# A Tractable, Transferable, and Empirically Consistent Fibrous Biomaterial Model

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Model Domain, Volume Fraction, and Spatial Distribution of Fibers

#### 2.2. Drawing Random Numbers from Probability Distributions

#### 2.3. Fiber Diameters and Lengths

#### 2.4. Fiber Orientations

#### 2.5. Fiber Tortuosity

#### 2.6. A Fiber as a Chain of Particles and Non-Penetration Conditions

_{c}are the stiffnesses of the bond, angle, and contact potential, respectively; ${\mathit{r}}_{ij}$ is the distance vector between particles $i$ and $j$; ${R}_{ij}$ is the distance between particles $i$ and $j$; ${R}_{0,ij}$ is the equilibrium distance between particles $i$ and $j$; ${R}_{c,ij}$ is the cutoff distance of the contact potential; ${\theta}_{ijk}=\mathrm{acos}\left({\mathit{r}}_{ij}\xb7{\mathit{r}}_{kj}/{R}_{ij}{R}_{kj}\right)$ is the angle between particles $i$, $j$, and k; and ${\theta}_{0,ijk}$ is the equilibrium angle between particles $i$, $j$, and $k$. The equilibrium distance between two bonded particles in the fiber was taken as ${\Delta}_{CS}$ of the smoothed VMF walk. The equilibrium angle between three consecutive particles was taken as the initial angle between them after the smoothed VMF walk. The contact potential cutoff distance between particles $i$ and $j$ was taken as $\frac{{d}_{f,i}+{d}_{f,j}}{2}$. The contact potential was not calculated between bonded particles or particles participating in a bond trio.

#### 2.7. Pore Size Measurement

## 3. Results and Discussions

#### 3.1. Relating de la Vallee Poussin Parameters Measured from 2D data to 3D Structures

#### 3.2. Relating ${\kappa}_{VMF}$ and ${N}_{VMF}$ to Fiber Tortuosity ${\tau}_{f}$

#### 3.3. Effect of Enforcing Non-Penetration on ${\tau}_{f}$, ${\widehat{\mathit{n}}}_{f}$, and ${l}_{f}$ Distributions

#### 3.4. Pore size Distribution Analysis of FM Models

#### 3.5. Limitations

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a.i**–

**a.iv**) Schematic of the de la Vallee Poussin distribution for (

**a.i**) transverse isotropic, (

**a.ii**) orthotropic, (

**a.iii**) and hexagonal symmetries; (

**a.iv**) shows how increasing the orientation strength parameter ${k}_{VP}^{3D}$ affects the shape of the distribution. (

**b.i**,

**b.ii**) Schematic detailing the smoothed VMF random walk procedure and showing examples of low- and high-tortuosity curves: (

**b.i**) A fiber is generated from a smoothed VMF random walk with step size ${\Delta}_{VMF}$, spline step size ${\Delta}_{CS}$, and preferred direction ${\widehat{\mathit{m}}}_{VMF}={\widehat{\mathit{n}}}_{f}$; (

**b.ii**) for a low orientation strength ${\kappa}_{VMF}$, the simulated fibers are highly tortuous, while for a high orientation strength ${\kappa}_{VMF}$, the fibers are less tortuous. (

**c.1**–

**c.iv**) Schematic of the forces defined for enforcing non-penetration conditions: (

**c.i**) bond force; (

**c.ii**) angle force; (

**c.iii**) contact force; (

**c.iv**) drag force. (

**d**) A graphic depicting the final result of the stochastic FM modeling procedures. (

**e**) A 2D schematic detailing the calculation of mean pore diameter $\langle {d}_{p}\rangle $ using the watershed method.

**Figure 2.**(

**a**) Schematic showing two VP distribution functions with identical orientation strength but different rotations with respect to an observer. The schematic also depicts an orientation drawn from either distribution being projected onto a plane from the observer’s perspective. (

**b**) Orientation distributions of both VP distribution functions in panel (

**a**) after projection and binning. (

**c**) The effects of ${k}_{VP}^{3D}$ and $\psi $ on the measured orientation strength ${k}_{VP}^{2D}$. (

**d**) The slope of ${k}_{VP}^{2D}$ vs. ${k}_{VP}^{3D}$ changes with the angle $\psi $ in subfigure (

**c**). (

**ei**–

**hii**) Curve fitting of empirical histograms and fiber orientation histograms from modeling with and without correcting for projection distortion for (

