Kung and Taylor (2011)

Development of a Physical Windkessel Module to Re-Create In VivoVascular Flow Impedance for In Vitro Experiments

ETHAN O. KUNG1 and CHARLES A. TAYLOR1,2

1Department of Bioengineering, Stanford University, James H. Clark Center, 318 Campus Drive, E350B, Stanford, CA 94305,USA; and 2Department of Surgery, Stanford University, Stanford, CA, USA

(Received 6 August 2010; accepted 5 November 2010; published online 20 November 2010)

Associate Editor Stephen B. Knisley oversaw the review of this article.

AbstractTo create and characterize a physical Windkesselmodule that can provide realistic and predictable vascularimpedances for in vitro ow experiments used for computa-tional uid dynamics validation, and other investigations ofthe cardiovascular system and medical devices. We developedpractical design and manufacturing methods for constructingow resistance and capacitance units. Using these units weassembled a Windkessel impedance module and dened itscorresponding analytical model incorporating an inductanceto account for uid momentum. We tested various resistanceunits and Windkessel modules using a ow system, andcompared experimental measurements to analytical predic-tions of pressure, ow, and impedance. The resistancemodules exhibited stable resistance values over wide rangesof ow rates. The resistance value variations of any partic-ular resistor are typically within 5% across the range of owthat it is expected to accommodate under physiologic owconditions. In the Windkessel impedance modules, themeasured ow and pressure waveforms agreed very favor-ably with the analytical calculations for four different owconditions used to test each module. The shapes andmagnitudes of the impedance modulus and phase agree wellbetween experiment and theoretical values, and also withthose measured in vivo in previous studies. The Windkesselimpedance module we developed can be used as a practicaltool to provide realistic vascular impedance for in vitrocardiovascular studies. Upon proper characterization of theimpedance module, its analytical model can accuratelypredict its measured behavior under different ow condi-tions.

KeywordsWindkessel, Vascular impedance, In vitro valida-

tion, Blood ow, Flow resistance, Flow capacitance, Flow

impedance, Flow inductance, Boundary condition, Phantom

outlet.

INTRODUCTION

Computational uid dynamics (CFD) is a power-ful tool for quantifying hemodynamic forces in thecardiovascular system. In CFD simulations, realisticoutow boundary conditions are necessary to repre-sent physical properties of the downstream vascula-ture not modeled in the numerical domain, and toproduce physiologic levels of pressure.15 While vari-ous types of boundary condition implementationsexist,2,5,7,12,13 previous studies showed that imped-ance-based boundary condition is the preferredapproach for coupling wave reections from thedownstream vasculature into the numericaldomain,15 and that simple lumped-parameter modelrepresentations can provide realistic impedancessimilar to those provided by a more complicatedmethod employing a distributed parameter model.2

The Windkessel model, due to its simplicity andability to provide physiologically realistic imped-ances,10,14,16,18 is a practical method of prescribingsuitable boundary conditions to the numericaldomain in CFD simulations.

The Windkessel model is represented as a circuitcontaining lumped elements of resistance, capacitance,and inductance. Although these elements are moregenerally interpreted in an electrical system, there is adirect analogy between the governing equations of anelectric circuit and those of a uid system, where theuid pressure, the uid volume, and the volumetricow rate directly parallels voltage, electrical charge,and electrical current, respectively. For example, therelationship between voltage and current related byelectrical resistance as described by the equationV = IR, can be directly modied into P = QR todescribe the relationship between pressure and owrate related by the uid resistance.

