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Unformatted text preview: otes the nonnegative part. We can interpret E as the sum of the squares of the nontarget excess doses. (a) Show that the treatment planning problem is convex. The optimization variable is b ∈ Rn ; the problem data are B max , A, T , Dtarget , and Dother . (b) Solve the problem instance with data given in the file treatment_planning_data.m. Here we have split the matrix A into Atarget, which contains the rows corresponding to the target voxels, and Aother, which contains the rows corresponding to other voxels. Give the optimal value. Plot the dose histogram for the target voxels, and also for the other voxels. Make a brief comment on what you see. Remark. The beam pattern matrix in this problem instance is randomly generated, but similar results would be obtained with realistic data. 17.3 Flux balance analysis in systems biology. Flux balance analysis is based on a very simple model of the reactions going on in a cell, keeping track only of the gross rate of consumption and production of various chemical species within the cell. Based on the known stoichiometry of the reactions, and known upper bounds on some of the reaction rates, we can compute bounds on the other reac...
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