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.

We propose a practical algorithm for the calculation of the relative entropy of entanglement (REE), defined as the minimum relative entropy between a state and the set of states with positive partial transpose. Our algorithm is based on a practical semi-definite cutting plane approach. In low dimensions the implementation of the algorithm in MATLAB provides an estimation for the REE with an absolute error smaller than 10 −3 .

This is a proof of principle study on an algorithm for optimizing external beam radiotherapy in terms of both photon beamlet energy and fluence. This simultaneous beamlet energy and fluence optimization is denoted modulated photon radiotherapy (XMRT). XMRT is compared with single-energy intensity modulated radiotherapy (IMRT) for five clinically relevant test geometries to determine whether treating beamlet energy as a decision variable improves the dose distributions. All test geometries were modelled in a cylindrical water phantom. XMRT optimized the fluence for 6 and 18 MV beamlets while IMRT optimized with only 6 MV and only 18 MV. CERR (computational environment for radiotherapy research) was used to calculate the dose deposition matrices and the resulting dose for XMRT and IMRT solutions. Solutions were compared via their dose volume histograms and dose metrics, such as the mean, maximum, and minimum doses for each structure. The homogeneity index (HI) and conformity number (CN) were calculated to assess the quality of the target dose coverage. Complexity of the resulting fluence maps was minimized using the sum of positive gradients technique. The results showed XMRT's ability to improve healthy-organ dose reduction while yielding comparable coverage of the target relative to IMRT for all geometries. All three energy-optimization approaches yielded similar HI and CNs for all geometries, as well as a similar degree of fluence map complexity. The dose reduction provided by XMRT was demonstrated by the relative decrease in the dose metrics for the majority of the organs at risk (OARs) in all geometries. Largest reductions ranged between 5% to 10% in the mean dose to OARs for two of the geometries when compared with both single-energy IMRT schemes. XMRT has shown potential dosimetric benefits through improved OAR sparing by allowing beam energy to act as a degree of freedom in the EBRT optimization process.

The curvature of a polytope, defined as the largest possible total curvature of the associated central path, can be regarded as the continuous analogue of its diameter. We prove the analogue of the result of Klee and Walkup. Namely, we show that if the order of the curvature is less than the dimension d for all polytope defined by 2d inequalities and for all d, then the order of the curvature is less that the number of inequalities for all polytopes.