Block-and-hole graphs: Constructibility and (3,0)(3,0)-sparsity

We show that minimally 3-rigid block-and-hole graphs, with one block or one hole, are characterised as those which are constructible from $K_3$ by vertex splitting, and also, as those having associated looped face graphs which are $(3,0)$-tight. This latter property can be verified in polynomial time by a form of pebble game algorithm. We also indicate connections to the rigidity properties of polyhedral surfaces known as origami and to graph rigidity in $\ell_p^3$ for $p\not=2$.Comment: 17 page

The generic rigidity of triangulated spheres with blocks and holes

A simple graph G = (V,E) is 3-rigid if its generic bar-joint frameworks in R^3 are infinitesimally rigid. Block and hole graphs are derived from triangulated spheres by the removal of edges and the addition of minimally rigid subgraphs, known as blocks, in some of the resulting holes. Combinatorial characterisations of minimal 3-rigidity are obtained for these graphs in the case of a single block and finitely many holes or a single hole and finitely many blocks. These results confirm a conjecture of Whiteley from 1988 and special cases of a stronger conjecture of Finbow-Singh and Whiteley from 2013

The rigidity of a partially triangulated torus

A simple graph is 3-rigid if its generic embeddings in R^3 are infinitesimally rigid. Necessary and sufficient conditions are obtained for the minimal 3-rigidity of a simple graph obtained from a triangulated torus by the deletion of edges interior to an embedded triangulated disc

Braced triangulations and rigidity

We consider the problem of finding an inductive construction, based on vertex splitting, of triangulated spheres with a fixed number of additional edges braces). We show that for any positive integer $b$ there is such an inductive construction of triangulations with $b$ braces, having finitely many base graphs. In particular we establish a bound for the maximum size of a base graph with $b$ braces that is linear in $b$. In the case that $b=1$ or $2$ we determine the list of base graphs explicitly. Using these results we show that doubly braced triangulations are generically) minimally rigid in two distinct geometric contexts arising from a hypercylinder in $\mathbb{R}^4$ and a class of mixed norms on $\mathbb{R}^3$

Risk algorithm using serial biomarker measurements doubles the number of screen-detected cancers compared with a single-threshold rule in the United Kingdom collaborative trial of ovarian cancer screening

PURPOSE: Cancer screening strategies have commonly adopted single-biomarker thresholds to identify abnormality. We investigated the impact of serial biomarker change interpreted through a risk algorithm on cancer detection rates. PATIENTS AND METHODS: In the United Kingdom Collaborative Trial of Ovarian Cancer Screening, 46,237 women, age 50 years or older underwent incidence screening by using the multimodal strategy (MMS) in which annual serum cancer antigen 125 (CA-125) was interpreted with the risk of ovarian cancer algorithm (ROCA). Women were triaged by the ROCA: normal risk, returned to annual screening; intermediate risk, repeat CA-125; and elevated risk, repeat CA-125 and transvaginal ultrasound. Women with persistently increased risk were clinically evaluated. All participants were followed through national cancer and/or death registries. Performance characteristics of a single-threshold rule and the ROCA were compared by using receiver operating characteristic curves. RESULTS: After 296,911 women-years of annual incidence screening, 640 women underwent surgery. Of those, 133 had primary invasive epithelial ovarian or tubal cancers (iEOCs). In all, 22 interval iEOCs occurred within 1 year of screening, of which one was detected by ROCA but was managed conservatively after clinical assessment. The sensitivity and specificity of MMS for detection of iEOCs were 85.8% (95% CI, 79.3% to 90.9%) and 99.8% (95% CI, 99.8% to 99.8%), respectively, with 4.8 surgeries per iEOC. ROCA alone detected 87.1% (135 of 155) of the iEOCs. Using fixed CA-125 cutoffs at the last annual screen of more than 35, more than 30, and more than 22 U/mL would have identified 41.3% (64 of 155), 48.4% (75 of 155), and 66.5% (103 of 155), respectively. The area under the curve for ROCA (0.915) was significantly (P = .0027) higher than that for a single-threshold rule (0.869). CONCLUSION: Screening by using ROCA doubled the number of screen-detected iEOCs compared with a fixed cutoff. In the context of cancer screening, reliance on predefined single-threshold rules may result in biomarkers of value being discarded

The cost-effectiveness of screening for ovarian cancer: results from the UK Collaborative Trial of Ovarian Cancer Screening (UKCTOCS)

Background: To assess the within trial cost-effectiveness of an NHS ovarian cancer screening (OCS) programme using data from UKCTOCS and extrapolate results based on average life expectancy. Methods: Within trial economic evaluation of no screening (C) versus either (1) an annual OCS programme using transvaginal ultrasound (USS) or (2) an annual ovarian cancer multimodal screening programme with serum CA125 interpreted using a risk algorithm (ROCA) and transvaginal ultrasound as a second line test (MMS), plus comparison of lifetime extrapolation of the no screening arm and the MMS programme using both a predictive and a Markov model. Results: Using a CA125-ROCA cost of £20, the within trial results show USS to be strictly dominated by MMS, with the MMS versus C comparison returning an Incremental Cost-Effectiveness ratio (ICER) of £91,452 per life year gained (LYG). If the CA125-ROCA unit cost is reduced to £15 the ICER becomes £77,818 per LYG. Predictive extrapolation over the expected lifetime of the UKCTOCS women returns an ICER of £30,033 per LYG, while Markov modelling produces an ICER of £46,922 per QALY. Conclusions: Analysis suggests that, after accounting for the lead-time required to establish full mortality benefits, a national OCS programme based on the MMS strategy quickly approaches the current NICE thresholds for cost-effectiveness when extrapolated out to lifetime as compared to the within trial ICER estimates. Whether MMS could be recommended on economic grounds would depend on the confirmation and size of the mortality benefit at the end of an ongoing follow-up of the UKCTOCS cohort

