Application of fractional calculus in modeling and solving the bioheat equation

Transcription

1 pplicaion of fracional calculus in modling and solving bioa quaion R. Magin 1, Y. Sagr & S. Borgowda 1,3 1 Dparmn of Bionginring, Univrsiy of Illinois a Cicago, US Dparmn of Mamaical Scincs, Florida lanic Univrsiy, US 3 Univrsiy of Cicago Gradua Scool of Businss, US bsrac Fracional calculus provids novl mamaical ools for modling pysical and biological procsss. T bioa quaion is ofn usd as a firs ordr modl of a ransfr in biological sysms. In is papr w dscrib formulaion of bioa ransfr in on dimnsion in rms of fracional ordr diffrniaion wi rspc o im. T soluion o rsuling fracional ordr parial diffrnial quaion rflcs inracion of sysm wi dynamics of is rspons o surfac or volum aing. n xampl an from a sudy involving pulsaing (on-off) cooling of a pripral issu rgion during lasr surgry is usd o illusra uiliy of mod. In fuur w op o inrpr pysical basis of fracional drivaivs using Consrucal Tory, according o wic, gomry biological srucurs volv as a rsul of opimizaion procss. ywords: bioa ransfr, diffusion, fracional calculus, modlling, lasr surgry, fracals, mpraur. 1 Inroducion T prsn papr considrs applicaion of fracional calculus o analysis of problms in bioa ransfr. T mods of fracional calculus, rviwd rcnly by Magin [1], ar dvlopd as basis for formulaion and soluion of bioa ransfr problm in pripral issu rgions. Invsigaors av sudid bioa ransfr using mamaical modls for mor an 5 yars [5-7]. In s modls issu cooling (or warming) is approximad by coupling issu prfusion o bul issu mpraur roug Nwon s law of cooling (or aing). In addiion o full body modls, r ar numrous modls in liraur 4 WIT Prss, ISBN

2 8 Dsign and Naur II a dscrib a ransfr mcanisms in a singl organ or a porion of body. In is rgard, an analyical modl dvlopd by llr and Silr xamins bioa ranspor pnomna wi a gnraion (mabolism) occurring in pripral issu rgions. T llr and Silr [8] modl was solvd numrically using paralll compurs o simula all possibl mods of bioa ransfr by Borgowda al. [9]. Rcnly a numbr of invsigaors [1-1] av applid bioa ransfr modl o priodic diffusion problms in localizd issu rgions suc as a wic occurs in sin wn lasr aing and/or cryogn cooling is applid. Fracional calculus is idally suid o addrss is ind of priodic aing or cooling, bu o our nowldg as no bn usd in modling bioa ransfr ir a issu, organ or wol body lvl. T prsn sudy dmonsras a fracional calculus can provid a unifid approac o xamin priodic a ransfr in pripral issu rgions. For xampl, in an xprimnal sudy conducd by Piula al. [13], cryogn spray cooling is uilizd o cool sin surfac during lasr sin surgry. gnralizd fracional calculus approac dvlopd by ulis and Lag [14-16] is adopd o modl localizd priodic bioa ransfr problms similar o on posd by Piula al. [13]. T on-dimnsional a flow problm can b complly solvd for wll dfind surfac mpraur or rmal flux boundary condiions by applying Laplac ransforms [17,18]. T soluion can also b xprssd as a fracional diffrnial quaion for smi-infini pripral issu rgion [14,15]. Furr, fracional diffrnial quaion can b solvd o compu a flux a boundary for diffrn priodic or on-off boundary condiions a closly rprsn aing and cooling of sin surfac during lasr surgry. T approac offrd by fracional calculus modls a larg class of biomdical problms a involv localizd puls aing and/or cooling. On advanag of is approac is a r is no nd o solv firs for mpraur in nir domain. Gnral formulaion T approac usd in is sudy is an approximaion o pysical modl dvlopd in sudy by Dng and Liu [1]. T rgion of inrs is boundary and is viciniy, and oal icnss is assumd o b larg, so a rcangular coordinas in on dimnsion can b usd for analysis. No a ourmos porion, sin, is considrd o b in so a is icnss is no xplicily incorporad ino modl. T localizd issu rgion a is rprsnd by is approxima pysical modl is sown in fig. 1. T gnralizd on-dimnsional bioa ransfr quaion for mpraur T ( x, in issu dvlopd by Pnns [] can b wrin as: T ( x, T ( x, ρ c = + ω c ( T T ( x, ) Q Q ( x, b ρb b a + m + r (1) x 4 WIT Prss, ISBN

