STUDY OF THE BIOHEAT EQUATION WITH A SPHERICAL HEAT SOURCE FOR LOCAL MAGNETIC HYPERTHERMIA

Transcription

1 XVI Cngess n Nueial Mehds and hei Aliains Códba, Agenina Obe -7, 7 STUDY OF THE BIOHEAT EQUATION WITH A SPHERICAL HEAT SOURCE FOR LOCAL MAGNETIC HYPERTHERMIA Gusav Guieez Mehanial Engineeing Deaen Univesiy f Pue Ri Mayaguez, PR e-ail: gguie@euedu Absa Hyeheia is a ye f ane eaen in whih ane ells ae exsed high eeaues (u C) Reseah has shwn ha high eeaues an daage and kill ane ells, by a lalized and nenaed heaing sue By killing ane ells and daaging eins and suues wihin ells, hyeheia ay shink us, wih inial injuy nal issues In addiin in vi and in viv sudies, ue siulain an be used undesand ans henena inside a u In his sudy a sheial egin naining a agnei aile ebedded in a issue is deled using he bihea equain wih he Penne s del f he heal ineain beween he issue and he efused bld Analyial ehniques ae used slve he bihea equain wih a in hea sue f nsan densiy we laed as he ene f a sheial dain The in hea sue del he hea geneaed by agnei ailes unde he effe f an alenaing agnei field, used in sudies f lal agnei hyeeia Paaei sudies f he eeaue files ae aied u sudy he effe f diffeen aaees like he hea geneain ae, efusin ae and diaee f he in sue n he axiu eeaue and n he eeaue file Se disussin abu ian aaees eseah issues in ane hyeeia ae als addessed Key wds: Hyeheia; lalized hea sue; he bihea equain NOMENCLATURA ρ, C, k, α : densiy, seifi hea, heal nduiviy and heal diffusiviy f he issue ρ, C b b : densiy and seifi hea f he bld q : eabli hea geneain q : efusin hea sue

2 ω : efusin ae ( /s f vluei bld flw e f issue) T aeial eeaue T, T : lal issue eeaue and diensinless lal issue eeaue Q( ) : in sue hea geneain, : adial dinae and diensinless adial dinae, R : adius f he inenal hea sue, axiu adius f he dain, : ie and diensinless ie I/, K / : dified Bessel funins f de ½ INTRODUCTION Bihea ansfe esses in living issues ae fen influened by he influene f bld efusin hugh he vasula newk n he lal eeaue disibuin When hee is a signifian diffeene beween he eeaue f he bld and he issue hugh whih i flws, nveive hea ans will u, aleing he eeaues f bh he bld and he issue Pefusin based hea ansfe ineain is iial a nube f hysilgial esses suh as heegulain and inflaain The bld/issue heal ineain is a funin f seveal aaees inluding he ae f efusin and he vasula anay, whih vay widely ang he diffeen issues, gans f he bdy, and ahlgy The lieaue nains an exensive ilain f efusin ae daa f any issues and gans The ae f efusin f bld hugh diffeen issues and gans vaies deending n fas suh as hysial aiviy, hysilgial siulus and envinenal ndiins Fuhe, any disease esses ae haaeized by aleains in bld efusin, and se heaeui inevenins esul in eihe an inease deease in bld flw in a age issue A gd efeene f he sudy f bihea ansfe an be fund in he CRC Handbk f Theal Engineeing, hae 44(Ed Fank Keih, ) Bagaia, and Jhnsn (Bagaia, and Jhnsn,, 5) sudied he he bihea equain nueially and analyially f hyeheia aliains f ane eaens The del sudies he e disibuin f agnei ailes hughu he u uld iniize he daage he suunding healhy issue while sill ainaining a heaeui eeaue in he u Hweve, his disibuin is defined aheaially bu is n feasible nl in aial aliains And he analyial sluin is liae and bsue Rsensweig, (Rsensweig, ) esen a del f heaing agnei fluid wih alenaing agnei field In his ae an analyial sudy is uied u slve he bihea equain wih a hea sue f nsan densiy we laed as he ene f a sheial dain The in hea sue del he hea geneaed by agnei ailes unde he effe f an alenaing agnei field, used in sudies f lal agnei hyeeia Paaei sudies f he eeaue files ae aied u sudy he effe f diffeen aaees like he hea geneain ae, efusin ae and diaee f he in sue n he axiu eeaue and n he eeaue file Se disussin abu ian aaees eseah issues in ane hyeeia ae addessed

