Volume Fraction

Volume fraction is a way to include cell-scale heterogeneities of the myocardium into tissue scale computations.

Citing [1]:

"The standard model used in cardiac electrophysiology is the bidomain model. It is an averaged model derived from the microscopic properties of the tissue. The bidomain model assumes that the electrically active myocytes are present uniformly everywhere in the heart. While this is a reasonable assumption for healthy hearts, it fails in some pathological cases where significant changes in the tissue structure occur, for example in ischaemic and rheumatic heart disease, inflammation, hypertrophy, or infarction. These tissue heterogeneities are often taken into account through an ad-hoc tuning of model parameters. [...]

We assume[...] a periodic alternation of healthy (bidomain model) and altered (diffusive inclusion) tissue patches. Such a model may be simulated directly, at the high computational cost of a very fine discretisation. Instead we derive[...] a homogenized model at the macroscopic scale, using a rigorous two-scale analysis. We recover[...] a bidomain-type model with modified conductivity coefficients. "

Volume fraction (VF) is the ratio between normal tissue and diffusive inclusions.

In order to add volume fraction in your computations (e.g. for scarred tissue), at least two data sources must be provided:

• a scalar array in which VF is defined on each mesh element, given in a .mesh or .vtk file. The file containing this array is specified in the main input file by the key

volume fraction : <file> <arrayName>

• the second data source are precomputed interpolation maps of conductivity tensors as a function of VF. These maps are generated by solving a PDE on the periodic cell containing the inclusion through another software that will be included in the CEPS environment soon. Different maps can be associated to different regions of the mesh. Here is the syntax for the main input file:

volume fraction map intracellular: file1 
volume fraction map extracellular: file2 

Each file is made of 10 columns: the volume fraction, then the modified values of conductivity tensors

Here is an example of interpolation map file

# VF g11 g12 g13 g22 g23 g33
0. 1.e-4 0. 0. 1.e-4 0. 1.e-4
0.5 3.e-4 0. 0. 1.e-4 0. 1.e-4
1. 0.004 0. 0. 0.001 0. 0.001

[1] Andjela Davidović. Multiscale mathematical modelling of structural heterogeneities in cardiac electrophysiology. PhD Thesis. Université de Bordeaux, 2016.