# Two-domain mechanics of a spherical, single chamber heart with applications to specific cardiac pathologies

- Steffan Puwal
^{1}Email author

**Received: **18 December 2012

**Accepted: **4 April 2013

**Published: **26 April 2013

## Abstract

Continuum approximations of tissue consider responses averaged over many cells in a region. This simplified approach allows consideration of macroscopic effects, such as deformation or action potential propagation. A bidomain (sometimes known as biphasic) approach retains the macroscopic character of a continuum approximation while allowing one to consider microscopic effects; novel behavior arising from interactions between the intracellular and extracellular spaces can also be noted. I consider a spherical, single chamber heart with the new mechanical bidomain model in four separate pathologies: hypertension, hypovolemic hypotension, and hypertrophic and dilational cardiomyopathies. Analytic solutions of intracellular and extracellular displacements and hydrostatic pressures are presented; the distributions describe elastic deformation and hydrostatic fluid pressure buildup of the extracellular collagen matrix and the intracellular muscle under simplified spherical geometry. Potential applications, such as stretch activated membrane channels, are also noted.

## Introduction

Continuum models of tissue average the behavior of interest over many cells. Typically this limits consideration to macroscopic effects. A bidomain approach to cardiac muscle is necessary to explain behavior that results from unequal intracellular and extracellular spaces (Tryanova et al. 1993). For example, proarrhythmic responses to external stimuli have been predicted with the electrical bidomain model when the intracellular and extracellular spaces have unequal anisotropy ratios for conductivity (Tryanova et al. 1993). At the sub-cellular and molecular levels of organization discrete models are necessary to describe the response of cardiac muscle. For example, Eshelby (Eshelby 1957) presented an early model useful for considering the elasticity of inclusion bodies (which, here, would be individual cells). Others (Ohayon & Chadwick 1988; Chadwick 1982; Chadwick 1981; Panfilov et al. 2005; Panfilov et al. 2007; Nash & Panfilov 2004; Bini et al. 2005; Glass et al. 1991; Guccione et al. 1991; Arts et al. 1979; Latimer et al. 2003) have presented continuum elastic models which are monodomain in the sense that, not only do they average over many cells, but they also average together both the intracellular and extracellular spaces. These models allow one to consider anisotropy, though they typically assume a strain linearly proportional to the applied stress. A background isotropic term is usually associated with the extracellular collagen mesh, while anisotropic terms represent the muscle fiber. Under sufficient stress, most elastic materials will behave nonlinearly, or even plastically deform. However, linear models such as these are useful for determining first order responses because we can find an analytic solution. Nonlinear models are typically limited to numerical results.

Puwal and Roth (Puwal & Roth 2010; Punal & Roth 2012; Puwal & Roth 2013) have previously presented a linear mechanical bidomain model that separately considers the intracellular and extracellular spaces as elastic continua coupled by a spring force whose magnitude is proportional to the displacement difference of the two spaces. (Note that the terms “monodomain” and “bidomain” are standard terms in the cardiac research literature. Those familiar with elasticity theory may recognize these terms as “monophasic” and “biphasic”.) The advantage of the bidomain approach is that we may consider a particular microscopic parameter averaged over many cells, while still discussing macroscopic effects. In this case, the microscopic parameter is the nature of the force coupling elastic deformations of the intracellular and extracellular spaces.

This new model may provide insight into how the diseased state may translate into large membrane stresses, thereby leading to ectopic electrical activity. Here, I consider a single chamber, spherical heart with a wall of finite thickness. Of course, the human heart is four chambered and not spherical. However, an analytic solution of elastic deformation is not likely to be found by considering the true geometry of the heart and a spherical model will retain the fundamental characteristics of the heart (e.g. increasing chamber pressure leads to an outward displacement of the muscle). The analytic solution presented here is intended only to approximate the response of the human heart. The displacement fields and hydrostatic pressures are found to be in general agreement with what we expect—namely, increasing active tension causes the heart to contract and increasing chamber pressure causes the heart to expand. The analytic solution allows us to determine the pressure and displacement fields and qualitatively comment on hearts with underlying pathologies. I conclude with a discussion of how these solutions may be applied to four specific pathologies in particular: hypertension, hypovolemic hypotension, hypertrophic cardiomyopathy, and dilational cardiomyopathy. My discussion of these pathologies is limited by the fact that I do not consider remodeling of the elastic parameters during progression of the diseased state.

## Methods

The results presented here are derived equations based on theory and, therefore, require no ethical considerations.

