Using Gmsh#
In this demo we will show how to generate Purkinje networks on an idealized bi-ventricular geometry generated with gmsh. In order to create this geometry with will use another library called cardiac-geometries which can be installed with pip, i.e
pip install cardiac-geometries
Note also that to create this geometry you need gmsh installed with OpenCASACADE. Normally this will come with gmsh if you install it from pypi but if you happen to be on a Mac M1, then you need to build this from source. In this case it might be easier to use the pre-built docker container that we have made.
First we will import the necessary libraries
import numpy as np
import logging
from pathlib import Path
import meshio # pip install meshio
from fractal_tree import generate_fractal_tree, FractalTreeParameters, Mesh
And set the logging level to INFO
logging.basicConfig(level=logging.INFO)
We will also set a seed for the random number generator so that the results we generate are reproducible.
np.random.seed(1234)
The mesh that we create from gmsh is not directly possible to visualize in Paraview, so in order to do that we create a simple helper function for converting the mesh to XDMF format which is possible to visualize in Paraview.
def create_mesh(mesh, cell_type):
cells = mesh.get_cells_type(cell_type)
cell_data = mesh.get_cell_data("gmsh:geometrical", cell_type)
out_mesh = meshio.Mesh(
points=mesh.points,
cells={cell_type: cells},
cell_data={"name_to_read": [cell_data]},
)
return out_mesh
We will now create the mesh. If the mesh already exist we skip this step. Here we need to import cardiac-geometries and call the biv_ellipsoid function from the gmsh sub-package. We alse set the characteristic length to 0.1 which will generate a finer mesh that what is the default.
path = Path("biv_ellipsoid.msh")
if not path.is_file():
# Generate mesh with cardiac geometries
import cardiac_geometries
cardiac_geometries.gmsh.biv_ellipsoid(path, char_length=0.1)
Info : Meshing 1D...
Info : [ 0%] Meshing curve 1 (BSpline)
Info : [ 10%] Meshing curve 3 (TrimmedCurve)
Info : [ 20%] Meshing curve 5 (BSpline)
Info : [ 20%] Meshing curve 6 (BSpline)
Info : [ 30%] Meshing curve 7 (BSpline)
Info : [ 30%] Meshing curve 8 (BSpline)
Info : [ 40%] Meshing curve 9 (BSpline)
Info : [ 40%] Meshing curve 10 (BSpline)
Info : [ 40%] Meshing curve 11 (BSpline)
Info : [ 60%] Meshing curve 14 (TrimmedCurve)
Info : [ 60%] Meshing curve 16 (BSpline)
Info : [ 70%] Meshing curve 17 (BSpline)
Info : [ 80%] Meshing curve 21 (TrimmedCurve)
Info : Done meshing 1D (Wall 0.152965s, CPU 0.153335s)
Info : Meshing 2D...
Info : [ 0%] Meshing surface 1 (BSpline surface, Frontal-Delaunay)
Info : [ 20%] Meshing surface 2 (Plane, Frontal-Delaunay)
Info : [ 30%] Meshing surface 3 (BSpline surface, Frontal-Delaunay)
Info : [ 40%] Meshing surface 4 (BSpline surface, Frontal-Delaunay)
Info : [ 50%] Meshing surface 5 (BSpline surface, Frontal-Delaunay)
Info : [ 70%] Meshing surface 6 (BSpline surface, Frontal-Delaunay)
Info : [ 80%] Meshing surface 7 (BSpline surface, Frontal-Delaunay)
Info : [ 90%] Meshing surface 8 (BSpline surface, Frontal-Delaunay)
Info : Done meshing 2D (Wall 1.02787s, CPU 1.02377s)
Info : Meshing 3D...
Info : 3D Meshing 1 volume with 1 connected component
Info : Tetrahedrizing 11659 nodes...
Info : Done tetrahedrizing 11667 nodes (Wall 0.219591s, CPU 0.208758s)
Info : Reconstructing mesh...
