Page 95 - Artificial Intelligence for Computational Modeling of the Heart

Chapter 2 Implementation of a patient-specific cardiac model 65

The matrix Γ H collects the gradients ∇φ e and does not require
extra computation. The matrices P and G contain coefficients de-
pending uniquely on the geometry of the epicardium and the
body torso meshes, and can therefore be precomputed. Their rows
correspond to different locations of the observation point on the
surface indicated by the first subscript (B:bodysurface, H:heart).
Similarly, their columns correspond to different locations on the
surface of integration, indicated by the second subscript.

Figure 2.19. Left Panel: general principle of geometry matrix definition using the
example of P HB (observation points on the heart and torso as integration
surface), see text for details. Right panel: actual implementation principle of P HB
calculation using triangular mesh representation of body surface. Different
triangles on the torso are represented using different shades of red (mid gray in
print version).

Following the formulation of [222], the matrices are con-
structed as follows:
• The P matrices contain coefficients regarding the potentials φ b
and φ e . For a given triangle k of the integration surface S,and
a particular observation point i on the observation surface, the
coefficient is expressed as the solid angle subtended by trian-
i i
gle k onto point i: P = k dΩ S .These P coefficients need to
k S k
be distributed onto the three vertices of triangle k,inorderto
compute element P ij of the desired matrix, as illustrated below.
• The G matrices contain coefficients regarding the normal com-
ponents of the gradients in Γ H . Using the same notation as
i dS i
before, we can define the coefficients G = ,where r
k S k r i
is the distance from the observer’s location to the respective
point on the integration surface. The integral is computed us-
ing the assumption that the triangle k can be approximated
i
by a circle sector that is perpendicular to the direction of r :
√
dS 2 2

S r = θ d + 2Aθ − θd,where A is theareaofthetriangle, d
the distance to its closest vertex with respect to the observation
point, and θ the angle between the two triangle edges sharing
the closest vertex.