Computational Mechanics

, Volume 55, Issue 3, pp 479–498 | Cite as

Fluid-structure interaction simulations of cerebral arteries modeled by isotropic and anisotropic constitutive laws

  • Paolo Tricerri
  • Luca Dedè
  • Simone Deparis
  • Alfio Quarteroni
  • Anne M. Robertson
  • Adélia Sequeira
Original Paper


This paper considers numerical simulations of fluid-structure interaction (FSI) problems in hemodynamics for idealized geometries of healthy cerebral arteries modeled by both nonlinear isotropic and anisotropic material constitutive laws. In particular, it focuses on an anisotropic model initially proposed for cerebral arteries to characterize the activation of collagen fibers at finite strains. In the current work, this constitutive model is implemented for the first time in the context of an FSI formulation. In this framework, we investigate the influence of the material model on the numerical results and, in the case of the anisotropic laws, the importance of the collagen fibers on the overall mechanical behavior of the tissue. With this aim, we compare our numerical results by analyzing fluid dynamic indicators, vessel wall displacement, Von Mises stress, and deformations of the collagen fibers. Specifically, for an anisotropic model with collagen fiber recruitment at finite strains, we highlight the progressive activation and deactivation processes of the fibrous component of the tissue throughout the wall thickness during the cardiac cycle. The inclusion of collagen recruitment is found to have a substantial impact on the intramural stress, which will in turn impact the biological response of the intramural cells. Hence, the methodology presented here will be particularly useful for studies of mechanobiological processes in the healthy and diseased vascular wall.


Material constitutive models Cerebral arterial tissue  Hypereslatic isotropic laws Hyperelastic anisotropic laws  Fluid-structure interaction Numerical simulations 



We acknowledge the support of the Swiss National Supercomputing Centre (CSCS) under the project ID s475 for providing the computational resources for the numerical simulations. P. Tricerri acknowledges the financial support of Fundação para Ciênca e a Tecnologia (FCT) of Portugal through the Research Center CEMAT-IST (under the grant SFRH/BD/51069/2010) and of the project EXCL/MAT-NAN/0114/2012. Prof. Quarteroni acknowledges the MATHCARD and the HP2C projects; Dr. Deparis thanks the HP2C project.


  1. 1.
    Augsburger L (2008) Fluid Mechanics of Cerebral Aneurysms and Effects of Intracranial Stents on Cerebral Aneurysmal Flow. Ph.D. thesis, École Polytechinque Fédérale de Lausanne, LausanneGoogle Scholar
  2. 2.
    Baek H, Jayaraman M, Richardson P, Karniadakis G (2010) Flow instability and wall shear stress variation in intracranial aneurysms. J R Soc Interface 7:967–988CrossRefGoogle Scholar
  3. 3.
    Balzani D (2006) Polyconvex Anisotropic Energies and Modeling of Damage Applied to Arterial Walls. Ph.D. thesis, University of Duisburg-Essen, EssenGoogle Scholar
  4. 4.
    Balzani D, Brinkhues S, Holzapfel G (2012) Constitutive framework for the modeling of damage in collagenous soft tissues with application to arterial walls. Comput Methods Appl Mech Eng 213–216:139–151CrossRefMathSciNetGoogle Scholar
  5. 5.
    Balzani D, Neff P, Schröder J, Holzapfel G (2006) A polyconvex framewok for soft biological tissues. Adjustment to experimental data. Int J Solids Struct 43:6052–6070CrossRefzbMATHGoogle Scholar
  6. 6.
    Balzani D, Schmidt T, Schriefl T, Holzapfel G (2013) Constitutive modeling of damage mechanisms in arterial walls and related experimental studies. In: XLI APM proceedings (advanced problems in mechanics), St. Petersburg, pp 17–25Google Scholar
  7. 7.
    Bazilevs Y, Hsu M, Zhang Y, Wang W, Kvamsdal T, Hentschel S, Isaksen J (2010) Computational vascular fluid-structure interaction: methodology and application to cerebral aneurysms. Biomech Model Mechanobiol 9:481–498CrossRefGoogle Scholar
  8. 8.
