Stem cells, branching processes and stochasticity in cancer

October 25, 2014 by Artem Kaznatcheev

When you were born, you probably had 270 bones in your body. Unless you’ve experienced some very drastic traumas, and assuming that you are fully grown, then you probably have 206 bones now. Much like the number and types of internal organs, we can call this question of science solved. Unfortunately, it isn’t always helpful to think of you as made of bones and other organs. For medical purposes, it is often better to think of you as made of cells. It becomes natural to ask how many cells you are made of, and then maybe classify them into cell types. Of course, you wouldn’t expect this number to be as static as the number of bones or organs, as individual cells constantly die and are replaced, but you’d expect the approximate number to be relatively constant. Thus number is surprisingly difficult to measure, and our best current estimate is around (Bianconi et al., 2013).

Both 206 and are just numbers, but to a modeler they suggest a very important distinction over which tools we should use. Suppose that my bones and cells randomly popped in and out of existence without about equal probability (thus keeping the average number constant). In that case I wouldn’t expect to see exactly 206 bones, or exactly 37200000000000 cells; if I do a quick back-of-the-envelope calculation then I’d expect to see somewhere between 191 and 220 bones, and between 37199994000000 and 37200006000000. Unsurprisingly, the variance in the number of bones is only around 29 bones, while the number of cells varies by around 12 million. However, in terms of the percentage, I have 14% variance for the bones and only 0.00003% variance in the cell count. This means that in terms of dynamic models, I would be perfectly happy to model the cell population by their average, since the stochastic fluctuations are irrelevant, but — for the bones — a 14% fluctuation is noticeable so I would need to worry about the individual bones (and we do; we even give them names!) instead of approximating them by an average. The small numbers would be a case of when results can depend heavily on if one picks a discrete or continuous model.

In ecology, evolution, and cancer, we are often dealing with huge populations closer to the number of cells than the number of bones. In this case, it is common practice to keep track of the averages and not worry too much about the stochastic fluctuations. A standard example of this is replicator dynamics — a deterministic differential equation governing the dynamics of average population sizes. However, this is not always a reasonable assumption. Some special cell-types, like stem cells, are often found in very low quantities in any given tissue but are of central importance to cancer progression. When we are modeling such low quantities — just like in the cartoon example of disappearing bones — it becomes to explicitly track the stochastic effects — although we don’t have to necessarily name each stem cell. In these cases we switch to using modeling techniques like branching processes. I want to use this post to highlight the many great examples of branching processes based models that we saw at the MBI Workshop on the Ecology and Evolution of Cancer.



Branching processes have a rich and varied history of application in biology (see Jagers, 1975; Kimmel & Axelrod, 2002; Kimmel’s slides provide a good overview) that includes modeling PCR amplification (Sun, 1995), gene amplification and drug resistance (Kimmel & Axelrod, 1990), stem cell proliferation with death and quiescence (Kimmel & Axelrod, 1991), complexity threshold in abiogenesis, and dynamic mutations (Gawel & Kimmel, 1996). Mathematical oncology, in particular, has benefited greatly from the application of branching processes, from early work on carcinogenesis (Kendall, 1960) to the evolution of chemotherapy resistance in tumours (Iwasa et al., 2006; Komarova, 2006). This approach is especially useful for hierarchical tissues — like in models of leukemia — where an initial population of stem cells differentiates into proginator cells which then produce mature cells (Wernet et al., 2011; 2014).

In the first talk of Thursday, September 18th, Arne Traulsen opened with a Michor et al.’s (2005) foundational paper on the (in)effectiveness of immatinib in killing the cancerous stem cells in chronic myeloid leukemia. This model assumed that there were 4 compartments (or types of cells) in the hierarchical tissue; but why exactly four? Traulsen discussed a mathematical model we could use to estimate the proper number of compartments from clinical data, and expected transition probabilities between the compartments. Once the number of compartments is known, it becomes important to determine at which level of the hierarchy the relevant mutations — the Philadelphia translocation in the case of CML — occur (Werner et al., 2011; 2013). In most cases, we believe that these oncogene mutations confer a fitness advantage to the cancer cells, but the extent of this is difficult to measure experimentally due to the co-occurrence of many mutations and the difficulty of experimentally recreating the in vivo micro-environment. An exception is CML where Traulsen et al. (2010) were able to apply their multi-compartment branching processes model to clinical data and estimate the relative fitness advantage of BCR-ABL expressing leukemic cells. A great example of moving from heuristic to abductive models.

