心血管系统中若干生物力学问题的不确定性量化分析

心血管疾病已成为世界范围内人类死亡的主要疾病，该疾病不仅危害生命健康，还给个体和整个社会经济带来了巨大的负担。如今，心血管建模已在临床治疗中广泛应用。这些模型可以辅助医生进行治疗策略的规划，但其中大量的不确定性却影响着模型预测的结果。因此对各种心血管模型进行不确定性量化分析已成为必要的工作。

研究表明，血管壁的应力分布会促使血管发生一系列生长和重构变化(包括疾病的发展)。这种应力分布可以通过对血管模型进行流固耦合分析来确定。因此，我们需要分析血管生长重构模型中的不确定性，同时也要考虑心血管模型中的不确定性。

本文采用蒙特卡罗方法和随机配点方法，分别研究了血管的生长重构模型中的生长，预拉伸以及材料参数和主动脉弓三维模型中的杨氏模量和血液黏度的不确定性，并进行敏感性分析得到这些参数的敏感性排名。研究的感兴趣的量包括生长重构模型中血管应力的变化和血管恢复稳态的时间，主动脉弓三维模型中各出口的流量和压强，以及整个主动脉弓模型上的时均壁面切应力以及震荡剪切指数的分布。

敏感性分析结果显示，生长重构模型中应力的变化对生长参数最为敏感，其次是预拉伸参数和材料参数。而恢复稳态的时间对预拉伸参数的敏感性远超其他参数。在主动脉弓三维模型中，杨氏模量对各出口流量和压强的影响大于血液黏度，血液黏度对这两个量影响很小。对时均壁面切应力以及震荡剪切指数来说，时均壁面切应力主要受血液黏度的影响，而震荡剪切指数对杨氏模量更为敏感。

[1] GUYTON A C, HALL J E. Guyton and Hall textbook of medical physiology[M]. Elsevier, 2011.
[2] OTTESEN J T, OLUFSEN M S, LARSEN J K. Applied mathematical models in human physiology[M]. Society for Industrial and Applied Mathematics, 2004.
[3] 柳兆荣. 心血管流体力学[M]. 复旦大学出版社, 1986.
[4] 胡 盛 寿 ,高 润 霖 ,刘 力 生 等 . 《 中 国 心 血 管 病 报 告 2018 》 概 要 [J]. 中 国 循 环 杂 志,2019,34(03):209-220.
[5] 中国心血管健康与疾病报告编写组. 中国心血管健康与疾病报告 2019 概要[J]. 中 华老年病研究电子杂志, 2020, 7(4):4-15.
[6] HARDING S, SILVA M J, MOLAODI O R, et al. Longitudinal study of cardiometabolic risk from early adolescence to early adulthood in an ethnically diverse cohort[J]. BMJ open, 2016, 6(12): e013221.
[7] TORO E F. Brain venous haemodynamics, neurological diseases and mathematical modelling. A review[J]. Applied Mathematics and Computation, 2016, 272: 542-579.
[8] HUMPHREY J D. Vascular adaptation and mechanical homeostasis at tissue, cellular, and sub-cellular levels[J]. Cell biochemistry and biophysics, 2008, 50: 53-78.
[9] DAS S, AIBA T, ROSENBERG M, et al. Pathological role of serum-and glucocorticoid-regulated kinase 1 in adverse ventricular remodeling[J]. Circulation, 2012, 126(18): 2208-2219.
[10] National Research Council, Division on Engineering, Physical Sciences, et al. The mathematical sciences in 2025[M]. National Academies Press, 2013.
[11] SMITH R C. Uncertainty quantification: theory, implementation, and applications[M]. Siam, 2013.
[12] DRAPER D. Assessment and propagation of model uncertainty[J]. Journal of the Royal Statistical Society Series B: Statistical Methodology, 1995, 57(1): 45-70.
[13] BEVEN K, BINLEY A. The future of distributed models: model calibration and uncertainty prediction[J]. Hydrological processes, 1992, 6(3): 279-298.
[14] YAO W, CHEN X, LUO W, et al. Review of uncertainty-based multidisciplinary design optimization methods for aerospace vehicles[J]. Progress in Aerospace Sciences, 2011, 47(6): 450-479.
[15] 汤涛, 周涛. 不确定性量化的高精度数值方法和理论[J]. 中国科学: 数学, 2015, 45(7): 891-928.
[16] FISHMAN G S. Monte Carlo: concepts, algorithms, and applications[M]. Springer Science & Business Media, 2013.
[17] XIU D, KARNIADAKIS G E. The Wiener--Askey polynomial chaos for stochastic differential equations[J]. SIAM journal on scientific computing, 2002, 24(2): 619- 644.
[18] XIU D, HESTHAVEN J S. High-order collocation methods for differential equations with random inputs[J]. SIAM Journal on Scientific Computing, 2005, 27(3): 1118- 1139.
[19] EULER L. Principia pro motu sanguinis per arterias determinando[J]. Opera postuma, 1862: 814-823.
[20] SAGAWA K, LIE R K, SCHAEFER J. Translation of Otto frank's paper" Die Grundform des arteriellen Pulses" zeitschrift für biologie 37: 483-526 (1899)[J]. Journal of molecular and cellular cardiology, 1990, 22(3): 253-254.
