zbMATH — the first resource for mathematics

A concurrent implementation of the surrogate management framework with application to cardiovascular shape optimization. (English) Zbl 1457.90158
Summary: The surrogate management framework (SMF) is an effective approach for derivative-free optimization of expensive objective functions. The SMF is typically comprised of surrogate-based infill methods (SEARCH step) coupled to pattern search optimization (POLL step). Although the latter is easy to parallelize, parallelization of the SEARCH step requires surrogate-based strategies that generate multiple candidates at each iteration. The impact of such SEARCH methods on SMF performance remains poorly explored. In this paper, we extend the SMF to incorporate concurrent evaluations at the SEARCH step by comparing two different infill approaches: single search multiple error sampling and expected improvement constant liar approaches. These variants are generalized to address non-linearly constrained problems by the filter method. The proposed methods are benchmarked for different infill sizes, while accounting for the variability in initialization. We then demonstrate the proposed methods on two shape optimization problems motivated by hemodynamically-driven surgical design. Surrogate-based multiple-infill strategies outperform their single-infill counterparts for a fixed computational time budget on bound constrained problems. Insights drawn from this study have implications not only on future instances of the SMF, but also for other surrogate-based and hybrid parallel infill methods for derivative-free optimization.
90C30 Nonlinear programming
Full Text: DOI
[1] Abbott, WM; Megerman, J.; Hasson, JE; L’Italien, G.; Warnock, DF, Effect of compliance mismatch on vascular graft patency, J Vasc Surg, 5, 2, 376-382 (1987)
[2] Abraham, F.; Behr, M.; Heinkenschloss, M., Shape optimization in unsteady blood flow: a numerical study of non-newtonian effects, Comput Methods Biomech Biomed Eng, 8, 201-212 (2005)
[3] Abramson, MA; Audet, C., Convergence of mesh adaptive direct search to second-order stationary points, SIAM J Optim, 17, 2, 606-619 (2006) · Zbl 1174.90877
[4] Abramson, MA; Audet, C.; Dennis, JE; Digabel, SL, OrthoMADS: a deterministic MADS instance with orthogonal directions, SIAM J Optim, 20, 2, 948-966 (2009) · Zbl 1189.90202
[5] Audet, C.; Dennis, JE Jr, A progressive barrier for derivative-free nonlinear programming, SIAM J Optim, 20, 1, 445-472 (2000) · Zbl 1187.90266
[6] Audet, C.; Dennis, JE Jr, Analysis of generalized pattern searches, SIAM J Optim, 13, 3, 889-903 (2003) · Zbl 1053.90118
[7] Audet, C.; Dennis, JE Jr, A pattern search filter method for nonlinear programming without derivatives, SIAM J Optim, 14, 4, 980-1010 (2004) · Zbl 1073.90066
[8] Audet C, Dennis Jr, J.E (2004b) Mesh adaptive direct search algorithms for constrained optimization. Tech Rep G-2004-04, Les Cahiers du GERAD, École Polytechnique de Montréal, Département de Mathématiques et de Génie Industriel, C.P. 6079, Centre-ville, Montréal (Québec), H3C 3A7 Canada
[9] Audet, C.; Dennis, JE Jr, Mesh adaptive direct search algorithms for constrained optimization, SIAM J Optim, 17, 1, 2-11 (2006)
[10] Audet, C.; Dang, CK; Orban, D., Efficient use of parallelism in algorithmic parameter optimization applications, Optim Lett, 7, 3, 421-433 (2011) · Zbl 1268.90083
[11] Audet, C.; Le Digabel, S.; Tribes, C., Dynamic scaling in the mesh adaptive direct search algorithm for blackbox optimization, Optim Eng, 17, 2, 333-358 (2016) · Zbl 1364.90360
[12] Bassiouny, HS; White, S.; Glagov, S.; Choi, E.; Giddens, DP; Zarins, CK, Anastomotic intimal hyperplasia: mechanical injury or flow induced, J Vasc Surg, 15, 4, 708-717 (1992)
[13] Beckley MC (2015) Comparison of sampling methods for kriging models. Ph.D. thesis, University of Pretoria
[14] Beiranvand, V.; Hare, W.; Lucet, Y., Best practices for comparing optimization algorithms, Optim Eng, 18, 4, 815-848 (2017) · Zbl 1390.90601
[15] Belitz P (2011) Applications on multi-dimensional sphere packings: derivative-free optimization. Ph.D. thesis, University of California, San Diego
[16] Booker AJ (2000) Well-conditioned Kriging models for optimization of computer models. Mathematics and Computing Technology Report 002, Boeing Phantom Works, Seattle, WA
[17] Booker, AJ; Dennis, JE Jr; Frank, PD; Serafini, DB; Torczon, V.; Trosset, MW, A rigorous framework for optimization of expensive functions by surrogates, Struct Optim, 17, 1, 1-13 (1999)
[18] Bossek, J., Smoof: single-and multi-objective optimization test functions, R Journal, 9, 1, 103-113 (2017)
[19] Box, GE; Hunter, JS; Hunter, WG, Statistics for experimenters: design, innovation, and discovery (2005), Hoboken: Wiley, Hoboken · Zbl 1082.62063
[20] Bozsak, F.; Gonzalez-Rodriguez, D.; Sternberger, Z.; Belitz, P.; Bewley, T.; Chomaz, JM; Barakat, AI, Optimization of drug delivery by drug-eluting stents, PLoS One, 10, 6, e0130182 (2015)
[21] Breiman, L.; Cutler, A., A deterministic algorithm for global optimization, Math Program, 58, 1-3, 179-199 (1993) · Zbl 0807.90103
[22] Brochu E, Cora VM, de Freitas N (2010) A tutorial on bayesian optimization of expensive cost functions, with application to active user modeling and hierarchical reinforcement learning. http://arxiv.org/abs/1012.2599
[23] Chiu, JJ; Chien, S., Effects of disturbed flow on vascular endothelium: pathophysiological basis and clinical perspectives, Physiol Rev, 91, 1, 327-387 (2011)
[24] Contal, E.; Buffoni, D.; Robicquet, A.; Vayatis, N., Parallel Gaussian process optimization with upper confidence bound and pure exploration, Joint European conference on machine learning and knowledge discovery in databases, 225-240 (2013), Berlin: Springer, Berlin
[25] Couckuyt, I.; Deschrijver, D.; Dhaene, T., Fast calculation of multiobjective probability of improvement and expected improvement criteria for pareto optimization, J Global Optim, 60, 3, 575-594 (2013) · Zbl 1303.90093
[26] Deb, K.; Thiele, L.; Laumanns, M.; Zitzler, E.; Abraham, A.; Jain, L.; Goldber, R., Scalable test problems for evolutionary multobjective optimization, Evolutionary multiobjective optimization, 105-145 (2005), London: Springer, London
[27] Diamond, P.; Armstrong, M., Robustness of variograms and conditioning of kriging matrices, J Int Assoc Math Geol, 16, 8, 809-822 (1984)
[28] Digabel, SL, Algorithm 909: NOMAD: nonlinear optimization with the MADS algorithm, ACM Trans Math Softw, 37, 4, 1-15 (2011) · Zbl 1365.65172
[29] Dolan, ED; Moré, JJ, Benchmarking optimization software with performance profiles, Math Program, 91, 2, 201-213 (2002) · Zbl 1049.90004
[30] Dyke, BV; Asaki, TJ, Using QR decomposition to obtain a new instance of mesh adaptive direct search with uniformly distributed polling directions, J Optim Theory Appl, 159, 3, 805-821 (2013) · Zbl 1284.90081
[31] Esmaily-Moghadam, M.; Bazilevs, Y.; Marsden, AL, A new preconditioning technique for implicitly coupled multidomain simulations with applications to hemodynamics, Comput Mech, 52, 5, 1141-1152 (2013) · Zbl 1388.76130
[32] Esmaily-Moghadam, M.; Bazilevs, Y.; Marsden, A., Impact of data distribution on the parallel performance of iterative linear solvers with emphasis on cfd of incompressible flows, Comput Mech, 55, 1, 93-103 (2015) · Zbl 1311.76057
[33] Esmaily-Moghadam, M.; Bazilevs, Y.; Marsden, AL, A bi-partitioned iterative algorithm for solving linear systems arising from incompressible flow problems, Comput Methods Appl Mech Eng, 286, 40-62 (2015) · Zbl 1423.76228
[34] Feinstein, JA; Benson, DW; Dubin, AM; Cohen, MS; Maxey, DM; Mahle, WT; Pahl, E.; Villafañe, J.; Bhatt, AB; Peng, LF, Hypoplastic left heart syndrome: current considerations and expectations, J Am College Cardiol, 59, 1, S1-S42 (2012)
[35] Ginsbourger, D.; Le Riche, R.; Carraro, L.; Tenne, Y.; Goh, C., Kriging is well-suited to parallelize optimization, Computational intelligence in expensive optimization problems. Adaptation learning and optimization (2010), Berlin: Springer, Berlin
[36] Ginsbourger D, Janusevskis J, Le Riche R (2011) Dealing with asynchronicity in parallel Gaussian Process based global optimization. Tech Rep hal-00507632
[37] Gould, N.; Scott, J., A note on performance profiles for benchmarking software, ACM Trans Math Softw (TOMS), 43, 2, 15 (2016) · Zbl 1369.65202
[38] Gramacy, RB; Gray, GA; Le Digabel, S.; Lee, HK; Ranjan, P.; Wells, G.; Wild, SM, Modeling an augmented lagrangian for blackbox constrained optimization, Technometrics, 58, 1, 1-11 (2016)
[39] Grechy, L.; Iori, F.; Corbett, R.; Shurey, S.; Gedroyc, W.; Duncan, N.; Caro, C.; Vincent, P., Suppressing unsteady flow in arterio-venous fistulae, Phys Fluids, 29, 10, 101901 (2017)
[40] Griffiths, G.; Nagy, J.; Black, D.; Stonebridge, P., Randomized clinical trial of distal anastomotic interposition vein cuff in infrainguinal polytetrafluoroethylene bypass grafting, Br J Surg, 91, 5, 560-562 (2004)
[41] Gropp, W.; Thakur, R.; Lusk, E., Using MPI-2: advanced features of the message passing interface (1999), Cambridge: MIT Press, Cambridge
[42] Haftka, RT; Villanueva, D.; Chaudhuri, A., Parallel surrogate-assisted global optimization with expensive functions-a survey, Struct Multidiscip Optim, 54, 1, 3-13 (2016)
[43] Haimovici, H.; Ascer, E.; Hollier, L.; Strandness, D. Jr; Towne, J., Haimovici’s vascular surgery (1996), Hoboken: Blackwell Science, Hoboken
[44] Hansen, N.; Lozano, JA; Larranaga, P.; Inza, I.; Bengoetxea, E., The CMA evolution strategy: a comparing review, Towards a new evolutionary computation, 75-102 (2006), Berlin: Springer, Berlin
[45] Hough, PD; Kolda, TG; Torczon, VJ, Asynchronous parallel pattern search for nonlinear optimization, SIAM J Sci Comput, 23, 1, 134-156 (2001) · Zbl 0990.65067
[46] How, T.; Rowe, C.; Gilling-Smith, G.; Harris, P., Interposition vein cuff anastomosis alters wall shear stress distribution in the recipient artery, J Vasc Surg, 31, 5, 1008-1017 (2000)
[47] Jamil, M.; Yang, XS, A literature survey of benchmark functions for global optimization problems, Int J Math Model Numer Optim, 4, 2, 150-94 (2013)
[48] Jansen, KE; Whiting, CH; Hulbert, GM, A generalized-\( \alpha\) method for integrating the filtered navier-stokes equations with a stabilized finite element method, Comput Meth Appl Mech Eng, 190, 3-4, 305-319 (2000) · Zbl 0973.76048
[49] Jin, OA; Chen, W.; Sudijianto, A., An efficient algorithm for constructing optimal design of computer experiments, J Stat Plan Inference, 134, 268-87 (2005)
[50] Jones, DR, A taxonomy of global optimization methods based on response surfaces, J Global Optim, 21, 4, 345-383 (2001) · Zbl 1172.90492
[51] Jones, DR; Perttunen, CD; Stuckman, BE, Lipschitzian optimization without the lipschitz constant, J Optim Theory Appl, 79, 1, 157-181 (1993) · Zbl 0796.49032
[52] Jones, DR; Schonlau, M.; Welch, WJ, Efficient global optimization of expensive black-box functions, J Global Optim, 13, 4, 455-492 (1998) · Zbl 0917.90270
[53] Koziel, S.; Michalewicz, Z., Evolutionary algorithms, homomorphous mappings and constrained parameter optimization, J Evol Comp, 7, 1, 19-44 (1999)
[54] Krige, D., A statistical approach to some mine valuations and allied problems in the Witwatersrand, J Chem Metall Min Soc South Africa, 52, 119-139 (1951)
[55] Ku, JP; Elkins, CJ; Taylor, CA, Comparison of CFD and MRI flow and velocities in an in vitro large artery bypass graft model, Ann Biomed Eng, 33, 3, 257-269 (2005)
[56] Lemson, M.; Tordoir, J.; Daemen, M.; Kitslaar, P., Intimal hyperplasia in vascular grafts, Eur J Vasc Endovasc Surg, 19, 4, 336-350 (2000)
[57] Levy, AV; Montalvo, A., The tunneling algorithm for the global minimization of functions, SIAM J Sci Stat Comput, 6, 1, 15-29 (1985) · Zbl 0601.65050
[58] Li, R.; Sudjianto, A., Analysis of computer experiments using penalized likelihood in gaussian kriging models, Technometrics, 47, 2, 111-120 (2005)
[59] Li, C.; Brezillon, J.; Görtz, S.; Dillmann, A.; Heller, G.; Krämer, E.; Kreplin, H.; Nitsche, W.; Rist, U., Efficient global optimization of a natural laminar airfoil based on surrogate modeling, New results in numerical and experimental fluid mechanics IX, 53-63 (2014), Berlin: Springer, Berlin
[60] Liu J, Han Z, Song W (2012) Comparison of infill sampling criteria in kriging-based aerodynamic optimization. In: 28th congress of the international council of the aeronautical sciences, pp 23-28
[61] Longest, P.; Kleinstreuer, C.; Archie, JP, Particle hemodynamics analysis of miller cuff arterial anastomosis, J Vasc Surg, 38, 6, 1353-1362 (2003)
[62] Marsden AL, Wang M, Dennis Jr JE (2003) Constrained aeroacoustic shape optimization using the surrogate management framework. In: Annual research briefs. Center for Turbulence Research, Stanford University, pp 399-412
[63] Marsden, AL; Feinstein, JA; Taylor, CA, A computational framework for derivative-free optimization of cardiovascular geometries, Comput Meth Appl Mech Eng, 197, 21-24, 1890-1905 (2008) · Zbl 1194.76296
[64] Marsden, AL, Optimization in cardiovascular modeling, Annu Rev Fluid Mech, 46, 519-46 (2014) · Zbl 1297.76202
[65] Marsden, AL; Esmaily-Moghadam, M., Multiscale modeling of cardiovascular flows for clinical decision support, Appl Mech Rev, 67, 30804, 1-11 (2015)
[66] McKay, MD; Conover, WJ; Beckman, RJ, A comparison of three methods for selecting values of input variables in the analysis of output from a computer code, Technometrics, 21, 239-245 (1979) · Zbl 0415.62011
[67] Miettinen, K., Nonlinear multiobjective optimization (1999), London: Kluwer Academic Publishers, London · Zbl 0949.90082
[68] Miller, J.; Foreman, R.; Ferguson, L.; Faris, I., Interposition vein cuff for anastomosis of prosthesis to small artery, Aust N Z J Surg, 54, 3, 283-285 (1984)
[69] Moghadam, ME; Marsden, TYHA, The assisted bidirectional Glenn: a novel surgical approach for first-stage single-ventricle heart palliation, J Thorac Cardio Surg, 149, 3, 699-705 (2015)
[70] Moghadam, ME; Bazilevs, Y.; Hsia, TY; Vignon-Clementel, I.; Marsden, A., A comparison of outlet boundary treatments for prevention of backflow divergence with relevance to blood flow simulations, Comput Mech, 48, 277-291 (2011) · Zbl 1398.76102
[71] Neville, RF; Tempesta, B.; Sidawy, AN, Tibial bypass for limb salvage using polytetrafluoroethylene and a distal vein patch, J Vasc Surg, 33, 2, 266-272 (2001)
[72] Niederreiter, H., Random number generation and quasi-Monte Carlo methods (1992), Philadelphia: SIAM, Philadelphia · Zbl 0761.65002