    Bazilevs Y, Hsu M, Zhang Y, Wang W, Liang X, Kvamsdal T, Brekken R, Isaksen J (2010) A fully-coupled fluid-structure interaction simulation of cerebral aneurysms. Comput Mech 46:3–16CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Bhole A, Flynn BP, Liles M, Saeidi N, Dimarzio CA, Ruberti JW (2009) Mechanical strain enhances survivability of collagen micronetworks in the presence of collagenase: implications for load-bearing matrix growth and stability. Philoso Trans Ser A Math Phys Eng Sci 367:3339–3362CrossRefGoogle Scholar
  10. 10.
    Brinkhues S, Klawonn A, Rheinbach O, Schröder J (2013) Augmented Lagrange methods for quasi-incompressible material–applications to soft biological tissue. Int J Numer Methods Biomed Eng 29:332–350CrossRefMathSciNetGoogle Scholar
  11. 11.
    Burton A (1954) Relation of structure to function of the tissue of the wall of blood vessels. Physiol Rev 34:619–642Google Scholar
  12. 12.
    Calvo B, Pẽna E, Martinez M, Doblaré M (2007) An uncoupled directional damage model for fibred biological soft tissues. Formulation and computational aspects. Int J Numer Methods Eng 69:2036–2057CrossRefzbMATHGoogle Scholar
  13. 13.
    Carew T, Vaishnav R, Patel D (1968) Compressibility of the arterial wall. Circ Res 23:61–68CrossRefGoogle Scholar
  14. 14.
    Castro M, Putman C, Cebral J (2006) Computational fluid dynamics modeling of intracranial aneurysms: effects of parent artery segmentation on intra-aneurysmal hemodynamics. Am J Neuroradiol 27:1703–1709Google Scholar
  15. 15.
    Cebral J, Castro M, Appanaboyina S, Putman C, Millan D, Frangi A (2005) Efficient pipeline for image-based patient-specific analysis of cerebral aneurysm hemodynamics: techniques and sensitivity. IEEE Trans Med Imaging 24:457–467CrossRefGoogle Scholar
  16. 16.
    Cebral J, Mut F, Weir F, Putman C (2011) Association of hemodynamics characteristics and cerebral aneurysms rupture. Am J Neuroradiol 32:264–270CrossRefGoogle Scholar
  17. 17.
    Chandra S, Raut S, Jana A, Biederman R, Doyle M, Muluk S, Finol E (2013) Fluid-structure interaction modeling of abdominal aortic aneurysms: the impact of patient-specific inflow conditions and fluid/solid coupling. J Biomech Eng 135:810011–8100114CrossRefGoogle Scholar
  18. 18.
    Chen H, Zhu L, Hou Y, Liu Y, Kassab G (2010) Fluid-structure interaction (FSI) modeling in the cardiovascular system. In: Guccione J, Kassab G, Ratcliffe M (eds) Computational cardiovascular mechanics, modeling and applications in heart failure. Springer, New York, pp 141–157CrossRefGoogle Scholar
  19. 19.
    Chen J, Wang S, Ding G, Yang X, Li H (2009) The effect of aneurismal-wall mechanical properties on patient-specific hemodynamic simulations: two clinical reports. Acta Mech Sin 25:677–688CrossRefzbMATHGoogle Scholar
  20. 20.
    Crosetto P (2011) Fluid-Structure Interaction Problems in Hemodynamics: Parallel Solvers, Preconditioners, and Applications. Ph.D. thesis, École Polytechinque Fédérale de Lausanne, LausanneGoogle Scholar
  21. 21.
    Crosetto P, Deparis S, Fourestey G, Quarteroni A (2011) Parallel algorithms for fluid-structure interaction problems in haemodynamics. SIAM J Sci Comput 33:1598–1622CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Crosetto P, Reymond P, Deparis S, Kontaxakis D, Stergiopulos N, Quarteroni A (2011) Fluid structure interaction simulations of physiological blood flow in the aorta. Comput Fluids 43:46–57CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Delfino A, Stergiopulos N, Moore J, Meister J (1997) Residual strain effects on the stress field in a thick wall finite element model of the human carotid bifurcation. J Biomech 30:777–786CrossRefGoogle Scholar
  24. 24.
    Devault K, Gremaud P, Novak V, Olufsen M, Vernières G, Zhao P (2008) Blood flow in the Circle of Willis: modeling and calibration. Multiscale Model Simulat 7:888–909CrossRefzbMATHGoogle Scholar
  25. 25.
    Draney M, Herfkens R, Hughes T, Pelc N, Wedding K, Zarins C, Taylor C (2002) Quantification of vessel wall cyclic strain using cine phase contrast magnetic resonance imaging. Ann Biomed Eng 30:1033–1045CrossRefGoogle Scholar
  26. 26.
    Ferguson G (1972) Direct measurements of mean and pulsatile blood pressure at operation in human intracranial saccular aneurysms. J Neurosurg 36:560–563CrossRefGoogle Scholar
  27. 27.
    Fernández M, Moubachir M (2005) A Newton method using exact jacobians for solving fluid-structure coupling. Comput Struct 83:127–142CrossRefGoogle Scholar
  28. 28.
    Flory P (1961) Thermodynamical relations for high elastic materials. Transa Farad Soc 57:829–838CrossRefMathSciNetGoogle Scholar
  29. 29.
    Ford M, Nikolov N, Milner J, Lownie S, Demont E, Kalata W, Loth F, Holdsworth D, Steinman D (2008) PIV-measured versus CFD-predicted flow dynamics in anatomically realistic cerebral aneurysm models. J Biomech Eng 130:1–15CrossRefGoogle Scholar
  30. 30.
    Formaggia L, Quarteroni A, Veneziani AE (2009) Cardiovascular mathematics, modeling and simulation of the circulatory system. MS & A, Springer-Verlag, BerlinzbMATHGoogle Scholar
  31. 31.
    Gambaruto A, João A (2012) Flow structures in cerebral aneurysms. Comput Fluids 65:56–65CrossRefMathSciNetGoogle Scholar
  32. 32.
    Gasser T, Ogden R, Holzapfel G (2006) Hyperelastic modelling of arterial layers with distributed collagen fiber orientations. J R Soc Interface 3:15–35CrossRefGoogle Scholar
  33. 33.
    Gasser T, Schulze-Bauer C, Holzapfel G (2002) A three-dimensional finite element model for arterial clamping. J Biomech Eng 124:355–363CrossRefGoogle Scholar
  34. 34.
    Giller C, Bowman G, Dyer H, Mootz L, Krippner W (1993) Cerebral arterial diameters during changes in blood pressure and carbon dioxide during craniotomy. Neurosurgery 32:737–741CrossRefGoogle Scholar
  35. 35.
    Gould RA, Chin K, Santisakultarm TP, Dropkin A, Richards JM, Schaffer CB, Butcher JT (2012) Cyclic strain anisotropy regulates valvular interstitial cell phenotype and tissue remodeling in three-dimensional culture. Acta Biomater 8:1710–1719CrossRefGoogle Scholar
  36. 36.
    Gupta V, Grande-Allen KJ (2006) Effects of static and cyclic loading in regulating extracellular matrix synthesis by cardiovascular cells. Cardiovasc Res 72:375–383CrossRefGoogle Scholar
  37. 37.
    Hartmann S, Neff P (2003) Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for nearly-incompressibility. Int J Solids Struct 40:2767–2791CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    den Heijer T, Skoog I, Oudkerk M, de Leeuw FE, de Groot J, Hofman A, Bretler M (2003) Association between blood pressure levels over time and brain atrophy in the elderly. Neurobiol Aging 24:307–313CrossRefGoogle Scholar
  39. 39.
    Hill M, Duan X, Gibson G, Watkins S, Robertson A (2012) A theoretical and non-destructive experimental approach for direct inclusion of measured collagen orientation and recruitment into mechanical models of the arterial wall. J Biomech 45:762–771Google Scholar
  40. 40.
    Hoi Y, Woodward S, Kim M, Taulbee D, Meng H (2006) Validation of CFD simulations of cerebral aneurysms with implication of geometric variations. J Biomech Eng 128:844–851CrossRefGoogle Scholar
  41. 41.
