Mathematical Model of Arterial Hemodynamics, Description, Computer Implementation, Results Comparison

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1 Appld Physcs Rsarch; Vol. 5, No. 3; 3 ISSN E-ISSN Publshd by Canadan Cntr of Scnc and Educaton Mathmatcal Modl of Artral Hmodynamcs, Dscrpton, Computr Implmntaton, Rsults Comparson Elshn Mhal Anatolvch & Guljav Jury Ptrovch Saratov Stat Unvrsty, Saratov, Russa Corrspondnc: Elshn Mhal Anatolvch, Saratov Stat Unvrsty, 83 Astrahansaja Strt, Saratov 46, Russa. Tl: E-mal: mchal_ma@mal.ru Rcvd: Fbruary 3, Accptd: March 6, Onln Publshd: Aprl, 3 do:.5539/apr.v5n3p9 URL: Abstract Analyss of basc paramtrs of a blood prssur n major artrs of human body s ssntal ssu nowadays. It can hlp to slct mor approprat way of surgcal ntrvnton for popl who ar ll wth pathologcal changs of artrs, for xampl athrosclross. Nw mathmatcal modl of blood prssur was dvlopd to do such nd of analyss. Th mathmatcal modl s on-dmnsonal n trms of spatal coordnats. It contans vrty of paramtrs that mas avalabl to adjust ths modl to artral bd of any xact cas of any xact patnt. Pc of softwar was mplmntd on th bass of th mathmatcal modl. Th softwar s amd to buld any partcular part of artral systm and calculat paramtrs of blood prssur n t. Comparson of rsults gathrd by dvlopd softwar and by fnt-lmnt analyss softwar ADINA showd a slght dffrnc. But th calculaton tm for th frst on s a consdrably lss. Kywords: bomchancs, vssl, mathmatc, flow, blood, mchancs, smulaton, modl. Introducton Th am of th study was to addrss mdcal and socal problms assocatd wth th optmzaton of surgcal tratmnt of blood crculaton dsordrs n gnral and as a partcular cas of lowr lmb blood flow dsordr. Statstcs says that cardovascular dsas (CVD) s on of th ladng causs of popl dsablty and prmatur dath n conomcally dvlopd countrs. Surgcal rconstructon s wdly usd for th tratmnt of such dsass. Howvr, thr ar stll no objctv ndcatons whch can hlp to choos a partcular typ of an mplant or a shunt, ts gomtrcal paramtrs, confguraton and typ of rconstructon. Bcaus of ths t s vtal to dvlop a mathmatcal modl, whch should proprly charactrz th actual ntracton of blood flow wth th vascular wall. Such modl should b fast (n trms of ts mplmntaton on a PC) and should hav a st of paramtrs, whch would allow adaptng t to th partcular patnt. In addton, thr s a ncssty to dvlop smpl usr-frndly softwar for prsonal computr (PC), usng th mathmatcal modl. Ths softwar should allow qucly crat a modl of a part of a vascular systm and calculat paramtrs of blood flow n ths part. Such nd of softwar would allow smulatng dffrnt varants of surgcal ntrvnton and would hlp to choos th most approprat varant for vry partcular cas and vry ndvdual patnt n advanc.. Mathmatcal Modl Exstng mathmatcal modls of blood flow do not hav th ncssary st of faturs and paramtrs. Bcaus of ths nw mathmatcal modl was ntroducd to match our rqurmnts. Th modl s lnar and on-dmnsonal. Th man systm of quatons of th modl can b solvd analytcally. In addton, ths modl tas nto account th ntal tnson n th vssl and th angls btwn th artrs n a bfurcaton or n a for of othr nd.. Man Systm of Equatons for Blood Flow Dynamcs Problm Consdr th systm of quatons that ncluds th followng rlatons: smplfd on-dmnsonal dffrntal quaton of a vscous ncomprssbl flud (Guljajv & Kossovch, ): Q R t P 8 z R 9 Q, (..)

2 Appld Physcs Rsarch Vol. 5, No. 3; 3 thr Q-volumtrc vlocty of a blood flow, P-prssur n blood, μ-blood vscosty, ρ-blood dnsty,r-radus of a partcular vssl, z-longtudnal coordnat, t-tm; contnuty quaton whch rlats th volumtrc flow rat Q wth th radal dsplacmnt of th vssl walls w: w Q ; (..) t R z dynamc quatons of moton of an axsymmtrc crcular cylndrcal shll of th mmbran thory (Pdl, 983): u S S T w h K u, (..3) t z R z w T T w h P w S Kw, (..4) t R R z whr h-thcnss of th vssl wall, u-longtudnal dsplacmnt of th vssl wall, K and K -coffcnts of th complanc of tssus n th radal and axal drctons, S and T-pulsaton componnts of axal and transvrs tnsl forc of th vssl, S and T -thr ntal valus; rlatons of dal lastcty of th vssl walls for gnral stat of plan strss Eh S u w, (..5) z R Eh u w T, (..6) z r thr E-lastcty modulus, v-posson's rato; or wll-nown rlatons that ta nto account th ansotropy of th vssl wall (Lhntsy, 977). Thus, w obtan a closd systm of sx Equatons (..) (..6) wth rspct to partal sx unnown valus u, w, Q, T, S, P. It s worth notng that thr as n most paprs blood s suggstd as a lqud that has a constant vscosty and can b modld as a Nwtonan flud (Brgr & Jou, ; Tambasco & Stnman, 3). At th sam tm Chn and Lu (6) found that th dffrnc n th valus obtand for th cas of Nwtonan and non-nwtonan fluds do not xcd % for larg vssls. In addton, Equaton (..3) (..6) allows tang nto account complanc of th vssl walls, whch s an mportant factor n modlng th flow of blood n th vssls (Ruttn, 998). Nxt stp s to smplfy somhow ths systm of quatons. At frst xprss prssur from (..4) and w wll hav followng formula w T T w P h w S Kw (..7) t R R z Now t s possbl to gt rd of th prssur n rlaton (..) usng ths formula for prssur 3 3 Q w w T w w R h T R R S R K 3 8 Q. (..8) t t z z z z z R Substtuton xprssons for forcs (..5) and (..6) n (..8) lads to followng 3 3 Q w w Eh u w w w R h T R R S R K 3 8 Q. (..9) t t z z z R z z z R Aftr groupng trms wth th qual drvatvs (..9) can b wrttn as: 3 3 Q w wt Eh Eh u w R h R RK R R S 8 Q. (..) 3 t t z z R R( ) z z R Th sam transformaton s appld to Equaton (..3) and t tas form of followng formula u Eh u w S T w h Ku. (..) t z R z R z

3 Appld Physcs Rsarch Vol. 5, No. 3; 3 Aftr groupng w hav fnal quaton u w Eh Eh u h S T Ku t R z. (..) z Aftr all dscrbd transformatons w hav shortr systm of quatons nstad of systm of sx quatons for sx unnown valus: u w Eh Eh u h S T Ku t R z, z 3 3 Q w wt Eh Eh u w R h R RK R R S 8 Q, (..3) 3 t t z z R R( ) z z R w Q. t R z It s a closd systm of thr partal dffrntal Equatons (..3) for thr unnown functons u ( z, t), w ( z, t) и Q ( z, t). Ths systm wll b usd as a bass for modlng th dynamcs of blood flow n th artral systm.. Analytcal Soluton for th Systm of Equatons Th unnown functons u, w, Q can b rprsntd n th form of complx Fourr srs: t t ~ u( z, t) U ( z) t, w( z, t) W ( z), Q( z, t) Q ( z),, (..) T thr T-th prod of blood crculaton. Nxt stp s to substtut (..) n systm of Equaton (..3) and thn omt subscrpt nar unnown functons. Aftr rducton by a factor t all quatons of th systm t can b wrttn as dw Eh Eh d U h U S T K U R 3 dw dw T Eh Eh d U d W Q R h R RK R R S 8 Q (..) 3 R R( ) R W ~ dq R Thn w solv ach quaton of systm (..) wth rspct to mmbrs contanng hghr drvatvs of unnown functons. Eh d U KU h U R dw Eh S T, 3 d W dw dw T Eh Eh d U RS 3 Q Rh R RK R 8 Q, (..3) R R( ) R ~ dq R d U In th scond quaton of th systm w can gt rd of th drvatv usng th frst quaton. Aftr that som smpl mathmatcal transformatons s appld and as a rsult followng systm of quatons s obtand W.

4 Appld Physcs Rsarch Vol. 5, No. 3; 3 d U ( K Eh h ) U EhR dw Eh S T, T T S 8 RK R h Q ( K h ) U, (..4) 3 dw dweh 3 4 RS R R R R S R S RS ~ dq R W. It s a systm of thr ordnary dffrntal quatons. It s worth notng that th unnown functons and coffcnts of th quatons ar complx functons and valus. Now w transform th systm of thr dffrntal quatons of scond, thrd, and frst-ordr (..4) to a systm of sx frst ordr dffrntal quatons. For ths purpos w ntroduc th auxlary functons as follows: du V, dw Z, (..5) dz Y. Substtuton of ths functons to th systm of Equaton (..4) lads to th followng rsult: du V dw Z dz Y dv Eh ( K h ) U S T Z Eh EhR dy Eh T T S 8 RK Rh Z Q ( K 4 h ) U RS R R R R S R S RS dq RW (..6) Followng constants ar ntroducd for brvty: A R, B ( K h ), Eh C EhR Eh S T, 8 D, (..7) 4 R S R S K h E T Eh S T, F RS RS KR Rh RS R R R R Assumng ths constants systm (..6) can b wrttn as

