Classification and stability of simple homoclinic cycles in R^5

The paper presents a complete study of simple homoclinic cycles in R^5. We find all symmetry groups Gamma such that a Gamma-equivariant dynamical system in R^5 can possess a simple homoclinic cycle. We introduce a classification of simple homoclinic cycles in R^n based on the action of the system symmetry group. For systems in R^5, we list all classes of simple homoclinic cycles. For each class, we derive necessary and sufficient conditions for asymptotic stability and fragmentary asymptotic stability in terms of eigenvalues of linearisation near the steady state involved in the cycle. For any action of the groups Gamma which can give rise to a simple homoclinic cycle, we list classes to which the respective homoclinic cycles belong, thus determining conditions for asymptotic stability of these cycles.Comment: 34 pp., 4 tables, 30 references. Submitted to Nonlinearit

The statistical analysis of a clinical trial when a protocol amendment changed the inclusion criteria

Abstract Background Sometimes, protocol amendments that change the inclusion and exclusion criteria are required in clinical trials. Then, the patient populations before and after the amendment may differ. Methods We propose to perform separate statistical tests for the different phases, i.e. for the patients recruited before and after the amendment, and to combine the tests using Fisher's combination test. After a significant combination test a multiple testing procedure can be applied to identify the phase(s) to which a proof of efficacy refers. We assume that the amendment(s) are not based on any type of unblinded data. The proposed method is investigated within a simulation study. Results The proposed combination approach is superior to the 'naïve' strategy to ignore the differences between the phases and pooling the data to perform just one statistical test. This superiority disappears when there are hardly any differences between the two phases. Conclusion When one or more protocol amendments change the inclusion and exclusion criteria, one should realize that the populations may differ. In this case, separate tests for the different phases together with a combination test are a powerful method that can be applied in a variety of settings. The (first) amendment should specify the combination test to be applied in order to combine the different phases.</p

Heterogeneous Pattern of Retinal Nerve Fiber Layer in Multiple Sclerosis. High Resolution Optical Coherence Tomography: Potential and Limitations

BACKGROUND: Recently the reduction of the retinal nerve fibre layer (RNFL) was suggested to be associated with diffuse axonal damage in the whole CNS of multiple sclerosis (MS) patients. However, several points are still under discussion. (1) Is high resolution optical coherence tomography (OCT) required to detect the partly very subtle RNFL changes seen in MS patients? (2) Can a reduction of RNFL be detected in all MS patients, even in early disease courses and in all MS subtypes? (3) Does an optic neuritis (ON) or focal lesions along the visual pathways, which are both very common in MS, limit the predication of diffuse axonal degeneration in the whole CNS? The purpose of our study was to determine the baseline characteristics of clinical definite relapsing-remitting (RRMS) and secondary progressive (SPMS) MS patients with high resolution OCT technique. METHODOLOGY: Forty-two RRMS and 17 SPMS patients with and without history of uni- or bilateral ON, and 59 age- and sex-matched healthy controls were analysed prospectively with the high resolution spectral-domain OCT device (SD-OCT) using the Spectralis 3.5mm circle scan protocol with locked reference images and eye tracking mode. Furthermore we performed tests for visual and contrast acuity and sensitivity (ETDRS, Sloan and Pelli-Robson-charts), for color vision (Lanthony D-15), the Humphrey visual field and visual evoked potential testing (VEP). PRINCIPAL FINDINGS: All 4 groups (RRMS and SPMS with or without ON) showed significantly reduced RNFL globally, or at least in one of the peripapillary sectors compared to age-/sex-matched healthy controls. In patients with previous ON additional RNFL reduction was found. However, in many RRMS patients the RNFL was found within normal range. We found no correlation between RNFL reduction and disease duration (range 9-540 months). CONCLUSIONS: RNFL baseline characteristics of RRMS and SPMS are heterogeneous (range from normal to markedly reduced levels)

Adaptive design methods in clinical trials – a review

In recent years, the use of adaptive design methods in clinical research and development based on accrued data has become very popular due to its flexibility and efficiency. Based on adaptations applied, adaptive designs can be classified into three categories: prospective, concurrent (ad hoc), and retrospective adaptive designs. An adaptive design allows modifications made to trial and/or statistical procedures of ongoing clinical trials. However, it is a concern that the actual patient population after the adaptations could deviate from the originally target patient population and consequently the overall type I error (to erroneously claim efficacy for an infective drug) rate may not be controlled. In addition, major adaptations of trial and/or statistical procedures of on-going trials may result in a totally different trial that is unable to address the scientific/medical questions the trial intends to answer. In this article, several commonly considered adaptive designs in clinical trials are reviewed. Impacts of ad hoc adaptations (protocol amendments), challenges in by design (prospective) adaptations, and obstacles of retrospective adaptations are described. Strategies for the use of adaptive design in clinical development of rare diseases are discussed. Some examples concerning the development of Velcade intended for multiple myeloma and non-Hodgkin's lymphoma are given. Practical issues that are commonly encountered when implementing adaptive design methods in clinical trials are also discussed

To add or not to add a new treatment arm to a multiarm study: A decision-theoretic framework.

Multiarm clinical trials, which compare several experimental treatments against control, are frequently recommended due to their efficiency gain. In practise, all potential treatments may not be ready to be tested in a phase II/III trial at the same time. It has become appealing to allow new treatment arms to be added into on-going clinical trials using a "platform" trial approach. To the best of our knowledge, many aspects of when to add arms to an existing trial have not been explored in the literature. Most works on adding arm(s) assume that a new arm is opened whenever a new treatment becomes available. This strategy may prolong the overall duration of a study or cause reduction in marginal power for each hypothesis if the adaptation is not well accommodated. Within a two-stage trial setting, we propose a decision-theoretic framework to investigate when to add or not to add a new treatment arm based on the observed stage one treatment responses. To account for different prospect of multiarm studies, we define utility in two different ways; one for a trial that aims to maximise the number of rejected hypotheses; the other for a trial that would declare a success when at least one hypothesis is rejected from the study. Our framework shows that it is not always optimal to add a new treatment arm to an existing trial. We illustrate a case study by considering a completed trial on knee osteoarthritis