Sensitivity to Heterogeneity in the Milieu of Propagation

In  the  presence  of  random  spatial  (structural)  or  temporal  heterogeneity,  analytical  strategies  might   return   velocity   values,   which   are   not   representative   of   the   whole   medium,   an   effect   that  

inadvertently   challenges   the   accuracy   of   the   obtained   result.   Using   simulated   data,   a   gradual   increase  of  spatial  heterogeneity  on  the  excitable  medium  is  implemented  and  manifests  itself  by  an   increase  in  the  severity  of  the  contour  lines’  deformations  (Figure  17a),  creating  therefore  patchy   configurations   where   the   isochronal   invaginations   and   the   protrusions   correspond   to   areas   of   slower  and  faster  conduction  respectively.  

These   simulations   tried   to   mimic   the   activations   observed   for   the   ΔKPQ   hearts   that   were   treated   with  a  NaV1.5-­‐blocker.  For  low  to  moderate  heterogeneity  level  (e.g.  σG  12  or  below),  the  LSEF  and   AF   seem   to   work   rather   well,   compared   to   the   PF.   The   velocity   vs.   sliding   interval   graph   (Figure   18b)  offers  the  possibility  to  detect  the  varying  transitions  in  the  estimation  of  velocity  at  constant   time   intervals   but   variable   spatial   distributions   (the   outward   emanating   wave   from   a   point   stimulation  progressively  increases  in  size  as  it  moves  away  towards  the  boundaries).  Each  colored   curve  corresponds  to  one  analytical  method  and  each  point  is  the  outcome  of  evaluating  the  velocity   at  a  particular  zone  of  the  map.  As  it  can  be  seen  from  the  graph  on  the  left,  the  LSEF  returns  a  Vmax   12%  less  than  the  other  two  methods  over  the  ROI  that  extends  from  the  4^th  to  the  8^th  isochrone.  

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Figure  17.  Comparison  of  LSEF  and   PF   methods   using   computational   modeling   for   heterogeneity.   a.  

Fitting   of   ellipses   and   planes   to   activation   maps   with   incremental   heterogeneity   level   (σG).   The   isochronal  distortions  are  introduced   by   locally   varying   the   diffusion   constant,   while   maintaining   a   constant   velocity   of   propagation.  

Irregularities   in   the   activation   patterns   lead   to   progressive   deviations   from   the   expected   anisotropic   behavior.   LSEF   becomes   an   inferior   choice   for   CV   evaluation   due   to   poor   incorporation   of   non-­‐

uniformity   in   the   calculated   CV   values   that   are   averaged   out   by   the   fitted   ellipses   and   the   linear   regression;   whereas   with   PF   the   gradual   effect   of   heterogeneity   is   well   demonstrated   with   the   vectors’  

local   divergence   in   direction   and   amplitude.   Horizontal   scaling   arrow  =  vector  magnitude  of  1  m.s^-­‐1.   Scale   bar  (in   all   maps)   =   1mm.   b.  

Anisotropy   estimated   by   the   closeness   of   the   isochronal   contours   to   the   elliptical   geometry   (left),   a   property  which  is  well-­‐monitored  by   the   cost   function,   which   returns   increasingly   higher   values   as   anisotropy   is   altered.   (Right)   The   increase   in   heterogeneity   is   incorporated   as   a   statistical   component   in   the   PF   method,   reflected   by   the   gradual   increase   in   the   SD   of   the   CV   within   each   bin,   which   signifies   loss   of   precision   in   determining  the  CV.  

Conduction  Slowing  in  Hearts  of  mdx-­‐Mice          77  

Excluding  this  point,  the  remaining  points  don’t  show  a  variation  that  exceeds  7%.  It’s  also  worth   mentioning  that  for  the  same  heterogeneity  level,  several  realizations  were  simulated  and  analyzed   accordingly,  in  principal  to  guarantee  dependence  of  the  curves  on  heterogeneity,  rather  than  on  a   particular   isochronal   configuration.   The   graph,   on   the   right   in   Figure   18b,   reveals   deviations   of   more   than   20%   among   the   mean   velocities   at   different   ROIs   with   maps   embodying   higher   heterogeneity   levels   (e.g.   σG   24).   At   this   level,   the   curves   do   not   show   consistency   among   the  

different   realizations.   The   behavior   is   expected,   where   experimentally   this   could   represent   completely   distinct   substrates.   Heterogeneity   and   uncertainty   due   to   the   associated   error   will   ultimately   influence   the   velocity   measured   and   its   significance.   A   global   estimate   of   velocity   will   deviate   from   the   local   one,   once   the   heterogeneity   in   the   medium   creates   an   activation   complex   enough  to  alter  the  morphology  of  the  propagation  pattern.  Since  only  PF  provides  local  calculations   of  velocity,  small  local  heterogeneities  are  readily  accessible  for  further  analysis,  where  the  mean   velocity  within  the  heterogeneous  bin  can  be  contrasted  to  the  average  velocity  in  the  medium.    To   further  explore  this  prevailing  hypothesis,  the  effect  of  heterogeneity  on  the  SD  is  plotted  in  Figure   17b.   The   PF   tends   to   incorporate   inhomogeneity   into   its   statistics.   Therefore   the   progressive   increase   in   SD   values   within   each   bin,   at   almost   all   orientations,   reflects   the   sensitivity   of   the   method   in   picking   up   heterogeneities   in   the   medium   (Figure   17b,   right).   The   LSEF   and   AF   can’t   provide   such   a   feature.   Nevertheless,   an   indirect   measure   of   heterogeneity   can   be   provided   as   deviations   of   the   isochrones   from   the   elliptical   geometry,   which   can   be   quantified   using   the   cost   function:  as  the  heterogeneity  strength  progressively  increases,  the  cost  function  faithfully  mirrors   this  effect  (Figure  17b,  left).  

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Figure  18.  Plot  of  the  maximal  and  minimal   velocities   vs.   sliding   interval.   a.   Methods   correspondence  with  increasing  noise.  Each   interval   [x,   x+4]   corresponds   to   one   ring   of   inner   circumference   formed   by   the   x^th   isochrone   and   outer   circumference   by   the   (x+4)^th   isochrone.   This   ring   is   a   well-­‐defined   ROI  where  the  minimal  and  maximal  CVs  were   calculated.     By   sliding   this   ring,   curves   that   scan   through   the   entire   activation   map   are   obtained,   revealing   the   deviations   from   the   ideal   configuration   observed   as   velocities   are   calculated   by   the   three   different   methods.  

Evidently,   all   methods   show   robustness   to   noise   with   very   comparable   behavior   (except   for   early   isochrones)   until   complete   misdetection   of   activation   times   at   SNR<0.03   for   LSEFP   and   PF   (Figure   16).  b.   Methods   correspondence   with   increasing   heterogeneity.   The   observed   fluctuations   in   velocity   values   reported   by   the   methods   are   due   to   the   continuous   increase   in   heterogeneity   level.   Discernible   divergence   among   the   values   is   detected   where   anisotropy  is  largely  disturbed  (right). Square   filled   symbol:   maximal   CV;  Circular   filled   symbol:   minimal   CV;   black:   LSEF;   blue:   PF;  

red:  AF.