IUTAM Symposium on C ellular, Molecular and Tissue Mechanics

IUTAM BOOKSERIES

Volume 16

Series Editors

G.M.L.Gladwell,University of Waterloo, Waterloo, Ontario, CanadaR. Moreau, INPG, Grenoble, France

Editorial Board

J. Engelbrecht, Institute of Cybernetics, Tallinn, EstoniaL.B. Freund, Brown University, Providence, USAA. Kluwick, Technische Universitat, Vienna, AustriaH.K. Moffatt, University of Cambridge, Cambridge, UKN. Olhoff, Aalborg University, Aalborg, DenmarkK. Tsutomu, IIDS, Tokyo, JapanD. van Campen, Technical University Eindhoven, Eindhoven,

The NetherlandsZ. Zheng, Chinese Academy of Sciences, Beijing, China

Aims and Scope of the Series

The IUTAM Bookseries publishes the proceedings of IUTAM symposiaunder the auspices of the IUTAM Board.

For other titles published in this series, go towww.springer.com/series/7695

�

Editors

IUTAM Symposium on C ellular,

Proceedings of the IUTAM ymposium

K rishna Garikipati Ellen M. Arruda

Molecular and Tissue Mechanics

held at W oods Hole, Mass., USA, June 18–21, 2008

s

123

K  rishna  GarikipatiDept. Mechanical  Engineering  University of  MichiganAnn Arbor, MichiganUSAkrishna@umich.edu

Ellen M. ArrudaDept. Mechanical EngineeringUniversity of  MichiganAnn Arbor, MichiganUSA

@ .eduarruda umich

ISSN 1875-3507 e-ISSN 1875-3493ISBN 978-90-481-3347-5 e-ISBN 978-90-481-3348-2DOI 10.1007/978-90-481-3348-2Springer Dordrecht Heidelberg London New York

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Editors

38830

Preface

Mechanics plays a central role in determining form and function in biology. Thisholds at the cellular, molecular and tissue scales.

At the cellular scale, mechanics influences cell adhesion, cytoskeletal dynamicsand the traction that the cell can generate on a given substrate. All of these in turn af-fect the cellular functions of migration, mitosis, phagocytosis, endocytosis and stemcell differentiation among others. Indeed, if cells do not develop the appropriatestresses, they are unviable and die. These aspects of cell mechanics are frequentlyused by mainstream biologists, as traditional mechanicians may be surprised tolearn. There is a growing view that many functions of the cell are mechanical innature even though chemical signals play crucial roles in the processes.

Free energy barriers control transitions between different conformations of virtu-ally every macromolecule including DNA, RNA, the adhesion protein integrin, themotor protein myosin, and the proteins vinculin and talin that link the cytoskeletonto focal adhesions. The strain energy can be a significant component of the total freeenergy barrier. For binding to take place, the macromolecules need to be in confor-mational states that expose chemical groups without steric hinderance. The kineticsof chemical reactions are therefore strongly influenced by the conformational strainenergy.

At the tissue level mechanics has obvious applications to understanding the pas-sive and active response of soft and hard tissue. This is seen in the wide use ofcontinuum mechanical models of muscle, tendon, cartilage and bone. Such modelsare not recent. However, the rise of tissue engineering in the past few years withits promise for tissue replacement therapies and the insights it holds to biomimeticstructures is a new frontier for the mechanics of tissue.

This peer-reviewed book of proceedings is one result of the International Unionof Theoretical and Applied Mechanics (IUTAM) Symposium on Cellular, Molecu-lar and Tissue Mechanics, which was held between June 18 and 21, 2008, at WoodsHole, Massachusetts. While held under the IUTAM umbrella, this meeting was un-usual for the active participation of biologists and biophysicists. At least a thirdof the participants had never previously been to a mechanics meeting of any kind.The unifying theme of the meeting and this book of proceedings is a focus on ex-plaining biological states, both normal and pathological, through mechanics, ratherthan merely viewing biology as a fertile playing ground for mechanicians. It was

v

vi Preface

our intent, as organizers, to build upon the success of the IUTAM Symposium onMechanics of Biological Tissue held in Graz, Austria, in 2004, and now as editorsto present a sequel to the corresponding book of proceedings. The IUTAM Sympo-sium on Cellular, Molecular and Tissue Mechanics, and this book of proceedings,as the titles suggest, shift the emphasis to include the cellular and molecular scalesat which the success or failure of a biological organism seems to be largely decided.

The book of proceedings has been organized into sections that roughly reflect thesessions at the Symposium:

� Tissue Mechanics� Cell–Substrate Interactions� Mechanics of DNA� Mechanics of Biopolymer Networks� Cell Adhesion� Growth� Poroelasticity of Bone

We wish to acknowledge the Scientific Committee of the Symposium: HuajianGao of Brown University, Eric van der Giessen of Rijksuniversiteit Groningen,Gerhard Holzapfel of Technische Universitat Graz, Ellen Kuhl of StanfordUniversity, Ray Ogden of University of Glasgow and Ulrich Schwarz of KarlsruheInstitut fur Technologie and Universitat Heidelberg. Karen Raab of University ofMichigan has provided indispensable administrative support over the 2 years thatwe have been involved in this effort. The US National Science Foundation providedmajor financial support that went toward this book of proceedings and the Sympo-sium. Both the NSF and IUTAM provided financial support for the participation ofa number of graduate students and post-doctoral scholars. Finally, we acknowledgeSpringer for its technical and financial support of this publication.

