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Planning Curvature and Torsion ConstrainedRibbons for Intracavitary Brachytherapy
Sachin Patil, Jia Pan, Pieter Abbeel, Ken GoldbergUC Berkeley EECS
Cancer Sites
Brachytherapy
Internal radiation therapy – Radioactive source travels in catheters to tumor vicinity
Intracavitary Brachytherapy
Intracavitary Brachytherapy
Limitations of current treatment options:
Lack of proximity to tumor Insufficient radiation to tumor volume
Undesirable radiation exposure to healthy tissue
Patient discomfort, no personalization
Tumor Coverage
Standard approach New approach
Multiple dose locations desired
proximal to tumor
3D Printing
Stratasys uPrint SE Plus
3D Systems ProJET HD 3000
3D Printed Implant
[Garg et al. 2013]
Customized 3D Printed Implants
CT Scan
3DModel
Channel Planning
3D Print
[Garg et al. 2013]
Channel ConstraintsCurvature constraints:
Finite dimensions of radioactive seed
Limited flexibility of catheters
Extraction of support material
Independent Channels
Infeasible for larger number of dose locations
Mutually collision free
Constraints on local/cumulative curvature
Ribbons
Ribbons
Improved arrangement Improved coverage
How do we create these implants?
Ribbon Kinematic Model
Consider ribbon cross-section:
Orient ribbon cross-section along binormal of Frenet-Serret frame [Frenet 1847; Serret 1851]
Ribbon Kinematic Model
Frenet-Serret equations:
Some manipulation yields:
Ribbon Kinematic Model
This gives the following model: Planning parameters:
: speed : curvature : torsion
Why Frenet-Serret Frame?
Different curvatures, lengths: Difficult to plan for
Same curvatures, lengths: Easier to plan for
Problem Specification
Input:
Implant volume conforming to patient anatomy from CT/MR scans
Dose dwell segment poses
Parameters of catheter and radioactive source channel width, curvature and torsion limits
Problem Specification
Objective: Compute ribbons such that:
Curvature and torsion constrained
Optimal – minimize energy
Mutually collision-free
Related Work Planning rigid body motions in SE(3)
without obstacles: Zefran et al. 1998; Belta et al. 2004; Goemans et al. 2005; Biggs et al. 2008; Cripps et al. 2012; etc.
Planning using physically-based models of curves/ribbons:Moll et al. 2006; Bretl et al. 2014; etc.
Planning for bevel-tip steerable needles:Alterovitz et al. 2006,2007; Hauser et al. 2009; Xu et al. 2009; Duindam et al. 2010; Van den Berg et al. 2010; Patil et al. 2012; etc.
Planning Challenges
Nonholonomic system Collision avoidance
Planning Approach
Two steps:
Sequential: Rapidly-exploring random trees (RRT) in SE(3) state space
Simultaneous: Local optimization using sequential quadratic programming (SQP)
RRT Planner
a
b
Sample random point in R3
Find nearest tree node that contains sample within reachable set
Connect
Add new node and edge to tree
Repeat till goal found or maximum
iterations exceeded
Collision detection
a
entry
dose dwell segment
For each dose dwell segment:
[Patil et al. 2012; Garg et al. 2013]
RRT Limitations
Non-smooth ribbons; unnecessary twists
No notion of optimality
(Simultaneous) Local OptimizationOptimization variables:
Minimize energy (rotational strain) :
subject to
Entry / initial pose constraint
Kinematic constraints
Bounds on curvature/torsion
Collision constraints
[Schulman et al. 2013]
Optimization on SE(3)SE(3) is not a vector space:
Locally parameterize SE(3) through its tangent space se(3)
Optimization on SE(3)1) Seed trajectory:
2) Solve: where and
3) Compute new trajectory:
[Saccon et al. 2013]
RRT + Local Optimization
Two steps:
Sequential: Rapidly-exploring random trees (RRT) in SE(3) state space
Simultaneous: Local optimization using sequential quadratic programming (SQP)
RRT + Local Optimization
Intracavitary Brachytherapy Scenario
RRT: Collision-free ribbons; unnecessary twists
RRT + Local optimization: final solution
Intracavitary Brachytherapy Scenario
46% improvement in coverage (metric as defined by Garg et al. 2013)
Limited to 18 channels Can include up to 36 channels
Performance
[single 3.5 Ghz Intel i7 processor]
Address global optimality of solutions [Bento et al. NIPS 2013s]
Automatic computation of dose dwell segments
Clinical studies (UC San Francisco Medical Center)
Future Work
Ribbons – Planning Applications
Source available at: https://github.com/panjia1983/channel_backward
Thank You
Contact: [email protected]@berkeley.edu
Narrow Passage Scenario
No probabilistic completeness guarantees