The Concepts and Applications of Fractional Order Differential Calculus in Modelling of Viscoelastic Systems: A primer

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1 The Concep nd Applicion of Frcionl Order Differenil Clculu in Modelling of Vicoelic Syem: A primer Mohmmd Amirin Mlob, Youef Jmli,2* Biomhemic Lborory, Deprmen of Applied Mhemic, Trbi Modre Univeriy, Irn 2 Compuionl phyicl Science Reerch Lborory, School of Nno-Science, Iniue for Reerch in Fundmenl Science (IPM), Tehrn, Irn Vicoeliciy nd reled phenomen re of gre impornce in he udy of mechnicl properie of meril epecilly, biologicl meril. Cerin meril how ome complex effec in mechnicl e, which cnno be decribed by ndrd liner equion (SLE) moly owing o hpe memory effec during deformion. Recenly, reercher hve been pplying frcionl clculu in order for probing vicoeliciy of uch meril wih high preciion. Frcionl clculu i powerful ool for modeling complex phenomenon. In hi uoril bed pper, we ry preen cler decripion of he frcionl clculu, i echnique nd i implemenion. The inenion i o keep he deil o minimum while ill conveying good ide of wh nd how cn be done wih hi powerful ool. We ry o expoe he reder o he bic echnique h re ued o olve he frcionl equion nlyiclly nd/or numericlly. More pecificlly, modeling he hpe memory phenomen wih hi powerful ool re udied from differen perpecive, well preened ome phyicl inerpreion in hi ce. Moreover, in order o how he relionhip beween frcionl model nd ndrd liner equion, frcl yem compriing pring nd dmper elemen i conidered, nd he coniuive equion i pproximed wih frcionl elemen. Finlly, fer brief lierure review, wo frcionl model re uilized o inveige he vicoeliciy of he cell, nd he comprion i mde mong hem, experimenl d, nd previou model. Verificion reul indice h no only doe he frcionl model mch he experimenl d well, bu i lo cn be good ubiue for previouly ued model. Keyword: Vicoeliciy, Frcionl clculu, Mechnicl properie, Cell biomechnic, Frcionl modeling, Frcl yem. Auhor Summry Frcionl Clculu i new powerful ool which h been recenly employed o model complex biologicl yem wih non-liner behvior nd long-erm memory.in pie of i compliced mhemicl bckground, frcionl clculu cme ino being of ome imple queion which were reled o he derivion concep; uch queion while he fir order derivive repreen he lope of funcion, wh hlf order derivive of funcion revel bou i? Finding nwer o uch queion, cieni mnged o open new window of opporuniy o mhemicl nd rel world, which h rien mny new queion nd inriguing reul. For exmple, he frcionl order derivive of conn funcion, unlike he ordinry derivive, i no lwy zero. In hi uorilbed pper i i ough o nwer he foremenioned queion nd o conruc comprehenive picure of wh frcionl clculu i, nd how i cn be uilized for modeliion purpoe. The focu of he reerch h been on vicoelic meril. Afer n exenive lierure review of he concep nd pplicion of hi poen ool, novel pplicion of hi ool i developed for imuling dynmic yem in order o inveige he mechnicl behvior of cell. *Correponding uhor. Tel: E-mil ddree: y.jmli@ipm.ir (Y.Jmli), m.mirinmlob@modre.c.ir (M.Amirin Mlob).

2 Conen Inroducion... 3 Mehod... 3 Frcionl clculu... 3 Definiion of frcionl clculu... 6 The inerpreion of frcionl clculu... 9 Lplcen inerpreion... 9 Anlyicl nd numericl mehod... Anlyicl mehod... Numericl oluion... 3 Vicoelic yem... 4 Creep e... 5 Periodic e... 6 Frcionl vicoelic model... 7 Periodic rin... 2 Relion beween frcionl equion nd SLE... 2 Applicion Two-dimenion frcionl nework model Dicuion Appendix Acknowledgmen Reference

3 Inroducion The concep of derivive i he min ide of clculu. I how he eniiviy o chnge of funcion i.e. he re or lope of quniy. The curren definiion of derivive w uggeed by Newon in 666. Newon wih phyicl viewpoin of derivive inerpreed he innneou velociy []. The inuiion of reercher from derivive nd inegrl i bed on heir geomericl or phyicl mening, e.g. he fir nd econd order derivive of diplcemen i clled velociy nd ccelerion repecively (lo jerk nd jounce for 3h nd 4h derivive). A well, inegrl of curve funcion men he re under he curve. Thi form of clicl clculu w developed exenively over four cenurie. Tody, cieni re ble o decribe nd model mny phyicl phenomen wih n ordinry differenil equion [2]. In mny ce, however, he clicl clculu i no ble o decribe excly hee complex phenomen. Under mll deformion, for exmple, he relionhip beween he force nd diplcemen in n idel pring (mll deformion in elic meril) i liner, i.e. force i reled o he zero derivive of diplcemen, while in n idel dmper, he force i proporionl o he velociy of exenion or compreion. In oher word, force i reled o he fir derivive of deformion. Which lw, however, govern he meril wih inermedie mechnicl properie (i.e. beween idel pring nd idel dmper)? Wih hi im, mny reercher ry o nwer he queion nd o model well o nlye he mechnicl behvior of hee non-liner yem by men of frcionl clculu [3]. Frcionl clculu i generlizion of ordinry differeniion nd inegrion o rbirry non-ineger order, bu wih hi definiion, mny inereing queion will rie; for exmple, if he fir derivive of funcion give you he lope of he funcion, wh i he geomericl mening of hlf derivive? In hlf order, which operor mu be ued wice o obin he fir derivive? The erly hiory of hi queion bck o he birh of frcionl clculu in 695 when Gofried Wilhelm Leibniz uggeed he poibiliy of frcionl derivive for he fir ime ] 2[. In hi ricle, we im o inroduce frcionl clculu new ool for modeling he complex yem, epecilly vicoelic meril. Fir, we briefly dicu he bic concep of frcionl clculu nd explin he eenil ep of he frcionlizion lgorihm. Nex, we preen n inerpreion of frcionl derivive nd elbore upon how frcionl equion could be olved nlyiclly. Then, we briefly look he modeling of vicoelic yem by he help of hi pproch. Ulimely, fer overviewing ome recen work, we preen n pplicion of he pproch in modeling biomechnicl properie of cell nd indice h he propoed model predic he cell behvior much beer hn he previou pring-dhpo model, well he model oupu re in good greemen wih experimenl d. To um, we re going o give he minimum need o ge reder fee we, o h reder cn quickly ge ino building frcionl clculu model for complex yem. Mehod Frcionl clculu In mhemic, mny complex concep developed from imple concep. For exmple, we cn refer o he exenion of nurl number o he rel one in ome mhemicl formule. Le' give n exmple o clrify: he fcoril of non-negive ineger n, denoed by n!, i he produc of ll poiive ineger le hn or equl o n. On he oher hnd, here i concep which nmed Gm funcion nd defined follow: Γ(x) = z e dx () 3

4 One propery of he funcion for n R + i Γ(n + ) = nγ(n) (2) Hence, hi funcion i equl o fcoril for he ineger number. A reul, he gmm funcion could be conidered n exenion of fcoril funcion o rel number. For innce, ccording o he bove formlim, fcoril of /2 cn be obined follow: ( 2 )! = Γ (3 2 ) = 3 2 Γ ( 2 ) = 3 π 2 Indeed, ccording o Wikipedi, he gmm funcion cn be een he oluion o find mooh curve h connec he poin (x, y) given by y = (x )! he poiive ineger vlue for x. Le ue hi pproch o exend he concep of derivive o non-ineger order; conider n h derivion of power funcion g(x): d n k! g(x) = dxn (3) g(x) = x k. x (4) (k n)! xk n = Γ( + k) Γ( + k n) xk n (5) which k nd n re rel ineger number repecively, nd k n. To generlize he bove equion, i could be poible o exend he ineger number n o rel vlue which nmed α: d α Γ( + k) g(x) = dxα Γ( + k α) xk α (6) Then for frcionl derivive of n rbirry funcion, expnd he funcion in power erie of x fir, nd hen by uing eq. 5, derive he expnion. For exmple, for derivive f(x) = e kx o α order, we rewrie f(x) funcion follow: 2 3 x x f ( x ) x 2! 3! (7) Hence [4]: d dx f x x x x ign xign xk e 2 3 kx (8) 2 kx, Th Γ( α, kx) i incomplee gmm funcion (dicued in Box 3). Of coure, hi i n rbirry wy o define frcionl derivive nd no he only wy; for exmple, i i poible o ue n exponenil funcion f(x) = e kx ined of power funcion, i.e. we cn define D α f(x) = k α e kx (9) The frcionl derivive of he exponenil funcion obined by Liouville in 832, nd he frcionl derivive of power funcion go by Riemnn in 847 [4]. By compring Eq. 8 wih 9, i i noiced h when he order i n ineger, he reul re me, bu he non-ineger order hey re differen! In fc, conrry o he ineger-order derivive in which he definiion nd he oupu under hee operor re me nd unique in he frcionl derivive, under differen operor, he reul i no he me nd no necerily unique. In oher word, here re muliple definiion for frcionl derivive nd ll of hem re mhemiclly correc. From he phyicl perpecive, ech definiion h i own pplicion nd inerpreion. 4

5 Frcionl derivive h mny inereing nd couner-inuiive properie; for exmple, he Riemnn derivive (Eq. 5) of conn i no zero! h i, d α dα dxα con = dx α x = Γ( α) x α () Puing i ll ogeher, here re mny wy o define frcionl derivive, provided h ech definiion pproch o ordinry derivive in he ineger order limi hi mehod known he frcionlizion lgorihm i.e.: lim f ( x ) n f ( x ), n,,... n Where Δ repreen n rbirry operor. A well, wo more exmple provided in Box which hve lo been ued in he following ecion. Box In hi box, bic definiion of boh he ordinry inegrl nd he ordinry derivive i preened by wo exmple, well i exenion o frcionl operor i propoed by he help of frcionlizion lgorihm (ref ecion 2); reuling we obin wo imporn frcionl funcion. Exmple. Auming f(x) [, b]. The fir order derivive of funcion f(x) i defined follow: () df f ( ) f ( h) f ( ) lim (B.) d h h by pplying derivive operor on he bove equion, we ge: 2 (2) d f f ( ) 2 f ( h) f ( 2 h) f ( ) lim (B.2) 2 h 2 d h by he help of inducion nd bove procedure, we gin: n n ( n) d f rn f ( ) lim ( ) f ( rh) 2 h n (B.3) d h r r Where b, h =, nd (n n r ) = n(n )(n 2) (n r+). r! Since he equion (B.3) hold for ll n N, we expnd n order o α R by he frcionlizion lgorihm; hereby chieving relly imporn equion known frcionl Grunwld-Lenikov differinegrl menioned in ecion 2. ( ) r fn ( ) lim ( ) f ( rh) h h r r (B.4) Exmple 2. Conider he following equion which i known he inegrl operor of order n: We cn rewrie i follow ccording o he Cuchy equion [6]: xn xn x n D f f ( x )dx dx n (B.5) Since he bove equion hold for ll n N, we pply he frcionlizion lgorihm o expnd he obined equion o frcionl equion. In oher word, if n α, we will ble o gin n inegrl known Riemnn-Liouville. Hence, i eem h frcionl clculu i n expnion of ineger clculion. x n ( ) ( ) n D f x f d ( n )! (B.6) 5

6 Definiion of frcionl clculu A menioned in he previou ecion, he frcionl order derivive i no necerily unique; in hi regrd, here re ome cceped nd common definiion in he lierure. In he following, we menioned number of imporn one. Grünwld Lenikov derivive Grünwld Lenikov derivive i bic exenion of he nurl derivive o frcionl one, which derived in Box (Eq. B.4). I w inroduced by Anon Krl Grünwld in 867, nd hen by Alekey Vilievich Lenikov in 868. Hence, i i wrien [5]: m D f ( ) lim ( ) f ( mh) h h m m () where n N, nd he binomil coefficien i clculed by he help of he Gmm funcion. ( )( 2)...( m ) m m! )2( Riemnn-Liouville frcionl derivive Riemnn-Liouville frcionl derivive cquiring by Riemnn in 847 i defined follow. d d d n f ( x) ( ) dx,( n [ ], ) n ( n ) d ( x) RL n ( n) D f ( ) ( ) ( D ) f ( ) (3) where α > ; hi operor i n exenion of Cuchy inegrl (Box ) from he nurl number o rel one. Bed on he frcionlizion lgorihm, i eem logicl o rech relion (3) by n order derivive of he Eq. B.6 In ddiion, ccording o he bove relion, if < α < hen he Riemnn-Liouville operor reduced o RL d f ( x) D f () dx ( ) d (4) ( x) I i worh noing h hi relion i he me of Eq. 5. [5], nd lo RL, α,, nd re he bbreviion of Riemnn-Liouville, frcionl order, he lower nd upper bound of he bove inegrl repecively. Cpuo derivive Since Riemnn-Liouville frcionl derivive filed in he decripion nd modeling of ome complex phenomen, Cpuo derivive w inroduced in 967 [6]. The Cpuo derivive of frcionl order α (n α < n ) of funcion f() defined D f( ) d ( n ) (5) ( ) n C D f () n 6

7 where D n i n h derivive operion, nd C repreen he Cpuo word. In oher word, bed on box 2, i cn be found h he Cpuo derivive i equl o he Riemnn Liouville inegrl of n h derivive of funcion. I i worh noing gin h he behvior of ll of he frcionl derivive, when he order i ineger re me; i, however, could be differen in he non-ineger order; for exmple, in he non-ineger order, he Cpuo derivive of conn, unlike he Riemnn Liouville derivive i zero. There re more frcionl derivive for more deiled informion, he inereed reder i referred o [4, 5, 7]. Since we ry o nlye nd o inveige vicoelic yem vi he Riemnn-Liouville nd Cpuo frcionl derivive, he min focu of he ricle i on hee wo ype of derivive. A previouly menioned, differen definiion for frcionl derivive wih he differen properie cn be propoed, which ll of hem re vlid nd mhemiclly ccepble. However, he min queion i which relion hould be pplied in modeling of pecific phenomenon? In oher word, which definiion would be more pproprie for pecific problem? A rule of humb, ince hey end o inerpre nurl phenomen, he definiion which i more conien wih he experimenl reul hve more privilege hn he oher frcionl definiion. BOX 2 In hi box, ome properie of frcionl Riemnn-Liouville differinegrl, uch commuive propery, diribuive lw, nd o forh re inveiged 2. Riemnn Liouville inegrl nd derivive To find profound undernding of Riemnn Liouville inegrl nd derivive, ome of he mo crucil properie of hi operor re menioned in he following. Lemm. Auming rbirry funcion f(x) nd m, n he following equion hold 3.. Semi-group propery: m n m n I I f I f 2. commuive propery: m n n m I I f ( x ) I I f ( x ) Lemm 2. Le f nd f 2 re wo funcion on [,b] well c, c 2 R, n >, nd m > n. Regrding hee, he following equion hold 4 :. Lineriy rule: RL D n ( 2) RL n RL n 2, RL n ( ) RL n f f D f D f D c f c D ( f ) 2. Zero rule: D f = f 3. Produc rule: q D ( fg ) D ( f ) D ( g ) RL q RL j RL q j j j 4. In he generl, emi-group propery doe no hold for Riemnn-Liouville frcionl derivive. Indeed, he following equion i no lwy rue. RL RL b RL b D D f D f 2 All he menioned properie in hi box hold lmo everywhere (.e) on [, b] nd lo, if (X, Σ, μ) i meure pce, quliy P i id o hold lmo everywhere in X if μ({x X: P(x)}) =. Alo, inegrl operor on inervl [, b] i defined I b. 3 f L [, b] where L {f: [, b] R; f i meurble on [, b] nd f(x) dx < }. b 4 Provided h D n f nd D n f 2 lmo everywhere define. 7

8 NB: To prove he bove lemm ee [8]. Cpuo derivive In hi pr, ome fundmenl properie reled o Cpuo operor re repreened bed on ref [5, 8]. Le f i enough differenible funcion, c, c 2 R, nd m > n. Conidering hee,. Cpuo derivive i he lef invere of Riemnn-Liouville inegrl. C n n DI f f Diribuive lw in Cpuo derivive. 4. Leibniz equion 5. D f ( ) I D f x f x x m k n C n k ( ) ( ) ( ) k k! D c f c f c D f c D f C n C n C n ( x ) n D fg ( x ) g ( ) f ( x ) f ( ) D g ( x ) f ( x ) I g ( x ) D f ( x ) n C n C n k n C k ( n) k k 5. The emi-group propery, he following equion, hold under ome pecil condiion 6. D D f D f C C C NB: The pr 5 of he bove Lemm) doe no hold for Riemnn Liouville derivive generlly. To prove hi clim, uming f funcion wih ideniy funcion f(x) =, well =, n =, nd ε = /2. Thu, if Riemnn Liouville operor i pplied o he lef ide, we will obin (D 2 f )(x) = D 2 =. On he oher hnd, by pplying hi operor o he righ ide, we will gin D f ( x) D I f ( x) x ( / 2) 3/2 2 /2 3/2 The relionhip beween frcionl inegrl nd frcionl derivive In hi pr, he relionhip beween frcionl inegrl nd frcionl derivive i inveiged. There re lwy everl properie in he clicl inegrl nd derivive re cceped conn role; hee properie, however, migh no hold lwy in he frcionl ene. The following equion, for innce, re lmo me in he clicl clculion, bu i h mjor difference in he frcionl clculion (m > α). m m m m d nd m D D D I dx d D D D I dx m m m m m Indeed, conn nd vrible erm in clculion repreen hi difference, which he conn nd vrible erm re reled o non-frcionl nd frcionl ene repecively. In oher word, uming funcion f: [, b] R where < < b < ; if he funcion f(x) on cloed inervl [, b] h been n order differenible, hen for ll x (, b), we hve: 5 Auming < n <, g nd f re nlyicl on ( h, + h) re eenil o hold he equion. 6 Specil condiion for exiing hi propery i f funcion mu belong o C k [, b] where C k [, b] {f: [, b] R; f h coninuou Kh derivive} for ome < b nd ome k N. Exiing l N wih l k nd n, n + ε [l, l] i eenil well (n, ε > ). NB: The exience of l wih he menioned properie i eenil. Oherwie, i i enough o ume h n = ε = 7/(i. e 7/ = n < < n + ε = 7/5), =, nd f(x) = x, cuing he righ ide of lemm become zero owing o C 7/5 3/5 3/5 D f ( x ) ( I f )( x ) I ; however, c 7/ D f(x)=. A reul, Γ(3/) x3/ he lef ide of lemm will obin follow: C 7/ C 7/ 2/5 D ( D f )( x ) x (3 / 5) 8

9 C ( D I f )( x ) f ( x ) In imple word, if he frcionl derivive wih order α R i pplied on he frcionl inegrl wih he me order, he oupu of h will be f(x). Thu, we cn y, Cpuo derivive i he lef invere of Riemnn-Liouville inegrl. Alo x n j C ( j ) ( I D f )( x ) f ( x ) ( f ( x ) )! x j j NB: To prove he menioned heorem ee [9]. Wih regrding he Riemnn-Liouville derivive operor, he following equion hold (n >, m > n): m nk n RL n ( x ) m k m n I D f ( x ) f ( x ) lim D I f ( z ) ( nk) z nd priculrly, for < n <, we hve: k n n RL n ( x ) n I D f ( x ) f ( x ) lim I f ( z ) ( n) z The inerpreion of frcionl clculu There i dozen of uggeion for inerpreion of frcionl clculu [4, 7]. However, mo of hem re lile bi brc nd give no phyicl inuiion. Hence, we ju preen mo ueful inerpreion of frcionl clculu. The core of hee inerpreion i memory concep. In generl, when he oupu of yem ech ime depend only on he inpu ime, uch yem re id o be memoryle yem. On he oher ide, when he yem h o remember previou vlue of he inpu in order o deermine he curren vlue of he oupu, uch yem re id non-memoryle yem, or memory yem. For exmple, ll of he Mrkov chin phenomen re memoryle [3, 4], nd humn deciion mking or hpe memory lloy re non-memoryle. Lplcen inerpreion Suppoe h Y() i quniy whoe vlue in erm of f() cn be chieved follow: ( ) ( ) Y() f ( ) d (6) ( ) i.e. he oupu Y() cn be viewed power-weighed um which ore he previou inpu of funcion f(). Bed on he bove definiion, uch yem i non-memoryle yem nd in uch yem, memory decy he re of w() = α /Γ(α). Applying he Cpuo derivive of order α o he boh ide of he l relion led o C D Y( ) f ( ) (7) A reul, he differenil equion governing he yem memory Y() i decribed by frcionl derivive. Therefore, he frcionl derivive i good cndide o explin he yem wih memory. The nure of weighed funcion deermine he ype of frcionl derivive which decribe yem memory. For exmple, If he weigh funcion of yem i defined by α Γ(α), he Riemnn- 9

10 Liouville elemen, nd by α θ(x )/Γ(α) he Cpuo elemen re ued--which θ i he Heviide funcion [4]. Since he convoluion operion repreening he inegrl of he produc of he wo funcion fer one i revered nd hifed, o if one of he funcion conidered weighed funcion, i my be climed h he k of hi funcion i collecing he yem memory over ime. For exmple, in he Riemnn- Liouville, "The relionhip indice h he informion funcion F() i memorized (orge) wih power low re funcion". In Ref [4] ome oher weigh funcion i reviewed. From he phyicl poin of view, wh he memory i nd how i i defined in yem depend on deep undernding of he phenomen. Alo, here i no cerin rule nd mehod o elec he frcionl ype in modeling. In oher word, ny definiion h i reul i more conien wih he experimenl d i he uible definiion o u. Anlyicl nd numericl mehod Anlyicl mehod There re differen mehod o olve frcionl differenil equion nlyiclly. One of he mo common nd widely ued mehod i he Lplce rnform. In he following, by n exmple, hi mehod i illured (for oher mehod ee reference [7]). Before proceeding, i i worh noing h in generl, he number of iniil condiion h re required for given ordinry differenil equion will depend upon he order of he differenil equion. However, in he frcionl differenil equion, number of iniil condiion equl o he ineger lower bound of order vlue (α) [, ]. Conider he following differenil equion D x( ) kx( ) f (8) Which x() i diplcemen, k, nd ξ re conn, well he frcionl derivive i lo Cpuo nd < α < --In wh follow, i h been hown h hi differenil equion model i he dynmic of purely elic pring nd vicoelic elemen connecing in prllel wih body of m m, which force f i pplied on body. In order o olve, he fir ep i o ke he Lplce rnform of boh ide of he originl differenil equion (he Lplce rnformion i conciely explined in Box 3). Bed on he equion B3.2 in Box 3, we hve f X() ( k / ) (9) where α nd re frcionl order nd Lplce domin vrible repecively. Alo, i i uppoed h x() =. To find he oluion, ll we need o do i o ke he invere rnform (Box 3.): f k x( ) E, ( ) (2) Which E α,α () i Mig-Leffler funcion [for more informion ee he Box 3 foonoe]

11 In he l exmple, if he pring i ignored, he Eq. 8 will be reduced o f = ξd α x() (2) Then bove, by ) king he Lplce rnform of boh ide of he equion, b) implifying lgebriclly he reul o olve he obined equion in erm of, nd c) finlly finding he invere rnform, we hve: x() = K α (22) Which K = f/ξγ( + α). Alhough he Lplce rnformion mehod i one of he imple nd prcicl mehod for olving he frcionl equion--me he ordinry differenil equion, mo of he frcionl equion could no be olved nlyiclly. In wh follow, we preen numericl echnique o olve Cpuo frcionl differenil equion. BOX 3 In hi box, he concep of Lplce Trnform i preened for everl frcionl definiion. To begin wih, le u bring up ome bic fc in hi regrd [7]. The funcion F() of he complex vrible i defined he following equion which i known he Lplce rnform of he funcion f(): F ( ) L{ f ( )} e f ( ) d ( B 3.) To exi inegrl (B3.), he funcion f() mu hve been n exponenil order α. In oher word, he exience of wo poiive conn, uch M nd T which ify he following condiion i eenil. e f ( ) M for ll T. Indeed, when, he funcion f() cnno grow fer hn cerin exponenil funcion. Wih he help of he invere Lplce rnform, he originl f() cn be gined from he Lplce rnform F() f ( ) L { F ( )} lim ( ) ( 3.2) 2i e F d B b ib where he inegrion i done long he vericl line Re() = in he complex plne uch h i greer hn he rel pr of ll ingulriie of F(), which gurnee h he conour ph i in he convergen region. If eiher ll ingulriie re in he lef hlf-plne or F() i mooh funcion on < Re() < (i.e., no ingulriie), hen cn be e o zero nd he bove invere inegrl formul become idenicl o he invere Fourier rnform. The direc clculion of he invere Lplce rnform (B3.2) i ofen ophiiced. Someime, however, i give ueful informion on he unknown originl f() behvior which we look for. The following formul eem o be noher ueful propery for he Lplce rnform of he derivive of n ineger order n of he funcion f(): ib n ( n ) n n n 2 ( n ) n n k ( k ) L{ f ( )} F ( ) f () f () f () F ( ) f () ( B 3.3) which cn be obined from he definiion (B3.). To fcilie pplying he Lplce rnform, we gher ome formule of vil funcion in he ble (B3.) for boh he ineger mode nd he non-ineger mode. k

12 Tble B3.. Lplce rnform ble of ome bic frcionl clculu. Frcionl order f() = L {F()} F() = L{f()} ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) e ( ) (, ) ( ) E ( ) E, ( ) E ( ) F ( ;; ) E, ( ) E, ( ), E, ( ) ( ; ; ) F ( ) f() = L {F()} k F ( ) bg ( ) n n L{ f ( )} f () n2 ( n) f () f () F ( c ) n ( n) ( ) F ( ) F ( ) G ( ) F ( u ) du F( ) Ineger order k F() = L{f()} (), k,2,3,... ( k ) f ( ) bg ( );, b ( n f ) ( ), n in( ) co( ) c e f () n f ( ), n,2,3,... f ( ) g ( ) d () f f ( ) d The Lplce rnform of frcionl operor In hi pr, he Lplce rnform of frcionl operor i repreened, nd ome formule of which i ummrized in ble (B3.2). We inroduced Riemnn- Liouville inegrl in Box follow: I f ( ) f ( ) d [ * f ( )] ( B 3. 4) ( ) ( ) Wih pplying he Lplce rnform on he Eq. B3.4, we obin (bed on Tble B3.): L { I f } { } { ( )} ( ) ( ) L L f F 7 incomplee gmm funcion 8 E α,β i denoed o Mige-Leffler funcion uch h z e d z (, ) k x E, ( x),, k ( k ) Alo, if α = β =, hen hi funcion cn be equled wih exponenil funcion. 9 Hypergeomeric funcion q p j j i p Fq ({ i };{ b j }; z ) p q n i i j ( b ) n ( i n) z ( ) ( b n) n! j 2

13 Similrly, oher Lplce rnform of frcionl operor will be gined by uilizing he propoed formule in Tble B3. [7]. Tble B3.2. The Lplce rnform of ome frcionl operor wih order α. Operor Riemnn- Liouville inegrl Riemnn- Liouville derivive Cpuo derivive Griinwld- Leinikov frcionl derivive Frcionl formul ( ) ( ) ( ) I f f d ( ) d f ( x ) ( n ) d ( x ) RL n D f ( ) ( ) n D f ( ) d ( n ) ( ) n C D f () n n m D f ( ) lim h ( ) f ( mh) m m h nh dx Lplce Trnform of he frcionl order operor F ( ) n k RL k ( ) ( ), k F D f n n n k ( k ) ( ) (), F f n n k F ( ), Numericl oluion Unil now, vriou numericl mehod hve been propoed for olving frcionl differenil equion which re bed on dicreizing he relionhip [2-7]. In hi ecion, new lgorihm which w recenly publihed in [8] i propoed dicreizing he Cpuo derivive. Conidering he Cpuo derivive of funcion u() wih order < α <, he following equion hold for dicreizing u() on [,T] in uniform wy. n j u ( ) ( n) ( n ) ( ) j j u d n u( ) u( ) ( ) (2 ) j n j d j j j n ( n ) d r, j u( ) u( ) r n j n j n, j, (23) Where d α,j = (j + ) α j α nd j =,,2,, n, well r n+ α,τ i locl runcion error. To uilize he bove lgorihm, equion (2), which ued in he l ecion imulion, i conidered. The following equion will be obined by pplying he equion (23) on equion (2). n n n, j n j n j j x( ) (2 )( f ) x( ) d ( x( ) x( )) (24) Comprion beween he numericl mehod nd he nlyicl nwer i hown in boh he figure () nd he ble () The Lplce rnform of he Griinwld- Lenikov frcionl derivive of order α > doe no exi in he clicl ene. 3