With our Symposium in 2008 and this book of proceedings, we hope to haveinitiated a forum for mechanicians venturing into biology to interact with biologistsand biophysicists who study mechanical influences on biological systems. We be-lieve that these steps will highlight and add momentum to the emergent conceptthat, while considerations of chemistry have hitherto dominated the explanations ofcausality in biology, mechanics often plays a role of at least equal importance.

University of Michigan, Ann Arbor Krishna GarikipatiMay, 2009 Ellen M. Arruda

Contents

Part I Tissue Mechanics

Experimental and Computational Investigationof Viscoelasticity of Native and Engineered Ligamentand Tendon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3J. Ma, H. Narayanan, K. Garikipati, K. Grosh, and E.M. Arruda

A Comparison of a Nonlinear and Quasilinear ViscoelasticAnisotropic Model for Fibrous Tissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19T.D. Nguyen

Hysteretic Behavior of Ligaments and Tendons:Microstructural Analysis of Damage, Softeningand Non-Recoverable Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31P. Ciarletta and M. Ben Amar

On Measuring Stress Distributions in Epithelia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45V.D. Varner and L.A. Taber

A Viscoelastic Anisotropic Model for Soft Collageneous TissuesBased on Distributed Fiber–Matrix Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55A.E. Ehret, M. Itskov, and G. Weinhold

Part II Cell-substrate Interactions

Chemical and Mechanical Micro-Diversityof the Extracellular Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69T. Volberg, J. Ulmer, J. Spatz, and B. Geiger

Tissue-to-Cellular Deformation Couplingin Cell-Microintegrated Elastomeric Scaffolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81J.A. Stella, J. Liao, Y. Hong, W.D. Merryman, W.R. Wagner,and M.S. Sacks

vii

viii Contents

Orientational Polarizability and StressResponse of Biological Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91S.A. Safran, R. De, and A. Zemel

Universal Temporal Response of Fibroblasts Adheringon Cyclically Stretched Substrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103S. Jungbauer, B. Aragues, J.P. Spatz, and R. Kemkemer

Part III Mechanics of DNA

Elastic and Electrostatic Model for DNA in Rotation–ExtensionExperiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113S. Neukirch, N. Clauvelin, and B. Audoly

Shape and Energetics of DNA Plectonemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123P.K. Purohit

Part IV Mechanics of Biopolymer Networks

Constitutive Models for the Force-Extension Behaviorof Biological Filaments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .141J.S. Palmer, C.E. Castro, M. Arslan, and M.C. Boyce

Small Strain Topological Effects of Biopolymer Networkswith Rigid Cross-Links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .161G. Zagar, P.R. Onck, and E. Van der Giessen

Part V Cell adhesion

An Observation on Bell’s Model for Molecular BondSeparation Under Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .173L.B. Freund

A Theoretical Study of the Thermodynamics and Kineticsof Focal Adhesion Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .181J.E. Olberding, M.D. Thouless, E.M. Arruda, and K. Garikipati

Tension-Induced Growth of Focal Adhesions at Cell–SubstrateInterface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .193J. Qian, J. Wang, and H. Gao

Pattern Formation and Force Generation by Cell Ensemblesin a Filamentous Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .203R. Paul and U.S. Schwarz

Contents ix

Mechano-Chemical Coupling in Shell Adhesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .215R.M. Springman and J.L. Bassani

Catch-to-Slip Bond Transition in Biological Bonds by Entropicand Energetic Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .227Y. Wei

Part VI Growth

Dilation and Hypertrophy: A Cell-Based ContinuumMechanics Approach Towards Ventricular Growthand Remodeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .237J. Ulerich, S. Goktepe, and E. Kuhl

A Morpho-Elastic Model of Hyphal Tip Growthin Filamentous Organisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .245A. Goriely, M. Tabor, and A. Tongen

Extracellular Control of Limb Regeneration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .257S. Calve and H.-G. Simon

Part VII Poroelasticity of Bone

Bone Composite Mechanics Relatedto Collagen Hydration State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .269M.L. Oyen and M. Galli

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .277

Part ITissue Mechanics

Experimental and Computational Investigationof Viscoelasticity of Native and EngineeredLigament and Tendon

J. Ma, H. Narayanan, K. Garikipati, K. Grosh, and E.M. Arruda

Abstract The important mechanisms by which soft collagenous tissues such asligament and tendon respond to mechanical deformation include non-linear elas-ticity, viscoelasticity and poroelasticity. These contributions to the mechanicalresponse are modulated by the content and morphology of structural proteins suchas type I collagen and elastin, other molecules such as glycosaminoglycans, andfluid. Our ligament and tendon constructs, engineered from either primary cellsor bone marrow stromal cells and their autogenous matricies, exhibit histologicaland mechanical characteristics of native tissues of different levels of maturity. Inorder to establish whether the constructs have optimal mechanical function for im-plantation and utility for regenerative medicine, constitutive relationships for theconstructs and native tissues at different developmental levels must be established.A micromechanical model incorporating viscoelastic collagen and non-linear elasticelastin is used to describe the non-linear viscoelastic response of our homogeneousengineered constructs in vitro. This model is incorporated within a finite elementframework to examine the heterogeneity of the mechanical responses of native lig-ament and tendon.

