Nondimensionalization of population dynamics ODE models but not pharmacokinetics ODE models

Hey folks, just a question based on curiosity. Although I am still relatively new to differential equations modelling, I do notice some of the variations in the methods and discourse in different fields. One such apparent difference that I have noticed, is that while the biology and population dynamics literature places great emphasis on non-dimensionalizing ODEs to reduce the number of parameters and understand the dynamics of these models, I don’t notice much import given to non-dimensionalization in pharmacokinetics or reaction dynamics models. Further, it seems like both population dynamics models and pharmacokinetics or reaction dynamics models are not so different in structure. Perhaps I am just reading a subset of papers, and so I missed deeper discussions on non-dimensionalization in pharmacokinetics models. My understanding is certainly limited by the breadth of my experience with these literatures, so if I am mistaken, then please correct me.

So I proceed assuming that there is some validity to my observation. As I understand it, mathematicians non-dimensionalize their models to reduce the number of parameters–or perhaps better isolate the critical parameters. Reducing the parameters makes it easier to explore the dynamics of the system by narrowing the search space. Indeed, many of the expositions on the virtues of non-dimensionalizing parameters come from population dynamics papers–such as those by Leah Segel “Simplification and Scaling,” or the book by Murray Mathematical Biology volume 1.

Pharmacokinetics models, or reaction dynamics models seem to have hundreds of parameters, and the focus in the literature seems to be on numerical estimation of these models. There is a lot of work on implicit solvers to avoid instabilities due to the presence of multiple time-scales, or conditioning issues for the ODE system due to large differences in the eigenvalues, etc. But I have not seen much discussion on non-dimensionalizing these large models to potentially understand their dynamics.

I don’t know why there seems to be such a difference in methods? Does non-dimensionalization have little to contribute to reaction dynamics or pharmcokinetics models?
My own sense of the answer is that biological models, with limited data and limited ability to track and monitor populations of animals, insects, etc., focused on developing models that qualitiatively match the behavior of systems, but are not necessarily predictive of those populations. Alternatively, pharmacokinetics models, with perhaps more experimental data and knowledge of the mechanisms, have a better ability to predict the evolution of the system vis-a-vis the experimental data, so the focus in those models is on prediction. This is just a guess and likely incorrect, but perhaps it serves as a starting point for a discussion.

Again, like I said, if I am just way off about the importance of non-dimensionalization in one field versus the other, then please set me straight. But otherwise, if anyone has some insights, it would really appreciate it.

You’re thinking too much. Nondimensionalization to make all of the numbers in a similar range is always helpful for reducing the dynamic range and improving numerical stability. But a lot of times it’s just not necessary, and keeping the model in the original units can just be easier without a noticeable difference in speed or stability.

Ahh okay, that makes sense. Yeah, I could not find much on dealing with larger numbers of parameters in any textbook treatments. But I was looking at the idea of sampling some observable from the dynamics using random parameter values, and then using mutual information as a way to understand which parameters contribute the most. If I can get that to work, then I can still examine the dynamics of the system without doing a lot of algebraic manipulation to just reduce 2 parameters. Thanks Chris.