Tly P. Appelmn Joseph Mizrhi Dror Seliktr e-mil: ror@m.tehnion..il Fulty of Biomeil Engineering, Tehnion - Isrel Institute of Tehnology, Hif 3, Isrel A Finite Element Moel of Cell-Mtrix Intertions to Stuy the Differentil Effet of Sffol Composition on Chonrogeni Response to Mehnil Stimultion Mehnilly inue ell eformtions hve een shown to influene honroyte response in 3D ulture. However, the reltionship etween the mehnil stimultion n ell response is not yet fully unerstoo. In this stuy finite element moel ws evelope to investigte ell-mtrix intertions uner unonfine ompression onitions, using tissue engineere enpsulting hyrogel seee with honroytes. Moel preitions of stress n strin istriutions within the ell n on the ell ounry were shown to exhiit spe-epenent responses tht vrie with sffol mehnil properties, the presene of periellulr mtrix (PCM), n the ell size. The simultions preite tht when the ells were initilly enpsulte into the hyrogel sffols, the ell size hrly ffete the mgnitue of the stresses n strins tht were rehing the enpsulte ells. However, with the inlusion of PCM lyer, lrger ells experiene enhne stresses n strins resulting from the mehnil stimultion. It ws lso note tht the PCM h stress shieling effet on the ells in tht the pek stresses experiene within the ells uring loing were signifintly reue. On the other hn, the PCM use the stresses t the ell-mtrix interfe to inrese. Bse on the moel preitions, the PCM moifie the sptil stress istriution within n roun the enpsulte ells y reireting the mximum stresses from the periphery of the ells to the ell nuleus. In tissue engineere rtilge expose to mehnil loing, the formtion of neo-pcm y enpsulte honroytes ppers to protet them from initilly exessive mehnil loing. Preitive moels n thus she importnt insight into how honroytes remoel their lol environment in orer to reistriute mehnil signls in tissue engineere onstruts. DOI: 0.5/.400334 Keywors: finite element moel, rtilge tissue engineering, mehnil stimultion, hyrogel, poly(ethylene glyol), mehnotrnsution, honroyte 2 3 4 5 6 7 8 9 0 2 3 4 5 6 7 Introution Crtilge ells re sensitive to their mehnil environment n ret iretly to mehnil stimultion. The ouple mehnil n iohemil ues t the ell surfe, where extrellulr signls re onverte to intrellulr ones, re the fol point of mehnotrnsution. Previous stuies hve shown tht the tivtion of mehnotrnsution pthwys use y ompressive strins n signifintly lter honroyte metolism 2 7. This tivtion, lthough still poorly unerstoo, ppers to inlue omintion of integrin-meite signling, strethtivte ion hnnel signling, n tin ytoskeleton eformtions 8. Other physil events pereive y enpsulte honroytes expose to mehnil loing my inlue whole ell eformtions, lol strin flututions, flui flows, osmoti hnges, n fixe hrge ensity ltertions. Theoretil moels of the honroyte miroenvironment re prtiulrly helpful in this ontext, s they enle quntittive preitions of events, Corresponing uthor. Mnusript reeive Mrh 30, 200; finl mnusript reeive Deemer 4, 200; epte mnusript poste Deemer 22, 200; pulishe online xxxxx-xxxxx-xxxxx. Asso. Eitor: Clrk T. Hung. whih re iffiult to mesure experimentlly. These moels require informtion on the onstitutive ehvior of the ells n the 9 8 sffols in whih they re emee or enpsulte. 20 In ntive rtiulr rtilge, the mehnilly loe honrotyes re enpsulte y periellulr mtrix PCM. The 22 2 PCM is istinguishe from the rest of the extrellulr mtrix 23 ECM y its unique omposition n mehnil properties. The 24 ell n PCM hve jointly een terme honron 2. The ECM 25 is omprise minly of type II ollgen n ggren proteoglyns while the PCM onsists of ollgen Types II, VI, n IX, 27 26 fironetin, ggren, hyluronn, n eorin 2,3. When isolte honroytes re enpsulte in sffols, they synthesize 29 28 tissue-speifi moleules tht initilly form PCM, whih therefter iffuses into the extrellulr spe to reinfore the sffol 3 30 n eomes prt of the eveloping ECM. 32 Our group hs previously reporte tht honroytes n respon ifferently when they re sujete to the sme ompres- 34 33 sive strin stimultion in ifferent enpsulting mteril 4. 35 To show this, we immoilize vrious proteins n proteoglyns 36 into poly ethylene glyol PEG se hyrogels to rete iosyntheti milieu for ultivting enpsulte honroytes uner 38 37 ynmi mehnil stimultion. The honroyte-enpsulting 39 onstruts were sujete to 5% ynmi ompressive strin 40 Journl of Biomehnil Engineering Copyright 20 y ASME APRIL 20, Vol. 33 / -
