Approximating the Maximum Overlap of Polygons under Translation

Let $P$ and $Q$ be two simple polygons in the plane of total complexity $n$, each of which can be decomposed into at most $k$ convex parts. We present an $(1-\varepsilon)$-approximation algorithm, for finding the translation of $Q$, which maximizes its area of overlap with $P$. Our algorithm runs in $O(c n)$ time, where $c$ is a constant that depends only on $k$ and $\varepsilon$. This suggest that for polygons that are "close" to being convex, the problem can be solved (approximately), in near linear time

Hyperpolarized 13C-pyruvate MRI detects real-time metabolic flux in prostate cancer metastases to bone and liver: a clinical feasibility study.

BackgroundHyperpolarized (HP) 13C-pyruvate MRI is a stable-isotope molecular imaging modality that provides real-time assessment of the rate of metabolism through glycolytic pathways in human prostate cancer. Heretofore this imaging modality has been successfully utilized in prostate cancer only in localized disease. This pilot clinical study investigated the feasibility and imaging performance of HP 13C-pyruvate MR metabolic imaging in prostate cancer patients with metastases to the bone and/or viscera.MethodsSix patients who had metastatic castration-resistant prostate cancer were recruited. Carbon-13 MR examination were conducted on a clinical 3T MRI following injection of 250 mM hyperpolarized 13C-pyruvate, where pyruvate-to-lactate conversion rate (kPL) was calculated. Paired metastatic tumor biopsy was performed with histopathological and RNA-seq analyses.ResultsWe observed a high rate of glycolytic metabolism in prostate cancer metastases, with a mean kPL value of 0.020 ± 0.006 (s-1) and 0.026 ± 0.000 (s-1) in bone (N = 4) and liver (N = 2) metastases, respectively. Overall, high kPL showed concordance with biopsy-confirmed high-grade prostate cancer including neuroendocrine differentiation in one case. Interval decrease of kPL from 0.026 at baseline to 0.015 (s-1) was observed in a liver metastasis 2 months after the initiation of taxane plus platinum chemotherapy. RNA-seq found higher levels of the lactate dehydrogenase isoform A (Ldha,15.7 ± 0.7) expression relative to the dominant isoform of pyruvate dehydrogenase (Pdha1, 12.8 ± 0.9).ConclusionsHP 13C-pyruvate MRI can detect real-time glycolytic metabolism within prostate cancer metastases, and can measure changes in quantitative kPL values following treatment response at early time points. This first feasibility study supports future clinical studies of HP 13C-pyruvate MRI in the setting of advanced prostate cancer

Elevated Tumor Lactate and Efflux in High-grade Prostate Cancer demonstrated by Hyperpolarized 13C Magnetic Resonance Spectroscopy of Prostate Tissue Slice Cultures.

Non-invasive assessment of the biological aggressiveness of prostate cancer (PCa) is needed for men with localized disease. Hyperpolarized (HP) 13C magnetic resonance (MR) spectroscopy is a powerful approach to image metabolism, specifically the conversion of HP [1-13C]pyruvate to [1-13C]lactate, catalyzed by lactate dehydrogenase (LDH). Significant increase in tumor lactate was measured in high-grade PCa relative to benign and low-grade cancer, suggesting that HP 13C MR could distinguish low-risk (Gleason score ≤3 + 4) from high-risk (Gleason score ≥4 + 3) PCa. To test this and the ability of HP 13C MR to detect these metabolic changes, we cultured prostate tissues in an MR-compatible bioreactor under continuous perfusion. 31P spectra demonstrated good viability and dynamic HP 13C-pyruvate MR demonstrated that high-grade PCa had significantly increased lactate efflux compared to low-grade PCa and benign prostate tissue. These metabolic differences are attributed to significantly increased LDHA expression and LDH activity, as well as significantly increased monocarboxylate transporter 4 (MCT4) expression in high- versus low- grade PCa. Moreover, lactate efflux, LDH activity, and MCT4 expression were not different between low-grade PCa and benign prostate tissues, indicating that these metabolic alterations are specific for high-grade disease. These distinctive metabolic alterations can be used to differentiate high-grade PCa from low-grade PCa and benign prostate tissues using clinically translatable HP [1-13C]pyruvate MR

Affine convex body semigroups

In this paper we present a new kind of semigroups called convex body semigroups which are generated by convex bodies of R^k. They generalize to arbitrary dimension the concept of proportionally modular numerical semigroup of [7]. Several properties of these semigroups are proven. Affine convex body semigroups obtained from circles and polygons of R^2 are characterized. The algorithms for computing minimal system of generators of these semigroups are given. We provide the implementation of some of them

Transmission Properties of the oscillating delta-function potential

We derive an exact expression for the transmission amplitude of a particle moving through a harmonically driven delta-function potential by using the method of continued-fractions within the framework of Floquet theory. We prove that the transmission through this potential as a function of the incident energy presents at most two real zeros, that its poles occur at energies $n\hbar\omega+\varepsilon^*$ ($0<Re(\varepsilon^*)<\hbar\omega$), and that the poles and zeros in the transmission amplitude come in pairs with the distance between the zeros and the poles (and their residue) decreasing with increasing energy of the incident particle. We also show the existence of non-resonant "bands" in the transmission amplitude as a function of the strength of the potential and the driving frequency.Comment: 21 pages, 12 figures, 1 tabl

Effect of Ce and Sb on Primary Graphite Growth in Cast Irons

It is well-known that if certain trace elements are present in cast iron melts the morphology of the graphite precipitates can be altered. In order to understand the effect of doping elements on primary growth of graphite, pure Fe–Sb alloys were prepared by induction melting. They were then placed in graphite crucibles and heated to a temperature above the Fe–C eutectic so that the charge became saturated in carbon and melted. To obtain Fe–Ce alloys, metallic Ce was added at the bottom of a graphite crucible and covered with iron, and then heated as for the Fe–Sb charge. In both cases, the melt was then cooled and held slightly above the eutectic temperature so that primary graphite crystals, which had nucleated on the crucible walls and then detached from it, could grow freely in the melt. The influence of the added elements on graphite growth was revealed by the change in the shape and distribution of the crystals compared to those obtained in similar experiments carried out with pure Fe. The experiments were made in air and vacuum so as to point out possible interactions between the elements present in the melt and oxygen