Address correspondence to Charles A. Taylor, Department of

Bioengineering, Stanford University, James H. Clark Center, 318

Campus Drive, E350B, Stanford, CA 94305, USA. Electronic mail:

[email protected]

Cardiovascular Engineering and Technology, Vol. 2, No. 1, March 2011 ( 2010) pp. 214DOI: 10.1007/s13239-010-0030-6

1869-408X/11/0300-0002/0 2010 Biomedical Engineering Society

2

When used to mimic vascular impedances, associa-tions exist between the lumped component values in aWindkessel model and in vivo physiological parame-ters. The resistance and inductance values are associ-ated with the density and viscosity of blood, and withthe geometry and architecture of the vasculature whichare functions of both the anatomy and the vasculartone. The capacitance value is most affected by thephysical properties and the vascular tone of the largearteries. Since the properties of blood, the blood vesselanatomy and physical properties, and the vasculartone do not vary signicantly within the time frames ofa cardiac cycle, it is the general practice to implementan analytical Windkessel model with xed componentvalues.

In order to validate CFD against experimental data,methods must be developed to reliably construct aphysical model of the Windkessel boundary conditionsuch that there is a direct parallel between the experi-mental setup and the CFD simulation. In this paper wepresent the theories, principles, practical design con-siderations, and manufacturing processes for physi-cally constructing the resistance and capacitancecomponents of a Windkessel impedance module. Thesemethods enable the construction of Windkessel com-ponents with values that are predictable and constantthroughout their operating ranges. We also present ananalytical model that describes the physical Windkes-sel module, and incorporates an inductance to accountfor uid momentum. We manufactured several resis-tance units and tested them independently in a owloop to verify their operations. Windkessel modulesthat mimic the thoracicaortic and renal impedanceswere then assembled and tested under physiologicpulsatile ow conditions, and experimental measure-ments were compared to analytical predictions ofpressure and ow.

METHODS

Determining Target Windkessel Component Values

Target values to aim for in the design and con-struction of the Windkessel components must rst bedetermined. The component value estimation may beperformed using a basic three element Windkesselmodel consisting of a proximal resistor (Rp), a capac-itor (C), and a distal resistor (Rd) as shown in Fig. 1.The target component values are those that would re-sult in the desired pressure and ow relationshipreecting the particular vascular impedance to bemimicked. For a periodic ow condition, the pressure

and ow is related by the equation in the frequencydomain:

Px QxZx 1where x is the angular frequency, Q is the volumetricow rate, and Z is the impedance of the three-elementWindkessel circuit:

Zx Rp Rd1 jxCRd 2

In previous reports, blood ow waveforms at vari-ous locations in the vascular tree have been obtainedwith imaging modalities such as ultrasound or phase-contrast magnetic resonance imaging,1,3,9 and pressurewaveforms have been obtained with pressure cuffs orarterial catheters.8 Using the available in vivo ow andpressure waveform data, together with Eqs. (1) and (2),an iterative process can be performed to nd the targetWindkessel component values for mimicking the invivo vascular impedance at a specic location. Webegin by using the ow data and initial guesses of thecomponent values as input parameter into Eqs. (1) and(2) to calculate a resulting pressure waveform. Thecomponent values can then be adjusted with the goalof matching the calculated pressure to the in vivomeasured pressure waveform. For any given inputow, the total resistance (sum of Rp and Rd) can beadjusted to vertically shift the calculated pressurewaveform, and the ratio of Rp/Rd as well as thecapacitance can be adjusted to modulate the shape andpulse amplitude of the calculated pressure waveform.Once we determine the component values which givethe desired pressure and ow relationship, we thenconsider them the target values in the design andconstruction of the components.

Flow Resistance Module

Theory and Construction Principles

In Poiseuilles solution for laminar ow in a straightcylinder, the relationship between the pressure drop

FIGURE 1. A basic three-element Windkessel model forcomponent value estimation purpose.

Development of a Physical Windkessel Module 3

across the cylinder (DP) and volumetric ow rate(Q) is:

DP 8llpr4

Q 3

The ow resistance dened as R = DP/Q is then:

R 8llpr4

4

where l is the dynamic viscosity of the uid, l is thelength of the cylinder, and r is the radius of the cyl-inder.