" e Fig. Histologil ross-setions of hyrogel onstruts stine for proteoglyns with Sfrnin O: fter 24 h, n fter 28 ys with stimultion in two ifferent types of hyrogels, n n e exmples of setions inluing ifferent ells sujete to the sme mehnil stimultion ut resulting in ifferent quntities of PCM seretion. Sle rs: 0 m. 4 stimultion t Hz frequeny for 28 ys, n the ellulr response ws mesure. Our finings inite tht the mtrix prop- 42 43 erties i.e., iotivity n moulus signifintly influene the 44 metoli tivity of the ells, presumly through hnges in the 45 mehnotrnsution pthwys. We sought to pply finite element moel FEM to our experimentl system in orer to gin 46 47 insight into how these smll hnges in the ECM properties n 48 result in ltertions to the mehnil stress n strin istriutions within n roun the enpsulte ells. 49 50 Finite element moeling of rtilge tissue ws previously pplie to simulte honroyte-mtrix intertions uner stey 5 52 stte 5,6, trnsient 7 9, n ynmi yli loing 53 20,2. Most of the work in this fiel hs een fouse on rtilge tissue explnts, either norml or osteorthriti, n hs s- 54 55 sume iphsi ehvior of the ifferent ell, PCM, n ECM 56 omponents. FEM simultions hve lso een use to lulte 57 the mehnil properties of either PCM or ell omponents y 58 inverse finite element pprohes omine with experimentl 59 t 5,22. Reently, FEM work reveling stresses n strins 60 for ifferent sffols use for rtilge tissue engineering ws 6 reporte y Blol n Bonssr 23. This work fouse on the 62 ifferenes in istriution n mgnitues of pressures, flui 63 flows, strins, n sher stresses in the entire onstrut, ut i not 64 ress istriutions of these fores t the ellulr level. In other 65 work reporte y Guilk n Mow 8, the iphsi simultion 66 prmeters were vrie to inlue grose n lginte hyrogels 67 sffols with enpsulte honroytes. Although ommonly 68 use for rtilge tissue engineering, oth lginte n grose re 69 very ifferent mterils when ompre with the PEG-protein hyrogels reporte y our group 4, n thus we pplie similr 70 7 FEM moeling pproh to our system in orer to etter unerstn our experimentl t. 72 73 The purpose of this stuy ws thus to evelop FEM pproh 74 to eluite the sequene of iomehnil n iohemil 75 events through whih pplie mehnil loing influenes 76 honroyte iotivity in tissue engineere rtilge in vitro. The 77 simultions were fouse on the pek mplitues uring ynmi 78 loing. We expet tht our results will provie further insight 79 into the omplex reltionship etween struture, iohemil 80 properties, metoli tivity, iomehnil performne, n tissue funtion; helping to gin etter unerstning of events 8 ssoite with tissue engineere rtilge evelopment in enpsulting PEG-se hyrogels. 82 83 2 Methos 2. Finite Element Simultions. A finite element nlysis 85 ws performe on hyrogel-enpsulte honroytes expose to 86 ompression using severl onstitutive moels. Speifilly, the 87 moeling ws se on honroytes enpsulte in four ifferent PEG-se hyrogel sffols, eh sujete to 5% y- 89 88 nmi ompressive strin stimultion t Hzfrequeny. The 90 mximum los t the pek 5% ompressive strin were ompre for the ifferent hyrogels 4. In other wors, the simul- 92 9 tions one were equivlent to the time point t the mximum 93 mplitue of 5%. The onstruts re initilly onsiere to e 94 ompose of sffol mteril n enpsulte rnomly istriute spheril honroytes. However, uring the ourse of 96 95 their evelopment, the ells proue PCM roun their outer 97 periphery s eviene y our reporte experimentl finings 98 4,24 Figs.. The volume frtion of honroytes 99 ws pproximtely.6%, se on ell rius of 5 m n 00 seeing ensity 30 0 6 ells per ui entimeter of hyrogel 0 mtrix. For our simultions volume frtion of 2% ws ssume. 02 The si unit ws tken s yliner with hrteristi height 03 of 32 m n rius 6 m, ssuming homogeneity of the 04 ells n the ove volume frtion. The enpsulte honroyte n its PCM n e moele s spheril inlusions with 06 05 ell rius of 5 m n periellulr lyer of thikness of 07 2.5 m 8. Due to xisymmetry roun the ell n symmetry 08 roun the r- plne refer to Fig. 2 for the oorinte system 09 the geometry use in the simultions ws reue to miroomin with rius of 6 m in the r iretion n height of 0 6 m in the z iretion Fig. 2. Vritions in the honroyte rius from the si 5 m were lso moele s follows: 3 2 rius of 7.5 m resulting in frtion volume of 6.8% Fig. 4 2, n rius of 0 m resulting in frtion volume of 5 6.3% Fig. 2. 6 For ll simultions, the xil-symmetry stress-strin pplition 7 moe of the ommeril finite element softwre COMSOL MULT- 8 84-2 / Vol. 33, APRIL 20 Trnstions of the ASME
z r Fig. 2 2D meshes use in the xisymmetry simultion in the se of ell rii of 5 m, 7.5 m, n 0 m. The oorinte system use in the simultions n e foun in. 9 20 2 22 23 24 25 26 27 28 29 30 IPHYSICS CM version 3.5, COMSOL, In., Burlington, MA ws use. The mesh onsiste of tringulr elements with Lgrnge qurti sis funtions in ll ses. The mesh ws refine t the ell n PCM ounries, where higher stress grients were ntiipte. A mesh of 3750 elements ws foun suffiient for onvergene purposes for 5 m ell rius. A finer mesh with 5,000 elements resulte in less thn 0.25% ifferene in mximum von Mises stresses, n prinipl strins. In those ses where the mteril ws moele s isotropi with lrge eformtions, finer mesh of 794 elements ws require to hieve onvergene mesh not shown. A summry of the ifferent element numers for the simulte geometries is foun in Tle. 3 2.2 Governing Equtions n Constitutive Moels. Negleting 32 33 34 the effet of oy fores, the governing eqution in eh of the suomins, the ell, the PCM, n the sffol ws given y 35 T =0 36 37 38 39 40 4 42 43 44 45 46 47 where T is the Cuhy stress. In ition to the xil-symmetry ounry onition roun the z xis n plne symmetry onition t z=0, onstnt presrie isplement equivlent to 5% ompressive strin ws pplie to the upper ege of the sffol suomin see Fig. 2 for oorinte system. Applying lulte ounry onitions from mrosopi moel to the miroomin Fig. 2, whih is known s the multisle pproh, ws