Biomedical Engineering Reference
In-Depth Information
intracellular calcium concentration change occurs. The dynamics is driven by a trav-
eling action potential, usually modeled by a reaction-diffusion equation, where the
ions species diffusion activates the ionic currents reaction, which eventually dictate
the depolarization and repolarization of the cells. Ionic currents depend on the jump
in electric potential according to Ohm's law whereas the conductance is typically
a highly nonlinear function of voltage described through gating variables or, more
recently, by means of Markov models. Such a nonlinearity is responsible for the
complex excitable behavior of the cardiac action potential cycle: rapid upstroke of
depolarization, followed by a plateau phase and a repolarization of the cells when
a voltage threshold is overcome. A variety of models exists in this respect, with in-
creasing detail in the description of ionic channels and intracellular reactions taken
into account (CellML,
2000
; Rudy and Silva,
2006
). Heuristic systems of equations,
that only reproduce a qualitative pattern of the voltage wave, are very useful in pro-
viding a framework simple enough to allow for mathematical analysis. However,
these kind of phenomenological models are not able to describe the correct behavior
of the cell in a pathological condition, or correctly describe drug interactions; fur-
thermore, the concentration of specific ions like intracellular calcium that induces
contractions and relaxations of cardiomyocytes, is typically not present. Therefore
a more detailed insight of several ionic currents is needed to provide the correct
physiological contractility.
The numerical simulation of these complex multiphysics and multiscale systems
poses a major challenge even if state-of-the-art computational techniques and com-
puter architectures are employed. Finite element formulations of nonlinear elasticity
for the myocardial tissue have been proposed since more than a decade (Nash and
Hunter,
2000
), followed by a series of works focusing on the integration of cardiac
systems including elasticity, electricity, perfusion models, and on the close connec-
tion of the proposed models with experimental observations (see a review in Kerck-
hoffs et al.,
2006
). If a certain level of accuracy of the geometrical description of a
patient specific model is desired and the solution is to be obtained within a reason-
able amount of time, there is no way around using parallel computers (and suitable
numerical techniques combined with scalable algorithms exploiting the underlying
architectures). The public availability of scientific computing libraries such as, e.g.,
LifeV (
2001
), Continuity (
2005
) and Chaste (Pitt-Francis et al.,
2009
), represents a
substantial step forward in this direction. Parallel algorithms capable of performing
cardiac mechano-electrical simulations have recently been implemented reporting
scalable behaviors in Chapelle et al. (
2009
); Reumann et al. (
2009
); Lafortune et al.
(
2012
); Nobile et al. (
2012
).
In this paper we aim at investigating some features of the active strain formu-
lation in cardiac electromechanics (Cherubini et al.,
2008
; Ambrosi et al.,
2011
;
Nobile et al.,
2012
). Such approach is based on the assumption that the mechani-
cal activation, laying in the core of the cell-level excitation-contraction mechanism,
may be represented as a virtual multiplicative splitting of the deformation gradi-
ent into a passive elastic response, and an active deformation depending directly
on the electrophysiology. Alternative options that avoid such decomposition at the
deformation level are active-stress descriptions (see, e.g., Nash and Panfilov,
2004
;
