Research Article Cartilage Dysfunction in ALS Patients as …downloads.hindawi.com/journals/bmri/2014/179070.pdf · · 2015-11-22Loss: 3D Mechano-Electrochemical Computational Model

Research ArticleCartilage Dysfunction in ALS Patients as Side Effect of MotionLoss 3D Mechano-Electrochemical Computational Model

Sara Manzano123 Eamonn A Gaffney4

Manuel Doblareacute123 and Mohamed Hamdy Doweidar123

1 Group of Structural Mechanics and Materials Modelling (GEMM) Aragon Institute of Engineering Research (I3A)University of Zaragoza 50018 Zaragoza Spain

2Mechanical Engineering Department School of Engineering and Architecture (EINA) University of ZaragozaMarıa de Luna sn Betancourt Building 50018 Zaragoza Spain

3 Biomedical Research Networking Center in Bioengineering Biomaterials and Nanomedicine (CIBER-BBN) 50018 Zaragoza Spain4Wolfson Centre for Mathematical Biology (WCMB) Mathematical Institute Oxford University Oxford OX1 2JD UK

Correspondence should be addressed to Mohamed Hamdy Doweidar mohamedunizares

Received 18 February 2014 Accepted 20 April 2014 Published 3 June 2014

Academic Editor Raquel Manzano

Copyright copy 2014 Sara Manzano et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Amyotrophic lateral sclerosis (ALS) is a debilitating motor neuron disease characterized by progressive weakness muscle atrophyand fasciculation This fact results in a continuous degeneration and dysfunction of articular soft tissues Specifically cartilageis an avascular and nonneural connective tissue that allows smooth motion in diarthrodial joints Due to the avascular natureof cartilage tissue cells nutrition and by-product exchange are intermittently occurring during joint motions Reduced mobilityresults in a change of proteoglycan density osmotic pressure and permeability of the tissue This work aims to demonstrate theabnormal cartilage deformation in progressive immobilized articular cartilage for ALS patients For this aim a novel 3D mechano-electrochemical model based on the triphasic theory for charged hydrated soft tissues is developed ALS patient parameters such astissue porosity osmotic coefficient and fixed anions were incorporated Considering different mobility reduction of each phase ofthe disease results predicted the degree of tissue degeneration and the reduction of its capacity for deformationThe present modelcan be a useful tool to predict the evolution of joints in ALS patients and the necessity of including specific cartilage protectorsdrugs or maintenance physical activities as part of the symptomatic treatment in amyotrophic lateral sclerosis

1 Introduction

Amyotrophic lateral sclerosis (ALS) is a neurodegenerativedisorder characterized by progressive loss of motor unitsleading to muscle atrophy weakness and immobilization[1] Despite substantial effort to decipher the etiology of thedisease the primary events triggering the pathology remainto be elucidated Currently there is no cure for the diseaseand its management is focused on symptomatic treatmentand nutritional supply to extend the life expectancy improvequality of life and help tomaintain the patientrsquos autonomy foras long as possible [2]

Important long term problems are facing ALS patientssuch as osteoarthritis and cartilage progressive degeneration

as a consequence of the chronic reduction in joints mobility[3 4]

Due to the avascular nature of cartilage cells nutritionand by-product exchange are diffusive mediated processesoccurring intermittently in the articular movements [5] Incase of ALS patients due to their inability to exercise and lackof continuous motion these diffusive processes are ceasedHowever no studies have looked into the correlation betweenprogressive loss of motion and cartilage degeneration in ALSpatients Such studies would help clinicians to determinethe need of specific cartilage treatments Cartilage tissueis mainly composed of fibrous network of proteins suchas collagen fibers proteoglycans (PGs) and elastin Thisnetwork is surrounded by synovial which is fluid mainly

Hindawi Publishing CorporationBioMed Research InternationalVolume 2014 Article ID 179070 13 pageshttpdxdoiorg1011552014179070

2 BioMed Research International

composed of water and ions (Na+ and Clminus among others)(Figure 1(a)) Since cells occupy a small fraction of the totalvolume of cartilage transport of fluid and solute substancesmainly depends on the properties of the extracellular matrix(ECM) [6]

Substantial numerical simulations have emerged toreproduce in vivo cartilage behavior [7ndash10] However mostof these models do not include ions role among the maincomponents of the tissue which is a key aspect whensimulating cartilage degenerative diseases [11]

In this sense capturing the influence of each component(ECM water and ions) within the macroscale behavior(deformation as a consequence of ion and water fluxes)requires the consideration of the cartilage as a multiphasicporoelastic material with three main components the liquidphase (water) the ion phase (Na+ and Clminus ions) and the solidphase collagen fibers PGs and proteins [12ndash14] Attachedto PGs there are glycosaminoglycan chains (GAGs) whichcontain sulphate and carboxyl groups PGs becomenegativelycharged in the physiological environment through interac-tionwith anions (Clminus)These ECM-attached anions so-calledldquofixed charge densityrdquo (FCD) present a coupled movementwith collagen fibers given rise to a swelling pressure in thetissue (Figure 1(b))

Under normal physiological conditions the fixed nega-tive charges in ECM are electrically neutralized by mobilecations contained in the interstitial fluid [15] The differencein ion concentration between the ECM and interstitial fluidgenerates an osmotic pressure (Donnan osmotic pressure)leading to the typical cartilage swelling and shrinking phe-nomena [16 17] These phenomena play a significant role incartilage behavior flexibility and stability required to supportloads and resist stretching since they are responsible for thefluid and ion content within the tissue [18] In ALS patientswhere the mobility is reduced the normal balance of thetissue is altered Thus the abnormal nutrient supplies to cellslead to progressive tissue degeneration [4]

Despite extensive efforts that focused on the developmentof computational models to predict in vivo cartilage behaviorand its application to clinical diagnosis and prognosis sub-stantial aspects nonetheless remain obscure

Most of those studies have been done in 1D or 2DHowever nowadays modern techniques of medical imagingsuch asMRI CTMR or PET generate 3D images containingaccurate information that cannot be captured by 1 or 2Dmodels [19ndash21]

Besides in vivo cartilage is influenced by real 3D bound-ary conditions [16 22 23] Several reports demonstrate thatcartilage properties (permeability diffusivity electric con-ductivity and electrochemical potentials) are tightly inter-connected with each other [24]

In this paper we present a 3D mechano-electrochemicalcomputational model of cartilage behavior based on thetriphasic theory of charged-hydrate soft tissues proposed byLai et al in 1991 [25] and lately extended by Sun et al in 1999[26]

The model has been used to the study of abnormalcartilage free swelling occurring in case of ALS patientsSimilar to other cartilage disorders in immobilized patients

the degeneration process starts in the tissue surface Whenthe disease gets advanced the degradation and subsequentdysfunction extend to deeper areasThe aim of this work is tofocus on the response of this primarily affected zone to studyits different behavior in healthy and pathological states [27ndash29]

The obtained results predict and quantify the impact ofALS immobilized cartilage properties into tissue deformationcapacity which would enable monitoring the progress ofcartilage tissue degeneration and evaluating the necessityof specific cartilage maintenance therapeutics during thedifferent stages of the disease

2 Material and Methods

The triphasic mechano-electrochemical theory [25] appliedhere considers a charged hydrated soft tissue as amixture thatconsists of a porous permeable charged solid phase (ECMcollagen fibers and PGs) a fluid phase (water) and an ionphase with two monovalent species (Na+ and Clminus) The mainhypothesis of such triphasic theory assumes that the differentphases exist simultaneously at each point in the space and areconsidered intrinsically incompressible

This theory is mainly based on the balance equations(mass momenta and charge density) for the solid matrixfluid (interstitial fluid) cations and anions Fluxes of fluidand ions depend on the gradients of their electrochemicalpotentials which in turn depend on fluid pressure and ionconcentration respectively

21 Model Formulation Four governing equations arerequired for solving the four basic unknowns in triphasictheory (u119904 the displacement of the solid matrix 120576

119908 themodified chemical potential of water and 120576

+ and 120576minus the

modified chemical potential for cations and anions resp[26]) These may be written as follows

Momentum balance equation of the mixture

nabla sdot 120590 = 0 (1)

Mass balance equation of the mixture

nabla sdot k119904 + nabla sdot J119908 = 0 (2)

Charge balance equation for each ion

120597 (Φ119908119888+)

120597119905

+ nabla sdot J+ + nabla sdot (Φ119908

119888+k119904) = 0 (3)

120597 (Φ119908119888minus)

120597119905

+ nabla sdot Jminus + nabla sdot (Φ119908

119888minusk119904) = 0 (4)

where 120590 is the mixture stress tensor and k119904 is the velocityof the solid matrix (small deformations has been consideredhere) specifically k119904 = 120597u119904120597119905 where 120597120597119905 is the temporalderivative 119888+ and 119888

minus are cation and anion concentrationrespectively andΦ

119908 represents the cartilage porosity J119908 is thewater flux while J+ and Jminus are the cation flux and anion flux

BioMed Research International 3

Chondrocytes

Proteoglycans complex

Collagen fibres

Ions in the interstitial fluid

Collagen fibrils

minus

minus

minus minus

minus

minus+

+

+

+

+

+

+

+

minus

(a)

Glycosaminoglycan chains

Fixed charges

Proteoglycan complex

Hyaluronan

minus

minusminus minus

minus

minus

minus

minus

minus minus

minus

minusminus

minus

minus

minus

minusminus

(b)

Figure 1 Schematic illustration of articular cartilage microstructure and its main components (a) Constituents of the cartilage tissuechondrocytes proteoglycans complex collagen fibers and ions dissolved in the interstitial fluid (b) Proteoglycan macromolecule aggregatewith charged-glycosaminoglycan side chains attached to a hyaluronic chain

respectively Importantly we consider in our formulationthe convective terms associated to ion-matrix coupling inthe last part of the balance equations for the ions (Φ119908119888+k119904)and (Φ119908119888minusk119904) This is neglected in most cartilage models ofthis type in literature [30 31] while they are essential forcapturing the different biological coupled movement of theions attached to collagen fibers through GAGs in healthy anddegenerated tissues

The constitutive equations for the state variables appear-ing in (1)ndash(4) 120576119908 120576+ and 120576

minus in terms of the basic variables ofthe problem may be also written as [15]

120590 = minus119875I + 120582119904120579I + 2120583

119904120598

120576119908

=

119875

119877119879

minus Φ (119888+

+ 119888minus

) +

119861119908

119877119879

120579

120576+

= 120574+

119888+ exp(

119865119888120595

119877119879

)

120576minus

= 120574minus

119888minus exp(minus

119865119888120595

119877119879

)

(5)

where 119865119888is the Faraday constant 120595 is the electrical potential

119861119908is the fluid-solid coupling coefficient Φ is the osmotic

coefficient 120574+ and 120574minus are the activity coefficient of anion

and cation respectively and I is the identity tensor 119877 is theuniversal gas constant and 119879 is the absolute temperature

Here 119875 is the fluid pressure 120579 = div us is the solidmatrix dilatation related to the infinitesimal strain tensor ofthe solid matrix 120582

119904and 120583119904are the Lame constants of the solid

matrix (a linear elastic isotropic material has been used forthe sake of simplicity) 120598 = (12)(nabla sdot us + nabla

