The biology and mathematical modelling of glioma invasion: a review

Abstract

Adult gliomas are aggressive brain tumours associated with low patient survival rates and limited life expectancy. The most important hallmark of this type of tumour is its invasive behaviour, characterized by a markedly phenotypic plasticity, infiltrative tumour morphologies and the ability of malignant progression from low- to high-grade tumour types. Indeed, the widespread infiltration of healthy brain tissue by glioma cells is largely responsible for poor prognosis and the difficulty of finding curative therapies. Meanwhile, mathematical models have been established to analyse potential mechanisms of glioma invasion. In this review, we start with a brief introduction to current biological knowledge about glioma invasion, and then critically review and highlight future challenges for mathematical models of glioma invasion.

Keywords: cell phenotypic plasticity; glioma invasion; infiltrative tumour morphology; malignant progression; mathematical modelling.

Figure 1.

Glioma cell migration. Schematic of the process of glioma cell invasion into host brain tissue. Invasion of glioma cells involves four distinct steps: (1) detachment of invading cells from the primary tumour mass, a process triggered by downregulation of cell–cell adhesion molecules and microenvironmental changes, (2) integrin-mediated adhesion to the extracellular matrix (ECM), (3) secretion of proteases, which locally degrade ECM components creating routes along which glioma cells invade the brain and (4) migration by extending a prominent leading cytoplasmic protrusion, followed by a burst of forward movement of the cell body. Figure adapted from [39].

Figure 2.

Invasive cell migration. Front interface for small (a) and high (b) effective cell–cell adhesion values. Shown are the simulations of a discrete stochastic lattice model; every black dot represents a cell, and every white dot corresponds to an empty site. The system size is 400 × 400 (in units of cell diameter). Figure reproduced with permission from [27].

Figure 3.

Phenotypic plasticity. Simulations of a lattice-gas cellular automaton (LGCA) model taking into account the ‘Go-or-Grow’ mechanism, cell–cell repulsion and a cell density-dependent switch between a migrating and a proliferative phenotype. Simulation results (in blue) against experimental data (in red). (a) Temporal evolution of the core radius (dotted line) and invasive radius (solid line). (b) Temporal evolution of the ratio of core to invasive radius. (c) Velocity field diagram indicating the direction of glioma cell movement. (d) Visualization of the spatial tumour structure, where the grey level refers to the number of cells per node. Figure reproduced with permission from [25].

Figure 4.

Infiltrative tumour morphology. Multiscale 3D computer model predicts gross morphologic features of a growing glioblastoma. (a) Viable (VT) and necrotic (NT) tissue regions and vasculature (MV, mature blood-conducting vessels in red; NV, new non-conducting vessels in blue) are shown. The time sequence (from left to right, over a period of 3 months) reveals that the morphology is affected by successive cycles of neovascularization, vasculature maturation and vessel cooption (VC). Scale bar, 250 μm. (b) Histology-like section of the last frame of the simulation in (a) (obtained by slicing horizontally through the simulated tumour) reveals viable tumour regions (white) surrounding necrotic tissue (dark). (c) Another view from simulation shown in (a), right. Figure reproduced with permission from [161].

Figure 5.

Malignant progression. Glioma grade as a function of cellular and microenvironmental compartments. Simulations of glioma grade II–IV varying the net proliferation rate of glioma cells and represented as a plot of density or concentrations of model variables (normoxic cells, hypoxic cells, necrotic tissue, vasculature and angiogenic factor) with respect to the distance from the centre of the in silico tumour. Figure reproduced with permission from [23].