The Haemodynamics of Asymmetric Stenoses

Transcription

1 Eur J Vsc Enovsc Surg 24, 332±337 (2002) oi: /ejvs , vilble online t on The Hemoynmics of Asymmetric Stenoses S. R. Dos Deprtment of Vsculr Surgery, Goo Hope Hospitl NHS Trust, Rectory Ro, Sutton Colfiel, West Milns, B75 7RR, U.K. Objective: Occlusive rteril isese is usully irregulr with both symmetric (concentric) n symmetric (eccentric) stenoses. Knowlege of the effect of stenosis geometry on stenosis hemoynmics is necessry to correctly interpret tests tht mesure stenosis severity ntomiclly. The inepenent hemoynmic effect of stenosis symmetry hs not been escribe. Design: In vitro flow-rig mesurement of the stey pressure/flow behviour of rigi, 20 mm long, squre-ene symmetric n symmetric stenoses in 6.7-mm imeter tube using bloo nlogue (40% glycerol). Results: All stenoses, irrespective of geometry, h liner resistnce (R) to flow (Q) reltionship such tht R ˆ k 1 k 2 Q where k 1 n k 2 re constnts (corr , p ). Asymmetry ws foun to hve significnt hemoynmic effect if the stenosis severity is expresse s imeter reuction but no effect if stenosis severity is expresse s n re reuction. The mximum flow for n inflow pressure of 90 mmhg (Q 90 ) fell from 2050 to 280 ml/min s stenosis re reuction increse from 80% to 96%. Conclusions: Arteril stenoses exhibit flow-epenent resistnce irrespective of their geometry. The effect of stenosis symmetry cn only be ignore if ntomicl severity is expresse s percentge re reuction. A cliniclly useful mesure of stenosis severity is the mximum flow for given inflow pressure. Key Wors: Occlusive rteril isese; Stenosis; Hemoynmics; Experimentl; Mthemticl moel. Introuction Since the observtions tht nrrowing n rtery mkes no mesurble ifference to the resting flow 1 or ownstrem pressure 2 until criticl egree of occlusion is reche, it hs been conventionl to gre the severity of rteril isese ntomiclly, either s imeter or n re reuction. Accurte ssessment of the functionl effect of occlusive rteril isese is crucil for ignosis n plnning tretment prticulrly when isese is present t severl sites. 3 In the evelopment n evlution of new ignostic n therpeutic techniques, it is recommene tht relible benchmrk for the objective mesurement of rteril isese severity is use 4 n n ntomicl metho of occlusive isese ssessment is convenient, s it cn be pplie to Doppler ultrsoun, ngiogrphy, n mgnetic resonnce imging. However, occlusive rteril isese is often irregulr n multi-focl n if the ngiogrphic ppernce of isese vessels is use s mesure of the functionl impirment, poor Plese ress ll corresponence to: Goo Hope Hospitl NHS Trust, Rectory Ro, Sutton Colfiel, West Milns, B75 7RR, U.K. inter-observer greement is foun. 4 Stenosis severity is usully escribe s percentge reuction reltive to the non-occlue vessel, but it is not lwys cler whether this refers to imeter or n re reuction. The ifference is importnt s stenosis symmetry ffects the reltionship between imeter n re reuction (Figs 1, 2). A combintion of pressure n flow mesurements is the most ccurte metho for ssessing the functionl effect of occlusive rteril isese 5 but requires n unerstning of the pressure±flow reltionship of prtilly occlue tube in orer to interpret the results n correlte them with the results of imging stuies. My et l. preicte tht the stey pressure rop (DP) vs volume flow rte (Q) reltionship of single, squre-ene, concentric stenosis shoul be of the form: DP ˆ k 1 Q k 2 Q 2, 1 where k 1 n k 2 re constnts relte to the stenosis severity (Appenix 1) n use this eqution to explin their in vivo experimentl results but i not vlite it in vitro. 6 This non liner pressure±flow reltionship hs subsequently been inepenently confirme in vitro for short, smooth, xisymmetric 1078±5884/02/ $35.00/0 # 2002 Elsevier Science Lt. All rights reserve.

