Page 130 - Advances in Biomechanics and Tissue Regeneration

126 7. MULTISCALE NUMERICAL SIMULATION OF HEART ELECTROPHYSIOLOGY

This approximation with different shape functions is admissible because it satisfies the finite element criteria of
integrability and completeness [52]. In the implementation, we have considered a nodal quadrature to evaluate
e
M with N i ¼ J i I, being J i the element Jacobian evaluated at node i.

7.3 VULNERABILITY IN REGIONALLY ISCHEMIC HUMAN HEART: EFFECT OF THE
EXTRACELLULAR POTASSIUM CONCENTRATION

Ventricular tachycardia and ventricular fibrillation are two types of cardiac arrhythmias that usually take place dur-
ing acute ischemia and frequently lead to sudden death. Proarrhythmic mechanisms of acute ischemia have been
extensively investigated, although often in animal models rather than in human ventricles. In this work, we investigate
how hyperkalemia affects the vulnerability window (VW) to reentry and the reentry patterns in the heterogeneous
substrate caused by acute regional ischemia using an anatomically and biophysically detailed human
biventricular model.
In recent years, mathematical modeling and computer simulations have been shown to be a useful tool in analyzing
electrophysiological phenomena. In particular, one of the major contributions of computer electrophysiology has been
the understanding of important relations between electrophysiological parameters [53]. For the ischemic heart, com-
puter models have allowed us to address the role of ischemic abnormalities in cardiac electrophysiological behavior
[54]. However, most of these simulations have been restricted to 2D [54, 55] or 3D simulations of the total ischemic
heart [56]. Little work has been carried out in modeling the entire heart subjected to acute ischemic conditions [57,
58]. In the work of Weiss et al., they considered heterogeneities caused by ischemia; their characterization of the
AP under acute ischemic conditions has mostly relied on the model characterized for guinea pigs [48]. Dutta et al.
[58] have investigated how reduced repolarization increases arrhythmic risk in the heterogeneous substrate caused
by acute myocardial ischemia. In their work, Dutta et al. [58] developed a human ventricular biophysically detailed
model with acute regional ischemia. Even tough macroreentries around the ischemic zone were reported; these reen-
tries self-terminated after three complete circuits of not being able to establish sustained reentry.
In this chapter, we investigate how hyperkalemia affects the VW to reentry and the reentry patterns in the hetero-
geneous substrate caused by acute regional ischemia using an anatomically and biophysically detailed human biven-
tricular model. The proposed mathematical model is based on the monodomain model [3] for simulating the
propagation of the electrical signal of the heart. For the biophysical description of the AP under normal and ischemic
conditions, the model proposed by ten Tusscher and Panfilov [8], TP06, is used. In this regard, the model has been
modified to account for ischemia by incorporating an ATP-sensitive potassium, I K(ATP) , current. By analyzing high
spatiotemporal resolution simulation data, we unravel the mechanisms associated with the observed reentrant pat-
terns in acutely ischemic ventricles.

7.3.1 Methods
7.3.1.1 Mathematical Model
The variation of the transmembrane potential, V, in the heart was modeled by means of the monodomain model [3]
∂V
r DrVÞ ¼ C m + J ion ðV,wÞ + J stm , (7.43)
∂t
ð
∂w
¼ fðw,V,tÞ, (7.44)
∂t
where D is the symmetric and positive definite conductivity tensor, C m the membrane capacitance, J ion (V, w) the trans-
membrane ionic current, J stm the stimulation current, w(w, V, t) is a vector of gating variables and concentrations, f is a
vector valued function, and t refers to time. Both J ion and f depend on the used cellular model. The boundary conditions
associated with this model are
n r D rVÞ ¼ 0, (7.45)
ð
where n is the outward normal.
The monodomain model represents an important simplification of the more complex bidomain model [3], with
important advantages for mathematical analysis and computation. Despite its simplicity, this model is adequate
for studying a number of electrophysiologic problems such as ventricular fibrillation or the onset of ischemia in
the electric behavior of the heart [8, 54–58].

I. BIOMECHANICS