# Multiscale composite model of fiber-reinforced tissues with direct representation of sub-tissue properties

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## Abstract

In many fiber-reinforced tissues, collagen fibers are embedded within a glycosaminoglycan-rich extrafibrillar matrix. Knowledge of the structure–function relationship between the sub-tissue properties and bulk tissue mechanics is important for understanding tissue failure mechanics and developing biological repair strategies. Difficulties in directly measuring sub-tissue properties led to a growing interest in employing finite element modeling approaches. However, most models are homogeneous and are therefore not sufficient for investigating multiscale tissue mechanics, such as stress distributions between sub-tissue structures. To address this limitation, we developed a structure-based model informed by the native annulus fibrosus structure, where fibers and the matrix were described as distinct materials occupying separate volumes. A multiscale framework was applied such that the model was calibrated at the sub-tissue scale using single-lamellar uniaxial mechanical test data, while validated at the bulk scale by predicting tissue multiaxial mechanics for uniaxial tension, biaxial tension, and simple shear (13 cases). Structure-based model validation results were compared to experimental observations and homogeneous models. While homogeneous models only accurately predicted bulk tissue mechanics for one case, structure-based models accurately predicted bulk tissue mechanics for 12 of 13 cases, demonstrating accuracy and robustness. Additionally, six of eight structure-based model parameters were directly linked to tissue physical properties, further broadening its future applicability. In conclusion, the structure-based model provides a powerful multiscale modeling approach for simultaneously investigating the structure–function relationship at the sub-tissue and bulk tissue scale, which is important for studying multiscale tissue mechanics with degeneration, disease, or injury.

## Keywords

Finite element modeling Multiscale modeling Structure-based Structure–function relationship Annulus fibrosus## 1 Introduction

Many soft tissues in the body include highly aligned collagen fibers embedded in a glycosaminoglycan-rich extrafibrillar matrix. The matrix allows for water and nutrient absorption, which is important for maintaining tissue homeostasis (Yang and O’Connell 2019), while fibers create anisotropic mechanical properties that allow the tissue to withstand large tensile loads. For example, tendons and ligaments have a single family of fibers, providing the tissue with greater stiffness along the primary in situ loading direction (Benjamin and Ralphs 1997). Meanwhile, tissues that undergo multiaxial loadings have more complex fiber networks, from two fiber populations, such as arterial walls and the annulus fibrosus (AF) of the intervertebral disk (Holzapfel et al. 2000; Adams and Roughley 2006), to randomly distributed fibers, such as skin (Cotta-Pereira et al. 1976).

Structural and mechanical behaviors of fibers and the matrix have been shown to change with degeneration, disease, and injury. For example, the AF has a cross-ply fiber structure (Cassidy et al. 1989; Marchand and Ahmed 1990), where collagen fibers can reorient under tensile loading. The amount of fiber reorientation has been shown to decrease with degeneration (Guerin and Elliott 2006), partly due to matrix stiffening and increased collagen cross-linking (Fujita et al. 1997; Wagner et al. 2006; O’Connell et al. 2011), which can lead to increased stress concentrations within the disk, triggering catabolic remodeling that can cause tissue failures (Antoniou et al. 1996; Adams and Roughley 2006). Failure of these fiber-reinforced tissues can cause a wide range of clinical issues, from mechanical dysfunctions of the disk to death (e.g., a ruptured aneurysm) (Juvela et al. 2000; Rubin 2007; Erwin and Hood 2014; O’Connell et al. 2015). Therefore, it is important to understand the role sub-tissue properties (e.g., fiber networks, matrix biochemical compositions, etc.) play on bulk tissue mechanics.

