Monday, 24 February 2014

Forecasting tumour growth

The concept of personalised oncology is often compared to weather prediction. The idea being that with increased amounts of data from patients (genetic sequencing, phenotypic characterisation of cancer cells, imaging etc.) and more advanced mathematical models, we will be able predict tumour progression and responses to therapy in individual patients with increased accuracy.

When making this comparison, the patient data is analogous to the current state of the atmosphere (temperature, air pressure, wind speed and direction etc.) used as input to fluid dynamics models, which can for example be used in order to predict the future course of a hurricane (or in terms of cancer predict the rate of growth under different therapies).

The current state of personalised medicine is however falling short on both accounts: the data acquired from patients is meager (although microarray data is 'big' it's pretty useless as input to computational models), and our present-day theoretical understanding of tumour growth is limited. Even if we had all the data we could dream of it would most likely be useless because of our ignorance.

Although the analogy seems to fall short it actually sheds some light on our inability to predict tumour growth, the reason being that meteorology was in a similar situation roughly a century ago. Before the advent of digital computers, weather prediction was a difficult business based on previous experience and certain rules of thumb. Or in the words of Lew Fry Richardson in the preface of "Weather prediction by numerical process" (1922):

The process of forecasting, which has been carried on in London for many years, maybe typified by one of its latest developments, namely Col. E. Gold's Index of Weather Maps. It would be difficult to imagine anything more immediately practical. The observing stations telegraph the elements of present weather. At the head office these particulars are set in their places upon a large-scale map. The index then enables the forecaster to find a number of previous maps which resemble the present one. The forecast is based on the supposition that what the atmosphere did then, it will do again now. There is no troublesome calculation, with its possibilities of theoretical or arithmetical error. The past history of the atmosphere is used, so to speak, as a full-scale working model of its present self.

This process of manually predicting the weather is similar to the way clinicians in the present day decide upon different cancer therapies. Treatment choices are based on sparse data and previous experience of similar patients.

The seminal work of Vilhelm Bjerknes and the above quoted book lay the foundation of quantitative weather prediction. What was still lacking was however the computational power, which held back large scale weather forecasting by another 30 years. The situation in mathematical oncology today is however reversed. We have all the computational power we need, but still lack the appropriate theoretical understanding. Hope fully one day that will change.

Thursday, 20 February 2014

The role of mathematical oncology

EDIT: I forgot to mention a paper on Chronic Myeloid Leukemia by Nowak et al. which possibly contradicts the point I'm trying to make. Thanks Heiko Enderling for pointing this out.

EDIT 2: Artem Kaznatcheev has written an excellent blog post arguing for the fact that the above mentioned paper does not contradict my argument.

The other day I was discussing the merits of mathematical oncology with some colleagues in the collaboratorium (a shared space at the IMO where scientific discussion blend with the smell of espresso) here at IMO. We came to the conclusion that our field of research still lacked that defining publication where the use of mathematical modelling was clearly tied to a clinical change benefiting patients. Or in other words, an instance where mathematical oncology has been proven to make a difference.

Other fields of mathematical biology have already had this pleasure. For example Ronald Ross developed a model of malaria dynamics in the 1910s (known as the SIR-model), which allowed for a completely new understanding of of malaria and opened up the field of epidemiology. A more recent example is the use of modelling in the discovery of the high turnover rates of HIV-particles (

Given the fact that more and more researchers work in mathematical oncology, isn't it just a matter of time before that landmark publication appears? Actually I think the answer is no.

The reason for me being so pessimistic pertains to the relation between the complexity of the problem and the amount of knowledge that you can fit into a standard publication (journal paper, conference proceeding etc.). Some of you might object and say the way we communicate our results ought to be secondary to the subject at hand, but I would like to argue that the politics and funding structure of science imposes a certain mode of communication that in turn influence how we approach research questions. What biologist can stick to a specific research agenda, work on it for 15+ years, and then publish a monograph on the topic? (The answer to this question is obviously Darwin, whose meticulous work couldn't have been carried out today.)