1 Introduction

Ligaments and tendons are soft tissues that support muscle and bone structures inthe body. The incidence of ligament and tendon rupture in the US has increaseddrastically in recent years; particularly acute among the pediatric population is theincreased incidence of knee ligament rupture. A common autograph approach to an-terior cruciate ligament (ACL) reconstruction uses a portion of the patient’s patellartendon as a graft. Previous investigations have shown differences in the viscoelasticresponses of ligaments and tendons suggesting limitations in the ultimate efficacy

J. Ma, H. Narayanan, K. Garikipati, K. Grosh, and E.M. Arruda (�)University of Michigan,e-mail: arruda@umich.edu

K. Garikipati and E.M. Arruda (eds.), IUTAM Symposium on Cellular, Molecularand Tissue Mechanics, IUTAM Bookseries 16, DOI 10.1007/978-90-481-3348-2 1,c� Springer Science+Business Media B.V. 2010

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4 J. Ma et al.

of a tendon as a ligament graft. Tendon allografts are also often used in ligamentreconstruction; these incur an additional risk of immune rejection. These limitationshave led to an increased urgency for engineered replacement tissues for ligament andtendon reconstructions. The goal of tissue engineering is to form viable constructsthat can replicate the biomechanical function of native tissue, with biomechanicallycompatible interfaces between the engineered musculoskeletal tissue and native tis-sue. This requires detailed understanding of the function of native tissue and tissueinterfaces, including growth and remodeling mechanics, structure – function rela-tionships and healing response mechanics, to design optimal structures for skeletaltissue replacement. Therefore, the goal of our current tissue engineering approachis to develop self-organized, scaffold-free constructs for skeletal tissue replacementor reconstruction with mechanically viable, biochemically relevant tissue interfacesfrom patient-harvested cells, and to compare their mechanics at various develop-mental stages to the mechanics of native tissue.

Our laboratory has previously created engineered ligament, tendon and bone invitro solely from bone marrow stromal cells (BMSC) or primary cells [6, 9, 15].In order to evaluate the compatibility and feasibility of our in vitro experimentalmodels, mechanical responses of native tissue under different conditions are inves-tigated. Furthermore, computational models based on the finite element frameworkare established to examine the mechanics of both native and engineered soft tis-sue. Various mechanisms by which tissues respond to mechanical deformation havebeen observed, including non-linear elasticity, viscoelasticity and poroelasticity.Contributions to the overall mechanical response involve various known and un-known factors. In our computational model, factors that modulate the mechanismsinclude the content and morphology of structural proteins such as type I collagenand elastin, other molecules such as glycosaminoglycans, and fluid. Sufficient ex-perimental data allow us to evaluate the accuracy and stability of the computationalmodel.

Recently our laboratory has developed a scaffold-less method to co-culturethree-dimensional (3D) ligament and bone constructs from rat BMSCs in vitroto engineer a bone-ligament-bone (BLB) construct [9]. Bone marrow was col-lected from rat femurs and tibias and cultivated to bone and ligament pathwaysusing specific growth factors. Both types of cells were plated onto laminin coatedculture dishes after the 3rd passage. After cells became confluent and the extra-cellular matrix that cells have synthesized was strong enough, bone monolayerswere cut into two pieces and pinned using minutien pins on top of the ligamentcell monolayers such that the proximal bone construct ends were 10 mm apart.Approximately 1 week following media change, the ligament monolayers rolledup around the bone constructs forming a 3-D BLB construct. These co-culturedconstructs were used for ligament replacement in a rat model and the mechanicsof these constructs were examined both prior to implantation and upon explanta-tion. Briefly, the native MCL was excised and holes were drilled at the originalMCL insertions on the bones. The engineered BLB construct replaced the nativeMCL by inserting its bone ends into corresponding holes. Four weeks of implanta-tion of our BLBs in a medial collateral ligament (MCL) replacement application

Experiments and Computations on Viscoelasticity of Ligament and Tendon 5

demonstrated that our in vitro engineered tissues initially grew and remodeledquickly in vivo to an advanced phenotype and functionality to restore structuralfunction to the knee [9]. Tangent moduli of the ligament portion of the BLB ex-plants were equivalent to those observed in 14-day-old neonatal rat MCLs andthis region stained positively throughout for crimped type I collagen and elastin.The explants also demonstrated viscoelastic and functionally graded responses thatclosely resembled those of native ligaments. We have also found that the averagemechanical response is not sufficient to fully characterize the mechanical proper-ties of ligament and tendon. Previous investigators have shown distinctly differentbending strain response along different portions of native MCL [1, 16, 18]. Theseworks have demonstrated higher strain levels near bone insertions compared tomid-ligament strains. Our investigations on native MCL have shown a heteroge-neous mechanical response in tension that is consistent with the previous results.Results from our implantation showed the engineered BLB constructs adapteda functionally graded mechanical response in vivo that matched the heterogene-ity of native MCL. Previously we have also shown functional inhomogeneity inrat tibialis anterior (TA) tendons [4]. Here we investigate this behavior in miceTA tendons from both adult and old animals. We develop a micromechanicalmodel of non-linear viscoelasticity and implement it into a computational frame-work. We examine the efficacy of our computational model [6] to describe thefunctionally graded viscoelastic responses of native and engineered ligament andtendon.

2 Experimental Methods

This section briefly explains the native tissue isolation and mechanical testing meth-ods. Details may be found in previous work [6, 9, 15].

2.1 Native Tissue Isolation

Fischer 344 rats were sacrificed at 14 days and 3 months following birth. The legswere dissected, removing the skin and muscle but maintaining the ligament connec-tions at the knee. The MCL was isolated by removing all other knee ligaments. Thetibia and femur were cut mid-bone to provide tissue for gripping during mechan-ical testing. C57Bl/6 mice were obtained at about 3 months and at 33–35 monthsand sacrificed. The feet were dissected and the TA tendon isolated as previouslydescribed [11]. The entire muscle–tendon–bone unit from the TA muscle to the firstmetatarsal bone was kept intact for gripping purposes so that the entire TA tendon,from the myotendinous junction (MTJ) to the enthesis, was in the field of view ofthe camera during mechanical testing.

6 J. Ma et al.

2.2 Mechanical Evaluation of Native Ligament and Tendon

Cyclic tensile tests on native MCLs and TA tendons were conducted to obtainporoviscoelastic responses to multiple load/unload cycles and examine the mechan-ical heterogeneity of the ligament and TA tendon. Cross sectional areas (CSA) weremeasured from multiple locations along the samples. Multiple CSA measurementswere taken at each region to obtain the average size of each region for future me-chanical property measurements. An in-house designed tensiometer was employedto conduct the cyclic tension. The device consisted of an optical force transducerof our own design with a force resolution of 0.2–200 mN, two uniaxial servomo-tors controlled using Labview, and a Basler digital video camera connected to aNikon (SMZ800) dissecting microscope [6]. Blue microsphere fiduciary markers(25 �m diameter) were brushed evenly on the surface of the samples for digitalimage correlation analysis of tissue displacements to provide highly accurate cal-culations of the tissue strain field along the entire sample. For strain reporting theTA tendon was partitioned into three approximately equal sub-regions, the distal ornear-bone region, the fibrocartilage (FC) region (mid-section) and the proximal ornear-muscle region. These three subsections are shown in Fig. 1. Ligament strainwas measured across 2mm lengths near the insertions (Regions I and III) and atthe mid-section (Regions II), as shown in Fig. 2. Samples were loaded in the de-vice under cyclic tension loading (0–10% strain, 0.01 Hz) and the synchronizedforce and image recordings were compiled and controlled by LabVIEW softwareon a Dell Precision 300 computer. Load-unload cycles were conducted to charac-terize the overall non-linear poroviscoelastic response based on the average strainalong the section length of the ligament and tendon. These same cyclic loadingdata were used with the local strain field measurements to examine the functionallygraded response of the engineered or native ligament. Smoothed strain data werecombined with the synchronized nominal stress (force over cross-sectional area)data to create the cyclic nominal stress vs. nominal strain response curves for eachspecimen.

Fig. 1 Subsections of mouse TA tendon for strain reporting purposes: the proximal end (bluearrow) attached to muscle, the fibrocartilagenous mid-section (red arrow) and the distal end at-tached to bone (green arrow). Note that the TA muscle and metatarsal are held in the grips and theentire TA tendon strain field, from the MTJ to the enthesis, is examined. Scale barD 1mm

Experiments and Computations on Viscoelasticity of Ligament and Tendon 7

Fig. 2 Native rat MCLmechanics at 3 months. (a)Strain analysis and (b) cyclicloading responses of I distal,II mid-section and IIIproximal regions

3 Mathematical Modeling of Mechanical Response

3.1 Micromechanical Modeling of Non-linear Viscoelasticity

The response of a homogeneous non-linear viscoelastic tissue is modeled in termsof the standard non-linear solid micromechanical model of Fig. 3. In this model, onenon-linear spring B and linear viscous dashpot in series is in parallel with anothernon-linear spring A. To model the engineered and native tissue, a Gaussian or neo-Hookean chain network (see Treloar for example [17]) is used for spring A and anetwork of MacKintosh chains [10] is utilized in spring B. In this figure (Fig. 3),the nonlinear spring represents the entire 8-chain network of chains which is shownin Fig. 4.

