\n \nAuthor: Martin Bishop martin.bishop@kcl.ac.uk\n\
<h2 id=\"tutorial_phase-singularities\">Phase Singularities\n
The most lethal\
\ types of cardiac arrhythmia are driven by reentrant wavefronts, which, in many\
\ circumstances, have an organising centre. In 2D, this is usually thought of as\
\ the tip of a rotating spiral wave (which reenters), whereas in 3D this becomes\
\ what is known as a filament which represents the centre-line of a scroll-wave.\
\ Filaments can therefore be thought of as lines of wave break around which a scroll\
\ wave rotates. A line of wave break therefore defines a region in which surfaces\
\ of activation and recovery intersect one another. In 2D a point of wave break\
\ is thus defined by intersecting lines of activation and recovery.
There are many different algorithms\
\ which can be used to detected filaments, which are summarised nicely in Clayton\
\ et al <a href=\"#fn1\" class=\"footnote-ref\" id=\"fnref1\" role=\"doc-noteref\"\
1.
\n
Locating filaments and phase singularities therefore\
\ relies on finding the lines or points of intersection of activation and recovery.\
\ This may be done explicitly or else by first transforming activation into phase.\
\ The term phase singularity refers to the fact that such a point is surrounded\
\ by tissue exhibiting all different values of phase; thus, that point itself does\
\ not have a defined phase value. In terms of activation and recovery intersection,\
\ you can think of a spiral wave core to be where the wavefront and waveback encroach\
\ on oneanother, which would be where activating tissue meets recovered (diastolic)\
\ tissue.
\n
The method used in CARPentry relies on explicitly finding the\
\ intersection of activation and recovery. The isosurface of activation is simply\
\ found by finding the surface corresponding to a certain membrane potential level,\
\ <span class=\"math inline\">\(V{\mathrm {thresh}}\). This is usually\
\ chosen to be around the point of activation of the sodium channel, so approximately\
\ -40mV. Note that in some complex cases of VF or when it is used to analysis simulated\
\ optical mapping signals, the action potential amplitude may well be significantly\
\ lower than during sinus rhythm, and thus this value may want to be lowered. The\
\ surface of recovery is usually defined similarly to be where <span class=\"math\
\ inline\">\(dV{m}/dt=0\), thus representing recovered and diastolic tissue.\
\ If this method is used then it is important to also define the time-window over\
\ which this is evaluated over. These two approaches are shown in the Figure below.\
\ The detection algorithm is implemented in GlFilament which can be\
\ used standalone or as a post-processing option.
Representation of a spiral\
\ wave (a) with (b) showing isolines of voltage (blue) and recovery (red). Panel\
\ (c) shows a zoomed-in region close to the spiral core, demonstrating how the intersection\
\ of activation and recovery successfully highlight a phase singularity. Taken from\
\ Clayton et al<a href=\"#fn2\" class=\"footnote-ref\" id=\"fnref2\" role=\"doc-noteref\"\
2
Phase singularity detection can be performed in either 2D or 3D with CARPentry.\
\ However, it is restricted to meshes that are triangles or tetrahedra.