(190ar) Modeling the Extensibility and Strain-Hardening Inelasticity of Fibrin Fibers during Coagulation

Fibrin bundles assemble during the early stages of the coagulation cascade to form the fibrinous networks that spread throughout a clot and help to maintain clot integrity in vivo under high arterial pressure [1]. Many mechanical properties of polymers can be measured from stress-strain curves, which are obtained by applying a stress, or a force per unit area, to the polymer and recording the resulting strain, or the stretch of the polymer normalized to its initial length. Experimental stress-strain curves obtained from single fibrin bundles reveal important properties of fibrin, including having extreme extensibility, inelasticity, and a modular elastic modulus [2]. The objective of this work is to develop an in silico particle-based mathematical model using the discrete element method (DEM) that captures the extremely extensible, strain-hardening, and inelastic behavior of individual fibrin bundles.

Fibrin bundles can exceed a 330% strain before rupturing, although prior to rupturing, the bundle surpasses an elastic limit and the fibrin bundle’s integrity is compromised [2]. To capture this extreme extensibility of fibrin in silico, we simulate a fibrin bundle as a chain of spherical particles connected by springs. The equilibrium spring length (distance between particle centers) is less than the diameter of a particle, which results in particle overlap at equilibrium, so as to mitigate physical gaps between particles at full extension. We use the force-strain curves obtained from an AFM experimental investigation performed by Liu et. al. as the basis for the structure and fitting of our model [2]. To capture the strain-hardening phenomena in our model, we have developed an incremental force model, which incorporates a modular spring coefficient of Hill-type functional form to calculate the force resulting from the spring between two adjacent particles within a single fibrin strand. As such, the modular spring coefficient, k, is a sigmoidal function of the strain of the spring. We use the extension portion of the force-strain data collected by Liu et. al. to fit the parameters of the Hill equation. To capture the inelastic behavior of fibrin that is observed on force-strain curves as a hysteresis within the extension and retraction portions, the Hill function parameters are modified upon retraction when the previous extension exceeds the elastic limit of fibrin.

The in silico reproduction of the force-strain curve generated by Liu et. al. for the single extension-retraction case demonstrates that our model is able to capture the mechanics of fibrin extensibility and strain-hardening inelasticity when we simulate a fibrin bundle composed of five particles. We are currently working to lengthen the in silico fibrin bundle by including 20-50 more particles per bundle in an effort to match the physiological length of fibrin bundles. After extending the in silico fibrin bundles, we expect to observe the same dynamics upon extension and retraction that we observe with the five-particle case. The cell-scale mechanistic model of fibrin mechanics presented herein is essential to our broader goal of capturing the micro-mechanical contributions in a multi-scale model of coagulation, with applications in trauma and coagulopathy.

[1] A. Undas. Fibrin clot properties and their modulation in thrombotic disorders. Thrombosis and Haemostasis, 112(1):32–42, 2014.

[2] W. Liu, C. R. Carlisle, E. A. Sparks, and M. Guthold. The mechanical properties of single fibrin fibers. Journal of Thrombosis and Haemostasis, 8(5):1030–1036, 2010.