2 Hemoynmics of Asymmetric Stenoses 333 n for short, smooth, symmetric stenoses where the reltionship between the geometric severity n the constnts k 1 n k 2 ws foun to vry with the prticulr stenosis shpe. 7 Dos et l. note tht My's eqution preicte tht stenosis resistnce (R) shoul epen on flow such tht R ˆ DP Q ˆ k 1 k 2 Q, 2 n confirme this preiction experimentlly for concentric, squre-ene stenoses. 8 The ims of this experimentl stuy re twofol: firstly to confirm tht the liner resistnce±flow reltionship is lso vli for rnge of squre-ene, rigi eccentric stenoses of ifferent geometries, n seconly see if stenosis symmetry hs n inepenent effect on the hemoynmic effect. Mterils n Methos Concentric Eccentric Offset Circulr Fig. 1. Schemtic igrm of the cross-sections of the three types of ielise mchine, rigi, stenoses use in the in vitro experiments: concentric ˆ circulr lumen on centrl xis; offset circulr ˆ circulr lumen offset from centrl xis; eccentric ˆ non-circulr lumen offset from centrl xis. The imeter of the lumen () is tken s the mximum imeter n the re of the lumen is clculte from the formule given in Appenix % The pressure±flow chrcteristics of rnge of ccurtely mchine stenoses ws mesure in vitro using the rrngement shown in Fig. 3 which consists of one metre lengths of 6.7-mm internl imeter silstic tubing ttche to custom-built computercontrolle pump cpble of supplying stey flow (0±2000 ml/min) of bloo nlogue (40% w/v glycerol, ensity r ˆ 1060 kg/m 3, viscosity m ˆ Ns/m 2 ). Ech mchine luminium plug ws fitte into specilly constructe connector n the intrluminl pressure ws mesure 2 cm upstrem n 20 cm ownstrem of the connector using strin guge physiologicl pressure trnsucers vi 18 guge blunt-ene neeles. The pressure trnsucer outputs were mplifie n smple t 100 Hz with n nlogue/igitl converter n the t numericlly verge to reuce noise n rnom fluctutions. The trnsucers were clibrte using mercury mnometer n their ccurcy foun to be in the orer of 1 mmhg over the pressure rnge of 0±100 mmhg. The computer controlle pump ws clibrte by time collection n foun to generte flows with n ccurcy better thn 10 ml/min over 90% 80% 100 cm 2 cm 20 cm Are Reuction 70% 60% 50% 40% 30% 20% Are=Dimeter Circulr Computer Controlle Pump Stenosis Pressure Trnsucers 10% Non circulr 0% 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Dimeter Reuction Fig. 2. Reltionship between percentge imeter n percentge re reuction for the circulr n the non-circulr stenoses shown in Fig. 1 using the formule given in Appenix 2. Computer Fig. 3. Schemtic igrm of the in vitro experimentl rrngement for mesuring the stey pressure±flow chrcteristics of experimentl stenoses. Eur J Vsc Enovsc Surg Vol 24, October 2002

3 334 S. R. Dos rnge of 0±2000 ml/min. Stenoses were fbricte from 20-mm long, 6.7-mm imeter, squre-ene luminium plugs over rnge of imeter reuctions between 50% n 95%. Three configurtions were compre (Fig. 1) n for the circulr-offset stenoses two sets of stenoses were me with offsets of 0.8 mm n 1.6 mm respectively. The re reuctions for the three configurtions were clculte from the mesure imensions using stnr formule (Appenix 2). For ech experiment, the pressure trnsucers were zeroe n the pump progrmme to generte sequence of flows over rnge tht generte pressure grient cross ech stenosis in the rnge of 1±100 mmhg up to mximum flow of 2000 ml/min. For ech flow the men pressure grient ws recore n the hyrulic resistnce clculte. The prmeters k 1 n k 2 for ech stenosis were clculte from the experimentl t using Eq. 2 by liner regression nlysis. The threshol for sttisticlly significnt vlue ws ssume s 0.05 n the gooness of fit between the experimentl t n the mthemticl moel expresse s the Person correltion coefficient. The overll hemoynmic effect of ech stenosis ws represente by the preicte flow for fixe inflow pressure of 90 mmhg (Q 90 ) using the mesure vlues of k 1 n k 2 n the stnr qurtic formul. All pressures re given in mmhg (1 mmhg ˆ 133 N/m 2 ), flows in ml/min (1 ml/min ˆ m 3 /s) n resistnces in peripherl resistnce units (PRU) where 1 PRU ˆ 1 mmhg/ml/min ( ˆ Ns/m 5 ) n 1 PRU ˆ 10 3 mpru ˆ 10 6 mpru. The inlet tube ws 1-metre long n the mximum men flow limite to less thn 2000 ml/min (i.e. below the criticl Reynols number of 2000) to ensure fully evelope lminr flow t the stenosis entrnce. Pressure n flow mesurements were foun to be reproucible to within