Using FEniCS

In this demo we will show how to generate Purkinje networks on a patient specific mesh from the paper Efficient estimation of personalized biventricular mechanical function employing gradient-based optimization. Instructions on how to get the data is described in more detail here. First you need to download the data. Next we will load the data with fenics-pulse

import numpy
import logging
import pulse
import dolfin
from fractal_tree import generate_fractal_tree, FractalTreeParameters, Mesh

Set logging level and a seed for reproducibility.

logging.basicConfig(level=logging.INFO)
numpy.random.seed(12)

Let us download the data from the paper

import urllib.request
import tarfile
import time
link = "https://www.dropbox.com/s/pxbx3ohix8e3jsx/data.tar?dl=1"
path = "data.tar"
print("Downloading data")
urllib.request.urlretrieve(link, path)
time.sleep(1.0)
print("Done downloading data. Extracting...")
data_file = tarfile.open(name=path, mode="r")
data_file.extractall()
print("Done extracting")

Downloading data

Done downloading data. Extracting...

Done extracting

When you download the data from the paper you will get a tarball with the data. We assume here that the data is extracted into a folder mesh. Here we will use the medium mesh for case 1

case = "CASE1_med"
path = f"data/mesh/{case}_60.h5"

We will load the data with pulse

geo = pulse.HeartGeometry.from_file(path)

2023-04-21 05:19:32,985 [192] INFO     pulse.geometry_utils: 
Load mesh from h5

We would like to visualize the mesh in Paraview afterwards, so let us save the mesh in an XDMF file.

with dolfin.XDMFFile(f"{case}_mesh.xdmf") as xdmf:
    xdmf.write(geo.mesh)

To load the mesh into fractal-tree we need to first get the coordinates

verts = geo.mesh.coordinates()

and the connectivities

connectivity = []
for facet in dolfin.facets(geo.mesh):
    if geo.ffun[facet] == geo.markers["ENDO_RV"][0]:
        connectivity.append(facet.entities(0))
connectivity = numpy.array(connectivity)

Alternatively we could save the facet function to .xdmf and load it with meshio

ffun_file = f"{case}_ffun.xdmf"
with dolfin.XDMFFile(ffun_file) as xdmf:
    xdmf.write(geo.ffun)
msh = meshio.read("data/mesh/CASE1_med_ffun.xdmf")
marker = 20  # ENDO RV
conn = msh.cells[0].data[msh.cell_data["f"][0].squeeze() == marker, :]
verts = msh.points

Next we will choose the initial node. Here you could open the mesh in Paraview and find the approximate coordinate to this node, and then find the the closest node

init_node = [1.0, 15.85, 15.36]
index = numpy.linalg.norm(numpy.subtract(verts, init_node), axis=1).argmin()

Now we can create the mesh for the fractal tree

mesh = Mesh(verts=verts, connectivity=connectivity, init_node=verts[index, :])

We set the fractal tree parameters and choose the initial direction as pointing in the positive \(x\)-direction

param = FractalTreeParameters(
    filename="case1-rv",
    N_it=300,
    length=0.3,
    initial_direction=numpy.array([1, 0, 0]),
)

and finally run the fractal tree algorithm

generate_fractal_tree(mesh, param)

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2023-04-21 05:20:02,044 [192] INFO     fractal_tree.tree: Finished growing, writing paraview file