[21] FLEETER C M, GERACI G, SCHIAVAZZI D E, et al. Multilevel and multifidelity uncertainty quantification for cardiovascular hemodynamics[J]. Computer methods in applied mechanics and engineering, 2020, 365: 113030.
[22] COPE F W. An elastic reservoir theory of the human systemic arterial system using current data on aortic elasticity[J]. The bulletin of mathematical biophysics, 1960, 22: 19-40.
[23] MCLEOD J. Physbe... a physiological simulation benchmark experiment[J]. Simulation, 1966, 7(6): 324-329.
[24] WOMERSLEY J R. Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known[J]. The Journal of physiology, 1955, 127(3): 553.
[25] GREEN N E, CHEN S Y J, MESSENGER J C, et al. Three-dimensional vascular angiography[J]. Current problems in cardiology, 2004, 29(3): 104-142.
[26] LEWIS M A. Multislice CT: opportunities and challenges[J]. The British journal of radiology, 2001, 74(885): 779-781.
[27] GRAVES M J. Magnetic resonance angiography[J]. The British Journal of Radiology, 1997, 70(829): 6-28.
[28] FENSTER A, DOWNEY D B, CARDINAL H N. Three-dimensional ultrasound imaging[J]. Physics in medicine & biology, 2001, 46(5): R67.
[29] FRAUENFELDER T, LOTFEY M, BOEHM T, et al. Computational fluid dynamics: hemodynamic changes in abdominal aortic aneurysm after stent-graft implantation[J]. Cardiovascular and interventional radiology, 2006, 29: 613-623.
[30] KUNG E, BARETTA A, BAKER C, et al. Predictive modeling of the virtual Hemi- Fontan operation for second stage single ventricle palliation: two patient-specific cases[J]. Journal of biomechanics, 2013, 46(2): 423-429.
[31] YANG W, MARSDEN A L, OGAWA M T, et al. Right ventricular stroke work correlates with outcomes in pediatric pulmonary arterial hypertension[J]. Pulmonary circulation, 2018, 8(3): 2045894018780534.
[32] YANG W, DONG M, Rabinovitch M, et al. Evolution of hemodynamic forces in the pulmonary tree with progressively worsening pulmonary arterial hypertension in pediatric patients[J]. Biomechanics and modeling in mechanobiology, 2019, 18(3): 779-796.
[33] RAMACHANDRA A B, KAHN A M, MARSDEN A L. Patient-specific simulations reveal significant differences in mechanical stimuli in venous and arterial coronary grafts[J]. Journal of cardiovascular translational research, 2016, 9(4): 279-290.
[34] LONG C C, MARSDEN A L, BAZILEVS Y. Shape optimization of pulsatile ventricular assist devices using FSI to minimize thrombotic risk[J]. Computational Mechanics, 2014, 54: 921-932.
[35] GUNDERT T J, MARSDEN A L, YANG W, et al. Identification of hemodynamically optimal coronary stent designs based on vessel caliber[J]. IEEE Transactions on Biomedical Engineering, 2012, 59(7): 1992-2002.
[36] MIGLIAVACCA F, PETRINI L, COLOMBO M, et al. Mechanical behavior of coronary stents investigated through the finite element method[J]. Journal of biomechanics, 2002, 35(6): 803-811.
[37] LAMBERT J W. Fluid flow in a nonrigid tube[M]. Purdue University, 1956.
[38] LAMBERT J W. On the nonlinearities of fluid flow in nonrigid tubes[J]. Journal of the Franklin Institute, 1958, 266(2): 83-102.
[39] HUGHES T J R. A study of the one-dimensional theory of arterial pulse propagation[M]. Structural Engineering Laboratory, University of California, 1974.
[40] HUGHES T J R, LUBLINER J. On the one-dimensional theory of blood flow in the larger vessels[J]. Mathematical Biosciences, 1973, 18(1-2): 161-170.
[41] OLUFSEN M S. Structured tree outflow condition for blood flow in larger systemic arteries[J]. American journal of physiology-Heart and circulatory physiology, 1999, 276(1): H257-H268.
[42] WAN J, STEELE B, SPICER S A, et al. A one-dimensional finite element method for simulation-based medical planning for cardiovascular disease[J]. Computer Methods in Biomechanics & Biomedical Engineering, 2002, 5(3): 195-206.
[43] CHEN P, QUARTERONI A, ROZZA G. Simulation‐based uncertainty quantification of human arterial network hemodynamics[J]. International journal for numerical methods in biomedical engineering, 2013, 29(6): 698-721.
[44] BLANCO P J, BULANT C A, MÜLLER L O, et al. Comparison of 1D and 3D models for the estimation of fractional flow reserve[J]. Scientific reports, 2018, 8(1): 17275.
[45] HUMPHREY J D, RAJAGOPAL K R. A constrained mixture model for growth and remodeling of soft tissues[J]. Mathematical models and methods in applied sciences, 2002, 12(03): 407-430.
[46] GLEASON R L, TABER L A, HUMPHREY J D. A 2-D model of flow-induced alterations in the geometry, structure, and properties of carotid arteries[J]. J. Biomech. Eng., 2004, 126(3): 371-381.
[47] ALFORD P W, HUMPHREY J D, TABER L A. Growth and remodeling in a thick- walled artery model: effects of spatial variations in wall constituents[J]. Biomechanics and modeling in mechanobiology, 2008, 7: 245-262.
[48] LAUBRIE J D, MOUSAVI J S, AVRIL S. A new finite‐element shell model for arterial growth and remodeling after stent implantation[J]. International journal for numerical methods in biomedical engineering, 2020, 36(1): e3282.
[49] GACEK E, MAHUTGA R R, BAROCAS V H. Hybrid discrete-continuum multiscale model of tissue growth and remodeling[J]. Acta Biomaterialia, 2023, 163: 7-24.
[50] LOH W L. On Latin hypercube sampling[J]. The annals of statistics, 1996, 24(5): 2058-2080.
[51] STEIN M. Large sample properties of simulations using Latin hypercube sampling[J]. Technometrics, 1987, 29(2): 143-151.
[52] NIEDERREITER H. Random number generation and quasi-Monte Carlo methods[M]. Society for Industrial and Applied Mathematics, 1992.
[53] Monte Carlo and quasi-Monte Carlo methods 1996: proceedings of a conference at the University of Salzburg, Austria, July 9-12, 1996[M]. Springer Science & Business Media, 1998.
[54] GILES M B. Multilevel monte carlo methods[J]. Acta numerica, 2015, 24: 259-328.
[55] NG L W T, WILLCOX K E. Multifidelity approaches for optimization under uncertainty[J]. International Journal for numerical methods in Engineering, 2014, 100(10): 746-772.
[56] GERACI G, ELDRED M S, Iaccarino G. A multifidelity multilevel Monte Carlo method for uncertainty propagation in aerospace applications[C]//19th AIAA non- deterministic approaches conference. 2017: 1951.
[57] GERSTNER T, GRIEBEL M. Dimension–adaptive tensor–product quadrature[J]. Computing, 2003, 71: 65-87.
[58] MA X, ZABARAS N. An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations[J]. Journal of Computational Physics, 2009, 228(8): 3084-3113.
[59] AGARWAL N, ALURU N R. A domain adaptive stochastic collocation approach for analysis of MEMS under uncertainties[J]. Journal of Computational Physics, 2009, 228(20): 7662-7688.
[60] SCHIAVAZZI D E, DOOSTAN A, IACCARINO G. Sparse multiresolution regression for uncertainty propagation[J]. International Journal for Uncertainty Quantification, 2014, 4(4).
[61] WAN X, KARNIADAKIS G E. Multi-element generalized polynomial chaos for arbitrary probability measures[J]. SIAM Journal on Scientific Computing, 2006, 28(3): 901-928.
[62] WAN X, KARNIADAKIS G E. An adaptive multi-element generalized polynomial chaos method for stochastic differential equations[J]. Journal of Computational Physics, 2005, 209(2): 617-642.
[63] SEO J, SCHIAVAZZI D E, KAHN A M, et al. The effects of clinically‐derived parametric data uncertainty in patient‐specific coronary simulations with deformable walls[J]. International journal for numerical methods in biomedical engineering, 2020, 36(8): e3351.
[64] BERTAGLIA G, CALEFFI V, PARESCHI L, et al. Uncertainty quantification of viscoelastic parameters in arterial hemodynamics with the a-FSI blood flow model[J]. Journal of Computational Physics, 2021, 430: 110102.
[65] YAN J, CAI S, CAI X, et al. Uncertainty quantification of microcirculatory characteristic parameters for recognition of cardiovascular diseases[J]. Computer Methods and Programs in Biomedicine, 2023, 240: 107674.
[66] BOCCADIFUOCO A, MARIOTTI A, Celi S, et al. Impact of uncertainties in outflow boundary conditions on the predictions of hemodynamic simulations of ascending thoracic aortic aneurysms[J]. Computers & Fluids, 2018, 165: 96-115.
[67] ANTONUCCIO M N, MARIOTTI A, FANNI B M, et al. Effects of uncertainty of outlet boundary conditions in a patient-specific case of aortic coarctation[J]. Annals of biomedical engineering, 2021, 49(12): 3494-3507.
[68] BRÜNING J, HELLMEIER F, YEVTUSHENKO P, et al. Uncertainty quantification for non-invasive assessment of pressure drop across a coarctation of the aorta using CFD[J]. Cardiovascular engineering and technology, 2018, 9: 582-596.
[69] FOSSAN F E, STURDY J, MÜLLER L O, et al. Uncertainty quantification and sensitivity analysis for computational FFR estimation in stable coronary artery disease[J]. Cardiovascular engineering and technology, 2018, 9: 597-622.
[70] KU D N. Blood flow in arteries[J]. Annual review of fluid mechanics, 1997, 29(1): 399-434.
[71] LOTH F, FISCHER P F, BASSIOUNY H S. Blood flow in end-to-side anastomoses[J]. Annu. Rev. Fluid Mech., 2008, 40: 367-393.
[72] SANKARAN S, HUMPHREY J D, MARSDEN A L. An efficient framework for optimization and parameter sensitivity analysis in arterial growth and remodeling computations[J]. Computer methods in applied mechanics and engineering, 2013, 256: 200-210.
[73] TRUESDELL C, NOLL W, TRUESDELL C, et al. The non-linear field theories of mechanics[M]. Springer Berlin Heidelberg, 2004.
[74] RAJAGOPAL K R, TAO L. Mechanics of mixtures[M]. World scientific, 1995.
[75] BAEK S, RAJAGOPAL K R, HUMPHREY J D. A theoretical model of enlarging intracranial fusiform aneurysms[J]. Journal of biomechanical engineering, 2006, 128(1): 142-149.
[76] VALENTIN A, HUMPHREY J D. Evaluation of fundamental hypotheses underlying constrained mixture models of arterial growth and remodelling[J]. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2009, 367(1902): 3585-3606.
[77] VALENTIN A, CARDAMONE L, BAEK S, et al. Complementary vasoactivity and matrix remodelling in arterial adaptations to altered flow and pressure[J]. Journal of The Royal Society Interface, 2009, 6(32): 293-306.
[78] RIZVI M A D, MYERS P R. Nitric oxide modulates basal and endothelin-induced coronary artery vascular smooth muscle cell proliferation and collagen levels[J]. Journal of molecular and cellular cardiology, 1997, 29(7): 1779-1789.
[79] RIZVI M A D, KATWA L, SPADONE D P, et al. The effects of endothelin-1 on collagen type I and type III synthesis in cultured porcine coronary artery vascular smooth muscle cells[J]. Journal of molecular and cellular cardiology, 1996, 28(2): 243-252.
[80] PRICE J M, DAVIS D L, KNAUSS E B. Length-dependent sensitivity in vascular smooth muscle[J]. American Journal of Physiology-Heart and Circulatory Physiology, 1981, 241(4): H557-H563.
[81] RACHEV A, HAYASHI K. Theoretical study of the effects of vascular smooth muscle contraction on strain and stress distributions in arteries[J]. Annals of biomedical engineering, 1999, 27: 459-468.
[82] BAEK S, VALENTÍN A, HUMPHREY J D. Biochemomechanics of cerebral vasospasm and its resolution: II. Constitutive relations and model simulations[J]. Annals of biomedical engineering, 2007, 35: 1498-1509.
[83] UEMATSU M, OHARA Y, NAVAS J P, et al. Regulation of endothelial cell nitric oxide synthase mRNA expression by shear stress[J]. American Journal of Physiology-Cell Physiology, 1995, 269(6): C1371-C1378.
[84] MALEK A, IZUMO S. Physiological fluid shear stress causes downregulation of endothelin-1 mRNA in bovine aortic endothelium[J]. American Journal of Physiology-Cell Physiology, 1992, 263(2): C389-C396.
[85] LANGILLE B L. Arterial remodeling: relation to hemodynamics[J]. Canadian journal of physiology and pharmacology, 1996, 74(7): 834-841.
[86] STENMARK K R, MECHAM R P. Cellular and molecular mechanisms of pulmonary vascular remodeling[J]. Annual review of physiology, 1997, 59(1): 89-144.
[87] RODRIGUEZ E K, HOGER A, MCCULLOCH A D. Stress-dependent finite growth in soft elastic tissues[J]. Journal of biomechanics, 1994, 27(4): 455-467.
[88] BAZILEVS Y, CALO V M, COTTRELL J A, et al. Variational multiscale residual- based turbulence modeling for large eddy simulation of incompressible flows[J]. Computer methods in applied mechanics and engineering, 2007, 197(1-4): 173-201.
[89] FRANCA L P, FREY S L. Stabilized finite element methods: II. The incompressible Navier-Stokes equations[J]. Computer Methods in Applied Mechanics and Engineering, 1992, 99(2-3): 209-233.
[90] FIGUEROA C A. A coupled-momentum method to model blood flow and vessel deformation in human arteries: applications in disease research and simulation-based medical planning[M]. Stanford university, 2006.
[91] BAZILEVS Y, GOHEAN J R, HUGHES T J R, 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[J]. Computer Methods in Applied Mechanics and Engineering, 2009, 198(45-46): 3534-3550.
[92] JANSEN K E, WHITING C H, HULBERT G M. A generalized-α method for integrating the filtered Navier–Stokes equations with a stabilized finite element method[J]. Computer methods in applied mechanics and engineering, 2000, 190(3-4): 305-319.
[93] CHUNG J, HULBERT G M. A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method[J]. 1993.
[94] MOGHADAM M E, VIGNON-CLEMENTEL I E, FIGLIOLA R, et al. A modular numerical method for implicit 0D/3D coupling in cardiovascular finite element simulations[J]. Journal of Computational Physics, 2013, 244: 63-79.
[95] LAN I S, LIU J, YANG W, et al. A reduced unified continuum formulation for vascular fluid–structure interaction[J]. Computer Methods in Applied Mechanics and Engineering, 2022, 394: 114852.
[96] LIU J, YANG W, LAN I S, et al. Fluid-structure interaction modeling of blood flow in the pulmonary arteries using the unified continuum and variational multiscale formulation[J]. Mechanics research communications, 2020, 107: 103556.
[97] UPDEGROVE A, WILSON N M, MERKOW J, et al. SimVascular: an open source pipeline for cardiovascular simulation[J]. Annals of biomedical engineering, 2017, 45: 525-541.
[98] LOÈVE M. Elementary probability theory[M]. Springer New York, 1977.
[99] BABUSKA I, TEMPONE R, ZOURARIS G E. Galerkin finite element approximations of stochastic elliptic partial differential equations[J]. SIAM Journal on Numerical Analysis, 2004, 42(2): 800-825.
[100] WIENER N. The homogeneous chaos[J]. American Journal of Mathematics, 1938, 60(4): 897-936.
