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

The mechanical bidomain model of cardiac muscle with curving fibers

In the heart, cardiac muscle fibers curve creating zones of membrane forces resulting in regions of mechanotransduction. This study uses the finite difference method to solve the mechanical bidomain equations numerically for a complex fiber geometry. The magnitude of the active tension T is constant but its direction makes an angle with the x-axis that varies with position. Differences between the intracellular and extracellular displacements result from the bidomain behavior of the tissue that gives rise to forces on the integrin proteins in the membrane. The long-term goal is to use the mechanical bidomain model to suggest experiments and make predictions about growth and remodeling in the heart.

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.

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.