Although experimental studies have provided important information regarding bulk tissue mechanics, there are few studies that have directly measured sub-tissue properties due to challenges in conducting tests on individual tissue subcomponents. Thus, many researchers have complemented experimental data with structure-based constitutive modeling (Spencer 1984) to investigate tissue structure–function relationships. Commonly, in these studies, phenomenological strain energy density functions developed based on the model are curve fit to experimental data of bulk tissue mechanics to calibrate for model parameters that describe the structural contributions of tissue subcomponents and their interactions. The structure-based constitutive models have been valuable for highlighting the importance of fiber–matrix interactions with respect to degeneration and different loading conditions (Wu and Yao 1976; Klisch and Lotz 1999; Elliott and Setton 2001; Bass et al. 2004; Wagner and Lotz 2004; Yin and Elliott 2005; Peng et al. 2006; Wagner et al. 2006; Guerin and Elliott 2007; Nerurkar et al. 2008, 2011; O’Connell et al. 2009, 2012). However, these models often include a large number of hypothesized invariant terms, generating nonunique model parameters that cannot be easily compared or applied across studies (Yin and Elliott 2005; Guo et al. 2012). Directly linking model parameters to tissue physical properties and measurable tissue compositional changes has also been difficult as most parameters are not physically interpretable (Yin and Elliott 2005; Eskandari et al. 2019).

Additionally, the constitutive models normally performed poorly in simultaneously predicting tissue mechanics under multiple test configurations, due to the commonly applied model parameter calibration approach. Typically, the models are calibrated by curve fitting to study-specific stress–strain curves, often from a single test configuration (Sun et al. 2005; Schmidt et al. 2006, 2007), resulting in a limited model accuracy and robustness under other loading modalities. For example, previous work showed that constitutive models calibrated to uniaxial tension data were not able to accurately predict mechanical behaviors under biaxial tension or simple shear (Bass et al. 2004; O’Connell et al. 2012). Simultaneous curve fitting to multiple loading modalities has also proved challenging, often resulting in relatively poor model fits (Klisch and Lotz 1999; Wagner et al. 2006).

To address some of these issues, there has been a growing interest in using finite element models (FEM) to study three-dimensional tissue deformations. So far, most bulk tissue-scale FEMs employ homogenization theory, where every model element includes a combined and homogenized description of tissue subcomponents (e.g., fibers and the matrix) (Bensoussan et al. 1978; Sanchez-Palencia and Zaoui 1987; Jones 1999; Yin and Elliott 2005). This approach has allowed researchers to study three-dimensional stress and strain distributions, which has been valuable for predicting peak strains at failure (Eberlein et al. 2001) and for directing experimental protocol designs (Jacobs et al. 2013; Werbner et al. 2017). Unfortunately, homogenization of tissue subcomponents does not accurately represent the heterogeneous architecture of native tissues, where fibers and the extrafibrillar matrix are distinct materials that occupy separate volumes. Therefore, these models are not capable of describing and explaining some recent experimental observations, including variations in collagen fibril diameter with osmotic loading and changes in interfibrillar strain field with mechanical loading (Han et al. 2012; Vergari et al. 2016).

To address the limitations of the discussed modeling approaches, the objective of this study was to develop and validate a structure-based FEM that can be used to investigate multiscale structure–function relationships of fiber-reinforced tissues. To do so, we developed a model based on the native heterogeneous structure of the human AF, where fibers and the extrafibrillar matrix were described as two distinct materials occupying separate volumes (SEP model). The model was calibrated and validated using a multiscale framework. Model parameters were calibrated to sub-tissue-scale mechanical test data (Holzapfel et al. 2005), while model was validated at the bulk scale by comparing model-predicted multiaxial mechanics of multi-lamellar structures with multi-lamellar experimental test data. Multi-lamellar models developed using homogenization theory (HOM models) were also created, and their validation results were compared to results from the multiscale structure-based models. The second objective of this study was to investigate the relationship between specimen geometry and bulk tissue mechanics using the validated multiscale structure-based model. Although this study was conducted using AF morphology, the approaches and techniques employed here are applicable to other fiber-reinforced biological tissues and composites.

## 2 Methods

### 2.1 Model development

A structure-based approach was employed during SEP model development to describe the AF as a fiber-reinforced composite containing distinct materials for the extrafibrillar matrix (matrix) and fiber bundles (SEP for “separate model”; Fig. 1b). Fiber bundles (fibers) were described as being uniformly distributed, full-length cylinders welded to the surrounding matrix (Shirazi-Adl et al. 1984; Goel et al. 1995; Michalek et al. 2009; Schollum et al. 2010). The radius of each fiber bundle was 0.06 mm, and interfibrillar spacing within each lamella was 0.22 mm (Marchand and Ahmed 1990). Fiber bundles were oriented at ± 30° (Fig. 1b—\(\theta = 30^\circ\)) to the transverse plane to represent specimens prepared from the middle-outer AF (Cassidy et al. 1989).

*k*) was described as being strain-dependent (Holmes–Mow description; Eq. 1):

In Eq. 1, \(J\) is the determinant of the deformation gradient tensor (* F*), \(k_{0}\) is represented hydraulic permeability in the reference state (\(k_{0} = 0.0064\) mm

^{4}/N s), \(\varphi_{0}\) represents the solid volume fraction (\(\varphi_{0} = 0.3\)), \(\alpha\) represents the power-law exponent (\(\alpha = 2\)), and \(M\) represents exponential strain-dependence coefficient (\(M = 4.8\)) (Mow et al. 1984; Antoniou et al. 1996; Iatridis et al. 1998; Gu et al. 1999; Beckstein et al. 2008; Cortes et al. 2014; O’Connell et al. 2015). Fixed charge density, which represents the tissue proteoglycan content and drives tissue swelling, was set to -100 mmol/L for the matrix (middle-outer AF) and 0 mmol/L for fibers (i.e., no active swelling in the fibers) (Urban and Maroudas 1979; Huyghe et al. 2003). The osmotic coefficient (0.927) was determined using a linear interpolation of the data reported in Robinson and Stokes (1949) and Partanen et al. (2017). Free diffusivity (\(D_{0} )\) and AF tissue diffusivity (

*D*

_{AF}) of Na

^{+}and Cl

^{−}were set based on data in Gu et al. (2004), and 100% ion solubility was assumed (\(D_{{0,{\text{Na}}^{ + } }} = 0.00116\) mm

^{2}/s; \(D_{{0,{\text{Cl}}^{ - } }} = 0.00161\) mm

^{2}/s; \(D_{{{\text{AF}},{\text{Na}}^{ + } }} = 0.00044\) mm

^{2}/s; \(D_{{{\text{AF}},{\text{Cl}}^{ - } }} = 0.00069\) mm

^{2}/s).

For FEMs that employed homogenization theory (HOM), a compressible hyperelastic Holmes–Mow material description was used to describe the ground matrix substance. Similar to SEP models, AF nonlinearity and anisotropy were incorporated by embedding a fiber description within the matrix. Fibers were described using a power-linear stress–strain relationship. Strain energy density functions for the ground matrix substance and fibers were identical to those used in the SEP models (Eqs. 3–6).

### 2.2 Multiscale model calibration and validation framework

Young’s modulus obtained from SEP and HOM model calibration compared to experimental data [EXP, average (standard deviation)].

Experimental data taken from Holzapfel et al. (2005)

EXP | SEP | HOM | |
---|---|---|---|

| 5.96(3.05) | 5.7 | 5.9 |

| 32.5(12.1) | 32.0 | 30.3 |

| 77.6(20.0) | 74.6 | 70.0 |

*gripped*,

*vertebrae*-

*attached*, and

*parallel*-

*plate*, Fig. 2). The gripped boundary condition represented sandpaper glued to specimens and used to interface with testing equipment (Acaroglu et al. 1995; Elliott and Setton 2001; Guerin and Elliott 2006; O’Connell et al. 2009, 2012). The vertebrae-attached boundary condition referred to the case where tissue testing was prepared with the adjacent vertebrae attached to the AF and used to interface with test equipment (Green et al. 1993; Żak and Pezowicz 2016). The parallel-plate boundary condition described the case where specimens were clamped between polystyrene parallel plates for simple shear testing (Fujita et al. 2000).

