2014 reinhardt

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James W. Reinhardt Department of Biomedical Engineering, The Ohio State University, 270 Bevis Hall, 1080 Carmack Rd., Columbus, OH 43210 Keith J. Gooch 1 Department of Biomedical Engineering, The Ohio State University, 270 Bevis Hall, 1080 Carmack Rd., Columbus, OH 43210; Dorothy M. Davis Heart & Lung Research Institute, The Ohio State University, 473 W. 12th Ave., Columbus, OH 43210 e-mail: [email protected] Agent-Based Modeling Traction Force Mediated Compaction of Cell-Populated Collagen Gels Using Physically Realistic Fibril Mechanics Agent-based modeling was used to model collagen fibrils, composed of a string of nodes serially connected by links that act as Hookean springs. Bending mechanics are imple- mented as torsional springs that act upon each set of three serially connected nodes as a linear function of angular deflection about the central node. These fibrils were evaluated under conditions that simulated axial extension, simple three-point bending and an end- loaded cantilever. The deformation of fibrils under axial loading varied <0.001% from the analytical solution for linearly elastic fibrils. For fibrils between 100 lm and 200 lm in length experiencing small deflections, differences between simulated deflections and their analytical solutions were <1% for fibrils experiencing three-point bending and <7% for fibrils experiencing cantilever bending. When these new rules for fibril mechan- ics were introduced into a model that allowed for cross-linking of fibrils to form a net- work and the application of cell traction force, the fibrous network underwent macroscopic compaction and aligned between cells. Further, fibril density increased between cells to a greater extent than that observed macroscopically and appeared simi- lar to matrical tracks that have been observed experimentally in cell-populated collagen gels. This behavior is consistent with observations in previous versions of the model that did not allow for the physically realistic simulation of fibril mechanics. The significance of the torsional spring constant value was then explored to determine its impact on remodeling of the simulated fibrous network. Although a stronger torsional spring con- stant reduced the degree of quantitative remodeling that occurred, the inclusion of tor- sional springs in the model was not necessary for the model to reproduce key qualitative aspects of remodeling, indicating that the presence of Hookean springs is essential for this behavior. These results suggest that traction force mediated matrix remodeling may be a robust phenomenon not limited to fibrils with a precise set of material properties. [DOI: 10.1115/1.4026179] Keywords: agent-based modeling, individual-based modeling, cell-based modeling, ECM, collagen fibril, mechanics, remodeling, alignment 1 Introduction Extracellular matrix (ECM) remodeling has been identified dur- ing many biological processes, including wound healing, develop- ment, fibrotic pathologies, and tissue engineering [16]. Despite its widespread occurrence and biological significance, a full understanding of this process remains incomplete. Common in vitro experimental models used to study ECM remodeling are cell-populated ECM gels [69]. Type I collagen and fibrin are popular and appropriate choices for these studies as these fibrous materials are the major structural protein and an important provi- sional matrix, respectively. As an alternative to experimentation in vitro, computational modeling is advantageous because it allows for inexpensive testing of hypotheses that may not be experimentally feasible and the development of a model provides an opportunity for further insight into traction force mediated remodeling of ECM. Traditional approaches to modeling this phenomenon include mathematical [7,10] and finite-element modeling (FEM) [11,12]. Most similar to the model presented here, perhaps, Aghvami et al. used an FEM-based multiscale model to simulate the reorganiza- tion of collagen gels due to simultaneous stretching and cell trac- tion force [12]. Cell traction was modeled simply by shortening fibers by a fixed percentage of their original length. We chose to use agent-based modeling as alternative approach because it can readily handle the dynamic and complex interaction between a cell and its surrounding fibrils. We previously developed an agent-based model of cell traction force induced remodeling of a fibrous network using NetLogo [13]. The goal of this initial “proof of principal” model was to determine if a relatively simple set of rules could give rise to the relatively complex matrix remodeling observed in experimental models. In this respect, the initial model was successful as many aspects of matrix remodeling observed experimentally emerged in the simulations from these simple rules, including (1) macroscopic compaction of the fibrous net- work, (2) greater compaction of fibrils in the pericellular region than in regions more distant from the cells, and (3) alignment and densification of fibrils in between pairs of cells. This initial model, however, had a number of limitations that can roughly be divided 1 Corresponding author. Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received September 11, 2013; final manuscript received November 26, 2013; accepted manuscript posted December 9, 2013; published online February 5, 2014. Editor: Victor H. Barocas. Journal of Biomechanical Engineering FEBRUARY 2014, Vol. 136 / 021024-1 Copyright V C 2014 by ASME Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 06/18/2014 Terms of Use: http://asme.org/terms

