I am attempting to get to grips with Thurston's hyperbolization theorem for Haken $3$--manifolds. In particular I was looking at the section related to gluing up along hierarchy surfaces in Otal, Jean-Pierre (1998), "Thurston's hyperbolization of Haken manifolds"
From what I understand we take hierarchy for a Haken $3$--manifold $M$ with corners (a decomposition of the manifold with corners along incompressible surfaces into a collection of $3$--balls), use something like Andreev's theorem to put a hyperbolic structure on the $3$--balls, and use an inductive argument to re-glue the manifold along the hierarchy surfaces in such a way that the original manifold carries a complete hyperbolic metric. I am speaking very loosely here since I can't say I understand all the details.
My question is as follows. Is it possible to re-formulate the this construction in terms of manifolds with boundary pattern (in the spirit of Johannson), and if so, if the boundary pattern on the collection of $3$--balls corresponds to the cuts along hierarchy surfaces, does the boundary pattern carry all the data needed for re-gluing?