3 Dsign and Naur II 9 wr ρ, c, and ar dnsiy, spcific a and rmal conduciviy of issu and ρ, c b b dnsiy and spcific a of blood, ωb is blood prfusion, T a is arrial blood mpraur (assumd o b consan, Qm is mabolic a gnraion and Q r ( x, a gnraion du o spaial aing in mdium. Figur 1: ssumd pysical modl of localizd issu rgion. T ( x, rprsns mpraur in issu, wil Φ ( dscribs surfac rmal flux a x =. W assum a problm as following boundary condiions: + T ( x, ) = Ti ( x,) iniial mpraur disribuion T (, Φ( = x surfac flux lim T ( x, = T consan cor mpraur x C If w iniially assum Q r o b zro w can solv is problm following Liu al. [1] in rms of T ( x, = T ( x, T ( x,) wr w av subracd i 4 WIT Prss, ISBN

4 1 Dsign and Naur II iniial mpraur disribuion T i (x,) wic is jus soluion of sady sa problm. pplying Laplac ransformaion o qn. (1) for givn boundary condiions w obain for = ω ρ c ρc and = ρc. ( x, ( s + ) ( x, = x b b b ( x, φ( s ) = x + T ( x, ) =, lim ( x, =, x Tis scond ordr ordinary diffrnial quaion as following soluion for spcifid boundary condiions x= ( x, = φ( x ( s+ ) s +. () If w considr only rlaionsip bwn flux and mpraur a x = boundary, n rsul can b wrin in rms of a Laplac convoluion ingral as T (, = τ Φ( τ) dτ = Φ( πτ, (3) wr w av usd Laplac ransform pair L 1 1 = s +. Tus, if surfac flux is modlld by Φ ( = Φu(, wr u ( is uni sp funcion n surfac mpraur will incras as T τ Φ Φ u (, d du Φ = τ = = rf πτ π ( ) x wr u = τ and rror funcion is dfind by rf ( x) = u du. π Howvr, convoluion ingral, qn. (3) can also b wrin as ( τ) Φ( τ) T (, = dτ = π( τ) π Φ( τ) τ τ dτ. (4) 4 WIT Prss, ISBN

5 Dsign and Naur II 11 wic can b wrin in rms of Rimann-Liouvill fracional ingral [19-] dfind by α = 1 τ τ Γ α F( ) α 1 u D F( d 1 ( ) ( τ) α, wr Γ( α) = u du. Tus, qn. (4), can b simply xprssd in rms of fracional ingraion by τ [ Φ( )] 1 T (, = D τ. If w assum a sp inpu in flux a x =, Φ ( = Φ u(, w can wri [ Φ ] T (, = D 1 wic sinc fracional ingral is a linar opraor and fracional drivaiv 1 D [ ] = rf ( ) [1], givs sam rsul for surfac mpraur as a obaind abov by invrsion of Laplac ransform. In cas of a spcifid surfac mpraur a surfac x =, a paralll analysis givs surfac flux in rms of fracional smidrivaiv of surfac mpraur, wic can b wrin [ T (, )] 1 Φ ( = D, (5) wr fracional drivaiv of ordr 1 is dfind [1] as D 1 d F( τ) F( = τ Γ d 1 (1 ) d ( τ) 1. Tis rsul can also b obaind using Babno s mod [,3]. Tus, for cas wr T (, = Tu(, a sp in surfac mpraur of T a x =, and using smidrivaiv of [1] w obain, 4 WIT Prss, ISBN

6 1 Dsign and Naur II T Φ = ( + rf ( ). (6) grap of is rsul is sown in fig. (). Figur : Grap of flux Φ (, ncssary o sablis a sp inpu in mpraur T u(, assuming = = =1. Two cass ar plod: on for bioa quaion wi = 1, and a scond for normal diffusion wiou blood flow cooling,.g., =. Sinc rlaionsip bwn flux and mpraur is assumd o follow from Fourir law for a flux i is valid a any poin in domain, no only a x = surfac. Trfor, for on-dimnsional problm of aing wi linar surfac cooling is allows us o wri our fracional ingral and drivaiv rsuls as 1 Φ ( x, = [ D T ( x, ], (7) and 1 (, ) [ (, )] T x = D Φ x. (8) 4 WIT Prss, ISBN

