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7.2 EQUATIONS THAT GOVERN THE ELECTRICAL ACTIVITY OF THE HEART 119

λ D i . With the following boundary condition:
where D ¼ 1+ λ
n r D rVÞ ¼ 0: (7.19)
ð
The monodomain model takes a numerical and computational complexity that is less than the bidomain. For this rea-
son, the monodomain model is often used to study the propagation of the AP in the heart.
Eq. (7.18) is a parabolic equation describing a reaction-diffusion phenomenon. The part associated with the reaction
is determined by the term J ion , which is governed by the cellular model. The diffusive (or conductive in this case) part
models the propagation of the AP in the tissue.

7.2.2 Myocardium Conductance
Within the heart, the fibers are organized transmurally with orientations varying from 60 degrees (regarding the
circumferential axis) to + 60 degrees from the epicardium to the endocardium [34]. The orientation of the muscle fibers
in each point of the myocardium can be obtained either by histology, or more recently by using magnetic resonance
imaging (MRI), in particular a technique known as diffusion tensor magnetic resonance imaging (DT-MRI) [35]. Born
for neuroimaging applications, diffusion tensor imaging (DTI), a special kind of the more general diffusion weighted
MRI, is an imaging method that uses the diffusion of water molecules to generate contrast in MR images. Because
diffusion of water in tissues is not free, but is affected by the interaction with obstacles such as fibers and heteroge-
neities in general, water molecule diffusion patterns can be used to identify details about tissue microstructure. DTI, in
particular, enables the measurement of the restricted diffusion of water in the myocardium. In DTI, each voxel contains
the rate of diffusion and the preferred directions of diffusion. Therefore, assuming that the diffusion is faster along the
fiber axis, the eigenvector corresponding to the largest diffusion tensor eigenvector defines the direction of the fiber
1
axis [35]. These are the data that we need to implement in the simulation of the human heart tissue. Even though the
heart tissue is truly orthotropic [36], for this work we consider it as transversely isotropic, with the direction of max-
imum conduction corresponding to the cardiac fiber direction. In the material fiber system, the conductivity tensor is
10 0
0 1
0 r 0
B C
D ¼ d o B C , (7.20)
00 r
@ A
where d o represents the conductance in the fiber direction and r 1 the conductivity ratio between the transversal and
longitudinal fibers. In Cartesian coordinates, under conditions of transverse anisotropy, the diffusion tensor can be
written as
Š, (7.21)
D ¼ d o ð1 rÞf
f + rI½
where f is the fiber orientation, I is the second-order identity tensor, and
indicates the tensorial product ((a
b) ij ¼
a i b j ). Expressing Eq. (7.21) in components, we obtain
0 1 0 1
10 0
f 1 f 1 f 1 f 2 f 1 f 3
01 0
f 2 f 1 f 2 f 2 f 2 f 3
B C B C
+ d o r : (7.22)
B C B C
D ¼ d o ð1 rÞ
@ f 3 f 1 f 3 f 2 f 3 f 3 A @ 00 1 A
7.2.3 Action Potential Models
Hodking and Huxley [37] in 1952 introduced the first mathematical model to reproduce the APs in the cell membrane.
Since then, many cardiac cell models, following the formulation established by these researchers, have been devel-
oped. AP models can be divided into two main families: (i) phenomenological models and (ii) electrophysiological
detailed models.
Phenomenological models macroscopically reproduced the behavior of the cell in terms of the shape and duration of
the AP, restitution properties, and CV. Electrophysiological detailed models offer a detailed description of the cellular
physiology. They not only include more currents, but also include pumps and exchangers as well as intracellular ion
concentration dynamics. Models have been developed for a number of species as well as cell types within the conduc-
tion system of the heart, as for example: Stewart et al. [38] for human Purkinjie cells, Maleckar et al. [39] and Nygren

1 See http://gforge.icm.jhu.edu/gf/project/dtmri_data_sets/.

I. BIOMECHANICS