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Publication# One dimensional and multiscale models for blood flow circulation

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Résumé

The aim of this work is to provide mathematically sound and computationally effective tools for the numerical simulation of the interaction between fluid and structures as occurring, for instance, in the simulation of the human cardiovascular system. This problem is global, in the sense that local changes can modify the solution far away. From the point of view of computing and modelling this calls for the use of multiscale methods, where simplified models are used to treat the global problem leaving to more accurate models the local description. Moreover it is characterised by the appearance of pressure waves inside the vessels. In large arteries the vessel wall dynamics can be described by a thin elastic membrane model (Navier equation) while the fluid motion can be represented by the Navier-Stokes equations for incompressible Newtonian fluids. Unfortunately, given the high levels of details furnished by this model, its computational complexity is dramatically high. Therefore reduced models have been developed. In particular, one-dimensional models, originally introduced by Euler, seem to be appropriate for the study in the time-space domain of pressure wave propagation induced by the interaction between the fluid and the vessel wall in the arterial tree. These reduced models are obtained after integrating the Navier-Stokes equations over a vessel section, supposed to be circular, and assuming an algebraic wall law to describe the relationship between pressure and wall deformation. They can be used in place of the more complex three dimensional fluid-structure models or in cooperation with them (multiscale approach). The first part of this work deals with one dimensional models. A reduced 1D model taken from literature is presented and analysed. Some extensions of the basic model, in particular with respect to vessel wall law (generalised string model) and more complex geometries (bifurcated and curved arteries), are also considered. Numerical schemes are proposed and some numerical results are presented. In the second part of this thesis we focus on a multiscale model. We consider a 55 arterial tree, described by the 1D model, coupled with lumped parameter models for heart and capillaries. In particular, specific attention has been devoted to the coupling between the left ventricle and the arterial system, whose physiopathological relevance is well known. This mathematical model gives good results in numerical tests and is able to describe the relevant features of the pressure wave propagation and reflections within the arterial system.

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Mathematical and numerical aspects of free surface flows are investigated. On one hand, the mathematical analysis of some free surface flows is considered. A model problem in one space dimension is first investigated. The Burgers equation with diffusion has to be solved on a space interval with one free extremity. This extremity is unknown and moves in time. An ordinary differential equation for the position of the free extremity of the interval is added in order to close the mathematical problem. Local existence in time and uniqueness results are proved for the problem with given domain, then for the free surface problem. A priori and a posteriori error estimates are obtained for the semi-discretization in space. The stability and the convergence of an Eulerian time splitting scheme are investigated. The same methodology is then used to study free surface flows in two space dimensions. The incompressible unsteady Navier-Stokes equations with Neumann boundary conditions on the whole boundary are considered. The whole boundary is assumed to be the free surface. An additional equation is used to describe the moving domain. Local existence in time and uniqueness results are obtained. On the other hand, a model for free surface flows in two and three space dimensions is investigated. The liquid is assumed to be surrounded by a compressible gas. The incompressible unsteady Navier-Stokes equations are assumed to hold in the liquid region. A volume-of-fluid method is used to describe the motion of the liquid domain. The velocity in the gas is disregarded and the pressure is computed by the ideal gas law in each gas bubble trapped by the liquid. A numbering algorithm is presented to recognize the bubbles of gas. Gas pressure is applied as a normal force on the liquid-gas interface. Surface tension effects are also taken into account for the simulation of bubbles or droplets flows. A method for the computation of the curvature is presented. Convergence and accuracy of the approximation of the curvature are discussed. A time splitting scheme is used to decouple the various physical phenomena. Numerical simulations are made in the frame of mould filling to show that the influence of gas on the free surface cannot be neglected. Curvature-driven flows are also considered.

The subject of this thesis is the numerical simulation of viscous free-surface flows in naval engineering applications. State-of-the-art numerical methods based on the solution of the Navier-Stokes equations are used to predict the flow around different classes of boats. We investigate the role of the Computational Fluid Dynamics in the design of racing boats, such as America's Cup yachts and Olympic class rowing hull. The mathematical models describing the different aspects of the physical problem, as well as the numerical methods adopted for their solution, are introduced and critically discussed. The different phases of the overall numerical simulation procedure, from grid generation through the solution of the flow equations to the post-processing of the results, are described. We present the numerical simulations that have been performed to investigate the role of different design parameters in the conception of America's Cup yachts and we describe how the results obtained from the simulations are integrated into the overall design process. The free-surface flow around an Olympic rowing boat is also considered. We propose a simplified approach to take into account the effect of the boat dynamics in the prediction of the hydrodynamic forces acting on the boat. Based on the results of the simulations, we propose a new design concept and we investigate its potential benefits on the boat performances. One of the aspects that is found to be not completely satisfactory, within the standard numerical methods adopted, is the modelling of complex free-surface flows. The second part of this thesis is devoted to a more theoretical and methodological investigation of this aspect. In particular, we present and analyse a new numerical method based on the level set approach for the solution of two-fluid flows. The numerical scheme based on a finite element discretization is introduced and different critical aspects of its implementation are discussed. In particular, we present and analyse a new technique for the stabilization of the advection equation associated to the level set problem. Moreover, we propose a new reinitialization procedure for the level set function which plays a crucial role in the accuracy of the algorithm. The convergence properties of this procedure are analysed and comparisons with more standard approaches are presented. Finally, the proposed method has been used to solve a variety of test cases concerning time dependent two-fluid viscous flows. The results of the simulation are presented and discussed.

