Comparison of sample size formulae for 2x2 cross-over designs applied to bioequivalence studies

We consider the comparison of two formulations in terms of average bioequivalence using the 2 × 2 cross-over design. In a bioequivalence study, the primary outcome is a pharmacokinetic measure, such as the area under the plasma concentration by time curve, which is usually assumed to have a lognormal distribution. The criterion typically used for claiming bioequivalence is that the 90% confidence interval for the ratio of the means should lie within the interval (0.80, 1.25), or equivalently the 90% confidence interval for the differences in the means on the natural log scale should be within the interval (-0.2231, 0.2231). We compare the gold standard method for calculation of the sample size based on the non-central t distribution with those based on the central t and normal distributions. In practice, the differences between the various approaches are likely to be small. Further approximations to the power function are sometimes used to simplify the calculations. These approximations should be used with caution, because the sample size required for a desirable level of power might be under- or overestimated compared to the gold standard method. However, in some situations the approximate methods produce very similar sample sizes to the gold standard method. Copyright © 2005 John Wiley & Sons, Ltd.

Noncompetitive plasma biokinetics of deuterium-labeled natural and synthetic alpha-tocopherol in healthy men with an apoE4 genotype

Previous studies comparing the biokinetics of deuterated natural (RRR) and synthetic (all-rac) α-tocopherol (vitamin E) used a simultaneous ingestion or competitive uptake approach and reported relative bioavailability ratios close to 2:1, higher than the accepted biopotency ratio of 1.36:1. The aim of the current study was to compare the biokinetics of deuterated natural and synthetic vitamin E using a noncompetitive uptake model both before and after vitamin E supplementation in a distinct population. Healthy men (n = 10) carrying the apolipoprotein (apo)E4 genotype completed a randomized crossover study, comprised of two 4-wk treatments with 400 mg/d (RRR-α-tocopheryl and all-rac-α-tocopheryl acetate) with a 12-wk washout period between treatments. Before and after each treatment period, the subjects consumed a capsule containing 150 mg deuterated α-tocopheryl acetate in either the PRR or all-rac form depending on their treatment regimen. Blood was obtained up to 48 h after ingestion, and tocopherols analyzed by LC/MS. After deuterated all-rac administration, plasma deuterated tocopherol maximum concentrations and area under the concentration vs. time curves (AUC) were lower than those following RRR administration. The RRR:all-rac ratios determined from the deuterated biokinetic profiles (maximum concentration; C-max) and AUCs were 1.35:1 &PLUSMN; 0.17 and 1.33:1 &PLUSMN; 0.18, respectively. The 4-wk supplementation with either PRR or all-rac significantly increased plasma a-tocopherol concentrations (P < 0.001), but decreased the plasma response to newly absorbed deuterated RRR or all-rac α-tocopherol. Using a noncompetitive uptake approach, the relative bioavailability of natural to synthetic vitamin E in apoE4 males was close to the currently accepted biopotency ratio of 1.36:1.

Characterization and classification of tumor lesions using computerized fractal-based texture analysis and support vector machines in digital mammograms

Objective: This paper presents a detailed study of fractal-based methods for texture characterization of mammographic mass lesions and architectural distortion. The purpose of this study is to explore the use of fractal and lacunarity analysis for the characterization and classification of both tumor lesions and normal breast parenchyma in mammography. Materials and methods: We conducted comparative evaluations of five popular fractal dimension estimation methods for the characterization of the texture of mass lesions and architectural distortion. We applied the concept of lacunarity to the description of the spatial distribution of the pixel intensities in mammographic images. These methods were tested with a set of 57 breast masses and 60 normal breast parenchyma (dataset1), and with another set of 19 architectural distortions and 41 normal breast parenchyma (dataset2). Support vector machines (SVM) were used as a pattern classification method for tumor classification. Results: Experimental results showed that the fractal dimension of region of interest (ROIs) depicting mass lesions and architectural distortion was statistically significantly lower than that of normal breast parenchyma for all five methods. Receiver operating characteristic (ROC) analysis showed that fractional Brownian motion (FBM) method generated the highest area under ROC curve (A z = 0.839 for dataset1, 0.828 for dataset2, respectively) among five methods for both datasets. Lacunarity analysis showed that the ROIs depicting mass lesions and architectural distortion had higher lacunarities than those of ROIs depicting normal breast parenchyma. The combination of FBM fractal dimension and lacunarity yielded the highest A z value (0.903 and 0.875, respectively) than those based on single feature alone for both given datasets. The application of the SVM improved the performance of the fractal-based features in differentiating tumor lesions from normal breast parenchyma by generating higher A z value. Conclusion: FBM texture model is the most appropriate model for characterizing mammographic images due to self-affinity assumption of the method being a better approximation. Lacunarity is an effective counterpart measure of the fractal dimension in texture feature extraction in mammographic images. The classification results obtained in this work suggest that the SVM is an effective method with great potential for classification in mammographic image analysis.

Euler's theorem under the microscope

In this article Geoff Tennant summarises the first half of Imre Lakatos's seminal 1976 book, "Proofs and refutations: the logic of mathematical discovery". Implications are drawn for the classroom treatment of proof.