Tsdot us) is the

solid matrix deformation and 119888+ and 119888

minus can be expressedin terms of modified electrochemical potentials and fixedcharged density as follows [26]

119888+

=

119888119865+ radic(119888

119865

0)2

+ (4120576+120576minus (120574+120574minus))

2

119888minus

=

minus119888119865+ radic(119888

119865

0)2

+ (4120576+120576minus (120574+120574minus))

2

(6)

where 1198881198650is the initial FCD the amount of charges which are

attached to collagen fibers within the GAGs


Regarding J119908 J+ and Jminus they can be expressed ascombination of the chemical potential gradients as follows[26]

J119908 = minus

119877119879Φ119908

120572

(nabla120576119908

+

119888+

120576+nabla120576+

+

119888minus

120576minusnabla120576minus

)

J+ = minus

119877119879Φ119908119888+

120572

nabla120576119908

minus [

Φ119908119888+119863+

120576+

+

119877119879Φ119908(119888+)

2

120572120576+

]nabla120576+

minus

119877119879Φ119908119888+119888minus

120572120576+

nabla120576minus

Jminus = minus

119877119879Φ119908119888minus

120572

nabla120576119908

minus [

Φ119908119888minus119863minus

120576minus

+

119877119879Φ119908(119888minus)

2

120572120576minus

]nabla120576minus

minus

119877119879Φ119908119888+119888minus

120572120576+

nabla120576+

(7)

where120572 is the drag coefficient between the solid and thewaterphase 119863+ and 119863

minus the ion diffusivities and Φ119908 the porosity

of the tissueThe intrafibrillar water within collagen fibers therefore

may be regarded as bound water and thus part of the solidphase Due to the intrinsic incompressibility for infinitesimalstrains Φ119908 can be expressed as

Φ119908

= 1 minus

120601119904

0

1 + 120579

(8)

where 120601119904

0is the solid volume fraction at a physicochemical

reference configuration (selected as the initial state) Thesecond part of this equation 120601119904

0(1 + 120579) corresponds to the

solid volume fraction of the tissue 120601119904 at any time along thedeformation process Note that in (7) the combination of theporosity and the drag coefficient of the material give rise tothe permeability of the tissue 119896 as follows 119896 = Φ

119908120572 [26]

Considering the electroneutrality condition betweencations and anions the following relation does exist [26]

119888+

= 119888minus

+ 119888119865

(9)

where 119888119865 is the FCD We assumed that the porous matrix is

negatively charged the fixed charges on the matrix remainunchanged and 119888

119865must be conserved during the tissue defor-mation According to experimental observations increasingthe volumetric deformation of the sample (eg tissue freeswelling) is followed by a decrease in the fixed charge ofthe tissue as it involves water entrance and therefore ionconcentration and fixed charges decrease in the matrixLikewise the opposite phenomenon also occurs when thevolumetric deformation decreases (eg the tissue shrinks)under the following considerations [26]

119888119865

=

119888119865

0

1 + (120579Φ119908

0)

(10)

where 1198881198650and Φ

119908

0are the initial water content in the tissue at

reference state

211 Differential Equation System Substituting the flux equa-tion (7) and the constitutive equation (5) into the governingequations (1)ndash(4) we find the following system of differentialequations

nabla sdot (120582119904120579I + 2120583

119904120598) minus nabla (119877119879120576

119908

+ 119877119879Φ119888119896

minus 119861119908120579) = 0

nabla sdot k119904 minus nabla sdot

119877119879

120572

[Φ119908

nabla120576119908

+

Φ119908119888+

120576+

nabla120576+

+

Φ119908119888minus

120576minus

nabla120576minus

] = 0

nabla sdot [minus

119877119879

120572

Φ119908

119888119865

nabla120576119908

] minus nabla

sdot [

Φ119908119888+119863+

120576+

+

119877119879

120572

Φ119908(119888+

)

2

120576+

minus

119877119879

120572

Φ119908119888+119888minus

120576+

]nabla120576+

+ nabla sdot [

Φ119908119888minus119863minus

120576minus

+

119877119879

120572

Φ119908(119888minus

)

2

120576minus

minus

119877119879

120572

Φ119908119888+119888minus

120576+

]nabla120576minus

= 0

120597 (Φ119908119888119896)

120597119905

minus nabla sdot [minus

119877119879

120572

Φ119908

119888119896

nabla120576119908

] minus nabla

sdot [

Φ119908119888+119863+

120576+

+

119877119879

120572

Φ119908(119888+

)

2

120576+

minus

119877119879

120572

Φ119908119888+119888minus

120576+

]nabla120576+

minus nabla

sdot [

Φ119908119888minus119863minus

120576minus

+

119877119879

120572

Φ119908(119888minus

)

2

120576minus

minus

119877119879

120572

Φ119908119888+119888minus

120576+

]nabla120576minus

+ nabla

sdot (Φ119908

119888119896k119904) = 0

(11)

where 119888119896 = 119888++ 119888minus

212 Solution Methods A finite element formulation wasused to obtain the solution of the governing system of(11) The primary unknowns of the model [u 120576119908 120576+ 120576minus] areinterpolated from nodal values through shape functions[26] The time derivatives are approximated with the Crank-Nicolson method [32] that yields an implicit approximationto the solution of the initial value problem 119910

1015840= 119891(119909 119910)

with 119910(1199090) = 119910

0at 119909 for a given time step ℎ To obtain the

fully coupled nonlinear system of equations describing thediscretized model (superscript lowast indicates the quantities inthe bathing solution) we first establish the weak formulationof the governing equations

intnabla120575u sdot 120590119889119881 = int

Γ

120575u120590lowastn 119889Γ

intnabla120575120576119908 sdot k119904 119889119881 + intnabla120575

120576119908 sdot J119908 119889119881 = minusint

Γ

120575120576119908J119908lowast

n 119889Γ

intnabla120575120576+ sdot J+ 119889119881 minus int

Ω

nabla120575120576+ sdot Jminus 119889119881 = int

Γ

120575120576+ (Jminus

lowast

minus J+lowast

) n 119889Γ


int120575120576minus

120597 (Φ119908119888119896)

120597119905

119889119881 + intnabla120575120576minus sdot J+119889119881 + intnabla120575

120576minus sdot Jminus 119889119881

+ intnabla120575120576minus sdot (Φ

119908

119888119896k119904) 119889119881 = minusint

Γ

120575120576minus (J+

lowast

+ Jminuslowast

)n 119889Γ

(12)

where n is the unit normal vector to the boundary and 120575u120575120576119908 120575120576+ and 120575

120576minus are the so-called test functions Trilinear

8-noded hexahedral elements with 2 times 2 times 2 Gaussianintegration points were used (see Supplementary Materialavailable online at httpdxdoiorg1011552014179070)

22 Numerical Implementation of the Mechano-Electrochem-ical Model The model presented above has been imple-mented within a commercial FE software ABAQUS via auser defined subroutine Initially the cartilage sample isin equilibrium with the external bath (initial equilibrium)Therefore the initial volumetric dilatation (120579) is equal to zeroand initial values for ion concentrations correspond to thosemeasured in the initial state of the external solution Aftermodifying the external solution electrochemical potentialsbetween the bath and the tissue are created (nabla120576119908 nabla120576+ andnabla120576minus) generating a new ion distribution within the cartilage

Due to this potential fluxes of water and ions tend to enterand leave the tissue respectively to regain equilibrium andthe tissue swells in absence of external loads Specific valuesfor displacement and ion concentrations are calculated inevery node and updated via a Lagrangian formulation (interms of the initial coordinates)The volumetric deformationat each finite element node is also computedThe full analysispresented in this work corresponds to a total time analysis(119905119891) of 3600 seconds coincident with usual experimental peri-

ods in cartilage free swelling assays [25]The implementationscheme of the 3D mechano-electrochemical model is shownin Figure 2

23 Experimental Example Firstly to validate the model theexperimental test described in [33] with healthy cartilagesample parameters is reproduce computationally Then theswelling of a cartilage sample of 05mm high and 15mmdiameter confined in a chamber immersed in a NaCl solutionwas analyzed for healthy and degenerated cartilage tissueincluding the properties associated to different stages of ALSpatient joints suffering a progressive loss of motion

231 Boundary Conditions The lateral sides of the sample aswell as the supporting surface are impermeable and confinethe lateral movement of the tissue (see Figure 2) Zero fluxeswere considered for ions and water through these surfaces ofthe sample (Figure 3(b)) Specifically for the lateral surfaces119869119908

119909119910= 119869+

119909119910= 119869minus

119909119910= 0 and for the lower surface 119869119908

119911= 119869+

119911=

119869minus

119911= 0 Furthermore it is assumed that displacements in

119909 and 119910 directions remain null on lateral surfaces 119906119909

=

119906119910

= 0 For the lower surface of the sample we constrainall displacements 119906

119909= 119906119910

= 119906119911

= 0 Regarding the freeupper surface of the sample we consider the electrochemicalpotential of water and ions identical to those measured in the

Initial equilibrium(i) Initial geometry(ii) Initial ions concentration

New ionsconcentration and

distribution

Electrochemical

effects

Electrochemical potentials(nabla120576w nabla120576+ nabla120576minus)

Swelling shrinking

Volumetricstrain (120579)

Final equilibrium(i) Final geometry(ii) Final ions concentrationand distribution

t = 3600 s

Mechanicaleffects

bath and tissue (Jw J+ Jminus)Fluxes between external

(iii) 120579 = 0

t = 0 s

t gt 0 s

Figure 2 Schematic diagram of the computational process

external bath 120576119908 = 120576119908lowast

120576+ = 120576+lowast

120576minus = 120576minuslowast

120576119908lowast

120576+lowast

and 120576minuslowast

in the external solution are kept constant in the whole process(no additional salt is added no dilution occurs in the bathingsolution)

232 Initial Conditions Initially it is considered that thesample is at equilibrium with the external bathing solutionwith a concentration 119888

lowast

0= 015M of monovalent ions (Na+

and Clminus) At 119905 = 0 s the concentration of the externalsolution is decreased to 0125M This reduction causes thetissue to swell to a new equilibrium state (final equilibrium)The initial modified chemical potential for water 120576119908

lowast

0and the

electrochemical potential for cations and ions 120576+lowast

0and 120576

minuslowast

0

inside the tissue are identical to their counterparts in theexternal solution 120576119908

0= 120576119908lowast

0 120576+0

= 120576+lowast

0 120576minus0

= 120576minuslowast

0 Choosing

this free swollen state as the reference configuration (initialequilibrium) the initial displacement is measured from thisreference state so 119906

1199090

= 1199061199100

= 1199061199110

= 0 The transientresponse of the solid displacement ion concentration fluidand ion velocities to this imposed ionic change is solvedhere by using our 3D model and is compared with previousnumerical results of this same example obtained in 1D by Sunet al (1999) [26]

24 Model Validation In absence of 3D mechano-electrochemical computational models of cartilage inthe literature we first reproduced the results obtained bySun et al in one-dimension (1D) for healthy cartilage tovalidate our model This 1D model includes an extensive


+

+

+

+

+

++

+

minus

minus

minus

minusminus

minus minusminus

Cartilage sample

Impermeable chamber

NaCl solutionNa+ Clminus

(a)

Upper surface sample(contact with bath solution)

120576w = 120576wlowast

120576+ = 120576+lowast


Y

Lateral surface sample (contact with chamber wall)

Lower surface sample (contact with bottom chamber)

Z

X

Jwxy = J+xy = Jminusxy = 0

ux = uy = 0

Jwz = J+z = Jminusz = 0

ux = uy = uz = 0

(b)

Figure 3 Schematic representation of the transient free swelling problem in Frijns et al [33] (a) A sample of cartilage of 05mm thickness and15mm diameter is immersed in a NaCl solution with an initial concentration of 015MThe sample is confined in an impermeable chamber(b) Boundary conditions of the cartilage sample applied in our computational simulations

study of the effects of cartilage parameter variations in tissuebehavior and clearly states the differences and limitationsfound between their model and the experimental results

Coincident with Sunrsquos 1D model the mesh of our modelpresents a total number of elements equal to 50 in depthTo accurately reproduce the 1D case of study we consider adiameter four orders of magnitude smaller than the height ofthe sample (0sam ≪ ℎsam) with only one element per rowThe selected average mesh size was of 001mm Finally theresulting total number of elements is 50 Similar to Sun et al(1999) [26] the displacements in 119909 and 119910 axis are constrainedinto the whole sample (119906

119909= 119906119910

= 0) and thereforeanalyzing only 119911 displacement (119906

119911) effects All parameters

used in this example are collected in Table 1 HoweverSunrsquos experimental results and its measurements variabilityare not indicated in their work they state that simulatedsurface displacement in free swelling test is consistent withtheir experimental observations [26] Their computed andexperimental results have been used to validate the presentedmechano-electrochemical model

25 3D Free Swelling Study of Healthy and ImmobilizedALS Patients Cartilage The model was applied to analyze

the effect of the variation of different properties governingcartilage behavior in patients with lack of mobility as inALS patients In this model application the experimentalconditions detailed in Frijns et al (1997) [33] againmotivatedthe geometry of the confined sample (0sam = 15mm)with simulations that particularly considered the interplaybetween all properties interacting in a 3D environmentResults corresponding to healthy cartilage were comparedto those yielded by Sunrsquos 1D model and biological studiesfor healthy cartilage behavior Furthermore we obtainedaccurate quantification of water and ion fluxes of thesesamples Then a comparative study between healthy anddegraded cartilage free swelling was performed

An interesting phenomenon of a delay in cartilageswelling associated with a small effectively zero amount ofshrinking was observed in our simulation during the initialphases when values of FCD into the tissue orΦ are coincidentwith those measured in unhealthy cartilage tissue [24] Thiseffect has not previously observed with 1D or 2D modelsapplying a wide variety of FCD for progressive immobilizedknees [26] which ranged from degenerated aged knees (eg119888119865

0= 011mEqmL) to tissues with other degenerative

diseases as osteoarthritis (eg 1198881198650= 006mEqmL)


Table 1 3D simplified model for healthy cartilage parameters

Description Symbol Value RefsInitial thickness of the sample ℎsam 05mm [26]Initial diameter of the sample 0sam 00005mm mdashNumber of mesh elements NELEM 50 [26]Number of mesh nodes NNODES 204 mdashYoungrsquos modulus 119864 385 sdot 10

5 Pa [15]Poisson coefficient ] 028 [15]Drag coefficient between thesolid and the water phase 120572 7 sdot 10

14Nsdotssdotmminus4 [26]

Diffusivity of the cations 119863+

5 sdot 10minus10msdotsminus1 [26]

Diffusivity of the anions 119863minus


Initial FCD 119888119865

002mEqsdotmLminus1 [26]

Activity coefficient of cations 120574+ 10 [26]

Activity coefficient of anions 120574minus 10 [26]

Gas constant 119877 8314 Jsdotmolminus1sdotKminus1 [26]Absolute temperature 119879 298K [26]Osmotic coefficient Φ 10 [15]Initial amount of water in thetissue Φ

119908

0075 [26]

3 Results and Discussion

The present model was used to simulate the behavior ofhealthy cartilage in free swelling conditions Then addi-tional simulation were performed varying physiologicaltissue properties (porosity osmotic coefficient and fixedanions into the tissue) to simulate progressive immobilizationprocess that take place in disabled ALS patientsThe resultingabnormal cartilage free swelling was compared with thatobtained with healthy tissue

Initially we analyzed the 119911-displacements of the uppersurface of the cartilage sample as well as cation distributionwithin the sample after changing the concentration of theexternal solution from 015M to 0125M As previously men-tioned for validation the 3D model was simplified reducingthe sample diameter to mimic 1D model conditions Resultsobtainedwere compared to those reported by Sun et al (1999)[26] with their 1D model Subsequently the same situationwas reanalyzed considering the 3D environment and applyingphysiological versus pathological cartilage properties realsample dimensions and three-dimensional confined condi-tions (see Table 2)

31 3D Simplified Model Under the assumption of fric-tionless chamber-impermeable walls and bottom surfacethe displacement of the upper surface is only affected byentrance of water and exit of ion fluxes The simplified 3Dmodel shows that after dilution and the associated massiveentrance of water into the sample the cartilage begins toswell reaching its maximal deformation at 900 seconds and0013mm height After this point sample dimensions remainconstant and enter a plateau stage (Figure 4(a)) The presentsimplified 3D model accurately reproduced the deformationevolution obtained by Sunwho found amaximal deformation

Table 2 3D free swelling model for healthy cartilage parameters

Description Symbol Value RefsInitial thickness of the sample ℎsam 05mm [26]Initial diameter of the sample 0sam 15mm [26]Number of mesh elements NELEM 760 [26]Number of mesh nodes NNODES 1026 mdashYoungrsquos modulus 119864 385 sdot 10


14 Nsdotssdotmminus4 [26]










119908

0075 [26]

of 00135mm height at 1000 seconds with the subsequentsteady state with that dimension

In these conditions the gradient of cations was alsomonitored after 100 seconds of this free swelling simulationand again compared well with the results reported by Sun1D model At that time the 1D model clearly showed a pro-gressive decrease of cation concentration towards the uppersurface of the sample (119909 = 05mm) ranging from 273molm3at 119909 = 0mm height (significantly lower compared to theinitial 280molm3 imposed by the external solution at time0 s) to 255molm3 at 119909 = 05mm height (Figure 4(b))

Our results closely resemble those obtained by Sun withan initial concentration of 274molm3 at 119909 = 0mm heightand 257molm3 at 119909 = 05mm height (Figure 4(b))

32 3D Free Swelling Study of Healthy and Immobilized ALSPatients Cartilage In this study the experiment described bySun et al was again reproduced considering a sample diam-eter of 15mm and the 3D confined environment reportedexperimentally by Frijns et al (1997) [33] together with thephysiological properties of cartilage later presented in Sun etal in 2004 [15] (Table 2)

Under these conditions fluxes of water cations andanions during the whole process were computed monitoredand analyzed In addition displacements of the free healthycartilage surface were again simulated as well as the effectsof the variation in the main cartilage properties (fixed chargedensity porosity of the tissue and ion diffusivities) accordingto the grade of mobility and total immobilization time of ALSpatients [34]

321 Fluxes of Water and Ions in Healthy Cartilage Concur-rent water and ions fluxes together with morphological


000E + 00

200E minus 03

400E minus 03

600E minus 03

800E minus 03

100E minus 02

120E minus 02

140E minus 02uz

(mm

)

0 200 400 600 800 1000 1200 1400 1600

Sun et al (1D)Simplified model (3D)

Time step (s)

(a)

250E + 02

255E + 02

260E + 02

265E + 02

270E + 02

275E + 02

280E + 02

c+(m

olm

3 )

0 01 02 03 04 05 06Depth (mm)


(b)

Figure 4 (a) Comparison of the upper surface displacement in the free swelling test from 1Dmodel of Sun et al (dotted line) and the present3D simplified model (continuous line) (b) Comparison of cation concentration distribution in the free swelling test from 1D model of Sun[26] and the present 3D simplified model after 100 seconds of swelling

changes of the tissue were simulated during 3600 secondsThe corresponding simulation results were divided into fivephases corresponding to the main events that take placeexperimentally [35] A detailed description of the status ofeach component during these phases is as follows

Phase I (initial equilibrium) In this stage the externalsolution is diluted to 0125M of NaCl which generates animbalance between the inner and outer medium of thesample At time 0 the sample remains equilibrated with the015M of NaCl initial solution 3D computational results didnot reveal any deformation (Figure 5(a) Phase I) as water(Figure 5(b) Phase I) and ion fluxes (Figures 5(c) and 5(d)Phase I) are minimal as well

Phase II This phase represents an abrupt deformation ofthe cartilage (Figure 5(a) Phase II) as a consequence of themassive entrance of water at 50 sdot 10minus9m3s (Figure 5(b)Phase II) to counteract the gradient induced in the previousstep At this stage outgoing ion fluxes would be expectedhowever these are clearly overwhelmed by the intake waterflux (Figures 5(c) and 5(d) Phase II) which leads to subse-quent substrate swelling during the next phase

Phase III During this phase high displacement of the uppersurface of the sample (Figure 5(a) Phase III) is observedat 200 seconds as already mentioned This is due to theprevious and continuous water flux uptake during this stage(Figure 5(b) Phase III) when the maximum water contentwithin the sample is reached Consistently ion fluxes for bothcation (Figure 5(c) Phase III) and ions (Figure 5(d) PhaseIII) remain at basal level

Phase IV After 800 seconds of simulation the substratedeformation is maintained (Figure 5(a) Phase IV) waterflux is drastically reduced (Figure 5(b) Phase IV) while

internal and external solutions regain the equilibrium statewhich enables the visualization of the previously impeded ionoutflow (Figures 5(c) and 5(d) Phase IV)

Phase V (final equilibrium) Finally this phase represents thecomplete restoration of electrochemical equilibrium betweenthe external and internal solution The cartilage sample doesnot show any further deformation (Figure 5(a) Phase V)in agreement with no external load applied on the sampleand the lack of a Na Cl ion gradient Consistently water(Figure 5(b) Phase V) and ion fluxes (Figures 5(c) and 5(d)Phase V) were not shown at this stage

322 Effects of Initial Fixed Charge Density (FCD) To fullyunderstand the effect of FCD on swelling the followingexperiment was designed Initial FCD measured experimen-tally in a young and aged healthy knees cartilage (119888119865

0=

02mEqmL and 119888119865

0= 0135mEqmL resp) [24] and

three different values of 119888119865

0in patient with osteoarthritis

and cartilage degenerative processes where the patient mustremain partially (early stage of ELA) or totally immobilized(advanced stage of ELA) [34 36] were introduced into thepresent 3D computationalmodel to compare the correspond-ing results

The results showed that physiological FCD values dis-played a maximal displacement of 0143mm for younghealthy knee and 00984mmfor agedhealthy kneewithin 200seconds of simulation subsequently these values remainedconstant for 3600 seconds The rest of FCD selected values(representative of pathologic cartilage conditions) offeredlower 119911-displacement ranging from 1679 sdot 10minus5mm to 8218sdot 10minus7mm (ie almost null swelling) after 900 secondsof simulation time (Figure 6(a)) In every case after tissueresearches its maximum 119911-displacement value this state wasmaintained until the end of the simulation (3600 seconds)