few per cent so points on the grphs represent the result of single experiments. Results Tble I summrises the complete set of experimentl results for these ielise 20-mm long stenoses. For ll stenoses the stey flow resistnce ws foun to vry linerly with flow (corr in ll experiments) confirming tht the liner resistnce±flow reltionship given in Eq. 2 is vli for the concentric, offset circulr n eccentric stenoses. The prmeters k 1 n k 2 were foun to be significnt (p ) for ll stenoses inicting tht both prmeters re require for complete representtion of the hemoynmic effect. The reltive contribution to the totl resistnce of the k 1 n the k 2 terms cn be clculte from Tble I using the eqution k 1 k 2 Q 3 Tble 1. Summry of experimentl results. In ll cses the vlues of k 1 n k 2 re clculte using liner regression nlysis n re significntly ifferent from zero (p ). % Dimeter reuction % Are reuction k 1 mpru k 2 mpru/ml/min Person correltion Flow t 90 mmhg ml/min Concentric (circulr, no offset) Circulr offset (0.8 mm offset) Circulr offset (1.6 mm offset) Eccentric Eur J Vsc Enovsc Surg Vol 24, October 2002

4 Hemoynmics of Asymmetric Stenoses 335 Q 90 ml/min % Reuction n for ll stenoses the effects of both terms re significnt for flows over resonble physiologicl rnge (300±2000 ml/min), inicting tht both terms re require for n ccurte mesurement of functionl effect. Tble I shows tht both prmeters k 1 n k 2 increse non-linerly with ntomicl severity expresse s either imeter or re reuction. There ws no significnt ifference between the functionl severity of the concentric n circulr-offset stenosis configurtions (p ). Figure 4 shows tht the hemoynmic severity of the stenoses s mesure by the Q 90 chnges rpily for stenoses over 80% re reuction n tht the reltionship between hemoynmic n ntomicl severity iffere between the circulr n non-circulr stenoses epening on the metho of expressing the ntomicl stenosis severity. When mesure s imeter reuction concentric stenosis is functionlly more severe thn n eccentric stenosis (p ), but when mesure s n re reuction there is no significnt ifference (p ). Discussion Circulr by Dimeter Eccentric by Dimeter Circulr byare Eccentric byare Fig. 4. Reltionship between stenosis functionl severity (Q 90 ˆ flow in ml/min for pressure grient of 90 mmhg) n ntomicl severity mesure by both imeter n re reuction for concentric n non-circulr, eccentric stenoses (men 95% confience intervl). In vivo the morphology of rteril isese is irregulr with both concentric n eccentric lesions n cler unerstning of the reltionship between stenosis geometry n hemoynmic effect is neee to juge the functionl severity of stenosis from informtion bout its ntomicl severity. Surprisingly, there is virtully no guince in stnr riologicl n surgicl texts or in the literture on how to interpret the effect of stenosis geometry n in prticulr the effect of stenosis eccentricity. The primry objective of this stuy therefore ws to evlute the effect of geometric symmetry on the hyrulic resistnce of moelle rteril stenoses. Figure 2 shows the reltionship between imeter n re reuction for the ielise circulr n non-circulr stenoses use in these experiments n illustrtes clerly tht the two mesures re not equivlent, prticulrly over the importnt 50±100% imeter reuction rnge. The impirment to flow tht stenosis presents cn be expresse s the stenosis resistnce R ˆ DP Q ˆ k 1 k 2 Q, 2 which sttes tht stenosis cn be consiere to be the sum of fixe resistnce (k 1 ) in series with flow-epenent resistnce tht increses linerly with the flow (k 2 Q). The hemoynmic effect of rteril stenoses is cuse by isturbnce of the norml lminr bloo flow n the cretion of regions of isorere n even turbulent flow, prticulrly ownstrem of the stenosis. The exct nture of this flow isturbnce is epenent on the geometry n the two constnts k 1 n k 2 represent the prtition between lminr flow (pressure rop is relte to bloo viscosity) n isorere flow (where the pressure rop is relte to the bloo ensity). My's eqution reltes specificlly to squre-ene concentric stenoses, but our results show tht the generl form of My's eqution (Eq. 1) is lso vli for ny stenosis geometry. At the flow rtes n vessel imensions use in these experiments, the effect of the non- Newtonin chrcteristics of bloo is miniml becuse the estimte rtes of sher re high n the use of glycerol solution of equivlent ensity n viscosity to bloo is