foun unneessry ue to the efinition of the suomins s isotropi homogenous n monophsi mterils n the ssumption tht the ounry effets re negligile. With x enoting the present position n X the referene position, the eformtion grient F ws efine s F= x/ X. The Tle Mesh prmeters Cell imeter m No. of elements No. of egrees of freeom 5 3750 6,562 7.5 4354 8,905 0 4905 20,948 volume rtio ws efine s J=et F n the right n left 48 Cuhy Green eformtion tensors were given, respetively, y 49 C=F T F n B=FF T. The first eformtion invrint ws given s 50 I =C:I=B:I. To seprte volume effets the following sling 5 ws introue: C =J 2/3 C where the first invrint of C ws 52 then Ī =J 2/3 I. 53 For inompressile mterils, the inompressiility onstrint 54 on the volume rtio ws impose s follows: 55 J = et F = 2 56 while, in generl, the seon Piol Kirhhoff stress n e foun 57 y 58 S =2 W s C = W s 3 E 59 In this se, the seon Piol Kirhhoff stress ws foun y 60 S = pjc +2 W s C = pjc + W s 4 E 6 where the hyrostti pressure p is the Lgrnge multiplier rising 62 from the inompressiility onstrint n E= /2 C I is the 63 Green Lgrngin strin. 64 The Cuhy stress ws otine, fter the seon Piol 65 Kirhhoff stress ws foun, through the following inverse Piol 66 trnsformtion: 67 T = J FSF T 5 68 The strin energy of the inompressile isotropi neo-hooken 69 NH moel ws 70 W s = C I 3 6 7 with the ition of Eq. 2. 72 It n e shown tht the Cuhy stress in this se ws given s 73 T = pi +2C B B:I I 3 7 74 The eformtion grient of n inompressile mteril sujete to unixil strin ws 75 76 Journl of Biomehnil Engineering APRIL 20, Vol. 33 / -3
77 78 79 0 0 0 0 F = 0 0 where is the streth rtio. This resulte in left Cuhy Green eformtion tensor of 8 80 0 0 0 B = 2 0 0 0 9 E sffol =0kP 8 n Cuhy stress of 82 T = 2C 2 3 2 3 2 p 0 0 p 0 2C 3 2 + 3 p 0 0 0 2C 3 2 + 3 83 0 84 Beuse T 22 =0, it n e shown tht T =2C 2 /. Beuse the time erivtive of the true strin ws efine 85 s 86 87 88 89 90 9 92 93 94 95 96 97 98 99 200 20 202 203 = the true strin ws therefore efine s =ln ; i.e., =e. The xil stress eme T =2C e 2 e. The erivtive with respet to strin t smll strins yiele C =E 0 /6, whih for n inompressile mteril orrespone to C = 0 /2. Here, E 0 n 0 re Young n sher mouli t smll eformtions, respetively. Experimentl mesurements of the sffol properties emonstrte liner true stress n true strin reltionship up to streth rtio of 0.7. The ompressive moulus use for the hyrogel ws se on the liner est fit in the strin rnge 0.05 0.2. To hieve etter pproximtion to the liner ehvior reporte in the experiments, moifition ws me to the NH moel y ing oeffiient A=.075, whih ws foun y minimizing the sum of squre error SSE in the rnge of streth =:0.8 Fig. 3. COMSOL simultions gve ientil results to these lultions. This enle us to vlite the implementtion of the ove NH moel moifition into COMSOL. The strin energy of the moifie NH moel MNH ws W s = A 0 2 I 3 204 2 205 206 In COMSOL, the following mixe formultion ws implemente for the neo-hooken inompressile mteril: W s = 0 2 Ī 3 p J p2 3 207 2 0 208 where 0 = E 0 /2 2 n p ws the hyrostti pressure Lgrngin multiplier. Therefore the moifie NH strin energy ws 209 20 implemente s 2 22 23 W s = A 0 2 Ī 3 p J p2 2 0 4 The ompressile NH moel ws implemente in COMSOL y the following strin energy: Fig. 3 Moifition of the neo-hooken moel for the sffol suomin. The moifition of the neo-hooken moel squres versus irles proue etter fit to the experimentl ehvior plin line oserve in the ompression experiments. The experimentl tren line ws se on moulus vlues tken iretly from the true stress-strin urves of the sffol uner ompression, s mesure y n Instron testing mhine 4. W s = 0 2 Ī 3 2 0 J 2 5 24 n ws use to represent the ell n PCM ehvior. 25 In orer to hek the effet of this moifition on the overll 26 results, set of simultions ws lso run with ll the three suomins, ell, PCM, n sffol moele s NH. 28 27 To further investigte the effet of the onstitutive equtions on 29 the simultion results, the three suomins were lso moele s 220 isotropi elsti mterils, while ssuming smll eformtions, using the liner reltion etween the Cuhy stress n the liner 222 22 strin through the liner elstiity tensor D. In ition, the three 223 suomins were lso moele s isotropi elsti mterils, 224 while ssuming lrge eformtions. In COMSOL, this ws implemente y ssuming tht the mteril ehves s Sint Vennt 226 225 Kirhhoff hyperelsti mteril, whih mens tht the strin energy n e written s 228 227 W s = 2 E:D:E 6 229 This results in liner reltion etween the seon Piol 230 Kirhhoff stress n the Green Lgrngin strin, through the liner elstiity tensor D, whih for the inompressile se res 23 232 S = pjc + W s E = pjc + D:E 7 233 In generl, the Sint Vennt Kirhhoff onstitutive eqution is 234 useful for ses tht the strins re smll n the eformtions re 235 lrge, euse t smll strins the Lgrngin strin n the 236 Cuhy liner strin re lmost equivlent to eh other, s re the 237 seon Piol Kirhhoff stress n the Cuhy stress. The vntge is tht oth S n E re ojetive, n rottions ue to lrge 239 238 eformtions re orretly tken into ount. One the seon 240 Piol Kirhhoff stress ws foun, the Cuhy stress ws lulte through the inverse Piol trnsformtion Eq. 5. 242 24 2.3 Moel Prmeters. The PEG-se sffols were moele s inompressile monophsi isotropi mterils with 244 243 ompressive moulus rnging from 2 kp to 0 kp s mesure 245 in previous experiments 4 Tle 2. The inompressiility ssumption 0.5 ws se on the reltion etween Young s 247 246 moulus mesure using n Instron testing mhine, n the sher 248-4 / Vol. 33, APRIL 20 Trnstions of the ASME
