Aggregation and travelling wave dynamics in a two-population model of cancer cell growth and invasion

Funding: Engineering and Physical Sciences Research Council (UK) grant numbers EP/L504932/1 (VB), EP/K033689/1 (RE).Cells adhere to each other and to the extracellular matrix (ECM) through protein molecules on the surface of the cells. The breaking and forming of adhesive bonds, a process critical in cancer invasion and metas- tasis, can be influenced by the mutation of cancer cells. In this paper, we develop a nonlocal mathematical model describing cancer cell invasion and movement as a result of integrin-controlled cell-cell adhesion and cell-matrix adhesion, for two cancer cell populations with different levels of mutation. The partial differential equations for cell dynamics are coupled with ordinary differential equations describing the extracellular matrix (ECM) degradation and the production and decay of integrins. We use this model to investigate the role of cancer mutation on the possibility of cancer clonal competition with alternating dominance, or even competitive exclusion (phenomena observed experimentally). We discuss different possible cell aggregation patterns, as well as travelling wave patterns. In regard to the travelling waves, we investigate the effect of cancer mutation rate on the speed of cancer invasion.Publisher PDFPeer reviewe

Cell-scale degradation of peritumoural extracellular matrix fibre network and its role within tissue-scale cancer invasion

Local cancer invasion of tissue is a complex, multiscale process which plays an essential role in tumour progression. Occurring over many different temporal and spatial scales, the first stage of invasion is the secretion of matrix degrading enzymes (MDEs) by the cancer cells that consequently degrade the surrounding extracellular matrix (ECM). This process is vital for creating space in which the cancer cells can progress and it is driven by the activities of specific matrix metalloproteinases (MMPs). In this paper, we consider the key role of two MMPs by developing further the novel two-part multiscale model introduced in [33] to better relate at micro-scale the two micro-scale activities that were considered there, namely, the micro-dynamics concerning the continuous rearrangement of the naturally oriented ECM fibres within the bulk of the tumour and MDEs proteolytic micro-dynamics that take place in an appropriate cell-scale neighbourhood of the tumour boundary. Focussing primarily on the activities of the membrane-tethered MT1-MMP and the soluble MMP-2 with the fibrous ECM phase, in this work we investigate the MT1-MMP/MMP-2 cascade and its overall effect on tumour progression. To that end, we will propose a new multiscale modelling framework by considering the degradation of the ECM fibres not only to take place at macro-scale in the bulk of the tumour but also explicitly in the micro-scale neighbourhood of the tumour interface as a consequence of the interactions with molecular fluxes of MDEs that exercise their spatial dynamics at the invasive edge of the tumour

Structured models of cell migration incorporating molecular binding processes

The dynamic interplay between collective cell movement and the various molecules involved in the accompanying cell signalling mechanisms plays a crucial role in many biological processes including normal tissue development and pathological scenarios such as wound healing and cancer. Information about the various structures embedded within these processes allows a detailed exploration of the binding of molecular species to cell-surface receptors within the evolving cell population. In this paper we establish a general spatio-temporal-structural framework that enables the description of molecular binding to cell membranes coupled with the cell population dynamics. We first provide a general theoretical description for this approach and then illustrate it with two examples arising from cancer invasion

Toward transient finite element simulation of thermal deformation of machine tools in real-time

Finite element models without simplifying assumptions can accurately describe the spatial and temporal distribution of heat in machine tools as well as the resulting deformation. In principle, this allows to correct for displacements of the Tool Centre Point and enables high precision manufacturing. However, the computational cost of FE models and restriction to generic algorithms in commercial tools like ANSYS prevents their operational use since simulations have to run faster than real-time. For the case where heat diffusion is slow compared to machine movement, we introduce a tailored implicit–explicit multi-rate time stepping method of higher order based on spectral deferred corrections. Using the open-source FEM library DUNE, we show that fully coupled simulations of the temperature field are possible in real-time for a machine consisting of a stock sliding up and down on rails attached to a stand

The Discontinuous Galerkin Finite Element Method for Solving the MEG and the Combined MEG/EEG Forward Problem

In Electro- (EEG) and Magnetoencephalography (MEG), one important requirement of source reconstruction is the forward model. The continuous Galerkin finite element method (CG-FEM) has become one of the dominant approaches for solving the forward problem over the last decades. Recently, a discontinuous Galerkin FEM (DG-FEM) EEG forward approach has been proposed as an alternative to CG-FEM (Engwer et al., 2017). It was shown that DG-FEM preserves the property of conservation of charge and that it can, in certain situations such as the so-called skull leakages, be superior to the standard CG-FEM approach. In this paper, we developed, implemented, and evaluated two DG-FEM approaches for the MEG forward problem, namely a conservative and a non-conservative one. The subtraction approach was used as source model. The validation and evaluation work was done in statistical investigations in multi-layer homogeneous sphere models, where an analytic solution exists, and in a six-compartment realistically shaped head volume conductor model. In agreement with the theory, the conservative DG-FEM approach was found to be superior to the non-conservative DG-FEM implementation. This approach also showed convergence with increasing resolution of the hexahedral meshes. While in the EEG case, in presence of skull leakages, DG-FEM outperformed CG-FEM, in MEG, DG-FEM achieved similar numerical errors as the CG-FEM approach, i.e., skull leakages do not play a role for the MEG modality. In particular, for the finest mesh resolution of 1 mm sources with a distance of 1.59 mm from the brain-CSF surface, DG-FEM yielded mean topographical errors (relative difference measure, RDM%) of 1.5% and mean magnitude errors (MAG%) of 0.1% for the magnetic field. However, if the goal is a combined source analysis of EEG and MEG data, then it is highly desirable to employ the same forward model for both EEG and MEG data. Based on these results, we conclude that the newly presented conservative DG-FEM can at least complement and in some scenarios even outperform the established CG-FEM approaches in EEG or combined MEG/EEG source analysis scenarios, which motivates a further evaluation of DG-FEM for applications in bioelectromagnetism

EXA-DUNE: Flexible PDE solvers, numerical methods and applications

In the EXA-DUNE project we strive to (i) develop and implement numerical algorithms for solving PDE problems efficiently on heterogeneous architectures, (ii) provide corresponding domain-specific abstractions that allow application scientists to effectively use these methods, and (iii) demonstrate performance on porous media flow problems. In this paper, we present first results on the hybrid parallelisation of sparse linear algebra, system and RHS assembly, the implementation of multiscale finite element methods and the SIMD performance of high-order discontinuous Galerkin methods within an application scenario

DUNEuro—A software toolbox for forward modeling in bioelectromagnetism

Accurate and efficient source analysis in electro- and magnetoencephalography using sophisticated realistic head geometries requires advanced numerical approaches. This paper presents DUNEuro, a free and open-source C++ software toolbox for the numerical computation of forward solutions in bioelectromagnetism. Building upon the DUNE framework, it provides implementations of modern fitted and unfitted finite element methods to efficiently solve the forward problems of electro- and magnetoencephalography. The user can choose between a variety of different source models that are implemented. The software’s aim is to provide interfaces that are extendable and easy-to-use. In order to enable a closer integration into existing analysis pipelines, interfaces to Python and MATLAB are provided. The practical use is demonstrated by a source analysis example of somatosensory evoked potentials using a realistic six-compartment head model. Detailed installation instructions and example scripts using spherical and realistic head models are appended