A Probabilistic Multiscale Approach to Hysteresis in Shear Wave Propagation in Biotissue

Abstract

Motivated by a problem involving wave propagation through viscoelastic biotissue, we present a theoretical framework for treating hysteresis as multiscale phenomena which must be averaged across distributions of internal variables. The resulting systems entail probability measure dependent partial differential equations for which we establish well-posedness in a framework that leads readily to computationally useful approximations.

Mathematical Models of Dividing Cell Populations: Application to CFSE Data

“…That is, one must use aggregate (or population level) data in an attempt to describe what are ultimately the dynamics of individual cells. This type of inverse problem is well known in mathematics, and successful mathematical models have been developed and fit to data in a variety of applications such as size-structured marine and insect population models [4,7,11,12], wave propagation models for viscoelastic solids [19], electromagnetic wave propagation [13,14], physiologically-based pharmacokinetics models [6,20], and HIV models [5]. In addition to these applications, theory for such inverse problems is well-developed [3,6,18].…”

A Brief Review of Some Approaches to Hysteresis in Viscoelastic Polymers

“…A benefit to using (6) as a constitutive equation is that, unlike simpler models for viscoelasticity, it allows for the consideration of a continuous spectrum (e.g., see the discussions in [38]) of relaxation times and frequencies (this is also true of the probabilistic-based internal variable approach developed in [23] and described below). (The need for a continuum of relaxation times in certain materials was observed many years ago [36,63,66,70].)…”

“…The probabilistic based internal variable alternative [23] to Fung's kernel involves a parameter dependent kernel with a continuous distribution of parameters and internal variables. In the case of a finite combination of Dirac δ distributions, one obtains a finite summation of exponential functions as the approximation kernel (see the discussions below).…”

“…Unfortunately, the QLV, as presented by Fung, leads to exceedingly difficult computations within full dynamical partial differential equations, especially in estimation and control problems. This motivated the development of the internal variable approach described in [3,23,48] (which permits discrete approximation to a continuum) in attempts to approximate well the corresponding dynamic responses even in cases where the stress-strain curves alone do not produce adequate approximations -see [38].…”

Material parameter estimation and hypothesis testing on a 1D viscoelastic stenosis model: Methodology

“…Accordingly, there is interest in examining other methods to determine the existence and location of stenosed vessels. Previous work [2,11,12,13,15,28,34] focused on developing a sensor device to be used with a physical model of a chest cavity, and then developing a mathematical model to describe the medium in which a stenosis-generated acoustic signal is propagated to the chest surface. After an interregnum of roughly five years between that earlier work and our current efforts, we have returned to the early ideas and have reformulated the problem to some extent.…”