Planning Curvature and Torsion Constrained Ribbons in 3D With Application to Intracavitary Brachytherapy

Abstract

Abstract. A "ribbon" is a surface traced out by sweeping a constant width line segment along a spatial curve. We consider the problem of planning multiple disjoint and collision-free ribbons of finite thickness along curvature and torsion constrained curves in 3D space. This problem is motivated by the need to route multiple smooth channels through a 3D printed structure for a healthcare application and is relevant to other applications such as defining cooling channels inside turbine blades, routing wires and cables, and planning trajectories for formations of aerial vehicles. We show that this problem is equivalent to planning motions for a rigid body, the cross-section of the ribbon, along a spatial curve such that the rigid body is oriented along the unit normal to the curve defined according to the Frenet-Serret frame. We present a two stage approach. In the first stage, we use sampling-based rapidly exploring random trees (RRTs) to generate feasible curvature and torsion constrained ribbons. In the second stage, we locally optimize the curvature and torsion along each ribbon using sequential quadratic programming (SQP). We evaluate this approach for a clinically motivated application: planning multiple channels inside 3D printed implants to temporarily insert high-dose radioactive sources to reach and cover tumors for intracavitary brachytherapy treatment. Constraints on the curvature and torsion avoid discontinuities (kinks) in the ribbons which would prevent insertion. In our experiments, our approach achieves an improvement of 46% in coverage of tumor volumes as compared to an earlier approach that generates each channel in isolation.

Additive manufacturing (3D printing) in superficial brachytherapy

“…The source paths are generally forward-planned, based on the expertise of radiation oncologist, medical physicist, or brachytherapist. Several groups have developed computer algorithms to optimize BT catheter paths as well as dwell positions, in the hope of eliminating the need for human oversight [75][76][77]. To date, however, inverse optimization of catheter paths have not been implemented for superficial BT.…”

Optimal under-actuated kinematic motion planning on the ϵ-group

“…Here we proceed as in § §2.2 by setting up the shooting function using the error matrix to give a shooting function of the form (17). However, instead of solving the set of equations ( 6), ( 21) iteratively, the semi-analytic approach is used where the equations ( 21) are solved analytically in terms of Jacobi elliptic functions as described in § §3.1 giving a solution h(t) of the form of one of the solutions in table (24) (or the limiting case as c i → 0 of one of these solutions if n < 3 in ( 1))…”

“…using the matlab function fminunc. The weight w in (50) must be fixed, since determining the minimum of (50) so that the boundary conditions are still met is a multi-objective problem requiring to minimize both the condition (17) and the performance metric value (49). In practice, it serves us best to minimize y(h 0 , t f ) for some low value of w to obtain an h 0 value for which cost is minimized, and then to decrease w incrementally to the case w = 0 to obtain the h 0 which initializes the low-cost extremal.…”

“…For example, the kinematics of various autonomous systems such as the attitude kinematics ( = 1, n = 3) of spacecraft [1][2][3], including those with velocity constraints [4] and those underactuated in control [5,6]. Systems such as wheeled robots, robotic grippers and slender underwater vehicles that exhibit a sliding-type constraint ( = 0, n = 2 or n = 3 ) [7][8][9][10][11][12][13][14] and in wider fields such as switched electrical networks [15], problems of quantum control [16] and planning curvature-and torsion-constrained 3D printed implants for facilitating radiation therapy [17]. Various methods have been developed to tackle the motion planning problem of left-invariant (respectively right) systems defined on matrix Lie groups of the form (1) where it is necessary to match the boundary condition g(0) = g 0 and g(T ) = g T .…”

Exact reachability analysis for planning skew-line needle arrangements for automated brachytherapy

“…We have worked on several aspects of the problem: treatment planning [4,13], needle configuration planning [5,14], robotic needle placement [6,15], and recently the use of custom guides for brachytherapy needle placement [16,17].…”