Constrained Decompositions of Integer Matrices and their - PowerPoint PPT Presentation

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

Aim of radiation therapy: Destroying the tumor(s) Preserving the organs located in the radiation field Céline Engelbeen Constrained Decompositions of Integer Matrices

Multileaf collimator (Saint-Luc, Belgium) Céline Engelbeen Constrained Decompositions of Integer Matrices

Leaves of the collimator Céline Engelbeen Constrained Decompositions of Integer Matrices

Planning in 3 steps Fixing the different radiation angles. 1 Determining the intensity function for each angle. 2 Modulating the radiation. 3 Céline Engelbeen Constrained Decompositions of Integer Matrices

Beam-On Time Problem (BOT): Given an intensity matrix I Decompose I as a combination I = α 1 S 1 + α 2 S 2 + · · · + α K S K where S i are C1 matrices α i ∈ N Aim: Minimize α 1 + α 2 + · · · + α K Céline Engelbeen Constrained Decompositions of Integer Matrices

  5 5 3 3 2 2 5 2   I =   5 3 3 2   2 5 5 3  1 1 1 1   1 1 1 1   1 1 0 0  1 1 1 1 0 0 1 0 0 0 1 0       = 2  + 1  + 2       1 1 1 1 1 1 1 0 1 0 0 0     1 1 1 1 0 1 1 1 0 1 1 0 α 1 + α 2 + α 3 = 2 + 1 + 2 = 5 Céline Engelbeen Constrained Decompositions of Integer Matrices

Question: How to optimally decompose  5 5 3 4  2 2 5 2   I =  ?   5 3 3 2  2 5 5 3 Céline Engelbeen Constrained Decompositions of Integer Matrices

Question: How to optimally decompose  5 5 3 4  2 2 5 2   I =  ?   5 3 3 2  2 5 5 3 Observation: rows are independent. Question: How to optimally decompose � � I 1 = 5 5 3 4 ? Céline Engelbeen Constrained Decompositions of Integer Matrices

Standard decomposition: ( 5 5 3 4 ) Céline Engelbeen Constrained Decompositions of Integer Matrices

Standard decomposition: ( 5 5 3 4 ) Céline Engelbeen Constrained Decompositions of Integer Matrices

Standard decomposition: ( 5 5 3 4 ) Céline Engelbeen Constrained Decompositions of Integer Matrices

Standard decomposition: ( 5 5 3 4 ) Céline Engelbeen Constrained Decompositions of Integer Matrices

Standard decomposition: ( 5 5 3 4 ) Céline Engelbeen Constrained Decompositions of Integer Matrices

Standard decomposition: ( 5 5 3 4 ) � � 1 1 0 0 Céline Engelbeen Constrained Decompositions of Integer Matrices

Standard decomposition: ( 5 5 3 4 ) � � 1 1 0 0 � � + 1 1 0 0 Céline Engelbeen Constrained Decompositions of Integer Matrices

Standard decomposition: ( 5 5 3 4 ) � � 1 1 0 0 � � + 1 1 0 0 � � + 1 1 1 1 Céline Engelbeen Constrained Decompositions of Integer Matrices

Standard decomposition: ( 5 5 3 4 ) � � 1 1 0 0 � � + 1 1 0 0 � � + 1 1 1 1 � � + 1 1 1 1 Céline Engelbeen Constrained Decompositions of Integer Matrices

Standard decomposition: ( 5 5 3 4 ) � � 1 1 0 0 � � + 1 1 0 0 � � + 1 1 1 1 � � + 1 1 1 1 � � + 1 1 1 1 Céline Engelbeen Constrained Decompositions of Integer Matrices

Standard decomposition: ( 5 5 3 4 ) � � 1 1 0 0 � � + 1 1 0 0 � � + 1 1 1 1 � � + 1 1 1 1 � � + 1 1 1 1 � � + 0 0 0 1 Céline Engelbeen Constrained Decompositions of Integer Matrices

