Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-29T06:04:24.529Z Has data issue: false hasContentIssue false
  • English
  • Français

Patient-specific Blood Flow Simulations: Setting DirichletBoundary Conditions for Minimal Error with Respect to Measured Data

Published online by Cambridge University Press:  31 July 2014

J. Tiago ,
A. Gambaruto  and
A. Sequeira
J. Tiago*
Affiliation:
Departamento de Matemática and CEMAT/IST Instituto Superior Técnico, Universidade de Lisboa Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
A. Gambaruto
Affiliation:
Computer Applications in Science & Engineering (CASE), Barcelona Supercomputing Center Nexus I - Campus Nord UPC, C/ Jordi Girona 2, 3a. Planta, 08034 Barcelona, Spain
A. Sequeira
Affiliation:
Departamento de Matemática and CEMAT/IST Instituto Superior Técnico, Universidade de Lisboa Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
*
⋆Corresponding author. E-mail: jftiago@math.ist.utl.pt
Get access

Abstract

We present a fully automatic approach to recover boundary conditions and locations of thevessel wall, given a crude initial guess and some velocity cross-sections, which can becorrupted by noise. This paper contributes to the body of work regarding patient-specificnumerical simulations of blood flow, where the computational domain and boundaryconditions have an implicit uncertainty and error, that derives from acquiring andprocessing clinical data in the form of medical images. The tools described in this paperfit well in the current approach of performing patient-specific simulations, where areasonable segmentation of the medical images is used to form the computational domain,and boundary conditions are obtained as velocity cross-sections from phase-contrastmagnetic resonance imaging. The only additional requirement in the proposed methods is toobtain additional velocity cross-section measurements throughout the domain. The toolsdeveloped around optimal control theory, would then minimize a user defined cost functionto fit the observations, while solving the incompressible Navier-Stokes equations.Examples include two-dimensional idealized geometries and an anatomically realisticsaccular geometry description.

