## 2 Materials and Methods
### Left Bundle Branch Block Simulation with Personalized Parameters  
An initial stimulus was set in the atrio-ventricular (AV) node, with an average of 60 particles, being identified from the earliest activated points in the EAM data of each case, to initiate the simulated electrical pattern over the two ventricles. The Purkinje (PK) system, which has fast conduction velocities, needs to be incorporated in the model for simulating a LBBB and disrupt the normal electrical propagation in the LV branch. Therefore, particles located in the lower (i.e., closer to the apex) half of the endocardial RV (around 500 particles) and the lower third of the LV (around 300 particles), if no scar was present, were labeled as Purkinje (see Figure 2), following the distribution of PK-myocardial junctions found in PK-based simulation studies [45]. Including PK particles in the LV, could seem contradictory for simulating LBBB electrical patterns. However, it was necessary to consider the LV retrograde activation due to the transmurality of the PK system in pigs [46], leading to latest activation points being located at the basal LV in several cases. As other participants at the CRT-EPiggy19 challenge [28; 29], we also added a second impulse in the electrophysiological simulations to replicate an early activation of the RV epicardium observed in the EAM data (see \(RV_{epi}\) initial activation point in Figure 3). Besides the expected activation induced by the AV node, the septomarginal trabecula may have a role in the fast activation of the RV that needs to be incorporated to replicate the  
Figure 2: Biventricular geometries with particles labelled as regular myocardial tissue (in blue) and Purkinje system (in red), used for the meshless solver. Top: point cloud representation. Bottom: mesh-based triangulation from applying the Delaunay algorithm to the cloud of points, for visualization purposes. electrophysiological measurements. The dynamics of the electrical pattern in LBBB cases are displayed in Figure 3, starting from the two stimulus (1 in the Figure 3), followed by the propagation to the biventricular apex (2 in the Figure 3), and the propagation from the LV apex to the base (3 in the Figure 3), with fast activation in the endocardi and slow in the epicardium. The local conduction velocity (CV) values defined in each geometry, guiding the wave propagation speed in the direction established by the modeled cardiomyocyte orientation, was one of the main parameters affecting the simulated electrical pattern. However, it is not simple to set up the number of heterogeneous conductivity regions: different values at each voxel would both be impractical (too many parameters to optimise) and does not make sense in relation with the sparsity of the available electonanatomical data; too few regions would not consider the existing CV heterogeneity (e.g., faster CV in PK system, complex electrical propagation in the septum due to discontinuities in cardiomyocyte orientation [41], presence of scar, etc.). Consequently, we performed a sensitivity analysis to determine the optimal number of different regions with local conductivities to optimize, from only a single region considering the whole biventricular geometry, to 21 regions including the 17 AHA segments. In total, the following seven regional CV configurations were tested:  
* 1 region (_whole biventricular geometry_). * Purkinje system  
- Purkinje system
- 17 LV_AHA segments  
The optimization of the CV distribution in each analyzed case was performed with the constrained non-linear Sequential Least Squares Programming algorithm. The cost function was based on minimizing the root mean square error of each particle activation time  
Figure 3: Electroanatomical maps of an infarcted and a non-infarcted training case in left bundle branch block condition. In (**a**,**b**), both initial activation points (1) in the RV epicardial layer (red circle) and LV endocardial one (yellow circle) are shown.