A 3-dimensional singular kernel problem in viscoelasticity: an existence result

Materials with memory, namely those materials whose mechanical and/or thermodynamical behaviour depends on time not only via the present time, but also through its past history, are considered. Specifically, a three dimensional viscoelastic body is studied. Its mechanical behaviour is described via an integro-differential equation, whose kernel represents the relaxation modulus, characteristic of the viscoelastic material under investigation. According to the classical model, to guarantee the thermodynamical compatibility of the model itself, such a kernel satisfies regularity conditions which include the integrability of its time derivative. To adapt the model to a wider class of materials, this condition is relaxed; that is, conversely to what is generally assumed, no integrability condition is imposed on the time derivative of the relaxation modulus. Hence, the case of a relaxation modulus which is unbounded at the initial time t = 0, is considered, so that a singular kernel integro-differential equation, is studied. In this framework, the existence of a weak solution is proved in the case of a three dimensional singular kernel initial boundary value problem.Comment: 15 page

Methods and Algorithms for Cardiovascular Hemodynamics with Applications to Noninvasive Monitoring of Proximal Blood Pressure and Cardiac Output Using Pulse Transit Time

Advanced health monitoring and diagnostics technology are essential to reduce the unrivaled number of human fatalities due to cardiovascular diseases (CVDs). Traditionally, gold standard CVD diagnosis involves direct measurements of the aortic blood pressure (central BP) and flow by cardiac catheterization, which can lead to certain complications. Understanding the inner-workings of the cardiovascular system through patient-specific cardiovascular modeling can provide new means to CVD diagnosis and relating treatment. BP and flow waves propagate back and forth from heart to the peripheral sites, while carrying information about the properties of the arterial network. Their speed of propagation, magnitude and shape are directly related to the properties of blood and arterial vasculature. Obtaining functional and anatomical information about the arteries through clinical measurements and medical imaging, the digital twin of the arterial network of interest can be generated. The latter enables prediction of BP and flow waveforms along this network. Point of care devices (POCDs) can now conduct in-home measurements of cardiovascular signals, such as electrocardiogram (ECG), photoplethysmogram (PPG), ballistocardiogram (BCG) and even direct measurements of the pulse transit time (PTT). This vital information provides new opportunities for designing accurate patient-specific computational models eliminating, in many cases, the need for invasive measurements. One of the main efforts in this area is the development of noninvasive cuffless BP measurement using patient’s PTT. Commonly, BP prediction is carried out with regression models assuming direct or indirect relationships between BP and PTT. However, accounting for the nonlinear FSI mechanics of the arteries and the cardiac output is indispensable. In this work, a monotonicity-preserving quasi-1D FSI modeling platform is developed, capable of capturing the hyper-viscoelastic vessel wall deformation and nonlinear blood flow dynamics in arbitrary arterial networks. Special attention has been dedicated to the correct modeling of discontinuities, such as mechanical properties mismatch associated with the stent insertion, and the intertwining dynamics of multiscale 3D and 1D models when simulating the arterial network with an aneurysm. The developed platform, titled Cardiovascular Flow ANalysis (CardioFAN), is validated against well-known numerical, in vitro and in vivo arterial network measurements showing average prediction errors of 5.2%, 2.8% and 1.6% for blood flow, lumen cross-sectional area, and BP, respectively. CardioFAN evaluates the local PTT, which enables patient-specific calibration and its application to input signal reconstruction. The calibration is performed based on BP, stroke volume and PTT measured by POCDs. The calibrated model is then used in conjunction with noninvasively measured peripheral BP and PTT to inversely restore the cardiac output, proximal BP and aortic deformation in human subjects. The reconstructed results show average RMSEs of 1.4% for systolic and 4.6% for diastolic BPs, as well as 8.4% for cardiac output. This work is the first successful attempt in implementation of deterministic cardiovascular models as add-ons to wearable and smart POCD results, enabling continuous noninvasive monitoring of cardiovascular health to facilitate CVD diagnosis

A Review on Fractional Differential Equations and a Numerical Method to Solve Some Boundary Value Problems