Blood flow in the arterial circulation induces hemodynamic forces that play an important role in various forms of vascular diseases. Temporal variation of the wall shear stress seems to play a significant role in atherogenesis and plaque stability. Flow induced wall shear stress has been linked to growth and possibly rupture of the aneurysm wall. Hemodynamic forces are patient-specific and difficult to assess in the clinic. At present, there is no in vivo measurement technique that enables measurement of hemodynamic forces to the degree of precision needed. However, when imaging modalities used frequently in clinical routine re-create high-definition, patient geometric quantification of the blood vessel, they can be employed as a base for creating predictive hemodynamic models. Which in the case of understanding healthy vs. pathologic blood flow within the cerebral or systemic circulation, renders this an interesting approach. First, we developed a "generic 1D" distributed model of the human arterial tree including the primary systemic arteries and coupled this to a heart model. The fluid mechanics equations were solved numerically to obtain pressure and flow throughout the arterial tree. A nonlinear viscoelastic constitutive law for the arterial wall was considered while distal vessels were terminated with a three-element Windkessel model. The coronary arteries were modeled assuming a systolic flow impediment proportional to ventricular varying elastance. The model predictions were validated with noninvasive measurements of pressure and flow performed in young volunteers. Flow in the large arteries was visualized with magnetic resonance imaging, cerebral flow detected with ultrasound Doppler and blood pressure measured with applanation tonometry. Model predictions at different arterial locations compared well to measured flow and pressure waves at the same anatomical points. Thus, the generic 1D model reflected the flow and pressure measurements of the "average subject" of our volunteer population. Following the same approach as the generic 1D model, we built and validated a patient-specific model. In this case, geometric data, flow and pressure measurements were obtained for one person. The model predicted pressure and flow waveforms in good agreement with the in-vivo measurements with regards to wave shape and features. Comparison with a generic 1-D model has shown that the patient-specific model better predicted pressure and flow at specific arterial sites. Overall, the patient-specific 1-D model was able to predict pressure and flow waveforms in the main systemic circulation, whereas this was not always the case for a generic 1-D model. The inherent underestimation of energy losses of the 1-D wave propagation model, due to bifurcations, non-planarity and complex geometry, were examined. The 1-D model was compared to a rigid wall 3-D computational fluid dynamic model. Newtonian and non-Newtonian blood properties were studied and the longitudinal pressure distribution along the arteries was compared with the 1-D patient-specific model mean pressure prediction. The results indicated that pressure drop is significant only in small diameter vessels such as the precerebral and cerebral arteries. In these vessels the 1-D model in comparison to 3-D models consistently underestimated pressure drop. The complex flow patterns resulting from asymmetry and bifurcation yield shear stresses in the 3-D model that were greater than the 1-D model. A 3-D unsteady fluid structure interaction simulation in a patient-specific model was performed to simultaneously capture the flow details, given by the 3-D model, and wave propagation phenomena, provided by the 1-D model. The 3-D unsteady fluid structure interaction approach is the most computationally intense and cumbersome, but it allows physiological simulations with a high level of detail and accuracy. For instance, this approach could be relevant to obtain blood flow details in regions that are prone to atherosclerotic plaques or development of aneurysms. The 3-D fluid structure interaction simulation was performed for a patient-specific aorta. Important clinical parameters such as wall shear stress were quantified and significant differences were found in comparison to the rigid wall 3-D simulation indicating the relevance of a fluid structure interaction approach. A comparison of the fluid structure interaction to an equivalent 1-D model resulted in good reproduction of the pressure and flow waveforms. The effect of a decreased compliance of the arterial tree on hemodynamical parameters has been assessed with the use of a 1-D model. Local, proximal aorta and global stiffening of the arterial tree were modeled and led to two different mechanisms that contribute to the increase in central pulse pressure. They probably both contribute to systolic hypertension and their relative contribution depends on the topology of arterial stiffening and geometrical alterations taking place in aging or in disease. All these patient-specific models are about to being in use in a clinical environment and will be useful for providing better diagnostics and treatment planning in a near future.