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

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)).

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).

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

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

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).

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.

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

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

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

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

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

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

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).

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

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

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

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

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

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)

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 }

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

Comparison of Results: Head & Neck Case for Corvus 4.0

Comparison of Results: Pancreas Case for Corvus 4.0

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.

Referenced Papers N. Boland, H. W. Hamacher, and F. Lenzen. “Minimizing beam-on time in cancer radiation treatment using multileaf collimators.” Networks, 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 T. Bortfeld, et. al. “Current IMRT optimization algorithms: principles, potential and limitations.” Massachusetts General Hospital, Harvard Medical School, Presentation 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 K. Engel, “A new algorithm for optimal multileaf collimator field segmentation.” University Rostock, Germany, March 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), P. Xia, L. J. Verhey, “Multileaf collimator leaf sequencing algorithm for intensity modulated beams with multiple static segments.” Medical Physics, 25 (8), 1998.