2.1.

Model of Cardiac Electrical Activity

The propagation of an action potential in cardiac tissue is modeled by the following reaction-diffusion equation:

We simulated a 3-D slab of the myocardial wall representing a portion of the porcine right ventricular wall. We defined the $z$ direction to be transmural from epi- to endocardium, and the $x$ and $y$ directions to be parallel with the epicardial surface. The planar $xy$ dimensions of the slab were $3.6\times 3.6\mathrm{cm}$ . We considered a flat epicardial surface and an irregular trabeculated endocardial surface (see Fig. 1), simulated by varying the thickness of the slab from 6 to $10\mathrm{mm}$ . In some simulations, we considered a regular slab with uniform thickness of $8\mathrm{mm}$ .

The diffusivity tensor $D$ was defined as in Ref. 24, to account for the anisotropic properties of cardiac tissue due to transmural fiber rotation. In all simulations, the diffusivity in the longitudinal direction $D_L$ was $1{\mathrm{cm}}^2∕\mathrm{s}$ and in the transverse direction $D_T$ was $0.11{\mathrm{cm}}^2∕\mathrm{s}$ . For these values, the conduction velocities of planar waves in longitudinal and transverse directions were 65 and $22\mathrm{cm}∕\mathrm{s}$ , respectively. The transmural fiber rotation angle $\theta$ was assumed to be a linear function of the depth: $\theta \left(z\right)=k(z-6)+90\mathrm{deg}$ . The rate of rotation $k$ was set at $15\mathrm{deg}∕\mathrm{mm}$ , based on experimental measurements.²⁵ Inside the trabeculae $(z>6\mathrm{mm}),\theta$ had the constant value of $90\mathrm{deg}$ . In some simulations, we considered a constant fiber orientation of $\theta =90\mathrm{deg}$ at all depths.

We obtained intramural scroll waves using an S1-S2 cross-field stimulation protocol. The depth of the filament $Z$ was controlled by varying the thickness of the S2 electrode. A transmural cross section of such a scroll wave is shown in Fig. 2 (upper row). A typical simulation lasted for 10 scroll wave rotation cycles. We did not simulate scroll waves inside the trabeculae.

Fig. 2

Electrical and optical signals during intramural reentrant activity: (a) two snapshots of the intramural scroll wave in a transmural cross section. The filament (white disk) is located at $2.6\mathrm{mm}$ from the epicardial surface and (b) the corresponding simulated “raw” optical images of the epicardium. Variations in brightness reflect the variations in thickness. The dashed line indicates the position of the filament. (c) The same images as in (b) after processing, which includes normalization, spatial and temporal filtering (see text for details), and subtraction of the background. (d) Optical action potentials in two points indicated by asterisks in (a), (b), and (c), respectively.

To integrate Eq. 1 we used an explicit scheme, as described in Ref. 26, using a time step of $0.01\mathrm{ms}$ and a space step of $0.18\mathrm{mm}$ . The phase-field method, as presented in Ref. 27, was used to ensure no-flux boundary conditions at the irregular endocardial surface. Simulations were carried out on a parallel cluster consisting of 16 dual AMD Athlon $\mathrm{MP}2200+$ processors running at $1.8\mathrm{GHz}$ . We used the MPI library and a “domain slicing” algorithm to parallelize the code.²⁸ A typical simulation took approximately $12\mathrm{h}$ of central processing unit time.

2.2.

Synthesis of Voltage-Sensitive Optical Signals

The light propagation in cardiac tissue is described by the time-independent diffusion equation: ^{10, 19, 29, 30}

The voltage-dependent changes of the absorption properties of the dyes were modeled through the source function $Q$ in Eq. 2. We assumed that the absorption at a given point in the cardiac tissue is proportional to the intensity of the light reaching that point and depends linearly on the transmembrane potential at that point:¹⁹

Light propagation in the red spectrum is modeled using the following values for the optical parameters: ${\mu }_s^{\prime }=1.4{\mathrm{mm}}^{-1}$ and ${\mu }_a=0.12{\mathrm{mm}}^{-1}$ , yielding $D_0=0.22\mathrm{mm}$ . These were determined experimentally at a wavelength of $630\mathrm{nm}$ characteristic for absorptive dyes.³¹ The chosen parameter values yield an attenuation length $\delta =(D_0∕{\mu }_a{)}^{1∕2}$ of $1.34\mathrm{mm}$ . For comparison, we also performed simulations of the widely used fluorescent dye DI-4-ANEPPS. In this case, we used the same parameter values as in Ref. 19.

