IJBEM, Vol. 5, No. 1, 2003

The Cable as a Basis for a Mechanoelectric Whole Heart Model

Edward J. Vigmond^a, David McQueen^b, and Charles S. Peskin^b

^aDepartment of Electrical and Computer Engineering, University of Calgary, Calgary, Canada
^bCourant Institute of Mathematical Sciences,New York University, New York, USA

Correspondence: Edward J Vigmond, Department of Electrical and Computer Engineering, University of Calgary, 2500 University Dr NW, Calgary, Canada T2N 1N4. E-mail: vigmond@ucalgary.ca, phone 403 210 3887, fax 403 282 6855

Abstract.The heart has the ability to self-regulate through a process termed mechano-electric feedback (MEF) whereby electrical events influence mechanical and vice versa. Intracellular Ca²⁺ handling and stretch activated channels provide the mechanisms of interaction. MEF can be arrhythmogenic in pathological hearts where abnormal stresses develop. Modelling such activity requires the whole organ as forces developed by any part of the myocardium are transmitted throughout by the blood. The Interconnected Cable Method and the Immersed Boundary Method have been successfully used in large-scale cardiac simulations to model electrical and mechanical events respectively. They both use as their basic functional units cables mimicing physiological fibres, and are, thus, ideally suited for synthesis into a complete electro-mechanical heart model. Linking of the methods is accomplished by a set of equations describing excitation-contraction coupling to compute tension in each myocyte and the inclusion of stretch-activated ion channels. Preliminary results showing the compatibility of the two methods are presented, demonstrating that a complete electro-mechanical heart model is feasible.

Keywords: Cardiac Tissue; Mechano-Electric Feedback; Electrophysiology; Computer Modelling

1. Introduction

The heart is a pump controlled by an electrical servomechanism. Mechano-Electric Feedback (MEF) is the process by which electrical and mechanical systems interact to allow for self-regulation. The primary mechanisms of MEF are stretch-activated ion channels (SAC) and intracellular [Ca²⁺] modulation. Contraction depends on [Ca²⁺] which is affected by (1) Ca²⁺ influx during the action potential, (2) uptake and release of Ca²⁺ by the sarcoplasmic reticulum, and (3) binding of Ca²⁺ to troponin. Contraction affects stretch-activated channels that alter the action potential and closes the feedback loop. MEF may be arrhythmogenic in pathological hearts where abnormal stresses may develop due to ischemia or dilation [Taggart, 1996].

The heart has a laminar structure where within each layer, myocytes are parallel and are coupled preferentially along their axes. This structure has been exploited by two modeling methods for different means: The IMmersed Boundary Method (IMBM) [McQueen, 2000; Kovacs et al, 2001] describes tissue as a series of cables with elastic properties immersed in a fluid and calculates the tension developed in each cable segment. These forces get transmitted to other cables via the fluid, producing movement. The InterConnected Cable Model (ICCM) [Vigmond and Leon, 1999] is a computationally efficient method for calculating electrical propagation in anisotropic tissue. It too relies on a domain composed of cables which are connected by transverse resistances, and has been used to model 3D slabs of ventricle[Vigmond and Leon,1999] and the complete atria[Vigmond et al., 2001].

This paper discusses the combining of these two methods to model MEF. This seems natural since both can use the same computational grid, one composed of cables which follow physiological fibre paths, to compute complementary quantities, electrical and mechanical.

2. Methods

The combined method is schematically outlined in Fig. 1. The computation must store the following variables: vm, transmembrane voltage; [Ca]; GV, state variables for ionic model; X, cable segment positions, and u, their velocity; z, available myosin; and T, tension. A force field density, F, must is be computed. The whole heart cable model [McQueen, 2001] is shown in Fig. 2.

Figure 1. Computational scheme showing linking of methods. Inputs are indicated in the boxes while outputs leave boxes.

Figure 2. Cable model of whole heart. 400 of 4000 cables are shown. Its jagged shape is due to a lack of blood pressure.

The ICCM divides the solution of the parabolic equation for vm into a semi-implicit particular part and a homogeneous part. Each of these is computationally simple, amenable to parallelization, and uses little memory. The ICCM computes the change in vm, GV and [Ca²⁺] due to ionic currents and the spread of current. Vm and X are then used to compute T in the myocytes [Hunter et al, 1998].

The IMBM computes the movement of the tissue, u, based on a Navier-Stokes description of an incompressible fluid subject to a force field derived from T. The new positions of the cable elements are used to calculate SAC currents [Kohl and Sachs, 2001] which produce changes in vm.

3. Results

The IMBM as used assigned mechanical activation functions with a simultaneous activation of the whole heart. As a first step in combining the two methods, the cable description of the heart was used by the ICCM to calculate propagation patterns to get a more realistic activation pattern. This involved resampling the cable grid for constant 100 mm spacing and determining the intercable connections which are not necessary for the IMBM. Regions corresponding to the SA node, point of attachment of Bachmann’s Bundle on the left atrium, the AV node, and the site of earliest activation in the left ventricular septal wall were defined. The electrical activation times were then successfully fed back to the IMBM code to be used as mechanical activation times.

4. Discussion

The successful synthesis of the two methods has been demonstrated. The integration of excitation-contraction coupling and SAC’s has yet to be performed but is straightforward. The computational demands of the IMBM portion of the problem are much greater than those of the ICCM. The ICCM can compute millions of nodes on a PC while the IMBM requires 700kB on a 16-processor SMP machine. The additional ICCM load will not add significantly to the computation.

Acknowledgements

David McQueen and Charles Peskin are supported by NSF Grant Number DMS-9980069 and Edward Vigmond is supported by the Natural Sciences and Engineering Research Council of Canada.

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