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
iTt
ttvti
∈∀ΦΩ= ∑ ∫=
∈,,,)(
1
)()()(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−
)(
)(
)(
t
t
vtxv
IGART Model: Adaptive ManagementIGART Model: Adaptive Management
Uv n
kiit TtVOIsvxG tvt
=
∈∈Ω=Φ ),|,(* )()()(
which optimizes the predefined objective function,
( ) ⎟⎠⎞⎜
⎝⎛ =∈∀ΦΩ∫∈ kiVOIsvxdF
iTtttvt ,,...1;|,, )()()(
v
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
vvv
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
( )mk
uk
itt
t
ii
−
−=∑=1
2)()(
)(
ˆˆ
μσ
Least Square Estimation Using },,{ )()( 1 ktt uu ⋅⋅⋅
( )⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=ΦΦΦΦ=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−
)(
)(
)(
)(12
111
;,
ˆ
ˆˆ
kk t
t
t
t
TT
m
u
uUU
a
aa
MvM
v
Mφ
φ
Process Parameter: Parametric EstimationProcess Parameter: Parametric Estimation
;ˆˆˆˆ )()(22)(11)( tmmttt aaa φφφμ ⋅+⋅⋅⋅+⋅+⋅=
[ ])()(2)(1)( ,,, tmttt φφφφ ⋅⋅⋅=v
Given an orthogonal base
)(2,1
)()(
2/)()( ˆ;ˆ t
mk
tt
mk,αttmk
kt σ
χσσμμ
α
⋅−
≤⋅≤−−−
−Residuals:
Parametric EstimationParametric Estimation
],,,[)(m
t tt1 ⋅⋅⋅=φv
0ˆ,...,0ˆ,0ˆand 21 ≈≈≠ maaaifExample:
,ˆˆ1
)(1 ∑
=
==k
i
it
kuaμ
1
)ˆ(ˆ 1
2)(
−
−=∑=
k
uk
iit μ
σ
Example: Prostate Isocenter DisplacementExample: Prostate Isocenter Displacement
-1.5
-1
-0.5
0
0.5
1
1.5
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
34
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
1r
k11ΦΩ++ΦΩ⋅≈
⋅ΦΩ= ∑ ∫=
∈
k
i
tt
n
iTt tt
xdxdkn
dtvxdvD
vL
v
v
Treatment Dose Construction & Estimation Treatment Dose Construction & Estimation
( ) dtxdvDn
iTt
ttvti
⋅ΦΩ= ∑ ∫=
∈1
)()()( ,,)( v
dtdExdxxAxTn
iTt E Rx
tvtEtxEtEi
n '),'(),,'(1
')()(),'(0)(
vvvvv
v ⋅Ω−⋅ΦΩ=∑∫ ∫ ∫=
∈ ∈
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⋅−⋅Φ⋅⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅= ∫
'
0
)(02
')'()(exp),'()(
x
tEwater
Ex
isoE
xtE dtx
llT
v
v
vv
vv ξρρμ
ρμ
ξ
))'(,( )( 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(2
]ˆ,[v
vvvv v
vv
xD
DxD rxpTrxrxp
Trp
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
-3
-2
-1
0
1
2
3
4
0 1 2 3 4 5 6 7 8 9 10
Repiratory Phase
Dos
e D
evia
tion%
),(),()(101 CT10CT1 Ω⋅++Ω⋅= xdwxdwxD v
Lvv
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
61
61.2
61.4
61.6
61.8
62
62.2
62.4
0 1 2 3 4 5 6 7 8 9 10
Treatment Fraction
EUD
(Gy)
PlannedDelivered
In Planning: TCP = 86% Tx Day: 85%, 84%, 83%, …………………………..…, 83%
Evaluation & DecisionEvaluation & Decision--makingmakingBrain Stem Cumulative Dose
36
36.5
37
37.5
38
38.5
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
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 10 20 30 40 50 60
Time (second)
it Ttu ∈;)(
PatientPatient--specific Target Marginspecific Target Margin(Parametric Approach)(Parametric Approach)
0
0.004
0.008
0.012
0.016
0.02
-0.1 0.1 0.3 0.5 0.7
Exhale --> Inhale
Prob
abili
ty
),( σμpdf
PatientPatient--specific Target Marginspecific Target Margin(Parametric Approach)(Parametric Approach)
Dose without motionDose with motion
σσ vv
v ⋅∂
∂⋅⋅=Δ 2
)(2
21
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
v
k
iik uurxvv
vvvvvv∫∑ ⋅⋅⋅−+= Φ+Φ=
3)ˆ,ˆ,(),()(),( )(
1
)(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
0
20
40
60
80
100
Dos
e &
Occ
upan
cy F
requ
ency
(%)
Target Density
Rectal Density
ConventionalInverse Planning(5 mm margin)
4D adaptiveInverse Planning
CTV1
CTV2
Mandible
L Parotid
Brain Stem
Cord
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
v
Uv
vvv ∫=
∈ΦΦ ⋅⋅=
1
)(),(),(ˆ
( )VOIsxDFMax xk ∈ΦΦ
vv |ˆ ),(
4D Adaptive Inverse Planning4D Adaptive Inverse Planning(Parametric Approach)(Parametric Approach)
)(1
ˆ ),()(),( )(11
ΦΦ +=
+⋅+
= ∑ vk
k
iik xvv dd
knD v
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
( )VOIsvDFMax vk ∈ΦΦ
|ˆ ),(
0
10
20
30
40
50
60
70
80
90
100
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
Dose (cGy)
Volu
me
(%)
0
10
20
30
40
50
60
70
80
90
100
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
Dose (cGy)
Volu
me
(%)
0
10
20
30
40
50
60
70
80
90
100
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
Dose (cGy)
Volu
me
(%)
0
10
20
30
40
50
60
70
80
90
100
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
Dose (cGy)
Volu
me
(%)
0
10
20
30
40
50
60
70
80
90
100
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
Dose (cGy)
Volu
me
(%)
0
10
20
30
40
50
60
70
80
90
100
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
Dose (cGy)
Volu
me
(%)
Day 1
Day 2
Day 3 Day 4
Day 5
Day 6
Online Adaptive
Plan
Rectal Wall
Prostate+SVOnlinePlan
O n lin e P la n
0
2 0
4 0
6 0
8 0
1 0 0
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 )
Volu
me
(%)
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
0
2 0
4 0
6 0
8 0
1 0 0
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 )
volu
me
(%)
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
vDi
ttvtk ,...,1,;
:,,~)(ˆ )()()(
v
⎭⎬⎫
⎩⎨⎧
∈∈ΦΩ
k
ttvt
TtVOIsvx
;:, )()(),(
v
( )VOIsvDFMax vk ∈ΦΦ
|ˆ ),(*Φ
( ) ?|)()(ˆ δ≥∈− VOIsvvDvDf pk
pD