00001

00001

00000200004

minus0001 minus0001

XY

Z

900E minus 05823E minus 05745E minus 05668E minus 05591E minus 05514E minus 05436E minus 05359E minus 05282E minus 05205E minus 05127E minus 05500E minus 06

(m) Phase Iequilibrium 1

Phase IIentrance of water

Phase IIIsaturation of sample

Phase IVflux of ions

Phase Vequilibrium 2

t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

uz

(a)

00001

00001

00000200004

minus0001 minus0001

XY

Z

Phase Iequilibrium 1





t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

500E minus 09450E minus 09400E minus 09350E minus 09300E minus 09250E minus 09200E minus 09150E minus 09100E minus 09500E minus 10000E + 00minus500E minus 10

w (m3s)J

(b)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

+ (mols)

Flujo posz200E minus 04168E minus 04136E minus 04105E minus 04727E minus 05409E minus 05909E minus 06minus227E minus 05minus545E minus 05minus864E minus 05minus118E minus 04minus150E minus 04

J

(c)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

minus (mols)

200E minus 04168E minus 04136E minus 04105E minus 04727E minus 05409E minus 05909E minus 06minus227E minus 05minus545E minus 05minus864E minus 05minus118E minus 04minus150E minus 04

J

(d)

Figure 5 (a) 119911-displacement (119906119911) (b) water (J119908) (c) cation (J+) and (d) anion (Jminus) fluxes obtained with the presented 3D computational

model during five phases of swelling (3600 seconds total time simulation) in healthy cartilage samples Note that negative fluxes for eachspecies studied correspond to the emergence of that component from the sample to the external solution Conversely positive fluxes refer tothe entrance of the different components (water or ions) into the sample (tissue gain of material)


100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00

0 500 1000 1500 2000 2500 3000 3500Time step (s)

uz

(mm

)

cF0 = 02mEqmLcF0 = 0135mEqmLcF0 = 0115mEqmL

cF0 = 011mEqmLcF0 = 006mEqmL

(a)

0 5 10 15 20 25 30 35 40 45 50 55

minus800E minus 11

minus600E minus 11

minus400E minus 11

minus200E minus 11

000E + 00

200E minus 11

400E minus 11

600E minus 11

800E minus 11

Time step (s)

uz

(mm

)



(b)

100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00

uz

(mm

)

0 500 1000 1500 2000 2500 3000 3500Time step (s)

Φw0 = 075

Φw0 = 077

Φw0 = 080

Φw0 = 085

(c)

100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00uz

(mm

)

0 500 1000 1500 2000 2500 3000 3500Time step (s)

Φ =

Φ =

Φ =

Φ =

10080

075070

(d)

Figure 6 (a) Surface displacement obtained with the 3D computational model as a function of tissue FCDThe rest of the model parametersare included in Table 2 (b) Details of the initial swelling delay phenomenon of cartilage associated with a nominal shrinking within the first40 seconds of computational simulation considering the parameters collected in Table 2 while the specific concentration of FCD shown in thesame figure (c) Surface displacement obtained with the 3D computational model as a function of tissue initial water content (Φ119908

0) (d) Surface

displacement obtained with the 3D computational model as a function of the osmotic coefficient

It is of interest to note that the present model shows adelay in subsequent swelling associated with a tiny effectivelyzero initial shrinking which appears when introducing thelower values of FCD content This phenomenon is illustratedin the three studies (variation of FCD porosity of thetissue and osmotic coefficient) (see Figure 6(b)) but was notobserved with higher FCD values

323 Effects of Initial Water Content The influence of initialporosity in the 119911-displacement suffered by the tissue was alsosimulated This porosity directly correlates with the initialamount of water in the cartilage sample since a fully saturated

sample is assumed For a normal human knee cartilage thisranges from 75 to 78 [37]

Results demonstrated that similarly to FCD influencea physiological initial porosity of the tissue (Φ119908

0= 75 of

water content) yielded the highest displacement at the uppersurface of the sample which exhibited a sharp incrementwithin the first 200 seconds of simulation and peaked at0143mmAgain the volume of the sample remained constantafter this time (Figure 6(c))

High initial porosities in the tissue (which imply higherwater content) leaded to proportional lower maximal dis-placements (Figure 6(c)) Interestingly several harmful con-ditions to the maintenance of cartilage such as the loss


of motion that ELA patients suffer involve the reductionin porosity [34] High water content in these tissues maylimit the capture of additional water molecules and thereforedownregulate in going fluxes as discussed below

324 Effects of Osmotic Coefficient If we now modify theosmotic coefficient of solid matrix and fix the rest of theparameters (Table 2) it is possible to quantify the influenceof this variable in the swelling of the cartilage sample Valuesbelow the physiological thresholdΦ = 08 fail to exhibit thenormal-range deformation of the substrate 0143ndash00954mmat 200 seconds of simulation (Figure 6(d)) Biologically thiscoefficient affects the velocity of water flow through theporous medium Thus a reduction of this coefficient oftenunder critical conditions such as joint large immobilizationperiods of time [38 39] entails a decrease of fluid flow intothe cartilage porous matrix and therefore implies smallerdeformations

4 Conclusion

In this work a 3D mechano-electrochemical cartilage behav-ior model is presented The model has been applied to studythe cartilage free swelling that takes place in case of ALSpatients by varying the main tissue properties (porosityosmotic coefficient and fixed anions)

Specifically in this study we analyze and quantify theeffect of FCD porosity of the ECM ionic diffusivity electro-chemical potentials and mechanical ECM properties in thebehavior of the cartilage tissue ranging fromhealthy cartilageproperties to those associated with joints from ALS patientsat different stages

Several simplifications have been made to develop thismodel Firstly we have neglected the PGs effects dueto FCD repulsion and the anisotropy in the mechanicalbehavior of cartilage [40] Including these aspects wouldvalidate the proposed model for wider purposes howeverit would also increase the number of model parametersand the associated validation Secondly the specific valuesfor the different variables included in the model (Table 2)were taken from experimental reports in literature whichgenerate a remarkably wide physiological range for eachvariable considering the very different measurement toolsused in each study and the technical limitations of thesequantifications Moreover parameters such as age sex orweight of the donor may severely influence these variables[24] Under these conditions the model has been validatedincluding different cases within the physiological range foreach variable those parameters that had been reported todrastically influence the value of a specific variable wereanalyzed separately (ie healthy versus Immobilized 119888

119865

0in

cartilage knees) (Figure 6(a))In absence of literature about previous models and accu-

rate free swelling measurements in a 3D geometry we vali-dated the presented model by comparing the results with the1D simulation of Sun et al (1999) [26] As expected similarswelling displacementswere obtained in both cases althoughin the full 3D model swelling occurs faster and the new

equilibrium (final equilibrium) after changing the externalsolution concentration is reached earlier These differencesmay arise from the extensive simplifications made by Sun etal such as neglecting the effect of osmotic coefficient and ionsactivities to get convergence In this sense even includingthe values reported by Sun et al our results closely parallelthose obtained by Frijns et al (1997) [33] who reported anearlier and faster swelling By including the same parametervalues reported in Frijnsrsquo experimental work within the 3Dmodel themaximum surface displacement slightly increasedtogether with the swelling velocity as reported by theseauthors

In the full 3D free swelling study applied to health andALS-degraded cartilage due to loss motion computationalresults showed how FCD ranging within the physiologicalvalue (02mEqmL and 0135mEqmL) results in sharpswelling However reducing FCD content according to ALSpatients with progressive loss motion it reduces deformationrates Small variations in the FCD content would lead to highdeviations of the swelling rate FCDs below the physiologicalthreshold have been described in arthritical degenerated orunhealthy cartilage [41] therefore tissue deformation underthese conditions would be impaired in agreement with ourcomputational observations

Due to its 3D consideration the modelling frameworkis sensitive to radial boundary conditions as occurs exper-imentally Specifically in the 3D model the initial swellingdelay is induced not only for values of FCD lower than005mEqmL as Sun et al mentioned but also for valuesof 011mEqmL The increased sensitivity of our model ispossibly due to the inclusion of 3D boundary conditionsIn any case it is remarkable that nominal shrinking and anassociated delay in significant net responsewas inducedwhenincluding small FCDsWe speculate that biologically in theseconditions higher amounts ofmobile ions remain uncoupledincreasing the fluxes of cations and anions Considering thatimmediately after dilution of the external solution to 0125Mhigh ion fluxes arise from the more concentrated internalsolution during the initial stages before water entranceoverwhelms this flux inducing a delay in the net cartilageresponse and a tiny effectively negligible shrinking Theeffect of initial porosity (Φ

119908

0) and osmotic coefficient (Φ)

on the swelling of the sample is also enhanced by the 3Dconditions Again upper surface cartilage decreases inducinga delay in the onset of swelling when the model adoptssubphysiological values of the porosity or osmotic coefficientΦ = 075 and Φ = 08 respectively This is explained byconsidering that all properties are interconnected within thecoupled 3D computational model hence the disturbance ofone of them highly influences the rest increasing the stiffnessof the tissue and impeding the normal damping of the charges[24]

Since swelling is a key process in cartilage tissue [18 4243] this event was thoroughly analyzed within the three-dimensional study of water and ion fluxes After changing theconcentration of the external solution initial equilibriumwasbroken and an incoming flux of water into the sample arisesto re-equilibrate both solutions The abrupt water entranceresults in sharp swelling of the sample for physiological


parameters During this stage water flux hides the com-paratively minor outgoing ion fluxes However during thelater stage this last flow becomes apparent as water currentdecreases

In summary the presented 3D model is able to decipherthe effects on cartilage deformation impairment for non-physiological values of the osmotic coefficient porosity andfixed charges density all of which aremeasurable parametersThe obtained results predict and quantify the impact ofALS immobilized cartilage properties into tissue deformationcapacity All these features together with the display of resultsinto clinically interpretable 3D images present this model asa helpful tool to predict prevent andmonitor the progress ofcartilage tissue degeneration in ALS patients and abrogatesfor the inclusion of cartilage protective therapy into themanagement of the disease to ameliorate the motion loss

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank Marıa Pilar VelascoPhD (Centro Universitario de la Defensa University ofZaragoza) for her valuable suggestionsThe authors gratefullyacknowledge the financial support from the SpanishMinistryof Science and Technology (FPU graduate research pro-gram AP20102557 and CYCIT DPI2010-20399-C04-01) andCIBER-BBN initiative CIBER-BBN is an initiative funded bythe VI National RampDampI Plan 2008ndash2011 Iniciativa Ingenio2010 Consolider Program CIBER Actions and financed bythe Instituto de Salud Carlos III with the assistance from theEuropean Regional Development Fund

References

[1] R H Brown Jr ldquoAmyotrophic lateral sclerosis recent insightsfrom genetics and transgenic micerdquo Cell vol 80 no 5 pp 687ndash692 1995

[2] B R Brooks ldquoEl escorial world federation of neurology criteriafor the diagnosis of amyotrophic lateral sclerosisrdquo Journal of theNeurological Sciences vol 124 pp 96ndash107 1994

