Image Guided Adaptive Radiotherapy Model4D Adaptive Inverse Planning (Nonparametric Approach)...

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Image Guided Adaptive Image Guided Adaptive Radiotherapy ModelRadiotherapy Model

Di Yan, D.Sc.Di Yan, D.Sc.Radiation OncologyRadiation Oncology

William Beaumont HospitalWilliam Beaumont Hospital

Reference plan Treatment DoseDelivery Process

Treatment Evaluation & Plan Modification Decision-Making

4D Adaptive Planning

Image (patient anatomy) & Machine Output Feedback

Daily & Cumulative Dose Construction & Estimation

IGART Process:IGART Process: A treatment process which includes the individual treatment information, such as patient anatomical variation and dose in organs of interest assessed during the therapy course, in the treatment evaluation and planning optimization

IGART Model: Treatment Dose ConstructionIGART Model: Treatment Dose Construction

( ) VOIsvxdvDn

∈∀ΦΩ= ∑ ∫=

∈,,,)(

)()()(v

Daily & Cumulative dose delivered to a organ subvolume v in organs of interest, VOIs, during the n fractions of treatment delivery Ti

Subvolume displacement at the time t (Process Variable)Patient global matter distribution represented by a CT image obtained at the time t (Process Variable)Machine output at the time t (Control Parameter)−Φ

−Ω−

IGART Model: Adaptive ManagementIGART Model: Adaptive Management

kiit TtVOIsvxG tvt

∈∈Ω=Φ ),|,(* )()()(

which optimizes the predefined objective function,

( ) ⎟⎠⎞⎜

⎝⎛ =∈∀ΦΩ∫∈ kiVOIsvxdF

iTtttvt ,,...1;|,, )()()(

Optimal mapping (control law), determined before any treatment delivery day k < n, from the space of patient anatomical variation (process variables), , to the space of therapy machine output (control parameter), ,

{ })(, ttx Ωv

)( tΦ

Lecture OutlineLecture Outline

1. Description of patient anatomical variation process

},{ )( ttx Ωv

}{ )( t∗Φ

2. Treatment dose construction & estimation

}ˆ{ tD

3. Treatment evaluation & adaptive planning optimization

Organ displacement during the treatment delivery can be completely described using a Random Process,

⎭⎬⎫

⎩⎨⎧

∈=∈=

+= VOIvTTtuxx in

ivtvrvt ;

1),()()( U

Patient Anatomical Variation ProcessPatient Anatomical Variation Process

Where subvolume displacement, , has a probability density function (pdf) characterized by the mean, , (the systematic displacement) & the standard deviation, , (the random displacement).

),( vtuv),( vtμv

),( vtσv

1. Stationary Process (i.e. daily setup induced variation)

),(~, )()(),( vvvt pdfuif VOIv σμ vvv∈∀

has the ‘time-invariant’ (constant) mean & standard deviation

2. Non-stationary Process (i.e dose response induced variation)

),(~ ),(),(),(, vtvtvt pdfuif VOIv σμ vvv∈∃

has the ‘time-variant’ mean or standard deviation

Patient Anatomical Variation ProcessPatient Anatomical Variation Process

−=∑=1

Least Square Estimation Using },,{ )()( 1 ktt uu ⋅⋅⋅

( )⎥⎥⎥

⎢⎢⎢

=⎥⎥⎥

⎢⎢⎢

=ΦΦΦΦ=

⎥⎥⎥⎥

⎢⎢⎢⎢

⎡−

Process Parameter: Parametric EstimationProcess Parameter: Parametric Estimation

;ˆˆˆˆ )()(22)(11)( tmmttt aaa φφφμ ⋅+⋅⋅⋅+⋅+⋅=

[ ])()(2)(1)( ,,, tmttt φφφφ ⋅⋅⋅=v

Given an orthogonal base

2/)()( ˆ;ˆ t

mk,αttmk

χσσμμ

⋅−

≤⋅≤−−−

−Residuals:

Parametric EstimationParametric Estimation

],,,[)(m

t tt1 ⋅⋅⋅=φv

0ˆ,...,0ˆ,0ˆand 21 ≈≈≠ maaaifExample:

,ˆˆ1

)(1 ∑

)ˆ(ˆ 1

−=∑=

iit μ

Example: Prostate Isocenter DisplacementExample: Prostate Isocenter Displacement

1 2 3 4 7 10 13 16 21 24 27 30 33 36 39 42 45

output 1st week 2nd Week 3rd Week 4th Week 5th Week

2321)( ˆˆˆˆˆ tatataat ⋅+⋅+⋅+=μ

Treatment (4D) Dose Construction & EstimationTreatment (4D) Dose Construction & Estimation

