Hi Diane,
There is an element of risk to all surgery, but it comes in the form of the anesthetic rather than the surgery itself, unless it's open heart and that sort.
Animal medicine and surgery utilizes many of the same medicines and operating room techniques as human hospitals. Only the patient and the procedure are the variables in many hospitals.
The science and art of anesthesia has improved from a good level to an excellent level today, using again, the same gasses and delivery equipment as people use. Granted, it may not have all the whistles and bells, but the technology is the same.
So, rest a bit easier. It's the anesthesia you need be anxious about, not the surgeon.
Also, consider how life will be without a surgery, for both you and your pet. I hope this is of help and things go well for you both.
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