Effect of spiral flow on the transport of oxygen

Effect of Spiral Flow on the Transport of Oxygen in the Aorta:A Numerical StudyX IAO L IU,Y UBO F AN,and X IAOYAN D ENGSchool of Biological Science&Medical Engineering,Beihang University,Beijing100191,China(Received18July2009;accepted15December2009)Associate Editor Gerald Saidel oversaw the review of this article.Abstract—To test the hypothesis that the spiralflow pattern in the human aorta may play a beneficial role in the transport of oxygen,hence sparing the ascending aorta from athero-sclerosis,a comparative study on the distribution of oxygen flux to the arterial wall was carried out numerically for three aorta models.Thefirst one(Model1)was constructed based on MRI images of a human aorta acquired in vivo.The second(Model2)was made the same as Model1but without the three branches.The third(Model3)was similar to Model2only with the aortic torsion removed.The results showed that without the torsion,theflow fashion in Model3 was very different from the swirlingflows in Models1and2 and exhibited the typical characteristics of Deanflow with two symmetrical helical structures.In Models1and2,the average Sherwood number(Sh)in the ascending aorta with the presence of a strong spiralflow was not only higher than that in the inner wall of the descending aorta where the spiralflow faded out,but also generally higher than that in Model3.The comparison between Models2and3demon-strated that the averaged Sh from ascending aorta to aortic arch was relatively higher for Model2.In conclusion,the spiral or swirlingflow may have certain physiological significance in the aorta and play a positive role in the transport of oxygen by enhancing oxygenflux to the arterial wall.Keywords—Oxygen transport,Atherosclerosis,Spiralflow, Aorta,Hypoxia,Shear stress.INTRODUCTIONIt has been well recognized that atherosclerotic lesions in the arterial wall develop at certain sites in the human arterial system such as along the inner walls of curved segments and the outer walls of arterial bifurcations.2,18,41This phenomenon is called the localization of atherosclerosis.Flow-induced shear stress is one of the most important hemodynamic factors involved in the localization of atherogenesis.28 Nevertheless,the underlying mechanism is still unclear.Oxygen is essential for organs including blood ves-sels to maintain their normal physiological functions. In1940s,Hueper15suggested that hypoxia was the precursor to atherosclerotic degeneration.Animal studies showed that hypoxia could increase the accu-mulation of lipids in the aortic wall21and this was caused by a direct effect on lipid accumulation in the aortic intima,not by an indirect action on blood lipid concentration.37In vitro studies using human mono-cyte-derived macrophages40demonstrated that mac-rophage-mediated LDL oxidation was significantly higher at hypoxia than at normoxia,and hypoxia could lead to enhanced loading of lipids into macro-phages,a key early event in atherogenesis.24Santilli et al.42observed in their experiment that oxygen ten-sion in the artery wall was decreased in the region of the canine carotid sinus compared with normal loca-tions,which might be a factor in the genesis and development of atherosclerotic lesions there.Recently, Sluimer et al.48directly observed the existence of hypoxia in human atherosclerotic carotid arteries for thefirst time.In contrary,hyperoxia had been shown to have the effect on the regression of atherosclerotic plaques.20,37Therefore,oxygenflux to the arterial wall probably is another important factor involved in the localization of atherogenesis.The aorta is the major blood vessel of complex geometry with such characteristics as branching, twisting,taper,and curvature.11It has been suggested that the special hemodynamic feature in the aorta was determined by a genetically programmed asymmetric development of the aortic arch.52Since bloodflow in the aorta is intricate,plenty of velocity measurements have been performed in the past40years using hot film,Doppler ultrasound,phase contrastmagnetic Address correspondence to Xiaoyan Deng,School of BiologicalScience&Medical Engineering,Beihang University,Beijing100191,China.Electronic mail:dengxy1953@Annals of Biomedical Engineering(Ó2009)DOI:10.1007/s10439-009-9878-8Ó2009Biomedical Engineering Societyresonance imaging and other tools.19,45,46These studies demonstrated that theflow pattern in the ascending aorta was skewed and helicalflows predominated in areas from the ascending aorta to the aortic arch.This form of bloodflow was a basic pattern for almost all the subjects no matter age and gender,3,14which was believed to