Binomial edge ideals of bipartite graphs

Binomial edge ideals are a noteworthy class of binomial ideals that can be associated with graphs, generalizing the ideals of 2-minors. For bipartite graphs we prove the converse of Hartshorne’s Connectedness Theorem, according to which if an ideal is Cohen–Macaulay, then its dual graph is connected. This allows us to classify Cohen–Macaulay binomial edge ideals of bipartite graphs, giving an explicit and recursive construction in graph-theoretical terms. This result represents a binomial analogue of the celebrated characterization of (monomial) edge ideals of bipartite graphs due to Herzog and Hibi (2005). Herzog J., Hibi T. Distributive lattices, bipartite graphs and Alexander duality J. Algebraic Combin., 22 (2005), pp. 289-30

Cohen-Macaulay binomial edge ideals and accessible graphs

The cut sets of a graph are special sets of vertices whose removal disconnect the graph. They are fundamental in the study of binomial edge ideals, since they encode their minimal primary decomposition. We introduce the class of accessible graphs as the graphs with unmixed binomial edge ideal and whose cut sets form an accessible set system. We prove that the graphs whose binomial edge ideal is Cohen-Macaulay are accessible and we conjecture that the converse holds. We settle the conjecture for large classes of graphs, including chordal and traceable graphs, providing a purely combinatorial description of Cohen-Macaulayness. The key idea in the proof is to show that both properties are equivalent to a further combinatorial condition, which we call strong unmixedness

A combinatorial characterization of S2S_2 binomial edge ideals

Several algebraic properties of a binomial edge ideal $J_G$ can be interpreted in terms of combinatorial properties of its associated graph $G$. In particular, the so-called cut-point sets of a graph $G$, special sets of vertices that disconnect $G$ in a minimal way, play an important role since they are in bijection with the minimal prime ideals of $J_G$. In this paper we establish the first graph-theoretical characterization of binomial edge ideals $J_G$ satisfying Serre's condition $(S_2)$ by proving that this is equivalent to having $G$ accessible, which means that $J_G$ is unmixed and the cut-point sets of $G$ form an accessible set system. The proof relies on the combinatorial structure of the Stanley-Reisner simplicial complex of a multigraded generic initial ideal of $J_G$, whose facets can be described in terms of cut-point sets. Another key step in the proof consists in proving the equivalence between accessibility and strong accessibility for the collection of cut sets of $G$ with $J_G$ unmixed. This result, interesting on its own, provides the first relevant class of set systems for which the previous two notions are equivalent

Cohen-Macaulay binomial edge ideals of small graphs

A combinatorial property that characterizes Cohen-Macaulay binomial edge ideals has long been elusive. A recent conjecture ties the Cohen-Macaulayness of a binomial edge ideal JG to special disconnecting sets of vertices of its underlying graph G, called cut sets. More precisely, the conjecture states that JG is Cohen-Macaulay if and only if JG is unmixed and the collection of the cut sets of G is an accessible set system. In this paper we prove the conjecture theoretically for all graphs with up to 12 vertices and develop an algorithm that allows to computationally check the conjecture for all graphs with up to 15 vertices and all blocks with whiskers where the block has at most 11 vertices. This significantly extends previous computational results

Powers of monomial ideals with characteristic-dependent Betti numbers

We explore the dependence of the Betti numbers of monomial ideals on the characteristic of the field. A first observation is that for a fixed prime p either the i-th Betti number of all high enough powers of a monomial ideal differs in characteristic 0 and in characteristic p or it is the same for all high enough powers. In our main results, we provide constructions and explicit examples of monomial ideals all of whose powers have some characteristic-dependent Betti numbers or whose asymptotic regularity depends on the field. We prove that, adding a monomial on new variables to a monomial ideal allows to spread the characteristic dependence to all powers. For any given prime p, this produces an edge ideal such that all its powers have some Betti numbers that are different over Q and over Zp. Moreover, we show that, for every r≥0 and i≥3 there is a monomial ideal I such that some coefficient in a degree ≥r of the Kodiyalam polynomials P3(I),…,Pi+r(I) depends on the characteristic. We also provide a summary of related results and speculate about the behavior of other combinatorially defined ideals

