Treatment Planning Optimization for VMAT, Tomotherapy and Cyberknife

Transcription

1 Treatment Planning Optimization for VMAT, Tomotherapy and Cyberknife Kerem Akartunalı Department of Management Science Strathclyde Business School Joint work with: Vicky Mak-Hau and Thu Tran 14 July 2015

2 Outline 1 Introduction 2 Solution Methods Improving the Formulation Methodologies 3 Computational Results

3 Outline 1 Introduction 2 Solution Methods Improving the Formulation Methodologies 3 Computational Results

4 VMAT: Gantry

5 VMAT: Multi-Leaf Collimator

6 3-D Imaging (Voxels) and Dose

7 IMRT vs. VMAT/Tomotherapy/Cyberknife: Overview Intensity-Modulated Radiation Therapy (IMRT) Technology widely used and been around for a while. Lots of literature, lots of solution methods. Beaming limited to a handful of selected angles only. Volumetric-Modulated Arc Therapy (VMAT), Tomotherapy and Cyberknife More recent technologies and limited literature. VMAT: Gantry rotates 360 degrees-single arc or dual arcs in co-planar manner. Beam on all the time. MLC leaf movement linear to gantry rotation. Tomotherapy: Rotation in a helical manner; simple binary MLC. Cyberknife: Source of radiation is mounted on a robotic arm; various collimators (such as circular).

8 Overview of Decision Problem A number of physical limitations. Shapes between consecutive leaves. Changes in shapes and dosage between consecutive angles. The aim is to optimize treatment plans Minimize treatment time, maximize radiation on tumor, etc. Not too much radiation on sensitive tissue/organs. But also need a minimum level of radiation to tumor. Motivation: Software currently in use do not even generate feasible treatment plans. Focus on minimizing error. Aim: To optimize the fluence and MLC apertures simultaneously. Incorporate all volume limits for dose upper bounds for organs at risk (OAR) and dose lower bounds for planning target volumes (PTV) as hard constraints, but allow to relax either if necessary.

9 Decisions to Make... There are a number of decisions to consider. When do we change shapes of MLC? How do the shapes of MLC look like? How does units of radiation applied change? Discretization of angles/snapshots. This eliminates the questions of when, given sufficient number of snapshots. Matrix-discretization of MLC. Rows already natural, columns also added.

10 Notation: Sets and Parameters - I = {1,..., m} be the index set of the MLC rows. - J = {1,..., n} be the index set of the MLC columns. Each cell (i, j) is called a beamlet or a bixel. Define J = {0, n + 1} J. - V be the index set of all voxels V t targeted tumor. V o organs/sensitive tissue. - L = {(l, r) l, r J, l < r}. - K be the index set of beam-angles. - Dijv k already defined. - L v prescription dose for voxel for v V t. - U v maximum dose allowed for voxel v V o.

11 Notation: Variables - yi(l,r) k {0, 1} 1 if the leaf position in the k-th snapshot and i-th row is (l, r). 0 otherwise. - z k : Number of radiation units used for snapshot k. Physical limitation of beamer: M z k 0 - D v : Total dose of radiation received by voxel v. More to come later about how we calculate this. - x v {0, 1} 1 if tumor voxel v receives at least target amount d of radiation. Note L v is a bare minimum, d is preferable. 0 otherwise.

12 Problem Constraints (l,r) L y k i,(l,r) = 1 i I, k K For each row on each snapshot, we can only choose one pair of right-left leaf positions. r δ 1 r=0 r 1 l=0 y k+1 + n+1 i( l, r) r 1 r=r+δ+1 l=0 i I, (l, r) L, k K y k+1 i( l, r) 1 y k i(l,r) Right-leaf position can move at most δ units between consecutive snapshots. Same is also true for left-leaf position...