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Page 1: 2014 Reinhardt

James W. ReinhardtDepartment of Biomedical Engineering,

The Ohio State University,

270 Bevis Hall, 1080 Carmack Rd.,

Columbus, OH 43210

Keith J. Gooch1

Department of Biomedical Engineering,

The Ohio State University,

270 Bevis Hall, 1080 Carmack Rd.,

Columbus, OH 43210;

Dorothy M. Davis Heart &

Lung Research Institute,

The Ohio State University,

473 W. 12th Ave.,

Columbus, OH 43210

e-mail: [email protected]

Agent-Based Modeling TractionForce Mediated Compaction ofCell-Populated Collagen GelsUsing Physically RealisticFibril MechanicsAgent-based modeling was used to model collagen fibrils, composed of a string of nodesserially connected by links that act as Hookean springs. Bending mechanics are imple-mented as torsional springs that act upon each set of three serially connected nodes as alinear function of angular deflection about the central node. These fibrils were evaluatedunder conditions that simulated axial extension, simple three-point bending and an end-loaded cantilever. The deformation of fibrils under axial loading varied <0.001% fromthe analytical solution for linearly elastic fibrils. For fibrils between 100 lm and 200 lmin length experiencing small deflections, differences between simulated deflections andtheir analytical solutions were <1% for fibrils experiencing three-point bending and<7% for fibrils experiencing cantilever bending. When these new rules for fibril mechan-ics were introduced into a model that allowed for cross-linking of fibrils to form a net-work and the application of cell traction force, the fibrous network underwentmacroscopic compaction and aligned between cells. Further, fibril density increasedbetween cells to a greater extent than that observed macroscopically and appeared simi-lar to matrical tracks that have been observed experimentally in cell-populated collagengels. This behavior is consistent with observations in previous versions of the model thatdid not allow for the physically realistic simulation of fibril mechanics. The significanceof the torsional spring constant value was then explored to determine its impact onremodeling of the simulated fibrous network. Although a stronger torsional spring con-stant reduced the degree of quantitative remodeling that occurred, the inclusion of tor-sional springs in the model was not necessary for the model to reproduce key qualitativeaspects of remodeling, indicating that the presence of Hookean springs is essential forthis behavior. These results suggest that traction force mediated matrix remodeling maybe a robust phenomenon not limited to fibrils with a precise set of material properties.[DOI: 10.1115/1.4026179]

Keywords: agent-based modeling, individual-based modeling, cell-based modeling,ECM, collagen fibril, mechanics, remodeling, alignment

1 Introduction

Extracellular matrix (ECM) remodeling has been identified dur-ing many biological processes, including wound healing, develop-ment, fibrotic pathologies, and tissue engineering [1–6]. Despiteits widespread occurrence and biological significance, a fullunderstanding of this process remains incomplete. Commonin vitro experimental models used to study ECM remodeling arecell-populated ECM gels [6–9]. Type I collagen and fibrin arepopular and appropriate choices for these studies as these fibrousmaterials are the major structural protein and an important provi-sional matrix, respectively. As an alternative to experimentationin vitro, computational modeling is advantageous because itallows for inexpensive testing of hypotheses that may not beexperimentally feasible and the development of a model providesan opportunity for further insight into traction force mediatedremodeling of ECM.

Traditional approaches to modeling this phenomenon includemathematical [7,10] and finite-element modeling (FEM) [11,12].Most similar to the model presented here, perhaps, Aghvami et al.used an FEM-based multiscale model to simulate the reorganiza-tion of collagen gels due to simultaneous stretching and cell trac-tion force [12]. Cell traction was modeled simply by shorteningfibers by a fixed percentage of their original length. We chose touse agent-based modeling as alternative approach because it canreadily handle the dynamic and complex interaction between acell and its surrounding fibrils. We previously developed anagent-based model of cell traction force induced remodeling of afibrous network using NetLogo [13]. The goal of this initial “proofof principal” model was to determine if a relatively simple set ofrules could give rise to the relatively complex matrix remodelingobserved in experimental models. In this respect, the initial modelwas successful as many aspects of matrix remodeling observedexperimentally emerged in the simulations from these simplerules, including (1) macroscopic compaction of the fibrous net-work, (2) greater compaction of fibrils in the pericellular regionthan in regions more distant from the cells, and (3) alignment anddensification of fibrils in between pairs of cells. This initial model,however, had a number of limitations that can roughly be divided

1Corresponding author.Contributed by the Bioengineering Division of ASME for publication in the

JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received September 11, 2013;final manuscript received November 26, 2013; accepted manuscript postedDecember 9, 2013; published online February 5, 2014. Editor: Victor H. Barocas.

