The recent development of intensity modulated radiation therapy (IMRT) allows the dose distribution to be tailored to match the tumour's shape and position, avoiding damage to healthy tissue to a greater extent than previously possible. Traditional treatment plans assume that the target structure remains in a fixed location throughout treatment. However, many studies have shown that because of organ motion, inconsistencies in patient positioning over the weeks of treatment, etc, the tumour location is not stationary. We present a probabilistic model for the IMRT inverse problem and show that it is identical to using robust optimization techniques, under certain assumptions. For a sample prostate case, our computational results show that this method is computationally feasible and promising-compared to traditional methods, our model has the potential to find treatment plans that are more adept at sparing healthy tissue while maintaining the prescribed dose to the target.

By analogy with the conjecture of Hirsch, we conjecture that the order of the largest total curvature of the central path associated to a polytope is the number of inequalities defining the polytope. By analogy with a result of Dedieu, Malajovich and Shub, we conjecture that the average diameter of a bounded cell of an arrangement is less than the dimension. We prove continuous analogues of two results of Holt-Klee and Klee-Walkup: we construct a family of polytopes which attain the conjectured order of the largest total curvature, and we prove that the special case where the number of inequalities is twice the dimension is equivalent to the general case. We substantiate these conjectures in low dimensions and highlight additional links. Continuous Analogue of the Conjecture of HirschLet P be a full dimensional convex polyhedron defined by m inequalities in dimension n. The diameter δ(P ) is the smallest number such that any two vertices of the polyhedron P can be connected by a path with at most δ(P ) edges. The conjecture of Hirsch, formulated in 1957 and reported in [2], states that the diameter of a polyhedron defined by m inequalities in dimension n is not greater than m − n. The conjecture does not hold for unbounded polyhedra. A polytope is a bounded polyhedron. No polynomial bound is known for the diameter of a polytope. Conjecture 1.1. (Conjecture of Hirsch for polytopes)The diameter of a polytope defined by m inequalities in dimension n is not greater than m − n.Intuitively, the total curvature [15] is a measure of how far off a certain curve is from being a straight line. Let ψ : [α, β] → R n be a C 2 ((α − ε, β + ε)) map for some ε > 0 with a non-zero derivative in [α, β]. Denote its arc length by l(t) = t α ψ (τ ) dτ , its parametrization by the arc lengthand its curvature at the point t by κ(t) =ψ arc (t). The total curvature is defined asThe requirementψ = 0 insures that any given segment of the curve is traversed only once and allows to define a curvature at any point on the curve.We present one useful proposition. Roughly speaking, it states that two similar curves might not differ greatly in their total curvatures either. This fact is used in Section 3 in proving the analogue of the d-step conjecture for the total curvature of the central path.Proposition 1

Summary. We consider a family of linear optimization problems over the ndimensional Klee-Minty cube and show that the central path may visit all of its vertices in the same order as simplex methods do. This is achieved by carefully adding an exponential number of redundant constraints that forces the central path to take at least 2 n −2 sharp turns. This fact suggests that any feasible path-following interior-point method will take at least O(2 n ) iterations to solve this problem, while in practice typically only a few iterations, e.g., 50, suffices to obtain a high quality solution. Thus, the construction potentially exhibits the worst-case iterationcomplexity known to date which almost matches the theoretical iteration-complexity bound for this type of methods. In addition, this construction gives a counterexample to a conjecture that the total central path curvature is O(n).

The dose-volume histogram (DVH) is a clinically relevant criterion to evaluate the quality of a treatment plan. It is hence desirable to incorporate DVH constraints into treatment plan optimization for intensity modulated radiation therapy. Yet, the direct inclusion of the DVH constraints into a treatment plan optimization model typically leads to great computational difficulties due to the non-convex nature of these constraints. To overcome this critical limitation, we propose a new convex-moment-based optimization approach. Our main idea is to replace the non-convex DVH constraints by a set of convex moment constraints. In turn, the proposed approach is able to generate a Pareto-optimal plan whose DVHs are close to, or if possible even outperform, the desired DVHs. In particular, our experiment on a prostate cancer patient case demonstrates the effectiveness of this approach by employing two and three moment formulations to approximate the desired DVHs.

Radiation therapy is an important modality in treating various cancers. Various treatment planning and delivery technologies have emerged to support intensity modulated radiation therapy (IMRT), creating significant opportunities to advance this type of treatment. However, one of the fundamental questions in treatment planning and optimization, 'can we produce better treatment plans relying on the existing delivery technology?' still remains unanswered, in large part due to the underlying computational complexity of the problem, which, in turn, often stems from the optimization model being non-convex. We investigate the possibility of including the dose prescription, specified by the dose-volume histogram (DVH), within the convex optimization framework for inverse radiotherapy treatment planning. Specifically, we study the quality of approximating a given DVH with a superset of generalized equivalent uniform dose (gEUD)-based constraints, the so-called generalized moment constraints (GMCs). As a bi-product, we establish an analytic relationship between a DVH and a sequence of gEUD values. The newly proposed approach is promising as demonstrated by the computational study where the rectum DVH is considered. Unlike the precise partial-volume constraints formulation, which is commonly based on the mixed-integer model and necessitates the use of expensive computing resources to be solved to global optimality, our convex optimization approach is expected to be feasible for implementation on a conventional treatment planning station.