Spectral analysis on infinite Sierpinski fractafolds

A fractafold, a space that is locally modeled on a specified fractal, is the fractal equivalent of a manifold. For compact fractafolds based on the Sierpinski gasket, it was shown by the first author how to compute the discrete spectrum of the Laplacian in terms of the spectrum of a finite graph Laplacian. A similar problem was solved by the second author for the case of infinite blowups of a Sierpinski gasket, where spectrum is pure point of infinite multiplicity. Both works used the method of spectral decimations to obtain explicit description of the eigenvalues and eigenfunctions. In this paper we combine the ideas from these earlier works to obtain a description of the spectral resolution of the Laplacian for noncompact fractafolds. Our main abstract results enable us to obtain a completely explicit description of the spectral resolution of the fractafold Laplacian. For some specific examples we turn the spectral resolution into a "Plancherel formula". We also present such a formula for the graph Laplacian on the 3-regular tree, which appears to be a new result of independent interest. In the end we discuss periodic fractafolds and fractal fields

Laplace Operators on Fractals and Related Functional Equations

We give an overview over the application of functional equations, namely the classical Poincar\'e and renewal equations, to the study of the spectrum of Laplace operators on self-similar fractals. We compare the techniques used to those used in the euclidean situation. Furthermore, we use the obtained information on the spectral zeta function to define the Casimir energy of fractals. We give numerical values for this energy for the Sierpi\'nski gasket

Identification of a novel susceptibility locus at 13q34 and refinement of the 20p12.2 region as a multi-signal locus associated with bladder cancer risk in individuals of European ancestry

Candidate gene and genome-wide association studies (GWAS) have identified 15 independent genomic regions associated with bladder cancer risk. In search for additional susceptibility variants, we followed up on four promising single-nucleotide polymorphisms (SNPs) that had not achieved genome-wide significance in 6911 cases and 11 814 controls (rs6104690, rs4510656, rs5003154 and rs4907479, P < 1 × 10−6), using additional data from existing GWAS datasets and targeted genotyping for studies that did not have GWAS data. In a combined analysis, which included data on up to 15 058 cases and 286 270 controls, two SNPs achieved genome-wide statistical significance: rs6104690 in a gene desert at 20p12.2 (P = 2.19 × 10−11) and rs4907479 within the MCF2L gene at 13q34 (P = 3.3 × 10−10). Imputation and fine-mapping analyses were performed in these two regions for a subset of 5551 bladder cancer cases and 10 242 controls. Analyses at the 13q34 region suggest a single signal marked by rs4907479. In contrast, we detected two signals in the 20p12.2 region—the first signal is marked by rs6104690, and the second signal is marked by two moderately correlated SNPs (r2 = 0.53), rs6108803 and the previously reported rs62185668. The second 20p12.2 signal is more strongly associated with the risk of muscle-invasive (T2-T4 stage) compared with non-muscle-invasive (Ta, T1 stage) bladder cancer (case–case P ≤ 0.02 for both rs62185668 and rs6108803). Functional analyses are needed to explore the biological mechanisms underlying these novel genetic associations with risk for bladder cancer

Propensity-matched analysis of patient-reported outcomes for neoadjuvant chemotherapy prior to radical cystectomy

© 2019, Springer-Verlag GmbH Germany, part of Springer Nature. Purpose: To evaluate patient-reported outcomes (PROs) for bladder cancer patients undergoing neoadjuvant chemotherapy (NAC) prior to radical cystectomy (RC) using longitudinal data and propensity-matched scoring analyses. Methods: 155 patients with muscle-invasive bladder cancer scheduled for RC completed the European Organization for Research and Treatment of Cancer questionnaires, EORTC QLQ-C30, EORTC QLQ-BLM30, Fear of Recurrence Scale, Mental Health Inventory and Satisfaction with Life Scale within 4 weeks of surgery. A propensity-matched analysis was performed comparing pre-surgery PROs among 101 patients who completed NAC versus 54 patients who did not receive NAC. We also compared PROs pre- and post-chemotherapy for 16 patients who had data available for both time points. Results: In propensity-matched analysis, NAC-treated patients reported better emotional and sexual function, mental health, urinary function and fewer financial concerns compared to those that did not receive NAC. Longitudinal analysis showed increases in fatigue, nausea and appetite loss following chemotherapy. Conclusion: Propensity-matched analysis did not demonstrate a negative effect of NAC on PRO. Several positive associations of NAC were found in the propensity-matched analysis, possibly due to other confounding differences between the two groups or actual clinical benefit. Longitudinal analysis of a small number of patients found small to modest detrimental effects from NAC similar to toxicities previously reported. Our preliminary findings, along with known survival and toxicity data, should be considered in decision-making for NAC

Survival of patients with nonseminomatous germ cell cancer: a review of the IGCC classification by Cox regression and recursive partitioning

