Page 94 - Artificial Intelligence for Computational Modeling of the Heart

64 Chapter 2 Implementation of a patient-specific cardiac model

using the following closed-form expression (Ω defines the com-
putational domain; |Ω| is its volume ) [99]:

λ 1
φ e (x,t) = [v(y,t) − v(x,t)]dy. (2.13)
1 + λ |Ω| Ω
In case the extra-cellular potential is computed on a Cartesian
grid, as done with the LBM method, it can be easily mapped back
onto the original epicardial mesh using tri-linear interpolation.

Boundary element model of torso potentials
The boundary element method (BEM) is commonly used to
map extracellular potentials onto a torso mesh by solving the
Laplace equation ∇· R T ∇φ = 0 with a Neumann boundary condi-
tion R T ∇φ · n = 0 on the torso mesh S B , and the Dirichlet bound-
ary condition φ = φ e on the epicardium S H .The parameterR T
denotes the tissue-dependent conductivity tensor.
Green’s second identity writes (A∇B − B∇A) · ndS =
S

(A B − B A) · dV for a volume V, its boundary surface S and
V
normal vector n,where A and B in this equation are scalar func-
tions of position. By defining A as the product of the electric
potential and the isotropic conductivity, and B as the term 1/r,
where r is the vector from a particular integration point to the po-
sition under investigation, Green’s second identity can be used to
analyze voltages on the surface of a conducting volume [222].
A personalized model that includes variable thoracic cavity
conductivity would be the ideal setup for accurate ECG simu-
lations. However, in a first approximation, one may assume an
isotropic conductivity between epicardium and body surface.
Therefore, the potential at any point x of the thoracic domain can
be formulated as follows:

1 r · n 1 r · n ∇φ e · n
φ(x) = φ b dS B − φ e + dS H .
4π r 3 4π r 3 r
S B S H
(2.14)
Here φ e denotes the previously computed extracellular potentials
at the epicardium, while φ b are the unknown body surface poten-
tials on the torso. If the surfaces S B (torso) and S H (epicardium)
are discretized into triangular meshes, the problem can be formu-
lated as the solution of a linear system of equations [222]:

P BB φ b + P BH φ e + G BH Γ H = 0,
P HB φ b + P HH φ e + G HH Γ H = 0.