Page 89 - Artificial Intelligence for Computational Modeling of the Heart

Chapter 2 Implementation of a patient-specific cardiac model 59

Tissue conductivity σ is modeled as a piecewise constant scalar
field over the Cartesian grid. Each grid point is at the center of a
voxel – a very small volume of tissue. The conductivity value as-
signed to each grid point ranges from normal to high based on
the volume fraction ψ of tissue in the voxel whose distance from
the endocardium is smaller than the threshold h. For every lattice
node x we have

σ(x) = ψσ purkinje + (1 − ψ) σ normal .

The volume fraction ψ is computed by a two-step algorithm. In
the first phase, a list of candidates, i.e., grid points that may have
a partial volume of high-speed conducting tissue, is built. Nodes
that are closer to the endocardium than the target distance h are
included in the list, as well as nodes that are further away but
may have at least part of the boundary within the high-speed con-
ducting tissue. In other words, all nodes whose distance from the
endocardium is less than an extended threshold h ext are selected,
corresponding to the thickness of the layer of high-speed conduct-
ing tissue plus the maximum distance between the barycenter of
the voxel and its boundary.
√
3
h ext = h + x
2
x being the spacing of the lattice. Given φ, level-set represen-
tation of the endocardial surface, its discretization φ x over the
Cartesian grid is computed. All lattice nodes
x such that φ x (
x)<
h ext are selected.
In the second phase, for each candidate we consider the voxel

v centered in
x anddefineasub-gridofnodes ξ ∈
v with uniform
spacing ξ < x. We compute
φ ξ = φ| , the discretization of the

v
level set function on the sub-grid and use it to classify the nodes
of the sub-grid as normal or high-speed conducting tissue, based
on their distance from the endocardium:

0 ≤
φ (ξ) ≤ h → High-speed conducting tissue
∀ ξ ∈
v :

φ ξ (ξ)>h → Normal tissue.
Finally, the partial volume ψ is obtained as the number of sub-grid
nodes belonging to the high-speed tissue over the total number
of sub-grid nodes. Algorithm 2 summarizes the main steps of the
method.
A key strength of this approach is that it allows an accurate
evaluation of the partial volume of tissue within a given distance
from the surface, especially when the spacing of the original lattice