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, PLoS Comp. Bio., 8(6) 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.Comment: Corrected error in the equation for the Jacobia

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 vs.\ well-mixed population we show that spatial structure restores selection for carrying capacity even at zero specificity, which a poses 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.Comment: Major revisions compared to previous version. The paper is now aimed at tumour modelling. We now start out with an agent-based model for which we derive a mean-field ODE-model. The ODE-model is further analysed using the theory of adaptive dynamic

Weak Selection and the Separation of Eco-evo Time Scales using Perturbation Analysis

We show that under the assumption of weak frequency-dependent selection a wide class of population dynamical models can be analysed using perturbation theory. The inner solution corresponds to the ecological dynamics, where to zeroth order, the genotype frequencies remain constant. The outer solution provides the evolutionary dynamics and corresponds, to zeroth order, to a generalisation of the replicator equation. We apply this method to a model of public goods dynamics and construct, using matched asymptotic expansions, a composite solution valid for all times. We also analyse a Lotka-Volterra model of predator competition and show that to zeroth order the fraction of wild-type predators follows a replicator equation with a constant selection coefficient given by the predator death rate. For both models, we investigate how the error between approximate solutions and the solution to the full model depend on the order of the approximation and show using numerical comparison, for [Formula: see text] and 2, that the error scales according to [Formula: see text], where [Formula: see text] is the strength of selection and k is the order of the approximation

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

The impact of cellular characteristics on the evolution of shape homeostasis

The importance of individual cells in a developing multicellular organism is well known but precisely how the individual cellular characteristics of those cells collectively drive the emergence of robust, homeostatic structures is less well understood. For example cell communication via a diffusible factor allows for information to travel across large distances within the population, and cell polarisation makes it possible to form structures with a particular orientation, but how do these processes interact to produce a more robust and regulated structure? In this study we investigate the ability of cells with different cellular characteristics to grow and maintain homeostatic structures. We do this in the context of an individual-based model where cell behaviour is driven by an intra-cellular network that determines the cell phenotype. More precisely, we investigated evolution with 96 different permutations of our model, where cell motility, cell death, long-range growth factor (LGF), short-range growth factor (SGF) and cell polarisation were either present or absent. The results show that LGF has the largest positive impact on the fitness of the evolved solutions. SGF and polarisation also contribute, but all other capabilities essentially increase the search space, effectively making it more difficult to achieve a solution. By perturbing the evolved solutions, we found that they are highly robust to both mutations and wounding. In addition, we observed that by evolving solutions in more unstable environments they produce structures that were more robust and adaptive. In conclusion, our results suggest that robust collective behaviour is most likely to evolve when cells are endowed with long range communication, cell polarisation, and selection pressure from an unstable environment

Bridging scales in cancer progression: Mapping genotype to phenotype using neural networks

In this review we summarize our recent efforts in trying to understand the role of heterogeneity in cancer progression by using neural networks to characterise different aspects of the mapping from a cancer cells genotype and environment to its phenotype. Our central premise is that cancer is an evolving system subject to mutation and selection, and the primary conduit for these processes to occur is the cancer cell whose behaviour is regulated on multiple biological scales. The selection pressure is mainly driven by the microenvironment that the tumour is growing in and this acts directly upon the cell phenotype. In turn, the phenotype is driven by the intracellular pathways that are regulated by the genotype. Integrating all of these processes is a massive undertaking and requires bridging many biological scales (i.e. genotype, pathway, phenotype and environment) that we will only scratch the surface of in this review. We will focus on models that use neural networks as a means of connecting these different biological scales, since they allow us to easily create heterogeneity for selection to act upon and importantly this heterogeneity can be implemented at different biological scales. More specifically, we consider three different neural networks that bridge different aspects of these scales and the dialogue with the micro-environment, (i) the impact of the micro-environment on evolutionary dynamics, (ii) the mapping from genotype to phenotype under drug-induced perturbations and (iii) pathway activity in both normal and cancer cells under different micro-environmental conditions

Fast and precise inference on diffusivity in interacting particle systems

Particle systems made up of interacting agents is a popular model used in a vast array of applications, not the least in biology where the agents can represent everything from single cells to animals in a herd. Usually, the particles are assumed to undergo some type of random movements, and a popular way to model this is by using Brownian motion. The magnitude of random motion is often quantified using mean squared displacement, which provides a simple estimate of the diffusion coefficient. However, this method often fails when data is sparse or interactions between agents frequent. In order to address this, we derive a conjugate relationship in the diffusion term for large interacting particle systems undergoing isotropic diffusion, giving us an efficient inference method. The method accurately accounts for emerging effects such as anomalous diffusion stemming from mechanical interactions. We apply our method to an agent-based model with a large number of interacting particles, and the results are contrasted with a naive mean square displacement-based approach. We find a significant improvement in performance when using the higher-order method over the naive approach. This method can be applied to any system where agents undergo Brownian motion and will lead to improved estimates of diffusion coefficients compared to existing methods

Model-based inference of metastatic seeding rates in de novo metastatic breast cancer reveals the impact of secondary seeding and molecular subtype

We present a stochastic network model of metastasis spread for de novo metastatic breast cancer, composed of tumor to metastasis (primary seeding) and metastasis to metastasis spread (secondary seeding), parameterized using the SEER (Surveillance, Epidemiology, and End Results) database. The model provides a quantification of tumor cell dissemination rates between the tumor and metastasis sites. These rates were used to estimate the probability of developing a metastasis for untreated patients. The model was validated using tenfold cross-validation. We also investigated the effect of HER2 (Human Epidermal Growth Factor Receptor 2) status, estrogen receptor (ER) status and progesterone receptor (PR) status on the probability of metastatic spread. We found that dissemination rate through secondary seeding is up to 300 times higher than through primary seeding. Hormone receptor positivity promotes seeding to the bone and reduces seeding to the lungs and primary seeding to the liver, while HER2 expression increases dissemination to the bone, lungs and primary seeding to the liver. Secondary seeding from the lungs to the liver seems to be hormone receptor-independent, while that from the lungs to the brain appears HER2-independent

The Impact of Elastic Deformations of the Extracellular Matrix on Cell Migration

The mechanical properties of the extracellular matrix, in particular its stiffness, are known to impact cell migration. In this paper, we develop a mathematical model of a single cell migrating on an elastic matrix, which accounts for the deformation of the matrix induced by forces exerted by the cell, and investigate how the stiffness impacts the direction and speed of migration. We model a cell in 1D as a nucleus connected to a number of adhesion sites through elastic springs. The cell migrates by randomly updating the position of its adhesion sites. We start by investigating the case where the cell springs are constant, and then go on to assuming that they depend on the matrix stiffness, on matrices of both uniform stiffness as well as those with a stiffness gradient. We find that the assumption that cell springs depend on the substrate stiffness is necessary and sufficient for an efficient durotactic response. We compare simulations to recent experimental observations of human cancer cells exhibiting durotaxis, which show good qualitative agreement.\ua0\ua9 2020, The Author(s)