These sort of multi-compartment (i.e. multi-type) branching processes are easy to simulate computationally or approximate by an ODE, but can be relatively hard to solve analytically. In the second to last talk of Tuesday, September 16th, Tibor Antal discussed his progress towards analytic solutions of certain kinds of multi-type branching processes. He traced the historic progression toward more fully stochastic solutions of branching processes in mathematical oncology from the early work of Kendall (1960) to modern treatments by Iwasa et al. (2006) and Durrett & Schmidt (2008), culminating in Antal & Krapinsky’s (2011) explicit solution of Kendall’s general two-type continuous time birth-death process. Continuing the mathematical trend, Kamran Kaveh constructed a Moran process based model with differentiation and environment-dependent plasticity, and solved for fixation times of cancerous mutants both analytically and through simulation. He showed that fixation probability is a strong function of plasticity rate and differentiation probabilities and compared their findings to Vermeulen et al.’s (2013) experimental work on stem cell dynamics of tumours in the mouse intestine.

In the talk right before Kaveh’s, Marc Ryser presented his recent work the clearance of the Human papillomavirus (HPV). HPV is an incredibly common sexually transmitted infection, with worldwide prevalence of 10%-30% and common to all socio-demographic groups. A person has an 80% lifetime risk of getting HPV, but in most cases it usually clears by itself, without much symptoms. It is also a common infectious cause of cancer (Parkin, 2006), and thus of interest to mathematical oncologists. However, Ryser did not cause on the carcinogenic aspects of HPV but on its interaction with the epithelial renewal process. By modeling epithelial renew as a branching-process, Ryser was able to argue for the importance of stochasticity in clearing HPV. In particular, since HPV is believed to have low copy number in the low layers (stem cells, or close to stem cells) of the renewal process, they can disappear due to chance events and then clear without major (or any) symptoms. He was able to estimate the clearance contribution as 83% due to stochasticity and 17% due to the immune system (which is known to play a factor based on studied of immunocompromised patients with HPV). What really set Ryser’s work apart for me — apart from that it didn’t directly deal with cancer — was that to calibrate his models, he did not use the standard pathological data that we usually rely on, but instead epidemiological data due to the very large studied available on the level of human populations. We usually think of the scales of cancer as progressing from molecules to the human body, so it was interesting to see the incorporation of a further scale: populations of humans.

Finally, in the second talk of Wednesday, September 17, Natalia Komarova presented a general calculus of stem cells. She focused on a particular shortcoming of typical branching processes: a lack of direct interaction between the model compartments. This assumption makes the models tractable, but removes any connection to ecology that were so heavily stressed by other workshop participants. In particular, to main homeostasis without interactions, a branching process has to be critical: the probability of the symmetric differentiation of a stem cell into two stem-cells has to be perfectly equal to the probability of differentiation into two progenitor cells. This is a rather non-robust requirement, and because of it a lot of people postulate feedback loops between the compartments. For example, in contrast to Traulsen’s work, we based our explorations of CML on Horn et al.’s (2013) feedback based model. Komarova’s innovation in her talk was to build an abstraction where she could classify all possible types of (simple) feedback loops between two, three, and four compartment models that maintain robust homeostasis. This provides a very useful model catalog for mathematical oncologists needing simple models of stem cells in homeostasis to be perturbed by cancer.

The depth of knowledge on branching-process at the MBI Workshop on the Ecology and Evolution of Cancer was really eye opening for me. It is a modeling technique that I have neglected to add to my theoretical toolbox, and these talks have let me realize the poverty of my perspective. I was also very happy to see the same tools used to span so many types of mathematical models: we saw abstractions, abductions, and heuristics. We also saw both computational and analytic results. I look forward to sharing more exploration of branching processes with you, dear reader, in the future.

This is my fourth post of a series on the MBI Workshop on the Ecology and Evolution of Cancer. The previous posts were: Colon cancer, mathematical time travel, and questioning the sequential mutation model; Experimental and comparative oncology: zebrafish, dogs, elephants; Ecology of cancer: mimicry, eco-engineers, morphostats, and nutrition. The graphic at the top of the post is from Antoniou et al. (2013).