7 Dsign and Naur II 13 Tus, givn flux or mpraur profils a a spcific locaion w can us is informaion o drmin corrsponding mpraur or flux. Tis approac could b usful in xprimnal siuaions wr alf-ordr fracional ingrals or drivaivs of nown funcions could b usd o drmin rquird inpu condiions ndd for dsird mpraur or flux oupus [4-6]. fw xampls ar lisd in abl 1, wic is adapd from Oldam and Spanir [1]. No a, if iniial mpraur disribuion T i (x,) is assumd o b uniform and consan,.g., T i ( x,) = T, n T ( x, = T ( x, T and flux xprssion, qn. (7), bcoms T [ T ( x, ] + rf ( ). Φ( x, = 1 D Tis quaion simplifis for = o T (, ) 1 Φ x = D T ( x, wic was prviously drivd by ulis and Lag [14]. ulis and Lag av rcnly applid fracional-diffusion ory o rmorflcanc masurmns of rmal propris of in films undr pulsd lasr aing. T currn bioa modl undr condiions of volumric as wll as surfac aing xnds ulis s rsuls, qn. (1) in [16], o yild Φ( x, = 1 1 [ D [ T ( x, ] ] + D [ P( x, ] P x wr P ( x, = P( x, P( x,) and P ( x, rprsns paricular soluion o Laplac domain inomognous ordinary diffrnial quaion. In is sor papr w av dscribd a fracional calculus approac o formulaion of bioa quaion. Tis mod provids a simpl xprssion for ir mpraur or flux undr xprimnal condiions ofn spcifid by lasr aing and cryogn-cooling procdurs. ddiional sudis ar ndd o dvlop a conncion bwn fracional ordr of opraors and marial srucur and propris of issu or subsa undr sudy. Rcn wor by Ws al. [7] and ors [8-3] is dircd oward sablising a srongr rol for fracional calculus in dscribing dynamic pnomna in complx marials. 4 WIT Prss, ISBN

8 14 Dsign and Naur II Tabl 1: Flux and mpraur for slcd inpu funcions. f (, > 1 Φ( x, = D f ( + rf ( ) ( ) + rf ( ) rf rfc ( ) rfc( ) rfc( ) + rfc( ) π I1 + I 1 T ( x, = D f ( 1 rf ( ) π daw( ) 1 [ 1 ] π 3 I1 I 1 1 [ rfc( )] [ rfc( ) ] π I π 3 I x wr daw (x) is Dawson s ingral dfind as daw( x) = d, I ( x), I1( x) ar yprbolic Bssl funcions and rfc (x) is complmnary rror funcion givn as rfc( x) = d = 1 rf ( x). π Rfrncs [1] Magin, R. L., Fracional Calculus in Bionginring, Criical Rviws in Bionginring in Prss, Dparmn of Bionginring, Univrsiy of Illinois, Cicago, IL, 3. [] Pnns, H. H., nalysis of issu and arrial blood mpraurs in rsing uman forarm. Journal of pplid Pysiology, 1(), pp. 93-1, [3] Gagg,. P., wo-nod modl of uman mpraur rgulaion in FORTRN. Bioasronauics Daa Boo, d. J. F. Parr and V. R. Ws, Wasingon D. C., pp , x x 4 WIT Prss, ISBN

