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Constrained Decompositions of Integer Matrices and their Applications to Intensity Modulated Radiation Therapy Cline Engelbeen Advisor: Samuel Fiorini Universit Libre de Bruxelles Facult des Sciences Dpartement de Mathmatique 15

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SLIDE 1

Constrained Decompositions of Integer Matrices and their Applications to Intensity Modulated Radiation Therapy

Céline Engelbeen Advisor: Samuel Fiorini

Université Libre de Bruxelles Faculté des Sciences Département de Mathématique

15 May 2008

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 2

Aim of radiation therapy: Destroying the tumor(s) Preserving the organs located in the radiation field

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 3

Multileaf collimator (Saint-Luc, Belgium)

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 4

Leaves of the collimator

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 5

Planning in 3 steps

1

Fixing the different radiation angles.

2

Determining the intensity function for each angle.

3

Modulating the radiation.

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 6

Beam-On Time Problem (BOT):

Given an intensity matrix I Decompose I as a combination I = α1S1 + α2S2 + · · · + αKSK where Si are C1 matrices αi ∈ N

Aim:

Minimize α1 + α2 + · · · + αK

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 7

I =     5 5 3 3 2 2 5 2 5 3 3 2 2 5 5 3     = 2     1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1    +1     1 1 1 1 1 1 1 1 1 1 1    +2     1 1 1 1 1 1     α1 + α2 + α3 = 2 + 1 + 2 = 5

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 8

Question: How to optimally decompose I =     5 5 3 4 2 2 5 2 5 3 3 2 2 5 5 3    ?

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 9

Question: How to optimally decompose I =     5 5 3 4 2 2 5 2 5 3 3 2 2 5 5 3    ? Observation: rows are independent. Question: How to optimally decompose I1 =

  • 5

5 3 4

  • ?

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 10

Standard decomposition:

( ) 5 5 3 4

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 11

Standard decomposition:

( ) 5 5 3 4

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 12

Standard decomposition:

( ) 5 5 3 4

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 13

Standard decomposition:

( ) 5 5 3 4

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 14

Standard decomposition:

( ) 5 5 3 4

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 15

Standard decomposition:

( ) 5 5 3 4

  • 1

1

  • Céline Engelbeen

Constrained Decompositions of Integer Matrices

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SLIDE 16

Standard decomposition:

( ) 5 5 3 4

  • 1

1

  • +
  • 1

1

  • Céline Engelbeen

Constrained Decompositions of Integer Matrices

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SLIDE 17

Standard decomposition:

( ) 5 5 3 4

  • 1

1

  • +
  • 1

1

  • +
  • 1

1 1 1

  • Céline Engelbeen

Constrained Decompositions of Integer Matrices

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SLIDE 18

Standard decomposition:

( ) 5 5 3 4

  • 1

1

  • +
  • 1

1

  • +
  • 1

1 1 1

  • +
  • 1

1 1 1

  • Céline Engelbeen

Constrained Decompositions of Integer Matrices

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SLIDE 19

Standard decomposition:

( ) 5 5 3 4

  • 1

1

  • +
  • 1

1

  • +
  • 1

1 1 1

  • +
  • 1

1 1 1

  • +
  • 1

1 1 1

  • Céline Engelbeen

Constrained Decompositions of Integer Matrices

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SLIDE 20

Standard decomposition:

( ) 5 5 3 4

  • 1

1

  • +
  • 1

1

  • +
  • 1

1 1 1

  • +
  • 1

1 1 1

  • +
  • 1

1 1 1

  • +
  • 1
  • Céline Engelbeen

Constrained Decompositions of Integer Matrices

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SLIDE 21

Standard decomposition:

( ) 5 5 3 4

  • 1

1

  • +
  • 1

1

  • +
  • 1

1 1 1

  • +
  • 1

1 1 1

  • +
  • 1

1 1 1

  • +
  • 1
  • 5

5 3 4

  • Céline Engelbeen

Constrained Decompositions of Integer Matrices

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SLIDE 22

Standard decomposition:

( ) 5 5 3 4

  • 1

1

  • +
  • 1

1

  • +
  • 1

1 1 1

  • +
  • 1

1 1 1

  • +
  • 1

1 1 1

  • +
  • 1
  • 5

5 3 4

  • Céline Engelbeen

Constrained Decompositions of Integer Matrices

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SLIDE 23

Standard decomposition:

( ) 5 5 3 4

  • 1

1

  • +
  • 1

1

  • +
  • 1

1 1 1

  • +
  • 1

1 1 1

  • +
  • 1

1 1 1

  • +
  • 1
  • 5

5 3 4

  • ∆ = (5

−2 1 −4)

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 24

∆ =

  • 5

−2 1 −4

  • Céline Engelbeen

Constrained Decompositions of Integer Matrices

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SLIDE 25

∆ =

  • 5

−2 1 −4

  • ✁

✁ ✁ ✁ ✁ ✁

∆+ =

  • 5

1

  • Céline Engelbeen

Constrained Decompositions of Integer Matrices

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SLIDE 26

∆ =

  • 5

−2 1 −4

  • ✁

✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ❆

∆+ =

  • 5

1

  • ∆− =
  • 2

4

  • Céline Engelbeen

Constrained Decompositions of Integer Matrices

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SLIDE 27

Two extra constraints

Interleaf motion constraint: Kalinowski (2005), Baatar, Hamacher, Ehrgott, Woeginger (2005)

  • k

bad

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 28

Two extra constraints

Interleaf motion constraint: Kalinowski (2005), Baatar, Hamacher, Ehrgott, Woeginger (2005)

  • k

bad Interleaf distance constraint For a constant c = 2:

  • k

bad 2 1 3 1

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 29

Two extra constraints

Interleaf motion constraint: Kalinowski (2005), Baatar, Hamacher, Ehrgott, Woeginger (2005)

  • k

bad Interleaf distance constraint For a constant c = 2:

  • k

bad 2 1 3 1

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 30

Key observation

When can we satisfy the interleaf distance constraint without increasing the BOT? Assume that each row of I has the same BOT T. Exemple 1: ∆ = 2 1 −2 1 −2 2 2 −2 −1 −1

  • The standard decomposition:

2 1 1 1 1

  • +

1 1 1 1 1

  • +

1 1

  • Céline Engelbeen

Constrained Decompositions of Integer Matrices

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SLIDE 31

Key observation

When can we satisfy the interleaf distance constraint without increasing the BOT? Assume that each row of I has the same BOT T. Exemple 2: ∆ = 2 1 −2 1 −2 4 −1 −1 −1 −1

  • The standard decomposition:

1 1 1

  • +

1 1 1 1

  • +

1 1 1 1 1 1

  • +

1 1 1 1 1

  • 1
  • k=1

δ+

2k > 1+2

  • k=1

δ+

1k

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 32

Key observation

When can we satisfy the interleaf distance constraint without increasing the BOT?

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 33

Key observation

When can we satisfy the interleaf distance constraint without increasing the BOT? Answer: If each row of I has the same BOT, let T, then the following propositions are equivalent: There exists a decomposition of I of time T which respects the interleaf distance constraint

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 34

Key observation

When can we satisfy the interleaf distance constraint without increasing the BOT? Answer: If each row of I has the same BOT, let T, then the following propositions are equivalent: There exists a decomposition of I of time T which respects the interleaf distance constraint The standard decomposition algorithm gives a decomposition wich respects the interleaf distance constraint

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 35

Key observation

When can we satisfy the interleaf distance constraint without increasing the BOT? Answer: If each row of I has the same BOT, let T, then the following propositions are equivalent: There exists a decomposition of I of time T which respects the interleaf distance constraint The standard decomposition algorithm gives a decomposition wich respects the interleaf distance constraint

j

  • n=1

δ+

mn

≤

j+c

  • n=1

δ+

m′n, ∀m, m′, j j

  • n=1

δ−

mn

≤

j+c

  • n=1

δ−

m′n, ∀m, m′, j

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 36

Idea for our model

Without constraint: With constraint:

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 37

BOT-IDC

min T s.t.

N+1

  • n=1

(δ+

mn + wmn) = T

∀m;

j

  • n=1

(δ+

mn + wmn) ≤ j+c

  • n=1

(δ+

m′n + wm′n)

∀j, m, m′

j

  • n=1

(δ−

mn + wmn) ≤ j+c

  • n=1

(δ−

m′n + wm′n)

∀j, m, m′ wmn ≥ 0 ∀m, n; wmn ∈ Z ∀m, n.

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 38

BOT-IDC’

max πN+2 − π0 s.t. πN+2 − πm,N+1 ≤ −

N+1

  • n=1

δ+

mn

∀m; πm,N+1 − πN+2 ≤

N+1

  • n=1

δ+

mn

∀m; πm′,j+c − πmj ≤

j+c

  • n=1

δ+

m′n − j

  • n=1

δ+

mn

∀j, m, m′; πm′,j+c − πmj ≤

j+c

  • n=1

δ−

m′n − j

  • n=1

δ−

mn

∀j, m, m′; πm1 − π0 ≤ ∀m; πmj − πm,j−1 ≤ ∀m, j > 1; πmj ∈ Z ∀m, j.

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 39

I = 3 4 2 1 2 2

  • W =

1 2

  • Céline Engelbeen

Constrained Decompositions of Integer Matrices

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Theorem The minimum beam-on-time problem under the interleaf distance constraint can be solved in time O(MN + KM).

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 41

Theorem The minimum beam-on-time problem under the interleaf distance constraint can be solved in time O(MN + KM). Theorem The minimum beam-on-time problem under the interleaf motion constraint can be solved in time O(M log(M)N + KM).

Céline Engelbeen Constrained Decompositions of Integer Matrices

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SLIDE 42

Céline Engelbeen Constrained Decompositions of Integer Matrices