M16: Robust and efficient modelling, simulation and validation techniques for space fractional models

Background

The traditional way of modelling electrical conduction in the heart is to represent the tissue as a continuous medium. However, the extracellular space is a complex heterogeneous media comprised of ground substance, blood vessels, connective tissue cells, collagen and ‘empty’ space. There have been recent attempts to incorporate this heterogeneity using discrete models of cardiac tissue; however, these models are computationally expensive and are still not able to reproduce some experimental characteristics of cardiac tissue that may play an important role in the development of an irregular heartbeat.

Techniques and Challenges

We will investigate new mathematical models capable of capturing the effects of tissue inhomogeneity at a reasonable computational cost. Fractional diffusion models have been proven to incorporate anomalous diffusion in a natural way [1]. They can be interpreted as a natural description of non-local transport when viewed as an integral operator with a broad propagator. However, since the integral operator is no longer local, the numerical discretisation of these models gives rise to large and full systems of equations, which can become computationally demanding. Alternatively, the use of spectral methods, which are by nature non-local, could yield a more efficient representation of such integral operators.

Results

Our preliminary numerical results indicate an extraordinary performance of these methods for fractional-in-space reaction-diffusion models in terms of computational accuracy and execution time in two- and three-dimensional spaces. From the experimental viewpoint, our simulations are able to reproduce characteristics not previously captured by standard models of propagation of cardiac electrical activity [2].

The Future

The results of the project may have important implications for our current understanding of heart rhythm, such as the success or failure of cardiac impulse propagation, the transition from ventricular tachycardia to ventricular fibrillation, or the success of defibrillation shocks, especially under diseased conditions. Finally, the outcomes of our methodology could be of broader interest to other research areas, such as hydrology, oil detection and extraction, or pattern generation in biological systems.

References and Related Publications

[12/100] Bueno-Orovio A., Kay D., Grau V., Rodriguez B., Burrage K.: Nonlocal models of electrical propagation in cardiac tissue: electrotonic effects and the modulated dispersion of repolarization:

[12/50] Bueno-Orovio A., Kay D., Burrage K.: Fourier spectral methods for fractional-in-space reaction-diffusion equations (Submitted to Journal of Computational Physics)

[1] Burrage K., Hale N., Kay D.: An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations: SIAM J. Sci. Comput.

[2] Badie N., Bursac N.: Novel micropatterned cardiac cell cultures with realistic ventricular microstructure: Biophys J 96, 3873-3885, 2009