Mechanics and microstructure of blood plasma clots in shear driven rupture.

Intravascular blood clots are subject to hydrodynamic shear and other forces that cause clot deformation and rupture (embolization). A portion of the ruptured clot can block blood flow in downstream vessels. The mechanical stability of blood clots is determined primarily by the 3D polymeric fibrin network that forms a gel. Previous studies have primarily focused on the rupture of blood plasma clots under tensile loading (Mode I), our current study investigates the rupture of fibrin induced by shear loading (Mode II), dominating under physiological conditions induced by blood flow. Using experimental and theoretical approaches, we show that fracture toughness, i.e. the critical energy release rate, is relatively independent of the type of loading and is therefore a fundamental property of the gel. Ultrastructural studies and finite element simulations demonstrate that cracks propagate perpendicular to the direction of maximum stretch at the crack tip. These observations indicate that locally, the mechanism of rupture is predominantly tensile. Knowledge gained from this study will aid in the development of methods for prediction/prevention of thrombotic embolization.

A continuum mechanical model of cell motion driven by a biphasic traction stress.

The aim of this paper is to place the cell locomotion problem within the general framework of classical continuum mechanics, and while doing so, to account for the deformation of the actin network in the cytoskeleton; the myosin activity on the lamellum including its effect on depolymerization at the trailing edge; model the stress-dependent driving forces and kinetic laws controlling polymerization at the leading edge, depolymerization at the trailing edge and ATP hydrolysis consistently with the dissipation inequality; and, based on the observations in Gardel et al. (Gardel et al. 2008 J. Cell Biol. 183, 999-1005 (doi:10.1083/jcb.200810060)), include a biphasic velocity-dependent traction stress acting on the actin network. While we chose certain specific models for each of these, in part to allow for an analytical solution, the generality of the framework allows one to readily introduce different constitutive laws to describe these phenomena as might be needed, for example, to study some different type of cells. As described in §5, the predictions of the model compare well with observations such as the magnitude of the very different actin retrograde speeds in the lamellum and lamellipodium including their jump at the interface, the magnitude of the cell speed, and the relative lengths of the lamellipodium and lamellum.

Cracks in tensile-contracting and tensile-dilating poroelastic materials.

Fibrous gels such as cartilage, blood clots, and carbon-nanotube-based sponges with absorbed oils suffer a reduction in volume by the expulsion of liquid under uniaxial tension, and this directly affects crack-tip fields and energy release rates. A continuum model is formulated for isotropic fibrous gels that exhibit a range of behaviors from volume increasing to volume decreasing in uniaxial tension by changing the ratio of two material parameters. The motion of liquid in the pores of such gels is modeled using poroelasticity. The direction of liquid fluxes around cracks is shown to depend on whether the gel locally increases or decreases in volume. The energy release rate for cracks is computed using a surface-independent integral and it is shown to have two contributions - one from the stresses in the solid network, and another from the flow of liquid. The contribution to the integral from liquid permeation tends to be negative when the gel exhibits volume decrease, which effectively is a crack shielding mechanism.

Mechanical fatigue testing in silico: Dynamic evolution of material properties of nanoscale biological particles.

