ABSTRACT
A nonlinear three-dimensional model for the amplification of a wave in the cochlea is analyzed. Using the long-slender geometry of the cochlea, and the relatively high frequencies in the hearing spectrum, an asymptotic approximation of the solution is derived for linear, but spatially inhomogeneous, amplification. From this, a nonlinear WKB approximation is constructed for the nonlinear problem, and this is used to derive an efficient numerical method for solving the amplification problem. The advantage of this approach is that the very short waves needed to resolve the wave do not need to calculated as they are represented in the asymptotic solution.
Subject(s)
Cochlea/anatomy & histology , Cochlea/physiology , Hearing/physiology , Models, Biological , Algorithms , Animals , Basilar Membrane/anatomy & histology , Basilar Membrane/physiology , Biomechanical Phenomena , Computer Simulation , Hair Cells, Auditory, Outer/physiology , Humans , Hydrodynamics , Imaging, Three-Dimensional , Mathematical Concepts , Nonlinear Dynamics , Organ of Corti/anatomy & histology , Organ of Corti/physiologyABSTRACT
Diabetes affects millions of Americans, and the correct identification of individuals afflicted with this disease, especially of those in early stages or in progression towards diabetes, remains an active area of research. The minimal model is a simplified mathematical construct for understanding glucose-insulin interactions. Developed by Bergman, Cobelli, and colleagues over three decades ago, this system of coupled ordinary differential equations prevails as an important tool for interpreting data collected during an intravenous glucose tolerance test (IVGTT). In this study we present an explicit solution to the minimal model which allows for separating the glucose and insulin dynamics of the minimal model and for identifying patient-specific parameters of glucose trajectories from IVGTT. As illustrated with patient data, our approach seems to have an edge over more complicated methods currently used. Additionally, we also present an application of our method to prediction of the time to baseline recovery and calculation of insulin sensitivity and glucose effectiveness, two quantities regarded as significant in diabetes diagnostics.
Subject(s)
Blood Glucose/metabolism , Diabetes Mellitus/metabolism , Insulin Resistance , Insulin/blood , Models, Biological , Computer Simulation , Humans , Metabolic Clearance RateABSTRACT
A nonlinear model for the mechanism responsible for the amplification of the sound wave in the ear is derived using the geometric and material properties of the system. The result is a nonlinear beam equation, with the nonlinearity appearing in a coefficient of the equation. Once derived, the beam problem is analyzed for various loading conditions. Based on this analysis it is seen that the mechanism is capable of producing a spatially localized gain, as required by any amplification mechanism, but it is also capable of increasing the spatial contrast in the signal.
