  Many new strong gravitational lensing (SGL) systems have been discovered in
the last two decades with the advent of powerful new space and ground-based
telescopes. The effect of the lens mass model (usually the power-law mass
model) on cosmological parameters constraints has been performed recently in
literature. In this paper, by using SGL systems and Supernovae type Ia
observations, we explore if the power-law mass density profile ($\rho \propto
r^{-\gamma}$) is consistent with the cosmic distance duality relation (CDDR),
$D_L(1+z)^{-2}/D_A=\eta(z)=1$, by considering different lens mass intervals. {
It has been obtained that the verification of the CDDR validity is
significantly dependent on lens mass interval considered: the sub-sample with
$\sigma_{ap} \geq 300$ km/s (where $\sigma_{ap}$ is the lens apparent stellar
velocity dispersion) is in full agreement with the CDDR validity, the
sub-sample with intermediate $\sigma_{ap}$ values ($200 \leq \sigma_{ap} <
300)$ km/s is marginally consistent with $\eta=1$ and, finally, the sub-sample
with low $\sigma_{ap}$ values ($\sigma_{ap} < 200$ km/s) ruled out the CDDR
validity with high statistical confidence. Therefore, if one takes the CDDR as
guarantee, our results suggest that using a single density profile is not
suitable to describe lens with low $\sigma_{ap}$ values and it is only an
approximate description to lenses with intermediate mass interval. }
