A Metaheuristic for IMRT Intensity Map Segmentation

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

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 • Apply radiation to tumor (target volume) sufficient to destroy it while maintaining the functionality of the surrounding organs (organs at risk) • Minimize amount of time patient spends positioned and fixed on the treatment couch. • 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. 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). Option 2: IMRT

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 3 Shape Matrix 2 Shape Matrix 4 + + + x 20 x 20 x 20 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=(ai,j) comprised of nonnegative integers • Output: • T aperture shape matrices dt (with entries dtij) • 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) gt = 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 Intensity Matrix = Sum of Shapes (Sk) times their weights (ak) K I = S ak Sk k=1 ak > 0 is time the linear accelerator is opened to release uniform radiation Sk is shape matrix

22. Multileaf Collimator (MLC) problem with minimal beam-on time min S at subject to S at St = I at >= 0 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 S at + (K - 1)Tc subject to S at St = I at >= 0 where t is an element of the index set of all possible shape matrices Tc 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. ptij= 1 if beam element in row i, column j is covered by the right leaf when the tth monitor unit is delivered = 0 otherwise ltijis similar for the right leaf dtij=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. zt= 1 when at least one beam element is exposed when the tth 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 dtij

29. Constraint: Monoshape No rows gaps are allowed: monoshapes are required • First determine which rows in each monitor unit are open to deliver radiation deliveryit=1 if the ith row is being used a time t = 0 otherwise Determine if the preceding row in the monitor unit delivers radiation dropit=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 jumpit=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

35. 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 3 Shape Matrix 2 Shape Matrix 4 + + + x 20 x 20 x 20 x 20

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

44. Recreate full shapes by matching upper shapes to lower shapes partition and match upper and lower shapes + + + x 20 x 20 x 20 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 • Partitioning- create a neighborhood - Partition subspace by identifying shapes • Random Sampling • Create random shapes • Use the random shape with a given probability • Promising Region • All solutions using the chosen shape • Valued based on Price of the best full solution • Backtrack - Disallow a shape to be used