NEW DEVELOPMENTS IN AN ANATOMICAL FRAMEWORK FOR MODELING CARDIAC ISCHEMIA

2003 ◽  
Vol 13 (12) ◽  
pp. 3717-3722 ◽  
Author(s):  
NICOLAS SMITH ◽  
CAREY STEVENS ◽  
ANDREW PULLAN ◽  
PETER HUNTER ◽  
PETER MULQUINEY
Keyword(s):  
Sheet Material ◽  
Ionic Current ◽  
Reaction Diffusion ◽  
Wall Stress ◽  
Ventricular Myocardium ◽  
The Body ◽  
Reaction Diffusion Equations ◽  
Constitutive Law ◽  
Cardiac Contraction ◽  
Sheet Structure

A new, anatomically accurate, mathematical model of the right and left porcine ventricular myocardium is described based on measurements of the geometry and fibrous-sheet structure. Passive and active properties of the myocardium are calculated using an orthotropic constitutive law based on the fibrous-sheet structure and a biophysical cellular based model of cardiac contraction. Using Galerkin finite element techniques, the equations of finite deformation are solved to determine deformation and regional wall stress through the heart cycle. The mechanics model is coupled via myocardial wall stress, to a one-dimensional coronary blood flow model embedded in the myocardium. Bidomain electrical activation of the myocardium is also modeled, with ionic current based electrophysiological equations and reaction–diffusion equations based on orthotropic conductivity tensors referred to the fibrous-sheet material axes. Metabolic models are used to couple energy supply to contraction and excitation in the heart, and at the body surface, a framework for quantifying the effect of ischemic heart disease is developed.

Effects of Catastrophic Anemia in an Intra-Host Model of Malaria

2014 ◽  
Vol 24 (07) ◽  
pp. 1450105
Author(s):  
Eddy Takoutsing ◽  
Samuel Bowong ◽  
David Yemele ◽  
Jurgen Kurths
Keyword(s):  
Blood Cells ◽  
Malaria Parasite ◽  
Reaction Diffusion ◽  
The Body ◽  
Reaction Diffusion Equations ◽  
Reproduction Rate ◽  
Spatiotemporal Model ◽  
Cell Reproduction ◽  
Temporal Model ◽  
The Impact

In this paper, we develop a mathematical model to assess the strength of the effects of catastrophic anemia level on the dynamical transmission of malaria parasite within the body of a host. We first consider a temporal model. The important mathematical features of the model are thoroughly investigated. We found that the model exhibits forward bifurcation. We also consider a spatiotemporal model using reaction–diffusion equations. The model is numerically analyzed to assess the impact of anemia on the dynamical transmission of malaria parasite within the body of a host. Through numerical simulation, we found that malaria can lead to a catastrophic anemia level even if the parasite is nonpersistent within the body of a host. Numerical results also suggest that to reduce or control the anemia level, the strategy should be to accelerate innate cell reproduction rate or should have the ability to clean parasitized red blood cells (PRBCs) with a high mortality rate.

scholarly journals Mix & Match: Phenotypic coexistence as a key facilitator of solid tumour invasion

2019 ◽  
Author(s):  
Maximilian A. R. Strobl ◽  
Andrew L. Krause ◽  
Mehdi Damaghi ◽  
Robert Gillies ◽  
Alexander R. A. Anderson ◽  
...  
Keyword(s):  
Alternative Hypothesis ◽  
Cell Movement ◽  
Reaction Diffusion ◽  
Cell Types ◽  
The Body ◽  
Reaction Diffusion Equations ◽  
Species Competition ◽  
Invasion Process ◽  
Malignant Tumours ◽  
Reaction Diffusion Model

AbstractInvasion of healthy tissue is a defining feature of malignant tumours. Traditionally, invasion is thought to be driven by cells that have acquired all the necessary traits to overcome the range of biological and physical defences employed by the body. However, in light of the ever-increasing evidence for geno- and phenotypic intra-tumour heterogeneity an alternative hypothesis presents itself: Could invasion be driven by a collection of cells with distinct traits that together facilitate the invasion process? In this paper, we use a mathematical model to assess the feasibility of this hypothesis in the context of acid-mediated invasion. We assume tumour expansion is obstructed by stroma which inhibits growth, and extra-cellular matrix (ECM) which blocks cancer cell movement. Further, we assume that there are two types of cancer cells: i) a glycolytic phenotype which produces acid that kills stromal cells, and ii) a matrix-degrading phenotype that locally remodels the ECM. We extend the Gatenby-Gawlinski reaction-diffusion model to derive a system of five coupled reaction-diffusion equations to describe the resulting invasion process. We characterise the spatially homogeneous steady states and carry out a simulation study in one spatial dimension to determine how the tumour develops as we vary the strength of competition between the two phenotypes. We find that overall tumour growth is most extensive when both cell types can stably coexist, since this allows the cells to locally mix and benefit most from the combination of traits. In contrast, when inter-species competition exceeds intra-species competition the populations spatially separate and invasion arrests either: i) rapidly (matrix-degraders dominate), or ii) slowly (acid-producers dominate). Overall, our work demonstrates that the spatial and ecological relationship between a heterogeneous population of tumour cells is a key factor in determining their ability to cooperate. Specifically, we predict that tumours in which different phenotypes coexist stably are more invasive than tumours in which phenotypes are spatially separated.

