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.     

Hybrid mathematical model of glioma progression

Objectives:  Gliomas are an important form of brain cancer, with high mortality rate. Mathematical models are often used to understand and predict their behaviour. However, using current modeling techniques one must choose between simulating individual cell behaviour and modeling tumours of clinically significant size.
Materials and Methods:  We propose a hybrid compartment-continuum-discrete model to simulate glioma growth and malignant cell invasion. The discrete portion of the model is capable of capturing intercellular interactions, including cell migration, intercellular communication, spatial cell population heterogeneity, phenotype differentiation, epigenetic events, proliferation, and apoptosis. Combining this with a compartment and continuum model allows clinically significant tumour sizes to be evaluated.
Results and Conclusions:  This model is used to perform multiple simulations to determine sensitivity to changes in important model parameters, specifically, the fundamental length parameter, necrotic cell degradation rate, rate of cell migration, and rate of phenotype transformation. Using these values, the model is able to simulate tumour growth and invasion behaviour, observed clinically. This mathematical model provides a means to simulate various tumour development scenarios, which may lead to a better understanding of how altering fundamental parameters can influence neoplastic progression.     

Are cancer cells really softer than normal cells?

Solid tumours are often first diagnosed by palpation, suggesting that the tumour is more rigid than its surrounding environment. Paradoxically, individual cancer cells appear to be softer than their healthy counterparts. In this review, we first list the physiological reasons indicating that cancer cells may be more deformable than normal cells. Next, we describe the biophysical tools that have been developed in recent years to characterise and model cancer cell mechanics. By reviewing the experimental studies that compared the mechanics of individual normal and cancer cells, we argue that cancer cells can indeed be considered as softer than normal cells. We then focus on the intracellular elements that could be responsible for the softening of cancer cells. Finally, we ask whether the mechanical differences between normal and cancer cells can be used as diagnostic or prognostic markers of cancer progression.     

Tensional Homeostasis in Single Fibroblasts

Adherent cells generate forces through acto-myosin contraction to move, change shape, and sense the mechanical properties of their environment. They are thought to maintain defined levels of tension with their surroundings despite mechanical perturbations that could change tension, a concept known as tensional homeostasis. Misregulation of tensional homeostasis has been proposed to drive disorganization of tissues and promote progression of diseases such as cancer. However, whether tensional homeostasis operates at the single cell level is unclear. Here, we directly test the ability of single fibroblast cells to regulate tension when subjected to mechanical displacements in the absence of changes to spread area or substrate elasticity. We use a feedback-controlled atomic force microscope to measure and modulate forces and displacements of individual contracting cells as they spread on a fibronectin-patterned atomic-force microscope cantilever and coverslip. We find that the cells reach a steady-state contraction force and height that is insensitive to stiffness changes as they fill the micropatterned areas. Rather than maintaining a constant tension, the fibroblasts altered their contraction force in response to mechanical displacement in a strain-rate-dependent manner, leading to a new and stable steady-state force and height. This response is influenced by overexpression of the actin crosslinker α-actinin, and rheology measurements reveal that changes in cell elasticity are also strain- rate-dependent. Our finding of tensional buffering, rather than homeostasis, allows cells to transition between different tensional states depending on how they are displaced, permitting distinct responses to slow deformations during tissue growth and rapid deformations associated with injury.     

Tensional Homeostasis in Single Fibroblasts

Adherent cells generate forces through acto-myosin contraction to move, change shape, and sense the mechanical properties of their environment. They are thought to maintain defined levels of tension with their surroundings despite mechanical perturbations that could change tension, a concept known as tensional homeostasis. Misregulation of tensional homeostasis has been proposed to drive disorganization of tissues and promote progression of diseases such as cancer. However, whether tensional homeostasis operates at the single cell level is unclear. Here, we directly test the ability of single fibroblast cells to regulate tension when subjected to mechanical displacements in the absence of changes to spread area or substrate elasticity. We use a feedback-controlled atomic force microscope to measure and modulate forces and displacements of individual contracting cells as they spread on a fibronectin-patterned atomic-force microscope cantilever and coverslip. We find that the cells reach a steady-state contraction force and height that is insensitive to stiffness changes as they fill the micropatterned areas. Rather than maintaining a constant tension, the fibroblasts altered their contraction force in response to mechanical displacement in a strain-rate-dependent manner, leading to a new and stable steady-state force and height. This response is influenced by overexpression of the actin crosslinker α-actinin, and rheology measurements reveal that changes in cell elasticity are also strain- rate-dependent. Our finding of tensional buffering, rather than homeostasis, allows cells to transition between different tensional states depending on how they are displaced, permitting distinct responses to slow deformations during tissue growth and rapid deformations associated with injury.     

A Hybrid Model of Tumor–Stromal Interactions in Breast Cancer

Ductal carcinoma in situ (DCIS) is an early stage noninvasive breast cancer that originates in the epithelial lining of the milk ducts, but it can evolve into comedo DCIS and ultimately, into the most common type of breast cancer, invasive ductal carcinoma. Understanding the progression and how to effectively intervene in it presents a major scientific challenge. The extracellular matrix (ECM) surrounding a duct contains several types of cells and several types of growth factors that are known to individually affect tumor growth, but at present the complex biochemical and mechanical interactions of these stromal cells and growth factors with tumor cells is poorly understood. Here we develop a mathematical model that incorporates the cross-talk between stromal and tumor cells, which can predict how perturbations of the local biochemical and mechanical state influence tumor evolution. We focus on the EGF and TGF-β signaling pathways and show how up- or down-regulation of components in these pathways affects cell growth and proliferation. We then study a hybrid model for the interaction of cells with the tumor microenvironment (TME), in which epithelial cells (ECs) are modeled individually while the ECM is treated as a continuum, and show how these interactions affect the early development of tumors. Finally, we incorporate breakdown of the epithelium into the model and predict the early stages of tumor invasion into the stroma. Our results shed light on the interactions between growth factors, mechanical properties of the ECM, and feedback signaling loops between stromal and tumor cells, and suggest how epigenetic changes in transformed cells affect tumor progression.     

