Radiation therapy planning for cancer treatment begins with computer images of several body tissues. A tumorous target is identified along with surrounding healthy tissues. The treatment goal is to get maximize the total radiation received by a target tumor t = 0 while avoiding damage to surrounding tissues t = 1,c, T by limiting the radiation allowed to fall on them. Radiation is provided from a large accelerator that can shoot beams from multiple angles j = 1,c, J around the patient’s body so as to spread the danger to healthy tissues while focusing on the tumor. The accelerator beam is relatively large, often approximately 10cm square. That is why Intensity Modulated Radiation Therapy (IMRT) adds precision to plans by treating each beam j as if made up of many beamlets k = 1,c, Kj with independently controllable intensities

(roughly exposure times) xj, k Ú 0.

It is important that both tumor and healthy tissue dose be spread fairly evenly across their respective volumes. This is modeled by dividing each tissue volume t into a larger number of mini-volumes called voxels i = 1,c, It. Then the impact of particular beamlets 1j, k2 on any voxel 1t, i2 can be estimated as aj, k, t, i per unit beamlet intensity. The total dose received at any voxel 1t, i2 can be assumed to be the sum of these contributions across all beamlets, that is, a J

j = 1 a KJ k = 1 aj, k, t, i xj, k. Then such total voxel doses within healthy tissues t = 1,c, T must be limited to a specified maximum safe doses bt.

Using the indicies and symbols defined above, formulate a linear program over decision variables xj, k to compute a treatment plan maximizing the average dose to target voxels while satisfying upper limits for all voxels of each surrounding healthy tissue. Be sure to explain and justify the meaning of the objective and all constraints.