Biological particles have evolved to possess mechanical characteristics necessary to carry out their functions. We developed a computational approach to "fatigue testing in silico", in which constant-amplitude cyclic loading is applied to a particle to explore its mechanobiology. We used this approach to describe dynamic evolution of nanomaterial properties and low-cycle fatigue in the thin spherical encapsulin shell, thick spherical Cowpea Chlorotic Mottle Virus (CCMV) capsid, and thick cylindrical microtubule (MT) fragment over 20 cycles of deformation. Changing structures and force-deformation curves enabled us to describe their damage-dependent biomechanics (strength, deformability, stiffness), thermodynamics (released and dissipated energies, enthalpy, and entropy) and material properties (toughness). Thick CCMV and MT particles experience material fatigue due to slow recovery and damage accumulation over 3-5 loading cycles; thin encapsulin shells show little fatigue due to rapid remodeling and limited damage. The results obtained challenge the existing paradigm: damage in biological particles is partially reversible owing to particle's partial recovery; fatigue crack may or may not grow with each loading cycle and may heal; and particles adapt to deformation amplitude and frequency to minimize the energy dissipated. Using crack size to quantitate damage is problematic as several cracks might form simultaneously in a particle. Dynamic evolution of strength, deformability, and stiffness, can be predicted by analyzing the cycle number (N) dependent damage, [Formula: see text] , where α is a power law and Nf is fatigue life. Fatigue testing in silico can now be used to explore damage-induced changes in the material properties of other biological particles. STATEMENT OF SIGNIFICANCE: Biological particles possess mechanical characteristics necessary to perform their functions. We developed "fatigue testing in silico" approach, which employes Langevin Dynamics simulations of constant-amplitude cyclic loading of nanoscale biological particles, to explore dynamic evolution of the mechanical, energetic, and material properties of the thin and thick spherical particles of encapsulin and Cowpea Chlorotic Mottle Virus, and the microtubule filament fragment. Our study of damage growth and fatigue development challenge the existing paradigm. Damage in biological particles is partially reversible as fatigue crack might heal with each loading cycle. Particles adapt to deformation amplitude and frequency to minimize energy dissipation. The evolution of strength, deformability, and stiffness, can be accurately predicted by analyzing the damage growth in particle structure.

Fracture toughness of fibrin gels as a function of protein volume fraction: Mechanical origins.

The mechanical stability of blood clots necessary for their functions is provided by fibrin, a fibrous gel. Rupture of clots leads to life-threatening thrombotic embolization, which is little understood. Here, we combine experiments and simulations to determine the toughness of plasma clots as a function of fibrin content and correlate toughness with fibrin network structure characterized by confocal and scanning electron microscopy. We develop fibrin constitutive laws that scale with fibrin concentration and capture the force-stretch response of cracked clot specimens using only a few material parameters. Toughness is calculated from the path-independent J* integral that includes dissipative effects due to fluid flow and uses only the constitutive model and overall stretch at crack propagation as input. We show that internal fluid motion, which is not directly measurable, contributes significantly to clot toughness, with its effect increasing as fibrin content increases, because the reduced gel porosity at higher density results in greater expense of energy in fluid motion. Increasing fibrin content (1→10mg/mL) results in a significant increase in clot toughness (3→15 N/m) in accordance with a power law relation reminiscent of cellular solids and elastomeric gels. These results provide a basis for understanding and predicting the tendency for thrombotic embolization. STATEMENT OF SIGNIFICANCE: Fibrin, a naturally occurring biomaterial, is the major determinant of the structural and mechanical integrity of blood clots. We determined that increasing the fibrin content in clots, as in some thrombi and fibrin-based anti-bleeding sealants, results in an increase in clot toughness. Toughness corresponds to the ability to resist rupturing in the presence of a defect. We couple bulk mechanical testing, microstructural measurements, and finite element modeling to capture the force-stretch response of fibrin clots and compute toughness. We show that increased fibrin content in clots reduces porosity and limits fluid motion and that fluid motion drastically alters the clot toughness. These results provide a fundamental understanding of blood clot rupture and could help in rational design of fibrin-containing biomaterials.

Biomechanical origins of inherent tension in fibrin networks.

Blood clots form at the site of vascular injury to seal the wound and prevent bleeding. Clots are in tension as they perform their biological functions and withstand hydrodynamic forces of blood flow, vessel wall fluctuations, extravascular muscle contraction and other forces. There are several mechanisms that generate tension in a blood clot, of which the most well-known is the contraction/retraction caused by activated platelets. Here we show through experiments and modeling that clot tension is generated by the polymerization of fibrin. Our mathematical model is built on the hypothesis that the shape of fibrin monomers having two-fold symmetry and off-axis binding sites is ultimately the source of inherent tension in individual fibers and the clot. As the diameter of a fiber grows during polymerization the fibrin monomers must suffer axial twisting deformation so that they remain in register to form the half-staggered arrangement characteristic of fibrin protofibrils. This deformation results in a pre-strain that causes fiber and network tension. Our results for the pre-strain in single fibrin fibers is in agreement with experiments that measured it by cutting fibers and measuring their relaxed length. We connect the mechanics of a fiber to that of the network using the 8-chain model of polymer elasticity. By combining this with a continuum model of swellable elastomers we can compute the evolution of tension in a constrained fibrin gel. The temporal evolution and tensile stresses predicted by this model are in qualitative agreement with experimental measurements of the inherent tension of fibrin clots polymerized between two fixed rheometer plates. These experiments also revealed that increasing thrombin concentration leads to increasing internal tension in the fibrin network. Our model may be extended to account for other mechanisms that generate pre-strains in individual fibers and cause tension in three-dimensional proteinaceous polymeric networks.

Humidity dependence of fracture toughness of cellulose fibrous networks.

Cellulose-based materials are increasingly finding applications in technology due to their sustainability and biodegradability. The sensitivity of cellulose fiber networks to environmental conditions such as temperature and humidity is well known. Yet, there is an incomplete understanding of the dependence of the fracture toughness of cellulose networks on environmental conditions. In the current study, we assess the effect of moisture content on the out-of-plane (i.e., z-dir.) fracture toughness of a particular cellulose network, specifically Whatman cellulose filter paper. Experimental measurements are performed at 16% RH along the desorption isotherm and 23, 37, 50, 75% RH along the adsorption isotherm using out-of-plane tensile tests and double cantilever beam (DCB) tests. Cohesive zone modeling and finite element simulations are used to extract quantitative properties that describe the crack growth behavior. Overall, the fracture toughness of filter paper decreased with increasing humidity. Additionally, a novel model is developed to capture the high peak and sudden drop in the experimental force measurement caused by the existence of an initiation region. This model is found to be in good agreement with experimental data. The relative effect of each independent cohesive parameter is explored to better understand the cohesive zone-based humidity dependence model. The methods described here may be applied to study rupture of other fiber networks with weak bonds.

Rheology of fibrous gels under compression.

A number of biological tissues and synthetic gels consist of a fibrous network infused with liquid. There have been a few experimental studies of the rheological properties of such gels under applied compressive strain. Their results suggest that a plot of rheological moduli as a function of applied compressive strain has a long plateau flanked by a steeply increasing curve for large compressive strains and a slowly decreasing curve for small strains. In this paper we explain these trends in rheological properties using a chemo-elastic model characterized by a double-well strain energy function for the underlying fibrous network. The wells correspond to rarefied and densified phases of the fibrous network at low and high strains, respectively. These phases can co-exist across a movable transition front in the gel in order to accommodate overall applied compression. We find that the rheological properties of fibrous gels share similarities with a Kelvin-Voigt visco-elastic solid. The storage modulus has its origins in the elasticity of the fibrous network, while the loss modulus is determined by the dissipation caused by liquid flow through pores. The rheological properties can depend on the number of phase transition fronts present in a compressed sample. Our analysis may explain the dependence of storage and loss moduli of fibrin gels on the loading history. We also point to the need for combining rheological measurements on gels with a microstructural analysis that could shed light on various dissipation mechanisms.

Wave manipulation using a bistable chain with reversible impurities.

We systematically study linear and nonlinear wave propagation in a chain composed of piecewise-linear bistable springs. Such bistable systems are ideal test beds for supporting nonlinear wave dynamical features including transition and (supersonic) solitary waves. We show that bistable chains can support the propagation of subsonic wave packets which in turn can be trapped by a low-energy phase to induce energy localization. The spatial distribution of these energy foci strongly affects the propagation of linear waves, typically causing scattering, but, in special cases, leading to a reflectionless mode analogous to the Ramsauer-Townsend effect. Furthermore, we show that the propagation of nonlinear waves can spontaneously generate or remove additional foci, which act as effective "impurities." This behavior serves as a new mechanism for reversibly programming the dynamic response of bistable chains.

Bacterial activity hinders particle sedimentation.

