1. Laura Goadrich13 Dec 2004 A Metaheuristic for IMRT Intensity Map Segmentation Laura D. Goadrich October 15, 2004 Supported with NSF Grant DMI-0400294

2. Contents Motivation Radiotherapy: Conformal vs. IMRT Intensity Map & Shape Matrices Program Outline –Constraints –Difference Matrix Results Improving Solvability –Partitioning –Condor –Nested Partitions Future Works References

3. Contents Motivation Radiotherapy: Conformal vs. IMRT Intensity Map & Shape Matrices Program Outline –Constraints –Difference Matrix Results Improving Solvability –Partitioning –Condor –Nested Partitions Future Works References

4. Radiation treatment of cancer: a bit of trivia…. Radiation has been used to treat cancer for more than 100 years. In fact, the first cancer patient was treated in Chicago in January, 1896, less than one month after the discovery of X-rays. Intensity modulated radiation therapy (IMRT) is a revolutionary type of external beam treatment that is able to conform radiation to the size, shape and location of a tumor.

5. Radiotherapy Motivation 1.2 million new cases of cancer each year in U.S., and many times that number in other countries Approximately 40% 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)).

6. Goals of Radiotherapy 1.Apply radiation to tumor (target volume) sufficient to destroy it while maintaining the functionality of the surrounding organs (organs at risk) 2.Minimize amount of time patient spends positioned and fixed on the treatment couch. 3.Minimize beam-on time (time in which radiation is applied to patient)

7. Planning Radiotherapy- Tumor Volume Contouring Isolating the tumor from the surrounding OAR using CAT scans is vital to ensure the patient receives minimal damage from the radiotherapy. Identifying the dimensions of the tumor is vital to creating the intensity maps (identifying where to focus the radiation).

8. Planning Radiotherapy- Beam Angles and Creating Intensity Maps Multiple angles are used to create a full treatment plan to treat one tumor.

9. Contents Motivation Radiotherapy: Conformal vs. IMRT Intensity Map & Shape Matrices Program Outline –Constraints –Difference Matrix Results Improving Solvability –Partitioning –Condor –Nested Partitions Future Works References

10. Option 1: Conformal Radiotherapy The beam of radiation used in treatment is a 10 cm square. Utilizes a uniform beam of radiation –ensures the target is adequately covered –however difficult to avoid critical structures except via usage of blocks

11. Option 2: IMRT Intensity Modulated Radiotherapy (IMRT) provides an aperture of 3mm beamlets using a Multi-Leaf Collimator (MLC), which is a specialized, computer-controlled device with many tungsten fingers, or leaves, inside the linear accelerator. Allows a finer shaped distribution of the dose to avoid unsustainable damage to the surrounding structures (OARs) Implemented via a Multi-Leaf Collimator (MLC) creating a time- varying aperture (leaves can be vertical or horizontal).

12. multileaf collimator

13. Contents Motivation Radiotherapy: Conformal vs. IMRT Intensity Map & Shape Matrices Program Outline –Constraints –Difference Matrix Results Improving Solvability –Partitioning –Condor –Nested Partitions Future Works References

14. IMRT: Planning- Intensity Map There is an intensity map for each angle –0 means no radiation –100 means maximum dosage of radiation Multiple beam angles spread a healthy dose A collection of apertures (shape matrices) are created to deliver each intensity map.

15. Delivery of an Intensity Map via Shape Matrices Original Intensity Map Shape Matrix 1 Shape Matrix 2 Shape Matrix 3 Shape Matrix 4 + + + x 20 =

16. Contents Motivation Radiotherapy: Conformal vs. IMRT Intensity Map & Shape Matrices Program Outline –Constraints –Difference Matrix Results Improving Solvability –Partitioning –Condor –Nested Partitions Future Works References

17. Program Input/Output Input: –An mxn intensity matrix A=(a i,j ) comprised of nonnegative integers Output: –T aperture shape matrices d t (with entries d t ij ) –Non-negative integers  t (t=I..T) giving corresponding beam-on times for the apertures –Apertures obey the delivery constraints of the MLC and the weight-shape pairs satisfy

18. Approach: Langer, et. al. Mixed integer program (MIP) with Branch and Bound by Langer, et. al. (AMPL solver) MIP: linear program with all linear constraints using binary variables Langer suggests a two-phase method where –First minimize beam-on time T is an upper bound on the number of required shape matrices –Second minimize the number of segments (subject to a minimum beam-on time constraint) g t = 1 if aperture changes = 0 otherwise

19. In Practice Langer, et. al. do not report times and we have found that computing times are impractical for many real applications. To obtain a balance between the need for a small number of shape matrices and a low beam-on time we seek to minimize numShapeMatrices*7 + beam-on time Initializing T close to the optimal number of matrices + 1 required reduces the solution space and solution time

20. Contents Motivation Radiotherapy: Conformal vs. IMRT Intensity Map & Shape Matrices Program Outline –Constraints –Difference Matrix Results Improving Solvability –Partitioning –Condor –Nested Partitions Future Works References

