47 research outputs found
Adaptive mesh reconstruction: Total Variation Bound
We consider 3-point numerical schemes for scalar Conservation Laws, that are
oscillatory either to their dispersive or anti-diffusive nature. Oscillations
are responsible for the increase of the Total Variation (TV); a bound on which
is crucial for the stability of the numerical scheme. It has been noticed
(\cite{Arvanitis.2001}, \cite{Arvanitis.2004}, \cite{Sfakianakis.2008}) that
the use of non-uniform adaptively redefined meshes, that take into account the
geometry of the numerical solution itself, is capable of taming oscillations;
hence improving the stability properties of the numerical schemes.
In this work we provide a model for studying the evolution of the extremes
over non-uniform adaptively redefined meshes. Based on this model we prove that
proper mesh reconstruction is able to control the oscillations; we provide
bounds for the Total Variation (TV) of the numerical solution. We moreover
prove under more strict assumptions that the increase of the TV -due to the
oscillatory behaviour of the numerical schemes- decreases with time; hence
proving that the overall scheme is TV Increase-Decreasing (TVI-D)
A mathematical insight in the epithelial-mesenchymal-like transition in cancer cells and its effect in the invasion of the extracellular matrix
Current biological knowledge supports the existence of a secondary group of
cancer cells within the body of the tumour that exhibits stem cell-like
properties. These cells are termed Cancer Stem Cells (CSCs}, and as opposed to
the more usual Differentiated Cancer Cells (DCCs), they exhibit higher
motility, they are more resilient to therapy, and are able to metastasize to
secondary locations within the organism and produce new tumours. The origin of
the CSCs is not completely clear; they seem to stem from the DCCs via a
transition process related to the Epithelial-Mesenchymal Transition (EMT) that
can also be found in normal tissue.
In the current work we model and numerically study the transition between
these two types of cancer cells, and the resulting "ensemble" invasion of the
extracellular matrix. This leads to the derivation and numerical simulation of
two systems: an algebraic-elliptic system for the transition and an
advection-reaction-diffusion system of Keller-Segel taxis type for the
invasion
An Extended Filament Based Lamellipodium Model Produces Various Moving Cell Shapes in the Presence of Chemotactic Signals
The Filament Based Lamellipodium Model (FBLM) is a two-phase two-dimensional
continuum model, describing the dynamcis of two interacting families of locally
parallel actin filaments (C.Schmeiser and D.Oelz, How do cells move?
Mathematical modeling of cytoskeleton dynamics and cell migration. Cell
mechanics: from single scale-based models to multiscale modeling. Chapman and
Hall, 2010). It contains accounts of the filaments' bending stiffness, of
adhesion to the substrate, and of cross-links connecting the two families.
An extension of the model is presented with contributions from nucleation of
filaments by branching, from capping, from contraction by actin-myosin
interaction, and from a pressure-like repulsion between parallel filaments due
to Coulomb interaction. The effect of a chemoattractant is described by a
simple signal transduction model influencing the polymerization speed.
Simulations with the extended model show its potential for describing various
moving cell shapes, depending on the signal transduction procedure, and for
predicting transients between nonmoving and moving states as well as changes of
direction
Mathematical modelling of cancer invasion : a review
Funding: MAJC gratefully acknowledges the support of EPSRC Grant No. EP/S030875/1 (EPSRC SofTMech∧MP Centre-to-Centre Award).A defining feature of cancer is the capability to spread locally into the surrounding tissue, with cancer cells spreading beyond any normal boundaries. Cancer invasion is a complex phenomenon involving many inter-connected processes at different spatial and temporal scales. A key component of invasion is the ability of cancer cells to alter and degrade the extracellular matrix through the secretion of matrix-degrading enzymes. Combined with excessive cell proliferation and cell migration (individual and collective), this facilitates the spread of cancer cells into the local tissue. Along with tumour-induced angiogenesis, invasion is a critical component of metastatic spread, ultimately leading to the formation of secondary tumours in other parts of the host body. In this paper we present an overview of the various mathematical models and different modelling techniques and approaches that have been developed over the past 25 years or so and which focus on various aspects of the invasive process.Postprin
Numerical study of cancer cell invasion dynamics using adaptive mesh refinement: the urokinase model
In the present work we investigate the chemotactically and proteolytically
driven tissue invasion by cancer cells. The model employed is a system of
advection-reaction-diffusion equations that features the role of the serine
protease urokinase-type plasminogen activator. The analytical and numerical
study of this system constitutes a challenge due to the merging, emerging, and
travelling concentrations that the solutions exhibit.