Each validation model was loaded in a two-step process. Free swelling in 0.15 M phosphate-buffered saline was simulated prior to mechanical loading to account for specimen hydration. For uniaxial tension, a 20% engineering strain was applied. For biaxial tension, corresponding strain was applied in the circumferential and axial directions to represent the relative strain ratios reported in the literature [circumferential/axial strain ratios = 1:1 (equibiaxial) and 1:0 (axial-fixed)]. The simulation for biaxial tension was terminated when strain in either direction reached 15%. For simple shear, a 10% shear strain was applied in either circumferential or axial direction. Linear-region, apparent, or shear modulus was calculated as the slope of the corresponding stress–strain curve in the linear region and compared to values reported in the literature. Valid SEP or HOM model predictions of multi-lamellar mechanics were defined as predicted bulk modulus being within one standard deviation of reported mean values (Green et al. 1993; Acaroglu et al. 1995; Fujita et al. 2000; Elliott and Setton 2001; Guerin and Elliott 2006; O’Connell et al. 2009; Żak and Pezowicz 2016).

^{−1}) (Green et al. 1993; Kasra et al. 2004). Mean and standard deviations for moduli were pooled across studies by calculating the weighted average of mean or standard deviations (\(E_{\text{pooled}} = \frac{{\mathop \sum \nolimits_{i = 1}^{s} n_{i} E_{i} }}{{\mathop \sum \nolimits_{i = 1}^{s} n_{i} }}\), \({\text{SD}}_{\text{pooled}} = \frac{{\sqrt {\mathop \sum \nolimits_{i = 1}^{s} n_{i} {\text{SD}}_{i}^{2} } }}{{\mathop \sum \nolimits_{i = 1}^{s} n_{i} - s}}\), where

*s*represents total number of studies included,

*n*represents study-specific sample size, and

*E*

_{i}and SD

_{i}represent the mean and standard deviation of the modulus reported in each study).

Summary of experimental data used for model validation, including sample size (*n*), tested specimen orientation, testing boundary condition, loading rate, reported modulus, and linearity of multi-lamellar stress–strain response (*NL* nonlinear; *PL* pseudo-linear). Bulk tissue mechanics reported as [average (standard deviation)] (*N.P.* not provided in study)

Uniaxial tension | Biaxial tension | Simple shear | ||||||
---|---|---|---|---|---|---|---|---|

Green et al. (1993) | Acaroglu et al. (1995) | Elliott and Setton (2001) | Guerin and Elliott (2006) | O’Connell et al. (2009) | Zak and Pezowicz (2016) | O’Connell et al. (2012) | Fujita et al. (2000) | |

| 9 | 15 | Ax.: 12; circ.: 20 | 8 | 7 | 18 | 16 | 20 |

Orientation tested | Ax. | Circ. | Ax.; circ. | Circ. | Ax.; circ. | Ax. | Ax.-circ. plane | Ax.; circ. |

Boundary condition | Vertebrae-attached | Gripped | Gripped | Gripped | Gripped | Vertebrae-attached | Gripped | Parallel-plate |

Loading rate | 4 mm/s | 0.0001 s | 0.0001 s | 0.0001 s | 0.0001 s | 0.5 mm/s | 0.0001 | N.P. |

Modulus (MPa) | Linear | Linear | Toe Linear Toe Linear | Toe Linear | Linear Toe Linear | Linear | N.P. | Shear Shear |

Linearity | NL | NL | Ax.: PL; circ.: NL | NL | Ax.: PL; circ.: NL | NL | NL | N.P. |

### 2.3 Effect of specimen geometry on tensile mechanics

Following model validation, the effect of specimen geometry on AF bulk mechanics was investigated, because experimental observations noted that modulus was sensitive to specimen geometry (Adams and Green 1993; Lechner et al. 2000; Werbner et al. 2017). Additional uniaxial multi-lamellar SEP models were created along the circumferential direction (*n* = 50 models; Fig. 1a). Specimen geometry for length was varied between 6 and 15 mm in 1 mm increments, and width was varied between 2 and 3 mm in 0.25 mm increments, resulting in length-to-width aspect ratios (AR) between 2.0 and 7.5. Uniaxial tension was applied as described above, and the predicted linear-region modulus was calculated. During loading, specimen top and bottom surfaces were constrained to restrict displacement in the loading direction.

*y*) and specimen geometry (\(x_{1}\): length; \(x_{2}\): 1/width; Eq. 7; R software, Foundation for Statistical Computing, Vienna, Austria). In Eq. 7, \(\beta_{i}\) represent regression parameters, which were determined using the least squares method, and \(\varepsilon\) represents errors in the statistical model. The effect of specimen width was represented as 1/width to incorporate aspect ratio as an interaction term (i.e., length/width or \(x_{1} x_{2}\)). If a parameter, \(\beta_{i}\), was determined to be statistically insignificant, it was removed from the model and the analysis was repeated with the reduced linear regression model. Significance was assumed for

*p*values ≤ 0.05. The relative contribution of specimen length, width, and aspect ratio to AF tensile modulus was calculated using the relaimpo package and reported as a percent (Grömping 2006).

## 3 Results

### 3.1 Multiscale model calibration and validation

Summary of calibrated model parameters for SEP and HOM models. Experimental data from sub-tissue mechanical tests are reported as [average (standard deviation)].

Experimental data taken from Fujita et al. (1997), Elliott and Setton (2001), Holzapfel et al. (2005), Van der Rijt et al. (2006), Shen et al. (2008), O’Connell et al. (2009), and Cao et al. (2009) (*N.A.* not applicable)

EXP | SEP | HOM | ||
---|---|---|---|---|

Matrix | Fibers | |||

| 0.2(0.19) | 0.22 | 0.22 | 0.22 |

| 0.59(0.35) | 0.3 | 0.3 | 0.3 |

| N.A. | N.A. | 1 | 1 |

| 0.86(0.45) | N.A. | 0.58 | 0.53 |

| N.A. | N.A. | 5.95 | 6 |

| 1.09(0.06) | N.A. | 1.07 | 1.09 |

SEP- and HOM-predicted linearity of multi-lamellar tissue stress–strain behavior (*Lin.* linearity; *NL* nonlinear; and *PL* pseudo-linear). Model-predicted moduli (mod.) were compared to pooled (if applicable) experimental (EXP) data in the literature [average (standard deviation)]

Uniaxial tension | Biaxial tension | Simple shear | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Circ. (gripped) | Ax. (gripped) | Ax. (vertebrae-attached) | Equibiax., circ. (gripped) | Equibiax., ax. (gripped) | Ax.-fixed, circ. (gripped) | Circ. (parallel-plate) | Ax. (parallel-plate) | ||||||||

Lin. | Toe mod. (MPa) | Linear mod. (MPa) | Lin. | Mod. (MPa) | Lin. | Mod. (MPa) | Lin. | Apparent mod. (MPa) | Lin. | Apparent mod. (MPa) | Lin. | Apparent mod. (MPa) | Shear mod. (kPa) | Shear mod. (kPa) | |

EXP | NL | 2.52(2.08) | 21.10(15.80) | PL | 0.67(0.57) | NL | 20.11(11.25) | NL | ~ 60 | NL | ~ 30 | NL | ~ 40 | 111(56) | 224(112) |

SEP | NL | 4.70 | 23.0 | PL | 0.60 | NL | 18.6 | NL | 58 | NL | 21 | NL | 31 | 163 | 1375 |

HOM | NL | 3.20 | 106.2 | PL | 0.70 | NL | 59.7 | PL | 33 | PL | 7 | PL | 16 | 326 | 1697 |

*E*

_{circ.}); in the axial direction, SEP and HOM models underestimated the apparent modulus by ~ 30% and ~ 70%, respectively (Fig. 7d—Equibiax.,

*E*

_{ax.}). Under the axial-fixed condition, SEP and HOM models underestimated the circumferential-direction apparent modulus by ~ 20% and ~ 60%, respectively (Fig. 7d—Ax.-fixed,

*E*

_{circ.}).

### 3.2 Effect of specimen geometry on tensile modulus

_{width = 7 mm}≈ 1.8 × slope

_{width = 15 mm}). This finding highlights the dependence of AF tensile modulus on the interaction between specimen length and width (i.e., aspect ratio), where tensile modulus decreased with an increase in aspect ratio (Fig. 9c). Moreover, it appeared that tensile modulus approached a horizontal asymptote as the aspect ratio exceeded 4.0 (ASTM guidelines for uniaxial test specimens (ASTM 2003, 2004); Fig. 9c—gray dots). Therefore, AF tensile modulus was a function of specimen length, width, and aspect ratio (Eq. 8; Supplementary Table). Lastly, based on the relative contribution analysis, AF tensile modulus was most sensitive to specimen width (48% contribution), followed by aspect ratio (36% contribution), and specimen length (16% contribution).