The recorded surface distribution of voltage-dependent signals was calculated using Fick’s law at the tissue boundary. To mimic the static background produced by dye molecules bound to membranes different from the cell membrane, we assumed a homogeneous voltage-independent distribution of absorption through a source $Q_{\mathrm{back}}\left({\mathit{r}}^{\prime }\right)=-A\beta \Phi \left({\mathit{r}}^{\prime }\right)$ . The background signal was calculated by solving Eq. 2 for this source term, and its amplitude was adjusted through the parameter $A$ to yield a signal-to-background ratio of 0.1, as was found in preliminary experimental studies⁴ using the absorptive dye RH-155. We also modeled noise in the optical signals through addition of a matrix of normally distributed random numbers. The SNR was set to 10, to mimic a typical optical mapping experiment.⁴

Equations 2, 3 were solved numerically using a relaxation technique.³² This was achieved by introducing a pseudotime variable $t_0$ in the optical model and a term $\partial \Phi ∕\partial t_0$ on the right-hand side of Eq. 2. An iterative scheme using finite differences (space step $0.18\mathrm{mm}$ ) was applied at each time step $t$ of the electrical model. The result ${\Phi }_n\left(\mathit{r}\right)$ of the $n$ ’th step with respect to the $t_0$ was used as an initial condition for the $n$ ’th $+1$ iteration. As an initial condition we set ${\Phi }_1\left(\mathit{r}\right)=V_m(\mathit{r},t)$ . The resulting series ${\Phi }_n\left(\mathit{r}\right)$ converged toward the solution of Eq. 2 and the iteration was stopped when the convergence criterion $\left[{\Phi }_{n-1}\right(\mathit{r})-{\Phi }_n(\mathit{r}\left)\right]∕{\Phi }_n\left(\mathit{r}\right)<0.0001$ was satisfied in all points $\mathit{r}$ of the slab. The number of iterations required varied from 50 to 3000 at different times $t$ of the wave propagation. The computation of the optical signals was implemented in $\mathrm{C}++$ and ran on a single Pentium 4 processor at $3.00\mathrm{GHz}$ .

2.3.

Analysis of Optical Signals

To mimic experiments we applied temporal and spatial filters with $3\mathrm{ms}$ and $0.9\mathrm{mm}$ kernel sizes, respectively. To detect the filament of intramural scroll waves, we used the following methods:

TSP method:^{4, 18} scroll wave activity was identified and localized by the presence of typical “zigzag” patterns. TSPs were obtained by plotting the optical signal over time along a line perpendicular to the intramural filament. The position of the filament was inferred from the area where left and right branches in the TSP intersect (see Sec. 3).

Amplitude maps:^{4, 21} in this study, we obtained amplitude maps by measuring the maximal amplitude of the optical signal in each pixel over one activation cycle. We normalized the maps for each depth of the intramural filament separately, to enable better comparison. The epicardial projection of the filament was identified as a region of minimal amplitude.

PDF maps:²² spectral analysis of the optical signal was performed using the fast Fourier transform (FFT) of the optical signal over 10 scroll wave rotations $\left(1\mathrm{s}\right)$ . The minimal PDF indicated the position of the filament’s projection on the epicardium.

3.1.

Absorptive Transillumination Optical Signal of an Intramural Scroll Wave

We began by simulating intramural scroll waves at different depths and calculating respective transillumination signals (see Sec. 2). Figure 2a shows two snapshots of membrane potential during scroll wave activity in a transmural cross section of the slab. The snapshots are separated by one half of the rotation period of the intramural scroll $\left(101\mathrm{ms}\right)$ . The filament of the scroll wave (indicated by a white disk) is located at $2.6\mathrm{mm}$ from the epicardial surface. The corresponding snapshots of the “raw” surface optical signal produced by this scroll are shown in Fig. 2b. One can see significant variations in the background signal: its intensity roughly follows the profile of the endocardial surface. It is largest where the tissue is thinnest, while the opposite is true for the thicker portions of the slab. The voltage-dependent part of the signal is unnoticeable, as both snapshots look almost identical. It can, however, be clearly observed in time recordings of individual pixels. Figure 2d shows electrical and optical action potentials recorded in a thick region (point $\mathbf{a}$ ) and a thinner region of the slab (point $\mathbf{b}$ ). The upper tracings show the electrical action potentials. The middle tracings show “raw” optical action potentials after subtraction of the background fluorescence. The effects of noise are much more pronounced in the thick portions of the slab, because of the lower signal amplitude.

