How do you create a mesh structure for boundary problems?

I'm trying to replicate a thermal finite element model from an academic paper. In the model heat is applied to one surface and when the temperature in a cell reaches a critical value the element is removed (melted).

The paper includes an images of a mesh which had been refined in the region where the heat is applied. It also contains a cell array:

{ {[x1 y1 z1] ; [N1x+;N1x-;N1y+...] ; [h1x+,h1x-,h1y+...]} ...}

Where the first element contains the dimensions of the cell, in the second element a 'ordinal' is included if one cell is a neighbor of another in a specified direction (i.e. whether the neighbor element has been melted or not) and the third element is the heat transfer coefficients in each direction.

My question is: does the cell array come first or the mesh? In other words is it possible for MATLAB to generate a mesh with a refined region from which I can extract values for the first and second elements of the cell array; or do I need to write functions in my cell array which can then act as input arguments to create a mesh?