Eindhoven Univeiy of echnology Depen of Bioedicl Engineeing Meil echnology Modeling of he heodiluion cuve fo coony flow een Cel Se U/e 9-6-7 BME7. pof.d.i. F.N. vn de Voe
Conen H Inoducion 3 H Mehod 5. Inoducion 5. he ixing che 5.3 he ue 8.3. Conevion euion 8.3. He lnce fo he wll 9.3.3 heodiluion euion.4 Soluion of he pole.4. Pole definiion.4. Scling.4.3 Soluion 3 H3 eul 5 H4 Concluion nd dicuion 7 efeence 8
H Inoducion he coony ciculion upplie he h wih lood. Evey ie he h e, ou 5 pecen of he oke volue goe o he coony eie. In eing iuion, when he cdic oupu i ou 4.5 l/in, hi yield n vege coony lood flow of 5 l/in []. When coony ey i enoed, he flow hough h ey will decee, depending on he eveiy of he enoi. Since inevenion y Pecune nluinl Coony Angioply PCA o yp ugey in ode o evculie he coony ciculion hould only e pefoed in evee ce, i i ipon o e le o ccuely deeine coony lood flow in ode o dignoe he eveiy of enoi. Mehod o indice he degee of deceed lood flow, e he een of he Coony Flow eeve CF nd he Fcionl Flow eeve FF []. he CF give he io eween flow duing xiu vodilion hypeeic flow nd flow in e eline flow. he FF i he io eween xiu flow in he enoed ey nd he xiu flow in helhy ey. Boh io e ued o deeine he heodynicl e of he ey nd he peiphel vcul ed. o deeine he flow in n ey, he ehod of heodiluion cn e ued. hi ehod ue olu injecion of cold line ino he loode nd eue he epeue chnge locion dil of he injecion. An exple of he kind of epeue pofile which i eing eued, i picued elow. hi cuve, whee he eued epeue i ploed gin he ie, i clled heodiluion cuve figue.. figue.: he heodiluion cuve d fo N.H.J. Pijl, ued wih peiion 3
If he line injecion nd he heodiluion cuve would e iple del-funcion, he flow velociy would e deived y dividing he veled dince of he line y he ie ken o vel h dince. Since he cuve i no del-funcion, en ni ie, n, i inoduced. n i he veged ie i ke fo he cold line o vel fo he poin of injecion o he poin of epeue eueen. n i clculed uing []: n. Hee epeen he epeue dop given. Fo n nd he locion of he epeue eueen he flow velociy cn e deeined. he gol of hi udy i o odel he heodiluion cuve. hi i done in ode o e le o e he lood flow oe pecie nd o coe o n uoic flow deecion lgoih. In chpe he ehod ued o odel he heodiluion cuve e explined. Afe n inoducion, he fi p of he odel i peened. Hee he ixing of he cold line wih he lood in fon of he ennce o he coony ey i epeened y ixing che. he euling ixue hen vel ino he coony ey. he ey i odeled ue in ecion.3 nd he effec on he epeue y diffuion, convecion nd he exchnge hough he wll e clculed uing euion fo conevion of nd enegy. he euling heodiluion euion i hen olved in ecion.4. Chpe 3 conin he eul nd he epo i concluded in chpe 4. 4
Mehod. Inoducion In ode o odel he heodiluion cuve, wo copen e defined [3]. he fi copen i he cul injecion of he cold line ino he lood. he line i injeced hough chee which end ju in fon of he oic vlve nd he ennce o he igh o lef coony eie. Hee he line i ued o e ixed wih he lood efoe hi ixue vel ino he econd copen of he odel, he coony ey. In he coony ey, he ixue of line nd lood vel fo he ennce of he ey, p he epeue euing locion nd fuhe ino he coony nche.. he ixing che he locion whee he cold line i injeced i efeed o he ixing che. Hee he line ixe wih he lood nd hi ixue ene he coony ey. he ixing che i defined depiced in figue.:, lood flow line flow flow o o flow o coony ey, V, lood epeue line epeue ixue epeue V volue of he ixing e, figue.: Definiion of he ixing che Coplince of he ixing che i negleced, o conevion euie:. Fo conn nd eul deniy nd conn nd eul pecific he conn peue fo oh lood nd line, he he euion i: 5
