Electromechanical Coupling and Wave Hypothesis

The analysis presented in this as well as the following chapter is based on the assumption that both electrical and mechanical activity in the heart are immediately electromechanically coupled entities with excitation-contraction coupling manifesting in two closely related spatial-temporal patterns. As already stated in the introduction of this chapter, in this study, electromechanical waves are under-stood to be composed of spreading waves of electrical excitation, immediately followed by a wave-like spreading of deformation, with elastic strain waves emerging in accordance with the spreading excitation. This section introduces the fundamental assumptions that were made in this thesis and explains the motivations that led to these assumptions, before conducting computer simulations and analyzing the experiments presented in chapter 6.

The biophysical mechanisms leading to excitation-contraction coupling are based on an interdepen-dence between transmembrane potential Vm of the cell and intracellular Calcium [Ca²⁺]i release, where the depolarization of the transmembrane potential triggers the release of intracellular Cal-cium, which in turn fuels the contraction of the cardiomyocyte, see figure 1.6 in the introduction. On a cellular level, coupled electromechanical waves are, as they are passing through each cell, a com-position of three subsequent waves, where an action potential (green curve) is followed by a calcium wave (black curve) and finally a wave-like onset and release of contraction (red curve). On the tis-sue level, however, this picture lacks to explain how the excitation-contraction coupling mechanism manifests inside an elastic continuum of cells, and accordingly needs to be extended, see figure 5.1.

The contraction of the cardiac muscle is a continuum mechanical quantity that arises from an entirety of cells exerting contractile forces. As a result, the deformation of the cardiac muscle can become far more complicated than the wave-like behavior of the contraction of a single cell, as depicted by the red curve in figure 1.6 or 5.1, suggests. First, the contractile forces occur along distinct preferred orientations that are given by the muscle fiber configuration inside the muscle. Next, the synchrony and coherence of the contractions and resultant shape of the deformation depend on the underlying

Chapter 5. Electromechancial Wave Pattern Reconstruction

100ms Vm/[Ca²⁺]i /F

t

Excitation Contraction

Coupling

AP

[Ca²⁺]_i-Transient Contraction

dilating contracting

exc. refr.

high conc. low conc.

Figure 5.1:Macroscopic, continuum mechanical manifestation of excitation-contraction coupling mechanism in cardiac tissue: action potential (greencurve) triggers release of intracellular calcium[Ca²⁺]i

(orangecurve), which in turn fuels contractions (redcurve) of the cell. On the tissue level, the contractions of many cells lead to a global deformation of the cardiac muscle, which can be char-acterized to be composed of elastic strain waves caused by spreading onset of contraction. The elastic strain waves are composed oftensileandcompressilestrains and rates of strain or defor-mation respectively and can be derived from the Cauch-Green or Green-Lagrangian defordefor-mation tensorsCorE.

excitation pattern orchestrating the contractile activity of the cells. Long range elastic effects may superimpose the local contractile behavior. Also, the underlying calcium cycling and active stress build-up and conversion into myofibril contraction may be heterogeneous. Lastly, in addition to mus-cle fiber anisotropy, the overall structural organization and heterogeneous elasticity of the cardiac muscle, on various hierarchical levels, may cause a highly anisotropic and heterogeneous elastic be-havior. These factors may severly constrain the applicability of the asumption postulated above.

To come to a better understanding of electromechanical wave propagation in cardiac tissue, it can be remunerative to modify and extent the elemental picture of excitation-contraction coupling, as it is depicted in figure 5.1. The figure is an adaptation fromBers,⁸³see also figure 1.6 in the introduction, and shows the basic mechanisms associated with excitation contraction-coupling inside a single car-diomyocyte, but furthermore also shows how this excitation-contraction coupling mechanism may be understood to manifest inside an elastic continuum of cells. Here, it is postulated that excitation-contraction coupling, on the tissue level, leads to a characteristic dynamical behavior of the elasticity and that this behavior can be thought of to constitute the wave-like character of electromechanical wave activity, which can also be used to describe and measure electromechanical wave propagation in cardiac tissue. Every site in the continuum of cells experiences its surrounding via long range elastic effects. That is, contractile activity at a distance may cause a tensile deformation of a local piece of tissue elsewhere. Specifically, while electrical wave activity spreads through the cardiac

