CS 326 A: Motion Planning

http://robotics.stanford.edu/~latombe/cs326/2003

Radiosurgical Planning

Radiosurgery

Minimally invasive procedure that uses an intense, focused beam of radiation as an ablative surgical instrument to destroy tumors

Tumor = bad Critical structures = good and sensitive Brain = good

The Radiosurgery Problem

Dose from multiple beams is additive

Treatment Planning for

• Determine a set of beam configurations that will destroy a tumor by cross firing at it • Constraints:

Radiosurgery

– Desired dose distribution – Physical properties of the

Critical Tumor

radiation beam – Constraints of the device delivering the radiation – Duration/fractionation of

Conventional Radiosurgical Systems

• Isocenter-based treatments • Stereotactic frame required

Gamma Knife LINAC System

Luxton

et al.

, 1993 Winston and Lutz, 1988

Isocenter-Based Treatments

• All beams converge at the isocenter • The resulting region of high dose is

spherical

Nonspherically shaped tumors are approximated by multiple spheres – “Cold Spots” where coverage is poor – “Hot Spots” where the spheres overlap – Over-irradiation of healthy tissue

Stereotactic Frame for Localization

• Painful • Fractionation of treatments is difficult • Treatment of extracranial tumors is impossible

Optimization Approach to Planning

• Treatment planning variables: – Isocenter locations – Treatment arcs – Collimator diameter – Beam weights • User specifies most planning variables • User defines constraints on dose to points and optimization function • Optimization technique determines beam weights [Bahr, 1968; Langer et al. 1987; Webb, 1992; Carol et al., 1992; Xing et al., 1998, …]

Motion-Planning Approach

Critical Structure Construction of free space

• Configuration space of the beam is a unit sphere • Construct free space by projecting critical structures onto the sphere • Search for longest arcs in free space [Schweikard, Adler, and Latombe, 1992, 1993]

Conformal Dose Distributions

X-Ray cameras

The CyberKnife

linear accelerator robotic gantry

Treatment Planning Becomes More Difficult

• Much larger solution space – Beam configuration space has greater dimensionality – Number of beams can be much larger – More complex interactions between beams • Path planning – Avoid collisions – Do not obstruct X-ray cameras  Automatic planning required (CARABEAMER)

Inputs to CARABEAMER

(1) Regions of Interest: Surgeon delineates the regions of interest CARABEAMER 3D regions generates

Inputs to CARABEAMER

(2) Dose Constraints: Dose to tumor Dose to critical structure

Tumor Critical

Falloff of dose around tumor Falloff of dose in critical structure (3) Maximum number of beams

Basic Problem Solved by CARABEAMER

• Given: – Spatial arrangement of regions of interest – Dose constraints for each region: a 400) • Find:  D  b – Maximum number of beams allowed: N (~100 – N beam configurations (or less) that generate dose distribution that meets the constraints.

Beam Configuration

• Position and orientation of the radiation beam z  y (x, y) x  • Amount of radiation or

beam weight

• Collimator diameter 

Find 6N parameters that satisfy the constraints

CARABEAMER’s Approach

1. Initial Sampling: Generate many (> N) beams at random, with each beam having a reasonable probability of being part of the solution.

2. Weighting: Use linear programming to test whether the beams can produce a dose distribution that satisfies the input constraints.

3. Iterative Re-Sampling: Eliminate beams with small weights and re-sample more beams around promising beams.

4. Iterative Beam Reduction: Progressively reduce the number of beams in the solution.

Initial Beam Sampling

• Generate even distribution of target points on the surface of the tumor • Define beams at random orientations through these points

Evenly Spacing Target Points on Tumor

• Turk [1992] • Normally distribute points on tumor surface • Use potential field to better distribute points

Deterministic Beam Selection is Less Robust

Curvature Bias

• Place more target points in regions of high curvature

Dose Distribution Before Beam Weighting

50% Isodose Surface 80% Isodose Surface

CARABEAMER’s Approach

1. Initial Sampling: Generate many (> N) beams at random, with each beam having a reasonable probability of being part of the solution.

2. Weighting: Use linear programming to test whether the beams can produce a dose distribution that satisfies the input constraints.

3. Iterative Re-Sampling: Eliminate beams with small weights and re-sample more beams around promising beams.

4. Iterative Beam Reduction: Progressively reduce the number of beams in the solution.

Beam Weighting

• Construct geometric arrangement of regions formed by the beams and the tissue structures • Assign constraints to each cell of the arrangement: – Tumor constraints – Critical constraints B1 B2 C T B4 B3

B1 B2

Linear Programming Problem

C B4 • •

2000



2000



2000



2000



2000



2000



2000



2000



2000



2000



Tumor



2200 B2 + B4



B4



2200 B3 + B4



2200 2200 B3



2200 B1 + B3 + B4



B1 + B4



2200 B1 + B2 + B4



B1



2200 B1 + B2



2200 2200 2200 0



0



Critical



B2



500 500

B3

Elimination of Redundant Constraints

•

2000 < Tumor < 2200 2000 < B2 + B4 < 2200 2000 < B4 < 2200 2000 < B3 + B4 < 2200 2000 < B3 < 2200 2000 < B1 + B3 + B4 < 2200 2000 < B1 + B4 < 2200 2000 < B1 + B2 + B4 < 2200 2000 < B1 < 2200 2000 < B1 + B2 < 2200

