The role of the variational formulation in the dimensionally-heterogeneous modelling of the human cardiovascular system

The modelling of the cardiovascular system entails dealing with different phenomena pertaining to different time, constitutive and geometrical scales. Specifically, the problem of integrating various geometrical scales can be understood from a kinematical point of view, which means to integrate models with different kinematics, and in particular different dimensionality. In this context, all the variational machinery can be employed to derive consistent variational formulations according to the underlying kinematical hypotheses that rule over the corresponding models. In this work we discuss the application of variational formulations to model the blood flow in the cardiovascular system making use of heterogeneous representations. Two examples of applications are used to show the capabilities and potentialities of the present approach.

[1]  Pablo J. Blanco,et al.  Black-box decomposition approach for computational hemodynamics: One-dimensional models , 2011 .

[2]  M. Olufsen,et al.  Numerical Simulation and Experimental Validation of Blood Flow in Arteries with Structured-Tree Outflow Conditions , 2000, Annals of Biomedical Engineering.

[3]  Alfio Quarteroni,et al.  Numerical Treatment of Defective Boundary Conditions for the Navier-Stokes Equations , 2002, SIAM J. Numer. Anal..

[4]  Fergal Boyle,et al.  A Numerical Methodology to Fully Elucidate the Altered Wall Shear Stress in a Stented Coronary Artery , 2010 .

[5]  H. Tran,et al.  Blood pressure and blood flow variation during postural change from sitting to standing: model development and validation. , 2005, Journal of applied physiology.

[6]  Thomas J. R. Hughes,et al.  On the one-dimensional theory of blood flow in the larger vessels , 1973 .

[7]  Patrick Jenny,et al.  Vascular Graph Model to Simulate the Cerebral Blood Flow in Realistic Vascular Networks , 2009, Journal of cerebral blood flow and metabolism : official journal of the International Society of Cerebral Blood Flow and Metabolism.

[8]  Rolf Rannacher,et al.  ARTIFICIAL BOUNDARIES AND FLUX AND PRESSURE CONDITIONS FOR THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS , 1996 .

[9]  S Glagov,et al.  The role of fluid mechanics in the localization and detection of atherosclerosis. , 1993, Journal of biomechanical engineering.

[10]  J. Mocco,et al.  Hemodynamic–Morphologic Discriminants for Intracranial Aneurysm Rupture , 2011, Stroke.

[11]  Ryutaro Himeno,et al.  Multi-scale modeling of the human cardiovascular system with applications to aortic valvular and arterial stenoses , 2009, Medical & Biological Engineering & Computing.

[12]  J. Tinsley Oden,et al.  Applied functional analysis , 1996 .

[13]  Alfio Quarteroni,et al.  Modeling dimensionally-heterogeneous problems: analysis, approximation and applications , 2011, Numerische Mathematik.

[14]  Hao Liu,et al.  Simulation of hemodynamic responses to the valsalva maneuver: an integrative computational model of the cardiovascular system and the autonomic nervous system. , 2006, The journal of physiological sciences : JPS.

[15]  Alejandro F. Frangi,et al.  Efficient pipeline for image-based patient-specific analysis of cerebral aneurysm hemodynamics: technique and sensitivity , 2005, IEEE Transactions on Medical Imaging.

[16]  Pablo J. Blanco,et al.  Assessing the influence of heart rate in local hemodynamics through coupled 3D‐1D‐0D models , 2010 .

[17]  Hao Liu,et al.  A Closed-Loop Lumped Parameter Computational Model for Human Cardiovascular System , 2005 .

[18]  P. Abbrecht,et al.  Digital computer simulation of human systemic arterial pulse wave transmission: a nonlinear model. , 1972, Journal of biomechanics.

[19]  M E Clark,et al.  A circle of Willis simulation using distensible vessels and pulsatile flow. , 1985, Journal of biomechanical engineering.

[20]  S. Rittgers,et al.  Hemodynamic factors at the distal end-to-side anastomosis of a bypass graft with different POS:DOS flow ratios. , 2001, Journal of biomechanical engineering.

[21]  D. Ku,et al.  Pulsatile Flow and Atherosclerosis in the Human Carotid Bifurcation: Positive Correlation between Plaque Location and Low and Oscillating Shear Stress , 1985, Arteriosclerosis.

[22]  A. Quarteroni,et al.  On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels , 2001 .

[23]  J. Mocco,et al.  MORPHOLOGY PARAMETERS FOR INTRACRANIAL ANEURYSM RUPTURE RISK ASSESSMENT , 2008, Neurosurgery.

[24]  R. Mark,et al.  Computational modeling of cardiovascular response to orthostatic stress. , 2002, Journal of applied physiology.

[25]  Alfio Quarteroni,et al.  Multiscale modelling of the circulatory system: a preliminary analysis , 1999 .

[26]  Toshio Kobayashi,et al.  Finite element simulation of blood flow in the cerebral artery , 2001 .

[27]  Pablo J. Blanco,et al.  Multidimensional modelling for the carotid artery blood flow , 2006 .

[28]  Y. Kivity,et al.  Nonlinear wave propagation in viscoelastic tubes: application to aortic rupture. , 1974, Journal of biomechanics.

