T-World 5: Problem 3 - Modelling L-type calcium current

The model of I_CaL that we had in ToR-ORd (based on the ORd formulation) had clear strengths, enabling nice early afterdepolarisations (EADs), for example. However, I also felt it had aspects I didn’t like that much. One is that the refractoriness under repeated activation was not that strong (and if made stronger, EADs would be lost). A second aspect was that the model produces slightly heavy-tailed current profile, which could probably inactivate more. One side-effect of this is that the current would produce nontrivial depolarizing current throughout the plateau, requiring quite a strong I_Kb to offset this and maintain good AP shape. Again, if the model was just reparametrized to be leaner, EADs would be lost. I therefore wanted a leaner-profile model that would be even more in line with data on refractoriness.

Having worked a bit with the 8-state cube model used e.g. in the Heijman-Rudy canine model, I thought it would be a very nice starting point. And in many ways, it was. I initially went over numerous published models of I_CaL, trying them, but often they would not be capable of nice EADs and/or recovery from refractoriness, even when I optimised their parameters using genetic algorithms. This 8-state model seemed clearly the most promising.

One hurdle was achieving a steep S1S2 restitution – the ToR-ORd itself had quite a flat one, which is a limitation. I could not quite get a great restitution steepness just with the 8-state model. However, analysing the Ten Tusscher 2006 model (a model well known for its capability to manifest steep restitution), I could see its steep restitution is mainly driven by a slow inactivation gate (f gate). So, I thought, ok, let’s just add this to our model, multiplying the Markov model output with an updated version of the f gate, with the idea that I can incorporate this in the Markov model more organically later. Nirvana, I could finally achieve steeper restitution. As a result, I had a version I was quite happy with for several years during the development.

…

Then I thought we’re almost done, and hence I can run a validation study confirming that in general, the longer the APD of a cell, the steeper, is the S1S2 restitution slope, as observed in a paper by Prof. Mike Shattock and also in other papers. This is a striking observation, and one that is very important. If we model drugs or diseases that change APD, they should have an appropriate effect on restitution slope. I remember running the simulation, not really expecting anything bad. However, the results were bad. They were BAD! The relationship was the opposite. I triple-checked the codes, ways of quantifying APD and restitution slope, raw traces, and I had to conclude this was the reality. The validation has failed. OK, with a trembling hand, I thought we’ll downgrade it to a calibration criterion, and that I’ll redevelop the model so that the trend is captured correctly. But I felt worried, intuitively feeling this might not be an easy fix. And indeed, I just could not achieve this through parametric changes, no matter what I tried.

Of course, we could have packaged the model as it was and state this as a limitation, but it was far too big an issue for me to be ignored. The complication with big issues is that you never know how big iceberg of a problem is under the surface, and just how many of model prediction in future it might invalidate.

Next stage was to understand where the issue is coming from. Is it I_CaL? Potassium currents? Restitution of calcium handling? In the end, I traced it to the f gate, the very feature that gave us the steep restitution. Writing this blog years after this has happened, I still get echoes of the desperation I was feeling. I then checked that the Ten Tusscher model has the same problem with the APD-slope relationship, meaning that while I was not alone having this problem, I also couldn’t simply learn how to fix it. I did then also see other warning signs – e.g. thanks to the f gate, the AP peak would recover quite slowly as the S2 coupling interval increases – much more slowly than in reality. Yet another indication that this mechanism of achieving steep restitution is not what I was looking for.

In the end I revisited the ORd/ToR-ORd model, made it work in its non-native calcium handling system of modified Bers/Grandi framework, and added a direct calcium-dependent inactivation gate. There was a lengthy literature review process at this stage, and I still am not sure we understand the interdependencies of voltage and calcium inactivation that amazingly… Anyway, developing this framework, we could in the end get a model that maintained EADs, enabled steep restitution, this restitution steepened with AP prolongation, and, going back to my original hopes, it had a leaner profile and showed decent refractoriness via P2P1 protocol (something we matched only qualitatively in ToR-ORd before). The last feature also played a part in the nicer rate-dependence of calcium handling of T-World, but that’s yet another story. It always feels good when a development direction enables multiple things to click into place, resolving several issues at once - it means a greater likelihood that the change is reasonably plausible.

One point to re-emphasize at the end, which is perhaps trivial: The fact that the development version with the 8-state Markov I_CaL model failed the validation on the positive APD-slope relationship and we changed the I_CaL model means that we had to stop considering this a validation criterion. While “calibration” often refers to parametric changes, in this case the change of cellular component to get better overall behaviour clearly constitutes a calibration step, which is why we downgraded the criterion on APD-slope relationship from validation to calibration.