Minimizing beam-on time during radiotherapy treatment by total-variation regularization

Transcription

1 Minimizing beam-on time during radiotherapy treatment by total-variation regularization Xing Lab, Department of Radiation Oncology, Stanford. Zhu, L. & Xing, L. (2009) Search for IMRT inverse plans with piecewise constant fluence maps, Med. Phys. 36(5). Zhu, L. et al. (2008) Using total-variation regularization for intensity modulated radiation therapy inverse planning with field specific numbers of segments, Phys. Med. Biol. 53(21). Rasmus Bokrantz

2 Rasmus Bokrantz 2 / 20 Outline 1. The trade-off between beam-on time and plan complexity. 2. Approaches for handling constraints on deliverability. 3. Treatment planning viewed as a signal processing problem. 4. Recovery of sparse signals. 5. Comparison between regularization approaches for a patient case.

3 Rasmus Bokrantz 3 / 20 Plan complexity vs. beam-on time Increased number of segments allows greater flexibility in shaping the dose distribution. Few and regular segments promotes robustness with respect to geometric uncertanties (e.g. organ motion). A decrease in number of segments reduces the risk of radiation-induced secondary cancers. Overall planning goal Minimize the number of segments while satisfying the clinical goals.

4 The treatment planning problem Rasmus Bokrantz 4 / 20 Perform a least squares fit to prescription dose minimize x R n Px ˆd 2 2 subject to x 0, x { deliverable apertures} where x beamlet weights P beamlet kernel matrix ˆd prescription dose

5 Rasmus Bokrantz 5 / 20 Approaches for handling the aperture constraint Beamlet weight optimization The aperture constraint is ignored. + Convex quadratic program. Regularization is needed (e.g. quadratic smoothing). Requires conversion into deliverable machine settings. Direct machine parameter optimization The aperture constraint is formulated excplicetily. + The optimized plan is directly deliverable. The aperture constraint is highly nonlinear and nonconvex.

6 Rasmus Bokrantz 6 / 20 Novel approach suggested by Zhu et al. View the treatment planning as a signal processing problem Optimal beam profiles are viewed as a signal to be detected. Deliverable fluence maps are piecewise constant. The gradient of a piecewise constant fluence map is sparse. Fluence maps can be constructed using compressed sensing: a technique for recovery of sparse signals.

7 Sparsity Rasmus Bokrantz 7 / 20 A signal is said to be sparse if a large number of coefficient are close or equal to zero when represented in some domain. f = n x i ψ i, where f R n, (signal) i=1 Ψ = (ψ 1, ψ 2,..., ψ n ). (ON basis)

8 Recovery of a sparse signal 1 (3) Rasmus Bokrantz 8 / 20 Nyquist-Shannon sampling theorem Perfect signal reconstruction requires a sampling frequencey at least twice the maximum frequency present in the signal.

9 Recovery of a sparse signal 2 (3) Rasmus Bokrantz 9 / 20 Compressed sensing A sparse signal can be reconstructed at a significanly lower sampling rate than the Nyquist rate. Algorithm: recover the signal by l 1 -norm minimization An n-dimensional signal f expressed in the representation basis Ψ can be reconstructed using m measurements y k = ϕ k, Ψx in the sensing basis Φ by solving the convex optimization problem minimize x R n x l1, subject to y k = ϕ k, Ψx, k = 1,..., m.

10 Recovery of a sparse signal 3 (3) Rasmus Bokrantz 10 / 20 If f is sufficiently sparse, recovery via l 1 minimization is exact with overwhelming probability. Theorem: Candes, E. & Romberg, J. (2007) Inverse Prob. 23(3). Let f R n and suppose the coefficient sequence x of f in the representation basis Ψ is S-sparse. Select m measures in the sensing domain Φ uniformely at random. Then if m Cµ 2 (Φ, Ψ)S log(n/δ), where C > 0, and µ(φ, Ψ) is the mutual coherence between Ψ and Φ, then, the probability of perfect reconstruction exceeds 1 δ. µ(φ, Ψ) n max 1 k,j n ϕ k, ψ j.

11 Noise contaminated signal l 1 -norm regularization promotes sparsity l 2 -norm regularization promotes smoothness Boyd & Vandenberghe (2009) Convex optimizaion Rasmus Bokrantz 11 / 20

12 Reformulatating the treatment planning problem Rasmus Bokrantz 12 / 20 Initial problem formulation minimize x R n Px ˆd 2 2 subject to x 0, x { deliverable apertures} Include a total-variaton regularization term and drop the aperture constraint minimize x R n Px ˆd β x 1 subject to x 0 This problem can be posed as a convex quadratic program using help variables.

13 Comparison between regularization approaches Rasmus Bokrantz 13 / 20 Total-variation regularization minimize x R n Px ˆd β x 1 subject to x 0 Quadratic smoothing minimize x R n Px ˆd β x 2 2 subject to x 0

14 Without regularization Rasmus Bokrantz 14 / 20 Highly irregular fluence profile which is ill-suited for conversion into deliverable machine settings.

15 Quadratic smoothing Rasmus Bokrantz 15 / 20 Smooth fluence profile which requires a relatively large number of segments for conversion.

16 Total-variation regularization Rasmus Bokrantz 16 / 20 Pieciewise continous fluence profile which can be converted into a small number of segments.

17 Rasmus Bokrantz 17 / 20 Finding the plan with minimal number of segments Re-solving at various regularization parameter values gives points on a two-dimensional Pareto surface in objective space. A plan can be identified as clinically satisfactory if it meets with a set of acceptance criteria. The authors selects the plan with smallest number of segments which also meets the acceptance criteria as the final solution.

18 Comparison to quadratic smoothing Rasmus Bokrantz 18 / 20 At a fixed number of segments, total-variation regularization yields clincally superior plans. At a fixed dose distribution performance, total-variation regularization results in a significant decreas in number of segments. Clinical performance is assessed using dose-volume histograms.

19 Rasmus Bokrantz 19 / 20 Comparison to direct aperture optimization No numerical comparison is performed due to difficulties in implementing direct machine parameter optimization. The authors claim that the suggested method is superior since global optimality is guaranteed and the treatment planning problem can be solved more efficiently. Direct aperture optimization requires prefixing the number of segments before optimization.

20 Rasmus Bokrantz 20 / 20 Summary Novel approach for beamlet weight optimization A sparsity objective is included in the problem formulation. Plan degeneracy is exploited for the benefit of reducing beam-on time. Conclusion Total-variation regularization allows for a reduction in beam-on time without compromising plan quality.