McMillan proved that any contractible 3-manifold is obtained as a union of handlebodies, each of which is homotopically trivial in the next (this generalizes the method of construction of Whitehead).
Later he proved that there are uncountably many topologically distinct contractible 3-manifolds.
There is a survey of these and other results by McMillan. Even though these are old results, they are essentially state-of-the-art. The outstanding open problem was whether one of these contractible manifolds could cover a closed 3-manifold, but this is now resolved by the geometrization theorem (any closed aspherical 3-manifold is covered by $R^3$). Robert Myers obtained partial results on this problem, extended by David Wright, but this is superseded by geometrization. Myers also studied the space of end-reductions for such manifolds.
Of course, the paper you are interested in needs a very minimal amount of 3-manifold topology (aside from the Poincare conjecture!), which seems to be contained in Lemma 4.1 (although I'm not sure what 3-manifold results are used in papers which they cite, in particular reference 14).