Minimizing Beam-On Time in Cancer Radiation Treatment Using Multileaf Collimators

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Minimizing Beam-On Time in Cancer Radiation
Treatment Using Multileaf Collimators

• Natashia Boland
• Horst W. Hamacher
• Frank Lenzen
• January 4, 2002

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GOALS
1. apply radiation to tumor (target volume)
sufficient to destroy it while maintaining
the functionality of the surrounding organs
(organs at risk)
2. Minimize amount of time patient spends
positioned and fixed on the treatment couch.
3. Minimize beam-on time (time in which radiation
is applied to patient)
3
Designing the treatment plan decisions to be
made

1. Location of tumor (target volume) and organs at
   risk
2. A discretization of the radiation beam head into
   bixels
3. A discretization of the tumor (target volume) and
   risk organs into voxels
4. Gantry stops
5. Amount of radiation released at each stop and in
   each bixel (intensity function)
6. How to achieve the intensity function using a
   multileaf collimator

4
bixel








5
Designing the treatment plan decisions to be
made

1. Location of tumor (target volume) and organs at
   risk
2. A discretization of the radiation beam head into
   bixels
3. A discretization of the tumor (target volume) and
   risk organs into voxels
4. Gantry stops
5. Amount of radiation released at each stop and in
   each bixel (intensity function)
6. How to achieve the intensity function using a
   multileaf collimator

6
The amount of radiation released at each stop and
in each bixel (the intensity function) can be
written as a system of linear equations
Px D
dosage vector Di is the radiation of each voxel i
accumulated as cumulative radiation from all
bixels j.
bixel-voxel unit radiation matrix Pij is amount
of radiation reaching voxel i if one unit of
radiation is released at bixel j.
xj is the amount of time radiation is sent off
at bixel j
Note this will later be written
as two-dimensional intensity matrix, I
Must satisfy contraints eg. lower bound - must
destroy cancer upper bound - maintain
functionality of
organs at risk Note in general, these
constraints are inconsistent and mathematical
programming Methods must be used to minimize
deviation From the bounds
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At this point we assume that 1-5 have been dealt
with. So all that remains is to decide on a
modulation of the uniform radiation.
For each stop of the gantry we have an intensity
function, I, where Iij is the amount of time
uniform radiation is released in bixel (i,j). For
example, if we have chosen a discretization of
the beam head into a 6x6 grid,
0 0 2 2 2 0
0 1 1 3 1 0
0 0 2 2 1 0
1 2 2 2 1 0
0 1 2 3 2 1
0 1 2 2 2 2
I
is a possible intensity matrix.
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This paper focus on using multileaf collimators
to achieve modulation. Here, each row of I has an
associated pair of leaves - a right leaf and a
left leaf. If I has n columns the left leaf may
be positioned in column 0,1,,n, and the right
leaf may be placed in columns 1,,n,n1, where
columns 0 and n1 are notational columns used to
represent the respective leafs fully retracted
position.
0 0 2 2 2 0
0 1 1 3 1 0
0 0 2 2 1 0
1 2 2 2 1 0
0 1 2 3 2 1
0 1 2 2 2 2
I
column n1
column 0
left leaf positions
right leaf positions
left leaf lt right leaf
9
This paper focus on using multileaf collimators
to achieve modulation. Here, each row of I has an
associated pair of leaves - a right leaf and a
left leaf. If I has n columns the left leaf may
be positioned in column 0,1,,n, and the right
leaf may be placed in columns 1,,n,n1, where
columns 0 and n1 are notational columns used to
represent the respective leafs fully retracted
position.
0 0 2 2 2 0
0 1 1 3 1 0
0 0 2 2 1 0
1 2 2 2 1 0
0 1 2 3 2 1
0 1 2 2 2 2
I
column n1
column 0
left leaf positions
right leaf positions
left leaf lt right leaf
10
This paper focus on using multileaf collimators
to achieve modulation. Here, each row of I has an
associated pair of leaves - a right leaf and a
left leaf. If I has n columns the left leaf may
be positioned in column 0,1,,n, and the right
leaf may be placed in columns 1,,n,n1, where
columns 0 and n1 are notational columns used to
represent the respective leafs fully retracted
position.
0 0 1 1 1 0
0 0 0 1 0 0
0 0 1 1 0 0
0 1 1 1 0 0
0 0 1 1 1 0
0 0 1 1 1 1
Shape matrix
S
column n1
column 0
left leaf positions
right leaf positions
left leaf lt right leaf
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K
I S ak Sk
k1
ak gt 0 is time the linear accelerator is opened
to release uniform radiation
Sk is shape matrix
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0 0 4 4 3 0
0 1 1 6 3 0
0 0 3 4 1 0
1 3 4 4 3 0
0 2 3 6 4 3
0 1 3 3 4 4
I
0 0 1 1 0 0
0 0 0 1 1 0
0 0 0 1 0 0
0 0 1 1 1 0
0 0 0 1 1 1
0 0 0 0 1 1
0 0 1 1 1 0
0 1 1 1 0 0
0 0 1 1 1 0
1 1 1 1 0 0
0 0 1 1 1 0
0 1 1 1 1 1
0 0 0 0 1 0
0 0 0 1 0 0
0 0 1 0 0 0
0 1 0 0 0 0
0 1 1 1 0 0
0 0 1 1 0 0
S1
S2
S3
I 3S1 1S2 2S3
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0 0 1 1 0 0
0 0 0 1 1 0
0 0 0 1 0 0
0 0 1 1 1 0
0 0 0 1 1 1
0 0 0 0 1 1
0 0 1 1 1 0
0 1 1 1 0 0
0 0 1 1 1 0
1 1 1 1 0 0
0 0 1 1 1 0
0 1 1 1 1 1
0 0 0 0 1 0
0 0 0 1 0 0
0 0 1 0 0 0
0 1 0 0 0 0
0 1 1 1 0 0
0 0 1 1 0 0
S1
S2
S3
0 L R 0
0 0 L R
0 0 L R 0
0 L R
0 0 L
0 0 0 L
0 L R
L R 0
0 L R
R
0 L R
L
0 0 0 L R
0 0 L R 0
0 L R 0 0
L R 0 0 0
L R 0
0 L R 0
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Multileaf Collimator (MLC) problem with minimal
beam-on time
min S at
subject to S at St I
at gt 0 where t is an
element of the index set of all possible shape
matrices
t
t
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Multileaf Collimator (MLC) problem with minimal
beam-on time
K
min S ak (K - 1)Tc
subject to S at St I
at gt 0
where t is an element of the index set of all
possible shape matrices
k 1
t
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Multileaf Collimator (MLC) problem with minimal
beam-on time
K
min S (ak c(Sk,Sk1))
subject to S at St I
at gt 0
where t is an element of the index set of all
possible shape matrices c(SK,SK1) 0
k 1
t
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D
1021
1131
1,1
1,0
1,2
1,3
1022
1132
2031
2131
2,1
2,0
2,2
2,3
2132
2032
3021
3031
3,3
3,0
3,1
3,2
3032
3022
4131
4021
4,3
4,1
4,0
4,2
4022
4132
D
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D
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D