•

2000 < Tumor < 2200 2000 < B4 2000 < B3 B1 + B3 + B4 < 2200 B1 + B2 + B4 < 2200 2000 < B1 2000 < B2 + B4 2000 < B4 B2 + B4 < 2200 B1 + B2 + B4 < 2200

Results of Beam Weighting

Before Weighting After Weighting 50% Isodose Curves 80% Isodose Curves

CARABEAMER’s Approach

1. Initial Sampling: Generate many (> N) beams at random, with each beam having a reasonable probability of being part of the solution.

2. Weighting: Use linear programming to test whether the beams can produce a dose distribution that satisfies the input constraints.

3. Iterative Re-Sampling: Eliminate beams with small weights and re-sample more beams around promising beams.

4. Iterative Beam Reduction: Progressively reduce the number of beams in the solution.

Iterative Re-Sampling

• The initial set of beam may not contain a solution.

• Find the best possible solution • Keep beams that are useful • Remove beams that are not useful • Re-sample

Reformulating the LP Problem …

A linear program is typically specified as:

Minimize: c 1 x 1 + c 2 x 2 + . . . + c n x n Subject to:

l

1



l

2



a 1,1 x 1 a 2,1 x 1 +a ...

1,2 x 2 +a 2,2 x 2 + . . . + a 1,n x n



+ . . . + a 2, n x n



u

1

u

2

l

m



a m,1 x 1 +a m,2 x 2 + . . . + a m, n x n



u

m

Reformulating the LP Problem …

Using slack variables , we can rewrite this:

Minimize: c 1 x 1 Subject to: + c 2 x 2 + . . . + c n x n

a 1,1 x 1 a 2,1 x 1 + a 1,2 x 2

...

+ . . . + a 1,n x n + s 1 + a 2,2 x 2 + . . . + a 2,n x n + s 2 = 0, = 0,

-u -u

1 2   s 1 s 2  

-l

1

-l

2 a m,1 x 1 + a m,2 x 2 + . . . + a m,n x n + s m = 0,

-u

m  s m 

-l

m

… to Solve for the Best Possible Solution

New slacks s 1 , …, s m : Minimize: Subject to: | s 1 | + | s 2 | + . . . + | s m | a 1,1 x 1 + a 1,2 x 2 + . . . + a 1,n x n + s 1 + s 1 = 0, a 2,1 x 1 + a 2,2 x

...

2 + . . . + a 2,n x n + s 2 + s 2 = 0,

-u

1  s 1 

-l

1

-u

2  s 2 

-l

2 a m,1 x 1 + a m,2 x 2 + . . . + a m,n x n + s m + s m = 0,

-u

m  s m 

-l

m The idea is to minimize the sum of the infeasibilities

Re-Sampling Step

Repeat until the constraints are met: 1. Run linear program to find closest possible solution 2. If some slack variables s 1 , …, s m  0 • Eliminate beams with low weights • Replace them with new beams: – Randomly generate beams in neighborhood of highly weighted beams – Randomly generate beams according to initial algorithm

CARABEAMER’s Approach

1. Initial Sampling: Generate many (> N) beams at random, with each beam having a reasonable probability of being part of the solution.

2. Weighting: Use linear programming to test whether the beams can produce a dose distribution that satisfies the input constraints.

3. Iterative Re-Sampling: Eliminate beams with small weights and re-sample more beams around promising beams.

4. Iterative Beam Reduction: Progressively reduce the number of beams in the solution.

Re-Sampling to Reduce Total # of Beams

Repeat until dose constraints are met with specified number N of beams: 1. If too many beams in the solution: • Eliminate beams with low weights • Generate smaller number of beams 2. If no solution: • Add more beams

Plan Review

• Calculate resulting dose distribution • Radiation oncologist reviews • If satisfactory, treatment can be delivered • If not...

– Add new constraints – Adjust existing constraints

Treatment Planning: Extensions

• Simple path planning and collision avoidance Critical Tumor

Evaluation on Sample Case

Linac plan 80% Isodose surface CARABEAMER ’s plan 80% Isodose surface

Another Sample Case

50% Isodose Surface LINAC plan 80% Isodose Surface CARABEAMER’s plan

Evaluation on Synthetic Data

2000



D T D C

 

2400, 500 X 2000



D T D C

 

2200, 500 X n = 500 n = 250 X Constraint Iteration Beam 10 X random seeds n = 100 Iteratio n 2000



D T D C

 

2100, 500

Dosimetry Results

Case #1 Case #2 80% Isodose Curve 90% Isodose Curve Case #3 80% Isodose Curve 90% Isodose Curve Case #4 80% Isodose Curve 90% Isodose Curve 80% Isodose Curve 90% Isodose Curve