5 Appld Physcs Rsarch Vol. 5, No. 3; 3 ~ dq AW dw Z du V dv BU CZ dz Y dy ~ DQ FZ EU (..8) Blow s a matrx form of th systm of Equatons (..8): thr Q W U H ( z) unnown vctor-functon, and V Z Y dh MH, (..9) A M matrx of coffcnts. B C D E F Th systm of dffrntal Equatons (..8) or th matrx dffrntal Equaton (..9) s th fnal rlatons that must b addrssd. To solv th rsultng systm of quatons w nd to form charactrstc quaton dt D A B E C F Calculaton of dtrmnant lads to a complx quaton of sxth dgr 6 4. (..) ( F B) ( BF CE AD) ABD. (..) Th charactrstc quaton s th sxth dgr quaton wth complx coffcnts. Snc th quaton contans only vn powrs of th varabl λ t can b convrtd to an quaton of thrd dgr: 3, (..) F B, BF CE AD, ABD. Thn vry soluton of (..) wll corrspond to two solutons of (..),,, 3,, 3, 4, 5, 6. Thus, w gt sx gnvalus of th systm of Equatons (..9). W can fnd th gnvctors for ach of th sx gnvalus, by solvng a followng systm of quatons 3

6 Appld Physcs Rsarch Vol. 5, No. 3; 3 4 ) ( E M, 6,, (..3) thr 6, 5, 4, 3,,, gnvctor corrspondng to th gnvalu. Equaton (..3) can b wrttn n a form of systm of sx algbrac quatons 5, 3,, 6, 6, 5, 5, 3, 4, 4, 3, 5,,,, F E D C B A (..4) An analytcal soluton of (..4) gvs th followng xprssons for th componnts of th gnvctors A A A 6, 3 5,,,, 6, 5,,, 6, 5,, (..5) B C 6, 4, 3, ) (, B C B C 6, 5,, 4. Th gnvctor for th gnvalu λ can b rprsntd as follows: ) ( 3 B C B C A. (..6)

7 Appld Physcs Rsarch Vol. 5, No. 3; 3 As a rsult unnown functons ar tan a form of followng formulas: U ( z) W ( z) ~ Q ( z) , z C,, z C, (..7), z C. Blow ar xprssons for sx unnown functons for systm of Equatons (..) (..6): Eh S( z, t) Eh T( z, t) u( z, t) w( z, t) Q( z, t) U ( z) W ( z) t t ~ Q ( z) u w Eh z R u w Eh z R t C C 6 6 C C 3,, z t, z t z t C (..8) 3, 3, C R C R,, z t z t w T T w T T w Pzt (,) h w S Kw K h ws t R R z R R z 6 T Eh Eh z t C, K h S 3, R R ( ) R( ) Th rlatons (..8) ar th soluton of th orgnal systm of quatons, but thy ar dtrmnd up to arbtrary constants of ntgraton. For ach artry t s ndd to dtrmn sx arbtrary constants..3 Boundary and Contact Condtons Arbtrary constants apparng n rlaton (..8) should b dtrmnd from th boundary and contact condtons. Undr contact condtons ar mant rlatons whch ar satsfd n a nod whr st of artrs ar connctd to ach othr (.g. bfurcaton or on to on conncton). Such nod n ths artcl s calld contact nod. Sx condtons should b st for ach vssl ncludd n th part of vascular bd that s undr consdraton. Th followng st of rlatons can b tan as such condtons (assum that th part undr consdraton has only on bgnnng (nput) artry and a numbr of ndng (output) artrs): - at th bgnnng of nput artry should b st nput volumtrc blood flow vlocty and fxaton condtons: Q(, t) Q ( t), u z u ( ), (.3.) z - at th output of ndng artrs should b st rlaton btwn prssur and volumtrc blood flow vlocty and fxaton condtons: * ; R Q( l) P( l), t 5

