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1. Use the hot keys (1-9 buttons) if you aren't already, they will improve your time much more than you would imagine.
2. The vacuum only works when you have the cursor over the moving red bar. (you probably knew this one already)
3. When you cut open poison sacks, try burning a few of the cuts closed before vacuuming up the poison. You only need about 6 seconds to soak up the poison blobs and the poison reinfects 30 seconds after they are cut open so you have time to work with. This tactic is particularly useful on this surgery because of how much poison is in Peter.
4. When you reach the inner most part of Peter, attack the spider nests before the eggs. I don't know why this works out better, but it does somehow, and in any case if the nests are the hardest part of the surgery so if you fail on them first there was no lost effort on the eggs.
5. The third nest is the tricky one because the spider will release an extra poison spot when killed. To succeed on this part, you must remove the poison that appears from opening the nest up (the first cut with the pizza cutter) before electricuting with the battery. There is just too much poison to deal with other wise, but if you spend too much time on that another poison spot will appear. So, cut open the first poison spots and soak up the poison fast (maybe burn some cuts closed), but don't bother to heal them fully; then kill the nest and the spider, cut open the poison spots, burn some wounds closed and then remove all the poison with the vacuum (remember above all that the priority is stopping the poison from spreading).
6. Try doing all of the health boosting you need for the entire surgery at the very beginning before you make the first cut. In general, if you need to boost health while the patient is open and bleeding, you are fighting a loosing battle, but at the same time, you don't really know how much health you'll need for the surgery so you run the risk of waisting valuable time. You'll just have to take a best guess.
Hope this helps, best of luck.
Someone's far less popular main.