Info : - Creating surface mesh
Info : - Identifying boundary edges
Info : - Recovering boundary
Info : Done reconstructing mesh (Wall 0.501053s, CPU 0.483253s)
Info : Found volume 6
Info : It. 0 - 0 nodes created - worst tet radius 36.9896 (nodes removed 0 0)
Info : It. 500 - 491 nodes created - worst tet radius 2.61427 (nodes removed 0 9)
Info : It. 1000 - 990 nodes created - worst tet radius 2.19749 (nodes removed 0 10)
Info : It. 1500 - 1490 nodes created - worst tet radius 1.96234 (nodes removed 0 10)
Info : It. 2000 - 1988 nodes created - worst tet radius 1.80497 (nodes removed 0 12)
Info : It. 2500 - 2488 nodes created - worst tet radius 1.70145 (nodes removed 0 12)
Info : It. 3000 - 2988 nodes created - worst tet radius 1.6309 (nodes removed 0 12)
Info : It. 3500 - 3488 nodes created - worst tet radius 1.55432 (nodes removed 0 12)
Info : It. 4000 - 3988 nodes created - worst tet radius 1.49442 (nodes removed 0 12)
Info : It. 4500 - 4488 nodes created - worst tet radius 1.44216 (nodes removed 0 12)
Info : It. 5000 - 4988 nodes created - worst tet radius 1.4001 (nodes removed 0 12)
Info : It. 5500 - 5488 nodes created - worst tet radius 1.36567 (nodes removed 0 12)
Info : It. 6000 - 5988 nodes created - worst tet radius 1.33222 (nodes removed 0 12)
Info : It. 6500 - 6488 nodes created - worst tet radius 1.304 (nodes removed 0 12)
Info : It. 7000 - 6987 nodes created - worst tet radius 1.27555 (nodes removed 0 13)
Info : It. 7500 - 7485 nodes created - worst tet radius 1.25059 (nodes removed 0 15)
Info : It. 8000 - 7985 nodes created - worst tet radius 1.22654 (nodes removed 0 15)
Info : It. 8500 - 8485 nodes created - worst tet radius 1.20464 (nodes removed 0 15)
Info : It. 9000 - 8984 nodes created - worst tet radius 1.18422 (nodes removed 0 16)
Info : It. 9500 - 9484 nodes created - worst tet radius 1.16676 (nodes removed 0 16)
Info : It. 10000 - 9983 nodes created - worst tet radius 1.1665 (nodes removed 0 17)
Info : It. 10500 - 10483 nodes created - worst tet radius 1.13336 (nodes removed 0 17)
Info : It. 11000 - 10983 nodes created - worst tet radius 1.11797 (nodes removed 0 17)
Info : It. 11500 - 11482 nodes created - worst tet radius 1.10346 (nodes removed 0 18)
Info : It. 12000 - 11981 nodes created - worst tet radius 1.09004 (nodes removed 0 19)
Info : It. 12500 - 12480 nodes created - worst tet radius 1.07649 (nodes removed 0 20)
Info : It. 13000 - 12979 nodes created - worst tet radius 1.06387 (nodes removed 0 21)
Info : It. 13500 - 13479 nodes created - worst tet radius 1.05069 (nodes removed 0 21)
Info : It. 14000 - 13979 nodes created - worst tet radius 1.03838 (nodes removed 0 21)
Info : It. 14500 - 14479 nodes created - worst tet radius 1.02799 (nodes removed 0 21)
Info : It. 15000 - 14978 nodes created - worst tet radius 1.0175 (nodes removed 0 22)
Info : It. 15500 - 15477 nodes created - worst tet radius 1.00736 (nodes removed 0 23)
Info : 3D refinement terminated (27501 nodes total):
Info : - 47 Delaunay cavities modified for star shapeness
Info : - 23 nodes could not be inserted
Info : - 137565 tetrahedra created in 1.11039 sec. (123888 tets/s)
Info : Done meshing 3D (Wall 2.25248s, CPU 2.2376s)
Info : Optimizing mesh...
Info : Optimizing volume 6
Info : Optimization starts (volume = 28.3238) with worst = 0.000172193 / average = 0.774877:
Info : 0.00 < quality < 0.10 : 307 elements
Info : 0.10 < quality < 0.20 : 868 elements
Info : 0.20 < quality < 0.30 : 1540 elements
Info : 0.30 < quality < 0.40 : 2531 elements
Info : 0.40 < quality < 0.50 : 3815 elements
Info : 0.50 < quality < 0.60 : 6448 elements
Info : 0.60 < quality < 0.70 : 14471 elements
Info : 0.70 < quality < 0.80 : 33208 elements
Info : 0.80 < quality < 0.90 : 50488 elements
Info : 0.90 < quality < 1.00 : 23889 elements
Info : 2676 edge swaps, 71 node relocations (volume = 28.3238): worst = 0.000384766 / average = 0.787162 (Wall 0.0969694s, CPU 0.097214s)
Info : 2698 edge swaps, 74 node relocations (volume = 28.3238): worst = 0.0182108 / average = 0.787253 (Wall 0.125742s, CPU 0.126024s)
Info : No ill-shaped tets in the mesh :-)
Info : 0.00 < quality < 0.10 : 1 elements
Info : 0.10 < quality < 0.20 : 1 elements
Info : 0.20 < quality < 0.30 : 5 elements
Info : 0.30 < quality < 0.40 : 2538 elements
Info : 0.40 < quality < 0.50 : 3667 elements
Info : 0.50 < quality < 0.60 : 6237 elements
Info : 0.60 < quality < 0.70 : 14421 elements
Info : 0.70 < quality < 0.80 : 33383 elements
Info : 0.80 < quality < 0.90 : 51082 elements
Info : 0.90 < quality < 1.00 : 23794 elements
Info : Done optimizing mesh (Wall 0.378676s, CPU 0.371995s)
Info : 27501 nodes 159050 elements
Info : Optimizing mesh (Netgen)...
Info : Optimizing volume 6
Info : CalcLocalH: 27501 Points 135129 Elements 23314 Surface Elements
Info : Remove Illegal Elements
Info : 929 illegal tets
Info : SplitImprove
Info : badmax = 146.709
Info : 149 splits performed
Info : SwapImprove
Info : 226 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : 495 illegal tets
Info : SplitImprove
Info : badmax = 146.709
Info : 138 splits performed
Info : SwapImprove
Info : 34 swaps performed
Info : SwapImprove2
Info : 3 swaps performed
Info : 93 illegal tets
Info : SplitImprove
Info : badmax = 620.627
Info : 30 splits performed
Info : SwapImprove
Info : 4 swaps performed
Info : SwapImprove2
Info : 1 swaps performed
Info : 2 illegal tets
Info : SplitImprove
Info : badmax = 620.627
Info : 1 splits performed
Info : SwapImprove
Info : 0 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : 0 illegal tets
Info : Volume Optimization
Info : CombineImprove
Info : 973 elements combined
Info : ImproveMesh
Info : Total badness = 181266
Info : Total badness = 173842
Info : SplitImprove
Info : badmax = 126.124
Info : 0 splits performed
Info : ImproveMesh
Info : Total badness = 173842
Info : Total badness = 172716
Info : SwapImprove
Info : 7772 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : ImproveMesh
Info : Total badness = 159217
Info : Total badness = 155746
Info : CombineImprove
Info : 210 elements combined
Info : ImproveMesh
Info : Total badness = 153873
Info : Total badness = 153420
Info : SplitImprove
Info : badmax = 118.446
Info : 0 splits performed
Info : ImproveMesh
Info : Total badness = 153420
Info : Total badness = 153345
Info : SwapImprove
Info : 1038 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : ImproveMesh
Info : Total badness = 152529
Info : Total badness = 151926
Info : CombineImprove
Info : 52 elements combined
Info : ImproveMesh
Info : Total badness = 151470
Info : Total badness = 151380
Info : SplitImprove
Info : badmax = 118.41
Info : 0 splits performed
Info : ImproveMesh
Info : Total badness = 151380
Info : Total badness = 151364
Info : SwapImprove
Info : 345 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : ImproveMesh
Info : Total badness = 151235
Info : Total badness = 151037
Info : Done optimizing mesh (Wall 12.0335s, CPU 12.0287s)
Info : Writing 'biv_ellipsoid.msh'...
Info : Done writing 'biv_ellipsoid.msh'
The generated mesh will be saved to the given path. We can now load this mesh using meshio.
msh = meshio.read(path)
For pure visualization purposes we will also save the mesh in .xdmf format so that we can visualize the geometry in together with the Purkinje networks in the end.
tetra_mesh = create_mesh(msh, "tetra")
meshio.write(path.with_suffix(".xdmf"), tetra_mesh)
For this example we will generate the Purkinje networks on the left ventricular endocardium. This mesh comes with some physical surfaces with different tags. These tags are stored in the field_ data attribute
print(msh.field_data)
{'BASE': array([1, 2]), 'ENDO_LV': array([2, 2]), 'ENDO_RV': array([3, 2]), 'EPI': array([4, 2])}
We will use the tag for the ENDO_LV
tag = msh.field_data["ENDO_LV"][0]
We also need to extract the relevant cells with this tag
inds = [i for i, x in enumerate(msh.cell_data["gmsh:physical"]) if x[0] == tag]
connectivity = np.vstack([msh.cells[i].data for i in inds])
The next task is to define the starting point for the network. We would like this to be a point close to the base of the left ventricular endocardial septum. You could use Paraview to find approximately what the coordinate of this point should be. In this case a suitable choice for the initial node would the \((0, 1, 0)\).
init_node = [0, 1, 0]
Now that we have the approximate coordinate, we need to find the closest node in the mesh. We can do this by selecting the point with the shortest distance to this node.
verts = msh.points
index = np.linalg.norm(np.subtract(verts, init_node), axis=1).argmin()
We create the mesh, and pass in the initial node
mesh = Mesh(verts=verts, connectivity=connectivity, init_node=verts[index, :])
We also set the number of generations to 15 and we set the initial direction to point in the positive \(x\)-direction which is the direction from the base towards the apex (this can also be found by visualizing the mesh in Paraview). We also specify the initial length to be 3. This is about the length of the septal wall, so that the first branch will represent the His bundle. We set the length to 0.2 which will be the length each successive branch.
param = FractalTreeParameters(
filename="biv-line-lv",
init_length=3.0,
N_it=15,
length=0.25,
initial_direction=np.array([1, 0, 0]),
)
Next we create the Purkinje networks for the LV
generate_fractal_tree(mesh, param)
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INFO:fractal_tree.tree:Finished growing, writing paraview file
FractalTreeResult(branches={0: <fractal_tree.branch.Branch object at 0x7f13e5aeb280>, 1: <fractal_tree.branch.Branch object at 0x7f13a9eb0cd0>, 2: <fractal_tree.branch.Branch object at 0x7f13a9eb0b20>, 3: <fractal_tree.branch.Branch object at 0x7f13a9ef4490>, 4: <fractal_tree.branch.Branch object at 0x7f13a9ef4610>, 5: <fractal_tree.branch.Branch object at 0x7f13a9ef4430>, 6: <fractal_tree.branch.Branch object at 0x7f13a9ef4640>, 7: <fractal_tree.branch.Branch object at 0x7f13a9ef46a0>, 8: <fractal_tree.branch.Branch object at 0x7f13a9ef45b0>, 