Sedimentation in active fluids has come into focus due to the ubiquity of swimming micro-organisms in natural and industrial processes. Here, we investigate sedimentation dynamics of passive particles in a fluid as a function of bacteria E. coli concentration. Results show that the presence of swimming bacteria significantly reduces the speed of the sedimentation front even in the dilute regime, in which the sedimentation speed is expected to be independent of particle concentration. Furthermore, bacteria increase the dispersion of the passive particles, which determines the width of the sedimentation front. For short times, particle sedimentation speed has a linear dependence on bacterial concentration. Mean square displacement data shows, however, that bacterial activity decays over long experimental (sedimentation) times. An advection-diffusion equation coupled to bacteria population dynamics seems to capture concentration profiles relatively well. A single parameter, the ratio of single particle speed to the bacteria flow speed can be used to predict front sedimentation speed.

Harnessing fluctuation theorems to discover free energy and dissipation potentials from non-equilibrium data.

The Jarzynski relation, as an equality form of the second law of thermodynamics, represents an exact thermodynamic statement that is valid arbitrarily far away from equilibrium. This remarkable relation directly links the equilibrium free energy difference between two states and the probability distribution of the work done along a process that drives the system from one state to the other. Here, we leverage the Jarzynski equality and a local equilibrium assumption, to go beyond the calculation of free energy differences and also extract the dissipation potential from additional measurements of kinematic field variables (strain and velocity fields). The proposed strategy is exemplified over pulling experiments of mass-spring models obeying overdamped Langevin dynamics, which is a prototype for nanorods, fibrous macro-molecules and the Rouse model of polymers. Different interaction potentials, fluid viscosities and bath temperatures are studied, so as to intrinsically control how close or far away the system is from equilibrium. The data-inferred continuum models are then validated against processes governed by different pulling protocols, thereby demonstrating their predictive capability. The methods presented here represent a first step toward full material characterization from non-equilibrium data of macroscopic observables, which could potentially be obtained from experimental observations.

Kinetics of self-assembly of inclusions due to lipid membrane thickness interactions.

Self-assembly of proteins on lipid membranes underlies many important processes in cell biology, such as, exo- and endo-cytosis, assembly of viruses, etc. An attractive force that can cause self-assembly is mediated by membrane thickness interactions between proteins. The free energy profile associated with this attractive force is a result of the overlap of thickness deformation fields around the proteins which can be calculated from the solution of a boundary value problem. Yet, the time scales over which two inclusions coalesce has not been explored, even though the evolution of particle concentrations on membranes has been modeled using phase-field approaches. In this paper we compute this time scale as a function of the initial distance between two inclusions by viewing their coalescence as a first passage time problem. The mean first passage time is computed using Langevin dynamics and a partial differential equation, and both methods are found to be in excellent agreement. Inclusions of three different shapes are studied and it is found that for two inclusions separated by about hundred nanometers the time to coalescence is hundreds of milliseconds irrespective of shape. An efficient computation of the interaction energy of inclusions is central to our work. We compute it using a finite difference technique and show that our results are in excellent agreement with those from a previously proposed semi-analytical method based on Fourier-Bessel series. The computational strategies described in this paper could potentially lead to efficient methods to explore the kinetics of self-assembly of proteins on lipid membranes.

A reduced-order model of the spine to study pediatric scoliosis.

The S-shaped curvature of the spine has been hypothesized as the underlying mechanical cause of adolescent idiopathic scoliosis. In earlier work, we proposed a reduced-order model in which the spine was viewed as an S-shaped elastic rod under torsion and bending. Here, we simulate the deformation of S-shaped rods of a wide range of curvatures and inflection points under a fixed mechanical loading. Our analysis determines three distinct axial projection patterns of these S-shaped rods: two loop (in opposite directions) patterns and one Lemniscate pattern. We further identify the curve characteristics associated with each deformation pattern, showing that for rods deforming in a Loop1 shape the position of the inflection point is the highest and the curvature of the rod is smaller compared to the other two types. For rods deforming in the Loop2 shape, the position of the inflection point is the lowest (closer to the fixed base) and the curvatures are higher than the other two types. These patterns matched the common clinically observed scoliotic curves-Lenke 1 and Lenke 5. Our S-shaped elastic rod model generates deformations that are similar to those of a pediatric spine with the same sagittal curvature characteristics and it can differentiate between the clinically observed deformation patterns.