    Holzapfel G (2000) Nonlinear solid mechanics: a continuum approach for engineering. Wiley, LondonGoogle Scholar
  42. 42.
    Holzapfel G, Gasser T, Ogden R (2000) A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elast 61:1–48CrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    Holzapfel G, Odgen R (2009) Constitutive modelling of passive myocardium: a structurally based framework for material characterization. Philos Trans R Soc A 367:3445–3475CrossRefzbMATHGoogle Scholar
  44. 44.
    Holzapfel G, Ogden R (2010) Constitutive modeling of arteries. Proc R Soc Lond A 466:1551–1597CrossRefzbMATHMathSciNetGoogle Scholar
  45. 45.
    Humphrey J (2002) Cardiovascular solid mechanics. Cells, tissues, and organs. Springer-Verlag, New YorkGoogle Scholar
  46. 46.
    Isaksen J, Bazilevs Y, Kvamsdal T, Zhang Y, Kaspersen J, Waterloo K, Romner B, Ingebrigtsen T (2008) Determination of wall tension in cerebral artery aneurysms by numerical simulation. Stroke 39:3172–3178CrossRefGoogle Scholar
  47. 47.
    Janela J, Moura A, Sequeira A (2010) Absorbing boundary conditions for a 3D non-Newtonian fluidstructure interaction model for blood flow in arteries. Int J Eng Sci 48:1332–1349CrossRefzbMATHMathSciNetGoogle Scholar
  48. 48.
    Jeong W, Rhee K (2012) Hemodynamics of cerebral aneurysms: computational analyses of aneurysm progress and treatment. Comput Math Methods Med 2012:1–11CrossRefGoogle Scholar
  49. 49.
    Kim C, Kiris C, Kwak D, David T (2006) Numerical simulation of local blood flow in the carotid and cerebral arteries under altered gravity. J Biomech Eng 128:194–202CrossRefGoogle Scholar
  50. 50.
    Li D, Robertson A (2009) A structural multi-mechanism constitutive equation for cerebral arterial tissue. Int J Solids Struct 46:2920–2928CrossRefzbMATHGoogle Scholar
  51. 51.
    Li D, Robertson A (2009) A structural multi-mechanism damage model for cerebral arterial tissue. J Biomech Eng 131:101013–101018CrossRefGoogle Scholar
  52. 52.
    Li D, Robertson A, Lin G, Lovell M (2012) Finite element modeling of cerebral angioplasty using a structural multi-mechanism anisotropic damage model. Int J Numer Methods Eng 92:457–474CrossRefMathSciNetGoogle Scholar
  53. 53.
    Lou J, Lee D, Morsi H, Mawad M (2008) Wall shear stress on ruptured and unruptured intracranial aneurysms at the internal carotid artery. Am J Neuroradiol 29:1761–1767CrossRefGoogle Scholar
  54. 54.
    Malossi A (2012) Partitioned Solution of Geometrical Multiscale Problems for the Cardiovascular System: Models, Algorithms, and Applications. Ph.D. thesis, École Polytechinque Fédérale de Lausanne, LausanneGoogle Scholar
  55. 55.
    Malossi A, Bonnemain J (2013) Numerical comparison and calibration of geometrical multiscale models for the simulation of arterial flows. Cardiovasc Eng Tecnol 4:440–463CrossRefGoogle Scholar
  56. 56.
    Mantha A, Karmonik C, Bendorf G, Strother C, Metcalfe R (2006) Hemodynamics in a cerebral artery before and after the formation of an aneurysm. Am J Neuroradiol 27:1113–1118Google Scholar
  57. 57.
    Marks M, Pelc N, Ross M, Enzmann D (1992) Determination of cerebral blood flow with a phase contrast cine MR imaging technique: evaluation of normal subjects and patients with an arteriovenous malformations. Radiology 182:467–476CrossRefGoogle Scholar
  58. 58.
