In this paper we address the problem of finding an "optimal" strategy for placing k biopsy needles, given a large number of possible initial needle positions. THis problem arises for example in guided, endoscopic needle biopsies, where the position of the endoscope's tip is known with some error. We consider tow variations of the problem: (1) Calculate the smallest set of needles, needed to guarantee a successful biopsy. (2) Given a number k, calculate k needles such that the probability of a successful biopsy is maximized. We formulate both problems in terms of two general, NP-hard optimization problems. Our solution to both problems is "optimal" with respect to the best approximative algorithm known for the respective NP-hard problem. For the latter problem there exists an approximative algorithm which requires virtually no implementation effort and is guaranteed to be within a factor of 1-1/e of the exact solution. For both variations of the problem we are able to provide success probabilities for each needle to the physician. We have implemented the approximative algorith, for the second variation. The resulting probabilities show that our approach can provide valuable decision support for the physician in choosing how many needles to place an how to place them.
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