# Cardiac Electrophysiology Meshfree Modeling through the Mixed Collocation Method
## 3 Results
### Electrical Propagation in 3D Biventricular Geometry with Irregular Nodal Distribution  
We simulated electrical propagation in a porcine cardiac biventricular model under healthy conditions. The biventricular model was generated from the segmentation of a diffusion weighted-magnetic resonance image (DW-MRI) with dimensions \(128\times 128\times 83\) mm\({}^{3}\) and voxel size \(1.09\times 1.09\times 1.2\) mm\({}^{3}\). Three labels were included in the segmentation to partition the ventricular wall in endocardial, midmyocardial, and epicardial regions with a 45:25:30 ratio. Median filter smoothing with 2 mm kernel size was applied on the segmented data to ensure a smooth boundary. A tetrahedral mesh (69,621 nodes and 380,299 elements) was generated using the iso2 mesh toolkit [37]. An irregular nodal distribution was obtained from the tetrahedral mesh of the DW-MRI segmentation. The direction of myocardial fibers was obtained by computing the diffusion tensor (DT) field for each voxel of the DW-MRI. In brief, a linear minimum mean squared error \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**h** & **P1** & **P2** & **P3** & **P4** & **P5** & **P6** & **P7** & **P8** & **C** \\ \hline \multicolumn{10}{c}{MCM-RPI activation time (ms)} \\
0.5 & 1 & 51 & 23 & 59 & 99 & 115 & 107 & 120 & 56 \\
0.2 & 1 & 34 & 11 & 36 & 37 & 53 & 42 & 58 & 26 \\
0.1 & 1 & 30 & 8 & 33 & 29 & 42 & 31 & 44 & 2 \\ \hline \multicolumn{10}{c}{MCM-MKI activation time (ms)} \\
0.5 & 1 & 49 & 24 & 60 & 99 & 111 & 101 & 119 & 58 \\
0.2 & 1 & 33 & 12 & 36 & 36 & 55 & 39 & 55 & 26 \\
0.1 & 1 & 30 & 8 & 32 & 29 & 42 & 31 & 42 & 2 \\ \hline \multicolumn{10}{c}{FEM activation time (ms)} \\
0.5 & 1 & 49 & 22 & 58 & 94 & 109 & 98 & 111 & 54 \\
0.2 & 1 & 31 & 11 & 35 & 35 & 51 & 39 & 54 & 25 \\
0.1 & 1 & 29 & 8 & 31 & 27 & 41 & 29 & 41 & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Activation times at the corners (P1–P8) and the center (C) of the 3D cuboid benchmark problem. Figure 4: Local activation time (LAT) map for the MCM-MKI solution of the Niederer benchmark [35] with space discretization \(h=0.1\) mm. (LMMSE) filter was applied to reduce Rician noise before computing the DT fields [38]. DT fields were computed by minimizing the Riemannian distance between an evaluated tensor spline and the DW-MRI data [39]. The principal eigenvector, which was obtained by solving the eigenvalue problem for the DT of each voxel, was considered as the myocardial fiber orientation of the corresponding voxel. The myocardial fiber orientations at the nodes of the tetrahedral mesh were obtained by interpolating the orientations of the image voxels using the MKI approximants. Electrical propagation was simulated using a value for the diffusion coefficient of \(d_{0}=0.002\) cm\({}^{2}\)/ms in the longitudinal direction of the myocardial fibers and a transverse-to-longitudinal conductivity ratio value of \(\rho=0.15\). The electrophysiology of the ventricular myocardial tissue was represented by the O'Hara et al. model, as in previous examples. The points of earliest activation across the ventricles were identified by coupling the biventricular model with a network of Purkinje fibers generated using a fractal-tree generation algorithm [40]. We applied stimuli with \(t_{d}=1\) ms, \(t_{T}=1\) s, and amplitude twice the diastolic threshold onto the Purkinje-myocardial junctions (PMJs) which were identified from the terminal nodes of the Purkinje network. Electrical impulse propagation was simulated using the MCM with RPI and MKI approximants as well as the FEM for a simulation time \(t_{s}=150\) ms.  
Rather than constructing dilated support domains, we chose to construct support domains using the nearest-neighbor approach due to the irregular distribution of the nodes in the tetrahedral mesh. For such distributions, dilated support domains require the inclusion of a very large number of neighboring nodes to ensure numerical stability.