Deriving the Bidomain Model of Cardiac Electrophysiology From a Cell-Based Model; Properties and Comparisons

The bidomain model is considered to be the gold standard for numerical simulation of the electrophysiology of cardiac tissue. The model provides important insights into the conduction properties of the electrochemical wave traversing the cardiac muscle in every heartbeat. However, in normal resolution, the model represents the average over a large number of cardiomyocytes, and more accurate models based on representations of all individual cells have therefore been introduced in order to gain insight into the conduction properties close to the myocytes. The more accurate model considered here is referred to as the EMI model since both the extracellular space (E), the cell membrane (M) and the intracellular space (I) are explicitly represented in the model. Here, we show that the bidomain model can be derived from the cell-based EMI model and we thus reveal the close relation between the two models, and obtain an indication of the error introduced in the approximation. Also, we present numerical simulations comparing the results of the two models and thereby highlight both similarities and differences between the models. We observe that the deviations between the solutions of the models become larger for larger cell sizes. Furthermore, we observe that the bidomain model provides solutions that are very similar to the EMI model when conductive properties of the tissue are in the normal range, but large deviations are present when the resistance between cardiomyocytes is increased.

Simulating Epileptic Seizures using the Bidomain Model

Epileptic seizures are due to excessive and synchronous neural activity. Extensive modelling of seizures has been done on the neuronal level, but it remains a challenge to scale these models up to whole brain models. Measurements of the brain’s activity over several spatiotemporal scales follow a power-law distribution in terms of frequency. During normal brain activity, the power-law exponent is often found to be around 2 for frequencies between a few Hz and up to 150 Hz, but is higher during seizures and for higher frequencies. The Bidomain model has been used with success in modelling the electrical activity of the heart, but has been explored far less in the context of the brain. This study extends previous models of epileptic seizures on the neuronal level to the whole brain using the Bidomain model. Our approach is evaluated in terms of power-law distributions. The electric potentials were simulated in 7 idealized 2D models and 3 MRI-derived 3D patient-specific models. Computed electric potentials were found to follow power-law distribtions with slopes ranging from 2 to 5 for frequencies greater than 10-30 Hz.

From Millimeters to Micrometers; Re-introducing Myocytes in Models of Cardiac Electrophysiology

Computational modeling has contributed significantly to present understanding of cardiac electrophysiology including cardiac conduction, excitation-contraction coupling, and the effects and side-effects of drugs. However, the accuracy of in silico analysis of electrochemical wave dynamics in cardiac tissue is limited by the homogenization procedure (spatial averaging) intrinsic to standard continuum models of conduction. Averaged models cannot resolve the intricate dynamics in the vicinity of individual cardiomyocytes simply because the myocytes are not present in these models. Here we demonstrate how recently developed mathematical models based on representing every myocyte can significantly increase the accuracy, and thus the utility of modeling electrophysiological function and dysfunction in collections of coupled cardiomyocytes. The present gold standard of numerical simulation for cardiac electrophysiology is based on the bidomain model. In the bidomain model, the extracellular (E) space, the cell membrane (M) and the intracellular (I) space are all assumed to be present everywhere in the tissue. Consequently, it is impossible to study biophysical processes taking place close to individual myocytes. The bidomain model represents the tissue by averaging over several hundred myocytes and this inherently limits the accuracy of the model. In our alternative approach both E, M, and I are represented in the model which is therefore referred to as the EMI model. The EMI model approach allows for detailed analysis of the biophysical processes going on in functionally important spaces very close to individual myocytes, although at the cost of significantly increased CPU-requirements.

Well-Posedness for a Modified Bidomain Model Describing Bioelectric Activity in Damaged Heart Tissues

We prove the existence and the uniqueness of a solution for a modified bidomain model, describing the electrical behaviour of the cardiac tissue in pathological situations. The leading idea is to reduce the problem to an abstract parabolic setting, which requires to introduce several auxiliary differential systems and a non-standard bilinear form. The main difficulties are due to the degeneracy of the bidomain system and to its non-standard coupling with a diffusion equation, accounting for the presence of the pathological zone in the heart tissue.