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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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 Fall '13
 F.Borrelli
 The Aeneid

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