## 4 Discussion

It is common for model parameter calibrations to be performed at the same scale as the study of interest in both constitutive and finite element modeling studies. For example, to study bulk tissue mechanics, model calibration would be conducted based on multi-lamellar test data from a dataset obtained from a single loading modality, often limiting the model’s ability to accurately predict tissue mechanics under other loading modalities (Bass et al. 2004). Moreover, findings from this study and experimental observations suggest that this curve-fitting approach is limited to specimens with a specific geometry constrained by a particular boundary condition, further restricting the predictive power and the robustness of the model (Adams and Green 1993; Sun et al. 2005; Jacobs et al. 2013; Werbner et al. 2017). Additionally, while these models are widely used to understand contributions of sub-tissue properties to bulk tissue mechanics, it has been difficult to establish relationships between model parameters and tissue physical properties (e.g., collagen stiffness) or biochemical compositions (e.g., cross-links) as model parameters can be nonunique and are purely mathematical coefficients without physical significance (Yin and Elliott 2005; Eskandari et al. 2019).

To address these limitations, we employed a unique multiscale framework for model calibration and validation in this study. Specifically, for both SEP and HOM model types, we considered multi-lamellar AF as a superposition of individual lamellae, which represented the fundamental structural unit (Holzapfel et al. 2005). While model calibration was performed at the sub-tissue scale using single-lamellar experimental data, model validation was performed at bulk tissue scale by predicting multiaxial mechanics using multi-lamellar models. This more rigorous approach ensured the accuracy and robustness of the model, if validated, such that the SEP model can be used to investigate tissue-level mechanics under multiple loading configurations and to understand the role of sub-tissue properties on tissue-level mechanics. Additionally, when developing SEP models, individual AF lamellae were modeled structure-based using known anatomical measurements, resulting in multi-lamellar models with a fibrous network that better resembled the native tissue.

The parameter calibration of single-lamellar SEP and HOM models resulted in a similar stress–strain response with almost identical computational moduli, suggesting that SEP and HOM models may predict similar mechanical behaviors for multi-lamellar specimens if the two modeling approaches shared a comparable accuracy and robustness. However, the SEP model type was rigorously validated (i.e., accurately predicted bulk tissue stress–strain response and corresponding moduli) in ten of 13 validation cases (> 75% passing rate), while the HOM model type was only validated in one validation case, proving the SEP model as a more accurate and robust modeling approach. Although the SEP model slightly overestimated the uniaxial tensile modulus in the axial direction as reported in O’Connell et al. (2009), we considered the SEP model prediction as acceptable, due to the relatively small difference in absolute values (difference between model-predicted modulus and experimental data = 0.19 MPa). Additionally, since only one representative stress–strain curve could be used for each biaxial tension validation case, it is also possible that the SEP model may be acceptable for describing axial-direction mechanics under equibiaxial loading (prediction was within 30% of the reported data) (O’Connell et al. 2012). However, it should be noted that the SEP model greatly overestimated the axial-direction shear modulus, which may be due to fibers being described as continuous bundles, resulting in an increased tissue stiffness due to the immediate engagement of the fiber bundles that extended between the parallel plates after the applied loading (Szczesny et al. 2015, 2017).

