evaluation of compliance of arterialvessel using coupled ... · evaluation of compliance of...

Copyright c© 2008 Tech Science Press MCB, vol.5, no.4, pp.229-245, 2008

Evaluation of Compliance of Arterial Vessel Using Coupled Fluid StructureInteraction Analysis

Abhijit Sinha Roy∗, Lloyd H. Back† and Rupak K. Banerjee‡

Abstract: The in vivo and ex vivo complianceof arteries are expected to be closely related andestimated. Fluid-structure interaction analysis canassess the agreement between the two compli-ances. To evaluate this hypothesis, a pulsatilefluid-structure interaction analysis of blood flowin femoral artery of a dog was conducted using:(1) measured in vivo mean pressure (72.5 mmHg),mean pressure drop (0.59 mmHg), mean velocity(15.1 cm/sec); and (2) ex vivo measurements ofnon – linear elastic properties of femoral artery.Additional analyses were conducted for physio-logical pressures (104.1 and 140.7 mmHg) andblood flow using a characteristic linear pressure– flow relationship. The computed compliancedecreased from 0.198% diameter change/mmHgat 72.5 mmHg to 0.145% diameter change/mmHgat 140.7 mmHg. The computed compliance tendsto match well with in vivo compliance of femoralartery at lower pressure but is overestimated athigher pressure. This suggests an alteration in thecompliance of the artery during ex vivo elasticitymeasurements.

Keyword: Hemodynamics, fluid-structure in-teraction, femoral artery, pressure-flow relation,compliance.

1 Introduction

The compliance and non-linear elasticity of ar-terial wall have been researched for many years.Researchers have developed experimental setups

∗ Cleveland Clinic Cole Eye Institute† Jet Propulsion Laboratory, California Institute of Technol-

ogy, Pasadena, CA, USA‡ Corresponding author. Department of Mechanical, Indus-

trial and Nuclear Engineering, 688 Rhodes Hall, PO Box210072, Cincinnati, OH 45221; Tel: 513-556-2124; Fax:513-556-3390; Email: [email protected]

to measure the circumferential, longitudinal mod-uli, anisotropy and the shear modulus of severalarteries both in humans and animals [1-4]. Sincearterial viscoelasticity is small, present modelingmethods treat arteries primarily as elastic in na-ture. Several models have been proposed to quan-tify non-linear, isotropic and a few on anisotropicbehavior of the arteries. Fluid-structure interac-tion is a valuable tool to analyze the effect offluid stresses on wall mechanics in healthy anddiseased arteries having plaque [5-14].

The issues of compliance mismatch and vessel in-jury are present in several clinical therapies, e.g.,arterio-venous fistula having different artery andvein elasticity [15], balloon angioplasty causingstretching of the artery [16,17], migration and en-doleaks in endovascular grafts [18,19], stents al-ter the compliance of the vessel [20,21]. To re-produce the in vivo hemodynamics and wall me-chanics, it is essential to know whether the ar-terial properties measured ex vivo are indicativeof properties that exist in vivo. Therefore, thisstudy aims to compare the in vivo complianceof an artery based on elastic properties measuredex vivo. In this study, this comparison has beenperformed for a range of systemic pressures andblood flow, which is characteristic of pressuredependent alterations in blood flow in femoralartery of dog. This study utilizes in vivo mea-surements of pressure, pressure drop, velocity andcross-sectional area in femoral artery of dog anda computational fluid structure-interaction modelusing non-linear, anisotropic wall properties forthe same vessel. The computed wall mechan-ics are compared with in vivo compliance of thefemoral artery of dog reported elsewhere.

230 Copyright c© 2008 Tech Science Press MCB, vol.5, no.4, pp.229-245, 2008

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Figure 1: (A) Experimental pressure and axial velocity measured in the tapered femoral artery of dog; (B)Measured in vivo pressure drop measured between the inlet and outlet of the tapered artery section.

Comparison of ex vivo and in vivo Arterial Compliance 231

2 Methods

2.1 In Vivo experiment

The experimental measurements are shown inFigure 1 [22]. Figure 2 shows a schematic dia-gram of the femoral artery, with dimensions andlocations of the pressure taps and flow cuff [22].The femoral artery tapers linearly along its length.At the inlet and outlet (figure 2), there was a smallbranch, which was used as a pressure tap and con-nected via tubing to the pressure transducer. Theflow was measured by an external Doppler flowcuff as shown in the figure. The time averagedpressure drop (Δp) measured experimentally be-tween the two pressure taps (inlet and outlet) was0.59 mmHg. The time and spatially averaged ax-ial velocity (uz) measured by the cuff was 15.1cm/sec. Mean blood pressure was 72.5 mmHg.The heart rate was 128 beats/min (period T =0.469 sec).