Keywords

boundary controlerror minimizationpatient-specificcomputational hemodynamicsdomain uncertaintyboundary condition uncertainty
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 6: Blood flows , 2014 , pp. 98 - 116
Copyright
© EDP Sciences, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Boas, F.E., Fleischmann, D.. CT artifacts: causes and reduction techniques. Imaging in Medicine, 4 (2012), 229-240. CrossRefGoogle Scholar
Bodnar, T., Sequeira, A., Prosi, M.. On the shear-thinning effects of blood flow under various flow rates. Applied Mathematics and Computation, Elsevier, 217 (2011), 5055-5067. CrossRefGoogle Scholar
Burkardt, J., Gunzburger, M., Peterson, J.. Insensitivite Functionals, Inconsistent Gradients, Spurious Minima and Regularized Funcionals in Flow Optimization Problems. International Journal of Computational Fluid Dynamics, 16 (2002), 171-185. CrossRefGoogle Scholar
Cebral, J.R., Castro, M.A., Appanaboyina, S., Putman, C.M., Millan, D., Frangi, A.F.. Efficient pipeline for image-based patient-specific analysis of cerebral aneurism hemodynamics: technique and sensitivity. IEEE Transactions on Medical Imaging, 344 (2005), 457-467. CrossRefGoogle Scholar
COMSOL Multiphysics, Users Guide, COMSOL 4.3, 2012.
Optimization Module, Users Guide, COMSOL 4.3, 2012.
M. D’ Elia, M. Perego, A. Veneziani. A Variational Data Assimilation Procedure for the Incompressible Navier-Stokes Equations in Hemodynamics. Journal of Scientific Computing, 53 (2011).
M. D’ Elia, A. Veneziani. A Data Assimilation technique for including noisy measurements of the velocity field into Navier-Stokes simulations. Proceedings of V European Conference on Computational Fluid Dynamics, ECCOMAS, June (2010).
De Los Reyes, J.C., Kunisch, K.. A semi-smooth Newton method for control constrained boundary optimal control of the Navier-Stokes equations. Nonlinear Anal., 62 (2005), no. 7, 1289-1316. CrossRefGoogle Scholar
De Los Reyes, J.C., Kunisch, K.. Optimal control of partial differential equations with affine control constraints. Control Cybernet., 38 (2009), no. 4, 1217-1249. Google Scholar
De Los Reyes, J.C., Yousept, Irwin. Regularized state-constraint boundary Optimal control of the Navier-Stokes equations., J-Math. Anal. Appl., 356 (2009), no. 1, 257-279. CrossRefGoogle Scholar
Deuflhard, P.. A Modified Newton Method for the Solution of Ill-conditioned Systems of Nonlinear Equations with Application to Multiple Shooting. Numer. Math., 22 (1974), 289-315. CrossRefGoogle Scholar
Formaggia, L., Gerbeau, J.F., Nobile, F., Quarteroni, A.. Numerical treatment of defective boundary conditions for the Navier-Stokes equations. SIAM J. Numer. Anal., 40 (2002) 376-401. CrossRefGoogle Scholar
Gambaruto, A.M., Doorly, D.J., Yamaguchi, T.. Wall shear stress and near-wall convective transport: Comparisons with vascular remodelling in a peripheral graft anastomosis. Journal of Computational Physics, 229 (2010), no. 14, 5339-5356. CrossRefGoogle Scholar
Garambuto, A., Janela, J., Moura, A. and Sequeira, A.. Sensitivity of hemodynamics in a patient specific cerebral aneurysm to vascular geometry and blood rheology. Mathematical Biosciences and Engineering, 8 (2011), no. 2, 409-423. Google Scholar
Gambaruto, A., Janela, J., Moura, A., Sequeira, A.. Shear-thinning effects of hemodynamics in patient-specific cerebral aneurysms. Mathematical Biosciences and Engineering, 10 (2013), no. 3, 649-665. CrossRefGoogle Scholar
Gambaruto, A.M., Peiro, J., Doorly, D.J., Radaelli, A.G.. Reconstruction of shape and its effect on flow in arterial conduits. International Journal for Numerical Methods in Fluids, 57 (2008), no. 5, 495-517. CrossRefGoogle Scholar
V. Girault, P.A. Raviart. Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Number 5 in Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 1986.
T. Guerra, J. Tiago, A. Sequeira. Optimal control in blood flow simulations. International Journal of Non-Linear Mechanics, Available online (2014), doi:10.1016/j.ijnonlinmec.2014.04.005.
Gunzburger, M.. Adjoint Equation-Based Methods for Control Problems in Incompressible, Viscous Flows. Flow, Turbulence and Combustion, 65 (2000), no. 3, 249-272. CrossRefGoogle Scholar
Gunzburger, M., Hou, L.S., Svobodny, T.P.. Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls. RAIRO - Modélisation mathémathique et analyse numérique, 25 (1991), no. 6, 711-748. Google Scholar
Gill, P., Murray, W., Saunders, M.A.. SNOPT: An SQP Algoritm for Large-Scale Constrained Optimization. Society for Industrial and Applied Mathematics, SIAM REVIEW, 47 (2005) no. I, 99-131. Google Scholar
Johnston, B.M., Johnston , P.R., Corney, S., Kilpatrick, D.. Non-Newtonian blood flow in human right coronary arteries: transient simulations. Journal of Biomechanics, 39 (2006), no. 6, 1116-1128. CrossRefGoogle ScholarPubMed
Kroon, M., Holzapfel, G.A.. Estimation of the distributions of anisotropic, elastic properties and wall stresses of saccular cerebral aneurysms by inverse analysis. Proc. R. Soc. A, 464 (2008), 807-825. CrossRefGoogle Scholar
Lee, K.L., Doorly, D.J., Firmin, D.N.. Numerical simulations of phase contrast velocity mapping of complex flows in an anatomically realistic bypass graft geometry. Medical Physics, 7 (2006), 2621-2631. CrossRefGoogle Scholar
Myers, J.G., Moore, J.A., Ojha, M., Johnston, K.W., Ethier, C.R.. Factors Influencing Blood Flow Patterns in the Human Right Coronary Artery. Annals of Biomedical Engineering, 29 (2001), no. 2, 109-120. CrossRefGoogle ScholarPubMed
Ogden, R.W., Saccomandi, G., Sgura, I.. Fitting hyperelastic models to experimental data. Computational Mechanics, 34 (2004), no. 6, 484-502. CrossRefGoogle Scholar
Pusey, E., Lufkin, R.B., Brown, R.K., Solomon, M.A., Stark, D.D., Tarr, R.W., Hanafee, W.N.. Magnetic resonance imaging artifacts: mechanism and clinical significance. Radiographics, 6 (1986), no. 5, 891-911. CrossRefGoogle ScholarPubMed
A. Quarteroni, L. Formaggia. Mathematical Modelling and Numerical Simulation of the Cardiovascular System, Modelling of living Systems. Handbook of Numerical Analysis Series. Elsevier, Amsterdam, 2002.
Ramalho, S., Moura, A., Gambaruto, A.M., Sequeira, A.. Sensitivity to outflow boundary conditions and level of geometry description for a cerebral aneurysm. International Journal for Numerical Methods in Biomedical Engineering, 28 (2012), no. 6-7, 697-713. CrossRefGoogle ScholarPubMed
Robertson, A.M., Sequeira, A., Kameneva, M.V.. Hemorheology. Hemodynamical Flows. Modeling, Analysis and Simulation. Oberwolfach Seminars. Birkhauser Verlag, Basel, 37 (2008), 63-120. Google Scholar
Schenk, O., Gärtner, K., Fichtner, W., Stricker, A.. PARDISO: A High-Performance Serial and Parallel Sparse Linear Solver in Semiconductor Device Simulation, Journal of Future Generation Computers Systems, 18 (2001), 69-78. CrossRefGoogle Scholar
T. Silva, A. Sequeira, R. Santos, J. Tiago. Mathematical Modeling of Atherosclerotic Plaque Formation Coupled with a Non-Newtonian Model of Blood Flow. Conference Papers in Mathematics, vol. 2013, Article ID 405914, 2013. doi:10.1155/2013/405914.
Smith, N.P., Pullan, A.J., Hunter, P.J.. An anatomically based model of coronary blood flow and myocardial mechanics. SIAM J. Appl. Math., 62 (2002), 990-1018. CrossRefGoogle Scholar
P. Tricerri. Mathematical and Numerical Modeling of Healthy and Unhealthy Cerebral Arterial Tissue. École Polytechnique Fédérale de Lausanne. Ph.D. thesis, 2014.
Y. Wei, S. Cotin, J. Dequidt, C. Duriez, J. Allard, E. Kerrien. A (Near) Real-Time Simulation Method of Aneurysm Coil Embolization http://dx.doi.org/10.5772/48635, INTECH, 2012.
C. Westbrook, C.K. Roth, J. Talbot. MRI in Practice. Wiley-Blackwell, 2011.