Analytical Calculations of the Characteristic Impedances in Arteries Using MAPLE

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1 ecet esearches i Mechaics Aalytical Calculatios of the Characteristic Impeaces i Arteries Usig MAPLE Daviso Castaño Cao Abstract At the begiig of the ivestigatios i health a specially i the cariovascular system, mathematical a physical moel have bee propose, oe of them is the electric aalogue represetatio of the arteries a bloo flow, for usig this it s importat to kow the characteristic impeace of the artery, a ofte this impeace oes ot have i accout the possible coitios the artery coul has. The Navier-Stokes is use with the bouary coitio, represetig Newtoia a o-newtoia fluis, i some represetative cases, maily itereste i Atherosclerosis isease. The Laplace trasform is use as metho of solutio, fiig aalytically the characteristic impeace for ifferet evets. The power of a computer algebra system like MAPLE is preset through this work. Keywors Arteries, Atherosclerosis, Bloo flow, Characteristic impeace, Computer algebra, No-ewtoia fluis. T I. INTODUCTION HE cariovascular system has as primary fuctio utriets a waste trasport through all boy, which is costitute by the veis, the arteries, the bloo vessels a the heart that pump the bloo for a etwork of brachig pipes, oig the istributio of oxyge a the collectio of carbo ioxie, betwee others fuctios. Bloo flow is a usteay pheomeo (use of ifferetial equatio are require), where a ormal arterial flow coul be cosiere as lamiar with secoary flows i the curves a braches. But i some cases, this flow coul tur turbulet, because i the cariovascular system, the eyols umber varies from i the small arterioles to 4 i the largest artery []. I the same way, the arteries varies epeig of the flow a pressure, these variatios create abormal coitios. This abormal coitio coul be prouce by problems ue to cholesterol, geeratig the Atherosclerosis isease. Both coitios (ormal a abormal) are preciously stuie for helpig to reuce the eaths maily i evelope coutries where the majority of eaths are the result of cariovascular iseases []. Similarly, the pressure a flow are pulsate, because of the heart cyclic fuctio, systolic a iastolic perios (i some special occasios the flow goes i the cotrary irectio of the pressure). As the viscosity is ot costat throughout the artery, it s ecessary to cosier the bloo (a flui compose by cells, proteis, lipoproteis) as a o-newtoia flui, a o-newtoia viscosity which is wiely stuie i the biorheology fiel. This ki of stuies will be useful for the preictio of particulars flow i each patiet, a the esigs of electroic evices that imitate of alter the bloo flow. For carryig out the stuies i hemoyamic pheomea [,4], it s easier to brig a electrical moel for the flui system, i which the pressure is aalogue of voltage a the flow is aalogue of curret, cosequetly, it s require a equivalet impeace for this moel [5]. I this work, we preset the calculatios base i the Navier-Stokes equatio for fiig the characteristic impeaces from ifferet possible cases i arterial pheomea as ormal artery comportmet or altere artery comportmet which is like with the atherosclerosis isease. II. POBLEM Hemoyamic flui systems coul be escribe by the Navier-Stokes equatio, which is presete i cylirical cooriates as a facility for the treatmet: v( r, t) P( z, t) v( r, t) v( r, t) () t z r r r where ρ is the esity of the bloo, μ is the viscosity, both costats. A v( r, t) a P( z, t) / z are the fuctios of spee a pressure graiet respectively. The omai of applicatio is a arterial cylirical sectio with raius a legth l. It s ecessary to have the iitial coitio a bouary coitios for obtaiig the complete solutio of a ifferetial equatio. As iitial coitio we will have a ull coitio for all cases treate i this stuy, v r, () Mauscript receive July 5,. This work was supporte by EAFIT Uiversity. D. Castaño Cao is Logic a Computatio Group at EAFIT Uiversity, Meellí, Colombia, South-America ( casta@eafit.eu.co). ISBN:

2 ecet esearches i Mechaics We have two bouary coitios, which represet a Newtoia or o-newtoia Flui, these coitios are presete i the below sectios. Moreover, i some cases we shoul use mathematical tools like the sigularity aalyze for gettig the complete solutio. A. Normal Newtoia Artery For a ormal artery, where the iertial forces are more importat that the viscous forces, we ca treat the flui yamics as a Newtoia flui, i the moel this is represete by the bouary coitio of isplacemet i the artery borer is zero B. Normal No-Newtoia Artery v, t () I the opposite irectio from the previous sectio, i the case of the viscous forces caot be eglecte, we have to cosier the flui as o-newtoia. For this case the bouary coitio epes o the spee a acceleratio of the flui. vr, t vr, t r r Where is the frictio factor betwee the bloo a the artery wall. C. Altere Artery with Liear Viscosity Shape For a artery with some ki of isease that is relate with the viscosity of the bloo, as example the atherosclerosis i which the cholesterol i the bloo chages his viscosity i a ot costat way, but we cosier that cholesterol istributio oes ot chages the flui esity. Because of the ocostat viscosity, the Navier-Stokes is moifie, a we have r vr, t Pz, t vr, t t z r r r vr, t r r r v r, t r as a first case we are goig to cosier the viscosity variatio as liear, with a viscosity i the cetre of the artery that icreases outwars. r (4) (5) r (6) We solve this ew equatio usig bouary cosieratios, Newtoia () a o-newtoia (4) coitio. Other kis of expressios for the atherosclerosis isease are presete by Wag [6]. D. Altere Artery with Quaratic Viscosity Shape As a extesio of the previous situatio we exte the viscosity fuctio ito a quaratic expressio a keep the Navier-Stokes (5). Equally we fi the impeace for the Newtoia coitio () as much as the o-newtoia coitio (4). The viscosity fuctio turs to: r r (7) This situatio coul represet a worse state of the isease tha previous oe because the viscosity icreases too much at the artery wall a ecrease the bloo flow. E. Altere Artery with Two Fluis Other way to aalyze the cariovascular isease as the atherosclerosis is to suppose that there are two fluis isie the artery, oe i the ier part a the other oe i the outer, which have ifferet characteristics as esity a viscosity, but costat for each oe. I this case, we have to solve the Navier-Stokes equatio () for each flui (this fluis coul be bloo a fat or lipi) a use a coitio i the iterface betwee them:,, v t v t i i (8) As previous problems, we make the calculatios for the Newtoia () a o-newtoia (4) bouary coitios. III. METHOD The process for solvig the ifferetial equatio ivolves the Laplace trasform, which take the equatio from the time omai to the frequecy omai. If we thik i the iput of our system, the heart, it has a perioic sigal, so, the Laplace omai makes the aalysis much easier. This metho is show i Fig.. Solvig the ifferetial equatio with the help of the iitial a bouaries coitios, we obtai the spee fuctio of the bloo ito the artery, for gettig the characteristic impeace, first we procee to itegrate the spee ito the artery cross sectio, obtaiig the flow Q a after we make the ivisio betwee a pressure ifferece a the flow, which is aalogue to the electric scheme. P V Z,that is aalogue to Z (9) Q I IV. ESULTS We preset the characteristic impeaces from the previously cases, respectig the same orer a havig the correspoig umeratio. A. Normal Newtoia Artery I this first case we preset all step as example of the metho, searchig the characteristic impeace. We start takig Laplace trasform to the Navier-Stokes equatio (): sv r vr, Pz z V r r r a we apply the iitial coitio () r V r i () ISBN:

3 ecet esearches i Mechaics sv r Pz z () V r V r r r r ow, we solve the system: Pz () s s V () r z r C r C () s where (x) is the Bessel fuctio of the first ki a (x) is the Bessel fuctio of the seco ki. After that, we make a series expasio of () arou r, lookig for sigularities. () Pz r s C l C z ( r ) () s here, we see that there is a logarithmic sigularity at r, if the costat C is ifferet of zero, so, C a Pz () s V () r z r C (4) s we fi the last costat usig the bouary coitio () i (4) C s Pz z s (5) we rewrite the solutio icluig the fou costat, obtaiig: s Pz r V( r) z s s (6) the term isie of the Bessel fuctio is relate to the Womersley umber which is (7) with the frequecy of pumpig, we rewrite Pz () / z yi Vr / i i r (8) with y / this result is similar to that presete by Womersley [7]. After we have the spee we procee to calculate the flow, itegratig the spee ito the artery cross sectio Q( s) V ( r) rr (9) icluig (8) i (9), we solve: s Pz Q( s) z () s s s where (x) is the moifie Bessel fuctio of the first ki a it coul be simplifie usig / same i ) Q s s Pz () z s (the () the graiet of pressure is suppose costat, alog the artery istace L we obtai so: Q s Pz () z P L () P sl usig the relatio (9) we obtai the characteristic impeace L s Zs () usig a recurrece relatio betwee Bessel fuctio it is simplifie to () (4) (5) Zs () L s B. Normal No-Newtoia Artery (6) As the previous result, we use the metho applyig the o- Newtoia bouary coitio (4), a we obtai: Zs () L s C. Altere Artery with Liear Viscosity Shape (7) Now, we solve the Navier-Stokes equatio with viscosity variable (5) with the viscosity fuctio (6), a we fi the impeace from its solutio. This first result is mae takig i accout the Newtoia flui cosieratio (): L s HeuC,,,,, z Zs () (8) ISBN:

4 ecet esearches i Mechaics with HeuC,,,,, z HeuC,,,,, zr s / a z / rr. This solutio has a itegral isie which is ot resolve yet, but for the applicatio we ca use umerical solutios which will provie results. Equally, the impeace is escribe by the Heu Cofluet fuctio HeuC which coul be reviewe i [8,9]. For the o-newtoia case (o-newtoia coitio)(4), the impeace expressio we fi is presete below: L s A Zs () (9) A Heu C,,,,, zr rr with A as follow: A HeuC,,,,, z HeuCPrime,,,,, z () i the last equatio appears HeuCPrime which is erivative of the Heu Cofluet fuctio. D. Altere Artery with Quaratic Viscosity Shape As the previous result, this oe is obtaie from the Navier- Stokes equatio with viscosity variable (5), but i this case with the viscosity fuctio (7). As first result we preset the characteristic impeace for a flui cosiere as Newtoia (): W L s Zs () W W ( r) rr () here appears the Legere Polyomials ( x ) where the egree is efie by s a the argumet, which is fuctio of the raius r W ( r) () r () For the o-newtoia flui cosieratio (4), the characteristic impeace we obtai is L s B Zs () B W ( r) rr with B as follow: 4 B m m W m W with a ew egree for the Legere polyomials m s m (4) (5) (6) E. Altere Artery with Two Fluis I this subsectio, we preset the characteristic impeace, which expressio is big so, we have to ecompose ito multiple parts, we start reefiig the argumets of the Bessel fuctios that appears: s s s Y Z (7) also we preset some expressio of Bessel fuctios compresse to get space: i iy Y i iy Y i i iy Y i i Y i i Y 4 for the Newtoia coitio (), we obtai: Zs () L s where C is efie as C C C C / (8) (9) C Z Z s (4) C ' is efie as follow C ' / 5/ Z s Z i i i C '' is escribe by s Z iy i s Z i i Y s (4) ISBN:

5 C ' Z s 4 Z i iy Y s Z Y i iy s Z Y i i iy Z Y i iy Y iy s s Now, we preset the solutio for the characteristic impeace for a o-newtoia flui (coitio (4)): L s D Zs () (4) (4) D C C D s s where D is efie as D ( ) Z s / / ( 4 ) s D ' is efie as follow / D ' Z s Z Z Y i i i iy 4 s 5/ V. DISCUSSION AND CONCLUSIONS ecet esearches i Mechaics (44) (45) MAPLE as other CAS (Computer Algebra System) have the last mathematical tools for helpig i the evelopmet of our aalytical moels, but i some cases the results are presete i a rustic way, so it is missig a capacity of factorize. ACKNOWLEDGMENT The author is greatly iebte to Prof. Jua F. Ospia for his costat ecouragemet a his costructive avices. The author woul like to thak Prof. Dr. Eg. Arés Sicar a Prof. Mario Elki Vélez for keepig compay i this work i the Logic a Computatio group. EFEENCES [] D.N. Ku, Bloo Flow i Arteries, Aual eview of Flui Mechaics, vol. 9, Ja. 997, pp [].M. Nerem, Vascular Flui Mechaics, the Arterial Wall, a Atherosclerosis, Joural of Biomechaical Egieerig, vol. 4, 99, pp [] B. Das a.l. Batra, No-Newtoia flow of bloo i a arteriosclerotic bloo vessel with rigi permeable walls, Joural of theoretical biology, vol. 75, 995, p.. [4] N. Mariamma a S. Majhi, Flow of a ewtoia flui i a bloo vessel with permeable wall A theoretical moel, Computers & Mathematics with Applicatios, vol. 4,, p [5] G.N. Jager, N. Westerhof, a a Noorergraaf, Oscillatory Flow Impeace i Electrical Aalog of Arterial System: epresetatio of Sleeve Effect a No-Newtoia Properties of Bloo., Circulatio research, vol. 6, Feb. 965, pp. -. [6] H.H. Wag, Aalytical moels of atherosclerosis., Atherosclerosis, vol. 59, Nov., pp. -7. [7] J. Womersley, Metho for the calculatio of velocity, rate of flow a viscous rag i arteries whe the pressure graiet is kow, The joural of physiology, vol. 7, 955, pp [8] S.Y. Slaviaov a W. Lay, Special fuctios: a uifie theory base o sigularities, New York: Oxfor Uiversity Press, USA,. [9] P.P. Fiziev, Novel relatios a ew properties of cofluet Heuʼs fuctios a their erivatives of arbitrary orer, Joural of Physics A: Mathematical a Theoretical, vol. 4,, pp We have suppose the pressure graiet is costat i the artery legth (), but if we o t o it, the characteristic impeace will rest ivariable with the exceptio of the appearace of L (as example the result (6) is similarly presete by Jager et al. [5]). We iclue L, which is alog z, because we coul chage it for a ifferetial of legth L a make the raius epeig of this: z, obtaiig i this irectio a impeace for the arteries with variable sectio what it is the most frequetly case. Usig the itegratio a mixig the impeace results we coul obtai the characteristic impeace of the complex arterial system. Mr. D. CASTAÑO CANO was bor i 989 i Meellí, Colombia. Where he attee i the basic scieces school for the Egieerig Physics egree at the EAFIT Uiversity i 7. Later he wet to Uiversité e Techologie e Compiège i Frace to get his master egree i mechaics a systems. Mr. Castaño Cao has worke as a assistat of the research i the stuies of the variable L circuits i his uiversity i. He has publishe Computer Algebra a Mechaize easoig i Mathematical Epiemiology (Proceeigs of the Worl Cogress o Egieerig a Computer Sciece, Sa Fracisco: 9, pp. 6-4), Desig, Moelig a Costructio of a liear ifusio of bomb ( the III Natioal Coferece of Physic, Sata Marta,Colombie,9)with Mr. J.M. Lopez a Geeralizatios i Mathematical Epiemiology: Usig Computer Algebra a Ituitive Mechaize easoig (Machie ISBN:

6 ecet esearches i Mechaics Learig a Systems Egieerig, S.-I. Ao, B. ieger, a M.A. Amouzegar, es., Dorrecht: Spriger Netherlas,, pp ). Mr. Castaño Cao has the memberships i IAENG. He also has publishe Diseases Propagatio Aalysis Due to Natural Disasters Usig Computatioal Mathematics ( IEEE Seco Iteratioal Coferece o Social Computig, IEEE,, pp. 6-). ISBN: