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.

Sensitivity of dynamical systems to Banach space parameters

“…We are currently pursuing such a theory in which the parameter space is no longer a Banach space, but rather a metric space that is based on a combination of the Prohorov metric topology (see [6]) and the L 2 topology (or possibly the weak L 2 topology for compatibility with the Prohorov metric-see [15]). …”

“…In particular, systems with probability measures embedded in the dynamics (problems involving aggregate dynamics as discussed in [6]) have become important in applications in biology [3,5,6], electromagnetics [7] and hysteretic [10,11,19,29,33] and polymeric [12,13,15] materials. These systems have the forṁ…”

A review of selected techniques in inverse problem nonparametric probability distribution estimation

“…There is a third possibility, that one has only an aggregate model (i.e., the dynamics depend explicitly on a distribution of parameters across the population) with aggregate data. Such examples arise in electromagnetic models with a distribution of polarization relaxation times for molecules (e.g., [15,16]); in biology with HIV cellular models [3][4][5]; and in wave propagation in viscoelastic materials such as biotissue [20]. The measure estimation problem for such examples is sufficiently similar to individual dynamics/aggregate data situation and accordingly we do not consider aggregate dynamics models as a separate case but only offer some explicit comments as warranted.…”

Model validation for a noninvasive arterial stenosis detection problem

“…The model here will continue from a previous line of work by Banks, et al, [8, 18, 19, 20, 21, 42, 50]. These models allow for a characterization of shear waves resulting from coronary stenoses, which will assist in uncovering the coronary artery sounds from the noisy background in the body.…”

“…The results in [8, 42, 50] demonstrate that the internal variable approach is valid and does appear to work as well as the continuum of times in the Fung kernel. A connection between the Fung’s kernel and the discrete kernel is provided by the work in [20]. The authors there form the kernel $$G\left(t\right)={\int }_{\tau }q(t;\tau )dP\left(\tau \right)$$ where τ = [ τ 1 , τ 2 ] ⊂ (0, ∞) is the set of admissible relaxation times, P ( τ ) is a probability measure on τ , and q ( t ; τ ) is a continuous function of relaxation times.…”