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

Abstract

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.

Analysis of non-contact acousto-thermal signature data

“…This task requires sophisticated data analysis techniques including hypothesis testing and information theoretic based model selection (such as the Akaike Information Criterion-AIC). Both hypothesis testing (see [3,4] for examples) and the AIC (an example is given in [5] and a thorough introduction may be found in [6]) require that parameter estimation be carried out. In order to carry out parameter estimation, a model (which depends on quantities of interest and the independent variables) and data are needed.…”

Model validation for a noninvasive arterial stenosis detection problem

“…To that end, we set up a similar framework as in the concept paper [13] and connect those results to the current work. We will require that the following assumptions hold:…”

“…Since r min > 0, there are no singularities in the final term in (6), and the kernel integral in the numerator of that term will converge in the same manner as the preceding kernel integral. Thus, the arguments from [13] apply in the case here, and we have the following theorem:…”

“…We develop this kernel from a different perspective than that given in [13], but the resulting form will be quite similar. Using the notation and parameter conventions of [12], we define the kernel in this work to be $$G(t;P)={\kappa }_r+K(t;P)$$ where κ r is a positive constant representing an instantaneous relaxation modulus (justified by the fact that the gel phantom acts partly as a solid) and K ( t ; P ) = $\int$ τ exp(− t/τ ) dP ( τ ) represents a continuum of distributed relaxation times with τ = [ τ 1 , τ 2 ] ⊂ (0, ∞) and where P ( τ ) is a probability measure on τ.…”

“…Here, we incorporate all of the aforementioned physical modeling concerns and continue the work of our concept paper [13] by focusing on wave propagation through a homogeneous viscoelastic medium. The constitutive relationship will be slightly more general than in our concept paper, and is based on Fung’s viscoelastic formulation that has been validated by actual tissue experimental data (more details below).…”