Microstructure-Oriented Modeling and Computational Remodeling of the Collagen Network in Corneo-Scleral Shells

Microstructure-Oriented Modeling and Computational Remodeling of the Collagen Network in Corneo-Scleral Shells Rafael Grytz and Günther Meschke Institute for Structural Mechanics Ruhr-University Bochum, Bochum, Germany rafael@grytz.de, guenther.meschke@rub.de Tissue adaptation and the mechanical condition within biological tissues are complex and mutually dependent phenomena. In this contribution, a computational model is presented to investigate the interaction between collagen fibril architecture and mechanical loading conditions in eye tissues. The biomechanical properties of eye tissues are derived from the single crimped fibril at the micro-scale via the collagen network of distributed fibrils at the meso-scale to the incompressible and anisotropic soft tissue at the macro-scale. Biomechanically induced remodeling of the collagen network is captured on the meso-scale by allowing for a continues reorientation of collagen fibrils. The remodeling process is introduced into an incompressible finite shell formulation. Finally, the presented approach is applied to a numerical human eye model considering the cornea and sclera. The predicted fibril morphology correlates well with experimental observations from x-ray scattering data. 1 Introduction The constitutive response of biological tissues existing in the human eye is mainly characterized by the elastic behavior of collagen fibrils. Several computational models have been proposed to predict the biomechanical properties of eye tissues: Hyperelastic anisotropic models for the cornea tissue have been proposed by several authors [1 3], where Pinsky et al. [1] considered x-ray scattering data to incorporate a locally dependent distribution of collagen fibril orientations. The quality of the biomechanical prediction was reported to be improved by considering anisotropy compared to isotropic models of the cornea [2]. From these numerical investigations, the necessity for a realistic consideration of collagen architecture in biomechanical models of the eye becomes evident. In order to provide reliable biomechanical simulations of the human eye one goal is to incorporate micro- and meso-structural information of the collagen network into a hyperelastic constitutive formulation.

2 Rafael Grytz and Günther Meschke = > Fig. 1. Scanning electron micrographs showing the architectures of collagen fibrils a) in the corneal stroma (reproduced from Meek and Fullwood [10]) and b) in the sclera (reproduced from Komai and Ushiki [11]). The growth, shape, mechanical properties and functionality of organs are completely dependent on remodeling of the extracellular matrix. Pathologic changes in a tissue may also be related to changes in the biosynthesis or degradation of the collagen network [4]. For example anomalies in the collagen fibril morphology due to pathophysiological adaptation may explain the changes in the mechanical behavior of the cornea observed in keratoconus [5]. Keratoconus is a noninflammatory disease characterized by thinning and scaring of the central portion of the cornea. To gain further insight into the complex biomechanical phenomena related to eye tissue remodeling a novel algorithm for fibrillar collagen reorientation is presented here. To this end, the investigations presented in this contribution start at the micro-level of individual collagen fibrils. It assumed that fibrillar collagen is the main load bearing constituent of extracellular matrix. Due to their huge aspect ratio, collagen fibrils embedded in a soft matrix crimp or buckle when the tissue is unloaded. Crimp usually occurs at the level of aggregated fibrils, e.g. at the level of fascicles in tendons [6] or at the level of lamellae in the corneal stroma [7]. The wavy structure is gradually reduced as the tissue is stretched. This straightening of crimped fibrils is the main reason for the nonlinear elastic response of soft tissues. Following ideas presented by Beskos and Jenkins [8], Freed and Doehring [9] the collagen fibril crimp is approximated by a cylindrical helix. The model is derived from the nonlinear axial force-stretch relationship of an extensible helical spring including the fully extension of the spring as a limit case. Organized collagen fibrils form fibrous networks on the meso-level. The threedimensional architecture of collagen fibrils differs substantially in the human cornea and sclera (see Fig. 1). In the corneal stroma collagen fibrils are aggregated in lamellae. There are about 300-500 lamellae across the thickness of the cornea and each lamella is composed of uniform-sized collagen fibrils running parallel to each other [11]. The distribution of collagen fibrils in the cornea have been investigated by measuring x-ray scattering patterns [12, 13]. In particular, Meek et al. [13] reported, using x-ray fibril diffraction, that the network of collagen fibrils in the central region of human corneas is char-

