
may require quite substantial amount of computing
time, and for a model to be widely useful long
runtimes are not practical. Models with long
runtimes are often defined as differential
equations (DES in NONMEM). In this work we aimed
to explore if updating parts of the functions
dependent on compartment amounts in the
differential equations at pre-specified intervals
could shorten model runtimes without loosing
model fit.
Methods and Materials
In NONMEM VI there is the possibility to update
the system at non-event times using a function
called MTIME. Different parts of the differential
equations in nine fairly complex models based on
differential equations 1-8 were moved from DES
to PK and MTIME was used to update PK at given
intervals. The intervals were increased to give
as short runtimes as possible but the intervals
were kept short enough to retain roughly the same
fit (OFV).
Figure 2. Relative bias in parameter estimates
for models using difference equations compared to
differential equations.
IF(TIME.EQ.0) STEP0 STEPSTEP0.1

MTIME(1)STEP MTDIFF1
PK
Results

• For five 1,4,6-8 of the nine models we were
  able to shorten the runtimes to a pronounced
  degree (59-96 reduction), see fig. 1.
• For the prolactin model 7 (Model 3) which had a
  runtime of over one month using FOCE the runtime
  dropped to 24 h.

• The fixed effects parameter estimates for four of
  the models which could be expedited were within
  12 from the estimates of the original model.
• For the last of the improved models 1 fixed
  effect parameter bias was slightly higher one
  parameter differed by 23 and one by 13, see
  fig. 2.

Figure 3. Population and individual predictions
for models using differntial equations vs using
difference equations.
Conclusions

• Moving parts of or whole equations from
  differential to difference equations using MTIME
  can shorten runtimes substantially.
• Model fit and parameter estimates are similar.
• This approach may for example be useful in
  covariate modeling and in exploring the random
  effects model (e.g. IIV, IOV and semi-parametric
  distributions 9).

References
1 Troconiz, I. F. et al Clin Pharmacol Ther
199864(1)106-16 2 Friberg, LE. et al. J Clin
Oncol 200220(24)4713-21 3 Hamrén, B. et al
Clin Pharmacol Ther. 2008 Mar 19. Epub ahead of
print 4 Lönnebo, A. et al. Br J Clin
Pharmacol. 200764(2)125-32 5 Ribbing, J. et
al. PAGE 2008 presentation 6 Silber, HE. et
al. J Clin Pharmacol. 200747(9)1159-71. 7
Friberg, L.E. et al. PAGE 2006 poster 8
Hamrén, B. Doctoral Thesis, 2008 Uppsala
University. 9 Petersson, K. et al. PAGE 2007
poster
4 6 7 8 1
4 6 7 8 1
Figure 1. Absolute and relative changes in
runtimes for five models where the runtimes could
be shortened by transforming whole or parts of
the differential equation system into difference
equations.