HEMODYNAMIC SIMULATION FOR CONGENITAL HEART DISEASE

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1 Seventh Internatonal Conference on CFD n the Mnerals and Process Industres CSIRO, Melbourne, Australa 9-11 December 2009 HEMODYNAMIC SIMULATION FOR CONGENITAL HEART DISEASE J L LIU 1, Y QIAN 1, 4*, M UMEZU 1, K ITATANI 2, 3 and K MIYAJI 2 1 Center for Advanced Bomedcal Scence, TWIns, Waseda Unversty, Toyo, Japan 2 School of Medcne, Ktasato Unversty, Kanagawa, Japan 3 Graduate School of Medcne, The Unversty of Toyo, Toyo, Japan 4 Australan School of Advanced Medcne, Macquare Unversty, Sydney, Australa *Correspondng author, E-mal address: sen11@aon.waseda.p ABSTRACT Hypoplastc left heart syndrome (HLHS) s one nd of congental heart dseases of newborn. The surgeres for HLHS have to be carred out at very early stage. However, currently there are no quanttatve standard to evaluate and predct the outcome of the therapy. In ths study, computatonal flud dynamcs (CFD) s ntroduced smulate the hemodynamcs after Norwood surgery; a frst stage surgery for HLHS. Blood flows derved from Echocardography measurements were used as boundary condtons for pulsatle calculaton. The crculatons of blood flows were observed and the flow dstrbuton n each vessel s estmated. Energy losses (EL), local pressure, and wall shear stress (WSS) n anastomoss were analyzed to estmate the result of HLHS treatment. The results ndcate that pulsatle smulaton s essental to quanttatvely evaluate the qualty of HLHS operaton, and the method of computatonal hemodynamcs analyss could be appled nto the process of the operatons for surgcal optmzaton. Keywords Hypoplastc left heart syndrome, Norwood operaton, computatonal flud dynamcs, hemodynamcs NOMENCLATURE α Womersley Number turbulent dsspaton rate μ dynamc vscosty ρ densty ω angular frequency Δ I dmenson of grd cell Δ t mamum tme step sze f body forces, coordnate aes turbulent netc energy p pressure t tme u tme averaged velocty u fluctuatng velocty C r Courant Number D characterstc length E energy EL energy loss Q volume flow rate Re Reynolds Number U velocty INTRODUCTION Hypoplastc left heart syndrome (HLHS) s one nd of serous congental heart dseases (CHD). Wth a small, underdeveloped left ventrcle, a heart sufferng from HLHS cannot effectvely supply enough blood flow to provde for the needs of the body. In order to mprove blood crculaton, surgery for HLHS has to be carred out at a very early stage. In general, three-stage pallatve surgcal management for newborns s now wdely accepted (Bove et al, 1996, McGur et al, 2005); Norwood, Glenn, and Fontan. Regardng to the Glenn and Fontan operaton, there are almost all publshed studes were focused on them not only n clnc but also CFD smulaton, such as Bove (2003), Mglavacca (1996) and Orlando (2006). Severs et al (1998) llustrated the turbulence at anastomoss of the total cavopulmonary connecton (TCPC) area was the reason of causng energy dsspaton, flow dsturbances and maldstrbuton of pulmonary blood flow by applyng the methodology of CFD. Whtehead et al (2007) studed the effects of eercse on TCPC power loss and varyng relatve blood flow to each lung on the power loss under eercse condtons. The analysng methods of CFD on mnmzng energy loss under restng condtons or eercse condtons through the connecton area to acheve an effcent crculaton and evaluate the hemodynamcs of operatons have made great contrbuton to the better desgns of Fontan operaton. The Fontan procedure and ts subsequent modfcatons over the past 30 years can be descrbed as a class of surgcal procedure (DeGroff, 2008). Compared wth Glenn and Fontan operaton, as the frst-stage surgery, Norwood operaton, although t plays an mportant role n the total treatment of HLHS and t has