Tle 2 Simultion prmeters Suomin Size m Type E kp Chonroyte r=5 0 Hyperelsti-neo-Hooken Isotropi-smll eformtions Isotropi-lrge eformtions 3 Results n Disussion 3.2 0.35 4 0.4 PCM h=2.5 Hyperelsti-neo-Hooken 40 0.04 Isotropi-smll eformtions 2 0 0.499 Isotropi-lrge eformtions 0 40 0.499 0.04 Sffol L=6 Hyperelsti-moifie neo-hooken Hyperelsti-neo-Hooken Isotropi-smll eformtions Isotropi-lrge eformtions 2 0 0.499 249 moulus of the mterils, whih ws mesure y strin-rte 250 ontrolle rheometer t not shown. These mesurements resulte in rtio of elsti moulus to sher moulus somewhere 25 252 in the rnge 3, whih orrespone to Poisson s rtio of 253 0.5. In ition, severl reports hve een pulishe with the 254 ssumption tht polymeri hyrogels, prtiulrly PEG-se hyrogels, ehve s inompressile solis 25,26. Chonroyte 255 256 n PCM t were tken from the relevnt literture. The PCM 257 ws moele s n isotropi mteril with Young s moulus of 40 258 kp n Poisson s rtio of 0.04 6. Chonroyte moulus ws 259 tken s 3.2 kp se on ompression tests of ells enpsulte 260 in lginte onstruts 27. Chonroytes were moele s n isotropi mteril with Poisson s rtio of 0.4 28,29. Even though 26 262 honroytes re known to exhiit visoelsti properties, they 263 were moele s elsti mterils for simpliity. This ws not n 264 unresonle ssumption given tht mehnil loing ws performe t Hz, n the honroytes hve relxtion time of 265 266 less thn 20 s 30. Furthermore, the sffols were sujete to 267.5 h of ontinuous stimultion, n previous ynmi simultions 268 hve shown tht stey stte ws hieve in less thn 5 min 269 23. 270 Given the vrious ssumptions n euse we were intereste 27 in preiting the pek vlues of stress n strin, the mximum 272 mplitue of the ell, the PCM, n the sffol were moele in 273 the nlysis. The moel preite how the mehnil loing 274 ws trnsmitte to the ells, oth t the ell-mtrix interfe n 275 within the entire honroyte, given speifie hnges in the mehnil properties of the ifferent suomins, the ell size, n 276 277 the presene or sene of the PCM. 278 279 3. Effets of Sffol Moulus. The sffol moulus ffete 280 28 282 283 284 285 286 287 288 289 290 29 292 293 294 295 296 297 298 the stresses, eformtions, isplements, n prinipl strins in ll suomins in oth ses: with the PCM n without the PCM Fig. 4. In generl, with the PCM Figs. 4 n 4, stiffer sffol resulte in n inrese in stresses throughout the suomins n in higher strins in the PCM n ellulr suomins. In ition, stiffer sffol exhiite more homogenous stress n strin fiel in the sffol suomin while softer sffol trnsferre less stresses n reue the strins rehing the ell n PCM. With the PCM, fivefol inrese in the sffol moulus use the mximum von Mises stress in the PCM suomin to inrese y ftor of 3.4 n in the sffol y ftor of 4. Without the PCM Figs. 4 n 4, stresses throughout the suomins lso inrese ue to the stiffer moulus. In the se of the ltter, inresing the sffol moulus y ftor of 5 use the mximum von Mises stress in the sffol suomin to inrese y ftor of 6. When the sffol moulus ws higher thn the ell moulus e.g., 0 kp, the ell ws sujete to higher isplements resulting in n inrese in strin reltive to tht of the sffol, whih n reh up to 9.7% strin, wheres when the ell moulus ws higher thn the sffol moulus e.g., 2 kp, the isplement fiel ppere to 300 299 wrp roun the ell, resulting in less strin to the ell t not 30 shown. The mximum von Mises stresses were preite to e in 302 the res immeitely outsie the ell ounry, lose to the ellmtrix interfe Figs. 4 n 4, irrespetive of the PCM. 304 303 Upon loser inspetion of the von Mises stresses t the ell 305 suomin, it ws evient tht the moulus of the sffol ffete oth the mgnitue n the istriution of the stresses Fig. 307 306 5. Our finings further reinfore the previously esrie oservtions tht the PCM introues stress shieling effet to the 309 308 ells 6. Aoring to our simultions, the mximum stress vlues erese y 20% n 52% in the presene of PCM with 3 30 sffol moulus of 0 kp n 2 kp, respetively, in omprison 32 to no PCM. Without the PCM, the stress grient ws in the z 33 iretion Figs. 5 n 5, wheres with the PCM, the stress 34 grient ws moifie to grient in the ril iretion with 35 higher vlues t the ell enter n lower vlues t the ounry 36 of the ell Figs. 5 n 5. In ition, the stresses were 37 istriute over wier rnge when the PCM ws present: 00 P 38 versus 20 P for sffol moulus of 0 kp n 60 P versus 39 3P for sffol moulus of 2 kp. Without the PCM, the 320 lotion of the pek stresses in the ell hnge s funtion of 32 sffol moulus. When the sffol moulus ws higher, the 322 mximum von Mises stress ws ner the lterl sie of the ell, 323 wheres when the sffol moulus erese elow the ell 324 moulus, the mximum von Mises stress ws lote ner the 325 pex of the ell. Interestingly, fter the PCM ws introue, the 326 mximum stress move to the enter of the ell where the honroyte nuleus ws lote. With the PCM in ple, oth the 328 327 mximum n minimum von Mises stresses inrese s the sffol moulus inrese. Beuse the moulus inrese i not 330 329 eliit hnge in the sptil istriution of stress, ut i roen 33 the rnge of the von Mises stresses, this implie tht the stress 332 grients insie the ell lso inrese. 