Non-invasive detection, localization and characterization of an arterial stenosis (a blockage or partial blockage in the artery) continues to be an important problem in medicine. Partial blockage stenoses are known to generate disturbances in blood flow which generate shear waves in the chest cavity. We examine a one-dimensional viscoelastic model that incorporates Kelvin-Voigt damping and internal variables, and develop a proof-of-concept methodology using simulated data. We first develop an estimation procedure for the material parameters. We use this procedure to determine confidence intervals for the estimated parameters, which indicates the efficacy of finding parameter estimates in practice. Confidence intervals are computed using asymptotic error theory as well as bootstrapping. We then develop a model comparison test to be used in determining if a particular data set came from a low input amplitude or a high input amplitude; this we anticipate will aid in determining when stenosis is present. These two thrusts together will serve as the methodological basis for our continuing analysis using experimental data currently being collected.Mathematics Subject Classification: 62F12; 62F40; 65M32; 74D05.
We illustrate the use of statistical tools (asymptotic theories of standard error quantification using appropriate statistical models, bootstrapping, model comparison techniques) in addition to sensitivity analysis that may be employed to determine the information content in data sets. We do this in the context of recent models [25] for nucleated polymerization in proteins, about which very little is known regarding the underlying mechanisms; thus the methodology we develop here may be of great help to experimentalists. We conclude that the investigated data sets will support with reasonable levels of uncertainty only the estimation of the parameters related to the early steps of the aggregation process.
In the formation of large clusters out of small particles, the initializing step is called the nucleation, and consists in the spontaneous reaction of agents which aggregate into small and stable polymers called nuclei. After this early step, the polymers are involved in a number of reactions such as polymerization, fragmentation and coalescence. Since there may be several orders of magnitude between the size of a particle and the size of an aggregate, building efficient numerical schemes to capture accurately the kinetics of the reaction is a delicate step of key importance. In this article, we propose a conservative scheme, based on finite volume methods on an adaptive grid, which is capable of simulating well the early steps of the reaction as well as the later chain reactions.
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