A Semi-Analytic Elastic Rod Model of Pediatric Spinal Deformity.

The mechanism of the scoliotic curve development in healthy adolescents remains unknown in the field of orthopedic surgery. Variations in the sagittal curvature of the spine are believed to be a leading cause of scoliosis in this patient population. Here, we formulate the mechanics of S-shaped slender elastic rods as a model for pediatric spine under physiological loading. Second, applying inverse mechanics to clinical data of the subtypes of scoliotic spines, with characteristic 3D deformity, we determine the undeformed geometry of the spine before the induction of scoliosis. Our result successfully reproduces the clinical data of the deformed spine under varying loads, confirming that the prescoliotic sagittal curvature of the spine impacts the 3D loading that leads to scoliosis.

Extensions of the worm-like-chain model to tethered active filaments under tension.

Intracellular elastic filaments such as microtubules are subject to thermal Brownian noise and active noise generated by molecular motors that convert chemical energy into mechanical work. Similarly, polymers in living fluids such as bacterial suspensions and swarms suffer bending deformations as they interact with single bacteria or with cell clusters. Often, these filaments perform mechanical functions and interact with their networked environment through cross-links or have other similar constraints placed on them. Here, we examine the mechanical properties-under tension-of such constrained active filaments under canonical boundary conditions motivated by experiments. Fluctuations in the filament shape are a consequence of two types of random forces-thermal Brownian forces and activity derived forces with specified time and space correlation functions. We derive force-extension relationships and expressions for the mean square deflections for tethered filaments under various boundary conditions including hinged and clamped constraints. The expressions for hinged-hinged boundary conditions are reminiscent of the worm-like-chain model and feature effective bending moduli and mode-dependent non-thermodynamic effective temperatures controlled by the imposed force and by the activity. Our results provide methods to estimate the activity by measurements of the force-extension relation of the filaments or their mean square deflections, which can be routinely performed using optical traps, tethered particle experiments, or other single molecule techniques.

A continuum model for the growth of dendritic actin networks.

Polymerization of dendritic actin networks underlies important mechanical processes in cell biology such as the protrusion of lamellipodia, propulsion of growth cones in dendrites of neurons, intracellular transport of organelles and pathogens, among others. The forces required for these mechanical functions have been deduced from mechano-chemical models of actin polymerization; most models are focused on single growing filaments, and only a few address polymerization of filament networks through simulations. Here, we propose a continuum model of surface growth and filament nucleation to describe polymerization of dendritic actin networks. The model describes growth and elasticity in terms of macroscopic stresses, strains and filament density rather than focusing on individual filaments. The microscopic processes underlying polymerization are subsumed into kinetic laws characterizing the change of filament density and the propagation of growing surfaces. This continuum model can predict the evolution of actin networks in disparate experiments. A key conclusion of the analysis is that existing laws relating force to polymerization speed of single filaments cannot predict the response of growing networks. Therefore, a new kinetic law, consistent with the dissipation inequality, is proposed to capture the evolution of dendritic actin networks under different loading conditions. This model may be extended to other settings involving a more complex interplay between mechanical stresses and polymerization kinetics, such as the growth of networks of microtubules, collagen filaments, intermediate filaments and carbon nanotubes.

Rupture of blood clots: Mechanics and pathophysiology.

