Seeing edge effects in tumour histology

Some of the hardest parts of working towards the ideal of a theorist, at least for me, are: (1) making sure that I engage with problems that can be made interesting to the new domain I enter and not just me; (2) engaging with these problems in a way and using tools that can be made compelling and useful to the domain’s existing community, and (3) not being dismissive of and genuinely immersing myself in the background knowledge and achievements of the domain, at least around the problems I am engaging with. Ignoring these three points, especially the first, is one of the easiest ways to succumb to interdisciplinitis; a disease that catches me at times. For example, in one of the few references to TheEGG in the traditional academic literature, Karel Mulder writes on the danger of ignoring the second and third points:

Sometimes scientists are offering a helping hand to another discipline, which is all but a sign of compassion and charity… It is an expression of disdain for the poor colleagues that can use some superior brains.

The footnote that highlights an example of such “disciplinary arrogance/pride” is a choice quote from the introduction of my post on what theoretical computer science can offer biology. Mulder exposes my natural tendency toward a condescension. Thus, to be a competent theorist, I need to actively work on inoculating myself against interdisciplinitis.

One of the best ways I know to learn humility is to work with great people from different backgrounds. In the domain of oncology, I found two such collaborators in Jacob Scott and David Basanta. Recently we updated our paper on edge effects in game theoretic dynamics of spatially structured tumours (Kaznatcheev et al., 2015); as always that link leads to the arXiv preprint, but this time — in a first for me — we have also posted the paper to the bioRxiv[1]. I’ve already blogged about the Basanta et al. (2008) work that inspired this and our new technical contribution[2], including the alternative interpretation of the transform of Ohtsuki & Nowak (2006) that we used along the way. So today I want to discuss some of the clinical and biological content of our paper; much of it was greatly expanded upon in this version of the paper. In the process, I want to reflect on the theorist’s challenge learning the language and customs of a newly entered domain.

As you might guess from the significant changes in this version of our paper, and from the long delay since we wrote the bulk of it, this version is a new submission. After a long review process, including one major changes resubmission, we were rejected from Proceedings of the Royal Society B[3], and have updated our paper for a more specialized journal. The reviews were mildly useful[4]. In particular, we have become much more specific about our language in light of them. Whereas before we spoke of ‘boundaries’ and ‘edges’ without any qualifiers, we now started to explicitly write ‘static boundary’ to remind the reader that our techniques do not apply to the dynamically edge where the tumour meets healthy tissue that it is growing into but only to static edges such as a blood-vessel, organ capsule, or basement membrane. We decided to illustrate this concept with some beautiful graphics:

Figure 1 from Kaznatcheev et al. 2015. On the left is a cartoon a hypothetical tumour, and on the right is a clinically produced micrograph of a sarcoma under low power with H&E staining.

Figure 1 from Kaznatcheev et al. 2015. On the left is a cartoon a hypothetical tumour, and on the right is a clinically
produced micrograph of a sarcoma under low power with H&E staining. The image on the left was drawn by Jacob Scott and the graphic on the right was produced by Marilyn Bui from the Moffitt Cancer Center.

We use the above figure to set up the main feature that we are focusing on: the difference in number of neighbours between cells in the bulk of the tumour — turquoise coloured cells in the cartoon, and cells inside the big black box in the bottom right on the stained image — and those at static boundaries like blood vessels, organelle capsules, or fibrous capsules — the pink cells in the cartoon, and the cells inside the angled rectangles in the bottom left and top right of the stained image. The key difference to notice, is that the cells in bulk tend to have more neighbours that those at boundaries. As you study the empirical image on the right of the above figure, remember that the white ‘gaps’ you see between stained cells is not an aspect of the cellular organization but an artifact of the imaging process; they are cracks from the sample drying on the slide.

Since we are aware of the importance of spatial structure but also wary of the heavy use of arbitrary assumptions in computational models of space, we took a minimalist approach to an analytic model. The standard procedure for analytic models of spatial structure seems to be a sort of double approximation. First researchers pick some detailed spatial structure that they assume approximates the real organization of the tissue, something like a grid, hex, Voronoi of random points in the plane, some other graph, or even free-particle model; this structure is almost always too complicated to work with analytically and can only be explored computationally. So, second, they create an analytic model that approximates this detailed computational model — hopefully this approximation is in some well-defined limit, but often it is another heuristic approximation. Instead, we argued that the pair-approximation technique (Matsuda et al., 1987; van Baalen, 2000; in particular, the transform of Ohtsuki & Nowak, 2006) can be used as a sort of direct first-order approximation for any spatial structure that we might have chosen[5].

By applying this technique to the Go-Grow game, we saw that there exists a parameter region where (in equilibrium) no motile cells would exists in the bulk, while the boundary would be polyclonal with motile and non-motile cells. Since these static boundaries (especially blood vessels) is where you least want to have motile cells, it is concerning that these regions — simply by having an architecture with fewer neighbours — increase the selective pressure for motile cells.

In the original version of the paper, we thought of this result as just a prediction or general word of warning; a heuristic to guide other modelers and maybe eventually experimentalists. This perspective changed surreptitiously on November 24th. I had spent the previous night selecting, studying, and editing the images that Marilyn Bui had provided us and had stumbled tired into Colm Morrissey’s talk. He was visiting from the University of Washington to discuss the rapid autopsy program they have developed — a very empirical talk full of content that I barely understood. However, since I had been focused on histology slides all night, I tuned into the pictures he was showing when something very similar to this image popped up in passing:

Right half of Figure 2 from Kaznatcheev et al. (2015) showing a section taken of bony metastatic deposits of prostatic adenocarcinoma. The staining is for SLUG, a marker of EMT.

Right half of Figure 2 from Kaznatcheev et al. (2015) showing a section taken of
bony metastatic deposits of prostatic adenocarcinoma. The staining is for SLUG, a marker of EMT. The image is courtesy of Colm Morrissey from the University of Washington.

If I had not bothered to learn the visual language of histology (mostly the night before — hence the serendipity), the above picture would have meant nothing to me; worse yet, I would not even have known to pay attention to at it and probably just registered it as ‘pretty pictures’ in a talk that was mostly not related to my own work. Instead, this image now looked to me as justification for our work. The epithelial-mesenchymal transition (EMT) is the process by which cells lose cell-cell adhesion and gain motility; so an EMT-cell is one that is motile[6]. This is not the only way for a cell to be a Go, motile, or invasive cell in our model, but EMT-cells do make good biological candidates. In the image above, you see a staining for SLUG, a marker of EMT; the darker cells with elongated bodies are those that are expressing the SLUG gene and thus are EMT-cells. The filmy (or flow-y — the word that spring to mind) transparent region on the left of the slide is a static boundary of bony trabecula. Up against this boundary, we can see a dark region of EMT-cells, followed by the tumour bulk that is cancerous (which can be seen from another slide that I am omitting in this post) but not composed of motile cells. This is more drastic than what we’d expect from our simple model, but it is encouraging to see that we are gesturing at a real effect[7] and one that has not been — at least to our knowledge — greatly explored, yet.

From a brief chat with Colm Morrissey, it also seems that the slide above is not exceptional. Most of the tissues he stains for EMT show a similar concentration of EMT-cells around static boundaries and absence in the bulk. This suggests some future modeling work closer to the data, in figuring out how to carefully measure these effects and work toward building an insilication instead of a heuristic model. But we’ll worry about this in the future, so stay tuned.

