Parallel Mapper

The construction of Mapper has emerged in the last decade as a powerful and effective topological data analysis tool that approximates and generalizes other topological summaries, such as the Reeb graph, the contour tree, split, and joint trees. In this paper, we study the parallel analysis of the construction of Mapper. We give a provably correct parallel algorithm to execute Mapper on multiple processors and discuss the performance results that compare our approach to a reference sequential Mapper implementation. We report the performance experiments that demonstrate the efficiency of our method

Scale-free random branching tree in supercritical phase

We study the size and the lifetime distributions of scale-free random branching tree in which $k$ branches are generated from a node at each time step with probability $q_k\sim k^{-\gamma}$. In particular, we focus on finite-size trees in a supercritical phase, where the mean branching number $C=\sum_k k q_k$ is larger than 1. The tree-size distribution $p(s)$ exhibits a crossover behavior when $2 < \gamma < 3$; A characteristic tree size $s_c$ exists such that for $s \ll s_c$, $p(s)\sim s^{-\gamma/(\gamma-1)}$ and for $s \gg s_c$, $p(s)\sim s^{-3/2}\exp(-s/s_c)$, where $s_c$ scales as $\sim (C-1)^{-(\gamma-1)/(\gamma-2)}$. For $\gamma > 3$, it follows the conventional mean-field solution, $p(s)\sim s^{-3/2}\exp(-s/s_c)$ with $s_c\sim (C-1)^{-2}$. The lifetime distribution is also derived. It behaves as $\ell(t)\sim t^{-(\gamma-1)/(\gamma-2)}$ for $2 < \gamma < 3$, and $\sim t^{-2}$ for $\gamma > 3$ when branching step $t \ll t_c \sim (C-1)^{-1}$, and $\ell(t)\sim \exp(-t/t_c)$ for all $\gamma > 2$ when $t \gg t_c$. The analytic solutions are corroborated by numerical results.Comment: 6 pages, 6 figure

Reproducibility of the lung anatomy under Active Breathing Coordinator control: Dosimetric consequences for scanned proton treatments.

Purpose/Objective The treatment of moving targets with scanning proton beams is challenging. By controlling lung volumes, Active Breathing Control (ABC) assists breath-holding for motion mitigation. The delivery of proton treatment fractions often exceeds feasible breath-hold durations, requiring high breath-hold reproducibility. Therefore, we investigated dosimetric consequences of anatomical reproducibility uncertainties in the lung under ABC, evaluating robustness of scanned proton treatments during breath-hold. Material/Methods T1-weighted MRIs of five volunteers were acquired during ABC, simulating image acquisition during four subsequent breath-holds within one treatment fraction. Deformation vector fields obtained from these MRIs were used to deform 95% inspiration phase CTs of 3 randomly selected non-small-cell lung cancer patients (Figure 1). Per patient, an intensity-modulated proton plan was recalculated on the 3 deformed CTs, to assess the dosimetric influence of anatomical breath-hold inconsistencies. Results Dosimetric consequences were negligible for patient 1 and 2 (Figure 1). Patient 3 showed a decreased volume (95.2%) receiving 95% of the prescribed dose for one deformed CT. The volume receiving 105% of the prescribed dose increased from 0.0% to 9.9%. Furthermore, the heart volume receiving 5 Gy varied by 2.3%. Figure 2 shows dose volume histograms for all relevant structures in patient 3. Conclusion Based on the studied patients, our findings suggest that variations in breath-hold have limited effect on the dose distribution for most lung patients. However, for one patient, a significant decrease in target coverage was found for one of the deformed CTs. Therefore, further investigation of dosimetric consequences from intra-fractional breath-hold uncertainties in the lung under ABC is needed

On the topological classification of binary trees using the Horton-Strahler index

The Horton-Strahler (HS) index $r=\max{(i,j)}+\delta_{i,j}$ has been shown to be relevant to a number of physical (such at diffusion limited aggregation) geological (river networks), biological (pulmonary arteries, blood vessels, various species of trees) and computational (use of registers) applications. Here we revisit the enumeration problem of the HS index on the rooted, unlabeled, plane binary set of trees, and enumerate the same index on the ambilateral set of rooted, plane binary set of trees of $n$ leaves. The ambilateral set is a set of trees whose elements cannot be obtained from each other via an arbitrary number of reflections with respect to vertical axes passing through any of the nodes on the tree. For the unlabeled set we give an alternate derivation to the existing exact solution. Extending this technique for the ambilateral set, which is described by an infinite series of non-linear functional equations, we are able to give a double-exponentially converging approximant to the generating functions in a neighborhood of their convergence circle, and derive an explicit asymptotic form for the number of such trees.Comment: 14 pages, 7 embedded postscript figures, some minor changes and typos correcte