Optimisation of Irradiation Directions in IMRT Planning Rick Johnston Matthias Ehrgott Department of Engineering Science University of Auckland M. Ehrgott, R. Johnston Optimisation of Irradiation Directions in IMRT Planning, OR Spectrum 25(2): , 2003

What is Radiotherapy?

 Intensity modulation - improves treatment quality  Inverse planning problem - conflicting objectives to irradiate tumour without damage to healthy organs IMRT

Model Formulation  Discretisation of Body and Beam gantry Voxels Bixels

Angle Discretisation  Linearises the problem  A number of LPs to be solved  Replicates physical setup

MOMIP Model Data  L 1 = lower bound in tumour  U k = upper bound in organ k  R = number of directions to be used Variables and functions  Intensity vector x = (x 11,...,x HN )  Direction choice vector y = (y 1,...,y H )  Deviation vector T = (T 1,...,T K )  Dose distribution vectors D k = (D k1,...,D kMk )

min (T 1,...,T K ) min (T 1,...,T K ) D 1 = P 1 x  (L 1 - T 1 )1 D 1 = P 1 x  (L 1 - T 1 )1 D k = P k x  (U k + T k )1, k=2,...,K D k = P k x  (U k + T k )1, k=2,...,K x hi  My h, h=1,…,H, i=1,…,N x hi  My h, h=1,…,H, i=1,…,N y y H  R y h  {0,1} h=1,...,H y h  {0,1} h=1,...,H T, x  0 T, x  0 To study effect of direction optimisation consider weighted sum min  1 T 1 +  2 T  K T K Extension of multicriteria model by Hamacher/Küfer

Solution Methods Two-phase Methods 3. Set Covering 4. LP Relaxation Integrated Methods 1. Mixed Integer Formulation 2. Local Search Heuristics

Integrated Methods  CPLEX 7.0  If R increases problem becomes easier, objective value improves  For small R and small angle discretisation often no feasible solution found MIP SOLVER 1

 Optimal solution of MIP problem Isodose curve pictures obtained with prototype software developed at ITWM, Kaiserslautern

Integrated Methods  Alter each gantry position in turn to find better angles  Steepest descent with randomised starting angles  Solve LP for each selection of angles LOCAL SEARCH 2

Local Search Movie

Two-phase Methods  Intuitive  Considers all angles  Relatively quick Fully irradiate every voxel in the tumour Avoid damage to healthy organs Benefits: SET COVERING 3

min C 1 y C S y S Ay  1 Ay  1 y {0,1} y {0,1} a ij =1 if and only if beam j hits voxel i  Weighted angle method C j is sum of  k /U k over all organs at risk and voxels in beam j  Dose deposition method C j is sum of  k P k (i,j)/U k over all voxels and all organs at risk

 Cost coefficients

 Set Covering Solution  MIP Solution

4 Two-Phase Methods LP RELAXATION Optimal solution of LP relaxation beams used

Results  All methods were successful in generating good treatment plans in a reasonable timeframe (10 min)  Optimal beams were often counterintuitive  Angle optimisation is important if few beams to be used

 Solution with equidistant beams  Solution with optimised beams

Comparison Objective Problem 1 3 headsProblem 1 4 headsProblem 2 3 headsProblem 2 4 headsProblem 3 3 heads Set Covering LP relaxation Local Search Mixed Integer

Objective vs. Time Objective Time (s) Local search improvement Set Covering Local Search LP relaxation Mixed Integer