Dose summation of subvolume v in VOI with using CT samples {CT1, …, CTk}, (assuming identical machine output in each treatment delivery)

( )),,(),,(

,),()(

rCTrCT

k11ΦΩ++ΦΩ⋅≈

⋅ΦΩ= ∑ ∫=

iTt tt

xdxdkn

dtvxdvD

Treatment Dose Construction & Estimation Treatment Dose Construction & Estimation

( ) dtxdvDn

⋅ΦΩ= ∑ ∫=

)()()( ,,)( v

dtdExdxxAxTn

iTt E Rx

tvtEtxEtEi

n '),'(),,'(1

')()(),'(0)(

v ⋅Ω−⋅ΦΩ=∑∫ ∫ ∫=

∈ ∈

⎟⎟

⎜⎜

⎛⋅−⋅Φ⋅⎟⎟

⎞⎜⎜⎝

⎛⋅= ∫

')'()(exp),'()(

tEwater

xtE dtx

vv ξρρμ

))'(,( )( vttE xxlEAA vv −⋅= ρEffective Density?

Treatment (4D) Dose EstimationTreatment (4D) Dose Estimation

udpdfuxDvD uRx rk v

vvv vvvv ⋅⋅ΦΩ+= ∫ ∈ )ˆ,ˆ,(

3),ˆ,()(ˆ

rrp )( σμ

Using “The Central Value Theory of Integral”, there exists ‘a single representation of the global matter distribution’ (the mean density) such that the spatial dose distribution calculated using the ‘mean density’ is time-invariant. In this case, the summation dose can be estimated convolving the spatial dose calculated with the mean density to the estimated pdf, such as

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

∂∂⋅⋅+∇⋅+≈

++ σσμ

μμ ˆˆ

21ˆ)( 2

]ˆ,[v

vvvv v

DxD rxpTrxrxp

Pelvic & Abdominal Region ¤ The planning CT image can most likely be used as the single

representation of the global matter distribution (if there is nolarge gas filling)

H&N Region¤ The planning CT can be applied if patient does not loss weight

significantlyChest Region & Skin Surface

¤ Using CT image obtained at a single breathing phase in cumulative dose evaluation can cause as much as 6% discrepancy. However, the weighted sum of 4D images (the average image) can be applied as the single representation of the global matter distribution to calculate the time-invariant spatial dose distribution for dose convolution

Treatment (4D) Dose EstimationTreatment (4D) Dose Estimation

0 1 2 3 4 5 6 7 8 9 10

Repiratory Phase

),(),()(101 CT10CT1 Ω⋅++Ω⋅= xdwxdwxD v

Example: Patient Respiratory MotionExample: Patient Respiratory Motion

=Ω−=Δ +⋅⋅⋅+ )ˆ,(ˆ)(10

101 CTCTxDxD vv 0.4%-0.3%-0.19%

,)(),(iCT xDxD vv −Ω=Δ

Application of adaptive management should not be limited by imaging modality & correction actions (online or offline)

Patient variation and treatment dose must be considered in the adaptive planning modification

Ideally, a continue optimal control from the space of patient anatomical variation (process parameters) to the space of therapy machine output (control parameters) should be applied

¤ However, it is either impossible or clinically impractical to perform such control mechanism

Adaptive ManagementAdaptive Management

Decide if the on going treatment plan needs to be modified

¤ Treatment evaluation based on dose-volume factor, EUD, NTCP or TCP model has been well established, and can be applied to determine the on going treatment quality and make a decision if the planning modification needs to be performed

Evaluation & DecisionEvaluation & Decision--makingmaking

Evaluation & DecisionEvaluation & Decision--makingmakingCTV Cumulative Dose

0 1 2 3 4 5 6 7 8 9 10

Treatment Fraction

PlannedDelivered

In Planning: TCP = 86% Tx Day: 85%, 84%, 83%, …………………………..…, 83%

Evaluation & DecisionEvaluation & Decision--makingmakingBrain Stem Cumulative Dose

0 1 2 3 4 5 6 7 8 9 10

Treatment Fraction

PlannedDelivered

In Planning: NTCP = 0.15% Tx Day: 0.21%, 0.18%, 0.22%, 0.22%, 1%, ……………… ., 1.2%

Image sampling during the treatment course

Estimate the individual systematic and random errors from the image registration (parametric approach) or ROI occupancy distribution(nonparametric approach)

