affil0affil0affiliationtext: Department of Biomedical Engineering,
Indian Institute of Technology Hyderabad, Kandi Sangareddy, Telangana, 502284,
India.

#  _In-silico_ modeling of the micromechanics of fibrous scaffolds and
stiffness sensing by cells

Dhruba Jyoti Mech  Mohd Suhail Rizvi

Corresponding author
Mohd Suhail Rizvi

Address
Office: C-214/C, IIT Hyderabad,

Kandi, Sangareddy, Telangana 502284, India.

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Abstract

> Mechanical properties of the tissue engineering scaffolds are known to play
> a crucial role in tissue regeneration. Here, we have utilized discrete
> network and finite element models to study fibrous scaffold mechanics and
> its dependence on structure. We have considered two loading conditions,
> first, uniaxial elongation (macroscopic), and second, localized cellular
> forces (microscopic). Using these two scenarios, we have tried to establish
> a link between scaffold micromechanics and its macroscopic mechanical
> properties. We have demonstrated that the macroscopic elastic modulus of
> fibrous scaffold is dependent on the sample shape, size and the degree of
> fiber fusion. Under microscopic loading conditions the deformation of the
> fibrous scaffolds is anisotropic with an orientation dependent decay with
> distance. Further, we explored the stiffness sensing under the conditions of
> fixed stress or fixed strain application by the cells. With fixed strain, we
> found that the stiffness sensed by a cell is proportional to scaffold’s
> macroscopic Young’s modulus. However, for fixed stress application, it also
> depends on cell’s elasticity with stiffness experienced by stiff and soft
> cells differing by an order of magnitude.
>
> The analyses demonstrate that there exists a gap between mechanical
> properties of the fibrous scaffolds as measured from the macroscopic
> testings and those sensed by the cells. This warrants a need for extreme
> care during the designing and reporting of experiments involving cell-
> scaffold interactions. The insights from this work will help in the
> designing of tissue engineering scaffolds for applications where mechanical
> stimuli are a critical factor.
>
> Keywords: micromechanics, scaffold, stiffness sensing, structure
>
> ##  1 Introduction
>
> Tissue engineering has made tremendous progress in the past couple of
> decades towards regeneration of tissues and organs in laboratory [1, 2, 3].
> Tissue regeneration involves cells, scaffolds for structural support to the
> cells, and appropriate biochemical factors and biomechanical stimuli [4, 5].
> The choice of cells in any particular tissue engineering application is
> quite limited as they are usually of the same type as the target tissue [6]
> or, in some cases, specific stem cells which differentiate to the desired
> cell type [4]. The bigger challenge is the design of a scaffold which can
> provide appropriate chemo-mechanical micro-environment, similar to native
> extracellular matrix (ECM), to the cells and also can give sufficient
> mechanical support in case of load bearing tissues [7, 8].
>
> For the fibrous structure is a ubiquitous feature of the ECM [9], fibrous
> scaffolds are utilized in tissue engineering quite extensively [10, 11].
> These scaffolds are either derived from the ECM of tissues by the process of
> de-cellularization [12] or are fabricated in lab from biological and
> synthetic materials using techniques such as electrospinning [13, 14] or
> three dimensional (3D) printing [15], with different techniques providing
> distinct control over scaffold architecture. These fibrous scaffolds have
> been shown to be extremely successful in the regeneration of articular
> cartilage [16], retina [17], inter-vertebral disc [18], skin [19], and
> several other tissue types [20, 21, 22].
>
> Owing to the importance of the mechanical forces in tissue engineering [23]
> the characterization of the mechanical properties of the fibrous scaffolds
> is a common feature in almost all tissue engineering exercises. It is
> usually performed by measuring its response against several loading
> conditions such as uniaxial elongation, biaxial elongation shear deformation
> or dynamic deformation for viscoelastic properties [24, 25, 26]. These
> measurements provide an insight into the overall macroscopic mechanical
> properties of the fibrous scaffold. This, however, does not ensure that the
> cells seeded onto the scaffold also experience the same mechanical response
> against microscopic cellular forces/deformations. It is therefore important
> to reliably predict the mechanical environment cells are going to encounter
> when seeded in a particular fibrous scaffold.
>
> In addition to the experimental methods, mathematical and computational
> approaches are extensively utilized to study the mechanical response of the
> fibrous scaffolds. One of the most prominent method for the mathematical
> modeling of the mechanical response of any material is the constitutive
> modeling under continuum assumption [27]. This approach has also been
> utilized for the fibrous materials to formulate structure based constitutive
> models of ECM [28, 29] and fibrous scaffolds [30]. These continuum based
> approaches, however, by definition, consider the microscopic length scale of
> the scaffold to be much smaller than the length scale of interest. This
> assumption is valid for the macroscopic uniaxial/biaxial or shear
> deformations of the scaffold but for the study of the cell-scaffold
> interactions continuum assumption is questionable [31]. This has led to the
> formulation of several statistical description based models of fibrous
> scaffolds [32, 33, 34] and discrete fiber network models [35, 36, 37, 38,
> 39, 40]. These works have looked at different aspects of the nonlinear
> mechanics of the fibrous networks including length scale dependent mechanics
> [37], effect of individual fiber mechanics [39], anomalous Poisson effects
> [40].
>
> Further, the effect of fusion of the fibers is not considered in the
> continuum constitutive models [29], whereas, in the discrete fiber models
> the all the fiber-fiber contacts are usually considered to be perfectly
> fused [35]. In some methods of fibrous scaffold fabrication, such as
> electrospinning, the fiber fusion can be controlled by the operating
> conditions [41]. Therefore, the study of the mechanical behavior of the
> mechanics of fibrous scaffolds need to take into account the degree of fiber
> fusion.
>
> Despite a lot of work on the macroscopic rheology and constitutive modeling
> under uniaxial and shear loading conditions, the fibrous scaffold mechanics
> at the cellular length scale (or microscopic scale) has remained less
> explored. Recently, there has been a lot of activity in this direction to
> understand the micromechanics of fibrous networks against localized cellular
> forces [31, 42, 43, 44, 45, 37, 36]. Experimental and modeling studies have
> shown non-affine [46] and anisotropic [44] deformations of fibrous scaffolds
> under cellular forces. Computational modeling has also shown force chain
> formations [45] and long range force transmission in the fiber networks
> under localized forces [47, 48, 49]. However, the nature of the sensing of
> the mechanical properties of the fibrous scaffolds, such as stiffness, by
> the cells has not been looked into in any considerable detail. One of the
> possible reason for this is the lack of mechanistic understanding of the
> mechanosensing by the cells where it is not yet clear if cells apply
> stresses or strains to the substrate [50, 51].
>
> In this work, we have looked at the micromechanics of fibrous scaffolds and
> its relationship with its macroscopic elastic properties and structure. We
> formulate a discrete fiber network computational model (DFM) using an
> approach motivated by the electrospinning process [52] and compare its
> results against experimental measurements and also continuum constitutive
> model of fibrous materials. We also looked at the stiffness sensing of the
> fibrous scaffolds by the cells and its dependence on the cell’s mechanical
> properties and the mechanosensing mechanism.
>
> ##  2 Model Description
>
> ###  2.1 Discrete fiber model
>
> Application of mechanical forces, either in mechanical testing setup or by
> cells via focal adhesions, results in the deformation of the fibrous
> scaffold. This scaffold deformation results in the deformation of the fibers
> in the form of their bending, straightening, and buckling. Further,
> depending on the fabrication conditions the fibers in the scaffold can also
> fuse with each other [41]. These inter-fiber fusion points can transmit the
> mechanical load from one fiber to another when the scaffold undergoes
> deformation. In order to study the mechanical behavior of fibrous scaffold
> as realistically as possible, therefore, it is essential to have its
> detailed description in terms of the fiber geometry and fiber-fiber fusion
> locations. Towards this goal, we describe the scaffold in terms of its
> constituent fibers as detailed next.
>
> ####  2.1.1 _In-silico_ fibrous scaffold generation
>
> Figure 1: Electrospinning and discrete fiber model. (A) Schematic of
> electrospinning setup and sample electron micrographs of (B) random
> (composed of Polycaprolactone/Gelatin fibers) and (C) aligned (composed of
> polystyrene fibers) electrospun fibrous matrices. (D) Generation of discrete
> fiber model by deposition of individual fibers in the form of straight line
> segments. (E) Random and (F) aligned discrete fiber matrices generated using
> two different probability distributions $p_{{}_{\theta}}$ (shown in
> respective insets).
>
> The discrete fiber model is motivated from the electrospinning method of
> fibrous scaffold fabrication. In electrospinning, a liquid jet is generated
> from an electrified droplet (Fig. 1A). With the help of the electric field
> the jet is stretched and elongated to generate fibers which are deposited at
> the collector surface [52, 53]. Owing to the bending instabilities in the
> extruded jet the deposited fibers at the collector are randomly arranged
> (see Fig. 1B for reference). However, the arrangement of the fibers in
> electrospinning can be modulated to an aligned arrangement (see Fig. 1C),
> upto a point though, by deploying a rotating collector [52].
>
> The electrospun fibrous matrices usually have a very small thickness [54].
> Therefore, we develop a two dimensional model of the fibrous scaffold by
> following the same approach and ‘deposit’ virtual fibers, in the form of
> straight line segments (for scaffold with straight fibers) onto a large
> region of dimensions $L_{mat}\times W_{mat}$ (Fig. 1D). These fibers are
> added to the virtual fibrous mat one-by-one until the total length of all
> the fibers does not reach a pre-specified threshold. This pre-specified
> threshold value of the fiber length corresponds to the duration of the
> electrospinning process or the amount of material used in the process. We
> define the orientation of a fiber as the angle $\theta$ made by the line
> joining its two end from a pre-specified direction, as shown in Fig. 1D.
> Further, for a fibrous scaffold with curved fibers we characterize the
> geometry of a fiber in terms of its normalized length $\xi=l/R$, the ratio
> of its contour length and the shortest distance between its two ends. This
> means for a scaffold with perfectly straight fibers, we have $\xi=1$ for all
> the fibers. For each fiber the orientation angle and length are taken from
> respective probability density functions (PDFs), $p_{{}_{\theta}}(\theta)$
> and $p_{{}_{\xi}}(\xi)$, which, for the results shown in this paper, are
> taken to be Gaussian distributions
>
>
> $p_{{}_{\theta}}(\theta)=C_{\theta}\exp{\left(-\frac{\left(\theta-\theta_{0}\right)^{2}}{2\sigma_{\theta}^{2}}\right)}\text{
> with }-\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2},$ (1)
>
> and
>
>
> $p_{{}_{\xi}}(\xi)=C_{\xi}\exp{\left(-\frac{\left(\xi-\xi_{0}\right)^{2}}{2\sigma_{\xi}^{2}}\right)}\text{
> with }\xi\geq 1$ (2)
>
> for fibers with no residual stresses. Here, $\theta_{0}$ and $\xi_{0}$ are
> the mean fiber orientation angle and mean normalized fiber length,
> respectively, $\sigma_{\theta}$ and $\sigma_{\xi}$ are the standard
> deviations of the respective PDFs, and $C_{i}$’s are the normalization
> coefficients to ensure
>
> $\int\limits_{-\pi/2}^{\pi/2}p_{{}_{\theta}}(\theta)d\theta=1\text{ and
> }\int\limits_{1}^{\infty}p_{{}_{\xi}}(\xi)d\xi=1.$ (3)
>
> Larger and smaller values of $\sigma_{\theta}$ result in random (Fig. 1E)
> and aligned (Fig. 1F) fiber systems, respectively. Further, the fusion among
> the fibers is described by a probability value, $p_{{}_{f}}$. $p_{{}_{f}}$
> fraction of all the fiber-fiber intersections are randomly selected in the
> model and the two participating fibers are considered to be fused with each
> other at those intersections. Therefore, $p_{{}_{f}}=0$ and $p_{{}_{f}}=1$
> stand for the fiber systems with no inter-fiber fusion and fully fused
> fibers, respectively. Similar to the electrospun fibrous mat, this process
> generates a 2D virtual fibrous mat of dimensions $L_{mat}\times W_{mat}$.
> However, the density of the fibers in this mat is not uniform with central
> region of the mat having higher number of fibers relative to the mat edges
> [54]. Therefore, we extract the samples of required dimensions $L\times W$
> (where $L,W\ll L_{mat},W_{mat}$) from the central region of the mat for
> further analysis. This discrete fiber model of the fibrous scaffold is,
> therefore, described in terms of the coordinates of the fiber fusion points
> and the geometry of the fibers connecting these points to each other and to
> the sample boundary (Fig. 2A).
>
> ####  2.1.2 Microscopic length scale of the scaffold
>
> Since in this work we are also comparing the mechanical responses of the
> fibrous scaffold as obtained from the discrete fiber model and continuum
> finite element models, it is important to define a microscopic length scale
> (akin to bond length or inter-particle distance in solids and mean free path
> in fluids [55]) which can help us ascertain the validity of the continuum
> assumption based models for fibrous scaffolds. For this purpose we look at
> the distances $l_{f}$ between adjacent fusion points on all the fibers. Fig.
> 2B shows the distribution of $l_{f}$ for different degrees of fiber fusion.
> We utilize $\ell=\langle l_{f}\rangle$, the mean value of $l_{f}$, as the
> microscopic length scale of fibrous scaffold which is smaller for higher
> degree of fiber fusion (inset in Fig. 2B).
>
> Figure 2: Microscopic length scale in fibrous scaffold. (A) Schematic
> showing fiber arrangement and fiber fusion points (marked with black dots).
> $l_{f}$ is defined as the length of the fiber segment between two
> consecutive fusion points. (B) Distributions of $l_{f}$, for different
> $p_{{}_{f}}$ for random fiber arrangement. Inset shows the dependence of
> $\ell=\langle l_{f}\rangle$, the characteristic microscopic length scale in
> a random fibrous scaffold, on fiber fusion probability $p_{{}_{f}}$.
>
> With this description of the fibrous scaffold geometry we will now describe
> the modeling of its mechanical response against macroscopic and microscopic
> loading conditions.
>
> ####  2.1.3 Mechanical response of a single fiber
>
> Deformation of the fibrous scaffold results in bending and stretching of the
> fibers. The combined effect of the bending (or straightening) and stretching
> of the fibers is usually written, in a phenomenological manner, as an
> exponential function of the fiber strain [56]. In contrast to the
> phenomenological approach, in the following, we are going to derive a
> constitutive law for the effective fiber while taking into account its
> geometrical properties.
>
> For this, we are first going to look at the fiber response under stretching
> and bending separately. The application of mechanical tension on a straight
> fiber results in its stretching. We model fiber stretching by neo-Hookean
> response which is described in terms of the energy
>
>
> $u_{e}(\xi,\lambda_{e})=\frac{1}{2}k_{e}\left(\lambda_{e}^{2}+\frac{2}{\lambda_{e}}-3\right)$
> (4)
>
> where $\lambda_{e}$ is the stretch ratio of the fiber and $k_{e}=EA$ is the
> product of the Young’s modulus $E$ of the fiber material and fiber cross-
> sectional area $A$. This gives the force required for the stretching of a
> straight fiber as
>
>
> $f_{e}(\lambda_{e})=k_{e}\left(\lambda_{e}-\frac{1}{\lambda_{e}^{2}}\right).$
> (5)
>
> On the other hand the deformation of any curved fiber or compression of a
> straight fiber results in the change of the fiber curvature. Since, in our
> description the fiber geometry is described in terms of a single parameter
> $\xi$, the normalized fiber length, we write its average curvature (in
> undeformed configuration) in terms of $\xi$ as (see Appendix A for details)
>
> $\langle\kappa_{0}(\xi)\rangle\approx 2\pi\sqrt{1-\sqrt{\frac{1}{\xi}}}.$
> (6)
>
> Similarly, for a deformed fiber the curvature is
>
> $\langle\kappa(\xi,\lambda_{b})\rangle\approx
> 2\pi\sqrt{1-\sqrt{\frac{\lambda_{b}}{\xi}}}$ (7)
>
> where $\lambda_{b}$ is the stretch ratio of the line joining the two ends of
> the fiber. This gives the bending energy of the fiber in the deformed
> configuration to be
>
>
> $u_{b}(\xi,\lambda_{b})=\frac{k_{b}}{2}\left(\langle\kappa(\xi,\lambda_{b})\rangle-\langle\kappa_{0}(\xi)\rangle\right)^{2}.$
> (8)
>
> This gives the force required for the fiber bending (or straightening) to be
>
>
> $f_{b}(\xi,\lambda_{b})\approx-\frac{k_{b}}{\xi}\frac{1}{\sqrt{\lambda_{b}\xi}}\left(1-\frac{\sqrt{\sqrt{\xi}-1}}{\sqrt{\sqrt{\xi}-\sqrt{\lambda_{b}}}}\right).$
> (9)
>
> This bending force also results in a requirement of a critical load
> (buckling) for a straight fiber under compression. It has to be noted that
> in this formulation both of these forces $f_{e}$ and $f_{b}$ act along the
> line joining the two ends of the fiber. In general, application of
> mechanical force results in fiber bending (or straightening) and stretching
> both. Both of these deformations can be combined by considering a series
> arrangement of two springs (one corresponding to the bending deformation and
> other for stretching) to obtain the relation [57]
>
> $\lambda=\lambda_{e}\lambda_{b}\left(f\lambda_{e}(f)\right)$ (10)
>
> where $\lambda$ and $f$ are the stretch ratio of the fiber and the total
> force required for its deformation, respectively. Here $\lambda_{e}(f)$ and
> $\lambda_{b}(f)$, the constitutive relations of the fiber stretching and
> bending, are given by equations (5) and (9), respectively. Using this we can
> define the effective fiber response as the force required for the
> deformation of and average fiber connecting any two points in the scaffold
> as
>
> $\langle
> f(\lambda)\rangle=\int\limits_{1}^{\infty}f(\xi,\lambda)p_{\xi}(\xi)d\xi.$
> (11)
>
> This expression relates the average force and stretch $\lambda$ in the
> effective fiber (see supplementary Fig. S1 for plots with different fiber
> geometries). We now can use this effective fiber response to estimate the
> mechanical behavior of the fibrous scaffold.
>
> ####  2.1.4 Mechanical response of the fibrous scaffold
>
> In the DFM the scaffold structure is described by the coordinates of the
> fiber fusion points $\mathbf{x}_{i}=$ ($x_{i}$, $y_{i}$) (as shown in Fig.
> 2A) and $\xi$ the normalized length of the fiber segments connecting the
> fusion points to each other and the sample boundary. We define the response
> of the fibrous scaffold against applied mechanical loads as the collective
> response of all the fibers. To estimate the mechanical resistance of the
> scaffold as a whole we calculate the forces on each fusion point in the
> deformed configuration by combining the forces arising due to each deformed
> fiber (equation (11)) at the fusion point. For $i^{\text{th}}$ fusion point,
> the force is, therefore, given by
>
> $\mathbf{f}_{i}=\sum\limits_{\alpha\in\mathcal{N}_{i}}\mathbf{f}_{\alpha}$
> (12)
>
> where $\mathbf{f}_{\alpha}$ is the tension in the $\alpha^{\text{th}}$ fiber
> segment and $\mathcal{N}_{i}$ denotes the set of all fiber segments
> connected to the $i^{\text{th}}$ fusion point. In the numerical simulation
> of the DFM we calculate the forces on each fiber fusion point. Assuming
> overdamped dynamics of the system we displace each fusion point by an amount
> proportional to the total force acting on it until forces on all fusion
> points are not below a pre-specified tolerance value.
>
> ###  2.2 Continuum finite element model of fibrous materials
>
> There are several constitutive models of the mechanical behavior of fibrous
> materials and composites which are based on the continuum assumption that
> the microscopic length scale is negligible as compared to the experimental
> length scale [28, 29]. These models are widely utilized for the study of the
> biological and non-biological applications [28, 58, 29]. In order to compare
> the properties of the discrete fiber model (DFM) we also studied the
> mechanics of fibrous scaffolds using a continuum constitutive model. It
> needs to be pointed out here that the continuum assumption may not exactly
> hold for the cell-scaffold interactions since the length scale $\ell$ can be
> comparable to the cell size. Still it will give us insights into the
> limitations of the continuum models for the study of cell-scaffold
> interactions.
>
> The model developed by Holzapfel, Gasser and Ogden [28] to study of the
> mechanics of fibrous biomaterials is very well known and has also been
> incorporated in finite element structural analysis softwares, such as
> ABAQUS. We also used this continuum model analyze the scaffold mechanics to
> compare with the discrete fiber model. It is worthwhile at this point to
> give a brief description of the continuum hyperelastic constitutive model of
> fibrous biomaterials (see [59] for more details) which is described by the
> energy density function in terms of the strain invariants as
>
>
> $U=C_{10}\left(\bar{I}-3\right)+\frac{k_{1}}{2k_{2}}\left(\mathrm{e}^{k_{2}E^{2}}-1\right)$
> (13)
>
> where $C_{10}$, $k_{1}$ and $k_{2}$ are the material parameters and
>
>
> $E=\kappa\left(\bar{I}_{1}-3\right)+\left(1-3\kappa\right)\left(\bar{I}_{4}-1\right)$
> (14)
>
> with $\kappa$ describing the nature of fiber arrangements with $\kappa=0$
> and $\kappa=1/3$ standing for perfectly aligned and random fiber
> orientations, respectively. Here $\bar{I}_{1}$ and $\bar{I}4$ are the first
> and fourth invariants of right Cauchy-Green tensor and the fiber orientation
> tensor $\mathbf{A}=\mathbf{a}\otimes\mathbf{a}$ where vector $\mathbf{a}$
> represents the fiber orientation. Please note that in this model one can
> incorporate multiple fiber families but for our purposes one fiber family
> suffices. Following standard analytical techniques one can obtain the
> expression for the stress tensor from the energy density function. It has to
> be pointed out that this particular model does not take into account any
> inter-fiber interactions and, therefore, the effect of fiber fusion can not
> be studied with this model. In this work we have used ABAQUS/Standard finite
> element package with CPS3 and CPS4R type elements to study the mechanical
> response of two dimensional fibrous scaffold against uniaxial elongation and
> localized forces. Similar to DFM, in the finite element (FE) model too, we
> kept only two loading boundaries fixed under uniaxial elongation loading and
> all found sample boundaries were fixed for the simulation of scaffold with
> localized point force.
>
> ###  2.3 Parameters values
>
> In this paper we have presented the results from discrete fiber model of
> fibrous scaffold with random and aligned fiber arrangements. For random
> fibers we have set $\xi_{0}=1.18$, $\sigma_{\xi}=0.2$ and
> $p_{{}_{\theta}}(\theta)=1/\pi$, that is uniform distribution of the fiber
> orientations. We obtained these parameter values by fitting the experimental
> data to the discrete fiber model (as shown in Fig. 3B). For aligned fibers,
> these parameters are set as $\xi_{0}=1.01$, $\sigma_{\xi}=0.01$ and
> $\sigma_{\theta}={\pi}/20$, that is very narrow distribution. The value of
> $\theta_{0}$ for aligned fibers is mentioned in respective figure captions.
>
> For the constitutive model of the bulk fiber material we have considered
> fibers to be composed of electrospun polystyrene with elastic modulus
> $E=3.2$GPa. These results, however, can be translated to fibrous scaffolds
> of other materials by appropriate rescaling of the Young’s modulus as long
> as the statistical description of the scaffold structure remains same. In
> tissue engineering application the fiber diameters are usually in the range
> of tens of nanometers to microns (for example [60]). Here we have taken the
> average fiber diameter to be $a=100$nm which sets $k_{e}=EA=2.5\times
> 10^{-5}$N. In calculations of the scaffold’s Young’s modulus we have taken
> the thickness of the sample to be $t=0.2$mm for $L_{m}=6\times 10^{6}$cm for
> a scaffold sample dimensions $L\times W=1$cm $\times 1$cm. For other values
> of $L_{m}$ the thickness is assumed to change linearly with $L_{m}$.