**e.i**,

**e.ii**) fibrin network, (

**f.i**,

**f.ii**) stone wool, (

**g.i**,

**g.ii**) cross-laid polyester, and (

**h.i**,

**h.ii**) collagen. ${n}_{f}^{exp}$ represents the experimental fiber orientation data; “VP fit” is the regression of Equation (7) to experimental data; the orange histogram is the fiber orientation distribution of simulated FM structures when projection distortion is accounted for; the white histogram is the fiber orientation distribution of simulated FM structures when directly using ${k}_{VP}^{2D}$ (from regression) as the orientation strength parameter ${k}_{VP}^{3D}$ (for drawing random fiber orientations).

**Figure 3.**Controlling the tortuosity of VMF random walks: (

**a**) Three randomly generated curves with varying tortuosity. Red spheres denote a step from the random walk, while blue spheres are points with added cubic splines. (

**b**) The relationship between expected fiber tortuosity and orientation strength ${\kappa}_{VMF}$. (

**c**) How the number of steps in the random walk affects the possible fiber tortuosity and the mean absolute percentage error between the resulting tortuosity and the expected tortuosity. (

**d.i**–

**f.ii**) Empirical fiber tortuosity distributions modelled using the VMF random walk and criteria for selecting ${\kappa}_{VMF}$ and ${N}_{VMF}$ ($\beta =20$ ) for (

**d.i**,

**d.ii**) collagen, (

**ei**,

**e.ii**) sintered metal fiber, and (

**f.i**,

**f.ii**) fiberboard. ${\tau}_{f}^{exp}$ represents the experimental data for the FMs collagen [70], sintered metal fiber [56], and fiberboard [58]; ${\tau}_{f}^{sim}$ is the measured fiber tortuosity distribution from the simulated FMs.

**Figure 4.**The effects of enforcing non-penetration conditions: (

**a.i**,

**a.ii**) An FM model before and after enforcing non-penetration conditions. (

**b**) Percent fiber overlap (PFO) during the simulation. (

**c**) Statistical characterization of mean absolute percentage error between the initial and final fiber lengths, tortuosities, and orientations.

**Figure 5.**Characterization of mean pore size vs. FM model structure: (

**a**–

**c**) Univariate analysis of mean pore diameter vs. mean fiber diameter, fiber volume fraction, and mean fiber tortuosity. (

**d.i**–

**e.ii**) Multivariate analyses of mean pore diameter. (

**d.i**,

**d.ii**) When all properties of the FM are variable, fiber diameter is still a strong predictor of mean pore size. (

**e.i**,

**e.ii**) Mean fiber diameter combined with volume fraction offers better predictive power than mean fiber diameter alone. (

**f.i**,

**f.ii**) Mean fiber tortuosity significantly increased the predictive power for this multivariate dataset; however, this dataset is not representative of typical FMs, since it includes a very large range of fiber tortuosities. (

**d.ii**–

**f.ii)**show the error between the measured mean pore diameter and the predicted values.

**Table 1.**Fitted parameters for experimental fiber orientation distributions and corrected 3D orientation strength parameters.

Fibrin | Stone wool | Polyester | Collagen | |
---|---|---|---|---|

${N}_{VP}$ | 1 | 3 | 2 | 2 |

${k}_{VP,i}^{2D}$ (Measured) | 0.98 | 5.25, 5.25, 5.25 | 2.50, 2.50 | 4.00, 2.20 |

${k}_{VP,i}^{3D}$ (Equation (18)) | 0.37 | 3.41, 3.41, 3.41 | 1.33, 1.33 | 2.12, 1.17 |

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**MDPI and ACS Style**

Filla, N.; Zhao, Y.; Wang, X.
A Tractable, Transferable, and Empirically Consistent Fibrous Biomaterial Model. *Polymers* **2022**, *14*, 4437.
https://doi.org/10.3390/polym14204437

**AMA Style**

Filla N, Zhao Y, Wang X.
A Tractable, Transferable, and Empirically Consistent Fibrous Biomaterial Model. *Polymers*. 2022; 14(20):4437.
https://doi.org/10.3390/polym14204437

**Chicago/Turabian Style**

Filla, Nicholas, Yiping Zhao, and Xianqiao Wang.
2022. "A Tractable, Transferable, and Empirically Consistent Fibrous Biomaterial Model" *Polymers* 14, no. 20: 4437.
https://doi.org/10.3390/polym14204437