[101] XIU D, KARNIADAKIS G E. The Wiener--Askey polynomial chaos for stochastic differential equations[J]. SIAM journal on scientific computing, 2002, 24(2): 619- 644.
[102] 王鹏, 修东滨. 不确定性量化导论[M]. 科学出版社, 2019.
[103] SMOLYAK S A. Quadrature and interpolation formulas for tensor products of certain classes of functions[C]//Doklady Akademii Nauk. Russian Academy of Sciences, 1963, 148(5): 1042-1045.
[104] BUNGARTZ H J, GRIEBEL M. Sparse grids[J]. Acta numerica, 2004, 13: 147-269.
[105] GAO F, GUO Z, SAKAMOTO M, et al. Fluid-structure interaction within a layered aortic arch model[J]. Journal of biological physics, 2006, 32: 435-454.
[1] GUYTON A C, HALL J E. Guyton and Hall textbook of medical physiology[M]. Elsevier, 2011.
[2] OTTESEN J T, OLUFSEN M S, LARSEN J K. Applied mathematical models in human physiology[M]. Society for Industrial and Applied Mathematics, 2004.
[3] 柳兆荣. 心血管流体力学[M]. 复旦大学出版社, 1986.
[4] 胡 盛 寿 ,高 润 霖 ,刘 力 生 等 . 《 中 国 心 血 管 病 报 告 2018 》 概 要 [J]. 中 国 循 环 杂 志,2019,34(03):209-220.
[5] 中国心血管健康与疾病报告编写组. 中国心血管健康与疾病报告 2019 概要[J]. 中 华老年病研究电子杂志, 2020, 7(4):4-15.
[6] HARDING S, SILVA M J, MOLAODI O R, et al. Longitudinal study of cardiometabolic risk from early adolescence to early adulthood in an ethnically diverse cohort[J]. BMJ open, 2016, 6(12): e013221.
[7] TORO E F. Brain venous haemodynamics, neurological diseases and mathematical modelling. A review[J]. Applied Mathematics and Computation, 2016, 272: 542-579.
[8] HUMPHREY J D. Vascular adaptation and mechanical homeostasis at tissue, cellular, and sub-cellular levels[J]. Cell biochemistry and biophysics, 2008, 50: 53-78.
[9] DAS S, AIBA T, ROSENBERG M, et al. Pathological role of serum-and glucocorticoid-regulated kinase 1 in adverse ventricular remodeling[J]. Circulation, 2012, 126(18): 2208-2219.
[10] National Research Council, Division on Engineering, Physical Sciences, et al. The mathematical sciences in 2025[M]. National Academies Press, 2013.
[11] SMITH R C. Uncertainty quantification: theory, implementation, and applications[M]. Siam, 2013.
[12] DRAPER D. Assessment and propagation of model uncertainty[J]. Journal of the Royal Statistical Society Series B: Statistical Methodology, 1995, 57(1): 45-70.
[13] BEVEN K, BINLEY A. The future of distributed models: model calibration and uncertainty prediction[J]. Hydrological processes, 1992, 6(3): 279-298.
[14] YAO W, CHEN X, LUO W, et al. Review of uncertainty-based multidisciplinary design optimization methods for aerospace vehicles[J]. Progress in Aerospace Sciences, 2011, 47(6): 450-479.
[15] 汤涛, 周涛. 不确定性量化的高精度数值方法和理论[J]. 中国科学: 数学, 2015, 45(7): 891-928.
[16] FISHMAN G S. Monte Carlo: concepts, algorithms, and applications[M]. Springer Science & Business Media, 2013.
[17] XIU D, KARNIADAKIS G E. The Wiener--Askey polynomial chaos for stochastic differential equations[J]. SIAM journal on scientific computing, 2002, 24(2): 619- 644.
[18] XIU D, HESTHAVEN J S. High-order collocation methods for differential equations with random inputs[J]. SIAM Journal on Scientific Computing, 2005, 27(3): 1118- 1139.
[19] EULER L. Principia pro motu sanguinis per arterias determinando[J]. Opera postuma, 1862: 814-823.
[20] SAGAWA K, LIE R K, SCHAEFER J. Translation of Otto frank's paper" Die Grundform des arteriellen Pulses" zeitschrift für biologie 37: 483-526 (1899)[J]. Journal of molecular and cellular cardiology, 1990, 22(3): 253-254.
[21] FLEETER C M, GERACI G, SCHIAVAZZI D E, et al. Multilevel and multifidelity uncertainty quantification for cardiovascular hemodynamics[J]. Computer methods in applied mechanics and engineering, 2020, 365: 113030.
[22] COPE F W. An elastic reservoir theory of the human systemic arterial system using current data on aortic elasticity[J]. The bulletin of mathematical biophysics, 1960, 22: 19-40.
[23] MCLEOD J. Physbe... a physiological simulation benchmark experiment[J]. Simulation, 1966, 7(6): 324-329.
[24] WOMERSLEY J R. Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known[J]. The Journal of physiology, 1955, 127(3): 553.
[25] GREEN N E, CHEN S Y J, MESSENGER J C, et al. Three-dimensional vascular angiography[J]. Current problems in cardiology, 2004, 29(3): 104-142.
[26] LEWIS M A. Multislice CT: opportunities and challenges[J]. The British journal of radiology, 2001, 74(885): 779-781.
