Biological tissues adapt to changed loading conditions through growth and remodelling (G&R) to reestablish a so-called homeostatic state. On the other hand, loading conditions above their physiological limits, as during trauma or surgical procedures, cause injury and can initiate pathological G&R. Herein, a modelling approach for G&R influenced by injury is presented combining the theories of plasticity and homogenised constrained mixtures. The results show that injury has a significant impact on the G&R behaviour and thus on the accomplishment of homeostasis

Gierig, Meike

Marino, Michele

Wriggers, Peter

English

Institutionelles Repositorium der Leibniz Universität Hannover

Received: 16 May 2019 Accepted: 6 June 2019DOI: 10.1002/pamm.201900259A Computational Model for Biological Tissues Considering the Influenceof Injury on Growth and RemodellingMeike Gierig1,∗, Michele Marino1, and Peter Wriggers11 Institute of Continuum Mechanics, Leibniz Universität Hannover, GermanyBiological tissues adapt to changed loading conditions through growth and remodelling (G&R) to reestablish a so-calledhomeostatic state. On the other hand, loading conditions above their physiological limits, as during trauma or surgical pro-cedures, cause injury and can initiate pathological G&R. Herein, a modelling approach for G&R influenced by injury ispresented combining the theories of plasticity and homogenised constrained mixtures. The results show that injury has asignificant impact on the G&R behaviour and thus on the accomplishment of homeostasis.c© 2019 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim1 IntroductionSoft biological tissues adapt continuously to their mechanical state by the addition of new mass (growth) to the tissue and massturnover (remodelling) within the tissue. In addition, tissues can also be damaged when unphysiologically loaded. Damagein collagen fibres is considered the most important since elastin deforms elastically up to high strains. Of several damagemechanisms in the fibres, the interstrand delamination, i.e. the permanent elongation of collagen molecular nanostructure,seems to play a crucial role [1]. The presented computational model considers the influence of injury on G&R. The interstranddelamination is modelled using the theory of elastoplasticity for fibre-reinforced matrices as presented by Gasser and Holzapfelin [2]. Growth and remodelling is added following the approach by Cyron et al. in [3] which is based on the homogenizedconstrained mixture theory.2 MethodsThe deformation gradient F is multiplicatively decomposed into elastic Fie, plastic Fip, remodelling Fir and growth Fig defor-mation gradients. Two constituents, collagen fibres (i = c) and the elastin matrix (i = m) are considered. Note that bothconstituents experience the same total deformation F but different elastic and inelastic deformations, see Fig. 1. Injury andTable 1: Material and simulation parametersµ 80 J/kg ρc0(0) 300 kg/m3k1 160 J/kg ρm0 650 kg/m3k2 3.5 ã0 {0.3, 0.5, 1.0}a0 ã0/||ã0||Injury [2] G&R [4]τ0 80 kPa σhom 60 kPah 50 kPa Tr 1.5 sτ∞ 200 kPa kσ 0.02a0 0.1Fig. 1: Decomposition of the deforma-tion gradient1 1.1 1.2 1.3 1.4 1.50200400600λ3σ33inkPawithout injurywith injuryFig. 2: Cauchy stresses with and with-out injury over stretchremodelling are assumed to occur only in the collagen fibres. Growth is the same for both constituents, Fcg = Fmg = Fg. Ananisotropic, invariant-based Helmholtz free energy functionΨ(Fce,Fme , a0, A, p, θ) = ρc0 (ψcσ(Fce, a0) + ψcA(A)) + ρm0 ψm(F̄me ) + ρ0ψL(p, θ) (1)is defined as a function of the elastic deformations, the fibre direction a0 in the reference configuration, an injury-relatedvariable A, a pressure p and strain variable θ. The free energy is additively decomposed into a free energy for collagen(superscript c), elastin (m) and a term accounting for the nearly incompressible behaviour of tissues (ΨL). A selective Hu-Washizu functional with the constraint for the volumetric deformation det(F)/det(Fg) = 1, which restricts the volumechange to the growth deformation, is employed. Since the reference mass density changes due to G&R, the free energies aredefined per unit mass and multiplied by the reference mass densities per unit volume of the whole tissue. The total referencemass density is ρ0 = ρc0 + ρm0 . To account for injury, the collagen free energy is again additively decomposed into an elastic∗ Corresponding author: e-mail gierig@ikm.uni-hannover.de, phone +49 511 762 19059, fax +49 511 762 5496This is an open access article under the terms of the Creative Commons Attribution License 4.0, which permits use,distribution and reproduction in any medium, provided the original work is properly cited.PAMM · Proc. Appl. Math. Mech. 2019;19:e201900259. www.gamm-proceedings.com 1 of 2https://doi.org/10.1002/pamm.201900259 c© 2019 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim2 of 2 Section 2: Biomechanics0 2 4 6 8 100.80.911.1t in s[F] 11[F]11[Fce]11[Fme ]11[Fcp]11[Fg]11[Fcr]110 2 4 6 8 100.80.911.1t in s[F] 11Fig. 3: Decomposed deformation gradient components [F]11 without (left) and with (right)injury over time0 2 4 6 8 100.9511.051.11.151.2t in sρc 0(t)/ρc 0(0)without injurywith injuryFig. 