Computational Modeling and Remodeling of Collagen Networks 3 acterized by two preferred directions, superior-inferior and nasal-temporal. Toward the limbus there was only one preferred direction tangential to the limbus, where a circumcorneal annulus of collagen fibrils has been observed [14, 15]. In contrast to the cornea, very little work has been carried out on scleral collagen. Also scleral fibrils form bundles, which are interwoven in a more complex pattern than those in the cornea. In the present contribution the meso-structure of the extracellular matrix is represented by a network of individually oriented collagen fibrils. It is assumed that the total amount of fibrils can be grouped into two families, where the distribution of fibril orientations within each family is represented by means of the two-dimensional von Mises density function. Following the idea of Gasser et al. [16] a generalized structure tensor is used for each fibril family with a single mesostructural parameter representing the fibril dispersion in a integral sense. The central focus of this paper is to capture, predict and explain basic trends involved in the reorientation process of fibrillar collagen (e.g. Type I) and its impact on the structural response of the corneo-scleral shell. In accordance to previous computational remodeling approaches [17 19] the present algorithm is stimulated by the stress environment at the macro or tissue level. In accordance to Hariton et al. [18] it is assumed that the mean directions of the two fibril families will situate in between the principal stress directions. The reorientation of the two mean directions is captured trough a three-dimensional rotational update procedure. Motivated by the hypothesis mentioned in Driessen et al. [20], in addition to the reorientation process of the two mean fibril directions also the angular distribution of fibril orientations of each family is altered. In the following the word remodeling is exclusively used with respect to collagen fibril reorientation stimulated by the stress environment within the tissue, while the fibril diameter and initial crimp shape as well as the collagen content is assumed to be constant. The proposed remodeling algorithm, characterized by the reorganization of the collagen fibril architecture at the meso-scale stimulated by the stress environment at the macro-scale, is implemented into an incompressible finite shell formulation [21] and employed for a computational model of the corneo-scleral shell. 2 Mechanics of crimped collagen fibrils at the micro-level In this section, the most relevant assumptions and results of the proposed constitutive model for crimped collagen fibrils are presented. For a detail derivation of the crimped collagen fibril model the authors refer to Grytz and Meschke [22]. On the micro-scale, collagen fibrils are assumed to crimp into a smooth three dimensional shape when the tissue is unloaded. As the nonlinear elastic re-

4 Rafael Grytz and Günther Meschke 4 0 2 B E > A * 2 B E > ) H Fig. 2. Reference configuration of a single crimped collagen fibril with fiber direction e 0. sponse of soft tissues is mainly caused by the straightening of the crimped fibrils [23] the hierarchical substructure of the fibrils is not considered. Let the undeformed body B 0 of a collagen fibril be a curved rod with circular cross section A 0 = πr 2 0 (Fig. 2). The mid-line of B 0 follows the curvilinear path of a helical space curve C 0. The geometry of C 0 can be specified by means of two of the following variables: the radius (or amplitude) R 0, the height (or wavelength) H 0, the arc length of one helical revolution L 0 or the crimp angle θ 0. For these geometrical variables of the undeformed state the relation ( ) 2 2πR0 + ( H0 ) 2 = sin 2 (θ 0 ) + cos 2 (θ 0 ) = 1 (1) L 0 L 0 holds. In a three-dimensional space the global orientation of the helical fibril can be specified by means of a unit vector e 0 representing the centerline of C 0. The basic assumptions used in the derivation of the hyperelastic model [22] are: The crimp of the collagen fibril has the shape of a cylindrical helix, also in the deformed state. The only load carried by a single fibril is the axial force P fib acting along the centerline of the helix e 0. The fibril material is considered as linear elastic in sense of St. Venant and incompressible. In contrast to existing helical crimped fibril models [8, 9], no additional assumptions concerning the extensibility of the fibril filament need to be incorporated. The hyperelastic model is derived from the nonlinear relation between the axial force P fib and the axial stretch λ fib = H H 0, where H is the wavelength of the deformed fibril. The solution of the boundary value problem at the micro-scale is used to formulate the 1. Piola-Kirchhoff axial stress of one single collagen fibril as an implicit function of the axial stretch P fib (λ fib ) = e 0 P fib (λ fib ) πr 2 0 cos(θ 0), (2)

