The surgeons handle this regularly, it's routine for them. You get to relax and let them manage everything.
I went through lung surgery a few years ago and coped by deciding that I could trust the surgeons (and other medical professionals) completely. They know what they are doing, and they have procedures in place to make sure everything goes well.
Learning to consciously relax and trust everyone around you is a something that takes constant effort, but in my opinion, it's worth it.
My surgery went well, no complications, and my recovery was exactly on schedule.