Fibrin is the three-dimensional mechanical scaffold of protective blood clots that stop bleeding and pathological thrombi that obstruct blood vessels. Fibrin must be mechanically tough to withstand rupture, after which life-threatening pieces (thrombotic emboli) are carried downstream by blood flow. Despite multiple studies on fibrin viscoelasticity, mechanisms of fibrin rupture remain unknown. Here, we examined mechanically and structurally the strain-driven rupture of human blood plasma-derived fibrin clots where clotting was triggered with tissue factor. Toughness, i.e., resistance to rupture, quantified by the critical energy release rate (a measure of the propensity for clot embolization) of physiologically relevant fibrin gels was determined to be 7.6 ± 0.45 J/m2. Finite element (FE) simulations using fibrin material models that account for forced protein unfolding independently supported this measured toughness and showed that breaking of fibers ahead the crack at a critical stretch is the mechanism of rupture of blood clots, including thrombotic embolization.

Statistical mechanics of a double-stranded rod model for DNA melting and elasticity.

The double-helical topology of DNA molecules observed at room temperature in the absence of any external loads can be disrupted by increasing the bath temperature or by applying tensile forces, leading to spontaneous strand separation known as DNA melting. Here, continuum mechanics of a 2D birod is combined with statistical mechanics to formulate a unified framework for studying both thermal melting and tensile force induced melting of double-stranded molecules: it predicts the variation of melting temperature with tensile load, provides a mechanics-based understanding of the cooperativity observed in melting transitions, and reveals an interplay between solution electrostatics and micromechanical deformations of DNA which manifests itself as an increase in the melting temperature with increasing ion concentration. This novel predictive framework sheds light on the micromechanical aspects of DNA melting and predicts trends that were observed experimentally or extracted phenomenologically using the Clayperon equation.

Strong, Ultralight Nanofoams with Extreme Recovery and Dissipation by Manipulation of Internal Adhesive Contacts.

Advances in three-dimensional nanofabrication techniques have enabled the development of lightweight solids, such as hollow nanolattices, having record values of specific stiffness and strength, albeit at low production throughput. At the length scales of the structural elements of these solids-which are often tens of nanometers or smaller-forces required for elastic deformation can be comparable to adhesive forces, rendering the possibility to tailor bulk mechanical properties based on the relative balance of these forces. Herein, we study this interplay via the mechanics of ultralight ceramic-coated carbon nanotube (CNT) structures. We show that ceramic-CNT foams surpass other architected nanomaterials in density-normalized strength and that, when the structures are designed to minimize internal adhesive interactions between CNTs, more than 97% of the strain after compression beyond densification is recovered. Via experiments and modeling, we study the dependence of the recovery and dissipation on the coating thickness, demonstrate that internal adhesive contacts impede recovery, and identify design guidelines for ultralight materials to have maximum recovery. The combination of high recovery and dissipation in ceramic-CNT foams may be useful in structural damping and shock absorption, and the general principles could be broadly applied to both architected and stochastic nanofoams.

Structure, mechanical properties, and modeling of cyclically compressed pulmonary emboli.

Pulmonary embolism occurs when blood flow to a part of the lungs is blocked by a venous thrombus that has traveled from the lower limbs. Little is known about the mechanical behavior of emboli under compressive forces from the surrounding musculature and blood pressure. We measured the stress-strain responses of human pulmonary emboli under cyclic compression, and showed that emboli exhibit a hysteretic stress-strain curve. The fibrin fibers and red blood cells (RBCs) are damaged during the compression process, causing irreversible changes in the structure of the emboli. We showed using electron and confocal microscopy that bundling of fibrin fibers occurs due to compression, and damage is accumulated as more cycles are applied. The stress-strain curves depend on embolus structure, such that variations in composition give quantitatively different responses. Emboli with a high fibrin component demonstrate higher normal stress compared to emboli that have a high RBC component. We compared the compression response of emboli to that of whole blood clots containing various volume fractions of RBCs, and found that RBCs rupture at a certain critical stress. We describe the hysteretic response characteristic of foams, using a model of phase transitions in which the compressed foam is segregated into coexisting rarefied and densified phases whose fractions change during compression. Our model takes account of the rupture of RBCs in the compressed emboli and stresses due to fluid flow through their small pores. Our results can help in classifying emboli as rich in fibrin or rich in red blood cells, and can help in understanding what responses to expect when stresses are applied to thrombi in vivo.