[27] GRAVES M J. Magnetic resonance angiography[J]. The British Journal of Radiology, 1997, 70(829): 6-28.
[28] FENSTER A, DOWNEY D B, CARDINAL H N. Three-dimensional ultrasound imaging[J]. Physics in medicine & biology, 2001, 46(5): R67.
[29] FRAUENFELDER T, LOTFEY M, BOEHM T, et al. Computational fluid dynamics: hemodynamic changes in abdominal aortic aneurysm after stent-graft implantation[J]. Cardiovascular and interventional radiology, 2006, 29: 613-623.
[30] KUNG E, BARETTA A, BAKER C, et al. Predictive modeling of the virtual Hemi- Fontan operation for second stage single ventricle palliation: two patient-specific cases[J]. Journal of biomechanics, 2013, 46(2): 423-429.
[31] YANG W, MARSDEN A L, OGAWA M T, et al. Right ventricular stroke work correlates with outcomes in pediatric pulmonary arterial hypertension[J]. Pulmonary circulation, 2018, 8(3): 2045894018780534.
[32] YANG W, DONG M, Rabinovitch M, et al. Evolution of hemodynamic forces in the pulmonary tree with progressively worsening pulmonary arterial hypertension in pediatric patients[J]. Biomechanics and modeling in mechanobiology, 2019, 18(3): 779-796.
[33] RAMACHANDRA A B, KAHN A M, MARSDEN A L. Patient-specific simulations reveal significant differences in mechanical stimuli in venous and arterial coronary grafts[J]. Journal of cardiovascular translational research, 2016, 9(4): 279-290.
[34] LONG C C, MARSDEN A L, BAZILEVS Y. Shape optimization of pulsatile ventricular assist devices using FSI to minimize thrombotic risk[J]. Computational Mechanics, 2014, 54: 921-932.
[35] GUNDERT T J, MARSDEN A L, YANG W, et al. Identification of hemodynamically optimal coronary stent designs based on vessel caliber[J]. IEEE Transactions on Biomedical Engineering, 2012, 59(7): 1992-2002.
[36] MIGLIAVACCA F, PETRINI L, COLOMBO M, et al. Mechanical behavior of coronary stents investigated through the finite element method[J]. Journal of biomechanics, 2002, 35(6): 803-811.
[37] LAMBERT J W. Fluid flow in a nonrigid tube[M]. Purdue University, 1956.
[38] LAMBERT J W. On the nonlinearities of fluid flow in nonrigid tubes[J]. Journal of the Franklin Institute, 1958, 266(2): 83-102.
[39] HUGHES T J R. A study of the one-dimensional theory of arterial pulse propagation[M]. Structural Engineering Laboratory, University of California, 1974.
[40] HUGHES T J R, LUBLINER J. On the one-dimensional theory of blood flow in the larger vessels[J]. Mathematical Biosciences, 1973, 18(1-2): 161-170.
[41] OLUFSEN M S. Structured tree outflow condition for blood flow in larger systemic arteries[J]. American journal of physiology-Heart and circulatory physiology, 1999, 276(1): H257-H268.
[42] WAN J, STEELE B, SPICER S A, et al. A one-dimensional finite element method for simulation-based medical planning for cardiovascular disease[J]. Computer Methods in Biomechanics & Biomedical Engineering, 2002, 5(3): 195-206.
[43] CHEN P, QUARTERONI A, ROZZA G. Simulation‐based uncertainty quantification of human arterial network hemodynamics[J]. International journal for numerical methods in biomedical engineering, 2013, 29(6): 698-721.
[44] BLANCO P J, BULANT C A, MÜLLER L O, et al. Comparison of 1D and 3D models for the estimation of fractional flow reserve[J]. Scientific reports, 2018, 8(1): 17275.
[45] HUMPHREY J D, RAJAGOPAL K R. A constrained mixture model for growth and remodeling of soft tissues[J]. Mathematical models and methods in applied sciences, 2002, 12(03): 407-430.
[46] GLEASON R L, TABER L A, HUMPHREY J D. A 2-D model of flow-induced alterations in the geometry, structure, and properties of carotid arteries[J]. J. Biomech. Eng., 2004, 126(3): 371-381.
[47] ALFORD P W, HUMPHREY J D, TABER L A. Growth and remodeling in a thick- walled artery model: effects of spatial variations in wall constituents[J]. Biomechanics and modeling in mechanobiology, 2008, 7: 245-262.
[48] LAUBRIE J D, MOUSAVI J S, AVRIL S. A new finite‐element shell model for arterial growth and remodeling after stent implantation[J]. International journal for numerical methods in biomedical engineering, 2020, 36(1): e3282.
[49] GACEK E, MAHUTGA R R, BAROCAS V H. Hybrid discrete-continuum multiscale model of tissue growth and remodeling[J]. Acta Biomaterialia, 2023, 163: 7-24.
[50] LOH W L. On Latin hypercube sampling[J]. The annals of statistics, 1996, 24(5): 2058-2080.
[51] STEIN M. Large sample properties of simulations using Latin hypercube sampling[J]. Technometrics, 1987, 29(2): 143-151.
[52] NIEDERREITER H. Random number generation and quasi-Monte Carlo methods[M]. Society for Industrial and Applied Mathematics, 1992.