Average Run Times

Case 1

Beam Constr

Case 2

Beam Constr

Case 3

Beam Constr

Case 4

Beam Constr 2000-2400 n = 500 n = 250 n = 100 2000-2200 n = 500 n = 250 n = 100 :20 :20 :35 :32 :29 :43 :41 :40 :51 :50 :59 :01:02 :03:30 :03:32 :04:28 :05:50 :05:50 :08:53 :05:23 :05:33 :07:19 :08:37 :08:42 :10:43 :04:36 :04:11 :05:03 :23:51 :24:44 :33:02 :06:45 :07:19 :07:06 :13:05 :12:16 :21:06 3:06:12 3:09:19 3:35:28 25:38:36 27:55:18 1:40:55 1:44:18 1:41:19 6:33:02 7:11:01 176:25:02 2000-2100 :01:34 :01:28 :02:21 :01:41 :01:31 :02:38 :48:54 :40:49 1:57:25 :27:39 :24:43 1:02:27 3:26:33 3:22:15 7:44:57 1:03:06 1:07:12 5:06:29 53:58:56 44:11:04 84:21:27

Evaluation on Prostate Case

50% Isodose Curve 70% Isodose Curve

Cyberknife Systems

Contact Stanford Report News Service

/ Press Releases

Stanford Report, July 25, 2001

Patients gather to praise minimally invasive technique used in treating tumors By MICHELLE BRANDT When Jeanie Schmidt, a critical care nurse from Foster City, lost hearing in her left ear and experienced numbing in her face, she prayed that her first instincts were off. “I said to the doctor, `I think I have an acoustic neuroma (a brain tumor), but I'm hoping I'm wrong. Tell me it's wax, tell me it's anything,'” Schmidt recalled. It wasn't wax, however, and Schmidt – who wound up in the Stanford Hospital emergency room when her symptoms worsened – was quickly forced to make a decision regarding treatment for her tumor. On July 13, Schmidt found herself back at Stanford – but this time with a group of patients who were treated with the same minimally invasive treatment that Schmidt ultimately chose: the CyberKnife. She was one of 40 former patients who met with Stanford faculty and staff to discuss their experiences with the CyberKnife – a radiosurgery system designed at Stanford by John Adler Jr., MD, in 1994 for performing neurosurgeries without incisions. “I wanted the chance to thank everyone again and to share experiences with other patients,” said Schmidt, who had the procedure on June 20 and will have an MRI in six months to determine its effectiveness. “I feel really lucky that I came along when this technology was around.” The CyberKnife is the newest member of the radiosurgery family. Like its ancestor, the 33-year-old Gamma Knife, the CyberKnife uses 3-D computer targeting to deliver a single, large dose of radiation to the tumor in an outpatient setting. But unlike the Gamma Knife – which requires patients to wear an external frame to keep their head completely immobile during the procedure – the CyberKnife can make real-time adjustments to body movements so that patients aren't required to wear the bulky, uncomfortable head gear. The procedure provides patients an alternative to both difficult, risky surgery and conventional radiation therapy, in which small doses of radiation are delivered each day to a large area. The procedure is used to treat a variety of conditions – including several that can't be treated by any other procedure – but is most commonly used for metastases (the most common type of brain tumor in adults), meningomas (tumors that develop from the membranes that cover the brain), and acoustic neuromas.

Since January 1999, more than

335 patients have been treated at Stanford with the CyberKnife.

Motion Planning

Where Are We?

Where Do We Go From Here?

Where Are We?

  Two main families of methods: Probabilistic sampling (PRM)  Criticality-based decomposition  Widely applicable, easy to implement, but remaining shortcomings Low-dimensional spaces, harder to implement Diverse problems have been studied: - Basic path planning - Dynamic, kinodynamic, stability, visibility constraints - Uncertainty (no much success)  Some applications: - GE: Disassembly planning for the maintenance of aircraft engines - Delmia: Graphic environment for robot programming - Accuray: Radiosurgical planning - Softimage: Animation of digital actors - Kineo: Transportation of Airbus 800 fuselage

Where Do We Go From Here?

 More complex constraints, e.g., aesthetics

Where Do We Go From Here?

  More complex constraints, e.g., aesthetics Dealing with deformable objects

Where Do We Go From Here?

   More complex constraints, e.g., aesthetics Dealing with deformable objects Thousands of DoFs and more …

Where Do We Go From Here?

    More complex constraints, e.g., aesthetics Dealing with deformable objects Thousands of DoFs and more … Better understanding of underlying theoretical problems methods : - Mathematical structure/geometry of configuration, state, motion, … spaces - Impact of constraints (e.g., kinodynamics, aesthetics) on connectivity - Combination of criticality-based and sampling

Most existing motion planning techniques have been motivated by robotics applications.

Robotics will remain important, but future progress in motion planning will be increasingly motivated by non-robotics applications, e.g., navigation in virtual worlds, digital actors, surgical planning, logistics, etc….