13 Problem Constraints (cont d) n+1 r=l+1 r 1 yi(l, r) k + l=0 r=1 l y k (i+1)( l, r) 1 i I, l J, k K

14 Problem Constraints (cont d) r 1 n+1 y k i( l,r) + l=0 r= l+1 l=r n y k (i+1)( l, r) 1 i I, r J, k K

15 Problem Constraints (cont d) z k z k+1 k K Number of unit radiation applied can be changed at most between consecutive snapshots. d v L v d v U v d v dx v v V t v V o v V t Lower- and upper-bounds on dose received by tumor and organs, resp.; is d achieved for tumor voxels?

16 Problem Objective max v V t x v Simply maximize number of tumor cells that receive at least d units of radiation. A more successful treatment plan Lesser number of treatments. There are other objectives used in practice. Minimize total treatment time (to comfort patients + use limited resources as much as possible). Minimize total number of shapes (to minimize technical effort). More to read on these in Ernst, Mak and Mason [2009,2010].

17 Defining Dose Received d v = k K i I j J 1 2 zk Dijv k (l,r) L l<j<r yi(l,r) k + zk+1 D k+1 ijv (l,r) L l<j<r y k+1 i(l,r)

18 Defining Dose Received d v = k K i I j J 1 2 zk Dijv k (l,r) L l<j<r yi(l,r) k + zk+1 D k+1 ijv (l,r) L l<j<r y k+1 i(l,r)

19 Outline 1 Introduction 2 Solution Methods Improving the Formulation Methodologies 3 Computational Results

20 Improving the Formulation: Linearizing D v d v = k K i I j J 1 2 zk Dijv k (l,r) L l<j<r yi(l,r) k + zk+1 D k+1 ijv (l,r) L l<j<r y k+1 i(l,r) All nonlinear terms are bilinear. Define a new variable z ij k to indicate radiation amount for beam angle k and (i, j) th location of MLC. d v = 1 ( ) Dijv k z k ij + D k+1 ijv z k+1 ij 2 k K i I j J

21 Improving the Formulation: Linearizing D v (cont d) To finalize the linearization, we add the following constraints: z ij k M k K, i I, j J (l,r) L l<j<r z ij k z k y k i(l,r) z ij k M 1 + z k ij 0 (l,r) L l<j<r yi(l,r) k + zk k K, i I, j J k K, i I, j J k K, i I, j J

22 Improving the Formulation: Valid Inequalities Proposition d v L v ( d L v )x v (1) is valid and dominates the constraint d v dx v.

23 Improving the Formulation: Valid Inequalities (cont d) Proposition Let Di,l+1,r 1,v k = r 1 j=l+1 Dk ijv. Then, the inequality z k z k ij (l,r) L l j r j min{ M, U v D k i,l+1,r 1,v is valid for the VMAT problem and dominates z ij k M 1 + (l,r) L l<j<r yi(l,r) k + zk }y k i(l,r) (2)

24 Improving the Formulation: Few Notes Linearization simplifies the problem significantly. Added constraints are nice. The proposed inequalities have no additional cost. They are tighter constraints and hence provide more compact formulation. They are facet-defining for some important subproblems (e.g., single-voxel, single-row subproblem). However, even the small problems might be still challenging to solve. So we need practical tools in addition to these theoretical tools.

25 Variable Neighborhood Search Heuristic (VNHS) Generate feasible shapes (y) for snapshots. Randomly assign radiation dosages (z) on each snapshot. Define neighborhoods around y and z. Penalize dose UB/LB violations. Apply guided-search: If # overdose > # underdose; then reduce z. If # underdose > # overdose; then increase z. Also can vary neighborhood size, e.g. proportional to % violated voxels. Random restarts after 50 iterations to avoid solution being trapped. Advantage: Very fast!

26 Lagrangian Relaxation A complex structure, a number of potential candidates, different expectations. LR 1 Relax constraints on max distance, dose lower/upper bounds, and max weight change K subproblems for each snapshot. LR 2 Relax constraints on interleaf, dose lower/upper bounds, linearization provides equal bound to the LP relaxation. LR 3 Relax linearization constraints two subproblems, one for y, and one for z and x. Subgradient optimization on its own is known to be computationally not very efficient. But maybe we can use these for some heuristics?