[3] C R Handy C Krudy N Boulis and T Federici ldquoPain inamyotrophic lateral sclerosis a neglected aspect of diseaserdquoNeurology Research International vol 2011 Article ID 4038088 pages 2011

[4] M J Palmoski R A Colyer and K D Brandt ldquoJoint motionin the absence of normal loading does not maintain normalarticular cartilagerdquo Arthritis and Rheumatism vol 23 no 3 pp325ndash334 1980

[5] M Ulrich-Vinther M D Maloney E M Schwarz R Rosierand R J OKeefe ldquoArticular cartilage biologyrdquo Journal of theAmerican Academy of Orthopaedic Surgeons vol 11 no 6 pp421ndash430 2003

[6] A Maroudas P D Weinberg K H Parker and C P WinloveldquoThe distributions and diffusivities of small ions in chondroitinsulphate hyaluronate and some proteoglycan solutionsrdquo Bio-physical Chemistry vol 32 no 2-3 pp 257ndash270 1988

[7] L P Li J Soulhat M D Buschmann and A Shirazi-AdlldquoNonlinear analysis of cartilage in unconfined ramp com-pression using a fibril reinforced poroelastic modelrdquo ClinicalBiomechanics vol 14 no 9 pp 673ndash682 1999

[8] N S Landinez-Parra D A Garzon-Alvarado and J CVanegas-Acosta ldquoA phenomenological mathematical modelof the articular cartilage damagerdquo Computer Methods andPrograms in Biomedicine vol 104 no 3 pp e58ndashe74 2011

[9] F Richard M Villars and S Thibaud ldquoViscoelastic modelingand quantitative experimental characterization of normal andosteoarthritic human articular cartilage using indentationrdquoJournal of the Mechanical Behavior of Biomedical Materials vol24 pp 41ndash52 2013

[10] R K Korhonen M S Laasanen J Toyras R Lappalainen HJ Helminen and J S Jurvelin ldquoFibril reinforced poroelasticmodel predicts specificallymechanical behavior of normal pro-teoglycan depleted and collagen degraded articular cartilagerdquoJournal of Biomechanics vol 36 no 9 pp 1373ndash1379 2003

[11] A Legare M Garon R Guardo P Savard A R Pooleand M D Buschmann ldquoDetection and analysis of cartilagedegeneration by spatially resolved streaming potentialsrdquo Journalof Orthopaedic Research vol 20 no 4 pp 819ndash826 2002

[12] A Maroudas M T Bayliss andM F Venn ldquoFurther studies onthe composition of human femoral head cartilagerdquoAnnals of theRheumatic Diseases vol 39 no 5 pp 514ndash523 1980

[13] V C Mow and C C B Wang ldquoSome bioengineering consid-erations for tissue engineering of articular cartilagerdquo ClinicalOrthopaedics and Related Research no 367 pp S204ndashS2231999

[14] X L Lu C Miller F H Chen X Edward Guo and V CMow ldquoThe generalized triphasic correspondence principle forsimultaneous determination of the mechanical properties andproteoglycan content of articular cartilage by indentationrdquoJournal of Biomechanics vol 40 no 11 pp 2434ndash2441 2007

[15] D D Sun X E Guo M Likhitpanichkul W M Lai andV C Mow ldquoThe influence of the fixed negative charges onmechanical and electrical behaviors of articular cartilage underunconfined compressionrdquo Journal of Biomechanical Engineer-ing vol 126 no 1 pp 6ndash16 2004

[16] L A Setton H Tohyama and V C Mow ldquoSwelling and curlingbehaviors of articular cartilagerdquo Journal of Biomechanical Engi-neering vol 120 no 3 pp 355ndash361 1998

[17] Y P Zheng J Shi L Qin S G Patil V C Mow and KY Zhou ldquoDynamic depth-dependent osmotic swelling andsolute diffusion in articular cartilage monitored using real-timeultrasoundrdquo Ultrasound in Medicine and Biology vol 30 no 6pp 841ndash849 2004

[18] S R Eisenberg and A J Grodzinsky ldquoSwelling of articularcartilage and other connective tissues electromechanochemicalforcesrdquo Journal of Orthopaedic Research vol 3 no 2 pp 148ndash159 1985

[19] D Burstein and M Gray ldquoNew MRI techniques for imagingcartilagerdquo Journal of Bone and Joint Surgery A vol 85 no 1 pp70ndash77 2003

[20] R R Regatte S V S Akella A J Wheaton A Borthakur JB Kneeland and R Reddy ldquoT1120588-relaxation mapping of humanfemoral-tibial cartilage in vivordquo Journal of Magnetic ResonanceImaging vol 18 no 3 pp 336ndash341 2003

[21] M T Nieminen J Rieppo J Silvennoinen et al ldquoSpatialassessment of articular cartilage proteoglycans with Gd-DTPA-enhanced T1 imagingrdquoMagnetic Resonance inMedicine vol 48no 4 pp 640ndash648 2002


[22] L A Setton W B Zhu and V C Mow ldquoThe biphasicporoviscoelastic behavior of articular cartilage role of thesurface zone in governing the compressive behaviorrdquo Journal ofBiomechanics vol 26 no 4-5 pp 581ndash592 1993

[23] R M Schinagl D Gurskis A C Chen and R L Sah ldquoDepth-dependent confined compression modulus of full-thicknessbovine articular cartilagerdquo Journal of Orthopaedic Research vol15 no 4 pp 499ndash506 1997

[24] A Maroudas ldquoPhysicochemical properties of cartilage in thelight of ion exchange theoryrdquo Biophysical Journal vol 8 no 5pp 575ndash595 1968

[25] W M Lai J S Hou and V C Mow ldquoA triphasic theory forthe swelling and deformation behaviors of articular cartilagerdquoJournal of Biomechanical Engineering vol 113 no 3 pp 245ndash258 1991

[26] D N Sun W Y Gu X E Guo W M Lai and V C MowldquoA mixed finite element formulation of triphasic mechano-electrochemical theory for charged hydrated biological softtissuesrdquo International Journal for Numerical Methods in Engi-neering vol 45 no 10 pp 1375ndash1402 1999

[27] C Ding P Garnero F Cicuttini F Scott H Cooley and GJones ldquoKnee cartilage defects association with early radio-graphic osteoarthritis decreased cartilage volume increasedjoint surface area and type II collagen breakdownrdquoOsteoarthri-tis and Cartilage vol 13 no 3 pp 198ndash205 2005

[28] K Hjelle E Solheim T Strand R Muri and M Brit-tberg ldquoArticular cartilage defects in 1000 knee arthroscopiesrdquoArthroscopy vol 18 no 7 pp 730ndash734 2002

[29] T M Link L S Steinbach S Ghosh et al ldquoOsteoarthritis MRimaging findings in different stages of disease and correlationwith clinical findingsrdquo Radiology vol 226 no 2 pp 373ndash3812003

[30] J Z Wu and W Herzog ldquoSimulating the swelling and defor-mation behaviour in soft tissues using a convective thermalanalogyrdquo BioMedical Engineering Online vol 1 article 8 2002

[31] E F Kaasschieter A J H Frijns and JMHuyghe ldquoMixed finiteelement modelling of cartilaginous tissuesrdquo Mathematics andComputers in Simulation vol 61 no 3ndash6 pp 549ndash560 2003

[32] J Crank and P Nicolson ldquoA practical method for numericalevaluation of solutions of partial differential equations of theheat-conduction typerdquo Mathematical Proceedings of the Cam-bridge Philosophical Society vol 43 no 1 pp 50ndash67 1947

[33] A J H Frijns J M Huyghe and J D Janssen ldquoA validation ofthe quadriphasic mixture theory for intervertebral disc tissuerdquoInternational Journal of Engineering Science vol 35 no 15 pp1419ndash1429 1997

[34] B Vanwanseele E Lucchinetti and E Stussi ldquoThe effects ofimmobilization on the characteristics of articular cartilagecurrent concepts and future directionsrdquo Osteoarthritis andCartilage vol 10 no 5 pp 408ndash419 2002

[35] R A Brown and P D Byers ldquoSwelling of cartilage and expan-sion of the collagen networkrdquoCalcified Tissue International vol45 no 4 pp 260ndash261 1989

[36] W F Enneking and M Horowitz ldquoThe intra-articular effects ofimmobilization on the human kneerdquo Journal of Bone and JointSurgery A vol 54 no 5 pp 973ndash985 1972

[37] J M Mansour Biomechanics of Cartilage edited by C A OatisLippincott Williams ampWilkins Baltimore Md USA 2003

[38] J A Buckwalter ldquoOsteoarthritis and articular cartilage use dis-use and abuse experimental studiesrdquo Journal of Rheumatologyvol 22 no 43 pp 13ndash15 1995

[39] F J Muller L A Setton D H Manicourt V C MowD S Howell and J C Pita ldquoCentrifugal and biochemicalcomparison of proteoglycan aggregates from articular cartilagein experimental joint disuse and joint instabilityrdquo Journal ofOrthopaedic Research vol 12 no 4 pp 498ndash508 1994

[40] M K Barker and B B Seedhom ldquoThe relationship of thecompressive modulus of articular cartilage with its deformationresponse to cyclic loading does cartilage optimize its modulusso as to minimize the strains arising in it due to the prevalentloading regimerdquo Rheumatology vol 40 no 3 pp 274ndash2842001

[41] T Aigner J Haag J Martin and J Buckwalter ldquoOsteoarthritisaging of matrix and cellsmdashgoing for a remedyrdquo Current DrugTargets vol 8 no 2 pp 325ndash331 2007

[42] D R Carter and M Wong ldquoModelling cartilage mechanobiol-ogyrdquo Philosophical Transactions of the Royal Society B BiologicalSciences vol 358 no 1437 pp 1461ndash1471 2003

[43] T Nagel and D J Kelly ldquoAltering the swelling pressures withinin vitro engineered cartilage is predicted to modulate theconfiguration of the collagen network and hence improve tissuemechanical propertiesrdquo Journal of the Mechanical Behavior ofBiomedical Materials vol 22 pp 22ndash29 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Anatomy Research International

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International Journal of

Volume 2014

Zoology


Molecular Biology International

GenomicsInternational Journal of


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BioinformaticsAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom

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Volume 2014

Stem CellsInternational



Enzyme Research



Microbiology


composed of water and ions (Na+ and Clminus among others)(Figure 1(a)) Since cells occupy a small fraction of the totalvolume of cartilage transport of fluid and solute substancesmainly depends on the properties of the extracellular matrix(ECM) [6]

Substantial numerical simulations have emerged toreproduce in vivo cartilage behavior [7ndash10] However mostof these models do not include ions role among the maincomponents of the tissue which is a key aspect whensimulating cartilage degenerative diseases [11]

In this sense capturing the influence of each component(ECM water and ions) within the macroscale behavior(deformation as a consequence of ion and water fluxes)requires the consideration of the cartilage as a multiphasicporoelastic material with three main components the liquidphase (water) the ion phase (Na+ and Clminus ions) and the solidphase collagen fibers PGs and proteins [12ndash14] Attachedto PGs there are glycosaminoglycan chains (GAGs) whichcontain sulphate and carboxyl groups PGs becomenegativelycharged in the physiological environment through interac-tionwith anions (Clminus)These ECM-attached anions so-calledldquofixed charge densityrdquo (FCD) present a coupled movementwith collagen fibers given rise to a swelling pressure in thetissue (Figure 1(b))

