Layered Elastomeric Fibrous Scaffolds: An In-Silico Study of the

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Layered elastomeric fibrous scaffolds: An in-silico study of the achievable range of mechanical behaviors James Carleton, G. J. Rodin, and Michael S Sacks ACS Biomater. Sci. Eng., Just Accepted Manuscript • DOI: 10.1021/acsbiomaterials.7b00308 • Publication Date (Web): 10 Aug 2017 Downloaded from http://pubs.acs.org on August 16, 2017

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ACS Biomaterials Science & Engineering

Layered Elastomeric Fibrous Scaffolds: An In-Silico Study of the Achievable Range of Mechanical Behaviors James B. Carleton,† Gregory J. Rodin,†,‡ and Michael S. Sacks∗,†,‡ Center for Cardiovascular Simulation, Institute for Computational Engineering and Sciences and the Department of Biomedical Engineering, The University of Texas at Austin, 201 East 24th Street, Austin TX 78712, and Department of Aerospace Engineering and Engineering Mechanics, 210 East 24th Street, Austin, TX 78712 E-mail: [email protected]

∗

To whom correspondence should be addressed Center for Cardiovascular Simulation, Institute for Computational Engineering and Sciences and the Department of Biomedical Engineering, The University of Texas at Austin, 201 East 24th Street, Austin TX 78712 ‡ Department of Aerospace Engineering and Engineering Mechanics, 210 East 24th Street, Austin, TX 78712 †

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ABSTRACT Our goal herein is to understand the mechanisms underlying soft tissue and scaffold behaviors by developing a physically-based micromechanical model as a means to connect the macroscopic behaviors to the underlying microstructural phenomena. Due to its well-documented capacity for generating elastomeric fibrous materials with a wide range of realizable architectures, the electrospun scaffold was used as the exemplar biomaterial. Fibrous network geometries based on a random walk algorithm were first generated to form the basis for subsequent micromechanical simulations. A basic understanding of randomly oriented fibrous network phenomena was then developed, and subsequently expanded on using networks with aligned fibers. Simulation results were then compared with experimental observations of electrospun scaffolds to evaluate the validity of the simulations. The effects of fiber alignment, tortuosity, and material properties on macroscopic mechanical behavior of the material have been presented both individually and in combination. We have seen that all three aspects of the scaffold network can have significant effects on the macroscopic behavior for different load cases. Overall, accurate representation of detailed fibrous network geometry permitted a greater understanding of the complex mechanisms underlying the macroscopic behavior unique to these biomaterials. Insights gained from such simulations can significantly aid the process of designing scaffold network geometries that result in engineered tissues that function as well as or better than the native tissues they are intended to replace.

Keywords scaffolds, micromechanics, simulations, tissue engineering

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INTRODUCTION The term "tissue engineering" was coined in 1988 by the great biomechanical engineering scientist Y.C. Fung. 1 In that early work, Fung underscored the importance of ...the application of principles and methods of engineering and life sciences toward a fundamental understanding of structure-function relationships in normal and pathologic mammalian tissues and the development of biological substitutes to restore, maintain, or improve tissue function. It is thus imperative that fundamental structure-function understanding guides the reproduction of native tissue if it is to emulate its native counterpart successfully. 2,3 Accurate simulations of the mechanical behavior of fibrous networks can be used to inform the design and manufacture of engineered tissues, especially with future fabrication technologies that enable control over important microstructural features. Moreover, while the current approach to manufacturing high quality scaffolds and functional engineered tissues remains somewhat of an art, current and future scaffold production would benefit enormously from the specification of target network geometries that produce desirable mechanical behavior. While the mechanical behavior of both hard and soft tissues can be quite complex, modeling of soft tissues is generally regarded as a more challenging task, due to major effects associated with large strains and significant microstructural changes. Traditionally, the mechanical behavior of soft tissues has been modeled using pseudo-hyperelastic phenomenological models. 4–8 But, like any natural or engineered biomaterial, the complex macroscopic mechanical behavior of soft tissues is a result of multi-scale interactions of the constituent phases. Those include collagen, elastin, nervous and muscular fibers, and, on finer scales, cells and extracellular matrix components such as glycosaminoglycans and proteoglycans. Thus relating the macroscopic mechanical behavior of soft tissues to their microstructure is an important step toward the development of a comprehensive multi-scale modeling framework.

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The fibrous microstructure of many natural soft tissues is often layered, especially for membrane structures (Figure 1-a). Native tissue structures can also include continuous graduations in fiber type, density, and organization within layers. In most native soft tissue models networks primarily act as very long and slender, mechanically non-interacting fibers. For example, we have recently developed a functional gradient long-fiber model (FGM) 9 that incorporated the transmural composition and fiber structure of the aortic valve (AV) leaflet. The FGM model predicted large stress variations both between and within the leaflet layers, suggesting that the continually varying structure of the AV leaflet has an important purpose with regard to valve function. We have also shown that explicit fiber-fiber interactions can be accounted for in related long-fiber models. 10,11 In attempting to mimic such structures, microstructures of engineered soft tissue scaffolds are often layered. As in native tissues, heterogeneity and anisotropy may significantly affect the macroscopic and microscopic mechanical behavior and functionality of synthetic biomaterials. However, in contrast to native tissues, the deformation of each fiber at the local scale is influenced by the positions and deformations of neighboring fibers through interconnections in synthetic biomaterials. 12 Thus understanding of relations between biological functionality and microstructure cannot be attained without characterizing relations between the microstructure and mechanical properties. Previously reported simulations have included three-dimensional assemblies of straight, unconnected fibers as the basis of a multiscale modeling approach in which the microscale geometry is regarded as a unit cell. 13 Volume averaging is used to relate the microscopic forces and displacements to the macroscopic continuum constitutive equation. The larger-scale structure is discretized with finite elements and the constitutive equations are evaluated at each integration point. The predicted macroscopic behavior agreed reasonably well with experimental measurements, but was unable to capture the effects of fiber straightening or interactions due to fiber-to-fiber intersections. Clearly, a more detailed understanding of these multiscale phenomena can only be gained using physically-realistic representations of the actual microstructures.

