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IMPT and optimization

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Content

• Mathematical aspects of treatment planning

• 3D conformal therapy and design of spread-out Bragg peaks, forward planning

• IMPT inverse planning as mathematical optimization problem and possible solutions

• Multicriteria optimization

• Robust optimization

• Intrafractional motion in treatment planning

• Accounting of radiobiological effects in IMPT

• Other applications optimization in proton therapy

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3D Planning and Spread-out Bragg peak

design

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Optimization of SOBP fields

• 3D conformal proton therapy

• Spread-out Bragg peak – uniform covering of large tumor volumes

• Usually iterative process of optimization

• SOBP is modulated according to the energy – optimization has to follow different ranges in tissue

• Intensity modulated SOBP field – temporally optimizing of beam current during the modulation cycle

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Forward planning with SOBP fields

• Manual optimization = current changes

• Disadvantages: depends on planner´s capabilitiesit is not systematic -> not an optimization in mathematical sense

choosing of angle, direction of beam

range and

SOBP modula

tion

forward

calculation

based on

assumed

beam fluence

adjusting

fluences and

weights of

beams

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IMPT as an optimization problem

• IMPT fields deliver typically nonuniform dose distribution => it helps achieve highest conformity of proton distribution

• Most common IMPT technique = 3D-modulation method• Individually weighted Bragg peaks spots are placed through the

target volume

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Important difference between IMRT and IMPT

• Spread-out Bragg peak -> modulation of its brings another degree of freedom to the proton treatment

• Despite this – optimization is same problem for both methods

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Mathematical approach

• Defining of objectives and constraints

• Volumes of interests (VOI) – include targets and critical organs OAR

• Total dose distribution from IMPT field – sum of contributions from static pencil beams of various voxels (dose influence matrix Dij)

• Total dose to any voxel:

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• Large number of pencil beams – optimization methods required

• Output of the plan optimization: set of beam weight distributions – FLUENCE MAP

• IMRT – 2D fluence maps (process uses different angles of irradiation)

• IMPT – separate map for every energy setting

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Objective function

• Optimization = process of looking for the minimal OF

• Quadratic penalty function:

• In general: Constraints have to be fulfilled in order to make plan acceptable

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General formulation of IMPT optimization problem

• Soft constraints setting

• On…objectivesαn…weighting factorsCm…constrainslm, um…lower and upper bounds

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Solving the optimization problem

• Variables = beam weights x (can have any nonnegative value, often discretized)

• Objectives and gradient of the objective can be often calculated analytically

• Two types: • Constrained -

• Unconstrained – no dosimetric hard constraints – all treatment goals defined as objectives (Newton method, gradient methods)

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Constrained optimization for IMPT

• Large number of variables and large number of voxels

• Linear programming (just linear objectives and constraints) vs. Sequential quadratic programming

• Convex (constraints and objectives are convex functions) and nonconvex (for example including radiobiological models; becoming more common in this time)

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Multicriteria Optimization

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Principle

• Single criterion optimization problem = ONE objective + ALL OTHER CONDITIONS are constraints

• Main objective = deliver prescribed dose to whole target volume

• Other objectives = to keep dose in OARs minimal

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Biological models

• Hypothetically – single criteria – maximizing TCP to the NTCPs

BUT

• Reality – pacientspecific trade-off: „How big gain do we get in TCP, when we allow NTCP for some organ in this amount?“

= MULTICRITERIUM!!!

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• Presently – standard commercial systems use single-criterion approach

• Disadvantage: • Longer process = iteration cycle depending on the treatment

planner

• Difficult to set weights and function parameters

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1) Prioritized optimization

• Each objective gets its own priority

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2) Pareto surface (PS) approach

• Instead of prioritizing of objectives treats each objective equally

• Yields not a single plan, but a set of optimal plans – trade off objectives in many ways

• PARETO OPTIMAL PLAN = plan which is feasible and there is not another plan, which is strictly better for at least one objective and it is not worse for any other

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Mathematical formulation PS

• X… all beamlet and dose constraintsN… number of objective functions

• Algorithm issues:• 1) How to compute a reasonable set of diverse PS plans

• 2) How to present resulting information to decision markers

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Main strategies of PS approach in radiotherapy

A) WEIGHTED SUM methods

Combining all the objectives into a weighted sum => solving of the resulting scalar optimization problem

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Main strategies of PS approach in radiotherapy

B) CONSTRAINT methods

Use objective functions as constraints (same as for prioritized optimization) – varying constraint levels (finding of different pareto-optimal solutions)

Problem - error measures are not natural part of algorithm or output

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Navigation on the PS based approach

• How to allow user to select plan from the set of Pareto-optimal soloutions?

1. N sliders, each for one objective

2. Allow to choose N sliders and 2 of N objectives -> picturing of the trade-off for those objectives

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Comparing Prioritized Optimization and PS-Based MCO

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Comparing Prioritized Optimization and PS-Based MCO

Prioritized optimizationProgrammable procedure – result is single Pareto-optimal plan

Only one plane presented to user in the end!