3 - MATHEMATICAL FORMULATION Pennes del (Pennes, 948) desibes he effes f eablis and bld efusin n he enegy balane wihin issue Basially, hese w effes ae inaed in he sandad heal diffusin equain and he equain is alled he bihea equain Hee, he dain f sudy will be a sheial dain wih a hea sue f adius a he ene f he sheial dain f adius R as is shwn sheaially in Figue f a healhy issue Figue Sheai f he sheial dain wih an inenal hea sue The bihea equain in sheial dinaes wih an inenal hea geneain a he ene f he shee an be wien as: q q ρc T T Q = k k k () Whee ρ, C, k ae he densiy, seifi hea and heal nduiviy f he issue, ρ, C ae densiy and seifi hea f he bld, q is he eabli hea geneain Pennes s del f ωρbc b he efusin hea sue is q = ( T T ) k, whee ω : is he efusin ae ( /s f vluei bld flw e f issue), T is he aeial eeaue and T is he lal issue eeaue Then, by inaing he Pennes s del in he diffusin equain () yields: ρc T T ωρbc b q Q = + ( T T ) + + () k k k k α T T Defining he fllwing diensinless aaees: =, = and T = Wiing R R T equain () in es f hese diensinless aaees, a diensinless equain is bained: b b T T RωρbC b Q R q R T = k kt kt ()

4 kt k By defining he fllwing nsans, Q = and C = ha diensinalize R R Q, q and ωρ C b b as: wien as: Q q Q = +, Q Q ωρ C =, he bihea equain in diensinless f an be C b b T T T Q = (4) Wih he fllwing iniial ndiin, eeaue us eain finie a he ene T (, = ) = and he fllwing bunday ndiins: he T (, ) = finie and he eeaue a he exenal sheial sufae is ainained a T, s he diensinless eeaue ( =, ) = T ANALYTICAL SOLUTION T bain an analyial sluin, le sli he diensinless hea geneain e nsan hea geneain due eablis ene q ( ) Q( ) = q + q Q in a q and a nsan hea sue f adius a he Then T T T q = q (5) Nw, le T T T = (, ) + and f nveniene le d he suesi T T T = q( ) T = T q (6) (7) Subje he fllwing bunday ndiins: T (, T (, (8) T T ( ) (9)

5 And he fllwing iniial ndiin: T (,) = T sine T T T (,) = (,) + = () Nw, le (, ) T (, ) = e U and subsiue in equain (6) U U = = (, ) q e g () Whee he sue e is nw a funin f and, g(, ) = q e Then, he bunday ndiins in es f U yields: U ( =, ) = and U ( =, ) = finie () And he iniial ndiin bees: U (, = ) = T () Then, equain (7) has be slved subje bunday ndiins (9) and equain () subje bunday ndiins () and iniial ndiin () Wih he fllwing hange f vaiables f H T =, equain (7) bees a dified Bessel equain in es f H and he sluin is bained in es f dified Bessel funins (see Aendix A) ( ) q I/ T = I/ The sluin equain () f U is bained using Geen funins wih an iniial ndiin given by I/ ( ) q U (,) = T = (5) I/ ( ) A sheial hea sue f adius laed a he ene f he sheial dain f inegain g(, ) = g e ( ) 4π δ (4) = β U (, ) = e sin ( β) 'sin ( β ')( T ( ')) d ' + + π sin ( ) sin ( ) τ = = β β τ τ e β β e g e dτ (6)

6 The inegal geneain τ = g ( β + ) τ e g dτ an be inegaed analyially f a nsan inensiy hea τ = ( β + ) τ ( ) ( ) e g β + β + τ e e g dτ = = g β + β + Ging bak T = Ue wih β = π, T (, ) akes he f ( ) ( ) g β + β + e T (, ) = e sin ( β) 'sin ( β ')( T( ') ) d ' + sin ( β) sin ( β ) π β + = = (7) The inegal ( β )( ) is T =T (,)+T () 'sin ' T ( ') d ' an be evaluaed nueially The final sluin f T q I/ I/ β+ q T (, ) = e sin ( β ) 'sin ( β ' ) d ' + + I / I/ = g e π β β + β + β sin ( β ) sin ( β ) = β (8) Whee β = π, =,,, And f he definiin f he diensinless eeaue, bained as T = T T + T T T T =, he eeaue T is T 5 - RESULTS AND DISCUSSION Hee, se yial values ae assued f he issue heal eies, he eabli hea geneain, he efusin ae and a adius f a sheial dain unde sudy k = 5 W ; ρb = kg ; C 6 J ; 7 ; 7 W, 5 ; = T = C q = w = R = C kg C s kt The adius f he inenal hea sue is = Wih hese values, Q = and R C = k R ae alulaed diensinalize qɺ and = wρbc b

