Tuning tissue growth with scaffold degradation inenzyme-sensitive hydrogels: a mathematical model
Umut Akalp, Stephanie J. Bryant and Franck J. Vernerey*
Despite tremendous advances in the field of tissue engineering, a number of obstacles remain that
hinder its successful translation to the clinic. One challenge that relates to the use of cells encapsulated
in a hydrogel is identifying a hydrogel design that can provide an appropriate environment for cells to
successfully synthesize and deposit new matrix molecules while providing a mechanical support that
can resist physiological loads at the early stage of implementation. A solution to this problem has been
to balance tissue growth and hydrogel degradation. However, identifying this balance is difficult due to
the complexity of coupling diffusion, deposition, and degradation mechanisms. Very little is known
about the complex behavior of these mechanisms, emphasizing the need for a rigorous mathematical
approach that can assist and guide experimental advances. To address this issue, this paper discusses
a model for interstitial growth based on mixture theory, that can capture the coupling between cell-
mediated hydrogel degradation (i.e., hydrogels containing enzyme-sensitive crosslinks) and the transport
of extracellular matrix (ECM) molecules released by encapsulated cells within a hydrogel. Taking
cartilage tissue engineering as an example, the model investigates the role of enzymatic degradation on
ECM diffusion and its impact on two important outcomes: the extent of ECM transport (and deposition)
and the evolution of the hydrogel’s mechanical integrity. Numerical results based on finite element
analysis show that if properly tuned, enzymatic degradation yields the appearance of a highly localized
degradation front propagating away from the cell, which can be immediately followed by a front of
growing neotissue. We show that this situation is key to maintaining mechanical properties (e.g.,
stiffness) while allowing for deposition of new ECM molecules. Overall, our study suggests a hydrogel
design that could enable successful tissue engineering (e.g., of cartilage, bone, etc.) where mechanical
integrity is important.
1. Introduction
The degeneration of tissue such as cartilage and bone due toinjury, aging or disease is a major source of disability, pain andeconomic burden in the U.S. and throughout the world. Currentsolutions range from prosthetic replacement to the autograft/allografts that replace the affected tissues, and thus alleviatingthe pain for a limited time. Tissue engineering presents apromising alternative whereby the damaged tissue is injectedwith a population of cells carried in a three-dimensional materialthat can stimulate repair mechanisms and restore normal function.Early successes have achieved skin regrowth,1 and recent progressin stem cell translation raises hope for the development ofpersonalized strategies that uses the patients’ own stem cellswithout requiring a tissue biopsy to obtain tissue specific cells.2
Despite these early successes, a number of obstacles still
hinders applications to a broader population with consistentoutcomes. A reason is that tissue growth requires multi-functionalscaffolds that can simultaneously provide a mechanical support tothe nascent tissue, convey adequate biological signals to implantedcells, enable mass transfer and degrade in time to providesufficient space for tissue development.3
Hydrogels have proven to be promising systems due to theircontrollable properties mimicking that of native tissues,4,5 theirease of (non-invasive) implantation by injection, as well astheir ability to support cell encapsulation and promote extracellularmatrix (ECM) synthesis.6,7 They also have an enormous potentialfor functionalization in terms of adhesion, microarchitecture anddegradation to guide the behavior of encapsulated cells. Tuningthe design of these complex hydrogels has been a significantchallenge because of our limited understanding, and thuscontrol, of how hydrogel structure and cell response evolveduring growth. In particular, contrary to more classical tissueengineering in porous scaffolds,8 a challenging issue for injectablehydrogel-based techniques is to ensure sustained tissue growthwhile maintaining the scaffold’s structural integrity during the
Department of Mechanical Engineering, Department of Chemical and Biological
Engineering, Program of Materials Science and Engineering, University of Colorado
development of engineered cartilage.3 Indeed, on the one hand,nondegradable hydrogels possess a network of cross-links thatinhibit the diffusion of most ECM molecules9,10 and restricttissue development to the immediate region surrounding thecell.11–13 On the other hand, degradable cross-links may solvethe problem in the short-term, but ultimately leads to the loss of thehydrogel’s load carrying capacity (and thus construct failure) beforethe neotissue is formed. Solutions have been suggested to addressthis issue, such as introducing cell-mediated degradation14,15 butthey often make the design more complex and harder to predictwithout theoretical guidance.
In contrast to most engineering materials, a major hindrance inhydrogel design has been the lack of theoretical and computationaldevelopments to assist and guide experimental efforts. One reasonis that in tissue engineering, hydrogels are not designed for theirinstantaneous properties, but for the temporal evolution of theirstructure, which interact with a biologically active component (cells).A fundamental question can therefore be cast as follows: is itpossible to finely tune the structure and degradation propertiesof a cell-laden hydrogel in order to enable tissue growth andcontinuous mechanical integrity during its transition to nativetissue? to explore this problem, an interdisciplinary approachwas taken; first, theoretical models of tissue growth, accountingfor mass transport and reaction kinetic, must be integrated withphysical model of hydrogels and their degradations. Second,these models must be combined with micro-macro analysis(sometimes called multi-scale analysis), in order to identifyhow phenomena occurring at the scale of the cell and polymerstructure yield an emerging behavior at the scale of the construct(scaffold + tissue). Third, these models must be validated andintegrated with experiments in the field of tissue engineering.While the literature on growth models is rich,16–24 research onthe interaction of growth mechanisms and the mechanics anddegradation of a polymer scaffold has been more elusive.Generally, the scaffold’s function is to provide a mechanicalsupport for cells and a material onto which new material can bedeposited. On the other hand, degradation not only is necessaryto ensure the transition from construct to tissue but it can alsobe used to facilitate the transport and macromolecules that laterbecomes the new tissue. Hydrogel design can therefore directthe temporal and spatial evolution of swelling, degradation andtherefore controls the way by which new tissue grows andacquires its mechanical properties.25,26 Few researchers haveinvestigated these processes, through modeling25,27–29 at thecell-hydrogel level. In this context, Dhote et al.26 built a singlecell model under the centro-symmetry assumption and showedthat localized degradation of the encapsulating scaffold helpsmaintain the mechanical integrity of the construct. Sengerset al.29 investigated the competition among ECM degradationor deposition and transport using a 2D model and its effect onthe overall construct stiffness. Trewenack et al.27 proposed amultispecies formulation of cell-mediated growth in cartilageconstructs, pointing out the distinct roles of advection of diffusionfluxes at the microscopic level. Finally, Haider et al.28 extendedthe phenomenological model of Wilson et al.30 to incorporateexperimentally measurable quantities, including the apparent
densities of the scaffold, the deposited and the unlinked ECM.Although these models provide a qualitative understandinghydrogel supported-growth, efforts are required to make simulationmore realistic and closer to experimental systems so that theybecome predictive and guide the current efforts in designingappropriate scaffolds that can be translated to the clinical.
