Nikolaos Sfakianakis

ENTROPY CONSERVATIVE SCHEMES AND ADAPTIVE MESH SELECTION FOR HYPERBOLIC CONSERVATION LAWS

We consider numerical schemes which combine non-uniform, adaptively redefined spatial meshes with entropy conservative schemes for the evolution step for shock computations. We observe that the resulting adaptive schemes yield approximations free of oscillations in contrast to known fully discrete entropy conservative schemes on uniform meshes. We conclude that entropy conservative schemes are transformed to entropy diminishing schemes when combined with the proposed geometrically driven mesh adaptivity.

A Multiscale Approach to the Migration of Cancer Stem Cells: Mathematical Modelling and Simulations

We propose a multiscale model for the invasion of the extracellular matrix by two types of cancer cells, the differentiated cancer cells and the cancer stem cells. We investigate the epithelial mesenchymal-like transition between them being driven primarily by the epidermal growth factors. We moreover take into account the transdifferentiation program of the cancer stem cells towards the cancer-associated fibroblast cells as well as the fibroblast-driven remodelling of the extracellular matrix. The proposed haptotaxis model combines the macroscopic phenomenon of the invasion of the extracellular matrix by both types of cancer cells with the microscopic dynamics of the epidermal growth factors. We analyse our model in a component-wise manner and compare our findings with the literature. We investigate pathological situations regarding the epidermal growth factors and accordingly propose “mathematical-treatment” scenarios to control the aggressiveness of the tumour.

A study on time discretization and adaptive mesh refinement methods for the simulation of cancer invasion

In the present work we investigate a model that describes the chemotactically and proteolytically driven tissue invasion by cancer cells. The model is a system of advection-reaction-diffusion equations that takes into account the role of the serine protease urokinase-type plasminogen activator. The analytical and numerical study of such a system constitutes a challenge due to the merging, emerging, and traveling 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 show 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, L∞ bounds for the solutions. We also studied 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.

Stable Length Distributions in Colocalized Polymerizing and Depolymerizing Protein Filaments

A model for the dynamics of the length distribution in colocalized groups of polar polymer filaments is presented. It considers nucleation, polymerization at plus-ends, and depolymerization at minus-ends and is derived as a continuous macroscopic limit from a discrete description. Its main feature is a nonlinear coupling due to competition of the depolymerizing ends for the limited supply of a depolymerization agent. The model takes the form of an initial-boundary value problem for a one-dimensional nonlinear hyperbolic conservation law, subject to a nonlinear, nonlocal boundary condition. Besides existence and uniqueness of entropy solutions, convergence to a steady state is proven. Technical difficulties are caused by the fact that the prescribed boundary data are not always assumed by entropy solutions.

Adaptive mesh reconstruction for hyperbolic conservation laws with total variation bound

We consider 3-point numerical schemes, that resolve scalar conservation laws, that are oscillatory either to their dispersive or anti-diffusive nature. The spatial discretization is performed over non-uniform adaptively redefined meshes. We provide a model for studying the evolution of the extremes of the oscillations. 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 behavior of the numerical schemes, decreases with time; hence proving that the overall scheme is TV Increase-Decreasing (TVI-D). We finally provide numerical evidence supporting the analytical results that exhibit the stabilization properties of the mesh adaptation technique. 1. Outline Mesh adaptation techniques have been employed by several authors in the past. It is worth noting the seminal works [DD87, For88, HH83, Luc85, Luc86, TT03], where several properties of mesh adaptation were studied. It was noticed in [AKM01, AMT04, AMS10, Sfa09] that proper use of non-uniform adaptively redefined meshes is capable of taming oscillations; hence improving the stability properties of the numerical schemes. To study the stabilization properties of mesh adaptation techniques we analyze the effect they have, on the oscillations that oscillatory/unstable numerical schemes produce. The setting is the one-dimensional scalar Riemann problem: ut+f(u)x = 0, x ∈ [a, b], with the flux function f being smooth and convex. For initial conditions we consider the single jump u0(x) = X[0,x0](x) with x, x0 ∈ (0, 1). We discretize spatially over a non-uniform adaptively redefined mesh. The mesh adaptation and the time evolution of the numerical solution are combined into the Main Adaptive Scheme: Definition 1.1 (Main Adaptive Scheme (MAS)). Given, at time step n, the mesh M x = {a = x1 < · · · < xN = b} and the approximations U = {u1 , . . . , uN}, the steps of the (MAS) are: 1. (Mesh Reconstruction). Construct new mesh: M x = {a = x 1 < · · · < x N = b}. 2. (Solution Update). Use the old mesh M x the approximations U n and the new mesh M x : Received by the editor September 19, 2009 and in revised form, September 2, 2011. 2010 Mathematics Subject Classification. Primary 65–XX. c ©2012 American Mathematical Society Reverts to public domain 28 years from publication

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, and as opposed to themore usual differentiated cancer cells, they exhibit highermotility, 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 cancer stem cells is not completely clear; they seem to stem from the differentiated cancer cells via a transition process related to the epithelial-mesenchymal transition 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.

Entropy dissipation of moving mesh adaptation

Non-uniform grids and mesh adaptation have become an important part of numerical approximations of differential equations over the past decades. It has been experimentally noted that mesh adaptation leads not only to locally improved solution but also to numerical stability of the underlying method. In this paper we consider nonlinear conservation laws and provide a method to perform the analysis of the moving mesh adaptation method, including both the mesh reconstruction and evolution of the solution. We moreover employ this method to extract sufficient conditions — on the adaptation of the mesh — that stabilize a numerical scheme in the sense of the entropy dissipation.

Numerical Treatment of the Filament-Based Lamellipodium Model (FBLM)

We describe in this work the numerical treatment of the Filament-Based Lamellipodium Model (FBLM). This model is a two-phase two-dimensional continuum model, describing the dynamics of two interacting families of locally parallel F-actin filaments. It includes, among others, the bending stiffness of the filaments, adhesion to the substrate, and the cross-links connecting the two families. The numerical method proposed is a Finite Element Method (FEM) developed specifically for the needs of this problem. It is comprised of composite Lagrange–Hermite two-dimensional elements defined over a two-dimensional space. We present some elements of the FEM and emphasize in the numerical treatment of the more complex terms. We also present novel numerical simulations and compare to in-vitro experiments of moving cells.

Energy intake functions and energy budgets of ectotherms and endotherms derived from their ontogenetic growth in body mass and timing of sexual maturation

Ectothermic and endothermic vertebrates differ not only in their source of body temperature (environment vs. metabolism), but also in growth patterns, in timing of sexual maturation within life, and energy intake functions. Here, we present a mathematical model applicable to ectothermic and endothermic vertebrates. It is designed to test whether differences in the timing of sexual maturation within an animals life (age at which sexual maturity is reached vs. longevity) together with its ontogenetic gain in body mass (growth curve) can predict the energy intake throughout the animals life (food intake curve) and can explain differences in energy partitioning (between growth, reproduction, heat production and maintenance, with the latter subsuming any other additional task requiring energy) between ectothermic and endothermic vertebrates. With our model we calculated from the growth curves and ages at which species reached sexual maturity energy intake functions and energy partitioning for five ectothermic and seven endothermic vertebrate species. We show that our model produces energy intake patterns and distributions as observed in ectothermic and endothermic species. Our results comply consistently with some empirical studies that in endothermic species, like birds and mammals, energy is used for heat production instead of growth, and with a hypothesis on the evolution of endothermy in amniotes published by us before. Our model offers an explanation on known differences in absolute energy intake between ectothermic fish and reptiles and endothermic birds and mammals. From a mathematical perspective, the model comes in two equivalent formulations, a differential and an integral one. It is derived from a discrete level approach, and it is shown to be well-posed and to attain a unique solution for (almost) every parameter set. Numerically, the integral formulation of the model is considered as an inverse problem with unknown parameters that are estimated using a series of empirical data.

Numerical Simulation of a Contractivity Based Multiscale Cancer Invasion Model

We present a problem-suited numerical method for a particularly challenging cancer invasion model. This model is a multiscale haptotaxis advection-reaction-diffusion system that describes the macroscopic dynamics of two types of cancer cells coupled with microscopic dynamics of the cells adhesion on the extracellular matrix. The difficulties to overcome arise from the non-constant advection and diffusion coefficients, a time delay term, as well as stiff reaction terms.