Homotopy bundle gerbes

In the previous talks by means of bundle gerbes a functor $bg$ from the category of compact topological spaces to the category of 2-groupoids has been constructed. Furthermore, using modules over bundle gerbes a functor $bg(X)-->Ab$ ("twisted $K$-theory") which is functorial on the base $X$ and coincides with $K^0(X)$ in case of trivial bundle gerbe has been defined.
The group of isomorphism classes of objects of the category $bg(X)$ is isomorphic to $H^3(X,\mathbb{Z})$, the third cohomology group of $X$ with integer coefficients. However twists from $H^3(X,\mathbb{Z})$ are not most general possible for K-theory. Therefore it seems natural to extend the above construction of twisted $K$-theory to more general twists. In order to do this, the notion of a bundle gerbe itself has to be generalized: first, in place of line bundles we have to consider arbitrary finite dimensional vector bundles, and second, replace isomorphisms by homotopies.