Biomedical Engineering Reference
In-Depth Information
Similarly, myocytes, for instance, are also capable of generating an electrical ac-
tivity as subjected to mechanical loading. Apart from these strong coupling ef-
fects, the polarized nature of the externally applied electric field and the intrinsi-
cally anisotropic, actively deforming micro-structure of the myocardium, results in
direction-dependent behavior (Nielsen et al.,
1991
; Rohmer et al.,
2007
). In the lit-
erature, the electromechanically coupled cardiac response is typically accounted for
at a constitutive level through an additional transmembrane potential-dependent ac-
tive stress term (Nash and Panfilov,
2004
; Keldermann et al.,
2007
; Niederer and
Smith,
2008
; Göktepe and Kuhl,
2010
).
In contrast to the active stress-based approaches, mentioned above, in this con-
tribution, we propose a new, general kinematic approach to the computational mod-
eling of electro-active materials. Inspired from the recent works of Cherubini et al.
(
2008
), Ambrosi et al. (
2011a
), Stålhand et al. (
2011
), we decompose the total de-
formation gradient into the active and passive parts. The active part is considered to
be dependent upon the electrical potential through a micro-mechanically motivated
evolution equation. In addition, the proposed kinematic framework incorporates the
inherently anisotropic, active architecture of the material. As opposed to the above
mentioned works where merely the active-passive split of the deformation gradient
has been utilized, we further additively decompose the free-energy function into pas-
sive and active parts. This decomposition allows us to recover the additive structure
of the stress response. Therefore, the proposed formulation can be considered as the
generalization of the approaches that employ either additive stress decomposition
or multiplicative split of the deformation gradient to account for excitation-induced
contraction. Furthermore, this kinematic setting is embedded in the recently pro-
posed, fully implicit, entirely finite-element-based coupled framework, which has
been originally developed in Göktepe and Kuhl (
2010
). The performance of the pro-
posed formulation is demonstrated through the fully coupled finite element analyses
of the nonlinear excitation-contraction of a generic heart model.
13.2 Coupled Cardiac Electromechanics
A coupled initial boundary-value problem of cardiac electromechanics within the
mono-domain setting is formulated in terms of the two primary field variables,
namely the placement
ϕ(
X
,t)
and the transmembrane potential
Φ(
X
,t)
. While
the latter refers to a potential difference between the intracellular medium and the
extracellular medium within the context of mono-domain formulations of cardiac
electrophysiology, see Keener and Sneyd (
1998
), the former is the nonlinear de-
formation map, depicted in Fig.
13.1
. Evolution of the primary field variables is
governed by two basic field equations: the balance of linear momentum and the
reaction-diffusion-type equation of excitation, which are introduced in Sect.
13.2.2
.