This is not to say that mathematical modelling does not contribute to our understanding of cancer, but rather that the insights gained from it arrive in smaller chunks and are absorbed by the experimental and clinical community. These insights and novel concepts then shapes their thinking and inspires them to perform new experiments or look at existing data in new ways.

In support of this thesis I would like to cite two ideas that today are ubiquitous in cancer research: networks and intra-tumoral heterogeneity.

The idea that intra-cellular signalling forms a network with feedback loops, crosstalk and robust properties does not emanate from mathematical oncology per se, but rather from complex systems theory and statistical physics. Nevertheless it is now part of the vocabulary of cancer biologists and helps in moving the subject forward.

The concept of tumour evolution was coined by Nowell in 1976,  and has been the subject of a large number of mathematical modelling papers. For a long time it was believed that this process was characterised by selective sweeps whereby a single clone would dominate the tumour cell population. However a number of theoretical studies suggested that spatial heterogeneity could give rise to a diversity of subclones, and also that tumour cells with similar phenotypes could harbour different genotypes, both processes contributing to intra-tumoral heterogeneity. When the technology arrived to measure genetic heterogeneity in cancers it was indeed shown that diversity can be large enough to classify different parts of a tumour into different subtypes.

Would these measurements have been made in the absence of theory supporting it? We can never know, but for one thing we can say that the theory facilitated the discovery.

In conclusion I think that mathematical oncology has an important role to play in advancing our knowledge of cancer, but I don't think it will happen through landmark discoveries, rather through piecemeal additions.

Wednesday, 5 February 2014

Preprint: Evolutionary dynamics of shared niche construction

I have just uploaded a new preprint on arXiv (and bioaRxiv, a new preprint repository for biology) that explores the evolutionary dynamics of shared niche construction.

In the model we assume that the carrying capacity of each species in the population consists of the sum of two parts: an intrinsic part, and a contribution from all species present in the system. If the constructed niche is highly specific, only the first part is included, while a non-specific niche construction corresponds to the second contribution dominating.

Now it turns out that the evolutionary dynamics of the system strongly depends on the specificity: when the carrying capacity is intrinsic, selection is almost exclusively for mutants with higher carrying capacity, while a shared carrying capacity yields selection purely on growth rate.

The below figure illustrates this fact. In the upper panel, where specificity is low, the invasion of a mutant can lead to a decrease in total population size, while in the lower panel, where carrying capacity is intrinsic, each successful invasion increases the total population size. 

Coming from a background in cancer I prefer interpret this result in the context of tumour growth. If you think of different types (or subclones) of cancer cells as being able to withstand and survive different cell densities (i.e. the niche is specific to each subclone) then growth rate of a rare mutant is irrelevant for determining if it spreads in a tumour populated at the maximal cell density of the resident subclone. Only if it can divide and survive at higher densities will it spread and take over the tumour.

The other extreme can be viewed in terms of diffusible factors, such as angiogenetic factors that attract new blood vessels to the growing tumour. The release of a factor benefits all cells (within a reasonable distance) and hence increases the carrying capacity of all subclones. Now a mutant that produces less factors compared to the resident will still receive the benefit, and if it divides faster, will spread in the population. This situation is analogous to the appearance of cheaters in the classical public goods game.


Many species engage in niche construction that ultimately leads to an increase in the carrying capacity of the population. We have investigated how the specificity of this behaviour affects evolutionary dynamics using a set of coupled logistic equations, where the carrying capacity of each genotype consists of two components: an intrinsic part and a contribution from all genotypes present in the population. The relative contribution of the two components is controlled by a specificity parameter $\gamma$, and we show that the ability of a mutant to invade a resident population depends strongly on this parameter. When the carrying capacity is intrinsic, selection is almost exclusively for mutants with higher carrying capacity, while a shared carrying capacity yields selection purely on growth rate. This result has important implications for our understanding of niche construction, in particular the evolutionary dynamics of tumor growth.