Utilizing IMRT Intensity Map Complexity Measures in Segmentation Algorithms. Multileaf Collimator (MLC) IMRT: Beamlet Intensity Matrices

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1 Radiation Teletherapy Goals Utilizing IMRT Intensity Map Complexity Measures in Segmentation Algorithms Kelly Sorensen August 20, 2005 Supported with NSF Grant DMI High Level Goals: Achieve prescribed radiation to tumor while minimizing dose to the surrounding organs Some Additional Goals: Minimize time patient spends on the treatment couch. Minimize beam-on time (time radiation beam in on ) Minimize treatment planning time IMRT: Beamlet Intensity Matrices Multileaf Collimator (MLC) Dose optimization yields beamlet intensity matrix at each selected beam angle varying beamlet intensities: 0 means no radiation, 100 units is max dose (5 intensity level example) Using this matrix as input, a segmentation method computes a set of apertures (shapes) needed to deliver the intensity map Angle I-Map

2 Delivery of an Intensity Map via Shape Matrices Intensity Map as Sum of Shapes Original Intensity Map = Given: Determine: Intensity Matrix (A)= Sum of Shapes (S k ) times their weights (" k ) K A =! " k S k=1 k Shape Matrix 1 Shape Matrix 2 Shape Matrix 3 Shape Matrix x 20 x 20 x 20 x 20 " k > 0 is time the linear accelerator is opened to release uniform radiation S k is the binary shape matrix Multileaf Collimator Problem With Minimal Beam-on Time min subject to! " t! " t S t = A " t >= 0 where t is an element of the index set of all possible shape matrices t t beam-on time What is Optimal? Given minimal beam-on time, minimize #segments Given minimal #segments, minimize beam-on time Minimize a weighted sum of beam-on time and #segments

3 Difference Matrix Difference Matrix example The difference matrix for an IMap is the column-forward matrix of differences. Creating the difference matrix, D: A # R mxn, D # R mxn+1 Expand A by adding a column of zeros to the left and to the right sides of A A 0 = 0 A n+1 = 0 Define D row-wise by the differences: D ij = A ij - A ij-1 i=1..m, j=1..n original IMap difference matrix Difference Matrix example A Couple of Single-Row Examples Angle original IMap IMap row profile of intensities steps up 1 step down lower bound of 6 segments difference matrix total intensity increase of lower bound of 80 units beam-on time

4 Some Basic Measures of Complexity numnonzeroimap A ij > 0 numnonzerodiff D ij " 0 numnonzerodiffvert V ij " 0 Note: V is a vertical version of the difference matrix. i.e. V is the row-wise forward matrix of differences. Complexity Which Might Affect #Segments: counts m ( ( )) summaxupdownsteps " max count j ( D ij > 0),count j D ij < 0 i=1 n ( ( )) summaxupdownstepsvert " max count i ( V ij > 0),count i V ij < 0 numvalleys count A ij : A ij#1 $ A ij $ A ij +1 j=1 ( ) ( ( )) maxnumvalleys max i count j A ij : A ij#1 $ A ij $ A ij +1 lower bound on #segments maxmaxupdownsteps max i max count j ( D ij > 0),count j D ij < 0 maxmaxupdownstepsvert max j max count i ( V ij > 0),count i ( V ij < 0) ( ( ( ))) ( ) ( ) Lower Bound on #Segments IMap row profile of intensities total intensity increase of steps up lower bound of 6 segments 1 step down lower bound of 80 units beam-on time Complexity Which Might Affect Beam-on Time: sums sumstepsupvert m n ( V ij ) + i=1 j=1 m "" "( ) + maxsumstepsupvert max j V ij sumstepsup lower bound on beam-on time "" i=1 m n ( D ij ) + i=1 j=1 n "( ) + maxsumstepsup max i D ij j=1

5 Lower Bound on Beam-on Time Difference Matrix example IMap row profile of intensities 6 steps up lower bound of 6 segments 1 step down original IMap total intensity increase of lower bound of 80 units beam-on time difference matrix Other Measures Considered: Poor Performers numdifferentstepsizes numdifferentvalues volume perimeter m n ( A ij ) i=1 j=1 "" ( ) + ( ) + ( ) + ( ) count A ij : A ij > 0 & A ij#1 = 0 count A ij : A ij > 0 & A ij +1 = 0 count A ij : A ij > 0 & A i#1 j = 0 count A ij : A ij > 0 & A i+1 j = 0 Random Intensity Map Generator: GenIMap Verified accuracy by: Comparing average values of complexity measures to those of clinical cases Comparing to a control set of IMaps generated using a uniform random distribution. Percent difference of complexity as compared to clinical cases: GenIMap algorithm: 24% difference Uniform Random Distribution: 117% difference * 50 IMaps were generated for each algorithm

6 bdiff Heuristic 1. Choose weight, ". " = min D ij first open position not covered by left leaf first position covered by right leaf Create candidate shapes. " a) create row-shapes where D il+1 = " and D ir = - " if a row has no such shapes, relax the rule to A ij! " $ l<j<r -" Shape Table Metaheuristic S 1 S 2 S 3 S 4 S 5 S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8 S 9 S 10 S 11 S 12 S 13 S 14 S 15 S 16 S 17 S 18 S 19 S 20 S 21 S 22 S 23 S 24 S 25 S 1 S 2 S 3 S 4 S 5 b) combine row-shapes into larger shapes 1. select the best single-row shapes expand by adding a row above, if possible otherwise expand by adding a row below 2. select the best 2-row shapes and expand 3. repeat with 3-row k-row shapes 3. Select best shape, remove it from the Intensity Map and repeat process until residual Intensity Map is 0. S 6 S 7 S 8 S 9 S 10 S 11 S 12 S 13 S 14 S 15 S 16 S 17 S 18 S 19 S 20 S 21 S 22 S 23 S 24 S 25 S 6 S 7 S 8 S 9 S 10 S 11 S 12 S 13 S 14 S 15 S 16 S 17 S 18 S 19 S 20 S 21 S 22 S 23 S 24 S 25 Scatter Plot Shows Linear Relationship #Nonzero in IMap: Makes Sense, But Has Poor Correlation SumStepsUp vs. BeamTime #NonzeroIMap vs. #Segments y = x R 2 = Linear () y = x R 2 = Linear ()

