HST.187: Physics of Radiation Oncology #9. Radiation therapy: optimization in the presence of uncertainty Alexei Trofimov, PhD [email protected] Jan.

HST.187: Physics of Radiation Oncology

#9. Radiation therapy: optimization in the presence of uncertainty

Alexei Trofimov, [email protected]

Jan Unkelbach, [email protected]

Dept of Radiation Oncology MGH

April 3, 2007

Uncertainties in RT• Intro

– Sources of uncertainty, e.g. -• Set-up, target localization (inter-fractional)• Intra-fractional motion

– Methods to counter the uncertainties• Volume definitions/ margins, treatment techniques

– Effect of uncertainties on the dose distribution

• Probabilistic planning techniques in the presence of uncertainties– Inter-fractional motion and set-up uncertainties – Proton range variations in tissue

• Handling of intra-fractional motion (respiratory) – Image-guided radiation therapy IGRT and “4D” planning– Probability-based motion-compensation– Intro to robust optimization

Target definition: inter-observer variation

Steenbakkers et al R&O 77:182 (2005)

Target motion (intra-fractional)

Targeting

Interplay between internal motion and the multi-leaf collimator sequence

JH Kung

P Zygmanski

Target motion (intra-fractional)

Targeting

Radiological depth changes

Inhale Exhale

Planned dose at exhale phase

Liver Tx plan, PA field Planned by J.Adams(TPS: CMS XiO)

As would be delivered at inhale

50%

Set-up uncertainties: day-to-day variation

Images: © 2007 Elsevier IncZhang et al IJROBP 67:620 (2007)

Variation over 8 weeks of treatment

Prostate treatment with protons

Compensatordesign

Variation In set-up

Compensatorsmear

Compensatorsmear

Intrafractional motion

Part 2:Probabilistic

approach to account for uncertainty in

IMRT/IMPT optimization

Content

• Motivation – interfractional random setup error

• Concept of probabilistic treatment planning

• Application to interfractional motion of the prostate

• Application to range uncertainties in IMPT

MotivationConsider inter-fractional random setup error in a

fractionated treatment

How can we achieve an improvement?

• Lower dose to regions where tumor is located rarely

• Have to compensate for it by higher dose to other regions

safety margin:

irradiate entire area where tumor may be with the full dose

Motivation

25 moving voxels

45 static voxels

Example:

Question?

Are there static dose fields that yield tumor coverage and improve healthy tissue sparing?

tumor voxels are at 5 different positions equally often

Motivation

Example:

integral dose: 40.8 (instead of 45.0)

MotivationDose in the moving tumor:

frequency for moving voxel i being at static voxel j

Have to solve system of linear equations to determine static dose field which yields D = 1

dose in moving tumorstatic dose field

Motivation

special solution (safety margin)

Set of solutions is affine subspace

kernel of the mapping P:

Set of static dose fields which preserve D = 1:

kernel dimension (number of static voxels) minus (number of tumor voxels)

Motivation

Intrinsic problems:• only handles predictable motion, not uncertainty

• cannot handle systematic errors

• cannot handle irreproducable breathing pattern

Method could in principle work if motion was predictable and treatment was infinitely long

Need more general method to handle uncertainty!

(having these ideas in mind)

Idea of probabilistic methodMain assumption:

The dose delivered to a voxel depends on a set of random variables

vector of random variables which parameterize the uncertainty

fluence map to be optimized

Assign probability distribution to random variables:

Idea of probabilistic methodApplications:

G = position of voxels

P(G) = Gaussian distribution

• Inter-fractional motion

• range uncertainty

• respiratory motion

G = amplitude, exhale position, starting phase

(note: P(G) unrelated to `breathing PDF`)

G = range shifts for all beamlets

Idea of probabilistic methodPostulate:

optimize the expectation value of the objective function

• incorporate all possible scenarios into the optimization with a weighting that corresponds to its probability of occurrence

Idea of probabilistic methodExample: quadratic objective

1st order term 2nd order term

variance of the dosedifference of expected and prescribed dose

expected dose:

Alternative formulations• In this talk: optimize expectation value

• most desireable might be something in between

(can be solved by robust optimization techniques in linear programming)

• alternative: optimization of the worst case

Application to prostate

Incorporating

inter-fractional motion

of the prostate

into

IMRT optimization

application to prostateUncertainty G: positions of voxels

Probability distribution P(G): Gaussian

application to prostate

static dose field (dose per fraction)