In this paper, we investigated the possibility of growing atissue in a cell-laden hydrogel through mathematical andnumerical modeling. Without lack of generality, we focus ourstudy on the case of cartilage cells (chondrocytes) embedded ina PEG-hydrogel. Due to their small mesh-size (few nanometers)compared to ECM molecules, growth in these systems hastypically been hindered when degradation does not occur. Theintroduction of hydrolytic (or bulk) degradation has howevernot solved the problem since the scaffold loses its mechanicalintegrity before growth can occur. We investigate here how bothissues can be resolved by using a scaffold that can be locallydegraded by enzymes produced by cells. For this, we build amultiphasic model of the growing construct that eventuallymaterializes through a system of nonlinear reaction-diffusionequations whose key parameters can be tuned with hydrogeldesign. We show, through three-dimensional numerical simulationof tissue development and scaffold degradation around cells thatgrowth and sustained mechanical integrity can be achieved in afairly narrow region of the gel design parameters. A discussion ofthese results, their impact on the field of tissue engineering and themotivation for new research direction are then provided.
2. Growth in enzyme degradablehydrogel scaffold
Tissue engineering starts with the encapsulation of chondro-cytes (cartilage cells) in a hydrogel, followed by their localsynthesis, diffusion and deposition of ECM molecules; in mostcases, these processes occur under physiological loads so thattheir evolution into a fully functional tissue depends on thecontinued mechanical integrity of the scaffold. A better under-standing of how scaffold design affects mechanics can begained by investigating the evolution of the ECM/scaffoldarchitecture at the cellular scale, and particularly the percolationof ECM into a viable neo-tissue. We concentrate here on aparticular model of degradation, or enzymatic degradation, inwhich cell-mediated enzymes can diffuse and locally disruptcross-links in the gel. More specifically, we concentrate on oneexample system, an enzyme-sensitive poly(ethylene glycol)hydrogel made by photo-polymerization of 8-arm PEG functionalizedwith norbornene and enzyme sensitive peptide crosslinks (Fig. 1).The hydrogel’s elastic modulus can be adjusted by changing themolecular weight of the monomers or formulation,31 while itsdegradation kinetics can be controlled by changing the aminoacids in the peptide.32 As shown in Fig. 2, cell-mediated scaffolddegradation and tissue growth can be summarized in a fewsteps. (a) The encapsulated cells release both enzyme and ECMmolecules through their cell membrane. (b) Due to their smallsize, enzymes can diffuse through the polymer network and
cleave existing cross-links following Michaelis–Menten kinetics.32
The change of cross-link density r, defined as the number of cross-links per volume of hydrogel in the dry state (i.e. when no solvent ispresent) can therefore be written:
DrDt¼ �k0ce
rrþ k00
(2.1)
where D/Dt represents the material time derivative, ce is theenzyme concentration while k0 = kcat(J0 � 1) is the degradationrate constant, k00 = Km(J0 � 1) is the cross-link density at whichthe reaction rate is at half-maximum, while kcat and Km areMichaelis–Menten kinetic constants,32,33 and J0 is the initialswelling ratio after cell encapsulation, but before degradationoccurs. Cross-link density will eventually reach the critical (orreverse-gelation) point rc at which the polymer network loses itsoverall connectivity. In other words, the polymer locally loses its
solid characteristics and its resistance to molecular diffusion.(c) The removal of cross-links enables the transport of large(unlinked) ECM molecules that can eventually link to form anew tissue. (d) A new tissue is formed when localized pockets oflinked ECM percolate and are able to sustain mechanical loadswithout the support of the hydrogel. It is important to note thatthe above scenario is difficult to achieve experimentally sincethe point of reverse gelation must be closely matched to pointwhen ECM percolation occurs. When reverse gelation occursbefore ECM percolation, the result is a mechanical collapse (ordissolution) of the hydrogel and an overall loss in mechanicalcompetence of the construct.34
As stated above, one of the key mechanism underlying tissuegrowth in a hydrogel is molecular transport, and particularlythe transport of enzyme and ECM molecules. The diffusivity ofa molecule (denoted here by the index a) depends on the ratio
Fig. 1 Schematic of a enzymatically-degradable hydrogel made by photo-polymerization of monomers with enzyme sensitive peptide crosslinkers.
Fig. 2 Cell mediated tissue growth in a porous scaffold. (a) The construct is composed of cells and the mixture of solid and fluid phases which are eithersecreted by cell or key elements of a hydrogel. (b) Tissue growth in a hydrogel at the micro-scale is shown. In this study we consider two cell-secretedmolecules: enzymes and ECM, which are essential to model tissue growth and hydrogel degradation. (c) Cartilage cells secreted extracellular matrix,shown by histological images36 stained for aggrecan (red), which increased as a function of time. By week 12, there was evidence of matrix connectivity inthe hydrogel. The compressive modulus of the construct decreased with culture time indicating that the hydrogel was undergoing bulk degradation.
