Quantitative characterization of viscoelastic behavior in tissue-mimicking phantoms and ex vivo animal tissues.

Viscoelasticity of soft tissue is often related to pathology, and therefore, has become an important diagnostic indicator in the clinical assessment of suspect tissue. Surgeons, particularly within head and neck subsites, typically use palpation techniques for intra-operative tumor detection. This detection method, however, is highly subjective and often fails to detect small or deep abnormalities. Vibroacoustography (VA) and similar methods have previously been used to distinguish tissue with high-contrast, but a firm understanding of the main contrast mechanism has yet to be verified. The contributions of tissue mechanical properties in VA images have been difficult to verify given the limited literature on viscoelastic properties of various normal and diseased tissue. This paper aims to investigate viscoelasticity theory and present a detailed description of viscoelastic experimental results obtained in tissue-mimicking phantoms (TMPs) and ex vivo tissues to verify the main contrast mechanism in VA and similar imaging modalities. A spherical-tip micro-indentation technique was employed with the Hertzian model to acquire absolute, quantitative, point measurements of the elastic modulus (E), long term shear modulus (η), and time constant (τ) in homogeneous TMPs and ex vivo tissue in rat liver and porcine liver and gallbladder. Viscoelastic differences observed between porcine liver and gallbladder tissue suggest that imaging modalities which utilize the mechanical properties of tissue as a primary contrast mechanism can potentially be used to quantitatively differentiate between proximate organs in a clinical setting. These results may facilitate more accurate tissue modeling and add information not currently available to the field of systems characterization and biomedical research

Recovery of cellular traction in three-dimensional nonlinear hyperelastic matrices

The traction exerted by a cell on the extra-cellular matrix (ECM) is critical to understanding and manipulating important biological processes such as stem cell differentiation, cancer cell metastasis, and embryonic morphogenesis. This traction is typically quantified through traction force microscopy (TFM). In TFM, the displacement of select markers inside the ECM is tracked, and is used in conjunction with an elasticity problem to reconstruct the traction field. Most applications of this technique thus far have assumed that the matrix behaves as a linear elastic solid that undergoes small deformation and infinitesimal strains. In this manuscript, we develop and implement a robust and efficient TFM methodology that overcomes these limitations by accounting for geometric and material nonlinearities in the ECM. We pose the TFM problem as an inverse problem and develop efficient adjoint-based minimization techniques to solve it. We test the effect of measurement noise on the proposed method, and examine the error incurred by not including nonlinear effects when solving the TFM problem. We present these results for in-silico traction fields that are applied to realistic geometric models of microglial and neuronal cells

A generalized computationally efficient inverse characterization approach combining direct inversion solution initialization with gradient-based optimization

A computationally efficient gradient-based optimization approach for inverse material characterization from incomplete system response measurements that can utilize a generally applicable parameterization (e.g., finite element-type parameterization) is presented and evaluated. The key to this inverse characterization algorithm is the use of a direct inversion strategy with Gappy proper orthogonal decomposition (POD) response field estimation to initialize the inverse solution estimate prior to gradient-based optimization. Gappy POD is used to estimate the complete (i.e., all components over the entire spatial domain) system response field from incomplete (e.g., partial spatial distribution) measurements obtained from some type of system testing along with some amount of a priori information regarding the potential distribution of the unknown material property. The estimated complete system response is used within a physics-based direct inversion procedure with a finite element-type parameterization to estimate the spatial distribution of the desired unknown material property with minimal computational expense. Then, this estimated spatial distribution of the unknown material property is used to initialize a gradient-based optimization approach, which uses the adjoint method for computationally efficient gradient calculations, to produce the final estimate of the material property distribution. The three-step [(1) Gappy POD, (2) direct inversion, and (3) gradient-based optimization] inverse characterization approach is evaluated through simulated test problems based on the characterization of elastic modulus distributions with localized variations (e.g., inclusions) within simple structures. Overall, this inverse characterization approach is shown to efficiently and consistently provide accurate inverse characterization estimates for material property distributions from incomplete response field measurements. Moreover, the solution procedure is shown to be capable of extrapolating significantly beyond the initial assumptions regarding the potential nature of the unknown material property distribution

Iterative Methods for the Elasticity Imaging Inverse Problem

Cancers of the soft tissue reign among the deadliest diseases throughout the world and effective treatments for such cancers rely on early and accurate detection of tumors within the interior of the body. One such diagnostic tool, known as elasticity imaging or elastography, uses measurements of tissue displacement to reconstruct the variable elasticity between healthy and unhealthy tissue inside the body. This gives rise to a challenging parameter identification inverse problem, that of identifying the Lamé parameter μ in a system of partial differential equations in linear elasticity. Due to the near incompressibility of human tissue, however, common techniques for solving the direct and inverse problems are rendered ineffective due to a phenomenon known as the “locking effect”. Alternative methods, such as mixed finite element methods, must be applied to overcome this complication. Using these methods, this work reposes the problem as a generalized saddle point problem along with a presentation of several optimization formulations, including the modified output least squares (MOLS), energy output least squares (EOLS), and equation error (EE) frameworks, for solving the elasticity imaging inverse problem. Subsequently, numerous iterative optimization methods, including gradient, extragradient, and proximal point methods, are explored and applied to solve the related optimization problem. Implementations of all of the iterative techniques under consideration are applied to all of the developed optimization frameworks using a representative numerical example in elasticity imaging. A thorough analysis and comparison of the methods is subsequently presented

