Simbiology example expression in the Tumor growth model

I am trying to understand one of the case study models in simbiology, "tumor_growth_fitPKPD.sbproj", explained at https://www.mathworks.com/help/simbio/ug/calculate-NCA-and-estimate-PKPD-parameters.html. I am wondering about the expression used for the total tumor weight in this example. Changes in tumor weight has been expressed using a dedicated reaction with a custom reaction rate. How the expression for the reaction rate has been derived? I guess, the original publication used wt(t) = x1(t) + x2(t) + x3(t) + x4(t).

Answer 1

[Arthur Goldsipe]

Hi,

We chose to model the tumor weight this way so that it would be possible to perform local sensitivity analysis using tumor weight as a sensitivity output. (Here's an example that does that.) This is basically due to a current technical limitation of SimBiology's local sensitivity analysis. If you don't care about local sensitivity analysis, then the more natural way to define tumor weight is probably to use a repeated assignment rule similar to what you wrote in your question.

-Arthur

[Anu]

Thanks Arthur!

Makes sense. It will be nice if you could further elaborate on the technical limitation of the local sensitivity analysis that you alluded to.

I also eventually found the derivations of the expression in Koch et al. 2009. Even though the current model implements the transition state idea from Simeoni et al., 2004, it utilises the rate equations from Kock et al. 2009, which is slightly different in its behavior. So, I though it would be great to add Koch et al. 2009 also to references in the example pages.

Best regards,

Anu

[Arthur Goldsipe]

Thanks for the feedback about the references. I'll pass that along to our documentation team.

Some of SimBiology's limitations for local sensitivity analysis are listed here. But it looks like that's an incomplete list, so I'll also work with our documentation team to address that gap. The relevant limitation here is that a variable determined by a repeated assignment rule cannot be used as a sensitivity input or sensitivity output.

[Anu]

Thanks for this information. I guess, this is because one cannot ensure that the output of a repeated assignment is a continuous function?

[Arthur Goldsipe]

It's not about continuity. SimBiology doesn't check the continuity of expressions. Today, you can attempt local sensitivity analysis on a model that has a discontinuous reaction rate, but you will likely encounter problems (for example, integration errors).

The last time I looked into this, my assessment was that doing this sensitivity analysis would require significant additions to our internal infrastructure. And the obvious way I would implement the functionality would likely slow down the calculations significiantly.

Here's a little more detail: The bulk of the sensitivity calculations are handled by the ODE solver itself. But the solver only calculates sensitivities of state variables. So repeated assignment rules need to be connected to these sensitivities. We had a similar challenge when we needed to take into account the affect of initial assignment rules on the sensitivity calculations. In that case, we only need to differentiate the rules at time=0. But for the case of repeated assignment rules as sensitivity outputs, we would need to differentiate the repeated assignment rules at every time reported in the simulation results.

[Anu]

Thanks, this insight is very helpful.