& $ " Computational Modeling and Remodeling of Collagen Networks 5 ) N E = I J H A I I 2 B E > 2 = 2 B E > - N F A H E A J 0 = I A A J = 0 A E N @ A ) I O F J J A 2 B E >! " # ) N E = I J H A J? D B E > Fig. 3. A fit of the helical crimp model to data taken from uniaxial extension experiments on fascicles from rat tail tendons [25]. where πr 2 0 cos(θ 0 ) is the projection of the undeformed area A 0 of the fibril to the plane perpendicular to e 0. Note, that a closed form representation of the presented axial stress function (2) can only be provided by introducing further assumptions into the helical crimp model such as the inextensibility of the filament [8]. The helical crimp model contains three independent parameters: One material parameter, the elastic modulus of the filament E; two microstructural parameters, the crimp angle θ 0 and the ratio of the amplitude of the helix to the radius of the filament cross-section R 0 /r 0. The microstructural parameters θ 0 and R 0 /r 0 have a clear physical interpretation and can be measured experimentally, e.g. from scanning electron microscope photographs [24]. Note, that for model parameters E > 0, R 0 /r 0 > 1 and 0 < θ 0 < 90 the stress function of the helical crimp model (2) represents a strictly convex function including compressive states and the almost linear region of fully extended fibrils, which is important to ensure both material stability and numerical stability for algorithms used in the finite elements method. The ability of the proposed microstructural model to reproduce the biomechanical response of fibrous tissue is demonstrated for uniaxial extension tests of fascicles from rat tail tendons performed by Hansen et al. [25]. Rat tail tendons have highly organized parallel fascicles of collagen fibrils with negligible dispersion of fibril orientations. The proposed helical crimp model replicates the typical J-type shape of experimental stress-stretch data including the linear region of almost fully extended fibril (Fig. 3). The model parameters have been identified by means of the nonlinear Levenberg-Marquardt algorithm as E = 476 MPa, θ 0 = 13.88 and R 0 /r 0 = 3.05. The physical relevance of the proposed model can be supported by comparing the microstructural model parameters to directly measured values of the microstructure of rat tail tendons. Diamant et al. [26] measured experimentally crimp angles between 12.5 and 15.4 for undulated fibrils of tendons taken

