The aim of this tutorial is to study\
\ the effect of fiber orientation on the electrical behavior of cardiac tissue.\
\ Generally, electrical properties of the myocardial tissue can be described by\
\ the bidomain model accounting for the intracellular and extracellular domain.\
\ The domains are of anisotropic nature, the intracellular domain even more so than\
\ the extracellular domain resulting in ''unequal anisotropy ratios'' affecting\
\ the heart's electrical behavior causing two effects ''rotation'' and ''distribution''.\
\ This tutorial focuses on the ''rotation'' effect, which occurs when two spaces\
\ have different degrees of anisotropy. If considering an isotropic tissue the electrical\
\ field <span class=\"math inline\">\(\mathbf{E}\) and current density\
\ <span class=\"math inline\">\(\mathbf{J}\) would point in the same direction.\
\ It follows that <span class=\"math inline\">\(\mathbf{J}=g\mathbf{E}\)\
\ with g being the scalar conductivity. In anisotropic tissue this is not the case\
\ as the conductiviy tensor rotates <span class=\"math inline\">\(\mathbf{J}\\
) towards the fiber direction relative to <span class=\"math inline\">\\
(\mathbf{E}\). As anisotropy ratios are unequal in the intra- and extracellular\
\ space current densities <span class=\"math inline\">\(\mathbf{J}_i\)\
\ and <span class=\"math inline\">\(\mathbf{J}_e\) are rotated by different\
\ amounts.<a href=\"#fn1\" class=\"footnote-ref\" id=\"fnref1\" role=\"doc-noteref\"\
Analytical\
\ vs. Numerical Setup: Given is a 2D strip of infinite length of width\
\ W=10mm in which fibers are oriented with <span class=\"math inline\">\(\alpha\
\ = 45^{\circ}\) (shown as grey dotted lines in the figure below). Conductivities\
\ are given as <span class=\"math inline\">\(g{iL} = 0.2\) S/m, <span\
\ class=\"math inline\">\(g{iT} = 0.02\) S/m, <span class=\"math inline\"\
\(g{eL} = 0.2\) S/m and <span class=\"math inline\">\(g{eT} = 0.08\\
) S/m where L and T indicate directions along and transverse to the fiber\
\ orientation, respectively (Anisotropy ratios are unequal!). The tissues membrane\
\ surface-to-volume ratio and the resistance are given as <span class=\"math inline\"\
\(\beta = 0.14\mu m^{-1}\) and <span class=\"math inline\">\(R_{m}\
\ = 0.91 \Omega m^{2}\) . For numerical analysis we assume that the length\
\ L of the strip is not infinite, but <span class=\"math inline\">\(L\gg\lambda\\
). For the data given choosing L=60mm should suffice. The spatial resolution\
\ should be used as height of the strip (only one layer of 3D elements). The analytical\
\ solutions should be used to verify the numerical results in 3D.
Model Settings: A hexahedral FE model\
\ with dimensions of 60mm x 10mm x 1mm shall be generated functioning as cardiac\
\ tissue strip. An orthotropic fiber setup is used with a sheet angle of <span class=\"\
math inline\">\(90^{\circ}\).
\n
Electrical Settings:\
\ As passive ionic model the modified Beeler-Router Druhard-Roberge model as implemented\
\ in LIMPET is used with <span class=\"math inline\">\(R{m}\) set to <span\
\ class=\"math inline\">\(0.91 \Omega m^{2}\) and <span class=\"math inline\"\
\(\beta = 0.14\mu m^{-1}\). Current <span class=\"math inline\">\(\\
mathbf{J}\) is injected by Electrode A at x=-L/2 and withdrawn by electrode\
\ B at x=+L/2. Electrode C is located at x=0 and can be gounded or not.
\n
Conductivity\
\ Settings: Unequal anisotropy is assigned using the following values: