Impact of transmural heterogeneities on arterial adaptation

Application to aneurysm formation
  • H. SchmidEmail author
  • P. N. Watton
  • M. M. Maurer
  • J. Wimmer
  • P. Winkler
  • Y. K. Wang
  • O. Röhrle
  • M. Itskov
Original Paper


Recent experimental and computational studies have shown that transmurally heterogeneous material properties through the arterial wall are critical to understanding the heterogeneous expressions of constituent degrading molecules. Given that expression of such molecules is thought to be intimately linked to local magnitudes of stress, modelling the transmural stress distribution is critical to understanding arterial adaption during disease. The aim of this study was to develop an arterial growth and remodelling framework that can incorporate both transmurally heterogeneous constituent distributions and residual stresses, into a 3-D finite element model. As an illustrative example, we model the development of a fusiform aneurysm and investigate the effects of elastinous and collagenous heterogeneities on the stress distribution during evolution. It is observed that the adaptive processes of growth and remodelling exhibit transmural variations. For physiological heterogeneous constituent distributions, a stress peak appears in the media towards the intima, and a stress plateau occurs towards the adventitia. These features can be primarily attributed to the underlying heterogeneity of elastinous constituents. During arterial adaption, the collagen strain is regulated to remain in its homoeostatic level; consequently, the partial stress of collagen has less influence on the total stress than the elastin. However, following significant elastin degradation, collagen plays the dominant role for the transmural stress profile and a marked stress peak occurs towards the adventitia. We conclude that to improve our understanding of the arterial adaption and the aetiology of arterial disease, there is a need to: quantify transmural constituent distributions during histopathological examinations, understand and model the role of the evolving transmural stress distribution.


Transmural Heterogeneity Remodelling Growth Artery Aneurysm 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Alberts B, Lewis AJJ, Raff M, Roberts K, Walter P (2008) Molecular biology of the cell. Garland Science, New YorkGoogle Scholar
  2. Alford PW, Humphrey JD, Taber LA (2008) Growth and remodelling in a thick-walled artery model: effects of spatial variations in wall constituents. Biomech Model Mechanobiol 7: 245–262CrossRefGoogle Scholar
  3. Arajou RP, McElwain DLS (2005) A mixture theory for the genesis of residual stresses in growing tissues i: a general formulation. SIAM J Appl Math 65(4): 1261–1284CrossRefMathSciNetGoogle Scholar
  4. Azeloglu EU, Albro MB, Thimmappa VA, Ateshian GA, Costa KD (2008) Heterogeneous transmural proteoglycan distribution provides a mechanism for regulating residual stresses in the aorta. Am J Physiol 294: H1197–H1205Google Scholar
  5. Baek S, Rajagopal KR, Humphrey JD (2006) A theoretical model for enlarging fusiform aneurysms. J Biomech Eng 142: 142–149CrossRefGoogle Scholar
  6. Barocas VH, Tranquillo RT (1997) An anisotropic biphasic theory of tissue-equivalent mechanics: the interplay among cell traction, fibril alignment, and cell contact guidance. J Biomech Eng 119: 541–553Google Scholar
  7. Bellousov LV (1998) The dynamic architecture of a developing organism: an interdisciplinary approach to the development of organisms. Kluwer, DordrechtGoogle Scholar