Notes and References

  1. Although ArXiv introduced a quantitative biology section back in 2003, the culture of preprints has not caught on in the biomedical fields like it has come to dominate physics, mathematics, and (theoretical) computer science. The new bioRxiv initiative from the Cold Spring Harbor Laboratory is aiming to change that by providing a wider and more experimentally focused range of categories than q-bio, and by making the submission process more friendly for non-LaTeX users — something that was a barrier for some biologists looking to use ArXiv, sine familiarity with TeX is not assumed in biomed. Of course, the wider cultural, institutional, and bureaucratic dependence on the traditional approach to publishing is a much bigger barrier than a choice of typesetting software, but hopefully with continued pressure from projects like PLoS One, F1000, and bioRxiv, even this barrier can be overcome.
  2. Note that there is a mistake in the main figure of the first draft of the paper (fig. 2 on pg. 6; now fig. 2 on pg. 7) that I propagated to the original TheEGG post summarizing the main findings. The mistake is in the form of the curve separating the red — all invasive — from the yellow — heterogeneous in invasive and non-invasive — regions of the phase space and stems from a mistake I made while factoring polynomials for the top-right entry of the payoff matrix in equation (3). I was, however, much more careful in checking the boundaries of the three regions that I described in the itemized text body of the article, and did not make the same mistake there. We corrected the issue without much fanfare in the second version of the preprint that we put out in January of last year. For more discussion, see footnote [4]. Although this mistake did not affect any of the main results, it was the most embarrassing part of this paper for me because I was in charge of the mathematical aspects of the paper and so I felt like I was letting Jacob and David down with such silly arithmetical mistakes.
  3. Just for fun, we also submitted to PNAS and (unsurprisingly) were rejected by the editors without review.
  4. Although “mildly” might be putting it too mildly, since there was an important omission in the review that scared me: the mistake I discussed in footnote [2] was not caught by the PRS-B reviewers. Neither the positive nor the negative review noticed the mistake, even though equation (3) was probably the most important equation specific to our results, neither of the reviewers bothered to compare it to the boundary conditions we derived (where I didn’t make the factoring mistake) nor tried to apply the ON-transform from equation (1) to the payoff matrix in equation (2) for themselves. This doesn’t seem to be an freak occurrence, either, I regularly notice math mistakes or typos in published papers. For example, one of my first contacts with David and Jacob involved pointing out a typo in Basanta et al. (2008) that had slipped past the reviewers, copy-editors, and the people that have cited the paper since in came out.
  5. In many cases — like regular latices, or graphs with slow mixing times — higher-order terms might play a more important role, but given that we don’t even have a good measure for the first-order terms; why speculate that far? The first-order approximation is the least we can introduce in order to state our question about neighborhood size — our question is not expressible in the 0-th order approximation of the inviscid model. Although Hilbe (2011) does provide an alternative that only counts the effect of different sized finite samples (and that could be used as a proxy for neighbourhood size), but that transform would yield the same qualitative dynamics as the inviscid case for the Go-Grow game, so you could say we moved to the second simplest model after seeing no novel effects in the simplest.
  6. Technically it should be called a mesenchymal cell, since EMT refers to the process of moving from an E state to an M state, but for some reason this more ambiguous terminology seems to be popular.
  7. Notice the explicitly weak language in “gestures”. I would not want to suggest that these sort of images serve as validation or confirmation of our models, since I don’t think that heuristics model can be validated. Instead, heuristic models are justified or made valuable by the ideas they raise in experimentalists and other modelers.

Basanta, D., Hatzikirou, H., & Deutsch, A. (2008). Studying the emergence of invasiveness in tumours using game theory. The European Physical Journal B, 63(3): 393-397.

Hilbe, C. (2011). Local replicator dynamics: a simple link between deterministic and stochastic models of evolutionary game theory. Bull Math Biol, 73: 2068-2097.

Kaznatcheev, A., Scott, J.G., & Basanta, D. (2015). Edge effects in game theoretic dynamics of spatially structured tumours. arXiv 1307.6914

Matsuda, H., Tamachi, N., Sasaki, A., & Ogida, N. (1987). A lattice model for population biology. In: Teramoto, E., Yamaguti, M. (Eds.), Mathematical topics in biology, morphogenesis and neurosciences. Spring Lecture Notes in Biomathematics 71: 154-161.

Ohtsuki, H., & Nowak, M.A. (2006). The replicator equation on graphs. Journal of Theoretical Biology, 243(1): 86-97.

van Baalen, M. (2000). Pair approximations for different spatial geometries. In: Dieckmann, U., Law, R., & Metz, J.A.J (Eds.), The geometry of ecological interactions: Simplifying spatial complexity. Cambridge University Press. 359-387.

About Artem Kaznatcheev
From the Department of Computer Science at Oxford University and Department of Translational Hematology & Oncology Research at Cleveland Clinic, I marvel at the world through algorithmic lenses. My mind is drawn to evolutionary dynamics, theoretical computer science, mathematical oncology, computational learning theory, and philosophy of science. Previously I was at the Department of Integrated Mathematical Oncology at Moffitt Cancer Center, and the School of Computer Science and Department of Psychology at McGill University. In a past life, I worried about quantum queries at the Institute for Quantum Computing and Department of Combinatorics & Optimization at University of Waterloo and as a visitor to the Centre for Quantum Technologies at National University of Singapore. Meander with me on Google+ and Twitter.

5 Responses to Seeing edge effects in tumour histology

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