ra/x of its hydrodynamic radius ra and the hydrogel mesh size xwhere the latter is defined as the distance between two oppositepolymer chains in the swollen ideal network (Fig. 1). Onewell accepted model to describe this relationship was derivedby Lustig and Peppas35 and displays a linear relationshipbetween the diffusivity and the relative particle radius ra/x ofthe form:
Da ra; r; Jð Þ ¼ D1a f J; J0ð Þ 1� ra
xðr; JÞ
� �; ra o xo xc (2.2)
where the mesh size x implicitly depends on cross-link densityr and equilibrium volumetric swelling ratio J and xc = x(rc, J) isthe mesh size when the gel reaches reverse gelation. Thefunction f ( J, J0) represents the probability for a solute moleculeto find a free volume for diffusion. Its expression can be derivedfrom Eyring’s equation37 in which the diffusivity in a swollennetwork differs from that in a pure solvent due to the entropiccontributions of polymer chains. Cohen and Turnbull38
and Peppas and Reinhart39 particularly showed that a goodapproximation of this probability is given by f ( J, J0) = exp(�1/( JJ0 � 1)). Note that once the hydrogel reaches reverse gelation,polymer chains are free to diffuse and assumed to quickly leavethe construct. This means that as r - rc, chains eventuallydisappear, and since this process is relatively faster than thedynamics of growth, one assumes that the function f ( J, J0)becomes unity as soon as r = rc. We further note that relation(2.2) is only valid over the defined range ra o x o xc; indeed,when xo ra, the small mesh size restrict diffusion and Da - 0.When x 4 xc, however, the mesh disappears (x - N) andthe diffusivity becomes that of a particle in a pure solvent, i.e.Da = DN
a as given by the Stokes–Einstein relation.40 While theradii of molecules cannot be changed, it is possible to tune
the hydrogel mesh size via its cross-link density through therelation:35
xðr; JÞ ¼ ‘J13
ffiffiffiffiffir0r
r; rc or (2.3)
where J is the swelling ratio of the gel from its dry state and thelumped length parameter c (unit of Angstrom) is given by
‘ ¼ l
ffiffiffiffiffiffiffiffiffiffiffiffiffi3rpCn
Mrr0
s(2.4)
Here, Cn is the characteristic ratio that defines the restrictedrotation of a single polymer chain, l is the average bond lengthin the polymer, Mr is the molar weight of the polymer repeatunit and rp is the density of the polymer. We note that eqn (2.3)is only valid before reverse gelation and diverges to infinity asr - rc. Two observations can be made based on eqn (2.2) and(2.3). First, enzymes, due to their small hydrodynamic radius(B60–85 Å33) may enter the hydrogel space but their diffusioncan be strongly hindered for tightly cross-linked gels. ECMmolecules (in particular collagen and aggrecans) are howevercharacterized by relatively large sizes, on the order of 200 Å andlarger, and are therefore unable to be transported within thegel. Second, as the gel reaches reverse gelation, the diffusivity ofECM suddenly reaches a finite value, allowing for matrixtransport. This means that diffusion of ECM molecules is onlypossible once the gel is fully degraded, a phenomenon that hasbeen a major hindrance to successful tissue growth. Usingeqn (2.2) and (2.3), Fig. 3a shows the dependency of thenormalized diffusion constant Da/DN
a on cross-link densityand its evolution during degradation for both enzyme andECM. One can clearly see the sharp rise in diffusivity at normal
Fig. 3 (a) Relationship between particle diffusivity and cross-link density for two different particle sizes (enzyme and ECM molecules), characterized bythe ratio r/x. (b) Restricted diffusion of two ECM building blocks (aggrecan and collagen) around chondrocytes encapsulated in 10% and 20% PEGhydrogels.41
gelation for large ECM particles in contrast to smaller enzymes.Fig. 3b confirms this trend by showing histological images ofECM (aggrecan and collagen) around the cartilage cells in twonon-degradable gels characterized by their polymer fraction(10 and 20%, respectively) at the time of hydrogel formation. Itis evident that due to their inability to diffuse, these moleculesremain confined in the pericellular space. One notes that astiffer gel (20% polymer fraction) provides a stronger confine-ment and thus a smaller pericellular matrix.
3. Model formulation
To explore the interactions between molecular transport, polymerdegradation and tissue growth, we propose here to develop amathematical model based on (multiphasic) mixture theory.42,43
More precisely, the construct is described as a continuousmixture of fluid and solid constituents whose respective volumefraction are denoted, respectively by f and ~f. Solid constituentsincluding the polymer ( ~fp) and newly deposited matrix ( ~fm)primarily participate to the mechanical integrity of the construct.Fluid-like constituents, including water (fw), enzymes (fe) andfreshly synthetized and unlinked matrix molecules (fm), play animportant role in degradation and transport. We note here thatthe latter two constituents are typically found in very smallproportions, such that fm { 1 and fe { 1. We further assumethat the above constituents constitute the majority of the inter-cellular space, yielding the saturation condition:
~fm + ~fp = ~f, fw E f, f + ~f = 1 (3.1)
where we omitted to specify the argument (x,t) for the sake ofclarity. It is however clear that these quantities vary both intime and space during the growth process. We show next thatthe evolution of these quantities in time and space, togetherwith the mechanical integrity of the mixture can be describedin terms of a coupled system of reaction-diffusion equationsand force balance over the physical domain of the construct.
3.1 Mass transfer
As the construct composition evolves, motion and mass trans-fer occurs in the inter-cellular space. In this dynamic process,all solid constituents typically follow the same displacement,denoted here as u(x,t) from their reference location x duringcell encapsulation. This concept allows us to consider a moreamenable mathematical treatment known as the constrainedmixture formulation.20 Mass transfer within the construct maythus be described in terms of the velocities na(x,t), (a = w, e, m)of each fluid-like constituent relative to the solid constituents.This description therefore attaches our point of observation tothe solid phase, such that the total velocity of a fluid phase ina fixed frame is n + :
u, where :u denotes the solid velocity.
Assuming that all constituents are incompressible, the continuityequation for the solvent takes the form:44
r� :u � r�(fvw) = 0 (3.2)
where r� :u is to be interpreted as the rate of solid swelling.Noting that the length scale of our analysis (on the order ofmicrons) is significantly larger than the pore scale (on the orderof nanometers), the fluid flow through the polymer networktypically falls within the Darcy regime.45–47 This means that thefluid velocity can be related to the water pressure p and theosmotic pressure p by fvw = �kr(p � p).48,49 The continuityequation for cell secreted molecules further takes the form of aconvection-diffusion-reaction system of the form:
Dca
Dt¼ r � Darcað Þ � r � vwcað Þ � car � _u� Ga a ¼ e; m
(3.3)
We recall here that the diffusivity Da is a nonlinear function ofcross-link density and swelling ratio as shown by eqn (2.2)and (2.3). Furthermore, the molar concentrations are relatedto volume fractions by ca = fa/na with na the molar volumeof constituent a, all of which remain constant due to theincompressibility assumption. The first term is purely diffusive,the second is the convective term from the motion of thesolvent, the third describes a change of concentration due topolymer swelling and the last is a reaction term that accountsfor the rate of production or consumption of the constituent.We assume here that enzyme only affects gel degradation anddoes not transform during the process, i.e. Ge = 0. By contrast,unlinked ECM molecules do transform into linked ECMmolecules and this process is described by the consumptionrate Gm. Mass balance of solid constituents (polymer andlinked matrix) may similarly be derived by cancelling thediffusion and convection terms in eqn (3.3), which leads to:
D~caDt¼ �~car � _uþ Ga; a ¼ p; m (3.4)
where again, the only non-vanishing reaction term is thetransformation rate Gm between unlinked and linked ECM.Motivated by the concept of product inhibition hypothesis,50
the state that a biological system tends to evolve towards aspecific target, we propose a model in which the rate of ECMlinkage is regulated by the level of existing linked ECM asfollows:
Gm ¼ km 1� ~cm~c0m
� �cm (3.5)
where c 0m is the target ECM concentration and km is the
intrinsic rate of linkage (i.e., the rate constant in the absenceof deposited ECM (c = 0)).
