Page 55 - Artificial Intelligence for Computational Modeling of the Heart

Chapter 1 Multi-scale models of the heart for patient-specific simulations 25

All the models mentioned above provide the active stress τ c
generated by the sarcomeres (Fig. 1.8). Following the spatial dis-
cretization scheme, the stress is integrated to obtain the nodal
forces f a that appear on the right-hand side of Newton’s law of mo-
tion (Eq. (1.13)). Section 2.3 presents a concrete example of active
stress and its numerical derivation in an FEM solver.

1.3.3 The virtual heart in its environment: boundary
conditions

Heart function depends on the constant interactions between
the myocardium, neighboring organs, intra-cardiac blood flow
and global circulation. These interactions are modeled as bound-
ary conditions to the biomechanical model described in the pre-
vious sections. As formalized in Eq. (1.13), two types of boundary
conditions need to be defined: the pressure force exercised by the
blood flow on the endocardial surfaces, denoted f bp , and the at-
tachments of the myocardium to the other organs, denoted f b .
Endocardial pressure
Intra-cardiac blood flow imposes stresses on the endocardium,
which can vary significantly throughout the four phases of the
cardiac cycle: isovolumic contraction, ejection, isovolumic relax-
ation, filling. Researchers investigated fluid-structure interaction
(FSI) methods to understand and quantify the 3D patterns of the
blood flow inside the chambers, along with the spatially varying
stresses on the endocardial surfaces [130,131]. To that end, a com-
plete anatomical model of the heart, with all chambers and valves,
must be available. These modeling approaches can be very de-
tailed and, correspondingly, complex to solve, with coupled sys-
tems controlled by large sets of parameters.
In order to achieve faster computations, researchers proposed
simplified FSI models in which the pressure is assumed uniform in
each chamber (i.e. it does not vary spatially). The computed pres-
sure is applied to the surface of the biomechanical model as a nor-
mal force (f bp in Eq. (1.13)). The time-varying pressure depends
then on 1) the force generated by the myocardium and on 2) the
circulatory system. The latter is modeled using lumped parame-
ters models (LPM) like those presented in [78,132,133](seenext
section), or directly prescribed by the user [45]. Two approaches
are possible to achieve the biomechanical/hemodynamics cou-
pling:
• Pressure-driven models: the pressure is an unknown of the
coupled system, and hence needs to be solved [121]