Mathematical Modelling and Numerical Simulation of the Cardiovascular System

Publisher Summary The development of mathematical models, algorithms and numerical simulation tools for the investigation of the human cardiovascular system has received a great impulse in the past years. This chapter addresses the problem of developing models for the numerical simulation of the human circulatory system. It particularly focuses on the problem of hemodynamics in large human arteries. There are several important aspects, which require the use of sophisticated mathematical and numerical tools, such as the reconstruction of geometries from medical data; the transport of biochemicals in blood and vessel wall tissue; the heart dynamics; and blood rheology. Besides, the need of validating the models calls for development of accurate in-vivo measurement techniques. The number and complexity of the mathematical, numerical and technological problems involved makes the development of tools for accurate, reliable and efficient simulations of the human cardiovascular system one of the challenges of the next decades.

[1]  Yvon Maday,et al.  Fluid-Structure Interaction: A Theoretical Point of View , 2000 .

[2]  G. Langewouters,et al.  The static elastic properties of 45 human thoracic and 20 abdominal aortas in vitro and the parameters of a new model. , 1984, Journal of biomechanics.

[3]  R. Ogden,et al.  A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models , 2000 .

[4]  J. Lumley,et al.  Mechanics of non-Newtonian fluids , 1978 .

[5]  Sunčica Čanić,et al.  Blood flow through compliant vessels after endovascular repair: wall deformations induced by the discontinuous wall properties , 2002 .

[6]  J. Lions,et al.  Problèmes aux limites non homogènes et applications , 1968 .

[7]  A. Quarteroni,et al.  One-dimensional models for blood flow in arteries , 2003 .

[8]  J. C. Simo,et al.  On a stress resultant geometrically exact shell model. Part III: computational aspects of the nonlinear theory , 1990 .

[9]  S. Čanić,et al.  Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi‐symmetric vessels , 2003 .

[10]  J. Lions,et al.  Inequalities in mechanics and physics , 1976 .

[11]  K. Thompson.  Time-dependent boundary conditions for hyperbolic systems, II , 1990 .

[12]  J. B. Perot,et al.  An analysis of the fractional step method , 1993 .

[13]  Lucia Gastaldi,et al.  A priori error estimates for the Arbitrary Lagrangian Eulerian formulation with finite elements , 2001, J. Num. Math..

[14]  Yvon Maday,et al.  A high-order characteristics/finite element method for the incompressible Navier-Stokes equations , 1997 .

[15]  Shu Chien,et al.  Handbook of Bioengineering , 1986 .

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

[17]  G. W. Hedstrom,et al.  Nonreflecting Boundary Conditions for Nonlinear Hyperbolic Systems , 1979 .

[18]  S. Giuliani,et al.  Time-accurate solution of advection-diffusion problems by finite elements , 1984 .

[19]  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..

[20]  F. Migliavacca,et al.  global mathematical modelling of the norwood circulation: a multiscale approach for the study of the pulmonary and coronary arterial perfusions , 2004, Cardiology in the Young.

[21]  Y. Fung,et al.  Biomechanics: Mechanical Properties of Living Tissues , 1981 .

[22]  P. Tallec,et al.  Fluid structure interaction with large structural displacements , 2001 .

[23]  Charbel Farhat,et al.  On the significance of the geometric conservation law for flow computations on moving meshes , 2000 .

[24]  A. Quarteroni,et al.  Factorization methods for the numerical approximation of Navier-Stokes equations , 2000 .

[25]  J. C. Simo,et al.  On stress resultant geometrically exact shell model. Part I: formulation and optimal parametrization , 1989 .

[26]  Alfio Quarteroni,et al.  From the global cardiovascular hemodynamics down to the local blood motion: preliminary applications of a multiscale approach , 2000 .

[27]  A. Prohl.  Projection and quasi-compressibility methods for solving the incompressible navier-stokes equations , 1997 .