9: <fractal_tree.branch.Branch object at 0x7f13a9ef45e0>, 10: <fractal_tree.branch.Branch object at 0x7f13a9ef4550>, 11: <fractal_tree.branch.Branch object at 0x7f13a9ef44c0>, 12: <fractal_tree.branch.Branch object at 0x7f13a9ef44f0>, 13: <fractal_tree.branch.Branch object at 0x7f13a9ef43d0>, 14: <fractal_tree.branch.Branch object at 0x7f13a9ef4370>, 15: <fractal_tree.branch.Branch object at 0x7f13a9ef4220>, 16: <fractal_tree.branch.Branch object 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(11533, 11742), (11330, 11743), (11743, 11744), (11744, 11745), (11745, 11746), (11746, 11747), (11330, 11748), (11748, 11749), (11749, 11750), (11582, 11751), (11751, 11752), (11752, 11753), (11753, 11754), (11754, 11755), (11755, 11756), (11756, 11757), (11757, 11758), (11758, 11759), (11759, 11760), (11760, 11761), (11761, 11762), (11762, 11763), (11763, 11764), (11764, 11765), (11765, 11766), (11766, 11767), (11767, 11768), (11768, 11769), (11769, 11770), (11770, 11771), (11771, 11772), (11772, 11773), (11773, 11774)])
Fig. 2 Idealized bi-ventricular geometry with generated Purkinje networks on the LV endocardium originating from the basal septum#
Once we have done this for the LV, doing it for the RV is completely the same. We first need to grab the correct tag
tag = msh.field_data["ENDO_RV"][0]
extract the relevant cells with this tag
inds = [i for i, x in enumerate(msh.cell_data["gmsh:physical"]) if x[0] == tag]
connectivity = np.vstack([msh.cells[i].data for i in inds])
and defining the initial node (again, we use Paraview to find this location).
init_node = [0, 1.5, 0.15]
verts = msh.points
index = np.linalg.norm(np.subtract(verts, init_node), axis=1).argmin()
mesh = Mesh(verts=verts, connectivity=connectivity, init_node=verts[index, :])
We use the same parameters as for the RV, expect for changing the filename to where it is saved.
param = FractalTreeParameters(
filename="biv-line-rv",
init_length=3.0,
N_it=15,
length=0.25,
initial_direction=np.array([1, 0, 0]),
)
generate_fractal_tree(mesh, param)
0%| | 0/15 [00:00<?, ?it/s]
13%|█▎ | 2/15 [00:00<00:00, 13.70it/s]
33%|███▎ | 5/15 [00:00<00:00, 21.43it/s]
60%|██████ | 9/15 [00:00<00:00, 28.29it/s]
100%|██████████| 15/15 [00:00<00:00, 42.04it/s]
INFO:fractal_tree.tree:Finished growing, writing paraview file
FractalTreeResult(branches={0: <fractal_tree.branch.Branch object at 0x7f13a25ed3f0>, 1: <fractal_tree.branch.Branch object at 0x7f13a9febee0>, 2: <fractal_tree.branch.Branch object at 0x7f13a9eb0e50>, 3: <fractal_tree.branch.Branch object at 0x7f13a25ecd00>, 4: <fractal_tree.branch.Branch object at 0x7f13a25eee00>, 5: <fractal_tree.branch.Branch object at 0x7f13a25ee380>, 6: <fractal_tree.branch.Branch object at 0x7f13a25eee30>, 7: <fractal_tree.branch.Branch object at 0x7f13a25eed70>, 8: <fractal_tree.branch.Branch object at 0x7f13a25edd50>, 9: <fractal_tree.branch.Branch object at 0x7f13a25eedd0>, 10: <fractal_tree.branch.Branch object at 0x7f13a25eeb90>, 11: <fractal_tree.branch.Branch object at 0x7f13a25eeec0>, 12: <fractal_tree.branch.Branch object at 