PS based MCOResult are all optimal options presented to user

Not useful for routine planning – user has to decide which one is best manually from large number of options

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Robust Optimization

methods for IMPT

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• Many uncertainties influencing the delivered dose

• The optimal plan needs to be robust

• Small deviations from the planed dose distribution don´t influent the treatment outcome

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• IMPT – inhomogeneous dose, more proton energies – combinations of dose distributions

• More variations – mismatches of doses from different fields

• Dose gradients - even bigger sensitivity for setup errors

• Hot and cold spots (dose in critical organs)

• More conformal dose -> more complex fluence map -> more sensitive plan for uncertainties

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Sensitivity of plan to setup errors

• The same plan, different error setup – big impact on the dose contribution of the posterior beam

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Robust optimization strategies

• Delivered dose distribution depends on set uncertain parameters

• Models of uncertainties:• Rigid setup error without rotation – parameter is 3D vector of

shifting in space λ

• Pencil beam simple model – overshooting and undershooting uncertainties of all beams

• More complicated pencil beam model – different errors for different pencil beams

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The probabilistic approach = Stochastic programming approach

• Dose distribution: d(x,λ), where x…beam spot weights to be optimized λ…uncertain parameter

• Objective function: O(d(x, λ)) … describes dose distribution

• Probability distribution: P(λ) … probability, that error λ occurs

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• We need to minimize the expected value of objective function

• The general goal:

To find the treatment plan, that is good for all possible errors, BUT: larger weights for scenarios with higher probability to occur, lower weights for scenarios with lower probability

• Typically – quadratics difference – minimizing of objective function in each voxel

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The robust approach

CONSTRAINTS

• Constraints have to be satisfied for every realization of the uncertain parameters

• We have constraints (example less than 50Gy for spinal chord) => robust approach sets less than that dose for every possible range of setup error!

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OBJECTIVES

= Worst-case optimization problem

• Result: as good as possible treatment plan for the worst case, that can occur

• Minimizing of maximum dose, which can be delivered for any possible range or setup error

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Optimization of the Worst-Case Dose Distribution

• Hypothetical

• Unphysical – every voxel is considered independently

• Principle: for every voxel the dose is defined as he worst dose value that can be realized for any error in the uncertainty model

• Primary objective function is done by sum of objective function in case of no errors and objective function in the worst-case

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Example of robust optimization

a) Conventional calculated plan optimized without accounting an uncertainty

b) Plan optimized for range and setup uncertainty using the probabilistic approach

(DVH for CTV for spinal cord)

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Dose distribution of individual beams

(A)Conventional IMPT

(B)Robust IMPT - Range uncertainty only

(C)Robust IMPT - Setup uncertainty only

(D)Robust IMPT - Considering both range and setup uncertainty

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4D Temporospatial Optimization

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Temporospatial (4D) Optimization

• What affect precision of IMPT?• Changes in setup

• Motions of target (respiration, peristaltic movements, gravity…)

• Degrading of gradients, increasing irradiation of healthy tissues

• IMPT requires to consider possible changes in radiological depth to target

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Avoiding to impact of intrafractional motions

• Many methods: compensator expansion, beam gating, field rescanning…

• Intrafractional motion as a part of beam weights optimization:• Requiring of geometrical

variation of patient anatomy = 4D CT for recording breathing cycle,

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Optimization based on a known motion probability density function

• Based on 4D CT scanning

• Delivering of inhomogeneous dose distribution to a static geometry

• Edge-enhancement = dose boosting at the edges of the target (most important part of target which is influenced by motions)

• PDF-based is strongly influent by reproducibility of target motions (motion deviates form expectation, significant deviation may occur)

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Accounting for Biological Effects

in IMPT Optimization

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Performation of optimization

1) RBE

For IMPT usually constant RBE -> treatment plan ptimization based on physical dose

2) LET

Higher LET -> increasing of radiation-induced cell-kills (at the end of proton beams range)

Objective function formulated in the terms of LET and RBE (f.e. linear-quadratic model)

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Other applications of optimization

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Scan path optimization

• In reality of 3D scanning, large number of beam spots have zero weight

• Spot scanning – steering of the beam in the zigzag pattern over entire grid including positions corresponding with zero weight

• Scan path optimization – avoiding regions with zero weight spots (simulated annealing principle)

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Beam current optimization for coninuous scanning

• Spot scanning: dose is delivered according to the optimized spot weight

• Continuous scanning: beam is constantly moving according to predefined pattern

• Optimization methods applied in the step of converting optimized spot positions at discrete positions to beam-current modulation with the same fluence

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References

• H Paganetti: Proton Therapy Physics, CRC Press, 2012

• F. Albertini: Planning and Optimizing Treatment Plans for Actively Scanned Proton Therapy: evaluating and estimating the effect of uncertainties, Disertation, ETH Zurich, 2011

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Thank you for your attention