As shown in Fig. 3 Fe is the elastic part of the deformation tensor and Fv is theviscous part. From compatibility, the total deformation F can be derived as F DFeFv. The left Cauchy Green Tensor B is FFT. From equilibrium of the system, thetotal Cauchy stress tensor can be derived as � D �AC �B , where �A is the Cauchystress tensor generated from spring A and �B is the Cauchy stress tensor generatedfrom spring B. The MacKintosh chain network of the micromechanical model isembedded within an initially isotropic or anisotropic 8-chain framework [2, 4, 5]as in Fig. 4 to mathematically model the mechanical behavior of ligaments and

8 J. Ma et al.

Fig. 3 Micromechanicalmodel of engineered andnative tissue

c

a

bN3

N1

N2

Fig. 4 An anisotropic representative volume element for a network of semi-flexible chains [4, 5]

tendons. The Cauchy stresses on each element can be represented as follows:

�A D nk�AAe � pI (1)

�B Dnk�B3a

r0

�c

1

4.1 � �c�0=Lc/2

�Lc=a � 6.1 � �cr0=Lc/

Lc=a � 2.1 � �cr0=Lc/

�B � pI;

�c Dp

tr.B/=3 (2)

In these constitutive equations, n is the chain density of the Gaussian or neo-Hookean and MacKintosh networks of chains, k is Boltzmann’s constant, � istemperature, and p is the hydrostatic pressure. For the MacKintosh chain network arepresents the persistence length, Lc represents the contour length, r0 is the initialvector chain length and �c is the chain stretch. The linear dashpot constitutive

Experiments and Computations on Viscoelasticity of Ligament and Tendon 9

equation is Dv D� 0B

��where Dv is the viscous shear strain rate, �� is the constant

shear viscosity and � 0B is the equivalent shear stress tensor. The network deforma-tion is assumed to be isochoric and incompressible.

A rate formulation is employed to compute the stress vs. strain responses of var-ious tissues to a cyclic load/unload test. Briefly, time and total stretch are prescribedso that Fv can be explicitly computed based on the rate of deformation of the vis-cous dashpot updated from the previous step. Fe is therefore updated in the currenttime step and then used to compute the stresses. Once the total stress is calculated,the rate of deformation is again updated for the viscous stretch computation in thenext step.

3.2 Governing Equations of the Computational Model

Previously we were able to establish a multi-phasic computational framework forgrowth and remodeling in tissues to model a nonlinear anisotropic elastic or vis-coelastic collagen network phase plus a fluid phase and various diffusing nutrientsand soluble factors. This multiple species approach necessitates that classical bal-ance laws are enhanced via fluxes of species relative to one another and sources e.g.of collagen to describe tissue growth [7, 12, 13]. In this way the model accounts forthe coupled transport of species in a microstructurally evolving system subject tomechanical and chemical signals. The computational framework is required for theheterogeneous mechanical response (functional gradient) seen in native ligamentand tendon as well as in engineered ligament explants.

The governing equations coupled with continuum balance equations describe thebehavior of soft tissue as shown in Fig. 5. Constitutive laws are derived to sat-isfy the governing equations and are used to establish the finite element framework

PN

x

j s n

Ω0

Ωt

n ·m

N·M

ql

g

X

l

l

Fig. 5 Interaction forces, tractions and body forces on a tissue

10 J. Ma et al.

to simulate the remodeling and ageing of soft tissue. Detailed derivations of themathematical equations may be found in previous work [12, 13].

As a result of mass transport and the inter-conversion of species, the mass balancefor an arbitrary species in the current configuration can be described as

@�l

@tD � l � rx �m

l (3)

where �l is the species concentration, � l is the species production rate andml is thespecies total flux. In soft tissues, the species production rate and flux are stronglydependent on the local state of stress. Therefore, the balance of linear momentum iscoupled to mass transport in the determination of the local state of strain and stresswhich is described as follows:

�l .gl C ql /Crx � �l � .rxvl /ml (4)

Quantities used in this equation are the gl body force, ql interaction force, � l partialCauchy stress and vl species velocity.

Summation over the rate of change of energy (1ST Law) for all species in this sys-tem gives a result that insures there is no net energy production mechanism internalto the system. The energy equation is combined with the entropy inequality (2ND

Law), resulting in the Clausius–Duhem inequality, or reduced dissipation, which,along with constitutive assumptions, provides the constitutive laws. For instance,the internal energy of a species may assumed to be of the form:

el D Oel .Fl ; �l ; �l / (5)

where Fl is the deformation gradient, �l is the entropy and the species concentrationis �l . Therefore the Clausius–Duhem inequality is derived as:

Xl

�l Pel � �l P�l � � l W grad

�vl�C

hl � grad./

!

CXl

��l .ql C grad.el / � grad

��l�/ � vl C � l

� l C

1

2k vl k2

��� 0

where l is Helmholtz free energy and hl is the partial heat flux. Thermody-namically consistent constitutive relationships therefore arise from the dissipationinequality. For an elastic or viscoelastic material, a sufficient condition to satisfy isto specify that the partial second Piola–Kirchhoff stress tensor Sc has the form

Sl D Fe�1

2@ O l

@CeFe�T

(6)

Experiments and Computations on Viscoelasticity of Ligament and Tendon 11

Fig. 6 The effect ofpersistence length on theforce vs. extension responseof a MacKintosh chain

20

a=L/2a=L/3a=L/4

10

f/kT

00 0.2

r/L

0.4 0.6 0.8 1

with a suitable energy equation O l for internal variables m of the collagen fibers.Since some compressible materials exhibit different bulk and shear responses, thefree energy function is therefore decomposed into volumetric and isochoric parts:

O l .Ce; 1; :::; m/ D Wvol .Je/CWiso. NC

e/C

mX˛D1

�˛. NCe; ˛/ (7)

where J e is the determinant of the elastic portion of the deformation gradient tensorand isochoric right Cauchy–Green deformation tensor NCe D J e

�2=3Ce . The aboveequation has included the volumetric and isochoric equilibrium response of the solidphase and the viscoelastic response, characterized from the last term [12].

