Simulation of nonlinear transient elastography: finite element model for the propagation of shear waves in homogeneous soft tissues

Linear, weakly nonlinear and fully nonlinear models for soft tissues: which ones provide the most reliable estimations of the stiffness?

Benign and malignant lesions in tissues or organs can be detected by elastographic investigations in which pathological regions are spotted from local alterations of the stiffness. As is known, the shear modulus provides a measure of the stiffness of an elastic material. Based on the classical theory of linear elasticity, an elastogram yields estimations of the linear shear modulus from measurements of the speed of small-amplitude transverse waves propagating in the medium tested. In this paper, we show that the estimation of the shear modulus can be improved significantly by employing the fourth-order weakly nonlinear theory of elasticity (FOE), and indicate how the stiffness can be assessed more precisely with the use of FOE. We discuss also why FOE provides more reliable results than the fully nonlinear theory of elasticity. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.

Modeling Ultrasound Propagation in the Moving Brain: Applications to Shear Shock Waves and Traumatic Brain Injury

Traumatic brain injury (TBI) studies on the living human brain are experimentally infeasible due to ethical reasons and the elastic properties of the brain degrade rapidly postmortem. We present a simulation approach that models ultrasound propagation in the human brain, while it is moving due to the complex shear shock wave deformation from a traumatic impact. Finite difference simulations can model ultrasound propagation in complex media such as human tissue. Recently, we have shown that the fullwave finite difference approach can also be used to represent displacements that are much smaller than the grid size, such as the motion encountered in shear wave propagation from ultrasound elastography. However, this subresolution displacement model, called impedance flow, was only implemented and validated for acoustical media composed of randomly distributed scatterers. Herein, we propose a generalization of the impedance flow method that describes the continuous subresolution motion of structured acoustical maps, and in particular of acoustical maps of the human brain. It is shown that the average error in simulating subresolution displacements using impedance flow is small when compared to the acoustical wavelength ( λ /1702). The method is then applied to acoustical maps of the human brain with a motion that is imposed by the propagation of a shear shock wave. This motion is determined numerically with a custom piecewise parabolic method that is calibrated to ex vivo observations of shear shocks in the porcine brain. Then the fullwave simulation tool is used to model transmit-receive imaging sequences based on an L7-4 imaging transducer. The simulated radio frequency data are beamformed using a conventional delay-and-sum method and a normalized cross-correlation method designed for shock wave tracking is used to determine the tissue motion. This overall process is an in silico reproduction of the experiments that were previously performed to observe shear shock waves in fresh porcine brain. It is shown that the proposed generalized impedance flow method accurately captures the shear wave motion in terms of the wave profile, shock front characteristics, odd harmonic spectrum generation, and acceleration at the shear shock front. We expect that this approach will lead to improvements in image sequence design that takes into account the aberration and multiple reflections from the brain and in the design of tracking algorithms that can more accurately capture the complex brain motion that occurs during a traumatic impact. These methods of modeling ultrasound propagation in moving media can also be applied to other displacements, such as those generated by shear wave elastography or blood flow.

Soft elastic hydrogel couplants for ultrasonography

Couplants play significant roles in ultrasonography. To ensure imaging quality, it is critical to maintain conformal contact of the couplant with both the skin surface and the ultrasound probe in clinical applications. In addition, either the probe or the couplant should not deform the skin surface significantly, which will result in an overestimated modulus of the tissue for elastography imaging. However, existing liquid gel couplants cannot bear external compressive force, while existing solid gel couplants cannot maintain a conformal contact with skin surface. Especially, the nonconformal contacts and deformation become more severe on body parts of locally high curvatures such as skin tumors, fingers, and elbows. Here we report a bilayer design of couplant for ultrasonography, composing of a stiff layer and a compliant layer of hydrogels. The bilayer hydrogel pad enables it to bear external compression, allowing the probe to move smoothly, conforming high curvature parts and releasing stress concentration. Our clinical experiments further show high quality imaging of thyroid nodules, skin tumors in elbows and fingers using the bilayer hydrogel pad, which represents a promising alternative for a range of applications in ultrasonic diagnosis.

Characterization of non-linear mechanical behavior of the cornea

The objective of this study was to evaluate which hyperelastic model could best describe the non-linear mechanical behavior of the cornea, in order to characterize the capability of the non-linear model parameters to discriminate structural changes in a damaged cornea. Porcine corneas were used, establishing two different groups: control (non-treated) and NaOH-treated (damaged) corneas (n = 8). NaOH causes a chemical burn to the corneal tissue, simulating a disease associated to structural damage of the stromal layer. Quasi-static uniaxial tensile tests were performed in nasal-temporal direction immediately after preparing corneal strips from the two groups. Three non-linear hyperelastic models (i.e. Hamilton-Zabolotskaya model, Ogden model and Mooney-Rivlin model) were fitted to the stress–strain curves obtained in the tensile tests and statistically compared. The corneas from the two groups showed a non-linear mechanical behavior that was best described by the Hamilton-Zabolotskaya model, obtaining the highest coefficient of determination (R² > 0.95). Moreover, Hamilton-Zabolotskaya model showed the highest discriminative capability of the non-linear model parameter (Parameter A) for the tissue structural changes between the two sample groups (p = 0.0005). The present work determines the best hyperelastic model with the highest discriminative capability in description of the non-linear mechanical behavior of the cornea.

Piecewise parabolic method for propagation of shear shock waves in relaxing soft solids: One-dimensional case

Shear shock waves can be generated spontaneously deep within the brain during a traumatic injury. This recently observed behavior could be a primary mechanism for the generation of traumatic brain injuries. However, shear shock wave physics and its numerical modeling are relatively unstudied. Existing numerical solvers used in biomechanics are not designed for the extremely large Mach numbers (greater than 1) observed in the brain. Furthermore, soft solids, such as the brain, have a complex nonclassical viscoleastic response, which must be accurately modeled to capture the nonlinear wave behavior. Here, we develop a 1D inviscid velocity-stress-like system to model the propagation of shear shock waves in a homogeneous medium. Then a generalized Maxwell body is used to model a relaxing medium that can describe experimentally determined attenuation laws. Finally, the resulting system is solved numerically with the piecewise parabolic method, a high-order finite volume method. The nonlinear and the relaxing components of this method are validated with theoretical predictions. Comparisons between numerical solutions obtained for the proposed model and the experiments of plane shear shock wave propagation based on high frame-rate ultrasound imaging and tracking are shown to be in excellent agreement.