>
> Even though the discrete fiber model is described here in two-dimensions the
> fibrous scaffolds are three dimensional structures. Therefore, the
> identification of the fiber fusion points by intersection of line segments
> in the virtual electrospinning is bound to overestimate the number of fiber-
> fiber contacts. In order to compensate for this overestimation, we do not
> consider very high values of $p_{f}$ and have taken the value of fiber
> fusion density to be in the range $p_{f}=0-0.2$ with its exact values
> mentioned in respective figure captions.
>
> For the continuum finite element model of the scaffold with random fiber
> arrangement we have set the parameters to be $C_{10}=0.8$MPa,
> $k_{1}=1400$MPa, $k_{2}=1000$ and $\kappa=1/3$.
>
> ##  3 Results
>
> ###  3.1 _In-silico_ mechanical testing
>
> ####  3.1.1 Uniaxial elongation
>
> Figure 3: Uniaxial elongation of the fibrous scaffold. (A) Schematic showing
> polystyrene scaffold in experiment, discrete fiber model and continuum
> finite element model. (B) Load-strain curves for scaffolds with random fiber
> arrangements as measured from mechanical testing of polystyrene scaffold
> (adapted from [32]), DFM ($p_{f}=0.02$) and FE models. Dependence of Young’s
> modulus of fibrous scaffold on (C) sample dimensions with $L_{m}=2\times
> 10^{7}$/unit area of sample, (D) degree of fiber fusion $p_{f}$, and (E)
> $L_{m}$, total fibers in the sample. The dashed lines in (D) show the fit of
> analytical expressions for the Young’s modulus, as obtained from equations
> (16) and (17). The inset in (D) shows the dependence of Young’s modulus on
> fiber fusion density $p_{f}$ with $W/\ell\approx 20$. The error bars in
> (C-E) are not shown for clarity.
>
> In experiments, the mechanical behavior of the fibrous scaffolds is usually
> assessed by measuring the resistance of the scaffold against uniaxial
> elongation [61, 62]. Therefore, we also first analyzed the mechanical
> response of the fibrous scaffolds under uniaxial elongation (Fig. 3A) as a
> function of its structural parameters. Fig. 3B shows the average mechanical
> response of fibrous scaffolds with random fiber arrangement as estimated
> using discrete fiber model and continuum finite element model. For small
> strain values, both the models demonstrate a nonlinear response of the
> scaffold (‘J-shaped’ curve, see [32]). Further, both of the models also
> showed good agreement with the experimental measurements on electrospun
> Polystyrene fibrous matrices for small strain values. Since we did not take
> fiber breakage into account in the models, at larger strains, the
> experimental measurements showed a deviation from the models. This
> disagreement at large strains is not of any major consequence here since the
> focus of this work is on the mechanical response of the fibrous scaffolds
> against cellular forces which do not result in fiber breakage.
>
> ####  3.1.2 Sample size effect
>
> Owing to the fact that the ratio of the microscopic length scale $\ell$ and
> the sample size $L$ is not very small (Fig. 2B) for the fibrous scaffolds,
> the sample size is expected to influence its mechanical response in DFM. In
> order to study this effect we estimated the Young’s modulus
> $\mathcal{Y}_{m}$ of the fiber scaffold with randomly arranged fibers for
> different sample dimensions (Fig. 3C). Here we observe mainly three types of
> behaviors. For $L\gg W$, the Young’s modulus of the fibrous scaffold sample
> is much smaller than that of the sample with $L\ll W$. For samples with
> aspect ratio close to $1$, that is $L\approx W$, the sample’s Young’s
> modulus is in between the two extremes. This effect of the sample size is
> more pronounced in the scaffold samples with low degree of fiber fusion and
> in sparse fibrous scaffolds (supplementary Fig. S2). We also observe the
> effect of sample size while changing the sample thickness by varying total
> fiber length $L_{m}$. There too the Young’s modulus of the scaffold
> increases with an increase in $L_{m}$ or sample thickness (Fig. 3E).
>
> The sample size effect on the elastic modulus can be intuitively explained
> for the fibrous scaffolds with no fiber fusion, that is $p_{f}=0$. For no
> fiber fusion, the resistance of the scaffold against uniaxial elongation can
> be written as the collective response of all the end-to-end fibers. For
> random arrangement of linear elastic straight fibers, the force required for
> uniaxial elongation of the sample follows (see Appendix B for details)
>
> $F_{y}(\varepsilon)\sim
> W\varepsilon\int\limits_{0}^{W}\int\limits_{x_{1}}^{x_{2}}\cos^{3}\theta
> p_{\theta}(\theta)d\theta dx$ (15)
>
> where $x_{1}=\tan^{-1}\left(-\cfrac{x}{L}\right)$ and
> $x_{2}=\tan^{-1}\left(\cfrac{W-x}{L}\right)$. For $W\ll L$, this simplifies
> to
>
> $F_{y}(\varepsilon)\sim\varepsilon W^{2},$ (16)
>
> and for $W\gg L$ we obtain
>
> $F_{y}(\varepsilon)\sim\varepsilon W.$ (17)
>
> This shows that for $W\ll L$ the Young’s modulus of the fibrous scaffold
> increases with the sample width (Fig. 3D). However, for the sample size with
> $W\gg L$, Young’s modulus attains a fixed value.
>
> In contrast to the DFM, the continuum FE model did not show any sample size
> effect (data not shown) on elastic modulus. This behavior is not surprising
> since continuum description of the scaffold assumes the microscopic length
> scale to be much smaller than the sample size. This demonstrates the
> limitation of continuum models for the scaffold mechanics where size of the
> cell is often comparable to microscopic length scale.
>
> ####  3.1.3 Effect of fiber fusion
>
> We also looked at the effect of the degree of fiber fusion on the mechanical
> response of the fibrous scaffolds against uniaxial elongation. For this we
> estimated the Young’s modulus of fibrous scaffolds with different fiber
> fusion density. We find that an increase in the fiber fusion results in
> stiffer scaffold (Fig. 3D, also reported in [35]). The fiber fusion also
> influences the sample size effect such that for higher fiber fusion the
> Young’s modulus attains saturation value for smaller sample widths. This is
> an expected outcome since for higher fiber fusion, due to smaller $\ell$,
> the ratio of microscopic and macroscopic length scale is very small.
>
> These results on the matrix response against uniaxial elongation as a
> function of structural parameters, which can also be called ‘macro-
> mechanical response’, have been studied extensively using experimentally as
> well as computational modeling. This, however, does not shed any light on
> whether the cells interacting with the fibrous matrices also experience the
> same mechanics.
>
> ###  3.2 Micro-mechanics of fibrous scaffold
>
> The mechanical interaction of the cells with scaffold is via the focal
> adhesions [63] which transfer localized microscopic cellular forces between
> the two. Therefore, in the following, we shift our attention to the scaffold
> micro-mechanics or its mechanical response against localized forces.
>
> ####  3.2.1 Scaffold deformation due to localized forces
>
> Towards this goal, we consider fibrous scaffold sample with all its
> boundaries fixed and apply a point force $f$ at its center (Fig. 4A). This
> loading condition is somewhat similar to the well studied Kelvin problem in
> three dimensional elasticity [64]. Fig. 4B shows the displacement fields in
> the fibrous scaffolds due to a localized point force as estimated from DFM.
> Displacement field reveal anisotropic deformation of the fibrous scaffolds,
> even for the samples with random fiber arrangement. The anisotropic
> displacement field in case of the samples with random fiber arrangement can
> be attributed to the very small bending stiffness $k_{b}$ of the fibers as
> compared to their stretching stiffness $k_{e}$. In case of aligned fibers
> the scaffold deformation depends on the direction of the applied point force
> relative to the fiber orientation angle (Fig. 4D-E). For a point force
> parallel to the fiber orientation the deformation field remains localized to
> a smaller region as compared to when the directions of force and fibers are
> normal to each other.
>
> Figure 4: Fibrous scaffold micromechanics. (A) Schematic showing application
> of localized point force. Displacement field in the fibrous scaffold with
> randomly arranged fibers as calculated from (B) discrete fiber model and (C)
> continuum finite element modeling. Displacement field in the fibrous
> scaffold with aligned fibers with point force (D) normal, and (E) parallel
> to the fiber alignment direction ($\theta_{0}=3\pi/4$ and $\pi/4$,
> respectively). (F) Displacement of the point at which force is applied as a
> function of fiber fusion density $p_{f}$ for random fiber arrangement. For
> (B), (D) and (E) $p_{f}=0.2$ and $L_{m}/W=2\times 10^{7}$.
>
> As opposed to DFM, the continuum FE model of the fibrous scaffold does not
> demonstrate anisotropy in the displacement field for the samples with random
> fiber arrangement (Fig. 4C).
>
> ####  3.2.2 Force transmission in fibrous scaffolds
>
> For a quantitative insight into the nature of scaffold deformation we
> plotted the displacement field in the matrix as a function of the distance
> $r$ from the point of force application for a sample with randomly arranged
> fibers. For this, we divide the scaffold into multiple regions (Fig. 5A) and
> the displacement plots for each region are shown in Fig 5B (note the log
> scaling on $y$-axis). In region-I, the displacement field decays
> exponentially with $r$ and by fitting
>
> $\delta(r)=\delta_{0}\exp\left(-r/r_{1}\right)$ (18)
>
> to the data we can estimate the characteristic decay length $r_{1}$ for
> region-I.
>
> Figure 5: Displacement decay and characteristic length of force
> transmission. (A) Schematic showing three regions relative to the applied
> point force $f$. Decay of displacement field as a function of $r$, the
> distance from the point force and, $\phi$, the position on the scaffold as
> obtained from (B) discrete fiber model ($p_{f}=0.2$) and (C) continuum FE
> model of scaffold with random fiber arrangement. (D) Characteristic decay
> lengths $r_{n}$ and $r_{f}$ for near and far field displacement,
> respectively, as a function of $\phi$ for different values of fiber fusion
> density $p_{f}$. For $\phi<\pi/2$ the far field displacement is too small
> for $r_{f}$ calculation. In (B) and (D) $p_{f}=0.02$ and $L_{m}/W=2\times
> 10^{7}$.
>
> In contrast, for region-II, even though the displacement field shows
> exponential decay, it shows two different characteristic decay lengths-
> $r_{2}$ for points close to the point force ($r\ll 1$), and $r_{1}$ (same as
> that in region-I) for points far from the force ($r\gg 1$). In region-III
> the scaffold response is somewhere in between that of the two extremes seen
> in regions-I and II. For region-II, the critical distance $r_{c}$ where
> transition between two behaviors takes place can be calculated as the
> intersection of two exponential decay curves. We also observe that this
> critical transition distance does not depend very strongly on the degree of
> fiber fusion (supplementary Fig. S3). This indicates that the anisotropy in
> the scaffold response is due to smaller bending stiffness of the fiber and
> not the fiber fusion.
>
> We further plotted the near-field ($r_{n}$ for $r<\ell$) and far-field
> ($r_{f}$ for $r>\ell$) characteristic decay lengths as a function of angle
> $\phi$ from the direction of point force (Fig. 5D). It shows that the far
> field decay length does not depend on $\phi$ indicating that the far-field
> displacement field is isotropic in nature. However, the near field decay
> length depends strongly on $\phi$. We observe that for small values of
> $\phi$ (region-II) $r_{n}\ll\ell$ whereas for $\phi\approx\pi$ (region-I)
> $r_{n}$ is large. This indicates the anisotropic nature of the force
> transmission in the fibrous scaffold. The fibers under tension (in region-I)
> transmit force to larger distance than the fibers under compression in
> region-II.
>
> Even though the exponential decay of the displacement field is seen in the
> continuum FE model as well (Fig. 5C) the anisotropy of the response is not
> observed (Fig. 5D). In fact, in the continuum model we obtain the
> characteristic decay length which does not depend on the direction. Even
> though there is no experimental data available for the scaffold deformation
> due to in-plane localized forces (as far as we know), the displacement field
> due to a polarized cell (see inset in Fig. 6B) shows the anisotropic nature
> of scaffold deformation. This shows that the continuum model predicts an
> isotropic force transmission in the fibrous scaffolds with random fibers.
>
> ####  3.2.3 Microscopic scaffold stiffness
>
> We can quantify the stiffness of the fibrous scaffold against the localized
> force as $E_{\mu}=(f/k_{e})/(\delta_{0}/L)$, the ratio of the magnitude of
> the force applied at the center and the displacement of that point [65].
> Therefore, for a fixed amount of applied force microscopic stiffness of the
> scaffold can be characterized by the displacement of the center point with
> stiffer scaffolds showing smaller displacements. Fig. 4F shows the
> displacement of center point $\delta_{0}$ as a function of the degree of
> fiber fusion $p_{f}$ for a fixed value of applied force. This shows an
> increase in the microscopic stiffness of the scaffold with increasing fiber
> fusion. Not only that, the numerical fit to the simulation data also reveals
> that the displacement of the center point scales as
> $\cfrac{\delta_{0}}{\ell}\sim\cfrac{1}{\sqrt{p_{f}}}$ for small values of
> $p_{f}$. It has to be pointed out that this scaling might not hold for
> larger values of $p_{f}$ since that would imply a rigid nature of scaffold
> for large $p_{f}$. This increase in microscopic stiffness with increasing
> fiber fusion also corresponds to the increase in the macroscopic stiffness
> (Fig. 3D). This estimate and scaling law gives us an insight which can be
> utilized in modulating the fiber fusion in the fibrous scaffolds for desired
> tissue engineering applications.
>
> ###  3.3 Stiffness sensing of a fibrous scaffold by a cell
>
> Figure 6: Scaffold deformation due to a cellular force dipole. (A) Schematic
> showing placement of cellular force dipole on fibrous scaffold. (B)
> Displacement field due to force dipole in fibrous scaffold with random fiber
> arrangement. Inset shows the displacement field due to an epithelial cell on
> a collagen scaffold (adapted from [66], with permission). Displacement field
> in fibrous scaffold with aligned fibers when the dipole is placed (C)
> parallel and (D) normal to the fiber orientations. Here $p_{f}=0.2$,
> $L_{m}/W=2\times 10^{7}$ for (B-D), and $\theta_{0}=\pi/4$ and $3\pi/4$ for
> (C) and (D), respectively.
>
> As mentioned earlier cells apply localized forces to the underlying
> substrate through the focal adhesion contacts to sense the mechanical
> properties of the scaffold [67]. Despite the extensive works on the
> biophysics of this process the exact mechanism of the stiffness sensing by
> the cells has remain elusive [50]. Since the exact measurement of the force
> acting on each focal adhesion is not straight forward, the forces applied by
> the cells are usually modeled as force dipoles [65, 50]. In its general
> form, a force dipole is defined as the second order tensor
>
> $\underline{\underline{\mathcal{P}}}=\int\mathbf{r}\otimes\mathbf{t}dl$ (19)
>
> where $\mathbf{r}$ and $\mathbf{t}$ are the position vectors of the focal
> adhesion and the localized force acting on them. Here the integration is
> performed along the boundary of the cell. Since the cells usually do not
> apply any toque to the substrate the force-dipole tensor is symmetric in
> nature with its principal directions (corresponding to the eigenvectors of
> the matrix representation) coinciding with the directions of maximum and
> minimum forces applied by the cells. Assuming the force in one principal
> direction to be much larger as compared to that in the other, the force-
> dipole model of a cell reduces down to two opposing point forces $f$
> separated by a distance $d$ (Fig. 6A). We looked at the deformation of the
> fibrous scaffold due to this force-dipole to understand the cell-scaffold
> interaction and stiffness sensing by a cell.
>
> ####  3.3.1 Scaffold deformation due to a cell
>
> Fig. 6B shows the displacement field of the fibrous scaffold with random
> fiber arrangement due to the forces applied by a force dipole (as shown in
> Fig. 6A). Here too, the displacement field shows anisotropy but that is due
> to the nature of the forces applied by the force-dipole. This displacement
> field is in qualitative agreement with the experimental measurements for an
> epithelial cell on a collagen scaffold (inset Fig. 6B). On the scaffold with
> aligned fibers the displacement field shows dependence on the cell
> orientation relative to fibers (Fig. 6C-D). Apart from the qualitative
> difference between the two aligned systems (Fig. 6C and Fig. 6D) the
> magnitude of the displacement (and thereby scaffold deformation) is smaller
> when the force dipole orientation is parallel to the fibers. This is due to
> primarily stretching and bending deformations of the fibers in Fig. 6C and
> Fig. 6D, respectively.