A "SHort course Accelerated RadiatiON therapy" (SHARON) During and Beyond the COVID-19 Pandemic

The current pandemic situation posed significant problems for radiotherapy (RT) services. In addition to the need to treat COVID-positive patients, it is important to protect health workers and healthy patients from the infection. Although some restrictions are being removed, it is not sure when the pandemic is actually going to be definitively over. Radiation oncologists (ROs) will be forced to face the pandemic for an unknown time interval (1). A recent guideline has been published on the possibility of adapting RT strategies in all settings (2). Particularly along the first months of pandemic spread, hypofractionated RT schedules adequately managing different clinical settings have been proposed to reduce the number of interactions and contacts in hospitals (for both patients–patients and patients–RT personnel), while delivering effective treatments (3–5). Only few were specifically dedicated to palliative RT or particularly oriented to relevant palliative presentations (e.g., bone metastases) (6). With the aim of decreasing hospital contacts, it has been proposed to omit, or delay, or modify the usual prescribed RT regimens (6), more often for palliative settings. However, in the field of palliative RT any omission and delay can dramatically worsen patients’ quality of life. In fact, the proposal to omit palliative radiotherapy during the COVID-19 pandemic has not been widely accepted, with some authors being worried by its clinical and ethical implications (7, 8). We would like to draw attention to a RT regimen tested in different settings. This scheme of SHort course Accelerated RadiatiON therapy: “SHARON” allows to complete a palliative RT course in four sessions and in only 2 days, using a double daily fractionation

Analisi Costo Minimizzazione delle preparazioni di Interferon Beta per il trattamento della Sclerosi Multipla

The multiple sclerosis (MS) is a neurologic disease that is characterized by a progressive demielinization of the white matter of the central nervous system. In the lasts decades, several therapies have been introduced after randomized, double-blind, placebo controlled trials. These trials supported the efficacy of Interferon-beta (INF- b) in reducing relapsing frequency and slowing the progressive disability, mainly in cases affected by relapsing- remitting MS course. In Italy four different preparations of INF-b are available for MS treatment having different INF-b types (i.e., INF-b1a e INF-b1b), different administration schemes, different INF-b doses and ways of administration. Recently, the biological activity of these preparations have been compared using the same assay system against the same INF-b standard. The aim of this study was to carry out a cost-minimization analysis, on the MS treatments in Italy comparing of the available preparations in terms of cost per microgram standardized by the level of biological activity. The economic evaluation has been conducted adopting the hospital perspective. Health resources have been valued considering euro currency during 2004. According to registered treatment protocol, the results showed that the micrograms per week of INF-b standardized by the level of biological activity ranged from 30mg of Avonex® to 132mg of Rebif44®. Under the same levels of biological activity, Rebif44® resulted the INF-b preparation with the lower cost per micrograms (1.95 euro), followed by Rebif®22 and Betaferon® that had a similar cost (2.90 e 2.97 respectively). Avonex® resulted the INF-b preparation with the highest cost per micrograms (6,37 euro), about three times higher than that of the preparation with the lowest cost

Personalized Automation of Treatment Planning for Linac-Based Stereotactic Body Radiotherapy of Spine Cancer

