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Special Issue on Mathematical Models and Methods in Biology, Medicine and Physiology. Guest Editors: Michele Piana, Luigi Preziosi

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"Special Issue on New Trends in Semi-Lagrangian Methods, Guest Editors: Luca Bonaventura, Maurizio Falcone and Roberto Ferretti

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Journal Details
Format
Journal
eISSN
2038-0909
First Published
15 Dec 2014
Publication timeframe
1 time per year
Languages
English

Search

Volume 11 (2020): Issue 1 (January 2020)

Journal Details
Format
Journal
eISSN
2038-0909
First Published
15 Dec 2014
Publication timeframe
1 time per year
Languages
English

Search

6 Articles
access type Open Access

Phase transitions of biological phenotypes by means of a prototypical PDE model

Published Online: 01 Feb 2020
Page range: 1 - 17

Abstract

Abstract

The basic investigation is the existence and the (numerical) observability of propagating fronts in the framework of the so-called Epithelial-to-Mesenchymal Transition and its reverse Mesenchymal-to-Epithelial Transition, which are known to play a crucial role in tumor development. To this aim, we propose a simplified one-dimensional hyperbolic-parabolic PDE model composed of two equations, one for the representative of the epithelial phenotype, and the second describing the mesenchymal phenotype. The system involves two positive constants, the relaxation time and a measure of invasiveness, moreover an essential feature is the presence of a nonlinear reaction function, typically assumed to be S-shaped. An identity characterizing the speed of propagation of the fronts is proven, together with numerical evidence of the existence of traveling waves. The latter is obtained by discretizing the system by means of an implicit-explicit finite difference scheme, then the algorithm is validated by checking the capability of the so-called LeVeque–Yee formula to reproduce the value of the speed furnished by the above cited identity. Once such justification has been achieved, we concentrate on numerical experiments relative to Riemann initial data connecting two stable stationary states of the underlying ODE model. In particular, we detect an explicit transition threshold separating regression regimes from invasive ones, which depends on critical values of the invasiveness parameter. Finally, we perform an extensive sensitivity analysis with respect to the system parameters, exhibiting a subtle dependence for those close to the threshold values, and we postulate some conjectures on the propagating fronts.

Keywords

  • phase transitions
  • reaction-diffusion systems
  • propagating fronts
  • finite difference schemes
  • wave speed approximations
access type Open Access

Subsampled Nonmonotone Spectral Gradient Methods

Published Online: 01 Feb 2020
Page range: 19 - 34

Abstract

Abstract

This paper deals with subsampled spectral gradient methods for minimizing finite sums. Subsample function and gradient approximations are employed in order to reduce the overall computational cost of the classical spectral gradient methods. The global convergence is enforced by a nonmonotone line search procedure. Global convergence is proved provided that functions and gradients are approximated with increasing accuracy. R-linear convergence and worst-case iteration complexity is investigated in case of strongly convex objective function. Numerical results on well known binary classification problems are given to show the effectiveness of this framework and analyze the effect of different spectral coefficient approximations arising from the variable sample nature of this procedure.

Keywords

  • spectral gradient methods
  • subsampling strategies
  • global convergence
  • nonmonotone line search
access type Open Access

Curvature Dependent Electrostatic Field in the Deformable MEMS Device: Stability and Optimal Control

Published Online: 31 Oct 2020
Page range: 35 - 54

Abstract

Abstract

The recovery of the membrane profile of an electrostatic micro-electro-mechanical system (MEMS) device is an important issue because, when applying an external voltage, the membrane deforms with the consequent risk of touching the upper plate of the device (a condition that should be avoided). Then, during the deformation of the membrane, it is useful to know if this movement admits stable equilibrium configurations. In such a context, our present work analyze the behavior of an electrostatic 1D membrane MEMS device when an external electric voltage is applied. In particular, starting from a well-known second-order elliptical semi-linear di erential model, obtained considering the electrostatic field inside the device proportional to the curvature of the membrane, the only possible equilibrium position is obtained, and its stability is analyzed. Moreover, considering that the membrane has an inertia in moving and taking into account that it must not touch the upper plate of the device, the range of possible values of the applied external voltage is obtained, which accounted for these two particular operating conditions. Finally, some calculations about the variation of potential energy have identified optimal control conditions.