6 Rafael Grytz and Günther Meschke B = B = " B = & B = $ B =! B = B = B = Fig. 4. Two-dimensional graphical representation of the distributed fibril orientations ρ famα e 0 of one fibril family. from 29 and 13 months old rats, respectively. The optimal crimp angle parameter of 13.88 predicted by the helical crimp model for the experimental data presented in Fig. 3 lies in the range of such directly measured values. Unfortunately, the physiological value of the second geometrical parameter R 0 /r 0 is 20-30 times large than the fitted one. Phenomena like fiber recruitment or other interactions between neighboring fibrils due to proteoglycan links might play an important role in understanding of the microstructural meaning of R 0 /r 0. 3 Mechanics of collagen fibril networks at the meso-level On the meso-scale, collagen fibrils form complex fibrous networks that introduce strong anisotropic and highly nonlinear attributes into the constitutive response of soft tissues. The mechanics of collagen networks is considered here through dispersed orientations e 0 of individual crimped collagen fibrils. As already discussed in the introduction the collagen network architecture varies significantly in different regions of the eye (Fig. 1). However, it is assumed that the overall material behavior of the collagen network in the cornea and sclera tissue can be approximated by means of two families of two-dimensional dispersed collagen fibrils. The orientations of individual collagen fibrils e 0 within each family (famα with α = 1, 2) is considered to be symmetrically dispersed by means of a normalized von Mises distribution function ρ famα in the plane spanned by the two vectors M famα 1 -M famα 2 of the orthonormal frame M famα j, where M famα 1 is the mean direction (Figure 4). Following the idea of Gasser et al. [16] a generalized structure tensor is introduced for each fibril family H famα = [(1 κ)m 1 M 1 + κm 2 M 2 ] famα (3) with a single dispersion parameter κ famα [0; 1/2] representing the twodimensional fibril dispersion in a integral sense κ famα = 1 π +π/2 ρ famα (φ) sin 2 (φ)dφ. (4) π/2

Computational Modeling and Remodeling of Collagen Networks 7 Let the strain energy density of eye tissues be composed of an isotropic part W mat and two anisotropic parts representing the energy contribution of the extrafibrillar matrix and of the two families of crimped collagen fibrils with dispersed orientations W = W mat + 2 W famα = c(i C 3) + α=1 2 α=1 IV famα C P fib (λ)dλ, (5) where I C = trc and IV famα C = H famα : C. Note, that in the present paper the incompressibility constraint III C = det C = 1 is enforced at the macroscale through elimination of displacement and strain variables as presented by Başar and Grytz [21]. Accordingly, no penalty term in W is needed to assure pure isochoric deformations. 1 4 Remodeling of the collagen fibril network To account for remodeling in the form of mesostructural rearrangement, individual collagen fibrils can rotate in response to the current mechanical stress environment of the macro-structure. The fundamental hypothesis of the proposed remodeling theory is that the orientation of individual collagen fibrils rotate such that after remodeling the collagen network can be again characterized by two generalized structural tensors of the form (3). Accordingly, the biomechanically induced remodeling process can be decomposed into the reorientation of the orthonormal frame M famα j and into the variation of the dispersion parameter κ famα of each collagen fibril family. The scalar function used for the definition of the stress based remodeling stimulus is postulated as { τ2 /τ Γ = 1 for τ 2 0 3 with τ = τ 0 for τ 2 < 0 i n i n i and τ 1 τ 2 τ 3. (6) Herein the spectral decomposition of the Kirchhoff stress tensor τ has been introduced, where τ i and n i are the corresponding eigenvalues and eigenvectors, respectively. The target directions M tarα j of the reorientation process of M famα j defined at the reference configuration are chosen such that at the current configuration all collagen fibrils tend to reorient into the n 1 -n 2 plane, while the mean fibril directions will be located between n 1 and n 2 [18] M tar1 M tar2 i=1 1 = F [cos(arctan Γ )n 1 + sin(arctan Γ )n 2 ] 1 = F [cos(arctan Γ )n 1 sin(arctan Γ )n 2 ] M tarα 2 = M tarα 3 M tarα 1, M tarα 3 = n 3 F/ n 3 F. In (7) the normalized pull-back operation M = F (m) = F 1 m F 1 m describes solely the change of orientation if applied to a vector m of unit length. The (7)