A very general definition of a cell would describe an interior space for metabolic reactions to take place which is separated from its surrounding extracellular space by a membrane (Guyton & Hall 2006; Nelson & Cox 2008). In muscle cells (myocytes), the sarcolemma consists of the typical phospholipid bilayer present in most cell membranes as well as a polysaccharide coat with collagen proteins (Guyton & Hall 2006; Nelson & Cox 2008). Cardiomyocytes have the additional distinguishing feature of intercalated disks which connect the intracellular spaces of adjacent cardiomyocytes to form a single continuous intracellular space (Guyton & Hall 2006). Our model of elasticity will consider the differing elasticity of components of the intracellular and extracellular spaces and discuss how mechanical disturbances in one space are translated into another.

### The extracellular space

The extracellular space is that portion of the tissue outside of the cell membrane. The cardiomyocyte extracellular space contains fluid (including large amounts of water), dissolved ions, polysaccharides and collagen (Guyton & Hall 2006; Nelson & Cox 2008). Important for our considerations will be the collagen matrix. Collagen is a protein found in all connective tissue as well as the extracellular space of cardiomyocytes (Nelson & Cox 2008). While there are several types of collagen, the structure of collagen generally involves three *α*-chain proteins very tightly twisted around each other and forming a coiled structure. This tight coil provides great tensile strength. The *α*-chains within a collagen fibril and across adjacent fibrils are linked by covalent bonding. The linking between adjacent collagen fibrils arranged at a random relative orientation forms a tightly woven mesh of collagen exhibiting relatively isotropic elasticity when compared to the intracellular space (Nelson & Cox 2008).

*G*is the shear elastic modulus we may write

where $\tilde{I}$ is the identity matrix and *q* is the extracellular hydrostatic pressure (Reismann & Pawlik 1980; Arya 1998; Arfken & Weber 2001). In general, collagen is quite resistant to elastic deformation and the shear modulus will be large—on the order of 10^{4} Pascals (Latimer et al. 2003; Puwal & Roth 2010).

### The intracellular space

The intracellular space of a cardiomyocyte is dominated by structures which facilitate the contraction of the muscle. Normal contraction is the result of actin and myosin protein filaments sliding past each other after an action potential triggers the release of calcium ions from the sarcoplasmic reticulum. The actin and myosin filaments are aligned in a side-by-side arrangement by an accessory protein called titin (Guyton & Hall 2006; Nelson & Cox 2008) (characterized as “very springy” (Guyton & Hall 2006)). These bundles of aligned contractile filaments (which may be visualized on an electron micrograph (Nelson & Cox 2008)) give the intracellular space a preferred direction of elasticity. We regard stresses which stretch the intracellular space as also stretching the longitudinally directed actin-myosin complexes and the covalently bonded titin proteins. While the intracellular space will exhibit some background isotropy, the relative displacements of two coupled isotropic spaces are not particularly interesting, so we will focus our attention on the anisotropic nature of the intracellular space.

*r, θ*) are the usual spherical coordinates shown in Figure 2,

*p*is the intracellular hydrostatic pressure,

*T*

_{0}is the active tension for the muscle fibers,

*μ*is the elastic modulus of the chamber wall in the diretion parallel to the wall itself (a term that is essentially a hybrid of the Young’s modulus and the shear modulus), $\tilde{\tau}$ is the intracellular stress tensor, and $\tilde{\in}$ is the intracellular strain tensor. The symmetry of the problem allows us to assert that the intracellular and extracellular displacement fields have no azimuthal or angular component and that the displacements are a function of the radial coordinate

*r*only (Reismann & Pawlik 1980).

*∂*

_{ r }is the partial derivative with respect to the radial coordinate (Reismann & Pawlik 1980). And, therefore, the off-diagonal terms of the stress tensors are found to be (Reismann & Pawlik 1980).

### Coupling between the intracellular and extracellular spaces and the equations of mechanical equilibrium

*K*(Puwal & Roth 2010; Punal & Roth 2012). The coupling force may be expanded in Taylor series; a linear spring constant is merely the lowest order term in the expansion. Taking advantage of the symmetry presented in the problem, the equations of mechanical equilibrium simplify to

These equations assert that non-zero relative displacements in the intracellular and the extracellular spaces will lead to a restoring force, the magnitude of which is proportional to the magnitude of the relative displacement (*u–w*) with the spring constant as that constant of proportionality. Consistent with Newton’s third law of motion, the force experienced by the extracellular space due to the intracellular space is equal in magnitude and opposite in sign to the force experienced by the intracellular space due to the extracellular space (Arya 1998), leading to the minus sign in the extracellular equation of mechanical equilibrium. Equations 15 and 16, along with Equations 7, 8, 10 and 11, are the coupled differential equations which we solve (subject to the appropriate boundary conditions) in order to obtain displacement fields u and w the hydrostatic pressure fields *p* and *q*.

