An anisotropic micro-sphere approach applied to the

An anisotropic micro-sphere approach applied to the modelling of soft biological tissues A. Menzel, T. Waffenschmidt and V. Alastrué

Abstract A three-dimensional model for the simulation of anisotropic soft biological tissues is discussed. The underlying constitutive equations account for large strain deformations and are based on a hyper-elastic form. As various soft biological tissues are nearly incompressible, we adopt the classical volumetric-isochoric split of the strain energy density. While its isotropic part is chosen to take a standard neo-Hookean form, its anisotropic part is determined by means of the so-called micro-sphere model. In this regard, physically sound one-dimensional constitutive models—as for instance the worm-like chain model—can be used and straightforwardly be extended to the three-dimensional case. As a key aspect, the microsphere model is extended to further capture remodelling. Such deformation-induced anisotropy is introduced by setting up evolution equations for the integration directions used to perform numerical integrations on the unit-sphere. The particular model proposed captures orthotropic material behaviour and additionally accounts for saturation effects combined with a visco-elasticity-type time-dependent anisotropy evolution.

A. Menzel Institute of Mechanics, Department of Mechanical Engineering, TU Dortmund, Leonhard-EulerStr. 5, D-44227 Dortmund, Germany, e-mail: [email protected] Division of Solid Mechanics, Lund University, P.O. Box 118, SE-22100 Lund, Sweden, e-mail: [email protected] T. Waffenschmidt Institute of Mechanics, Department of Mechanical Engineering, TU Dortmund, Leonhard-EulerStr. 5, D-44227 Dortmund, Germany, e-mail: [email protected] V. Alastrué Group of Structural Mechanics and Materials Modelling, Arago´ n Institute of Engineering Research (I3A), University of Zaragoza, Mar´ıa de Luna, 3 E-50018 Zaragoza, Spain, e-mail: victorav@ unizar.es

1

2

A. Menzel, T. Waffenschmidt and V. Alastrué

1 Introduction Apart from biological and chemical effects affecting the behaviour of soft biological materials such as ligaments, tendons, muscles, and skin—to name just a few examples—biological tissues in general possess a pronounced composite-type multi-scale structure together with strongly anisotropic mechanical properties. The local mechanical response of these tissues is typically determined by elastin and collagen fibre-bundels. In this regard, adaptation of these fibre networks influenced by mechanical loading is a main biomechanical phenomenon occurring in hard as well as soft biological tissues. In general, adaptation processes can include changes in mass and internal structure, whereas this paper exclusively focuses on the latter, which we denote as remodelling—the related processes often being denoted as fibre reorientation or rather turnover. The computational remodelling approach proposed in the following is partly motivated by the investigations reported in [3], where a fibroblast-populated collagen lattice was tested. As a result, macroscopically tension-type mechanical loads cause the initially unstructured collagen fibre network to reorient with the local dominant stretch direction and thus showing transversely isotropic characteristics. However, as many biological tissues—for example arteries—show fibre alignment with more than one single direction, we here extend the remodelling formulation proposed in [6] for transversal isotropy to orthotropic material behaviour. The paper is organised as follows: section 2 briefly reviews essential kinematic relations, based on which key aspects of the micro-sphere model are outlined in section 3. Section 4 constitutes the main part of this contribution, wherein the remodelling formulation is introduced. In section 5 a numerical example is discussed, before the paper closes with a short summary in section 6.

2 Essential kinematics Let x = ϕ (X,t) : B 0 × T → Bt describe the motion of a body mapping position vectors X ∈ B 0 from the material configuration to their spatial counterpart x ∈ B t . The local deformation is characterised by the common deformation gradient tensor F = ∇X ϕ with the Jacobian J = det(F) > 0 and the corresponding right CauchyGreen strain tensor C = Ft · F, while their isochoric counterparts are represented as 1 ¯ = F¯ t · F. ¯ F¯ = J − 3 F¯ and C In view of the computational micro-sphere-scheme used later on, additional kinematic relations referring to the underlying unit-sphere U2 are introduced. In this 2 regard, an affine stretch in the direction of a referential unit-vector √ r ∈ U can be ¯ via λ¯ = r · C ¯ · r. However, determined by the macroscopic deformation tensor C it is well-known that the affinity assumption is not in agreement with experimental observations for cross-linked polymer-type materials. For this reason, according to

An anisotropic micro-sphere approach applied to the modelling of soft biological tissues

3

[7], we make use of a non-affine stretch taking the form 1 λ= λ¯ p dA 4 π U2

Z

1/p

.

(1)

Obviously the non-affine stretch can be interpreted as an averaged stretch over the unit-sphere, with p defining a non-affine stretch parameter. Moreover, inspired by [7], a single collagen chain is additionally constrained by another micro-kinematic variable, namely the contraction ν of the cross-section of a micro-tube that contains the related chain. Therefore, analogous to relation (1), one could also introduce a non-affine area stretch 1/q Z 1 q ¯ dA ν= ν , (2) 4 π U2 where q denotes a non-affine tube parameter. At this stage, however, we restrict ourselves to account only for the non-affine stretch-contributions λ .

3 Hyper-elastic micro-sphere model Apart from the remodelling approach discussed later on, we make use of a hyperelastic form of the strain energy. In this regard, we adopt the well-established volumetric-isochoric split and decompose the isochoric part into an isotropic and an anisotropic contribution, namely ¯ + Ψani (λ (C, ¯ ri )) ; Ψ(C, ri ) = Ψvol (J) + Ψiso (C)

(3)

see [1] in view of an affine anisotropic part. Due to the almost incompressible response of soft biological tissues, we assume a nearly-incompressible neo-Hooke model to account for the volumetric and isotropic isochoric part of the strain energy, i.e. 1 ¯ −3], (4) Ψvol = D [ J − 1 ]2 and Ψiso = µ [ I : C 4 with D defining a penalty parameter, µ being a material parameter and I representing the second-order identity tensor. According to the highly anisotropic material properties of the type of biological tissue we are interested in, the strain energy function (3) is assumed to depend not only on the right Cauchy-Green strain tensor C but also on a finite number of referential direction vectors or rather integration directions ri defined on the microsphere U2 . In this regard, a one-dimensional constitutive equation is applied for every integration direction ri . To be specific, we make use of the micro-mechanically motivated worm-like chain model, which takes the representation

4


ψ

ani

2 2 i r¯ Kθ L r¯ ln(λ˜ 4r0 ) h r0 1 1 ˜ −1 − ψc , (5) 2 2+ 4 + − − (λ ) = 4A L 4 r0 L L [1 − rL0 ]2 1 − Lr¯ L

for λ˜ ≥ 1 while ψ ani (λ˜ ) = 0 is assumed for compression, i.e. λ˜ < 0. Herein the Boltzmann constant is denoted by K = 1.38 × 10−23 [JK−1 ], θ is the absolute temperature, and A is established as persistent contour length. While r0 characterises the length of the chain for the undeformed state, the actual representative chain length follows from r¯ = λ˜ r0 ∈ [0, L). The extension of this one-dimensional constitutive law (5) to the threedimensional macroscopic level is performed by means of the micro-sphere formulation. Characteristic for this approach is a finite number of unit vectors ri to be considered for the numerical integration over the unit sphere U2 , which yields the total anisotropic contribution Ψani to be computed by means of the fibre-related strain energy ψ ani via ¯ ri )) = 1 Ψani (λ (C, 4π

Z

U2

¯ ri )) dA . ψ ani (λ (C,

(6)

4 A remodelling formulation for orthotropic material behaviour The key aspect of this contribution consists in incorporating remodellingphenomena by setting up deformation-driven evolution equations for the integration directions ri , which means that these are not constant but evolve in time. To be specific we directly relate the integration directions—now taking the interpretation as internal variables—to the numerical framework, i.e. the integration of equation (6), which, algorithmically, leads to a summation over a finite number of integration directions m

¯ ri )) ≈ ∑ wi ψ (λ¯ i ) , Ψani (λ (C,

(7)

i=1

with wi denoting integration factors, which depend on the particular integration scheme. Since various biological tissues show fibre alignment with more than one single direction, we subsequently propose a remodelling formulation reflecting macroscopically orthotropic behaviour. An analogous approach for the transversely isotropic case has recently been discussed in detail, see [6]. In view of the reorientation criterion, a crucial point consists in the identification of the deformation-dependent mean directions l1,2 , which on the one hand should determine the alignment of the integration directions and on the other hand is here assumed to reflect extremal states of strain energy. In this context, one could align the integration directions ri with respect to the principal stretch directions or alternatively such that the directions, according to which the integration unit-vector are aligned with, share identical angles with the principal directions. As a special case of the latter, we make use of two particular directions reported in [4]: the so-called


limiting directions can be calculated via the relation q q ¯ ¯ ¯ ¯ λ1C nC2 ± λ2C nC1 ¯ ¯ q l1,2 = for λ1C > λ2C > 1 ¯ ¯ λ1C + λ2C

5

(8)

¯

C and principal directions nC of the isochoric right using the principal values λ1,2 1,2 ¯ Practically speaking, these limiting directions suffer Cauchy-Green strain tensor C. the maximum shear in the considered plane of tension.

Fig. 1 Graphical illustration of the evolution equation (9): construction of the direction of the rate of ri according to an alignment with respect to the closest limiting direction l1,2 PSfrag replacements

ri [ri · l] ri

l1

r˙ i / f

l2 = l

U2

As a result, the evolution of ri is motivated by its alignment with the limiting directions l1,2 —see figure 1—as reflected by r˙ i = f sign (ri · l) [l − [ri · l] ri ]

so that

r˙i · ri = 0 ,

(9)

where the integration direction ri aligns either with l2 in case of ri being closer to l2 or with l1 else, i.e. ( l2 if |ri · l1 | ≤ |ri · l2 | . (10) l= l1 else Unlike the approach used in [6] and due to the present assumption of orthotropy with two mean directions, in this case two second-order generalised structural tensors are introduced as m

A1,2 = ∑ wi ri ⊗ ri = i=1

3

1,2 1,2 ∑ A1,2 j nj ⊗nj

j=1

∀

ri → l1,2 ,

(11)

1,2 1,2 1,2 1 where A1,2 1 ≥ A2 ≥ A3 ≥ 0 with ∑ j A j = 1/2. To give an example, A will be calculated for those integration directions ri , which—due to the deformation—align with the first mean direction l1 . In order to later on visualise local anisotropic ma1 2 terial properties the orientation-distribution-type function ρ A = ρ A ∪ ρ A is intro1,2 duced with ρ A = e · A1,2 · e and e ∈ U2 denoting a unit-vector.

6


The remaining task consists in particularising the factor f occurring in the evolution equation (9). As the adaptation of biological tissues is usually bounded by certain biological limits, the evolution equation for ri should account for saturation effects. On the one hand, the evolution saturates for ri aligning with one of the limiting directions l1,2 . On the other hand, we restrict the maximum degree of anisotropy by assuming ri to evolve as long as the difference between the largest and smallest eigenvalue of A1,2 remain smaller than a pre-defined limit value A∆ . In addition, a relaxation parameter t ∗ is incorporated and we also set r˙ i to zero in case the difference of the related fibre stretches remain smaller than a certain threshold λ¯ c . In summary, the proportionality factor f introduced in equation (9) is assumed as  1,2 1,2   A∆ − [ A1 − A3 ] if λ C¯ > 1 and λ C¯ − λ C¯ > λ¯ c 2 1 2 t ∗ A∆ . (12) f=   0 else

5 Numerical example and results The model is now investigated for homogeneous biaxial tension with the corre −1 sponding deformation gradient F = λ1U e1 ⊗ e1 + λ2U e2 ⊗ e2 + λ1U λ2U e3 ⊗ e 3 . The particular loading history considered is based on linearly increasing the representative loading parameters λ1U and λ2U within a time period of 20 time steps and then fixing its value for a time period of 380 steps; see figure 2(a). Special emphasis is thereby placed on the evolution of deformation-induced anisotropy, which is illustrated by means of A1,2 in terms of the odf-type function ρ A and via the difference between its respective maximal and minimal principal 1,2 values, A1,2 1 − A3 . Diagram 2(b) shows the saturation behaviour of the anisotropy evolution by 1,2 means of visualising the degree of anisotropy A1,2 1 − A3 . Obviously it takes place in a viscous manner as the loading is fixed after 20 steps and the graph of A11,2 − A1,2 3 continues to increase. The anisotropy evolution is additionally visualised in a more descriptive odf-type manner by figure 2(c). We observe for the different states of deformation, that the anisotropy evolves in time as the odf deviates from a spherical distribution. The entire information can be obtained by directly displaying the integration directions in figure 2(d). We see that these directions indeed align according to two limiting directions. Note that in this case we used 162 integration directions for the integration on the unit sphere, as discussed, e.g., in [5].


(a)

(c)

(b)

(d)

7

PSfrag replacements

1,2 Fig. 2 Biaxial tension: (a) applied loading history; (b) evolution of A1,2 1 − A3 ; (c) odf-type funcA tion ρ300 at load step 300; (d) integration directions at load step 300.

6 Summary In this work, remodelling is understood as a process that renders the internal substructure of the material to adapt to the local loading conditions. Such an alignment of fibres is often also denoted as reorientation or, from the biological point of view, as turnover. The model developed directly combines this remodelling with the computational micro-sphere approach. To be specific, the respective directions introduced to perform the numerical integration over the unit-sphere are reoriented. As a result, the formulation accounts for deformation-induced anisotropy evolution. In order to capture orthotropic material behaviour, the evolution equation describing the reorientation, was assumed to align the integration directions with respect to two particular mean line elements—the so-called limiting directions. Saturation effects are on the one hand naturally included by a stopping remodelling process as soon as a direction is aligned with the particular limiting direction. On the other hand, an additional saturation value has been introduced to be able to further limit the maximal degree of anisotropy of the tissue. The numerical example investigated showed the basic algorithmic applicability of the modelling framework and captured the fundamental reorientation and remodelling effects observed for soft biological tissues. Moreover—even though not shown here—the formulation can be applied to the simulation of general boundary value problems, as based on, for instance, iterative finite element approaches.

8


References 1. V. Alastrué, M.A. Mart´ınez, M. Doblaré, and A. Menzel. Anisotropic micro-sphere-based finite elasticity applied to blood vessel modelling. J. Mech. Phys. Solids, 57:178–203 (doi:10.1016/j.jmps.2008.09.005), 2009. 2. V. Alastrué, M.A. Mart´ınez, A. Menzel and M. Doblaré. On the use of non–linear transformations for the evaluation of anisotropic rotationally symmetric directional integrals. Application to the stress analysis in fibred soft tissues. IJNME, doi: 10.1002/nme.257, 2009. 3. M. Eastwood, V.C. Mudera, D.A. McGrouther, and R.A. Brown. Effect of precise mechanical loading on fibroblast populated collagen lattices: morphological changes. Cell Motil. Cytoskeleton, 40:1321, 1998. 4. Ph. Boulanger, M. Hayes. On finite shear. Arch. Ration. Mech. Anal. 151, 125-185, 2000. 5. I. Kurzhöfer. Mehrskalen–Modellierung polykristalliner Ferroelektrika basierend auf diskreten Orientierungsverteilungsfunktionen, Universität Duisburg–Essen, Institut für Mechanik, Bericht Nr. 4, 2007. 6. A. Menzel, T. Waffenschmidt. A micro-sphere-based remodelling formulation for anisotropic biological tissues. Phil. Trans. R. Soc. A, 367(1902):3499-3523 (doi: 10.1098/rsta.2009.0103), 2009. 7. C. Miehe, S. Göktepe, and F. Lulei. A micro-macro approach to rubber-like materials - Part I: the non-affine micro-sphere model of rubber elasticity. J. Mech. Phys. Solids, 52:2617–2660, 2004.