8 Rafael Grytz and Günther Meschke temporal evolution of the frames M famα j and of the dispersion parameters κ famα are expressed by first order rate equations Ṁ famα j = ω famα M famα j with ω famα = ω tarα κ famα = 1 t (κ tarα κ famα ) with κ tarα = Γ/2, κ t ω N tarα ω (8) where ω tarα = ω tarα N tarα ω is the Rodrigues rotation vector of the rotation tensor R tarα = M tarα j M j famα. In (8) t ω and t κ can be interpreted as time relaxation parameters of the reorientation process. Equation (6) implies that for uniaxial loading conditions the stimulus function tends to Γ 0 and that according to (8) the generalized structure tensors (3) tend to H famα F (n 1 ) F (n 1 ), which represents the reorientation of all individual fibrils of both fibril families to the pulled-back major loading direction F (n 1 ) at the reference configuration. In case of an equibiaxial loading condition (τ 1 = τ 2 ) the stimulus function becomes Γ 1 and the remodeling of the collagen network leads to a uniform distribution (κ famα = 0.5) of all fibrils within the F (n 1 ) F (n 2 ) plane, which is represented by H famα F (n β ) F (n β ). For the temporal discretization of (8) an explicit rotational update of the mean fibril directions is suggested here. The updated-rotation formulation is widely used in shell analysis to define finite rotations of the inextensible shell director [21, 27, 28] and has been recently applied to fiber reorientation formulations with only one predominant orientation [29, 30]. This approach is used for the exact rotation of the vector sets (M famα j interval t = t k+1 t k ) k into (M famα j ) k+1 within the time (M famα j ) k+1 = R famα (M famα j ) k. (9) The rotation tensor R famα of Rodrigues type can be given according to (8) as ( ) ( ( )) t R famα ωtarα N tarα t ωtarα N tarα = I + sin ω + 1 cos ω N tarα ω, where N tarα ω t ω is an abbreviation for N tarα ω t ω (10) ( ) = N tarα ω ( ). Finally, equations (9) together with the time discretization of the fibril dispersion parameters can be used to update the generalized structure tensor (3). Note, that sufficiently small time steps t have to be chosen to yield a stable iterative solution process. 5 Numerical example For the numerical analysis of the human eye a bi-linear finite shell element with a quadratic kinematic assumption in thickness direction is used [21]. The

E Computational Modeling and Remodeling of Collagen Networks 9 ) N E I > K > E ) F A N + H A = 0 + 0 + ) 0 5 4 + +, + 5 5 4 5 E > K I 5? A H = - G K = J H > K > E 0 5 - E! E 0 5 2 E E! + H A = C A A J H O 4 + % # $ 0 + ) # # 0 + $ #, + 5 # 5? A H = C A A J H O 4 5 # 0 5 & 0 5 - # # 0 5 2 2 I J A H E H F A Fig. 5. Geometry and finite element discretization of the human eyeball model. Table 1. Initial constitutive parameters of the human eye model. c [MPa] E [MPa] θ 0 [ ] R 0 /r 0 [-] κ famα [-] Sclera 0.01 18.71 5.09 1.04 0.5 Cornea 0.01 11.56 6.32 1.48 0 geometry of the human eye globe is approximated by means of two spherical shells representing the corneo-scleral shell (Fig. 5). The intersection line between the corneal shell and the scleral shell represents the limbus. The finite element discretization derived by means of the blending function method is presented in Fig. 5, where due to the symmetry of the model only one fourth of the eye globe is considered for the numerical analyses. The initial fibril architecture of the cornea is approximated by means of perfectly aligned fibrils in superior-inferior and nasal-temporal directions, while scleral fibrils are assumed to be randomly oriented parallel to the shell midsurface. The constitutive parameters of the corneo-scleral shell presented in Table 5 have been identified by means of the nonlinear Levenberg-Marquardt algorithm from inflation experiments performed by Woo et al. [31]. Both fibril families are assumed to have identical constitutive parameters. The proposed remodeling strategy is applied to the collagen network of the human eye model subjected to intraocular pressure. The scaled time steps are held fixed t/t = 0.01 throughout the analysis with the time relaxation parameters t = t ω = t κ, which account for the speed of adaptation. In the first 32 time steps the intraocular pressure is increased incrementally to the physiological value p IOP = 16 mmhg. Then, the pressure is held constant while the collagen network is allowed to remodel progressively at 0.32 < t/t 20 until biological equilibrium occurs. Due to a numerical thickness integration of the considered shell element formulation the proposed adaptation of the meso-

10 Rafael Grytz and Günther Meschke J J J J! J J J J J J # J J B = = # # #!! # " " # # Fig. 6. Evolution of the deformed configuration of the corneal and the adjacent scleral tissue, the dispersion parameters κ famα and the mean fibril orientations M famα 1 of both fibril families (deformations 10 exaggerated). structure represented by (8) is evaluated at the 2 2 2 Gaussian points of each element. Fig. 6 shows selected snapshots of the remodeling process of the corneal and adjacent scleral tissue. In Fig. 6 the mean fibril directions M famα 1 of both colla-

# 3 3 # 3 # 3 Computational Modeling and Remodeling of Collagen Networks 11 = > J J J J # # # J J J J # J J # J J Fig. 7. Evolution of the distributed fibril orientations of both corneal fibril families [ρ fam1 + ρ fam2 ]e 0 a) at the limbus and b) at the apex. J J J J gen fibril families (α = 1, 2) and the contour plot of the dispersion parameters κ famα are plotted onto the deformed configurations, where deformations are 10-fold magnified. Note, that due to identical initial values the dispersion parameters κ famα of family fam1 and fam2 coincide at any time. Throughout the whole remodeling process the mean fibril directions of the sclera and of the central cornea region remain almost orthogonal, while the dispersion parameters tend to the upper limit κ famα 0.5 of randomly oriented fibrils. Accordingly, this region of the eye shell is mainly subjected to an equibiaxial membrane state. The corneal fibrils near the limbus experience a substantial change in their morphology. Due to the different curvatures of the scleral sphere and the corneal sphere the tensile stress in circumferential direction is much higher than in meridional direction at the limbus. Consequently, the mean fibril directions of both corneal fibril families near the limbus evolve toward the circumferential direction, while the dispersion parameters evolve towards the low value κ famα 0.19. The different evolution histories of the mesostructural changes in the collagen fibril architecture at a Gaussian point near the limbus and at a Gaussian point near the apex are illustrated in Fig. 7, respectively. In Fig. 7a the development from two perfectly aligned orientations at t/t = 0 via the still visible overlapping densities of the two distributed orientations at t/t = 1 towards a concentrated distribution of fibrils in the circumferential direction Θ 1 at the biological equilibrium state (t/t = 20) is clearly visible. Conversely, due to the almost equibiaxial stress state the collagen fibrils reorient toward an almost isotropic distribution parallel to the shell surface at the apex (see Fig. 7b). The mesostructural changes in the collagen fibril architecture lead to significant changes in the deformation response of the macro-structure (see Fig. 6). The unrealistic changes of the corneal curvature at the beginning of the remodeling process (t/t = 0.32) are almost fully eliminated at the biological equilibrium state (t/t = 20).

