BoMedcal Engneerng OnLne BoMed Central Research The role of venous valves n pressure sheldng Constantnos Zervdes*, Andrew J Narracott, Patrca V Lawford and Davd R Hose Open Access Address: Academc Unt of Medcal Physcs, School of Medcne and Bomedcal Scences, The Unversty of Sheffeld, Sheffeld, UK Emal: Constantnos Zervdes* - c.zervdes@hotmal.co.uk; Andrew J Narracott - a.j.narracott@shef.ac.uk; Patrca V Lawford - p.lawford@shef.ac.uk; Davd R Hose - d.r.hose@shef.ac.uk * Correspondng author Publshed: 15 February 008 BoMedcal Engneerng OnLne 008, 7:8 do:10.1186/1475-95x-7-8 Receved: November 007 Accepted: 15 February 008 Ths artcle s avalable from: 008 Zervdes et al; lcensee BoMed Central Ltd. Ths s an Open Access artcle dstrbuted under the terms of the Creatve Commons Attrbuton Lcense (http://creatvecommons.org/lcenses/by/.0), whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Abstract Background: It s wdely accepted that venous valves play an mportant role n reducng the pressure appled to the vens under dynamc load condtons, such as the act of standng up. Ths understandng s, however, qualtatve and not quanttatve. The purpose of ths paper s to quantfy the pressure sheldng effect and ts varaton wth a number of system parameters. Methods: A one-dmensonal mathematcal model of a collapsble tube, wth the faclty to ntroduce valves at any poston, was used. The model has been exercsed to compute transent pressure and flow dstrbutons along the ven under the acton of an mposed gravty feld (standng up). Results: A quanttatve evaluaton of the effect of a valve, or valves, on the sheldng of the ven from peak transent pressure effects was undertaken. The model used reported that a valve decreased the dynamc pressures appled to a ven when gravty s appled by a consderable amount. Concluson: The model has the potental to ncrease understandng of dynamc physcal effects n venous physology, and ultmately mght be used as part of an nterventonal plannng tool. Background The motvaton behnd ths study was a desre to understand the physologcal effects of compresson cuff therapy for preventon of deep ven thromboss. It s generally accepted [1-4], that deep ven thromboss s assocated wth flow stass, partcularly n and around the venous valves and ther snuses. From a survey of the lterature, t rapdly became apparent that the role and quanttatve performance of venous valves, even n the normal physologcal state s poorly understood. Texts on venous physology always dentfy the role of the valves as the control of reverse flow [4-7]; most often n the context of muscle pump acton to mantan flow n the drecton of the heart and sometmes n the context of postural changes and of exercse. The purpose of ths paper s to explore the effects of gravty on the pressure and flow dstrbuton n a smple representaton of a ven n the leg, and n partcular to quantfy the role of the valves n pressure sheldng under the acton of standng. The effects of a range of parameters on the sheldng performance of the valves are examned. It s demonstrated that the effects depend not only on the dstrbuton, locaton and performance of the valves themselves, but also on the geometrc and mechancal characterstcs of the vens. It s antcpated that ths nfor- Page 1 of 10
BoMedcal Engneerng OnLne 008, 7:8 maton wll be of drect nterest to the vascular surgeon because t provdes an ndcaton of the lkely effect of nterventons, ncludng removal or repar of valves as well as nserton of bypass grafts, on peak pressure dstrbutons n the perpheral vasculature. A person who stands, nactve, for a perod of tme wll be subjected to the full hydrostatc pressure gradent n the venous system and the pressure n the vens n the foot wll reach somethng of the order of 100 mmhg [8,9,7]. Ths s confrmed by Pollack [10] and by Arnold [11]. The presence of a valve or valves cannot sheld aganst ths statc pressure whch wll be assocated wth the physologcal phenomenon of blood poolng, and related to oedema through the Starlng equaton [7], but t can allevate the transent maxmum that wll occur as posture s changed. In the absence of valves, a smple analyss would suggest that the transent pressure peak experenced n the ven mght be double the fnal standng pressure. Neglectng nertal and vscoelastc effects n the vessel wall, the stress n the wall of the ven s proportonal to the nstantaneous appled pressure, and t s postulated that ncompetent valves do not provde adequate transent pressure sheldng and thus mght be strongly mplcated n the formaton of varcostes [9,1,13]. A prmary quanttatve measure of valve, or rather system, performance s gven by ts effectveness n reducng the pressure peaks assocated wth the transent response. A second measure mght be one of the effectveness of a postural change, and the assocated acton of gravty on valve openng and closure characterstcs and thus on the 'wash out' of the snuses and dsplacement of statonary pockets of blood. Although a full three-dmensonal analyss s requred to address ths queston n detal, the one-dmensonal model presented n ths paper s used to examne whether the propertes of the system are such that there mght be suffcent backflow to close the valve for realstc geometres. Materals and Methods The methodology adopted n ths study nvolves the constructon and analyss of a numercal model that s able to capture the spatal and temporal pressure dstrbutons n a collapsble tube that ncludes valves. The ven s represented as an analogous electrcal crcut as llustrated n Fgure 1. For smplcty, and because the focus of ths study s to capture the characterstc effect of the valves, the ven s represented as a straght collapsble tube. The chosen geometrcal confguraton s representatve of a ven from the lower extremtes. The resstve component represents the vscous resstance of the blood, the nductve component the nerta of the blood, and the capactve component the elastcty (and thus storage capacty) of the ven. A perfect valve s assumed to allow no backflow, and can be represented as a dode (n the analogous Schematc representaton of tube model Fgure 1 Schematc representaton of tube model. The resstve component represents the vscous resstance of the blood, the nductve component the nerta of the blood, and the capactve component the elastcty of the ven. A perfect valve s represented by a dode. electrc crcut a dode allows current to flow n one drecton and blocks t from flowng n the opposte drecton). A real valve wll permt some backflow, assocated partly wth the swept volume of the valve leaflets durng the closure phase, and ths s represented by allowng a fnte volume of blood to pass through the valve before closure. A 'leaky' valve s assumed to allow a fxed rate of leakage, although ths could readly be modfed to make the leakage proportonal to the pressure drop across the valve segment. A seres of ordnary dfferental equatons are wrtten to represent the electrcal system. The nonlnear elastc propertes of the ven, ncludng those assocated wth collapse, are represented by a tube law [14-17]. The man purpose of the tube law s to capture the ven's flexblty at small negatve pressures as collapse s ntated, whlst mantanng the propertes of a stffenng response for hgher negatve or postve pressures. A penalty of ths formulaton s that t does not reduce to the standard lnear approxmaton at small postve pressures, and for the current work, a modfcaton has been mplemented to remedy ths defcency. A number of numercal technques are avalable for soluton of the derved equatons [18]. The one adopted for the current study s a Lax Wendroff formulaton, whch s accurate to second order n tme and space. For completeness, the governng equatons and the numercal dscretsaton are lsted n Appendx 1. Ths formulaton has been adopted n other studes of the cardovascular system [19], although Brook [16,0], has dentfed condtons under whch numercal nstabltes mght be manfest. Numercal testng has ndcated that the system s stable under the pertnent condtons for the current study. One of the mportant propertes of the system that wll have sgnfcant nfluence on the results s the boundary condtons appled at the proxmal and dstal ends of the ven segment. For the purposes of the current study, a constant atmospherc pressure boundary condton has been appled at the proxmal end and a constant flow boundary condton at the dstal end. It s recognsed that the prescrbed boundary condtons mght represent a gross sm- Page of 10
BoMedcal Engneerng OnLne 008, 7:8 plfcaton of physologcal flow n the venous segments of nterest. The mportant feature of the proxmal pressure boundary condton s that t allows unmpeded reverse flow nto the ven segment as gravty acts. It would be possble to apply a negatve pressure representatve of that n the thoracc cavty, but as a constant offset ths would not sgnfcantly affect the results. A transent thoracc pressure representatve of respraton could also be appled, but prmary focus n ths paper s on relatvely short term events assocated wth a near-nstantaneous applcaton of gravty. The relatvely low frequency respratory cycle would not sgnfcantly modfy the results. It has further been assumed that there s a constant flow nto the 'bottom' (dstal