As shown in Fig. 6, variations in persistence lengths lead to differences in me-chanical response. A smaller persistence length results in a longer toe region witha relatively compliant initial mechanical response, while a larger persistence lengthleads to a shorter toe region and a relatively stiffer response.

4 Results

4.1 Engineered Ligament In Vitro,In Vivo and Young Animal MCL

Uniaxial cyclic load/unload tests were conducted using in vitro and in vivo en-gineered BLB constructs and native neonatal rat MCL. The parameters of mi-cromechanical model described previously were determined from the experimentalresults. The numerical and experimental results are shown in Fig. 7. By varying thestiffnesses of two nonlinear springs and the viscosity of the dashpot, the microme-chanical model is robust enough to capture the mechanical responses of in vitro andin vivo engineered constructs and native MCLs.

12 J. Ma et al.

Fig. 7 Micromechanical modeling of engineered in vitro BLB constructs (a), engineered 1-monthin vivo BLB constructs (b) and 14day old rat neonatal MCL (c)

4.2 Native Ligament and TA Tendon Mechanics

As shown in Fig. 2, our investigations of native MCL have found that the nativeligament exhibits a heterogeneous mechanical response. Near either bone inser-tion the ligament is more compliant and more extensible and it exhibits appreciable

Experiments and Computations on Viscoelasticity of Ligament and Tendon 13

hysteresis (or viscous loss) during cyclic loading whereas along the mid-section theligament is stiffer and less extensible and little hysteresis is seen. Functional het-erogeneity is also found in adult mouse TA tendons. As shown in Fig. 8a, overall,the tendon demonstrates a viscoelastic response. Locally, the distal end is stiffer and

Fig. 8 Local mechanical response from experimental (a) & (b) and computational (c) and (d)results of adult and old TA tendons. (a) and (c) adult TA tendon, (b) and (d) old TA tendon

14 J. Ma et al.

Fig. 8 (Continued)

less extensible than the proximal end, which is very compliant and extensible. Boththe mid-section and the proximal end exhibit hysteresis, indicating a time dependentor viscoelastic behavior, whereas little hysteresis is seen at the distal end. A similartest protocol was conducted on old mouse TA tendon and the responses are shownin Fig. 8b. Ageing results in a leftward shift or stiffening in the response of the

Experiments and Computations on Viscoelasticity of Ligament and Tendon 15

mid-section and proximal end and therefore a decrease in the functionally gradedmechanical response in old tendons. Hysteresis is also reduced in the mid-sectionand proximal end of old tendons.

4.3 Computational Results

The mathematical formulation developed for soft tissue has been implemented intoa finite element framework using COMSOL Multiphysics, a computational en-vironment for solving coupled systems of partial differential equations [12]. Forsimplification, the model is set up in a two dimensional structure assuming a stateof plane strain. Triangular elements are used to characterize and estimate the dis-placement field of the soft tissue. Representative model geometries at the initial(undeformed) state are shown in Fig. 9. The mid-section of the adult TA tendonhas the largest cross-sectional area whereas in the old TA tendon, this is the small-est section. Model geometries were chosen to approximate the cross-sectional areadata. In order to model the functionally graded response of an adult TA tendon, thepersistence length a was allowed to vary linearly in the simulation from the prox-imal end to the distal end whereas the contour length Lc , and the initial length r0were assumed to be constant along the tendon. In the old TA tendon much of thefunctionally graded extensibility has been lost and overall, the TA response is stiffer.This is modeled by a third order polynomial variation in a, a linear variation in lp ,a linear variation in r0, and constant contour length.

Parameters were fit from the experimental results and by varying the persis-tence lengths for different portions of TA tendons accordingly, the computationalmechanical responses of adult and old TA tendons are obtained and shown in Fig. 8cand d. Compared to the corresponding experimental results, the model accuratelycaptured several features of the overall mechanical response and the functional gra-dient of adult and old TA tendons with the same patterns that have been shown inthe experimental data.

Fig. 9 TA tendon specimens from adult (a) and old (b) mice and representative initial modelgeometries for adult (c) and old (d) mouse TA tendons

16 J. Ma et al.

5 Conclusion

Experimental results show the mechanical response of ligaments and tendons isnon-linear, poroviscoelastic and functionally graded. Moreover, engineered liga-ments used as an MCL replacement develop a functional gradient in vivo. Ourcomputational model of connective tissue has been used to explore the rich me-chanical response of native and engineered tendons and ligaments such as thatof the TA tendon described above. Ageing was used an as example of how thiscomputational model may also be used to examine constitutive property changesin tendon with disease and pathology. To replicate the complicated mechanical re-sponse of soft connective tissue with engineered materials is a challenge for tissueengineering.