0x7f13a25eefe0>, 13: <fractal_tree.branch.Branch object at 0x7f13a25eef20>, 14: <fractal_tree.branch.Branch object at 0x7f13a25eef80>, 15: <fractal_tree.branch.Branch object at 0x7f13a25eef50>, 16: <fractal_tree.branch.Branch object at 0x7f13a25ed690>, 17: <fractal_tree.branch.Branch object at 0x7f13a25eddb0>, 18: <fractal_tree.branch.Branch object at 0x7f13a25edfc0>, 19: <fractal_tree.branch.Branch object at 0x7f13a25edd80>, 20: <fractal_tree.branch.Branch object at 0x7f13a25ef190>, 21: <fractal_tree.branch.Branch object at 0x7f13a25eee60>, 22: <fractal_tree.branch.Branch object at 0x7f13a25ee410>, 23: <fractal_tree.branch.Branch object at 0x7f13a25edb70>, 24: <fractal_tree.branch.Branch object at 0x7f13a25ed540>, 25: <fractal_tree.branch.Branch object at 0x7f13a25eccd0>, 26: <fractal_tree.branch.Branch object at 0x7f13a25ee980>, 27: <fractal_tree.branch.Branch object at 0x7f13a25ed750>, 28: <fractal_tree.branch.Branch object at 0x7f13a25ed600>}, nodes=array([[ 0. , 1.49283182, 0.14646892],
[ 0.0099979 , 1.49267214, 0.14645311],
[ 0.01999434, 1.49251248, 0.14643723],
...,
[ 0.65925009, 0.62524571, -1.32303276],
[ 0.65356095, 0.62518229, -1.32369751],
[ 0.64786993, 0.62512121, -1.32436353]]), end_nodes=[499, 421, 445, 445, 493, 493, 534, 581, 586, 610, 636, 665, 721, 713, 738], lines=[(0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8), (8, 9), (9, 10), (10, 11), (11, 12), (12, 13), (13, 14), (14, 15), (15, 16), (16, 17), (17, 18), (18, 19), (19, 20), (20, 21), (21, 22), (22, 23), (23, 24), (24, 25), (25, 26), (26, 27), (27, 28), (28, 29), (29, 30), (30, 31), (31, 32), (32, 33), (33, 34), (34, 35), (35, 36), (36, 37), (37, 38), (38, 39), (39, 40), (40, 41), (41, 42), (42, 43), (43, 44), (44, 45), (45, 46), (46, 47), (47, 48), (48, 49), (49, 50), (50, 51), (51, 52), (52, 53), (53, 54), (54, 55), (55, 56), (56, 57), (57, 58), (58, 59), (59, 60), (60, 61), (61, 62), (62, 63), (63, 64), (64, 65), (65, 66), (66, 67), (67, 68), (68, 69), (69, 70), (70, 71), (71, 72), (72, 73), (73, 74), (74, 75), (75, 76), (76, 77), (77, 78), (78, 79), (79, 80), (80, 81), (81, 82), (82, 83), (83, 84), (84, 85), (85, 86), (86, 87), (87, 88), 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(173, 174), (174, 175), (175, 176), (176, 177), (177, 178), (178, 179), (179, 180), (180, 181), (181, 182), (182, 183), (183, 184), (184, 185), (185, 186), (186, 187), (187, 188), (188, 189), (189, 190), (190, 191), (191, 192), (192, 193), (193, 194), (194, 195), (195, 196), (196, 197), (197, 198), (198, 199), (199, 200), (200, 201), (201, 202), (202, 203), (203, 204), (204, 205), (205, 206), (206, 207), (207, 208), (208, 209), (209, 210), (210, 211), (211, 212), (212, 213), (213, 214), (214, 215), (215, 216), (216, 217), (217, 218), (218, 219), (219, 220), (220, 221), (221, 222), (222, 223), (223, 224), (224, 225), (225, 226), (226, 227), (227, 228), (228, 229), (229, 230), (230, 231), (231, 232), (232, 233), (233, 234), (234, 235), (235, 236), (236, 237), (237, 238), (238, 239), (239, 240), (240, 241), (241, 242), (242, 243), (243, 244), (244, 