Using Evolution Against Itself
Initiation of Re-entry in an Excitable Medium: A Structural Investigation of Cardiac Tissue using a Genetic Algorithm by
S. Scarle & R. H. Clayton, Department of Computer Science, University of Sheffield, Regent Court, 211 Portobello Street, Sheffield, S1 4DP, UK.
Introduction
In the heart, propagating waves of electrical excitation initiate contraction of the cardiac muscle. Their abnormal propagation can lead to cardiac arrhythmias; with the most dangerous of these being ventricular tachycardia (VT) and ventricular fibrillation (VF). These arrhythmias can result in sudden cardiac death, which is the largest categorical cause of death in the industrialized world.
After injury to the heart both the chances of arrhythmias and the percentage of fibrotic tissue increase. Critically, there appears to be a relationship between the type of injury and the structural arrangement of this tissue. Whereas after infarction it increases locally and appears in the form of long fibrous strands, during ageing fibrotic tissue is present in a more diffuse form, forming small patches. Further, different patterns of fibrosis effect Action Potential (AP) propagation differently: For example, a patchy structure can result in a large conduction delay, whilst the same amount of fibrosis but with a more diffuse structure may barely affect conduction. We therefore used an excitable medium model of AP propagation to carry out a systematic investigation of the arrhythmgenic role played by the structural layout of fibrotic tissue. Fibrotic tissue was modelled as regions of non-excitable but conductive tissue.
The more general problem of diffuse heterogeneity in excitable media is of great interest and has been widely studied, and of particular importance in this field is the problem of heterogeneities in the electrical properties of cardiac tissue. In general previous modelling studies have imposed a random arrangement of normal and abnormal regions. In this study we took a different approach, using a genetic algorithm (GA) to evolve arrhythmgenic structures. This GA essentially applied evolution against itself, as its fitness value selected for more arrhythmgenic simulated tissue structures. Structures which in real life would be far less fit. The overall aim of this study was to investigate patterns of normal and diseased tissue that favour the development of re-entry, and how they can be discovered and classified.
GA
GAs are biologically inspired optimization algorithms. They borrow heavily on the concepts and nomenclature of natural selection, evolution and inheritance. In a GA, a population of possible solutions to a problem are encoded as a set of chromosomes each made of genes. These solutions are then measured against some fitness metric as to how well they solve the problem.
Using this metric as a guide the individuals of the population are bred together to produce a new population. The aim being to use various selection criteria to ensure that on average more of the genes of a fitter individual than a weaker one are present in the next and subsequent generations. Additionally, so that the solution space is fully explored, mutation operators are applied during the breeding process to randomize some of the new population's genes.
In this way a GA implements a model version of the evolutionary process, and as such although it cannot be assured that it will find the/an optimal solution for a given problem, it can usually be assured to improve on an arbitrary starting population.
In our case our population described the tissue type of each site of a set of simulated tissue samples. Each simulated tissue sample was then presented with a regular train of stimuli and the simulation was then continued until all sites returned to their rest-state, the amount of simulated time taken to do this was taken as the basis of our fitness measure.
The computational model used for the cardiac tissue was the FitzHugh-Nagumo (FHN) model. The FHN model is a simplification of the famous Hodgkin-Huxley model. It describes qualitatively the response of a nerve membrane to stimuli. Importantly, it exhibits the main behavioural features of real neurons and cardiac cells. (i.e. a recovery mechanism, enhanced and depressed excitability, and refractory states depending on time since stimulation.)
Geometries
We ran the GA with each of the following structural layouts of mutable and non-mutable sites. By mutable, we mean capable of being mutated and by non-mutable we mean fixed to being excitable.
- NORMAL Entire sheet mutable and all boundary conditions zero-flux. This geometry was chosen to simulate cardiac tissue with diffuse fibrosis throughout.
- SIDE 5 site non-mutable strip down the two edges parallel to impulse train and zero-flux boundary conditions on these edges. The other two edges were linked by periodic boundary conditions. This geometry was intended to be topologically equivalent to a ring and therefore representative of a slice through a single ventricle.
- BOX 5 site non-mutable strip round all edges and all boundary conditions zero-flux. This geometry was chosen to represent cardiac tissue with a specific region of fibrosis.
Results
The GA produced structures which underwent self sustaining re-entry in many of the combinations for stimulus regime and geometry we presented it with. We show here animations of the passage of the electrical excitation in the system without abnormal cells, and then from a re-entrant tissue for each of the three geometries.
Firstly, the simulation without any abnormal tissue.
[Click on the image to see an animation (mpeg) of each geometry]
In the Normal geometry case we see the production of a spiral wave re-entry.
In the Box geometry case the entire mutable zone acts as the centre of a anatomical re-entry.
Finally, in the Side geometry the incoming excitation is turned by 90o to exploit the periodic boundary conditions.
Conclusions
We have shown that a GA is capable of producing re-entrant structures within the FHN model. The mechanism that produced re-entry in this study was delayed conduction, or block of the final beat. This corresponds with experimental observation. The arrangement of mutable cells, i.e. geometry, was important. Normal geometry tended to produce functional re-entry. Whereas the Side geometry produced re-entry through the periodic boundary conditions, and the Box geometry favoured anatomical re-entry around the box of fibrotic tissue. In each case, the GA was able to expose a different type of arrhythmia.
We would therefore expect isolated areas of fibrotic tissue surrounded by interconnected excitable tissue to be more arrhythmgenic than a wide spread scattering of such cells.
Throughout this work we used a technique novel to modelling damage to cardiac tissue and excitable media, the GA. This has proved to be a highly useful tool in producing re-entrant structures in situations where more random methodologies fail. Also, by their very nature, GAs conveniently give us a progression of more arrhythmgenic structures as they evolve through progressive generations. It should be noted that the effects/methodology studied here are relevant not only for cardiac tissue but also for wave propagation in other excitable media.
Acknowledgments
We would like to thank the British Heart Foundation for funding this work (PG/03/102/1582). We are also grateful to the White Rose Grid (www.wrgrid.org.uk) and the Integrative Biology e-Science project (EPSRC grant ref GR/572023/01) for providing computational resources. This work is to be published in the journal Chaos.