8 Appld Physcs Rsarch Vol. 5, No. 3; 3 u z thr l-th lngth of th vssl. - at contact nods should b st followng condtons: u ( l), (.3.) zl n Q ;, (total volumtrc blood flow vlocty of ncomng and outgong artrs s zro) P P, n, u u, n, w w, n, (.3.3) n Sl S n l, n Tl T n l, thr n-numbr of artrs connctd n th nod, l -crcumfrnc of cross-scton of -th vssl n th contact nod. Rlatons (.3.), (.3.) and (.3.3) form a systm of quatons. Thr w hav thr condtons at th bgnnng and thr condtons at th ach ndng of th part of artral systm that s undr consdraton. To ma th systm closd t s ncssary to hav thr condtons for ach of th artrs at ach of contact nod. Indd, w hav a balanc quaton for volumtrc blood flow vlocty, th two avragd quatons of qualty of longtudnal and transvrs forcs and n quatons for ach prssur and both of dsplacmnts: 3 + 3(n ) = 3n,.., for n artrs n contact nod w hav 3n quatons, n othr words 3 quatons for ach artry. Ths mans that w hav a closd systm of algbrac quatons to fnd all arbtrary constants for ach artry ncludd n th consdrd part of artral systm. To solv ths systm of algbrac quatons frst rlaton that dscrbs nput volumtrc blood flow vlocty should b dcomposd nto a complx Fourr srs (.3.) Q ( t) a a ( a ( a cos t b b ) t sn t) Q q t. (.3.4) Now w ar tryng to fnd soluton for th pulsatng flow sub-problm (only pulsatng componnt of flow s undr consdraton) and bcaus of ths w can dscard th constant trm n (.3.4). Ths trm wll b tan nto account latr durng solvng of stady-stat flow sub-problm. Nxt stp s to substtut nto th boundary and contact condtons xprssons (..8). Thn quat th trms at th sam frquncs ω and dvd by t. Aftr all ths transformatons w hav followng rlatons for th boundary condtons 6

9 Appld Physcs Rsarch Vol. 5, No. 3; 3 at th nput: - at th outputs: 6, C a b, 6 6 3, C, (.3.5) 3, C ; T Eh Eh l RC C K h S, 6 6 * l,, 3, R R ( ) R( ) 6 6 l 3, C, (.3.6) l 3, C. Condtons at th contact nods should b transformd n th sam way, but thr form s dpndd on th confguraton of partcular contact nod, th numbr of ncomng and outgong vssls, thr mchancal and gomtrcal paramtrs. For a complt soluton of th problm t s ndd to solv stady-stat flow problm. Evntually th fnal rsults should b th sum of corrspondnt paramtrs calculatd for rsults pulsatng flow sub-problm and for stady-stat flow sub-problm..4 Smplfcaton of th Modl (for Pulsatng Flow Problm) It s possbl to smplfy th soluton of th problm by rducng th numbr of arbtrary constants of ntgraton n ach artry ncludd n th part of th artral systm. Ths can b don for rasons of lmtng n som rspcts. For xampl, f mass dnsty of th matral of th vascular wall tnds to zro th two roots of quaton (..) wll also tnd to zro. Such passag to th lmt corrsponds to nglctng th nrta forcs of th vssl walls. In ths cas, w choos four gnvalus at ach artry, whch dos not tnd to zro. Boundary and contact condtons can b tan l blow: - at th nput: Q(, t) Q ( t), - at th outputs: w ( ) ; * R Q( l) P( l), - at contact nods: w ( ) ; n Q, P P, u u, n wl w n l. 7

10 Appld Physcs Rsarch Vol. 5, No. 3; 3 Thus, w hav two condtons for ach artry at th bgnnng and ndngs ponts, and two condtons for ach artry at ach contact nod. In othr words, w hav 4 quatons for ach artry. That mans that w hav a closd systm of algbrac quatons for th arbtrary constants. In th smplst cas, only two gnvalus of th four can b lft. Whn longtudnal tnsl forc of th vssl walls S tnds to zro only two gnvalus rman fnt. Thn th boundary and contact condtons can b tan as followng: - at th nput: Q(, t) Q ( t) ; - at th outputs: - at contact nods: * R Q( l) P( l) ; n Q P P. In ths cas w hav two condtons for ach artry to dtrmn th two arbtrary constants whch mans that w hav a closd systm of quatons..5 Soluton for Stady-Stat Flow Problm Ths soluton tang nto consdraton angls btwn artrs n contact nods. At frst w hav to calculat th rsstanc to flow n contact nods tang nto account th angls btwn th ncomng and outgong artrs. As basc rlatons w ta th dynamc contact condtons (Pdl, 983). Ths quatons wr obtand from th consrvaton quatons of a momntum of a contnuum: 8, Q Q Q cos cos Q Q cos Q cos t t t F F F3 Q Q 3 Q Q 3 F F3 3 PF PF PF F F 8 F F3 sn sn cos cos, Q Q sn sn Q sn Q sn 3 t t F F3 Q Q 3 Q Q 3 F F3 3 PF PF F F F F3 cos cos sn sn. 8 8 W can lav n ths quatons only trms corrspondng to th stady-stat flow: Q Q cos Q cos F F F Q Q 3 Q Q 3 F F3 3 PF PF PF F F 8 F F3 sn sn cos cos, Q sn Q sn F 3 3 F3 Q Q 3 Q Q 3 F F3 3 PF PF F F 8 F F3 cos cos sn sn. Lt s consdr th cas whn thr ar only two artrs connctd n th nod: on s ncomng and anothr s outgong. For ths w nd just dscard trms corrspondng to th thrd artry. In addton should b tan n to account fact that bcaus thr ar now only two artrs both of thm hav th sam volumtrc blood flow vlocty ( Q Q ). Evntually w hav followng quatons:

11 Appld Physcs Rsarch Vol. 5, No. 3; 3 Q F Q F Q Q cos F sn P F P F cos, 8 F F Q Q Q sn F cos P F sn. F 8 F F Now w multply frst quaton by cosα and scond on by snα and sum rsultng quatons. Aftr such transformaton w hav nxt formula: Thn w xprss Q from ths t: Q F Q cos P F cos P F. F Q P F cos P F (.5.) cos F F Nxt w consdr lctrodynamc analogy for outgong artry (mthod of lctrodynamc analogy for blood flow analyss was frst usd n study (Guljajv & Kossovch, ) for th cas of ntrnal mammary artry). Elctrcal schma for ths analogy s shown on th Fgur.5.. Thr P -prssur at th nd of th ncomng artry, P -prssur at th bgnnng of outgong artry, P -prssur at th nd of outgong artry, R y -rsstanc of contact nod, R π -Posull rsstanc of th outgong artry, Q -volumtrc vlocty of blood flow. Prssur s an quvalnt for lctrc potntal, rsstanc s an quvalnt for lctrc rsstanc and volumtrc vlocty s an quvalnt for lctrcal currnt. Du to th fact that ths schm rprsnts squncng crcut currnt n ach pont of t s th sam. Accordng to th Ohm's law w hav followng rlatons: P P P P P P Q (.5.) RУ RП RУ RП In trms of Fgur.5. Formula (.5.) for part wth rsstanc R У can b wrttn as: Substtuton of (.5.3) nto (.5.) lads to: Q P P R У PF cos PF (.5.3) cos F F PF cos PF cos F F Thn w transform Formula (.5.4). Blow s a rsult of ths transformaton. P P R У P P F cos F P cos F F (.5.4) (.5.5) Snc all prssurs n formulas hr nsgnfcantly dffr from th normal atmosphrc prssur w can assum Н 5 Н that P 35 and ntroduc coffcnt м м P (.5.6) P Aftr substtuton (.5.6) nto (.5.5) w xprss R y from rsultng rlaton 9

12 Appld Physcs Rsarch Vol. 5, No. 3; 3 R У ( ) cos 5 F F F cos F (.5.7) 3 thr ~ du to th small prssur drop n th blood vssls rlatv to normal atmosphrc prssur. To calculat paramtrs of stady-stat flow w nd to combn a systm of quaton for whol part of artral systm that s undr consdraton. Ths systm should contan followng quatons: - for ach artry rlaton of volumtrc vlocty of blood flow and prssur (analogy for rlaton of lctrcal currnt and potntal) accordng to Formula (.5.) P Q R, У P R, П, (.5.8) thr -numbr of ncomng artry n contact nod; -numbr of outgong artry; P, -prssur at th nd of -th artry, n othr words, prssur rght bfor contact nod; P, -prssur at th nd of -th artry. For th frst P (nput) artry of consdrd part of artral systm w hav rlaton, P, Q. RП - for ach contact nod Q, (.5.9) thr -st of numbrs of artrs connctd n th nod - at th bgnnng of nput artry w st ntal volumtrc vlocty of flow Q = Q, thr Q -constant trm of th srs (.3.4). - at th ndngs of output artrs w st prssur P j, =, thr j -st of numbrs of ndng (output) artrs for th part of artral systm that s undr consdraton (thr P, and P, -prssur at th bgnnng and at th ndng of -th artry corrspondngly). Thus w hav closd systm of quatons to calculat paramtrs of stady-stat flow n th artral systm tang nto account th angls btwn th vssls n contact nod. Fgur.5.. Elctrodynamc analogy.6 Exampl of Systm of Equatons for Stady-Stat Flow Problm Thr s shown on a partcular xampl how to combn such systm of quaton. Exampl artral systm part for th xampl s shown on a Fgur.6.. Th part of artral systm undr consdraton conssts of thr artrs and on contact nod. Th contact nod has on ncomng artry and two outgong artrs. For such vssls bd w hav lctrodynamc analogy that s shown on th Fgur.6.. Nxt stp s to analyz ths schma accordng to Formula (.5.8). In frst part w hav prssur drop (potntal) qual to P, P.. Rsstanc to flow n ths part s qual to Posull rsstanc R R П du to th fact that th frst part s an nput and dos not com out from any contact nod: P, P, Q (.6.) R П 3