9 Dsign and Naur II 15 [4] Solwij, J.. J., Mamaical Modl of Pysiological Tmpraur Rgulaion in Man. NS Tcnical Rpor No. NS CR-1855, [5] Wisslr, E. H., Mamaical simulaion of uman rmal bavior using wol body modls (Capr 13 ). Ha Transfr in Mdicin and Biology, Vol. 1, d.. Sizr and R. C. Ebrar, Plnum Prss: Nw Yor, pp , [6] Smi, C. E., Transin Tr-Dimnsional Modl of Human Trmal Sysm, P.D. Tsis, ansas Sa Univrsiy, Manaan, ansas, May [7] Fu, G., Transin, Tr-Dimnsional Mamaical Trmal Modl for Clod Human, P.D. Tsis, ansas Sa Univrsiy, Manaan, ansas, May [8] llr,. H., and Silr, Jr. L., n analysis of pripral a ransfr in man. Journal of pplid Pysiology, 3(5), pp , [9] Borgowda, S. C., Morris, J. D., and Tiwari, S. N., Evaluaion of Effciv Trmal Conduciviy in Pripral Tissu Rgions, Cambridg Univrsiy Prss: Nw Yor,. [1] Dng, Z-S., & Liu, J., nalyical sudy on bioa ransfr problms wi spaial or ransin aing on sin surfac or insid biological bodis. Journal of Biomcanical Enginring, 14, pp ,. [11] nvari, B., Milnr, T. E., Tannbaum, B. S., iml, S., Svaasand, L. O., & Nlson, J., S., Slciv cooling of biological issus: applicaions for rmally mdiad rapuic procdurs. Pysics in Mdicin and Biology, 4, pp. 41-5, [1] Liu, J., Zou, Y-X., & Dng, Z-S., Sinusoidal aing mod o noninvasivly masur issu prfusion. IEEE Transacions on Biomdical Enginring, 49, pp. 41-5, [13] Piula, P. B., Tunnll, J. W., and nvari, B., Modology for caracrizing a rmoval mcanism in uman sin during cryogn spray cooling. nnals of Biomdical Enginring, 31, pp , 3. [14] ulis, V. V., & Lag, J. L., Fracional-diffusion soluions for ranspor local mpraur and a flux. SME Journal of Ha Transfr, 1, pp ,. [15] ulis, V. V., & Lag, J. L., Fracional-diffusion soluions for ransin mpraur and a ransfr. SME Journal of Ha Transfr, 1(), pp ,. [16] ulis, V. V., & Lag, J. L., fracional-diffusion ory for calculaing rmal propris of in films from surfac ransin rmorflcanc masurmns. SME Journal of Ha Transfr, 13, pp , 1. [17] Bjan,., Ha Transfr, Jon Wily and Sons: Nw Yor, [18] Pouliaos, D., Conducion Ha Transfr, Prnic-Hall: Englwood Cliffs, NJ, [19] ocnff, E., & Sagr, Y., Conjuga mpraurs. Journal of pproximaion Tory, 7, pp , WIT Prss, ISBN

10 16 Dsign and Naur II [] Talr, J., Tory of ransin xprimnal cniqus for surfac a ransfr. Inrnaional Journal of Ha and Mass Transfr, 39(17), pp , [1] Oldam,. B., & Spanir, J., T Fracional Calculus, cadmic Prss: Nw Yor, [] Podlubny, I. V., Fracional Diffrnial Equaions, cadmic Prss: San Digo, [3] Babno, Y. I., Ha and Mass Transfr, imiya: Lningrad, [4] Tunnll, J.W., Torrs, J. H. & nvari, B., Modology for simaion of im-dpndn surfac flux du o Cryogn spray cooling. nnals of Biomdical Enginring, 3, pp ,. [5] Torrs, J. H., Tunnll, J. W., Piula, B. M., & nvari, B., n analysis of a rmoval during cryogn spray cooling and ffcs of simulanous airflow applicaion. Lasrs in Surgry and Mdicin, 8, pp , 1. [6] nvari, B., Milnr, B. S., Tannbaum, B. S., & Nlson, J. S., comparaiv sudy of uman sin rmal rspons o sappir conac and cryogn spray cooling. IEEE Transacions on Biomdical Enginring, 45, pp , [7] Ws, B. J., Bologna, M., Grigolini, P., Pysics of Fracal Opraors, Springr: Nw Yor, 3. [8] Hilfr, R., (d). pplicaions of Fracional Calculus in Pysics, World Scinific: Singapor,. [9] Carpinri,., & Mainnardi, F., Fracals and Fracional Calculus in Coninuum Mcanics, Springr Win: Nw Yor, [3] Bar-Yam, Y., Dynamics of Complx Sysms, Prsus: Rading, WIT Prss, ISBN