A Metaheuristic for IMRT Intensity Map Segmentation

1. A Metaheuristic for IMRT Intensity Map Segmentation Athula Gunawardena, Warren D’Souza, Laura D. Goadrich, Kelly Sorenson, Robert Meyer, and Leyuan Shi University of Wisconsin-Madison October 15, 2004 Supported with NSF Grant DMI-0400294

2. 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)).

3. 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).

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

5. 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

6. 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

7. 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.

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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).

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

16. 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

17. 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

18. 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

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

20. 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)

21. 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 }

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

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

24. Comparison of Results:Pancreas Case for Corvus 4.0

25. 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.

26. 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.