14 Fig. A chemic comprion of nlyicl nd numericl nwer wih he repreened meh in Tble. Tble. Comprion beween he numericl reul, equion 24, nd he nlyicl nwer of equion (25) wih f = pn re (ξ nd α re choen from ble 4). Meh poin (τ) Numericl reul in = 3 Anlyicl reul in = 3 Difference Percenge error Vicoelic yem Eliciy i he biliy of meril o rei dioring or deforming force nd reurn o i originl form when he force i removed. According o he clicl heory in he infinieiml deformion, mo elic meril, bed on Hooke' Lw (Eq. 25), cn be decribed by liner relion beween he re σ nd rin ε [9]. where E i conn known he elic or Young' modulu. σ() = Eε() (25) The vicoiy of fluid i quniy which decribe i reince o deformion under her re or enile re. In he idel vicoe fluid, ccording o he Newonin fluid (Eq. 26) he re i proporionl o he locl her velociy (i.e. he re of deformion over ime): where η i he her modulu. σ() = η dε() d (26) However, mny liquid briefly repond like elic olid when ubjeced o brup re. On he conrry, mny olid will flow like liquid, lbei very lowly, under mll re. Such meril poeing boh eliciy (recion o deformion) nd vicoiy (recion o he re of deformion) re known vicoelic meril. When hee meril ubjeced o inuoidl re, he correponding rin i neiher in he me phe he pplied re (like n idel elic meril) nor in he π/2 ou of phe (like n idel vicoe meril). In hee meril, ome pr of inpu energy i ored nd recover in ech cycle nd he oher pr of he energy i diiped he. Meril whoe behvior exhibi hee chrcerized re clled vicoelic. If boh rin nd re be no only infinieiml bu dependen upon he ime well, rin-re relion (coniuive equion) cn be decribed by liner differenil equion wih conn coefficien i.e. he meril which exhibi liner vicoelic behvior. One of he min conequen of hi umpion i he re (rin) repone o ucceive rin (re) imuli re ddiive. In oher word, in he creep experimen coni of pplying ep re σ, ( re incremen ime =, which i kep conn for >), nd meuring he correponding in repond ε(), he coniuive equion i ε() = σ J(), where J() which ermed creep complince i he rin ime owing o uni re incremen ime 4

15 . Suppoe h vriou re incremen Δσ occur he ucceive ime inervl Δ bed on he uperpoiion of effec, we ge: Afer N ep, we hve ( ) J( ) J( ) (27) N ( ) J( ) J( n) n (28) n Thi relion recognized Bolzmnn uperpoiion principle which e h he repone of meril o given lod i independen of he repone of he meril o ny lod, which i lredy on he meril [9]. Fig 2. A grph veru ime of () ep-plu incremen re pplied o liner vicoelic meril nd (b) he reuling rin. in he limi Δ d ( ) ( ) J ( ) J ( ) d d (29) Thi relion i ermed herediry or uperpoiion inegrl. Similrly, wih pplying ep rin, we ge Where G() i relxion modulu. ( ) ( ) ( ) ( ) d G G d d (3) In he mechnicl cience, o evlue mechnicl properie of meril, fer udden re nd dynmic (periodic) experimen, reercher implemen cerin e, uch creep e. In he following, we will briefly dicu hee wo e. Creep e One of he mo common e on he vicoelic yem i creep e. In hi e, yem i ubjeced o conn enion nd he moun of deformion i recorded over ime. For vicoelic meril creep v. ime digrm how power-lw behvior (figure 3). ( ) J ( ) (3) 5

16 Where J() = /G() i ermed creep complince. Fig 3. The profile of her creep experimenl. Periodic e In hi common experimen, yem ubjeced o cyclic pplied re. Therefore, menioned bove, for vicoelic yem, rin in repone o n impoed periodic re of ngulr frequency ω i lo periodic wih he me frequency bu in he differen phe (figure 4). Bed on relion (3), if he periodic re ε = ε in (ω) pplied, we expec h he re follow periodic repone wih he me frequency nd differen phe: co( ) co in in co (32) And by exenion of relxion modulu, we define wo her orge modulu G, nd her lo modulu G follow G ( / )co G ( / )in G / G n (33) Fig 4. Geomery nd ime profile of imple her experimen wih inuoidlly vrying her [9]. 6

17 Frcionl vicoelic model There re differen model o evlue nd o predic he coniuive equion in he vicoelic yem, hee model commonly, compoing of differen combinion of pring nd dmper elemen [2, 2]. For exmple, wo well-known model o nme re Mxwell nd Kelvin model (Tble 2). In hi ecion, model which i known frcionl order elemen (FOE) wih equion (39) i preened in order o inveige he mechnicl repone of vicoelic yem from differen poin of view. I i indiced h hi frcionl model cn be good ubiue for modeling he vicoelic meril hn SLE model. Wih hi im, le u ke cloer look Mxwell model; bed on hi model, he coniuive equion will be follow: dσ E d + σ η = dε d (34) where E( + ) = η( + ). Alo, σ, ε, η, nd E re re, rin, her modulu, nd Young modulu repecively. The following equion will be obined by king Lplce rnformion nd conidering ep rin funcion. Where > nd G() i relxion modulu. G() = Ee ( τ ) (35) In nure, he G() vlue in vicoelic meril doe no uully follow imple exponenil behvior, nd more reliic expreion for i behvior i power-low repone [3]: G() = E α Γ( α) ( τ ) (36) Tble 2. Vicoelic model, which σ, ε nd τ re re, rin nd η/e repecively; lo, X α = E α η α nd < α <. Nme Model Coniuive equion Relxion modulu (G()) Mxwell fluid ( ) D ( ) ED ( ) / Ee Voig fluid ( ) E ( ) D ( ) E () FOE ( ) X D ( ) E ( ) A i menioned in he previou ecion, he vlue of re in pure elic yem, n idel pring, i proporionl o he derivive zero diplcemen. On he oher hnd, i i proporionl o he fir-order derivive of diplcemen in he pure vicou yem. I i expeced h he vicoelic yem hve behvior beween hee wo yem, n elic nd vicou meril, reuling o conclude h re i proporionl o frcionl derivive whoe order i beween zero nd one (Fig 5). Hence, from he mhemicl poin of view, hi phyicl inerpreion cn be modelled he following equion. σ() = X α D α ε() (37) Where α nd X α re he conn coefficien o mke n equl relion beween re nd rin he α index repreen he dependence of he conn X o expnded order. 7