245), (245, 246), (246, 247), (247, 248), (248, 249), (249, 250), (250, 251), (251, 252), (252, 253), (253, 254), (254, 255), (255, 256), (256, 257), (257, 258), (258, 259), (259, 260), (260, 261), (261, 262), (262, 263), (263, 264), (264, 265), (265, 266), (266, 267), (267, 268), (268, 269), (269, 270), (270, 271), (271, 272), (272, 273), (273, 274), (274, 275), (275, 276), (276, 277), (277, 278), (278, 279), (279, 280), (280, 281), (281, 282), (282, 283), (283, 284), (284, 285), (285, 286), (286, 287), (287, 288), (288, 289), (289, 290), (290, 291), (291, 292), (292, 293), (293, 294), (294, 295), (295, 296), (296, 297), (297, 298), (298, 299), (299, 300), (300, 301), (301, 302), (302, 303), (303, 304), (304, 305), (305, 306), (306, 307), (307, 308), (308, 309), (309, 310), (310, 311), (311, 312), (312, 313), (313, 314), (314, 315), (315, 316), (316, 317), (317, 318), (318, 319), (319, 320), (320, 321), (321, 322), (322, 323), (323, 324), (324, 325), (325, 326), (326, 327), (327, 328), (328, 329), (329, 330), (330, 331), (331, 332), (332, 333), (333, 334), (334, 335), (335, 336), (336, 337), (337, 338), (338, 339), (339, 340), (340, 341), (341, 342), (342, 343), (343, 344), (344, 345), (345, 346), (346, 347), (347, 348), (299, 349), (349, 350), (350, 351), (351, 352), (352, 353), (353, 354), (354, 355), (355, 356), (356, 357), (357, 358), (358, 359), (359, 360), (360, 361), (361, 362), (362, 363), (363, 364), (364, 365), (365, 366), (366, 367), (367, 368), (368, 369), (369, 370), (370, 371), (371, 372), (372, 373), (373, 374), (374, 375), (375, 376), (376, 377), (377, 378), (378, 379), (379, 380), (380, 381), (381, 382), (382, 383), (383, 384), (384, 385), (385, 386), (386, 387), (387, 388), (388, 389), (389, 390), (390, 391), (391, 392), (392, 393), (393, 394), (394, 395), (395, 396), (396, 397), (348, 398), (398, 399), (399, 400), (400, 401), (401, 402), (402, 403), (403, 404), (404, 405), (405, 406), (406, 407), (407, 408), (408, 409), (409, 410), (410, 411), (411, 412), (412, 413), (413, 414), (414, 415), (415, 416), (416, 417), (417, 418), (418, 419), (419, 420), (420, 421), (348, 422), 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(671, 672), (672, 673), (673, 674), (674, 675), (675, 676), (676, 677), (677, 678), (678, 679), (679, 680), (680, 681), (681, 682), (682, 683), (683, 684), (684, 685), (685, 686), (686, 687), (687, 688), (688, 689), (689, 690), (690, 691), (691, 692), (692, 693), (693, 694), (694, 695), (695, 696), (696, 697), (697, 698), (698, 699), (699, 700), (700, 701), (701, 702), (702, 703), (703, 704), (704, 705), (705, 706), (706, 707), (707, 708), (708, 709), (709, 710), (710, 711), (711, 712), (712, 713), (689, 714), (714, 715), (715, 716), (716, 717), (717, 718), (718, 719), (719, 720), (720, 721), (713, 722), (722, 723), (723, 724), (724, 725), (725, 726), (726, 727), (727, 728), (728, 729), (729, 730), (730, 731), (731, 732), (732, 733), (733, 734), (734, 735), (735, 736), (736, 737), (737, 738)])
Fig. 3 Idealized bi-ventricular geometry with generated Purkinje networks on both LV and RV#