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Media and Motion

Authors: Amal Alphonse, Simon Bignold, Yuchen Pei
Supervisors: Dwight Barkley, Andreas Dedner

Contents

Below is an overview of the material that can be found on this website.

We give our motivation and aims in the Media and Motion topic and present the model that we use throughout our project.

A detail of the numerical methods that are needed to run the simulation is given in this page. We use the finite element method for the spatial discretisation and a semi-implicit method for the time discretisation.

The first task is to consider the simulation of the spiral wave on a 2D square. Both the finite element method and the finite difference method are used so that the results obtained using both methods can be compared.

We model spiral waves on a unit sphere next. One can consider the sphere to be a part of the surface of the heart.

Wave patterns in the heart can be disturbed by areas of reduced conductivity. This may be because of the presence of holes caused either by the entry of vessels such as veins into the heart, or because of the presence of damaged cardiovascular tissue. The first case is simulated by filtering out elements of the sphere to create a sphere with a hole. In the second case the diffusion coefficient is reduced which is equivalent to reducing the conductivity of the material. We consider several ways of doing this.

To incorporate the pulsation and movement of the heart into our experiments, we simulate the Barkley model on a moving surface. We specifically look at a unit sphere which oscillates to become an ellipsoid (and back) with varying magnitudes of oscillation.

We slighltly alter the Barkley equation and its initial conditions to show that we can produce a pulse on an oscillating sphere.

There is still a lot of work that we would like to have completed had we had more time. Here we detail some possibilities for further work, for example we consider stability estimates for more complicated surfaces including more realistic models. We also consider more complicated models for which there is evidence in the literature that they might model observed behaviour.

  • Other Media

For a more detailed analysis of our work, see the (PDF Document)report we compiled. A brief synopsis is found in the(PDF Document) poster we made.

Introduction

Motivation

Our motivation stems from understand the mechanisms of the heart. During normal heart activity, a pulse of the action potential travels through the heart which causes cells in the heart to contract and expand. This is what pushes blood around the body. However, if there are inhomogeneities in the heart (such as damaged tissues or holes), the heart rhythm is disrupted: this is called an arrhythmia. These arrhythmias can be fatal, so it is important to study and understand them and the regimes in which they operate. Specifically, we concentrate on the case where spiral waves are produced in the heart as a result of the deformities.

The Barkley Model

To simulate the formation of spiral waves one starts with a system of coupled partial differential equations (PDEs) that lie on a surface \(\Gamma\)

\begin{align*} \dot {u} +u\nabla_\Gamma\cdot\mathbf v -a \Delta_\Gamma u &=f(u,v) \textrm{ in } \mathcal G_T:=\bigcup_{t\in[0,T]}\{t\}\times \Gamma_t \\ \dot {v}+v\nabla_\Gamma\cdot\mathbf v &=g(u,v) \textrm{ in } \mathcal G_T \end{align*}

where

\begin{align*} f(u,v) &= {1 \over \epsilon } u \left (1-u \right) \left( u - {{v+b } \over c} \right) \\ g(u,v) &= u-v \end{align*}

with initial conditions
\begin{align*} u \left( 0, \cdot \right) & = u_0 \left( \cdot \right) \\ v \left( 0 , \cdot \right) &= v_0 \left( \cdot \right) \end{align*}

and model parameters \(a, b, c \textrm{ and } \epsilon\). Here, \( \mathbf{ v} \) is the velocity of the surface which comes into play when \(\Gamma\) is a moving surface (otherwise we take it to be identically zero). When the surfaces on which we implement this model have no boundary (like spheres), obviously boundary conditions are not required; when they do have a boundary (for example, spheres with holes), we use zero Neumann boundary conditions

\begin{equation*}\nabla_\Gamma u(x) \cdot \nu (x) = 0 \qquad \forall x \in \partial\Gamma,\end{equation*}

where \( \nu(x) \) is the normal to the surface. This particular model is know as the Barkley model. We use this model because it has been shown to produce spiral waves and is relatively simple and numerically efficient.

Acknowledgements

We acknowledge and thank the help of our supervisors.
We would also like to acknowledge the funding body EPSRC and the support from MASDOC CDT.