18 Fig 5. A chemic repreenion of vicoelic regime nd re-rin relion from phyicl poin of view. To obin he coefficien, he eenil condiion of frcionlizion lgorihm (ecion 2) i pplied fir. Thu, he equion (4) mu conin boh Hook lw nd Newon fluid in boundry poin, i.e. α = nd α =. In oher word lim X E, lim X (38) One of he pecific ce which cn ify he menioned boundry condiion i X α = E α η α. In wh follow, i i hown h he propoed X α i good uggeion. To mke ufficien condiion in frcionlizion lgorihm, he phyicl poin of view mu be conidered nd o hi end, he equion (37) wih he umed coefficien i rewrien. σ() = E α η α dα ε() (39) dα By pplying he Lplce rnform o he bove FOE he following equion (4) will be obined derivive i Cpuo. ( ) E ( ) (4) A reul, he relxion modulu will be chieved by conidering uni ep rin funcion. G() = E α Γ( α) ( τ ) Where τ = E/η. Thu, we gined he me equion (4) by he help of phyicl inerpreion nd mhemicl mehod which hve beer mch wih he experimenl reul. In equion (43), α nd X α re quniie which repreen he yem propery. In oher word, i could be poible o conider he E α η α coefficien n independen quniy which i inroduced by vrible X α i men h whever deermine he quniy nd he rucure of he yem re ju he vlue of α nd X α ; indeed, i i poible h yem wih E nd η modulu h equl dynmicl behvior noher yem which h differen modulu E nd η i.e. η η nd E E, bu E ( α) η α = E ( α) η α. Hence, he produc of wo moduli which expreed in equion (4) will be ju one independen quniy. If α, he equion (4) will led o he Hook lw nd if α, hen he equion urn o Newon fluid model. A reul, bed on he phyicl inerpreion, i look logicl o conclude h he propoed model cn be proper ool for inveiging he mechnicl properie of vicoelic meril. To ke more preciely look he FOE equion in he vicoelic yem, in wh follow, he formlim of inherince inegrl for he frcionl e i inveiged. Equion (4) i obined by king pril inegrl from equion (29). dj ( ) ( ) ( ) J ( ) ( ) d (4) d( ) 8

19 Alo, he creeping e wih uni ep re funcion will be follow bed on FOE equion. J ( / ) () E ( ) (42) By king derivive from equion (42) nd puing he obined reul in equion (4), equion (43) will be chieved. ( ) ( ) J ( ) ( ) ( ) d E ( ) (43) Which he equion inide he brcke i Riemnn-Liouville inegrl; hu, by rewriing he bove equion, we hve ( ) ( ) J ( ) I ( ) (44) E According o he bovemenioned ubjec, in equion (44), he previou even re embedded in he frcionl kernel funcion α /Γ(α); which he obined reul h mch wih he propoed inerpreion of ecion (3). Regrding hoe inerpreion, he informion of yem, which i reled o deformion, re oring wih power-lw funcion which h frcionl order α from he iniil ep ( + = ) unil he preen momen () which he frcionl order i he indicion of ving order informion. According o he kernel funcion α /Γ(α) in equion (44), α = nd α = re equl wih memory-le yem nd full-memory yem in creeping e repecively owing o Γ() =. Indeed, in pure vicoe yem, yem poiion compleely depend on hi fc h how re pplie on he yem over ime. In oher word, he nure of re nd he moun of which h direc impc on he preen poiion of he yem. However, owing o hi fc h he effec of memory differen ime h uni weigh funcion, he yem i ble o ve ll he prior hiory of ielf preciely nd eqully. In conr, in he pure elic yem, he preen re i n effecive elemen on he moun of diplcemen, no previou ree, becue he weigh funcion i zero. Similrly, he bove procedure cn be followed for relxion modulu. The following equion will be obined by puing equion (36) in (3). E d( ) ( ) G( ) ( ) d ( ) d (45) A reul, by he help of equion (5) nd rewriing he bove equion, coniuive equion i gined follow. ( ) G( ) E D ( ) C Bed on he equion (45), in he relxion e, α = nd α = repreen full-memory yem nd memory-le yem repecively Γ() =. In oher word, in he pure elic yem, he diplcemen of n objec in prior ime compleely keep in he memory of objec owing o Becue of oring diplcemen chnge over ime by n uni funcion. known weigh funcion over ime, nd here i no difference in very fr nd very ner ime. A reul, ll he previou diplcemen will hve been n effec on he vlue of re. In he pure vicoe yem, however, uch memory he menioned one doe no exi, well he previou diplcemen will no hve impced on he preen re of yem; hu, he yem i boluely devoid of ny memory in uch condiion. Thi i relly inriguing, fcining inerpreion of FOE model wih equion (39) becue on he one, 9

20 hnd he frcionl order αcn repreen yem which i devoid of ny memory, i.e. memory-le yem, when he order i zero, bu on he oher hnd, i cn be he repreenion of full-memory yem. I i indiced h he frcionl model re vluble ool for decribing he dynmic properie of rel meril, pecificlly, in polymer which ue for conrolling he ound nd vibrion [22-33]. Alo, he combinion of frcionl model wih oher liner elemen cn mke inriguing rucure; for exmple, i cn be referred o frcionl Mxwell nd Kelvin model which re known pring-po model (Tble 2). Since SLE re no lwy preciely ucceful in inveiging mny vicoelic yem, reercher pplied pring-po model o ddre he problem, well by virue of hee frcionl model nd he combinion of hem, hey chieved he onihing reul [33-42] o more deil ee he Applicion ecion. Tble 3. Frcionl order Voig Model. Model Coniuive equion Relxion modulu (G()) ( ) E D ( ) E( ) E ( ) E ( ) E D ( ) D( ) E ( ) () ( ) E D ( ) E D ( ) 2 2 E E ( ) ( ) 2 Tble 4. Frcionl order Mxwell Model Model Coniuive equion Relxion modulu (G()) ( ) D ( ) E D EE ( ) ( ) D ( ) 2D ( ) E E, ( ) D ( ) E D E E, ( ) Auming he frcionl Mxwell model which mde of wo FOE model wih (α, E, τ ) nd (β, E 2, τ 2 ) elemen, where < α, β <. Regrding hi, he coniuive equion, in relxion e, will be follow [o prove, ee Appendix]: G() E E, (46) 2

21 Where E α,β i Mig-Lefler funcion [o ee he definiion of hi funcion, ee Appendix]. Above equion will be reduced o he relxion modulu of clicl Mxwell by uming α = nd β =. In oher word, hi equion include no only he clicl relxion e bu lo furher expnion compred o he SLE (more frcionl model i preened in he ble 3 nd 4). Periodic rin A previouly deiled, by impoing periodic rin upon vicoelic objec, re i neiher excly in he me phe wih rin nor 9 o ou of phe. In hi ecion, he ruene of hi fc i inveiged for FOE model. Auming FOE i ubjeced o inuoidl rin, i.e, ε = ε in (ω), hen he re of which will be obined follow by he help of equion (39) nd Cpuo derivive (5). ( ) E ( ) co( ) d ( ) )74( A reul, he following equion hold [o prove he equion, ee Appendix]. ( ) E i in( )exp i (2 ) 2 2 )74( In equion (74), if α =, re will be 9 o ou of phe wih rin pure vicou objec, nd if α =, i will be excly in he me phe wih rin pure elic objec. Therefore, when < α <, hen re will neiher excly in he me phe wih rin nor 9 o ou of phe. Hence, he reul of equion (74) horoughly mche wih phyicl inerpreion. Relion beween frcionl equion nd SLE In hi ecion, wih he help of new frcl yem which mde of quie few pring nd dmper (Fig 6), he relion beween frcionl equion nd SLE i inveiged. Increing he number of elemen in he SLE i one pproch o improving he ccurcy of model in he vicoelic yem. Clculion of nd work wih uch ophiiced yem re no uully ey. Thu, in order o inveige he coniuive equion in infinie yem, which mde of mny boh pring nd dmper, differen model were derived o fr, nd heir oupu were reduced o frcionl equion [24,48,49]. Regrding hi, new frcl yem i conidered nd fer mive mhemicl clculion, i revel h he coniuive equion i equl o pring-po. Conider he pring-dmper circui in Fig 6, o equion (74) will be obined wih he help of governing equion on he re/rin elemen. σ () = η ( ) (74) ε d () + E (+η ) η ) Where ε () = L{ε()}, σ () = L{σ()}. E ( + E (+η 2 η 2 E2 ( + E n 2 η n (+ η n En +η n ) From he mhemicl poin of view, by impoing locl force o he propoed yem, n equivlen opology for inveiging he relionhip beween re nd rin over ime i found bed on he mhemicl formule. In oher word, he govern equion in figure 9 i in R pce wih he Eucliden norm, while he oupu of he umed yem i loced in L pce. L {f: [, b] R; f i meurble on [, b] nd f(x) dx < } b 2