4: Reference collagen densities rela-tive to initial reference density over timeterm ψcσ and a plastic term ψcA associated with the permanent elongation due to interstrand delamination. Plastic flow isgoverned by a combined linear-exponential hardening [2] with the internal variable A as a hardening variable. G&R governsthe change of the reference total mass density ρ̇0 = ρ̇c0 + ρ̇m0 , where the collagen density rate ρ̇c0 is governed by the differencebetween the current Cauchy stress and a homeostatic stress value, whereas the elastin density is assumed to be constant intime (i.e., ρ̇m0 = 0). The detailed approach is presented by Braeu et al. in [4]. Following the slow-growth assumption, thetime-scales of growth and elastoplastic deformations can be clearly separated. As a consequence, the plastic deformation andG&R are implemented in a staggered way.3 Numerical ExamplesThe features of the model are investigated by a uniaxial tension test on an H1P0 one-element cube with edge length 1 cm.The maximum stretch λ3 = 1.5 of the top surface is achieved after 1 s and held constant afterwards. G&R is activated after1.5 s and isotropic growth is considered. Two simulations are compared: in the first simulation, the material deforms onlyelastically before G&R whereas in the second, it is damaged during loading. The stresses are derived using an exponentialfunction for collagen and a Neo-Hookean approach for the matrixψcσ =k12k2(ek2(I4−1)2 − 1), ψm =µ2(Ī1 − 3), (2)where k1, k2, µ are material parameters, I4 the quadratic stretch of collagen in fibre direction and Ī1 the first invariant of theisochoric part of the elastic matrix deformation gradient. The material parameters are listed in Tab. 1.For both simulations, the Cauchy stresses progressively decrease at λ = 1.5 due to G&R, converging towards the homeo-static state, see Fig. 2. However, before G&R, the evolution of the stresses differ between both simulations. This differencecauses different evolutions of growth and remodelling and thus total deformations, see Fig. 3. The evolution of the referencedensities of collagen relative to the initial reference density is depicted in Fig. 4. The density changes with injury 9.7 % andwithout 18.0 %. The results show that the impact of injury on G&R is evident and hence should be considered in furtherresearch. The approach has also been applied to idealized axisymmetric artery models. These results and modifications ofthe presented approach will be discussed in future publications. In particular, the biochemical and cellular reaction to injurydriving G&R is an essential ingredient to be modelled and incorporated.Acknowledgements This project is carried out within the framework of the Masterplan SMART BIOTECS, alliance between the TechnicalUniversity of Braunschweig and the Leibniz University of Hannover. This initiative is financially supported by the Ministry of Science andCulture (MWK) of Lower Saxony, Germany.References[1] M. Marino, M. I. Converse, K. L. Monson, and P. Wriggers, J Mech Behav Biomed Mat, doi:10.1016/j.jmbbm.2019.04.022 (2019).[2] T. C. Gasser and G. A. Holzapfel, Comput Mech 29(4-5), 340–360 (2002).[3] C. J. Cyron, R. C. Aydin, and J. D. Humphrey, Biomech Model Mechanobiol 15(6), 1389–1403 (2016).[4] F. A. Braeu, A. Seitz, R. C. Aydin, and C. J. Cyron, Biomech Model Mechanobiol 16(3), 889–906 (2017).c© 2019 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim www.gamm-proceedings.com 16177061, 2019, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/pamm.201900259 by Technische Informationsbibliothek, Wiley Online Library on [17/10/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

A Computational Model for Biological Tissues Considering the Influence of Injury on Growth and Remodelling

https://www.repo.uni-hannover.de/bitstream/123456789/15147/1/A-Computational-Model-for-Biological.pdf

A Computational Model for Biological Tissues Considering the Influence of Injury on Growth and Remodelling

Abstract

Similar works

Full text

Available Versions

Institutionelles Repositorium der Leibniz Universität Hannover