[73] Norberto, JJ; Sidawy, AN; Trad, KS; Jones, BA; Neville, RF; Najjar, SF; Sidawy, MK; DePalma, RG, The protective effect of vein cuffed anastomoses is not mechanical in origin, J Vasc Surg, 21, 4, 558-566 (1995)
[74] Norwood, W.; Kirklin, J.; Sanders, S., Hypoplastic left heart syndrome: experience with palliative surgery, J Thorac Cardiovasc Surg, 82, 511-9 (1981)
[75] Norwood, W.; Lang, P.; Castaneda, A.; Campbell, D., Experience with operations for hypoplastic left heart syndrome, J Thorac Cardiovasc Surg, 82, 511-9 (1981)
[76] Panneton, JM; Hollier, LH; Hofer, JM, Multicenter randomized prospective trial comparing a pre-cuffed polytetrafluoroethylene graft to a vein cuffed polytetrafluoroethylene graft for infragenicular arterial bypass, Ann Vasc Surg, 18, 2, 199-206 (2004)
[77] Park, C.; Haftka, RT; Kim, NH, Remarks on multi-fidelity surrogates, Struct Multidiscip Optim, 55, 3, 1029-1050 (2017)
[78] Passerini, AG; Milsted, A.; Rittgers, SE, Shear stress magnitude and directionality modulate growth factor gene expression in preconditioned vascular endothelial cells, J Vasc Surg, 37, 1, 182-190 (2003)
[79] Peherstorfer, B.; Willcox, K.; Gunzburger, M., Survey of multifidelity methods in uncertainty propagation, inference, and optimization, SIAM Rev, 60, 3, 550-591 (2018) · Zbl 1458.65003
[80] Peng, CY; Wu, C., On the choice of nugget in kriging modeling for deterministic computer experiments, J Comput Graph Stat, 23, 1, 151-168 (2014)
[81] Powell, MJ, The newuoa software for unconstrained optimization without derivatives, Large-scale nonlinear optimization, 255-297 (2006), Berlin: Springer, Berlin · Zbl 1108.90005
[82] Ramachandra, AB; Sankaran, S.; Humphrey, JD; Marsden, AL, Computational simulation of the adaptive capacity of vein grafts in response to increased pressure, J Biomech Eng, 137, 3, 031009 (2015)
[83] Rasmussen, CE; Williams, CKI, Gaussian processes for machine learning (Adaptive computation and machine learning) (2005), Cambridge: The MIT Press, Cambridge
[84] Rozza, G., On optimization, control and shape design for an arterial bypass, Int J Numer Methods Fluids, 47, 1411-1419 (2005) · Zbl 1155.76439
[85] Sacks, J.; Welch, WJ; Mitchell, TJ; Wynn, HP, Design and analysis of computer experiments, Stat Sci, 4, 4, 409-423 (1989)
[86] Sankaran, S., Stochastic optimization using a sparse grid collocation scheme, Prob Eng Mech, 24, 3, 382-396 (2009)
[87] Santner, TJ; Williams, BJ; Notz, WI, The design and analysis of computer experiments (2003), New York: Springer, New York
[88] Sasena, M.J.: Flexibility and efficiency enhancements for constrained global design optimization with kriging approximations. Ph.D. thesis, University of Michigan, Ann Arbor (2002)
[89] Saxena, DK; Duro, JA; Tiwari, A.; Deb, K.; Zhang, Q., Objective reduction in many-objective optimization: linear and nonlinear algorithms, IEEE Trans Evol Comput, 17, 1, 77-99 (2012)
[90] Schroeder, WJ; Lorensen, B.; Martin, K., The visualization toolkit: an object-oriented approach to 3D graphics (2004), Clifton Park: Kitware, Clifton Park
[91] Seo, J.; Schiavazzi, D.; Marsden, A., Performance of preconditioned iterative linear solvers for cardiovascular simulations in rigid and deformable vessels, Comput Mech, 64, 3, 717-739 (2019) · Zbl 1465.74166
[92] Shang, JK; Esmaily, M.; Verma, A.; Reinhartz, O.; Figliola, RS; Hsia, TY; Feinstein, JA; Marsden, AL, Patient-specific multiscale modeling of the assisted bidirectional Glenn, Ann Thorac Surg, 107, 4, 1232-39 (2019)
