An edge clique cover of an undirected graph $G$ is a set of cliques such that every edge of $G$ belongs to some clique in the set. The edge clique cover number $\theta(G)$ is the minimum size of edge clique cover of $G$.

Let's define $k$-restricted edge clique cover of $G$ as a set of cliques such that every edge of $G$ belongs to at least one but not more the $k$ cliques in the set. The $k$-restricted edge clique cover number $\theta_k(G)$ is the minimum size of $k$-restricted edge clique cover of $G$.

My questions are following: is anything known about this concept? I'm most interested in upper bound on $\theta_k(G)$ in terms of $\theta(G)$, but any reference would help.