    Marzo A, Singh P, Reymond P, Stergiopulos N, Patel U, Hose R (2009) Influence of inlet boundary conditions on the local haemodynamics of intracranial aneurysms. Comput Methods Biomech Biomed Eng 12:431–444CrossRefGoogle Scholar
  59. 59.
    Moireau P, Xiao N, Astorino M, Figueroa C, Chapelle D, Taylor C, Gerbeau J (2012) External tissue support and fluid-structure simulation in blood flows. Biomech Model Mechanobiol 11:1–18CrossRefGoogle Scholar
  60. 60.
    Nichols W, O’Rourke M (1998) McDonald’s blood flow in arteries: theoretical, experimental, and clinical principles. Arnold, LondonGoogle Scholar
  61. 61.
    Nobile F (2001) Numerical Approximation of Fluid-Structure Interaction Problems with Application to Haemodynamics. Ph.D. thesis, École Polytechinque Fédérale de Lausanne, LausanneGoogle Scholar
  62. 62.
    Oshima M, Sakai H, Torii R (2005) Modelling of inflow boundary conditions for image-based simulation of cerebrovascular flow. Int J Numer Methods Fluids 47:603–617CrossRefzbMATHMathSciNetGoogle Scholar
  63. 63.
    Quarteroni A, Sacco R, Saleri F (2007) Numerical mathematics. Springer, BerlinzbMATHGoogle Scholar
  64. 64.
    Quarteroni A, Valli A (1999) Domain decomposition methods for partial differential equations. Oxford University Press, OxfordzbMATHGoogle Scholar
  65. 65.
    Quarteroni A, Valli A (1999) Numerical approximation of partial differential equations. Springer-Verlag, BerlinGoogle Scholar
  66. 66.
    Reymond P, Bohraus Y, Perren F, Lazeyras F, Stergiopulos N (2010) Validation of a patient-specific one-dimensional model of the systemic arterial tree. Am J Physiol 301:1173–1182Google Scholar
  67. 67.
    Roach M, Burton A (1957) The reason for the shape of the distensibility curves of arteries. Can J Biochem 35:681–690CrossRefGoogle Scholar
  68. 68.
    Robertson A, Sequeira A, Kameneva M (2008) Hemorheology. In: Galdi G, Rannacher R, Robertson A, Turek S (eds) Hemodynamical flows, oberwolfach seminars, vol 37. Springer-Verlag, Basel, pp 63–120CrossRefGoogle Scholar
  69. 69.
    Robertson A, Watton P (2013) Mechanobiology of the arterial wall. In: Becker S, Kuznetsov A (eds) Modeling of transport in biological media. Elsevier, New York, pp 275–347CrossRefGoogle Scholar
  70. 70.
    Roy S, Boss C, Rezakhaniha R, Stergiopulos N (2010) Experimental characterization of the distribution of collagen fiber recruitment. J Biorheol 24:84–93CrossRefGoogle Scholar
  71. 71.
    Saad Y (2003) Iterative methods for sparse linear systems. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  72. 72.
    Schröder J, Neff P (2003) Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions. Int J Solids Struct 40:401–445CrossRefzbMATHGoogle Scholar
  73. 73.
    Scott S, Ferguson G, Roach M (1972) Comparison of the elastic properties of human intracranial arteries and aneurysms. Can J Physiol Pharmacol 50:328–332CrossRefGoogle Scholar
  74. 74.
    Sforza D, Putman C, Cebral J (2009) Hemodynamics of cerebral aneurysms. Annu Rev Fluid Mech 41:91–107CrossRefGoogle Scholar
  75. 75.
    Spencer A (1984) Constitutive theory of strongly anisotropic solids. In: Spencer A (ed) Continuum theory of the mechanics of fibre-reinforced composites. Springer-Verlag, Wien, pp 1–32Google Scholar
  76. 76.
    Takizawa K, Bazilevs Y, Tezduyar T, Long C, Marsden A, Schjodt K (2014) Patient-specific cardiovascular fluid mechanics analysis with the ST and ALE-VMS methods. In: Idelsohn S (ed) Numer Simulat Coupled Probl Eng, vol 33. Springer, Heidelberg, pp 71–102CrossRefGoogle Scholar
  77. 77.