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into limitations of scope (e.g., using a 2D model of a 3D systemor being able to only model a very small region of a cell-populated gel), limitation of knowledge (e.g., having to choosemodel parameters when only limited experimental data are avail-able to inform such choices), and limitations of the accuracy orappropriateness of the governing equations of the model. To vari-ous degrees, these limitations are common in most analytical andcomputational models. As we begin to improve upon our initialagent-based model, we have considered each of these three typesof limitations but have paid special attention to the governingequations of our model. Since one of the primary benefits of anagent-based model is that it allows one to explore behaviors thatemerge from a set of rules (i.e., governing equations), we believethat this attention to the accuracy of the rules is appropriate.

In our initial model, we relied on the built-in layout-springfunction in NetLogo for the exertion, transmission, and responseto mechanical forces by cells and fibrils. A layout-spring, basedupon the Fruchterman–Reingold algorithm, is a force-directed lay-out algorithm whose designed purpose was for force-directedgraph drawing [14]. By using the built-in layout-spring function,in our initial model spring-like behavior was modeled usingHooke’s law and repulsion was modeled using Coulomb’s law.While there are solid experimental data suggesting that Hooke’slaw describes some aspects of collagen fibril deformation [15,16],there is no experimental evidence to suggest that long-range re-pulsive forces embodied in Coulomb’s law play an important rolein fibril mechanics. Instead, long-range repulsive forces wereincluded in the initial model as a convenient method to enablefibrils to resist bending. The use of a number of nodes connectedwith layout-spring allowed modeled fibrils to reproduce aspects ofECM fibrils (e.g., force transmission in tension and compressionand elastic deformation in the axial direction as well as bending),however, the absence of realistic fibril mechanics calls into ques-tions the validity of our reported results. Here, we implement a re-vised and improved model based on physically realistic fibrilmechanics. As before, Hookean springs are used to resist axialdeformation, but we no longer rely on the black box provided bylayout-spring for how this behavior is implemented. Motion isnow calculated using appropriate spring constants, masses, andformulae from Newtonian mechanics. In addition, torsionalsprings, as opposed to long-range repulsive forces, are used sothat fibrils may resist bending. The choice of torsional springs ispartially motivated by the work of Sander et al. who reported thatsimulations that treat collagen networks as fibers connected bytorsional springs at the cross-links reproduce aspects of themechanical properties of collagen under macroscopic loads [17].The ability of these new fibril mechanics to accurately predict thebehavior of individual fibrils under load is first evaluated by com-paring simulation results to their analytical solutions. The impactof implementing these new fibril mechanics into an agent-basedmodel of cell-fibril interactions is then assessed by the ability ofthe model to qualitatively predict experimentally observed matrixremodeling. We hypothesize that these new fibril mechanics willallow for a model in which cells can significantly remodel theECM as measured by macroscopic compaction and alignment offibrils between cells.

2 Materials and Methods

2.1 General Model Components. Agent-based models wereconstructed in NetLogo [18]. The initial distribution of fibrousECM is generated as previously described [13]. Briefly, fibrils aregenerated as roughly linear strands and overlapping (Fig. 1(a)) oradjacent fibrils (nodes<�3.5 lm apart) may be cross-linked. Theinteraction between nodes on a fibril is now governed by two setsof mechanical rules. Experimental data from collagen fibrils inaxial tension suggest that their behavior can be approximated aslinearly elastic [15,19]. Accordingly, in the new model, one setof mechanical forces acting between pairs of adjacent nodes in

collagen fibrils are consistent with Hookean springs (Fh¼ khDL),where DL is the distance between the two linked nodes minus theequilibrium length of the link (Fig. 1(b)). The second set of me-chanical rules governing the interaction of nodes on fibrils isbased on the concept of torsional springs. Torsional springs areintended to handle the bending behavior of fibrils, which in theprevious model was dictated by the long-range repulsive compo-nent of layout-spring. Each set of three serially connected nodes istreated as a joint acted upon by a torsional spring (Fig. 1(c)). Themagnitude of the moment (Mt) generated by the torsional spring islinearly dependent upon the angle of deflection (180 deg-h).Therefore, Mt¼ kt(180 deg-h) where kt is the torsional spring con-stant. The force required to generate this moment (Ft) is calculatedas Ft¼Mt/d where d is the distance between adjacent nodes.Based on the observation that fibrils generally experience axialstrains less than 2% in our models of cell-fibril interactions, fibrillength was considered a constant when calculating the forces forthe torsional springs. This force is then applied to each adjacentnode (Fig. 1(c)). A complementary force (Ftc) is also applied tothe central node such that when it is summed with the forces onthe two adjacent nodes, the total force acting on the set of threenodes is zero. Since the sum of the forces is zero, the bendingforces can move the relative position of the three nodes but do notmove the center of mass.

Cross-links are modeled by treating each angle formed by thisintersection of links, and not subdivided by another link, as a tor-sional spring (Fig. 1(d)). A consequence of cross-linking is thatfor the nodes connected to the common node the distance to thecommon node usually no longer matches the standard restinglength of 5 lm. Conceptually, this means there is strain present inthese regions of the fibrous network at initial conditions. Beforecell traction is initiated, this strain is relieved by allowing thefibrous network to relax for 100 time steps (10 s).

The mass of each node on a fibril was calculated by treating thenode as representative of a fibril segment, or link. This segment wasassumed to be solid cylinder 100 nm in diameter and 5 lm in lengthwith a specific gravity of 1. As detailed in the supplement that willbe linked here April 2014, the mass of each cell component was cal-culated by dividing the mass of this representative slice of the cellby the number of nodes used to construct a cell. Calculated massesfor the cell components and fibril nodes were 4.30� 10�15kg and3.93� 10�17kg, respectively. Once the forces due to Hookean andtorsional springs are determined, the acceleration, velocity, and posi-tion of each agent is calculated using Newton’s second law, F¼ma,and a fourth order Runge–Kutta method. A difference in massbetween different types of agents results in a difference in the rela-tive motion each node would experience under the same force.

While the primary focus of this study is a more physically real-istic model of matrix fibrils, the modeling of the cells was alsoupdated to replace the layout-spring with Hookean springs. Inaddition, several minor changes were made to the rules for cell-fibril interactions. These changes are detailed in the supplementalmaterial to be linked here April 2014.

2.1.1 Fibril Density. The average fibril density is calculatedby dividing the number of fibril segments, or links, by the area ofthe fibrous network. This area was determined by starting with thetotal number of patches in the simulated region and excludingempty patches along the edge of the simulated region along withall contiguous, empty patches. By this method, empty patcheswithin the fibrous network are considered part of the fibrous net-work while empty patches bordering the edge of the fibrous net-work are not. Density is also calculated in a defined regionbetween the two cells as previously described [13]. In short, thereis a rectangle-shaped region of interest in which the number offibril segments is divided by the area of the rectangle.

2.1.2 Fibril Alignment. Alignment was quantified by calculat-ing the anisotropy index and angle of alignment of fibrils in theintercellular region as previously described [13].

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2.1.3 Statistics. Quantitative data is reported as mean 6 stan-dard error from 10 replicates (n¼ 10). Statistical differences weremeasured using the paired Student’s t test with Bonferroni correc-tion for multiple comparisons. P values below 0.05 were consid-ered statistically significant.

3 Results

3.1 Modeling the Deformation of Single Fibrils UnderLoad

3.1.1 Model Stability and Achieving Steady-State Solutions.A simulation performed of an undamped 100 lm end-loaded can-tilever with an initial deflection of 20 lm resulted in the fibriloscillating with a frequency of �2 Hz. In order for the simulationto be stable, a step size on the order of 10�6 s was necessary.Instability caused by using a larger time step resulted from largeaxial forces developed as a result of axial strain during deflection.These forces caused the nodes to oscillate about an equilibriumposition with increasing distance in a positive feedback fashion.In comparison to the small requisite time step, cells extendand contract membrane processes on the time scale of minutes(�102 s) and large-scale matrix remodeling occurs on the timescale of hours (�104 s). To accurately and stably model the rapidfluctuations of fibrils using a time step on the order of 10�6 s overthe time scale required for large-scale matrix remodeling wouldrequire �1010 iterations, a number that is not computationally fea-sible. To bridge this multiscale problem, a time step of 0.1 s was

chosen and stability was achieved by slowing the time scale offibril motion to that of cell-fibril interactions, implemented bydividing the calculated acceleration by 109, a minimized value.This means that the temporal dynamics of potentially rapid fibrilmotion is lost while preserving their long-term behavior, withthe latter being of primary importance for these long-termsimulations.

