Modified Navier-Stokes equations for the outflow boundary conditions in hemodynamics

Abstract We present a new approach for the outflow boundary conditions of Navier–Stokes equations in hemodynamics. We first describe some existing 3D–0D coupling methods and highlight benefits and disadvantages of each of them. We then introduce a new method that consists in adding a 3D artificial part where the Navier–Stokes equations are modified to obtain an equivalent energy balance to a standard coupling with a 3-element Windkessel model. We investigate theoretically the stability of the system and compare it to previously introduced methods. Finally we compare these coupling methods for numerical simulations of blood flow in three patient-specific models, which represent different flow regimes in the pulmonary and systemic circulations. The new method, especially in its hybrid form, is a possible alternative to existing methods. It can be in particular convenient in codes that do not allow users to implement non-standard boundary conditions.

[1]  Giancarlo Pennati,et al.  Pulmonary Hemodynamics Simulations Before Stage 2 Single Ventricle Surgery: Patient-Specific Parameter Identification and Clinical Data Assessment , 2015, Cardiovascular engineering and technology.

[2]  A. Marsden,et al.  An integrated approach to patient-specific predictive modeling for single ventricle heart palliation , 2014, Computer methods in biomechanics and biomedical engineering.

[3]  P. Fabrie,et al.  EFFECTIVE DOWNSTREAM BOUNDARY CONDITIONS FOR INCOMPRESSIBLE NAVIER-STOKES EQUATIONS , 1994 .

[4]  R A Corley,et al.  A bidirectional coupling procedure applied to multiscale respiratory modeling , 2013, J. Comput. Phys..

[5]  Thomas J. R. Hughes,et al.  Patient-specific isogeometric fluid–structure interaction analysis of thoracic aortic blood flow due to implantation of the Jarvik 2000 left ventricular assist device , 2009 .

[6]  Pablo J. Blanco,et al.  A dimensionally-heterogeneous closed-loop model for the cardiovascular system and its applications. , 2013, Medical engineering & physics.

[7]  Miguel A. Fernández,et al.  Group-wise construction of reduced models for understanding and characterization of pulmonary blood flows from medical images , 2014, Medical Image Anal..

[8]  Ryo Torii,et al.  Patient-specific modeling and multi-scale blood simulation for computational hemodynamic study on the human cerebrovascular system. , 2012, Current pharmaceutical biotechnology.

[9]  A. Marsden,et al.  A Primer on Computational Simulation in Congenital Heart Disease for the Clinician , 2010, 1101.3726.

[10]  I. Vignon-Clementel,et al.  Three-dimensional simulations in Glenn patients: clinically based boundary conditions, hemodynamic results and sensitivity to input data. , 2011, Journal of biomechanical engineering.

[11]  Anamika Prasad,et al.  Computational Analysis of Stresses Acting on Intermodular Junctions in Thoracic Aortic Endografts , 2011, Journal of endovascular therapy : an official journal of the International Society of Endovascular Specialists.

[12]  Mahmoud Ismail,et al.  Adjoint-based inverse analysis of windkessel parameters for patient-specific vascular models , 2013, J. Comput. Phys..

[13]  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.

[14]  Alfio Quarteroni,et al.  On the physical consistency between three-dimensional and one-dimensional models in haemodynamics , 2013, J. Comput. Phys..

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

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

[17]  Kenneth E. Jansen,et al.  Developing computational methods for three-dimensional finite element simulations of coronary blood flow , 2010 .

[18]  Alison L. Marsden,et al.  Optimization of shunt placement for the Norwood surgery using multi-domain modeling. , 2012, Journal of biomechanical engineering.

[19]  A Porpora,et al.  Numerical treatment of boundary conditions to replace lateral branches in hemodynamics , 2012, International journal for numerical methods in biomedical engineering.

[20]  Justine Fouchet-Incaux.  Artificial boundaries and formulations for the incompressible Navier–Stokes equations: applications to air and blood flows , 2014 .

[21]  Alison L. Marsden,et al.  A modular numerical method for implicit 0D/3D coupling in cardiovascular finite element simulations , 2013, J. Comput. Phys..

[22]  Nicolas Meunier,et al.  OUTLET DISSIPATIVE CONDITIONS FOR AIR FLOW IN THE BRONCHIAL TREE , 2005 .

[23]  Sanjay Pant,et al.  A Multiscale Filtering-Based Parameter Estimation Method for Patient-Specific Coarctation Simulations in Rest and Exercise , 2013, STACOM.

[24]  Alfonso Caiazzo,et al.  A tangential regularization method for backflow stabilization in hemodynamics , 2014, J. Comput. Phys..

[25]  O. Frank,et al.  Die grundform des arteriellen pulses , 1899 .

[26]  P. Moireau,et al.  Sequential parameter estimation for fluid–structure problems: Application to hemodynamics , 2012, International journal for numerical methods in biomedical engineering.

[27]  P. Fabrie,et al.  New efficient boundary conditions for incompressible Navier-Stokes equations : a well-posedness result , 1996 .

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

[29]  Sansuke M. Watanabe,et al.  Identification of vascular territory resistances in one-dimensional hemodynamics simulations. , 2012, Journal of biomechanics.

[30]  Alison L. Marsden,et al.  Airflow and Particle Deposition Simulations in Health and Emphysema: From In Vivo to In Silico Animal Experiments , 2014, Annals of Biomedical Engineering.

[31]  Giancarlo Pennati,et al.  Respiratory effects on hemodynamics in patient-specific CFD models of the Fontan circulation under exercise conditions , 2012 .

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

[33]  Pablo J. Blanco,et al.  On the continuity of mean total normal stress in geometrical multiscale cardiovascular problems , 2013, J. Comput. Phys..

[34]  Maxim A. Olshanskii,et al.  A finite element solver and energy stable coupling for 3D and 1D fluid models , 2013, 1301.3958.

[35]  Giancarlo Pennati,et al.  Numerical blood flow simulation in surgical corrections: what do we need for an accurate analysis? , 2014, The Journal of surgical research.

[36]  A Comerford,et al.  A stable approach for coupling multidimensional cardiovascular and pulmonary networks based on a novel pressure‐flow rate or pressure‐only Neumann boundary condition formulation , 2014, International journal for numerical methods in biomedical engineering.

[37]  A. Marsden,et al.  A comparison of outlet boundary treatments for prevention of backflow divergence with relevance to blood flow simulations , 2011 .

[38]  A. Marsden,et al.  Hepatic blood flow distribution and performance in conventional and novel Y-graft Fontan geometries: a case series computational fluid dynamics study. , 2012, The Journal of thoracic and cardiovascular surgery.

[39]  Charles A. Taylor,et al.  Tuning Multidomain Hemodynamic Simulations to Match Physiological Measurements , 2010, Annals of Biomedical Engineering.

[40]  Olivier Pironneau,et al.  A nouveau sur les équations de Stokes et de Navier-Stokes avec des conditions aux limites sur la pression , 1987 .

[41]  J-F Gerbeau,et al.  A methodological paradigm for patient‐specific multi‐scale CFD simulations: from clinical measurements to parameter estimates for individual analysis , 2014, International journal for numerical methods in biomedical engineering.

[42]  Volker Gravemeier,et al.  A novel formulation for Neumann inflow boundary conditions in biomechanics , 2012, International journal for numerical methods in biomedical engineering.

[43]  L. Formaggia,et al.  On the stability of the coupling of 3D and 1D fluid-structure interaction models for blood flow simulations , 2007 .