have certain beneficial effects.3,19,33It was also found that carotid atheromatous disease was associated with a reduction in prevalence of systolic spiralflow pattern in the aortic arch.14In our previous study,we showed that it was mainly the torsion of the aorta that induced the helicalflow in the ascending aorta and the helicalflow had a great influence on the transport or spatial distribution of atherogenic lipids such as LDLs.26We believe that the helicalflow observed in the aorta may also have a great impact on oxygen transport in the lumen of the aorta, hence on theflux of oxygen into the aortic wall.We therefore hypothesize that without the torsion feature, oxygen transport in the aorta could have been impeded and oxygenflux into the arterial wall would have been reduced.To verify this hypothesis,in the present study, we carried out a comparative numerical study on oxygen transport in the aorta using three aortic models:one was an aorta model constructed based on in vivo MRI images of a human aorta;to eliminate the effect of branches,the second one was made the same as thefirst one but without branches;the third one constructed with the torsion of the aorta removed.In addition,the progression of helicalflow in the aorta was characterized by area-weighted average of helicity density and the effect of hemoglobin on the distribu-tion of oxygenflux to the aortic wall was also discussed.METHODSMR Imaging and3-D Geometrical Reconstruction Three models as shown in Fig.1were created from Images from40transverse slices of the thorax acquired from a healthy male volunteer of38years old using a 1.5Tesla imager(Signa,GE Medical System),which has been described in detail previously.26Thefield of view was36936cm2,the image resolution was5129 512pixels and the slice thickness was5mm.Model1:To mimic the human aorta with all itsgeometrical features intact.Model2:To investigate the influence of aorticbranching on oxygenflux,Model2was made thesame as Model1but without the branches.Model3:In order to investigate the effect ofaortic torsion on oxygen transport,Model3wasconstructed with the torsion removed.The cen-terline curvature of Model3was basically thesame as that of Model2.Numerical ApproachesAssumptionsIn a recent study by O’Callaghan et al.,35they summarized the constitutive equations representing the shear rate dependence of blood viscosity proposed by various researchers.According to their summary, when shear rate is larger than10sÀ1which is the case in most regions of the arteries,blood can be treated approximately as Newtonianfluid.In the present study,blood was considered as ahomogeneous, FIGURE1.(a)Model1was constructed from the acquired images with all the geometrical features of the aorta intact.(b)Model2 was made the same as Model1but without the branches.(c)Model3was a planar curving model without the aortic torsion.(d)The eight representative slices were used to presentflow patterns in the aorta models and quantify the helicalflow in the aorta.Letter a: anterior wall,Letter i:inner wall,Letter p:posterior wall,and Letter o:outer wall.L IU et al.incompressible Newtonianfluid.23Arterial wall was assumed to be no-slip rigid.10,32The simulations were carried out under steady-stateflow conditions. Governing EquationsTheflow simulation was based on the three-dimensional incompressible Navier–Stokes equations:qð~uÁrÞ~uþr pÀl D~u¼0ð1ÞrÁ~u¼0ð2Þwhere~u and p represent,respectively,thefluid velocity vector and the pressure.q and l were the density and viscosity of blood(q=1050kg/m3,l=3.59 10À3kg/m s).Oxygen transport inflowing blood could be described by311þ½HbadSdPO2~uÁr PO2¼rÁD b1þ½HbaD cD bdSdPO2r PO2!ð3ÞPO2was the plasma oxygen tension.[Hb]was the total oxygen carrying capacity of hemoglobin in blood ([Hb]=0.2mL O2/mL blood1).a was the solubility of oxygen in plasma(a=2.5910À5mL O2/mL blood/ mmHg43).D b and D c represent,respectively,the dif-fusivities of free oxygen and oxyhemoglobin in blood (D b=1.2910À9m2/s,49D c=1.5910À11m2/s6). S,the saturation function,defined as the ratio of oxyhemoglobin to total hemoglobin,was approxi-mated by the Hill equation:S¼PO n2PO n2þPO n50ð4Þwhere n=2.7and PO50=26.6mmHg.31In order to investigate the contribution of hemo-globin to oxygen transport,two cases were considered.Case1:With hemoglobin,i.e.,oxygen transportinflowing blood consists of both free oxygentransport and the oxygen carried by hemoglobin.Case2:Without hemoglobin,i.e.,[Hb]=0.Boundary ConditionsThe boundary conditions for the Navier–Stokes equations were as follows:BC-A:For Model1,at each of the three aorticarch branches,5%offlow volume was allowedto be ejected.47BC-B:For all models,aflat