Simulations of single fibrils showed that forces applied to fibrilscaused oscillation about a new fibril equilibrium length (axialloading) or magnitude of deflection (three-point bending and end-loaded cantilever). In order to allow for a stable, steady-state solu-tion, a damping component was included and implemented as areduction of the calculated velocity by a fixed percentage (�10%)each time step. This damping compounded such that the change invelocity of a node during one time step would have a decreasingcontribution to that node’s velocity with time. Stated differently,in the absence of force, velocity has a half-life of �0.8 s. Includ-ing this damping component was also necessary when simulatingcell traction force induced remodeling of the ECM. For example,in a hypothetical simulation containing a single cell in a fibrousnetwork, traction forces applied to the network over a finite pe-riod of time would result in overall movement of the fibrous net-work inward towards the cell. Without damping, if traction forceswere turned off, compaction of the fibrous network would stillcontinue to occur due to the inertia of each node. With the levelof damping proposed, the nodes would quickly decelerate withina few seconds and inward compaction due to inertia would rap-idly cease.

Fig. 1 (a) A simplified representation of the fibrous ECM that allows for discus-sion of the rules governing fibril mechanics. (b) A pair of nodes that are connectedby a link behaves according to Hooke’s Law. If the link is stretched or compressed,a force (Fh) proportional to the difference between the link’s length and its restinglength will act at each node in an effort to restore the resting length of the link. (c)Each set of three serially connected nodes behaves as a torsional spring. The mag-nitude of the force (Ft) that acts upon the outer nodes is proportional to the angleformed by the three nodes and acts in the direction perpendicular to the link thatconnects the outer node to the central node. A complementary force (Ftc) also actsupon the central node such that the sum of the forces acting upon the system iszero. (d) A cross-link is treated as multiple torsional springs. Each torsional springshares the common node as its central node and a unique, nondivided angleformed by two links. In this example there are four torsional springs each identifiedby its angle, h1–h4, and the corresponding forces that act upon its nodes. Comple-mentary forces are not labeled on this figure.

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3.1.2 Single-Fibril Results. To validate the fibril model basedon Hookean springs that resist axial strain and torsional springsthat resist bending, the simulated deformations of single fibrilsunder different loading conditions were compared to their analyti-cal solutions using infinitesimal strain theory. Based on the lineartheory of elasticity, the force required to lengthen a fibril a givendistance (DL) is F¼EA(DL/L) where E is the elastic modulus, Ais the cross-sectional area, and L is the length. In comparison, forHooke’s law F¼ khDL. Thus, to model the axial deformation of afibril as a Hookean spring, kh is set to EA/L. Based on considera-tions explained in the supplemental material to be linked here April2014, an elastic modulus of 0.5 GPa and diameter of 100 nm werechosen for collagen fibrils. Consistent with that expected from elas-tic mechanics, pure axial loading of a simulated straight fibrilresulted in axial deformation proportional to the magnitude of theapplied force with a disagreement <0.001% between deformationscalculated analytically and from the agent-based simulation(Fig. 2). The model predictions were independent of the number offibril segments connected in series. For example, a 5 lm fibril mod-eled by two nodes connected by one Hookean spring will give anidentical prediction for strain due to axial loading as a 100 lm fibrilmodeled using 21 nodes and 20 Hookean springs.

To capture the shape of a bent fibril, it was necessary to use asufficient number of nodes to model the fibril. At the lowerextreme, a fibril consisting of two nodes cannot bend and one withthree nodes can only adopt a V-shape, while many nodes allowthe fibril to adopt a more realistic shape. Here, fibrils were createdby spacing nodes 5 lm apart. To consider bending properties of

the fibril, results from a simulated fibril undergoing three-pointbending were compared to the analytical solution for a simplysupported beam with a central load. First, the analytical solutionfor the maximum deflection (dmax), dmax¼FL [3]/48EI, fromEuler–Bernoulli beam theory was used to inform the value of thetorsional spring constant. Specifically, the value of kt was selectedsuch that the deflection from a simulation for a 100 lm fibrilmatched its analytical solution with dmax equal to 1 lm. Havingselected the value of kt to be used in all subsequent simulations,the effect of varying load on dmax for 100 lm fibrils (Fig. 3b) andvarying fibril length on dmax at a constant load (Figs. 3c and 3d)was simulated with good agreement between analytical and com-putational values. The shape of the simulated fibril was then com-pared to the analytical solution, which is d¼Fx(4x2-3L2)/48EI forone-half of the symmetric curve. Consistent with the analyticalsolution, the fibril adopted a parabolic-like shape (Fig. 3e). Alsoconsistent with the analytical solution, d scales with L3 for a con-stant load (or force) (Fig. 3f). Similar comparisons were madebetween the deflection predicted analytically and from simulationsfor fibrils loaded as cantilevers with again good correlationbetween the results (Fig. 4).

