Given a differential graded algebra $(A_\bullet,d)$, is there a well-defined notion of a K-injective, K-projective, K-flat dimension of a differential graded module, or even of the category of differential graded modules?

Moreover, if there is a well-defined notion, when are each of them finite? For example, does the Koszul complex $K^\bullet_R(M;f_1,\ldots,f_k)$ have computable K-dimensions?