  The sensitivity of a refractive index sensor based on an optical bound state
in the continuum is considered. Applying Zel'dovich perturbation theory we
derived an analytic expression for bulk sensitivity of an all-dielectic sensor
utilizing symmetry protected in-$\Gamma$ optical bound states in a dielectric
grating. The upper sensitivity limit is obtained. A recipe is proposed for
obtaining the upper sensitivity limit by optimizing the design of the grating.
The results are confirmed through direct numerical simulations.