Equation (3) holds true in a laminar ow condition,where the resistance is constant and independent ofow rate. In turbulent ow, however, the additionalenergy loss leads to the pressure drop across the owchannel becoming proportional to the ow ratesquared (DP Q2), implying that the total effectiveresistance as dened by R = DP/Q is proportional tothe ow rate (R Q). Since the goal is to create aconstant resistance that is independent of ow rate, itis thus important to avoid turbulence and maintainlaminar ow. An approximate condition for laminarow in a circular cylinder is the satisfaction of thefollowing equation for Reynolds number:

Re vrt Q

ptr

thin-walled glass capillary tubes (Sutter Instrument,CA) inside a plexiglass cylinder as shown in Fig. 3a.We applied a small amount of silicone rubber adhesivesealant (RTV 102, GE Silicones, NY) in between thecapillary tubes around their middle section to adherethe tubes to one another, and to block uid passage-ways through the gaps in between the tubes. We thenapplied a small amount of epoxy (5 Minute Epoxy,Devcon, MA) between the plexiglass surface and thebundle of capillary tubes to secure the capillary tubesinside the plexiglass cylinder.

The theoretical resistance of the resistance module isgiven by:

R 8llpNr4

14

where l is the dynamic viscosity of the working uid,l is the length of the capillary tubes, r is the insideradius of each individual capillary tube, and N is thetotal number of capillary tubes in parallel.17

For a standard capillary tube length of 10 cm,Fig. 2b shows the relationship between the number of

tubes and the resulting resistance for various standardcapillary tube sizes that can be readily purchased.

Using the same principle of parallel channels, Fig. 3bshows a method for creating a switchable resistancemodule where the resistance value can be changed dur-ing an experiment. Multiple resistance modules can beplaced in parallel, with control valves that open andclose to add in or remove parallel resistor(s) in order todecrease or increase the effective total resistance.

The resistance module must be connected to tubingat each end. It is important to ensure that laminar owis maintained throughout the connection tubing, andthat diameter changes at the connection junctions areminimized to avoid the creation of turbulence. Weconstructed Table 1 to aid the design process ofchoosing an appropriate combination of a standardcapillary tube size and connection tubing size, suchthat the resistance module can connect smoothly to itsinlet and outlet tubing, and that the connection tubingitself can also accommodate the maximum ow raterequired. The maximum laminar ow for any partic-ular ow conduit diameter can be calculated from

FIGURE 2. (a) Maximum laminar flow rate vs. number of parallel channels for various resistance values. (b) Resistance vs.number of parallel channels for various standard capillary tube inside diameters (ID). Calculated using: Fluid dynamic viscos-ity 5 0.046 g/cm s. Capillary tube length 5 10 cm.

FIGURE 3. (a) Capillary tube resistance module construction. (b) Switchable resistance setup.


Eq. (5), and is listed beside each conduit diameter inthe table. Note that the Reynolds number within thecapillary tubes is much lower than that in the con-nection tubing (due to the smaller diameter of thecapillary tubes), thus the critical factor in maintaininglaminar ow is the connection tubing diameter. FromTable 1, the optimal capillary tube size for construct-ing the resistance module is determined by identifyinga resistance value that is close to the desired targetvalue, in combination with a conduit diameter that canaccommodate the maximum expected ow. Once thecapillary tube size is determined, a circle packingalgorithm11 can then be used to determine the preciseplexiglass cylinder diameter required to house thespecic number of capillary tubes needed for obtainingthe desired resistance. Upon completing the actualconstruction of the resistance module, we manuallycount the number of capillary tubes in the plexiglasscylinder, and use the resulting count, together with themeasured dynamic viscosity of the working uid andEq. (14), to determine the theoretical resistance of themodule.