Standard decomposition: ( 5 5 3 4 ) � � 1 1 0 0 � � + 1 1 0 0 � � + 1 1 1 1 � � + 1 1 1 1 � � + 1 1 1 1 � � + 0 0 0 1 � � 5 5 3 4 Céline Engelbeen Constrained Decompositions of Integer Matrices

Standard decomposition: 0 ( 5 5 3 4 ) 0 � � 1 1 0 0 � � + 1 1 0 0 � � + 1 1 1 1 � � + 1 1 1 1 � � + 1 1 1 1 � � + 0 0 0 1 � � 5 5 3 4 Céline Engelbeen Constrained Decompositions of Integer Matrices

Standard decomposition: 0 ( 5 5 3 4 ) 0 � � 1 1 0 0 � � + 1 1 0 0 � � + 1 1 1 1 � � + 1 1 1 1 � � + 1 1 1 1 � � + 0 0 0 1 � � ∆ = ( 5 0 − 2 1 − 4 ) 5 5 3 4 Céline Engelbeen Constrained Decompositions of Integer Matrices

� � ∆ = 5 0 − 2 1 − 4 Céline Engelbeen Constrained Decompositions of Integer Matrices

� � ∆ = 5 0 − 2 1 − 4 ✁ ✁ ✁ ✁ ✁ ✁ ∆ + = � � 5 0 0 1 0 Céline Engelbeen Constrained Decompositions of Integer Matrices

� � ∆ = 5 0 − 2 1 − 4 ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ∆ + = ∆ − = � � � � 5 0 0 1 0 0 0 2 0 4 Céline Engelbeen Constrained Decompositions of Integer Matrices

Two extra constraints Interleaf motion constraint: Kalinowski (2005), Baatar, Hamacher, Ehrgott, Woeginger (2005) ok bad Céline Engelbeen Constrained Decompositions of Integer Matrices

Two extra constraints Interleaf motion constraint: Kalinowski (2005), Baatar, Hamacher, Ehrgott, Woeginger (2005) ok bad Interleaf distance constraint For a constant c = 2: 2 1 3 1 ok bad Céline Engelbeen Constrained Decompositions of Integer Matrices

Two extra constraints Interleaf motion constraint: Kalinowski (2005), Baatar, Hamacher, Ehrgott, Woeginger (2005) ok bad Interleaf distance constraint For a constant c = 2: 2 1 3 1 ok bad Céline Engelbeen Constrained Decompositions of Integer Matrices

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: � 1 � 0 � 0 � � � 1 0 0 1 1 1 0 0 1 2 + + 1 1 0 0 0 1 1 0 0 1 0 0 Céline Engelbeen Constrained Decompositions of Integer Matrices

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 � 0 � � � 1 0 0 1 0 0 1 1 1 + + 1 0 0 0 1 1 0 0 1 1 1 0 � 0 � 0 0 1 + 1 1 1 1 1 1 + 2 � � δ + δ + 2 k > 1 k k = 1 k = 1 Céline Engelbeen Constrained Decompositions of Integer Matrices

Key observation When can we satisfy the interleaf distance constraint without increasing the BOT? Céline Engelbeen Constrained Decompositions of Integer Matrices

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

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

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 j + c � � δ + δ + m ′ n , ∀ m , m ′ , j ≤ mn n = 1 n = 1 j j + c � � δ − δ − m ′ n , ∀ m , m ′ , j ≤ mn n = 1 n = 1 Céline Engelbeen Constrained Decompositions of Integer Matrices

Idea for our model Without constraint: With constraint: Céline Engelbeen Constrained Decompositions of Integer Matrices

BOT-IDC min T N + 1 � ( δ + s.t. mn + w mn ) = T ∀ m ; n = 1 j j + c � � ( δ + ( δ + ∀ j , m , m ′ mn + w mn ) ≤ m ′ n + w m ′ n ) n = 1 n = 1 j j + c � � ( δ − ( δ − mn + w mn ) ≤ m ′ n + w m ′ n ) ∀ j , m , m ′ n = 1 n = 1 w mn ≥ 0 ∀ m , n ; w mn ∈ Z ∀ m , n . Céline Engelbeen Constrained Decompositions of Integer Matrices