# Optimized fiducial marker placement using B‐spline surface modeling and graph theory for Cyberknife stereotactic body radiotherapy for superficial tumors

## TL;DR

This paper introduces a new algorithm for placing fiducial markers in CyberKnife radiotherapy for superficial tumors, improving efficiency and accuracy.

## Contribution

A novel algorithm for fiducial marker placement using B-spline surfaces and graph theory, significantly faster than brute-force methods.

## Key findings

- The algorithm uses B-spline surfaces and graph theory to optimize marker placement for superficial tumors.
- The proposed method reduces time complexity from O(n³) to O(mlogm + m² + 3n³).
- Experimental results show efficient planning of fiducial marker placements for CyberKnife treatments.

## Abstract

CyberKnife, an established noninvasive stereotactic radiotherapy technology, has been extensively utilized to treat various malignancies because of its high precision and conformal dose delivery. The success of CyberKnife treatment is crucially dependent on optimal fiducial marker placement. This study introduces a novel fiducial marker placement planning algorithm tailored for superficial tumors, which are located 20–50 mm beneath the epidermis. A retrospective analysis was performed on the data collected from three patients with thymus, breast, and submandibular gland tumors. This algorithm generated potential implantation sites by constructing and optimizing a B‐spline surface around the tumor. Candidate points were filtered using multi‐criteria constraints: (1) a minimum of 18‐mm inter‐marker distance, (2) angular separation >30°, and (3) nonoverlapping visibility in 45° oblique digital reconstructed radiographs. To enhance the computational efficiency, a kd‐tree spatial indexing structure was integrated with graph theory, specifically the Bron–Kerbosch algorithm for maximal clique detection. The proposed method achieved a time complexity of O(mlogm+m2+3n3), demonstrating a significant improvement over the brute‐force O(n3) approach. The experimental results showed that our algorithm could efficiently plan fiducial marker placement, thereby simplifying the planning process and providing valuable technical support for CyberKnife treatments.

This study introduces a Fiducial marker placement planning algorithm, specifically for superficial tumors that are located 20‐50 mm beneath the epidermis. The algorithm proposes numerous potential implantation sites by constructing and optimizing a B‐spline surface. The time complexity of the algorithm proposed in this study is O(mlogm+m2+3(n/3)), significantly faster than the brute‐force O(n3) approach. Experimental outcomes show that our algorithm can efficiently plan Fiducial marker placements, simplifying the planning process and providing valuable technical support for CyberKnife treatments.