[28]  Fabio Nobile,et al.  A Stability Analysis for the Arbitrary Lagrangian Eulerian Formulation with Finite Elements , 1999 .

[29]  Philippe G. Ciarlet,et al.  Introduction to Linear Shell Theory , 1989 .

[30]  James Serrin,et al.  Mathematical Principles of Classical Fluid Mechanics , 1959 .

[31]  J. Womersley.  Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known , 1955, The Journal of physiology.

[32]  L. Segel,et al.  Mathematics Applied to Continuum Mechanics , 1977 .

[33]  A. Quarteroni,et al.  Coupling between lumped and distributed models for blood flow problems , 2001 .

[34]  Yvon Maday,et al.  NUMERICAL ANALYSIS OF SOME DECOUPLING TECHNIQUES FOR THE APPROXIMATION OF THE UNSTEADY FLUID STRUCTURE INTERACTION , 2001 .

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

[36]  Charbel Farhat,et al.  Partitioned procedures for the transient solution of coupled aeroelastic problems , 2001 .

[37]  H. B. Veiga.  On the Existence of Strong Solutions to a Coupled Fluid-Structure Evolution Problem , 2004 .

[38]  C. Farhat,et al.  Mixed explicit/implicit time integration of coupled aeroelastic problems: Three‐field formulation, geometric conservation and distributed solution , 1995 .

[39]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuscka-Brezzi condition: A stable Petrov-Galerkin formulation of , 1986 .

[40]  Alfio Quarteroni,et al.  Domain Decomposition Methods for Partial Differential Equations , 1999 .

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

[42]  Alfio Quarteroni,et al.  A One Dimensional Model for Blood Flow: Application to Vascular Prosthesis , 2002 .

[43]  C. Taylor,et al.  Predictive medicine: computational techniques in therapeutic decision-making. , 1999, Computer aided surgery : official journal of the International Society for Computer Aided Surgery.

[44]  C. Farhat,et al.  Two efficient staggered algorithms for the serial and parallel solution of three-dimensional nonlinear transient aeroelastic problems , 2000 .

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

[46]  K Perktold,et al.  Pulsatile albumin transport in large arteries: a numerical simulation study. , 1996, Journal of biomechanical engineering.

[47]  H Handa,et al.  Stiffness and elastic behavior of human intracranial and extracranial arteries. , 1980, Journal of biomechanics.

[48]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[49]  Alfio Quarteroni,et al.  Mathematical and Numerical Modeling of Solute Dynamics in Blood Flow and Arterial Walls , 2001, SIAM J. Numer. Anal..

[50]  Alfio Quarteroni,et al.  Analysis of the Yosida Method for the Incompressible Navier-Stokes Equations , 1999 .

[51]  Carl D. Meyer,et al.  Matrix Analysis and Applied Linear Algebra , 2000 .

[52]  Jean-Luc Guermond,et al.  Convergence Analysis of a Finite Element Projection/Lagrange-Galerkin Method for the Incompressible Navier-Stokes Equations , 2000, SIAM J. Numer. Anal..

[53]  W. P. Timlake,et al.  A theory of fluid flow in compliant tubes. , 1966, Biophysical journal.

[54]  Jean-Luc Guermond,et al.  Un résultat de convergence d'ordre deux en temps pour l'approximation des équations de Navier-Stokes par une technique de projection incrémentale , 1999 .

[55]  B. Reddy,et al.  Introductory Functional Analysis: With Applications to Boundary Value Problems and Finite Elements , 1997 .

[56]  R. Aris.  Vectors, Tensors and the Basic Equations of Fluid Mechanics , 1962 .

[57]  Andrew J. Pullan,et al.  An Anatomically Based Model of Transient Coronary Blood Flow in the Heart , 2002, SIAM J. Appl. Math..

[58]  J. C. Simo,et al.  On a stress resultant geometrically exact shell model. Part II: the linear theory; computational aspects , 1989 .