    Takizawa K, Bazilevs Y, Tezduyar T (2012) Space-time and ALE-VMS techniques for patient-specific cardiovascular fluid-structure interaction modeling. Arch Comput Methods Eng 19:171–225CrossRefMathSciNetGoogle Scholar
  78. 78.
    Takizawa K, Takagi H, Tezduyar T, Torii R (2014) Estimation of element-based zero-stress state for arterial FSI computations. Comput Mech 54:895–910CrossRefzbMATHMathSciNetGoogle Scholar
  79. 79.
    Tezduyar T, Takizawa K, Brummer T, Chen P (2013) Spacetime fluidstructure interaction modeling of patient-specific cerebral aneurysms. Numer Methods Biomed Eng 27:1665–1710CrossRefMathSciNetGoogle Scholar
  80. 80.
    Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar T (2006) Fluid-structure interaction modeling of anuerysmal conditions with high and normal blood pressures. Comput Mech 38:482–490CrossRefzbMATHGoogle Scholar
  81. 81.
    Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar T (2007) Numerical investigation of the effect of hypertensive blood pressure on cerebral aneurysm–dependence of the effect on the aneurysm shape. Int J Numer Methods Fluids 54:995–1009CrossRefzbMATHMathSciNetGoogle Scholar
  82. 82.
    Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar T (2008) Fluidstructure interaction modeling of a patient-specific cerebral aneurysm: influence of structural modeling. Comput Mech 43:151–159CrossRefzbMATHGoogle Scholar
  83. 83.
    Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar T (2009) Fluid-structure interaction modeling of blood flow and cerebral aneurysm: significance of artery and aneurysm shapes. Comput Method Appl Mech Eng 198:3613–3621CrossRefzbMATHMathSciNetGoogle Scholar
  84. 84.
    Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar T (2010) Influence of wall thickness on fluid-structure interaction computations of cerebral aneurysms. Int J Numer Methods Biomed Eng 26:336–347CrossRefzbMATHMathSciNetGoogle Scholar
  85. 85.
    Tricerri P, Dedè L, Quarteroni A, Sequeira A (2013) Numerical validation of isotropic and transversely isotropic constitutive models for healthy and unhealthy cerebral arterial tissue. Technical Report 39.2013, MATHICSE Report EPFLGoogle Scholar
  86. 86.
    Valencia A, Burdiles P, Ignat M, Mura J, Bravo E, Rivera R, Sordo J (2013) Fluid structural analysis of human cerebral aneurysm using their own wall mechanical properties. Comput Math Methods Med
  87. 87.
    Valencia A, Solis F (2006) Blood flow dynamics and arterial wall interaction in a saccular aneurysm model of the basilar artery. Comput Struct 84:1326–1337CrossRefGoogle Scholar
  88. 88.
    Weisbecker H, Pierce D, Holzapfel G (2011) Modeling of damage-induced softening for arterial tissue. In: Proceedings of the 2011 SCATh joint workshop on new tecnologies for computer/robot assisted surgery, Graz, pp 1–4Google Scholar
  89. 89.
    Wulandana R, Robertson A (2005) An inelastic multi-mechanism constitutive equation for cerebral arterial tissue. Biomech Model Mechanobiol 4:235–248CrossRefGoogle Scholar
  90. 90.
    Zakaria H, Robertson A, Kerber C (2008) A parametric model for studies of flow in arterial bifurcations. Ann Biomed Eng 36:1515–1530CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Paolo Tricerri
    • 1
    • 2
  • Luca Dedè
    • 1
  • Simone Deparis
    • 1
  • Alfio Quarteroni
    • 1
    • 3
  • Anne M. Robertson
    • 4
  • Adélia Sequeira
    • 2
  1. 1.École Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.Department of Mathematics and CEMAT, Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal
  3. 3.MOX - Modeling and Scientific Computing, Dipartimento di Matematica “F. Brioschi”Politecnico di MilanoMilanoItaly
  4. 4.Department of Mechanical Engineering and Materials Science, Department of BioengineeringUniversity of PittsburghPittsburghUSA

Personalised recommendations