& $ " "! " # "! % " 12 Rafael Grytz and Günther Meschke. E > H E H E A J = J E @ A I E J O F F F F F F F F - K A H E = = C A H = @ H N H = O E J A I E J O = J I K F A H E H E > K I H N H = O E J A I E J O = J E B A H E H E > K I H N H = O E J A I E J O = J = F A N. - @ A @ A I E J O = J E > K I J J. - @ A @ A I E J O = J = F A N J J Fig. 8. Corneal fibril orientation density ρ fam1 + ρ fam2 predicted by the FE-model at the biological equilibrium (t/t = 20) and normalized x-ray scattered intensity data from preferentially aligned collagen fibrils at the limbus and at the apex (from Aghamohammadzadeh et al. [14]). 6 Discussion A continuum approach of remodeling for the collagen network architecture of incompressible soft biological tissue has been proposed with a particular focus on eye tissue. Herein, remodeling is attributed to the reorientation of collagen fibrils in the three-dimensional space as a natural consequence of changes in the mechanical stress conditions. The biomechanical properties of eye tissue are derived from the single crimped fibril at the micro-scale via the collagen network of distributed fibrils at the meso-scale to the incompressible and anisotropic soft tissue at the macro-scale. On the fibril level, a physiologically motivated constitutive formulation has been presented based on the observation that fibrils crimp into the shape of a helix, when the tissue is unloaded. Only one material parameter (the elastic modulus of the fibril) and two microstructural parameters (the crimp angle θ 0 and the ratio R 0 /r 0 between the radii of the helical wave form and of the fibril cross section), which may directly be determined from the morphology of crimped collagen fibrils, are required. The proposed stress function is strictly convex, includes compressive states and allows for a smooth transition from a crimped to a fully elongated state of collagen fibrils. On the extracellular matrix level, the collagen network is represented by means of two families of fibrils with statistical distributed orientations. Following the idea of Gasser et al. [16] each family of fibrils is represented by means of a generalized structure tensor assuming a two-dimensional symmetrical distribution of fibril orientations. Only one mesostructural parameter per fibril family (the dispersion parameter κ famα ) and one orthonormal vector set characterize the collagen network architecture. The numerical findings presented here agree qualitatively with experimental observations. The concentration of the collagen fibril orientations towards the circumferential direction at the limbus predicted by the remodeling algorithm

Computational Modeling and Remodeling of Collagen Networks 13 can be interpreted as the development of a fibrillar annulus. From the synchrotron x-ray diffraction data the existence of a circumcorneal annulus of collagen fibrils at the limbus is well known [15]. The density of corneal fibril orientations of both families ρ fam1 (ϕ) + ρ fam2 (ϕ) predicted by the remodeling algorithm at the biological equilibrium (t/t = 20) is compared in Fig. 8 to x-ray scattered intensities ρ x ray from preferentially aligned collagen fibrils experimentally measured by Aghamohammadzadeh et al. [14] at the limbus and at the apex. For comparative reasons the experimental data have been normalized such that the relation holds 2π 0 (ρ fam1 + ρ fam2 )dϕ = 2π 0 ρ x ray dϕ = 4π. (11) Obviously, the predicted concentration of fibril orientations in circumferential direction ϕ = 1/2π, 3/2π at the limbus is in very good agreement with the experimental data. At the central part of cornea the remodeling algorithm predicts a uniform distribution of fibrils, but x-ray scattering data reveal the existence of two preferred orientations at ϕ = 1/2π, 3/2π and ϕ = 0, π, respectively. Daxer and Fratzl [12] noted that the two predominant orientations in the central cornea correspond to the directions of the insertion of the four musculi recti oculi. They assume that the preferentially oriented fibrils in the central cornea might appear as biomechanical elements of the complex opteokinetic system capable to withstand the mechanical forces of the musculi recti along the corneal trajectories. The forces introduced through the musculi recti onto the corneo-scleral shell have not been considered here. Furthermore, other non-mechanical factors might be of importance to assure for example the transparency of the cornea. The exact stimulus for the arrangement of collagen fibrils in the corneal stroma remains unknown. Although the constitutive behavior of individual fibrils and of the collagen network are micro- and mesostructural motivated the proposed remodeling algorithm remains phenomenological since cell level phenomena like mechanotransduction or biochemically coupled processes are not addressed. Nevertheless, the computational remodeling analysis of the human eye predicted that an annulus of collagen fibrils encircle the limbus in accordance to experimental observations and demonstrated the mechanical sensibility of the macro-structure to geometrical changes of the underlying collagen morphology at the meso-scale. Acknowledgments The authors would like to thank Prof. JB. Jonas (University of Heidelberg, Mannheim, Germany) for several helpful discussions on the physiology of human eye tissues.

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