Using the Mechanical Bidomain Model to Analyze the Biomechanical Behavior of Cardiomyocytes

During the decade of 2010–2020, my research shifted from bioelectricity and biomagnetism to biomechanics and mechanotransduction. I took the bidomain model of cardiac electrophysiology—described in Chapter 7 of Intermediate Physics for Medicine and Biology— and adapted it to describe growth and remodeling in response to mechanical forces. In other words, I traded resistors for springs. This effort was not entirely successful, but I think it provided some useful insights.

In 2015 I described the mechanical bidomain model in a chapter of Cardiomyocytes: Methods and Protocols. This book was part of the series Methods in Molecular Biology, and each chapter had a unusual format. The research was outlined, with the details relegated to an extensive collection of endnotes. A second edition of the book was proposed, and I dutifully submitted an updated chapter. However, the new edition never come to pass. Rather than see my chapter go to waste, I offer it to you, dear reader. You can download a draft of my chapter for the second edition here. For those of you who have time only for a summary, below is the abstract.

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 for solving the model equations, to predict where the membrane forces acting on integrin proteins coupling the intracellular and extracellular spaces are large, and to suggest experiments to test the model predictions.

The main difference between the chapter in the first edition and the one submitted for the second was a new section called “Experiments to Test the Mechanical Bidomain Model.” There I describe how the model can reproduce data obtained when studying colonies of embryonic stem cells, sheets of engineered heart tissue, and border zones between normal and ischemic regions in the heart. The chapter ends with this observation:

The most important contribution of mathematical modeling in biology is to make predictions that can be tested experimentally. The mechanical bidomain model makes many predictions, in diverse areas such as development, tissue engineering, and hypertrophy.

I particularly like a new figure in the second edition. It’s a revision of a figure created by Xavier Trepat and Jeffrey Fredberg that compares mechanobiology to a game of tug-of-war. I added the elastic properties of the extracellular space (the green arrows), saying “It is as if the game of tug-of-war is played on a flexible surface, such as a flat elastic sheet.” In other words, tug-of-war on a trampoline. 

Enjoy!

The “tug-of-war” model of tissue biomechanics, adapted from an illustrationby Trepat and Fredberg. Top: the intracellular (yellow), extracellular (green) and integrin (blue) forces acting on a monolayer of cells. Middle: The analogous forces among the players of a game of tug-of-war. This figure is extended beyond that of Trepat and Fredberg by allowing both the intracellular and extracellular spaces to move. Bottom: Representation of the mechanical bidomain model by a ladder of springs.