Keywords

  • MEMS DEVICE
  • ELECTROSTATIC ACTUATORS
  • BOUNDARY SEMI-LINEAR ELLIPTC MODELS
  • CURVATURE
access type Open Access

A new two-component approach in modeling red blood cells

Published Online: 31 Oct 2020
Page range: 55 - 71

Abstract

Abstract

This work consists in the presentation of a computational modelling approach to study normal and pathological behavior of red blood cells in slow transient processes that can not be accompanied by pure particle methods (which require very small time steps). The basic model, inspired by the best models currently available, considers the cytoskeleton as a discrete non-linear elastic structure. The novelty of the proposed work is to couple this skeleton with continuum models instead of the more common discrete models (molecular dynamics, particle methods) of the lipid bilayer. The interaction of the solid cytoskeleton with the bilayer, which is a two-dimensional fluid, will be done through adhesion forces adapting e cient solid-solid adhesion algorithms. The continuous treatment of the fluid parts is well justified by scale arguments and leads to much more stable and precise numerical problems when, as is the case, the size of the molecules (0.3 nm) is much smaller than the overall size (≃ 8000 nm). In this paper we display some numerical simulations that show how our approach can describe the interaction of an RBC with an exogenous body as well as the relaxation of the shape of an RBC toward its equilibrium configuration in absence of external forces.

Keywords

  • Red blood cell
  • Mathematical modeling
  • Biological fluid mechanics
  • Lipid membrane
  • cytoskeleton
  • Fluid-solid interactions
  • Cell biology
access type Open Access

Topological graph persistence

Published Online: 06 Dec 2020
Page range: 72 - 87

Abstract

Abstract

Graphs are a basic tool in modern data representation. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological constructions can be used to gain information otherwise concealed by the low-dimensional nature of graphs. We do this by extending previous work in homological persistence, and proposing novel graph-theoretical constructions. Beyond cliques, we use independent sets, neighborhoods, enclaveless sets and a Ramsey-inspired extended persistence.

Keywords

  • Clique
  • independent set
  • neighborhood
  • enclaveless set
  • Ramsey
6 Articles
access type Open Access

Phase transitions of biological phenotypes by means of a prototypical PDE model

Published Online: 01 Feb 2020
Page range: 1 - 17

Abstract

Abstract

The basic investigation is the existence and the (numerical) observability of propagating fronts in the framework of the so-called Epithelial-to-Mesenchymal Transition and its reverse Mesenchymal-to-Epithelial Transition, which are known to play a crucial role in tumor development. To this aim, we propose a simplified one-dimensional hyperbolic-parabolic PDE model composed of two equations, one for the representative of the epithelial phenotype, and the second describing the mesenchymal phenotype. The system involves two positive constants, the relaxation time and a measure of invasiveness, moreover an essential feature is the presence of a nonlinear reaction function, typically assumed to be S-shaped. An identity characterizing the speed of propagation of the fronts is proven, together with numerical evidence of the existence of traveling waves. The latter is obtained by discretizing the system by means of an implicit-explicit finite difference scheme, then the algorithm is validated by checking the capability of the so-called LeVeque–Yee formula to reproduce the value of the speed furnished by the above cited identity. Once such justification has been achieved, we concentrate on numerical experiments relative to Riemann initial data connecting two stable stationary states of the underlying ODE model. In particular, we detect an explicit transition threshold separating regression regimes from invasive ones, which depends on critical values of the invasiveness parameter. Finally, we perform an extensive sensitivity analysis with respect to the system parameters, exhibiting a subtle dependence for those close to the threshold values, and we postulate some conjectures on the propagating fronts.

