Page 101 - Artificial Intelligence for Computational Modeling of the Heart

Chapter 2 Implementation of a patient-specific cardiac model 71

which receive pressures as inputs from both sides (ventricle/artery
or ventricle/atrium) and output the flow across the valves, as
described in more detail in section 2.4. The valves are pressure-
driven, meaning that they open and close according to the pres-
sure gradient through them. The last term of Eq. (2.22)isan“ar-
tificial compressibility” term [227] controlled by the capacitance
parameter μ. That term transforms the minute temporal changes
in the ventricular volume V(t), computed by the biomechanical
solver during isovolumic stages, into temporal pressure gradients.
The final balance of these flows (Eq. (2.22)) with the ventricular
flow rate dV/dt, computed directly from the finite element mesh,
allows direct computation of the ventricular pressure by integrat-
ing dp/dt. The boundary face traction is finally updated using the
computed pressure f p =−pn,where n is the endocardial surface
normal.

Attachment to atria and arteries
The effect of arteries and atria on the ventricular motion is sim-
ulated by connecting the vertices of the valve plane to springs
whose stiffness is k base . The fixed extremity of the springs corre-
sponds to the rest position of the nodes, taken at mid diastasis,
when the heart is at rest. The spring stiffness k base is anisotropic to
allow free in-plane motion (e r ,e c ) while minimizing the longitu-
dinal motion of the base along the long axis e l of the heart. Under
these definitions, the basis stiffness force writes:

⎛ ⎞
k base,l 00
f base = M −1 ⎝ 0 k base,r 0 M(x − x 0 ) (2.23)
⎠
00 k base,c
where M is the transformation matrix going from the global coor-
dinate system to the coordinate system defined by the LV long axis
and the short axis plane, as illustrated in Fig. 2.22.

Modeling the effect of the pericardium bag
In [6,135], a contact-based model of the pericardium has been
proposed to mimic the effects of the neighboring organs and of
the pericardium bag on the cardiac motion. The idea consists in
reducing the motion of the epicardial nodes in the radial direction
towards the outside of the pericardium, while allowing friction-
free sliding. For this, the pericardial domain is first estimated as a
signed distance map Π(x) from the detected cardiac epicardium
at end-diastole (Fig. 2.23, left panel). The interior and exterior of
the pericardium bag are negative and positive regions respectively