333 An inrese in sffol moulus use n inrese in the 334 mximum von Mises stress oth in the ell omin Figs. 6 n 335 6 n t the ell-mtrix ounry Figs. 6 n 6, irrespetive of the PCM. Interestingly, with the PCM, this inrese 337 336 epene on the ell size Figs. 6 n 6, wheres without 338 the PCM, the ell size i not ffet these vlues Figs. 6 n 339 6. The simultions showe tht lrger ell with the sme 340 PCM thikness evelope higher stresses uner the sme presrie isplements uring loing. Therefore, thiker PCM 342 34 woul e require to result in the sme effetive stress shieling 343 for lrger ell. This ws importnt euse the ell size, whih 344 ws ffete y the iotivity of the sffol 3, my pt in 345 response to its surrouning n therey nee to synthesize itionl PCM, even though it is sujete to the sme mehnil 347 346 strin uner loing. Aoring to these simultions, ell emee insie sffol will initilly e sujete to stresses 349 348 without eing ffete y the reltive ell size. However, one the 350 Journl of Biomehnil Engineering APRIL 20, Vol. 33 / -5
Simultions With PCM Von-Mises Stress, E Sffol = [2 kp] [P] Simultions Without PCM Von-Mises Stress, E Sffol = [2 kp] [P] ell egins to synthesize PCM, the stresses t the ell-mtrix ounry will inrese ue to the inrese stiffness of the PCM reltive to tht of the sffol. On the other hn, the stress shieling effet of the PCM will use the overll stresses in the ell to erese when the rius of the ell is 5 m or 7.5 m. These internl stresses epen on ell size; if the ell rius is 0 m the stresses without PCM re slightly lower thn with PCM, mening tht in this se, the PCM with thikness of 2.5 m is not suffiient to use stress shieling. Higher stresses in the ell my stimulte itionl PCM synthesis s to use higher stress shieling effets. Interestingly, our previous experimentl results onur tht ells hve ifferent PCM sizes, whih my e orrelte with the ell size 4,24 Figs. n e. 3.2 Cell Size n Cell Deformtion. Compression of the enpsulting mtrix evokes signifint ell eformtions, s quntifie y the reltive ell rius hnge RRC prmeter Fig. 7. RRC prmeter ws efine s the eforme rius ivie y the initil rius. Compressive eformtion inrese the RRC prmeter in the r iretion n erese it in the z iretion. The ell eformtion inrese inrese r-rrc n erese z-rrc with n inrese in the sffol moulus, irrespetive of the PCM. With the PCM, the initil ell rius ffete the RRC, oth in the z n r iretions Figs. 7 n 7. Aoring to these simultions, ell with smller initil rius unergoes Von-Mises Stress, E Sffol = [0 kp] [P] Von-Mises Stress, E Sffol = [0 kp] [P] Fig. 4 Sffol moulus effets on the von Mises stresses with eformtions: n with PCM n n without PCM, for sffol moulus of n 2 kp n n 0 kp. Eh set of mthing figures hve the sme sle n olor r to simplify omprison. PCM prmeters re E=40 kp, =0.04, ell prmeters re E=3.2 kp, =0.4. 35 352 353 354 355 356 357 358 359 360 36 362 363 364 365 366 367 368 369 370 37 372 373 374 smller eformtions in oth iretions ompre with lrger 375 ell. Without PCM, the reltive ell rius in the z iretion erese s the sffol moulus inrese, though the influene of 377 376 the initil ell rius ws more evient t higher sffol moulus Fig. 7. A lrger ell use less eformtion in the z 379 378 iretion when mesure reltive to the originl rius. Without 380 the PCM, the ell rius in the r iretion inrese s the sffol 38 moulus inrese, ut hrly ny ifferene ws note etween 382 the ifferent initil ell rii Fig. 7. 383 In ells with initil rii of 5 m n 7.5 m, the PCM 384 use strin shieling irrespetive of the sffol moulus. 385 Strin shieling ws note when the overll eformtion of the 386 ell ws reue, ompre with the se without the PCM Fig. 387 7. In the r iretion only, there ws lso strin shieling for ells 388 hving n initil rius of 0 m. Interestingly, t tht ell rius, 389 the presene of 2.5 m PCM lyer i not reue the reltive 390 eformtion in the z iretion, ut rther, slightly inrese it 39 when ompre with the equivlent sffols without PCM. 392 3.3 Constitutive Eqution Effets. The vrious onstitutive 393 equtions evlute showe tht the moel use in the simultion 394 n gretly ffet the preite stresses n strins within the 395 suomins. In generl there ws similr tren etween the 396 ifferent simultions in eh moel type; however, ifferenes in 397 the preite mximum vlues mong the moels were more thn 398-6 / Vol. 33, APRIL 20 Trnstions of the ASME
