Assigning Properties for Electrical Conductivity
One should note that DTI measures the effective tensor of water diffusivity in tissue which
is sensitive to the underlying tissue structure and G measures intracellular conductivity
properties. A strong correlation is assumed between the eigenvectors of the water diffusion
tensor and the eigenvectors of the electrical conductivity tensor based on tissue microstructure in
order to assign electrical propagation directionality. This 'cross property' relationship has been
previously studied and assigned by Tuch et al.25
DTI data was processed to assign fiber orientation to the electrical conductivity tensor
along the longitudinal, transverse and normal directions of the tissue using a customized Matlab
subroutine. This subroutine scanned every point of the DTI data and calculated the eigenvalues
and eigenvectors at every location. After segmenting the tissue as mention above, the electrical
conductivity tensor components in the local coordinate system of the heart were assigned to each
node by sorting them in descending order and creating the matrix
g1 0 0
G= 0 g22 0 (3-2)
0 0 g33
where gii are the conductivity eigenvalues and were obtained from Eason et al.26 Electrical
conductivity was assigned values in the global coordinate system using
G '=PGP (3-3)
where P is the transformation matrix and the columns of P are equivalent to the eigenvectors of
the water diffusion tensor at each point. In this way, the principal directions of the water
diffusion tensor provided the fiber orientation and the direction of maximum intracellular
conductivity in the global coordinate system.