Elastic strips

Motivated by the problem of finding an explicit description of a developable narrow Moebius strip of minimal bending energy, which was first formulated by M. Sadowsky in 1930, we will develop the theory of elastic strips. Recently E.L. Starostin and G.H.M. van der Heijden found a numerical description for an elastic Moebius strip, but did not give an integrable solution. We derive two conservation laws, which describe the equilibrium equations of elastic strips. In applying these laws we find two new classes of integrable elastic strips which correspond to spherical elastic curves. We establish a connection between Hopf tori and force--free strips, which are defined by one of the integrable strips, we have found. We introduce the P--functional and relate it to elastic strips.Comment: 21 pages, 2 figure

A multistate model for early decision making in oncology

The development of oncology drugs progresses through multiple phases, where after each phase a decision is made about whether to move a molecule forward. Early phase efficacy decisions are often made on the basis of single arm studies based on RECIST tumor response as endpoint. This decision rules are implicitly assuming some form of surrogacy between tumor response and long-term endpoints like progression-free survival (PFS) or overall survival (OS). The surrogacy is most often assessed as weak, but sufficient to allow a rapid decision making as early phase studies lack the survival follow up and number of patients to properly assess PFS or OS. With the emergence of therapies with new mechanisms of action, for which the link between RECIST tumor response and long-term endpoints is either not accessible yet because not enough data is available to perform a meta-regression, or the link is weaker than with classical chemotherapies, tumor response based rules may not be optimal. In this paper, we explore the use of a multistate model for decision making based on single-arm early phase trials. The multistate model allows to account for more information than the simple RECIST response status, namely, the time to get to response, the duration of response, the PFS time and time to death. We propose to base the decision on efficacy on the OS hazard ratio (HR), predicted from a multistate model based on early phase data with limited survival follow-up, combined with historical control data. Using three case studies and simulations, we illustrate the feasibility of the estimation of the OS HR using a multistate model based on limited data from early phase studies. We argue that, in the presence of limited follow up and small sample size, and on assumptions within the multistate model, the OS prediction is acceptable and may lead to better decisions for continuing the development of a drug