Attributed to the multiscale calibration framework, the majority of SEP model parameters (six of eight parameters) could be directly linked to tissue physical properties. The parameters included modulus and Poisson’s ratio of the ground matrix substance (*E*_{matrix}, *ν*_{matrix}), collagen fiber modulus (*E*_{lin.}), and transition strain (\(\lambda_{0}\)). Additionally, all calibrated values agreed well with reported values (Table 2 and Fig. 4) (Fujita et al. 1997; Elliott and Setton 2001; Holzapfel et al. 2005; Van der Rijt et al. 2006; Shen et al. 2008; Cao et al. 2009; O’Connell et al. 2009). This suggests that the SEP model parameters represent intrinsic tissue properties, broadening the model’s ability to study the effect of degeneration, disease, or injury on tissue mechanics. Particularly, the effect of tissue degeneration and regeneration can be investigated by adjusting fixed charge density of the extrafibrillar matrix, which is indicative to tissue degeneration in native tissues or tissue growth in engineered constructs (Adams and Roughley 2006; Nerurkar et al. 2007). The effect of disease can be investigated by varying fiber modulus, which has been shown to increase with greater fiber cross-linking with diabetes (Li et al. 2013; Svensson et al. 2018). Lastly, the effect of injury, which has been found to be rate dependent, can be investigated by changing the computational loading rate (Wang et al. 2000; Kasra et al. 2004).

To further demonstrate the predictive power of the SEP model, we evaluated the relationship between specimen geometry and AF tensile modulus, based on experimental observations that reported modulus sensitivity to specimen width (Adams and Green 1993; Werbner et al. 2017). A multivariate linear regression model was used to characterize AF tensile modulus as a function of specimen geometry, where specimen length and width were investigated as main factors and aspect ratio was evaluated as an interaction term. The regression analysis suggested that AF tensile modulus was a function of specimen length, width, and aspect ratio. Specifically, AF tensile modulus increased with specimen width and decreased with specimen length and aspect ratio, with specimen width being the most dominant factor. Therefore, unlike traditional engineering materials, AF tensile modulus may not be considered an intrinsic material property due the composite heterogeneous structure of the tissue, and it may be necessary to account for differences in specimen geometry when comparing data across studies. Our findings also suggest that obtaining consistent bulk tissue properties along the circumferential direction may be possible by using specimens with large aspect ratios and a smaller width, which agrees with recent work on meniscus, tendons, and ligaments (Wren et al. 2001; Peloquin et al. 2016; Creechley et al. 2017). Interestingly, Adams and Green (1993) and Werbner et al. (2017) both observed an increase in modulus as the midlength width relative to the grip width decreased. While the midlength-to-grip width ratio was not varied in this study, differences in trends may be due to a difference in fiber engagement, which can be directly evaluated with the SEP model, but not the HOM model.

A few assumptions were made to simplify the current SEP model. First, the fiber network did not include fiber dispersion (Guo et al. 2012), potential fibers in the radial direction (Marchand and Ahmed 1990), variation in fiber diameter or length (Marchand and Ahmed 1990; Han et al. 2012), or fiber–fiber interactions (e.g., cross-links). Particularly, cross-links have been shown to play an important role in tissue subfailure and failure mechanics and will be included in future iterations of the model (Moore et al. 1996; Elliott and Setton 2001; Adams and Roughley 2006; Guerin and Elliott 2006; Provenzano and Vanderby 2006; Roeder et al. 2009; O’Connell et al. 2009; Isaacs et al. 2014). Second, the current model did not investigate different mechanisms for fiber–matrix interactions, which have been suggested to be important for stress distribution during loading (Bruehlmann et al. 2004; Szczesny et al. 2015, 2017; Vergari et al. 2016).

In this study, we developed and validated a multiscale structure-based finite element model that accurately and robustly predicted AF bulk tissue mechanics under multiple loading configurations. Modeling fibers and the extrafibrillar matrix as separate materials, based on the native tissue architecture, resulted in uniquely determined model parameters with physical interpretations. Applying a multiscale framework for model calibration and validation resulted in a rigorous validation process that ensured and improved model accuracy and robustness. In conclusion, the multiscale structure-based modeling approach allows for studies that simultaneously investigate tissue- and sub-tissue-scale mechanics, which will be important for studying multiscale tissue mechanics with degeneration, disease, and injury (Iatridis and Gwynn 2004; Iatridis et al. 2005).

## Notes

### Acknowledgements

Funding was provided by National Science Foundation (Grant No. 1760467).

### Compliance with ethical standards

### Conflict of interest

The authors declare no conflict of interest.

## Supplementary material

## References

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