inletflow velocityprofile was used assuming the time-averageRe=790(velocity is0.1m/s)based the mea-surements.13The assumption for aflat inletvelocity profile could be justified by the in vivohotfilm anemometry measurements on variousanimal models,34which indicated that bloodflowat the inlet of the aorta only consisted of a weakhelical component.The boundary conditions for the mass transport equation were as follows31:BC-1inlet:PO2inlet¼85mmHgð5ÞBC-2outlet:@PO2=@n¼0ð6ÞBC-3wall:PO2wall¼60mmHgð7ÞBC-3was based on the in vivo measurements of oxygen wall tension in dogs by Buerk and Goldstick.4 n indicated the direction normal to the boundary.Computation ProceduresThe numerical calculations were carried out using a validatedfinite volume-based algorithm Fluent(Ansys. A)with a user-defined C-like function(UDF). The UDF was used to solve the oxygen transport equation and to do post-processing and validated by the results from Moore and Ethier.31The grid struc-tures used in the computational study consisted of hybrid grids.Due to the intensive variation in oxygen tension within a thinfluid layer adjacent to the vessel wall,high quality hexahedral cells were used near the wall.Other regions werefilled with low skew tetrahe-drons.In order to assure that the results were mesh independent,the grid-adaptation technique was used, which refined the grid based on the geometric and numerical solution data.Mesh independence was considered to be achieved when the averaged difference in oxygenflux between two successive simulations was less than3%.Mesh independence was reached at 1,592,670,1,260,059,and906,572cells with nodes of 1,052,009,741,498,and617,104for the three models, respectively.RESULTSFlow PatternsThe numerical results onflow patterns in the three models have been presented in our previous study.26 The results showed that theflow patterns of Model1 were very similar to those of Model2,but theflow patterns in Model3were very different from those in Models1and2.For Model1,at the middle portion of the ascending aorta(S2),2asymmetric vortices formed along the anterior and the posterior walls.When theEffect of Spiral Flow on the Transport of Oxygen in the Aortaflow moved into the aortic arch,the anterior wall vortex grew bigger squeezing the posterior wall vortex to a narrow region.The anterior wall vortex (the big-ger one)exhibited apparent characteristics of swirling flows (S3–S5)and attenuated in the descending aorta (S6–S8).For Model 3,two symmetrical helical structures characteristic of the Dean-type flow pattern were seen at all eight representative sections,being more appar-ently so downstream from section S4.26Helicity DensityIn order to better characterize the swirling or helical flow in the aorta,area-weighted average of helicity density H a was calculated as follows:H a ¼1S Z H d dSð8Þwhere S was the cross-sectional area,H d the helicity density (the scalar product of velocity and vorticity in the flow field),25,30defined by Eq.(9):H d ¼V !Ár ÂV ! :ð9ÞFor Model 1,as shown in Fig.2,H a at S1wasslightly over zero,which indicates the initiation of swirling flow with the formation of secondary flow there.At the middle portion of the ascending aorta (S2),the swirling flow grew stronger with increased H a .As blood flowed along the ascending aorta,H a increased gradually and reached its peak at the exit part of the aortic arch (S5).Then,the helicity density decreased dramatically and dropped close to zero at the middle part of the descending aorta (S8).Effect of BranchingFor Model 2without branches,as shown in Fig.2,H a was almost equal to that of Model 1from the front part of ascending aorta to the middle of the aortic arch (S1–S4).At the exit part of the arch (S5),the helicity density also reached its maximum,simi-larly to Model 1.In general,the distribution of helicity density for both Models 1and 2was very similar,indicating that the swirling flow in the aorta was induced largely by the torsion and the branching had limited effect on it.Effect of TorsionFor the planar Model 3,the helicity density at all eight representative sections (S1–S8)was close to zero (Fig.2).Therefore,the result indicated that without the aortic torsion,the fashion of blood flow in Model 3was indeed very different from the swirling blood flow in Model 2.Sherwood Number DistributionsIn the present study,Sherwood number (Sh )was used as an indication of oxygen flux to the arterial wall.The definition of Sh was as follows 31,50:Sh ¼kaDð10ÞIn Eq.