I suspect that when you talk about cartesian coordinates, you are referring to row, column, layer coordinates.
Note that typically for images, what you care about is coordinates relative to corner of the volume, rather than absolute coordinates. In such cases, you can typically translate coordinate systems just by knowing the x, y, and z resolution. However, you specifically asked for world coordinates, so you will have to pay attention to the Image Position (0020,0032) and Image Orientation (0020,0037). I have to say, though, that it is very rare to see an imaging scanner with an attached GPS, as would be necessary to work with world coordinates.