Estimation of surface area and volume of a nematode from morphometric data

Nematode volume and surface area are usually based on the inappropriate assumption that the animal is cylindrical. While nematodes are approximately circular in cross section, the radius varies longitudinally. We use standard morphometric data to obtain improved estimates of volume and surface area based on (i) a geometrical approach and (ii) a Bezier representation of the nematode. These new estimators require only the morphometric data available from Cobb's ratios, but if fewer coordinates are available the geometric approach reduces to the standard estimates. Consequently, these new estimators are better than the standard alternatives

Estimation of Surface Area and Volume of a Nematode from Morphometric Data

Nematode volume and surface area are usually based on the inappropriate assumption that the animal is cylindrical. While nematodes are approximately circular in cross section, the radius varies longitudinally. We use standard morphometric data to obtain improved estimates of volume and surface area based on (i) a geometrical approach and (ii) a Bézier representation of the nematode. These new estimators require only the morphometric data available from Cobb's ratios, but if fewer coordinates are available the geometric approach reduces to the standard estimates. Consequently, these new estimators are better than the standard alternatives

Multiple novel prostate cancer susceptibility signals identified by fine-mapping of known risk loci among Europeans

Genome-wide association studies (GWAS) have identified numerous common prostate cancer (PrCa) susceptibility loci. We have fine-mapped 64 GWAS regions known at the conclusion of the iCOGS study using large-scale genotyping and imputation in 25 723 PrCa cases and 26 274 controls of European ancestry. We detected evidence for multiple independent signals at 16 regions, 12 of which contained additional newly identified significant associations. A single signal comprising a spectrum of correlated variation was observed at 39 regions; 35 of which are now described by a novel more significantly associated lead SNP, while the originally reported variant remained as the lead SNP only in 4 regions. We also confirmed two association signals in Europeans that had been previously reported only in East-Asian GWAS. Based on statistical evidence and linkage disequilibrium (LD) structure, we have curated and narrowed down the list of the most likely candidate causal variants for each region. Functional annotation using data from ENCODE filtered for PrCa cell lines and eQTL analysis demonstrated significant enrichment for overlap with bio-features within this set. By incorporating the novel risk variants identified here alongside the refined data for existing association signals, we estimate that these loci now explain ∼38.9% of the familial relative risk of PrCa, an 8.9% improvement over the previously reported GWAS tag SNPs. This suggests that a significant fraction of the heritability of PrCa may have been hidden during the discovery phase of GWAS, in particular due to the presence of multiple independent signals within the same regio

Ten-year mortality, disease progression, and treatment-related side effects in men with localised prostate cancer from the ProtecT randomised controlled trial according to treatment received

Background The ProtecT trial reported intention-to-treat analysis of men with localised prostate cancer randomly allocated to active monitoring (AM), radical prostatectomy, and external beam radiotherapy. Objective To report outcomes according to treatment received in men in randomised and treatment choice cohorts. Design, setting, and participants This study focuses on secondary care. Men with clinically localised prostate cancer at one of nine UK centres were invited to participate in the treatment trial comparing AM, radical prostatectomy, and radiotherapy. Intervention Two cohorts included 1643 men who agreed to be randomised and 997 who declined randomisation and chose treatment. Outcome measurements and statistical analysis Analysis was carried out to assess mortality, metastasis and progression and health-related quality of life impacts on urinary, bowel, and sexual function using patient-reported outcome measures. Analysis was based on comparisons between groups defined by treatment received for both randomised and treatment choice cohorts in turn, with pooled estimates of intervention effect obtained using meta-analysis. Differences were estimated with adjustment for known prognostic factors using propensity scores. Results and limitations According to treatment received, more men receiving AM died of PCa (AM 1.85%, surgery 0.67%, radiotherapy 0.73%), whilst this difference remained consistent with chance in the randomised cohort (p = 0.08); stronger evidence was found in the exploratory analyses (randomised plus choice cohort) when AM was compared with the combined radical treatment group (p = 0.003). There was also strong evidence that metastasis (AM 5.6%, surgery 2.4%, radiotherapy 2.7%) and disease progression (AM 20.35%, surgery 5.87%, radiotherapy 6.62%) were more common in the AM group. Compared with AM, there were higher risks of sexual dysfunction (95% at 6 mo) and urinary incontinence (55% at 6 mo) after surgery, and of sexual dysfunction (88% at 6 mo) and bowel dysfunction (5% at 6 mo) after radiotherapy. The key limitations are the potential for bias when comparing groups defined by treatment received and changes in the protocol for AM during the lengthy follow-up required in trials of screen-detected PCa. Conclusions Analyses according to treatment received showed increased rates of disease-related events and lower rates of patient-reported harms in men managed by AM compared with men managed by radical treatment, and stronger evidence of greater PCa mortality in the AM group. Patient summary More than 95 out of every 100 men with low or intermediate risk localised prostate cancer do not die of prostate cancer within 10 yr, irrespective of whether treatment is by means of monitoring, surgery, or radiotherapy. Side effects on sexual and bladder function are better after active monitoring, but the risks of spreading of prostate cancer are more common

Functional and quality of life outcomes of localised prostate cancer treatments (prostate testing for cancer and treatment [ProtecT] study)

Objective To investigate the functional and quality of life (QoL) outcomes of treatments for localised prostate cancer and inform treatment decision-making. Patients and Methods Men aged 50–69 years diagnosed with localised prostate cancer by prostate-specific antigen testing and biopsies at nine UK centres in the Prostate Testing for Cancer and Treatment (ProtecT) trial were randomised to, or chose one of, three treatments. Of 2565 participants, 1135 men received active monitoring (AM), 750 a radical prostatectomy (RP), 603 external-beam radiotherapy (EBRT) with concurrent androgen-deprivation therapy (ADT) and 77 low-dose-rate brachytherapy (BT, not a randomised treatment). Patient-reported outcome measures (PROMs) completed annually for 6 years were analysed by initial treatment and censored for subsequent treatments. Mixed effects models were adjusted for baseline characteristics using propensity scores. Results Treatment-received analyses revealed different impacts of treatments over 6 years. Men remaining on AM experienced gradual declines in sexual and urinary function with age (e.g., increases in erectile dysfunction from 35% of men at baseline to 53% at 6 years and nocturia similarly from 20% to 38%). Radical treatment impacts were immediate and continued over 6 years. After RP, 95% of men reported erectile dysfunction persisting for 85% at 6 years, and after EBRT this was reported by 69% and 74%, respectively (P < 0.001 compared with AM). After RP, 36% of men reported urinary leakage requiring at least 1 pad/day, persisting for 20% at 6 years, compared with no change in men receiving EBRT or AM (P < 0.001). Worse bowel function and bother (e.g., bloody stools 6% at 6 years and faecal incontinence 10%) was experienced by men after EBRT than after RP or AM (P < 0.001) with lesser effects after BT. No treatment affected mental or physical QoL. Conclusion Treatment decision-making for localised prostate cancer can be informed by these 6-year functional and QoL outcomes

A simple confidence band for the Michaelis-Menten equation

Analysis of enzyme kinetic data requires more than just comparisons of Kms and Vmaxs using the corresponding error estimates of the parameters. This approach is often employed, but it can prompt contradictory and misleading inferences that might be avoided using a confidence band. We derive expressions for the confidence band for the Michaelis-Menten rate equation that rely on estimates and variances of Km and Vmax. These can be expressed in terms of the substrate concentration or the rate of reaction. While these equations are simple, they are nonlinear, which reinforces the need to consider both parameters simultaneously. The equations show that the amplitude of the confidence interval (ε) passes through a maximum if the variance of the Km is sufficently large compared with the variance of Vmax. We illustrate the value of the expressions by applying them to comparisons of the kinetics of enzymes involved in nitrogen metabolism in parasites. These examples confirm that (i) the variance of the estimate of the Km has a particularly significant effect on ε and (ii) comparisons among Kms and among Vmaxs are not necessarily sufficient to determine the significance of differences in activity

The kinetics of enzyme mixtures

Even purified enzyme preparations are often heterogeneous. For example, preparations of aspartate aminotransferase or cytochrome oxidase can consist of several different forms of the enzyme. For this reason we consider how different the kinetics of the reactions catalysed by a mixture of forms of an enzyme must be to provide some indication of the characteristics of the species present. Based on the standard Michaelis-Menten model, we show that if the Michaelis constants (Km) of two isoforms differ by a factor of at least 20 the steady-state kinetics can be used to characterise the mixture. However, even if heterogeneity is reflected in the kinetic data, the proportions of the different forms of the enzyme cannot be estimated from the kinetic data alone. Consequently, the heterogeneity of enzyme preparations is rarely reflected in measurements of their steady-state kinetics unless the species present have significantly different kinetic properties. This has two implications: (1) it is difficult, but not impossible, to detect molecular heterogeneity using kinetic data and (2) even when it is possible, a considerable quantity of high quality data is required

Confidence band approximation using interval arithmetic

Confidence bands are commonly obtained from linear regression, but they are rarely shown for nonlinear functions. In part this is due to mathematical complexity. A simple, intuitive and easily automated alternative is to employ interval arithmetic to approximate the confidence band. We illustrate the method by applying it to a straight line, and then apply it to the rate equation of a Michaelis-Menten enzyme and a general polynomial. In each case the approximation is generally larger (and never smaller) than the corresponding standard confidence band, but in at least some instances the upper bound of the discrepancy is about 40%

What happens when the wrong equation is fitted to data?

A common variant of the Michaelis-Menten model of enzyme kinetics involves inhibition by excess substrate. This phenomenon is known as substrate inhibition and the mathematical description of it requires an inhibition constant (Ki) as well as the usual kinetic parameters (Km and Vmax). Fitting the 3-parameter substrate inhibition expression to data that might reasonably be described by the 2-parameter Michaelis-Menten model yields biased estimates of Km and Vmax. Numerical simulations demonstrate that the extent of the bias is related to the magnitude of the estimated Ki. The quality of the data is particularly important in determining the size of Ki and, therefore, in the bias of the other parameters. Consideration of the residuals, statistical justification of the inclusion of extra parameters and reporting of the estimated values should be matters of routine. The estimates of Km and Vmax obtained from a three-parameter substrate inhibition model can only be compared with the corresponding estimates from the two-parameter Michaelis-Menten model with caution