A modified error in constitutive equation approach for frequency-domain viscoelasticity imaging using interior data

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Abstract

This paper presents a methodology for the inverse identification of linearly viscoelastic material parameters in the context of steady-state dynamics using interior data. The inverse problem of viscoelasticity imaging is solved by minimizing a modified error in constitutive equation (MECE) functional, subject to the conservation of linear momentum. The treatment is applicable to configurations where boundary conditions may be partially or completely underspecified. The MECE functional measures the discrepancy in the constitutive equations that connect kinematically admissible strains and dynamically admissible stresses, and also incorporates the measurement data in a quadratic penalty term. Regularization of the problem is achieved through a penalty parameter in combination with the discrepancy principle due to Morozov. Numerical results demonstrate the robust performance of the method in situations where the available measurement data is incomplete and corrupted by noise of varying levels.

Introduction

Inverse characterization of viscoelastic properties is of high relevance in many areas of science, engineering, and medicine. In particular, in the medical field, it is well-known that viscoelastic properties are correlated to tissue pathology  [1]. This observation has spurred the development of a group of techniques, commonly referred to as elastography or elasticity imaging techniques, whose goal is to identify the elastic or viscoelastic properties of tissue non-invasively (see  [2], [3], [4], [5], [6] for reviews). Other applications abound and include the nondestructive evaluation of structural systems such as buildings, bridges, and aircraft components.

In this work, we are concerned with the quantification of viscoelastic material properties using noisy interior data, in situations where traction and/or displacement boundary conditions are unknown or uncertain. This problem is of high relevance in the field of elasticity imaging where displacement or velocity fields are obtained using ultrasound or MRI, and there is a high degree of uncertainty regarding the magnitude and nature of the excitation sources. Different techniques have been developed that address this problem, including algebraic direct inversion  [7], [1], [8] and the adjoint weighted equations (AWE) methods  [9], [10]. These techniques have the advantage of being non-iterative but need the evaluation of derivatives of the data, making them very sensitive to noise. Moreover, optimization approaches  [11], [12], [13], [14], [15], which have the advantage of handling sparse and imperfect data, have received limited or no attention for problems with interior data that lack knowledge of the boundary conditions due to the complication of the forward problem then being ill-posed.

Our goal in this work is to develop a methodology for reconstructing viscoelastic properties from imperfect interior data and underspecified boundary conditions. The proposed methodology is based on the minimization of a modified error in constitutive equation (MECE) functional [16], [13], [17]. MECE functionals combine the error in constitutive equation (ECE) [18], which measures the discrepancy in the constitutive equations that connect kinematically admissible strains and dynamically admissible stresses, and a quadratic error term that incorporates the measurement data. MECE-based approaches have found applications in model updating with vibrational data [16], [19], time-domain formulations  [20], [13], and large scale identification problems in both elastodynamics [17] and coupled acoustic-structure systems [21]. MECE-based identification methods investigated thus far assume situations where well-posed boundary conditions are available. By contrast, one of the main features of the present approach is the fact that the proposed MECE formulation leads to optimality systems that are invertible, subject to a solvability condition that is easily met in practice, even in cases where (i) boundary conditions are (totally or partially) underspecified, and (ii) interior data is available only in a portion of the solid under investigation. This result is moreover achieved by exploiting classical first-order optimality conditions only. Some of the limitations inherent to other optimization-based formulations are therefore avoided. In addition, we adapt previous ideas regarding the solution strategy for the minimization problem to the realms of viscoelasticity, where positivity requirements are enforced using inequality constraints.

The rest of this article is organized as follows. Section  2 describes the steady-state viscoelasticity problem and the inverse problem of interest. The MECE-based minimization strategy is derived in Section  3, which also addresses the adjustment of its main parameters. Section  4 is devoted to a set of numerical experiments designed to demonstrate the capabilities of the method. Concluding remarks are finally given in Section  5.

Section snippets

Problem setting

Governing equations of motion. Let a solid viscoelastic body occupy a bounded and connected domain ΩRd(1d3) with boundary Γ. The time-harmonic motion of this body is governed by the balance equations σ+b=ρω2uin   Ω,σn=ton   ΓN, where u is the displacement field, ω represents the specified angular frequency, ρ denotes the known mass density, b is a given body force density, σ represents the stress tensor, t and ΓNΓ are the given surface force density (traction) and its support,

MECE functional

Following the approach presented in  [17], [21], the inverse problem addressed in this work is formulated as an optimization problem in which the unknown constitutive tensor C is estimated by minimizing an objective function that additively combines two error terms: (1) an error in constitutive equation (ECE) functional  [18] that has been adapted for viscoelastic materials and that measures the discrepancy in the constitutive equation that connects kinematically admissible strains and

Numerical experiments

This section is devoted to a series of numerical experiments, inspired by the field of biomedical imaging, that are intended to showcase the capabilities of the methodology of Section  3. They consist in imaging a viscoelastic inclusion (with unknown location, geometry, and characteristics) embedded in a viscoelastic background medium (all materials being assumed to be isotropic) by reconstructing the complex bulk and shear moduli fields B and G, treated as completely unknown. In particular,

Conclusions

The intended contributions of this work, devoted to the imaging of linearly viscoelastic heterogeneous moduli using interior data, include (i) the development of an algorithm exploiting a generalized error in constitutive equation functional which handles in a very natural way situations where boundary conditions are not known and (ii) a demonstration of its feasibility on 2D and 3D synthetic imaging problems involving up to tens of thousands of unknown viscoelastic moduli and using incomplete

Acknowledgment

Wilkins Aquino and Manuel Diaz were partially supported by NIH Grant # R01CA174723.

References (30)

  • T.E. Oliphant et al.

    Complex-valued stiffness reconstruction for magnetic resonance elastography by algebraic inversion of the differential equation

    Magnetic resonance in Medicine

    (2001)
  • E. Park et al.

    Shear modulus reconstruction in dynamic elastography: time harmonic case

    Phys. Med. Biol.

    (2006)
  • Y. Zhang et al.

    Solution of the time-harmonic viscoelastic inverse problem with interior data in two dimensions

    Int. J. Numer. Methods Eng.

    (2012)
  • M. Doyley et al.

    Evaluation of an iterative reconstruction method for quantitative elastography

    Physics in Medicine and Biology

    (2000)
  • A.A. Oberai et al.

    Solution of inverse problems in elasticity imaging using the adjoint method

    Inverse Problems

    (2003)
  • Cited by (0)

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