[53] Monte Carlo and quasi-Monte Carlo methods 1996: proceedings of a conference at the University of Salzburg, Austria, July 9-12, 1996[M]. Springer Science & Business Media, 1998.
[54] GILES M B. Multilevel monte carlo methods[J]. Acta numerica, 2015, 24: 259-328.
[55] NG L W T, WILLCOX K E. Multifidelity approaches for optimization under uncertainty[J]. International Journal for numerical methods in Engineering, 2014, 100(10): 746-772.
[56] GERACI G, ELDRED M S, Iaccarino G. A multifidelity multilevel Monte Carlo method for uncertainty propagation in aerospace applications[C]//19th AIAA non- deterministic approaches conference. 2017: 1951.
[57] GERSTNER T, GRIEBEL M. Dimension–adaptive tensor–product quadrature[J]. Computing, 2003, 71: 65-87.
[58] MA X, ZABARAS N. An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations[J]. Journal of Computational Physics, 2009, 228(8): 3084-3113.
[59] AGARWAL N, ALURU N R. A domain adaptive stochastic collocation approach for analysis of MEMS under uncertainties[J]. Journal of Computational Physics, 2009, 228(20): 7662-7688.
[60] SCHIAVAZZI D E, DOOSTAN A, IACCARINO G. Sparse multiresolution regression for uncertainty propagation[J]. International Journal for Uncertainty Quantification, 2014, 4(4).
[61] WAN X, KARNIADAKIS G E. Multi-element generalized polynomial chaos for arbitrary probability measures[J]. SIAM Journal on Scientific Computing, 2006, 28(3): 901-928.
[62] WAN X, KARNIADAKIS G E. An adaptive multi-element generalized polynomial chaos method for stochastic differential equations[J]. Journal of Computational Physics, 2005, 209(2): 617-642.
[63] SEO J, SCHIAVAZZI D E, KAHN A M, et al. The effects of clinically‐derived parametric data uncertainty in patient‐specific coronary simulations with deformable walls[J]. International journal for numerical methods in biomedical engineering, 2020, 36(8): e3351.
[64] BERTAGLIA G, CALEFFI V, PARESCHI L, et al. Uncertainty quantification of viscoelastic parameters in arterial hemodynamics with the a-FSI blood flow model[J]. Journal of Computational Physics, 2021, 430: 110102.
[65] YAN J, CAI S, CAI X, et al. Uncertainty quantification of microcirculatory characteristic parameters for recognition of cardiovascular diseases[J]. Computer Methods and Programs in Biomedicine, 2023, 240: 107674.
[66] BOCCADIFUOCO A, MARIOTTI A, Celi S, et al. Impact of uncertainties in outflow boundary conditions on the predictions of hemodynamic simulations of ascending thoracic aortic aneurysms[J]. Computers & Fluids, 2018, 165: 96-115.
[67] ANTONUCCIO M N, MARIOTTI A, FANNI B M, et al. Effects of uncertainty of outlet boundary conditions in a patient-specific case of aortic coarctation[J]. Annals of biomedical engineering, 2021, 49(12): 3494-3507.
[68] BRÜNING J, HELLMEIER F, YEVTUSHENKO P, et al. Uncertainty quantification for non-invasive assessment of pressure drop across a coarctation of the aorta using CFD[J]. Cardiovascular engineering and technology, 2018, 9: 582-596.
[69] FOSSAN F E, STURDY J, MÜLLER L O, et al. Uncertainty quantification and sensitivity analysis for computational FFR estimation in stable coronary artery disease[J]. Cardiovascular engineering and technology, 2018, 9: 597-622.
[70] KU D N. Blood flow in arteries[J]. Annual review of fluid mechanics, 1997, 29(1): 399-434.
[71] LOTH F, FISCHER P F, BASSIOUNY H S. Blood flow in end-to-side anastomoses[J]. Annu. Rev. Fluid Mech., 2008, 40: 367-393.
[72] SANKARAN S, HUMPHREY J D, MARSDEN A L. An efficient framework for optimization and parameter sensitivity analysis in arterial growth and remodeling computations[J]. Computer methods in applied mechanics and engineering, 2013, 256: 200-210.
[73] TRUESDELL C, NOLL W, TRUESDELL C, et al. The non-linear field theories of mechanics[M]. Springer Berlin Heidelberg, 2004.
[74] RAJAGOPAL K R, TAO L. Mechanics of mixtures[M]. World scientific, 1995.
[75] BAEK S, RAJAGOPAL K R, HUMPHREY J D. A theoretical model of enlarging intracranial fusiform aneurysms[J]. Journal of biomechanical engineering, 2006, 128(1): 142-149.
[76] VALENTIN A, HUMPHREY J D. Evaluation of fundamental hypotheses underlying constrained mixture models of arterial growth and remodelling[J]. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2009, 367(1902): 3585-3606.
[77] VALENTIN A, CARDAMONE L, BAEK S, et al. Complementary vasoactivity and matrix remodelling in arterial adaptations to altered flow and pressure[J]. Journal of The Royal Society Interface, 2009, 6(32): 293-306.
[78] RIZVI M A D, MYERS P R. Nitric oxide modulates basal and endothelin-induced coronary artery vascular smooth muscle cell proliferation and collagen levels[J]. Journal of molecular and cellular cardiology, 1997, 29(7): 1779-1789.
