Abstract
The aim of the present work is to address the closure problem for hemodynamic simulations by developing a flexible and effective model that accurately distributes flow in the downstream vasculature and can stably provide a physiological pressure outflow boundary condition. To achieve this goal, we model blood flow in the sub-pixel vasculature by using a non-linear 1D model in self-similar networks of compliant arteries that mimic the structure and hierarchy of vessels in the meso-vascular regime (radii \(500 {-} 10\,\mu {\text{m}}\)). We introduce a variable vessel length-to-radius ratio for small arteries and arterioles, while also addressing non-Newtonian blood rheology and arterial wall viscoelasticity effects in small arteries and arterioles. This methodology aims to overcome substantial cut-off radius sensitivities, typically arising in structured tree and linearized impedance models. The proposed model is not sensitive to outflow boundary conditions applied at the end points of the fractal network, and thus does not require calibration of resistance/capacitance parameters typically required for outflow conditions. The proposed model convergences to a periodic state in two cardiac cycles even when started from zero-flow initial conditions. The resulting fractal-trees typically consist of thousands to millions of arteries, posing the need for efficient parallel algorithms. To this end, we have scaled up a Discontinuous Galerkin solver that utilizes the MPI/OpenMP hybrid programming paradigm to thousands of computer cores, and can simulate blood flow in networks of millions of arterial segments at the rate of one cycle per 5 min. The proposed model has been extensively tested on a large and complex cranial network with 50 parent, patient-specific arteries and 21 outlets to which fractal trees where attached, resulting to a network of up to 4,392,484 vessels in total, and a detailed network of the arm with 276 parent arteries and 103 outlets (a total of 702,188 vessels after attaching the fractal trees), returning physiological flow and pressure wave predictions without requiring any parameter estimation or calibration procedures. We present a novel methodology to overcome substantial cut-off radius sensitivities.
Similar content being viewed by others
References
Alastruey, J., K. H. Parker, J. Peiró, and S. J. Sherwin. Lumped parameter outflow models for 1D blood flow simulations: effect on pulse waves and parameter estimation. Commun. Comput. Phys. 4:317–336, 2008.
Blanco, P. J., S. M. Watanabe, and R. A. Feijóo. Identification of vascular territory resistances in one-dimensional hemodynamics simulations. J. Biomech. 45:2066–2073, 2012.
Cassot, F., and F. Lauwers, S. Lorthois, P. Puwanarajah, V. Cances-Lauwers, and H. Duvernoy. Branching patterns for arterioles and venules of the human cerebral cortex. Brain Res. 1313:62–78, 2010.
Cousins, W. Boundary Conditions and Uncertainty Quantification for Hemodynamics. Ph.D. thesis, North Carolina State University, 2013.
Cousins, W., and P.A. Gremaud. Boundary conditions for hemodynamics: The structured tree revisited. J. Comput. Phys. 231:6086–6096, 2012.
Cousins, W., P. A. Gremaud, and D. M. Tartakovsky. A new physiological boundary condition for hemodynamics. SIAM J. Appl. Math. 73:1203–1223, 2013.
Craiem, D. O., F. J. Rojo, J. M. Atienza, R. L. Armentano, and G. V. Guinea. Fractional-order viscoelasticity applied to describe uniaxial stress relaxation of human arteries. Phys. Med. Biol. 53:4543, 2008.
Doehring, T. C., A. D. Freed, E. O. Carew, I. Vesely, et al. Fractional order viscoelasticity of the aortic valve cusp: an alternative to quasilinear viscoelasticity. J. Biomech. Eng.-T. ASME 127:700, 2005.
Formaggia, L., A. Quarteroni, and A. Veneziani. Cardiovascular Mathematics: Modeling and Simulation of the Circulatory System, vol. 1, Springer Verlag, 2009.
Fung, Y. Biomechanics: Mechanical Properties of Living Tissues. Springer-Verlag, 1993.
Grinberg, L., E. Cheever, T. Anor, J. R. Madsen, and G. E. Karniadakis. Modeling blood flow circulation in intracranial arterial networks: a comparative 3D/1D simulation study. Ann. Biomed. Eng. 39:297–309, 2011.