• expected quadratic objective function

• 30 fractions

• large amplitude of motion ( 8mm AP, 5mm LR/CC)

application to prostateexpected dose in the moving tumor coordinate system

• Best estimate for the dose delivered to a voxel

application to prostateProblem: Uncertainty implies that we don‘t know the dose distribution which will be delivered

standard deviation: assess uncertainty of the dose in each point

treatment plan evaluation difficult

application to prostateProbability for the delivered dose to be below/within/above a 3% interval around the prescribed dose

below abovewithin

(D Maleike, PMB 2006)

application to prostatePrototype GUI to view probabilities for over/under dosage

(D Maleike, PMB 2006)

• user may select dose intervals of interest

Application to prostate

• Incorporate organ motion in IMRT planning to overcome the need of defining safety margins

• resemble the idea of inhomogeneous dose distributions on static targets in order to achieve better healthy tissue sparing

• control the sacrifice of guaranteed tumor homogeneity

Probabilistic approach can ...

Application to range uncertainties

Handling range uncertainty

in

IMPT optimization

Application to range uncertainties

degraded dose distribution if the actual range differs from the assumed range

assumed range + 5 mm - 5 mm

Conventional IMPT treatment plans may be sensitive to range variations

Application to range uncertaintiesWhy? Because ...

• pencil beams stop in front of an OAR

• dose distributions of individual beams are inhomogeneous

Application to range uncertainties

Range uncertainty assumptions for probabilistic optimization:

• 5 mm uncertainty (SD) of the bragg peak location for each beam spot

• Gaussian distribution for the range shifts

• is considered a systematic error (no averaging over different range realizations in different fractions)

Application to range uncertainties

assumed range + 5 mm - 5 mm

• Probabilistic optimization can significantly reduce the sensitivity to range variations

convetional plan

Application to range uncertaintiesWhy? Because ...

• lateral fall-off of the pencil beam is used

• dose distributions of individual beams are more homogeneous in beam direction

convetional plan

Application to range uncertaintiesPrice of robustness:• lateral fall-off is more shallow

convetional plan

plan quality for the assumed range is slightly compromised

- higher dose to OAR or reduced target coverage

probabilistic plan

Application to range uncertainties

• take advantage of the characteristic features of the proton beam and the many degrees of freedom in IMPT to make treatment plans robust with respect to range variations

(which cannot be achieved by other known heuristics)

Probabilistic approach can ...

Part 3: Intrafractional motion

Continuous irradiation

IMRT delivery to a moving target

Int map no motion motion 1 fraction motion 4 fx

The effect of target motion on dose distribution

Coverage assured with planning margins

Gated Tx at MGH

Varian RPM-system

marker block with IR-reflecting dots

IR-source + CCD camera

External-internal correlation Tsunashima et al IJROBP 2003

Gierga et al IJROBP 2004:

correlation differs between markers

Phase shift

HHoisak et al IJROBP 2003

External-internal correlation

• Generally well-correlated, but…

• Not necessarily linear• Phase shift has been observed, not

necessarily constant on different days

• Proportionality coefficients, phase may vary with – marker position– respiratory “discipline” (e.g. compliance with

breath-training/coaching)

(“Fast”) tracking delivery

Inverse optimization

• Dose calculation using (Dij) matrix:

beamlet jx

voxel i

4D- influence matrix (D-ij) approach

• Dij ’s are precalculated for all beams and all instances of geometry (4D-CT phases)

• At instance (phase) k we have

k = 1, …, 5: breathing phase

beamlet jx

voxel i

Eike Rietzel, GTY Chen “Deformable registration of 4D CT data” Med Phys 33:4423 (2006)

• Determine voxel displacement vector field between Pk and P0 (reference phase)

P0 (inhale) P4 (exhale)

• Deformations are then applied to all pencil beams in Dij matrix

pencil beam in P4 (exhale)

same pencil beam transformed to P0(inhale)

x

x

A Trofimov et al PMB 50:2779 (2005)

Continuous irradiation: instantaneous dose distribution

From a different prospective: a moving instant. dose in a fixed reference geometry

Approaches to temporo-spatial optimization of IMRT

(1) Planning with optimal margins (Internal Target Volume)

(2) Planning with Motion kernel(a) Uniform approach (motion PDF)(b) Adaptive approach (sum influence matrix)

(3) Gating / Unoptimized tracking – plans optimized separately, 1 best plan chosen out of several or all delivered dynamically

(4) Optimized tracking – several plans optimized simultaneously, delivered dynamically

A Trofimov et al PMB 50:2779 (2005)

