Planar Electromechanical Wave Propagation

Planar electromechanical wave propagation was simulated in cubic-shaped bulk media by setting the excitatory variableuof all units on one side of the cube above threshold. As a result, a planar elec-trical wave would propagate across the medium, see figures 5.3 and 5.4. In the following figures, the shape of the deforming medium is indicated by gray edges. The electrical wave is indicated in green using a ray-casting volume rendering technique that renders opaque green color at locations with u > 0.1and transparent empty space otherwise. The muscle fiber alignment is indicated by grey arrows. The internal Green-Lagrangian strain E is indicated using the red-white-blue color-code presented in figure 5.1 and the data reductionEˆ = g(E)as discussed in the previous section.

Similarly as for the excitatory variableu, ray-casting volume rendering is used to indicate dilating material segments in blue and contracting material segments in red and undeformed material seg-ments as transparent empty space respectively. The visualization technique allows to display both time-varying, three-dimensional scalar fields of the electrical activityu(x, y, z, t)and the mechani-cal activityE(x, y, z, t).ˆ

Figure 5.3 shows planar electromechanical wave propagation in a cubic-shaped bulk medium with uniform linearly transverse muscle fiber alignment along the horizontale_x-axis (gray arrows). The electrical wave (green) starts spreading into the cube from the left face of the cube and traverses the cube along theex-axis as a planar wave with the wavelength being comparably large to the di-mensions of the cube. Note that the wave propagation direction coincides with the direction of the fiber alignment. As the wave traverses the cube it deforms accordingly. The tissue within and be-hind the electrical wave front experiences shortening due to cardiac cells contracting. As a result,

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Figure 5.3:Planar electromechanical wave propagation in bulk medium with uniform, linearly transverse mus-cle fiber alignment and wave propagation direction coinciding with fiber alignment: dynamic be-havior of the contraction exhibits strong gradients between contractile and dilated regions inside the tissue matching the orientation and propagation direction of electrical wave (electrical wave:

greennormalized units of excitation[0,1]→[−80mV,20mV], dilation:blue, contraction:red).

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Figure 5.4:Planar electromechanical wave propagation in bulk medium with rotating muscle fiber alignment;

most fibers perpendicular to the propagation direction of the wave (electrical wave:green normal-ized units of excitation[0,1]→[−80mV,20mV], dilation:blue, contraction:red).

one can observe a compressile strain wave front inside and behind the electrical wave front, indi-cated in red. Tissue in front of the approaching electrical wave front, experiencing long range elastic effects, is being pulled towards the wave front and dilates. One can observe a tensile strain wave front, indicated in blue, in front of the electrical wave. Overall, one can observe two clearly distinct tissue regions, one within and one in front of the wave, experiencing compressile and tensile strain respectively, with a strong gradient between the two regions. This effect is particularly strong as the contraction is directed uniformly along theex-axis. Note that the deformation pattern corresponds to the picture postulated in figure 5.1. Figure 5.4 shows planar electromechanical wave propagation in a cubic-shaped bulk medium with rotational anisotropic muscle fiber alignment. Again, the electrical wave (green) starts spreading into the cube from the left face of the cube and traverses the cube along thee_x-axis as a planar wave with the wavelength being comparably large to the dimensions of the cube. However, this time the wave propagation direction does not coincides with the fiber alignment.

The fiber configuration is not uniform, but rotates and is aligned at most locations perpendicularly to the propagation direction of the electrical wave. Accordingly, the deformation is very different from the one shown in figure 5.3, and, in particular, the strong gradient between dilating and contracting regions of the tissue can not be observed. Nevertheless, one can observe a compressile strain wave front inside and behind the electrical wave front, indicated in red.

Planar electromechanical wave propagation can be observed in the heart during pacing, using

stim-Chapter 5. Electromechancial Wave Pattern Reconstruction

Figure 5.5:Elastic behavior of central material volume element during trespassing of planar electrical wave (green) in elastic excitable cubic-shaped bulk medium with uniform linearly transverse (left, see also figure 5.3) and rotating muscle fiber alignment (right, see also figure 5.4): the dynamic be-havior of the elasticity (red) allows to estimate the trespassing of the wave (black bar).

ulation electrodes and applying stimulation pulses at one site on the ventricular surface to induce electrical wave activity that spreads over the ventricles away from that site. During sinus activity, the electrical activity starts spreading from multiple sites, which correspond to the ejection sites of the Purkinje system. The simulations were conducted with cubic-shaped bulk media of size41×31×41 particles or 40×30 ×40 hexaedral cells using equations 2.1.13-2.1.14. Parameters were set to a= 0.05,b= 0.5,µ1= 0.1,µ2= 0.3,kT = 1.9,k= 8.0,ε= 0.01,D= 0.001,dt= 0.02. Simu-lations were run for 3000 time steps. Note, that the wavelength is comparably large as the dimensions of the cube.

5.2.1 Muscle Fiber Anisotropy

The two figures 5.3 and 5.4 demonstrate that deformations can differ significantly from each other if the tissues retain different underlying muscle fiber configurations. In particular, the deformations can differ from each other even though the underlying electrical pattern causing the deformation is the exact same pattern, compare 5.3 and 5.4. The muscle fiber anisotropy of the cardiac substrate determines the deformation and its time-varying behavior during the contraction, compare also with the work by Otani et al.¹⁹² This behavior strongly determines how deformation patterns emerge inside the cardiac muscle and affects how the pattern can be analyzed to reconstruct the underlying

electrical wave pattern, see following section. The simulations show that the angle between local muscle fiber orientation and the wave normal of the electrical wave determines the strength of the deformation around that area inside the tissue.

5.2.2 Wave Front Detection from Dynamic Features of Elasticity

Figure 5.5 shows the according traces of different deformation measures obtained from the central material volume element of each of the two bulk media that retain differing muscle fiber configu-rations. In addition to the deformation state, that is the strain derived from the primary eigenvalue or the squared length of the deformation tensorE, see also section 5.1.1, the rate of deformation or strain-rate as well as the strain-rate acceleration are shown (from top to bottom). All three traces exhibit strongly dynamic behavior with the trespassing of the electrical wave, for the case of uniform linearly transverse fiber alignment as well as for the case with rotational muscle fiber alignment. The strain undergoes a transition from dilated to compressed deformation states. The strain-rate accord-ingly switches sign from dilating to contracting rates of deformation at the peak of the dilation with high rates of change and a negative peak in the strain-rate acceleration accordingly. This dynamic elastic behavior allows the automatic detection of the trespassing of the electrical wave as the average time of the peak of dilation and the zero-crossing of the strain-rate from dilating to contracting rates of deformation and the fastest contracting acceleration rates of deformation coincides very well with the upstroke of the electrical wave front, see black bars in figure 5.5. In the following sections, this techniques is used to detect electromechanical wave fronts during arbitrary electromechanical wave activity. For every transition of an electrical wave through each material point inside the tissue, it is possible to observe a sequence of stretching, shortening and relaxation of the tissue with the rate of deformation becoming extremal with the onset of contraction. Accordingly, the above described detection technique reveals the isochronal spatial-temporal structure of periodic activation pattern.

Note that the order of magnitude of the rate of deformation is smaller in the case of rotational muscle fiber alignment. As the wave propagation direction and the local muscle fiber alignment differ, the aforementioned effects are not as pronounced and the detection of the trespassing of an electrome-chanical wave may become superimposed by other elastic activity if analyzing noisy data obtained in experiments with real cardiac tissue preparations.