7 W 5 7 C kt C W q 7 W / wρbc b Q ɺ = = 85 ; q 78 ; 6 = = = = = = R ( ) Q 85 W / C k R α The aaee diensinalize he ie R W 5 α k C = = = 9 R ρc kg J R 6 ( s ) kg C T evaluae equain (8), an analysis f de f agniude is efed fis The fis e f equain (8) ges ze when ie ges infiniy Thus, T eesens he seady sae eeaue file due a nsan eabli hea geneain Assuing nly eabli hea geneain, he axiu eeaue will u a he ene f he sheial dain a he seady sae The eeaue ise due yial values f eabli hea geneain is negligible aed wih he eeaue ise due he hea sue ha have be geneaed by he agnei ailes, in hyeheia aliains Then, negleing he fis w es in equain (8), he eeaue file geneaed by he hea sue f adius a he ene f he sheial dain f adius R will be disussed nw Figue shws he eeaue files geneaed by an inenal hea sue a he ene f he shee f g = W / The axiu eeaue us a he ene and is axiaely 55ºC The seady sae eeaue disibuin is shwn and i an be seen he eeaue gadiens a he ene ae vey see be able diffuse he hea geneaed Sine he nduiviy f he issue is lw and he ss seinal aea in he egin lse he hea sue is als sall, f Fuie law f nduin he eeaue gadiens have be high As he adius is ineasing, he hea fluxes ae eduing and hea an be diffusive wih lwe gadiens This ilies ha in he egin nex he hea sue, a vey see eeaue file an be ainained f a lng ie geneaing highe eeaue in a vey sall egin and lwe eeaues a a elaively sh disane f he hea sue, wihu affeing healhy ells This is vey ian in hyeheia eaens, sine is desiable iniize senday effes F aaei sudies, he seady sae file is eahed in a vey sh ie (uh less han ne send) A a disan f he eeaue is less han 9 C As an be seen, he highe eeaues ae nenaed a sh disane f he hea sue Figue shws he eeaue file f a adius f he inenal hea sue f = F he sae vluei hea geneain g = W / he hea we is uh highe sine he vlue inease as he ubi f he adius bu he eeaues geneaed ae lwe beause hea an diffuse easie in his ase Figue 4 shws he eeaue files f a adius f he inenal hea sue f = as in Figue bu a vluei hea geneain f g = 7 W / I an be seen ha he eeaue eneaes fuhe in he dain

8 Teeaue ( C) g = W / = 4 - P = π g = 49 W Radial dinae () Figue Seady sae eeaue file f = 9 Teeaue ( C) g = W / = 4-9 P = π g = 49 W Radial dinae () Figue Seady sae eeaue file f =

9 5 Teeaue ( C) g = 7 W / = 4-8 P = π g = 9 W Radial dinae () Figue 4 Seady sae eeaue file f = and g =7 W/ Finally, f he analyial sluin (8), he effe f he efusin ae is quanified by he e β F yial values f in he de f 6, /π is sall and he effe f efusin an be negleed f hyeheia aliains 6 - CONCLUSIONS An analyial sudy f he bihea equain has been aied u An analyial sluin was bained f he ase f eabli hea geneain in a sheial dain and a nenaed hea sue a he ene f he shee F aaei sudies, eabli hea geneain an be negleed beause he eeaue aise ha will geneae is sall aed wih he yial eeaue inease geneaed in hyeheia aliains f ane eaens Als, he effe f he efusin ae is n signifian F vey nenaed hea sues, he eeaues gadiens ae vey high and highe eeaues ae geneaed in a sall egin lse he hea sue A sh disane f he sue, eeaues an be ainained aially a a issue eeaue wihu affeing healhy ells If he size f he hea sue ineases, he hea diffuses easie and he eeaue file eneae fahe in he healhy ells

10 7 - ACKNOWLEDGMENT D Gusav Guieez aeiaes he su f he NSF-NIRT ga and he Univesiy f Pue Ri-Mayaguez f he finanial su his wk 8 - REFERENCES Keih F, Tiehaus K, Li N, Shaw H, Shah RK, Bell K J, The CRC Handbk f Theal Engineeing Ed Fank Keih, Ba Ran: CRC Pess LLC, H G Bagaia and D T Jhnsn, "Tansien sluin he bihea equain and iizain f agnei fluid hyeheia," Inenainal Junal f Hyeheia, vl, 57-75, 5 Bagaia, H and Jhnsn, DT, Analyial and Nueial Sluin a Cneni Shee Mdel and Oiizain f Magnei Fluid Hyeheia Teaen Inenainal Junal f Hyeheia (5) Bagaia, H and Jhnsn, DT Nueial Sluin he Magnei Fluid Hyeheia Heaing f Pailes Absbed n a Cell Wall Peedings f he AIChE Cnfeene, San Fanis, CA Nainal Cane Insiue Hyeheia in Cane Teaen: Quesins and Answes wwwninihgv Rsensweig, R E "Heaing agnei fluid wih alenaing agnei field," Junal f Magneis and Magnei Maeials, vl 5, 7-74, Pennes, H H "Analysis f issue and aeial bld eeaues in he esing huan fea," Junal f Alied Physilgy, vl, 9-, 948 APENDIX A T slve equain (A), given belw T = T q (A) The fllwing hange f vaiable dt = H ' H d / / T H( ) = is ade Then, equain (A) ansfs

11 ' d = d / H H / / H q Mulilying by and ding q = we bained he esnding hgeneus equain ' = d / / H H / H d By exanding / / / / / H ' + H '' H H ' H = 4 Mulilying by / H '' + H ' H = This is he dified Bessel equain f de ½ The sluin is bained in es f he dified Bessel funins I/, K / H = BI / + BK/ T saisfy he ndiin ha H has be finie B = Then I ( ) / = B + Paiula sluin T If q is nsan, hen he aiula sluin is T Then q = I ( ) q / = B + T B is fund f he ndiin T ( ) =, whih ily ha B = = I and he sluin f T is / ( ) q I/ T = I/ q