Let $M$ be a von Neumann with separable predual. It well known that one can write $M$ as a direct sum $M=M_I\oplus M_{II} \oplus M_{III}$ of von Neumann algebras of types $I$, $II$ and $III$. It is also known that one can write $M$ as a direct integral of factors $$ M=\int_X^\oplus M(x) d\mu(x) $$ and obtains that $Z(M)= L^\infty(X,\mu)$. Clearly one can partition $X$ into three parts $X_I$, $X_{II}$ and $X_{III}$ so that $M_j = \int_{X_j}^\oplus M(x) d\mu(x)$ where $j=I,II,III$. My question is whether a strengthening in the following sense is possible:

Is it possible to find factors $M_i$ and measure spaces $(X_i,\mu_i)$ indexed by some (necessarily countable) index set $I$, such that $$ M = \bigoplus_{i\in I} L^\infty(X_i,\mu_i) \bar\otimes M_i, \qquad \text{such that } i\ne j\implies M_i \not\simeq M_j ? $$ It would follow that $(X,\mu) \cong \bigoplus_{i\in I} (X_i,\mu_i)$ with $(X,\mu)$ as above.

My hope is that one can construct such a decomposition using the Choi-Effros Borel structure on the set of von Neumann algebras on a fixed separable Hilbert space $H$. Also, if the answer is yes I would be grateful for a reference.