Fractional differential equations can describe the dynamics of several complex and nonlocal systems with memory. They arise in many scientific and engineering areas such as physics, chemistry, biology, biophysics, economics, control theory, signal and image processing, etc. Particularly, nonlinear systems describing different phenomena can be modeled with fractional derivatives. Chaotic behavior has also been reported in some fractional models. There exist theoretical results related to existence and uniqueness of solutions to initial and boundary value problems with fractional differential equations; for the nonlinear case, there are still few of them. In this work we will present a summary of the different definitions of fractional derivatives and show models where they appear, including simple nonlinear systems with chaos. Existing results on the solvability of classical fractional differential equations and numerical approaches are summarized. Finally, we propose a numerical scheme to approximate the solution to linear fractional initial value problems and boundary value problems

Investigation of Flow Disturbances and Multi-Directional Wall Shear Stress in the Stenosed Carotid Artery Bifurcation Using Particle Image Velocimetry

Hemodynamics and shear forces are associated with pathological changes in the vascular wall and its function, resulting in the focal development of atherosclerosis. Flow complexities that develop in the presence of established plaques create environments favourable to thrombosis formation and potentially plaque rupture leading to stroke. The carotid artery bifurcation is a common site of atherosclerosis development. Recently, the multi-directional nature of shear stress acting on the endothelial layer has been highlighted as a risk factor for atherogenesis, emphasizing the need for accurate measurements of shear stress magnitude as well direction. In the absence of comprehensive patient specific datasets numerical simulations of hemodynamics are limited by modeling assumptions. The objective of this thesis was to investigate the relative contributions of various factors - including geometry, rheology, pulsatility, and compliance – towards the development of disturbed flow and multi-directional wall shear stress (WSS) parameters related to the development of atherosclerosis An experimental stereoscopic particle image velocimetry (PIV) system was used to measure instantaneous full-field velocity in idealized asymmetrically stenosed carotid artery bifurcation models, enabling the extraction of bulk flow features and turbulence intensity (TI). The velocity data was combined with wall location information segmented from micro computed tomography (CT) to obtain phase-averaged maps of WSS magnitude and direction. A comparison between Newtonian and non-Newtonian blood-analogue fluids demonstrated that the conventional Newtonian viscosity assumption underestimates WSS magnitude while overestimating TI. Studies incorporating varying waveform pulsatility demonstrated that the levels of TI and oscillatory shear index (OSI) depend on the waveform amplitude in addition to the degree of vessel constriction. Local compliance resulted in a dampening of disturbed flow due to volumetric capacity of the upstream vessel, however wall tracking had a negligible effect on WSS prediction. While the degree of stenosis severity was found to have a dominant effect on local hemodynamics, comparable relative differences in metrics of flow and WSS disturbances were found due to viscosity model, waveform pulsatility and local vessel compliance

Dynamic radial deformations of nonlinear elastic structures. On the influence of constitutive modeling

Mención Internacional en el título de doctorThe objective of this dissertation is to develop a comprehensive theoretical approach to the role of the constitutive model on the dynamic radial deformations of nonlinear elastic structures. Using 1D and 2D models, cylindrical and spherical thick-walled shells are considered. These geometries are representative of man-made and natural structures that can be found in a wide variety of engineering applications and biological systems. Lead-rubber bearings, vibration isolators, peristaltic pumps, rubber bushings, saccular aneurysms or arteries are examples of nonlinear elastic structures with spherical and cylindrical geometries that are constantly subjected to all kinds of vibrational and dynamic loads. The research, which starts by considering isotropic, incompressible and rate independent constitutive models, is based on the systematic incorporation of compressibility, viscosity and anisotropy in the description of the mechanical response of the material. We investigate free and forced vibrations using different initial and boundary conditions: (1) ab initio elastic stored and kinetic energies, (2) constant radial pressures, (3) linearly time dependent radial pressures and (4) harmonic time dependent radial pressures. While the isotropic and incompressible 1D elastic structures subjected to constant pressure admit an analytical closed-form solution, all the other cases need to be solved numerically. To this end, we have developed in this work a number of specific numerical schemes. The overall outcome of this dissertation is to make it plain that the constitutive model used to describe the mechanical behavior of thick-walled shells plays a fundamental role in the nonlinear dynamics of such structures. In particular, we have demonstrated the influence of the constitutive model on: (1) the loss of oscillatory behavior of the structure, (2) the transition from periodic motions to quasi-periodic and chaotic motions, (3) the nonlinear resonances of the shells, (4) the propagation of shock waves within the structure and (5) the onset and development of cavitation instabilities.Programa Oficial de Doctorado en Ingeniería Mecánica y de Organización IndustrialPresidente: Ignacio Romero Olleros.- Secretario: Massimo Ruzzene.- Vocal: Antonio Morass