Solution of the nonlinear elasticity imaging inverse problem: The incompressible case

We have recently developed and tested an efficient algorithm for solving the nonlinear inverse elasticity problem for a compressible hyperelastic material. The data for this problem are the quasi-static deformation fields within the solid measured at two distinct overall strain levels. The main ingredients of our algorithm are a gradient based quasi-Newton minimization strategy, the use of adjoint equations and a novel strategy for continuation in the material parameters. In this paper we present several extensions to this algorithm. First, we extend it to incompressible media thereby extending its applicability to tissues which are nearly incompressible under slow deformation. We achieve this by solving the forward problem using a residual-based, stabilized, mixed finite element formulation which circumvents the Ladyzenskaya-Babuska-Brezzi condition. Second, we demonstrate how the recovery of the spatial distribution of the nonlinear parameter can be improved either by preconditioning the system of equations for the material parameters, or by splitting the problem into two distinct steps. Finally, we present a new strain energy density function with an exponential stress-strain behavior that yields a deviatoric stress tensor, thereby simplifying the interpretation of pressure when compared with other exponential functions. We test the overall approach by solving for the spatial distribution of material parameters from noisy, synthetic deformation fields.     

Time assignment speech interpolation system—a microprocessor based design

A normal synchronous time multiplexed system has low capacity utilisation of a trunk channel, owing to the ON-OFF nature of speech in human conversation. A time assignment speech interpolation (TASI) system nearly doubles this efficiency, at relatively small increase in hardware cost. In TASI, during the silent period of speech, the channel is allotted to some other ‘active’ subscriber on a first-come-first-serve basis. A microprocessor based system offers a very cost effective solution from hardware count. Intel 8085 A has been selected for the purpose. The microprocessor functions in a distributed processing mode together with the main CPU controlling the stored program exchange. The system uses digital dynamic speech detectors for detecting transitions in speech over a channel, as they show considerably superior performance over amplitude detectors. An assembly language program for the system has been developed.     

Linear and nonlinear elastic modulus imaging: an application to breast cancer diagnosis

We reconstruct the in vivo spatial distribution of linear and nonlinear elastic parameters in ten patients with benign (five) and malignant (five) tumors. The mechanical behavior of breast tissue is represented by a modified Veronda-Westmann model with one linear and one nonlinear elastic parameter. The spatial distribution of these elastic parameters is determined by solving an inverse problem within the region of interest (ROI). This inverse problem solution requires the knowledge of the displacement fields at small and large strains. The displacement fields are measured using a free-hand ultrasound strain imaging technique wherein, a linear array ultrasound transducer is positioned on the breast and radio frequency echo signals are recorded within the ROI while the tissue is slowly deformed with the transducer. Incremental displacement fields are determined from successive radio-frequency frames by employing cross-correlation techniques. The rectangular regions of interest were subjectively selected to obtain low noise displacement estimates and therefore were variables that ranged from 346 to 849.6 mm2 . It is observed that malignant tumors stiffen at a faster rate than benign tumors and based on this criterion nine out of ten tumors were correctly classified as being either benign or malignant.     

Material parameter optimization for interior and exterior fluid-structure acoustic problems

We present an efficient adjoint-based framework for computing sensitivities of quantities of interest with respect to material parameters for coupled fluid-structural acoustic systems with explicit interface coupling. The fluid is modeled using the Helmholtz equation and the structure is modeled using the Navier-Cauchy equations. Sensitivities are used to drive a gradient based optimization algorithm to solve important problems in structural acoustics, viz noise minimization and vibration isolation. For each problem, we consider two different priors: one where the optimal solution has a smooth variation and another with a bimaterial distribution. These priors are imposed with the help of suitable regularization terms. The effectiveness of this approach is demonstrated on both interior and exterior structural acoustic problems.     

Adjoint‐weighted variational formulation for the direct solution of inverse problems of general linear elasticity with full interior data

We describe a novel variational formulation of inverse elasticity problems given interior data. The class of problems considered is rather general and includes, as special cases, plane deformations, compressibility and incompressiblity in isotropic materials, 3D deformations, and anisotropy. The strong form of this problem is governed by equations of pure advective transport. The variational formulation is based on a generalization of the adjoint‐weighted variational equation (AWE) formulation, originally developed for flow of a passive scalar. We describe how to apply AWE to various cases, and prove several properties. We prove that the Galerkin discretization of the AWE formulation leads to a stable, convergent numerical method, and prove optimal rates of convergence. The numerical examples demonstrate optimal convergence of the method with mesh refinement for multiple unknown material parameters, graceful performance in the presence of noise, and robust behavior of the method when the target solution is C^∞, C⁰, or discontinuous. Copyright © 2009 John Wiley & Sons, Ltd.     

Spectral analysis of turbulence based on the DNS of a channel flow

The development and assessment of spectral turbulence models requires knowledge of the spectral turbulent kinetic energy distribution as well as an understanding of the terms which determine the energy distribution in physical and wave number space. Direct numerical simulation (DNS) of turbulent channel flow yields numerical “data” that can be, and was, analyzed using a spatial Fast Fourier Transform (FFT) to obtain the various spectral turbulent kinetic energy balance terms, including the production, dissipation, diffusion, and the non-linear convective transfer terms.     

Direct error in constitutive equation formulation for inverse heat conduction problem

We present a mixed numerical formulation that handles discontinuities well for scalar hyperbolic partial differential equations. The formulation is based on a least‐square error in the constitutive equation. It is motivated by scalar inverse diffusion problems with interior data and applies to convection of a passive scalar in a discontinuous compressible flow field. We motivate the need for a mixed formulation by discretizing using an irreducible finite element method and discuss some of the limitations of that approach. We then develop and prove that the mixed formulation is well posed and verify that it works for problems with continuous and discontinuous thermal conductivity distributions.     

Three‐dimensional hydrogel encapsulated embryonic stem and carcinoma cells as culture platforms for cytotoxicity studies

The utility of alginate hydrogels for three‐dimensional (3‐D) culture of mouse embryonic stem cells (mESCs) and future development of 3‐D stem cell culture‐based in vitro screens of toxicity is described. Using alginate hydrogels of various stiffness, we first evaluated the impact of substrate modulus on mESC viability, proliferation, as well as expression of pluripotency and germ‐layer markers and observed that low concentration alginate hydrogels (0.5% and 1% alginate) were most suitable for long‐term culture of mESCs. These results were not unique to mESCs; long‐term viability and proliferation of mouse embryonic carcinoma cells (mECCs) was also best supported by similar conditions. Finally, we determined cytotoxic responses of alginate encapsulated cells to commercially available chemicals and interestingly observed similar responses for mESCs and mECCs, thereby suggesting that mECCs can predict stem cell responses to chemicals. These studies will facilitate future design of optimal stem cell‐based platforms of organ‐specific and developmental toxicity. © 2015 American Institute of Chemical Engineers AIChE J, 61: 3180–3184, 2015