# Cardiac Electrophysiology Meshfree Modeling through the Mixed Collocation Method
## 4 Discussion  
In this study, we derived the mixed collocation method (MCM) to solve the monodomain model for the numerical simulation of cardiac electrophysiology. We considered two different interpolating trial functions, the radial point interpolation (RPI) and the moving kriging interpolation (MKI). We solved several numerical examples in 2D and 3D domains, comparing the MCM solution with a solution obtained by the FEM. The accuracy of MCM solutions was found to be similar for both RPI and MKI approximants. However, MKI was more efficient since, in contrast to RPI, it does not require polynomial enrichment. It should be noted that the optimal values of the parameters for the RPI and MKI approximations were obtained manually in this study. However, optimization techniques such as those based on minimizing a predictor function could be applied to obtain the values of these parameters automatically. For further details on the topic we refer the reader to [43]. In all the numerical examples in 2D tissue sheets and 3D tissue slabs, good agreement was found between MCM and FEM solutions. The convergence analysis in 3D tissue slabs demonstrated that the MCM solution improved for larger support domains as the number of included collocation points was increased (see Figure 3). However, as expected, as the size of the support domain increased, the execution time was also increased, as can be seen in Figure 2. Therefore, a compromise between accuracy and execution time must be made. The value \(a_{sd}=2.8\) represents a good compromise. In the case of the biventricular anatomy model, including the 150 nearest neighbors of each field node in its support domain was found to be an optimal choice balancing accuracy and memory footprint. Local activation time (LAT) maps generated with the MCM were shown to be in good agreement with those obtained by the FEM. The mean LAT difference between the MCM and the FEM was reduced from 18% for irregular nodal distribution to 4% for immersed grid nodal distribution. Normalized histogram intersection with the FEM solution remained similar for both nodal distributions (over 0.93). These results, in combination with the simplicity of field nodes generation with the immersed grid approach, demonstrate an important advantage of the MCM over the FEM, especially for clinical applications where clinical image data could be treated directly as a meshfree model by the MCM, while the FEM requires the generation of a mesh. In addition, the FEM requires mesh preprocessing prior to the simulation to define parameters such as ventricular tissue partition and myocardial fiber orientation. The MCM with the immersed grid approach does not require any preprocessing since there is a one-to-one relationship between image voxels and field nodes. Anatomic partition data and myocardial fiber orientation defined on the voxels of the image are assigned directly to the corresponding immersed grid nodes. The flexibility of the MCM renders it a useful numerical method that could be extended to other applications, such as the investigation of soft tissue mechanical response [17]. In terms of execution efficiency, the simulation time for the MCM was comparable to the FEM for the 2D tissue sheet simulations. In the simulation of the biventricular anatomy model, the MCM presented better performance, with reductions in computational time of up to 28% compared to the FEM for both irregular and immersed grid nodal distributions. This was due to the size of the critical time step that was used during the adaptive explicit integration using the DAETI algorithm. In realistic models of complex geometry, such as the biventricular anatomy, small mesh elements restricting the size of the critical time step may arise. This limitation is relaxed when meshfree approximants are used. Therefore, the MCM presents higher efficiency despite the higher number of support domain nodes. For more information regarding the choice of the critical time step in DAETI, we refer the reader to [33].