13 Appld Physcs Rsarch Vol. 5, No. 3; 3 For scond part drop of prssur s qual to P, P,. Rsstanc n ths cas conssts of two componnts, Posull rsstanc and rsstanc of contact nod and qual to R RУ RУ. Rlaton for volumtrc vlocty can b wrttn as: P, P, Q. (.6.) RУ RП Smlarly w buld rlaton btwn volumtrc vlocty and prssur for thrd part: P, P3, Q3. (.6.3) R3 У R3П Rsstancs of contact nod R У ar dfnd by (.5.7). Posull rsstanc can b calculatd usng followng formula (Guljajv & Kossovch, ): 8l RП, 4 r thr μ vscosty coffcnt of blood, l lngth of vssl, r vssl radus. Accordng to (.5.9) w hav followng rlaton for contact nod: W nd to st nput volumtrc vlocty: Also w nd to st prssur at ndngs: Q Q Q3. (.6.4) Q Q. (.6.5) P P, 3,,. (.6.6) Fnally w hav a systm of algbrac Equatons (.6.) (.6.6) that can b wrttn as: Q Q Q Q Q3 P, P, Q R П P, P Q RУ R P, P Q R3У R3 P, P3, Ths s a closd systm of svn algbrac quatons wth svn unnown valus Q, Q, Q 3, P, P, P, P 3. Ths systm can b solvd usng dffrnt mathmatcal approachs.g. usng Gauss mthod. W can fnd all ndd paramtrs of stady-stat flow by solvng ths systm of quatons. Thus, th problm of th pulsaton of blood flow s fnally rsolvd. Th fnal rsults ar th sum of th stady-stat and fluctuaton componnts., 3, П П 3

14 Appld Physcs Rsarch Vol. 5, No. 3; 3 Fgur.6.. Exampl of vssls bd Fgur.6.. Exampl of lctrodynamc schm.7 Dtrmnaton of Excssv Blood Volum n th Vssls On of th most mportant paramtr of th blood crculaton s an xcssv volum of blood n vssls. To dtrmn ths paramtr w consdr quaton of contnuty for -th vssl. Thr R R w and R w t Q. R z -ntal radus of vssl. Ths quaton can b rwrttn as R Q. t R z Aftr smpl transformaton w hav followng formula: R t Intgratng by th last rlaton, w obtan xprsson Q z. Thn tang nto accordanc l l R Q (, t) Q ( l, t) t. R V (thr V actual volum of vssl) w hav quaton 3

15 Appld Physcs Rsarch Vol. 5, No. 3; 3 V t Q (, t) Q ( l, t). Intgratng ths by quaton by tm gv us formula for xcssv volum of blood n vssl n xact prod of tm t V ( t) V ( t) V () [ Q (, t) Q ( l, t)] dt. (.7.) Substtuton of rlaton (..8) nto formula (.7.) fnally gvs us followng xprsson 3. Th Softwar Program 6 t V ( t) C, ( ). (.7.) Nw pc of softwar has bn dvlopd to prform th actual calculatons usng a mathmatcal modl dscrbd abov. Th softwar program s dsgnd to smulat blood flow n th consdrd part of th human artral systm. Th softwar program allows you to graphcally buld vssls bd to st th paramtrs of blood and ach vssl sparatly. Algorthm assmbl at run-tm systms of quatons for gvn vssls bd. Aftr all systms of quatons ar assmbld algorthm solv thm and output soluton as plots of dffrnt nd nto mult-wndows usr ntrfac. Blow ar lsts of nput and output paramtrs of th softwar program. Input paramtrs: Gomtry of artral systm (vssls bd confguraton). Mchancal paramtrs of blood (dnsty, vscosty). Mchancal charactrstcs of vssls (Young's modulus, Posson's rato, th ntal tnson). Volumtrc vlocty of blood flow at th bgnnng of nput artry Q (t). Th paramtrs rlatng th volumtrc vlocty of blood flow wth a prssur at ndngs Output data: Blood flow prssur n artral bd. Volumtrc blood flow vlocty n artral bd. l Lnar vlocty of blood flow (avrag ovr th cross scton). Excssv volum of blood n artrs. Rsults ar rprsntd as plots of rlatonshp btwn dffrnt paramtrs of blood flow and longtudnal coordnat and tm. Thr s ablty to gt two-dmnsonal and thr-dmnsonal plots. Two-dmnsonal plots show th chang of paramtr along on of varabl wth th scond on fxd. Also t s possbl to vw anmatd graphcs whr ach fram shows plot for partcular valu of fxd varabl. On mor fatur s that t s possbl to vw plot for partcular artry or a st of plots for vry artry n systm n on wndow. Scrnshots of th softwar program usr ntrfacs ar shown on Fgurs

