A Contemporary Look at Biomechanical Models of Myocardium

Abstract

Understanding and predicting the mechanical behavior of myocardium under healthy and pathophysiological conditions are vital to developing novel cardiac therapies and promoting personalized interventions. Within the past 30 years, various constitutive models have been proposed for the passive mechanical behavior of myocardium. These models cover a broad range of mathematical forms, microstructural observations, and specific test conditions to which they are fitted. We present a critical review of these models, covering both phenomenological and structural approaches, and their relations to the underlying structure and function of myocardium. We further explore the experimental and numerical techniques used to identify the model parameters. Next, we provide a brief overview of continuum-level electromechanical models of myocardium, with a focus on the methods used to integrate the active and passive components of myocardial behavior. We conclude by pointing to future directions in the areas of optimal form as well as new approaches for constitutive modeling of myocardium.

Keywords: constitutive behavior; electromechanics; growth and remodeling; myocardial tissue; structural models.

Figure 1

Structural hierarchy involved in myocardium tissue. Understanding of the tissue behavior at the mesoscale can better inform both the organ-scale behavior and the cellular and subcellular behavior. Abbreviations: ATP, adenosine triphosphate; c, capillary lacunae; G&R, growth and remodeling; m, myofiber lacunae; polyGAG, polyglycosaminoglycan. Images of the human heart, cardiac wall, collagen, and myofiber network are from References , , , and , respectively. The image of the layered structure is an unpublished image taken at the Willerson Center at the University of Texas, Austin.

Figure 2

(a) Cutaway view of a cardiac myocyte, including several sarcomere units. (b) Machinery schematic for active muscle contraction via cross-bridge cycling. The central blue rods represent several myosin molecules (thick filament) with the actin binding site projecting out, surrounded by thin purple filaments. Titin proteins support the thick filament. Panel a adapted from Reference . Panel b adapted with permission from Pearson Education, Inc. Copyright 2004.

Figure 3

Different representations of the microstructural details of the collagen fiber arrangement in myocardium. (a) Arrangement of endomysial and perimysial collagen fibers with respect to myofibers. (b) Sheath-like arrangement of fine endomysial collagen fiber network. (c) Strut and weave representations of collagen fibers around myofibers (M) and capillary vessels (C). Panel a adapted from Reference . Panel b reprinted from Reference .

Figure 4

Schematic of cardiac microstructure showing transmural fiber orientation and branching sheet structures. Abbreviations: TR, transverse; AT, axial-transmural; TN, tangential planes. Combined figure adapted from References , , and .

Figure 5

(a) Active contraction in fiber and cross-fiber directions observed by Lin & Yin (45) (using barium contracture in rabbit left ventricular tissue). (b) Stress–strain curve showing three distinct stiffness regions—myofiber dominated, transitioning, and collagen dominated—in the passive biomechanical response of myocardium. E_lb and E_ub represent the lower and upper bound strains of collagen recruitment, respectively. Panel b adapted from Reference .

Figure 6

(a) Schematic representation of the fiber–sheet–normal microstructure. The mean myofiber direction is denoted by the vector f, the direction transverse to the fiber axis within the layers is denoted by s, and the direction normal to the layers is denoted by n. (b) Model of the heart wall as a layered structure, with each layer exhibiting transverse isotropy. Other vectors: c, circumferential direction; l, longitudinal direction; t, transmural direction. Panel a inspired by Reference . Panel b inspired by Reference .

Figure 7

Schematic representations of active contraction model. (a) Active stress model consisting of a contractile element in parallel with the passive spring element. (b) The active strain model consisting of a passive spring element in series with a contractile element. (c) Hill’s three-element model consisting of two spring elements in parallel and in series with a contractile element. Abbreviations: a and act, active; e, elastic; pas, passive; S and F, total stress and deformation gradient tensors, respectively.