22 Bed on equion (5), fer rewriing equion (74) o (5), he following equion i obined 2. x(x + ) α = x + σ () = η ε d () ( c/ +.(+α).(2 α) ( α)x x x ( γ)c/ + 2.(+α) 3.4 x + (n )(n γ) (2n )(2n 2) c/ 2.(3 α) 4.5 x + (5) ) (2) Where < x, α < nd c = E /η. The equion (5) will be reduced o he following equion by uming c when n. σ () ε d () η ( (c /) γ ) Fig. 6 Digrm of he finie mechnicl rrngemen pplied o imule he generlized dmper. By uming E = nd rewriing he bove equion, we hve [o follow he proof in deil, ee Appendix]: σ d () = 2η dε d () d d η 2γ ε d () d 2γ (7) In oher word, he oupu of he conidered yem i combinion of FOE nd dmper connecing in erie wy when < γ <. Alo, bed on he chieved reul, derivive in he bove equion i 2 Cpuo. 2 Noion b + b b n b n 2 = b + b 2 + n b n 22

23 Applicion A he bovemenioned ecion, frcionl clculu i powerful ool for modeling complex yem, pecificlly for vicoelic meril. In hi ecion, fir, brief review of he pplicion of frcionl clculu in he modeliion of vicoelic meril in he vriey field re preened. In he nex ep, one-dimenionl frcionl model for inveiging cell deformion in creeping e i propoed, nd hen 2D frcionl model for imuling he nework of cin filmen on he cellulr urfce i propoed. Owing o he perfec mch wih experimenl d nd imuled model, we logiclly conclude h frcionl model cn be gre replcemen for previou model, which re mde of pring nd dmper, in hi regrd. Frcionl clculu h been employed for modelling vicoelic yem which cover mny field nd ubjec. In ref [43], Djordjević, V.D., e l. uilized frcionl clculion in four prmeric model wih equion (5) o inveige cell vicoeliciy in he rnge of 2 2 Hz for periodic e. The oupu of hi model w compred wih experimenl d obined from he mgneic ocillory cyomery mehod, nd i reveled h he propoed model highly mched wih experimenl d. ( ) ( ) D ( ) ( ) 2 )5( I i noeworhy Riemnn-Liouville (frcionl) derivive w conidered in equion (5). Inveigion of phl mixure propery during heir ervice life i rel chllenge owing o i complexiy nd eniiviy o environmenl nd loding condiion. I h been hown h phl mixure behve liner vicoelic meril when i i ubjeced o loding condiion. Trdiionlly, he vicoelic meril i modeled vi creep/recovery funcion by he help of pring nd dmper model. The oupu of hee model hve hown n exponenil behvior, in nure, however, he mechnicl repone of hee meril hve power-lw behvior rher hn n exponenil. Conidering hi fc, pring-po model were propoed for predicing creep/recovery behvior of phl mixure. Finlly, he reul of propoed model were compred wih experimenl d, howing h hey hve in good greemen wih he d [39]. Scholr believe h ome pulmonry diee re reled o lung vicoeliciy. Thu, profound undernding of vicoelic model in hi field cn bring bou new progre in lung phology nd rum. A reul, re relxion e w pplied on pig lung, nd he oupu of hi e howed power-lw behvior hn n exponenil. Regrding hi, wo ineger ndrd liner olid generlized Mxwell model nd Frcionl ndrd liner olid model were propoed for predicing he mechnicl behvior of yem. The reul of hi inveigion indiced h he frcionl model by fr beer mche wih experimenl d hn ineger model [4]. To inveige he vicoelic repone of humn bre iue cell, he frcionl Zener model w recenly propoed [44]. The uhor demonred h he propoed frcionl model h beer reul compred o h of he non-frcionl model howed for probing he mechnicl repone in relxion e. According o he recen udie [42], frcionl Mxwell nd Kelvin model were propoed for nlyzing nd inerpreing he mechnicl repone of polymer meril, nd he reul of which hve hown h he frcionl Mxwell model w no ble o inveige he polymer behvior. On he oher hnd, he frcionl Kelvin model i good proper model wih hi im [42]. In 28, frcionl model w uilized o inveige he reril vicoeliciy. In h reerch, he oupu of he frcionl model nd SEL model w pinpoined by he help of le-quire. The reul 23

24 h vlidion howed h he frcionl model i ble o nlyze he yem mechnicl repone o enile perfecly [35]. Two-dimenion frcionl nework model In he l ecion of he pper, o inveige he mechnicl repone of cell, wo (new) frcionl model re uggeed. In he fir ep, one-dimenion frcionl model i propoed, nd in he econd, more reliic model i pu forwrd in wo dimenion by exending he fir model. We define cell he mlle vil uni of he configurion of ny live exience. The cell i mde of differen pr, uch cyoplm, nucleu, cyokeleon (CSK), ec. The CSK ply remrkbly eenil role in he inrcellulr force rnmiion, cellulr conrciliy, well he rucurl inegriy of cell no only in he ic bu in dynmic e well. The biologicl funcion of cell, uch growh, differeniion, nd popoi re ocied wih chnge in he cell hpe, re reled o he mechnicl behvior of he CSK [45-47]. The CSK rucure re compoed of hree min elemen: cin filmen, inermedie filmen, nd microubule. The rucurl chnge in he CSK, including deformion nd rerrngemen, re reled o chnge in he enile force h re pplied o he cin filmen [45, 46, 48-5]; reul, he cin filmen hve minly enion-bering role. The vicoelic properie of cell hve been found o ply n imporn role in diinguihing dignoic, em cell nd ec. Regrding hi, everl wo-dimenionl (2D) nd hree-dimenionl (3D) CSK nework model hve been propoed o clcule he mechnicl repone of cell [56,58-6]. The vicoeliciy of cell under ome pecific exernl force w meured wih he help of Mgneic Bed Microrheomery mehod in creeping e wih F = pn [5]. To inveige he gined reul, he uhor uilized model which mde of wo pring nd wo dmper (Fig 7). The reul of hi model i compred wih he experimenl d (Fig 8). Fig. 7. Buch model h been propoed o inveige cell vicoeliciy under deformion [5]. The mechnicl repone of living cell i quie inrice. The complex, heerogeneou chrceriic of cellulr rucure cue h imple liner model of vicoeliciy cnno predic he mechnicl properie of cell under he deformion quie well, epecilly under minucule deformion. Regrding hi, frcionl model re pplied o ddre he poin menioned. In he fir ep, FOE model wih Eq. 39 i propoed for modeling cell deformion in he creeping e. To vlide he propoed model, he model reul i compred wih he experimenl d which expreed in he ref [5]. In order o mke he me condiion experimenl d, fer olving equion (39) by regrding he creep e, equion (22) i obined. According o Fig 8, i i cler h he FOE model i gre greemen wih he experimenl d. By he help of imuled nneling opimizion mehod, he be coefficien α, E, nd η in equion (22) i gined well (Tble 5). Hence, i look logicl o conclude h he frcionl model cn be uible replcemen for modeling he mechnicl properie of cell hn he Buch model. 24