ganed popularty as a means of pallatng many lesons by creatng new local blood crculaton to and from lungs around some of the defectve areas of the heart, there are currently no quanttatve standard to evaluate and predct the outcome of ths therapy. Due to the dffculty of studyng the outcome drectly n vvo, mage-based computatonal flud dynamcs (CFD) was ntroduced to smulate the blood flow of the Norwood operaton. In present study, a computatonal hemodynamc analyss system (Qan et al, 2008) whch was developed to estmate the rupturng process of cerebral aneurysms was ntroduced. The promoted CFD methodology to analyse blood flow based on patent-specfc geometry, wth flow boundary pressure and flow condtons measured n-vvo n Ktasato Unversty Hosptal by usng a catheter wth Copyrght 2009 CSIRO Australa 1

2 pressure sensor and echocardography. A seres of n-vtro verfcaton and valdaton were carred out, ncludng grd ndependent and calculaton doman nfluence. The am of the study was not only to confrm the applcablty of the computatonal hemodynamc analyss system n the HLHS surgcal optmzaton but to dsclose the characterstcs of local hemodynamcs at the area of anastomoss on predctng the outcomes after Norwood operaton n clnc by applyng the methodology of CFD. METHODS The obectve of ths study s to nvestgate the relatonshps between characterstcs of local hemodynamcs at the area of anastomoss after the Norwood operaton and optmzaton of the operatons for surgcal treatment of HLHS by usng the computatonal hemodynamc analyss system developed n our prevous studes. The mage-based analyss approach employed by the system nvolves the followng four steps. Frst, the patent-specfc clncal data after operatons are acqured by applyng some medcal equpment ncluded the geometry of the vessels whch wll be studed by CT or MRI (Magnetc resonance magng) scan, the nformaton of physologcally pulsatle blood flow velocty profles at the nlets by MRI or Echocardography, and the dstrbuton of blood pressure at the outlets by catheter wth pressure sensors. Second, the three-dmenson (3-D) geometry of patent-specfc vessels are reconstructed by usng the geometry functon of our analyss system from DICOM fles and also converted them nto 3-D numercal model at the same tme. A number of fnte elements are appled to dscretze the numercal model area after the converson for CFD smulaton. Thrd, blood flow smulaton wth the boundary condtons at the nlets and outlets obtaned by the frst step are carred out to acqure a set of converged soluton for the flow feld. Last but not least, several methods of vsualzaton and analyss are appled to analyse the hemodynamc propertes of blood flow n patent-specfc vessels after operatons and to gve some suggestons and optmzaton for the future treatment. The setch of above process could be shown n Fgure 1. Fgure 1: Computatonal hemodynamc analyss process Clncal Treatment A female patent was dagnosed as Hypoplastc Left Heart Syndrome (HLHS), Aortc Stenoss, Hypoplastc Aortc Arch, Coarctaton of Aorta, Ventrcular Septal Defect (VSD), wth echocardography. She was born after 40- wee pregnancy, and was n a shoc state due to ecessve pulmonary flow on 4 days old. She underwent Norwood procedure (aortc arch repar, Damus-Kay-Stansel anastomoss), rght ventrcle to pulmonary artery (RV-PA) condut, and atral septal defect (ASD) enlargement on 8 days old. Wth the agreement of the parents and the protocol whch had prevously receved the approval of the local nsttutonal revew board and the regonal research ethcs commttee, our patent-specfc assocated studes were approved. Numercal Analyss Blood Flow Governng Equatons The blood flow performed by the computatonal analyss system of equatons s the Naver-Stoes (N-S) equaton and contnuty equaton (1) that