applied sciences Article  
Neural Tract Avoidance Path-Planning Optimization:  
Robotic Neurosurgery  
[PERSON],*(c), [PERSON](c), [PERSON](c) and [PERSON](c)  
Footnote 1  
those may affect more tissue due to longer paths. However, curved needle navigation would allow for avoidance of high-risk regions, increasing safety at the cost of less intuitive planning and execution, in addition to the need to consider more constraints, thus posing a bigger challenge to neurosurgeons. Robotic neurosurgery could help with these challenges, and it can also take benefits from the use of these new instruments, with the ability to perform curved trajectories to achieve a surgical goal. However, the difficulty of calculating these trajectories is significantly greater than straight trajectories, which primarily involve evaluating the entry point's position, entry angle, and segment length. In the context of nonrigid tools, additional parameters must be considered, such as the inherent curvature limitations (bend limits) of the tool and the possibility of avoiding critical anatomical structures. When discussing methodologies for nonrigid needle path-planning, the literature presents a variety of studies. Some investigations focus on helping neurosurgeons determine the optimal approaches to deep brain stimulation (DBS) [10; 11] and assisting electrode implantation [12] and electrochemotherapy treatment planning and navigation [13]; another approach has focused on resection interventions [14] and biopsy procedures [15], as well as the automatic calculation of entry points in minimally invasive surgery (MIS) [16]. Regarding the methodological approach, Markov models are investigated as an alternative for motion planning in the presence of motion uncertainty [17]. Other authors use rapidly exploring random tree (RRT) algorithms. [PERSON] et al. [15] suggested a distance metric for the incremental growth of RRT algorithms to improve planning performance, and [PERSON] et al. [11] proposed an RRT-based planning method that considers flexible and anatomical needle restrictions. Multi-objective optimization (MOP) has also been explored; [PERSON] et al. [18] exposed a population evolutionary algorithm to solve three different objectives to acquire precise and effective solutions, and [PERSON] et al. [19] implemented a follow-the-leader, flexible path-planning method using patient-specific image data. Other methods, such as the A* path-finding algorithm [20] or automatic path planning using the GA3C reinforcement learning algorithm [21], have been published; in addition, modern studies have combined methods to address the problem, such as the bi-directional tree proposed in [22], which handles partially observable Markov decision processes (POMDPs) in a continuous state, aiming to improve the calculation of efficiency, and [23], in which they use the Dijkstra A* algorithms and their variants to find optimal trajectories with risk scores based on segmentations carried out by surgeons. Most of these works rely on the Pareto optimization method to find (among the multiple trajectories) the optimal path based on multiple constraints [16]. Our approach is based on the application of a modified version of the rapidly exploring random tree algorithm for multi-trajectory generation and optimization, with a cost function that evaluates the different entry points and uses the information of the MRI images as segmented binary maps. As a novelty, an avoidance module for the critical neuronal tracts defined by the neurosurgeon is included in the optimization process. The structure of the paper is as follows: the \"Materials and Methods\" section presents details of the algorithm used. First, the modeling of geometric constraints is presented, and the software tools used are detailed. This section describes the implemented cost function and specifies the different terms and weights assigned.