>
> ####  3.3.2 Scaffold stiffness sensing by a cell
>
> It is still a matter of debate [51] whether cells apply a fixed amount of
> force to the substrate and sense stiffness based on the magnitude of the
> substrate deformation or they apply a fixed amount of strain and stiffness
> is sensed in terms of the resistance force offered by the substrate. Here we
> test both of these possibilities to identify if the stiffness sensed by a
> cell depends on cellular mechanism and how does stiffness sensed by the cell
> depend on the cell’s own stiffness.
>
> In both cases, in principle, the cell can sense the scaffold stiffness in
> terms of the difference between its deformation with and without substrate
> interaction. Therefore, for small magnitude of cell deformation the scaffold
> stiffness sensed by the cell is
>
> $\mathcal{E}_{m}=\frac{\sigma_{0}}{\epsilon_{c}}-\mathcal{E}_{c}$ (20)
>
> where $\sigma_{0}=f/A_{\text{FA}}$ is the stress defined as the ratio of the
> force by the cell and the cross-sectional area of a focal adhesion,
> $\epsilon_{c}$ is the strain in the cell when it is attached to the
> scaffold. Here the cell stiffness $\mathcal{E}_{c}$ is an effective
> parameter which takes into account the contributions from the cytoskeleton,
> focal adhesion, and other cellular components. For the scenario when cell
> applies a fixed amount of force (force-dipole) $f$ the strain in the cell
> $\epsilon_{c}$ is the unknown which we have to calculate. On the other hand,
> when the fixed strain is applied by the cell (strain-dipole), we have to
> estimate the force $f$ required by the cell. We performed numerical
> simulations for fiber scaffolds with random fiber arrangements and force
> dipole as a model of cell in both of these conditions.
>
> Fig. 7A shows the stiffness of the fibrous scaffold with random fiber
> arrangement as sensed by the cell as a function of the cell stiffness. It
> can be seen that the scaffold stiffness sensed by the cell does not depend
> on the cell stiffness when the cell controls the deformation and not force
> (dashed curve in Fig. 7A). However, when cell applies a constant amount of
> force the stiffness sensed by the cell decreases with increasing cell
> stiffness.
>
> In order to understand this interesting behavior we looked at this more
> closely using a simplified lumped-parameter model of stiffness sensing. Such
> models have previously been used to study the cell-substrate interactions
> [68]. In the lumped-parameter model we consider stiffness sensing by a cell
> in one dimension where a cell (elastic modulus $\mathcal{E}_{c}$, length
> $d$) is attached to a 1D scaffold (length $2l+d$) on two sides and the two
> ends of the substrate are held fixed. We consider that the stress-strain
> relation for this 1D scaffold is nonlinear and is given by function
> $\sigma_{m}(\epsilon_{m})$ (with
> $\left.\cfrac{d\sigma_{m}}{d\epsilon_{m}}\right|_{\epsilon_{m}\rightarrow
> 0}=\mathcal{Y}_{m}$) where $\epsilon_{m}$ is the strain in the scaffold. For
> $\epsilon_{c}$ strain in the cell the strain in the scaffold on each side of
> the cell is $\epsilon_{m}=\cfrac{\epsilon_{c}d}{2l}$. The stress applied by
> the cell to generate this deformation is, therefore, given by
>
>
> $\sigma_{0}=\mathcal{E}_{c}\epsilon_{c}+\sigma_{m}\left(\frac{\epsilon_{c}d}{2l}\right).$
> (21)
>
> Therefore when cell applies a constant amount of strain the stiffness of the
> scaffold sensed by the cell can be calculated from equation (20) and is
> given by
>
>
> $\mathcal{E}_{m}=\left.\frac{1}{\epsilon_{c}}\sigma_{m}\left(\frac{\epsilon_{c}d}{2l}\right)\right|_{\epsilon_{c}\rightarrow
> 0}\sim\mathcal{Y}_{m}.$ (22)
>
> This shows that in the strain controlled scenario the stiffness sensed by
> the cell is proportional to the macroscopic stiffness and does not depend on
> the stiffness of the cell. This is in agreement with the numerical
> simulation of the scaffold model (Fig. 7A).
>
> On the other hand, when cell applies a constant force we need to solve
> equation (21) for $\epsilon_{c}$ which is unknown in this case. This will
> require the knowledge of the exact form of the constitutive relation of the
> scaffold. For our purposes, since the fibrous scaffold shows nonlinear
> relation between stress and strain (Fig. 3B), we consider a small degree of
> nonlinearity in the scaffold constitutive model and assume that it has the
> form
>
>
> $\sigma_{m}(\epsilon_{m})=\mathcal{Y}_{m}\epsilon_{m}+\alpha\epsilon_{m}^{2}$
> (23)
>
> where $\alpha$ is a small phenomenological parameter. We have chosen a small
> degree of nonlinearity ($\alpha\ll 1$) to keep the analysis tractable. In
> principle, the force-displacement curve due to the localized force can be
> utilized for more accurate calculation. Following a perturbative approach,
> we can now solve equation (21) to obtain
>
>
> $\epsilon_{c}=\frac{\sigma_{0}}{\mathcal{S}}\left(1-\frac{\alpha\sigma_{0}}{\mathcal{S}^{2}}\right)$
> (24)
>
> where $\mathcal{S}=\mathcal{E}_{c}+\mathcal{Y}_{m}\cfrac{d}{2l}$. We can
> substitute this in equation (20) to obtain
>
>
> $\mathcal{E}_{m}=\mathcal{Y}_{m}\frac{d}{2l}+\frac{\alpha}{\left(\mathcal{E}_{c}+\mathcal{Y}_{m}\cfrac{d}{2l}\right)}\sigma_{0}$
> (25)
>
> which shows the dependence of scaffold stiffness sensed by the cell on
> $\mathcal{E}_{c}$. Fig. 7A-B shows that this relation fit very well with the
> numerical results obtained for the fibrous scaffold.
>
> Figure 7: Stiffness sensing of fibrous scaffold by a cell. (A) Dependence of
> stiffness of the fibrous scaffold with random fiber arrangement, as sensed
> by a cell by applying constant force (force dipole) and constant deformation
> (strain dipole), on the cell stiffness. The error bars in the strain dipole
> case are not shown for clarity. (B) Stiffness sensing of fibrous scaffold
> with aligned fibers by a cell via force dipole. The gray curve for the
> scaffold with random fiber arrangement (same as in panel A) is also shown
> for reference. (C) Estimated stiffness sensing of fibrous scaffolds (with
> randomly arranged fibers) composed of materials with different elastic
> properties by different cell types. In (C) $E$ is the Young’s modulus of the
> bulk material of the fibers. The exact values of the elastic properties of
> materials and cells shown in (C) are listed in supplementary information.
> For all panels $p_{f}=0.02$ and $L_{m}/W=2\times 10^{7}$.
>
> The above analysis also demonstrates that if $\alpha=0$, that is scaffold
> response is linear elastic, then the scaffold stiffness sensed by the cell
> is independent of the cell stiffness and also does not depend on the details
> of the cellular machinery (constant strain vs constant stress). This
> dependence of the cell sensed stiffness on $\alpha$, the non-linearity in
> the scaffold response, can be intuitively understood in the following
> manner. Considering the forces applied by the cells to be same the
> deformation in a stiffer cell (a cell with high $\mathcal{E}_{c}$) is
> smaller. This results in smaller deformation of the scaffold as well. Owing
> to the non-linearity in the scaffold constitutive response we get smaller
> $\mathcal{E}_{m}$ for smaller deformations, resulting in decrease in
> $\mathcal{E}_{m}$ with increasing $\mathcal{E}_{c}$. This mechanism is
> further visible when we consider strain applied by the cell instead of
> stress as the model for the cell forces (Fig. 7A). In this scenario, the
> deformation in the scaffold is not dependent on cell stiffness and therefore
> the resultant $\mathcal{E}_{m}$ is independent of $\mathcal{E}_{c}$.
