Optimization of Gamma Knife Radiosurgery

Michael FerrisUniversity of Wisconsin, Computer Sciences

David ShepardUniversity of Maryland School of Medicine

Overview

• Details of machine and problem• Formulation

– modeling dose– shot / target optimization

• Results– Two-dimensional data– Real patient (three-dimensional) data

The Leksell Gamma Knife

Problem characteristics

• Machine has 201 radiation sources focussed on one location

• Very accurate dose delivery• Benefits of computer solution

– uniformity of treatment plan– better treatment plan– faster determination of plan

Problem outline

• Target volume (from MRI or CT)• Maximum number of shots to use

– Which size shots to use– Where to place shots– How long to deliver shot for

– Conform to Target (50% isodose curve)– Real-time optimization

Two-dimensional example

Ideal Optimization

mints;xs;ys

Dose(NonTarget)

subject to Dose(i; j ) =X

s2S

tsD(xs;ys; i; j )

1ô Dose(Target) ô 2ts õ 0Sj j ô N

Dose calculation

• Measure dose at distance from shot center

• Fit a nonlinear curve to these measurements

• Functional form from literature, 6 parameters to fit via least-squares

1à m1 erf ( û1

xà r1) à m2 erf ( û2

xà r2)

8mm shot

Nonlinear ApproachLet xs;ys bevariable locations

s = 1;2;. . .;ND(xs;ys; i; j ) is nasty nonlinear function

What width shot touseat xs;ys?

s;w =1 if shot s is width w0 else

ú

T s;w ô ts;w ô T s;wPw s;w ô 1

Two-stage approach

• Approximate via “arctan”

• First, solve with approximation, then fix shot widths and reoptimize

-30 -20 -10 0 10 20 30-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

8s 2 SPwarctan(ts;w) ô 2

ù

3D slice image

Slice + 3

Axial slice Manual Computer Optimized

Axial slice Manual Computer Optimized

Coronal slice Manual Computer Optimized

Sagittal slice Manual Computer Optimized

Challenges

• Integration into real system• Reduction of optimization time• What if scenarios?

– Improve the objective function– Change number of shots

• Global versus local solutions