# Optimal Path Planning of a Multi-tube Bent Cannula for Deep Brain Surgery

**Figure 1.**For deep brain stimulation or biopsy, a straight cannula enters the skull through a small hole to reach the target position. Figure courtesy of

*Augsburger Allgemeine Zeitung*.

**Figure 2.**A series of multiple nested interacting curved tubes comprise a curved cannula with a common backbone shape that navigates the brain when no straight path is available. Figure courtesy of Matthias Hoffmann.

Hoffmann relied on the Rucker model for optimal path planning. This geometrically exact, nonlinear, quasi-static model computes an approximation of a cannula’s backbone based on given tubes and their \(\alpha\) and \(\beta\) values. Neglected effects like contact forces, friction, and material nonlinearities mean that some errors will be unavoidable; nevertheless, the Rucker model serves as a good starting point. “We want to compute an optimal solution for \(\alpha\) and \(\beta\) and adapt it to reach the target point by modifying \(\alpha\) and \(\beta\) based on measurements of the system,” Hoffmann said.

A photogrammetry system allowed Hoffmann to triangulate the position of the cannula tip; this process occurs outside of the patient because the system is robust to changes (see Figure 3). He conducted optimization-based trajectory planning to obtain the \(\alpha\) and \(\beta\) values, measured the position of the cannula tip, and computed the tip position’s sensitivity for \(\alpha\) and \(\beta\) via the Rucker model. Hoffmann then used sensitivity-based ILC to change \(\alpha\) and \(\beta\) and convert to the actual target. “We want the tip to end up in the target at the end,” he said. He was able to change the \(\alpha\) and \(\beta\) levels again and again based on the measurements. The photogrammetry system confirmed that the solution converges.

In the difficult case of especially bent cannula tubes, the algorithm reduced the target error from 37 mm to less than 0.2 mm, which meets the aforementioned surgical requirements. “Sensitivity information for ILC, while a rough approximation, is a valuable tool,” Hoffmann said. The solution reached a minimum in 25 iteration steps and began to deteriorate, but Hoffmann stored it and can return to it once performance worsens. If the curvature is not strong enough, the algorithm reduces the error over time but never reaches the target point. In this case, one must replace the tubes and start again.

**Figure 3.**Diagram of the photogrammetry system that triangulates the position of the cannula tip. Figure courtesy of Matthias Hoffmann.

Next, Hoffmann briefly overviewed results from data-based ILC, for which he trained neural networks (NNs) using PyTorch. Results from this method were not as good as those from the sensitivity-based approach; the error reduced from 32 mm to six mm in nine iterations, which does not meet the necessary requirements for surgery. Future steps for this project involve working to improve the data-based outcomes. Hoffmann and his team also intend to employ higher-order sensitives, combine sensitivity and data-based approaches, address constraints, and explore NN-based ILC techniques as a possible initialization for reinforcement learning.

Lina Sorg is the managing editor of SIAM News. |