Under normal physiological conditions the fixed nega-tive charges in ECM are electrically neutralized by mobilecations contained in the interstitial fluid [15] The differencein ion concentration between the ECM and interstitial fluidgenerates an osmotic pressure (Donnan osmotic pressure)leading to the typical cartilage swelling and shrinking phe-nomena [16 17] These phenomena play a significant role incartilage behavior flexibility and stability required to supportloads and resist stretching since they are responsible for thefluid and ion content within the tissue [18] In ALS patientswhere the mobility is reduced the normal balance of thetissue is altered Thus the abnormal nutrient supplies to cellslead to progressive tissue degeneration [4]

Despite extensive efforts that focused on the developmentof computational models to predict in vivo cartilage behaviorand its application to clinical diagnosis and prognosis sub-stantial aspects nonetheless remain obscure

Most of those studies have been done in 1D or 2DHowever nowadays modern techniques of medical imagingsuch asMRI CTMR or PET generate 3D images containingaccurate information that cannot be captured by 1 or 2Dmodels [19ndash21]

Besides in vivo cartilage is influenced by real 3D bound-ary conditions [16 22 23] Several reports demonstrate thatcartilage properties (permeability diffusivity electric con-ductivity and electrochemical potentials) are tightly inter-connected with each other [24]

In this paper we present a 3D mechano-electrochemicalcomputational model of cartilage behavior based on thetriphasic theory of charged-hydrate soft tissues proposed byLai et al in 1991 [25] and lately extended by Sun et al in 1999[26]

The model has been used to the study of abnormalcartilage free swelling occurring in case of ALS patientsSimilar to other cartilage disorders in immobilized patients

the degeneration process starts in the tissue surface Whenthe disease gets advanced the degradation and subsequentdysfunction extend to deeper areasThe aim of this work is tofocus on the response of this primarily affected zone to studyits different behavior in healthy and pathological states [27ndash29]

The obtained results predict and quantify the impact ofALS immobilized cartilage properties into tissue deformationcapacity which would enable monitoring the progress ofcartilage tissue degeneration and evaluating the necessityof specific cartilage maintenance therapeutics during thedifferent stages of the disease

2 Material and Methods

The triphasic mechano-electrochemical theory [25] appliedhere considers a charged hydrated soft tissue as amixture thatconsists of a porous permeable charged solid phase (ECMcollagen fibers and PGs) a fluid phase (water) and an ionphase with two monovalent species (Na+ and Clminus) The mainhypothesis of such triphasic theory assumes that the differentphases exist simultaneously at each point in the space and areconsidered intrinsically incompressible

This theory is mainly based on the balance equations(mass momenta and charge density) for the solid matrixfluid (interstitial fluid) cations and anions Fluxes of fluidand ions depend on the gradients of their electrochemicalpotentials which in turn depend on fluid pressure and ionconcentration respectively

21 Model Formulation Four governing equations arerequired for solving the four basic unknowns in triphasictheory (u119904 the displacement of the solid matrix 120576

119908 themodified chemical potential of water and 120576

+ and 120576minus the

modified chemical potential for cations and anions resp[26]) These may be written as follows

Momentum balance equation of the mixture

nabla sdot 120590 = 0 (1)

Mass balance equation of the mixture

nabla sdot k119904 + nabla sdot J119908 = 0 (2)

Charge balance equation for each ion

120597 (Φ119908119888+)

120597119905

+ nabla sdot J+ + nabla sdot (Φ119908

119888+k119904) = 0 (3)

120597 (Φ119908119888minus)

120597119905

+ nabla sdot Jminus + nabla sdot (Φ119908

119888minusk119904) = 0 (4)

where 120590 is the mixture stress tensor and k119904 is the velocityof the solid matrix (small deformations has been consideredhere) specifically k119904 = 120597u119904120597119905 where 120597120597119905 is the temporalderivative 119888+ and 119888

minus are cation and anion concentrationrespectively andΦ

119908 represents the cartilage porosity J119908 is thewater flux while J+ and Jminus are the cation flux and anion flux


Chondrocytes


Collagen fibres


Collagen fibrils

minus

minus

minus minus

minus

minus+

+

+

+

+

+

+

+

minus

(a)


Fixed charges


Hyaluronan

minus

minusminus minus

minus

minus

minus

minus

minus minus

minus

minusminus

minus

minus

minus

minusminus

(b)





120590 = minus119875I + 120582119904120579I + 2120583

119904120598

120576119908

=

119875

119877119879

minus Φ (119888+

+ 119888minus

) +

119861119908

119877119879

120579

120576+

= 120574+

119888+ exp(

119865119888120595

119877119879

)

120576minus

= 120574minus


119865119888120595

119877119879

)

(5)








Tsdot us) is the



119888+

=

119888119865+ radic(119888

119865

0)2

+ (4120576+120576minus (120574+120574minus))

2

119888minus

=

minus119888119865+ radic(119888

119865

0)2

+ (4120576+120576minus (120574+120574minus))

2

(6)





J119908 = minus

119877119879Φ119908

120572

(nabla120576119908

+

119888+

120576+nabla120576+

+

119888minus


)

J+ = minus

119877119879Φ119908119888+

120572

nabla120576119908

minus [

Φ119908119888+119863+

120576+

+

119877119879Φ119908(119888+)

2

120572120576+

]nabla120576+

minus

119877119879Φ119908119888+119888minus

120572120576+

nabla120576minus

Jminus = minus

119877119879Φ119908119888minus

120572

nabla120576119908

minus [

Φ119908119888minus119863minus

120576minus

+

119877119879Φ119908(119888minus)

2

120572120576minus

]nabla120576minus

minus

119877119879Φ119908119888+119888minus

120572120576+

nabla120576+

(7)





Φ119908

= 1 minus

120601119904

0

1 + 120579

(8)

where 120601119904





119908120572 [26]


119888+

= 119888minus

+ 119888119865

(9)




119888119865

=

119888119865

0

1 + (120579Φ119908

0)

(10)

where 1198881198650and Φ

119908


reference state


nabla sdot (120582119904120579I + 2120583

119904120598) minus nabla (119877119879120576

119908

+ 119877119879Φ119888119896

minus 119861119908120579) = 0


119877119879

120572

[Φ119908

nabla120576119908

+

Φ119908119888+

120576+

nabla120576+

+

Φ119908119888minus

120576minus

nabla120576minus

] = 0

nabla sdot [minus

119877119879

120572

Φ119908

119888119865

nabla120576119908

] minus nabla

sdot [

Φ119908119888+119863+

120576+

+

119877119879

120572

Φ119908(119888+

)

2

120576+

minus

119877119879

120572

Φ119908119888+119888minus

120576+

]nabla120576+

+ nabla sdot [

Φ119908119888minus119863minus

120576minus

+

119877119879

120572

Φ119908(119888minus

)

2

120576minus

minus

119877119879

120572

Φ119908119888+119888minus

120576+

]nabla120576minus

= 0

120597 (Φ119908119888119896)

120597119905


119877119879

120572

Φ119908

119888119896

nabla120576119908

] minus nabla

sdot [

Φ119908119888+119863+

120576+

+

119877119879

120572

Φ119908(119888+

)

2

120576+

minus

119877119879

120572

Φ119908119888+119888minus

120576+

]nabla120576+

minus nabla

sdot [

Φ119908119888minus119863minus

120576minus

+

119877119879

120572

Φ119908(119888minus

)

2

120576minus

minus

119877119879

120572

Φ119908119888+119888minus

120576+

]nabla120576minus

+ nabla

sdot (Φ119908

119888119896k119904) = 0

(11)

where 119888119896 = 119888++ 119888minus


1015840= 119891(119909 119910)

with 119910(1199090) = 119910




Γ

120575u120590lowastn 119889Γ


120576119908 sdot J119908 119889119881 = minusint

Γ

120575120576119908J119908lowast

n 119889Γ


Ω


Γ

120575120576+ (Jminus

lowast

minus J+lowast

) n 119889Γ



120597 (Φ119908119888119896)

120597119905




119908

119888119896k119904) 119889119881 = minusint

Γ

120575120576minus (J+

lowast

+ Jminuslowast

)n 119889Γ

(12)









119909119910= 119869+

119909119910= 119869minus


119911= 119869+

119911=

119869minus



=

119906119910


119909= 119906119910

= 119906119911




distribution

Electrochemical

effects


Swelling shrinking



t = 3600 s

Mechanicaleffects


(iii) 120579 = 0

t = 0 s

t gt 0 s



120576+ = 120576+lowast


120576119908lowast

120576+lowast




lowast



lowast

0and the


0and 120576

minuslowast

0


0= 120576119908lowast

0 120576+0

= 120576+lowast

0 120576minus0

= 120576minuslowast

0 Choosing


1199090

= 1199061199100

= 1199061199110




+

+

+

+

+

++

+

minus

minus

minus

minusminus

minus minusminus

Cartilage sample

Impermeable chamber


(a)


120576w = 120576wlowast

120576+ = 120576+lowast


Y



Z

X


ux = uy = 0


ux = uy = uz = 0

(b)




119909= 119906119910























119908

0075 [26]


















119908

0075 [26]








000E + 00

200E minus 03

400E minus 03

600E minus 03

800E minus 03

100E minus 02

120E minus 02

140E minus 02uz

(mm

)

0 200 400 600 800 1000 1200 1400 1600


Time step (s)

(a)

250E + 02

255E + 02

260E + 02

265E + 02

270E + 02

275E + 02

280E + 02

c+(m

olm

3 )

0 01 02 03 04 05 06Depth (mm)


(b)










0=

02mEqmL and 119888119865







00001

00001

00000200004

minus0001 minus0001

XY

Z







t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

uz

(a)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s


w (m3s)J

(b)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

+ (mols)


J

(c)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

minus (mols)


J

(d)




100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00

0 500 1000 1500 2000 2500 3000 3500Time step (s)

uz

(mm

)



(a)

0 5 10 15 20 25 30 35 40 45 50 55

minus800E minus 11

minus600E minus 11

minus400E minus 11

minus200E minus 11

000E + 00

200E minus 11

400E minus 11

600E minus 11

800E minus 11

Time step (s)

uz

(mm

)



(b)

100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00

uz

(mm

)

0 500 1000 1500 2000 2500 3000 3500Time step (s)

Φw0 = 075

Φw0 = 077

Φw0 = 080

Φw0 = 085

(c)

100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00uz

(mm

)

0 500 1000 1500 2000 2500 3000 3500Time step (s)

Φ =

Φ =

Φ =

Φ =

10080

075070

(d)


0) (d) Surface






0= 75 of






4 Conclusion




119865

0in






119908









Acknowledgments


References




















































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


Chondrocytes


Collagen fibres


Collagen fibrils

minus

minus

minus minus

minus

minus+

+

+

+

+

+

+

+

minus

(a)


Fixed charges


Hyaluronan

minus

minusminus minus

minus

minus

minus

minus

minus minus

minus

minusminus

minus

minus

minus

minusminus

(b)