In all simulations of bending 100 lm fibrils, the percent differ-ences between the analytical and computational values of d werethe smallest with small deflections with percent errors onlyexceeding �10% when the deformation was greater than �20%of the fibril length. The only cases where a large percent error wasobserved for small deformations were for cases with very shortfibrils. For instance, at the smallest possible length of 10 lm,

Fig. 2 (a) A diagram representing how axial forces were applied to a single fibril. Fibril mechanical properties were testedunder tensile (b) and compressive (c) loads and agreed with analytically predicted values <0.001%.

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simulated fibril deflection was greater than the analytical solutionby 49% for a fibril undergoing three-point bending and 46% dur-ing end-loaded bending. While the percent error may be large forvery short fibrils, the absolute values of the deformation for boththe analytical and simulation prediction were less than 0.01 lm.To explore the impact of simulating a 10 lm long fibril under-going larger deformation, results of agent-based models werecompared to analytical solutions when the torsional spring was

bent by 14.8 deg, the average torsional strain seen in models ofcell-fibril interactions. In this case, the percent error was 43%.Thus, the major source of error was due to approximating theshort continuous fibril as just three nodes and not due to theamount of bending. Thus, these results indicate that an agent-based model of a fibril based on Hookean and torsional springsapproximates the linear analytical solution for fibrils undergoingaxial loading, even for large deformation, as well as small

Fig. 3 (a) A diagram representing how three-point bending of a fibril was performed. A single force was applied at the midpointof a fibril. The ends of the fibril were free to move in the horizontal but not vertical direction. (b) A plot for the deflection of a100 lm fibril in response to a range of different loads. (c,d) The deflections for fibrils of different lengths were plotted inresponse to a constant applied load. The shape of a 100 lm fibril under various loads (e) and of fibrils of different lengths inresponse to a constant load (f) were plotted in comparison to their analytical solutions.

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deformation bending, except when the continuous fibril was com-posed of just a few nodes.

3.2 Integrating New Fibril Mechanics With an Agent-Based Model of Cell-Populated Fibrous Network

3.2.1 Computational Time. One unexpected impact of replac-ing layout-spring with a combination of Hookean and torsionalsprings is that the time to complete each iteration of the simula-tions decreased dramatically. Additionally, the computationaltime appeared to scale linearly with the number of fibril segments,in contrast to the previous model in which computational timescaled exponentially. This increase in computational efficiency issignificant since NetLogo code does not parallelize effectively, sousing more powerful desktop central processing units with multi-ple cores or mainframe computers does not significantly decreasethe time required to complete a single simulation. The currentstudies capitalized on this increased computational efficiency toincrease the spatial and temporal resolution of the model byincreasing the number of agents (8000 versus 1400), number ofiterations per simulations (72,000 versus 22,500) and size of the

space modeled (197,025 lm2 versus 45,375 lm2) relative to previ-ous simulations that relied upon layout-springs.

3.2.2 Comparison of Simulation Results to QualitativeObservations Seen in Multiple Experimental Systems. Across vari-ous experimental models of cell-populated fibrous ECM, severalforms of matrix remodeling are consistently observed with theextent of matrix remodeling dependent on the system and experi-mental conditions. Consistent forms of remodeling include globalcompaction [6–9] of the matrix with greater matrix densificationin the regions around a cell [20] (i.e., pericellular region) orbetween nearby cells [21] (i.e., intercellular region). In addition,the fibrils in the intercellular region align with a line segmentconnecting the cells. Some authors refer to the dense and alignedmatrix between cells as matrical tracks [22] or matrix guidancepathways [23]. Previously, we demonstrated that an agent-basedmodel of cell-fibril interactions based on layout-spring qualita-tively captures each of these aspects of matrix remodeling [13].To explore the impact of building an agent-based model of cell-fibril interactions around more physically realistic Hookean andtorsional springs, we quantified and averaged the simulated global

Fig. 4 (a) A diagram representing how fibrils were treated as end-loaded cantilevers. A singleforce was applied at one end of a fibril. The other end of the fibril was fixed in the horizontal andvertical directions. (b) A plot for the deflection of a 100 lm fibril in response to a range of differ-ent loads. (c,d) The deflection for fibrils of different lengths was plotted in response to a con-stant applied load. The shape of a 100 lm fibril under various loads (e) and of fibrils of differentlengths in response to a constant load (f) were plotted in comparison to their analyticalsolutions.

