For a 4-manifold M and a knot $k:{\mathbb{S}}^1\hookrightarrow \partial M$ with dual sphere $G:{\mathbb{S}}^2\hookrightarrow \partial M$, we compute the set $\mathbb{D}(M;k)$ of smooth isotopy classes of neat embeddings ${\mathbb{D}}^2\hookrightarrow M$ with boundary k using an invariant going back to Dax. Moreover, we construct a group structure on $\mathbb{D}(M;k)$ and show that it is usually neither abelian nor finitely generated. We recover all previous results for isotopy classes of spheres with framed duals and relate the group $\mathbb{D}(M;k)$ to the mapping class group of M.