120590 = minus119875I + 120582119904120579I + 2120583

119904120598

120576119908

=

119875

119877119879

minus Φ (119888+

+ 119888minus

) +

119861119908

119877119879

120579

120576+

= 120574+

119888+ exp(

119865119888120595

119877119879

)

120576minus

= 120574minus


119865119888120595

119877119879

)

(5)








Tsdot us) is the



119888+

=

119888119865+ radic(119888

119865

0)2

+ (4120576+120576minus (120574+120574minus))

2

119888minus

=

minus119888119865+ radic(119888

119865

0)2

+ (4120576+120576minus (120574+120574minus))

2

(6)





J119908 = minus

119877119879Φ119908

120572

(nabla120576119908

+

119888+

120576+nabla120576+

+

119888minus


)

J+ = minus

119877119879Φ119908119888+

120572

nabla120576119908

minus [

Φ119908119888+119863+

120576+

+

119877119879Φ119908(119888+)

2

120572120576+

]nabla120576+

minus

119877119879Φ119908119888+119888minus

120572120576+

nabla120576minus

Jminus = minus

119877119879Φ119908119888minus

120572

nabla120576119908

minus [

Φ119908119888minus119863minus

120576minus

+

119877119879Φ119908(119888minus)

2

120572120576minus

]nabla120576minus

minus

119877119879Φ119908119888+119888minus

120572120576+

nabla120576+

(7)





Φ119908

= 1 minus

120601119904

0

1 + 120579

(8)

where 120601119904





119908120572 [26]


119888+

= 119888minus

+ 119888119865

(9)




119888119865

=

119888119865

0

1 + (120579Φ119908

0)

(10)

where 1198881198650and Φ

119908


reference state


nabla sdot (120582119904120579I + 2120583

119904120598) minus nabla (119877119879120576

119908

+ 119877119879Φ119888119896

minus 119861119908120579) = 0


119877119879

120572

[Φ119908

nabla120576119908

+

Φ119908119888+

120576+

nabla120576+

+

Φ119908119888minus

120576minus

nabla120576minus

] = 0

nabla sdot [minus

119877119879

120572

Φ119908

119888119865

nabla120576119908

] minus nabla

sdot [

Φ119908119888+119863+

120576+

+

119877119879

120572

Φ119908(119888+

)

2

120576+

minus

119877119879

120572

Φ119908119888+119888minus

120576+

]nabla120576+

+ nabla sdot [

Φ119908119888minus119863minus

120576minus

+

119877119879

120572

Φ119908(119888minus

)

2

120576minus

minus

119877119879

120572

Φ119908119888+119888minus

120576+

]nabla120576minus

= 0

120597 (Φ119908119888119896)

120597119905


119877119879

120572

Φ119908

119888119896

nabla120576119908

] minus nabla

sdot [

Φ119908119888+119863+

120576+

+

119877119879

120572

Φ119908(119888+

)

2

120576+

minus

119877119879

120572

Φ119908119888+119888minus

120576+

]nabla120576+

minus nabla

sdot [

Φ119908119888minus119863minus

120576minus

+

119877119879

120572

Φ119908(119888minus

)

2

120576minus

minus

119877119879

120572

Φ119908119888+119888minus

120576+

]nabla120576minus

+ nabla

sdot (Φ119908

119888119896k119904) = 0

(11)

where 119888119896 = 119888++ 119888minus


1015840= 119891(119909 119910)

with 119910(1199090) = 119910




Γ

120575u120590lowastn 119889Γ


120576119908 sdot J119908 119889119881 = minusint

Γ

120575120576119908J119908lowast

n 119889Γ


Ω


Γ

120575120576+ (Jminus

lowast

minus J+lowast

) n 119889Γ



120597 (Φ119908119888119896)

120597119905




119908

119888119896k119904) 119889119881 = minusint

Γ

120575120576minus (J+

lowast

+ Jminuslowast

)n 119889Γ

(12)









119909119910= 119869+

119909119910= 119869minus


119911= 119869+

119911=

119869minus



=

119906119910


119909= 119906119910

= 119906119911




distribution

Electrochemical

effects


Swelling shrinking



t = 3600 s

Mechanicaleffects


(iii) 120579 = 0

t = 0 s

t gt 0 s



120576+ = 120576+lowast


120576119908lowast

120576+lowast




lowast



lowast

0and the


0and 120576

minuslowast

0


0= 120576119908lowast

0 120576+0

= 120576+lowast

0 120576minus0

= 120576minuslowast

0 Choosing


1199090

= 1199061199100

= 1199061199110




+

+

+

+

+

++

+

minus

minus

minus

minusminus

minus minusminus

Cartilage sample

Impermeable chamber


(a)


120576w = 120576wlowast

120576+ = 120576+lowast


Y



Z

X


ux = uy = 0


ux = uy = uz = 0

(b)




119909= 119906119910























119908

0075 [26]


















119908

0075 [26]








000E + 00

200E minus 03

400E minus 03

600E minus 03

800E minus 03

100E minus 02

120E minus 02

140E minus 02uz

(mm

)

0 200 400 600 800 1000 1200 1400 1600


Time step (s)

(a)

250E + 02

255E + 02

260E + 02

265E + 02

270E + 02

275E + 02

280E + 02

c+(m

olm

3 )

0 01 02 03 04 05 06Depth (mm)


(b)










0=

02mEqmL and 119888119865







00001

00001

00000200004

minus0001 minus0001

XY

Z







t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

uz

(a)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s


w (m3s)J

(b)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

+ (mols)


J

(c)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

minus (mols)


J

(d)




100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00

0 500 1000 1500 2000 2500 3000 3500Time step (s)

uz

(mm

)



(a)

0 5 10 15 20 25 30 35 40 45 50 55

minus800E minus 11

minus600E minus 11

minus400E minus 11

minus200E minus 11

000E + 00

200E minus 11

400E minus 11

600E minus 11

800E minus 11

Time step (s)

uz

(mm

)



(b)

100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00

uz

(mm

)

0 500 1000 1500 2000 2500 3000 3500Time step (s)

Φw0 = 075

Φw0 = 077

Φw0 = 080

Φw0 = 085

(c)

100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00uz

(mm

)

0 500 1000 1500 2000 2500 3000 3500Time step (s)

Φ =

Φ =

Φ =

Φ =

10080

075070

(d)


0) (d) Surface






0= 75 of






4 Conclusion




119865

0in






119908









Acknowledgments


References




















































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology



J119908 = minus

119877119879Φ119908

120572

(nabla120576119908

+

119888+

120576+nabla120576+

+

119888minus


)

J+ = minus

119877119879Φ119908119888+

120572

nabla120576119908

minus [

Φ119908119888+119863+

120576+

+

119877119879Φ119908(119888+)

2

120572120576+

]nabla120576+

minus

119877119879Φ119908119888+119888minus

120572120576+

nabla120576minus

Jminus = minus

119877119879Φ119908119888minus

120572

nabla120576119908

minus [

Φ119908119888minus119863minus

120576minus

+

119877119879Φ119908(119888minus)

2

120572120576minus

]nabla120576minus

minus

119877119879Φ119908119888+119888minus

120572120576+

nabla120576+

(7)





Φ119908

= 1 minus

120601119904

0

1 + 120579

(8)

where 120601119904





119908120572 [26]


119888+

= 119888minus

+ 119888119865

(9)




119888119865

=

119888119865

0

1 + (120579Φ119908

0)

(10)

where 1198881198650and Φ

119908


reference state


nabla sdot (120582119904120579I + 2120583

119904120598) minus nabla (119877119879120576

119908

+ 119877119879Φ119888119896

minus 119861119908120579) = 0


119877119879

120572

[Φ119908

nabla120576119908

+

Φ119908119888+

120576+

nabla120576+

+

Φ119908119888minus

120576minus

nabla120576minus

] = 0

nabla sdot [minus

119877119879

120572

Φ119908

119888119865

nabla120576119908

] minus nabla

sdot [

Φ119908119888+119863+

120576+

+

119877119879

120572

Φ119908(119888+

)

2

120576+

minus

119877119879

120572

Φ119908119888+119888minus

120576+

]nabla120576+

+ nabla sdot [

Φ119908119888minus119863minus

120576minus

+

119877119879

120572

Φ119908(119888minus

)

2

120576minus

minus

119877119879

120572

Φ119908119888+119888minus

120576+

]nabla120576minus

= 0

120597 (Φ119908119888119896)

120597119905


119877119879

120572

Φ119908

119888119896

nabla120576119908

] minus nabla

sdot [

Φ119908119888+119863+

120576+

+

119877119879

120572

Φ119908(119888+

)

2

120576+

minus

119877119879

120572

Φ119908119888+119888minus

120576+

]nabla120576+

minus nabla

sdot [

Φ119908119888minus119863minus

120576minus

+

119877119879

120572

Φ119908(119888minus

)

2

120576minus

minus

119877119879

120572

Φ119908119888+119888minus

120576+

]nabla120576minus

+ nabla

sdot (Φ119908

119888119896k119904) = 0

(11)

where 119888119896 = 119888++ 119888minus


1015840= 119891(119909 119910)

with 119910(1199090) = 119910




Γ

120575u120590lowastn 119889Γ


120576119908 sdot J119908 119889119881 = minusint

Γ

120575120576119908J119908lowast

n 119889Γ


Ω


Γ

120575120576+ (Jminus

lowast

minus J+lowast

) n 119889Γ



120597 (Φ119908119888119896)

120597119905




119908

119888119896k119904) 119889119881 = minusint

Γ

120575120576minus (J+

lowast

+ Jminuslowast

)n 119889Γ

(12)









119909119910= 119869+

119909119910= 119869minus


119911= 119869+

119911=

119869minus



=

119906119910


119909= 119906119910

= 119906119911




distribution

Electrochemical

effects


Swelling shrinking



t = 3600 s

Mechanicaleffects


(iii) 120579 = 0

t = 0 s

t gt 0 s



120576+ = 120576+lowast


120576119908lowast

120576+lowast




lowast



lowast

0and the


0and 120576

minuslowast

0


0= 120576119908lowast

0 120576+0

= 120576+lowast

0 120576minus0

= 120576minuslowast

0 Choosing


1199090

= 1199061199100

= 1199061199110




+

+

+

+

+

++

+

minus

minus

minus

minusminus

minus minusminus

Cartilage sample

Impermeable chamber


(a)


120576w = 120576wlowast

120576+ = 120576+lowast


Y



Z

X


ux = uy = 0


ux = uy = uz = 0

(b)




119909= 119906119910























119908

0075 [26]


















119908

0075 [26]








000E + 00

200E minus 03

400E minus 03

600E minus 03

800E minus 03

100E minus 02

120E minus 02

140E minus 02uz

(mm

)

0 200 400 600 800 1000 1200 1400 1600


Time step (s)

(a)

250E + 02

255E + 02

260E + 02

265E + 02

270E + 02

275E + 02

280E + 02

c+(m

olm

3 )

0 01 02 03 04 05 06Depth (mm)


(b)










0=

02mEqmL and 119888119865







00001

00001

00000200004

minus0001 minus0001

XY

Z







t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

uz

(a)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s


w (m3s)J

(b)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

+ (mols)


J

(c)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

minus (mols)


J

(d)




100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00

0 500 1000 1500 2000 2500 3000 3500Time step (s)

uz

(mm

)