>
> For the scaffold with aligned fibers also we find similar behavior of cell
> stiffness dependent sensing of the scaffold stiffness (Fig. 7B). For
> scaffold with aligned fibers the orientation of the cell relative to the
> fiber direction influences the stiffness sensed by the cell. This results in
> higher and lower values of $\mathcal{E}_{m}$ for cell oriented parallel and
> normal to the fiber direction, respectively. This effect is due to the fact
> that for a force-dipole oriented parallel to the fibers the scaffold
> deformation is mainly due to the stretching of the fibers as opposed to the
> fiber bending, which offers lesser resistance as compared to fiber
> stretching, when force-dipole is orientation normal to the fibers.
>
> From this insight into the effect of cell stiffness on scaffold stiffness
> sensing via force-dipole we created a map of combinations of fibrous
> scaffolds of different materials and their stiffness sensing by several cell
> types (Fig. 7C). This maps shows that the cells such as leukocytes (with
> stiffness of the order of $0.07-0.2$kPa [69]) and cardiocytes (with
> stiffness of the order of $100$kPa [70]) will sense difference stiffness of
> the same fibrous scaffold. As shown in the figure that this cell type
> dependence on the sensed stiffness can be very large upto multiple orders of
> magnitude. Further, the effect of cell type is also dependent on the
> scaffold material itself. For an example, for fibrous scaffold composed of
> relatively soft materials (such as fibrin with Young’s modulus $\approx
> 15$MPa [71]) the variation in the sensed stiffness by different cells is
> very small (see Fig. S4). On the other hand, for scaffold made up of stiffer
> materials (for example polystyrene with Young’s modulus $\approx 3$GPa [72])
> this variation is multiple orders of magnitude (Fig. S4). This demonstrates
> a quite intricate relationship between the cell type and scaffold material
> on the stiffness sensing by the cells.
>
> ##  4 Discussion
>
> Mechanical properties of tissue engineering scaffolds have been known to be
> critical for their application in tissue regeneration applications. This has
> propelled the research on understanding their mechanics experimentally as
> well as by mathematical and computational means, extensively. This has
> resulted in extensive works in this direction but most of the insights into
> the mechanical behavior of biomaterials in general and fibrous scaffolds in
> particular have been from the macroscopic view point [25, 26]. The
> understanding of the mechanical response of these materials at the length
> scale of a single cell has been quite limited. The present work is an
> attempt to fill this gap using a computational model of fibrous scaffold
> mechanics.
>
> A consideration of the microscopic length scale of scaffold ($\ell$ in this
> work) not very small as compared to the sample size or the size of a cell
> results in some very interesting properties. For example, we observe the
> effect of the sample size on the mechanical response of scaffold under
> uniaxial elongation for $\ell\not\ll L,W$. Even though this sample size
> effect is very well known in other materials with a well defined
> microstructure [73] its effect on fibrous biomaterials has not been very
> well studied [34]. It illustrates that during experimental testing of
> fibrous scaffolds under uniaxial or biaxial loading conditions a care is
> warranted. It is required that during these mechanical testing conditions
> materials are tested with multiple sample sizes so that the effects of
> microscopic length scale (if any) can also be taken into account during
> elastic moduli calculations. For instance for fibrous scaffolds with random
> fiber arrangement sample with large width should be used in uniaxial
> elongation testing for a consistent measurement of the Young’s modulus (Fig.
> 3D-E).
>
> Further, the sample size effect can also be used for indirect estimation of
> the extent of fiber fusion in the scaffolds. Since in scaffolds with higher
> fiber fusion the effect of sample width is smaller (Fig. 3D) we can estimate
> the microscopic length by performing uniaxial mechanical testing with
> different sample widths. This information can then be used with Fig. 2B to
> quantify the degree of fiber fusion in scaffold.
>
> This effect of the microscopic length scale, however, does not reflect in
> the finite element analysis of the continuum homogenization based models
> [28] of the fibrous scaffold under same condition as that in DFM. This is
> not surprising since the formulation of the continuum model (such as HGO
> model used in this work) is based on the assumption that the microscopic
> length scale is very small as compared to the sample size. This also means
> that for the situations where the cell size is comparable to the microscopic
> length, which is often the case in tissue engineering application, the
> utilization of continuum modeling is not completely justifiable. Another
> important advantage of the discrete fiber modeling over continuum approach
> is due to the fact that from DFM one can extract the mechanical response of
> a particular scaffold sample with a known (possibly from a non-destructive
> imaging technique such as computed tomography) fiber architecture. Continuum
> models of fibrous scaffolds, by virtue of dealing with average mechanical
> quantities, usually do not provide information about the variability in the
> scaffold response which becomes important when the microscopic length scale
> is not too small.
>
> The dependence of the sensitivity of the stiffness sensing by the cell on
> the cell’s own elastic properties presents another challenge to the practice
> of macroscopic mechanical testing (uniaxial/biaxial elongation or shear
> deformation) for the characterization of scaffold mechanics for tissue
> engineering. The use of macroscopic elastic moduli as a measure of the
> mechanical behavior of the scaffold are justifiable as long as the
> force/deformation applied by the cell is small enough where the constitutive
> relation of the substrate can be linear. The linear constitutive relation,
> even for small deformations, however, is not always correct for the fibrous
> materials [33]. It is possible to have two structurally very different
> fibrous scaffolds which can show identical stress-strain curves under
> uniaxial elongation tests but provide completely different micro-mechanical
> environment to the cells [34]. This limitation of the macroscopic mechanical
> tests has been acknowledged and has resulted in the use of atomic force
> microscopy (AFM) for the estimation of local mechanical properties of the
> biomaterials [74]. This, however, does not resolve the issue completely
> since the nature of the mechanical forces applied by the AFM (usually normal
> to the substrate surface) and the forces applied by the cells (tangential to
> the substrate surface) are not the same. Therefore, until a new experimental
> technique is devised which can assess scaffold mechanics under loading
> condition similar to that by a cell one has to apply combine mathematical
> modeling with the experimental measurements for assessing the micro-
> mechanical environment experienced by the cells.
>
> One of the many functions of the force application by the cells is also the
> probing the mechanical properties of the underlying substrates [67, 75]. The
> debate on the exact mechanism of stiffness sensing by the cells is still not
> settled [50, 51]. As mentioned previously, it, however, has been distilled
> to a handful of mechanisms involving sensing of resistance force by the cell
> or sensing of deformation or, more recently reported sensing of the stored
> energy [51]. In this work we have estimated the fibrous scaffold’s stiffness
> as a function of cell’s own elasticity which shows that the deformation
> sensing by the cell (after application of a fixed amount of force) can show
> cell type dependent response where the difference between the stiffness of a
> scaffold sensed by a stiff cell (such as a cardiocyte) and that by a soft
> cell (such as a leukocyte) can be as high as an order of magnitude (Fig.
> 7C). Further, we have also seen that the dependence of stiffness sensing (by
> force-dipole) on cell stiffness can be attributed to the smaller overall
> deformation of the system by stiffer cells. This also implies that the
> stiffness sensed by the cells may also depend on the magnitude of the cell
> forces where a cell applying small force will experience a softer scaffold.
> This can be an important factor since all the cells do not apply same amount
> of force [76, 77, 78].The force sensing by the cell after application of a
> fixed amount of deformation, however, does not depend on the mechanical
> properties of cell. Even though these observations require experimental
> validation they also suggest possible avenues to look into to understand the
> mechanisms of mechanosensing. It is also not unlikely that different cell
> types use different mechanisms for the same goal of mechanosensing and,
> therefore, we may not have a single ‘winner’ among different possibilities.
>
> Furthermore, here we have assumed that the cell applies a fixed amount of
> force irrespective of the scaffold stiffness. This may appear to be in
> conflict with the fact that cells undergo cytoskeletal remodeling [67, 79]
> depending on the scaffold stiffness and, therefore, the force/deformation
> applied by the cells can depend on the scaffold stiffness. Therefore, it
> needs to be pointed out here that the mechanosensing we are studying in this
> work is only limited to the very early stages of the cell contact with the
> scaffold when the cytoskeletal remodeling has not yet taken place. Of
> course, in order to look at the long term response of the cell as a function
> of scaffold stiffness we have to take into account the cytoskeletal
> remodeling and focal adhesion dynamics [80].
>
> Finally, it is also important to keep an eye on the limitations of the
> discrete fiber model presented here. First, the consideration of the
> scaffold to be a two dimensional structure puts a limitation on its
> applicability. This assumption, however, is justified in case of the
> electrospun fibrous scaffolds where the sample thickness is almost always
> very small as compared to its other dimensions [54].
