REPORT 0910-1: Inference for Comparing Two Treatments Using Kernel Density Estimation
Sibabrata Banerjee
Schering-Plough Research Institute, Kenilworth, NJ 07033
Sunil Dhar
Department of Mathematical Sciences, Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102
Farid Kianifard
Biometrics, U.S. Clinical Development, Novartis Pharmaceuticals, East Hanover, NJ 07936
Abstract:
In randomized clinical trials, nonparametric approaches are considered when assumptions of a parametric approach are not reasonable. Nonparametric approaches have typically concentrated on hypothesis testing and, unlike parametric approaches, have not been amenable to providing measures of treatment efficacy. If X and Y denote the random variables representing the responses on two treatments A and B, respectively, P(Y>X) is an intuitive measure of efficacy. We consider point and interval estimation of P(Y>X) using kernel density estimation and bootstrapping. We illustrate this methodology on a data set, where comparison is made with point and interval estimates obtained by inverting the nonparametric Wilcoxon-Mann-Whitney test.
REPORT 0910-2: Development of a Recursive Finite Difference Pharmacokinetic Model from an Exponential Model: Application to a Propofol Infusion
Glen Atlas
Department of Anesthesiology, University of Medicine and Dentistry of New Jersey, Newark, NJ and Department of Chemistry, Chemical Biology and Biomedical Engineering, Stevens Institute of Technology, Hoboken, NJ
Sunil Dhar
Department of Mathematical Sciences, Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102
Abstract:
Pharmacokinetic models, using recursive finite difference equations (RFDEs), can be derived directly from traditional exponential models. This method has been successfully applied to propofol infusion data. Furthermore, this technique yields identical accuracy, on a subject-specific basis, as the exponential model from which each RFDE model was derived. Specifically, these infusion models are based upon an inhomogenous RFDE: P(k+3) = A·P(k+2) + B·P(k+1) + C·P(k) + R, where A, B, C, and R are non-zero constants and P represents plasma propofol levels for each kth unit of time. When applied to propofol infusions, RFDE modeling has advantages, over traditional exponential models, in that fewer coefficients are needed and patient-to-patient variation of these coefficients is reduced. However, initial conditions for RFDEs have to be specified. These characteristics, of RFDE modeling of propofol infusions, are similar to those for RFDE modeling of propofol boluses. Based on these findings, as well as those of our prior study, RFDE pharmacokinetic modeling can be applied to both infusion and bolus data of propofol. Further research, on the applications of RFDEs in pharmacokinetics, appears warranted.
REPORT 0910-3: Homeomorphisms and Fredholm Theory for Perturbations of Nonlinear Fredholm Maps of Index Zero with Applications
P.S. Milojevic
Department of Mathematical Sciences and CAMS, New Jersey Institute of Technology, Newark, NJ 07102
Abstract:
We develop a nonlinear Fredholm alternative theory involving k-ball and k-set perturbations of general homeomorphisms as well as of homeomorphisms that are nonlinear Fredholm maps of index zero. Various generalized first Fredholm theorems are given and finite solvability of general (odd) Fredholm maps of index zero is also studied. We apply these results to the unique and finite solvability of potential and semilinear problems with strongly nonlinear boundary conditions and to quasilinear elliptic equations. The basic tools used are the Nussbaum degree and the recent degree theories for nonlinear Fredholm maps of index zero and their perturbations.
REPORT 0910-4: An Efficient Algorithm for the Evaluation of Certain Convolution Integrals with Singular Kernels
Shidong Jiang
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102
Abstract:
An efficient algorithm is developed for the evaluation of a broad class of convolution integrals with singular kernels. The algorithm is then applied to study the Havriliak-Negami model for dielectric medias.
REPORT 0910-5: A Hybrid Numerical Method for Interfacial Fluid Flow with Soluble Surfactant
M.R. Booty and M. Siegel
Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102, USA
Abstract:
We address a significant difficulty in the numerical computation of fluid interfaces with soluble surfactant that occurs in the physically representative limit of large bulk Peclet number Pe. At the high values of Pe in typical fluid-surfactant systems, there is a transition layer near the interface in which the surfactant concentration varies rapidly, and large gradients at the interface must be resolved accurately to evaluate the exchange of surfactant between the interface and bulk flow. We use the slenderness of the layer to develop a fast and accurate `hybrid' numerical method that incorporates a separate, singular perturbation analysis of the dynamics in the transition layer into a full numerical solution of the interfacial free boundary problem. The accuracy and efficiency of the method is assessed by comparison with a more `traditional' numerical approach that uses finite differences on a curvilinear coordinate system exterior to the bubble, without the separate transition layer reduction. The traditional method implemented here features a novel fast calculation of fluid velocity off the interface.