In the general case, this is the point where medicine arrive at both a medical and ethical problem. The biopsy and further tests both involve some non-insignificant risk to the patient, as does overtreatment. If 99% or even higher number of patients are false positive, and the biopsy has a risk of causing significant negative sides, at what point is testing doing more harm that good?

This is the general argument against testing for prostate cancer in non-symptomatic patients. If you only test patients that has symptoms then the rate of false positives goes down. The downside is that you only catch late stage patients where treatment might be too late. The alternative (which is used for mammography) is that for every person saved, an order of magnitude more will be permanent harmed by unnecessary surgery.