therefore ssume not to introuce significnt errors. Mesurements using stey flows re justifie in chrcterising the functionl severity of rigi stenoses s previous in vivo stuies hve shown tht such stey-flow pproximtions cn be pplie to unstey pressures n flows 6,9 but further work is neee to confirm this for unstey flow through eccentric stenoses in complint vessels. This stuy i not ttempt to ress the importnt issue of the mechnicl properties of the stenosis itself, in prticulr the conitions tht le to plque rupture, nor the effect of ngioplsty on plque geometry n structure but the results of this stuy o show tht the hemoynmic effect of criticl stenosis cn be effectively eliminte by reltively smll chnges in the re of the resiul lumen. In vivo the criovsculr reflexes re esigne to mintin constnt Eur J Vsc Enovsc Surg Vol 24, October 2002

5 336 S. R. Dos men rteril pressure n occlusive rteril isese cuses symptoms by restricting the mximum bloo flow, prticulrly when emn increses such s uring exercise. Therefore, more pproprite wy to represent the functionl severity of stenosis is to solve Eq. 1 using the stnr qurtic formul to give the mximum flow tht stenosis will permit for given men inflow pressure. Using this metho, the functionl effect of stenosis symmetry becomes cler (Fig. 4). The flow isturbnce crete by n symmetric stenosis will be ifferent from tht crete by concentric stenosis, n it is very useful to fin tht the hemoynmic effect is the sme provie tht ntomicl severity is expresse s n re reuction. This mens tht inirect mesures of re reuction such s uplex ultrsoun velocity rtios my in fct be more ccurte estimte of functionl effect thn the usul ``gol stnr'' imeter reuction mesure from n ngiogrm. The vntge of using functionl stenosis mesurement is tht it cn be use to mke preictions of pressures n flows uner hypotheticl conitions, such s uring exercise n fter surgicl proceures. The 6.7-mm imeter tube use in these experiments is pproximtely the imeter of the humn common femorl rtery n, ssuming tht the men femorl rtery pressure is roun 90 mmhg, Fig. 4 shoul give estimtes of mximum flow tht gree closely with in vivo t. Lewis et l. mesure the resting men femorl bloo flow in norml subjects n cluicnts t 400 ml/min n showe tht in norml subjects the femorl flow cn increse to more thn five times the resting flow uring rective hyperemi. 10 Figure 4 shows tht single stenosis of roun 80% re reuction woul limit the mximum flow to bout five times the resting flow n is therefore consistent with the well-known observtion tht single femorl rtery stenosis must rech this severity before symptoms of cluiction re prouce. The men pressure grient cross n 80% re stenosis t flow of 400 ml/min is preicte from Tble 1 to be roun 5 mmhg, figure tht grees well with Archie et l., who mesure resting men ilic pressure grients in vivo. 11 These t inicte clerly tht non-circulrity of the stenosis lumen is n importnt fctor in etermining its functionl effect, n it is interesting to note tht Brice et l. observe comprble effect in simulte croti rtery stenoses but i not ttempt to quntify the pressure±flow reltionship. 12 The reltionship between the ntomicl severity n functionl effect shown in Fig. 4 implies tht ccurte ngiogrphic mesurements of both imeter reuction n egree of non-circulrity woul be require to preict the hemoynmic significnce of single stenosis between 80% n 95% re reuction. The use of single plne ngiogrphy s metho for vliting other techniques such s uplex ultrsoun is therefore put in question. A goo pproximtion for the pressure rop cross suen expnsion in tube is given by the moifie Bernoulli eqution tht expresse in terms of volume flow is: DP& 1 r Q where A s is the re of the stenosis lumen n r is the ensity of the flui. 7 This eqution is commonly use to estimte the pressure grient cross stenotic hert vlve using n ultrsoun mesurement of the jet velocity. 13 However, when pplie to isese ilic rteries, Kohler et l. foun this metho gve very vrible pressure grient estimtes n conclue tht it ws not useful. 