3.2 Evolution of construct mechanics
As discussed above, an important aspect of hydrogel assistedgrowth is the evolution of mechanical properties, and especiallythe transition from a cell-laden hydrogel to a tissue. Since theconstruct possesses a heterogeneous structure during growth,this transition depends on both the spatial distribution and theload carrying capacity of each phase (hydrogel and linked ECM)in time. To characterize this evolution, let us consider that at a giventime, the construct is subjected to a macroscopic deformation,
which at the level of the intercellular space, yields a hetero-geneous displacement field u(x,t) around cells. Material deformationcan therefore be characterized by the deformation gradientF = I + qu/qx, or alternatively the nonlinear strain e = (1/2)(FTF � I), where I is the identity tensor. The change in materialvolume (or swelling ratio) from its initial equilibrium valueprior to any degradation is then measured by J = det(F). Foran elastic material, strains give rise to the appearance of stresses,represented by the tensor r, that must verify the balance of linearmomentum:
r � r� pIð Þ ¼ 0 and r ¼ @c@F
FT
J(3.6)
More precisely, r is the elastic stress supported by the hydrogeland the newly deposited ECM while p is the internal pressure inthe solvent. When no external loads are applied to the construct,this pressure is equal to the osmotic pressure p in the hydrogel:24,51
p = RT/nw(ln(f) + (1 � f) + w(1 � f)2) (3.7)
The elastic stress is further related to the deformation gradientthrough a strain energy function, which we assume to beadditively decomposed52 into polymer and ECM components as:
c(r,cm,e) = cp(r,e) + cm(Cm,e) (3.8)
The energy cp stored in the polymer is expressed in terms of theclassical Rubber elasticity theory as:51
cp r;Fð Þ ¼ 1
2rRT J0
2=3tr FTF� �
� 3� ln J0Jð Þh i
; r4rc (3.9)
where R and T are the gas constant and the absolute temperature,respectively, and J0 is the initial equilibrium swelling ratio of thehydrogel prior to degradation. The latter term implies that thepolymer is in a stressed state, balanced by the osmotic pressureof the solvent, eqn (3.7), after cell encapsulation and equili-bration with the solvent. The consequence of this phenomenonis that the stored elastic energy remains positive, even in theabsence of deformation F. One can note from eqn (3.9) that adrop in cross-link density r tends to decrease the elastic energystored in the gel and hence its load carrying capacity. Importantly,when the gel reaches reverse gelation (r = rc), it loses all of itselasticity and:
cp(r,F) = 0, r o rc (3.10)
This behavior creates a sharp discontinuity in the material’sresponse as the hydrogel undergoes reverse gelation; thisfeature is most likely responsible for the sudden failure of thepolymer network before a functional tissue can be obtained.Unlike the hydrogel, the elastic energy stored in the newlydeposited matrix has not been well characterized.20,22,24
However, for convenience and consistency with previous work,24
we choose here to express it in terms of a Saint-Venant strainenergy function, that is a generalization of Hooke’s law in thecontext of finite deformation. The corresponding strain energyfunction reads:
cmðeÞ ¼ ~Cml2½trðeÞ�2 þ m tr e2
� �� �(3.11)
where l and m are the Lame constants, expressed in units of energyper mole of linked ECM. This definition of material parametersimplies that the stiffness of the material increases the apparentdensity of linked ECM.28 In this study, we introduce the constraintl = 0.8m in order to satisfy the porous network behavior53,54 oflinked ECM similar to the that of polymer. Furthermore, since thedeformation of the hydrogel and neo-tissue remains relativelysmall during the growth process investigated in this work, thestress–strain relations in (3.6) can be simplified according to thesmall strain theory for which one can use the approximationFTF E FT + F � I, e = 1/2(FT + F) � I and J = tr(e) + 1. The totalelastic stress at any point within the extra-cellular spacer = rp + rm can therefore be written as:
rp ¼ rH r� rcð Þ RT
trðeÞ þ 1J0
2=3ð2eþ IÞ � 1
2I
� �(3.12)
rm = Cm(l[tr(e)]I + 2me) (3.13)
In which the Heaviside function H(r � rc) ensures that thepolymer stress vanishes when r = rc.