The spatial excitation patterns become apparent after subtraction and normalization by the background intensity $(\mathrm{\Delta }A∕A)$ . The effects of noise can be reduced by applying spatial and temporal filters. Figure 2c shows the optical images after such processing: in the first snapshot $\left(25\mathrm{ms}\right)$ the optical signal is strongest left of the filament (dashed line), while in the second snapshot $\left(75\mathrm{ms}\right)$ the optical wave has propagated to the right side of the slab. This is consistent with the overall wave propagation depicted in the transmural cross sections [Fig. 2a].

These instantaneous optical images are insufficient to reveal the presence of an intramural scroll wave. To detect those, the analysis of the whole temporal sequence of images is required. In the following, we compare the sensitivities of three different methods for the detection of intramural scroll waves in absorptive and fluorescent transillumination.

We considered scroll waves with rectilinear filaments parallel to the epicardial surface. Figure 3a depicts $V_m$ amplitude maps (maximal amplitude over one rotation) in transmural cross sections for scroll waves at three selected depths: 1.8, 3.2 and $4.6\mathrm{mm}$ . The region of low amplitude (dark region) corresponds to the filament’s cross section or core. The core size and its shape change significantly as the scroll wave’s depth varies. The variations in core size and shape are the result of the transmural fiber rotation: in the deeper layers of the slab, the fibers are more aligned with the filament, resulting in a smaller and more circular core. Note also that the filament is located in different portions of the slab with different local thicknesses. The lateral shift of the filament is a consequence of the specific protocol used for the initiation of scroll waves (see Sec. 2).

Fig. 3

Detection of scroll waves using amplitude maps: (a) the amplitude maps of the electrical activity in a transmural cross section of the slab, the bottom tracings show the amplitude profile along the dotted line in the amplitude maps, and (b) the corresponding normalized amplitude maps of the epicardial optical signal. White indicates maximal amplitude of 1.0, while black encodes for the lowest amplitudes (different in each image). A white dashed line indicates the position of the intramural filament. The bottom tracings show the amplitude profile along the dotted line in the amplitude maps.

3.2.

Detection Using Amplitude Maps and TSPs

An image of the filament’s projection onto the epicardial surface can be obtained by calculating the amplitude map of the optical signal (see Sec. 2). Figure 3b shows amplitude maps of the optical signal for the scroll wave filaments depicted in Fig. 3a. In each map, a region of low amplitude can be distinguished, right above the filament. At $Z=1.8\mathrm{mm}$ , the dark band is quite broad and becomes narrower as the depth of the filament increases, as can be seen from the normalized amplitude profiles obtained along the dotted line in the corresponding amplitude maps. The latter can be quantified by measuring the full width at half minimum (FWHM) and the difference between minimal and maximal intensities of the amplitude map $\left(\mathrm{\Delta }\mathrm{AMP}\right)$ , as depicted in the middle tracing.

Figure 4 depicts FWHM and $\mathrm{\Delta }\mathrm{AMP}$ as a function of the filament depth (solid line). FWHM shows a significant decrease from the epicardium up to depths of $3.5\mathrm{mm}$ , followed by a plateau and slight increase at larger depths [Fig. 4a]. In contrast, $\mathrm{\Delta }\mathrm{AMP}$ has a pronounced peak in the middle of the slab $(Z=3.6\mathrm{mm})$ , whereas closer to the epi or endocardial surface $\mathrm{\Delta }\mathrm{AMP}$ is significantly smaller.

Fig. 4

Characteristics of the filament “image” on the amplitude map: (a) how the FWHM of the filament image varies with the depth $Z$ of the filament. The two curves represent simulations with fiber rotation (solid line) and without fiber rotation (dotted line). The numbers indicate the core size for each depth of the intramural scroll wave in the irregular slab. (b) How the difference between maximal and minimal amplitude $\left(\mathrm{\Delta }\mathrm{AMP}\right)$ depends on $Z$ . Simulations were performed in the irregular slab model described in Sec. 2 (solid line) and in a regular slab of uniform thickness of $8\mathrm{mm}$ (dotted line). The numbers indicate the local tissue thickness in millimeters for each depth of the filament in the irregular slab.

