Addressing medical issues in grad school applications For most of university, I usually got around a ~3.5, with my GPA in my fourth year being around a 3.8.

However, during the first semester of my first year, I had some medical issues that made it really challenging to focus on school. Because of that, my GPA that semester was around a 1.7.

This one semester 4 years ago is completely destroying my GPA. Because of it my cumulative GPA is around a 3.35 instead of the 3.6 it should be. 

I'm applying to PhD programs right now, and I'm afraid that most grad schools will take a quick look at the 3.35 and immediately dismiss my application because it's so low... I've done as much as I can to mitigate the effect of it, including doing quite well my fourth year, publishing a research paper and doing several research internships throughout university.

I was hoping for some advice on the best way to deal with this. I'll definitely mention this in my SOP, but I don't know if that will be sufficient. I'm also not sure how many details I'm expected to share about the medical issue. It's obviously very personal and I'd prefer to not share too much, but if I absolutely need to I guess that's okay...
I would think that the one semester is an anomaly. As such, since it was your first semester, it likely won't have much impact as long as you explain it adequately. People do get sick. Other people understand that. On the other hand some people do tend to party a lot and others may not be so understanding. But an explanation should suffice. If appropriate (it isn't always) be able to back up your statements with some evidence - say a recommendation from a doctor that the issue is in the past. 

Some people won't even blink since your trend line is positive. 

If you can also show the the poor grades early on didn't leave you with a hole in your education, all the better. That seems unlikely in this case, of course. 

But of course, my analysis won't be universal. Some people will look harshly, but it should be rare most places.