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Simulations based on directly capturing microstructural details have elucidated the relationship between scaffold fiber deformations and the aspect ratios of cells filling the pore spaces. 14,15 Those results also reveal that the highly loaded fibers in the network form long, straight effective fibers that are reminiscent of the structure of soft tissues. While these simulations provide an accurate geometric representation of actual scaffolds, the generated geometries are limited to specific scaffolds that have been imaged. Thus extending this approach to statistical studies and other microstructures, especially in 3-D, is somewhat problematic. Clearly, better geometric representations are needed for more accurate macroscopic predictions and for evaluation of quantities of interest related to the geometry of the evolving microstructure. Moreover, the ability to parametrically investigate the ranges of possible microstructures that elucidate what can be potentially achievable mechanically at the macro level remains an unmet goal in biomaterials design. It is clear that existing simulation approaches to fibrous networks lack robust geometric representations necessary for more reliable predictions of the macroscopic response and evolving microstructure. Such predictions would enable parametric studies of fibrous microstructures with the objective of identifying optimal microstructures that realize desired macroscopic properties. This mostly unmet objective is central to biomaterials design. Our goal herein is thus to understand the mechanisms underlying scaffold behaviors. For this purpose, physically-based structural models are preferable to phenomenological models since they offer a means to connect the macroscopic behaviors to the underlying microstructural phenomena. Structural models have indicated that the highly nonlinear and anisotropic macroscopic behavior of soft tissues can be explained by alignment and straightening of undulated fibers at the microscale, in addition to material nonlinear effects. 7,16–20 These predictions are based on the assumption of affine (homogeneous) fiber deformations, as supported by recent work on planar collagenous tissues. 21 Due to its widely-documented capacity of generating elastomeric, fibrous materials with a wide range of realizable, controlled architectures, 22–28 the electrospun scaffold was used as the exemplar biomaterial. Fibrous network

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METHODS Approach. In the present study, we first describe the geometric and mechanical models for generalized, planar fibrous networks. These models were based on a random walk algorithm (Figure 1-b) 29 and a mechanical model, which is a variant of Cosserat’s rod theory.Âă We then present simulation results aimed at quantifying the macroscopic response and evolving fibrous microstructure for both initially isotropic and anisotropic networks.Âă Simulation results were then compared with experimental observations of electrospun scaffolds to evaluate the validity of the simulations and are used to explore the effects of different microstructural features on macroscopic behavior. Network simulations were seen to be a useful tool for supplementing experiments when attempting to understand phenomena that are difficult to measure. Finally, we discuss various implications of our simulation results on characterization and design of biomaterials.

Source Material and Geometric Model. In this section, we describe a simulation methodology for quantifying the mechanical response of fibrous networks intended for modeling electrospun polymeric scaffolds, 30,31 whose structural and mechanical properties have been thoroughly characterized previously. 26,28,30,32 In previous studies, cytocompatible poly (ester urethane) urea (PEUU) was synthesized from polycaprolactone diol and 1,4-diisocyanatobutane with subsequent chain extension by putrescine. By syringe pump into a stainless-steel capillary suspended 13-cm vertically over a 4.5 cm diameter aluminum mandrel 5-wt% PEUU solution in hexaïňĆuoroisopropanol was fed at 1.0 mL/h. PEUU was charged with +12 kV and the aluminum target with -7 kV using high voltage generators. Aligned PEUU ïňĄbers were formed by electrospinning onto the target rotating at speeds ranging from 0.0 to 13.8 m/s. Structural and biaxial mechanical characterization were then performed. 26,28,30,32 7



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Constructing an accurate representation of the resulting scaffold network geometry is essential to performing high-fidelity simulations that provide insights into the microstructural mechanisms underlying complex macroscopic mechanical behavior. Electrospun scaffolds have been observed to be layered structures. 30,31 Therefore, they are modeled as 3-D fibrous networks formed by stacked layers. The fibrous geometry of each layer is idealized as a collection of random walks of the total length Lt confined to a square LB × LB box. 29 These random walks are characterized by the line density ξ :=