Couple stresses and discrete potentials in the vertex model of cellular monolayers

The vertex model is widely used to simulate the mechanical properties of confluent epithelia and other multicellular tissues. This inherently discrete framework allows a Cauchy stress to be attributed to each cell, and its symmetric component has been widely reported, at least for planar monolayers. Here, we consider the stress attributed to the neighbourhood of each tricellular junction, evaluating in particular its leading-order antisymmetric component and the associated couple stresses, which characterise the degree to which individual cells experience (and resist) in-plane bending deformations. We develop discrete potential theory for localised monolayers having disordered internal structure and use this to derive the analogues of Airy and Mindlin stress functions. These scalar potentials typically have broad-banded spectra, highlighting the contributions of small-scale defects and boundary layers to global stress patterns. An affine approximation attributes couple stresses to pressure differences between cells sharing a trijunction, but simulations indicate an additional role for non-affine deformations.

     

Nonlinear simulation of tumor necrosis,neo-vascularization and tissue invasion via an adaptive finite-element/level-set method

We present a multi-scale computer simulator of cancer progression at the tumoral level, from avascular stage growth, through the transition from avascular to vascular growth (neo-vascularization), and into the later stages of growth and invasion of normal tissue. We use continuum scale reaction-diffusion equations for the growth component of the model, and a combined continuum-discrete model for the angiogenesis component. We use the level set method for describing complex topological changes observed during growth such as tumor splitting and reconnection, and capture of healthy tissue inside the tumor. We use an adaptive, unstructured finite element mesh that allows for finely resolving important regions of the computational domain such as the necrotic rim, the tumor interface and around the capillary sprouts. We present full nonlinear, two-dimensional simulations, showing the potential of our virtual cancer simulator. We use microphysical parameters characterizing malignant glioma cells, obtained from recent in vitro experiments from our lab and from clinical data, and provide insight into the mechanisms leading to infiltration of the brain by the cancer cells. The results indicate that diffusional instability of tumor mass growth and the complex interplay with the developing neo-vasculature may be powerful mechanisms for tissue invasion.     

The role of mechanical host-tumour interactions in the collapse of tumour blood vessels and tumour growth dynamics

A mathematical model of residual stress evolution in a growing vascular tumour is presented, in an attempt to elucidate the poorly understood phenomenon of vascular collapse. Whereas earlier studies in this area have neglected the effects of mechanical interactions between the tumour and the surrounding host tissue, the significance of these interactions for the long-term development of a tumour is now considered. The model predicts tumour stress distributions which reflect the distinctive patterns of vascular collapse reported in experimental studies. Moreover, while neglecting mechanical host/tumour interactions results in the eventual complete regression of the tumour to its avascular dormant size in the event of vascular collapse, this new model points to the possibility of oscillations in the tumour's size in the long term.     

Evolutionary game theory elucidates the role of glycolysis in glioma progression and invasion

Abstract. Objectives: Tumour progression has been described as a sequence of traits or phenotypes that cells have to acquire if the neoplasm is to become an invasive and malignant cancer. Although genetic mutations that lead to these phenotypes are random, the process by which some of these mutations become successful and cells spread is influenced by tumour microenvironment and the presence of other cell phenotypes. It is thus likely that some phenotypes that are essential in tumour progression will emerge in the tumour population only with prior presence of other different phenotypes. Materials and methods: In this study, we use evolutionary game theory to analyse the interactions between three different tumour cell phenotypes defined by autonomous growth, anaerobic glycolysis, and cancer cell invasion. The model allows us to understand certain specific aspects of glioma progression such as the emergence of diffuse tumour cell invasion in low‐grade tumours. Results: We have found that the invasive phenotype is more likely to evolve after appearance of the glycolytic phenotype which would explain the ubiquitous presence of invasive growth in malignant tumours. Conclusions: The result suggests that therapies, which increase the fitness cost of switching to anaerobic glycolysis, might decrease probability of the emergence of more invasive phenotypes.     

The mathematical modelling of tumour angiogenesis and invasion

In order to accomplish the transition from avascular to vascular growth, solid tumours secrete a diffusible substance known as tumour angiogenesis factor (TAF) into the surrounding tissue. Endothelial cells which form the lining of neighbouring blood vessels respond to this chemotactic stimulus in a well-ordered sequence of events comprising, at minimum, of a degradation of their basement membrane, migration and proliferation. Capillary sprouts are formed which migrate towards the tumour eventually penetrating it and permitting vascular growth to take place. It is during this stage of growth that the insidious process of invasion of surrounding tissues can and does take place. A model mechanism for angiogenesis is presented which includes the diffusion of the TAF into the surrounding host tissue and the response of the endothelial cells to the chemotactic stimulus. Numerical simulations of the model are shown to compare very well with experimental observations. The subsequent vascular growth of the tumour is discussed with regard to a classical reaction-diffusion pre-pattern model.