Philip Gerlee

The Model Muddle: In Search of Tumor Growth Laws

In this article, we will trace the historical development of tumor growth laws, which in a quantitative fashion describe the increase in tumor mass/volume over time. These models are usually formulated in terms of differential equations that relate the growth rate of the tumor to its current state and range from the simple one-parameter exponential growth model to more advanced models that contain a large number of parameters. Understanding the assumptions and consequences of such models is important, as they often underpin more complex models of tumor growth. The conclusion of this brief survey is that although much improvement has occurred over the last century, more effort and new models are required if we are to understand the intricacies of tumor growth.

Evolution of cell motility in an individual-based model of tumour growth

Tumour invasion is driven by proliferation and importantly migration into the surrounding tissue. Cancer cell motility is also critical in the formation of metastases and is therefore a fundamental issue in cancer research. In this paper we investigate the emergence of cancer cell motility in an evolving tumour population using an individual-based modelling approach. In this model of tumour growth each cell is equipped with a micro-environment response network that determines the behaviour or phenotype of the cell based on the local environment. The response network is modelled using a feed-forward neural network, which is subject to mutations when the cells divide. With this model we have investigated the impact of the micro-environment on the emergence of a motile invasive phenotype. The results show that when a motile phenotype emerges the dynamics of the model are radically changed and we observe faster growing tumours exhibiting diffuse morphologies. Further we observe that the emergence of a motile subclone can occur in a wide range of micro-environmental growth conditions. Iterated simulations showed that in identical growth conditions the evolutionary dynamics either converge to a proliferating or migratory phenotype, which suggests that the introduction of cell motility into the model changes the shape of fitness landscape on which the cancer cell population evolves and that it now contains several local maxima. This could have important implications for cancer treatments which focus on the gene level, as our results show that several distinct genotypes and critically distinct phenotypes can emerge and become dominant in the same micro-environment.

The Impact of Phenotypic Switching on Glioblastoma Growth and Invasion

The brain tumour glioblastoma is characterised by diffuse and infiltrative growth into surrounding brain tissue. At the macroscopic level, the progression speed of a glioblastoma tumour is determined by two key factors: the cell proliferation rate and the cell migration speed. At the microscopic level, however, proliferation and migration appear to be mutually exclusive phenotypes, as indicated by recent in vivo imaging data. Here, we develop a mathematical model to analyse how the phenotypic switching between proliferative and migratory states of individual cells affects the macroscopic growth of the tumour. For this, we propose an individual-based stochastic model in which glioblastoma cells are either in a proliferative state, where they are stationary and divide, or in motile state in which they are subject to random motion. From the model we derive a continuum approximation in the form of two coupled reaction-diffusion equations, which exhibit travelling wave solutions whose speed of invasion depends on the model parameters. We propose a simple analytical method to predict progression rate from the cell-specific parameters and demonstrate that optimal glioblastoma growth depends on a non-trivial trade-off between the phenotypic switching rates. By linking cellular properties to an in vivo outcome, the model should be applicable to designing relevant cell screens for glioblastoma and cytometry-based patient prognostics.

A mathematical model of tumour self-seeding reveals secondary metastatic deposits as drivers of primary tumour growth

Two models of circulating tumour cell (CTC) dynamics have been proposed to explain the phenomenon of tumour ‘self-seeding’, whereby CTCs repopulate the primary tumour and accelerate growth: primary seeding, where cells from a primary tumour shed into the vasculature and return back to the primary themselves; and secondary seeding, where cells from the primary first metastasize into a secondary tissue and form microscopic secondary deposits, which then shed cells into the vasculature returning to the primary. These two models are difficult to distinguish experimentally, yet the differences between them is of great importance to both our understanding of the metastatic process and also for designing methods of intervention. Therefore, we developed a mathematical model to test the relative likelihood of these two phenomena in the subset of tumours whose shed CTCs first encounter the lung capillary bed, and show that secondary seeding is several orders of magnitude more likely than primary seeding. We suggest how this difference could affect tumour evolution, progression and therapy, and propose several possible methods of experimental validation.

Modelling evolutionary cell behaviour using neural networks: Application to tumour growth

In this paper, we present a modelling framework for cellular evolution that is based on the notion that a cells behaviour is driven by interactions with other cells and its immediate environment. We equip each cell with a phenotype that determines its behaviour and implement a decision mechanism to allow evolution of this phenotype. This decision mechanism is modelled using feed-forward neural networks, which have been suggested as suitable models of cell signalling pathways. The environmental variables are presented as inputs to the network and result in a response that corresponds to the phenotype of the cell. The response of the network is determined by the network parameters, which are subject to mutations when the cells divide. This approach is versatile as there are no restrictions on what the input or output nodes represent, they can be chosen to represent any environmental variables and behaviours that are of importance to the cell population under consideration. This framework was implemented in an individual-based model of solid tumour growth in order to investigate the impact of the tissue oxygen concentration on the growth and evolutionary dynamics of the tumour. Our results show that the oxygen concentration affects the tumour at the morphological level, but more importantly has a direct impact on the evolutionary dynamics. When the supply of oxygen is limited we observe a faster divergence away from the initial genotype, a higher population diversity and faster evolution towards aggressive phenotypes. The implementation of this framework suggests that this approach is well suited for modelling systems where evolution plays an important role and where a changing environment exerts selection pressure on the evolving population.

The evolution of carrying capacity in constrained and expanding tumour cell populations

Cancer cells are known to modify their micro-environment such that it can sustain a larger population, or, in ecological terms, they construct a niche which increases the carrying capacity of the population. It has however been argued that niche construction, which benefits all cells in the tumour, would be selected against since cheaters could reap the benefits without paying the cost. We have investigated the impact of niche specificity on tumour evolution using an individual based model of breast tumour growth, in which the carrying capacity of each cell consists of two components: an intrinsic, subclone-specific part and a contribution from all neighbouring cells. Analysis of the model shows that the ability of a mutant to invade a resident population depends strongly on the specificity. When specificity is low selection is mostly on growth rate, while high specificity shifts selection towards increased carrying capacity. Further, we show that the long-term evolution of the system can be predicted using adaptive dynamics. By comparing the results from a spatially structured versus well-mixed population we show that spatial structure restores selection for carrying capacity even at zero specificity, which poses a solution to the niche construction dilemma. Lastly, we show that an expanding population exhibits spatially variable selection pressure, where cells at the leading edge exhibit higher growth rate and lower carrying capacity than those at the centre of the tumour.

Evolving homeostatic tissue using genetic algorithms.

Multicellular organisms maintain form and function through a multitude of homeostatic mechanisms. The details of these mechanisms are in many cases unknown, and so are their evolutionary origin and their link to development. In order to illuminate these issues we have investigated the evolution of structural homeostasis in the simplest of cases, a tissue formed by a mono-layer of cells. To this end, we made use of a 3-dimensional hybrid cellular automaton, an individual-based model in which the behaviour of each cell depends on its local environment. Using an evolutionary algorithm (EA) we evolved cell signalling networks, both with a fixed and an incremental fitness evaluation, which give rise to and maintain a mono-layer tissue structure. Analysis of the solutions provided by the EA shows that the two evaluation methods gives rise to different types of solutions to the problem of homeostasis. The fixed method leads to almost optimal solutions, where the tissue relies on a high rate of cell turnover, while the solutions from the incremental scheme behave in a more conservative manner, only dividing when necessary. In order to test the robustness of the solutions we subjected them to environmental stress, by wounding the tissue, and to genetic stress, by introducing mutations. The results show that the robustness very much depends on the mechanism responsible for maintaining homeostasis. The two evolved cell types analysed present contrasting mechanisms by which tissue homeostasis can be maintained. This compares well to different tissue types found in multicellular organisms. For example the epithelial cells lining the colon in humans are shed at a considerable rate, while in other tissue types, which are not as exposed, the conservative type of homeostatic mechanism is normally found. These results will hopefully shed light on how multicellular organisms have evolved homeostatic mechanisms and what might occur when these mechanisms fail, as in the case of cancer.

Diffusion-limited tumour growth: simulations and analysis.

The morphology of solid tumours is known to be affected by the background oxygen concentration of the tissue in which the tumour grows, and both computational and experimental studies have suggested that branched tumour morphology in low oxygen concentration is caused by diffusion-limited growth. In this paper we present a simple hybrid cellular automaton model of solid tumour growth aimed at investigating this phenomenon. Simulation results show that for high consumption rates (or equivalently low oxygen concentrations) the tumours exhibit branched morphologies, but more importantly the simplicity of the model allows for an analytic approach to the problem. By applying a steady-state assumption we derive an approximate solution of the oxygen equation, which closely matches the simulation results. Further, we derive a dispersion relation which reveals that the average branch width in the tumour depends on the width of the active rim, and that a smaller active rim gives rise to thinner branches. Comparison between the prediction of the stability analysis and the results from the simulations shows good agreement between theory and simulation.

PRODUCTIVITY AND DIVERSITY IN A CROSS-FEEDING POPULATION OF ARTIFICIAL ORGANISMS

Cross‐feeding interactions are a common feature of many microbial systems, such as colonies of Escherichia coli grown on a single limiting resource, and microbial consortia cooperatively degrading complex compounds. We have studied this phenomenon from an abstract point of view by considering artificial organisms that metabolize binary strings from a shared environment. The organisms are represented as simple cellular automaton rules and the analog of energy in the system is an approximation of the Shannon entropy of the binary strings. Only organisms that increase the entropy of the transformed strings are allowed to replicate. This system exhibits a large degree of species diversity, which increases when the flow of binary strings into the system is reduced. Investigating the relation between ecosystem productivity and diversity we find that diversity is negatively correlated with biomass production and energy uptake, while it correlates positively with energy‐uptake efficiency. By performing invasion experiments, we show that the source of diversity is negative frequency‐dependent selection acting among the different species, and that some of these interactions are intransitive, another mechanism known to promote diversity.

Travelling wave analysis of a mathematical model of glioblastoma growth

In this paper we analyse a previously proposed cell-based model of glioblastoma (brain tumour) growth, which is based on the assumption that the cancer cells switch phenotypes between a proliferative and motile state (Gerlee and Nelander, 2012). The dynamics of this model can be described by a system of partial differential equations, which exhibits travelling wave solutions whose wave speed depends crucially on the rates of phenotypic switching. We show that under certain conditions on the model parameters, a closed form expression of the wave speed can be obtained, and using singular perturbation methods we also derive an approximate expression of the wave front shape. These new analytical results agree with simulations of the cell-based model, and importantly show that the inverse relationship between wave front steepness and speed observed for the Fisher equation no longer holds when phenotypic switching is considered.