FractalTreeResult(branches={0: <fractal_tree.branch.Branch object at 0x7f4e3d7f4730>, 1: <fractal_tree.branch.Branch object at 0x7f4e749831c0>, 2: <fractal_tree.branch.Branch object at 0x7f4e749817e0>, 3: <fractal_tree.branch.Branch object at 0x7f4e3d7fc040>, 4: <fractal_tree.branch.Branch object at 0x7f4e3d7fe140>, 5: <fractal_tree.branch.Branch object at 0x7f4e3d7fd060>, 6: <fractal_tree.branch.Branch object at 0x7f4e3d7fe770>, 7: <fractal_tree.branch.Branch object at 0x7f4e3d7fe620>, 8: <fractal_tree.branch.Branch object at 0x7f4e3d7fda20>, 9: <fractal_tree.branch.Branch object at 0x7f4e3d7fd150>, 10: <fractal_tree.branch.Branch object at 0x7f4e3d7fdea0>, 11: <fractal_tree.branch.Branch object at 0x7f4e3d7fd600>, 12: <fractal_tree.branch.Branch object at 0x7f4e3d7fdde0>, 13: <fractal_tree.branch.Branch object at 0x7f4e3d7fe530>, 14: <fractal_tree.branch.Branch object at 0x7f4e3d7fecb0>, 15: <fractal_tree.branch.Branch object at 0x7f4e3d7fe740>, 16: <fractal_tree.branch.Branch object at 0x7f4e3d7fd9f0>, 17: <fractal_tree.branch.Branch object at 0x7f4e3d7fe650>, 18: <fractal_tree.branch.Branch object at 0x7f4e3d7fe590>, 19: <fractal_tree.branch.Branch object at 0x7f4e3d7fe320>, 20: <fractal_tree.branch.Branch object at 0x7f4e3d7fda50>, 21: <fractal_tree.branch.Branch object at 0x7f4e3d7fe170>, 22: <fractal_tree.branch.Branch object at 0x7f4e3d7fe290>, 23: <fractal_tree.branch.Branch object at 0x7f4e3d7fe6b0>, 24: <fractal_tree.branch.Branch object at 0x7f4e3d7fec80>, 25: <fractal_tree.branch.Branch object at 0x7f4e3d7fd540>, 26: <fractal_tree.branch.Branch object at 0x7f4e3d7fd810>, 27: <fractal_tree.branch.Branch object at 0x7f4e3d7fdf90>, 28: <fractal_tree.branch.Branch object at 0x7f4e3d7fee00>, 29: <fractal_tree.branch.Branch object at 0x7f4e3d7fdff0>, 30: <fractal_tree.branch.Branch object at 0x7f4e3d7feb90>, 31: <fractal_tree.branch.Branch object at 0x7f4e3d7fdc90>, 32: <fractal_tree.branch.Branch object at 0x7f4e3d7fea70>, 33: <fractal_tree.branch.Branch 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(15139, 15140), (15140, 15141), (14810, 15142), (15142, 15143), (15143, 15144), (15144, 15145), (15145, 15146), (15146, 15147), (15147, 15148), (15148, 15149), (15149, 15150), (15150, 15151), (15151, 15152), (15152, 15153), (15153, 15154), (15154, 15155), (15155, 15156), (15156, 15157), (15157, 15158), (15158, 15159), (15159, 15160), (15160, 15161), (15161, 15162), (15162, 15163), (15163, 15164), (15164, 15165), (15165, 15166), (15166, 15167), (15167, 15168), (15168, 15169), (15169, 15170), (14955, 15171), (15171, 15172), (15172, 15173), (15173, 15174), (15174, 15175), (15175, 15176), (15176, 15177), (15177, 15178), (15178, 15179), (15179, 15180), (15180, 15181), (15181, 15182), (15182, 15183), (15183, 15184), (15184, 15185), (15185, 15186), (15186, 15187), (15187, 15188), (15188, 15189), (15189, 15190), (15190, 15191), (15191, 15192), (15192, 15193), (15193, 15194), (15194, 15195), (15195, 15196), (15196, 15197), (15197, 15198), (15198, 15199), (14984, 15200), (15200, 15201), (15201, 15202), (15202, 15203), (15203, 15204), (15204, 15205), (15205, 15206), (15206, 15207), (15207, 15208), (15208, 15209), (15209, 15210), (15210, 15211), (15211, 15212), (15212, 15213), (15213, 15214), (15214, 15215), (15215, 15216), (15216, 15217), (15217, 15218), (15218, 15219), (15219, 15220), (15220, 15221), (15221, 15222), (15222, 15223), (15223, 15224), (15224, 15225), (15225, 15226), (15226, 15227), (15227, 15228), (14984, 15229), (15229, 15230), (15230, 15231), (15231, 15232), (15232, 15233), (15233, 15234), (15234, 15235), (15235, 15236), (15236, 15237), (15237, 15238), (15238, 15239), (15239, 15240), (15240, 15241), (15241, 15242), (15242, 15243), (15243, 15244), (15244, 15245), (15245, 15246), (15246, 15247), (15247, 15248), (15248, 15249), (15249, 15250), (15250, 15251), (15251, 15252), (15252, 15253), (15253, 15254), (15254, 15255), (15255, 15256), (15256, 