Thought Experiment #$2$: Brain Replacement

$$Suppose that I need surgery. All of my brain cells have a defect which, in time, would be fatal. But a surgeon can replace all these cells. He can insert new cells that are exact replicas of the existing cells except that they have no defect. We can distinguish two cases.

$$In Case One, the surgeon performs a hundred operations. In each of these, he removes a hundredth part of my brain, and inserts a replica of this part.

$$In Case Two, the surgeon follows a different procedure. He first removes all of the parts of my brain, and then inserts all of their replicas.

$$The difference between the cases is merely... a difference in the ordering of removals and insertions. In Case One, the surgeon alternates between removing and inserting. In Case Two he does all the removing before all the inserting. Can this be the difference between life and death? Can it be so important, for my survival, whether the new parts are, for a time, joined to the old parts?$$