C doesn't start with a modifier the way the other answer choices do.
In A, we have "Unlike transplants..., all patients"
In B, we have "Besides transplants..., all patients"
In D we have "Aside from a transplant..., all patients"
and in E, we have "Other than transplants..., all patients"
In C, however, we have a different construction altogether.
Let's look at a similar construction in a different sentence:
Jimmy rides through the street, spraying gravel behind his bike.
The first part is the main subject and verb, and the second part is subordinated.
Or this one:
I sat at the table with my dinner, thinking of all the other food I'd rather eat.
Make more sense this way? Sometimes, the length of the sentences on the GMAT make them unwieldy, and thus harder to analyze grammatically...
Hope this helps,