>
> Furthermore, here we take into account the fusion between the fibers (also
> studied in few other works [35]) but fusion points are modeled only to
> transfer mechanical fibers among the participating fibers. This
> simplification is justifiable for the fibrous scaffolds where the fiber
> deformation is stretching dominated which is the case in this work. For the
> ‘soft’ fibrous scaffolds with highly curved fibers (that is $\xi\gg 1$) the
> deformation is mainly fiber bending and there we have to also take into
> account the deformation of the fiber-fiber junctions. Here we have also
> assumed the fiber fusion to be perfect, that is the fused fibers do not
> detach from each other, which may not be the case when large deformations
> are imposed on the scaffolds. The microscopic cellular forces, however, are
> not usually large enough to cause breaking of fibers or fiber junctions.
> Furthermore, in addition to the fiber fusion the mechanical interaction
> among fibers can also be due to the friction between [81] the two fibers in
> contact which has also been ignored here.
>
> Here, in order to keep the system as simple as possible we modeled the
> cellular forces by a force or strain dipole. Even with this simple
> representation the displacement filed obtained has a qualitative similarity
> with that in experiments [66]. For a more detailed understanding we also
> need to model the cell with more details where cell geometry, ligand-
> receptor based cell-scaffold adhesion, cytoskeletal remodeling and focal
> adhesion dynamics are also taken into account. Among these factors the focal
> adhesion dynamics as a function of the substrate stiffness is very important
> since it also governs the short and long term response of the cell on
> fibrous scaffold. Similarly, for fibers also more detailed Euler-Bernoulli
> [82] or Timoshenko [83] beam based models should be taken into account if
> the cell size becomes comparable to that the fiber dimensions.
>
> ##  5 Conclusions
>
> In this work we have looked at the mechanics of the tissue engineering
> fibrous scaffolds using mathematical modeling. Modeling of the mechanical
> behavior of fibrous scaffolds present a unique challenge since their
> microscopic length scale is usually comparable to physical length or size of
> the cells. We have explained that this property results in sample size
> dependent macroscopic mechanical properties. This work also presents the
> microscopic mechanics of fibrous scaffold due to localized cellular forces
> which generate anisotropic deformation which decays exponentially with
> distance. Finally, the mathematical model is also used to study the
> mechanism of stiffness sensing by the cells. This shows that the stiffness
> of the scaffold sensed by the cell can depend very strongly on the cellular
> machinery (force dipole vs strain dipole) and cell stiffness. These
> demonstrations will help not only in designing of tissue engineering
> scaffolds with specific application but also in formulating the experimental
> strategies for the study of the mechanosensing by the cells.
>
> ## Appendix A: Estimation of average fiber curvature
>
> Curvature of a fiber (considering it as a two dimensional curve for 2D DFM)
> is a characteristic property which is dependent on its exact shape and is
> usually defined at each location along the fiber. In the discrete fiber
> model, however, we consider only a statistical description of the fibers in
> terms of their normalized length $\xi=l/R$ (a single scalar quantity for
> each fiber) and do not take into account the exact shape of each fiber (Fig.
> A.1). Therefore, for the constitutive modeling of the bending of the fibers
> we have to estimate the average fiber curvature in terms of $\xi$.
>
> Figure A.1: Schematic showing average curvature estimation for a fiber.
>
> As shown in Fig. A.1 the exact geometry of fiber in 2D can be described in
> terms of the tangent angle $\beta(s)$ made by the fiber with its director
> vector (vector joining its two ends). This angle is a function of arclength
> parameter $s$ defined along the fiber length. Therefore we have
>
> $\frac{1}{\xi}=\frac{R}{l}=\int\limits_{0}^{1}\cos(\beta(s))ds.$ (A.1)
>
> In order to approximate the shape of the curved fiber we assume that the
> tangent angle $\beta$ is given by
> $\beta(\tilde{s})=\sum\limits_{n=1}^{\infty}\beta_{n}=\sum\limits_{n=1}^{\infty}a_{n}\sin(n\pi\tilde{s})$
> where $n$ is an integer and $\tilde{s}=s/l$ is the normalized arclength
> parameter. Therefore, the amplitude of the $n^{\text{th}}$ mode $a_{n}$ can
> be obtained from the equation (A.1). If we assume that the fibers are not
> too curved, that is $\xi\not\gg 1$, we can approximate the fiber shape by
> only first mode to obtain
>
> $a_{1}=2\sqrt{2}\sqrt{1-\frac{1}{\sqrt{\xi}}}.$ (A.2)
>
> Therefore, the average root mean square curvature of the fiber can be
> written as
>
>
> $\langle\kappa_{0}(\xi)\rangle=\left(\int\limits_{0}^{1}\left(\frac{d\beta}{d\tilde{s}}\right)^{2}d\tilde{s}\right)^{1/2}\approx
> 2\pi\sqrt{1-\frac{1}{\sqrt{\xi}}}.$ (A.3)
>
> ## Appendix B: Sample size effect on Young’s modulus
>
> For a rectangular sample of dimensions $L\times W$ (Fig. B.1) with straight
> fibers, where fiber orientation is given by probability density function
> $p_{{}_{\theta}}(\theta)$, the strain in a fiber oriented at angle $\theta$
> when $\varepsilon\ll 1$ elongation strain is applied on the sample is given
> by
>
>
> $\varepsilon_{f}=\frac{L\sqrt{\tan^{2}\theta+(1+\varepsilon)^{2}}-L\sqrt{\tan^{2}\theta+1}}{L\sqrt{\tan^{2}\theta+1}}\approx\varepsilon\cos^{2}\theta$
> (B.1)
>
> If the fiber material is linear elastic with Young’s modulys $Y_{f}$ and
> cross-sectional are $A_{f}$ then the resistance force due to this one fiber
> in the direction of elongation is
>
> $f_{y}(\theta)=Y_{f}\varepsilon_{f}A_{f}\cos\theta\approx
> Y_{f}A_{f}\varepsilon\cos^{3}\theta.$ (B.2)
>
> Writing the resistance of the scaffold as the collective response of all the
> fibers, we get
>
>
> $F_{y}=N_{0}\int\limits_{0}^{W}\left(\int\limits_{x_{1}}^{x_{2}}f_{y}(\theta)p_{{}_{\theta}}(\theta)d\theta\right)dx,$
> (B.3)
>
> where $N_{0}$ is the number of end-to-end fibers per unit width of the
> sample. For $W\ll L$, we have $x_{1}\approx-x/L$, $x_{2}\approx(W-x)/L$ and
> $\cos\theta\approx 1$ which give
>
> $F_{y}\approx
> N_{0}Y_{f}A_{f}\varepsilon\int\limits_{0}^{W}\int\limits_{-x/L}^{(W-x)/L}p_{{}_{\theta}}(0)d\theta
> dx=\frac{N_{0}Y_{f}A_{f}\varepsilon p_{{}_{\theta}}(0)W^{2}}{L}.$ (B.4)
>
> On the other hand, for $W\gg L$ with uniform probability density
> $p_{{}_{\theta}}(\theta)=1/\pi$ we obtain
>
>
> $F_{y}\approx\frac{N_{0}Y_{f}A_{f}\varepsilon}{\pi}\int\limits_{0}^{W}\int\limits_{x_{1}}^{x_{2}}\cos^{3}\theta
> d\theta dx=\frac{4}{3}\frac{N_{0}Y_{f}A_{f}\varepsilon W}{\pi}.$ (B.5)
>
> Calculating Young’s modulus of the scaffold as
> $\mathcal{Y}_{m}=\cfrac{F_{y}}{W\varepsilon t}$, with $t$ being sample
> thickness, we obtain the dependence of sample width on Young’s modulus.
>
> Figure B.1: Schematic showing uniaxial elongation of a rectangular fibrous
> scaffold sample.
>
> ## Acknowledgements
>
> We are grateful to Dr. Prasoon Kumar for providing electron micrographs of
> electrospun scaffolds for Fig. 1. DJM thanks the MoE of scholarship and MSR
> was supported by Institute Seed Grant from Indian Institute of Technology
> Hyderabad.
>
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