end) of the ven, based on average steady state dranage nto the femoral ven. It s unlkely that there wll be sgnfcant backflow through the dstal end durng gravty applcaton, due to the hgher resstance of the smaller vessels. More sophstcated descrptons of transent flow waveforms measured under a range of condtons can be found n the lterature. Of most drect nterest s the study reported by Raju S et al [1], who descrbe flow condtons under ambulatory condtons but not under frst applcaton of the gravty feld, whlst Neglen and Raju [] also focus on the measurement of ambulatory pressures n ndvduals wth sgns of chronc venous defcency. Wlleput R et al [3] and Abu-Yousef M [4] focus on rest and respratory condtons. Agan, t s suggested that the frequences assocated wth these temporal varatons are relatvely low compared wth those assocated wth the phenomenon addressed n ths paper. Furthermore, the startng condton for the analyss s a steady flow through the system (equal to the dstal end flow), wth no gravty appled. Ths paper focuses on the transent pressure and flow dstrbutons n a ven segment, wth and wthout valves, under a near-nstantaneous applcaton of gravty. The system s consdered passve, and effects of the muscle pump are not ncluded: smlarly other relatvely low frequency external load factors are neglected. A body force s appled, n the opposte drecton to flow, representng the acton of gravty under a change of posture from horzontal to vertcal: ths force s sgmodal n tme, so that there s smooth transton from zero to the full gravty force, whch s then held constant. Results and Dscusson Baselne condton, no valve A seres of numercal tests were performed, to ensure that the model performed properly and returned accurate results for smple condtons, ncludng for example usng a lnear tube law, for whch analytcal comparsons were avalable, and for other condtons for whch numercal results have been publshed [17,19]. Once these tests were passed, a frst baselne analyss was performed usng the followng parameters: ven dameter 1.19 cm [5], ven thckness-to-dameter rato 0. [6], ven length 1 m, wall stffness 1 MPa [7], blood vscosty 0.004 Pa.s, blood densty 1000 kg/m 3, dstal (nlet) flow 15.1 ml/s [5], proxmal (outlet) pressure 0 mmhg (0 Pa), near nstantaneous body-force applcaton (gravty ncreased from zero to 9.8 m/s over 0.01 mllseconds). These values are gven n convenent unts: all analyses were performed n SI unts. The ntal condton, pror to the applcaton of gravty, was a steady flow n the opposte drecton to that n whch gravty would be appled (.e from dstal to proxmal end of the tube). Dscretsaton-ndependence tests were performed to ensure that the results dd not depend ether on the number of elements used to represent the geometry of the vessel or on the smulaton tme-step. Results for the baselne condton and for several parameter varatons are reported n Table 1. One of the most mportant results s the 'dynamc pressure rato'. Ths s defned as the rato of the peak dynamc pressure to the unavodable statc pressure that wll be reached when the system has stablsed. Also reported n Table 1 are the magntudes of the frst and second pressure peaks recorded as the system oscllates (to gve an ndcaton of how quckly the overall peak s reached), the tme for whch the valve remans closed durng the frst oscllatory phase, and measures of the peak postve and negatve area changes as the ven expands and collapses. Fgure presents the computed pressure aganst tme at the dstal end of the vessel. The system s oscllatory (the only dampng n ths system s that due to the vscosty of the blood t s recognsed that the real system wll have addtonal dampng due to the vscoelastc propertes of the vessel wall, and probably more mportantly of the surroundng tssues), but after a perod of approxmately 1 s the pressure at the dstal end remans wthn % of the steady state value at 74.5 mmhg (9936.3 Pa), consstent wth the hydrostatc force appled plus the (small) pressure drop assocated wth the supermposed steady flow. The overshoot assocated wth the dynamc system produces a peak pressure of 136.5 mmhg (18151 Pa), representng a dynamc pressure rato of 1.83 (a smple frst prncples analyss wthout dampng would suggest a rato of.00 [8], so ths result s plausble). Fgure 3 llustrates the pressure and flow aganst tme at a pont halfway along the ven. The pressure exhbts smlar characterstcs to that at the dstal end, oscllatng about ts hydrostatc condton of one-half of the dstal end value. The flow starts from the ntal condton, oscllates n response to the sudden applcaton of gravty, and returns to the steady condton after approxmately 1 s. It s noted that there s very sgnfcant reverse flow ( 30 ml/s) Page 3 of 10