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and intercellular remodeling for ten simulations. Two cells ini-tially 200 lm apart were simulated within a 555 lm by 355 lmregion. Initially the cells are within a randomly oriented fibrousnetwork with minimal stresses (Fig. 5(a)). As the cells exertedtraction on the fibrous network, stresses propagated through the

network, which can be visualized by the change in colors of thefibrils (Fig. 5(b)). With additional simulation time (Fig. 5(c)),there was global compaction as well as densification and align-ment of fibrils in the intercellular region characterizing matricaltracks. After 2 h of simulated time, the complete ECM had

Fig. 5 Simulations were performed with two cells initially 200 lm apart (n 5 10). (a) 0 h. (b) 1 h. (c) 2 h. Scale bar,100 lm. (d) Percent change in the average density of the fibrous network, measured in links per patch. (e) Per-cent change in the density of the fibrous network between the two cells. Alignment of fibrils between the twocells was measured by both the anisotropy index (f) and overall direction of the fibrils with respect to a line con-necting the nuclei of the two cells (g). *p < 0.05 compared to initial conditions.

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compacted by more than 10% (Fig. 5(d)) and the density in theregion between cells increased by more than 50% (Fig. 5(e)).Fibrils also aligned between cells, a region where the greatestfibril strain was observed reaching up to �0.015. Fibrils outsideof this region typically did not exceed a strain of 0.008. Over 2 hthere was approximately doubling of the anisotropy index forfibrils between cells (Fig. 5(f)) and overall they became highlyaligned with a line that connected the nuclei of the two cells (Fig.5(g)). For comparison, at the end of the simulation shown in Figs.5(a)–5(c), the average deflection experienced by torsional springswas 14.8 deg 6 20.6 deg.

3.2.3 Exploring the Importance of the Torsional SpringConstant Value. Although we have taken great care with respectto the implementation of realistic fibril mechanics and parameterssuch as fibril diameter and elastic modulus, other parameters havenot currently been implemented at equally realistic values. As anexample, the value used for the initial average fibril density wasrestricted by computational cost and underestimates the range thatwould be expected from representative collagen gels. Accord-ingly, internodal distances in the simulated fibrous network(�15 lm) are greater than that suggested by electron micrographsof collagen gels (�1 lm or less) [24,25]. Since longer fibrilswould have a decreased effective stiffness, the ability of fibrils toresist bending in our simulations is likely less than would beexpected for fibrils in a collagen gel. To explore the impact ofincreasing fibril stiffness on fibrous network remodeling, simula-tions were conducted of two cells separated by 100 lm in a fibrousnetwork with identical initial fibril positions and a torsional springconstant 1, 10, or 100 times that used in previous simulations.Hookean spring constants were not varied. At the start of eachsimulation, in the intercellular region the anisotropy index andangle of alignment were 0.52 and 24 deg, respectively. After 1 h,the angle of alignment had decreased in all cases to 6.8 deg (1X),5.8 deg (10X), and 8.5 deg (100X). At the 1X value of the tor-sional spring constant the anisotropy index measured in the inter-cellular region increased to 0.69. Increasing the torsional springconstant tenfold had only a modest impact on the measured ani-sotropy index (0.64). However, increasing the torsional springconstant 100-fold resulted in a decrease in the anisotropy index to0.49. Furthermore, in simulations with the 100X torsional springconstant, fibrils retained more of their initial shape while fibrils insimulations with a 1X or 10X torsional spring constant werestraightened by cell traction forces (Fig. 6(a) versus 6(b)).

Additional simulations were performed using a wider range oftorsional spring constant values (data not shown). With respect tothe average fibril density, in simulations in which torsional springswere not included or a relatively weak torsional spring constantwas used, network compaction occurred steadily until the cellsappeared to internalize the fibrous network in its entirety. Whenusing a relatively strong torsional spring constant, the fibrous net-work could elastically resist cell traction forces and network den-sification slowed until a steady state was reached at a partial levelof compaction.