4. Model analysis and solution
In this study, we concentrate on the conditions by which thecompetition between hydrogel degradation and ECM depositionenables a smooth transition between scaffold and tissue properties.As shown in Fig. 4, for a homogeneous distribution of cells atsmall density, the analysis simplifies to the problem of a periodicrepresentative volume of characteristic length L, centered on asingle cell of radius Rc. The average spacing L between cells isfurther related to their volume fraction f in the construct as:
L ¼ Rc
ffiffiffiffiffiffiffiffiffiffiffiffiffi4p=3f3
p(4.1)
To specify proper assumption and boundary conditions, onemust first specify the external conditions surrounding theconstruct during the growth process. We first consider thatthe specimen is free of external loads (neither constant norcyclic) at any time; consequence of this is that (a) externalstresses on the external boundary of the represented domainvanish and (b) solvent permeation becomes negligible, i.e. vw =0. This means that the convective term in eqn (3.3) is negligibleand the transport of enzyme and ECM is described by a coupledsystem of diffusion-reaction equations. Further assuming thatthe hydrogel reaches its new equilibrium significantly fasterthan the dynamics of degradation and enzyme diffusion, theosmotic pressure can be shown to follow expression (3.7). Basedon the periodicity assumption of the representative domain,boundary conditions (B.C.) for g = {u, p, Ce, Cm} become:55
gjSþ� gjS�¼ Lrg (4.2)
where S+ and S� are two opposite boundaries of the representative
domain whilerg is macroscopic gradient of the field g over thedomain. In particular, the absence of loading imposes that
rp ¼ rCa ¼ 0, while ru is directly related to the overall strain
experienced (such as swelling) by the domain during growth.This quantity is calculated by ensuring stress-free boundaryconditions as described in ref. 56. Boundary conditions on thecell surface Scell can further be specified in terms of enzyme andECM productions as:
qajScell¼ q0a 1� ca
c0a
� �(4.3)
where q0a is the production rate of constituent a (= e, m). This
simple relation indicates that cells produce enzyme and ECMmolecule until they reach a homeostatic state, given by a targetconcentration c0
a.28
To analyze the solution, we first apply the technique of non-dimensionalization that consists of rescaling all variables toreference dimensions and times as:
x� ¼ x
L; t� ¼ tD1m
L2(4.4)
This operation aims to simplify the original equations andreduce the number of physically meaningful parameters for abetter interpretation of the system’s response. Using the abovescaling relations, the new dimensionless variables become:
u� ¼ u
L; ca
� ¼ ca
c0a; p� ¼ p
RT; r� ¼ r
r0(4.5)
where the characteristic length scale L is determined by theaverage spacing between cells, energies are scaled with thethermal energy RT and all concentrations are defined in relationto their threshold value c0
a defined in eqn (4.2). Substitutingthese variables in the original formulation, our final systempreserves the structure of the original transport and mechanics
eqn (3.3) and (3.6), for which the critical parameters areexpressed as follows. For the transport phenomena, normalizedpermeabilities are written as a fraction of the maximum valueDN
m for ECM molecules and are expressed in terms of swellingratio, molecule radius and cross-link density as:
Da� J; ra
�; rð Þ ¼ DaJ�1=3 exp � 1
JJ0 � 1
� �J1=3 � ra
� ffiffiffiffiffir�
p (4.6)
with Da = DN
a /DN
m and
ra* = ra/c (4.7)
It is clear here that ra* is interpreted as the relative size of anenzyme or ECM molecule compared to the hydrogel mesh size.Similarly, the normalized rate of degradation and rate of ECMdeposition read:
ke� ¼k0ce0D1m r0
L2; km� ¼km
D1mL2 (4.8)
They characterize, respectively, the competition between enzymedegradation and diffusion and the competition between matrixdeposition and diffusion. The flux boundary condition at the cellboundary is similarly normalized as q0�a ¼ q0aL
2�
D1m c0m� �
and theproperties of the ECM are written in relation to the polymerstiffness and RT as:
m� ¼ C0mmRT
; l� ¼ lm
(4.9)
where Poisson’s ratio is n = 0.5(1 + l*). Note here that the non-dimensional critical cross-link density rc* is also known as thenetwork connectivity parameter.33,57 This parameter is importantto the degradation process as decrease in its value would implyan increase in the number of crosslink to be cleaved before reversegelation. The parameters used in the non-dimensionalization,along with their value and corresponding references are listedin Table 1. These values are key as they enable the mapping ofall non-dimensional results obtained in the next section intophysical quantities.
5. Results
The above equations were solved numerically using a nonlinearfinite element scheme, whose details are given in appendix.Our approach relies on investigating the spatial and temporalevolution of the hydrogel and ECM locally around cells andrelating this to the temporal evolution in the overall constructmechanical properties. For clarity, we decompose our approachin three steps; in the first example, we investigate the role ofhydrogel design on the local hydrogel degradation dynamics,without transport of ECM molecules. The second example theninvestigates how hydrogel degradation and transport of ECMmolecules interact to enable localized growth. We finally explorehow various ECM growth/hydrogel degradation dynamics influencethe mechanical integrity of the construct in time. This exerciseallows us to identify a region in the hydrogel design that
enables an optimized combination of ECM growth andmechanical integrity in time.
5.1 Degradation around a single cell
We first investigate the coupled mechanisms of enzyme transport-hydrogel degradation-enzyme diffusion without ECM production.For this problem, the relevant non-dimensional parameters can bereduced to the relative enzyme size re, and the enzymatic relativedegradation ke*. The effect of polymer mesh size and cross-linkdensity are implicitly contained in these quantities (eqn (4.6) and(4.7)). For the sake of illustration, we consider a construct with lowcell density (L/rc E 20), b = 0.8 (see Table 1) and a constant enzymeproduction equal to P0
e = 6 � 10�16 mole per cell per week.58
Degradation front. We show in Fig. 5 that, according to thevalue of parameters re* and ke*, the nonlinear diffusion-reaction equation yields an enzyme concentration profile thatranges from a diffusion-like appearance to a more narrow andpropagating wave. The corresponding evolution in cross-linkdensity exhibits similar features (Fig. 5a–c) between the fullydegraded region (r* = b) and the intact region (r* = 1).Importantly, we note that this transition region or ‘‘degradationfront’’ travels away from the cell surface in time. In this study,this front is characterized in two ways: its speed is defined asthat of the boundary (r* = b) that separates the completelydegraded and non-degraded gel regions while its width isdefined as the distance between the boundary (r = rc) and
Table 1 Parameters used in non-dimensionalization
Parameter Definition Value Unit Ref.
P0m ECM production rate 2.3 � 10�15 Mole per cell per week 29
P0e Enzyme production rate 6 � 10�16 Mole per cell per week 58
c0m ECM concentration at homeostasis 6.9 � 10�15 Mole 29
c0e Enzyme concentration at homeostasis 1.8 � 10�15 Mole 58
rcell Cell radius 5 mm 25b Network connectivity 0.8 33DeN Diffusion of enzyme in pure solvent 6.6 � 10�7 mm2 s�1 25
DmN Diffusion of ECM in pure solvent 1.3 � 10�7 mm2 s�1 59
rm Hydrodynamic radius of ECM molecules 20 nm 26T Absolute temperature 310 KR Gas constant 8.314 J K�1 mol�1
L Length scale (cell spacing) 400 mm
Fig. 5 Characteristics of degradation dynamics as a function of the normalized enzyme size re and degradation rate ke*. Plots (a–c) show the distributionof cross-link density in terms of the distance x from the cell surface for three specific times; one observes a transition from a diffusion-like profile to anarrow moving degradation front as re* and ke* increase. The relationship between the width and velocity of this traveling front and the parameters(re*, ke*) can be visualized on two-dimensional maps represented in (d) and (e), respectively.
the point at which r* = 0.99. Similar observations are reportedboth numerically and experimentally in ref. 33.
Diffusion-dominated and diffusion-limited systems. Fig. 5a–cclearly shows that the width and speed of the degradation front,can be controlled by varying ke* and re*. For large enzyme radius(or small hydrogel mesh size), the diffusivity becomes so smallthat it is the rate-limiting step. Such a diffusion-limited systemtypically exhibits a very sharp degradation front followed by aregion of intact polymer (Fig. 5c). By contrast, when the enzymesize becomes small (or alternatively, when the hydrogel mesh sizebecomes large), the diffusivity is close to that experienced in apure solvent and hydrogel degradation becomes the rate limitingstep. In this case, the system is diffusion-dominated and theenzyme concentration and cross-link density both display adiffusion-like profile away from the cell surface (Fig. 5a).
Characterization of the degradation dynamics. At first sight,our results seem to imply that sharp fronts are faster than theirwider counterpart. This can be seen in Fig. 5a–c by comparingthe evolution of the point r* = 0 (shaded regions) for three gelsexhibiting wide (I), intermediate (II) and sharp (III) fronts,respectively. To better understand this trend, we further inves-tigated the role of the relative enzyme size and degradation onthe width and speed of the degradation front. This was done bynumerically scanning the space (re*,ke*) and estimating thevalues of front width (w) and velocity (v) for each simulation.Results, recorded in the form of maps (Fig. 5d and e) clearlyshow nonlinear relationships between the hydrogel design andthe degradation behavior, but generally suggest that increasingre* and ke* results in a sharper and faster degradation front. Wenote here that these results cannot be generalized as differenttrends may be observed for different values of the networkconnectivity b.33 A more general understanding of this systemmay be found by studying the role of b on the width and speedof the degradation front, yielding three-dimensional maps inthe (re*, ke*, b) space. This is however, beyond the scope of thisstudy. Overall these results indicate that it is possible to tunethe hydrogel design and specifically its mesh size, cross-linkdensity as well as the sensitivity of degradable links to enzymesto yield a variety of localized degradation dynamics aroundcells. The way by which such dynamics influence growth isdiscussed next.
5.2 The role of hydrogel degradation on the nature of ECMdeposition
In this work, tissue growth is defined as cell-mediated (a)synthesis, (b) release, (c) transport and (d) deposition of ECMmolecules within a hydrogel. Since the size of these macro-molecules (whether it is collagen, GAGs or other key constituentsof native tissue) is usually a few times larger that of the hydrogelmesh size, their diffusion is strongly hindered in intact hydrogelsas can be seen in Fig. 3. This means that growth is limited in non-degradable hydrogels. However, degradable hydrogels mayenable transport via the depletion of their cross-links and theincrease in mesh-size. This effect is especially important as thepolymer crosses the reverse gelation point, due to the suddendisappearance of a mesh and the sudden increase in diffusivity.
Here, we study the interplay between hydrogel degradationand diffusion of ECM molecules and their deposition in anenzymatically degradable hydrogel through the solution of twocoupled reaction-diffusion eqn (2.1) and (3.3) of enzyme andECM transport/deposition. We concentrate on the effect ofthree key parameters, re*, ke* and km*. Note that since therelative ECM size rm* is typically much larger than unity inapplications (i.e., larger than the mesh size of the hydrogel atany point during degradation), we keep it fixed in all simulations.Furthermore, for the sake of clarity, we consider a pair ofparameters re* and ke*, which display, respectively, a wideand sharp degradation and for each case, investigate the effectof small and large rates of ECM deposition. Results are shownin Fig. 6. Subfigures (a) and (b) illustrate the case of a slow rateof ECM deposition coupled with a (a) wide and (b) sharpdegradation front. In the first case, one can observe that thehydrogel undergoes significant degradation before ECM can bedeposited. In the second case, free ECM can easily diffuse in thecavity left by the degradation front around a cell, but because ofits slow conversion to solid ECM, growth remains concentratedin the peri-cellular region. By contrast, Fig. 6c and d display thesituation where the rate of ECM deposition is fast compared tothe rate of degradation. When the degradation front is wide(Fig. 6c), degradation is fairly uniform around cells, and thepoint of reverse gelation propagates slowly away from the cellboundary. Because of its relatively slow speed, this degradationfront is immediately followed by a reservoir of free ECMmolecules that quickly convert into a solid phase. Eventually,the construct is made up of a composite of deposited ECMsurrounded by a region of non-degraded gel. In the case of asharp degradation front, however, the region between intactand fully degraded gel is extremely thin, and the degradationprofile may be thought of as an expanding sphere centeredaround a cell. Within this sphere, ECM can freely diffuse anddeposit at a high rate. This yields a situation where thedegradation front is immediately followed by a deposition frontand a construct that resembles a hydrogel matrix filled withexpanding inclusions of ECM. Interestingly, this case maximizesthe growth to degradation ratio, that is, we observe a significantamount of growth for a minimal level of gel degradation. Althoughall of the above situations allow for growth in time, one mayask whether they could maintain a continuous mechanicalintegrity in time.
5.3 Evolution of the construct’s mechanical integrity duringcombined hydrogel degradation and ECM growth
The primary function of a tissue engineering scaffold is toprovide a temporary mechanical support to cells as the newtissue grows. In vivo, this means that the construct must be ableto resist physiological load at all time during its transition fromscaffold to tissue. As discussed earlier, current hydrogel-basedstrategies that are used for cell encapsulations often lead todissolution and failure of the construct before the ECM canbear any load. Here, we propose to numerically investigatewhether the concept of localized degradation has the potentialto switch this paradigm and allow both tissue growth and
continuity of the construct’s mechanical integrity. For this, weuse techniques of numerical homogenization that allows us toestimate the overall stress–strain response of the unit cell(Fig. 4) in time as the spatial composition of the constructevolves. More precisely, an overall state of deformation
�F is
applied to the domain O and the corresponding displacementfield u(x) and stress �r are calculated as:60
uðxÞ ¼ F � xþ ~uðxÞ; r ¼ 1
V0
ð�Gt� XdO
� �� F
T
�J(5.1)
where u(x) are fluctuations in displacement that remain periodicon the domain’s boundary, t is the traction force vector on Gand %J = det %F. The construct’s Young’s modulus may then beestimated by the secant modulus at a 5% strain for a uniaxialdeformation:
�EðtÞ ¼ rxxðtÞexx
¼ ryyðtÞeyy
¼ rzzðtÞezz
(5.2)
where �exx = �eyy = �ezz = 5% and the last two equalities stem fromthe symmetry of the unit cell. To understand how this moduluscorrelates with the evolution of the hydrogel and ECM compositionaround a cell, we further introduce a measure of connectivity asfollows. A solid phase, polymer or ECM, is defined as connectedif any two points within that space can be connected by a
continuous path. We note here that polymer and ECM areconsidered mechanically non-existent if r r rc and cm* = 0,respectively. Furthermore, since the unit cell is periodic, acontinuous path can cross an external boundary and reappearon its opposite side.
To compare the stiffness of the gel to that of the linked ECM,we define a new dimensionless parameter E* = EECM/Egel whereEgel = rRT(1.05J0
�1/3 � (2.1J0)�1) is the secant modulus for 5%strain31 and EECM = m(3l + 2m)/(l + m). The evolution of theconstruct’s properties is now affected by the three non-dimensional parameters re*, ke*, km* and the ECM propertiesm* and l*. However, to clarify the analysis, we show the fourcharacteristic cases considered in Fig. 6 and for each, explorethe effect of different ECM participation to the overall stiffness;specifically, we consider E* = 0.5 (the ECM is weaker thanthe gel), E* = 1 (the ECM stiffness is comparable to the gel) andE* = 1.5 (the ECM is stronger than the gel). Key results arepresented in Fig. 7 for (a) a wide degradation front & slowdeposition rate, (b) a sharp degradation front & slow depositionrate, (c) a wide degradation front & fast deposition rate and (d) asharp degradation front & fast deposition rate. For each case,the figure shows the evolution of the Young’s modulus (curve)and the connectivity of each phase (shaded blue for hydrogeland shaded red for ECM). Three dimensional contours of
Fig. 6 Profile of polymer cross-link density, free ECM concentration and linked ECM concentration as a function of distance from the cell surface.Results are shown for three characteristic times during growth t = 3.5, 7 and 10.5 day. (a) Slow ECM deposition in a hydrogel scaffold with a wide and slowdegradation front; (b) slow ECM deposition in a hydrogel scaffold with a sharp and fast degradation front; (c) fast ECM deposition in a hydrogel scaffoldwith a wide and slow degradation front and finally (d) fast ECM deposition in a hydrogel scaffold with a sharp and fast degradation front.
polymer cross-link density (blue) and linked ECM concentration(red) are also depicted at three characteristic times during theconstruct’s evolution.
When deposition is slow compared to degradation (Fig. 7aand b), results show that the mechanical integrity of theconstruct monotonically drops with time until it completelydissolves ( %E = 0). A closer look at the ultrastructure evolutionclearly shows that regardless of the sharpness and speed of thedegradation front, ECM deposition lags behind and is unableto produce a well-connected phase before the hydrogel fullydegrades. Interestingly, we note that a sharper degradationfront yields a faster loss in construct’s stiffness; a phenomenonthat can be attributed to the fact that sharper fronts moverelatively faster than wide ones (Fig. 5b and d). When ECMdeposits fast (relative to degradation) one predicts that it canreach connectivity before the hydrogel is completely degraded.As a consequence, even in the case of a wide degradation front,the model suggests that the construct does not completely loseits mechanical integrity. Indeed, while there exists a time intervalfor which none of the phases are connected, their mechanical
interactions allow for a load transfer between them and an overallnon-zero (although small) elastic modulus. In the situation of asharp degradation front however, ECM can effectively grow withinthe empty interstices left by the propagating front. This eventuallyleads to a situation in which both polymer and ECM are connectedand an optimized continuity of the overall construct’s modulus.
6. Discussion and concluding remarks
In summary, we have constructed a multiphasic mixture modelto represent the combined cell-mediated hydrogel degradationand tissue growth. The resulting model has the form of a coupledsystem of two reaction-diffusion equations corresponding toenzyme diffusion/hydrogel degradation and ECM diffusion/linkage.We have shown that according to the design of the hydrogel (inparticular, its initial cross-link density and degradation kinetics), thesystem’s behavior ranges from diffusion-dominated to diffusion-limited. The latter situation is associated with the appearance of alocalized, spherical degradation front propagating outwards
Fig. 7 Evolution of construct modulus over time. The mechanical integrity is preserved in the case of sharp degradation front and fast deposition relativeto the front speed.
from each cell. This space creates pockets of unhindered spaceenabling the diffusion of large ECM molecules. If the rate ofECM linkage is significantly faster than the rate of hydrogeldegradation (and therefore the degradation front velocity), themodel suggests that spherical bodies of solid ECM may growwithin cavities left by degradation and eventually connect intime. In this case, the construct structure displays a doubleconnected network of ECM and hydrogel, which eventuallyallows a smooth transition between hydrogel and tissue andcontinuous mechanical integrity. In a nutshell, the model thereforepoints out that continuous mechanical integrity of the construct canbe achieved by tuning the hydrogel design to achieve both a sharpand slow moving degradation front.
Although it does not appear explicitly, the model alsocaptures the role of cell density during growth. Indeed, due tothe non-dimensionalization procedure, the typical width w = w/Lof the front is measured relative to the cell spacing L. Since Lincreases with decreasing cell density f as shown in eqn (4.1), adegradation front appears sharper, and thus more favorable tothe growth process for low cell densities. This also means thatthe predicted mechanism may be difficult to achieve for highcell density systems. Interestingly, experimental studies61 haveshown that strategies based on hydrolytic (bulk) degradationshow a better potential when higher cell densities are employed.This suggests that optimized growth and degradation can beachieved with a controlled combination of bulk (hydrolytic) andlocalized (enzymatic) degradation kinetics. This dosage wouldmove towards preferably hydrolytic in high cell density systemto enzymatic in low cell-density systems. A quantitative analysisof these dynamics will be the object of future studies.
On a final note, it is important to mention that the scenariohighlighted by our study may not be straightforward to reproduceexperimentally due to the number of uncertainties and imper-fections that characterizes both the polymer structure and thebehavior of embedded cells. A number of effects may indeedaffect the above model predictions. First of all, the homo-geneous spatial distribution of cells is often not verified experi-mentally. Histology images (Fig. 3b) indeed typically display aheterogeneous distribution containing local clusters of high celldensity separated by low density regions. This distribution canfurther evolve in time via cell division, migration and death.62
This may affect the predicted mechanism by only allowingtissue growth in localized regions. To capture this mechanisms,the presented model would need to consider a large number ofcells and their distribution, making the computational problemprohibitively costly. This issue can potentially be avoided by usingmulti-scale techniques based on homogenization as discussed inref. 63 and 64. Another uncertainty not captured by our modelpertains to the behavior of cells, particularly in terms of enzymeand ECM production rate in time. Tissue-producing cells isolatedfrom either tissue or derived from stem cells typically constitutea heterogeneous population that highly reactive to changes(chemical and mechanical) in their environment. The mechanismsdiscussed here could therefore display variations in space and timeand be affected by the type of cell used. In addition to these, theenzymatic activity, i.e. degradation rate, can vary spatially due to
the chemical and mechanical changes during the remodelingof the construct.32,65 Finally, cell proliferation is not explicitlyincluded in the model. Although cell proliferation significantlyinfluences the overall growth of neo-tissue, the increase intissue growth on a per cell basis is less dramatic.66,67 Furthermore,the effect of cell proliferation on tissue synthesis will be capturedwithin the rate of ECM synthesis per cell that used as an inputparameter in the model. Therefore, the model, which describesECM growth per cell, should provide a reasonable assessment ofECM growth. Further experimental characterizations are needed toaddress these questions.