14 The wiely quote Poiseuille n Bernoulli stey pressure±flow pproximtions re therefore specil cses (k 2 ˆ 0 n k 1 ˆ 0 respectively) tht represent the extremes of flow behviour. Our results confirm tht, in generl, lminr n isorere flow co-exist in prtilly occlue peripherl vessels n neither of these simplifictions is ccurte enough for clinicl use. A 2 s Conclusions All stenoses, irrespective of their geometry, emonstrte liner reltionship between hyrulic resistnce n flow tht is expresse s two prmeters: fixe resistnce k 1 tht represents lminr flow losses n flow-epenent resistnce coefficient k 2 tht represents isorere or turbulent flow. The functionl effect of stenosis cn be conveniently expresse s mximum possible flow for given men inflow pressure. Rigi non-circulr stenoses crete the sme hyrulic resistnce to flow s circulr stenosis of the sme re reuction but this equivlence is lost if stenosis severity is expresse s imeter reuction. This reltionship between stenosis geometry n the hemoynmic effect mens tht n estimte of functionl impirment bse on imeter reuction mesure from n ngiogrm is likely to be inccurte, especilly for stenoses over 60% imeter reuction. Acknowlegements The uthor woul like to thnk Dr N. K. Bourne of the Cvenish Lbortory, Cmbrige for technicl ssistnce n Miss J Wlton n Mr Tony Birch of Southmpton University Hospitl for ssistnce with the experimentl work. Eur J Vsc Enovsc Surg Vol 24, October 2002

6 Hemoynmics of Asymmetric Stenoses 337 References 1 Mnn FC, Herrick JF, Essex HE, Bles EJ. Effect on bloo flow of ecresing the lumen of bloo vessel. Surgery 1938; 4: 249± Himovici H. Stenosing rteril thrombosis: n experimentl physiopthologicl stuy. Surgery 1954; 36: 1075± Hrris PL. Mngement of combine segment isese. Eur J Vsc Surg 1987; 1: 367± Bruins Slot H, Strijbosch L, Greep JM. Interobserver vribility in single-plne ortogrphy. Surgery 1981; 90: 497± Lorentsen E, Hoel BL, Hol R. Evlution of the functionl importnce of therosclerotic obliterns in the ortoilic rtery by pressure/flow mesurements. Act Me Scn 1972; 191: 339± My AG, DeWeese JA, Rob CG. Hemoynmic effects of rteril stenosis. Surgery 1963; 53: 513± Young DF, Tsi FY. Flow chrcteristics in moels of rteril stenoses ± I. Stey flow. J Biomechnics 1973; 6: 395± Dos SR, Bourne NK, Chnt ADB. The effect of flow on the resistnce of moelle femorl rtery stenoses. Br J Surg 1996; 83: 957± Young DF, Cholvin NR, Roth AC. Pressure rop cross rtificilly inuce stenoses in the femorl rteries in ogs. Circ Res 1975; 36: 735± Lewis P, Psil JV, Morgn RH, Dvies WT, Woocock JP. Common femorl rtery volume flow in peripherl vsculr isese. Br J Surg 1990; 77: 183± Archie JP, Felmn RW. Intropertive ssessment of the significnce of ilic n profun femoris rtery stenosis. Surgery 1981; 90: 876± Brice JG, Dowsett DJ, Lowe RD. The effect of constriction on croti bloo flow n pressure grient. Lncet 1964; i: 84± Holen J, Asli R, Lnmrk K, Simonsen S. Determintion of pressure grient in mitrl stenosis with non invsive ultrsoun Doppler technique. Act Me Scn 1976; 199: 455± Kohler TR, Nicholls SC, Zierler E, Bech KW, Schubrt PJ, Strnness DE. Assessment of pressure grient by Doppler ultrsoun: Experimentl n clinicl observtions. J Vsc Surg 1987; 5: 460±469. Appenix 1 Eq. A1.1 cn be re-written s DP ˆ pm 8L 4:8r s A 2 Q r s A 2 Q 2, A1:2 s where r s is the rius of the circulr lumen n Eq. A1.2 cn be written in generl form s DP ˆ k 1 Q k 2 Q 2, A1:3 where k 1 n k 2 re constnts relte only to the geometry of the stenosis n the viscosity n ensity of the flui. Appenix 2 The percentge imeter reuction for circulr stenosis is 1 /D where is the imeter of the lumen n D is the imeter of the unocclue tube n the corresponing percentge re reuction is given by 1 (/D) 2. With reference to Fig. A2.1, the percentge imeter reuction for the eccentric stenosis shown in Fig. 1 is 1 /2r where r is the rius of the unocclue tube n is the mximum imeter of the stenosis. The re of the segment S cn esily be shown to be where S ˆ r 2 y r r sin y, A2:1 The pressure±flow eqution suggeste by My et l. for concentric, squre-ene stenosis ws cos y ˆ r =r A2:2 DP ˆ 8mL r 2 n A 2 4:8m A s A 1:5 n rn 2 A 2 r A s A s A1:1 where r is the rius of the unstenose lumen, A is the re of the unstenose rtery, v is the velocity of the bloo in the prestenotic rtery, A s is the re of the resiul lumen of the stenosis n L is the length of the stenosis, n where m n r re the viscosity n ensity of the flui respectively. 15 Using the reltionships n Q ˆ n A, A s ˆ pr 2 s, Fig. A2.1. S so the percentge re reuction of the eccentric stenosis is the rtio of the re of the occlue segment of lumen to the re of the unocclue tube or 1 S/pr 2. Accepte 14 June 2002 θ r Eur J Vsc Enovsc Surg Vol 24, October 2002