A Nitsche-based cut finite element method for a fluid--structure interaction problem

We present a new composite mesh finite element method for fluid--structure interaction problems. The method is based on surrounding the structure by a boundary-fitted fluid mesh which is embedded into a fixed background fluid mesh. The embedding allows for an arbitrary overlap of the fluid meshes. The coupling between the embedded and background fluid meshes is enforced using a stabilized Nitsche formulation which allows us to establish stability and optimal order \emph{a priori} error estimates, see~\cite{MassingLarsonLoggEtAl2013}. We consider here a steady state fluid--structure interaction problem where a hyperelastic structure interacts with a viscous fluid modeled by the Stokes equations. We evaluate an iterative solution procedure based on splitting and present three-dimensional numerical examples.Comment: Revised version, 18 pages, 7 figures. Accepted for publication in CAMCo

Bilinear modeling and nonlinear estimation

New methods are illustrated for online nonlinear estimation applied to the lateral deflection of an elastic beam on board measurements of angular rates and angular accelerations. The development of the filter equations, together with practical issues of their numerical solution as developed from global linearization by nonlinear output injection are contrasted with the usual method of the extended Kalman filter (EKF). It is shown how nonlinear estimation due to gyroscopic coupling can be implemented as an adaptive covariance filter using off-the-shelf Kalman filter algorithms. The effect of the global linearization by nonlinear output injection is to introduce a change of coordinates in which only the process noise covariance is to be updated in online implementation. This is in contrast to the computational approach which arises in EKF methods arising by local linearization with respect to the current conditional mean. Processing refinements for nonlinear estimation based on optimal, nonlinear interpolation between observations are also highlighted. In these methods the extrapolation of the process dynamics between measurement updates is obtained by replacing a transition matrix with an operator spline that is optimized off-line from responses to selected test inputs

2-D and 3-D high frame-rate Pulse Wave Imaging for the characterization of focal vascular disease

Cardiovascular diseases are major causes of morbidity and mortality in Western-style populations. Atherosclerosis and Abdominal Aortic Aneurysms (AAAs) are two prevalent vascular diseases that may progress without symptoms and contribute to acute cardiovascular events such as stroke and AAA rupture, which are consistently among the leading causes of death worldwide. The imaging methods used in the diagnosis of these diseases, have been reported to present several limitations. Given that both are associated with mechanical changes in the arterial wall, imaging of the arterial mechanical properties may improve early disease detection and patient care. Pulse wave velocity (PWV) refers to the velocity at which arterial waves generated by ventricular ejection travel along the arterial tree. PWV is a surrogate marker of arterial stiffness linked to cardiovascular mortality. The foot-to-foot method that is typically used to calculate PWV suffers from errors of distance measurements and time-delay measurements. Additionally, a single PWV estimate is provided over a relatively long distance, thus inherently lacking the capability to provide regional arterial stiffness measurements. Pulse Wave Imaging (PWI) is a noninvasive, ultrasound-based technique for imaging the propagation of pulse waves along the wall of major arteries and providing a regional PWV value for the imaged artery. The aim of this work was to enable PWI to provide more localized PWV and stiffness measurements within the imaged arterial segment and to further extend it into a 2-D and 3-D technique for the detection and monitoring of focal vascular disease at high temporal and spatial resolution. The improved modality was integrated with blood flow imaging modalities aiming to render PWI a comprehensive methodology for the study of arterial biomechanics in vivo. Spatial information was increased with the introduction of piecewise PWI. This novel technique was used to measure PWV within small sub-regions of the imaged vessel in murine aneurysmal (n = 8) and atherosclerotic aortas (n = 11) in vivo. It provided PWV and stiffness maps while capturing the progressive arterial stiffening caused by atherosclerosis. PWI was further augmented with a sophisticated adaptive algorithm, enabling it to optimally partition the imaged artery into relatively homogeneous segments, automatically isolating arterial stiffness inhomogeneities. Adaptive PWI was validated in silicone phantoms consisting of segments of varying stiffness and then tested in murine aortas in vivo. Subsequently, the conventional tradeoff between spatial and temporal resolution was addressed with a plane wave compounding implementation of PWI, allowing the acquisition of full field of view frames at over 2000 Hz. A GPU-accelerated PWI post-processing framework was developed for the processing of the big bulk of generated data. The parameters of coherent compounding were optimized in vivo. The optimized sequences were then used in the clinic to assess the mechanical properties of atherosclerotic carotids (n=10) and carotids of patients after endarterectomy (n=7), a procedure to remove the plaque and restore blood flow to the brain. In the case of atherosclerotic patients undergoing carotid endarterectomy, the results were compared against the histology of the excised plaques. Investigation of the mechanical properties of plaques was also conducted for the first time with a high-frequency transducer (18.5 MHz). Additionally, 4-D PWI was introduced, utilizing high frame rate 3-D plane wave acquisitions with a 2-D matrix array transducer (16x16 elements, 2.5 MHz). A novel methodology for PWV estimation along the direction of pulse wave propagation was implemented and validated in silicone phantoms. 4-D PWI provided comprehensive views of the pulse wave propagation in a plaque phantom and the results were compared against conventional PWI. Finally, its feasibility was tested in the carotid arteries of healthy human subjects (n=6). PWVs derived in 3-D were within the physiological range and showed good agreement with the results of conventional PWI. Finally, PWI was integrated with flow imaging modalities (Color and Vector Doppler). Thus, full field-of-view, high frame-rate, simultaneous and co-localized imaging of the arterial wall dynamics and color flow as well as 2-D vector flow was implemented. The feasibility of both techniques was tested in healthy subjects (n=6) in vivo. The relationship between the timings of the flow and wall velocities was investigated at multiple locations of the imaged artery. Vector flow velocities were found to be aligned with the vessel’s centerline during peak systole in the common carotid artery and interesting flow patterns were revealed in the case of the carotid bifurcation Consequently, with the aforementioned improvements and the inclusion of 3-D imaging, PWI is expected to provide comprehensive information on the mechanical properties of pathological arteries, providing clinicians with a powerful tool for the early detection of vascular abnormalities undetectable on the B-mode, while also enabling the monitoring of fully developed vascular pathology and of the recovery of post-operated vessels