Heterogeneity and proliferation of invasive cancer subclones in game theory models of the Warburg effect

OBJECTIVES: The Warburg effect, the switch from aerobic energy production to anaerobic glycolysis, promotes tumour proliferation and motility by inducing acidification of the tumour microenvironment. Therapies that reduce acidity could impair tumour growth and invasiveness. I analyse the dynamics of cell proliferation and of resistance to therapies that target acidity in a population of cells under the Warburg effect. MATERIALS AND METHODS: The dynamics of mutant cells with increased glycolysis and motility is analysed in a multi-player game with collective interactions in the framework of evolutionary game theory. Perturbations of the level of acidity in the microenvironment are used to simulate the effect of therapies that target glycolysis. RESULTS: The non-linear effects of glycolysis induce frequency-dependent clonal selection leading to the coexistence of glycolytic and non-glycolytic cells within the tumour. Mutants with increased motility can invade such polymorphic population and spread within the tumour. While reducing acidity may produce a sudden reduction in tumour proliferation, frequency-dependent selection enables the tumour to adapt to the new conditions and can enable the tumour to restore the original levels of growth and invasiveness. CONCLUSIONS: The acidity produced by glycolysis acts as a non-linear public good that leads to the coexistence of cells with high and low glycolysis within the tumour. Such heterogeneous population can easily adapt to changes in acidity. Therapies that target acidity can only be effective in the long term if the cost of glycolysis is high, that is, under non-limiting oxygen concentrations. Their efficacy, therefore, is reduced when combined with therapies that impair angiogenesis

Cellular adaptations to hypoxia and acidosis during somatic evolution of breast cancer

Conceptual models of carcinogenesis typically consist of an evolutionary sequence of heritable changes in genes controlling proliferation, apoptosis, and senescence. We propose that these steps are necessary but not sufficient to produce invasive breast cancer because intraductal tumour growth is also constrained by hypoxia and acidosis that develop as cells proliferate into the lumen and away from the underlying vessels. This requires evolution of glycolytic and acid-resistant phenotypes that, we hypothesise, is critical for emergence of invasive cancer. Mathematical models demonstrate severe hypoxia and acidosis in regions of intraductal tumours more than 100 m from the basement membrane. Subsequent evolution of glycolytic and acid-resistant phenotypes leads to invasive proliferation. Multicellular spheroids recapitulating ductal carcinoma in situ (DCIS) microenvironmental conditions demonstrate upregulated glucose transporter 1 (GLUT1) as adaptation to hypoxia followed by growth into normoxic regions in qualitative agreement with model predictions. Clinical specimens of DCIS exhibit periluminal distribution of GLUT-1 and Na+/H+ exchanger (NHE) indicating transcriptional activation by hypoxia and clusters of the same phenotype in the peripheral, presumably normoxic regions similar to the pattern predicted by the models and observed in spheroids. Upregulated GLUT-1 and NHE-1 were observed in microinvasive foci and adjacent intraductal cells. Adaptation to hypoxia and acidosis may represent key events in transition from in situ to invasive cancer

Nonlinear modelling of cancer: bridging the gap between cells and tumours

Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis