1. Physical optimization criteria Target objective function N t = number of voxels in target D i = dose to voxel i D presc = prescribed dose to target D min = minimum dose to voxel i D max = maximum dose to voxel i c t,min = penalty associated with underdosage c t,max = penalty associated with overdosage Heaviside function* The target sub-function includes the target prescription dose and dose- uniformity limits

2. Physical optimization criteria OAR objective function N O = number of voxels in OAR D max = maximum dose to voxel i D dv = maximum dose to voxel i, iєN dv c O,max = penalty weight for overdosage c O,dv = penalty weight for violation of dose-volume constraint N dv = number of voxels in OAR whose dose must be below the DVH constraint The OAR sub-functions include maximum dose and DVH-limits

3. Radiobiology Linear-quadratic (LQ) model The cell survival curve describes the relationship between the radiation dose and the proportion of cells that survives The surviving fraction of target cells SF(d), after a single radiation dose d can be fitted to experimental data using an exponential function with the parameters  and . LQ model used to compare the biological effect of different treatment fractionation schedules Radiation dose (Gy) dd 0.01 0.1 1 0.001  d2d2 Surviving fracions of cells 12164

4. Radiobiology gEUD gEUD is based on the concept of a generalized mean dose, and is a means to reduce a complex 3D dose distribution to a single, biologically representative dose value The a parameter is tissue specific and describes the volume effect of the tissue under consideration D = total dose a = tissue specific volume parameter N = number of voxels in tissue D i = dose in voxel i - a < 0: minimum dose (tumour) - a ≈ 1: mean dose (parallel tissue) - a → ∞: maximum dose (serial tissue) gEUD is not a sigmoid function and does not predict a response, however, it is a convex function…

5. Biological Optimization Criteria Objective function example (iii) The objective function includes combinations of the Logistic function and the gEUD function for both target and OARs Target sub-function includes the gEUD prescription dose and a relative importance factor OAR sub-functions include the gEUD limit dose and relative importance factors