I googled the title on the internet, and arrived at the following result

$$HH_k(D)\cong H_{DR}^{2n-k}(M).$$

Here $M$ is a smooth manifold of dimension $n$, and $D$ is the ring of differential operators on $M$. My first question is

(A): is there a known chain map realizing the above isomorphism?

I think the above isomorphism remains true if we replace $D$ by the ring of differential operators on $M$ with coefficients in a vector bundle $E$. But I am not so sure. If this is indeed the case, my second question is

(B): question (A) with coefficients in $E$?