Classical numerical methods applied to this system necessitate very fine
discretization grids to resolve these dynamics in an accurate way. To reduce
the computational cost without sacrificing the accuracy of the solution, we
apply adaptive mesh refinement techniques, in particular h-refinement. Extended
numerical experiments exhibit that this approach provides with a higher order,
stable, and robust numerical method for this system. We elaborate on several
mesh refinement criteria and compare the results with the ones in the
literature.
We prove, for a simpler version of this model, bounds for the
solutions, we study the stability of its conditional steady states, and
conclude that it can serve as a test case for further development of mesh
refinement techniques for cancer invasion simulations
Multiscale modeling of glioma invasion : from receptor binding to flux-limited macroscopic PDEs
Funding: The work of the first and fourth authors was supported by the German Federal Ministry of Education and Research (BMBF) through project GlioMaTh 05M2016. The work of the third author was supported by the Postdoctoral Fellowship for Research in Japan (Standard) of the Japan Society for the Promotion of Science.We propose a novel approach to modeling cell migration in an anisotropic environment with biochemical heterogeneity and interspecies interactions, using as a paradigm glioma invasion in brain tissue under the influence of hypoxia-triggered angiogenesis. The multiscale procedure links single-cell and mesoscopic dynamics with population level behavior, leading on the macroscopic scale to flux-limited glioma diffusion and multiple taxis. We verify the nonnegativity of regular solutions (provided they exist) to the obtained macroscopic PDE-ODE system and perform numerical simulations to illustrate the solution behavior under several scenarios.Publisher PDFPeer reviewe
Stochastic differential equation modelling of cancer cell migration and tissue invasion
Funding: Engineering and Physical Sciences Research Council (EPSRC) EP/S030875/1.Invasion of the surrounding tissue is a key aspect of cancer growth and spread involving a coordinated effort between cell migration and matrix degradation, and has been the subject of mathematical modelling for almost 30 years. In this current paper we address a long-standing question in the field of cancer cell migration modelling. Namely, identify the migratory pattern and spread of individual cancer cells, or small clusters of cancer cells, when the macroscopic evolution of the cancer cell colony is dictated by a specific partial differential equation (PDE). We show that the usual heuristic understanding of the diffusion and advection terms of the PDE being one-to-one responsible for the random and biased motion of the solitary cancer cells, respectively, is not precise. On the contrary, we show that the drift term of the correct stochastic differential equation scheme that dictates the individual cancer cell migration, should account also for the divergence of the diffusion of the PDE. We support our claims with a number of numerical experiments and computational simulations.Publisher PDFPeer reviewe
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A hybrid multiscale model for cancer invasion of the extracellular matrix
The ability to locally degrade the extracellular matrix (ECM) and interact with the tumour microenvironment is a key process distinguishing cancer from normal cells, and is a critical step in the metastatic spread of the tumour. The invasion of the surrounding tissue involves the coordinated action between cancer cells, the ECM, the matrix degrading enzymes, and the epithelialto-mesenchymal transition. This is a regulatory process through which epithelial cells (ECs) acquire mesenchymal characteristics and transform to mesenchymal-like cells (MCs). In this paper, we present a new mathematical model which describes the transition from a collective invasion strategy for the ECs to an individual invasion strategy for the MCs. We achieve this by formulating a coupled hybrid system consisting of partial and stochastic di erential equations that describe the evolution of the ECs and the MCs, respectively. This approach allows one to reproduce in a very natural way fundamental qualitative features of the current biomedical understanding of cancer invasion that are not easily captured by classical modelling approaches, for example, the invasion of the ECM by self-generated gradients and the appearance of EC invasion islands outside of the main body of the tumour