Homogenization of the wave equation with non-uniformly oscillating coefficients

The focus of our work is a dispersive, second-order effective model describing the low-frequency wave motion in heterogeneous (e.g., functionally graded) media endowed with periodic microstructure. For this class of quasi-periodic medium variations, we pursue homogenization of the scalar wave equation in [Formula: see text], [Formula: see text], within the framework of multiple scales expansion. When either [Formula: see text] or [Formula: see text], this model problem bears direct relevance to the description of (anti-plane) shear waves in elastic solids. By adopting the lengthscale of microscopic medium fluctuations as the perturbation parameter, we synthesize the germane low-frequency behavior via a fourth-order differential equation (with smoothly varying coefficients) governing the mean wave motion in the medium, where the effect of microscopic heterogeneities is upscaled by way of the so-called cell functions. In an effort to demonstrate the relevance of our analysis toward solving boundary value problems (deemed to be the ultimate goal of most homogenization studies), we also develop effective boundary conditions, up to the second order of asymptotic approximation, applicable to one-dimensional (1D) shear wave motion in a macroscopically heterogeneous solid with periodic microstructure. We illustrate the analysis numerically in one dimension by considering (i) low-frequency wave dispersion, (ii) mean-field homogenized description of the shear waves propagating in a finite domain, and (iii) full-field homogenized description thereof. In contrast to (i) where the overall wave dispersion appears to be fairly well described by the leading-order model, the results in (ii) and (iii) demonstrate the critical role that higher-order corrections may have in approximating the actual waveforms in quasi-periodic media.

Results of studying the azimuthal anisotropy of the paleozoic basement with vsp technique at the Bystrovka vibroseismic test site

At the Bystrovka vibroseismic test site (Novosibirsk region) 3-component refracted waves profiling was performed at three intersecting lines. Shear waves analysis made possible to detect anisotropy of the Paleozoic basement occurring at depth of about 10 m and to suggest symmetry elements of this medium along with their orientation. Compressional waves data were used to construct depth sections estimating head waves velocities. These velocities demonstrate significant variation in lines of different orientation. The results obtained agree with previously performed VSP.

Cardiac shear wave elastography can distinguish healthy and scarred myocardium in patients with conduction delays

Background Cardiac resynchronization therapy (CRT) is an established therapy for patients suffering from heart failure and left bundle branch block (LBBB) conduction delays. Despite its proven beneficial effects, CRT is associated with a high percentage of non-response. Since CRT has shown to be less effective in patients with ischemic cardiomyopathy, determining the presence of myocardial scar before implantation could help to improve the response-rate. However, the gold standard to assess myocardial scar, magnetic resonance imaging (MRI), cannot be used in every patient, due to already implanted devices and/or reduced renal function. Recently introduced shear wave elastography (SWE) allows the non-invasive assessment of myocardial stiffness. Natural shear waves are excited by mitral valve closure (MVC) and travel through the heart with a speed directly related to tissue stiffness. SWE has previously been proven to be able to detect myocardial scar, however this has never been shown in the presence LBBB. Purpose The aim of this study was to evaluate the capability of SWE as a novel method to determine myocardial scar in patients with conduction delays. Methods We included 24 heart failure patients (age: 68±10; 50% males) with ischemic (n=8) and non-ischemic (n=16) cardiomyopathy. The CRT device was set to AAI mode in order to obtain native ventricular conduction. For patients with ischemic cardiomyopathy, the presence and location of scar was determined by MRI or scintigraphy. All ischemic patients had septal scar only. For SWE, left ventricular parasternal long-axis views were acquired with an experimental high frame rate ultrasound scanner (average frame rate: ±1200 Hz). Shear waves were visualized in M-modes of the septum, colour coded for tissue acceleration. The slope of the shear waves in the M-mode represents their propagation speed (Figure A). Results There was no significant difference between the ischemic and non-ischemic patients in QRS width after CRT (149±31 ms vs 144±26 ms), systolic blood pressure blood pressure (135±11 mmHg vs 135±23 mmHg), diastolic blood pressure (74±9 mmHg vs 70±11 mmHg) and heart rate (58±4 bpm vs 63±9 bpm) (all p>0.05). Ejection fraction (33±8% vs 45±10%), end-diastolic volume (196±34 ml vs 129±64 ml) and global longitudinal strain (−9.8±3.1% vs −14.1±4.1%) differed significantly between the groups (all p<0.05). Shear wave speed after MVC was significantly higher in patients with septal scar compared to non-ischemic patients (8.2±1.9 m/s vs 5.5±1.2 m/s; p<0.01) (Figure B). Conclusion In the presence of scar, we found markedly elevated shear wave propagation speed compared to non-ischemic patients. These results indicate that SWE is able to identify scarred myocardium even in patients with LBBB. We therefore believe that SWE could be a novel easy and non-invasive method to evaluate septal myocardial scarring in patients before CRT implantation. FUNDunding Acknowledgement Type of funding sources: Public grant(s) – National budget only. Main funding source(s): FWO - Research Foundation Flanders

Mechanics Of Ultrasonic Neuromodulation In A Mouse Subject

Ultrasound neuromodulation (UNM), where a region in the brain is targeted by focused ultrasound (FUS), which, in turn, causes excitation or inhibition of neural activity, has recently received considerable attention as a promising tool for neuroscience. Despite its great potential, several aspects of UNM are still unknown. An important question pertains to the off-target sensory effects of UNM and their dependence on stimulation frequency. To understand these effects, we have developed a finite-element model of a mouse, including elasticity and viscoelasticity, and used it to interrogate the response of mouse models to focused ultrasound (FUS). We find that, while some degree of focusing and magnification of the signal is achieved within the brain, the induced pressure-wave pattern is complex and delocalized. In addition, we find that the brain is largely insulated, or ‘cloaked’, from shear waves by the cranium and that the shear waves are largely carried away from the skull by the vertebral column, which acts as a waveguide. We find that, as expected, this waveguide mechanism is strongly frequency dependent, which may contribute to the frequency dependence of UNM effects. Our calculations further suggest that off-target skin locations experience displacements and stresses at levels that, while greatly attenuated from the source, could nevertheless induce sensory responses in the subject.

Approximations to a generalized real ray-tracing method for heterogeneous anisotropic viscoelastic media

The real raytracing approach leads to an effective solution in the real space domain using a homogenous ray velocity vector. However, it fails to yield solutions for quasi-shear waves, which suffer triplication of the wavefronts. To address this challenging problem, a generalized real ray-tracing method and its new approximations are presented to solve the complex ray equation. The numerical results show that the generalized ray-tracing method is superior to the real ray-tracing method in the presence of triplications of the quasi-shear waves in the computation of ray velocity, ray attenuation, and ray quality factors, as well as the reflection and transmission coefficients in viscoelastic anisotropic media. Based on the assumptions of the real slowness direction and real polarization vectors, two new approximations of the generalized real ray-tracing method are developed for directly computing the homogeneous complex ray velocity vectors of three wave modes (qP, qS1, and qS2). These approximations significantly improve the computational efficiency by avoiding the iterative process required by the generalized real ray-tracing method that is inherited from the real ray-tracing method. The computational accuracies are verified through transversely isotropic models and orthorhombic models with different strengths of attenuation and anisotropy. The incorporation of the new approximation into the shortest-path method turned out to be an efficient and accurate method for seismic ray tracing in heterogeneous viscoelastic and transversely isotropic media with a vertical axis of symmetry, even in the presence of strong attenuation and anisotropy.