Effective mathematical modelling of fractional-diffusion in cardiac electrophysiology

The Hodgkin-Huxley model for the generation of the nerve action potential is one of the most successful mathematical models of a complex biological process that has ever been formulated. Alan Hodgkin and Andrew Huxley described the model in 1952 to explain the ionic mechanisms underlying the initiation and propagation of action potentials in the squid giant axon. The typical Hodgkin–Huxley model treats each component of an excitable cell as an electrical element resulting in a set of nonlinear differential equations that approximates the electrical characteristics of excitable cells such as neurons and cardiac myocytes. It is a continuous-time dynamical system. Hodgkin and Huxley received the 1963 Nobel Prize in Physiology or Medicine for this work.

The starting point of the original Hodgkin-Huxley model is that the nerve membrane of the squid giant axon contains three different types of ion channels. Cardiac electrophysiological modelling dates back to classical works based on this Hodgkin-Huxley formalism looking for balance-laws for ion transport across the cell membrane of cardiac excitable cells. Due to the extreme complexity of cardiac tissue and its intrinsic nonlinear dynamics, mathematical and computational modelling played (and continue to play) a crucial role in unveiling multiscale emerging phenomena and explaining both regular, and irregular behaviour, up to the description of life-threatening arrhythmias and fibrillation patterns.

Open questions remain: the early prediction of fibrillation onset based on microscale tissue features is an example. One general approach is to go down to the finer details: accounting at the subcellular level for the effect of localized extracellular electrical fields. Another one is generalizing the concept of diffusivity at the macroscopic scale through the introduction of nonlinear porous-medium-based operators. A third way is using non-local fractional calculus-based operators.

Fractional calculus, generalizing classical differentiation concepts and account-ing for long-range interactions and system memory effects, has been considered asa possible modelling approach for various phenomena in a variety of fields. Space-fractional differential operators allow accounting for the multi-scale nature of the observed phenomenon, without the need of a detailed resolution of the spatial scale all the way down to the microscopic level.

Bueno-Orovio et al. ¹ have shown that a fractional model can be interpreted as a smooth transition from the case of a perfectly homogeneous medium to a domain with increasing degree of heterogeneity as the order of the fractional operator decreases. In addition to offering a valid biophysical justification for the use of space-fractional operators in the study of cardiac tissue electrophysiology dynamics, their work provides promising results: by comparing numerical tests with and without space-fractional diffusion against a set of selected experimental data they have shown that the non-local model can often capture better than the local counterpart important features of the observed data. Nevertheless, the proposed approach relied on some simplifying assumptions and did not investigate (or simply did not highlight) some important features of the spatio-temporal dynamics obtained as a result of the underlying non-locality assumption.

Now, a new study ² addresses some key features of the non-local diffusion operator focusing on the intrinsic spatio-temporal coupling of such a complex excitable media as cardiac tissue.

The researchers provide a series of classical tests for the experimental-based model tuning, showing that the introduction of non-local operators definitely affects the overall predictive power of the model in terms of purely spatial, temporal, and spatio-temporal features. An extensive computational study in terms of genetic algorithm parameter tuning constraining the model dynamics to reproduce the correct physiological conduction velocity is also produced.

The scientists stress the importance of considering boundary conditions as a critical unsolved problem within the last 60 years of literature in computational cardiology. In this context, they perform a preliminary numerical investigation on the effect of boundary conditions in one-dimensional domains on two determining factors in the study of arrhythmias and alternans dynamics, namely, on spatial dispersion of action potential duration and on model electrotonic couplings. They find that the imposition of simplified boundary conditions in the non-local settings hinders the reliability and the predictive power of the models themselves.

Even though care ought to be taken in the selection of the ionic description, that is likely to require an ad hoc reparametrisation in the fractional setting, the researchers conclude that a properly tuned phenomenological model might become an extremely powerful tool, offering several computational advantages because of its simplified description.

Author: César Tomé López is a science writer and the editor of Mapping Ignorance

Disclaimer: Parts of this article may be copied verbatim or almost verbatim from the referenced research paper.