Page 49 - Artificial Intelligence for Computational Modeling of the Heart

Chapter 1 Multi-scale models of the heart for patient-specific simulations 19

research interest is the study of long-term myocardium remodel-
ing due to growth or pathologies. We refer the readers to [108–111],
and references therein, as notable examples.

1.3.1 The passive myocardium
Experimental testing showed that the passive myocardium is
an essentially incompressible orthotropic material, characterized
by distinct material responses in three mutually orthogonal planes
defined by the fibers and fiber sheets. A large variety of mod-
els have been proposed to simulate these properties [43,112,113].
Constitutive laws are often formulated as an energy-strain func-
tional in polynomial or exponential form [102,113]. In [114], the
authors proposed an early model of the myocardium using a
transverse isotropic strain energy density function. This model,
referred to as the Guccione law, is still used in benchmarks [115].
More complex models have been proposed to include the effects
of myocardial sheets, assumed to be involved in myocardium
thickening during systole. A first category of models are those
based on the pole-zero technique, originally proposed by Hunter
and colleagues [116] and then extended by Niederer et al. [117]. In
this approach, the microstructure of the myocardium is modeled
with three independent axes (fibers, sheets and normals axes). A
pole-zero formulation is then used to model each axis with an in-
dependent, separate pole, thus accounting for the different strain
behavior in each axis with a formulation shown to be more numer-
ically stable than traditional alternatives. In [118], the authors an-
alyzed closely the fiber/fiber sheet mechanism and proposed an
exponent law, now referred to as the Costa law and largely adopted
in the community. In [119], the authors presented a quantitative
comparison of the most common models at the time, in terms of
prediction accuracy with respect to ex-vivo experiments. In these
experiments, the Costa law tended to outperform the other mod-
els. More recently, the structurally-based Holzapfel–Ogden (HO)
model [120] has gained popularity among the community, ar-
guably making it the current state-of-the-art. Contrary to the phe-
nomenological constitutive laws of Costa and Guccione, the HO
model is derived from considerations on the microstructure of the
tissue, and not by fitting exponential functions to stress-strain re-
lationship observed experimentally. It is also easier to implement
and the parameters are directly related to the structural function
of the muscle. Other groups rely on more standard models like
Mooney–Rivlin [103,121] or corotational linear elasticity [6], both
simpler and more computationally efficient. However, advanced
numerical schemes now allow fast computation of hyper-elastic
models, making the less accurate linear models obsolete.