1. Fig 6. Procedure for generating NURBS mesh at a junction of fibers
Modeling Fibrin Network using 3D spatial Euler-Bernoulli Beam Soham Mane, Dr. Manuel Rausch   Department of Aerospace Engineering & Engineering Mechanics The University of Texas at Austin W. R. Woolrich Laboratories, C0600 ,210 East 24th Street, Austin, Texas, ,
Abstract
Thrombus is vital to our well-being as blood coagulation prevents bleeding after vascular injury. However, thrombus role is diametrical in that pathological coagulation also causes deep vein thrombosis, heart attacks, and strokes. Thrombus’ propensity to detach and emboli, and thus cause havoc, is strongly linked to its mechanical properties, which are primarily determined by its fibrin backbone. To understand how thrombus macromechanics (such as its failure behavior) are linked to its microstructure, we have developed a numerical model of thrombus’ 3D microstructure. Specifically, in a representative volume element (RVE), we model each individual fibrin fiber as a 3D spatial Euler-Bernoulli beam, whose behavior is driven by St Venant-Kirchhoff material law. We implement and solve the resulting formulation in an Isogeometric framework. We have validated our formulation for single fiber against three benchmark problems. A technique for discretizing multiple connected fibers using NURBS shape functions is developed. We use Voronoi tessellations to generate random networks and discretize them using the technique developed.
1. Introduction
Fibrin network determines strength of thrombus. Fibrin is a protein which polymerizes to form a stable network which provides scaffold for forming a thrombi. Along with the mechanical properties of fibrin, the geometry of the fibrin network governs the overall mechanical properties of thrombi. We have developed a numerical model to study role of network geometry and mechanical properties of individual fiber in determining overall mechanical properties of the network. Each fiber is modelled as a 3D spatial Euler-Bernoulli beam. Finite deformations are considered and Green-Lagrange strain measure is used. The structural element considered is based on Bernoulli kinematics, which assume that cross sections remain orthogonal to the centerline after deformation and there are no changes of the cross sectional dimensions. Torsion is considered while warping is neglected. These assumptions hold true because a typical fibrin fiber has a diameter orders smaller than its length. The structural element developed here has four degrees of freedom (DOFs) where three DOFs define positions of the centerline while fourth DOF defines relative rotation around the centerline. We have considered isotropic elastic material for the time being. St. Venant-Krichhoff material law is used here, which require only two material parameters, Young’s Modulus and Poisson’s Ratio namely. This element is developed in Isogeometric frame work, wherein NURBS shape functions are used to discretize the weak form and implemented in MATLAB. Validation of the numerical model for a single fiber is discussed in the second section while technique for connecting multiple fibers and generating a 2D random network based on Voronoi tessellations is discussed in the third section.
2. Validation with benchmark problems
We have tested our formulation on three popular benchmark problems widely used in the literature. The first example is a cantilevered beam with point moment applied at the tip. For a particular value of the moment, the beam curls up into a circle. The second example is a 45 degree arch clamped on one end and loaded with out of plane point load on the other. This example test the formulation for bending and torsion interaction for initially curved beams. Tip displacement values from the literature are compared against our results. The third example includes a shallow arch loaded at the crown with a point load. Displacement controlled simulation is performed to capture the load-deflection curve of the arch. Our formulation is capable of capturing the snap-through behavior and closely agrees with the curve from the literature. This example is tried with NURBS of different order and it is found that lower order NURBS (2nd & 3rd) require large number of elements as compared to higher order NURBS (5th & 6th) however using second order or third order NURBS with higher number of elements is computationally efficient that using higher order NURBS with less elements.
3. Random network
We have developed a technique to connect multiple fibers at a point. This is required to generate a random network in order to test it under different mechanical loading scenarios. We have used Voronoi tessellations for generating 2D random networks. These can simulate 2D cross-linked fibrin networks. The assumption is that wherever we have crosslinking, we have perfect bonding between the fibers. The process for generating a NURBS mesh for a network is descripted in the figure (add figure no) below. The fibers in the randomly generated network are straight, however our formulation is capable of simulating response of initially curved fibers. One of the reason for selecting Voronoi tessellations for generating 2D random networks is the feature that no more than 3 fibers intersect in a 2D Voronoi unit cell. In a real fibrin network it is observed that a single fiber branches out into two fibers. Thus this geometrical feature is capture in our random network model.
Fig 3 Reference and Deformed configurations of cantilever beam with tip moment applied on one end. (The beam curls into a circular for the applied value of the tip moment)
Fig 4. Geometry, Deformed Shape and Load-displacement curve for shallow arch for order three NURBS and different number of elements & by S. H. Lo
Fig 1. SEM micrographs of fibrin clots: (A) Clot with thick fibers and few branch points. (B) Clot with thin fibers and many branch points.
Fig degree circular arch cantilevered at one end
Fig 2. Transmission electron micrographs of negatively contrasted fibrin fibers that show the substructure of branch points.
Fig 6. Procedure for generating NURBS mesh at a junction of fibers
Typical junction with 3 fibers intersecting at a point
Separate the fibers by making cut from the point of intersection
Find control points for defining circular arc between fibers
Define arcs connecting fibers and bending strips connecting fibers to arcs
Fig 7. Random 2D Voronoi Network with 50 seeds
Fig 8. NURBS mesh for the randomly generated 2D Voronoi network ( control points of fibers in red & fibers shown in blue)
Fig 9. Deformed shapes of 50 seed network for 5%, 10%, and 15% strain and Load Vs Displacement curves upto 10% and 15% strain
Proceedings of the 2018 ASEE Gulf-Southwest Section Annual Conference The University of Texas at Austin
April 4-6, 2018