[79] RIZVI M A D, KATWA L, SPADONE D P, et al. The effects of endothelin-1 on collagen type I and type III synthesis in cultured porcine coronary artery vascular smooth muscle cells[J]. Journal of molecular and cellular cardiology, 1996, 28(2): 243-252.
[80] PRICE J M, DAVIS D L, KNAUSS E B. Length-dependent sensitivity in vascular smooth muscle[J]. American Journal of Physiology-Heart and Circulatory Physiology, 1981, 241(4): H557-H563.
[81] RACHEV A, HAYASHI K. Theoretical study of the effects of vascular smooth muscle contraction on strain and stress distributions in arteries[J]. Annals of biomedical engineering, 1999, 27: 459-468.
[82] BAEK S, VALENTÍN A, HUMPHREY J D. Biochemomechanics of cerebral vasospasm and its resolution: II. Constitutive relations and model simulations[J]. Annals of biomedical engineering, 2007, 35: 1498-1509.
[83] UEMATSU M, OHARA Y, NAVAS J P, et al. Regulation of endothelial cell nitric oxide synthase mRNA expression by shear stress[J]. American Journal of Physiology-Cell Physiology, 1995, 269(6): C1371-C1378.
[84] MALEK A, IZUMO S. Physiological fluid shear stress causes downregulation of endothelin-1 mRNA in bovine aortic endothelium[J]. American Journal of Physiology-Cell Physiology, 1992, 263(2): C389-C396.
[85] LANGILLE B L. Arterial remodeling: relation to hemodynamics[J]. Canadian journal of physiology and pharmacology, 1996, 74(7): 834-841.
[86] STENMARK K R, MECHAM R P. Cellular and molecular mechanisms of pulmonary vascular remodeling[J]. Annual review of physiology, 1997, 59(1): 89-144.
[87] RODRIGUEZ E K, HOGER A, MCCULLOCH A D. Stress-dependent finite growth in soft elastic tissues[J]. Journal of biomechanics, 1994, 27(4): 455-467.
[88] BAZILEVS Y, CALO V M, COTTRELL J A, et al. Variational multiscale residual- based turbulence modeling for large eddy simulation of incompressible flows[J]. Computer methods in applied mechanics and engineering, 2007, 197(1-4): 173-201.
[89] FRANCA L P, FREY S L. Stabilized finite element methods: II. The incompressible Navier-Stokes equations[J]. Computer Methods in Applied Mechanics and Engineering, 1992, 99(2-3): 209-233.
[90] FIGUEROA C A. A coupled-momentum method to model blood flow and vessel deformation in human arteries: applications in disease research and simulation-based medical planning[M]. Stanford university, 2006.
[91] BAZILEVS Y, GOHEAN J R, HUGHES T J R, 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[J]. Computer Methods in Applied Mechanics and Engineering, 2009, 198(45-46): 3534-3550.
[92] JANSEN K E, WHITING C H, HULBERT G M. A generalized-α method for integrating the filtered Navier–Stokes equations with a stabilized finite element method[J]. Computer methods in applied mechanics and engineering, 2000, 190(3-4): 305-319.
[93] CHUNG J, HULBERT G M. A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method[J]. 1993.
[94] MOGHADAM M E, VIGNON-CLEMENTEL I E, FIGLIOLA R, et al. A modular numerical method for implicit 0D/3D coupling in cardiovascular finite element simulations[J]. Journal of Computational Physics, 2013, 244: 63-79.
[95] LAN I S, LIU J, YANG W, et al. A reduced unified continuum formulation for vascular fluid–structure interaction[J]. Computer Methods in Applied Mechanics and Engineering, 2022, 394: 114852.
[96] LIU J, YANG W, LAN I S, et al. Fluid-structure interaction modeling of blood flow in the pulmonary arteries using the unified continuum and variational multiscale formulation[J]. Mechanics research communications, 2020, 107: 103556.
[97] UPDEGROVE A, WILSON N M, MERKOW J, et al. SimVascular: an open source pipeline for cardiovascular simulation[J]. Annals of biomedical engineering, 2017, 45: 525-541.
[98] LOÈVE M. Elementary probability theory[M]. Springer New York, 1977.
[99] BABUSKA I, TEMPONE R, ZOURARIS G E. Galerkin finite element approximations of stochastic elliptic partial differential equations[J]. SIAM Journal on Numerical Analysis, 2004, 42(2): 800-825.
[100] WIENER N. The homogeneous chaos[J]. American Journal of Mathematics, 1938, 60(4): 897-936.
[101] XIU D, KARNIADAKIS G E. The Wiener--Askey polynomial chaos for stochastic differential equations[J]. SIAM journal on scientific computing, 2002, 24(2): 619- 644.
[102] 王鹏, 修东滨. 不确定性量化导论[M]. 科学出版社, 2019.
[103] SMOLYAK S A. Quadrature and interpolation formulas for tensor products of certain classes of functions[C]//Doklady Akademii Nauk. Russian Academy of Sciences, 1963, 148(5): 1042-1045.
[104] BUNGARTZ H J, GRIEBEL M. Sparse grids[J]. Acta numerica, 2004, 13: 147-269.
[105] GAO F, GUO Z, SAKAMOTO M, et al. Fluid-structure interaction within a layered aortic arch model[J]. Journal of biological physics, 2006, 32: 435-454.