Lung: CTV vs Internal Target Volume (ITV)

Planning with “Internal” margins - ITV

App. 1: Optimal margins (ITV): lung

DVH for ITV plan recalculated for different geometries (CT phases): lung

Approaches to Temporo-Spatial Optimization of IMRT

(1) Planning with expanded margins (ITV)

(2) Planning with modified dose kernel (b) Uniform approach (motion PDF)(a) Adaptive approach (sum influence matrix)

(3) Gating / Unoptimized tracking – plans optimized separately, 1 best plan chosen out of several or all delivered dynamically

(4) Optimized tracking – several plans optimized simultaneously, delivered dynamically

Motion probability distribution function (PDF)

Motion-compensation in IMRT treatment planning• If the motion (PDF) is known

(reproducible), the dosimetric effect can be reduced – Deconvolution of intensity map– Planning with “smeared” beams

– .

Reduction of integral dose with motion-adaptive planning

.

Motion kernel: “one-size-fits-all” vs. “custom-made”

Original beamlet

=

Convolved “motion” beamlet Sum of deformed beamlets

IMRT with motion-compensated Tx PlanInt map no motion motion 1 fraction motion 4 fx

Patient data

lung liver

App. 2a: Motion kernel plan, DVH recalculated for 5 ph’s

MK plan: DVH recalculated for diff phases

App. 2b: with averaged Dij-matrices (liver)

App. 2b: with averaged Dij-matrices (liver)

App. 2b: with averaged Dij-matrices (liver)

Inhale (recalc’d to reference) Exhale (reference)

Inhomogeneous “per-phase” doses are designed so that the some conforms to the prescription

Approaches to Temporo-Spatial Optimization of IMRT

(1) Planning with expanded margins (ITV)

(2) Planning with modified dose kernel (Motion kernel)(a) Uniform approach (motion PDF)(b) Adaptive approach (sum influence matrix)

(3) Gating / Unoptimized tracking – plans optimized separately, 1 best plan selected for gated delivery or all delivered dynamically

(4) Optimized tracking – several plans optimized simultaneously, delivered dynamically

App. 3: Gating / Unoptimized tracking (liver)

App. 3: Gating / Unoptimized tracking (lung)

Approaches to Temporo-Spatial Optimization of IMRT

(1) Planning with optimal margins (ITV)

(2) Planning with modified dose kernel (Motion kernel)(a) Uniform approach (motion PDF)(b) Adaptive approach (sum influence matrix)

(3) Gating / Unoptimized tracking – plans optimized separately, 1 best plan selected for gated delivery or all delivered dynamically

(4) Optimized tracking – several plans optimized simultaneously, delivered dynamically

App. 4: optimized tracking (lung)

App. 4: Optimized tracking (lung)

DVH comparison for the lung case

DVH comparison for liver case

Ideal case for tracking delivery (vs gating)

DVH and dose for different “gated” (single phase) plans for the lung case

Sources of delay: RPM: 60-90 ms , 75 ms averageSystem response time : < 5 msWait for the next modulation cycle: 0-100 ms

Total delay: 65-195 ms, average 130 ms

Delivery of gated proton treatment : Timing

Delivery restricted to complete modulation cycles: on/off at the stop block position only

100 ms

Hsiao-Ming Lu

Residual motion with gating Probability distribution

Inter-fractional variability

Liver-2

Cardiac-1

Cardiac-2

Cardiac-1

Time

Positio

nPositio

n

Variability between patients

Lung-2

Liver-2Cardiac-2

Robust formulation for probabilistic treatment planning:

– Tim Chan et al: Phys Med Biol 51:2567 (2006)

– Outcome will be “acceptable” as long as the realized motion is within the expected “limits”

PDFuncertaintybounds

Realized PDF

Realized PDFPlanning PDF

Planning PDF

Dose to moving target

Summary• (Some) sources of uncertainty in RT:

imaging, target definition, dose calc, set-up, inter-, intra-fractional motion

• Margin/ITV approach is the most robust for target coverage, but substantially increases dose to healthy tissue

• Image-guided RT improves dose conformity, reduced irradiation of healthy tissues, BUT relatively complex delivery, not error-proof

• Probabilistic motion-adaptive treatment planning in combination with image-guided delivery may be the optimal solution

Acknowledgements

J Adams

T Bortfeld, PhDT Chan, PhDS Jiang, PhDJ Kung, PhDHM Lu, PhDH Paganetti, PhD

E Rietzel, PhD

C Vrancic