# Further Work

## Stability of more complicated objects

The following table summarises the results we have achieved and highlights areas for further work. First, we propose investigating the stability of our spiral waves on a more general moving deformed sphere. Then we propose looking at numerical stability on static surfaces that more realistically resemble the heart. Finally, we suggest that the numerical stability of these objects should be investigated after they have been given an oscillatory motion. This is obviously important work as we will then be able to assess whether this method is viable on a more realistic model of the heart and if so for what parameters.

\( \Gamma \) | \(a\) | Inhomogeneity | Method | Result |

\( \Omega_{150} \) | 1 | None | FD | Stable |

\( \Omega_{3.5} \) | 1 /1790.49 | None | FD | Stable |

\( \Omega_{3.5} \) | 1 /179.049 | None | FD | Stable |

\( \Omega_{3.5} \) | 1 /1790.49 | None | FE | Unstable |

\( \Omega_{3.5} \) | 1 /179.049 | None | FE | Stable |

Unit sphere | 1 /179.049 | None | FE | Stable |

Unit sphere | 1 /1790.49 | None | FE | Unstable |

Unit sphere | 2/1790.49 | None | FE | Unstable |

Unit sphere | 1/500 | None | FE | Unstable |

Unit sphere | 1/350 | None | FE | Unstable |

Unit sphere | 1/250 | None | FE | Unstable |

Ellipsoid | 1/179.049 | None | FE | Stable |

Unit sphere | 1 /179.049 | Zero conductivity | FE | Unstable |

Unit sphere | 1 /179.049 | Reduced conductivity 1 | FE | Stable |

Unit sphere | 1 /179.049 | Reduced conductivity 2 | FE | Unstable |

Unit sphere | 1 /179.049 | Continuous conductivity | FE | Stable |

Unit sphere | 1 /179.049 | Physical hole | FE | Stable |

Moving surface (0.1,2) | 1 /179.049 | None | FE | Unstable |

Moving surface (0.01,2) | 1 /179.049 | None | FE | Unstable |

Moving surface (0.1,0.2) | 1 /179.049 | None | FE | Stable |

Moving surface (0.1,0.4) | 1 /179.049 | None | FE | Unstable |

Moving surface (0.2,0.2) | 1 /179.049 | None | FE | Stable |

Moving surface (0.3,0.2) | 1 /179.049 | None | FE | Stable |

Moving surface (0.4,0.2) | 1 /179.049 | None | FE | Stable |

Moving surface (0.5,0.2) | 1 /179.049 | None | FE | Stable |

Moving surface | 1/179.049 | Any Type | FE | TODO |

Fixed heart geometry | Any | Physical holes and diseased tissues | FE | TODO |

Pulsating heart geometry | Any | Physical holes and diseased tissues | FE | TODO |

Where "Moving surface (a,b) " is the moving surface with deformation \( \alpha= \) a and period given by \( \beta= {{\textrm{b}} \over 2} \). "Reduced conductivity 1" means a hole centered at (0.15,0) in the upper hemi-sphere with radius 0.1 where the conductivity factor is reduced by a factor of one hundred and "Reduced conductivity 1" means a hole centered at (0.35,0) in the upper hemi-sphere with radius 0.3 where the conductivity factor is reduced by a factor of one hundred.

## More physically accurate models

In the heart spiral waves often break up to form turbulent waves. We hope a simple modification to our equations may simulate this. The equations used are exactly the same as those used in the Barkley model except that we alter \(g\) the function that defines the rate of change of \( v \) so that it is now

\begin{equation*} g(u,v)=u^3 -v. \end{equation*}

This is still linear in \( v\) and thus the numerical implementation of the solution is not much more difficult. We believe that this equation is inherently less stable then our previous equation and thus could lead to spiral break-up which is a more realistic model of arrhythmia in certain regimes.

We demonstrate the principle by choosing the parameters

\begin{equation*} a={1 \over { 179.049}} \: \: \; \epsilon =0.06 \: \: \; b=0.06 \: \: \; c=0.75 \end{equation*}

and initial conditions

\begin{align*} u_0 (x,y) &= \mathbb I _{y>0.3} \\ v_0 ( x,y ) &= {3 \over 8} \mathbb I _{x<0 } \end{align*}

that lead to the formation of a stable spiral.

We obtain the following simulation.

We can see that the wave forms in a different way and it can be seen that it takes longer to form. However we have demonstrated the formation of spiral wave -- but there is none of the expected turbulence. We hope that turbulence may be achieved if the surface is made to oscillate, and implementing this model on an oscillating surface is another avenue for further work.

## Other work

### Stability

From the table we can see that one direction of possible further research is to fix the instability problems in various simulations; for example the implementation of thinner spirals, i.e., larger diffusion coefficient \(a\).

### Period of oscillation

Another example is to simulate a faster oscillating rate of the moving surface because the current rate defined by \(\beta =0.1 \) is a bit too low.

### Parameter testing

For our simulations to truly model the heart accurately, we need to set the parameters of the Barkley model correctly so that they reflect the properties of the heart. This would require the study of biological data and collaboration with experts in the field.