Lt A

and the probability density

function f (ϕ) for the fiber orientation angle ϕ, or simply the orientation density function (ODF). For the purposes of mechanical simulations, it is imperative to rely on periodic random walks. That is, if a walk intersects an edge of the confinement box, it is forced to reenter the box on the opposite side; for details we refer to our previous work. 29 This previously reported algorithm 29 allows one to match a target fiber density ξ t and a target ODF f t (ϕ), including those determined experimentally for the scaffolds. Since the algorithm mimics the actual electro-spinning process, it is capable of capturing many geometric and topological features of the scaffold (Figure 1-b). In addition to the prescribed targets, ξ t and f t (ϕ), the algorithm requires the following input parameters 1. nf - number of random walks per layer. 2. ns - number of segments per random walk. 3. sˆ - segment length normalized by ξ −1 . ˆ B - box edge length normalized by ξ −1 . 4. L 5. α - the maximum orientation change between two sequential segments. These parameters are constrained by nf ns sˆ = 1, ˆ 2B L

8


(1)

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so that only three of these parameters must be specified. Small values of α result in nearly straight fibers and large α values result in highly tortuous fibers. This geometric feature may be quantified by defining the tortuosity of a fiber chain, which consists of all random walk segments between two adjacent intersections, as Lc τ := ¯ , Lc

(2)

¯ c is the length of the chain’s end-to-end vector. where Lc is the arc length of the chain, and L Although the average tortuosity of the network also depends on ξ, the parameter α allows for tortuosity control. Random walks become fibers once they are assigned a diameter D. In this assignment, we neglect out-of-plane O(D) deflections of the fibers occurring at fiber intersections. Thus each layer is assumed to be planar and of thickness D. With this provision, the 3-D network is naturally formed by stacking the planar layers. The stacking results in forming additional, inter-layer, fiber intersections. The fiber volume fraction φ for the individual layers and the entire 3-D network is φ :=

π ξD. 4

(3)

Mechanical Model and Numerical Implemention. Following the standard practice in continuum mechanics, we model fibers as rods. That is, we introduce simplifications allowing us to describe the fiber deformation as a system of equations defined on the fiber centerline. There is a long history of rod theories going back to Bernoulli and Euler. Modern continuum rod theories, accounting for large strains and rotations, and nonlinear constitutive equations, are classified as either asymptotic or direct approaches. 33 In asymptotic approaches, the rod equilibrium and constitutive equations are derived from the underlying 3-D continuum theory, by exploiting the rod slenderness. In direct approaches, the governing equations are formulated without any reference to 3-D

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theories. The result of both approaches is a set of governing equations expressed in terms of the position of a point on the centerline and the shape and orientation of the cross section. Theories using a set of vectors to describe the orientation of the cross section are classified as Cosserat rod theories, and the vectors are referred to as the directors. Rod theories of interest to this work result in nonlinear governing equations. Consequently numerical implementations of rod theories are nontrivial, and usually involve tradeoffs among accuracy, efficiency, and robustness. For example, the ABAQUS Euler-Bernoulli beam element uses an asymptotic approach, which includes large 3-D rotations and strains. 34 While in theory this approach is ideal for our simulations, ABAQUS is not an ideal tool for our purposes. A serial implementation may be adequate for purely 2-D fiber networks 35 or 3-D networks with straight fiber geometry between intersections, 36 but this limitation is exceedingly restrictive for large-scale simulations. For single layer simulations considered here, problems with hundreds of thousands of degrees of freedom are common, partly because multiple fiber segments are required to capture the curved fiber geometry. 29 We also needed to perform multilayer simulations 37 to assess the sufficiency of single layer simulations. Multilayer simulations of complete heart valve leaflets could easily require hundreds of millions of degrees of freedom. For a problem with 2.5 million degrees of freedom, the peak parallel speedup (serial time over parallel time) for our code was 6.35 on 64 processors (Figure 2).

Figure 2: Parallel speedup plots for problems with 300,000, 600,000, and 2,500,000 degrees of freedom (dof).

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Our assumptions and approach closely follow those adopted for the ABAQUS EulerBernoulli beam element, but there are some differences, which we highlight here. A complete description of our approach is provided in our previous work. 37 In our approach, the cross section is defined by a position vector and two orthonormal director vectors. We assume that the rod deforms such that plane sections remain plane, no deformations occur within the cross section, and the cross sections remain normal to the centerline. In contrast to the ABAQUS approach, in which the primary kinematical variables are the displacement and rotation vectors at each node, we relied on the position and director vectors the primary kinematical variables. Our approach is computationally less efficient, as it involves more equations and unknowns, but the governing equations are much simpler. We also differ from ABAQUS in the way we enforce the kinematic constraint that the cross section remains normal to the centerline. In the ABAQUS Euler-Bernoulli beam element, this constraint is enforced by direct substitution into the virtual work statement. As a result of this substitution, second derivatives of the position vector with respect to arc length appear in the governing equations. In our approach, we delay applying this constraint and apply it only after performing the finite element discretization, as has been done by others. 38 One advantage of this approach is that only first derivatives of the position vector appear in the governing equations. Another consequence of this choice is that extra terms due to shear deformation appear in our version, which do not appear in the virtual work statement of the ABAQUS Euler-Bernoulli beam element. Terms due to shear deformation do appear in the virtual work statement of the ABAQUS Timoshenko beam element, but we seek to model an Euler-Bernoulli beam, and the applied constraints remove the virtual strain energy due to these terms. One other minor difference is that ABAQUS uses the Green Lagrange strain in the virtual work term due to axial deformation, whereas we choose to use the stretch instead. As in our previous study, 14 an incompressible Yeoh material 39 is used for modeling the axial response, whereas a linear elastic material model used for fiber bending, torsion, and shear. There are many ways of modelling fiber intersections. We simply assumed that at all