2.2 Compliant artery – blood flow model

2.2.1 Geometry

The mean inner wall diameter at the inlet (di)andoutlet(do) is 3.8 and 3.6 mm, respectively. The insitu axial distance between the pressure taps is 5.2cm, such that the ratio of axial length to diameteris around 13.7 [22].

2.2.2 Blood flow model

The blood flow through the femoral artery is lam-inar, incompressible, viscous, pulsatile and non-Newtonian. An axi-symmetric form of the gov-erning equations is solved. The following equa-tions are used:

∇.u = 0 (continuity equation) (1)

ρ(∂u/∂ t +((u−um).∇u) = −∇p+∇.(μ∇u)(momentum equation) (2)

The Carreau model of non-Newtonian blood vis-cosity is used [16]:

μ = μ∞ +(μ0 −μ∞) .(

1+(β γ)2)(n−1)/2

(3)

where μ∞ = 0.0345 Poise, μ0 =0.56 Poise,β =3.313 sec and n = 0.3568. The density (ρ)of blood is 1.05 gm/cm3.

The boundary conditions are as follows (Figure2):

p at the inlet = pi(t) (4)

p at the outlet = po(t) (5)

ur at the axis = 0 (6)

∂u/∂ z = 0 at the inlet and outlet (7)

where um, ur and u is the mesh velocity, radial ve-locity and velocity vector, respectively. At lumenand arterial wall interface, no slip boundary con-dition is applied:

u = dS (8)

where dS is the time derivative of displacement atblood – inner wall (of the artery) interface.

2.2.3 Arterial wall model

The arterial wall is assumed to be homogenous,hyperelastic and incompressible. The equilibriumequations and boundary conditions for the arterialwall are as follows:

σSαβ ,β = 0 (9)

dS = dF at the inner wall (10)

σSαβ .nβ = 0 at the outer wall (11)

σSαβ .nβ = σF

αβ .nβ at the fluid-solid interface

(12)

where dS, dF , σSαβ and σF

αβ are the displacementand stress tensors for the arterial wall and bloodflow, respectively, and n is a unit vector normalto the boundary. The radial displacement of theaxis is zero. The in vivo length of the artery is5.2 cm. The cauchy circumferential and longitu-dinal stress – stretch ratio data for the dog femoralartery are used as shown in figure 3A [23]. Onlythe average values of ex vivo stresses were re-ported by Attinger [23], measured from cylin-drical samples of dog femoral artery of 6 cm insitu length. The Mooney-Rivlin model for incom-pressible solid has been used. The strain energy


Axial length = 5.2 cm

Inlet OutletDoppler cuff location

Lumen

Inner wall of artery

~ 1.7 cm

or~ = 0.18 cm

ir~ =0.19 cm

Outer wall of artery

r

z CL

Location of proximal pressure port

Location of distal pressure port

Figure 2: Geometry of tapered femoral artery of dog. Dimensional values reported are mean values obtainedfrom angiographic images. Figure shows an axi-symmetric model of the artery.

density function (W : eq. 13) is expressed as func-tion of I1 and I2 (see eq. 13), where I1 and I2 arethe first and second strain invariants, respectively.

W = C1 (I1 −3)+C2 (I2 −3)+C3 (I1 −3)2

+C4 (I1 −3) (I2 −3)+C5(I2 −3)2 (13)

From eq. 13, the cauchy (σ cαα) principal stresses

are given by:

σ cαα = λα (∂W/∂λα) (14)

Here, Cj’s are the material constants obtained bycurve fitting the experimental data [17]. For anincompressible artery, λrλθ λz = 1. For curvefitting the circumferential stress – stretch ratiodata, it has been assumed that λr = λθ = λ andλz = λ−2 where λ is the corresponding circum-ferential stretch ratio. For curve fitting the longi-tudinal stress – stretch ratio data, it has been as-sumed that λz = λ ′ and λr = λθ = λ ′−(1/2) whereλ ′ is the corresponding longitudinal stretch ratio.These assumptions describe the implemented ex-perimental protocols by Attinger [23]. The datawas obtained from the no – load configuration,i.e., when transmural pressure is zero and corre-sponding stresses in the artery are zero, assumingresidual stresses are not present in the wall [23].

