Data-driven spatio-temporal modelling of glioblastoma

Abstract

Mathematical oncology provides unique and invaluable insights into tumour growth on both the microscopic and macroscopic levels. This review presents state-of-the-art modelling techniques and focuses on their role in understanding glioblastoma, a malignant form of brain cancer. For each approach, we summarize the scope, drawbacks and assets. We highlight the potential clinical applications of each modelling technique and discuss the connections between the mathematical models and the molecular and imaging data used to inform them. By doing so, we aim to prime cancer researchers with current and emerging computational tools for understanding tumour progression. By providing an in-depth picture of the different modelling techniques, we also aim to assist researchers who seek to build and develop their own models and the associated inference frameworks. Our article thus strikes a unique balance. On the one hand, we provide a comprehensive overview of the available modelling techniques and their applications, including key mathematical expressions. On the other hand, the content is accessible to mathematicians and biomedical scientists alike to accommodate the interdisciplinary nature of cancer research.

Keywords: Bayesian inference; agent-based modelling; data-driven modelling; glioblastoma; reaction–diffusion equations.

Figure 1.

Schematic overview of the concepts behind LGCA models. (a) The cell dynamics that are handled as stochastic processes in the single-cell CA models discussed in §3.1.1: cell migration, death (and lysis), growth and cell division. Filled (orange) circles denote occupied sites, while there are no cells in the empty circles. (b) LGCA models attribute a set of velocity channels to each location, subdividing each site. The cells can shift between these channels following stochastic rules. Again, filled (orange) circles denote occupied channels, while empty channels are indicated by empty circles. Here, there are five velocity channels per location: four channels that lead to migration and one channel at which the cell is at rest. For LGCA models, the time evolution takes place in two steps. First, cell death, growth, division and migration between velocity channels occur as stochastic events. Cell migration between lattice sites, however, is no longer included in this step. Instead, cell migration takes place in the second step based on the velocity channels that the cells occupy. In this second step, cell migration complies with momentum and mass conservation.

Figure 2.

Schematic overview of CPMs. Each cell (and the ECM) has a unique identifier. Here, we consider cells 1, 2 and 3 and the ECM denoted by 0. Each cell occupies several lattice sites. Randomly selected sites at the cell borders (asterisks) might swap affiliation depending on the energy change associated with this swap, allowing the cells to grow and move.

Figure 3.

Schematic overview of off-lattice models. (a) Spherical cells in a CBM, highlighting forces on cell 1. Due to the overlap (δ₂₁) between cells 1 and 2, cell 1 is repulsed by cell 2. Overlapping cells forming a dumb-bell also occur during cell division. F₃₁ exemplifies an adhesive force occurring without direct contact. In an isotropic environment, the drag force will be proportional to the velocity, v₁. (b) A cell made of multiple nodes in a DCM. Node i at the cell membrane is subject to forces from the surrounding nodes (F_ji). Constraints on areas and volumes might lead to additional forces (${\mathbf{F}}_{△i}$), while the movement of the node (v_i) gives rise to viscous forces. The cell has a cytoskeleton (CSK, dashed lines).