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details are provided in our previous work. 37 The finite element method with linear shape functions and a reduced quadrature scheme was used to reformulate the problem as a system of nonlinear algebraic equations, which were solved numerically using Newton’s method. The fiber intersection constraints and periodic boundary conditions were enforced using translational and rotational penalties. The parallel finite element code was developed using PETCSc. 40–42

Model Calibration. In order to make useful observations about material mechanical behavior, one must conduct simulations on sufficiently large networks, which accurately represent the material response. Such networks are referred to as Representative Volume Elements (RVEs). For 3-D networks, ˆ B and (ii) the two sizes are important: (i) the minimum normalized in-plane box size L minimum number of layers. The process of identifying the RVEs is purely empirical. The box size is increased (Figure 3-b) until the error in a quantity of interest falls below an acceptable magnitude. Once the quantity of interest and the corresponding error tolerance have been specified, parametric studies are conducted to determine the minimum sizes. Here we summarize key findings of those studies; for details we refer to our previous work. 37 • The quantity of interest chosen was the strain energy density induced by the macroscopic deformation gradient  

F= 

1.50

0

0

1.25



 . 

(4)

• The relative error tolerance was set at 3%. That is the relative difference in the strain energy density between two RVEs could not exceed 3%. ˆ B ≥ 100. • For single-layered RVEs L

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• For in-plane loadings considered in this paper, the difference between multi-layer and single-layer networks was minimal, and therefore it is sufficient to consider single-layer RVEs. The last point is somewhat surprising and deserves an explanation. To this end, consider a single layer characterized by Ni fiber intersections. Let us embed this layer into a 3-D multi-layer network. Then the number of fiber intersection for the layer increases roughly by a factor of three, as fibers within the layer can now have intersections with fibers of the neighboring layers. Thus a characteristic spacing between fiber intersections in the isolated layer is roughly three times longer than that in the embedded layer. This spacing reduction significantly increases the bending stiffness but not the axial stiffness of the individual fiber segments connecting neighboring intersections. The ratio of intersection spacing to fiber diameter determines whether the network strain energy will be dominated by stretch or bending deformations in the fibers. 43,44 As a matter of fact, one can argue that the axial stiffness of the isolated and embedded layers must be roughly the same because the embedment does not change the total length of the fibers within the layer. Thus the only scenario under which the overall stiffnesses of the isolated and embedded layers are equal is when the strain energy due to bending is negligible in comparison to the strain energy due to axial deformation. Indeed, this statement held true not only for simulations used for calibrating the model but in entire set of simulations conducted in this study.

RESULTS Macroscopic Material Behavior In this subsection, we present simulation results showing the macroscopic mechanical response of initially isotropic networks. All results were obtained using the data compiled in Table (1). We present results using the first Piola-Kirchhoff macroscopic stress versus the macroscopic stretch curves obtained for several stretch-controlled and stress-controlled 14


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kinematics, including straightening and rotation effects, is most easily studied using data for strain-controlled loadings. The nearly linear nature of the stress-stress curves for these loadings suggests that fibers that are initially nearly aligned with the applied load, and thus undergo small rotations, quickly rotate and form a dominant load bearing subnetwork. Other fibers that are not aligned with the applied load rotate more gradually and carry very little load. The nonlinearity of the stress-strain curves under stress-controlled loading may be understood as a non-proportional traversal of the stretch space. This observation is an agreement with tissue 45 and needled non-woven scaffold models, 46 in which any significant network stretching under uniaxial stress-controlled conditions is preceded by lateral contraction of the network. This phenomenon can be explained by the rotation of fibers toward the applied load direction, as described in the next section. In the 3:1 and 4:1 stress controlled loading cases, the initial negative slope of the y-direction stress vs. stretch curves can be viewed as a typical Poisson’s ratio effect observed in a uniaxial tension test, in which axial extensions correspond to lateral contractions. As more load is applied and more fibers rotate toward the x direction, the effective y-direction stiffness decreases, and the network becomes less capable of resisting the load in that direction. Consequently, the stress vs. stretch curve slope becomes positive and the network begins to expand in the y direction. This and other effects associated with microstructural changes are best described in terms of the evolving ODF, as it is done in the next subsection.

Evolution of Microscopic Quantities of Interest. To elucidate some of the more subtle aspects of the macroscopic mechanical material behavior, we examined quantities of interest pertaining to the underlying microstructure. Just as we could not load the network in a state of pure uniaxial stress due to fiber instabilities, pure shear strain loading proved equally difficult to achieve. Instead, we loaded the network in a state of pure shear strain with some equibiaxial strain superimposed. Starting with the

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fiber stretch gives a sense of evolving network mechanics, and shows how the applied load is distributed among the fibers. The average stretch is clearly maximum along the principal loading direction and minimum along the normal direction (Figure 6). Also, the simulation results are in very close agreement with the affine predictedions. The chain tortuosity in the unloaded configuration is roughly uniform for all orientation angles (Figure 6). In the loaded state, the mean tortuosity of chains along the principal load direction is almost exactly equal to one, as these chains become almost completely straight. Along the normal direction, the tortuosity is large, as these chains compress and just begin to buckle.

Comparison with mechanical responses. To compare the simulation and experimental results for actual scaffold configurations, fiber networks were generated to match the properties of electrospun Poly (ester urethane) urea (ES-PEUU) scaffolds that have been previously tested 30 and analyzed. 14,32 Importantly, experimental trends in the relationships between fiber material properties, network geometry, and macroscopic mechanical behavior are comparable with simulation results. This suggests that we have captured the key aspects of the scaffold mechanics. Three scaffolds with different degrees of fiber alignment were studied. In the electrospinning process, greater mandrel speeds produce more highly aligned scaffolds. 28 The fiber diameter D and scaffold fiber volume fraction φ are also affected by mandrel speed, so scaffolds with different degrees of alignment have different fiber diameters and volume fractions. 30 The experimentally measured 30 orientation density functions for the three scaffolds show the trend of increasing geometric anisotropy with increasing mandrel speed (Figure 7a). The scaffold stress vs. stretch plots reported previously 14,30 are for equibiaxial stress loading. We see that mechanical anisotropy and nonlinearity tend to increase with increasing fiber alignment (Figure 7-b). Single-layer networks (Table 2) were generated with target geometric configurations matching the experimental data. As explained in Section , a single layer is sufficient to represent the 3-D scaffold. 19




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behavior. This effect was explored by generating networks with both isotropic and aligned fiber ODFs (Table 3 and Figure 9-a,b). The alignment direction is π2 , so the ODF can be characterized by computing the dispersion about π2 , which is defined as 47 Z π π π D( ) = [1 − cos(ϕ − )]f (ϕ) dϕ 2 2 0

For a purely isotropic network (f (ϕ) = π1 ), D( π2 ) = 1 − aligned to

π , 2

D( π2 ) = 0. The dispersions about

π 2

2 π

(6)

≈ 0.363. For a network perfectly

of the generated isotropic and aligned

networks are 0.367 and 0.203 respectively. Table 3: Target geometry, random walk parameters, and fiber material properties for fiber alignment simulations D(µm) 0.33

φt 0.18

ˆB L 100

sˆ 2.0

nf 100

α π 10

Ef (MPa) 7.5

The macroscopic stress versus stretch curves for both equibiaxial stress and 3:1 biaxial stress loading (Figure 9-c,d) show the significant differences between the isotropic and aligned networks. For the aligned network, there are more fibers along the vertical (y) direction, so under equibiaxial stress, the slope of the stress vs. stretch curve is greater for this direction. Fibers rotate toward the horizontal (x) direction, and the stiffening effect along this direction allows the network to traverse a non-proportional load path in strain space. The resulting stress vs. stretch curve is nonlinear. These effects are even more pronounced for nonequibiaxial load case. The aligned network geometry causes the stress versus stretch curves to become significantly nonlinear, and the contraction in the y direction becomes significant. Another key finding was that fiber tortuosity has a moderate effect on the nonlinearity of the macroscopic stress vs. stretch curves for strain controlled loading. This effect was explored by generating isotropic networks with three different levels of tortuosity, which was controlled by varying the maximum random walk angle (Table 4 and Figure 10-a). The 22


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Another important finding was that fiber material properties have a direct effect on the resulting network properties, so large differences between fiber materials translate into large differences between the identical networks. Two fiber material models were compared: a linear axial force versus displacement model and an incompressible Yeoh model. 39 The Yeoh model constants were taken from extant AFM measurements (c1 = 1.25M P a, c2 = 4.0M P a, c3 = 0.0M P a). 14,32 For the linear model, Young’s modulus was chosen to be equal to the initial slope of the Yeoh model force versus stretch curve. It is clear that that curves are quite different (Figure 11-a) in the interval of typical fiber stretches. The resulting network stress vs. stretch curves under equibiaxial strain loading (Figure 11-b) look similar to the fiber axial force versus stretch curves.

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network of extracellular matrix (collagen and elastin) fibers. These soft tissue networks can have a wide variety of geometric structures, ranging from highly isotropic to highly aligned (Figure 1-a) along the circumferential direction in heart valve layers 9,52? or helical in chordae tendinae 53 and arteries. 54 In this work, we introduced a method for modeling materials that have a layered, fibrous network microstructure. A primary application is in the field of tissue engineering, where high fidelity models of electrospun scaffolds are needed to better understand how the network geometry affects the mechanical and biological function of the tissues that are grown on the scaffolds. Scaffold geometry has a strong influence on the tissue’s macroscopic mechanical behavior, cell proliferation and attachment, nutrient and waste flows, and extracellular matrix (ECM) generation. This work was focused on characterizing scaffold network geometry and elucidating the impact of geometry on macroscale mechanics.The effects of fiber alignment, tortuosity, and material properties on macroscopic mechanical behavior of the material have been presented both individually and in combination. We have seen that all three aspects of the scaffold network can have significant effects on the bulk behavior for different load cases. Understanding these effects is essential to the process of designing electrospun scaffolds that have optimal material properties for functional engineered tissues. The ultimate goal is to design better engineered tissues that function mechanically and biologically as well as or better than the native tissue. The random walk algorithm 37 used to generate the scaffold geometry mimics the electrospinning process. The 3D, layered geometry that it generates is a reasonable representation of the actual scaffold microstructure. While the generated geometry does not accurately capture fiber undulations through the thickness or interlayer penetrations, it does capture interlayer intersections as well as many in-plane geometric features of each layer, such as fiber curvature, orientation distribution, and intersections, and the fidelity of this geometric representation represents a significant improvement over current models. The network fibers are modeled as Cosserat rods, in which director vectors are used to keep track of the