References

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3. Arruda EM, Mundy K, Clave SC, Baar K (2006) Regional variation of tibialis anterior tendonmechanics is lost following denervation. J Appl Phys 53(4):1113–1117

4. Bischoff JE, Arruda EM, Grosh K (2002a) A microstructurally based orthotropic hyperelasticconstitutive law. J Appl Mech 69:570–579

5. Bischoff JE, Arruda EM, Grosh K (2002b) Orthotropic hyperelasticity in terms of an arbitrarymolecular chain model. J Appl Mech 69(4):198–201

6. Calve SC, Dennis RG, Kosnik P, Baar K, Groash K, Arruda EM (2004) Engineering of func-tional tendon. Tissue Eng 10(5,6):755–761

7. Garikipati K, Arruda EM, Grosh K, Narayanan H, Calve SC (2004) A continuum treatmentof growth in biological tissue: Mass transport coupled with mechanics. J Mech Phys Solids52(7):1595–1625

8. Larkin LM, Calve SC, Kostrominova TY, Arruda EM (2006) Structure and functional evalua-tion of tendon-skeletal muscle constructs engineered in vitro. Tissue Eng 12(11):3149–3158

9. Ma J, Goble K, Smietana M, Kostrominova T, Larkin L, Arruda EM (2008) Morphologicaland functional characteristics of three-dimensional engineered bone-ligament-bone constructsfollowing implantation. J Biomech Eng (submitted)

10. MacKintosh FC, Kas J, Janmey PA (1995) Elasticity of semiflexible biopolymer networks.Phys Rev Lett 75:4425

11. Mendias CL, Bakhurin KI, Faulkner JA (2001) Tendons of myostatin-deficient mice are small,brittle, and hypocellular. PNAS 105(1):388–393

12. Narayanan H (2007) Ph.D. Thesis: A continuum theory of multiphase mixtures for modellingbiological growth, in Mechanical Engineering, University of Michigan, Ann Arbor

13. Narayanan H, Arruda EM, Grosh K, Garikipati K (2004) The micromechanics of fluid–solidinteractions during growth in porous soft biological tissue. J Mech Phys Solid 52:1595–1625

14. Palmer JS, Boyce MC (2008) Constitutive modeling of the stress–strain behavior of F-actinfilament networks. Acta Biomater 4:597–612

15. Syed-Picard FN, Larkin LM, Shaw CM, Arruda EM (2009) Three-dimensional engineeredbone from bone marrow stromal cells and their autogenous extracellular matrix. Tissue EngPart A 15(1):187–195

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16. Thomopoulos S, Marquez JP, Weinberger B, Birman V, Genin GM (2006) Collagen fiber ori-entation at the tendon to bone insertion and its influence on stress concentrations. J Biomech39:1842

17. Treloar LRG (2005) The physics of rubber elasticity. Oxford University Press18. Warren LF, Marshall JL, Girgus F (1974) The prime static stabilizer of the medial side of the

knee. J Bone Jt Surg 56(A):665

A Comparison of a Nonlinear and QuasilinearViscoelastic Anisotropic Modelfor Fibrous Tissues

T.D. Nguyen

Abstract This paper presents a nonlinear and quasilinear viscoelasticity model forthe behavior of soft fibrous tissues that incorporates the effects of matrix and fiber-level viscoelasticity. Both models treat the tissue as a composite of N fiber familiesin an isotropic matrix. For the nonlinear model, the anisotropic contribution ofthe fibers to the constitutive behavior of the tissue is developed by first definingthe stress response and viscous flow response of the fiber families, then averagingby the fiber orientation. Similarly, the anisotropic contribution of the quasilinearmodel is developed from a hereditary integral formulation for the stress responseof the fiber families. The stress relaxation response of the nonlinear and quasilinearmodels are compared for different applied strains. As expect, the time-dependent re-sponse of the two formulations are nearly identical for small strain, but they exhibitsignificant differences at large strain.

1 Introduction

Soft fibrous tissues such as the cornea and tendons have a unique combination ofmechanical properties that enable them to perform important structural, protec-tive, and energy-absorbing applications. Because of their fibrous microstructure,these tissues are extraordinarily stiff and strong for their weight. They also possessa unique combination of flexibility and toughness that is exploited for energy-absorbing and protective applications. The toughness of these tissues arises fromtheir ability to dissipate energy through a variety of mechanisms, including poro-plastic flow, fiber-matrix interactions, and matrix and fiber viscoelasticity. For theselatter mechanisms, we have developed a microstructure-based constitutive modelfor the anisotropic, nonlinear viscoelastic, behavior of soft tissues that incorporatesthe effects of matrix and fiber-level viscoelasticity.