References

Antal, T., & Krapivsky, P. L. (2011). Exact solution of a two-type branching process: models of tumor progression. Journal of Statistical Mechanics: Theory and Experiment, 2011(08): P08018.

Antoniou, A., Hébrant, A., Dom, G., Dumont, J.E, & Maenhaut, C. (2013). Cancer stem cells, a fuzzy evolving concept. Cell Cycle, 12(24): 3743-3748.

Bianconi, E., Piovesan, A., Facchin, F., Beraudi, A., Casadei, R., Frabetti, F., … & Canaider, S. (2013). An estimation of the number of cells in the human body. Annals of Human Biology, 40(6): 463-471.

Durrett, R., & Schmidt, D. (2008). Waiting for two mutations: with applications to regulatory sequence evolution and the limits of Darwinian evolution. Genetics, 180(3): 1501-1509.

Gawel, B., & Kimmel, M. (1996). The iterated Galton-Watson process. Journal of Applied Probability, 949-959.

Horn, M., Glauche, I., Müller, M.C., Hehlmann, R., Hochhaus, A., Loeffler, M., & Roeder, I. (2013). Model-based decision rules reduce the risk of molecular relapse after cessation of tyrosine kinase inhibitor therapy in chronic myeloid leukemia. Blood, 121(2): 378-84.

Iwasa, Y., Nowak, M.A., & Michor, F. (2006). Evolution of resistance during clonal expansion. Genetics, 172(4): 2557-2566.

Jagers, P. (1975). Branching processes with biological applications. Wiley, New York.

Kendall, D. G. (1960). Birth-and-death processes, and the theory of carcinogenesis. Biometrika, 13-21.

Kimmel, M., & Axelrod, D.E. (1990). Mathematical models of gene amplification with applications to cellular drug resistance and tumorigenicity. Genetics, 125(3): 633-644.

Kimmel, M., & Axelrod, D. E. (1991). Unequal cell division, growth regulation and colony size of mammalian cells: a mathematical model and analysis of experimental data. Journal of Theoretical Biology, 153(2): 157-180.

Kimmel, M., & Axelrod, D.E. (2002). Branching Processes in Biology. Interdisciplinary Applied Mathematics 19. Springer, New York.

Komarova, N. (2006). Stochastic modeling of drug resistance in cancer. Journal of Theoretical Biology, 239(3): 351-366.

Michor, F., Hughes, T., Iwasa, Y., Branford, S., Shah, N., Sawyers, C., & Nowak, M.A. (2005). Dynamics of chronic myeloid leukaemia. Nature, 435(7046): 1267-1270

Parkin, D.M. (2006). The global health burden of infection-associated cancers in the year 2002. Int. J. Cancer, 118(12): 3030–44.

Sun, F. (1995). The polymerase chain reaction and branching processes. Journal of Computational Biology, 2(1): 63-86.

Traulsen, A., Pacheco, J., & Dingli, D. (2010). Reproductive fitness advantage of BCR–ABL expressing leukemia cells Cancer Letters, 294 (1), 43-48 DOI: 10.1016/j.canlet.2010.01.020

Vermeulen, L., Morrissey, E., van der Heijden, M., Nicholson, A. M., Sottoriva, A., Buczacki, S., … & Winton, D. J. (2013). Defining stem cell dynamics in models of intestinal tumor initiation. Science, 342(6161): 995-998.

Werner, B., Lutz, D., Brümmendorf, T. H., Traulsen, A., & Balabanov, S. (2011). Dynamics of resistance development to imatinib under increasing selection pressure: a combination of mathematical models and in vitro data. PLoS ONE, 6(12): e28955.

Werner, B., Dingli, D., & Traulsen, A. (2013). A deterministic model for the occurrence and dynamics of multiple mutations in hierarchically organized tissues. Journal of The Royal Society Interface, 10(85): 20130349.

Werner, B., Gallagher, R. E., Paietta, E. M., Litzow, M. R., Tallman, M. S., Wiernik, P. H., … & Dingli, D. (2014). Dynamics of Leukemia Stem-like Cell Extinction in Acute Promyelocytic Leukemia. Cancer Research, 74(19): 5386-5396.