Flow Capacitance Module

The capacitance of a uid system is dened asC = DV/DP where DV and DP are the changes in

volume and pressure. In a closed system at constanttemperature, an ideal gas exhibits the behaviorPV = (P + DP)(V DV), where P and V are thereference pressure and volume. The capacitance of apocket of air is then:

Ca V DV=P 15We constructed the capacitance module with a

plexiglass box that can trap a precise amount of air,which acts as a capacitance in the system (Fig. 4a).Equation (15) indicates that, as uid enters thecapacitor and compresses the air, the capacitance ofthe module would decrease. For small changes involume relative to the reference volume, however, areasonably constant capacitance can be maintained. Asuid enters and exits the box, the vertical level of theuid in the box rises and falls slightly. The varyinguid level contributes to an additional capacitance thatis in series with the capacitance due to air compression.The pressure change in the uid due to the varyinguid level under the effects of gravity and uid mass is:

DP qgDh qgDV=A 16where q is the uid density, g is the gravitationalconstant, and A is the area of the uid/air inter-face (assuming a column of uid with constant

TABLE 1. Estimated resistance values (and numbers of capillary tubes) resulting from various combinations of conduit diameter(maximum laminar flow rate), and capillary tube size.

Capillary tubes

OD/IDa (mm) 1 (200 cc/s) 3/4 (150 cc/s) 5/8 (125 cc/s) 1/2 (100 cc/s) 3/8 (75 cc/s) 1/4 (50 cc/s)

2/1.56 231 (137) 410 (77) 591 (54) 923 (34) 1,641 (19) 3,693 (9)

1.5/1.1 525 (244) 934 (137) 1,345 (95) 2,101 (61) 3,735 (34) 8,404 (15)

1.2/0.9 750 (381) 1,334 (214) 1,920 (149) 3,000 (95) 5,334 (54) 12,002 (24)

1/0.78 923 (548) 1,641 (308) 2,364 (214) 3,693 (137) 6,566 (77) 14,773 (34)

1/0.75 1,080 (548) 1,920 (308) 2,765 (214) 4,321 (137) 7,681 (77) 17,282 (34)

Calculated using fluid dynamic viscosity = 0.046 g/cm s.

Fluid density = 1.1 g/mL.

Capillary tube length = 10 cm.

Circle packing density = 0.85 by area.aOD/ID stands for outside diameter/inside diameter.

FIGURE 4. (a) Capacitance module construction. (b) Capacitor inlet contour.

E. O. KUNG AND C. A. TAYLOR6

cross-sectional area). The capacitance due to thevarying uid level is then:

Cv A=qg 17Since Cv is in series with Ca the overall capacitance

C can be approximated by Ca alone if Cv Ca:For Cv Ca:

C 1Ca 1Cv

1 CaCvCa Cv Ca 18

In the actual construction of the capacitance mod-ule, we designed the box to be large enough so that theapproximation in Eq. (18) is true. We also designed asmooth contour for the inlet of the capacitance module(Fig. 4b) in order to minimize ow turbulences andthus avoid parasitic resistances. In addition, two accessports are included at the top of the capacitance modulefor air volume modulation and pressure measure-ments, and a graduated scale on the sidewalls for airvolume measurement (Fig. 4a).

Flow Inductance

The ow inductance is an inherent parameter of auid system resulting from the uid mass. It describeshow a force, manifest as a pressure dierential, isrequired to accelerate a body of uid. The inductancein a uid system creates a pressure drop in response toa change in ow as described by the equation:

DP LdQdt

19

where L is the inductance value, which can be calcu-lated from the uid density and the geometry of theow conduit:

L ql=A 20where l and A are the length and the cross-sectionalarea of the conduit.