T.D. Nguyen (�)Johns Hopkins University, Baltimore, MD 21218, USAe-mail: vicky.nguyen@jhu.edu

K. Garikipati and E.M. Arruda (eds.), IUTAM Symposium on Cellular, Molecularand Tissue Mechanics, IUTAM Bookseries 16, DOI 10.1007/978-90-481-3348-2 2,c� Springer Science+Business Media B.V. 2010

19

20 T.D. Nguyen

Microstructure-based approaches have been applied to model the anisotropicbehavior of many fibrous tissues, including tendinous tissues [11, 22], arterial andvalvular tissues [7, 20], irregular tissues like skin [1, 12], and the cornea [14–16].One-dimensional nonlinear viscoelastic models have been developed for tendons,but most three-dimensional viscoelastic models for irregular and planar tissues usea quasilinear approach. However, recent studies have shown that quasilinear modelscannot reproduce the nonlinear time-dependent behavior, such as stress-dependentcreep rates and strain-dependent stress relaxation rates, observed for tendons, lig-aments, and cornea [3, 9, 17]. Here, we briefly present a generalized constitutiveframework for the anisotropic behavior of soft fibrous tissues that incorporates theeffects of nonlinear viscoelasticity of the matrix and at the fiber-level. A more de-tailed presentation can be found in [14,15]. We also present an analogous quasilinearmodel that is similar in all details to the nonlinear model except for its quasilineartreatment of the fiber viscoelasticity. The uniaxial stress relaxation response of bothmodels are compared for an orthotropic material to demonstrate important featuresof the nonlinear formulation.

2 Model Development

This section briefly presents the nonlinear and quasilinear viscoelastic models. Thestress relaxation response of the two models are compared in the following sectionfor an orthotropic material.

2.1 Anisotropic Nonlinear Viscoelastic Model

The fibrous tissue is modelled as a composite of N fiber families (e.g., collagenfibers) in an isotropic matrix (e.g., peptidoglycan). The orientation of the fiber fam-ilies are described in the reference configuration using structure tensors M˛ D

P˛ ˝ P˛ , where P˛ is the unit orientation vector for the fiber family F˛ [21]. Todescribe the time-dependent response, it is assumed that the deformation gradientcan be decomposed into viscous and elastic parts F D Fe

FFvF D Fe

MFvM. Assuming

that the fibers deform affinely with the continuum, the total and viscous fiber stretchcan be defined as,

�˛ Dp

C WM˛; �v˛ D

pCv

F WM˛; (1)

where C D FTF and CvF D FvT

F FvF . The structure tensor of the fiber families are

mapped to the intermediate configuration using the viscous deformation gradientof the fiber phase, eP˛ D 1

�v˛

FvFP˛ [18]. This allows the elastic fiber stretch to be

defined as,

�e˛ D

qCe

F WeM˛; (2)

where CeF D FeT

F FeF .

Comparison of Models for Fibrous Tissues 21

The free energy density of the composite is modelled as the sum of an isotropicpart WM for the matrix and an anisotropic part for the fiber phase. The latter is thesum of the free energy density WF˛ of the fiber families and depends only on thefiber stretches. To model time-dependent behavior, bothWM andWF˛ are split intoan equilibrium part that depends only on the total deformation and a disequilibriumpart that depends only on the internal deformation Ce

M D FeT

MFeM of the matrix and

internal stretch �e˛ of the fiber,

W�

C;M˛;CvM;Cv

F�D W

eqM . I1; I2; I3/CW

neqM

�I eM1; I e

M2; I e

M3

�

C

NX˛D1

�W

eqF˛ . �˛/CW

neqF˛

��e˛

��: (3)

The variables I1; I2; I3 are the isotropic invariants of C and I eM1; I e

M2; I e

M3are the

isotropic invariants of CeM. The second Piola–Kirchhoff stress is computed from the

free energy density using the standard definition S D 2 @W@C as,

S D 2@W

eqM

@I11C

@WeqM

@I2. I11 � C/C

@WeqM

@I3I3C�1

„ ƒ‚ …SeqM

C 2@W

neqM

@I eM1

Cv�1M C

@WneqM

@IM2

�I eM1

Cv�1M � Cv�1

M CCv�1M

�C@W

neqM

@I eM3

I eM3

C�1

„ ƒ‚ …SneqM

C

NX˛D1

seqf˛. �˛/M˛

„ ƒ‚ …SeqF

C

NX˛D1

sneqf˛

��e˛

� M˛

CvF WM˛„ ƒ‚ …

SneqF

; (4)

where seqf˛. �˛/ and sneq

f˛

��e˛

�are the equilibrium and disequilibrium stress of the

fiber family F˛ given by,

seqf˛. �˛/ D

1

�˛

@WeqF˛ . �˛/

@�˛; s

neqf˛

��e˛

�D

1

�e˛

@WneqF˛

��e˛

�@�e˛

: (5)

Note that the anisotropic part of the stress response, SneqF , depends on the total

and viscous fiber stretch and the fiber structure as defined by M˛ . To completethe model, we must define constitutive relations for the viscous deformation of thematrix and fiber families. For the fibers families, we first define a flow rule for theviscous fiber stretch as,

P�v˛

�v˛

D1

�f˛s

neqf˛�e2˛ ; (6)

where �˛ is the viscosity of the fiber family F˛ . In general, �˛ can depend on thefiber flow stress, sneq

f˛, and the total and elastic fiber stretch. The following nonlinear