### Incompressibility

*ad hoc*assumption

and so the displacements *u* and *w* are proportional to 1/r^{2}.

### The cardiac cycle, time dependence of parameters, and quasi-static equilibrium

The contraction (systole) and relaxation (diastole) of the chambers of the heart are coordinated with each beat in a sequence of events that constitute the cardiac cycle (Guyton & Hall 2006). The stresses which the intracellular and extracellular spaces undergo vary over the course of the cardiac cycle. For example, the active tension is less in magnitude in cardiac tissue during the diastolic phase of the cardiac cycle than it is during the systolic phase. Therefore, the stresses, strains, active tension, and hydrostatic pressures are all functions of time. Chadwick (Chadwick 1982; Chadwick 1981) suggests *a priori* that the tissue is in quasi-static equilibrium at each instant of time. The validity of this assumption is supported by two observations. First, the elastic moduli are large in magnitude and displacement waves travel at relatively high speeds in media with large elastic moduli (Latimer et al. 2003; Lowrie 2007). Second, *post hoc* results predicated on the assumption of quasi-static equilibrium predict that displacements and hydrostatic pressures are very small in magnitude; perturbations of these quantities over the course of the cardiac cycle are even smaller. Small changes in displacements and hydrostatic pressures will, therefore, occur at a very rapid rate, essentially achieving quasi-static equilibrium.

We are interested in the average response of the heart tissue over the course of the cardiac cycle. We will, therefore, concern ourselves with the time average of the active tension *T*
_{0} = <*T*(*t*)>, the time average of the chamber pressure *q*
_{
0
} = <*q*
_{
c
}(*t*)>, and the time average of the pericardial pressure *q*
_{1} = <*q*
_{
p
}(t)>. Elastic moduli and the coupling spring constant are assumed to be time-independent parameters.

### Geometry of the left ventricle

*a*= 3.14

*cm*, outer radius

*b*= 4.24

*cm*, a time averaged outside (pericardial) pressure

*q*

_{1}= 2 mmHg, and a time averaged inside (chamber) pressure

*q*

_{0}= 100 mmHg (see Figure 2). A summary of parameters used is given in Table 1.

**Table of parameters**

Parameter | Value | Comment |
---|---|---|

Chamber Pressure |
| A time-average value on the order of the chamber pressure during ventricular systole |

Pericardial Pressure |
| |

Extracellular Shear Modulus |
| |

Ratio of Elastic Moduli |
| Based on Ohayan, et al. Implies |

Active Tension |
| A time-average value on the order of systole for isolated myocytes (Chen & Varghese 2010) |

Radius of Inner Chamber |
| Implies an inner chamber volume similar to the human left ventricle |

Radius of Spherical Heart |
| Implies a wall thickness similar to the human left ventricle |

Estimate of Coupling Parameter |
| Based on experimental data of 5 to 6 |

### Permanent deformations

We have assumed that we are in the region of reversible elasticity—that is, when the deforming stress is removed the elastic deformations return to zero. This is not always the case. The extracellular collagen mesh is formed from cross-linking covalent bonds. Each individual covalent bond, as we have said, may be approximated by a spring and the elastic continuum approach is based on a single preferred bond point. Under increasing stresses the cross-linking bonds may be broken, new bonds re-formed, and the deforming stress plastically deforms the collagen mesh. Active tension, pericardial pressure, and chamber pressure are not generally of sufficient magnitude to cause plastic deformation (Guyton & Hall 2006; Peskin 1975; Morgan et al. 1965).