4 Discussion

The major findings of this work are that (1) steady-state bend-ing of single linearly elastic fibrils under load can be modeled bya series of nodes connected by Hookean and torsional springs and(2) when a network of such fibrils is combined with a model ofcell-fibril interactions, fibrous network remodeling consistent withthat frequently reported in experimental models is observed. De-spite these strengths of this agent-based model, it still has severalnotable limitations.

An agent-based model of a fibril based on Hookean and tor-sional springs approximates the linear analytical solution forfibrils undergoing axial loading, even for large deformation, aswell as small deformation bending, except when the fibril wascomposed of just a few nodes. Better agreement between the

analytical and model predictions at small deformations is notunexpected given that the analytical equations were derivedassuming small deflections of fibrils undergoing bending. Asnoted previously, to capture the shape of a bent fibril, it was nec-essary to use a sufficient number of nodes to model the fibril. Inthe case of the shortest single fibrils that could be experiencesimulated bending, their 10 lm length was modeled by threenodes, two Hookean springs, and one torsional spring. Thus, it isnot surprising that this system that can only adopt a V-shape andfailed to accurately agree with the predicted deflection for a fibril.Decreasing the distance between nodes on a fibril (e.g., from thecurrent 5 lm to 1 lm) would allow better estimates of the mechan-ical properties fibril segments that are on the order of 10 lm.While we modeled collagen fibrils as linearly elastic as suggestedby some reports [15,19], others have suggested that the stress-strain curve may follow an exponential function [19,26,27].Exploring the impact of nonlinear stress-strain behaviors on mod-els of cell-fibril interactions could be accomplished by altering therules for fibril mechanics in the existing agent-based model.

When the new fibril mechanics were incorporated into a modelallowing for cross-linking of fibrils and cell traction force, themodel supported qualitative characteristic features of ECMremodeling (i.e., change in average matrix and intercellular den-sities, as well as intercellular alignment). Notably, our previousmodel that used a layout-spring to approximate fibril mechanicsalso yielded qualitatively similar fibrous network remodeling. Thefact that these characteristic features of ECM remodeling occur inagent-based models with different rules for the mechanical prop-erties of fibrils as well as a variety of experimental models sug-gests that remodeling is a robust phenomenon that is notnecessarily dependent on precise material properties or matrixarchitectures. Comparing simulation results with varying torsionalspring constants illustrates this concept. Whether the spring con-stant is increased 100-fold or reduced to zero, the simulation stillqualitatively reproduces key behaviors observed experimentally in

Fig. 6 Simulations that demonstrate the impact of the tor-sional spring constant value on fibrous network remodeling.Simulations were performed using identical initial conditionsbut either the standard 1X value for the torsional spring con-stant (a) or a torsional spring constant that has been increasedby a factor of 100 (b). Results are shown after 1 h. Scale bar,20 lm.

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cell-populated gels, including the alignment of fibrils in the regionbetween two adjacent cells. While incorporating the resistance tofibril bending embodied in the torsional springs is not required toqualitatively capture fibril alignment, varying the torsional springconstant does influence the degree of fibril alignment, as seen inFig. 6, as well as other quantitative aspects of fibrous networkremodeling such as the average fibril density. These results con-trast with the importance of the Hookean springs to remodeling ofthe fibrous network. When Hookean springs were not included,traction forces could not be propagated through the fibrils andremodeling was limited to the region just surrounding the cell.This suggests that the Hookean springs, but not the torsionalsprings, are essential for characteristic remodeling to occur.

While this agent-based model was intended to be a step towardsa physically realistic model of fibril mechanics, it still has a num-ber of limitations. For example, the mechanical rules for fibrilmechanics do not explicitly consider the drag that would occur asthey move through an aqueous environment. Conceptually, vis-cous drag would impose a force that would act against the direc-tion of movement of the fibrils. While damping was crudely andconveniently included in our model to create stability, its effect issimilar to that which might be anticipated from the inclusion ofviscous drag. In addition, the existing simulation is a two-dimensional representation of a three-dimensional system. Withthe previous model based on a layout-spring where computationaltime appeared to scale with the number of agents squared, it waslikely that it would not be possible to conduct 3D simulations.However, with the physically based rules of fibril mechanicswhere computational time appears to scale linearly with the num-ber of agents, 3D simulations of cell-fibril interactions maybe fea-sible within the NetLogo environment. Despite these and othercurrent limitations, agent-based modeling appears to be well-suited to study aspects of cell-mediated fibrous network remodel-ing and additional refinement of such models appears merited.

Acknowledgment

This work was supported by NSF CMMI-1334757.

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