Piecewise smooth systems near a co-dimension 2 discontinuity manifold: can one say what should happen?

We consider a piecewise smooth system in the neighborhood of a co-dimension 2 discontinuity manifold $\Sigma$. Within the class of Filippov solutions, if $\Sigma$ is attractive, one should expect solution trajectories to slide on $\Sigma$. It is well known, however, that the classical Filippov convexification methodology is ambiguous on $\Sigma$. The situation is further complicated by the possibility that, regardless of how sliding on $\Sigma$ is taking place, during sliding motion a trajectory encounters so-called generic first order exit points, where $\Sigma$ ceases to be attractive. In this work, we attempt to understand what behavior one should expect of a solution trajectory near $\Sigma$ when $\Sigma$ is attractive, what to expect when $\Sigma$ ceases to be attractive (at least, at generic exit points), and finally we also contrast and compare the behavior of some regularizations proposed in the literature. Through analysis and experiments we will confirm some known facts, and provide some important insight: (i) when $\Sigma$ is attractive, a solution trajectory indeed does remain near $\Sigma$, viz. sliding on $\Sigma$ is an appropriate idealization (of course, in general, one cannot predict which sliding vector field should be selected); (ii) when $\Sigma$ loses attractivity (at first order exit conditions), a typical solution trajectory leaves a neighborhood of $\Sigma$; (iii) there is no obvious way to regularize the system so that the regularized trajectory will remain near $\Sigma$ as long as $\Sigma$ is attractive, and so that it will be leaving (a neighborhood of) $\Sigma$ when $\Sigma$ looses attractivity. We reach the above conclusions by considering exclusively the given piecewise smooth system, without superimposing any assumption on what kind of dynamics near $\Sigma$ (or sliding motion on $\Sigma$) should have been taking place.Comment: 19 figure

On the Injectivity of Mean Value Mapping between Convex Quadrilaterals

We prove that Mean Value mapping between convex quadrilaterals is injective, affirmatively proving a conjecture stated in M. S. Floater and J. Kosinka, On the injectivity of Wachspress and mean value mappings between convex polygons, Adv. in Comp. Math. 32 (2010), 163-174

