A stochastic model of stroma: interweaving variability and compressed fibril exclusion

How to Cite

Vasta M, Gizzi A, Pandolfi A. A stochastic model of stroma: interweaving variability and compressed fibril exclusion. MAIO [Internet]. 2018 Jun. 18 [cited 2024 May 22];2(2):58-63. Available from: https://www.maio-journal.com/index.php/MAIO/article/view/73

Copyright notice

Authors who publish with this journal agree to the following terms:

  1. Authors retain copyright and grant the journal right of first publication, with the work twelve (12) months after publication simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work’s authorship and initial publication in this journal.

  2. After 12 months from the date of publication, authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.


compressive fibril exclusion; hyperelasticity; lamellar branching; second order structure tensors


Hyperelastic constitutive models of the human stroma accounting for the stochastic architecture of the collagen fibrils and particularly suitable for computational applications are discussed. The material is conceived as a composite where a soft ground matrix is embedded with collagen fibrils characterized by non-homogeneous spatial distributions typical of reinforcing stromal lamellae. A multivariate probability density function of the spatial distribution of the fibril orientation is used in the formulation of the lamellar branching observed on the anterior third of the stroma, selectively excluding the contribution of compressed fibrils. The physical reliability and the computational robustness of the model are enhanced by the adoption of a second order statistics approximation of the average structure tensors typically employed in fiber reinforced models.



Renna A, Pandolfi A, Martinez FC, Aliò JL. Myopic surface ablation in asymmetrical topographies: refractive results and theoretical corneal elastic response. Am J Ophtalmol. 2017;177:34-43.

Pandolfi A, Manganiello F. A model for the human cornea: constitutive formulation and numerical analysis. Biomech Mod Mechanobiol. 2006;5:237-246.

Simonini I, Pandolfi A. Customized finite element modelling of the human cornea. PLoS One. 2015;10:e0130426.

Petsche SJ, Pinsky PM. The role of 3-d collagen organization in stromal elasticity: a model based on x-ray diffraction data and second harmonic-generated images. Biomech Mod Mechanobiol.2013;12:1101-1113.

Pandolfi A, Vasta M. Fiber distributed hyperelastic modeling of biological tissues. Mech Mater. 2012;44:151-162.

Gizzi A, Pandolfi A, Vasta M. A generalized statistical approach for modelling fiber-reinforced materials. J Eng Math. 2017;1-16. doi:10.1007/s10665-017-9943-5.

Abass A, Hayes S, White N, Sorensen T, Meek KM. Transverse depth-dependent changes in corneal collagen lamellar orientation and distribution. J Roy Soc Interface. 2015;12:20140717.

Sacks MS. Incorporation of experimentally-derived fiber orientation into a structural constitutive model for planar collagenous tissues. J Biomech Eng – Trans ASME. 2003;125:280-287.

Lanir Y. Constitutive equations for fibrous connective tissues. J Biomech. 1983;16:1-12.

Organ CO, Saccomandi G. A new constitutive theory for fiber-reinforced incompressible nonlinearly elastic solids. J Mech Phys Solids. 2005;53:1985-2015.

Federico S, Herzog W. Towards an analytical model of soft tissues. J Biomech. 2008;41:3309-3313.

Gasser TC, Ogden RW, Holzapfel GA. Hyperelastic modeling of arterial layers with distributed collagen fibre orientations. J Roy Soc Interface. 2006;3:15-35.

Federico S, Gasser TC. Nonlinear elasticity of biological tissues with statistical fibre orientation. J Roy Soc Interface. 2010;7:955-966.

Vasta M, Gizzi A, Pandolfi A. On three-and two-dimensional fiber distributed models of biological tissues. Prob Eng Mech. 2014;37:170-179.

Gizzi A, Pandolfi A, Vasta M. Statistical characterization of the anisotropic strain energy in soft materials with distributed fibers. Mech. Mater. 2016;92:119-138.

Hashlamoun K, Grillo A, Federico S. Efficient evaluation of the material response of tissues reinforced by statistically oriented fibres. ZAMP. 2016;67:113.

Holzapfel GA, Niestrawska JA, Ogden RW, Reinisch AJ, Schriefl AJ. Modelling non-symmetric collagen fibre dispersion in arterial walls. J Roy Soc Interface. 2015;12:20150188.

Gizzi A, Vasta M, Pandolfi A. Modeling collagen recruitment in hyperelastic bio-material models with statistical distribution of the fiber orientation. Int J Eng Sci. 2014;78:48-60.

Latorre M, Montáns FJ. On the tension-compression switch of the Gasser–Ogden–Holzapfel model: Analysis and a new pre-integrated proposal. J Mech Behav Biomed Mater. 2016;57:175-189.

Li K, Ogden RW, Holpzafel GA. Computational method for excluding fibers under compression in modeling soft fibrous solids. Eur J Mech A/Solids. 2016;57:178-193.

Maceri F, Marino M, Vairo G. A unified multiscale mechanical model for soft collagenous tissues with regular fiber arrangement. J Biomech. 2010;43:355-363.

Marino M, Vairo G. Stress and strain localization in stretched collagenous tissues via a multiscale modelling approach. Comput Method Appl Mech Eng. 2014;17:11-30.