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orientation of the fiber cross section. This approach allows large, 3D rotations and axial stretches to be handled accurately. Proper treatment of these deformations is important since the large macroscopic strains experienced by tissues during normal function result in large rotations and stretches of the network fibers at the microscale. The individual fibers are connected using constraints at the points of intersection and at the periodic boundary. Periodic boundary conditions are applied to complete the specification of the network boundary value problem. The finite element method is used to reformulate boundary value problem as a system of nonlinear algebraic equations that is solved using Newton’s method. These equations form the basis of the nonlinear, parallel finite element code that was used to perform the mechanical simulations. The ability to run the simulations in parallel allowed problems involving sufficiently large networks with sufficiently intricate representations of fibrous geometry details to be solved in a reasonable time period. A more complete understanding of the mechanisms underlying the macroscopic behavior of scaffolds can be gained using micromechanical simulations than can be gained from analytical models alone. Insight into the relationship between microstructural evolution and mechanical behavior was gained by monitoring microscopic quantities of interest (QOIs). We first developed an understanding of basic fibrous network phenomena for initially isotropic network geometries. The effects of periodic box size, number of samples, and number of layers on the uncertainty in geometric and mechanical QOIs was quantified. Predictions were found to be reasonably accurate for moderate box and sample sizes. For the in-plane loadings considered in this study, multilayer scaffolds were found to be accurately modeled by a single network layer for the purpose of capturing the macroscopic stress vs. stretch response, despite the fact that the monolayer contains one third the number of fiber-to-fiber contacts as the real scaffold. This finding suggests that the fiber interactions are secondary to the effects of fiber volume fraction and ODF. This conclusion is also supported by the finding that the bending energy contribution to the total strain energy in the network is negligible.

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Observations of the loaded network geometry and analyses all support the conclusions that affine fiber kinematics can be used to predict both the orientations and stretches of the fibers. Most of the networks of interest to the field of tissue engineering have aligned fibers in order to duplicate the properties of native tissues, and simulations of aligned fibrous networks based on realistic geometries from tissue engineering applications are used to build on the isotropic network results. Thus, all network geometries and material properties presented here are designed to be good representations of typical electrospun Poly (ester urethane) urea (ES-PEUU) scaffolds, and the investigations presented here are directed toward the ultimate goal of designing scaffolds having optimal properties for functional tissue engineering. Despite the inability of analytical models to capture the details of local deformations, it was found that the assumption of affine kinematics is a reasonable description of average fiber deformations and can be used to predict the evolution of the fiber ODF and average stretch at each orientation with great accuracy. Fibers always rotated toward the direction of the principal eigenvector of the left stretch tensor and always straighten and bear the most load along this direction. We underscore that the transition to affinity as observed in native tissues 21 is a result of the transition to long fiber-like behavior observed in the present simulations. We believe this is a key result, as it will help with the applications to developing simulated native tissue like behaviors. The effects of fiber alignment, tortuosity, and material model, both individually and in combination, on the macroscopic stretch vs. stretch behavior were explored. For moderately tortuous networks of linear fibers, increasing fiber alignment significantly increases mechanical anisotropy, and significantly increases nonlinearity under stress-controlled loading. For isotropic networks of linear fibers, increasing tortuosity moderately increases nonlinearity at small strains. A linear fiber was compared with a fiber modeled as an incompressible Yeoh hyperelastic material. Large differences in the nonlinearity of fiber response translate into large differences in the network response. These simulations allow the effects of geometry and fiber material properties on macroscale behavior to be studied systematically in a way

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that is virtually impossible using current experimental techniques. The effects of both high alignment and high tortuosity were then investigated together by constructing an aligned network with high tortuosity (Figure 12-a). For aligned networks under equibiaxial stress using a linear fiber material model, the effect of tortuosity is significant. Using a Yeoh material model, subjected to equibiaxial stress, we observed that increasing tortuosity significantly increases material nonlinearity, and the added degree of nonlinearity due to tortuosity is comparable to that of using the Yeoh model. The effects of fiber alignment, tortuosity, and material properties on macroscopic mechanical behavior of the material were presented both individually and in combination. We saw that all three aspects of the scaffold network can have significant effects on the bulk behavior for different load cases. Understanding these effects is essential to the process of designing electrospun scaffolds that have optimal material properties for functional engineered tissues.

Implications and broader considerations. Many of the reasons for poor implant performance or failure of prosthetic biomaterials and engineered tissues remain ill-defined, often being a combination of inadequate or mismatched mechanical properties and biological complexities. Despite the multitude of challenges, many early engineered tissue approaches have shown promising results. For example, heart valve prostheses derived from bovine pericardium or porcine aortic valves have long been used to enhance survival and improve the quality of life of patients presenting with a variety of valvular diseases. 55 Similarly, engineered dermal grafts have successfully been used clinically to treat severe burns or wounds that would otherwise be unable to close and heal properly. 56 Decellularized extracellular matrix scaffolds have also shown successes in regenerating organized tissue after severe musculoskeletal tissue loss or injury. 57 In addition to providing invaluable educational experience to guide future efforts, this incremental progress moves the field ever closer towards the ultimate goal of developing technologies for safer and more efficacious tissue repairs and replacements. 26 31



