Image Guided Adaptive
Radiotherapy Model
Di Yan, D.Sc.
Radiation Oncology
William Beaumont Hospital
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
4D Adaptive
Planning
Reference plan
Treatment Dose
Delivery Process
Image (patient anatomy) &
Machine Output Feedback
Treatment Evaluation & Plan
Modification Decision-Making
Daily & Cumulative Dose
Construction & Estimation
IGART Model: Treatment Dose Construction
Daily & Cumulative dose delivered to a organ subvolume v in organs
of interest, VOIs, during the n fractions of treatment delivery Ti
D (v ) =
n
∑∫
i =1
t ∈T i
v
d ( x t ( v ) , Ω ( t ) , Φ ( t ) ),
∀ v ∈ VOIs
v
xt ( v ) −
Subvolume displacement at the time t (Process Variable)
Ω(t ) −
Patient global matter distribution represented by a CT image
obtained at the time t (Process Variable)
Φ (t ) −
Machine output at the time t (Control Parameter)
IGART Model: Adaptive Management
Optimal mapping (control law), determined before any treatment
delivery day k < n, from the space of patient anatomical variation
v
(process variables), { x t , Ω ( t )}, to the space of therapy machine
output (control parameter), Φ ( t ) ,
Φ * (t )
v
= G ( x t ( v ) , Ω ( t ) | v ∈ VOIs ),
n
t ∈ U Ti
i=k
which optimizes the predefined objective function,
v
⎛
F ⎜ ∫ d ( x t ( v ), Ω ( t ) , Φ ( t ) ) | ∀ v ∈ VOIs ; i = 1,... , k ⎞⎟
⎠
⎝ t∈Ti
Lecture Outline
1. Description of patient anatomical variation process
v
{ x t , Ω ( t )}
2. Treatment dose construction & estimation
{ Dˆ t }
3. Treatment evaluation & adaptive planning optimization
{ Φ ∗ (t) }
Patient Anatomical Variation Process
Organ displacement during the treatment delivery can be
completely described using a Random Process,
n
⎧v
⎫
v
v
⎨ xt ( v ) = xr ( v ) + u ( t , v ) t ∈ T = U Ti ; v ∈ VOI ⎬
i =1
⎩
⎭
v
Where subvolume displacement, u ( t , v ) , has a probability density
v
function (pdf) characterized by the mean, μ ( t , v ), (the systematic
v
displacement) & the standard deviation, σ(t, v) , (the random
displacement).
Patient Anatomical Variation Process
1. Stationary Process (i.e. daily setup induced variation)
if ∀ v ∈ VOI ,
v
v
v
u ( t , v ) ~ pdf ( μ ( v ) , σ ( v ))
has the ‘time-invariant’ (constant) mean & standard deviation
2. Non-stationary Process (i.e dose response induced variation)
if ∃ v ∈ VOI ,
v
v
v
u ( t , v ) ~ pdf ( μ ( t , v ) , σ ( t , v ))
has the ‘time-variant’ mean or standard deviation
Process Parameter: Parametric Estimation
Least Square Estimation Using {u( t1 ) , ⋅ ⋅⋅, u( tk ) }
v
Given an orthogonal base φ
(t)
= [ φ 1 ( t ) , φ 2 ( t ) , ⋅ ⋅⋅ , φ m
(t)
∑ (u
k
μˆ( t ) = aˆ1 ⋅ φ1 ( t ) + aˆ2 ⋅ φ2 ( t ) + ⋅ ⋅ ⋅ + aˆm ⋅ φm ( t ) ;
σˆ ( t ) =
v
⎡ aˆ1 ⎤
⎡φ ( t1 ) ⎤
⎡ u( t1 ) ⎤
⎢ aˆ ⎥
⎢ 2 ⎥ = (Φ T Φ )−1 Φ T U , Φ = ⎢ M ⎥ ; U = ⎢ M ⎥
⎢v ⎥
⎥
⎢
⎢ M ⎥
⎢φ ( t k ) ⎥
⎢ u( t ) ⎥
⎢ˆ ⎥
⎣ k ⎦
⎦
⎣
a
m
⎣ ⎦
]
i =1
ˆ ( ti )
( ti ) − μ
k −m
)
2
Parametric Estimation
Residuals:
μˆ ( t ) − μ ( t ) ≤ tα / 2 ,k − m ⋅
σ (t )
k
;
σ (t ) ≤
k −m
χ
2
1−α , k − m
⋅ σˆ ( t )
Example:
v
φ
(t)
= [ 1 , t , ⋅ ⋅⋅ , t m ] and if aˆ1 ≠ 0, aˆ2 ≈ 0 ,..., aˆm ≈ 0
k
k
u ( ti )
μˆ = aˆ1 = ∑
,
i =1 k
σˆ =
2
ˆ
(
u
−
)
μ
(
)
t
∑ i
i =1
k −1
Example: Prostate Isocenter Displacement
output
1st week
2nd Week
3rd Week
4th Week
5th Week
1.5
1
0.5
0
-0.5
-1
1
2
3
4
7
10
13
16
21
24
27
30
33
-1.5
2
3
ˆ
ˆ
ˆ
ˆ
ˆ
μ ( t ) = a1 + a2 ⋅ t + a3 ⋅ t + a4 ⋅ t
36
39
42
45
Treatment (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)
D (v ) =
n
∑∫
i =1
(
t∈Ti
v
d ( x t ( v ), Ω t , Φ r )⋅ dt
n
v
v
≈ ⋅ d ( x t1 , Ω CT1 , Φ r ) + L + d ( x tk , Ω CTk , Φ r )
k
)
Treatment Dose Construction & Estimation
D (v ) =
n
∑∫
v
d ( x t ( v ) , Ω ( t ) , Φ ( t ) )⋅ dt
∫∫
v
v v
v
0 v
TE ( x ' , Ω( t ) , Φ E ( x ' , t )) ⋅ AE ( x '− xt ( v ), Ω( t )) ⋅ dx ' dE dt
n
i =1
n
= ∑∫
i =1
t ∈T i
v
t∈Ti E x '∈R
v
x'
⎛
v⎞
⎛ liso ⎞
v
μ
0 v
⎟⎟ ⋅ Φ E ( x ' , t ) ⋅ exp⎜ − (
TE = ( v ) E ⋅ ⎜⎜
) E ⋅ ∫ ρ t (ξ )dξ ⎟
⎜ ρ water
⎟
ρ t ( x ' ) ⎝ l xv ' ⎠
0
⎝
⎠
v v
AE = A( E , ρ t ⋅ l ( x ' − x t ( v )))
μ
2
Effective Density?
Treatment (4D) Dose Estimation
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
Dˆ k (v ) = ∫v
x∈R
v
v ˆ
v
v vˆ vˆ
Dp ( xr ( v ) + u , Ω r , Φ r ) ⋅ pdf (u , μ , σ ) ⋅ du
3
⎛ vT
v
≈ D p ( xr ) + ⎜⎜ μˆ ⋅ ∇ D p [ xv r , x r
⎝
2
∂
Dp ( xr
1 vˆ T
vˆ
⋅σ ⋅
+ μ] +
v2
∂x
2
v
+ μˆ )
vˆ ⎞⎟
⋅σ ⎟
⎠
Treatment (4D) Dose Estimation
Pelvic & Abdominal Region
¤ The planning CT image can most likely be used as the single
representation of the global matter distribution (if there is no
large gas filling)
™ H&N Region
¤ The planning CT can be applied if patient does not loss weight
significantly
™ Chest 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
™
Example: Patient Respiratory Motion
v
v
v
D( x) = w1 ⋅ d ( x , ΩCT1 ) + L+ w10 ⋅ d ( x , ΩCT10 )
v
v
Δ = D( x , ΩCTi ) − D( x ) ,
4
Dose Deviation%
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
-1
-2
-3
RepiratoryPhase
v
v ˆ
ˆ
Δ = D ( x ) − D ( x , Ω CT1 + ⋅⋅⋅ + CT10 ) =
10
0.4%
-0.3%
-0.19%
Adaptive Management
™
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
Evaluation & Decision-making
™
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 & Decision-making
CT V Cumulative Dose
62.4
62.2
Planned
Delivered
EUD (Gy)
62
61.8
61.6
61.4
61.2
61
0
1
2
3
4
5
6
7
8
T reatment Fraction
In Planning: TCP = 86%
Tx Day: 85%, 84%, 83%, …………………………..…, 83%
9
10
Evaluation & Decision-making
Brain Stem Cumulative Dose
38.5
38
37.5
Planned
Delivered
37
36.5
36
0
1
2
3
4
5
6
7
8
9
Treatment Fraction
In Planning: NTCP = 0.15%
Tx Day: 0.21%, 0.18%, 0.22%, 0.22%, 1%, ……………… ., 1.2%
10
Adaptive Management: Patient-specific Target Margin
™ Image
sampling during the treatment course
the individual systematic and random
errors from the image registration (parametric
approach) or ROI occupancy distribution
(nonparametric approach)
™ Estimate
™ 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
Patient-specific Target Margin
(Parametric Approach)
0.7
0.02
0.6
P r o b a b ilit y
0.5
0.4
0.3
0.2
0.012
0.008
0.004
0.1
0
-0.1
0.016
0
10
20
30
40
Time (second)
u(t ) ; t ∈Ti
50
60
0
-0.1
0.1
0.3
0.5
Exhale --> Inhale
pdf ( μ , σ )
0.7
Patient-specific Target Margin
(Parametric Approach)
2
D ( xref ) v
∂
1 vT
⋅σ
Δ = ⋅σ ⋅
v
2
2
∂x
Respiratory Motion: 2.0 cm
Dose without motion
Dose with motion
Adaptive Management: 4D Inverse Planning
™ 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
4D Adaptive Inverse Planning
(Parametric Approach)
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,
(
Max F Dˆ k ( v ,
Φ
Φ ) |v
∈ VOIs
)
k
v
v
v
v vˆ vˆ
ˆ
Dk (v, Φ) = ∑di (v) + (n − k ) ⋅ ∫ 3 d ( xr ( v ) + u, Φ) ⋅ pdf (u, μ , σ ) ⋅du
i =1
R
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
Dose & Occupancy Frequency(%)
4D Inverse Planning
for Prostate Cancer
100
Rectal Density
80
60
40
20
0
31
Target Density
29
27
25
23
21
19
Anterior-Posterior Position (cm)
17
15
Conventional
Inverse Planning
(5 mm margin)
4D adaptive
Inverse Planning
CTV1
CTV2
Mandible
Brain Stem
Cord
R Parotid
L Parotid
4D Adaptive Inverse Planning
(Nonparametric Approach)
Offline:
(
v
v
ˆ
Max F Dk ( x , Φ ) | x ∈ VOIs
Φ
v
ˆ
Dk ( x , Φ ) = ∫v
k
UVOIi
x∈
)
v
v v
d ( x , Φ ) ⋅ w( x ) ⋅dx
i =1
Baum C, Alber M, Birkner, Nusslin F (2006) Robust treatment
planning for intensity modulated radiotherapy of prostate cancer
based on coverage probability. Radiotherapy & Oncology, 78:27-35
4D Adaptive Inverse Planning
(Parametric Approach)
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
(
Max F Dˆ k ( v ,
Φ
Dˆ k ( v , Φ ) =
Φ ) | v ∈ VOIs
)
k
n
⋅ ( ∑ d i ( v ) + d k +1 ( xv ( v ),
k + 1 i =1
Φ ))
90
80
80
70
70
60
60
100
90
Volume (%)
90
Volume (%)
Day 2
Day 4
100
50
Day 5
100
50
90
40
40
30
30
80
80
70
70
Volume (%)
20
Volume (%)
20
60
10
10
50
0
1000
2000
3000
4000
5000
6000
7000
8000
0
9000
60
50
40
0
0
40
1000
2000
3000
4000
5000
6000
7000
8000
9000
Dose (cGy)
Dose(cGy)
30
30
20
20
10
10
0
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
Dose (cGy)
0
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
Dose (cGy)
Day 6
Day 1
100
100
90
90
Prostate+SV
80
80
70
Online
Plan
70
Rectal Wall
60
Volume (%)
Volume (%)
Day 3
100
50
Online
Adaptive
Plan
60
50
40
40
30
30
20
20
10
10
0
0
0
1000
2000
3000
4000
5000
Dose (cGy)
6000
7000
8000
9000
0
1000
2000
3000
4000
5000
Dose (cGy)
6000
7000
8000
9000
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
O n lin e P la n
100
Day_1
Day_2
80
Day_3
Volume (%)
O n lin e A d a p t iv e P la n
100
volume (%)
80
60
Day_4
D ay_1
Day_5
D ay_2
40
Day_6
D ay_3
D ay_4
60
20
D ay_5
D ay_6
40
0
1000
20
2000
3000
4000
5000
D o s e (c G y )
0
1000
2000
3000
4000
5000
d o s e (c G y )
6000
7000
8000
6000
7000
8000
IGART Model: Summary
(
)
MaxF Dˆ k ( v, Φ) | v ∈VOIs
Φ
Φ*
Treatment Dose Delivery
Patient Variation Process
v
⎧ xt ( v ), Ω(t ) , Φ(t ) :⎫
⎨
⎬
⎩ v ∈VOIs ; t ∈ Tk ⎭
Dp
(
)
f Dˆ k (v) − Dp (v) | v ∈VOIs ≥ δ ?
v
⎧d (xt (v), Ω(t) , Φ(t)) :
⎫
ˆ
Dk (v) ~ ⎨
⎬
⎩v ∈VOIs; t ∈Ti , i = 1,..., k ⎭