25 Fig 8. Comprion beween he FOE model nd he Buch model. The red poin nd he blck line repreen he experimenl nd he repone of he FOE model repecively. Which he righ ide reled o he FOE model nd he lef ide repreen he Buch model. Tble 5. The opimized coefficien of he FOE model which obined by nneling opimizion mehod. Frcionl order (α) Young modulu (P) Sher modulu (P. α ) MSE e To inveige he model error wih experimenl d, normlized roo-men-qure deviion (NRMSD) mehod i pplied. NRMSD MSE y y mx Alo, NRMSE w clculed.2% nd 3.6% for Buch model nd FOE model by he help of bove formul repecively, howing he frcionl model i highly ccure for probing nd predicing he mechnicl properie of ubjeced cell. mx Fig 9. A chemic repreenion of he prepoed nework model. Which he blck poin re fixed nd, he oher re dynmic. Alo, he line beween wo poin repreen he cin filmen connecing by cro-linking proein (in cell corex). In he nex ep, wo-dimenion frcionl model, o-clled 2D frcionl nework model, i propoed o inveige he cell vicoeliciy. The cell cn del wih he exernl nd inercellulr force impoed on i by virue of CSK. Regrding hi, hee proein filmen, CSK, i conidered red poin of mking 2D model. In mo cell, cin filmen re inenely concenred in lyer ju beneh he plm membrne, even hough hey re found hroughou he cyoplm of eukryoic cell. In hi region, which i clled 25

26 he cell corex, hee filmen re linked by cin-binding proein ino mehwork h give mechnicl rengh o cell nd uppor he ouer urfce of hem well. A implificion, wihou lo of generliy, imging he whole mechnicl properie of cell i reled o cin filmen [52]. Conidering hi, we cn propoe dynmic nework model, hown in Fig. 8, for probing he cell mechnicl behvior. In 23, model he propoed nework model in hi pper w uggeed predicing he mechnicl repone of cell [53]. The mechnicl properie of cin filmen were pproximed wih pure elic meril bed on n experimenl pper [54]; however, ccording o he experimenl d, hee filmen hve hown n elic repone in milliecond cle under deformion. Neglecing hi cle implificion my cue remrkble error in long ime imulion. In oher word, he deformion of ny meril under minuculr re i divided ino hree phe which he fir one i reled o he elic limi. Alo, he mechnicl repone of n cin filmen w repored in hi phe, nd by overgenerlizing hi poin, pure elic nework model which every cin w pproximed wih pring w propoed in he cle of ec for probing he whole mechnicl behvior of cell [53]. Conequenly, he elic nework model, o-clled pre-reed cble nework model, w filed o predic he experimenl d from bed micromery mehod [52]. Regrding hi, new model wih FOE elemen i uggeed in hi pper. Two ignificn meri of which, conneced o he yem memory nd he vicoiy erm in rbirry cle ime, re embedded in propoed model; reuling he frcionl model i ble o highly mch wih he experimenl d. The frcionl nework model i bed on he principle h cin filmen, ccording o ref [52, 55], hve hown vicoelic behvior. Therefore, he FOE model cn be proper model o inveige he mechnicl repone of ech cin filmen. By expnding he FOE model ino 2D model wih he me order o ech elemen, he frcionl nework model, which depic in he Fig. 9, will be obined. Fig. A chemic repreenion of 2D dynmic frcionl imuled nework. Since he direc obervion of l (he lengh of n cin filmen) i no exi from living cell, l i pproximed from CSK pore (D p ) in n dhere cell bed on equion D p = 3l /3 pper [53]. In oher word, bed on he obervion, he vlue of D p i conidered in he middle of experimenl d h i D p = nm; reuling l = 73.2nm. The pce of nework i pproximed nm 2 bed on l, which i lmo me wih he repored vlue in he ref [53]. To vlide he imulion reul nd o compre i wih he previou model, we imule he FOE nd pring nework, Fig how he behvior of hee wo model under he me re. Conidering he 26

27 reul, he frcionl model i beer mched wih he experimenl d. Hence, i eem h we could conclude h 2D frcionl nework model i good cndide for modeling nd predicing CSK in cell. Fig. A comprion beween he 2D dynmic frcionl imuled nework, he lef ide figure, nd he elic model which repreened in ref [53], he righ ide figure. Experimenl d i obined from ref [5]. Dicuion The frcionl clculu i powerful ool for decribing he complex phyicl yem which hve longerm memory nd long-erm pil inercion. In hi pper, differen pec of hi powerful ool re inroduced hrough concie lierure review o provide he reder wih picure of wh frcionl clculu i. Then, wih he help of developed concep, i i ough o find relion beween frcionl nd vicoelic yem nd o employ he obined relion for inveiging he mechnicl propery of cell. In he fir ep, o inveige he mechnicl repone of elic, vicou, nd vicoelic meril, cholr ypiclly pply pring, dmper model nd ome combinion of hem repecively. In nure, pure elic or vicou yem cn rrely found nd moreover, he mechnicl behvior of hem re modelled by he help of he Hook lw nd Newonin fluid. Regrding minucule deformion, he repone of meril i modelled by hee lw, hereby mking ome error in boh he imulion nd he clculion of heoreicl model pecificlly, bio-meril uch cell nd iue. In pie of neglecing he rien error in ome ce, cholr re uully mking highly complex model for improving he ccurcy nd reducing he error in minue yem. Conidering hi fc, propoing new mehod o cope well wih he ophiiced problem cn be worhwhile ool. Bed on he inveigion which i done in hi pper, i i indiced h frcionl clculu i uible cndide o hi end. To prove hi clim, frcl yem, mde of he combinion of quie few pring nd dmper on lrge cle, w conidered, well he oupu of he propoed yem w pproximed only wih he imple frcionl model. In he nex ep, verion of frcionl model, known FOE, w employed o model he mechnicl repone of cell in one-dimenion. By virue of he chieved reul, 2D model o-clled 2D frcionl nework model w propoed o inveige he mechnicl repone of cin filmen, exi in CKS nd cover cellulr ouer lyer. The creeping e w conidered in boh model, nd he reul of which w vlided by no only he experimenl de bu previou model well. The reul reveled h in he one-dimenionl model, d vrince relive o previou model preened in ref [5] w bou %, while in he FOE model, i w eimed 3%. In oher word, he FOE model preciion h evlued 97% in comprion wih experimenl d. Furhermore, he imulion of wo-dimenion elic lice, repreened in ref [53], for cin filmen on he cell urfce, i olly invlid in 27