  8. Cacho F, Elbischger PJ, Rodriguez JF, Doblare M, Holzapfel GA (2007) A constitutive model for fibrous tissues considering collagen fiber crimp. Non Linear Mech 42: 391–402CrossRefGoogle Scholar
  9. Cardamone L, Valentn A, Eberth JF, Humphrey JD (2009) Origin of axial prestretch and residual stress in arteries. Biomech Model Mechanobiol. doi: 10.1007/s10237-008-0146-x
  10. Chien S (2008) Effects of disturbed flow on endothelial cells. Ann Biomed Eng 36(4): 554–562CrossRefGoogle Scholar
  11. Chiquet M, Renedo AS, Huber F, Flück M (2003) How do fibroblasts translate mechanical signals into changes in extracellular matrix production? Matrix Biol 22: 73–80CrossRefGoogle Scholar
  12. Connolly ES, Fiore AJ, Winfree CJ, Prestigiacomo CJ, Goldman JE, James E, Solomon RA (1997) Elastin degradation in the superficial temporal arteries of patients with intracranial aneurysms reflects changes in plasma elastase. Neurosurg Online 40: 903–909Google Scholar
  13. Davies PF (1995) Flow-mediated endothelial mechanotransduction. Physiol Rev 75: 519–560Google Scholar
  14. Delfino A, Stergiopolous NJEM Jr, Meister J-J (1997) Residual strain effects on the stress field in a thick–walled finite element model of the human carotid artery. J Biomech 30: 777–786CrossRefGoogle Scholar
  15. Driessen NJB, Bouten CVC, Baaijens FPT (2005) A structural constitutive model for collagenous cardiovascular tissues incorporating the angular fiber distribution. J Biomech Eng 127: 494– 504CrossRefGoogle Scholar
  16. Driessen NJB, Wilson W, Bouten CVC, Baaijens FPT (2004) A computational model for collagen fibre remodelling in the arterial wall. J Theor Biol 226: 53–64CrossRefGoogle Scholar
  17. Ehret AE, Itskov M (2008) Modeling of anisotropic softening phenomena: application to soft biological tissues. Int J Plast 25: 901–919CrossRefGoogle Scholar
  18. Ehret AE, Itskov M, Schmid H (2009) Numerical integration on the sphere and its effect on the material symmetry of constitutive equations—a comparative study. Int J Numer Methods. doi: 10.1002/nme.2688
  19. Fisher GM, Llaurado JG (1966) Collagen and elastin content in canine arteries selected from functionally different vascular beds. Circ Res 19: 394–399Google Scholar
  20. Fonck E, Prodhom G, Roy S, Augsburger L, Rüfenacht DA, Stergiopulos N (2007) Effect of elastin degradation on carotid wall mechanics as assessed by a constituent-based biomechanical model. Am J Physiol 292: H2754–H2763Google Scholar
  21. Frösen J, Piippo A, Paetau A, Kangasniemi M, Niemelä M, Hernesniemi J, Jääskeläinen J (2004) Remodeling of saccular cerebral artery aneurysm wall is associated with rupture—histological analysis of 24 unruptured and 42 ruptured cases. Strike 35: 2287–2293CrossRefGoogle Scholar
  22. Fung YC (1991) What are residual stresses doing in our blood vessels? Ann Biomed Eng 19: 237–249CrossRefMathSciNetGoogle Scholar
  23. Gasser TC, Ogden RW, Holzapfel GA (2006) Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. J R Soc Interface 3: 15–35CrossRefGoogle Scholar
  24. Gleason RL, Humphrey JD (2004) A 2-d model of flow-induced alterations in the geometry, structure, and properties of carotid arteries. J Biomech Eng 126: 371–381CrossRefGoogle Scholar
  25. Gleason RL, Humphrey JD (2005) A 2d constrained mixture model for arterial adaptations to large changes in flow, pressure and axial stretch. Math Med Biol 22: 347–369CrossRefzbMATHGoogle Scholar
  26. Gundiah N, Ratcliffe MB, Pruitt LA (2009) The biomechanics of arterial elastin. J Mech Behav Biomed Mater 2: 288–296CrossRefGoogle Scholar
  27. Guo X, Lanir Y, Kassab GS (2007) Effect of osmolarity on the zero–stress state and mechanical properties of. Am J Physiol 293: H2328–H2334CrossRefGoogle Scholar
  28. Guo X, Lu X, Kassab GS (2005) Transmural strain distribution in the blood vessel wall. Am J Physiol 288: H881–H886CrossRefGoogle Scholar
  29. 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
  30. Hansen F, Mangell P, Sonesson B, Lanne T (1995) Diameter and compliance in the human common carotid artery—variations with age and sex. Ultrasound Med Biol 21: 1–99CrossRefGoogle Scholar
  31. Hariton I, deBotton G, Gasser TC, Holzapfel GA (2007) Stress-driven collagen fiber remodeling in arterial walls. Biomech Model Mechanobiol 6: 163–175CrossRefGoogle Scholar
  32. He CM, Roach M (1993) The composition and mechanical properties of abdominal aortic aneurysms. J vasc surg 20(1): 6–13Google Scholar
  33. Holtz J, Forstermann U, Pohl U, Giesler M, Bassenge E (1984) Flow–dependent, endothelium-mediated dilation of epicardial coronary arteries in conscious dogs: effects of cyclooxygenase inhibition. J Cardiovasc Pharmacol 6: 1161–1169Google Scholar
  34. Holzapfel GA (2006) Determination of material models for arterial walls from uniaxial extension tests and histological structure. J Elast 238: 290–302MathSciNetGoogle Scholar
  35. Holzapfel GA, Gasser TC, Ogden RW (2000) A new constitutive framework for arterial wall mechanics and a comparative study for material models. J Theor Biol 61: 1–48MathSciNetzbMATHGoogle Scholar
  36. Horgan CO, Saccomandi G (2003) A description of arterial wall mechanics using limiting chain extensibility constitutive models. Biomech Model Mechanobiol 1: 251–266CrossRefGoogle Scholar
  37. Humphrey JD (1999) Remodeling of a collageneous tissue at fixed length. J Biomech Eng 121: 591–597CrossRefGoogle Scholar
  38. Humphrey JD, Na S (2002) Elastodynamics and arterial wall stress. Ann Biomed Eng 30: 509–523CrossRefGoogle Scholar
  39. Humphrey JD, Rajagopal KR (2002) A constrained mixture model for growth and remodeling of soft tissues. Math Models Methods Appl Sci 12(3): 407–430CrossRefMathSciNetzbMATHGoogle Scholar
  40. Humphrey JD, Rajagopal KR (2003) A constrained mixture model for arterial adaptations to a sustained step change in flow. Biomech Model Mechanobiol 2: 109–126CrossRefGoogle Scholar
  41. Johnson P (1981) The myogenic response. In: Bohr DF, Somlyo AP, Sparks HV Jr (eds) Handbook of physiology. The cardiovascular system: vascular smooth muscle Sec. II. Am Physiol Soc, 409–442Google Scholar
  42. Kim YS, Galis ZS, Rachev A, Han H-C, Vito RP (2009) Matrix metalloproteinase-2 and −9 are associated with high stresses predicted using a nonlinear heterogeneous model of arteries. J Biomech Eng 131: 1–10Google Scholar
  43. Korshunov VA, Berk BC (2008) Smooth muscle apoptosis and vascular remodelling. Curr Opin Hematol 143: 250–254CrossRefGoogle Scholar
  44. Kroon M, Holzapfel GA (2007) A model for saccular cerebral aneurysm growth by collagen fibre remodelling. J Theor Biol 247: 775–787CrossRefMathSciNetGoogle Scholar
  45. Kroon M, Holzapfel GA(2008) Modelling of saccular aneurysm growth in a human middle cerebral artery. J Biomech Eng 130(5): 10CrossRefGoogle Scholar
  46. Kuhl E, Holzapfel GA (2007) A contiunuum model for remodeling in living structures. J Mater Sci 42: 8811–8823CrossRefGoogle Scholar