15257), (14839, 15258), (15258, 15259), (15259, 15260), (15260, 15261), (15261, 15262), (15262, 15263), (15263, 15264), (15264, 15265), (15265, 15266), (15266, 15267), (15267, 15268), (15268, 15269), (15269, 15270), (15270, 15271), (15271, 15272), (15272, 15273), (15273, 15274), (15274, 15275), (15275, 15276), (15276, 15277), (15277, 15278), (15278, 15279), (15279, 15280), (15280, 15281), (15281, 15282), (15282, 15283), (15283, 15284), (15284, 15285), (15285, 15286), (14839, 15287), (15287, 15288), (15288, 15289), (15289, 15290), (15290, 15291), (15291, 15292), (15292, 15293), (15293, 15294), (15294, 15295), (15295, 15296), (15296, 15297), (15297, 15298), (15298, 15299), (15299, 15300), (15300, 15301), (15301, 15302), (15302, 15303), (15303, 15304), (15304, 15305), (15305, 15306), (15306, 15307), (15307, 15308), (15308, 15309), (15309, 15310), (15310, 15311), (15311, 15312), (15312, 15313), (15313, 15314), (15314, 15315), (15199, 15316), (15316, 15317), (15317, 15318), (15318, 15319), (15319, 15320), (15320, 15321), (15321, 15322), (15322, 15323), (15323, 15324), (15324, 15325), (15325, 15326), (15326, 15327), (15327, 15328), (15328, 15329), (15329, 15330), (15330, 15331), (15331, 15332), (15332, 15333), (15333, 15334), (15334, 15335), (15335, 15336), (15336, 15337), (15337, 15338), (15338, 15339), (15339, 15340), (15340, 15341), (15341, 15342), (15342, 15343), (15343, 15344), (15228, 15345), (15345, 15346), (15346, 15347), (15347, 15348), (15348, 15349), (15349, 15350), (15350, 15351), (15351, 15352), (15352, 15353), (15353, 15354), (15354, 15355), (15355, 15356), (15356, 15357), (15357, 15358), (15358, 15359), (15359, 15360), (15360, 15361), (15361, 15362), (15362, 15363), (15363, 15364), (15364, 15365), (15365, 15366), (15366, 15367), (15367, 15368), (15368, 15369), (15369, 15370), (15370, 15371), (15371, 15372), (15372, 15373), (15228, 15374), (15374, 15375), (15375, 15376), (15376, 15377), (15377, 15378), (15378, 15379), (15379, 15380), (15380, 15381), (15381, 15382), (15382, 15383), (15383, 15384), (15384, 15385), (15385, 15386), (15386, 15387), (15387, 15388), (15388, 15389), (15389, 15390), (15390, 15391), (15391, 15392), (15392, 15393), (15393, 15394), (15394, 15395), (15395, 15396), (15396, 15397), (15397, 15398), (15398, 15399), (15399, 15400), (15400, 15401), (15401, 15402), (15286, 15403), (15403, 15404), (15404, 15405), (15405, 15406), (15406, 15407), (15407, 15408), (15408, 15409), (15409, 15410), (15410, 15411), (15411, 15412), (15412, 15413), (15413, 15414), (15414, 15415), (15415, 15416), (15416, 15417), (15417, 15418), (15418, 15419), (15419, 15420), (15420, 15421), (15421, 15422), (15422, 15423), (15423, 15424), (15424, 15425), (15425, 15426), (15426, 15427), (15427, 15428), (15428, 15429), (15429, 15430), (15430, 15431), (15286, 15432), (15432, 15433), (15433, 15434), (15434, 15435), (15141, 15436), (15436, 15437), (15437, 15438), (15438, 15439), (15439, 15440), (15440, 15441), (15441, 15442), (15442, 15443), (15443, 15444), (15444, 15445), (15445, 15446), (15446, 15447), (15447, 15448), (15448, 15449), (15449, 15450), (15450, 15451), (15451, 15452), (15452, 15453), (15453, 15454), (15454, 15455), (15455, 15456), (15456, 15457), (15457, 15458), (15458, 15459), (15459, 15460), (15460, 15461), (15461, 15462), (15462, 15463), (15463, 15464), (15464, 15465), (15465, 15466), (15466, 15467), (15467, 15468), (15468, 15469), (15469, 15470), (15470, 15471), (15471, 15472), (15472, 15473), (15464, 15474), (15474, 15475), (15475, 15476), (15476, 15477), (15477, 15478), (15478, 15479), (15479, 15480), (15480, 15481), (15481, 15482), (15482, 15483), (15483, 15484), (15484, 15485), (15485, 15486), (15486, 15487), (15487, 15488), (15488, 15489), (15489, 15490), (15490, 15491), (15491, 15492), (15492, 15493), (15493, 15494), (15494, 15495), (15495, 15496), (15496, 15497), (15497, 15498), (15498, 15499), (15402, 15500), (15500, 15501), (15501, 15502), (15502, 