Regression and ANOVA are in the most important senses the same thing. The difference here: if you do factorial ANOVA in the manner I'm describing, you're allowing the possibility of a group X scanner interaction. If you do the regression approach you outlined, you're not; you're only adding the main effect of scanner to the model, but not the interaction. (Note, however, that in a regression context you can also model the interaction.)
While it looks like you'd be splitting up the groupings based on diagnosis, it's actually just assuming the interaction might be nontrivial.
If you look at the Stonnington et al. article that Chris mentioned, they explicitly consider a group-by-scanner interaction. I only skimmed it just now, but I think in that paper they ended up saying that that interaction wasn't significant. (From the abstract: "There was no significant interaction of scanner with disease group.") Based on what I recall from standard stats texts, you could optionally at that point drop the interaction term from your model.
Finally, you might claim _a priori_ that, based on your understanding of the concrete situation (diagnosis as one factor, scanner as another), there's no possibility of an interaction, but that's not so clear to me. Of course, that might become more important if one of your four cells is underpopulated, though at this point I'll stop blithering and recommend you consult a stats reference text.
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