The rapid drop in FWHM with $Z$ in Fig. 4a is a result of the changing core size due to the fiber rotation (indicated in millimeters for each data point). When we eliminated the fiber rotation, we did not see any drop in FWHM, but only a slight increase due to the increased optical scattering. The dotted line in Fig. 4a shows FWHM when the fiber angle $\theta$ was set at the constant value of $90\mathrm{deg}$ for all layers. This yields a core size of $0.6\mathrm{mm}$ at all depths. In this case, FWHM increases from 3.1 to $4.1\mathrm{mm}$ as the depth of the filament increased from 1.8 to $4.6\mathrm{mm}$ , solely due to scattering. Both curves in Fig. 4a coincide at large depths, confirming our hypothesis on the role of core size and optical diffusion on FWHM of amplitude maps.

Significant variations of $\mathrm{\Delta }\mathrm{AMP}$ are primarily caused by local variations in thickness. The dotted line in Fig. 4b shows the results of simulations utilizing a regular slab of uniform thickness of $8\mathrm{mm}$ . Although $\mathrm{\Delta }\mathrm{AMP}$ shows some variations in the regular slab ( $\mathrm{\Delta }\mathrm{AMP}$ is reduced near the epicardial surface due to loss of photons at the boundary), the effect is rather small when compared to the irregular slab (solid line). The solid curve (varying thickness) is always above the dotted one (constant thickness) when the local tissue thickness, indicated in millimeters for each data point, is lower than $8\mathrm{mm}$ and is below the dotted curve when the local tissue thickness is higher than $8\mathrm{mm}$ . Both curves intersect at local tissue thicknesses of $8\mathrm{mm}$ . This strong correlation supports our hypothesis on the effects of tissue thickness on $\mathrm{\Delta }\mathrm{AMP}$ .

Figure 5a shows TSPs derived from the same data set as in Fig. 3. They were obtained by visualizing the spatiotemporal behavior of the optical signal along a line perpendicular to the filament’s projection. The rotating excitation gives rise to a zigzag pattern with two mutually phase-shifted branches (left and right), which is a signature of rotating waves. The center of the zigzag pattern (dashed line) is a good approximation to the filament’s location, which is depicted in Fig. 5b using $V_m$ amplitude maps in a transmural cross section. The filament is readily identifiable for all investigated depths [compare Figs. 5a and 5b].

Fig. 5

Detection of scroll wave filaments using the TSP method: (a) optical TSPs for three selected depths of the filament (dashed line); 1.8, 3.2, and $4.6\mathrm{mm}$ , where the TSPs were obtained along a line perpendicular to the filament’s epicardial projection, and (b) the respective amplitude maps of the electrical activity in a transmural cross section.

3.3.

Detection Based on PDF Maps

We discovered that PDF maps of the optical signal, which were previously used for the analysis of fibrillation,²² can also be used for the detection of intramural filaments. Figure 6a shows normalized power of the dominant frequency (PDF) maps for the optical signals emanating from intramural scroll waves at several depths (top row). In all cases, the dominant frequency was $10\mathrm{Hz}$ , which is the activation frequency of the scroll wave. For each depth of the intramural scroll wave, a narrow region of low power of the dominant frequency is observed above the filament.

Fig. 6

Detection of intramural filament using PDF maps. (a) Normalized power of the dominant frequency (PDF) maps for selected depths of the filament: 1.8, 2.5, 3.2, 3.9 and 4.6 mm. White stands for maximal power, black for minimal power. The filament appears as a region of low power. (b) Power spectra for each depth of the scroll wave in a point directly above (point $\mathbf{a}$ ) and far away from the filament (point $\mathbf{b}$ ), respectively. (c) Optical action potentials above the filament (point $\mathbf{a}$ ) and far from the filament (point $\mathbf{b}$ ) during two scroll wave rotations. The power spectra were obtained by a FFT over 10 scroll wave rotations.

Analysis of the underlying spectra [Fig. 6b] shows that the drop in power of the dominant frequency occurs due to an increase of the power in the second harmonic at $20\mathrm{Hz}$ , right above the filament (point $\mathbf{a}$ ). These effects are most pronounced for the scroll wave located at $3.2\mathrm{mm}$ below the epicardial surface, roughly in the middle of the slab. In this case, the second harmonic is comparable to the first harmonic at point $\mathbf{a}$ . The overall power is decreased by a factor of 30. Note the almost nonexistent second harmonic and large spectrum power away from the filament (point $\mathbf{b}$ ).

The increased power of the second harmonic near the filament is explained by the occurrence of dual-humped optical action potentials directly above the filament. The two humps reflect the wave crossing the line-of-sight twice over one scroll wave rotation within the tissue volume that contributes to the optical signal.^{7, 8} Figure 6c shows optical recordings during two scroll wave rotations directly above the filament (point $\mathbf{a}$ ) and away from the filament (point $\mathbf{b}$ ). Directly above the filament dual-humped action potentials are observed, whereas normal action potentials were found far from the filament. The second hump observed in the optical action potential close to the reentrant core affects the power spectrum of the optical signal by making a substantial contribution to the second harmonic at $20\mathrm{Hz}$ , thus increasing its power relative to the first harmonic at $10\mathrm{Hz}$ .

