arXiv subjects

Christian Richter

Publications and source records attributed to Christian Richter.

23 records · Page 2Linked to original sources

arXiv2016

Range prediction for tissue mixtures based on dual-energy CT

The use of dual-energy CT (DECT) potentially decreases range uncertainties in proton and ion therapy treatment planning via determination of the involved physical target quantities. For eventual clinical application, the correct treatment of tissue mixtures and heterogeneities is an essential feature, as they naturally occur within a patient's CT. Here, we present how existing methods for DECT-based ion-range prediction can be modified in order to incorporate proper mixing behavior on several structural levels. Our approach is based on the factorization of the stopping-power ratio into the relative electron density and the relative stopping number. The latter is confined for tissue between about 0.95 and 1.02 at a therapeutic beam energy of 200 MeV/u and depends on the I-value. We show that convenient mixing and averaging properties arise by relating the relative stopping number to the relative cross section obtained by DECT. From this, a maximum uncertainty of the stopping-power ratio prediction below 1% is suggested for arbitrary mixtures of human body tissues.

physics.med-ph↗

arXiv2016

Ball Convex Bodies in Minkowski Spaces

The notion of ball convexity, considered in finite dimensional real Banach spaces, is a natural and useful extension of usual convexity; one replaces intersections of half-spaces by suitable intersections of balls. A subset $S$ of a normed space is called ball convex if it coincides with its ball hull, which is obtained as intersection of all balls (of fixed radius) containing $S$. Ball convex sets are closely related to notions like ball polytopes, complete sets, bodies of constant width, and spindle convexity. We will study geometric properties of ball convex bodies in normed spaces, for example deriving separation theorems, characterizations of strictly convex norms, and an application to complete sets. Our main results refer to minimal representations of ball convex bodies in terms of their ball exposed faces, to representations of ball hulls of sets via unions of ball hulls of finite subsets, and to ball convexity of increasing unions of ball convex bodies.

math.MG↗

arXiv2014

Minsum Location Extended to Gauges and to Convex Sets

One of the oldest and richest problems from continuous location science is the famous Fermat-Torricelli problem, asking for the unique point in Euclidean space that has minimal distance sum to n given (non-collinear) points. Many natural and interesting generalizations of this problem were investigated, e.g., by extending it to non-Euclidean spaces and modifying the used distance functions, or by generalizing the configuration of participating geometric objects. In the present paper, we extend the Fermat-Torricelli problem in a two-fold way: more general than for normed spaces, the unit balls of our spaces are compact convex sets having the origin as interior point (but without symmetry condition), and the n given objects can be general convex sets (instead of points). We combine these two viewpoints, and the presented sequence of new theorems follows in a comparing sense that of theorems known for normed spaces. Some of these results holding for normed spaces carry over to our more general setting, and others not. In addition, we present analogous results for related questions, like, e.g., for Heron's problem. And finally we derive a collection of results holding particularly for the Euclidean norm.

math.MG↗

arXiv2013

Illuminating and covering convex bodies

Covering numbers of convex bodies based on homothetical copies and related illumination numbers are well-known in combinatorial geometry and, for example, related to Hadwiger's famous covering problem. Similar numbers can be defined by using proper translates instead of homothets, and even more related concepts make sense. On these lines we introduce some new covering and illumination numbers of convex bodies, present their properties and compare them with each other as well as with already known numbers. Finally, some suggestive examples illustrate that these new illumination numbers are interesting and non-trivial.

math.MG↗

arXiv2012

Deliverable navigation for multicriteria step and shoot IMRT treatment planning

We consider Pareto surface based multi-criteria optimization for step and shoot IMRT planning. By analyzing two navigation algorithms, we show both theoretically and in practice that the number of plans needed to form convex combinations of plans during navigation can be kept small (much less than the theoretical maximum number needed in general, which is equal to the number of objectives for on-surface Pareto navigation). Therefore a workable approach for directly deliverable navigation in this setting is to segment the underlying Pareto surface plans and then enforce the mild restriction that only a small number of these plans are active at any time during plan navigation, thus limiting the total number of segments used in the final plan.

physics.med-ph↗