MANAGEMENT OF ADVANCED BREAST CANCER: HOW TO INTEGRATE SCIENTIFIC DATA AND CLINICAL JUDGMENT

Recent epidemiological data have shown a significant decline in breast cancer mortality over the past 15 years, as a result of screening programs, better education, and the introduction of more effective adjuvant treatments1. However, about 20-30% of the patients eventually relapse while approximately 5-7% of cases present with metastatic disease at diagnosis2. Metastatic breast cancer is still largely incurable: the median survival time is generally in the range of 2 to 4 years3. In the metastatic setting, treatment goals can be quite different depending on patient and tumor characteristics. There are patients for whom the main objective is symptom control to improve or maintain quality of life, cases with life-threatening disease for whom a rapid tumor shrinkage is required, asymptomatic patients with slowly growing disease for whom a prolonged progression-free survival (PFS) duration is the desirable target; finally, some patients can obtain an important survival prolongation and a few of them might be cured4. The selection of treatment depends on several factors, including patient characteristics, aggressiveness of the disease, response to previous therapies, time since last exposure, agents used in the past and cumulative doses. Availability and regulatory approval of various anticancer agents further diversify treatment patterns in different part of the world. A rapidly growing pool of effective treatment options for advanced breast cancer has increased response rates and outcome. First, many new cytotoxic drugs are in development or have recently been approved in this setting, such as ixabepilone, eribulin and nab-paclitaxel. For instance, in the phase III trial EMBRACE, eribulin mesylate improved overall survival (median 13.1 months, 95% CI 11.8-14.3), compared to treatment of physician’s choice (median 10.6 months, CI 9.3-12.5; HR 0.81 95% CI 0.66-0.99, p=0.041), in patient who had received two to five prior chemotherapy regimens, including an anthracycline and a taxane for advanced breast cancer.5 In clinical studies, 3-weekly nab-paclitaxel has been shown to increase both the safety and the efficacy of 3-weekly paclitaxel in patients with advanced breast cancer (median time to progression 23 vs 16.9 weeks, hazard ratio 0.75, p=0.006).6 Weekly nab-paclitaxel produced meaningful results even in taxanes pre-treated patients (ORR 14% and 16% in the 100 and 125 mg/sqm cohorts, respectively; median PFS of 3 and 3.5 months, respectively).7 At the same time, research efforts are directed to implement the pool of targeted therapies, in order to offer more individualized options to breast cancer patients. In fact, the molecular breast cancer subtype is a fundamental determinant of treatment choice both in early and advanced setting. Breast cancer consists of at least three different diseases: hormone-sensitive breast cancer, the human epidermal growth factor receptor (HER2)-positive subtype, and triple-negative disease. Each molecular subtype has distinct biological features and a distinct clinical course: hormone receptor–positive (HR+) disease is generally characterized by a more indolent course, with a long disease-free interval (DFI) and a tendency to relapse in the bone or soft tissues; amplification of the HER-2 gene confers a more aggressive clinical behavior to the HR+ subgroup, with a higher propensity for visceral relapses. Both triple-negative breast cancer and hormone receptor–negative (HR-)/HER-2+ breast cancer are aggressive subtypes, with early visceral or central nervous system metastases. Each molecular subtype requires distinct therapeutic approaches. In HR+ tumors, endocrine manipulation is the cornerstone of therapy. Treatment choice depends on many factors such as menopausal status and disease-free interval. For postmenopausal women many agents are available: non-steroidal and steroidal aromatase inhibitors (AI), tamoxifen and fulvestrant; however no definitive recommendation for the optimal cascade can be given. For premenopausal patients, the data on aromatase inhibitors or fulvestrant are more scanty8. In case of life-threatening and rapidly-growing disease, or in case of failure of various endocrine agents, chemotherapy has to be considered. Yet, recent studies have shown that HR+ positive tumors do also derive benefit from additional targeted agents: data from the BOLERO-2 trial showed an impressive improvement in progression free-survival with the addition of everolimus to exemestane vs exemestane alone as first- or second-line treatment for HR+ advanced breast cancer patients, after failure of a non-steroidal AI in the adjuvant or metastatic setting (median PFS 10.6 vs 4.1 months according to central assessment, HR 0.36;95% CI 0.27-0.47, p<0.001)9. Thus, overcoming endocrine resistance by combined targeting of redundancy pathways will be one of the key issues in the near future. In this context, even the association of trastuzumab or lapatinib to endocrine agents is an important option for HR+/HER2+ patients. Targeting HER2 in HR+ breast cancer has been explored as a means of improving endocrine responsiveness. The randomized phase II TAnDEM trial included 207 patients with known ER+/HER2+ metastatic breast cancer and reported a doubling of progression-free survival with the addition of trastuzumab over anastrozole alone (hazard ratio 0.63; 95% CI, 0.47 to 0.84; median PFS, 4.8 v 2.4 months; p =0016)10. Finally, results from a phase III trial of 1,286 patients with metastatic ER+ breast cancer who were randomized to receive either letrozole alone or letrozole combined with lapatinib have been published. In patients with known ER+/HER2+ tumors (n=219), the addition of lapatinib to letrozole significantly reduced the risk of progression as compared to letrozole alone: median PFS was 8.2 v 3.0 months, respectively (HR 0.71; 95% CI, 0.53 to 0.96; p=0.019)11. In HR-/HER2+ tumors, the incorporation of trastuzumab has substantially reversed the negative prognostic impact of HER-2 overexpression/amplification12. However, due to the approval of trastuzumab as standard adjuvant therapy for early HER2+ breast cancer and the emergency of resistance to this drug, the need of new anti-HER2 agents has emerged, as well as the need to clarify the role of continuing trastuzumab beyond progression, with different cytotoxic agents. Lapatinib, combined with capecitabine, has been approved for the treatment of HER2+ metastatic breast cancer patients, previously treated