The Non-Affine Network Solver: A Multiscale Material for Finite Element Analysis

Introduction:: Many biological materials are complex multiscale micro-architected materials. Biologic tissues, for example, have a geometrically complex shape driven by their function, and are composed of a complex fibrous microstructure. Much research has been performed on how to represent these networks and their growth and remodeling in a continuum mechanics context1–5 by simplifying the microstructure and obscuring much of the discrete network detail. Further, many existing codes do not have straightforward means to predict growth, remodeling, and failure from one simulation framework. To overcome these barriers, we develop a discrete fiber network plugin framework that works universally as a material model in finite-element analysis which will allow development of coupled growth, remodeling, and failure simulations with applications ranging from tissue engineering and material development to creating patient specific models for clinical risk evaluation.

Materials and Methods:: The implementation of a material model into a finite-element code requires a stress calculation and an elasticity (or tangent stiffness) calculation from a given deformation.
For any general structure, the Cauchy stress can be written as the integral of the stresses in all components over the domain which can be written as a sum for a discrete number of elements as in Eq. 1 or converted to second Piola-Kirchhoff stress in Eq. 2. Thus, we need a constitutive model for the fibers and a domain in which we place fibers. In this case, we use a periodic cube of fibers generated via Delaunay tessellation (Fig. 1). For a network of 1D trusses that meets the Maxwell stability criteria, we can reliably use an iterative method (e.g., Newton-Raphson) to solve for equilibrium of the structure.
The material elasticity tensor is given in Eq. 3. The two critical pieces, the derivative of stretch with the right Cauchy-Green tensor and the derivative of PK2 stress with deformation are given in Eq. 4 and Eq. 5. As a part of Eq. 5, the fiber force is calculated using a linear approximation (assuming the fiber stiffness is k=df^e/dλ^e) and assuming affine deformation (Eq. 6), which can be differentiated with deformation to give Eq. 7.
This framework is implemented as a material plugin in FEBio4 utilizing the network material in conjunction with a neo-Hookean ground matrix. The neo-Hookean material provides incompressibility to the tissue as networks will tend to flatten when stretched.

Results, Conclusions, and Discussions:: To demonstrate the utility of our simulation framework, we simulate two common biomedical materials: reconstituted gels and whole tissues.
Reconstituted Gel Mechanics: To demonstrate the use of multiple fiber type as well as multiple fiber network architectures, we developed three representative case studies: a linear, highly connected fibrous network (i.e., synthetic fiber network like polycaprolactone), an actively contractile, lower connectivity network (i.e., actomyosin network), and a loosely connected, nonlinear fiber network (i.e., collagen) shown in Fig. 1A, 1B, and 1C respectively. The equibiaxial behavior of the tissue is shown in Fig. 1D, 1E, and 1F for the linear networks, the active contractile networks, and the non-linear networks, respectively. The yellow arrow shows the basal contraction of the active contractile network and the red arrow shows the initiation of lockout for the non-linear network.
Whole Tissue Mechanics: To demonstrate the use of interacting networks of fibers, we simulate the microarchitecture of an aortic medial lamellar unit composed of collagen, elastin, and cells (which themselves are composed of networks of actin) as shown in Fig. 2A,B. An example of a cruciform biaxial test on an artery shows Cauchy stress over the sample at 1.5 stretch in Fig. 2C and the averaged PK1 stress for the sample for all stretches in Fig. 2E. A cylindrical inflation test and the principal stretches at 100mmHg are shown in Fig. 2D along with the associated inner and outer pressure-diameter curves in Fig. 2F.
The examples in Fig. 1 represents a large cross-section of materials encountered in biomedical engineering. Further, in Fig. 2, we show biomechanics applied to arteries with a cruciform biaxial stretch and an inflation experiment and report the results one would obtain experimentally (i.e., PK1 stress vs. stretch and pressure vs. diameter).
This work is an open source, easily implemented plugin for the popular FEBio software allowing users to perform multiscale simulations using fiber networks of various types and topologies. Further, our previous works6,7 have used these network simulations in the context of growth, remodeling, and failure making the avenue to applying this next level of detail to the model system relatively straightforward.

Acknowledgements (Optional): : This work was supported by the National Institutes of Health under grants R01 HL164800 and R01 NS126762, and by the National Science Foundation under grant CMMI-1563198. RRM acknowledges support by the American Heart Association Postdoctoral Fellowship Award #827242.This work is an independent publication and is neither affiliated with, nor authorized, sponsored, or approved by Microsoft Corporation, Eigen, or FEBio. The authors acknowledge many helpful discussions on computational implementation and model creation with Elizabeth Gacek.

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