λ₁(E) = 0

tr(E) λ₁(E)>0

tr(E)

λ₁(E)<0

tr(E)

dX dY dZ

P^(ijk)

Figure 5.2:Deformation measures used for the visualization of two- or three-dimensional deformation pat-terns and elastic strain waves in elastic excitable media during coupled electromechanical wave activity: tensileandcompressile deformation or rates of deformation derived from deformation tensorsEorCof material volume element or material prism, see also section 2.2 in chapter 2.

muscle and causes locally parts of the tissue to contract, other parts of the tissue experience this ac-tivity by being pulled towards these sites that exert contractile forces. Now, if the electrical acac-tivity approaches a site, which may be the site depicted in figure 5.1, and causes also this site to actively contract, it is this very moment that reveals the trespassing of an electrical wave when monitoring the deformation state of the tissue. Shortly before the electrical wave front arrives, the tissue experiences a dilation or, if in an already contracted deformed state, is dilating. As the electrical wave front passes through, the tissue starts to contract and experiences a swift change from dilating to contracting rates of deformation, to become possibly contracted, if previously in a dilated state. This situation is de-picted by the horizontal colorbars in the lower half in figure 5.1. The horizontal bar showing the blue-white-red color scheme indicates the deformation or rate of deformation of the local piece of tissue. The blue side of the bar corresponds to tissue lying in front of the approaching electrical wave front experiencing tensile strains or strain-rates. The red side of the bar corresponds to tissue lying inside the electrical or electromechanical wave experiencing compressile strains or strain-rates. The white part in between blue and red sides of the bar marks the transition from dilated or dilating tissue over a neutral or undeformed state to contracted or contracting tissue. This region coincides with the upstroke or front of the electrical wave that is also depicted by the green horizontal bar. In the following sections, this picture is used to depict the time-varying elastic deformation patterns that result in simulations of coupled electromechanical wave activity.

5.1.1 Deformation Measures

Mechanical deformation is a complex measure typically expressed in terms of deformation tensors of second order with3×3entries, see section 2.2 in chapter 2. These tensors describe the local state of deformation of a material volume element or material prism as depicted in figure 5.2. To be able to visualize two- or three-dimensional tensor fields, which describe the full deformation state of a deforming continuum body, it is necessary to reduce^{72, 81}this data to scalar-valued data:

E −→^g Eˆ (5.1.1)

The scalar-valued measureEˆ allows to visualize two- or three-dimensional patterns of deformation following, for instance, the color-scheme shown in figure 5.1, see also section 2.2 in chapter 2. Figure 5.2 shows the corresponding deformation measures that were used to depict dilating or contracting

Chapter 5. Electromechancial Wave Pattern Reconstruction

pieces of tissue throughout this thesis. The primary eigenvalueλ₁(E)as well as the squared length of the material volume element or prismtr(E)was used as an estimate for the deformation state, in particular, to detect whether a material volume element was dilated withλ1(E)>0ortr(E)>0or compressed withλ₁(E)<0ortr(E)<0:

Eˆ =

(dilated for tr(E)>0, λ₁(E)>0

contracted for tr(E)<0, λ1(E)<0 (5.1.2) Accordingly, the derivative of the scalar-valued deformation measure ∂_tEˆ was used to estimate the rate of deformation of the material volume element:

∂_tEˆ =

(dilating for tr(E)>0, λ1(E)>0

contracting for tr(E)<0, λ₁(E)<0 (5.1.3) Note that here the volume was considered to be preserved withdet(F) = 0. This chapter presents three-dimensional time-varying or four-dimensional simulation data. The data is scalar-valued due to the above described data reduction. For convenience, the use of capital letters(X, Y, Z)for de-scription of the material coordinates is omitted. see also section 2.2 in chapter 2. Throughout this chapter, the coordinate(x, y, z)refers to the material coordinate(X, Y, Z)of the continuum body if not stated otherwise.