with trastuzumab. Many other anti-HER2 agents are being developed such as T-DM1, neratinib and pertuzumab. In T-DM1 trastuzumab is conjugated with an antimicrotubule drug maytansinoid. Activation of cytotoxicity of this conjugate requires internalization into the cell after binding to HER2. A single-arm, phase II trial (n = 112 MBC patients whose disease progressed on trastuzumab) showed at a follow-up of ≥12 months a median PFS of 4.6 months (95% CI, 3.9 to 8.6) and an overall response rate of 26%. Hypokalemia, thrombocytopenia, and fatigue were the most common observed adverse events. No dose-limiting cardiotoxicity was reported13. T-DM1 is undergoing further testing in the context of several other studies. An open-label, phase III randomized trial (EMILIA) is comparing single-agent T-DM1 with the combination of capecitabine and lapatinib in patients whose HER2-positive disease has progressed on trastuzumab. In another phase III trial MARIANNE, T-DM1 monotherapy is being compared to trastuzumab plus a taxane. Neratinib/HKI-272 is an oral, irreversible, small molecule inhibitor of EGFR/HER1, HER2, and HER4. In an open-label, phase II study, patients with advanced HER2-positive BC with and without prior trastuzumab treatment received neratinib. The 16-week PFS was 59% for patients with prior trastuzumab (n = 63) and 78% for those without (n = 64); median PFS were 22.3 and 39.6 weeks, respectively. The most frequent AEs were diarrhea, nausea, vomiting, and fatigue. Grade 3 or 4 diarrhea occurred in 30% of patients with prior trastuzumab therapy, leading to neratinib dose reduction in 29% of this cohort14. A phase III randomized study of paclitaxel with either neratinib or trastuzumab in MBC is ongoing, as is a randomized phase II study of neratinib alone versus the combination of capecitabine and lapatinib. Pertuzumab is a first-in-class recombinant, humanized monoclonal antibody that binds to domain II of the HER2 receptor, thus inhibiting HER2 heterodimerization with HER1, HER3, and HER4. Recent data from a randomized phase III trial showed that the combination of trastuzumab, pertuzumab and docetaxel as first-line treatment for HER2+ advanced breast cancer patients, significantly improves progression-free survival, with a gain of 6 months in median progression-free survival, as compared to the combination of trastuzumab and docetaxel (PFS 18.5 vs 12.4 months, HR 0.62; 95% CI, 0.51 to 0.75; P<0.001)15. These results contribute to increase the interest in dual HER2 blockade that derived from early breast cancer trials. In this context, the combination of trastuzumab and lapatinib in trastuzumab-pretreated patients resulted in a more prolonged PFS as compared to lapatinib alone (HR 0.73; 95%CI 0.57-0.93, p=0.008)16. Lastly, chemotherapy is the only available option so far for the triple-negative (TNBC) subtype, which is characterized by the absence of hormone receptors and HER-2 negativity. At this time, there are no targeted agents that are specifically approved for the treatment of this breast cancer subtype. Bevacizumab appears to prolong progression-free survival when added to chemotherapy for patients with TNBC (as it does for those with HR+/HER2- disease), but does not enhance survival17. Moreover, neoadjuvant trials provide conflicting results on the role of bevacizumab for TNBC18,19. Although there was great enthusiasm based on phase II data for the combination of carboplatin, gemcitabine and the PARP-inhibitor iniparib, the phase III results did not support the preliminary data20. A variety of other targeted therapies, including the PI3K inhibitors and a number of agents that inhibit DNA repair are under active investigation. Nowadays, more patients are likely to be diagnosed with oligo-metastatic disease, asymptomatic and in good performance status, due to the use of more sophisticated imaging techniques and the information derived from serum-markers dosage; therefore more selective therapeutic strategies that include a multimodality approach and local therapies are becoming more and more important. A substantial improvement in multimodality treatments, including, but not limited to, stereotactic radiosurgery, percutaneous radiofrequency ablation, and minimally invasive surgery, has increased the chance for disease control in selected patients with limited and indolent metastatic disease. In this context, surgery on primary tumor in case of oligometastatic disease has been suggested to improve survival, but further data are needed and a randomized trial addressing this issue is ongoing. Furthermore, an interesting field of research is the molecular characterization of metastatic disease. Biopsies of recurrent sites are not routinely performed and treatment decision for metastatic disease is mainly based on the receptor status of the primary tumor. However, discordance rates in HR and/or HER2 status between primary and recurrent tumors have been reported in the range of 10% to 35%21. Reasons for discordance may include: test artifacts, tumor heterogeneity, genetic drift during progression, selective pressure of adjuvant therapies. Nevertheless, recent reports suggest that the change in the receptor status during tumor progression may have also a prognostic impact22. It is therefore critical to incorporate all disease and patient information to assure the best treatment strategy for a given patient. The choice of the best treatment for metastatic disease has become even more difficult because the more efficacious agents have been progressively incorporated into the management of earlier stages. As a consequence, even if the number of patients who experience disease recurrence is gradually decreasing, treatment options for recurring patients are more and more influenced by prior exposure to adjuvant therapy. Unfortunately, most trials conducted in advanced breast cancer do not take into account all these factors that are necessary for an appropriate decision-making process. A deeper insight into tumor biology and mechanisms of resistance to established therapies will allow to develop new cytotoxic or targeted drugs or new combinations of available drugs for the treatment of metastatic disease. Key preclinical studies are needed, in order to guide the choice of which combination or single-agent deserve to be tested in early phase clinical trials. Moreover, well-designed trials that take into account the critical issues that frequently present in clinical routine are needed, in order to allow for a better translation of scientific results into daily practice