With PCM Without PCM E Sffol = [0 kp] [P] E Sffol = [0 kp] [P] E Sffol = [2 kp] [P] 3.4 Effets of Cell Moulus. When etermining whih prmeters to use in our moels se on reent pulitions in the relevnt literture, we note tht rnge of more thn one orer of mgnitue hs een reporte for ell moulus, epening on the E Sffol = [2 kp] [P] Fig. 5 Von Mises stresses within the ell spe n with n n without PCM for the ses tht the sffol moulus ws n 0 kp n n 2 kp. Chnges n e note in oth sptil istriution n mgnitue. 399 40% in some ses. The MNH moel preite the highest 400 stresses Figs. 8 n 9 n the lrgest RRC exept in the z 40 iretion without PCM Fig. 0. Assuming lrge eformtion 402 with smll strins n thus using isotropi onstitutive equtions 403 for lrge eformtions woul hve resulte in very lrge unerestimtion of the stresses n most of the ell eformtions. In- 404 405 terestingly in the z iretion without the PCM, this moel preite the smllest eformtions with the softer sffol mterils, 406 407 ut ultimtely preite the lrgest eformtion for the stiffer mterils. In the MNH simultion only the sffol ws moele s 408 409 MNH mteril, while the PCM n ell were ssume to e neo- 40 Hooken. This smll moifition resulte in notiele stress n 4 strin hnges throughout the ifferent suomins. Consequently, if we ssume isotropi mteril ehvior with smll e- 42 43 formtions, this woul result in loser preition to the MNH 44 thn with the isotropi lrge eformtion theory for the stresses 45 n strins. 46 47 48 49 ell origin n mesurement tehnique 22,27,28,32 34. For 420 most of the simultions, we hose ell moulus of 3.2 kp, 42 vlue estimte y Knight et l. 27 from ompression tests in 422 lginte. The effet of ell moulus on simultion ws lso onsiere, n simultions were thus performe with ifferent vl- 424 423 ues rnging from 0.35 kp to 4 kp Fig.. In the se where 425 PCM ws present, n low sffol moulus of 2 kp ws 426 hosen, hnging the ell moulus resulte in reltively liner 427 hnge in stresses n strins. The stresses in the sffol were 428 only slightly reue 88 82 kp y the inrese in ell moulus. With the inrese in ell moulus, the stresses in the ell 430 429 inrese, while the stresses in the PCM erese Figs., 43 f, n g. Deformtion in the z n r iretions erese 432 with the inrese in ell moulus Figs. n. The 433 mximum von Mises stress on the ell-pcm ounry lso erese with the inrese in ell moulus Fig. e. In this se, 435 434 ll the mximum vlues hnge in ner-liner pttern. 436 3.5 Effets of the PCM. The effet of PCM synthesis ws 437 lso moele using FEM nlysis. This ws simulte y inresing the moulus of the PCM suomin 0 40 kp while reu- 439 438 ing Poisson s rtio 0.499 0.04 Fig. 2. The stresses in the 440 Journl of Biomehnil Engineering APRIL 20, Vol. 33 / -7
Mx Von Mises Stress in Cell Domin [N/m 2 ] 800 700 Mx Von Mises Stress on Cell-PCM Bounry [N/m 2 ] 5 0 4 4000 3 3000 2 600 400 300 Simultions With PCM Cell Rius 5µm Cell Rius 7.5µm Cell Rius 0µm 200 E Sffol [KP] Mx Von Mises Stress in Cell Domin [N/m 2 ] 800 750 700 650 600 550 450 Simultions Without PCM Cell Rius 5µm Cell Rius 7.5µm Cell Rius 0µm 400 E Sffol [KP] E Sffol [KP] Cell Rius 5µm Cell Rius 7.5µm Cell Rius 0µm Mx Von Mises Stress on Cell-Sffol Bounry [N/m 2 ] 800 600 400 200 800 600 400 E Sffol [KP] Cell Rius 5µm Cell Rius 7.5µm Cell Rius 0µm AQ: # Fig. 6 Mximum von Mises stresses in the n ell suomin n n on the ell-mtrix ounry s funtion of sffol moulus in the se of three ifferent ell sizes, efore n fter the PCM ws rete Simultions With PCM Reltive Cell Rius Chnge Simultions Without PCM Reltive Cell Rius Chnge Fig. 7 The effet of sffol moulus on the reltive ell rius hnge RRC, efine s the eforme rius ivie y the initil rius, in the z n r iretions, with n without PCM, for three ell sizes -8 / Vol. 33, APRIL 20 Trnstions of the ASME
MxVonMisesStress[N/m 2 ] 9000 8000 7000 6000 0 4000 3000 Simultions With PCM Sffol-Moife Neo-Hooken PCM-Moife Neo-Hooken Cell-Moife Neo-Hooken Sffol-Isotropi, Smll eformtion PCM-Isotropi, Smll eformtion Cell--Isotropi, Smll eformtion Sffol- Neo-Hooken PCM- Neo-Hooken Cell- Neo-Hooken Sffol-Isotropi, Lrge eformtion PCM-Isotropi, Lrge eformtion Cell-Isotropi, Lrge eformtion MxVonMisesStress[N/m 2 ] 2 Simultions Without PCM Sffol-Moife Neo-Hooken Cell-Moife Neo-Hooken Sffol-Isotropi, Smll eformtion Cell-Isotropi, Smll eformtion Sffol- Neo-Hooken Cell- Neo-Hooken Sffol-Isotropi, Lrge eformtion Cell-Isotropi, Lrge eformtion 0 E Sffol [KP] 0 E Sffol [KP] Fig. 8 The effet of ifferent moeling tehniques on the mximum von Mises stress in the ifferent suomins s funtion of sffol moulus, with n without PCM. Cell rius ws tken s 5 m. 44 442 443 444 445 446 PCM suomin inrese with the inrese in PCM moulus Fig. 2. This susequently use erese in the mximum stresses rehing the ell suomin. On the other hn, the inrese moulus of the PCM use n inrese in the stresses t the ell-pcm ounry, prtiulrly t the lterl sie of the ell Fig. 2 where the mximum stresses in the PCM lso our. 447 3.6 Implitions on Mehnotrnsution. Signling meite 448 449 450 45 452 453 454 through integrin hesion reeptors regultes severl hon- royte funtions inluing ifferentition, viility, mtrix remoeling, n responses to mehnil stimuli 35. Integrins, primrily integrins, re responsile for honroyte tthment to the ECM. Iniviul integrin-firinogen ons hve imeter of 5 nm 36 n hve een evlute s hving rupture fore of f=00 pn 37,38. Clulting the verge single on rupture norml stress, 4f/ 2 =imeter, yiels vlue of 566 kp. 455 Integrin-fironetin on fores hve een evlute s 20 pn 456 39, therefore resulting in lulte norml rupture stress of 457 3 kp. These evlute stresses re higher thn the simulte 458 mximum stresses t the ell-mtrix ounry, whih o not exee 5 kp for ll the simultions performe. It shoul e note 460 459 tht ompressive stresses n rise up to 0 20 MP within the 46 joint 40, whih is orers of mgnitue higher thn the stresses 462 we report; however, ntive ult rtiulr rtilge lso hs 463 muh higher moulus when ompre with the PEG-se hyrogels use s sffol for the tissue engineere onstruts. 