27 A Heuristic Related to LR 1 (Heur1) Output: K candidate solutions for k K do Solve the problem P1: min{ v V t (L v d v ) + + v V (dv Uv )+ (y, z, z) X rel } ; where X rel is the LR 1 subproblem, for only k. ; Fix shapes for k ; for k = k + 1 to k = K and k = k 1 to k = 1 do Solve the problem P1 with inter-snapshot constraints for only k. ; Fix shapes for k ; end Now all shapes fixed, solved the original problem. ; end

28 A Heuristic Related to LR 3 (Heur2) Idea: Combine subgradient optimization and heuristics. Initialize Lagrangian multipliers α k ij and β k ij using LPR duals. We obtain two separate problems. Pr.1 over x, z, z variables. Pr.2 over y variables. Apply subgradient optimization to update the multipliers. Also take the valid solution of Pr.2, fix all these shapes and run the original problem. Already know this heuristic runs fast.

29 A Centering-Based Heuristic (Heur3) Number of y variables and inter-leaf constraints prohibitive for real-size problems Idea 1: If we knew the location of the center of the opening in a row, we could have simply defined n binary variables (one for each location, either left or right). Idea 2: If we extend this to the whole snapshot (i.e., one center for the whole snapshot), then we can eliminate inter-leaf constraints. Advantage: A significant reduction in the number of variables and constraints (crucial for real-size problems). Disadvantage: The candidate opening patterns for a snapshot limited to only centered patterns (might cause infeasibilities).

30 Outline 1 Introduction 2 Solution Methods Improving the Formulation Methodologies 3 Computational Results

31 Generating Random Problems It is important to test different methodologies on as many different problems as possible. Real test cases are important, but they are scarce. Most papers published have only one or two cases. Fully exact methods suffer on any real-size problem. Optimization is practically impossible. Hence no insight on the performance of a method. Hence we created a random problem generator. We can solve some of these problems to optimality. We can run a lot of quick tests and comparisons. We can also see how difficult problems can get. Available at:

32 Random Problems: An Overview of Complexity p-mlc Size - # Voxels - # Snapshots d = 0.5 max{uv } d = max{uv } p LPR 0.02s LPR 0.02s Opt 0.05s Opt 2.11s p LPR 0.11s LPR 0.14s Opt 5.97s Opt s p LPR 0.52s LPR 12.3s Opt Opt > 8h p LPR 2.2s LPR 3.39s Opt s Opt NA In reality, MLCs discretization 5mm, voxels 4mm, and 5-8 degrees between snapshots. For a 10cm treatment field: MLC: 20x20; Treatment field: 25x25x25; Number of Snapshots:

33 Overall Comparison of Heuristic Methods Problem Set # best overall solutions (max MLC size) Heur1 Heur2 Heur3 GVNS Small (8 8) Medium (12 12) Large (15 15) Very Large Problem Set # instances with no solutions (max MLC size) Heur1 Heur2 Heur3 GVNS Small (8 8) Medium (12 12) Large (15 15) Very Large

34 GVNS: Huge Data Instances 1st sol Overall m n w h d K # Di,j,v k > 0 Time # ITER Time # ITER , 400, , 500, , , 500, , , 687, 500 7, , , 000, , , 000, , , 000, , , , 500, 000 5, , , 000, , , 400, , Best solution with original objective recorded 3 times before 1,000 iterations reached. In comparison to Peng et al. (2012), where highest # Di,j,v k > 0 is 114, 315, 187.

35 Conclusions A unified MIP formulation for VMAT, TomoTherapy, and CyberKnife. Heur 2 is very efficient for small problems (but fails otherwise). Heur3 and GVNS often fail for small problems. Heur1 and Heur3 have similar performance for medium and large problems. GVNS is powerful for larger problems (hence usable in real-life applications). A unified mixed-integer programming model for simultaneous fluence weight and aperture optimization in VMAT, Tomotherapy, and Cyberknife, Computers & Operations Research, Vol. 56, April 2015.