# Edge effects on the invasiveness of solid tumours

Careful readers might have noticed that, until last night’s post, the blog was silent for an atypically long 10 days (17 days since I last posted). As usual, the primary culprit is laziness, but this time it is not alone! After my fun visit to the Integrated Mathematical Oncology Department of the Moffit Cancer Research Center, I have been working closely with Jacob Scott and David Basanta to finish up our first joint paper. The last week was particularly busy as we pushed the paper out for submission and posted a draft to the ArXiv.

We look at the effect of spatial structure, in particular a spatial boundary, on the evolutionary dynamics of motility in cancer. For a tumor, one of the key steps in going from a benign to malignant is gaining the ability to spread from one organ to another non-adjacent organ. To achieve this, a cancer cell has to transition from simple proliferative cells (AG) to a motile ones (INV). However, motility usually involves a cost to the organism. We show that spatial structure can lower this cost, and smaller neighborhood size at an edge can promote motile cells at the boundary even when they are absent in the tumour body.

I’ve discussed the motile-proliferate model previously, and Basanta, Hatzikirou, & Deutsch (2008) encode the game for motility in the payoff matrix:

$\begin{pmatrix} \frac{b}{2} + \frac{1}{2}(b - c) & b - c \\ b & \frac{b}{2} \end{pmatrix}$

Whenever $b > c$, this game is a social dilemma with invasive cells as cooperators. Hence, it is natural to expect that spatial structure will help with invasiveness. This is summarized in the main figure:

in the above, you see the level of viscosity $1/(k - 2)$ (where $k$ is the number of neighbours a cell has, the inviscid case is $k \rightarrow \infty$) versus relative cost of motility $c/b$. The parameter space is divided into three regions with qualitatively different dynamics:

• Red — population evolves toward all INV,
• Yellow — population evolves toward a mixed equilibrium of INV and AG,
• Green — population remains benign with all AG.

The lowest horizontal dotted line marks $k = 8$ and corresponds to a cell in the middle of a 2D tumor having all of its Moore neighbors. The second horizontal dotted line marks $k = 5$ and corresponds to a cell at the edge of a 2D tumor, and thus having 3 of its neighbors eliminated by the boundary. The vertical dotted line marks an example of a fixed game $c/b = 0.53$ where the population in the body of the tumor ($k = 8$) is benign, but the population at the boundary ($k = 5$) is malignant with around 8% of the cells invasive.

From the medical perspective, this can add yet another source of sampling bias to tissue biopsies and
suggests that the architectural, not just the molecular, context is important. For instance, consider an idealized fine-needle aspiration biopsy, assuming the standard 0.7 mm needle samples a perfect column around 20 cells in diameter of tumour cells right next to a critical boundary such as a capillary. In our example of $c/b = 0.53$ this would result in the sample containing only about $0.4\%$ invasive cells since 19 out of every 20 cells are not at the boundary, and so contain no invasive phenotypes. This is below the detection levels of the state of the art medical practice. However, the critical 1 out of every 20 cells at the boundary, would have a dangerous $8\%$ invasive cells. Thus, an oncologist performing a diagnostic fine-needle aspiration biopsy would misdiagnose a malignant tumour as benign because she is using a technique that destroys the structure of the tumour and mixes the cells in the critical tumour boundary with the (in this case) irrelevant tumour body.

Biologically, this also offers an alternative to the Warburg shift in tumor progression. An invasive phenotype can be favored without glycolytic metabolism by the simple change in local neighborhood size at the boundary. Our future work will combine this analysis with a game-theoretic model of the Warburg effect, and paint a fuller picture of how acid mediated invasion and edge effects interact. The early results are promising, and I look forward to a continuing collaboration.

Although this is my ninth paper, in many ways it feels like a first. It is the first paper to come from this blog; much of the analysis comes from my July 5th post and relies on the Ohtsuki-Nowak transform that I detailed in October. It is my first experience of an online contact (I met David on Twitter in September, and Jacob in October; for a more detailed overview, see Jacob’s post) growing into one of my closest collaborations (highlighted by Jacob and I sharing first authorship). Finally, it is my first submission to the ArXiv! You should give it a read.

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.

Kaznatcheev, Artem, Scott, Jacob G., & Basanta, David (2013). Edge effects in game theoretic dynamics of spatially structured tumours ArXiv arXiv: 1307.6914v1