Step 1: The question wants one modality that can sweep the whole body to hunt deposits. The ideal tool gives excellent tissue and marrow contrast while sparing the patient radiation.
Step 2: Whole-body $MRI$ fits this perfectly. With a rolling table and rapid T2 turbo spin-echo plus 3D gradient-echo sequences, it images the body in stations and detects marrow and soft-tissue metastases with high accuracy.
Step 3: Rule out the others on principle: angiography and venography are targeted vessel studies, not body-wide surveys, and CT delivers a heavy cumulative radiation load with poorer marrow contrast for this purpose.
\[\boxed{Magnetic\ Resonance\ Imaging}\]