On a Nonlinear System of Reaction-Diffusion Equations

2006 ◽  
Vol 11 (2) ◽  
pp. 115-121 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Weak Solution ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Elliptic System ◽  
Positive Parameter ◽  
Nonlinear Elliptic ◽  
Positive Weak Solution

The aim of this article is to study the existence of positive weak solution for a quasilinear reaction-diffusion system with Dirichlet boundary conditions,− div(|∇u1|p1−2∇u1) = λu1α11u2α12... unα1n,   x ∈ Ω,− div(|∇u2|p2−2∇u2) = λu1α21u2α22... unα2n,   x ∈ Ω, ... , − div(|∇un|pn−2∇un) = λu1αn1u2αn2... unαnn,   x ∈ Ω,ui = 0,   x ∈ ∂Ω,   i = 1, 2, ..., n,  where λ is a positive parameter, Ω is a bounded domain in RN (N > 1) with smooth boundary ∂Ω. In addition, we assume that 1 < pi < N for i = 1, 2, ..., n. For λ large by applying the method of sub-super solutions the existence of a large positive weak solution is established for the above nonlinear elliptic system.

scholarly journals Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

scholarly journals Integrodifference master equation describing actively growing blood vessels in angiogenesis

2020 ◽  
Vol 21 (7-8) ◽  
pp. 705-713 ◽  
Author(s):  
Luis L. Bonilla ◽  
Manuel Carretero ◽  
Filippo Terragni
Keyword(s):  
Master Equation ◽  
Blood Vessels ◽  
Transition Probabilities ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Birth Process ◽  
Sink Term ◽  
System Of Particles ◽  
Vessel Network ◽  
Tip Cells

AbstractWe study a system of particles in a two-dimensional geometry that move according to a reinforced random walk with transition probabilities dependent on the solutions of reaction-diffusion equations (RDEs) for the underlying fields. A birth process and a history-dependent killing process are also considered. This system models tumor-induced angiogenesis, the process of formation of blood vessels induced by a growth factor (GF) released by a tumor. Particles represent vessel tip cells, whose trajectories constitute the growing vessel network. New vessels appear and may fuse with existing ones during their evolution. Thus, the system is described by tracking the density of active tips, calculated as an ensemble average over many realizations of the stochastic process. Such density satisfies a novel discrete master equation with source and sink terms. The sink term is proportional to a space-dependent and suitably fitted killing coefficient. Results are illustrated studying two influential angiogenesis models.

Nonlinear solution of the reaction–diffusion equation using a two-step third–fourth-derivative block method

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Oluwaseun Adeyeye ◽  
Ali Aldalbahi ◽  
Jawad Raza ◽  
Zurni Omar ◽  
Mostafizur Rahaman ◽  
...  
Keyword(s):  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Numerical Approach ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Block Method ◽  
Wide Range ◽  
Higher Derivatives ◽  
Fourth Derivative ◽  
Improved Accuracy

AbstractThe processes of diffusion and reaction play essential roles in numerous system dynamics. Consequently, the solutions of reaction–diffusion equations have gained much attention because of not only their occurrence in many fields of science but also the existence of important properties and information in the solutions. However, despite the wide range of numerical methods explored for approximating solutions, the adoption of block methods is yet to be investigated. Hence, this article introduces a new two-step third–fourth-derivative block method as a numerical approach to solve the reaction–diffusion equation. In order to ensure improved accuracy, the method introduces the concept of nonlinearity in the solution of the linear model through the presence of higher derivatives. The method obtained accurate solutions for the model at varying values of the dimensionless diffusion parameter and saturation parameter. Furthermore, the solutions are also in good agreement with previous solutions by existing authors.

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