Purpose/Objective(s)Stereotactic ablative body radiotherapy (SBRT) for vertebral metastases is a challenging treatment process. Planning automation has recently reported the potential to improve plan quality and increase planning efficiency. We performed a dosimetric evaluation of the new Personalized engine implemented in Pinnacle3 for full planning automation of SBRT spine treatments in terms of plan quality, treatment efficiency, and delivery accuracy. Materials/MethodsThe Pinnacle3 treatment planning system was used to reoptimize six patients with spinal metastases, employing two separate automated engines. These two automated engines, the existing Autoplanning and the new Personalized, are both template-based algorithms that employ a wishlist to construct planning goals and an iterative technique to replicate the planning procedure performed by skilled planners. The boost tumor volume (BTV) was defined as the macroscopically visible lesion on RM examination, and the planning target volume (PTV) corresponds with the entire vertebra. Dose was prescribed according to simultaneous integrated boost strategy with BTV and PTV irradiated simultaneously over 3 fractions with a dose of 30 and 21 Gy, respectively. Dose-volume histogram (DVH) metrics and conformance indices were used to compare clinically accepted manual plans (MP) with automated plans developed using both Autoplanning (AP) and Personalized engines (Pers). All plans were evaluated for planning efficiency and dose delivery accuracy. ResultsFor similar spinal cord sparing, automated plans reported a significant improvement of target coverage and dose conformity. On average, Pers plans increased near-minimal dose D98% by 10.4% and 8.9% and target coverage D95% by 8.0% and by 4.6% for BTV and PTV, respectively. Automated plans provided significantly superior dose conformity and dose contrast by 37%-47% and by 4.6%-5.7% compared with manual plans. Overall planning times were dramatically reduced to about 15 and 23 min for Pers and AP plans, respectively. The average beam-on times were found to be within 3 min for all plans. Despite the increased complexity, all plans passed the 2%/2 mm gamma-analysis for dose verification. ConclusionAutomated planning for spine SBRT through the new Pinnacle3 Personalized engine provided an overall increase of plan quality in terms of dose conformity and a major increase in efficiency. In this complex anatomical site, Personalized strongly reduce the tradeoff between optimal accurate dosimetry and planning time

Pain Relief after Stereotactic Radiotherapy of Pancreatic Adenocarcinoma: An Updated Systematic Review

Severe pain is frequent in patients with locally advanced pancreatic ductal adenocarcinoma (PDCA). Stereotactic body radiotherapy (SBRT) provides high local control rates in these patients. The aim of this review was to systematically analyze the available evidence on pain relief in patients with PDCA. We updated our previous systematic review through a search on PubMed of papers published from 1 January 2018 to 30 June 2021. Studies with full available text, published in English, and reporting pain relief after SBRT on PDCA were included in this analysis. Statistical analysis was carried out using the MEDCALC statistical software. All tests were two-sided. The I-2 statistic was used to quantify statistical heterogeneity (high heterogeneity level: >50%). Nineteen papers were included in this updated literature review. None of them specifically aimed at assessing pain and/or quality of life. The rate of analgesics reduction or suspension ranged between 40.0 and 100.0% (median: 60.3%) in six studies. The pooled rate was 71.5% (95% CI, 61.6-80.0%), with high heterogeneity between studies (Q(2) test: p < 0.0001; I-2 = 83.8%). The rate of complete response of pain after SBRT ranged between 30.0 and 81.3% (median: 48.4%) in three studies. The pooled rate was 51.9% (95% CI, 39.3-64.3%), with high heterogeneity (Q(2) test: p < 0.008; I-2 = 79.1%). The rate of partial plus complete pain response ranged between 44.4 and 100% (median: 78.6%) in nine studies. The pooled rate was 78.3% (95% CI, 71.0-84.5%), with high heterogeneity (Q(2) test: p < 0.0001; I-2 = 79.4%). A linear regression with sensitivity analysis showed significantly improved overall pain response as the EQD2 alpha/beta:10 increases (p: 0.005). Eight papers did not report any side effect during and after SBRT. In three studies only transient acute effects were recorded. The results of the included studies showed high heterogeneity. However, SBRT of PDCA resulted reasonably effective in producing pain relief in these patients. Further studies are needed to assess the impact of SBRT in this setting based on Patient-Reported Outcomes