An optimized approach for robust spot placement in proton pencil beam scanning.

Maintaining a sharp lateral dose falloff in pencil beam scanning (PBS) proton therapy is crucial for sparing organs at risk (OARs), especially when they are in close proximity to the target volume. The most common approach to improve lateral dose falloff is through the use of physical beam shaping devices, such as brass apertures or collimator based systems. A recently proposed approach focuses on proton beam spot placements, moving away from traditional grid-based placements to concentric-contours based schemes. This improves lateral dose falloff in two ways: (1) by better conforming all spots to the tumor boundary and (2) allowing for 'edge enhancement', where boundary spots deliver higher fluence than more central spots, thereby creating a steeper lateral dose falloff. However, these benefits come at the expense of maintaining uniformity of spot distribution inside the target volume. In this work we have developed a new optimized spot placement scheme that provides robust spot distributions inside the target. This approach achieves the boundary conformity of a concentric-contours based approach and uses a fast-iterative method to distribute the interior spots in a highly uniform fashion in an attempt to improve both the lateral dose falloff and uniformity. Furthermore, we quantified the impact of this new approach through direct comparison with grid, contour, and hybrid spot placements schemes, showing improvements for this new approach. The results were validated in homogeneous medium for two different target shapes having concave and convex geometry.

Discrete conformal methods for cortical brain flattening.

Locations and patterns of functional brain activity in humans are difficult to compare across subjects because of differences in cortical folding and functional foci are often buried within cortical sulci. Unfolding a cortical surface via flat mapping has become a key method for facilitating the recognition of new structural and functional relationships. Mathematical and other issues involved in flat mapping are the subject of this paper. It is mathematically impossible to flatten curved surfaces without metric and area distortion. Nevertheless, "metric" flattening has flourished based on a variety of computational methods that minimize distortion. However, it is mathematically possible to flatten without any angular distortion--a fact known for 150 years. Computational methods for this "conformal" flattening have only recently emerged. Conformal maps are particularly versatile and are backed by a uniquely rich mathematical theory. This paper presents a tutorial level introduction to the mathematics of conformal mapping and provides both conceptual and practical arguments for its use. Discrete conformal mapping computed via circle packing is a method that has provided the first practical realization of the Riemann Mapping Theorem (RMT). Maps can be displayed in three geometries, manipulated with Möbius transformations to zoom and focus on particular regions of interest, they respect canonical coordinates useful for intersubject registration and are locally Euclidean. The versatility and practical advantages of the circle packing approach are shown by producing conformal flat maps using MRI data of a human cerebral cortex, cerebellum and a specific region of interest (ROI).

Cortical cartography using the discrete conformal approach of circle packings.

Cortical flattening algorithms are becoming more widely used to assist in visualizing the convoluted cortical gray matter sheet of the brain. Metric-based approaches are the most common but suffer from high distortions. Conformal, or angle-based algorithms, are supported by a comprehensive mathematical theory. The conformal approach that uses circle packings is versatile in the manipulation and display of results. In addition, it offers some new and interesting metrics that may be useful in neuroscientific analysis and are not available through numerical partial differential equation conformal methods. In this paper, we begin with a brief description of cortical "flat" mapping, from data acquisition to map displays, including a brief review of past flat mapping approaches. We then describe the mathematics of conformal geometry and key elements of conformal mapping. We introduce the mechanics of circle packing and discuss its connections with conformal geometry. Using a triangulated surface representing a cortical hemisphere, we illustrate several manipulations available using circle packing methods and describe the associated "ensemble conformal features" (ECFs). We conclude by discussing current and potential uses of conformal methods in neuroscience and computational anatomy.