DDEs in NONMEM

Dear all, I have a problem regarding delay differential equations (DDEs) in NONMEM. So far, I have been unable to find a way of successfully using DDEs in NONMEM, due to there being no in-built DDE solver (as far as I know). To me, it seems that the problem is that there is no way of recalling the value of the dependant variable at an earlier time point within the $DES block (u(t-T), where T is some fixed value, possibly to be estimated). I think that the problem is not a simple one to solve as NONMEM uses an adaptive step size when performing the integration, so even if there was a table output of the past values of the dependant variable, the value at the exact time (u(t-T)) might not exist. Has anybody been able to successfully implement DDEs in NONMEM? I would really appreciate some help with this. Note: I am talking about DDEs in a general sense, I know that certain specific cases of DDEs can be solved in NONMEM (ref: Modelling of delays in PKPD, classical approaches and a tutorial for delay differential equations, G Koch, W. Krzyzanski, J. J. Perez-Ruixo, Johannes schropp, JPKPD). Many thanks, Jon Moss, PhD Modeller BAST Inc Limited Loughborough Innovation Centre Charnwood Wing Holywell Park Ashby Road Loughborough, LE11 3AQ, UK Tel: +44 (0)1509 222908

Dear Jon, Thera are two aspects of DDEs that make them different from ODEs: presence of delays states and need to define a past (history of the states). Implementation of DDEs in software that does not have a DDE solver but has an ODE one is (in general) not a trivial task, but it is possible. The idea is to convert a system of DDEs to a system of ODEs. This can be done for any DDE system with fine number of delays that needs to be solved on over a finite time interval 0<t<tend. The technique to apply is methods of steps. The methods of steps is particularly efficient for category 2.5 DDEs examples of which you found in the tutorial by Koch et al you cited. Bob Bauer implemented the method of steps in S-ADAPT which allows this program to solve arbitrary DDEs with non-constant past (Bauer et al., Computer Methods and Programs in Biomedicine 111:715-734 (2013).doi: 10.1016/j.cmpb.2013.05.026). A brief introduction for solving of DDEs by the method of steps has been published by Perez-Ruixo (J. Pharmacokin. Pharmacodyn. 32: 767-793 (2005)). The methods of steps has one serious limitation (among others), the number of ODEs is roughly speaking determined by the number of steps needed to reach tend. This number (for a system with one delay Tdel) is the product of the number of DEs in your model and the ratio tend/Tdel. This generally introduces of hundreds of ODES even for small 3-4 compartment DDE models. Coding manually this many equations provides a challenge. I believe the current version of NONMEM allows to code for such large dimensional models. One will need also a large number of ALAGs. Because of these challenges, it takes an experience NONMEM user to correctly apply the methods of steps, and I DO NOT RECOMMEND IT. NONMEM should have a DDE solver not to make NONMEM users suffer if they want to apply DDEs. Please contact me directly if you want to apply the methods of steps in NONMEM. I have done it for several lifespan based indirect response models. Regards, Wojciech