  A recently introduced analytical model for the nuclear density profile[1] is
implemented in the Extended Thomas-Fermi (ETF) energy density functional.
  This allows to (i) shed a new light on the issue of the sign of surface
symmetry energy in nuclear mass formulas, which is strongly related to the
non-uniformity of the isospin asymmetry in finite nuclei, as well as to (ii)
evaluate the in-medium corrections to the nuclear cluster energies in
thermodynamic conditions relevant for the description of the (proto)-neutron
star crust. The ground state configurations of the model are compared to
Hartree-Fock calculations in spherical symmetry for some selected isotopic
chains, and systematic errors are quantified. The in-medium modification of the
nuclear mass due to the presence of a gas component is shown to strongly depend
both on the density and the asymmetry of the nucleon gas. This shows the
importance of accounting for such effects in the realistic modelizations of the
equation of state for core-collapse supernovae and proto-neutron stars.