Ovarian cancer screening and mortality in the UK Collaborative Trial of Ovarian Cancer Screening (UKCTOCS): a randomised controlled trial

Background Ovarian cancer has a poor prognosis, with just 40% of patients surviving 5 years. We designed this trial to establish the eff ect of early detection by screening on ovarian cancer mortality. Methods In this randomised controlled trial, we recruited postmenopausal women aged 50–74 years from 13 centres in National Health Service Trusts in England, Wales, and Northern Ireland. Exclusion criteria were previous bilateral oophorectomy or ovarian malignancy, increased risk of familial ovarian cancer, and active non-ovarian malignancy. The trial management system confirmed eligibility and randomly allocated participants in blocks of 32 using computergenerated random numbers to annual multimodal screening (MMS) with serum CA125 interpreted with use of the risk of ovarian cancer algorithm, annual transvaginal ultrasound screening (USS), or no screening, in a 1:1:2 ratio. The primary outcome was death due to ovarian cancer by Dec 31, 2014, comparing MMS and USS separately with no screening, ascertained by an outcomes committee masked to randomisation group. All analyses were by modified intention to screen, excluding the small number of women we discovered after randomisation to have a bilateral oophorectomy, have ovarian cancer, or had exited the registry before recruitment. Investigators and participants were aware of screening type. This trial is registered with ClinicalTrials.gov, number NCT00058032. Findings Between June 1, 2001, and Oct 21, 2005, we randomly allocated 202 638 women: 50 640 (25·0%) to MMS, 50 639 (25·0%) to USS, and 101 359 (50·0%) to no screening. 202 546 (>99·9%) women were eligible for analysis: 50 624 (>99·9%) women in the MMS group, 50 623 (>99·9%) in the USS group, and 101 299 (>99·9%) in the no screening group. Screening ended on Dec 31, 2011, and included 345 570 MMS and 327 775 USS annual screening episodes. At a median follow-up of 11·1 years (IQR 10·0–12·0), we diagnosed ovarian cancer in 1282 (0·6%) women: 338 (0·7%) in the MMS group, 314 (0·6%) in the USS group, and 630 (0·6%) in the no screening group. Of these women, 148 (0·29%) women in the MMS group, 154 (0·30%) in the USS group, and 347 (0·34%) in the no screening group had died of ovarian cancer. The primary analysis using a Cox proportional hazards model gave a mortality reduction over years 0–14 of 15% (95% CI –3 to 30; p=0·10) with MMS and 11% (–7 to 27; p=0·21) with USS. The Royston-Parmar fl exible parametric model showed that in the MMS group, this mortality eff ect was made up of 8% (–20 to 31) in years 0–7 and 23% (1–46) in years 7–14, and in the USS group, of 2% (–27 to 26) in years 0–7 and 21% (–2 to 42) in years 7–14. A prespecified analysis of death from ovarian cancer of MMS versus no screening with exclusion of prevalent cases showed significantly diff erent death rates (p=0·021), with an overall average mortality reduction of 20% (–2 to 40) and a reduction of 8% (–27 to 43) in years 0–7 and 28% (–3 to 49) in years 7–14 in favour of MMS. Interpretation Although the mortality reduction was not signifi cant in the primary analysis, we noted a signifi cant mortality reduction with MMS when prevalent cases were excluded. We noted encouraging evidence of a mortality reduction in years 7–14, but further follow-up is needed before firm conclusions can be reached on the efficacy and cost-eff ectiveness of ovarian cancer screening

Impact on mortality and cancer incidence rates of using random invitation from population registers for recruitment to trials

Background: Participants in trials evaluating preventive interventions such as screening are on average healthier than the general population. To decrease this 'healthy volunteer effect' (HVE) women were randomly invited from population registers to participate in the United Kingdom Collaborative Trial of Ovarian Cancer Screening (UKCTOCS) and not allowed to self refer. This report assesses the extent of the HVE still prevalent in UKCTOCS and considers how certain shortfalls in mortality and incidence can be related to differences in socioeconomic status.Methods: Between 2001 and 2005, 202 638 postmenopausal women joined the trial out of 1 243 312 women randomly invited from local health authority registers. The cohort was flagged for deaths and cancer registrations and mean follow up at censoring was 5.55 years for mortality, and 2.58 years for cancer incidence. Overall and cause-specific Standardised Mortality Ratios (SMRs) and Standardised Incidence Ratios (SIRs) were calculated based on national mortality (2005) and cancer incidence (2006) statistics. The Index of Multiple Deprivation (IMD 2007) was used to assess the link between socioeconomic status and mortality/cancer incidence, and differences between the invited and recruited populations.Results: The SMR for all trial participants was 37%. By subgroup, the SMRs were higher for: younger age groups, extremes of BMI distribution and with each increasing year in trial. There was a clear trend between lower socioeconomic status and increased mortality but less pronounced with incidence. While the invited population had higher mean IMD scores (more deprived) than the national average, those who joined the trial were less deprived.Conclusions: Recruitment to screening trials through invitation from population registers does not prevent a pronounced HVE on mortality. The impact on cancer incidence is much smaller. Similar shortfalls can be expected in other screening RCTs and it maybe prudent to use the various mortality and incidence rates presented as guides for calculating event rates and power in RCTs involving women