To summarize, promoting tissue growth in a hydrogel is acomplex problem that relies on a deep understanding of thephysical, chemical and biological interaction between cells andhydrogels. In this complexity, an integrated approach mergingmodeling and experimental research will be crucial to identifythe dominating mechanisms that control growth, degradationand continuous mechanical integrity of the tissue construct.This study is a step towards identifying these mechanisms. Assuch, we point to a direction in which experimental effortsshould be oriented.
AppendixA. Methods: finite element implementation
In order to describe the tissue growth in porous scaffolds weuse three balance law; balance of linear momentum (eqn (3.6)),balance of mass (eqn (3.3)) and the equation of osmoticpressure (eqn (3.7)):
r�(rp + rm � pI) = 0 (A.1)
Dcm�
Dt�¼ r � Dm
�rcm�ð Þ � cm�r � _u� � km� 1� ~cm
�
~c�0m
� �cm� (A.2)
Dce�
Dt�¼ r � De
�rce�ð Þ � ce�r � _u� (A.3)
lnfw + (1 � fw) + w(1 � fw)2 + p*nw = 0 (A.4)
These governing equations are solved in three-dimension forthe displacement field u, the pore pressure p, the concentrationof enzyme ce and ECM cm. Moreover we treat the crosslinkingdensity r and concentration of deposited ECM cm as internalstate variables for which we use degradation/depositionkinetics law (eqn (2.1) and (3.5)). We use finite element methodto solve the system of equations for which the weak form iswritten as follows:ð
Note that these equations are coupled and nonlinear in termsof the field variables. Also note that if the deformation field issmall (5–10% strain) then the nonlinearities due to deformationand material behavior vanishes which leads the linear thebalance of linear momentum equation. The linearized formsare as follows:ð
V
rwT1 �
@rp
@eþ @rm
@e
� �@e
@u�du� þ Idp�
� �dV
¼ðS0
wT1 t0dS (A.9)
ðV
wT2
@dcm�
@t�þ rwT
2 � d _u�cm� þ _u�dcm�ð Þ
� �dV
þðV
rwT2 Dm
�ð Þrdcm�dV
þðV
wT2 km
� 1� ~cm�
~cm�
� �dcm�dV ¼
ðS0
wT2 �QmdS
(A.10)
ðV
wT2
@dce�
@t�þ rwT
2 � d _u�ce� þ _u�dce�ð Þ
� �dV
þðV
rwT2 De
�ð Þrdce�dV ¼ðS0
wT2 �QedS
(A.11)
ðV
wT2
1� fw
fw
� 2w 1� fwð Þ� �
1
J2
� �@J
@u�du� þ nwdp�
� �dV ¼ 0
(A.12)
where the left and right hand sides are called internal Fint andexternal Fext force vectors respectively. After linearization oursolution scheme becomes iterative in which we are solving fordd = [du*dp*dce*dcm*] with the stopping criterion |R|NE 0 whereR = Fext � Fint. As the iterative method we chose Newton–Raphson, and in order to capture the transient behavior weimplemented this method in backward Euler time integrationscheme:
_d _d i t� þ Dt�ð Þ
¼ C i�1 t� þ Dt�ð Þ þ K i�1 t� þ Dt�ð ÞDt�� �1
Ri�1 t� þ Dt�ð Þ(A.13)
:di(t* + Dt*) =
:di�1(t* + Dt*) +d
:di(t* + Dt*) (A.14)
d(t* + Dt*) =:d(t* + Dt*)Dt* + d(t*) (A.15)
where C, K, R, are the damping matrix, stiffness matrix, andresidual vector respectively. In order to calculate these matricesand vector we use the FE discretization as follows. We used a
mixed u–p formulation, that is 27-node element for u* and8-node element for p*.68 Moreover for the concentration fieldscm* and ce* 8-node element formulation is used. The testfunctions w1 and w2 and the interpolation of the fields thenbecomes:
w1 = N27 %w1 and w2 = N8 %w2 u* = N27 %u*,
p* = N8 %p*, cm* = N8%cm*
and ce* = N8%ce* (A.16)
Note that same interpolation rules are used for the rates ofthese fields. The stiffness and damping matrices become:
Verification of the model is performed by comparing thesolution with that of 1D degradation-diffusion model presentedin ref. 33 (Fig. 8c). In order to check the consistency of themodel, we investigated the spatial and temporal change ofcrosslinking density r along the diffusion direction. Note fromeqn (2.1) that any numerical error in the enzyme transportaffects the calculated crosslinking density. The error in cross-linking density depending on the element mesh size (Fig. 8a)and time step (Fig. 8b) is calculated using L2 norm:
er2 ¼
ðV
rkþ1 � rk� �2
dV
� �1=2(A.33)
The convergence rate of the error is determined by fitting aleast square curve to the calculated errors. The convergence ratewith respect to the mesh size is quadratic, in agreement withtheoretical predictions.68 Moreover the convergence of theEuler time integration scheme is linear as expected.69 For theanalysis in this paper we chose a mesh size and time step givenby h* = 0.08 (shown in red circle in Fig. 8a) and Dt* = 0.006(shown in red circle in Fig. 8b), which provides reasonableaccuracy (Fig. 8c).
Acknowledgements
Research reported in this publication was supported by theNational Institute of Arthritis and Musculoskeletal and SkinDiseases of the National Institutes of Health under AwardNumber 1R01AR065441 to S. J. B. and F. J. V. The content issolely the responsibility of the authors and does not necessarilyrepresent the official views of the National Institutes of Health.F. J. V. acknowledge the support of the National ScienceFoundation under the CAREER award 1350090. Work by UAwas partially supported by the Department of Civil, Environmental& Architectural Engineering at the University of Colorado Boulderthrough a Doctoral Assistantship for Completion of Dissertation.
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