7 R-squared values for linear fit numshapes beamtime #Levels Basic Measures of Complexity numnonzeroimap numnonzerodiff numnonzerodiffvert L.B Complexity Affecting #Segments: counts maxmaxupdownsteps summaxupdownsteps maxvalleys totalvalleys maxupdownstepsvert summaxupdownstepsvert Complexity Affecting #Segments: sums maxsumstepsupvert L.B sumstepsupvert Complexity Affecting Beam-on Time: sums maxsumstepsup sumstepsup Results: Fewer Segments! SumStepsUp vs. #Segments y = x R 2 = Linear (Original bdiff) Results: Trades-off Beam-on Time Side-by-Side Comparison SumStepsUp vs. BeamTime 1550 y = x SumStepsUp vs. #Segments SumStepsUp vs. BeamTime R 2 = Linear (Original bdiff) y = x R 2 = Linear (Original bdiff) y = x R 2 = Linear (Original bdiff)

8 Results: s Follow Similar Pattern SumMaxUpDownSteps vs. #Segments y = x R 2 = Linear (Original bdiff) Summary of Results 100-level discretization: 8% improvement! 20-level discretization: 4% improvement 10-level discretization: 4% improvement Future Work Dose Optimization use complexity measures to predict treatment times use penalty function to produce IMaps with lower predicted treatment times discretize IMaps more intelligently Referenced Papers K. Engel, A new algorithm for optimal multileaf collimator field segmentation. University Rostock, Germany, March S. Luan, C. Wang, S.A. Naqvi, D.Z. Chen, X.S. Hu, C.L. Lee, and C.X. Yu. A New MLC Segmentation Algorithm/Software for Step and Shoot IMRT. Medical Physics, 31(4): , D. Baatar, H.W. Hamacher, M. Ehrgott, and G.J. Woeginger, Decomposition of Integer Matrices and Multileaf Collimator Sequencing. Discrete Applied Mathematics. Accepted for publication, R.K. Ahuja and H. Hamacher, A network flow algorithm to minimize beam-on time for unconstrained multileaf collimator problems in cancer radiation therapy. Networks 45: 36-41, N. Boland, H. W. Hamacher, and F. Lenzen. Minimizing beam-on time in cancer radiation treatment using multileaf collimators. Networks, M. Langer, V. Thai, and L. Papiez, Improved leaf sequencing reduces segments or monitor units needed to deliver IMRT using multileaf collimators. Medical Physics, 28(12), 2001.

9 Random Intensity Map Generator: GenIMap Purpose: Larger sample size of variable intensity shapes Reduce over-tuning on our small sample set (7 IMaps!) Goal: Produce a realistic representation of an intensity map considering 8x24 bixel IMap Tumor area 3 or more tumors (I used 5) Organs at risk (3+) overlapping the tumor area Generating the Tumor Area Five dense circles are created randomly within the center of IMap (8 bixels x 24 bixels with a 1-2 bixel margin on all sides). Each circle contains a variable radius with decreasing intensity values along the radius. To ensure equal spacing for these circles, they are alternately placed in the left half and right half of the IMap. Representing the OAR Three smaller circles are generated and placed randomly throughout the IMap. These smaller circles contain a smaller variable radius with decreasing OAR density along the radius. To create larger, more complex problems, increase the area for the "tumor circles" and increase the number of "OAR circles." Keeping the element of randomness for the position and radius of each circle helps to ensure each created intensity map is distinct from others. Ensuring Appropriateness Verified accuracy by: Comparing average values of complexity measures to those of clinical cases Comparing to a control set of IMaps generated using a uniform random distribution. Percent difference of complexity as compared to clinical cases: GenIMap algorithm: 24% difference Uniform Random Distribution: 117% difference * 50 IMaps were generated for each algorithm

10 Radiation has been used to treat cancer for more than 100 years. Radiotherapy Motivation Emil Herman Grubbé first treats a breast cancer patient using radiation - less than one month after the public announcement of Roentgen s discovery of X-rays Gösta Fossel first reports on the use of brachytherapy to treat 32 patients Mark P. Carol, M.D. develops initial idea for IMRT Computer technology has advanced sufficiently that Carol begins working on a prototype of the Peacock IMRT system FDA approval for Carol s system More than 1.3 million new cases of cancer each year in U.S., and many times that number in other countries Approximately 60% of U.S. patients with cancer have radiation therapy sometime during the course of their disease Organ and function preservation are important aims (minimize radiation to nearby organs at risk (OAR)). bdiff heuristic 1. Choose weight, ". " = min D ij 2. Create candidate shapes. 3. Select best shape, remove it from the Intensity Map and repeat process until residual Intensity Map is 0. bdiff heuristic 1. Choose weight, ". " = min D ij 2. Create candidate shapes. " a) create row-shapes where D il+1 = " and D ir = - " if a row has no such shapes, relax the rule to A ij! " $ l<j<r b) combine row-shapes into larger shapes 1. select the best single-row shapes expand by adding a row above, if possible otherwise expand by adding a row below 2. select the best 2-row shapes and expand 3. repeat with 3-row k-row shapes Select best shape, remove it from the Intensity Map and repeat process until residual Intensity Map is 0. -"