  Analytical expressions for the Transverse Momentum Dependent (TMD, or
unintegrated) gluon and sea quark densities in nuclei are derived at leading
order of QCD running coupling. The calculations are performed in the framework
of the rescaling model and Kimber-Martin-Ryskin (KMR) prescription, where the
Bessel-inspired behavior of parton densities at small Bjorken $x$ values,
obtained in the case of flat initial conditions in the double scaling QCD
approximation, is applied. The derived expressions are used to evaluate the
inclusive heavy flavor production in proton-lead collisions at the LHC. We find
a good agreement of our results with latest experimental data collected by the
CMS and ALICE Collaborations at $\sqrt s = 5.02$ GeV.