V d. enging nd dividing y V give: d V V.3 o oin, he genel oluion fo fi ode diffeenil euion i ued: if y p x y x, wih h p x dx hen [ ] h h y x e e dx c Fo he ixing che diffeenil euion hi yield, wih : V V e e V.4 If he line injecion i ingle ue wve h egin nd end 4 figue., hn he coeponding epeue pofile, ed on he given pee vlue, i depiced in figue.3. Hee, edy ce i ued. o o figue.: he inpu flow; he injecion i epeened y he olid line. 6
76 l/ 4 l/ 7 l/ l/ V 8 l 37 C C figue.3: he ixing che oupu epeue he oupu epeue will e he inpu epeue fo he econd copen of he odel, which i he coony ey. 7
.3 he ue he epeue of he ixue, while veling hough he coony ey, chnge ove ie due o he convecion, diffuion nd he exchnge he wll. hee fco e ued o clcule he chnge in epeue of he ixue ove cein dince nd ie in n ey. o iplify he pole, he ey i defined ue wih pecificion given in figue.4. u d wih: u velociy in -diecion ue diu d wll hickne figue.4: Definiion of he ue.3. Conevion euion o clcule he chnge in epeue, conevion euion fo enegy nd e needed. he enegy euion wihou inenl he poducion nd vicou diipion uing cylindicl coodine i defined : ρc u θ λ[ ] θ θ p u u.5 wih lood deniy ρ, he cpciy of lood c p nd he diffuion coefficien of lood λ. he conevion euion i defined : ρ ρ v.6 ρ Blood i ued o e incopeile, o nd ρ. hi led o v.7 8
Wien in cylindicl coodine hi give: uθ u v u.8 θ Becue fully developed flow i ued, he velociie h e no in he diecion of u he longiudinl xi of he ue e eo: u u θ, o fo.8:. Suiuing hi ino he enegy euion.5 give: u, fo.9 λ wih:, he he euliing coefficien fo lood. ρ c p.3. He lnce fo he wll When uing.9 o deeine he epeue pofile in he wll of he ue, he he diffuion in -diecion cn e negleced, ecue << in he wll. hi led o he following euion fo he he lnce in he wll: w, fo < d. wih w he he euliing coefficien of he wll If he peneion deph of he he in he wll i uch lle hen he wll hickne, hen he e i eo. Since w i conn hi led o: d d d d. Inegion of. wih d he oue wll diee, whee i he ody epeue, nd he inne wll diee, whee : ln ln d. ln ln d wih: epeue in dil diecion epeue on he ouide of he wll epeue on he inide of he wll d wllhickne king he deivive of. give he e: 9
d ln ln.3 which cn e ued in.9..3.3 heodiluion euion Ineging.9 ove dil lice now give: d d d u d π π π π.4 ln ln d u π π π π.5 divide y π : ln ln d u.6 Now define ε : ln ln ln d c d p ρ λ ε.7 nd ince u i conn nd coepond wih he conn flow defined in he ixing che, A u. Hee A i he co-ecionl e of he ue. he euion fo he epeue chnge, o heodiluion euion, ecoe: o A ε, fo >, >.8
.4 Soluion of he pole.4. Pole definiion In ecion.3.3 he heodiluion euion i defined : A ε, fo >, >.9 o olve hi euion n iniil condiion cn e pecified:,. nd he ixing che oupu epeue, in ecion., i he oundy condiion fo : V V V e e F,. I i convenien o cle.9, in ode o iplify he pole..4. Scling, nd e he vile h need o e de dienionle, ince ll ohe e conn. hey e de dienionle uing:. Suiue hee in.9 nd enge: A ε.3 divide y nd uiue ε : A ε ε.4
Now uiue : εa εa.5 εa And wih D c, he euion ecoe: D c.6 o iplify he ue of he euion, he index i dopped nd he dienionle heodiluion euion ecoe: D c.7 Since he e in.7 e cled nd O, he vlue of D c i evlued uing [3]: ε. / A 7.e-6.4e-7 / 4e-6 3 / D c h vlue of ppoxiely 4.8e-8. hi en i i poile o dicd he econd ode e in.7:.8 In ode o olve.8 he vile nd e de dienionfull gin uing:, wih ε nd, wih εa I i convenien o leve dienionle fo now, u he index i einoduced o e le o dicen eween he dienionle nd dienionfull. Euion.8 now ed fe enging:
A ε.9 o iplify hi, he he euliing coefficien fo he wll, ε, i ued eo. hi leve he line dvecion euion: A.3 Suiuing he dienionle epeue, ccoding o only eul in he lo of he ccen:, ino.3 A.3.4.3 Soluion he oluion of.3 cn e oined wih he ehod of eping vile. wo new vile eplce, ccoding o, X Y. Suiuion in.3 nd enging yield: Y Y AX X.3 Euion.3 only hold if oh ide of he euion e eul o he e conn. In hi ce he conn i defined k nd wo epe euion e oined: Y Y k nd AX X k.33 Soluion of.33 nd einoducion of, led o: ln Y k C nd ln X k C A.34 Y k C e nd X A k C e.35, A k X Y Ce.36 he conn C nd k cn e olved uing he iniil nd oundy condiion fo. nd.. 3
4 Coining. nd.36 led o: k A Ce k A e C.37 Coining. nd.37 led o: k Ce F A k e F ln A F k.38 Now he vlue fo C nd k cn e uiued in.36: ln, A F e.39 o iplified: A A F,.4 wih: V V V e e F.4
3 eul Uing.4, epeue cn e clculed fo ny poiive nd. hi en h fo given dince fo he ennce of he ue, epeue pofile cn e oined. An exple i hown in figue 3., whee 5 c nd he ixing che oupu fo figue.3 i ued inpu. 76 l/ 4 l/ 7 l/ l/ V 8 l 37 C C figue 3.: he heodiluion oupu epeue he pofile h i oined i lo excly he e h of he ixing che oupu. Howeve, he pofile doe flen lighly owd 37 o C fo inceing eueen dince fo he ennce of he ue. Figue 3. how he ppoxiion of he pien d fo figue. y he heodiluion odel. he ppoxiion w de y viion of he pee fo.4. Mo of he pee e known nd hey cn e ued o deeine he wo eining unknown pee. hee unknown pee h need o e djued o oin pope ppoxiion e he coony flow,, nd he ixing che conol volue, V. he pee ued o fi he pien d in figue 3. e: 8 l/ 4 l/ 5 l/ 37 C C 5
5 l/ V 6 l figue 3.: heodiluion cuve i fied o pien d 6
H4 Concluion nd dicuion A odel w de o poduce heodiluion cuve. In he ixing che copen of he odel, he ixing of he injeced cold line wih he lood i odeled. In he econd copen, he npo of he ixue hough ue i odeled, euling in.8. he ue of.4 oundy condiion fo olving.8 finlly euled in.4. hi euion decie he epeue of he conveced line nd lood ixue. o e le o eie coony flow, evel eueen pecific pee ued in.4 nd.4 need o e defined. Uing hee pee, cuve cn e ched o pien d y djuing he unknown coony flow,, nd he ixing che conol volue, V. Howeve, hi ching pocedue need o e fuhe evlued efoe i cn e pplied. Fo he oined heodiluion cuve, i ppe h pien d cn e ppoxied wih he cuen odel. Whehe he ppoxiion yield ule vlue fo he coony flow h no een inveiged in hi udy. Alo, he odel w iplified o line dvecion pole, which poly educe he ccucy of he eiion of he coony flow. Howeve, if he npo of he ixue cn e decied y line dvecion euion, hen he inpu epeue pofile he ennce of he ue i highly defining fo he finl ppoxiion of he heodiluion cuve. hi en h fuue wok involving he heodiluion cuve pehp hould concene on oe dvnced odeling of he line injecion cuve inpu fo he ixing che in ed of he ep funcion ued in hi epo. hi could eul in ee heodiluion cuve ppoxiion nd coneuenly oe ccue coony flow clculion. 7
efeence [] A.C. Guyon, J.E. Hll, exook of Medicl Phyiology, W.B. Sunde Copny, [] B. De Buyne, N.H.J. Pijl, L. Sih, M. Wievegg, G.. Heyndickx, Coony heodiluion o Ae Flow eeve, Ciculion ; 4:3-6 [3] J.A.M. Schuun, D nlyi of heodiluion fo coony flow, Meepo M.6, 8