(10),k ¼ÀD @PO 2@n wPO 2inlet ÀPO 2wallð11Þwhere k was the mass transfer coefficient;a was the representative diameter which was the mean diameter of all cross sections of the aorta;and D wastheFIGURE 2.Plots of area-weighted average of helicity density at eight representative slices for the three models.L IU et al.diffusivity of oxygen in plasma.For Case2(without hemoglobin),D=D b.For Case1(with hemoglobin),D¼D b1þ½Hba D cD bdSdPO2;@PO2@nwwas the directionalderivative normal to the vessel wall.Whenever Sh number was below the Damkholer number(the dimensionless metabolic consumption rate)of oxygen within the wall,hypoxia would occur there.37,51The Damkholer number is defined as _QTa=ðKDPbÞ;51where_Q is the tissue oxygen con-sumption rate;a is the representative diameter of the vessel which is the mean diameter of all cross sections of the aorta;T is the tissue thickness(the distance from the surface to the minimum tissue oxygen concentra-tion);K is Henry’s law constant;D is the oxygen dif-fusivity;and P b is the bulk oxygen partial pressure.For the aorta,the value of the dimensionless metabolic consumption rate of oxygen or Damkholer number was approximately185.Figure3shows the distribution of Sh for Model1 with hemoglobin(Case1).In the ascending aorta,Sh varied in a narrow range from300to500except for Area D where Sh was200–300and the region near the aortic arch where Sh was approximately600.In the aortic arch,the distribution of Sh was quite uneven with a variation from100to3441.The Sh number at the spot of Area B on the inner wall of the arch was quite low(from100to200),but for majority of the aortic arch it was above600.In the descending aorta, the Sh number along most of the inner wall(Area C) was below300,meanwhile its outer wall had Sh number well above400.Sh number in the three bran-ches of the aorta was quite high(above500),except for Area A in the entry area of the brachiocephalic artery where Sh was below200.Effect of BranchingAs shown in Fig.3,the distributions of Sh for Models1and2with hemoglobin(Case1)were very similar except some small discrepancies.At the front part of the ascending aorta(Area D),the value ofSh FIGURE3.Distribution of Sherwood Number(Sh)in the three models.In Model1,Areas A,B,and C have a Sh number below200. For most part of the ascending aorta(Areas E),the Sh number is higher than that of the corresponding area in Mode3.Effect of Spiral Flow on the Transport of Oxygen in the Aortanumber was relatively higher for Model2when com-pared with Model1.However,in Area E,the situation was reversed.The averaged Sh of the ascending aorta over the region from Slice1to Slice3was approxi-mately360and355for Models1and2,respectively.In the descending aorta,Model2had less area with low Sh than Model1,which was possibly caused by the reduction of bloodflow rate in the descending aorta of Model1due to the diversion of bloodflow into the three branches of the aorta.Effect of TorsionFigure3also gives the comparison between Models2 and3with hemoglobin(Case1)showing the effect of aortic torsion on oxygen transport in the aorta.It can be seen from thefigure that for Model2,the value of Sh in Area E on the ascending aorta was over400, while for Model3,the Sh number at the corresponding location(Area E)was below350.When the torsion was removed(Model3),the Sh number in the exit area of the ascending aorta(Area H)was reduced to450 from the value of over500for Model2.The compu-tation also showed that without the aortic torsion (Model3),the two regions with low Sh value(Areas B and C)were obviously larger when compared with Model2.At the end part of the descending aorta(Area F),the Sh number for Model2was significantly higher than that for Model3.To better evaluate the role of the torsion in oxygen transfer in the aorta,the values of Sh were averaged over different locations of the aorta.The results showed that for most areas,the values of the averaged Sh in Model2were higher than those in Model3(Fig.4).The results indicated that the swirlingflow induced by the aortic torsion enhanced oxygenflux to the aortic wall.Effect of HemoglobinFigure5shows the distribution of Sh for both cases (with and without hemoglobin)in Model 1.The numerical simulation revealed that basically for Model1,hemoglobin only affected the absolute value of Sh,but not the distribution of Sh.For both cases (with and without hemoglobin),the highest Sh in Model1were located at theflow divider of the brachiocephalic-left common carotid branch,which were3441for Case1and1557for Case2,respectively (not shown in Fig.5).Correlation of Sh Distribution with WSS Distribution Figure6is the distribution of wall shear stress (WSS)in the three models,which had been reported in our previous study.26In general,the comparison of Fig.3with Fig.6showed a positive correlation between Sh and WSS.Nevertheless,the results also revealed that although WSS in the ascending aorta was relatively low when compared with that in the descending aorta,the average value of Sh inthe FIGURE4.The averaged Sherwood Number(Sh)for differ-ent sites of Models2and3.The abscissa represents differentlocations of the Model.For example,S1–S3stands for the areabetween Slices1and3.FIGURE5.Effect of hemoglobin on the distribution of Sh.Note that the scales of the color-coding in thefigure are dif-ferent for the upper panel(a)and the bottom panel(b).L IU et al.ascending aorta was higher than that in the descending aorta,especially much higher than the average value of Sh along the inner wall of the descending aorta.Therefore,the present study indicated that WSS was not the only factor that determines the distribution of Sh .It was affected by the flow pattern itself in the aorta as well.DISCUSSIONIn past years,oxygen transport was investigated numerically for different locations in the circulation including arterial stenoses,the coronary artery,and the carotid bifurcation etc.31,38,50These studies employed arterial models that were either 2-or 3-dimensional but symmetrical.As most of the vessels in the human arterial tree were non-planar,5recently oxygen transfer modeling was carried in an idealized 3-dimensionalartery by Coppola and Caro.7Their result 7showed that 3-dimensionality with non-planar feature could facilitate oxygen transport and enhance oxygen flux to the arterial wall.However,to our knowledge,there has been no study on oxygen transport using a physio-logically mimic model of the human aorta.We believe that the swirling or spiral motion of blood flow in the human aortic arch is a typical example of ‘form follows function’in the vascular system 26and hypothesize that it is the swirling blood flow that eliminates stagnation flow regions and pro-vides guaranties for the ascending aortic wall to get even and sufficient oxygen supply from flowing blood,therefore atherosclerotic plaques can hardly form in the area of the ascending aorta.To substantiate this hypothesis,in the present study we simulated numeri-cally the flow of blood in the aorta,characterized the spiral flow by helicity density and investigated the effect of spiral flow on oxygen flux to the arterialwall.FIGURE 6.Wall shear stress distribution in the three models.26In Model 1,the lowest WSS are located in Region A on the outer wall of the brachiocephalic artery and in Region B on the inner wall at the exit of the aortic arch.WSS is relatively high at anterior wall of the aortic arch (Region G).Effect of Spiral Flow on the Transport of Oxygen in the AortaWe also studied the influence of hemoglobin on oxygen transport.The study showed that theflow in Model1(the true model of the aorta)was a swirling or spiralflow with changing helicity strength along the aorta.First,H a increased along the vessel,then declined after the exit of the aortic arch andfinally dropped to zero in the descending aorta where the spiralflow faded out.The variation trend of H a in Model2without branches was almost the same as that in Model1except that after the branches,H a was relative higher for Model1,which indicated that the three branches can introduce some swirling elements in the bloodflow,but this effect was relatively weak when compared with the influence of the torsion.On the other hand,without the aortic torsion(Model3),bloodflow in the planar model showed the typical characteristics of a Deanflow with symmetrically paired counter-rotating helices.There-fore,the area-weighted average of helicity density,H a in the whole model was close to zero.Kilner et al.19examined theflow patterns in the aortic arches of10volunteers.The aortic arches of9of the10subjects are twisted,but in one subject,the aorta was unusuallyflat,with nearly the entire ascending aorta including part of the sinus of Valsalva,lying in the arch plane.Theflow pattern of this subject was a typical Deanflow that was very different from theflow patterns of others.The presentflow simulation was in good agreement with theirfindings.Our numerical simulation also showed that in Model1,the average Sh number in the ascending aorta with the presence of a strong spiralflow was not only higher than that in the inner wall of descending aorta where the spiralflow faded out,but also higher than that in the planar curvature model(Model3)with a Deanflow.The comparison between Models1and2 indicated that the three branches on the aortic arch had small influences on the distribution of Sh.The comparison between Models2and3demonstrated that averaged Sh from ascending aorta to aortic arch was relatively higher for Model2.The study therefore suggests that spiralflow may facilitate oxygen trans-port inflowing blood and enhance oxygenflux to the arterial wall.In contrast to this,the relatively low Sh predicted in Region A of the brachiocephalic branch, in Region B on the inner wall of the aortic arch,and along the whole inner wall of the descending aorta (Region C),may cause hypoxia in these regions.Hypoxia has been shown to cause damage to the endothelial barrier and result in inter-endothelial gaps,36therefore leading to increased permeability of the arterial wall to macromolecule.Hypoxia also induces endothelial cell apoptosis,which can increase LDL transport through leaky junctions.29In addi-tion,hypoxia can up-regulate the release of vascular endothelial cell growth factor(VEGF)by vascular cells and affect endothelial permeability.51Deng et al.8 demonstrated that increased permeability could wor-sen concentration polarization of atherogenic lipids such as LDLs.In addition,our previous study indi-cated that LDL concentration polarization were severe in these regions.26Beyond the role of accelerating lipid accumulation in the arteries,hypoxia can promote inflammation,ATP depletion,and angiogenesis in the development of advanced atherosclerotic lesions.16All the above may explain why atherosclerotic plaques are localized in regions A,B,and C.9,12,39The present study revealed that low Sh tended to be located in areas with very low WSS.Nevertheless,this correlation between WSS and Sh was not clear-cut.For instance,the outer wall of the ascending aorta had lower WSS than other parts of the ascending aorta and even lower than most parts of the descending aorta.But Sh in the ascending aorta distributed relatively uniformly and was generally much higher than that in the descending aorta.The inner wall of aortic arch had quite high WSS, but there was a narrow low Sh region.The results therefore indicate that oxygenflux to the arterial wall depends not only on local wall shear stress but also depends on both the global and localflow patterns.Using a2-D stenosis model,Moore and Ethier31 demonstrated that neglecting oxygen binding with hemoglobin led to significant errors in the simulation of oxygen transport.The present study further proved the importance of hemoglobin in oxygen transport in flowing blood.The value of Sh in Case1with hemo-globin was over twice of that in Case2without hemoglobin.Nevertheless,the distribution of Sh in the aorta was almost the same for the2cases(with and without hemoglobin).Therefore,ourfinding on the effect of hemoglobin supports the postulation sug-gested by others7,50that hemoglobin only affects the absolute value of Sh,but not the general distribution of Sh,which has not been verified previously.By studying oxygen transfer in both rigid and dis-tensible tubes,Schneiderman et al.44demonstrated that bloodflow pulsation negligibly affected oxygen trans-port to the arterial wall.Furthermore,the study by Ma et al.27revealed that the time-averaged mass transfer in a sudden expansion region under pulsatileflow con-ditions was similar to the steadyflow mass transfer. Therefore,although the steady-stateflow assumption in our study was different from the in vivo pulsatile flow conditions,the results still can shed some lights on oxygen transport in the aorta.In the present study,the compliance and motion of the aorta was neglected,which may affect the accuracy of our prediction on the absolute value of Sh,but not the general tendency of the Sh distribution in the aorta.22,50It has been shown that the compliance andL IU et al.motion of the aorta can reinforce the spiralflow in the aortic arch,17which may be in favor of oxygen trans-port to the arterial wall.CONCLUSIONThe spiral or swirlingflow may play a positive role in the transport of oxygen by enhancing oxygenflux to the arterial wall,and therefore has certain physiologi-cal significance in the aorta with possible tendency of sparing the arterial wall from atherosclerotic plaques. The parameter of area-weighted average of helicity density can be used to describe the spiralflow in the aorta.The modeling without considering hemoglobin only affects the absolute value of Sh,but not the gen-eral distribution of Sh in the model.ACKNOWLEDGMENTSThis work was supported by Grants-in-Aid from the National Natural Science Foundation of China (No.10632010,30670517).REFERENCES1Back,L.H.,J.R.Radbill,and D.W.Crawford.Analysis of oxygen transport from pulsatile viscous bloodflow to diseased coronary arteries of man.J.Biomech.10:763–774, 1977.2Berceli,S. A.,V.S.Warty,R. A.Sheppeck,W. A. 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