  47. Lanir Y (2009) Mechanisms of residual stress in soft tissues. J Biomech Eng 131: 1–5CrossRefGoogle Scholar
  48. Lillie MA, Gosline JM (2007) Limits to the durability of arterial elastic tissue. Biomaterials 28: 2021–2031CrossRefGoogle Scholar
  49. MacSweeney S, Young G, Greenhalgh R, Powell J (1992) Mechanical properties of the aneurysmal aorta. Br J Surg 79: 1281–1284CrossRefGoogle Scholar
  50. McAnulty RJ (2007) Fibroblasts and myofibroblasts: their source, function and role in disease. Int J Biochem Cell Biol 39: 666–671CrossRefGoogle Scholar
  51. Meng H, Wang Z, Hoi Y, Gao L, Metaxa E, Swart DD, Kolega J (2007) Complex hemodynamics at the apex of an arterial bifurcation induces vascular remodelling resembling cerebral aneurysm initiation. Stroke 38: 1924–1931CrossRefGoogle Scholar
  52. Menzel A (2007) A fibre reorientation model for orthotropic multiplicative growth. configurational driving stresses, kinematics-based reorientation, and algorithmic aspects. Biomech Model Mechanobiol 6: 303–320CrossRefGoogle Scholar
  53. Monson KL, Goldsmith W, Barbaro NM, Manley GT (2003) Axial mechanical properties of fresh human cerebral blood vessels. J Biomech Eng 125: 288–294CrossRefGoogle Scholar
  54. Mulvany J (1999) Vascular remodelling of resistance vessels: can we define this? Cardiovasc Conundra Ser 41: 9–13Google Scholar
  55. Murray CJL, Lopez AD (1997) Amortality by cause for eight regions of the world: global burden of disease study. Lancet 349(9061): 1269–1276CrossRefGoogle Scholar
  56. Nash MP, Hunter PJ (2000) Computational mechanics of the heart. J Elast 61: 113–141CrossRefMathSciNetzbMATHGoogle Scholar
  57. Nissen R, Cardinale GJ, Udenfriend S (1978) Increased turnover of arterial collagen in hypertensive rats. Proc Natl Acad Sci USA 75: 451–453CrossRefGoogle Scholar
  58. O’Connell MK, Murthy S, Phan S, Xu C, Buchanan J, Spilker R, Dalman R, Zarins CK, Denk W, Taylor CA (2008) The three-dimensional micro- and nanostructure of the aortic medial lamellar unit measured using 3d confocal and electron microscopy imaging. Matrix Biology (in press)Google Scholar
  59. Oden JT (1972) Finite elements of nonlinear continua—an introduction, vol 1. McGraw–Hill, New YorkGoogle Scholar
  60. Papadaki M, Eskin S (1997) Effects of fluid shear stress on gene regulation of vascular cells. Biotechnol Prog 13: 209–221CrossRefGoogle Scholar
  61. Rachev A (2000) A model of arterial adaptation to alterations in blood flow. J Elast 61: 83–111CrossRefMathSciNetzbMATHGoogle Scholar
  62. Rachev A, Hayashi K (1999) Theoretical study of the effects of vascular smooth muscle contraction on strain and stress distributions in arteries. Ann Biomed Eng 27: 459–468CrossRefGoogle Scholar
  63. Raghavan ML, Trivedi S, Nagaraj A, McPherson DD, Chandran K (2004) Three-dimensional finite element analysis of residual stress in arteries. Ann Biomed Eng 32:257–263CrossRefGoogle Scholar
  64. Schmid H, Nash MP, Young AA, Röhrle O, Hunter PJ (2007) A computationally efficient optimization kernel for material parameter estimation procedures. J Biomech Eng 129(2): 279–283CrossRefGoogle Scholar
  65. Seliktar D, Nerem RM, Galis ZS (2003) Mechanical strain-stimulated remodeling of tissue-engineering blood vessel constructs. Tissue Eng 9: 657–666CrossRefGoogle Scholar
  66. Skalak R, Dusgupta G, Moss M, Otten E, Dullemejer P, Vilmann H (1982) Analytical description of growth. J Theor Biol 94: 555–577CrossRefGoogle Scholar
  67. Smith NP, Nickerson DP, Crampin EJ, Hunter PJ (2004) Multiscale computational modelling of the heart. Acta Numerica 13:371–431CrossRefMathSciNetGoogle Scholar
  68. Stalhand J, Klarbring A, Karlsson M (2004) Towards in vivo aorta identification and stress estimation. Biomech Model Mechanobiol 2: 169–186CrossRefGoogle Scholar
  69. Stopak D, Harris AK (1982) Connective tissue morphogenesis by fibroblast traction. i. tissue culture observations. Dev Biol 90(2): 383–398CrossRefGoogle Scholar
  70. Taber LA (2008) Theoretical study of beloussov’s hyper-restoration hypothesis for mechanical regulation of morphogenesis. Biomech Model Mechanobiol 7: 427–441CrossRefGoogle Scholar
  71. Taber LA, Humphrey JD (2001) Stress-modulated growth, residual stress, and vascular heterogeneity. J Biomech Eng 123: 528–535CrossRefGoogle Scholar
  72. Takamizawa K, Hayashi K (1987) Strain energy density function and uniform strain hypothesis for arterial mechanics. J Biomech Eng 20: 7–17CrossRefGoogle Scholar
  73. Takamizawa K, Hayashi K (1988) Uniform strain hypothesis and thin-walled theory in arterial mechanics. Biorheology 25: 555–565Google Scholar
  74. Tsamis A, Stergiopulos N (2007) Arterial remodelling in response to hypertension using a constituent based model. Am J Physiol 293: H3130–H3139Google Scholar
  75. Vena P, Gastaldi D, Socci L, Pennati G (2008) An anisotropic model for tissue growth and remodeling during early development of cerebral aneurysms. Comput Mater Sci 43: 565–577CrossRefGoogle Scholar
  76. Vito RP, Dixon SA (2003) Blood vessel constitutive models 1995–2002. Ann Rev Biomed Eng 5: 413–439CrossRefGoogle Scholar
  77. Watton PN, Hill NA (2009) Evolving mechanical properties of a model of abdominal aortic aneurysm. Biomech Model Mechanobiol 8: 25–42CrossRefGoogle Scholar
  78. Watton PN, Hill NA, Heil M (2004) A mathematical model for the growth of the abdominal aortic aneurysm. Biomech Model Mechanobiol 3: 98–113CrossRefGoogle Scholar
  79. Watton PN, Ventikos Y (2009) Modelling evolution of saccular cerebral aneurysms. J Strain Anal 44(5): 375–389CrossRefGoogle Scholar
  80. Watton PN, Raberger NB, Ventikos Y, Holzapfel GA (2009a) Coupling the hemodynamic environment to the growth of cerebral aneurysms: computational framework and numerical examples. J Biomech Eng 131(10): 14CrossRefGoogle Scholar
  81. Watton PN, Ventikos Y, Holzapfel GA (2009b) Modelling the growth and stabilization of cerebral aneurysms. Math Med Biol 26(2): 133–164CrossRefzbMATHGoogle Scholar
  82. Watton PN, Ventikos Y, Holzapfel GA (2009c) Modelling the mechanical response of elastin for arterial tissue. J Biomech 42: 1320–1325CrossRefGoogle Scholar
  83. Wulandana R, Robertson AM (2005) An inelastic multi-mechanism constitutive equation for cerebral arterial tissue. Biomech Model Mechanobiol 4(4):235–248CrossRefGoogle Scholar
  84. Zulliger MA, Fridez P, Hayashi K, Stergiopolous N (2004) A strain energy function for arteries accounting for wall composition and structure. J Biomech 37: 989–1000CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • H. Schmid
    • 1
    Email author
  • P. N. Watton
    • 2
  • M. M. Maurer
    • 1
  • J. Wimmer
    • 1
  • P. Winkler
    • 1
  • Y. K. Wang
    • 3
  • O. Röhrle
    • 4
  • M. Itskov
    • 1
  1. 1.Department of Continuum MechanicsRWTH Aachen UniversityAachenGermany
  2. 2.Institute of Biomedical EngineeringUniversity of OxfordOxfordUK
  3. 3.Auckland Bioengineering InstituteThe University of AucklandAucklandNew Zealand
  4. 4.Institute für Mechanik (Bauwesen)University of StuttgartStuttgartGermany

Personalised recommendations