Nonnegative moment coordinates on finite element geometries

In this paper, we introduce new generalized barycentric coordinates (coined as {\em moment coordinates}) on nonconvex quadrilaterals and convex hexahedra with planar faces. This work draws on recent advances in constructing interpolants to describe the motion of the Filippov sliding vector field in nonsmooth dynamical systems, in which nonnegative solutions of signed matrices based on (partial) distances are studied. For a finite element with $n$ vertices (nodes) in $\mathbb{R}^2$, the constant and linear reproducing conditions are supplemented with additional linear moment equations to set up a linear system of equations of full rank $n$, whose solution results in the nonnegative shape functions. On a simple (convex or nonconvex) quadrilateral, moment coordinates using signed distances are identical to mean value coordinates. For signed weights that are based on the product of distances to edges that are incident to a vertex and their edge lengths, we recover Wachspress coordinates on a convex quadrilateral. Moment coordinates are also constructed on a convex hexahedra with planar faces. We present proofs in support of the construction and plots of the shape functions that affirm its properties

Exceptional and Durable Responses to TDM-1 After Trastuzumab Failure for Breast Cancer Skin Metastases: Potential Implications of an Immunological Sanctuary

Breast Cancer (BC) skin metastases represent a challenging clinical scenario. Although they usually arise when other distant metastases are already present, they may also represent a form of locoregional recurrence (LRR). Systemic therapy in this setting may have a role both in case a radical locoregional approach is unfeasible in order to achieve disease control, and as adjuvant strategy after radical removal of cutaneous lesions, in order to prevent or delay subsequent disease spread. Systemic therapy for HER2+ metastatic BC (MBC) currently relies on anti-HER2 targeted agents. In this context TDM1 is an option in trastuzumab-resistant patients.Here we present 2 cases of isolated skin metastases in patients with HER2+ BC progressing during or early after trastuzumab-based therapy, showing impressive responses to TDM1. We hypothesize that the unique properties of skin immune microenvironment may explain the failure of trastuzumab, which exerts its action also through immunological mechanisms, and the subsequent outlier responses to TDM1, that relies on a partially different mechanism of action

Nonnegative moment coordinates on finite element geometries

In this paper, we introduce new generalized barycentric coordinates (coined as moment coordinates) on convex and nonconvex quadrilaterals and convex hexahedra with planar faces. This work draws on recent advances in constructing interpolants to describe the motion of the Filippov sliding vector field in nonsmooth dynamical systems, in which nonnegative solutions of signed matrices based on (partial) distances are studied. For a finite element with $n$ vertices (nodes) in $\mathbb{R}^2$, the constant and linear reproducing conditions are supplemented with additional linear moment equations to set up a linear system of equations of full rank $n$, whose solution results in the nonnegative shape functions. On a simple (convex or nonconvex) quadrilateral, moment coordinates using signed distances are identical to mean value coordinates. For signed weights that are based on the product of distances to edges that are incident to a vertex and their edge lengths, we recover Wachspress coordinates on a convex quadrilateral. Moment coordinates are also constructed on a convex hexahedra with planar faces. We present proofs in support of the construction and plots of the shape functions that affirm its properties