16 Appld Physcs Rsarch Vol. 5, No. 3; 3 Fgur 3.. Usr ntrfac ntndd to buld artral bd and st artrs paramtrs Fgur 3... Dalog wndow ntndd to st nput volumtrc vlocty and mchancal paramtrs of blood Fgur Output of calculaton rsults. Plot of rlatonshp btwn avrag vlocty and tm at th bgnnng of frst artry 34

17 Appld Physcs Rsarch Vol. 5, No. 3; 3 Fgur Output of calculaton rsults. Thr-dmnsonal plot of rlatonshp btwn volumtrc vlocty and tm and coordnat Fgur Output of calculaton rsults. St of two-dmnsonal plots of rlatonshp btwn xtnsv volum and tm 4. Comparson of th Rsults To undrstand whthr a partcular mathmatcal mthod or program gvs adquat rsults or not t s usful to compar ts rsults wth xprmntal data or wth data obtand by any othr mthods or othr programs. In ths rsarch w compar rsults obtand usng dscrbd modl and softwar program wth a rsults obtand usng th fnt lmnt mthod. ADINA Systm 8. softwar program was chosn as a fnt lmnt mthod mplmntaton. Thr-dmnsonal modl was bult n a CAD systm Sold Wors Input Paramtrs (Charactrstcs of th Modl) Bd of th fmoral artry of th author was chosn for calculatons (as objct for smulaton). All paramtrs of th artral bd wr obtand by duplx ultrasound dvc Toshba Xaro. A gomtrc modl of th artral systm of th fmoral artry has th paramtrs shown n Tabl 4... Blow s a lst of mchancal paramtrs of artral walls: Matral of th wall consdrd as sotropc wth Young's modulus E. МПа (Bgun & Shulo, 5), Posson's rato v =.5 (Vssl wall s ncomprssbl). Th boundary condtons. Th nputs and outputs cross sctons ar rgdly fxd. Th nnr surfac s th surfac of contact wth blood. No-slp condtons ar slctd on th contact surfac (th flud vlocty matchs th vlocty of th wall on ths surfac). 35

18 Appld Physcs Rsarch Vol. 5, No. 3; 3 Fnt lmnt s a ttrahdron wth dg lngth. m. As a rsult aftr parttonng w hav about fnt lmnts. Blood modl has followng mchancal paramtrs: Dnsty: Vscosty: кг 5 м 3..5 Па с (vscosty can b masurd usng rotatonal vscomtr at vlocty qual to c - ). Th boundary condtons. Th latral surfac s th surfac of contact wth th vssls wall. No-slp condtons ar slctd on th contact surfac. Front surfacs at outputs ar consdrd as fr surfacs. Ths mans that prssur hr s qual to zro. At nput front surfac w st rlatonshp btwn ntal blood flow vlocty and tm (Fgur 4..8) (m/c c). Fnt lmnt s a ttrahdron wth dg lngth. m. As a rsult aftr parttonng w hav about fnt lmnts. Intal volumtrc blood flow vlocty at nput s slctd to b as t s shown on Fgur 4... Fgur 4... Plot of rlatonshp btwn vlocty and tm (m/c c). Doubl prod of pulsaton Tabl 4... Gomtrcal paramtrs of th modl Artry nam Lngth (m) Wall wdth (m) Radus at th bgnnng Radus at th nd (m) (m) Artra fmorals Profunda fmors artry Suprfcal artra fmorals Popltal artry Pronal trun Calculatd Rsults As a rsult of calculatons w hav plots of dpndnc of avrag vlocty on tm at nput and output cross-sctons of th artral systm that s tan nto consdraton. Ths plots ar shown on th Fgurs Nxt stp s to prform th sam smulaton usng th mathmatcal modl and th softwar program dscrbd n ths artcl. For ths purpos w us th sam nput paramtrs (sam part of artral systm, sam blood and vssls paramtrs) as for smulatng usng fnt lmnt mthod. Rsults of calculaton can b found on th Fgurs

19 Appld Physcs Rsarch Vol. 5, No. 3; 3 Fgur 4... Plot of dpndnc of vlocty on tm n th nput cross-scton of th common fmoral artry Fgur 4... Plot of dpndnc of vlocty on tm n th output cross-scton of th dp fmoral artry Fgur Plot of dpndnc of vlocty on tm n th output cross-scton of th popltal artry 37

20 Appld Physcs Rsarch Vol. 5, No. 3; 3 Fgur Plot of dpndnc of vlocty on tm n th output cross-scton of th pronal trun Fgur Plot of dpndnc of vlocty on tm n th nput cross-scton of th common fmoral artry (m/c c) Fgur Plot of dpndnc of vlocty on tm n th output cross-scton of th dp fmoral artry 38