[29]  Alfio Quarteroni,et al.  Analysis of a Geometrical Multiscale Blood Flow Model Based on the Coupling of ODEs and Hyperbolic PDEs , 2005, Multiscale Model. Simul..

[30]  M. P. Spencer,et al.  The Square-Wave Electromagnetic Flowmeter: Theory of Operation and Design of Magnetic Probes for Clinical and Experimental Applications , 1959 .

[31]  Pablo J. Blanco,et al.  A variational approach for coupling kinematically incompatible structural models , 2008 .

[32]  Charles A. Taylor,et al.  On Coupling a Lumped Parameter Heart Model and a Three-Dimensional Finite Element Aorta Model , 2009, Annals of Biomedical Engineering.

[33]  P. Blanco,et al.  A unified variational approach for coupling 3D-1D models and its blood flow applications , 2007 .

[34]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[35]  P. Blanco,et al.  Sensitivity analysis in kinematically incompatible models , 2009 .

[36]  Thomas J. R. Hughes,et al.  Finite Element Modeling of Three-Dimensional Pulsatile Flow in the Abdominal Aorta: Relevance to Atherosclerosis , 2004, Annals of Biomedical Engineering.

[37]  G. Baselli,et al.  Model of arterial tree and peripheral control for the study of physiological and assisted circulation. , 2007 .

[38]  Charles A. Taylor,et al.  Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries , 2006 .

[39]  Alfio Quarteroni,et al.  Computational vascular fluid dynamics: problems, models and methods , 2000 .

[40]  F. Hoppensteadt,et al.  Modeling and Simulation in Medicine and the Life Sciences , 2001 .

[41]  J. Karemaker,et al.  Mathematical modeling of gravitational effects on the circulation: importance of the time course of venous pooling and blood volume changes in the lungs. , 2006, American journal of physiology. Heart and circulatory physiology.

[42]  K. Perktold,et al.  Computer simulation of local blood flow and vessel mechanics in a compliant carotid artery bifurcation model. , 1995, Journal of biomechanics.

[43]  G. Strang,et al.  An Analysis of the Finite Element Method , 1974 .

[44]  F. Migliavacca,et al.  Multiscale modelling in biofluidynamics: application to reconstructive paediatric cardiac surgery. , 2006, Journal of biomechanics.

[45]  R. Schroter,et al.  Atheroma and arterial wall shear - Observation, correlation and proposal of a shear dependent mass transfer mechanism for atherogenesis , 1971, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[46]  A. Avolio,et al.  Multi-branched model of the human arterial system , 1980, Medical and Biological Engineering and Computing.

[47]  Pablo J. Blanco,et al.  Iterative strong coupling of dimensionally heterogeneous models , 2009 .

[48]  C M Putman,et al.  Hemodynamics and Bleb Formation in Intracranial Aneurysms , 2010, American Journal of Neuroradiology.

[49]  Rainald Löhner,et al.  Applications of patient‐specific CFD in medicine and life sciences , 2003 .

[50]  K. Parker,et al.  Wave propagation in a model of the arterial circulation. , 2004, Journal of biomechanics.

[51]  Charles A. Taylor,et al.  Augmented Lagrangian method for constraining the shape of velocity profiles at outlet boundaries for three-dimensional finite element simulations of blood flow , 2009 .

[52]  R. Keynton,et al.  Intimal hyperplasia and wall shear in arterial bypass graft distal anastomoses: an in vivo model study. , 2001, Journal of biomechanical engineering.

[53]  P. Blanco,et al.  On the potentialities of 3D-1D coupled models in hemodynamics simulations. , 2009, Journal of biomechanics.

[54]  T. Korakianitis,et al.  Numerical simulation of cardiovascular dynamics with healthy and diseased heart valves. , 2006, Journal of biomechanics.

[55]  R Pietrabissa,et al.  Multiscale modelling as a tool to prescribe realistic boundary conditions for the study of surgical procedures. , 2002, Biorheology.

[56]  Alfio Quarteroni,et al.  Analysis of a Geometrical Multiscale Model Based on the Coupling of ODE and PDE for Blood Flow Simulations , 2003, Multiscale Model. Simul..

[57]  Giuseppe Pontrelli,et al.  A Multiscale Approach for Modelling Wave Propagation in an Arterial Segment , 2004, Computer methods in biomechanics and biomedical engineering.

[58]  M. Anliker,et al.  Theoretical analysis of arterial hemodynamics including the influence of bifurcations , 2006, Annals of Biomedical Engineering.

[59]  C A Taylor,et al.  Outflow boundary conditions for 3D simulations of non-periodic blood flow and pressure fields in deformable arteries , 2010, Computer methods in biomechanics and biomedical engineering.

[60]  G E Karniadakis,et al.  LARGE‐SCALE SIMULATION OF THE HUMAN ARTERIAL TREE , 2009, Clinical and experimental pharmacology & physiology.

[61]  D. F. Young,et al.  Computer simulation of arterial flow with applications to arterial and aortic stenoses. , 1992, Journal of biomechanics.

[62]  N. Stergiopulos,et al.  Validation of a one-dimensional model of the systemic arterial tree. , 2009, American journal of physiology. Heart and circulatory physiology.