A perturbation solution of the mechanical bidomain model

Bidomain modeling of electrical and mechanical properties of cardiac tissue

Throughout the history of cardiac research, there has been a clear need to establish mathematical models to complement experimental studies. In an effort to create a more complete picture of cardiac phenomena, the bidomain model was established in the late 1970s to better understand pacing and defibrillation in the heart. This mathematical model has seen ongoing use in cardiac research, offering mechanistic insight that could not be obtained from experimental pursuits. Introduced from a historical perspective, the origins of the bidomain model are reviewed to provide a foundation for researchers new to the field and those conducting interdisciplinary research. The interplay of theory and experiment with the bidomain model is explored, and the contributions of this model to cardiac biophysics are critically evaluated. Also discussed is the mechanical bidomain model, which is employed to describe mechanotransduction. Current challenges and outstanding questions in the use of the bidomain model are addressed to give a forward-facing perspective of the model in future studies.

Using the mechanical bidomain model to analyze the biomechanical behavior of cardiomyocytes

The mechanical bidomain model provides a macroscopic description of cardiac tissue biomechanics and also predicts the microscopic coupling between the extracellular matrix and the intracellular cytoskeleton of cardiomyocytes. The goal of this chapter is to introduce the mechanical bidomain model, to describe the mathematical methods required to solve the model equations, and to predict where the membrane forces acting on integrin proteins coupling the intracellular and extracellular spaces are large.

A numerical solution of the mechanical bidomain model

INTRODUCTION

The mechanical bidomain model predicts forces on integrin proteins in the membrane. It has been solved analytically for idealized examples, but a numerical algorithm is needed to address realistic problems.

METHODS

The bidomain equations are approximated using finite differences. An ischemic region is modeled as a circular area having no active tension, surrounded by normal tissue.

RESULTS

The membrane force is large in the ischemic border zone, but is small elsewhere. Strain is distributed widely throughout the ischemic region and surrounding tissue.

CONCLUSION

This calculation provides a testable prediction for the mechanism of mechanotransduction and remodeling in cardiac tissue.

Boundary Layers and the Distribution of Membrane Forces Predicted by the Mechanical Bidomain Model

The mechanical bidomain model is a mathematical description of the elastic properties of cardiac tissue. The unique feature of the bidomain model is that it is a macroscopic continuum representation of tissue that nevertheless accounts for the intracellular and extracellular spaces individually, thereby focusing on mechanical forces arising across the cell membrane. In this paper, the mechanical bidomain model describes a two-dimensional sheet of cardiac tissue undergoing a uniform active tension. At the boundary, the tissue sheet is free to move. Analytical solutions are found for the intracellular and extracellular displacements and pressures. The model predicts that membrane forces, which may be responsible for phenomena such as mechanotransduction and remodeling, are large near the tissue boundary, and fall off rapidly with distance from the boundary.

Two-domain mechanics of a spherical, single chamber heart with applications to specific cardiac pathologies

Continuum approximations of tissue consider responses averaged over many cells in a region. This simplified approach allows consideration of macroscopic effects, such as deformation or action potential propagation. A bidomain (sometimes known as biphasic) approach retains the macroscopic character of a continuum approximation while allowing one to consider microscopic effects; novel behavior arising from interactions between the intracellular and extracellular spaces can also be noted. I consider a spherical, single chamber heart with the new mechanical bidomain model in four separate pathologies: hypertension, hypovolemic hypotension, and hypertrophic and dilational cardiomyopathies. Analytic solutions of intracellular and extracellular displacements and hydrostatic pressures are presented; the distributions describe elastic deformation and hydrostatic fluid pressure buildup of the extracellular collagen matrix and the intracellular muscle under simplified spherical geometry. Potential applications, such as stretch activated membrane channels, are also noted.

Monodomain shear wave propagation and bidomain shear wave dispersion in an elastic model of cardiac tissue

Cardiac tissue elastically deforms under an applied stress, permitting shear waves to propagate through the heart. Traditionally, this behavior has been modeled with a monodomain approach, in which the mechanical properties of the intracellular and extracellular spaces are averaged together. We consider a mechanical bidomain model of cardiac tissue in which the mechanics of the intracellular and extracellular spaces are considered individually with the two spaces coupled by a spring constant. We find two normal modes of oscillation: one in which the intracellular and extracellular spaces oscillate together (a monodomain mode) and the other in which they oscillate in opposition (a bidomain mode). These two modes have unique dispersion relationships. In the extreme approximation of equal shear moduli and mass densities of the intracellular and extracellular spaces, the dispersion in the bidomain mode depends on the spring constant, while it does not in the monodomain mode.

The Mechanical Bidomain Model: A Review

The mechanical bidomain model is a new mathematical description of the elastic behavior of cardiac tissue. Its primary advantage over previous models is that it accounts for forces acting across the cell membrane arising form differences in the displacement of the intracellular and extracellular spaces. In this review, I describe the development of the mechanical bidomain model. I emphasize new predictions of the model, such as the existence of boundary layers at the tissue surface where the membrane forces are large, and pressure differences between the intracellular and extracellular spaces. Although the theoretical analysis is quite mathematical, I highlight the types of experiments that could be used to test the model predictions. Finally, I present open questions about the mechanical bidomain model that may be productive future directions for research.