Algorithms for Design of Continuum Robots Using the Concentric Tubes Approach: A Neurosurgical Example

Abstract

We propose a novel systematic approach to optimizing the design of concentric tube robots for neurosurgical procedures. These procedures require that the robot approach specified target sites while navigating and operating within an anatomically constrained work space. The availability of preoperative imaging makes our approach particularly suited for neurosurgery, and we illustrate the method with the example of endoscopic choroid plexus ablation. A novel parameterization of the robot characteristics is used in conjunction with a global pattern search optimization method. The formulation returns the design of the least-complex robot capable of reaching single or multiple target points in a confined space with constrained optimization metrics. A particular advantage of this approach is that it identifies the need for either fixed-curvature versus variable-curvature sections. We demonstrate the performance of the method in four clinically relevant examples.

Fig. 1

Ventricular system (VS) of the brain consists of four cerebral ventricles: the paired lateral ventricles, and the midline third and fourth ventricles. Source: Mayfield Clinic ©.

Fig. 2

ETV procedure. Oblique view (A) demonstrating typical location of the burr hole and trajectory; midsagittal view (B) demonstrating location of ventriculostomy. Source: Neurosug Focus ©.

Fig. 3

Concentric tube robot comprised of four telescoping sections that can be rotated and translated with respect to each other.

Fig. 4

Geometric models of the VS of healthy (A) and hydrocephalic (B) subjects. The models were reconstructed from T1-weighted MR images.

Fig. 5

Parameterization definition: we assume that the entry and exit points x⃗₀ and x⃗ _t, as well as the direction n⃗₀ are defined. For any arbitrarily selected x⃗₁ there exists only one circular arc passing through it, so by fixing the location of x⃗₁ we fully define the first section (A). Since specifying x⃗_i also defines the direction n⃗_i for the next section x⃗_{i + 1}, we can recursively specify points x⃗₁, …, x⃗_N−1, with x⃗_N = x⃗_t and thus fully define a unique N- sectioned robot (B).

Fig. 6

Example 1: the objective is to navigate from the entry point (blue dot) to the target point at tip of the temporal horn of the lateral ventricle (red dot). For a family of three sectioned robots, only up to 83% of the robot can be contained within the ventricle (A, B); the algorithm found a solution for a robot with four sections where the robot is wholly contained within the ventricle (C, D).

Fig. 7

Example 2 - the objective is to navigate inside the VS of a hydrocephalic ventricle and to approach the base of the choroid plexus (red dot) from a predefined direction specified by the green vector. The algorithm succeeded in identifying a three sectioned robot with its tip direction aligned almost perfectly (red arrow) with the green arrow.

Fig. 8

Example 3 – the objective is to identify a single robot with minimal number of segments capable of reaching all six target points. A robot with only three segments was found by the algorithm. The design parameters for this robot are listed in Table I.

Fig. 9

Example 4 – the objective is to identify a single robot with minimal number of segments capable of reaching all three target points. A robot with three (two constant- and one variable- curvature) segments was found by the algorithm. The design parameters for this robot are listed in Table II.