d = Dx subject to Stochastic programming methods for

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1 Inverse planning based on probability distributions 3 Figure 1. Prostate patient who had five CT scans before treatment. Bladder and rectum filling differs from day to day and hence the location and shape of the CTV. Stochastic programming methods for constant in time and that the geometry one fraction is uncorrelated tothe geometry handling the previous uncertainty fraction. This meens we do not consider time and trends during motion treatment, in e.g. weight loss or radiation effects on the tumor and healthy tissues. IMRT planning 2.2. The model To gain some basic understanding of inverse planning based on probability distributions of patient geometries we consider a model of idealised geometry that originates from rotation therapy with high Jan energyunelbach photons. We consider the planar irradiation of a circularly shaped CTV of radius RT with a rotating gantry. The CTV issurrounded by a healthy tissue of radius R (see Fig. 2). Organ movements are simulated by rigid translations of the entire body for simplicity. Industry collaborations: RaySearch, Philips Medical Systems This allows us to trac each point of the body during movement and to calculate the cumulative dose in each point. The geometry of the patient can then be parameterised by a single vector r = ( r, ϕ) denoting the position of the center of mass. In the following text we denote vectors in two spacial dimensions by bold italic characters and scalars by normal italic characters. For simplicity, we assume that the displacements ( r, ϕ) follow a Gaussian probability distribution Content A. Stochastic programming in IMRT planning B. What is the advantage over a PTV approach? 1. Systematic positioning errors Balancing target coverage and normal tissue sparing 2. Reducing normal tissue dose through horns 3. Range uncertainty in proton therapy Breadown of the static dose cloud approximation Fluence map optimization in IMRT Minimize dose-based objective function minimize subject to f ( d) d = Dx x ³ 0 1

2 Including motion and uncertainty Assume a discrete set of errors can occur Delivered dose depends on the error scenario d = D x Assign probabilities to errors: p Minimize expected value of objective function minimize ( ) åp f d Including motion and uncertainty Quadratic objective function ( ) = d i - d pres f d ( ) 2 iît å + å d i iîn ( ) 2 2 å p ( d i - d pres ) 2 æ = çå p d i - d pres è ø deviation of prescribed and expected dose ö æ ö + p d å ç i -å p d i è ø variance 2 Systematic errors Systematic errors (Setup errors or internal deformation) Inverse planning based on probability distributions 3 Figure 1. Prostate patient who had five CT scans before treatment. Bladder and rectum filling differs from day to day and hence the location and shape of the CTV. 2

3 dose [%] dose [%] Systematic errors Gaussian p σ = 10 cutoff at ± 2σ 40 scenarios Ref: Löf 1995, Inv Prob position Systematic errors reproduces a PTV-lie plan may yield a smooth falloff position See also: Sir 2006, PMB Systematic errors Benefit: Automation: no explicit PTV definition necessary Could optimally balance target coverage and OAR sparing Stochastic programming natural with TCP/NTCP minimize ( ) åp TCP d marginalization of a TCP model over the uncertain dose distribution subject to åp NTCP d ( )

4 Motion tumor size = 20 amplitude = 20 ITV plan exhale inhale Can normal tissue dose be reduced? Tumor accumulates dose in different breathing phases d = n å i=1 w i D i x n å w i =1 i=1 Idea: reduce dose to regions where the tumor is rarely deliver higher dose to regions always occupied by tumor 4

5 4D optimization Assume predictable breathing motion dose pea where there is tumor most of the time w i dose reduction at the edge of ITV Problem: dose will degrade if breathing pattern varies Stochastic programming: Allow different breathing patterns w with probability p d = n å w i D i x i=1 n å w i =1 i=1 exhale inhale Account for uncertainty in breathing pattern larger uncertainty in w gradually yields more ITV-lie plans special case w i = 1 models systematic error Ref: McQuaid 2011 AAPM summerschool 5

6 Dose delivered to moving tissue (nominal trajectory) Realistic cases Assume predictable motion (g) (h) (i) (j) Dose on exhale () Accumulated dose (l) 6

7 Respiratory 96 motion (g) (a)(h) (j) (b)(i) ()( (j) (d)() (e)(l) ( No uncertainty medium motion Motion modeled as uncertainty Figure 5.5: systematic 3D distributions error for tr (a-c) optimized for compensator b IMRT (g-i) optimizat ion of the exp optimized Ref: Heath for an 2009 infinitesimal Med Phys sho the static dose distribution, (b,e,h standard deviation for an infinites 5.8c,d apply. Benefit: 4D optimization yields dose horns Normal tissue dose reduction compared to PTV Stochastic programming can account for breathing variations Find the balance between robustness and normal tissue sparing through horns Proton therapy Range uncertainty in IMPT 7

8 Proton therapy Proton therapy Robustness analysis: 5 mm range overshoot nominal plan Proton therapy Stochastic programming: Assume 3 scenarios: nominal scenario p 1 = mm range overshoot p 2 = mm range undershoot p 3 =

9 Sensitivity analysis 5 mm range overshoot conventional plan generated using stochastic programming Motivation How is robustness achieved? conventional plan generated using stochastic programming Commercial implementations Proton therapy led to the first implementation of probabilistic / robust planning in commercial TPS Examples: Before that: IMPT Pinnacle (in development) (implements a probabilistic approach) RayStation v4.5 (implements a minimax approach) (Ref: Fredrisson 2011, Med Phys) Hyperion (in-house TPS in Tübingen, Germany) (coverage probability method to account for positioning errors in prostate treatments) (Ref: Baum 2006, R&O) 9

10 Raystation 4.5 User interface Setup uncertainty Range uncertainty Raystation 4.5 User can robustify important objectives Raystation 4.5 Plan evaluation define error scenario 10

11 Summary Stochastic programming for handling uncertainty: optimize expected value of the objective function general purpose method applicable to many uncertainties Advantage over a PTV depends on type of uncertainty: Automating target expansions (systematic positioning errors) Normal tissue dose reduction through horns (respiratory motion) Mitigate beam misalignments riss (IMPT) Status in practice Range and setup uncertainty in IMPT: Fundamental limitations of the PTV concept led to the first commercial implementations Dose accumulation relies on deformable registration Computationally intensive Setup errors, inter-fraction organ motion Qualitatively similar to PTV plans Magnitude of the error reduced through image guidance 11