Iterative regularization in intensity-modulated radiation therapy optimization. Carlsson, F. and Forsgren, A. Med. Phys. 33 (1), January 2006.

Transcription

1 Iterative regularization in intensity-modulated radiation therapy optimization Carlsson, F. and Forsgren, A. Med. Phys. 33 (1), January 2006.

2 2 / 15 Plan

3 3 / 15 to paper The purpose of the paper is to explain how the IMRT problem can be solved in few iterations using a quasi-newton (QN) method. This is done by studying the iterative regularization approach commonly used in image reconstruction and verify that similar behaviour is observed when the QN method is applied to radiation therapy problems.

4 4 / 15 Intensity-modulated radiation therapy The patient is irradiated with photon beams. The goal is to kill all clonogenic tumor cells while sparing healthy tissue. Variable fluence over the beam cross-section is used to conform the dose to the shape of the target. The patient is discretized into voxels and the beams into beamlets.

5 5 / 15 Procedure 1. Find the beamlet weights (fluence map) that that minimize the difference between the delivered dose d and the prescribed dose ˆd. This is a large-scale optimization problem. 2. Convert the beamlet weights into multi-leaf collimator (MLC) segments that can be delivered. 3. Post-process by optimizing the segment weights (the time each segment is used). The more jagged the fluence profiles are, the more segments are needed more leakage radiation

6 6 / 15 Continuous case Fredholm integral equation of the first kind: find the nonnegative fluence x such that x(s)p(r, s) ds = ˆd(r), r V, S where r V is a point in the patient volume, s S is a point on a isocenter plane of the beams, ˆd is prescribed dose, and p(r, s) is a pencil beam kernel. Ill-posed problem since kernels p are blurry and have smoothing effect on x. Similar to problems in, e.g., image reconstruction.

7 7 / 15 Discrete case (beamlet weight optimization) In the simplest case, solve the problem minimize x 0 Px ˆd 2 2, where P is a matrix in which the element p ij describes how much dose is delivered to voxel i from beamlet j. The closer to the optimal solution, the more jagged fluence maps. Degenerate problem: many solutions produce almost identical objective value. Hessian has few large eigenvalues and many small eigenvalues: ill-conditioned problem. Truncated SVD is not satisfactory.

8 8 / 15 es Variational methods: objective term that penalizes nonsmooth fluence maps. Filtering methods: smoothen fluence maps prior to conversion. Iterative methods: considered in the paper. Iterative regularization Idea: Solve the problem directly. Iterate long enough to find a solution with objective value close to the optimal objective value, but terminate the optimization before the beam profiles get too jagged.

9 9 / 15 Unconstrained quadratic programming The quadratic program (QP) 1 minimize x R n 2 x T Hx + c T x, where H = P T P and c = P T ˆd, has solution x = H 1 c. Exact solution is time-consuming and leads to jagged beam profiles use e.g. conjugate gradient (CG) method and stop after a few iterations. Conjugate directions, i.e., p T k Hp l = 0 when k l, where p k denotes the search direction in iteration k. Initially proceeds in directions of the largest singular values (but does not truncate). The regularization parameter is the number of iterations.

10 10 / 15 Steps in CG Figure: Left: Eigenvectors to the Hessian of the QP. Right: Steps expressed as linear combinations of eigenvectors when the QP is solved by a QN method with BFGS update (equivalent to CG) and the identity matrix as initial Hessian estimate.

11 Steps i CG For the even simpler quadratic problem 1 n minimize λ i ξi 2, (1) ξ 2 the CG iterates become i=1 λ ξ (0) ξ (1) ξ (2) ξ (3) ξ (4) ξ (5) i = i = i = i = i = Table: The vectors ξ (k), k = 0,..., 5, are the eigenvector weights in iteration k for a problem with eigenvalues λ = (2, 1.5, 1, 0.1, 0.01) T solved with CG from ξ (0) = (1, 1, 1, 1, 1) T. 11 / 15

12 12 / 15 Solving IMRT problems IMRT problems are not unconstrained QPs. There are often penalties to fractions of the organs. This makes the problems nonconvex. IMRT problems are typically solved using a quasi-newton (QN) method. CG and QN (with BFGS update) are equivalent for QP with exact line-search, but there is no theoretical equivalence for fluence map problems. The empirical results in the paper indicate that the optimization method initially proceeds in directions of dominant singular values for the IMRT problems.

13 13 / 15 Fluence profiles Figure: Beam profiles for two patient cases.

14 14 / 15 Objective value Figure: Objective value as a function of iteration number, (a) before conversion, (b) after conversion, (c) after segment weight optimization.

15 15 / 15 Conclusions Empirically, the iterates of the QN method tend to proceed along directions that initially solve main conflict of plan, while the fluence map is kept smooth. These properties are crucial when performing iterative regularization. It is not necessary to determine the regularization parameter exactly, as the segment weight optimization diminishes the differences in treatment quality. Important not to over-optimize the plans, since both treatment quality and planning time can be gained.