(a)

0 5 10 15 20 25 30 35 40 45 50 55

minus800E minus 11

minus600E minus 11

minus400E minus 11

minus200E minus 11

000E + 00

200E minus 11

400E minus 11

600E minus 11

800E minus 11

Time step (s)

uz

(mm

)



(b)

100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00

uz

(mm

)

0 500 1000 1500 2000 2500 3000 3500Time step (s)

Φw0 = 075

Φw0 = 077

Φw0 = 080

Φw0 = 085

(c)

100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00uz

(mm

)

0 500 1000 1500 2000 2500 3000 3500Time step (s)

Φ =

Φ =

Φ =

Φ =

10080

075070

(d)


0) (d) Surface






0= 75 of






4 Conclusion




119865

0in






119908









Acknowledgments


References




















































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology



120597 (Φ119908119888119896)

120597119905




119908

119888119896k119904) 119889119881 = minusint

Γ

120575120576minus (J+

lowast

+ Jminuslowast

)n 119889Γ

(12)









119909119910= 119869+

119909119910= 119869minus


119911= 119869+

119911=

119869minus



=

119906119910


119909= 119906119910

= 119906119911




distribution

Electrochemical

effects


Swelling shrinking



t = 3600 s

Mechanicaleffects


(iii) 120579 = 0

t = 0 s

t gt 0 s



120576+ = 120576+lowast


120576119908lowast

120576+lowast




lowast



lowast

0and the


0and 120576

minuslowast

0


0= 120576119908lowast

0 120576+0

= 120576+lowast

0 120576minus0

= 120576minuslowast

0 Choosing


1199090

= 1199061199100

= 1199061199110




+

+

+

+

+

++

+

minus

minus

minus

minusminus

minus minusminus

Cartilage sample

Impermeable chamber


(a)


120576w = 120576wlowast

120576+ = 120576+lowast


Y



Z

X


ux = uy = 0


ux = uy = uz = 0

(b)




119909= 119906119910























119908

0075 [26]


















119908

0075 [26]








000E + 00

200E minus 03

400E minus 03

600E minus 03

800E minus 03

100E minus 02

120E minus 02

140E minus 02uz

(mm

)

0 200 400 600 800 1000 1200 1400 1600


Time step (s)

(a)

250E + 02

255E + 02

260E + 02

265E + 02

270E + 02

275E + 02

280E + 02

c+(m

olm

3 )

0 01 02 03 04 05 06Depth (mm)


(b)










0=

02mEqmL and 119888119865







00001

00001

00000200004

minus0001 minus0001

XY

Z







t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

uz

(a)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s


w (m3s)J

(b)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

+ (mols)


J

(c)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

minus (mols)


J

(d)




100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00

0 500 1000 1500 2000 2500 3000 3500Time step (s)

uz

(mm

)



(a)

0 5 10 15 20 25 30 35 40 45 50 55

minus800E minus 11

minus600E minus 11

minus400E minus 11

minus200E minus 11

000E + 00

200E minus 11

400E minus 11

600E minus 11

800E minus 11

Time step (s)

uz

(mm

)



(b)

100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00

uz

(mm

)

0 500 1000 1500 2000 2500 3000 3500Time step (s)

Φw0 = 075

Φw0 = 077

Φw0 = 080

Φw0 = 085

(c)

100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00uz

(mm

)

0 500 1000 1500 2000 2500 3000 3500Time step (s)

Φ =

Φ =

Φ =

Φ =

10080

075070

(d)


0) (d) Surface






0= 75 of






4 Conclusion




119865

0in






119908









Acknowledgments


References




















































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


+

+

+

+

+

++

+

minus

minus

minus

minusminus

minus minusminus

Cartilage sample

Impermeable chamber


(a)


120576w = 120576wlowast

120576+ = 120576+lowast


Y



Z

X


ux = uy = 0


ux = uy = uz = 0

(b)




119909= 119906119910























119908

0075 [26]


















119908

0075 [26]








000E + 00

200E minus 03

400E minus 03

600E minus 03

800E minus 03

100E minus 02

120E minus 02

140E minus 02uz

(mm

)

0 200 400 600 800 1000 1200 1400 1600


Time step (s)

(a)

250E + 02

255E + 02

260E + 02

265E + 02

270E + 02

275E + 02

280E + 02

c+(m

olm

3 )

0 01 02 03 04 05 06Depth (mm)


(b)










0=

02mEqmL and 119888119865







00001

00001

00000200004

minus0001 minus0001

XY

Z







t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

uz

(a)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s


w (m3s)J

(b)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

+ (mols)


J

(c)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

minus (mols)


J

(d)




100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00

0 500 1000 1500 2000 2500 3000 3500Time step (s)

uz

(mm

)



(a)

0 5 10 15 20 25 30 35 40 45 50 55

minus800E minus 11

minus600E minus 11

minus400E minus 11

minus200E minus 11

000E + 00

200E minus 11

400E minus 11

600E minus 11

800E minus 11

Time step (s)

uz

(mm

)



(b)

100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00

uz

(mm

)

0 500 1000 1500 2000 2500 3000 3500Time step (s)

Φw0 = 075

Φw0 = 077

Φw0 = 080

Φw0 = 085

(c)

100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00uz

(mm

)

0 500 1000 1500 2000 2500 3000 3500Time step (s)

Φ =

Φ =

Φ =

Φ =

10080

075070

(d)


0) (d) Surface






0= 75 of






4 Conclusion




119865

0in






119908









Acknowledgments


References




















































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology















119908

0075 [26]


















119908

0075 [26]








000E + 00

200E minus 03

400E minus 03

600E minus 03

800E minus 03

100E minus 02

120E minus 02

140E minus 02uz

(mm

)

0 200 400 600 800 1000 1200 1400 1600


Time step (s)

(a)

250E + 02

255E + 02

260E + 02

265E + 02

270E + 02

275E + 02

280E + 02

c+(m

olm

3 )

0 01 02 03 04 05 06Depth (mm)


(b)










0=

02mEqmL and 119888119865







00001

00001

00000200004

minus0001 minus0001

XY

Z







t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

uz

(a)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s


w (m3s)J

(b)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

+ (mols)


J

(c)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

minus (mols)


J

(d)




100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00

0 500 1000 1500 2000 2500 3000 3500Time step (s)

uz

(mm

)



(a)

0 5 10 15 20 25 30 35 40 45 50 55

minus800E minus 11

minus600E minus 11

minus400E minus 11

minus200E minus 11

000E + 00

200E minus 11

400E minus 11

600E minus 11

800E minus 11

Time step (s)

uz

(mm

)



(b)

100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00

uz

(mm

)

0 500 1000 1500 2000 2500 3000 3500Time step (s)

Φw0 = 075

Φw0 = 077

Φw0 = 080

Φw0 = 085

(c)

100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00uz

(mm

)

0 500 1000 1500 2000 2500 3000 3500Time step (s)

Φ =

Φ =

Φ =

Φ =

10080

075070

(d)


0) (d) Surface






0= 75 of






4 Conclusion




119865

0in






119908









Acknowledgments


References




















































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


000E + 00

200E minus 03

400E minus 03

600E minus 03

800E minus 03

100E minus 02

120E minus 02

140E minus 02uz

(mm

)

0 200 400 600 800 1000 1200 1400 1600


Time step (s)

(a)

250E + 02

255E + 02

260E + 02

265E + 02

270E + 02

275E + 02

280E + 02

c+(m

olm

3 )

0 01 02 03 04 05 06Depth (mm)


(b)










0=

02mEqmL and 119888119865







00001

00001

00000200004

minus0001 minus0001

XY

Z







t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

uz

(a)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s


w (m3s)J

(b)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

+ (mols)


J

(c)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

minus (mols)


J

(d)




100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00

0 500 1000 1500 2000 2500 3000 3500Time step (s)

uz

(mm

)



(a)

0 5 10 15 20 25 30 35 40 45 50 55

minus800E minus 11

minus600E minus 11

minus400E minus 11

minus200E minus 11

000E + 00

200E minus 11

400E minus 11

600E minus 11

800E minus 11

Time step (s)

uz

(mm

)



(b)

100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00

uz

(mm

)

0 500 1000 1500 2000 2500 3000 3500Time step (s)

Φw0 = 075

Φw0 = 077

Φw0 = 080

Φw0 = 085

(c)

100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00uz

(mm

)

0 500 1000 1500 2000 2500 3000 3500Time step (s)

Φ =

Φ =

Φ =

Φ =

10080

075070

(d)


0) (d) Surface






0= 75 of






4 Conclusion




119865

0in






119908









Acknowledgments


References




















































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


00001

00001

00000200004

minus0001 minus0001

XY

Z







t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

uz

(a)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s


w (m3s)J

(b)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

+ (mols)


J

(c)

00001

00001

00000200004

minus0001 minus0001

XY

Z






t = 0 s t = 67 s t = 200 s t = 800 s t = 3600 s

minus (mols)


J

(d)




100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00

0 500 1000 1500 2000 2500 3000 3500Time step (s)

uz

(mm

)



(a)

0 5 10 15 20 25 30 35 40 45 50 55

minus800E minus 11

minus600E minus 11

minus400E minus 11

minus200E minus 11

000E + 00

200E minus 11

400E minus 11

600E minus 11

800E minus 11

Time step (s)

uz

(mm

)



(b)

100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00

uz

(mm

)

0 500 1000 1500 2000 2500 3000 3500Time step (s)

Φw0 = 075

Φw0 = 077

Φw0 = 080

Φw0 = 085

(c)

100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00uz

(mm

)

0 500 1000 1500 2000 2500 3000 3500Time step (s)

Φ =

Φ =

Φ =

Φ =

10080

075070

(d)


0) (d) Surface






0= 75 of






4 Conclusion




119865

0in






119908









Acknowledgments


References




















































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00

0 500 1000 1500 2000 2500 3000 3500Time step (s)

uz

(mm

)



(a)

0 5 10 15 20 25 30 35 40 45 50 55

minus800E minus 11

minus600E minus 11

minus400E minus 11

minus200E minus 11

000E + 00

200E minus 11

400E minus 11

600E minus 11

800E minus 11

Time step (s)

uz

(mm

)



(b)

100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00

uz

(mm

)

0 500 1000 1500 2000 2500 3000 3500Time step (s)

Φw0 = 075

Φw0 = 077

Φw0 = 080

Φw0 = 085

(c)

100E minus 20

100E minus 18

100E minus 16

100E minus 14

100E minus 12

100E minus 10

100E minus 08

100E minus 06

100E minus 04

100E minus 02

100E + 00uz

(mm

)

0 500 1000 1500 2000 2500 3000 3500Time step (s)

Φ =

Φ =

Φ =

Φ =

10080

075070

(d)


0) (d) Surface






0= 75 of






4 Conclusion




119865

0in






119908









Acknowledgments


References




















































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology




4 Conclusion




119865

0in






119908









Acknowledgments


References




















































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology






Acknowledgments


References




















































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology








Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology

Documents

Research Article Cartilage Dysfunction in ALS Patients as …downloads.hindawi.com/journals/bmri/2014/179070.pdf · · 2015-11-22Loss: 3D Mechano-Electrochemical Computational Model