  We consider the asymptotic shape of clusters in the Eden model on a
d-dimensional hypercubical lattice. We discuss two improvements for the
well-known upper bound to the growth velocity in different directions by that
of the independent branching process (IBP). In the IBP, each cell gives rise to
a daughter cell at a neighboring site at a constant rate. In the first
improvement, we do not allow such births along the bond connecting the cell to
its mother cell. In the second, we iteratively evolve the system by a growth as
IBP for a duration $\Delta$ t, followed by culling process in which if any cell
produced a descendant within this interval, who occupies the same site as the
cell itself, then the descendant is removed. We study the improvement on the
upper bound on the velocity for different dimensions d. The bounds are
asymptotically exact in the large-d limit. But in $d =2$, the improvement over
the IBP approximation is only a few percent.