descrbe the most general movement of flud medum. These equatons are defned below. p ( ) ( ) ρu + ρuu = + μ + + f t ρ + ( ρu ) = 0 (1) t where, =1,2,3, 1, 2, 3 means coordnate aes, u,u and p are the velocty vector and the pressure n the pont of the flud doman, ρ and μ are blood densty and vscosty, t s tme. The term f epresses the acton of body forces. Due to the relatvely large sze of the vessels compared to ndvdual blood cells (McDonald, 1974) and shear rates n lager arteres are typcally suffcently large (Fung, 1981), the blood flows were assumed to be a Newtonan flud, wth constant densty ( ρ = 1060 g/m 3 ) and vscosty ( μ = Pa s) (Lnderamp et al, 1982, Long et al, 2005, Macntosh, 1973), and the body forces of blood was omtted. The Reynolds Number (Re) whch s epressed the moton of flud and Womersley Number (α ) whch s standed for blood flow pulsatle frequency were evaluated respectvely by: ρu D D ρ ω Re = α = (2) μ 2 μ where the angular frequency (ω ) based on the tme scale of nterest s defned as ω = 2π t. The typcal Reynolds number s appromately 4000 n the aorta (Ku, 1997). Therefore, turbulence calculaton would be used n the study. For analyss of turbulent flow, nstead of N-S equatons (1), Reynolds equatons (3) were used: t ( ρu ) + ( ρu u ) + ( ρu u ) p = + μ + (3) where u, 1 u, 2 u are tme averaged velocty components; 3 u, 1 u, 2 u are fluctuatng velocty components. Dfferent 3 turbulence models are used for a seres of these equatons. Because of the relatve hgh-reynolds number calculated above, the standard - model, developed by Launder and Sharma (Launder and Sharma, 1974), was used n the current study. The standard - model has been very successful n a large varety of dfferent flow stuatons. In ths model, Boussnesq s assumpton (4) was employed. The terms ( ρ u u ) n equatons (3) are defned by followng epresson: u u ρuu μ 2 = t + + ρδ 3 (4) Copyrght 2009 CSIRO Australa 2

3 The moton of turbulent flud could be descrbed by new forms of equatons (5) wth other two dfferental equatons of turbulent netc energy (6) and turbulent dsspaton rate (7) whch are defned below. p ( ) ( ) ( ) 2 ρu + ρu u = + μ + μt + ρδ t 3 (5) ( ρ ) ( ρ ) (6) + u = Γ + ρ P t ( ρ ) ( ρu ) + = Γ + ( C 1P ρc 2 ) (7) t where the term P epresses generaton of energy, μ t μt P = ρu u Γ = μ + Γ = μ + (8) σ σ Parameters and μ are defned as follows: t 2 2 μ = μt = ρc (9) μ ρ the constants of - model are as follows: C μ =0.09, C =1.44, 1 C =1.92, 2 σ =1.0, σ =1.3. (Launder and Spaldng, 1974) The energy loss (EL) equaton (10) was used to verfy the grds ndependence n the CFD smulaton and also to evaluate the Norwood operaton, whch s epressed as below: EL = Pnlet + ρunlet Qnlet Poutlet + ρuoutlet Qoutlet nlet 2 outlet 2 (10) Geometry Reconstructon The cross-sectonal mages n clnc were acqured by 16- slce mult-detector row enhanced computerzed tomography (CT) (BrghtSpeed Elte, GE Medcal System, Toyo, Japan). Each slce thcness was mm. The seres of orgnal cross-sectonal CT mages was reconstructed by commercal software, RealINTAGE R. A non-shrnng smoothng technque was employed to generate the numercal model for CFD smulaton. The accuracy of the reconstructed model had been checed aganst the true geometry wth an eacted measurement. The geometry after smoothng etracted from patent CT mages was shown n Fgure 2(a). Mesh Generaton The commercal software, ANSYS R -ICEM 11.0, was appled to the grd generaton. In order to accurately measure WSS at near-wall-regon, the boundary-ftted prsm layers were generated at the boundares to mprove the resoluton of the relevant scales n flud moton. The meshes were shown n Fgure 2(a). In present study, there were fve layers generated wth an average nodal space, ncreasng by a rato of 1.3. The dstance of the frst layer to the vessel surface was fed at 0.02 mm. The total thcness of the layers was changed wth the dfferent branches from 1.3% to 14.5% dependng on the average local-vessel dameters. Because the accuracy of CFD results reles heavly on the grd resoluton and boundary condtons, grd qualty tests and boundary doman etenson tests were carred out usng ANSYS R Fluent As shown n Fgure 2(b), when the grd number was around 1,000,000, the energy loss at the condton of systolc pea, created at calculated doman, started to converge nto a constant. (a) (b) Fgure 2:(a)Geometry wth grds (b)grd ndependent test Boundary Condtons The boundary condtons, at ascendng aorta(aao), pulmonary artery(pa), coronary artery(ca), descendng aorta(da), and nec vessels(nv), ncludng nnomnate anonyma (IA), common carotd artery (CCA), and left subclavan artery (SA), were respectvely measured nvvo by usng an ntracardac catheter wth pressure sensor and echoardography n real-tme wth ECG (Electrocardogram). Detaled nformaton of nlets ncluded AAo and PA was dsplayed as curves showed n Fgure 3. Furthermore, a relatvely long length of the nlet blood vessels etended 20 tmes of correspondng vessels dameter allows for fully developng the flow boundary layer at the nlet of the secton of AA and PA. The outlet doman for the smulaton was etended to be appromately tmes to suffcently recover the blood pressure at the outlet. The velocty profle at each nlet of AAo and PA and zero pressure gradents at the outlets of the each etended doman were appled to solve the governng equatons n present study. The vessels were presumed as rgd surfaces, ncludng the etenson parts. Fgure 3: Clncal nformaton as boundary condtons Calculaton Two numercal solver methods were used n the study separately, a steady state solver and a transent flow Copyrght 2009 CSIRO Australa 3

4 solver. The steady state method solves the non-tme dependant form of the N-S equatons. For the transent soluton method, the Euler method was appled to solve the tme-dependent N-S equatons. Moreover, the moton of blood flow n the study s descrbed below. For the steady soluton, the tme average velocty was 0.41 m/s wth a mamum velocty of 0.97 m/s occurrng at 0.08s at anastomoss. The characterstc length ( D ) s m at anastomoss and the average Re was wth the correspondng mamum Re was The Womersley Number (α ) based on the average cross sectonal area was Flow n the doman s manly turbulence. For the transent soluton, because of the pulsatng flow, the turbulence may occur for a Reynolds Number much large that epected for steady flow. Ths s due to the fact that an acceleratng flow s more stable than steady flow, and also the deceleratng flow s more unstable than steady flow (Fung, 1997, Hart, 1997). A crtcal Reynolds Number (Re c ) for unsteady flow was found by Nerem et al.(1972) Ths taes the form of Re c =constant α, wth the constant of proportonal rangng from 250 to 1000.( Nerem et al, 1972) In present research, Re c ranges from 2850 to The mamum Re n our study s n the crtcal range. Therefore, the flow n ths study was assumed as unsteady and transton. Because of the relatve hgh- Reynolds number, - turbulence model was employed. The smulaton was performed for three consecutve cycles at 120 BPM to reach a cyclcally repeated soluton, and the Courant Number defned below was used to calculate how many tme steps would be subdvded n each cycle. Δt C r = u (11) ΔI where u s average velocty, Δt s mamum tme step sze, and ΔI s the dmenson of grd cell. As a general rule the soluton s deemed unstable f the Courant Number eceeds 1.0. In present study, equal tme step sze was appled, Δ t = s, and each cycle was subdvded nto tme steps. The Courant Number ranged from to As the convergence crtera, the relatve varaton of the calculated quanttes between two successve teratons was smaller than the preassgned mamum resdence RESULTS Fgure 4 dsplays the results of velocty patterns and streamlnes at the systolc phase and dastolc phase. Close eamnaton of the nstantaneous velocty pattern, at the systolc phase, t shows that hgh velocty was observed near the outsde wall of the AA-PA connecton and then turned nto the nsde of the aortc arch when the blood flow passed the steeply curved aortc arch. At the dastolc phase, the phenomenon stll ests. The results are accordant wth the potental flow theory, whch predcts a sewng of the velocty profle toward the nner wall of the bend, such as the ascendng arch. It s ndcated that flow hghly changed drecton at near the aortc arch. The energy loss and oscllaton of the WSS may be caused, whch could gve a sgnfcant nfluence on the endothelal layer permeablty drectly. Contour plots of the total pressure and WSS dstrbuton at s temporal nstances, ncludng the systolc phase and dastolc phase, are llustrated n Fgure 5. It s obvous that total pressure of the blood flud s hgh near the mddle poston of the aortc arch, whereas a promnent low-pressure area s formed at anastomoss. Fgure 4: Streamlnes of systole and dastole On the contrary, the dstrbuton of WSS dsplayed opposte characterstcs. Relatve hgh WSS was occurred at the AA-PA connecton durng the whole cycle. Furthermore, the sze and locaton of the low-pressure area and hgh WSS area are both varable n one cardac cycle shown n Fgure 5. In addton, the pressure results calculated at the ste of AAo, DA, CA and NV were dsplayed n Fgure 6 (a), (b), (c) and (d), respectvely. Both the tme-varyng energy loss and the average energy loss n one heart cycle were dsplayed n Fgure 7. The mamum energy loss was appromately 0.112W whch was generated at the systolc pea, and the average value of the energy loss from pulsatle calculaton was W. The result was about 1.5 tmes of the energy loss whch calculated by steady calculaton. Fgure 7: The energy loss n one heart cycle Furthermore, n one heart cycle the percentage of blood flow dstrbuton at each branch vessels was shown n Table 1. Copyrght 2009 CSIRO Australa 4

5 Percentage of Blood Flow Dstrbuton (%) Calculaton DA IA CCA SA CA Steady Pulsatle Table 1: Percentage of blood flow dstrbuton Fgure 5: Dstrbuton of nstantaneous pressure and wall shear stress (WSS) T 3 =0.08s the blood flow arrved at the systolc pea (a) (b) (c) (d) Fgure 6: Pressure dstrbuton at AAo (a), DA (b), CA (c) and NV (d) DISCUSSION In present study, the CFD system whch was verfed and valdated n-vvo and n-vtro n our prevous studes was developed for the systemc quantfcaton and evaluaton of the Norwood operaton, whch was calculated based on the patent-specfc reconstructed vessels geometry wth physologcally realstc pulsatle flow condtons n vvo. The obect of ths study s to show the capablty and the mportance of computatonal hemodynamc analyss on understandng the local characterstcs of hemodynamcs n the step-by-step treatment of HLHS for future surgeres. Meanwhle, a well-establshed analyss approach of CFD based on the clncal mages was ntroduced, whch was shown n Fgure 1. In the smulaton, computatonal hemodynamcs analyss was used to obtan the blood flow nformaton n detals, ncludng WSS, pressure dstrbuton, energy loss, and so on. Furthermore, due to the strong dependence of flow characterstcs on the vessel geometry, physologcal flow features observed by the hemodynamc reproduced n vtro may dsplay more clearly than that n smplfed or dealzed arteral models, especally for the surgcal optmzaton. Therefore, the accuracy of the 3-D geometry reconstructon on patent-specfc vessels s the ey factor n the total numercal analyss. The geometry based on the CT mages used n the study s accurate enough satsfyng the need of analyss. The pulsatal flow and steady flow smulaton are both been studed n present research. Wth the comparson of energy loss n the pulsatle smulaton and steady smulaton n Fgure 7, t shows that the average energy loss of pulsatle calculaton s as much as two tmes of steady calculaton and the pulsatle flow smulated s more mportant and necessary than steady flow for analysng local blood energy loss on the quanttatve evaluaton and predcton of surgcal outcomes n detals. However, n the practce of clnc, the steady flow nformaton of blood s much easer to be obtaned by some technques, such as MRI, and also brng lttle pan to the patents than usng the Catheter pressure measurement to get the nformaton of realstc pulsatle blood flow. Therefore, to fnd the relatonshp of the energy loss between the pulsatle flow and steady flow n predcton surgcal outcomes has greatly sgnfcant meanngs n clnc. The nformaton regardng spatal and temporal patterns of blood flow, ncludng velocty patterns, total pressure and WSS, were contaned n the computatonal results. For the purposes of analyss, total pressure dstrbuton and WSS at some selected tme pont n one cardac cycle were dsplayed. The results of low-pressure area wth hgh WSS on the surface of anastomoss ndcated that the energy loss whch may be created at these area. Wall shear stress, the Copyrght 2009 CSIRO Australa 5

6 frctonal load from the blood vessel wall, s beleved to nfluence the functon of the endothelal cells. Moreover, although vessel s elastcty s an mportant ssue n hemodynamc analyss, especally at the aortas, and the pressure waves calculated perhaps appear a lttle of delay varyng n dfferent parts of the arteral system, the rgd vessel walls were assumed n the smulaton and the wall complance was not consdered because of not only a small space n the thora for chldren s heart movement but also the postoperatve dense adheson around the reconstructed aortc arch. In addton, from the technque pont of vew, the smulaton of rgd vessels can be fnshed n a relatve short tme compared wth the smulaton of complance vessels. Ths maes t possble to apply our methodology drectly n the clncal applcaton. The smulaton of complance vessels by usng Flud Structure Interacton (FSI) method taes a huge CPU tme. Furthermore, currently, there are no any proper avalable methodologes whch have been valdated. Therefore, t s reasonable to smplfy the blood vessels as rgd surface n the calculaton. In present study, the purpose of our research s to show the capablty and the mportance of computatonal hemodynamc analyss on understandng the local characterstcs of hemodynamcs at the area of anastomoss after the Norwood operaton and optmzaton of the operatons for surgcal treatment of HLHS. Hence, the factors of elastcty affect the characterstcs of blood flow and pressure dd not consder and assumed these arteres as thn-rgd vessels wthout any movement durng the cardac cycle n present study. In the future, the further study relatve to the effects of vessels elastcty on hemodynamcs of blood flow and pressure waves wll be carred on. The mportant mplcaton contaned n the results of ths study was very essental for future correlatve studes n the outcomes predcton and optmzaton of HLHS surgeres. Although the patent-specfc geometry of vessels before or after operaton are dfferent from person to person, the methodology used by our computatonal hemodynamc analyss system s feasble that would allow the applcaton of CFD analyss nto clncal treatment and the process of surgcal optmzaton to establsh the quanttatve standard n the further studes. CONCLUSION Based on the analyss of current study, we can come to the concluson that the system to be able to quanttatvely estmate the qualty of CHD surgery has been developed. In-vtro verfcaton and valdaton processes for grd and boundary condton ndependent have been carred out. Due to the grd numbers beng reduced to an effcent sze, the smulaton process s able to be performed usng a personal computer. In a future study, the wor wll centre on further valdaton not only by the eperment studes whch are presently underway n our laboratory but also the methodology employed n our hemodynamc analyss system. A seres of before and after cases regardng Norwood surgery wll be analysed, as well as systems to be developed for further stage treatments; Glenn and Fontan treatments. ACKNOWLEDGMENTS We than Japanese Mnstry of Educaton, Culture, Sports, Scence and Technology (MEXT) for supportng ths research. Grant number: A REFERENCES BOVE EL, DE LEVAL MR, MIGLIAVACCA F, GUADAGNI G, DUBINI G, (2003), Computatonal flud dynamcs n the evaluaton of hemodynamc performance of cavo-pulmonary connecton after the Norwood procedure for hypoplastc left heart syndrome, J Thorac Cardovasc Surg,126: BOVE E. L., AND LLOYD T.R., (1996), Staged reconstructon for hypoplastc left heart syndrome: Contemporary results, Annals of Surgery 224(3): KU D.N, (1997), Blood flow on arteres, Annual Revew of Flud Mechancs, 29: DEGROFF C.G, (2008), Modelng the Fontan Crculaton: Where We Are and Where We Need to Go, Pedatr Cardal, 29: 3-12 FUNG Y. C., (1981) Bomechancs, Sprnger- Verlag FUNG Y. C., (1997), Bomechncs Crculaton, Sprnger, New Yor, Second Edt, Chap 3. HART, J.D., (1997), Nonparametrc Smoothng and Lac-of-Ft Tests, Sprnger-Verlag, New Yor, Inc, frst edton LAUNDER. B.E. AND SHARMA. B.I. (1974), Applcaton of the energy dsspaton model of turbulence to the calculaton of flow near a spnnng dsc. Letters n Heat Mass Transfer, 1: LINDERKAMP O, PAUL Y. K. WU, AND HERBERT J. MEISELMAN, (1982), Defomablty of Densty Separated Red Blood Cells n Normal Newborn Infants and Adults, Pedatr. Res. 16: LONG J. A, AKIF UNDAR, KEEFE B. MANNING, AND STEVEN DEUTSCH, (2005), Vscoelastcty of Pedatrc Blood and ts Implcatons for the Testng of a Pulsatle Pedatrc Blood Pump, Amercan Socety of Artfcal Internal Organs, MACKINTOSH TF., AND WALKER CHM., (1973), Blood vscosty n the newborn, Archves of Dsease n Chldhood. 48: MCDONALD DA., (1974), Blood flow n arteres, Edward Arnold Ltd. MCGUIRK S.P., GRISELLI M., STUMPER O.F., RUMBALL E.M., MILLER P, DHILLON R, GIOVANNI J.V., WRIGHT J.G., AND ET AL. (2005), Staged surgcal management of hypoplastc left heart syndrome: a sngle nsttuton 12-year eperence, Heart 92: MIGLIAVACCA F, DE LEVAL MR, DUBINI G, PIETRABISSA R (1996) A computatonal pulsatle model of the bdrectonal cavopulmonary anastomoss: The nfluence of pulmonary forward flow, J Bomech Eng 118: NEREM. R. M., SEED. W. A., AND WOOD. N. B., (1972), An epermental study of the velocty dstrbuton and transton to turbulence n the aorta, J. Flud Mech. 52: Copyrght 2009 CSIRO Australa 6

7 ORLAMDO W, SHANDAS R, DEGROFF C (2006), Effcency dfferences n computatonal smulaton of the total cavopulmonary crculaton wth and wthout complant vessel walls. Compu Meth Prog Bomed 81: QIAN Y, TAKAO H, FUKUI K, UMEZU M, ISHIBASHI T AND MURAYAMA Y, (2008), Computatonal Rs Parameter Analyss and Geometrc Estmaton for Cerebral Aneurysm Growth and Rupture, Stroe: A Journal of the Amercan Heart Assocaton 39: SIEVERS HH, GERDES A, KUNZE J, PFISTER G (1998), Superor hydrodynamcs of a modfed cavopulmonary connecton for the Norwood operaton, Ann Thorac Surg 65: WHITEHEAD KK, PEKKAN K, KITAJIMA H.D, PARIDON S.M, YOGANATHAN A.P AND FOGEL M.A (2007), Nonlnear Power Loss Durng Eercse n Sngle-Ventrcle Patents After the Fontan: Insght From Computatonal Flud Dynamcs, Crculaton, 116: Copyrght 2009 CSIRO Australa 7