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Healthy native tissues undergo intricate, multi-scale modes of deformation which work synergistically with biochemical stimuli to drive physiological responses. In order to mimic native tissue structure and organization, scaffolds characteristic lengths on a scale comparable to those observed in nature need to be produced and controllable techniques to achieve those are necessary. Electrospun scaffolds have been employed extensively in tissue engineering as they are able to replicate some of the multi-scale properties that are important characteristic features of native tissues (e.g. among many others, annulus fibrosus, 58 native cartilage, 59 and anistropic cardiac tissues. 60 Electrospun constructs are amenable to modification during and after fabrication to introduce functionality or modify microstructure and mechanical response. 27 Functional groups of peptides can be introduced through surface treatments after fabrication or grafted to the polymer chains prior to solvent processing. Fibers can be patterned or aligned to encourage contact guidance of seeded cells and to produce tunable tensile properties and structural and mechanical anisotropy. However, while electrospinning permits fabrication of biodegradable matrices that resemble the scale, architecture, and mechanical behavior of the native ECM, 30 this morphology also results in pore sizes that are generally smaller (< 5µm) and more irregular than those produced by other non-fibrous production methods. Thus, achieving high cellular density and infiltration remains challenging and several approaches have been proposed to overcome this limitation such as the alteration of manufacturing parameters (e.g. fiber diameter and packing), inclusion of native ECM proteins, labile porogens, or sacrifial fiber populations, and concurrent electrospining of scaffold fibers and electrospraying of cell populations suspended in culture media. 61,62 Fiber diameter and alignment are able to modulate cellular morphology, i.e. projected cell area, aspect ratio, and long axis length. 22 Sub-micron fiber diameters induce diminished cell adhesion and spreading, possibly due ineffective focal adhesions (which can be larger than 1µm). In contrast, increased fiber alignment promoted increased cell spreading. Adhesion and spreading of anchorage-dependent cells is a prerequisite for cell viability and

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proliferation 63 as cells perceive and react to their mechanical environment through adhesion and cellular deformations induced by and into the surrounding mechanical environment. These complex interactions likely play a critical role in the mechanotransduction of proteins required for cell viability and proliferation, and have direct implications towards the development of engineered tissues. 15 Extracellular matrix micro-structure, mechanical properties and cell-ECM interactions play an important role in diverse biological processes, e.g. pathological remodeling of cancer development, 64 or in the vasculature, in atherogenesis and atherosclerosis. 65 Substrate stiffness is well-known to play a substantial role in directing cell differentiation and cell general behavior. 66 Strains imposed to the cells at the cellular and tissue scales (either in vivo or in vitro engineered tissue systems) are propagated into the cell acting through the cytoskeleton and lead to altered stress levels on the nucleus, induce nuclear shape change, and ultimately affect cell function and causing modifications in gene expression and protein synthesis. 67–69 Experimental observations of tissues grown in bioreactors have shown that tissue composition, morphology, and mechanical properties can all be altered by controlling static and dynamic compression and hydrodynamic conditions. 70 Macro-scale coupled multi-physics simulations of tissues grown in the bioreactor environment have also confirmed these effects on tissue growth, as well as the effects of bending loading and nutrient and oxygen concentrations in the fluid phase. 71 The micro-scale scaffold simulations presented herein could in principal be extended to include the coupled mechanical, chemical, and biological phenomena present in bioreactors or even in vivo. The geometry, stiffness, and deformations of the scaffold fibers all provide mechanical cues to the attached cells. These cues can be used to control stem cell differentiation and matrix biosynthesis, 26,72 but the micro-scale mechanisms are still poorly understood. The role of trans-membrane proteins in the adhesion of cells to matrix and scaffold fibers, the transmission of stress to cell nuclei through the cytoskeleton, the chemical, structural, hydrodynamic effects on protein syntheses, and the migration of cells through the scaffold are all phenomena that can be studied in detail using micro-scale

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models. These models could be combined with macro-scale models of tissue function to create complete multiscale, multiphysics models of the complex phenomena that occur during engineering tissue growth. Alternatively, it would be possible to perform direct numerical simulations of features that extend to the full tissue length scale. Networks containing many layers and reaching the thickness of a heart valve leaflet could be loaded in flexure, torsion, or out of plane shear. The importance of scaffold-cell mechanical coupling becomes readily apparent in the rational design of engineered tissues. Moreover, the unique micromechanics of various scaffolds induce different cell deformation responses, which could correlate to substantial changes in cell proliferation and function. To summarize, the understanding of how microstructure influences cell behavior is important for the following 1. Defining fabrication variables and microstructural features to obtain better performing tissue surrogates that are able to recapitulate native tissues behavior at multiple lengthscales (cellular, tissue and organ), not only structurally and mechanically (in particular, their anisotropy), but also biochemically and biologically. 2. Establishing two-way coupling strategies and multi-scale models to study not only the effects of cells and ECM surrounding the scaffold in the micro- and macro-mechanics of the engineered tissue composite, but also the propagation of macro-scale mechanical stimuli to the micro-scale environment to trigger biological responses that distinguish implant evolution either into healthy integration or pathological rejection/failure. 3. Developing better design tools and methodologies to predict time-evolving scaffold properties and structure of scaffolds (which are desirably biodegradable), not only to tailor the scaffold with sufficient properties to function acutely, but also stress-shielding and smooth transition of load from the degrading scaffold to the developing de novo tissues.

