Take a look at Example 6 in [1] which computes $rk \,\pi_2 = 22$, $rk\: \pi_3= 252$ and $rk \: \pi_4 = 3520$ simply using the fact that a quartic in $\mathbb{P}^3$ is a K3 surface

[1]Terzić, Svjetlana. On rational homotopy of four-manifolds. Contemporary geometry and related topics, 375–388, World Sci. Publ., River Edge, NJ, 2004.