21. Intensity Map as Sum of Shapes I =  k S k K k=1  k > 0 is time the linear accelerator is opened to release uniform radiation S k is shape matrix Intensity Matrix = Sum of Shapes (S k ) times their weights (  k )

22. Multileaf Collimator (MLC) problem with minimal beam-on time min  t subject to  t S t = I  t  where t is an element of the index set of all possible shape matrices t t

23. Multileaf Collimator (MLC) problem with minimal beam-on time min  t + (K - 1)T c subject to  t S t = I  t  where t is an element of the index set of all possible shape matrices T c is set-up time K is the number of shapes used t t

24. Mechanical Constraints After receiving the intensity maps, machine specific shape matrices must be created for treatment. There are numerous types of IMRT machines currently in clinical use, with slightly different physical constraints that determine the possible leaf positions (hence the possible shape matrices). Each machine has varying aperture setup times that can dominate the radiation delivery time. To limit patient discomfort and patient motion error: reduce the time the patient is on the couch. Goals: –Minimize beam-on time –Minimize number of different shapes

25. Constraint: Right and Left Leaves Cannot Overlap To satisfy the requirement that leaves of a row cannot override each other implies that one beam element cannot be covered by the left and right leaf at the same time. p t ij = 1 if beam element in row i, column j is covered by the right leaf when the t th monitor unit is delivered = 0 otherwise l t ij is similar for the right leaf d t ij =1 if bixel is open

26. Constraint: Full Leaves and Intensity Matrix Requirements Every element between the leaf end and the side of the collimator is also covered (no holes in leaves).

27. Constraint: No Leaf Collisions Due to mechanical requirements, in adjacent rows, the right and left leaves cannot overlap

28. Accounting and Matching Constraints The total number of shape matrices used is tallied. z t = 1 when at least one beam element is exposed when the t th monitor unit in the sequence is delivered = 0 otherwise I is the number of rows J is the number of columns Must sum to the intensity matrix. is the intensity assigned to beam element d t ij

29. Constraint: Monoshape No rows gaps are allowed: monoshapes are required First determine which rows in each monitor unit are open to deliver radiation delivery it =1 if the i th row is being used a time t = 0 otherwise Determine if the preceding row in the monitor unit delivers radiation drop it =1 if the preceding row (i-1) in a shape is non-zero and the current row (i) is 0 = 0 otherwise

30. Constraint: Monoshape Determine when the monoshape ends jump it =1 if the preceding row (i-1) in a shape is zero and the current row (i) is nonzero = 0 otherwise There can be only one row where the monoshape begins and one row to end

31. Complexity of Problem The complexity of the constraints results in a large number of variables and constraints.

32. Contents Motivation Radiotherapy: Conformal vs. IMRT Intensity Map & Shape Matrices Program Outline –Constraints –Difference Matrix Results Improving Solvability –Partitioning –Condor –Nested Partitions Future Works References

33. Diff: Heuristic Fast heuristics use a difference matrix Transformation: Given an mxn intensity matrix M, define the corresponding mx(n+1) difference matrix D –Expand M by adding a column of zeros to the left and to the right sides of M –Define D row-wise by the differences: D(i, j)= M(i, j+1) - M(i, j)

34. Difference Matrix example

36. Diff in Practice Variables: –Delta: generates difference matrix –Count: counts nonzero rows –Frequency(D,v): counts appearances of v or -v in matrix D Algorithm D = delta(M) // generate initial difference matrix while (count(D) > 0){ find d > 0 that maximizes frequency(D,d) // choose intensity d call create_shape_matrix(S,d) // create shape matrix S D= D - d*delta(S) // update the difference matrix }

37. Contents Motivation Radiotherapy: Conformal vs. IMRT Intensity Map & Shape Matrices Program Outline –Constraints –Difference Matrix Results Improving Solvability –Partitioning –Condor –Nested Partitions Future Works References

38. Comparison of Results: Prostate Case for Corvus 4.0 Weighted Score = numShapeMatricies*7 + beam-on time

39. Comparison of Results: Head & Neck Case for Corvus 4.0

40. Comparison of Results: Pancreas Case for Corvus 4.0

41. Contents Motivation Radiotherapy: Conformal vs. IMRT Intensity Map & Shape Matrices Program Outline –Constraints –Difference Matrix Results Improving Solvability –Partitioning –Condor –Nested Partitions Future Works References

42. Delivery of an Intensity Map via Shape Matrices Original Intensity Map Shape Matrix 1 Shape Matrix 2 Shape Matrix 3 Shape Matrix 4 + + + x 20 =

43. Improving computation time via divide-and-conquer partition and match upper and lower shapes + + + x 20

44. Recreate full shapes by matching upper shapes to lower shapes partition and match upper and lower shapes + + + x 20

45. Contents Motivation Radiotherapy: Conformal vs. IMRT Intensity Map & Shape Matrices Program Outline –Constraints –Difference Matrix Results Improving Solvability –Partitioning –Condor –Nested Partitions Future Works References