15503), (15503, 15504), (15504, 15505), (15505, 15506), (15506, 15507), (15507, 15508), (15508, 15509), (15509, 15510), (15510, 15511), (15511, 15512), (15512, 15513), (15513, 15514), (15514, 15515), (15515, 15516), (15516, 15517), (15517, 15518), (15518, 15519), (15519, 15520), (15520, 15521), (15521, 15522), (15522, 15523), (15523, 15524), (15524, 15525), (15525, 15526), (15526, 15527), (15527, 15528), (15402, 15529), (15529, 15530), (15530, 15531), (15531, 15532), (15532, 15533), (15533, 15534), (15534, 15535), (15535, 15536), (15536, 15537), (15537, 15538), (15538, 15539), (15539, 15540), (15540, 15541), (15541, 15542), (15542, 15543), (15543, 15544), (15544, 15545), (15545, 15546), (15546, 15547), (15547, 15548), (15548, 15549), (15549, 15550), (15550, 15551), (15551, 15552), (15552, 15553), (15553, 15554), (15554, 15555), (15555, 15556), (15556, 15557), (15344, 15558), (15558, 15559), (15559, 15560), (15560, 15561), (15561, 15562), (15562, 15563), (15563, 15564), (15564, 15565), (15565, 15566), (15566, 15567), (15567, 15568), (15568, 15569), (15569, 15570), (15570, 15571), (15571, 15572), (15572, 15573), (15573, 15574), (15574, 15575), (15575, 15576), (15576, 15577), (15577, 15578), (15578, 15579), (15579, 15580), (15580, 15581), (15581, 15582), (15582, 15583), (15583, 15584), (15584, 15585), (15585, 15586), (15344, 15587), (15587, 15588), (15588, 15589), (15589, 15590), (15590, 15591), (15591, 15592), (15592, 15593), (15593, 15594), (15594, 15595), (15595, 15596), (15596, 15597), (15597, 15598), (15598, 15599), (15599, 15600), (15600, 15601), (15601, 15602), (15602, 15603), (15603, 15604), (15604, 15605), (15605, 15606), (15606, 15607), (15607, 15608), (15608, 15609), (15609, 15610), (15610, 15611), (15611, 15612), (15612, 15613), (15613, 15614), (15614, 15615), (15431, 15616), (15616, 15617), (15617, 15618), (15618, 15619), (15619, 15620), (15620, 15621), (15621, 15622), (15622, 15623), (15623, 15624), (15624, 15625), (15625, 15626), (15626, 15627), (15627, 15628), (15628, 15629), (15629, 15630), (15630, 15631), (15631, 15632), (15632, 15633), (15633, 15634), (15634, 15635), (15635, 15636), (15636, 15637), (15637, 15638), (15638, 15639), (15639, 15640), (15640, 15641), (15641, 15642), (15642, 15643), (15643, 15644), (15431, 15645), (15645, 15646), (15646, 15647), (15647, 15648), (15648, 15649), (15649, 15650), (15650, 15651), (15651, 15652), (15652, 15653), (15653, 15654), (15654, 15655), (15655, 15656), (15656, 15657), (15657, 15658), (15658, 15659), (15659, 15660), (15660, 15661), (15661, 15662), (15662, 15663), (15663, 15664), (15664, 15665), (15665, 15666), (15666, 15667), (15667, 15668), (15668, 15669), (15669, 15670), (15670, 15671), (15671, 15672), (15672, 15673), (15673, 15674), (15674, 15675), (15675, 15676), (15676, 15677), (15677, 15678), (15678, 15679), (15679, 15680), (15680, 15681), (15681, 15682), (15682, 15683), (15683, 15684), (15684, 15685), (15685, 15686), (15686, 15687), (15687, 15688), (15688, 15689), (15689, 15690), (15690, 15691), (15691, 15692), (15692, 15693), (15693, 15694), (15694, 15695), (15695, 15696), (15696, 15697), (15697, 15698), (15698, 15699), (15699, 15700), (15700, 15701), (15701, 15702), (15673, 15703), (15703, 15704), (15704, 15705), (15705, 15706), (15706, 15707), (15707, 15708), (15708, 15709), (15709, 15710), (15710, 15711), (15711, 15712), (15712, 15713), (15713, 15714), (15714, 15715), (15715, 15716), (15716, 15717), (15717, 15718), (15718, 15719), (15719, 15720), (15720, 15721), (15721, 15722), (15722, 15723), (15723, 15724), (15724, 15725), (15725, 15726), (15726, 15727), (15727, 15728), (15728, 15729), (15729, 15730), (15730, 15731), (15615, 15732), (15732, 15733), (15733, 15734), (15734, 15735), (15702, 15736), (15736, 15737), (15737, 15738), (15738, 15739), (15739, 15740), (15740, 15741), (15741, 15742), (15742, 15743), (15743, 15744)])

Fig. 4 Purkinje networks on a patient specific biventricular mesh