A mathematical model for preoperative planning of radiofrequency ablation of hepatic tumors

Background: Radiofrequency ablation (RFA) is rapidly evolving as an effective minimally invasive technique for the treatment of small and unresectable liver tumors. A potential cause of treatment failure is the inability to determine the optimum number of overlapping ablations needed to completely destroy tumors larger than the size of a single ablation. To clarify this relationship, we performed a mathematical evaluation that enables us to accurately estimate the number of ablations needed to completely ablate larger tumors. Methods: This estimation is based on the assumptions that complete ablation of the surface of a target tumor, including its blood supply, would completely destroy the tumor and that the tumor and ablations produced are perfectly spherical. The smallest possible number of partially overlapping ablations that would completely cover the surface of the target tumor is the same as the number of faces on a regular polyhedron that has a circumscribing diameter equal to or greater than the diameter of the target sphere. Results: This mathematical analysis shows that for a 5-cm ablation device, tumors with diameters ranging between 3.01 and 3.30 cm will require at least four ablations. Tumors between 3.31 and 4.12 cm require six overlapping ablations, and tumors between 4.13 and 6.23 cm require 12 overlapping ablations. The number of ablations needed for larger tumors and for 3-, 4-, 6-, and 7-cm ablation devices are also determined. Conclusion: The smallest number of ablations required to completely ablate a spherical target tumor larger than the size of the ablation sphere increases dramatically as tumor size increases. Because this model is geometrically optimized, even a small change in the position of the ablation spheres with respect to the target sphere can leave potentially unablated tumor and thus result in treatment failure.