Skip to main content
SpringerLink
Log in

A computationally efficient gradient-enhanced healing model for soft biological tissues

  • Original Paper
  • Published:
Biomechanics and Modeling in Mechanobiology Aims and scope Submit manuscript

Abstract

Soft biological tissues, such as arterial tissue, have the ability to grow and remodel in response to damage. Computational method plays a critical role in understanding the underlying mechanisms of tissue damage and healing. However, the existing healing model often requires huge computation time and it is inconvenient to implement finite element simulation. In this paper, we propose a computationally efficient gradient-enhanced healing model that combines the advantages of the gradient-enhanced damage model, the homeostatic-driven turnover remodeling model, and the damage-induced growth model. In the proposed model, the evolution of healing-related parameters can be solved explicitly. Additionally, an adaptive time increment method is used to further reduce computation time. The proposed model can be easily implemented in Abaqus, requiring only a user subroutine UMAT. The effectiveness of proposed model is verified through a semi-analytical example, and the influence of the variables in the proposed model is investigated using uniaxial tension and open-hole plate tests. Finally, the long-term development of aneurysms is simulated to demonstrate the potential applications of the proposed model in real biomechanical problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

References

  • Abaqus 6.14 documentation (2014)

  • Braeu F, Seitz A, Aydin R, Cyron C (2017) Homogenized constrained mixture models for anisotropic volumetric growth and remodeling. Biomech Model Mechanobiol 16:889–906

    Article  Google Scholar 

  • Braeu FA, Aydin RC, Cyron CJ (2019) Anisotropic stiffness and tensional homeostasis induce a natural anisotropy of volumetric growth and remodeling in soft biological tissues. Biomech Model Mechanobiol 18:327–345

    Article  Google Scholar 

  • Comellas E, Gasser TC, Bellomo FJ, Oller S (2016) A homeostatic-driven turnover remodelling constitutive model for healing in soft tissues. J R Soc Interface 13:20151081

    Article  Google Scholar 

  • Cyron CJ, Humphrey JD (2017) Growth and remodeling of load-bearing biological soft tissues. Meccanica 52:645–664

    Article  MathSciNet  Google Scholar 

  • Cyron CJ, Aydin RC, Humphrey JD (2016) A homogenized constrained mixture (and mechanical analog) model for growth and remodeling of soft tissue. Biomech Model Mechanobiol 15:1–15

    Article  Google Scholar 

  • Dimitrijevic B, Hackl K (2008) A method for gradient enhancement of continuum damage models. Tech Mech 28:43–52

    Google Scholar 

  • Dimitrijevic B, Hackl K (2011) A regularization framework for damage-plasticity models via gradient enhancement of the free energy. Int J Numer Methods Biomed Eng 27:1199–1210

    Article  MathSciNet  Google Scholar 

  • Ebinger T, Diebels S, Steeb H (2007) Numerical homogenization techniques applied to growth and remodelling phenomena. Comput Mech 39:815–830

    Article  Google Scholar 

  • Eichinger JF, Haeusel LJ, Paukner D, Aydin RC, Humphrey JD, Cyron CJ (2021) Mechanical homeostasis in tissue equivalents: a review. Biomech Model Mechanobiol 20:833–850

    Article  Google Scholar 

  • Fereidoonnezhad B, Naghdabadi R, Sohrabpour S, Holzapfel GA (2017) A mechanobiological model for damage-induced growth in arterial tissue with application to in-stent restenosis. J Mech Phys Solids 101:311–327

    Article  MathSciNet  Google Scholar 

  • Frank CB, Shrive NG, Hiraoka H, Nakamura N, Kaneda Y, Hart DA (1999) Optimisation of the biology of soft tissue repair. J Sci Med Sport 2:190–210

    Article  Google Scholar 

  • Frank CB, Hart DA, Shrive NG (1999) Molecular biology and biomechanics of normal and healing ligaments—a review. Osteoarthr Cartil 7:130–140

    Article  Google Scholar 

  • Fung YC (1995) Stress, strain, growth, and remodeling of living organisms. In: Casey J, Crochet MJ (eds) Theoretical, experimental, and numerical contributions to the mechanics of fluids and solids: a collection of papers in honor of Paul M. Naghdi. Springer, pp 469–482

    Chapter  Google Scholar 

  • Ganghoffer JF, Boubaker MB (2016) Micromechanical analysis of volumetric growth in the context of open systems thermodynamics and configurational mechanics. application to tumor growth. Contin Mech Thermodyn 29:1–27

    MathSciNet  Google Scholar 

  • Ganghoffer JF, Boubaker MB (2017) Micromechanical analysis of volumetric growth in the context of open systems thermodynamics and configurational mechanics. application to tumor growth. Continuum Mech Thermodyn 29:429–455

    Article  MathSciNet  Google Scholar 

  • Ganghoffer J-F, Rahouadj R (2017) Thermodynamic formulations of continuum growth of solid bodies. Math Mech Solids 22:1027–1046