21 Appld Physcs Rsarch Vol. 5, No. 3; 3 Fgur Plot of dpndnc of vlocty on tm n th output cross-scton of th popltal artry Fgur Plot of dpndnc of vlocty on tm n th output cross-scton of th pronal trun 4.3 Compar Rsults To ma comparson of rsults mor llustratv rsults ar xportd nto Mcrosoft Offc Excl. Thn w buld dagrams for th rsults acqurd for both mthods n th sam cross scton on th sam coordnat plan. Rsult of such data massagng ar shown on Fgurs Ths dagrams show that th rsults obtand by fnt lmnt mthod and mthod dscrbd n ths artcl ar clos to ach othr. Th maxmum dvaton btwn rsults of ths two mthods w hav at th momnt of systol and ths dffrnc dos not xcd %. Elswhr n th dagrams curvs dffr slghtly or vn concd. Th curvs n Fgur 4.4. concd compltly du to th fact that t rprsnts nput cross scton and w st up flow vlocty hr manually as an nput paramtr for both of mthods (t s a boundary condton). Thus, th dvlopd mathmatcal modl and softwar systm can b usd to analyz th ovrall, global stat of th prodc lamnar vscous flow n lastc tubs and as a varant of th blood flow n th artral bd. Howvr, snc that s on-dmnsonal mathmatcal modl, t dos not allow us to analyz th dstrbuton of a paramtr n a small nghborhood of a pont (or ara) wth a strong gomtrc or physcal htrognty, for xampl n th small nghborhood of th for. Also ths modl allows calculatng th flow vlocty avragd ovr th cross scton, but t dos not allow obtanng th vlocty profl n a partcular scton. Such problms ar asr to solv usng th fnt-lmnt smulatng. 39

22 Appld Physcs Rsarch Vol. 5, No. 3; 3 Fgur Plot of dpndnc of vlocty on tm n th nput cross-scton of th common fmoral artry Fgur Plot of dpndnc of vlocty on tm n th output cross-scton of th dp fmoral artry 4

23 Appld Physcs Rsarch Vol. 5, No. 3; 3 Fgur Plot of dpndnc of vlocty on tm n th output cross-scton of th popltal artry Fgur Plot of dpndnc of vlocty on tm n th output cross-scton of th pronal trun 5. Rsults and Conclusons ) A on-dmnsonal, lnar mathmatcal modl of th prodc flow of blood. Th modl s applcabl to vascular tr of an arbtrary confguraton. Th systm of quatons of th modl has an analytc soluton, bcaus of ths fact softwar program that mplmnts th mathmatcal modl s fast. ) It s shown that rsults obtand usng on-dmnsonal modl dffr slghtly from th rsults of calculatons obtand usng thr-dmnsonal modl. Howvr, snc that s on-dmnsonal mathmatcal modl, t dos not allow us to analyz th dstrbuton of a paramtr n a small nghborhood of a pont (or ara) wth a strong gomtrc and physcal htrognty, for xampl n th nghborhood of th vssls for or nar athrosclrotc plaqus. Also, th modl dos not allow analyzng th vlocty dstrbuton ovr th cross scton and dos not account bndng of th vssls. 4

24 Appld Physcs Rsarch Vol. 5, No. 3; 3 3) Th nw softwar program was dvlopd. Ths softwar program mplmnts dscrbd mathmatcal modl. Ths softwar program can smulat a wd rang of th vascular systm confguratons and asly customzd to a spcfc cas. 4) Th dscrbd mathmatcal modl and softwar program can b rgardd as a bass for furthr clncal studs to substantat th slcton of th mthod and opton of rconstructon ncludng typ and shap of th plastc matral to sut th ndvdual charactrstcs of artrs of ach patnt. Rfrncs Bgun, P. I., & Shulo, Ju. A. (5). Bomchancs. Moscow, 5. Brgr, S. A., & Jou, L. D. (). Flows n Stnotc Vssls. Annu Rv Flud Mch., 3, Chn, J., & Lu, X. Y. (6). Numrcal Invstgaton of th non-nwtonan Pulsatl Blood Flow n a Bfurcaton Modl Wth a Non-Planar Branch. Journal of Bomchancs, 39, Guljajv, Ju. P., & Kossovch, L. Ju. (). Mathmatcal modls of bomchancs n mdcn. Saratov: Saratov Stat Unvrsty. Lhntsy, S. G. (977). Thory of lastcty of ansotropc sold. Moscow: Naua. Pdl, T. (983). Hydrodynamcs of larg blood crculaton vssls. Moscow: Mr. Ruttn, M. (998). Flud-Sold Intracton n Larg Artrs. Endhovn Unvrsty of Tchnology, Nthrlands. Tambasco, M., & Stnman, D. A. (3). Path-Dpndnt Hmodynamcs of th Stnosd Carotd Bfurcaton. Annals of Bomdcal Engnrng, 3,