From heuristics to abductions in mathematical oncology

As Philip Gerlee pointed out, mathematical oncologists has contributed two main focuses to cancer research. In following Nowell (1976), they’ve stressed the importance of viewing cancer progression as an evolutionary process, and — of less clear-cut origin — recognizing the heterogeneity of tumours. Hence, it would seem appropriate that mathematical oncologists might enjoy Feyerabend’s philosophy:

[S]cience is a complex and heterogeneous historical process which contains vague and incoherent anticipations of future ideologies side by side with highly sophisticated theoretical systems and ancient and petrified forms of thought. Some of its elements are available in the form of neatly written statements while others are submerged and become known only by contrast, by comparison with new and unusual views.

If you are a total troll or pronounced pessimist you might view this as even leading credence to some anti-scientism views of science as a cancer of society. This is not my reading.

For me, the important takeaway from Feyerabend is that there is no single scientific method or overarching theory underlying science. Science is a collection of various tribes and cultures, with their own methods, theories, and ontologies. Many of these theories are incommensurable.

The three cultures of particular relevance to cancer research are the (molecular) biologists, clinicians, and mathematical (and computational) modelers. The first two are empirical fields and separated by many levels of description and the amount of experimental control. Biologists tend to focus on molecular pathways, only occasionally extending to the level of cell-cell interactions (Calvo & Sahai, 2011), and rarely getting to simple tissues. Clinicians on the other hand, usually have a focus on patient statistics, occasionally individual patients, and only rarely getting down to the level of complex tissues. If you are working in molecular medicine, and trying to bridge these two cultures, it is nearly impossible to do so without giant holes where things are assumed to scale in a straightforward way from molecular pathways to the patient. Usually, this mental model is more or less equivalent to just treating a human as a sack of simple pathways.

Mathematical modeling can help open our eyes to how such simplifications mislead. This is the basic premise of complex systems: show how microscopic rules that we think we understand lead to macroscopic behavior that we do not expect, or how some complicated macroscopic properties can emerge from simple microscopic rules. However, due to the fact that our mathematical models and our experiments and clinical data come from different cultures with different languages, there is no reason to expect that the things we call by the same words actually capture the same aspects of physical reality in the same ways. We cannot directly compare the mental models of experimentalists that give meaning to their theory-laden measurements, to the mathematical models that give meaning to the theory-laden parameters. In other words, mathematical oncologists can build heuristic models to train and test the intuitions of experimentalists, but they can’t just take the values that are empirically measure and stick them into their models ‘where it feels right’ until experiment and mathematical modeling are united within a single overarching framework and culture, as is the case in fields like physics. In other words, in my classification of models, mathematical oncologists can build heuristics but not insilications.

Or so I thought, until the IMO workshop and our team’s work on chronic myeloid leukemia.

An advertised goal of mathematical oncology is to bridge the gap between the molecular world and the patient. First, this means designing biological experiments of microdynamics in the same framework as the mathematical model so that we can be confident that the measurements mean the same thing as the model parameters. Second, using those models to calculate resulting macroscopic effects in the same framework as the clinical measurements and testing the results with new data. This would be an insilication, and I still don’t think anything of this sort is even close on the horizon. Even in CML, the best current attempts are still heuristics built to lend support to purely empirical conclusions.

The mistake I made in thinking that only heuristics were possible was not in eliminating insilications, but forgetting abductions in my classification. Thankfully, Ishanu Chattopadhyay caught my omission. An abduction is the standard paradigm in machine learning, and a great way to bridge frameworks. It can serve as an automated interpreter between cultures. We fix some hypothesis class using the language and culture of the mathematical oncologists, and we use the language and culture of the clinician (or the biologist) to give the training data. From these two pieces we can do two things. First, we can fit the data and see how closely the values of parameters in our model correspond to the values we think they are proxies for in the experimental literate. Does the C++ variable that we are calling the growth rate, or the ODE parameter we are calling the diffusion coefficient have a similar value to what the experimentalists call growth rates or diffusion coefficients? Second, and more important, right from the get-go we can not even worry about the meaning of our hypothesis class parameters, and just use the hypothesis the machine learns to directly generate a prediction for future clinical parameters.

In some ways, the second approach is a lot like the blind data view that I denounced earlier, so I have to slice off and eat a healthy portion of humble pie. The only way to really convince a clinical and experimental audience that is skeptical of and untrained in mathematics, is to show them results that are directly interpret-able in their world-view. The abductions can provide this. Then after we have the doctors and biologists on-board, we can point out that the guts of our models happened to use mechanistic mathematical (or computational) models, and ask for their help to build a bridge between their shore of science and ours.

References

Calvo, F., & Sahai, E. (2011). Cell communication networks in cancer invasion. Current Opinion in Cell Biology, 23(5): 621-629.

Nowell, P. (1976). The clonal evolution of tumor cell populations Science, 194 (4260), 23-28 DOI: 10.1126/science.959840