    Article  MathSciNet  Google Scholar 

  • Gasser TC (2017) Damage in vascular tissues and its modeling. In: Avril S, Evans S (eds) Material parameter identification and inverse problems in soft tissue biomechanics, pp 85–118

  • Gierig M, Wriggers P, Marino M (2023) Arterial tissues and their inflammatory response to collagen damage: a continuum in silico model coupling nonlinear mechanics, molecular pathways, and cell behavior. Comput Biol Med 158:106811

    Article  Google Scholar 

  • Himpel G (2007) Computational modeling of biomechanical phenomena—remodeling, growth and reorientation. Cambridge University Press

    Google Scholar 

  • Horvat N, Virag L, Holzapfel GA, Sori J, Karaj I (2019) A finite element implementation of a growth and remodeling model for soft biological tissues: verification and application to abdominal aortic aneurysms. Comput Methods Appl Mech Eng 352:586–605

    Article  MathSciNet  Google Scholar 

  • Horvat N, Virag L, Holzapfel GA, Sorić J, Karšaj I (2019) A finite element implementation of a growth and remodeling model for soft biological tissues: verification and application to abdominal aortic aneurysms. Comput Methods Appl Mech Eng 352:586–605

    Article  MathSciNet  Google Scholar 

  • Humphrey JD (2003) Continuum biomechanics of soft biological tissues. Proc R Soc Lond Ser A Math Phys Eng Sci 459:3–46

    Article  MathSciNet  Google Scholar 

  • Humphrey J (2021) Constrained mixture models of soft tissue growth and remodeling-twenty years after. J Elast 145:49–75

    Article  MathSciNet  Google Scholar 

  • Humphrey JD, Rajagopal K (2002) A constrained mixture model for growth and remodeling of soft tissues. Math Models Methods Appl Sci 12:407–430

    Article  MathSciNet  Google Scholar 

  • Kuhl E, Menzel A, Steinmann P (2003) Computational modeling of growth. Comput Mech 32:71–88

    Article  Google Scholar 

  • Latorre M, Humphrey JD (2018) Critical roles of time-scales in soft tissue growth and remodeling. APL Bioeng 2:026108

    Article  Google Scholar 

  • Latorre M, Humphrey JD (2020) Numerical knockouts-in silico assessment of factors predisposing to thoracic aortic aneurysms. PLoS Comput Biol 16:e1008273

    Article  Google Scholar 

  • Loerakker S, Ristori T (2020) Computational modeling for cardiovascular tissue engineering: the importance of including cell behavior in growth and remodeling algorithms. Curr Opin Biomed Eng 15:1–9

    Article  Google Scholar 

  • Lubliner J (1984) A maximum-dissipation principle in generalized plasticity. Acta Mech 52:225–237

    Article  MathSciNet  Google Scholar 

  • Nolan DR, Gower AL, Destrade M, Ogden RW, McGarry J (2014) A robust anisotropic hyperelastic formulation for the modelling of soft tissue. J Mech Behav Biomed Mater 39:48–60

    Article  Google Scholar 

  • Ostwald R, Kuhl E, Menzel A (2019) On the implementation of finite deformation gradient-enhanced damage models. Comput Mech 64:847–877

    Article  MathSciNet  Google Scholar 

  • Rao IJ (2011) Modeling of growth and remodeling in soft biological tissues with multiple constituents. Mech Res Commun 38:24–28

    Article  Google Scholar 

  • Schulte R, Ostwald R, Menzel A (2020) Gradient-enhanced modelling of damage for rate-dependent material behaviour—a parameter identification framework. Materials 13:3156

    Article  Google Scholar 

  • Simo JC, Ju J (1987) Strain-and stress-based continuum damage models—I. Formulation. Int J Solids Struct 23:821–840

    Article  Google Scholar 

  • Skalak R (1981) Growth as a finite displacement field. In: Proceedings of the IUTAM symposium on finite elasticity. Springer, pp 347–355

  • Waffenschmidt T, Polindara C, Menzel A, Blanco S (2014) A gradient-enhanced large-deformation continuum damage model for fibre-reinforced materials. Comput Methods Appl Mech Eng 268:801–842

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The research leading to this paper is funded by Department of Education of Liaoning Province [JYTQN2023010] and Dalian Science and Technology Innovation Fund [2023R063].

Author information

Authors and Affiliations

  1. Department of Engineering Mechanics, Dalian Jiaotong University, Dalian, 116028, People’s Republic of China

    Di Zuo, Mingji Zhu, Daye Chen & Qiwen Xue

Authors
  1. Di Zuo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mingji Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Daye Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Qiwen Xue
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Di Zuo.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zuo, D., Zhu, M., Chen, D. et al. A computationally efficient gradient-enhanced healing model for soft biological tissues. Biomech Model Mechanobiol (2024). https://doi.org/10.1007/s10237-024-01851-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10237-024-01851-5

Keywords

Search

Navigation