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: sachinpatil@berkeley.edugoldberg@berkeley.edu

Narrow Passage Scenario

No probabilistic completeness guarantees