465 464 Reent experiments showe tht hrveste humn honroytes, whih were seee on rtilge tissue, ethe t sher 467 466 stresses of 70 P, 80 h postseeing 4. In this se, integrin 468 Mx Von Mises Stress on Cell-PCM Bounry [N/m 2 ] 0 4 4000 3 3000 2 Moife Neo-Hooken Simultions With PCM Isotropi, Smll eformtion Neo-Hooken Isotropi, Lrge eformtion E Sffol [KP] Mx Von Mises Stress on Cell-Sffol Bounry [N/m 2 ] 800 600 400 200 800 600 400 Moife Neo-Hooken Simultions Without PCM Isotropi, Smll eformtion Neo-Hooken Isotropi, Lrge eformtion 200 E Sffol [KP] Fig. 9 The effet of ifferent onstitutive moels on the mximum von Mises stress t the ell-mtrix ounry s funtion of sffol moulus, with n without PCM. Cell rius ws tken s 5 m. Journl of Biomehnil Engineering APRIL 20, Vol. 33 / -9
z iretion 0.95 0.9 0.85 Reltive Cell Cell Dimeter Rius Chnge Chnge - With PCM With PCM Moife Neo-Hooken Isotropi, Smll eformtion Neo-Hooken Isotropi, Lrge eformtion 0.8 ziretion 0.95 0.9 0.85 0.8 Reltive Cell Cell Dimeter Rius Chnge Chnge - Without Without PCM PCM Moife Neo-Hooken Isotropi, Smll eformtion Neo-Hooken Isotropi, Lrge eformtion 0.75.4..08.2 r iretion.06.04 riretion..08.02.06.04 ESffol[KP] ESffol[KP] Fig. 0 The effet of ifferent onstitutive moels on the reltive ell rius hnge RRC, efine s the eforme rius ivie y the initil rius, in the z n r iretions s funtion of sffol moulus, with n without PCM. The initil ell rius ws tken s 5 m. Suomin Type E [kp]! Chonroyte NH 0.35-4 0.4 3000 Sffol 2 PCM Cell f PCM NH 40 0.04 Sffol MNH 2 0.499 g MxVonMisesStress[N/m 2 ] e Mx Von Mises Stress on Cell-PCM ounry [N/m 2 ] 480 460 440 420 400 380 360 340 320 300 280 0 0.5.5 2 2.5 3 3.5 4 ECell[KP] z iretion r iretion 0 0 0.5.5 2 2.5 3 3.5 4 E Cell [KP] Reltive Cell Dimeter Cell Rius Chnge Chnge 0.95 0.945 0.94 0.935 0.93 0.925 0 0.5.5 2 2.5 3 3.5 4.026.025.024.023.022 0 0.5.5 2 2.5 3 3.5 4 ECell[KP] Fig. The effet of ell moulus on mximum von Mises stresses in the suomins, reltive ell rius hnge, efine s the eforme rius ivie y the initil rius, in the z iretion, reltive ell rius hnge. In the r iretion, n e mximum von Mises stress t the ell-pcm ounry prmeters use re summrize in. The sptil von Mises stresses n eformtion re emonstrte for the se of ell moulus of f 0.35 kp n g 4 kp. -0 / Vol. 33, APRIL 20 Trnstions of the ASME
MxVonMisesStress[N/m 2 ] vpcm 0.499 9000 0.42 0.35 0.27 0.9 0.2 0.04 Sffol 8000 PCM Cell 7000 6000 0 4000 3000 0 0 5 20 25 30 35 40 EPCM[KP] Von-Mises Stress on Cell-Mtrix Bounry [P] 0 4 4000 3 3000 2 E PCM 0 [kp], v PCM 0.499 E PCM 3 [kp], v PCM 0.454 E PCM 6 [kp], v PCM 0.408 E PCM 9 [kp], v PCM 0.362 E PCM 22 [kp], v PCM 0.36 E PCM 25 [kp], v PCM 0.27 E PCM 28 [kp], v PCM 0.224 E PCM 3 [kp], v PCM 0.78 E PCM 34 [kp], v PCM 0.32 E PCM 37 [kp], v PCM 0.086 E PCM 40 [kp], v PCM 0.04 0 2 3 4 Ar-length 5 6 7 8 x0-6 Fig. 2 PCM seretion effets on the mximum von Mises stresses in the ifferent suomins n the von Mises stresses on the ell-mtrix ounry plotte oring to the r length 469 470 47 472 473 474 475 476 477 478 479 480 48 482 483 484 485 486 487 488 489 490 49 492 493 494 495 496 497 498 499 50 502 503 504 505 506 507 508 509 50 5 52 53 ons re roken ue to high sher stress levels. In our simultions, ll von Mises stresses t the ell-mtrix interfe were preite to e in the rnge etween 330 P n 0 P, therefore exeeing the experimentl effetive sher stress reporte y Kurtis et l. 4, ut lower thn the lulte on rupture stress. Therefore, the possiility tht integrin on rupturing ws involve in the ifferenes in ell mehnotrnsution ws not supporte y the moel preitions. 3.7 Limittions n Other Consiertions. It shoul e note tht the present moels hve severl limittions ue, in prt, to simplifying ssumptions. In the experiments, honroytes emee in the hyrogels sometimes exhiite nonspheril geometry, off-xis lignment, lustering of ells surroune y single PCM, n uneven PCM thikness. The ssumption of perfet sphere ws me primrily to enle us to employ n xisymmetril simplifition. This is unlike mture helthy rtilge, where the ells re firly well ligne with the iretion of tissue epth 42 n where the ell s spet rtio n e tken into ount in the efinition of the xisymmetry simultions 43. Although, initilly, the sffol n e onsiere isotropi, with the inlusion of ells n synthesize PCM, the mteril eomes inresingly nisotropi. In ition, enzymes egre the sffol in iffusion epenent mnner wy from the ell ounry; wheres hyrolysis egres the sffol in ulk. For simpliity, we ssume isotropy euse of this highly omplite egrtion profile within the hyrogel. Unfortuntely, given the hosen omin size, whih ws itte y the seeing ensity, prt of the solutions exhiite presene of ounry lyers ner the omin eges. One of the inherent ssumptions ws tht perturtions in the eformtion ue to the inlusion of the ell vnish in the fr fiel ege of the omin. Therefore, when