Inverse material identification in coupled acoustic-structure interaction using a modified error in constitutive equation functional

International audienceThis work focuses on the identification of heterogeneous linear elastic moduli in the context of frequency-domain, coupled acoustic-structure interaction (ASI), using either solid displacement or fluid pressure measurement data. The approach postulates the inverse problem as an optimization problem where the solution is obtained by minimizing a modified error in constitutive equation (MECE) functional. The latter measures the discrepancy in the constitutive equations that connect kinematically admissible strains and dynamically admissible stresses, while incorporating the measurement data as additional quadratic error terms. We demonstrate two strategies for selecting the MECE weighting coefficient to produce regularized solutions to the ill-posed identification problem: 1) the discrepancy principle of Morozov, and 2) an error-balance approach that selects the weight parameter as the minimizer of another functional involving the ECE and the data misfit. Numerical results demonstrate that the proposed methodology can successfully recover elastic parameters in 2D and 3D ASI systems from response measurements taken in either the solid or fluid subdomains. Furthermore, both regularization strategies are shown to produce accurate reconstructions when the measurement data is polluted with noise. The discrepancy principle is shown to produce nearly optimal solutions, while the error-balance approach, although not optimal, remains effective and does not need a priori information on the noise level

Shape Analysis Based Strategies for Evaluation of Adaptations in In Vivo Right Ventricular Geometry and Mechanics as Effected by Pulmonary Hypertension

Pulmonary hypertension (PH) is a deadly disease, which as it progresses over time alters many aspects of the afflicted heart, and particularly the right ventricle (RV), such as its size, shape, and mechanical material properties. However, due to the limitations of what can be measured noninvasively in a standard clinical setting and the difficulty caused by the intrinsic complexity of the human RV, there has been little success to-date to identify clinically obtainable metrics of RV shape, deformation, or material properties that are quantitatively linked to the onset and progression of PH. Towards addressing this challenge, this work proposes the use of the shape and shape change of the RV, which is measurable from standard clinical imaging, along with statistical analysis and inverse material characterization strategies to identify new metrics of RV mechanical function that will be uniquely predictive of the state of the heart subject to PH. Thus, this thesis can be broken into two components: the first is statistical shape analysis of the RV, and the second is inverse characterization of heart wall mechanical material properties from RV shape change and measurable hemodynamics. For the statistical shape analysis investigation, a custom approach using harmonic mapping and proper orthogonal decomposition is applied to determine the fundamental components of shape (i.e., modes) from a dataset of 50 patients with varying states of PH, including some without PH at all. For the inverse characterization work, a novel method was developed to estimate the heterogeneous properties of a structure, given only the target shape of that structure, after a known excitation is applied to deform the structure. Lastly, the inverse characterization algorithm was extended to be applicable to actual in vivo cardiac data, particularly through the inclusion of a registration step to account for the organ-scale rotation and translation of the heart. Future work remains to expand on the computational efficiency of this inverse solution estimation procedure, and to further evaluate and improve upon the consistency and clinical interpretability of the material property estimates

Large Scale Parameter Estimation Problems in Frequency-Domain Elastodynamics Using an Error in Constitutive Equation Functional

International audienceThis paper presents the formulation and implementation of an Error in Constitutive Equations (ECE) method suitable for large-scale inverse identification of linear elastic material properties in the context of steady-state elastodynamics. In ECE-based methods, the inverse problem is postulated as an optimization problem in which the cost functional measures the discrepancy in the constitutive equations that connect kinematically admissible strains and dynamically admissible stresses. Furthermore, in a more recent modality of this methodology introduced by Feissel and Allix (2007), referred to as the Modified ECE (MECE), the measured data is incorporated into the formulation as a quadratic penalty term. We show that a simple and efficient continuation scheme for the penalty term, suggested by the theory of quadratic penalty methods, can significantly accelerate the convergence of the MECE algorithm. Furthermore, a (block) successive over-relaxation (SOR) technique is introduced, enabling the use of existing parallel finite element codes with minimal modification to solve the coupled system of equations that arises from the optimality conditions in MECE methods. Our numerical results demonstrate that the proposed methodology can successfully reconstruct the spatial distribution of elastic material parameters from partial and noisy measurements in as few as ten iterations in a 2D example and fifty in a 3D example. We show (through numerical experiments) that the proposed continuation scheme can improve the rate of convergence of MECE methods by at least an order of magnitude versus the alternative of using a fixed penalty parameter. Furthermore, the proposed block SOR strategy coupled with existing parallel solvers produces a computationally efficient MECE method that can be used for large scale materials identification problems, as demonstrated on a 3D example involving about 400,000 unknown moduli. Finally, our numerical results suggest that the proposed MECE approach can be significantly faster than the conventional approach of L2 minimization using quasi-Newton methods