TITLE

Dose-volume requirements modeling for radiotherapy optimization

ABSTRACT

Radiation therapy is an important modality in cancer treatment. To find a good treatment plan, optimization models

and methods are typically used, while dose-volume requirements play an important role in plan’s quality evaluation.

We compare four different optimization approaches to incorporate the so-called dose-volume constraints into the

fluence map optimization problem for intensity modulated radiotherapy. Namely, we investigate (1) conventional

emph{Mixed Integer Programming} (MIP) approach, (2) emph{Linear Programming} (LP) approach to partial volume

constraints, (3) emph{Constrained Convex Moment} (CCM) approach, and (4) emph{Unconstrained Convex Moment

Penalty} (UCMP) approach. The performance of the respective optimization models is assessed using anonymized data

corresponding to eight previously treated prostate cancer patients. Several benchmarks are compared, with the goal

to evaluate the relative effectiveness of each method to quickly generate a good initial plan, with emphasis on

conformity to DVH-type constraints, suitable for further, possibly manual, improvement.

BIO

Dr. Zinchenko received his PhD from Cornell University on 2005 under supervision of Prof. James Renegar.

From 2005 to 2008 he held a PDF position at the Advanced Optimization Lab at McMaster University, working

with Prof. Tamas Terlaky and Prof. Antone Deza, and spent portion of his fellowship with radiation oncology group

at the Princess Margaret Hospital in Toronto. Currently, Yuriy is an Associate Professor of Mathematics & Stat at the

University of Calgary.

Dr. Zinchenko’s primary research interest lies in convex optimization, and particularly, the curvature of the central path for

interior-point methods, and applications. Yuriy’s work on optimal radiotherapy design was recognized by 2008 MITACS

Award for Best Novel Use of Mathematics in Technology Transfer, and in 2012-2015 he served as one of the PIs for

PIMS Collaborative Research Group grant on optimization.

Smooth Actions of Compact Lie Groups on $S^2$ are Smoothly Equivalent to Linear Actions:

**Abstract:**

Mathematicians have been interested in group actions on spheres since before the algebraic description of a group was defined. The rotational and reflective symmetries of the circle and of S^2 were naturally among the first to be considered. When we restrict attention to a compact topological group, there is a classic theorem of Kerekjártó to the effect that for S^2 , these are essentially the only actions:

Theorem 1 (Kerekjártó, [3]). Every continuous, effective action of a compact topological group G on S^2 is topologically conjugate to a linear action (to the standard action of a subgroup of O(3) on S^2 as a subset of R^3 ).

Thus in the topological category, in order to understand all effective actions of compact groups on S^2 , it is enough to understand the subgroups G ≤ O(3) and their actions via matrix multiplication on S^2 ⊆ R^3 (so called linear actions).

The goal of this paper is to extend this result to the smooth category. In other words, we show that every smooth, effective action of a compact Lie group, on S^2 is smoothly conjugate (i.e. conjugate through a diffeomorphism) to a linear action.

The main theorem of the paper is helpful for studying the topology of the space of actions of compact Lie groups on S^2 and we present a corollary as an example of this. We also explicitly determine all compact subgroups of O(3) up to conjugacy within O(3), and use this information to construct explicit G-CW decompositions for preferred representatives of each of these conjugacy classes. The G-CW decompositions are useful in other areas for example in the classification of G-equivariant vector bundles over S^2, and in determining whether such bundles have algebraic models.

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