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Future work. Overall, the simulations presented here represent an important advancement in the field of tissue engineering. The accurate representation of detailed fibrous network geometry permits a greater understanding of the complex mechanisms underlying the macroscopic behavior unique to these biomaterials. Furthermore, simulations of scaffold fiber networks form the basis for understanding how interactions with cellular and ECM phases contribute to the growth, remodeling, and ultimate mechanical and biological behavior of the entire engineered tissue. Insights gained from such simulations can significantly aid the process of designing scaffold network geometries that result in engineered tissues that function as well as or better than the native tissues they are intended to replace. Looking forward, the current work may be expanded and enhanced in many ways. The Cosserat rod theory used in this work to model the individual fibers includes the effects of large rotations and large stretches of the fibers, as well as the effects of bending and torsion. 29 However, scaffold networks of practical interest have been found to be axial dominated, with negligible contributions from bending and torsion to the total strain energy density. While fiber bending stiffness is needed to ensure a nonsingular stiffness matrix, it is possible that formulations based simplified rod theories may be used without affecting the accuracy of the quantities of interest. For example, for the terms associated with bending in the expression for the virtual strain energy density, the assumption of small strains could be invoked while still accounting for large fiber rotations and stretches. Since the equations in the current formulation are quite complex, the use of simplified rod theories that still capture the important phenomena would be justified. The basic parameters, such as volume fraction and ODF, that characterize the geometry could be viewed as state variables that evolve as the scaffold network is loaded and the fibers deform. Simulations could be used to help develop equations that describe the evolution of these variables. Many phenomenological 73 and structural 7,16,18 constitutive models of soft tissues have been developed to describe and explain the complex behavior of soft tissues. 35



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The Mesoscale Structural Constitutive Model (MSSCM) 16,19 is known to work well for soft tissues partly because it is a good representation of the microstructure and functioning of these materials . Although electrospun scaffolds are physically very different from soft tissues, the MSSCM has been shown to predict the bulk mechanical behavior of these materials with good fidelity. 30 Simulations could be used to explore the correspondence between soft tissue models and fibrous networks in an attempt to help explain why soft tissue models work as well as they do in predicting the macroscopic mechanical behavior of electrospun scaffolds. In order to better explain how the MSSCM model predictions relate to the simulation results, it would be helpful to consider the correspondence between the parameters of the MSSCM and the parameters that describe the real network geometry. Making these connections would also be helpful to the process of scaffold design. Given MSSCM model parameters that result in a known, desired macroscale behavior, it would be useful to relate those parameters to a corresponding network fiber geometry and material that has the same desired behavior. This network description would provide a target geometry and material for scaffold manufacturers. Another logical step would be to include cellular and ECM phases in the model. Tissue growth and remodeling is known to be strongly influenced by applied stress fields. Ideal engineered heart valve tissues have a dense, organized ECM phase made of aligned collagen and elastin fibers. Replicating the cyclic loading conditions found in vivo may be important in producing these ideal structures. Experimental observations of tissues grown in bioreactors have shown that tissue composition, morphology, and mechanical properties can all be altered by controlling static and dynamic compression and hydrodynamic conditions. 74 Macroscale coupled multiphysics simulations of tissues grown in the bioreactor environment have also confirmed these effects on tissue growth, as well as the effects of bending loading and nutrient and oxygen concentrations in the fluid phase. 71 The microscale scaffold simulations presented herein could in principal be extended to include the coupled mechanical, chemical, and biological phenomena present in bioreactors or even in vivo. The geometry, stiffness, and deformations of the scaffold fibers all provide

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mechanical cues to the attached cells. These cues can be used to control stem cell differentiation and matrix biosynthesis, 62,72 but the microscale mechanisms are still poorly understood. The role of transmembrane proteins in the adhesion of cells to matrix and scaffold fibers, the transmission of stress to cell nuclei through the cytoskeleton, the chemical, structural, hydrodynamic effects on protein syntheses, and the migration of cells through the scaffold are all phenomena that can be studied in detail using microscale simulations. These simulations could be combined with macroscale simulations that treat the tissue as a continuum to create complete multiscale, multiphysics models of the complex phenomena that occur during tissue growth. Alternatively, it would be possible to perform direct numerical simulations of features that extend to the full tissue length scale. Networks containing many layers and reaching the thickness of a heart valve leaflet could be loaded in flexure, torsion, or out of plane shear. The methods introduced may be applied to materials other than electrospun scaffolds. Hydrogel and needled nonwoven tissue scaffolds also have a fibrous network microstructure, and this work could be even more broadly applicable to other materials, such as the extracellular matrix of soft tissues, the actin cytoskeleton, some engineering composite materials, and many textiles. Finally, formal methods could be employed to solve the inverse problem of finding the optimal microstructure that produces materials with desired properties. As biomaterial manufacturing technologies improve, it may one day be possible to produce tissues repeatably that are superior to those produced by nature. Network mechanics simulations will undoubtedly play an important role in understanding and designing the intricate microstructural details of such advanced materials.

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Acknowledgement The authors gratefully acknowledge NIH grant HL068816, and Dr. Joao S. Soares for careful review of the manuscript.

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