[93] Shephard, MS; Georges, MK, Automatic three-dimensional mesh generation by the finite octree technique, Int J Numer Methods Eng, 32, 4, 709-749 (1991) · Zbl 0755.65116
[94] Siegman, F., Use of the venous cuff for graft anastomosis, Surg Gynecol Obstet, 148, 6, 930-930 (1979)
[95] Stonebridge, P.; Prescott, R.; Ruckley, C., Randomized trial comparing infrainguinal polytetrafluoroethylene bypass grafting with and without vein interposition cuff at the distal anastomosis, J Vasc Surg, 26, 4, 543-550 (1997)
[96] Tamisier, D.; Vouhe, P.; Vernant, F.; Leca, F.; Massot, C.; Neveux, J., Modified blalock-taussig shunts: results in infants less than 3 months of age, Ann Thorac Surg, 49, 797-801 (1990)
[97] Taylor, CA; Hughes, TJR; Zarins, CK, Finite element modeling of blood flow in arteries, Comput Meth Appl Mech Eng, 158, 155-196 (1998) · Zbl 0953.76058
[98] Torczon, V.; Trosset, MW, From evolutionary operation to parallel direct search: pattern search algorithms for numerical optimization, Comput Sci Stat, 29, 396-401 (1998)
[99] Towns, J.; Cockerill, T.; Dahan, M.; Foster, I.; Gaither, K.; Grimshaw, A.; Hazlewood, V.; Lathrop, S.; Lifka, D.; Peterson, GD, XSEDE: accelerating scientific discovery, Comput Sci Eng, 16, 5, 62-74 (2014)
[100] Updegrove, A.; Wilson, NM; Merkow, J.; Lan, H.; Marsden, A.; Shadden, S., Simvascular: an open source pipeline for cardiovascular simulation, Ann Biomed Eng, 45, 3, 525-541 (2016)
[101] Veith, FJ; Gupta, SK; Ascer, E.; White-Flores, S.; Samson, RH; Scher, LA; Towne, JB; Bernhard, VM; Bonier, P.; Flinn, WR; Astelford, P.; Yao, JS; Bergan, JJ, Six-year prospective multicenter randomized comparison of autologous saphenous vein and expanded polytetrafluoroethylene grafts in infrainguinal arterial reconstructions, J Vasc Surg, 3, 1, 104-114 (1986)
[102] Verma, A.; Esmaily, M.; Shang, J.; Figliola, R.; Feinstein, JA; Hsia, TY; Marsden, AL, Optimization of the assisted bidirectional glenn procedure for first stage single ventricle repair, World J Pediatr Congenit Heart Surg, 9, 2, 157-170 (2018)
[103] Vignon-Clementel, IE; Figueroa, CA; Jansen, KE; Taylor, CA, Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries, Comput Meth Appl Mech Eng, 195, 3776-3796 (2006) · Zbl 1175.76098
[104] Whiting, CH; Jansen, KE, A stabilized finite element method for the incompressible Navier-Stokes equations using a hierarchical basis, Int J Numer Meth Fluid, 35, 1, 93-116 (2001) · Zbl 0990.76048
[105] Wild, S.; More, J., Benchmarking derivative-free optimization algorithms, SIAM J Optim, 20, 1, 172-191 (2009) · Zbl 1187.90319
[106] Wilson, N.; Wang, K.; Dutton, R.; Taylor, CA, A software framework for creating patient specific geometric models from medical imaging data for simulation based medical planning of vascular surgery, Lect Notes Comput Sci, 2208, 449-456 (2001) · Zbl 1041.68783
[107] Wong K, Brown L, Coan J, White D (2014) Distributive interoperable executive library (DIEL) for systems of multiphysics simulation. In: 2014 15th international conference on parallel and distributed computing, applications and technologies. IEEE
[108] Yang, W.; Feinstein, JA; Marsden, AL, Constrained optimization of an idealized Y-shaped baffle for the Fontan surgery at rest and exercise, Comput Methods Appl Mech Eng, 199, 2135-2149 (2010) · Zbl 1231.74357
[109] Yang, W.; Vignon-Clementel, IE; Troianowski, G.; Reddy, VM; Feinstein, JA; Marsden, AL, Hepatic blood flow distribution and performance in conventional and novel y-graft fontan geometries: a case series computational fluid dynamics study, J Thorac Cardiovasc Surg, 143, 5, 1086-1097 (2012)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.