Grinberg, L., D. Fedosov, and G. E. Karniadakis. Parallel multiscale simulations of a brain aneurysm. J. Comput. Phys. 244:131–147, 2013.
Grinberg, L., and G. E. Karniadakis. Outflow boundary conditions for arterial networks with multiple outlets. Ann. Biomed. Eng. 36:1496–1514, 2008.
Ito, H., I. Kanno, H. Iida, J. Hatazawa, E. Shimosegawa, H. Tamura, and T. Okudera. Arterial fraction of cerebral blood volume in humans measured by positron emission tomography. Ann. Nucl. Med. 15:111–116, 2001.
Lombardi, D. Inverse problems in 1D hemodynamics on systemic networks: a sequential approach. Int. J. Numer. Methods Biomed. Eng. 30:160–179, 2014.
Mainardi, F. Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, 2010.
McDonald, D. A. Blood flow in arteries 1974.
Melani, A. and A. Quarteroni. Adjoint-Based Parameter Estimation in Human Vascular one Dimensional Models. Ph.D. thesis, EPFL, 2013.
Olufsen, M. S. Structured tree outflow condition for blood flow in larger systemic arteries. Am. J. Physiol.- Heart Circ Physiol. 276:H257–H268, 1999.
Perdikaris, P., and G. E. Karniadakis. Fractional-order viscoelasticity in one-dimensional blood flow models. Ann. Biomed. Eng. 42:1012–1023, 2014.
Pries, A. R., D. Neuhaus, P. Gaehtgens, et al. Blood viscosity in tube flow: dependence on diameter and hematocrit. Am. J. Physiol. 263:H1770–H1770, 1992.
Rengachary, S. S., and R. G. Ellenbogen. Principles of Neurosurgery. Elsevier Mosby, 2005.
Reymond, P., Y. Bohraus, F. Perren, F. Lazeyras, and N. Stergiopulos. Validation of a patient-specific one-dimensional model of the systemic arterial tree. Am. J. Physiol.-Heart Circ. Physiol. 301:1173–1182, 2011.
Sherwin, S. J., V. Franke, J. Peiro, and K. H. Parker. One-dimensional modelling of a vascular network in space-time variables. J. Eng. Math. 47:217–250, 2003.
Steele, B. N., M. S. Olufsen, and C. A. Taylor. Fractal network model for simulating abdominal and lower extremity blood flow during resting and exercise conditions. Comput. Methods Biomech. Biomed. Eng. 10:39–51, 2007.
VanBavel, E., and J. A. Spaan. Branching patterns in the porcine coronary arterial tree. Estimation of flow heterogeneity. Circ. Res. 71:1200–1212, 1992.
Watanabe, S. M., P. J. Blanco, and R. A. Feijóo. Mathematical model of blood flow in an anatomically detailed arterial network of the arm. ESAIM Math. Model. Numer. Anal. 47:961–985, 2013.
Xiao, N., J. D. Humphrey, and C. A. Figueroa. Multi-scale computational model of three-dimensional hemodynamics within a deformable full-body arterial network. J. Comput. Phys. 244:22–40, 2013.
Zamir, M. et al. On fractal properties of arterial trees. J. Theor. Biol. 197:517–526, 1999.
Acknowledgments
This work received partial support by the Air Force Office of Scientific Research under Grant No. FA9550-12-1-0463, and the National Institutes of Health under Grant No. 1U01HL116323-01. We also thank Dr. Pablo J. Blanco, LLNC, Brazil, for providing the arm arterial model. Large scale computations have utilized resources at the Argonne Leadership Computing Facility at Argonne National Laboratory, through support by the DOE/INCITE program.
Author information
Authors and Affiliations
Corresponding author
Additional information
Associate Editor Andreas Anayiotos oversaw the review of this article.
Rights and permissions
About this article
Cite this article
Perdikaris, P., Grinberg, L. & Karniadakis, G.E. An Effective Fractal-Tree Closure Model for Simulating Blood Flow in Large Arterial Networks. Ann Biomed Eng 43, 1432–1442 (2015). https://doi.org/10.1007/s10439-014-1221-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10439-014-1221-3