3.4.

Comparison with Fluorescent Transillumination

While absorptive transillumination gives good results for all investigated depths of the filament, this is not the case for fluorescent dyes.

Figure 7 illustrates the filament detection methods applied to fluorescent (fluo) and absorptive (abs) transillumination for an intramural scroll wave at 1.8 and $4.6\mathrm{mm}$ depth. Close to the epicardial surface $(Z=1.8\mathrm{mm})$ , the quality of the filament visualization is significantly lower in the case of fluorescent transillumination, independent of the detection method. Specifically, the TSP does not exhibit the typical zigzag pattern and the amplitude and PDF maps show only a faint and smeared projection of the filament, as opposed to the sharper images obtained using absorptive transillumination. However, closer to the illuminated endocardial surface $(Z=4.6\mathrm{mm})$ , fluorescent transillumination succeeds in detecting intramural filaments as well as absorptive transillumination does.

Fig. 7

Filament detection methods applied to fluorescent (fluo) and absorptive (abs) transillumination for an intramural scroll wave at 1.8 (left) and $4.6\mathrm{mm}$ (right) depths. Profiles of the amplitude and PDF maps were obtained along the dotted line. One can see that close to the epicardium $(Z=1.8\mathrm{mm})$ fluorescent transillumination fails to detect the intramural scroll wave, whereas close to the endocardium $(Z=4.6\mathrm{mm})$ both methods give similar results.

We quantified the quality of the filament detection using amplitude maps by investigating how the contrast of the image changed as a function of the depth of the scroll wave. Contrast was defined as ratio $\mathrm{\Delta }\mathrm{AMP}∕\mathrm{FWHM}$ . Figure 8 compares the contrast versus $Z$ for absorptive (abs) and fluorescent (fluo) transillumination. Close to the endocardial surface $(\mathrm{Z}>4.0\mathrm{mm})$ , fluorescent transillumination yields higher contrasts. However, throughout the bulk of the myocardial wall, absorptive transillumination provides the sharpest filament projections. Indeed, as the filament gets closer to the epicardium, the quality of the amplitude map deteriorates rapidly for fluorescent transillumination. This observation is in accordance with previous experimental and theoretical studies,^{9, 10} showing a skew of the optical weight toward the illuminated surface.

Fig. 8

Contrast of the amplitude maps $(\mathrm{\Delta }\mathrm{AMP}∕\mathrm{FWHM})$ versus depth of filament $Z$ (in millimeters) for absorptive (abs) and fluorescent (fluo) transillumination. The contrast was measured at a 50% level of the amplitude maps.

4.1.

Limitations of the Model

For the purpose of this study, we focused on linear intramural filaments parallel to the epicardium. In reality, filaments are likely to have more complex 3–D shapes.¹² However, experimental studies of the BZ reaction have shown that complex filament shapes can be visualized as long as the complete filament projection can be recovered off the surface.^{21, 38} In this study, our data show that in the case of absorptive transillumination all tissue layers contribute almost equally to the optical signal. This indicates the feasibility of the proposed methods for different filament shapes.

We used a monodomain approach for the reaction-diffusion properties of cardiac tissue, neglecting its bidomain structure consisting of intracellular and extracellular media. Boundary conditions at the tissue-bath and tissue-air interfaces, which can only be properly addressed using a bidomain model, have been shown to influence the dynamics of intramural filaments that are close to a boundary.³⁹ In this study, however, we investigated intramural scroll waves that were sufficiently distant from the epi- and endocardial surface (at least $1\mathrm{mm}$ ). For such deeply located filaments, it is reasonable to use a monodomain model.

Our optical model assumed Dirichlet boundary conditions at the tissue surfaces. This approach does not take into account reflection of light at the tissue-bath interface due to refractive index mismatches. This would require the use of so-called Robin or mixed boundary conditions.^{40, 41} However, our group recently showed that differences in optical resolution resulting from the use of different boundary conditions were only noticeable for electrical sources close to the tissue surfaces.¹¹ The shape of the optical action potential remained unaffected in all cases. Therefore, it is to be expected that our results obtained for deeply located intramural scroll waves should hold unchanged with Robin boundary conditions as well.