Assembled Windkessel Module and CorrespondingAnalytical Model

We assembled the Windkessel impedance module byputting together two resistors and one capacitor as

shown in Fig. 5a. Note that in such a physical setupthe reference pressure of the capacitor is the initialpressure within the capacitor when the system is in no-ow equilibrium, and thus the capacitor is consideredto be connected to the ground. In the analyticalmodel, inductive effects of the uid body is taken intoaccount14 and the impedance module is represented asan LRCR circuit as shown in Fig. 5b. Note that eventhough there is an inductance associated with thedownstream resistance Rd, since the ow through Rd istypically nearly constant, the presence of the induc-tance is transparent to the operation of the impedanceunit. Incorporating only the upstream inductance inthe analytical model is sufcient to fully capture thebehavior of the physical impedance module.

EXPERIMENTAL TESTING AND DATAANALYSIS

Resistance Module

We tested the operation of the resistance moduleswith a setup depicted in Fig. 6. We used a 1/12 horse-power, 3,100 RPM, steady ow pump (Model 3-MD-HC, Little Giant Pump Co., OK) to drive ow throughthe resistance module. The working uid in the owsystem was a 40% glycerol solution with a dynamicviscosity similar to that of blood. For data acquisition,we used an ultrasonic transit-time ow probe tomonitor the ow through the system. We placed theexternally clamped ow probe (8PXL, Transonic

FIGURE 5. (a) Assembled impedance module. (b) Final analytical model of impedance module.

FIGURE 6. Resistance module steady flow testing setup.


Systems, NY) around a short section of Tygon tubingR3603, and sent the signals from the probe into aowmeter (TS410, Transonic Systems, NY). Forpressure measurements, we inserted catheter pressuretransducers (Mikro-Tip SPC-350, Millar Instru-ments, Huston, TX) into the ow conduit immediatelyupstream and downstream of the resistance module tocapture instantaneous pressure readings, and obtainthe pressure drop across the resistor. We sent the sig-nals from each catheter pressure transducer into apressure control unit (TCB-600, Millar Instruments,TX) which produces an electrical output of 0.5 V per100 mmHg of pressure. We recorded the data from theow meter and the pressure control units at a samplerate of 2 kHz using a data acquisition unit (USB-6259,National Instruments, Austin, TX) and a LabVIEWprogram (LabVIEW v.8, National Instruments, Aus-tin, TX). We averaged 8,000 samples of ow andpressure (effectively, 4 s of ow and pressure) to obtaineach data point. We then divided the measured pres-sure drop across the resistor by the measured volu-metric ow rate through the resistor to obtain theresistance value.

The ow control for the steady pump consisted of aLabVIEW program that directed the data acquisitionunit to send a voltage to an isolation amplier (AD210,Analog Devices, MA), which then produced the samecontrol voltage to feed into a variable frequency drive(Stratus, Control Resources Inc., MA) that drove theow pump to produce dierent constant ow ratesthrough the ow loop. The purpose of including theisolation amplier in the signal chain was to electron-ically de-couple the high-power operation of the vari-able frequency pump drive from the data acquisitionunit to avoid signal interference.

In addition to the resistance modules, we also testedthe resistance of a partially closed ball valve, which has

commonly been used as a method to produce owresistance in previous literatures.4,6 We adjusted therelative resistance of the ball valve by adjusting theproportion that the valve was closed.

Assembled Windkessel Module

We tested the assembled Windkessel impedancemodules using a setup depicted in Fig. 7. A custom-built, computer-controlled pulsatile pump in parallelwith a steady ow pump produced physiological-level,pulsatile, and cyclic ow waveforms into the Wind-kessel module. Two ultrasonic transit-time owprobes (8PXL & 6PXL, Transonic Systems, NY) wereused to monitor the volumetric ows through Rp andRd. For pressure measurements, we inserted catheterpressure transducers into the ow conduits and intothe capacitor chamber to capture the pressure wave-forms at three points in the circuit. The ow andpressure data were recorded at a sample rate of96 samples per second. We averaged approximately50 cycles of ow and pressure data to obtain onerepresentative cycle of ow and pressure waveforms.We used the pressures measured at P3 as the groundreference, and subtracted it from the pressures mea-sured at P1 and P2, to obtain the true pressurewaveforms at P1 and P2.

