Excitation-contraction coupling and cardiac contractile force: by Donald M. Bers, Kluwer Academic Publishers, 1991, $96.50 (xviii + 258 pages) ISBN 0 7923 1186 8

A Mathematical Treatment of Integrated Ca Dynamics within the Ventricular Myocyte

Substrate Stiffness Affects the Functional Maturation of Neonatal Rat Ventricular Myocytes

Finite elasticity theory combined with finite element analysis provides the framework for analysing ventricular mechanics during the filling phase of the cardiac cycle, when cardiac cells are not actively contracting. The orthotropic properties of the passive tissue are described here by a “pole–zero” constitutive law, whose parameters are derived in part from a model of the underlying distributions of collagen fibres. These distributions are based on our observations of the fibrous-sheet laminar architecture of myocardial tissue. We illustrate the use of high order (cubic Hermite) basis functions in solving the Galerkin finite element stress equilibrium equations based on this orthotropic constitutive law and for incorporating the observed regional distributions of fibre and sheet orientations. Pressure–volume relations and 3D principal strains predicted by the model are compared with experimental observations. A model of active tissue properties, based on isolated muscle experiments, is also introduced in order to predict transmural distributions of 3D principal strains at the end of the contraction phase of the cardiac cycle. We end by offering a critique of the current model of ventricular mechanics and propose new challenges for future modellers.

https://link.springer.com/content/pdf/10.1023/A:1011084330767.pdf

Computational Mechanics of the Heart

We introduce the concept of a contracting excitable medium that is capable of conducting non-linear waves of excitation that in turn initiate contraction. Furthermore, these kinematic deformations have a feedback effect on the excitation properties of the medium. Electrical characteristics resemble basic models of cardiac excitation that have been used to successfully study mechanisms of reentrant cardiac arrhythmias in electrophysiology. We present a computational framework that employs electromechanical and mechanoelectric feedback to couple a three-variable FitzHugh-Nagumo-type excitation-tension model to the non-linear stress equilibrium equations, which govern large deformation hyperelasticity. Numerically, the coupled electromechanical model combines a finite difference method approach to integrate the excitation equations, with a Galerkin finite element method to solve the equations governing tissue mechanics. We present example computations demonstrating various effects of contraction on stationary rotating spiral waves and spiral wave break. We show that tissue mechanics significantly contributes to the dynamics of electrical propagation, and that a coupled electromechanical approach should be pursued in future electrophysiological modelling studies.

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Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias.

Recent morphological studies have demonstrated a laminar (sheet) organization of ventricular myofibers. Multiaxial measurements of orthotropic myocardial constitutive properties have not been reported, but regional distributions of three-dimensional diastolic and systolic strains relative to fiber and sheet axes have recently been measured in the dog heart by Takayama et al. [30]. A three-dimensional finite-deformation, finite element model was used to investigate the effects of material orthotropy on regional mechanics in the canine left ventricular wall at end-diastole and end-systole. The prolate spheroidal model incorporated measured transmural distributions of fiber and sheet angles at the base and apex. Compared with transverse isotropy, the orthotropic model of passive myocardial properties yielded improved agreement with measured end-diastolic strains when: (1) normal stiffness transverse to the muscle fibers was increased tangent to the sheets and decreased normal to them; (2) shear coefficients were increased within sheet planes and decreased transverse to them. For end-systole, orthotropic passive properties had little effect, but three-dimensional systolic shear strain distributions were more accurately predicted by a model in which significant active systolic stresses were developed in directions transverse to the mean fiber axis as well as axial to them. Thus the ventricular laminar architecture may give rise to anisotropic material properties transverse to the fibers with greater resting stiffness within than between myocardial laminae. There is also evidence that intact ventricular muscle develops significant transverse stress during systole, though it remains to be seen if active stress is also orthotropic with respect to the laminar architecture.

Effect of Laminar Orthotropic Myofiber Architecture on Regional Stress and Strain in the Canine Left Ventricle

We coupled a continuum model of impulse propagation with a three-dimensional model of regional ventricular mechanics. Equations for action potential propagation, myofilament activation and active contraction were solved in an anatomically detailed finite element model of canine left and right ventricular geometry, muscle fiber architecture and Purkinje fiber network anatomy. Finite element equations for time-dependent excitation and recovery variables were assembled using a collocation method and solved using adaptive Runge---Kutta integration. Resting tissue mechanics were modeled as nonlinear, orthotropic and hyperelastic. A Windkessel model for arterial impedance was coupled to ventricular pressure and volume to compute hemodynamic boundary conditions during ejection. Ventricular volume constraints were imposed during the isovolumic phases. This model showed good agreement in the time course of regional systolic strains with experimental measurements during normal sinus rhythm and demonstrates the importance of the Purkinje fiber system in determining the mechanical activation sequence.

Computational model of three-dimensional cardiac electromechanics

Computational modeling provides a potentially powerful way to integrate structural properties measured in vitro to physiological functions measured in vivo. Focusing on the various scales (cell-tissue-organ-system), we give an overview of the importance and applications of numerical models of ventricular anatomy, electrophysiology, mechanics, and circulatory models. The integration of these models in one multiscale model of cardiac electromechanics is discussed in the light of applications to hypothesis generation, diagnosis, surgery(planning, training, and outcome of interventions), and therapies. Special attention is paid to practical use in terms of computational demand. Because of growing computer power and the development of efficient algorithms, we expect that real-time simulations with multiscale models of cardiac electromechanics become feasible in 2008 (despite the increasing complexity of models due to data accumulation on molecular and cellular mechanisms).

Computational Methods for Cardiac Electromechanics

Introduction: Asynchronous electrical activation can cause abnormalities in perfusion and pump function. An electromechanical model was used to investigate the mechanical effects of altered cardiac activation sequence.



Methods and Results: We used an anatomically detailed three-dimensional computational model of the canine ventricular walls to investigate the relationship between regional electrical activation and the timing of fiber shortening during normal and ventricular paced beats. By including a simplified Purkinje fiber network and anisotropic impulse conduction in the model, computed electrical activation sequences were consistent with experimentally observed patterns. Asynchronous time courses of regional strains during beats stimulated from the left or right ventricular epicardium showed good agreement with published experimental measurements in dogs using magnetic resonance imaging tagging methods. When electrical depolarization in the model was coupled to the onset of local contractile tension development by a constant time delay of 8 msec, the mean delay from depolarization to the onset of systolic fiber shortening was 14 msec. However, the delay between the onset of fiber tension and initial shortening varied significantly; it was as late as 60 msec in some regions but was also as early as −50 msec (i.e., 42 msec before depolarization) in other regions, particularly the interventricular septum during free-wall pacing.



Conclusion: The large variation in delay times was attributable to several factors including local anatomic variations, the location of the site relative to the activation wavefront, and regional end-diastolic strain. Therefore, we conclude that these factors, which are intrinsic to three-dimensional ventricular function, make the regional sequence of fiber shortening an unreliable surrogate for regional depolarization or electromechanical activation in the intact ventricles. (J Cardiovasc Electrophysiol, Vol. 14, pp. S196-S202, October 2003, Suppl.)

Taras P. Usyk

Papers

Effect of Laminar Orthotropic Myofiber Architecture on Regional Stress and Strain in the Canine Left Ventricle

Computational model of three-dimensional cardiac electromechanics

Computational Methods for Cardiac Electromechanics

Relationship between regional shortening and asynchronous electrical activation in a three-dimensional model of ventricular electromechanics.

Electromechanical model of cardiac resynchronization in the dilated failing heart with left bundle branch block.