The International Germ Cell Consensus (IGCC) classification identifies good, intermediate and poor prognosis groups among patients with metastatic nonseminomatous germ cell tumours (NSGCT). It uses the risk factors primary site, presence of nonpulmonary visceral metastases and tumour markers alpha-fetoprotein (AFP), human chorionic gonadotrophin (HCG) and lactic dehydrogenase (LDH). The IGCC classification is easy to use and remember, but lacks flexibility. We aimed to examine the extent of any loss in discrimination within the IGCC classification in comparison with alternative modelling by formal weighing of the risk factors. We analysed survival of 3048 NSGCT patients with Cox regression and recursive partitioning for alternative classifications. Good, intermediate and poor prognosis groups were based on predicted 5-year survival. Classifications were further refined by subgrouping within the poor prognosis group. Performance was measured primarily by a bootstrap corrected c-statistic to indicate discriminative ability for future patients. The weights of the risk factors in the alternative classifications differed slightly from the implicit weights in the IGCC classification. Discriminative ability, however, did not increase clearly (IGCC classification, c=0.732; Cox classification, c=0.730; Recursive partitioning classification, c=0.709). Three subgroups could be identified within the poor prognosis groups, resulting in classifications with five prognostic groups and slightly better discriminative ability (c = 0.740). In conclusion, the IGCC classification in three prognostic groups is largely supported by Cox regression and recursive partitioning. Cox regression was the most promising tool to define a more refined classification

A phase I/II study of gemcitabine and fractionated cisplatin in an outpatient setting using a 21-day schedule in patients with advanced and metastatic bladder cancer

A randomised phase III trial of MVAC (methotrexate, vincristine, doxorubicin, cisplatin) vs gemcitabine and cisplatin (GC) (G 1000 mg m(-2) days 1, 8, and 15 plus C 70 mg m(-2) day 2, q 4 wks) indicated GC had similar efficacy and lower toxicity (JCO 2000). Significant haematologic toxicities in the GC arm occurred on day 15, necessitating dose adjustments in 37% of cycles. We conducted a phase I/II dose escalation trial using GC on a 21-day cycle, with G and C split between days 1 and 8. The objective of the study to define maximum-tolerated dose and dose-limiting toxicity (DLT), objective response rate, and overall survival. In all, 32 patients with locally advanced, relapsed, or metastatic disease received: dose level 1, G/C 1000/35; level 2, 1100/35; level 3, 1200/35; level 4, 1200/45 mg m(-2) (G and C given on days 1 and 8 every 3 wks). A total of 19 patients had glomerular filtration rate <60 ml min(-1) and 19 patients had metastatic disease. Dose-limiting toxicity was haematologic (grade 4 thrombocytopenia) at dose level 2. Of 151 cycles, at day 15, platelets were <100 in 61 cycles; neutrophils <0.5, platelets <50 in 26 cycles. Only seven cycles were deferred due to haematological toxicity; four for renal toxicity (chemotherapy instituted posthydration). Overall response rate was 65.5% on an intention-to-treat analysis (75% [21/28] for assessable patients), with four complete responses (12.5%) and 17 partial responses (53%). After the median follow-up of 17.2 months (range 13.1-32.4 months), 12 patients remain alive. The overall median survival was 16 months (range 10.1-26.6 months). G plus C every 3 weeks is active and well tolerated in an outpatient setting, even in patients receiving prior platinum-based regimens and with poor renal reserve

Phase II multicentre study of docetaxel plus cisplatin in patients with advanced urothelial cancer

A multicentre phase II trial was undertaken to evaluate the activity and toxicity of docetaxel plus cisplatin as first-line chemotherapy in patients with urothelial cancer. Thirty-eight patients with locally advanced or metastatic transitional-cell carcinoma of the bladder, renal pelvis or ureter received the combination of docetaxel 75 mg m−2 and cisplatin 75 mg m−2 on day 1 and repeated every 21 days, to a maximum of six cycles. The median delivered dose-intensity was 98% (range 79–102%) of the planned dose for both drugs. There were seven complete responses and 15 partial responses, for and overall response rate of 58% (95% CI, 41–74%). Responses were even seen in three patients with hepatic metastases. The median time to progression was 6.9 months, and the median overall survival was 10.4 months. Two patients who achieved CR status remain free of disease at 4 and 3 years respectively. Grade 3–4 granulocytopenia occurred in 27 patients, resulting in five episodes of febrile neutropenia. There was one toxic death in a patient with grade 4 granulocytopenia who developed acute abdomen. Grade 3–4 thrombocytopenia was rare (one patient). Other grade 3–4 toxicities observed were anaemia (three patients), vomiting (five patients), diarrhoea (four patients), peripheral neuropathy (two patients) and non-neutropenic infections (seven patients). Docetaxel plus cisplatin is an effective and well-tolerated regimen for the treatment of advanced urothelial cancer, and warrants further investigation