  Pb doping effect in the Bi$_2$Sr$_2$Ca$_2$Cu$_3$O$_{10}$ compound (Bi2223) on
the structural and electronic properties were investigated, using the Local
Density (LDA) and Virtual Crystal (VCA) approximations within the framework of
the Density Functional Theory (DFT), taking as reference the procedure
implemented by H.Lin {\it et al.} in the Bi2212 compound [{\it Phys. Rev.
Lett.} {\bf 96} (2006) 097001]. Results show that, the incorporation of
Pb-dopant in Bi2223 lead a rigid displacement of the Bi/Pb-O bands toward
higher energies, with a null contribution at the Fermi level, around the high
symmetry point $\overline{\text{M}}$ in the irreducible Brillouin zone, for Pb
doping concentration equal to or more than 26\%, avoiding the presence of the
so-called Bi-O {\it pockets} in the Fermi surface, in good agreement with
angle-resolved photoemission spectroscopy (ARPES) and nuclear magnetic
resonance (NMR) experiments, although a slight metallic character of the Bi-O
bonds is still observed which would disagree with some experimental reports.
The calculations show that the changes on the structural properties are
associated to the presence or absence of the Bi-O {\it pockets} in the Fermi
surface