We tested two impedance modules, one mimickingthe in vivo thoracicaortic impedance, and the othermimicking the in vivo renal impedance, using fourdifferent input ow waveforms approximately simu-lating physiological ows for each module. Weincluded input ow waveforms with different periods,as well as considerably different shapes, to investigatethe impedance behavior of each module across a widerange of ow conditions.

FIGURE 7. Impedance module pulsatile flow testing setup.


The impedance of the analytical Windkessel circuitin Fig. 7 can be represented by the equation:

Zx jxL Rp Rd1 jxCRd 21

By prescribing the measured input ow waveform,and the values of the lumped components, we calcu-lated the theoretical pressure waveform at P1 usingEqs. (1) and (21). We then calculated the theoreticalpressure waveform at P2 and the ow waveform Qdusing the equation DP = QR.

RESULTS AND DISCUSSIONS

Resistance Module

Figure 8 presents results of resistance vs. ow ratefor two of the resistance modules, and for a partiallyclosed ball valve. In Fig. 8a, the theoretical resistanceof the resistance module is 500 Barye s/cm3. Themeasured resistance is very close to the expected the-oretical value, and the resistance module exhibits rel-atively constant resistance values over the range ofow rates tested. The variation in the resistance valuebetween ow rates of 20 and 100 cc/s is approximately5%. The ball valve on the other hand, exhibitsa resistance that varies linearly with the ow rate.Figure 8b shows results of a resistance module withtheoretical resistance of 6,700 Barye s/cm3, and a ballvalve adjusted to produce a higher ow resistance. Wesee similar results at this higher regime of resistancevalues. The value variation of the resistance modulebetween ow rates of 20 and 60 cc/s is approximately7%. Note that a resistance unit with resistance in thehigher regime typically only needs to accommodate

relatively low ows in its actual operation. If placedwithin a Windkessel module under physiologic ows,the expected maximum ow through such a resistor inFig. 8b would be approximately 30 cc/s. All of theother resistor modules we have tested (but not shownhere) also exhibited similar behaviors of relativelyconstant resistance values over the range of ow ratesthey are expected to accommodate. The resistance ofthe ball valve showing a linear dependence on owrate, and extrapolated value of zero at zero ow, sug-gests that it is a result of turbulence alone as discussedin Flow Resistance Module section. For the resis-tance modules, the slight increases in the resistancevalue with ow rate suggest that there is a smallamount of turbulence present in the modules.

The resistance variations at the low ow regions arelikely due to measurement imprecision, but not due tothe actual resistance change or instability in the resis-tance module. Very low ows and small pressure dropsacross the resistor result in low signal-to-noise for boththe ultrasonic ow probe and the pressure transducers,and hence diculties in obtaining precise measure-ments. Fortunately, the fact that the pressure dropacross the resistor is insignicant during very low ows,means that the resistance value also have minimalimpact during that period. The accuracy of experi-mental conrmation of resistance values during thevery low ow regions is thus of minimal importance.

At a xed ow rate, we found that the resistancevalue of a resistance module may decrease over time byup to 5%. The decrease may be due to trapped airbubbles being purged out of the capillary tubes overtime with ow (since the presence of air bubbles in thetubes would obstruct the uid passage and result inelevated resistance). This source of resistance variationcan be minimized with careful removal of air from the

FIGURE 8. Resistance vs. flow rate for resistance module with theoretical resistance of (a) 500 Barye s/cm3, (b) 6,700 Barye s/cm3, and a partially closed ball valve.


ow system during setup to minimize the amount of airthat would be trapped in the resistor during operation.