BoMedcal Engneerng OnLne 008, 7:8 Table 1: Baselne condton, valve performance and gravty applcaton tme results Parameters No valve Perfect valve "Real" valve Leaky valve Gravty test Dameter (cm) 1. 1. 1. 1. 1. Length (m) 1 1 1 1 1 Young's modulus (kpa) 1000 1000 1000 1000 1000 Posson's rato 0.5 0.5 0.5 0.5 0.5 Blood vscosty (mpas) 4 4 4 4 4 Blood densty (Kgm -3 ) 1000 1000 1000 1000 1000 Gravty applcaton tme (s) Near nstantaneous Near nstantaneous Near nstantaneous Near nstantaneous 0.1 Valve dstrbuton No valve One One One One Valve locaton from nlet (m) No valve 0.5 0.5 0.5 0.5 Valve performance No valve Perfect "Real" Leaky Perfect Frst peak (mmhg) 136.9 (18. kpa) 60. (8.01 kpa) 60.3 (8.0 kpa) 65.4 (8.69 kpa) 50.3 (6.69 kpa) Second peak (mmhg) 17.1 (16.9 kpa) 81.9 (10.9 kpa) 81.9 (10.9 kpa) 115.8 (15.4 kpa) 7.6 (9.66 kpa) Maxmum pressure (mmhg) 136.9 (18. kpa) 93.3 (1.4 kpa) 93.9 (1.4 kpa) 115.8 (15.4 kpa) 99.3 (13. kpa) Dynamc pressure rato 1.83 1.5 1.5 1.51 1.33 Valve closed tme (s) No valve 0.17 0.17 0.11 0.16 Maxmum collapse (%) 0.0 0. 0.4 3.57 8.17 Maxmum expanson (%) 15.1 9.89 9.89 1.5 10.6 occurrng approxmately 80 ms after gravty s appled. The reverse flow acts to fll the more dstal sectons of the ven as t dstends under the ncreased (gravtatonal) pressure. As demonstrated later, ths reverse flow would be enough to close a competent valve and consequently to sheld the lower parts of the ven and reduce peak dynamc pressures. If there were suffcent nflow from below (unlkely n the human system unless the process of standng s done very slowly) then the ven could be flled (dstended) entrely by the nflow, and reverse flow mght not occur. The analyss of the baselne condton gves some confdence n the operaton of the model, and also provdes quanttatve nformaton on the peak pressure that can be expected at the dstal end of the ven n the absence of protecton from venous valves. Further confdence has been developed by comparson of the results wth those from three dmensonal models usng a commercal fnte ele- Baselne the Fgure vessel condton pressure aganst tme at the dstal end of Baselne condton pressure aganst tme at the dstal end of the vessel. The system s oscllatory but after approxmately 1 s the pressure at the dstal end remans wthn % of the steady state value at 74.5 mmhg (9936.3 Pa), consstent wth the hydrostatc force appled plus the (small) pressure drop assocated wth the supermposed steady flow. tme Fgure Baselne at the 3condton mdpont pressure of the vessel aganst tme and flow rate aganst Baselne condton pressure aganst tme and flow rate aganst tme at the mdpont of the vessel. The pressure exhbts smlar characterstcs to that at the dstal end, oscllatng about ts hydrostatc condton of one-half of the dstal end value. The flow starts from the ntal condton, oscllates n response to the sudden applcaton of gravty, and returns to the steady condton after approxmately 1 s. Page 4 of 10
BoMedcal Engneerng OnLne 008, 7:8 ment code, but the reportng of these results s beyond the scope of ths paper. Effect of perfect, real and ncompetent valves The model was next used to evaluate and to provde quanttatve nformaton about a hypothess often expressed n text book descrptons of venous physology, e.g. Browse [13]: 'The venous valves normally protect the wall of the ven below each valve from the pressure n the ven above t.' A perfect (no reverse flow) valve was ntroduced nto the ven at a pont halfway along ts length, and a smulaton performed to llustrate the effect of the valve on the peak pressures and flows n the system. The oscllatory nature of the system s such that the valve wll open and close several tmes before fnally settlng n the open state n the hydrostatc condton. Attenton s focused on the early phase, from the tme of gravty applcaton through to the tme of peak pressure n the system. Fgure 4 llustrates the pressure at the dstal end aganst tme durng the frst second after applcaton of gravty, together wth the results for the case wth no valve. The dstnctve saw-tooth appearance of the oscllatons s due to the summaton of pressure waves as they reflect from the doman boundares. The presence of the valve sgnfcantly changes the form of the dynamc response. Wthout the valve, the system undergoes relatvely hgh-ampltude oscllatons about the hydrostatc pressure value, gradually dampng towards the steady state. Wth the valve, the approach Pressure wthout Fgure 4perfect aganst valve tme at the dstal end of the vessel wth and Pressure aganst tme at the dstal end of the vessel wth and wthout perfect valve. The presence of the valve sgnfcantly changes the form of the dynamc response and the approach towards the steady state s reasonably asymptotc, wth relatvely lower pressure oscllatons supermposed on the asymptote. towards the steady state s reasonably asymptotc, wth relatvely lower pressure oscllatons supermposed on the asymptote. The peak pressure for the system wth a valve s 93.0 mmhg (1370 Pa), representng a dynamc pressure rato of 1.5. Perhaps, therefore, the most mportant observaton s that, consstent wth the hypothess, the valve has provded a very sgnfcant sheldng effect (over 45 mmhg (5985 Pa) reducton n peak pressure). Fgure 5 llustrates pressure and flow at the secton mmedately dstal to the valve, together wth the no-valve system. Here the effect of the valve s very clearly ndcated. The analyses are dentcal up to the pont at whch flow reversal occurs. At ths pont, the valve closes and the flow s set to zero untl forward flow s re-establshed, partly by the constant nflux from the dstal boundary and partly by wave reflecton. The segment of ven dstal to the valve now acts as a closed cylnder (at least n the porton mmedately dstal to the valve whch takes some tme to be affected by the constant nflux from the (relatvely dstant) dstal boundary). Blood contnues to fall towards the dstal end but at a sgnfcantly reduced rate, and the segment mmedately dstal to the valve reduces n dameter and starts to collapse. The degree of collapse s determned by the rate of applcaton of gravty and the physcal characterstcs of the system. In the model reported here, the area reducton mmedately dstal to the valve s approxmately 0%. Fgure 6 llustrates the rato of cross-sectonal area to undeformed cross-sectonal area along the length of the ven at dfferent ponts n tme and Fgure 7 llustrates the pressure varaton along the length of the ven at dfferent ponts n tme, showng clearly the pressure dscontnuty at the valve. The presence of waves reflectng from proxmal and dstal boundares s also apparent. After approxmately 6.3 s, the system settles to wthn % of the hydrostatc state. The above results llustrate the effect of a perfect valve on pressure sheldng. A real valve must permt some reverse flow as t s swept to the closed poston. A frst estmate of the volume of flow reversal (neglectng the reverse flow as the valve s frst entraned) can be made by measurng the swept volume of the valve durng closure. An estmate of ths volume, based on conc sectons s 0.1 ml. Results based on ths approxmaton are llustrated n Fgure 8, together wth those for the perfect valve. Although the 'real' valve allows some regurgtaton as the leaflets are swept to closure, the volume assocated wth ths event s small and the effect s neglgble; ths mght not be true f a larger regurgtant volume were to be admtted, reflectng the entranment of the leaflets n the reversng flow. An ncompetent valve mght be expected to le further towards the no-valve condton. To test ths hypothess, an analyss has been performed n whch the negatve flow rate through the valve has been lmted to 0 ml/s. In ths Page 5 of 10
BoMedcal Engneerng OnLne 008, 7:8 Pressure pont Fgure of 5the aganst vessel tme wth and and flow wthout rate aganst a perfect tme valve at the md- Pressure aganst tme and flow rate aganst tme at the mdpont of the vessel wth and wthout a perfect valve. Here the effect of the valve s very clearly ndcated snce the analyses are dentcal up to the pont at whch flow reversal occurs. condton, the pressure sheldng effect s substantally reduced, and the dynamc pressure rato s ncreased (from 1.5 for the perfect valve) to 1.51. In a fnal test on the reference confguraton wth a perfect valve, the tme over whch gravty was appled was ncreased from near nstantaneous to 100 ms (more consstent wth the lkely tme taken to stand up). As Pressure pressure Fgure 7dscontnuty along the ven at length the valve for ncreasng locaton tme showng a Pressure along the ven length for ncreasng tme showng a pressure dscontnuty at the valve locaton. expected, the frst and second pressure peaks were lower (by the order of 10%) but perhaps surprsngly, the absolute peak was a lttle hgher. Ths was due to dfferent nteractons of the wave reflectons n the system, but t does not affect the overall shape of the response, nor ndeed the conclusons. Parameter studes The model permts the examnaton of the effect of change of the geometrcal and mechancal characterstcs of the system on the pressure-sheldng phenomenon. Results for varatons of a number of parameters are presented n Rato tonal Fgure of area 6cross-sectonal along the ven area length over for undeformed ncreasng tme cross sec- Rato of cross-sectonal area over undeformed cross sectonal area along the ven length for ncreasng tme. Flow rate vs tme through a perfect and "real" valve Fgure 8 Flow rate vs tme through a perfect and "real" valve. Page 6 of 10
BoMedcal Engneerng OnLne 008, 7:8 Table : Parameter varaton test results Parameters Perfect valve Dameter test Stffness test Dstrbuton test Length test Locaton test Dameter (cm) 1. 0.8 1.6 1. 1. 1. 1. Length (m) 1 1 1 1 0.50 0.75 1 Young's modulus (kpa) 1000 1000 500 000 1000 1000 1000 Posson's rato 0.5 0.5 0.5 0.5 0.5 0.5 Blood vscosty (mpas) 4 4 4 4 4 4 Blood densty (Kgm -3 ) 1000 1000 1000 1000 1000 1000 Gravty applcaton tme (s) Near nstantaneous Near nstantaneous Near nstantaneous Near nstantaneous Near nstantaneous Near nstantaneous Valve dstrbuton One One One Two One One Valve locaton from nlet (m) 0.5 0.5 0.5 0.5 and 0.75 0.5 0.5 0.75 Valve performance Perfect Perfect Perfect Perfect Perfect Perfect Frst peak (mmhg) 60. (8.01 kpa) 66.3 (8.81 kpa) 56.8 (7.55 kpa) 56.3 (7.48 kpa) 63. (8.4 kpa) 3.6 (4.34 kpa) 33. (4.41 kpa) 46.8 (6.3 kpa) 3.6 (4.33 kpa) 87. (11.6 kpa) Second peak (mmhg) 81.9 (10.9 kpa) 114.3 (15. kpa) 74.9 (9.97 kpa) 74.6 (9.9 kpa) 95.5 (1.7 kpa) 55.6 (7.39 kpa) 55.9 (7.44 kpa) 57 (7.58 kpa) 99.3 (13. kpa) 90. (1 kpa) Maxmum pressure (mmhg) 93.3 (1.4 kpa) 115.8 (15.4 kpa) 89.5 (11.9 kpa) 90.9 (1.1 kpa) 104.5 (13.9 kpa) 9.9 (1.36 kpa) 57. (7.61 kpa) 77.5 (10.31 kpa) 110.5 (14.7 kpa) 105.3 (14 kpa) Dynamc pressure rato 1.5 1.47 1. 1.1 1.41 1.5 1.53 1.39 1.49 1.41 Valve closed tme (s) 0.17 0.11 0.19 0.6 0.11 0.09 and 0.17 0.06 0.11 0.16 0.15 Maxmum collapse (%) 0. 3.64 4.5 48. 6.06 35.6 3.58 10.6 3.5 4.3 Maxmum expanson (%) 9.89 1.1 9.87 1. 5.41 9.89 5.9 7.19 11.9 11.5 Page 7 of 10
BoMedcal Engneerng OnLne 008, 7:8 Table. Reducng the undeformed dameter of the ven by one-thrd ncreases the dynamc pressure rato to 1.47 n the presence of the valve, and ncreasng the dameter of the ven by one thrd reduces the peak pressure rato to 1.. Smlarly, doublng the ven stffness causes an ncrease to 1.41, and halvng t causes a decrease to 1.1. Each of these results s qualtatvely consstent wth expectatons based on an understandng of the physcs phenomena: the model provdes quantfcaton of the effect. Changng the overall length of the system (whlst mantanng the poston of the valve at 0.5 m from the nlet), or changng the poston of the valve along the length of the 1 m ven, ncreased the dynamc pressure rato, suggestng that the optmal poston for a valve n a ven wth the mposed boundary condtons s near to the mdpont. Fnally, a test wth two valves, one at one-quarter length and one at three-quarters length produced a pressure sheldng of the same magntude as that obtaned wth a sngle valve halfway along the ven. Concluson The one-dmensonal model reported n ths paper permts the quanttatve evaluaton of the effects of venous valves on the loads and geometrcal changes nduced by the acton of gravty. It s an mportant frst step n a longer-term study of venous valves, venous dseases and ther preventon. Wth refnements to the venous valve descrpton, appled tube law and boundary condtons, a more physologcally realstc model can be created n an equvalent form to the Westerhof arteral model [9]. Ths wll enable the model to be valdated aganst physologcal data. For the purposes of ths paper though the greatest nterest s n the dynamc pressure rato, whch provdes a measure of the ncrease of the peak local pressure n the system (due to dynamc effects) over the correspondng hydrostatc pressure. It s demonstrated that, for a confguraton typcal of the femoral ven, the dynamc pressure rato wthout a valve s 1.83, and that wth a perfect valve located halfway along the ven s 1.5. The absolute pressure reducton s over 40 mmhg (530 Pa). The model has been used to nvestgate the quanttatve nfluence of varaton of a number of parameters. Followng extensve n vtro and n vvo valdaton, ths model mght be used to evaluate the effects of valve ncompetence on venous pressure dstrbutons and could have mplcatons for the understandng of the progresson of dsease n the context of varcose vens. It mght also be used as part of an nterventonal plannng tool. The reported study s entrely theoretcal. Valdaton aganst other reported numercal studes has been performed to gve confdence n the numercal mplementaton, but valdaton aganst expermental data s an mportant next step. Some expermental data does exst for collapsble tubes wth gravty effects, for example of the fllng under gravty of an ntally collapsed tube [30], but none of drect relevance to the current study. A prelmnary expermental model that can be used drectly to valdate the current model has been reported by Potter [31] and Burnett [3] but ths s not yet suffcently mature for detaled comparatve evaluaton. Valdaton aganst physologcal data, such as that presented n the works referenced n the secton on boundary condtons, wll requre frst the constructon of an mproved numercal model wth a more complex network representaton of the venous crculaton n the lower lmb. In the longer term a detaled three dmensonal (3D) model s requred to compute the haemodynamc characterstcs n the regon of the valve, and to evaluate the effects of local geometrc and materal varatons. The 1D model descrbed n ths paper provdes mportant mutual valdaton data for such a 3D model, as well as the potental to provde local boundary condtons for t n the regon of the valve. Competng nterests The author(s) declare that ths work was funded by the Brtsh Heart Foundaton whch also fnanced ths manuscrpt. Authors' contrbutons CZ carred out all of the computatonal work reported n ths manuscrpt. DRH drafted the framework of the manuscrpt, ntegrated the computatonal results provded by CZ and drafted the conclusons. DRH, PVL and AJN conceved the program of work, supervsed ts progresson and provded dstnct and separate ntellectual nput. All authors partcpated n the crtcal revew and revson of ths manuscrpt pror to submsson. Appendx: Equatons and dscretsaton A1: Governng equatons The contnuty equaton used was: where da n C = l = dp 10Kp The momentum equaton used was: C P + Q = 0 (A.1) t z ( A n ) n 1 9 A 10 o ( A 15. o ) + 15. Kp ( A n ) (A.) Page 8 of 10
BoMedcal Engneerng OnLne 008, 7:8 where L Q + P t = RQ z (A.3) frst one s easly found from the contnuty equaton and s: C P t + Q P Q = t = 1 0 C (A.10) R = ( A n ) n 8 πµ (A.4) In order to fnd the second tme dervatve of pressure the frst tme dervatve of pressure must be dfferentated n tme. Ths gves: L = n ρ ( A n ) (A.5) P = 1 Q P = 1 Q P 1 Q = t C x t t C t x t C t x (A.11) The tube law used was: where A n P P Kp n e = Ao E Kp = 1 1 σ A n Ao The varables used n equatons 1 7 were: A 0 undeformed vessel area-, A n vessel area at pont along the vessel length at tme n-, µ flud vscosty-, ρ flud densty-, E vessel wall Young's modulus, σ-vessel Posson's rato, h vessel wall thckness and fnally r vessel radus-. All varables used were descrbed n SI unts. A: Lax-Wendroff dscretsaton To fnd the two equatons of nterest for pressure and flow usng the Lax-Wendroff technque the followng two equatons have to be used. The above two equatons are the Taylor seres expanson of second order accuracy. Based on the above two equatons and the smplfed verson of the mass and momentum conservaton equatons above, the equatons needed for pressure and flow usng the Lax-Wendroff technque can be found. Frstly for pressure to be found the frst and second tme dervatves of pressure have to be found. The z h r ( ) 3 1. 5 P t P P t + t P t = + t + Q t Q Q t + t Q t = + t + (A.6) (A.7) (A.8) (A.9) Thus to fnd the second tme dervatve of pressure Q t x must be found. Ths s done by dfferentatng the momentum equatons n space. Ths gves: L Q P Q P R RQ t x t L x L Q Q P ( ) + = = 1 1 = t L R L x Q Q = 1 P R Q t x L L ( ) (A.1) Now by substtutng 1 n 11 the second tme dervatve of pressure s found to be: P 1 Q = P 1 1 P R Q = t C t x t C L x L x (A.13) Snce both the frst and second tme dervatve of pressure are found, equaton 8 can be solved. Ths gves: P t P P t + t P t = + t + P t + t = P t 1 Q t + t + 1 1 P C C L R Q L + t Q t = + P P P C C L + t R t t t Q x C L t P P 1 1 1 CL P P P t t + t t = + t t + t + C Q t t R + Q + CL For flow to be found the same methodology as used to fnd the pressure equaton s used and gves: Q t 1 t Q t 1 (A.14) tr t Q Q L L P P t t + t t = 1 t + t LC Q t + Q ( )+ t 1 1 1 Q t t R + P t P t Q t t R + 1 ( + 1 LL )+ L (A.15) Page 9 of 10
BoMedcal Engneerng OnLne 008, 7:8 Acknowledgements Ths study was funded by the Brtsh Heart Foundaton under BHF PhD studentshp FS/05/086/1946. Many useful synerges arose between ths and a parallel study on deep ven thromboss, EPSRC GR/S86464/01, funded by the Engneerng and Physcal Scences Research Councl. References 1. Gbbs NM: Venous thromboss of the lower lmbs wth partcular reference to bed-rest. Br J Surg 1957, 45:09-36.. McLachln AD, McLachln JA, Jory TA, Rawlng EG: Venous stass n the lower extremtes. Ann Surg 1960, 15:678-685. 3. Sevtt S: The structure and growth of valve-pocket thromb n femoral vens. J Cln Pathol 1974, 7:517-58. 4. Gottlob R, May R, Geleff S: Venous valves: morphology, functon, radology, surgery. New York: Sprnger-Verlag; 1986. 5. Tooke JE, Lowe GO: A textbook of vascular medcne. London, Arnold; 1996. 6. Guyton AC, Hall JE: Textbook of medcal physology. Phladelpha, Saunders; 000. 7. Klabunde RE: Cardovascular physology concepts. Phladelpha, Lppncott Wllams and Wlkns; 005. 8. Stck C, Jaeger H, Wtzleb E: Measurements of volume changes and venous pressure n the human lower leg durng walkng and runnng. J Appl Physol 199, 7:063-068. 9. Tbbs DJ: Varcose vens and related dsorders. Boston, Butterworth-Henemann; 199. 10. Pollack AA, Wood EH: Venous pressure n the saphenous ven at the ankle n man durng exercse and changes n posture. Journal of Appled Physology 1949, 1:649-66. 11. Arnold CC: Venous pressure n the leg of healthy human subjects at rest and durng muscular exercse n the nearly erect poston. Acta Chr Scand 1965, 130:570-583. 1. Tbbs DJ: Varcose vens, venous dsorders, and lymphatc problems n the lower lmbs. New York, Oxford Unversty Press; 1997. 13. Browse NL: Dseases of the vens. New York, Arnold ; 1999. 14. Shapro AH: Steady Flow n Collapsble Tubes. J Bomech Eng 1977, 99:16-147. 15. Elad D, Kamm RD, Shapro AH: Chokng phenomena n a lunglke model. J Bomech Eng 1987, 109:1-9. 16. Brook BS: The effect of gravty on the haemodynamcs of the graffe jugular ven. Unversty of Leeds, Department of Appled Mathematcal Studes; 1997. 17. Brook BS, Pedley TJ: A model for tme-dependent flow n (graffe jugular) vens: unform tube propertes. J Bomech 00, 35:95-107. 18. Anderson JD: Computatonal flud dynamcs : the bascs wth applcatons. New York, McGraw-Hll; 1995. 19. Mlsc V, Quarteron A: Analyss of lumped parameter models for blood flow smulatons and ther relaton wth 1D models. Esam-Mathematcal Modellng and Numercal Analyss-Modelsaton Mathematque Et Analyse Numerque 004, 38:613-63. 0. Brook BS, Falle S, Pedley TJ: Numercal solutons for unsteady gravty-drven flows n collapsble tubes: evoluton and rollwave nstablty of a steady state. Journal of Flud Mechancs 1999, 396:3-56. 1. Raju S, Green AB, Fredercks RK, Neglen PN, Hudson CA, Koeng K: Tube collapse and valve closure n ambulatory venous pressure regulaton: studes wth a mechancal model. J Endovasc Surg 1998, 5:4-51.. Neglen P, Raju S: Ambulatory venous pressure revsted. J Vasc Surg 000, 31:106-113. 3. Wlleput R, Rondeux C, De Troyer A: Breathng affects venous return from legs n humans. J Appl Physol 1984, 57:971-976. 4. Abu-Yousef MM, Mufd M, Woods KT, Brown BP, Barloon TJ: Normal lower lmb venous Doppler flow phascty: s t cardac or respratory? AJR Am J Roentgenol 1997, 169:171-175. 5. Fronek A, Crqu MH, Denenberg J, Langer RD: Common femoral ven dmensons and hemodynamcs ncludng Valsalva response as a functon of sex, age, and ethncty n a populaton study. J Vasc Surg 001, 33:1050-1056. 6. Da G, Gertler JP, Kamm RD: The effects of external compresson on venous blood flow and tssue deformaton n the lower leg. J Bomech Eng 1999, 11:557-564. 7. Buxton GA, Clarke N: Computatonal Phlebology: The Smulaton of a Ven Valve. Journal of Bologcal Physcs 006, 3:507-51. 8. Roark RJ, Young WC: Formulas for stress and stran. London, McGraw-Hll; 001. 9. Bert G, Lonsdale G, Schmdt JG, Benker S, Hose DR, Fenner JW, Jones DM, Mddleton SE, Wollny G: Grd smulaton servces for the medcal communty. Internatonal Journal of Computatonal Methods 005. 30. Fullana JM, Cros F, Flaud P, Zalesk S: Fllng a collapsble tube. Journal of Flud Mechancs 003, 494:85-96. 31. Potter K: The effect of gravty on valve closure and ven collapse n the deep vens of the leg. Pressure and flow analyss. The Unversty of Sheffeld, Department of Medcal Physcs and Clncal Engneerng; 006. 3. Burnett J: The effect of gravty on valve closure and ven collapse n the deep vens of the leg. Image analyss. The Unversty of Sheffeld, Department of Medcal Physcs and Clncal Engneerng; 006. Publsh wth BoMed Central and every scentst can read your work free of charge "BoMed Central wll be the most sgnfcant development for dssemnatng the results of bomedcal research n our lfetme." Sr Paul Nurse, Cancer Research UK Your research papers wll be: avalable free of charge to the entre bomedcal communty peer revewed and publshed mmedately upon acceptance cted n PubMed and archved on PubMed Central yours you keep the copyrght BoMedcentral Submt your manuscrpt here: http://www.bomedcentral.com/nfo/publshng_adv.asp Page 10 of 10