## Results

*a*and the outer radius is

*b*; the muscle wall is in the region

*a < r < b*. The pericardial sac is represented by the region

*r > b*and an average hydrostatic pressure

*q*

_{1}is maintained over the course of the cardiac cycle. In the heart chamber

*r < a*an average hydrostatic pressure

*q*

_{0}is maintained. (We note that

*q*

_{0}>

*q*

_{1}) The fluid in the spaces

*r < a*and

*r > b*are assumed to be continuous with the extracellular space, while there is, of course, no intracellular space in these regions. To find the displacement fields and hydrostatic pressures associated with these boundary pressures, we seek an analytical solution to Equations 15 and 16 that is consistent with the boundary conditions

^{10}

*N/m*

^{3}). Within the heart wall, a plot of pressures (Figure 4) versus

*G/K*obeys a general relation (

*A*+

*BG/K*)/(

*C*+

*DG/K*), where

*A, B, C,*and

*D*are constants. It should be noted, with positive elastic moduli that $\left(1-\mathrm{\Delta \Pi}\right)<0;\phantom{\rule{0.5em}{0ex}}\mathit{ln}\left(\frac{a}{b}\right)$ is less than zero as well, since

*a < b*.

*u*= 0) of the intracellular space.

*u*(

*r*=

*a*) = –

*a*. (Similarly, the extracellular space will collapse toward the origin with increasing active tension, though as we saw above there is an outward force, caused by the higher chamber pressure, that is greater in the extracellular space than in the intracellular space.) The upper bound on tension, therefore, is found when

*u*(

*a*) = –

*a*. (But, of course, systemic physiological responses will resist the chamber collapsing, so the region of validity for these equations is narrower than this.)

Further increases in active tension are accompanied by increases in hydrostatic pressure and cannot be accompanied by continued inward collapse; however, the exact functional form of the hydrostatic pressures will differ from that given by Equations 30 and 32.

### Hypertension

*t*.

where *τ* is the characteristic time for the remodeling mechanisms.

As a simplified model of elastic deformation under hypertension we shall assume the deformation is early in the course of the condition (such that *t* ≪ *τ*) and, therefore, hypertension is considered to simply be an increase in *q*
_{0}. (Since the pericardial sac is sufficiently isolated from systemic circulation, the pericardial pressure negligibly increases with the increased blood pressure (Guyton & Hall 2006; Morgan et al. 1965; Libby et al. 2008). Observe the displacement fields and the hydrostatic pressures are linearly proportional to this chamber pressure *q*
_{0} so that, for all *r* in the range *a < r < b* there is an increase in the intracellular and extracellular displacements and hydrostatic pressures that is proportional to the increase of the chamber pressure. Thus, early in the course of illness for a hypertensive patient there is an outward directed force due to the high chamber pressure; later compensatory hypertrophy may then be considered as being in direct response to this outward directed force.

A bidomain approach allows us, in our description of strain, to separately consider the collagen matrix and the intracellular actin-myosin complex. With a mathematical description of the strains in both spaces we may subsequently consider, for example, whether the process of compensatory hypertrophy is centrally concerned with the intracellular space and remodeling of the extracellular collagen matrix is incidental (i.e. that there is a single process at work) or whether there are two compensatory processes that respond to strain separately, but simultaneously in each space.

### Hypovolemic hypotension

Hypovolemia, or decreased blood volume, is almost always accompanied by hypotension, or decreased blood pressure. Hypovolemic hypotension will present with decreased chamber pressure (Guyton & Hall 2006; Libby et al. 2008). Thus, the hypotensive elastic response predicted is for a smaller outward directed force on the extracellular collagen matrix and functionally similar displacement and pressure fields. Equations 29 and 31 describe the displacement fields; observe that with a sufficiently small chamber pressure, an increased inward contraction of the muscle wall will result. With the same pericardial pressures and active tensions, the inward contraction of the muscle wall will be of greater magnitude than that experienced by the non-hyoptensive patient.

### Hypertrophic cardiomyopathy

Hypertrophic Cardiomyopathy (HCM) is the most common genetically transmitted cardiovascular disease. HCM presents with a constellation of symptoms, including sudden cardiac death. In HCM there is a non-dilational thickening of the wall of the left ventricle (LV). Postmortem assessment shows LV wall thickness nearly doubles, though the thickening is often asymmetrical (Guyton & Hall 2006; Libby et al. 2008). A non-dilational thickening of the wall may be approximated as symmetrical and taken as in increase in the outer wall radius *b* in Equations 29, 30, 31 and 32. Observe in the displacement fields that the factor *ln(b)* is multiplied by the active tension *T*
_{0}; we conclude that smaller active tensions are predicted to be necessary in HCM hearts to cause similar deformations as non-HCM hearts. In other words, it is predicted that a more massive muscle can cause the same deformation with less required tension. Meanwhile, in the hydrostatic pressures the geometric shorthand variables *l*
^{2} and *z* are multiplied by (*q*
_{
0
} – *q*
_{
1
}); larger pressure differences between the chamber and pericardium are, thus, predicted to be necessary in HCM hearts to cause the same hydrostatic pressures within the tissue as non-HCM hearts. In other words, it is necessary to put a bigger pressure differential across a more massive heart to achieve the same distribution of pressure within. Note that these predictions presume the elastic parameters of HCM and non-HCM hearts are identical. This may not be a valid assumption after the tissue is remodeled.

### Dilated cardiomyopathy

*a*

_{0}, the initial outer radius is

*b*

_{0}, and the dilated inner radius is

*a*

_{1}, then the dilated outer radius will be

*b*

_{1}where

Quantities such as $-\mathit{ln}\left(\frac{a}{b}\right)$, *z*, and *l*
^{2} increase with increasing *a*
_{1}. Equations 29, 30, 31 and 32 predict that similar displacements will occur for smaller active tension *T*
_{0} than in a non-DCM heart. However, again note that these predictions presume the elastic parameters of DCM and non-DCM hearts are identical. This may not be a valid assumption after the tissue has remodeled.

## Conclusion

The two-domain approach to elastic deformation of cardiac muscle allows us to separately analyze displacements and hydrostatic pressures of the intracellular and extracellular spaces. We considered the deformations and pressures applied to four important pathologies: hypertension, hypotension, HCM, and DCM. The human heart is not a spherical, single chamber heart. Here, the geometry, elastic properties of the muscle, and deforming stresses of the pathology exhibit angular and azimuthal symmetry; this simplification is intended to facilitate the determination of an analytic solution that describes the effects of radially directed pressures and tangential active tension in the muscle may be analyzed. In reality, the largest chamber of the human heart is the left ventricle, which is approximately conical in shape and resembling a folded sheet of cardiac tissue. Intercalated disks bridge the cells in the radial direction as well as the tangential direction, providing background isotropy of the intracellular space and a radially directed active tension. Atria empty into the ventricles and the ventricles, in turn, empty into arteries; thus, the stresses are not uniform over the outer surface of the heart (pericardial sac). The great stresses around the aorta, for example, are not considered here.

In spite of these limitations, the results presented here are generally consistent with our expectations. In the hypertensive example, for instance, the deforming stress is a relative net outward force (greater chamber pressure, unchanged pericardial pressure) on the extracellular collagen mesh; the spring coupling between the two continua translates this outward force to the intracellular muscle. It is not surprising, therefore, that the extracellular collagen mesh deforms to a greater extent than the intracellular space. I propose that one method of measuring the coupling spring constant is to image the microstructure of pre- and early post-hypertensive cardiac chambers. By determining the relative extent of deformation of the intracellular and extracellular spaces Eqs. 29 and 31 can be employed to provide an approximate magnitude of *K*.

Under long term deforming stress (e.g. untreated hypertension) there is a sustained hydrostatic pressure differential across the membrane (Eqs. 30 and 32 are related to the true pressures by the volume fraction of the two spaces). Given long term pressure differences, there is the potential for bulk fluid flow between the two spaces. This can potentially be a mechanism by which the tissue remodels itself (i.e. how the magnitudes of the elastic moduli change).

One significant future application of this model is likely to require numerical evaluation: It is known that a two-domain treatment of the electrical behavior of heart muscle may exhibit pro-arrhythmic behavior at a distance with electrical stimulation. There is the potential for mechanical perturbations (e.g. multiple ischemic regions distributed through the muscle) to translate their deforming stresses along the collagen matrix and combine in a non-ischemic region, thereby activating stretch-activated membrane channels. The activation of these stretch-activated channels in an otherwise healthy region of tissue will lead to ectopic beats and, if activated in a quasi-refractory state, may trigger rotary wave activity (i.e. fibrillation). Thus, the chief benefit of a two-domain elastic model is to permit us to consider the mechanical behavior of the muscle averaged over many cells (the tissue approach) while still separately considering the separate spaces (the cellular approach). It is with such a model that we are able to consider whole tissue behavior and predict cellular-level dynamics.

## Declarations

### Acknowledgements

I would like to thank my colleague Natalya Melkus, B.S. at Oakland University for her very helpful discussions and suggestions regarding tissue histology and medical applications. I would also like to thank Dr. Bradley J. Roth at Oakland University for the use of his lab space, many helpful suggestions, and his postdoctoral support for this research under a grant from the U.S. National Institutes of Health (Grant ID R01EB008421).

## Authors’ Affiliations

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