Assembled Windkessel Module

Table 2 shows the theoretical Windkessel compo-nent values as calculated from their physical con-structions, where the values of L calculated from thegeometry of the physical system as described in FlowInductance section, the values of resistances calcu-lated from their construction details as described inFlow Resistance Module section, and the values ofC calculated from the operating pressure and air vol-ume in the capacitors as described in Flow Capaci-tance Module section. The experimental componentvalues in Table 2, unless otherwise noted, were deter-mined from the experimentally measured pressure andow data, using a method similar to that described inDetermining Target Windkessel Component Valuessection. For the thoracicaortic impedance module,the inductance and resistances behaved as theoreticallypredicted, where the observed capacitance in the actualexperiment was larger than the theoretical expectation.For the renal impedance module, we experimentallydetermined the resistance values from steady ow testsof the impedance module, and found that the actualresistances were about 15% less than theoretical. Theinductance value, on the other hand, was higher thantheoretical. The capacitance value was consistent withthe theoretical prediction. The differences between thetheoretical and experimental component values may beattributed to variations in the physical construction ofthe components, as well as to the connection parts inbetween the components.

We prescribed the experimental component valuesfrom Table 2 in the analytical calculations of pressureand ow. Figure 9 shows pressure and ow compari-sons between experimental measurements and analyt-ical calculations for the impedance module mimickingthe in vivo aortic impedance at the thoracic level. Forall of the four different input ow waveforms tested,the measured pressure waveforms at P1 and P2 (asdenoted in Fig. 7), and the ow waveform through Rd,

all agree extremely well with the analytical calculationsin their shapes, phases, and magnitudes. The maximumdifference between measured and calculated pressure(P1 & P2) and ow (Qd) is 6 and 8%, respectively.Note that two different cyclic periods (1 and 0.75 s)were included in the test and analysis, and theimpedance module performed predictably under owconditions with both period lengths. Figure 10 showssimilar results for the impedance module mimickingthe renal impedance. There is the same excellent matchbetween experimental measurements and analyticalcalculations of pressure and ow waveforms for all ofthe four different ow conditions tested. The maxi-mum difference between measured and calculatedpressure (P1 & P2) and ow (Qd) is 8 and 15%,respectively.

In Figs. 9 and 10, the ow waveforms show thatmuch of the pulsatility in the input ow is absorbed bythe capacitor, and the ow through the downstreamresistor is fairly constant. This implies that for anygiven input ow waveform, the proximal resistor Rpneeds to be able to accommodate the peak ow of theinput waveform, where the downstream resistorRd only needs to accommodate approximately theaveraged ow of the input waveform.

By subtracting Qd from Q, we can calculate the owinto the capacitor, which then can be integrated to ndthe change in uid volume inside the capacitor overeach cardiac cycle. From calculations of pressure andvolume with Eq. (15), we conrmed that the variationof capacitance value due to the volume change overeach cardiac cycle is less than 3% from the referencevalue for both impedance modules.

Figure 11 shows the impedance modulus and phaseas derived from the analytical model, and as calculatedfrom the four sets of experimental pressure and owdata for each module. For both impedance modules,there is close agreement between the theoreticalimpedance modulus and phase, and those determinedfrom the experimental data of all of the four differentow conditions. This further shows that the impedancemodules behave very consistently even when the owconditions were changed. The general shapes and

TABLE 2. Theoretical and experimental Windkessel component values for the thoracicaortic andrenal impedance modules.

Thoracicaortic Renal

Theoretical Experimental Theoretical Experimental

L (Barye s2/cm3) 7 7 16 26

Rp (Barye s/cm3) 245 245 3,050 2,522

C (cm3/Barye) 2.3 e4 4.0 e4 1.3 e4 1.3 e4Rd (Barye s/cm

3) 4,046 4,046 5,944 5,221


magnitudes of the impedance modulus and phase alsocompare well with those measured in vivo in previousstudies.10,14,17,18

