Minimizing Beam-On Time in Cancer Radiation Treatment Using Multileaf Collimators Natashia Boland Horst W. Hamacher Frank Lenzen January 4, 2002

1. apply radiation to tumor (target volume) sufficient to destroy it while maintaining the functionality of the surrounding organs (organs at risk) GOALS 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)

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

bixel

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

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 bixel-voxel unit radiation matrix P ij is amount of radiation reaching voxel i if one unit of radiation is released at bixel j. x j is the amount of time radiation is sent off at bixel j 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 dosage vector D i is the radiation of each voxel i accumulated as cumulative radiation from all bixels j. Note: this will later be written as two-dimensional intensity matrix, I

I = For each stop of the gantry we have an intensity function, I, where I ij 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, 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. is a possible intensity matrix.

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,n+1, where columns 0 and n+1 are notational columns used to represent the respective leaf’s fully retracted position I = left leaf positions right leaf positions column 0 column n+1 left leaf < right leaf

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,n+1, where columns 0 and n+1 are notational columns used to represent the respective leaf’s fully retracted position I = left leaf positions right leaf positions column 0 column n+1 left leaf < right leaf

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,n+1, where columns 0 and n+1 are notational columns used to represent the respective leaf’s fully retracted position S = left leaf positions right leaf positions column 0 column n+1 left leaf < right leaf Shape matrix

I =  k S k K k=1  k > 0 is time the linear accelerator is opened to release uniform radiation S k is shape matrix

I = S 1 =S 2 =S 3 = I = 3S 1 + 1S 2 + 2S 3

S 1 =S 2 =S 3 = 0LR0 00LR 00LR0 0LR 00L 000L 0LR LR0 0LR R 0LR L 000LR 00LR0 0LR00 LR000 LR0 0LR0

Multileaf Collimator (MLC) problem with minimal beam-on time min  t subject to  t S t = I  t  where t is an element of the index set of all possible shape matrices t t

Multileaf Collimator (MLC) problem with minimal beam-on time min  k + (K - 1)T c subject to  t S t = I  t  where t is an element of the index set of all possible shape matrices k = 1 t K

Multileaf Collimator (MLC) problem with minimal beam-on time min  k + c(S k,S k+1 )) subject to  t S t = I  t  where t is an element of the index set of all possible shape matrices c(S K,S K+1 ) = 0 k = 1 t K

D D’ 1,11,01,21, ,1 3,33,03,1 4,34, ,02,22, , ,04,

D D’