46. Condor: Increasing Throughput Created by UW-Madison CS department –Software and documentation is Free –Supports Unix, Linux, Windows Workload management system for compute-intensive jobs Runs on clusters- using idle computers Provides: –Job queuing mechanism –Scheduling policy –Priority scheme –Resource monitoring –Resource management Allows serial or parallel jobs

47. Condor: www.cs.wisc.edu/condor Only need a Submission file and a Code file (with any input files- stdin & file input) –Sample Submission file

48. Condor: timely response Sample execution of 5 programs submitted simultaneously to Condor

49. Contents Motivation Radiotherapy: Conformal vs. IMRT Intensity Map & Shape Matrices Program Outline –Constraints –Difference Matrix Results Improving Solvability –Partitioning –Condor –Nested Partitions Future Works References

50. Nested Partitions 1.Partitioning- create a neighborhood - Partition subspace by identifying shapes 2.Random Sampling -Create random shapes -Use the random shape with a given probability 3.Promising Region -All solutions using the chosen shape -Valued based on Price of the best full solution 4.Backtrack -Disallow a shape to be used

51. A shape is created by choosing –Each row and a value –Growing each row upward and downward Use all possible columns in each new row Eg. The optimal result is using 2 shapes Red is the starting cell. NP: random shapes =+ =++

52. Selecting each cell individually would result in the worst case scenario of having 8 different shapes A good random shape can be created by –Choosing a random (non-zero) starting row –Choosing a random starting and ending column (could be the same column) without holes –Growing the row up and down (storing each new shape). NP: random shape

53. NP: storing shapes Heap benefits –Quick, easy retrieval –Quick sorting of price (best price at top of tree) –Can get a flat sample (eg. every 10th shape) Heap disadvantages –No easy way to select a biased sample –Gives no feel for the amount of different types of shapes (size, variety of prices) Therefore, for selecting random shape, need a Bucket sorter

54. NP: bucket sorter Benefit: have information on each bucket within easy access –Amount of shapes,type,etc. Heap would have to keep a record of the types and amounts of types entered –Looses the heap benefits of speed –Too much overhead

55. NP: bucket sorter Can get a biased sample –Uses the knowledge that shapes with better prices turn out to create good solutions. Linear Exponential: weighted distribution n(n+1) 2 Amount from each bucket = 60% from best price 30% from next best 10% from next best

56. NP: regions Initial Solution S 1 1, S 2 1, S 3 1, … Promising Region (1) Force S 1 1 New Solution: S 1 2, S 2 2, S 3 2, … Complementary Region (1) Disallow S 1 1 New Solution: S 1 3, S 2 3, S 3 3, … Shapes organized best (price) to worst by increasing subscript. … ……… PR1 > CR1 PR1 <= CR1

57. NP: promising region Promising Region (1) Force S 1 1 New Solution: S 1 2, S 2 2, S 3 2, … Promising Region(2) Force S 1 2, S 2 2 New Solution: S 1 4, S 2 4, S 3 4, … Complementary Region(2) Allow one of S 1 2 or S 2 2 New Solution: S 1 5, S 2 5, S 3 5, … … ……… PR2 <= CR2PR2 > CR2

58. NP: complementary region Promising Region (3) Force S 1 3, S 2 3 New Solution: S 1 6, S 2 6, S 3 6, … Complementary Region (3) Allow one of S 1 3 or S 2 3 New Solution: S 1 7, S 2 7, S 3 7, … Complementary Region (1) Disallow S 1 1 New Solution: S 1 3, S 2 3, S 3 3, … … ……… PR3 > CR3PR3 <= CR3

59. Future Work Incorporate the Nested Partitions method into our shape matrix method to take advantage of randomized strategies. Partition the more complicated shapes into two smaller shapes which can be handled quickly and easily. Then merge the resulting segments using the marriage algorithm to give a solution to the original problem.

60. Referenced Papers N. Boland, H. W. Hamacher, and F. Lenzen. “Minimizing beam-on time in cancer radiation treatment using multileaf collimators.” Networks, 2002. T.R. Bortfeld, D.L. Kahler, T.J Waldron and A.L.Boyer, “X-ray field compensation with multileaf collimators.” International Journal of Radiation Oncology Biology 28 (1994), pp. 723-730. T. Bortfeld, et. al. “Current IMRT optimization algorithms: principles, potential and limitations.” Massachusetts General Hospital, Harvard Medical School, Presentation 2000. D. Dink, S.Orcun, M. P. Langer, J. F. Pekny, G. V. Reklaitis, R. L. Rardin, “Importance of sensitivity analysis in intensity modulated radiation therapy (IMRT).” EuroInforms Presentation 2003. K. Engel, “A new algorithm for optimal multileaf collimator field segmentation.” University Rostock, Germany, March 2003. 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. P. Xia, L. J. Verhey, “Multileaf collimator leaf sequencing algorithm for intensity modulated beams with multiple static segments.” Medical Physics, 25 (8), 1998.