efining the suomin ounry onition, the presene of suh itionl ounry lyers my prelue this si ssumption. These ouple effets ner the ounries of the sffol omin were intensifie s the ell size ws inrese n the ell-to-sffol volume frtion inrese. Inresing the ury of the simultions woul neessitte in this se more omplex representtion of lrger omin inluing more ells, whih is eyon the sope of this stuy. As mentione efore, the visoelsti properties of the ell n PCM were neglete s were oy fores. Crtilge hs een previously moele s iphsi mteril 44. This present stuy iffers in two mjor respets from the work of Guilk n Mow 8 : first, the rpi moe of loing n, seon, the ft tht the meium simulte is loser to hyrogel thn to rtilge. In this respet our experimentl mesurements hve showe tht the mehnil ehvior of the hyrogels ws very 54 lose to tht of n elsti mteril n for this reson monophsi moel ws juge suffiient for this stuy. The moe of lo- 56 55 ing for whih the simultions were me in this stuy is onsiere in rtiulr rtilge to e short-term eformtion n is 58 57 thought to e ssoite with rpi movement n rerrngements 59 of the tissue uner onstnt volume 45. This is onsierly 520 ifferent from long-term eformtion, whih involves the movement of flui reltive to the ollgen network through the very 522 52 fine pores forme y the proteolgyln moleules 46. Due to the 523 ssumptions of homogenous monophsi mterils n negligile 524 ounry effets, multisle moel eme unneessry n ws 525 not implemente. Moreover, honroytes hve een reporte to 526 e sensitive to the type of loing onitions: Different loing 527 frequenies hve een shown to ffet glyominoglyns 528 GAGs synthesis wheres stti loing hs een shown to hve 529 negtive effet on GAG synthesis 3. It woul pper tht 530 more omplex FEM shoul e evelope for this type of system, 53 tking into ount the ove experimentl finings. The ell, 532 PCM, n sffol were efine not to llow ny slippge or 533 reltive movement, n onstnt tthment etween the suomins ws ssume. Cell tthment to its surrouning is elieve 535 534 to our t istint fol ontt points. Thus, iotive n permissive sffols use in our experimentl stuy were elieve to 537 536 hve evoke very ifferent fol hesion profiles. It woul e 538 interesting to investigte how ifferent ounry onitions etween the ell n its surrouning ffet the mehnil stimuli 540 539 rehing the ell, ut this is eyon the sope of the urrent report. The PCM, while eing synthesize, hnges its mehnil 542 54 properties n geometry. In this work we only resse the 543 hnge in mehnil properties. Further work shoul inlue the 544 investigtion of the effet of the hnge in the PCM thikness, 545 euse notiele hnges were oserve in experimentl finings 24. 546 547 4 Conlusions 548 A numeril FE moel ws evelope n use to simulte 549 tests tht were previously performe in unonfine ompression 550 uner vrious onfigurtions 4. The stresses n eformtions 55 of honroytes within ifferent sffol types were preite for 552 the pek phse of yli unonfine ompression loing. Four 553 ifferent moeling methos were ompre to evlute their effets on the preite vlues. The simultions presente in this 555 554 stuy emonstrte epenene of the preite stresses in the 556 ellulr environment on the ifferent onstitutive moels use for 557 Journl of Biomehnil Engineering APRIL 20, Vol. 33 / -
558 559 560 56 562 563 564 the three ifferent suomins: ell, PCM, n sffol. In ition, it ws emonstrte tht the mteril properties of the sffol n PCM, s well s ell size influene stress n strin istriutions. The numeril simultions showe tht the PCM plys ritil role in regulting the iomehnil signls rehing the ell, oth in mgnitue n in sptil istriution. Interestingly, the effets of ell size prove signifint only one the 565 PCM ws intt. These simultion results provie further insight 566 into the vritions in the metoli responses previously reporte 567 y our group, foun with ifferent PEG-se ell-seee hyrogel onstruts sujete to the sme mehnil stimultion. 568 569 Aknowlegment 570 The uthors grtefully knowlege the finnil support of the 57 Unite Sttes-Isrel Bintionl Siene Fountion Grnt No. 572 200384 n the Russell Berrie Nnotehnology Institute - Tehnion. The uthors grtefully knowlege the useful isussions 573 574 with M. Ruin. Referenes 575 576 577 578 579 Wng, N., Butler, J. P., n Inger, D. E., 993, Mehnotrnsution Aross the Cell Surfe n Through the Cytoskeleton, Siene, 260 5, pp. 24 27. 2 Bushmnn, M. D., Gluzn, Y. A., Grozinsky, A. J., n Hunziker, E. B., 580 995, Mehnil Compression Moultes Mtrix Biosynthesis in 58 582 583 584 585 586 587 588 589 590 59 592 593 594 595 596 597 598 599 600 60 602 603 604 605 606 607 608 609 60 6 62 63 64 65 66 67 68 69 620 62 622 623 624 625 626 627 628 629 630 63 632 633 634 635 636 637 Chonroyte/Agrose Culture, J. Cell Si., 08 4, pp. 497 508. 3 Lee, D. A., n Ber, D. L., 997, Compressive Strins t Physiologil Frequenies Influene the Metolism of Chonroytes Seee in Agrose, J. Orthop. Res., 5 2, pp. 8 88. 4 Free, L. E., Vunjk-Novkovi, G., n Lnger, R., 993, Cultivtion of Cell-Polymer Crtilge Implnts in Bioretors, J. Cell. Biohem., 5 3, pp. 257 264. 5 Muk, R. L., Seyhn, S. L., Ateshin, G. A., n Hung, C. 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