Correct the systematic error by adjusting patient position or beam aperture

Construct the patient–specific CTV-to-PTV margin to compensate for the random error and the residuals

Adaptive Management: Adaptive Management: PatientPatient--specific Target Marginspecific Target Margin

0 10 20 30 40 50 60

Time (second)

it Ttu ∈;)(

PatientPatient--specific Target Marginspecific Target Margin(Parametric Approach)(Parametric Approach)

-0.1 0.1 0.3 0.5 0.7

Exhale --> Inhale

),( σμpdf

PatientPatient--specific Target Marginspecific Target Margin(Parametric Approach)(Parametric Approach)

Dose without motionDose with motion

σσ vv

v ⋅∂

∂⋅⋅=Δ 2

xD refxT Respiratory Motion: 2.0 cm

Image sampling during the treatment course

Deformable organ registration (parametric approach) or ROI occupancy distribution (nonparametric approach)

Include organ motion information directly in the objective function for optimal plan search

Adaptive Management: 4D Inverse Planning Adaptive Management: 4D Inverse Planning

4D Adaptive Inverse Planning4D Adaptive Inverse Planning(Parametric Approach)(Parametric Approach)

udpdfdkndDR

iik uurxvv

vvvvvv∫∑ ⋅⋅⋅−+= Φ+Φ=

3)ˆ,ˆ,(),()(),( )(

)(ˆ σμ

Birkner M, Yan D, Alber M, Liang J, Nusslin F (2003) Adapting inverse planning to patient and organ geometrical variation: algorithm and implementation. Med Phys, 30(10):2822-2831

( )VOIsvDFMax vk ∈ΦΦ

|ˆ ),(

Offline: Determine the optimal beam intensity map, Φ∗, for the rest of treatment delivery (k+1 to n) with using the patient anatomical position and treatment dose observed during the k previous dose deliveries,

4D Inverse Planning for Prostate Cancer

Anterior-Posterior Position (cm)31 29 27 25 23 21 19 17 15

Target Density

Rectal Density

ConventionalInverse Planning(5 mm margin)

4D adaptiveInverse Planning

Mandible

L Parotid

Brain Stem

R Parotid

4D Adaptive Inverse Planning4D Adaptive Inverse Planning(Nonparametric Approach)(Nonparametric Approach)

Offline:

Baum C, Alber M, Birkner, Nusslin F (2006) Robust treatment planning for intensity modulated radiotherapy of prostate cancerbased on coverage probability. Radiotherapy & Oncology, 78:27-35

xdwdD k

iiVOIxk xxx

vvv ∫=

∈ΦΦ ⋅⋅=

)(),(),(ˆ

( )VOIsxDFMax xk ∈ΦΦ

vv |ˆ ),(

4D Adaptive Inverse Planning4D Adaptive Inverse Planning(Parametric Approach)(Parametric Approach)

ˆ ),()(),( )(11

ΦΦ +=

= ∑ vk

iik xvv dd

Online: Determine the optimal beam intensity map, Φ∗, for the k+1th

treatment delivery with using the instant patient anatomical position as well as all previous observations (anatomy and dose) during the k previous dose deliveries

|ˆ ),(

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Dose (cGy)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Dose (cGy)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Dose (cGy)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Dose (cGy)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Dose (cGy)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Dose (cGy)

Day 3 Day 4

Online Adaptive

Rectal Wall

Prostate+SVOnlinePlan

O n lin e P la n

1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0 7 0 0 0 8 0 0 0D o s e (c G y )

D a y _ 1D a y _ 2D a y _ 3D a y _ 4D a y _ 5D a y _ 6

O n lin e A d a p t iv e P la n

1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0 7 0 0 0 8 0 0 0

d o s e ( c G y )

D a y _ 1D a y _ 2D a y _ 3D a y _ 4D a y _ 5D a y _ 6

Day 1 Day 2 Day 3 Day 4 Day 5 Day 6

IGART Model: Summary IGART Model: Summary

Treatment Dose Delivery Patient Variation Process

( )⎭⎬⎫

⎩⎨⎧

=∈∈ΦΩ

kiTtVOIsvxd

ttvtk ,...,1,;

:,,~)(ˆ )()()(

⎭⎬⎫

⎩⎨⎧

∈∈ΦΩ

TtVOIsvx

;:, )()(),(

|ˆ ),(*Φ

( ) ?|)()(ˆ δ≥∈− VOIsvvDvDf pk

Image Guided Adaptive Radiotherapy Model4D Adaptive Inverse Planning (Nonparametric Approach)...

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