Gender influence on professional satisfaction and gender issue perception among young oncologists. A survey of the Young Oncologists Working Group of the Italian Association of Medical Oncology (AIOM)

Background The professional gender gap is increasingly recognised in oncology. We explored gender issues perception and gender influence on professional satisfaction/gratification among young Italian oncologists. Methods Italian oncologists aged ≤40 years and members of the Italian Association of Medical Oncology were invited to participate in an online survey addressing workload/burnout, satisfaction in professional abilities and relations, relevant factors for professional gratification, and gender barriers. ‡ 2 test for general association or ‡ 2 test for trend was used to analyse the data. Results 201 young oncologists participated in the survey: 67% female, 71% aged 30-40 years, 41% still in training and 82% without children. Women and men were equally poorly satisfied by the relations with people occupying superior hierarchical positions. There was heterogeneity between women and men in current (p=0.011) and expected future (p=0.007) satisfaction in professional abilities: women were more satisfied by current empathy and relations with colleagues and were more confident in their future managerial and team leader skills. The most important elements for professional gratification indicated by all participants were, in general, work-life balance (36%) and intellectual stimulation/research (32%); specifically for women, work-life balance (48%) and intellectual stimulation/research (20%); and specifically for men, career (29%) and social prestige/recognition (26%). Heterogeneity within the same gender emerged. For example, the elements indicated by men as the most important were intellectual stimulation/research (39%) and work-life balance (21%) in general, versus social prestige/recognition (24%) and career (24%), respectively, specifically for men (p&lt;0.0001). More women versus men perceived gender issue as an actual problem (60% vs 38%, p=0.03); men underestimated gender barriers to women's career (p=0.011). Conclusions Satisfaction in professional abilities varied by gender. Work-life balance is important for both women and men. Stereotypes about gender issues may be present. Gender issue is an actual problem for young oncologists, mostly perceived by women

oncological outcome of fat grafting for breast reconstruction after cancer

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Identification of RNA polymerase III-transcribed Alu loci by computational screening of RNA-Seq data

Of the 3c1.3 million Alu elements in the human genome, only a tiny number are estimated to be active in transcription by RNA polymerase (Pol) III. Tracing the individual loci from which Alu transcripts originate is complicated by their highly repetitive nature. By exploiting RNA-Seq data sets and unique Alu DNA sequences, we devised a bioinformatic pipeline allowing us to identify Pol III-dependent transcripts of individual Alu elements. When applied to ENCODE transcriptomes of seven human cell lines, this search strategy identified 3c1300 Alu loci corresponding to detectable transcripts, with 3c120 of them expressed in at least three cell lines. In vitro transcription of selected Alus did not reflect their in vivo expression properties, and required the native 5'-flanking region in addition to internal promoter. We also identified a cluster of expressed AluYa5-derived transcription units, juxtaposed to snaR genes on chromosome 19, formed by a promoter-containing left monomer fused to an Alu-unrelated downstream moiety. Autonomous Pol III transcription was also revealed for Alus nested within Pol II-transcribed genes. The ability to investigate Alu transcriptomes at single-locus resolution will facilitate both the identification of novel biologically relevant Alu RNAs and the assessment of Alu expression alteration under pathological conditions

Exceptional and Durable Responses to TDM-1 After Trastuzumab Failure for Breast Cancer Skin Metastases: Potential Implications of an Immunological Sanctuary

Breast Cancer (BC) skin metastases represent a challenging clinical scenario. Although they usually arise when other distant metastases are already present, they may also represent a form of locoregional recurrence (LRR). Systemic therapy in this setting may have a role both in case a radical locoregional approach is unfeasible in order to achieve disease control, and as adjuvant strategy after radical removal of cutaneous lesions, in order to prevent or delay subsequent disease spread. Systemic therapy for HER2+ metastatic BC (MBC) currently relies on anti-HER2 targeted agents. In this context TDM1 is an option in trastuzumab-resistant patients.Here we present 2 cases of isolated skin metastases in patients with HER2+ BC progressing during or early after trastuzumab-based therapy, showing impressive responses to TDM1. We hypothesize that the unique properties of skin immune microenvironment may explain the failure of trastuzumab, which exerts its action also through immunological mechanisms, and the subsequent outlier responses to TDM1, that relies on a partially different mechanism of action