CONCLUSION

We showed that using the methods presented here,we can construct ow resistance units with stableresistance values over wide ranges of ow rates. This isa signicant advancement from the common practiceof using a partially closed valve to create ow resis-tances. The resistance value of the units we constructedcan both be theoretically determined from construc-tion details, and experimentally conrmed from pres-sure and ow measurements. We further showed thatthe impedance module assembled from individual

resistor and capacitor components performs veryconsistently across dierent ow conditions, and thatthe corresponding analytical model faithfully capturesthe behavior of the physical system. When actuallyemploying the physical Windkessel module in otherexperimental applications, whenever possible, ow andpressure data should be used to conrm or adjust thelumped component value assignments in the corre-sponding analytical model. We have shown that uponproper characterization of a particular impedancemodule, the analytical model can then accurately pre-dict its behavior under dierent ow conditions.

Compared to the Windkessel module previouslypresented by Westerhof et al.,17 the methods presentedhere offer simpler and more robust construction, andinclude considerations for minimizing turbulence inorder to minimize parasitic resistances and resistance

FIGURE 9. Comparisons between measured (solid lines) and calculated (dots) pressure and flow waveforms for the thoracicaortic impedance module under four different flow conditions. Note that P1, P2, Qd, and Q are the pressures and flows in theimpedance module as depicted in Fig. 7.


variations across different ow rates. The analyticalmodel presented here also includes the physical effectsof inductance, offering a more complete description ofthe physical system.

In conclusion, the Windkessel impedance modulewe developed can be used as a practical tool forin vitro cardiovascular studies. Implementing theWindkessel module in a physical setup enables theexperimental system to replicate realistic blood pres-sures under physiologic ow conditions. The ability toconstruct in vitro physical systems to mimic in vivoconditions can aid in the direct physical testing ofimplantable cardiovascular medical devices such asstents and stent grafts, and enable reliable measure-ments of how the in vivo forces and tissue motionswill interact with the devices. In the area of CFD

validation, well-characterized physical Windkesselmodules connected to the outlets of a physical phan-tom will allow prescriptions of the same outletboundary condition in the computational domain,such that the boundary condition prescription in silicois representative of the physical reality. Furthermore,the ability to implement realistic impedances in vitroenables experimental studies involving deformablematerials, where realistic pressures are absolutelyessential for obtaining proper uidsolid interactions.These studies will be useful for investigating the pul-satile motions of blood vessels, and wave propaga-tions in the cardiovascular system. The workpresented here serves as a basis to contribute towardsmore rigorous cardiovascular in vitro experimentalstudies in the future.

FIGURE 10. Comparisons between measured (solid lines) and calculated (dots) pressure and flow waveforms for the renalimpedance module under four different flow conditions. Note that P1, P2, Qd, and Q are the pressures and flows in the impedancemodule as depicted in Fig. 7.


ACKNOWLEDGMENTS

The authors would like to thank Lakhbir Johal andChris Elkins for assistance with the physical con-struction of the Windkessel modules and with the owexperiments. This work was supported by the NationalInstitutes of Health (grants P50 HL083800, P41RR09784, and U54 GM072970).

REFERENCES

1Bax, L., C. J. Bakker, W. M. Klein, N. Blanken, J. J.Beutler, and W. P. Mali. Renal blood ow measurements

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FIGURE 11. Comparisons between theoretical and experimental flow impedance modulus and phase for the (a) thoracicaortic,and (b) renal, impedance module.


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Development of a Physical Windkessel Module to Re-Create In Vivo Vascular Flow Impedance for In Vitro ExperimentsAbstractIntroductionMethodsDetermining Target Windkessel Component ValuesFlow Resistance ModuleTheory and Construction PrinciplesPractical Design and Construction Methods

Flow Capacitance ModuleFlow InductanceAssembled Windkessel Module and Corresponding Analytical Model

Experimental Testing and Data AnalysisResistance ModuleAssembled Windkessel Module

Results and DiscussionsResistance ModuleAssembled Windkessel Module

ConclusionAcknowledgmentsReferences

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Kung and Taylor (2011)