Keywords

  • phase transitions
  • reaction-diffusion systems
  • propagating fronts
  • finite difference schemes
  • wave speed approximations
access type Open Access

Subsampled Nonmonotone Spectral Gradient Methods

Published Online: 01 Feb 2020
Page range: 19 - 34

Abstract

Abstract

This paper deals with subsampled spectral gradient methods for minimizing finite sums. Subsample function and gradient approximations are employed in order to reduce the overall computational cost of the classical spectral gradient methods. The global convergence is enforced by a nonmonotone line search procedure. Global convergence is proved provided that functions and gradients are approximated with increasing accuracy. R-linear convergence and worst-case iteration complexity is investigated in case of strongly convex objective function. Numerical results on well known binary classification problems are given to show the effectiveness of this framework and analyze the effect of different spectral coefficient approximations arising from the variable sample nature of this procedure.

Keywords

  • spectral gradient methods
  • subsampling strategies
  • global convergence
  • nonmonotone line search
access type Open Access

Curvature Dependent Electrostatic Field in the Deformable MEMS Device: Stability and Optimal Control

Published Online: 31 Oct 2020
Page range: 35 - 54

Abstract

Abstract

The recovery of the membrane profile of an electrostatic micro-electro-mechanical system (MEMS) device is an important issue because, when applying an external voltage, the membrane deforms with the consequent risk of touching the upper plate of the device (a condition that should be avoided). Then, during the deformation of the membrane, it is useful to know if this movement admits stable equilibrium configurations. In such a context, our present work analyze the behavior of an electrostatic 1D membrane MEMS device when an external electric voltage is applied. In particular, starting from a well-known second-order elliptical semi-linear di erential model, obtained considering the electrostatic field inside the device proportional to the curvature of the membrane, the only possible equilibrium position is obtained, and its stability is analyzed. Moreover, considering that the membrane has an inertia in moving and taking into account that it must not touch the upper plate of the device, the range of possible values of the applied external voltage is obtained, which accounted for these two particular operating conditions. Finally, some calculations about the variation of potential energy have identified optimal control conditions.

Keywords

  • MEMS DEVICE
  • ELECTROSTATIC ACTUATORS
  • BOUNDARY SEMI-LINEAR ELLIPTC MODELS
  • CURVATURE
access type Open Access

A new two-component approach in modeling red blood cells

Published Online: 31 Oct 2020
Page range: 55 - 71

Abstract

Abstract

This work consists in the presentation of a computational modelling approach to study normal and pathological behavior of red blood cells in slow transient processes that can not be accompanied by pure particle methods (which require very small time steps). The basic model, inspired by the best models currently available, considers the cytoskeleton as a discrete non-linear elastic structure. The novelty of the proposed work is to couple this skeleton with continuum models instead of the more common discrete models (molecular dynamics, particle methods) of the lipid bilayer. The interaction of the solid cytoskeleton with the bilayer, which is a two-dimensional fluid, will be done through adhesion forces adapting e cient solid-solid adhesion algorithms. The continuous treatment of the fluid parts is well justified by scale arguments and leads to much more stable and precise numerical problems when, as is the case, the size of the molecules (0.3 nm) is much smaller than the overall size (≃ 8000 nm). In this paper we display some numerical simulations that show how our approach can describe the interaction of an RBC with an exogenous body as well as the relaxation of the shape of an RBC toward its equilibrium configuration in absence of external forces.

Keywords

  • Red blood cell
  • Mathematical modeling
  • Biological fluid mechanics
  • Lipid membrane
  • cytoskeleton
  • Fluid-solid interactions
  • Cell biology
access type Open Access

Topological graph persistence

Published Online: 06 Dec 2020
Page range: 72 - 87

Abstract

Abstract

Graphs are a basic tool in modern data representation. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological constructions can be used to gain information otherwise concealed by the low-dimensional nature of graphs. We do this by extending previous work in homological persistence, and proposing novel graph-theoretical constructions. Beyond cliques, we use independent sets, neighborhoods, enclaveless sets and a Ramsey-inspired extended persistence.

Keywords

  • Clique
  • independent set
  • neighborhood
  • enclaveless set
  • Ramsey

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