  In this paper, we prove a quantitative regularity theorem and a blow-up
criterion of classical solutions for the three-dimensional Navier-Stokes
equations. By adapting the strategy developed by Tao in [20], we obtain an
explicit blow-up rate in the setting of critical Lorentz spaces $L^{3,
q_{0}}(\mathbb R^3)$ with $3 \leq q_0 < \infty $. Our results improve the
previous regularity in critical Lebesgue spaces $L^3(\mathbb R^3)$ in [20] and
quantify the qualitative result by Phuc in [16].