Thus, the error function ε has been minimized us-ing the Nelder – Mead minimization algorithm toobtain the material constants:

ε2 =n

∑i=1

[(σθθ −σ∗

θθ )2i +

(σzz −σ∗

zz

)2i

](15)

where σ and σ∗ are the curve fit and measuredex vivo values [23] of stresses, respectively. Theindex i represents the ith of n data points. As ameasure of goodness of fit, the R2 value of 0.98has been achieved for both the circumferentialand longitudinal stress data (figure 3A). The curvefitting procedure provided the following materialconstants: C1 = 296220 dynes/cm2, C2 = -221950dynes/cm2, C3 = 27166 dynes/cm2, C4 = 35681dynes/cm2, C5 = -3686 dynes/cm2. To check thestability of this curve fit, the contours of W havebeen plotted as function of both stretch ratio andGreen strain (Figure 3B). As seen in figure 3B, theconvexity of contour plot of W ensures the stabil-ity of the compliant wall – blood flow computa-tions performed in this study [24].

2.2.4 Hemodynamic pressure – flow relation forthe femoral artery of dog

The linear pressure – flow relationship for femoralartery of dog at resting and vasodilated flow [25-27] is used to estimate the blood flow at differentpressures (po). Ehrlich et al. [25] have measureda zero – flow pressure of 35.4 mmHg at restingflow in femoral artery of dogs, .i.e., when pres-sure approaches 35.4 mmHg, blood flow in theartery ceases. Using our in vivo data and the zero– flow pressure of 35.4 mmHg, a linear line is re-gressed (figure 4A). Two different pressure pro-files are used with mean values (po) of 104.1 [22]and 140.7 mmHg [29] to compute the complianceat higher po. The time period of elevated pressurewaveforms is 0.469 sec [22]. At the elevated po,the corresponding flow rate is obtained by extrap-


0.0E+00

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Stretch Ratio

Cau

chy

Stre

ss (d

ynes

/cm

)

Expt. Circumferential: Attinger (1968)Mooney-Rivlin curve fit: CircumferentialExpt. Longitudinal: Attinger (1968)Mooney-Rivlin curve fit: Longitudinal

2

3A

olating the linear pressure – flow line (Figure 4A),and is 188.2 and 289.8 ml/min, respectively.

2.2.5 Computation procedure

The details of the computation procedures are asfollows:

a) In vivo measurements: Baseline case

At the outlet, the time varying pressure is pre-scribed as normal traction. However, the exper-imental velocity profile, measured distal to the in-let, does not account for radial variation of thelocal axial velocity. However, both a tempo-ral and spatially varying velocity uz(r, t) profileis required as a boundary condition. To spec-ify the boundary conditions similar to the in vivomeasurements, the following procedure has beenadopted:

(1) First in a rigid artery model with dimen-sions as shown in Figure 2, the pressure drop,Δp(t), is computed by specifying temporaland spatial dependent axial velocity profile atthe inlet and our measured in vivopo(t), hav-ing mean value of 72.5 mmHg, at the outlet.At any time t, a parabolic velocity profile for a

Newtonian liquid is used to describe the spa-tial variation of axial velocity.

(2) Since Δp in a compliant vessel is differentthan Δp in a rigid vessel, a constant number isadded to each instantaneous computed Δp(t)obtained from the rigid artery model. Thisprovides an adjusted time averaged pressuredrop (Δp), which is higher than that com-puted from the rigid artery model. The pos-itive number is chosen using an iterative pro-cedure where multiple computations are per-formed till uz = 15.1 cm/sec, similar to our invivo measurements (Figure 1), is obtained atthe cuff location.

(3) Then, the computed (from the rigid arterymodel) and incremented (after adding thepositive number) Δp(t) is added to the in vivopressure po(t) at the outlet to obtain the pres-sure pi(t) at the inlet.

(4) The resultant pi(t) is specified as normal trac-tion at the inlet while our measured in vivopo(t) [22] is prescribed as normal traction atthe outlet for the computation.


Longitudinal Green Strain

Circ

umfe

rent

ial G

reen

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in

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× 106

× 106

3BFigure 3: (A) Non-linear circumferential and longitudinal stress–stretch ratio data. The Mooney-Rivlinmodel, regressed to this experimental data, is shown as a solid line; (B) Plot showing convexity of Mooney-Rivlin material model using regressed material constants for the femoral artery of dog. Color scale units aredynes/cm2.


738

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Time (sec)

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eloc

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t cuf

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Numerical: Mean Outlet Pressure = 72.5 mmHgNumerical: Mean Outlet Pressure = 104.1 mmHg Numerical: Mean Outlet Pressure = 140.7 mmHgExperiment: Mean Outlet Pressure = 72.5 mmHg

Mean Axial Velocity = 29.9 cm/sec (Numerical) Mean Axial Velocity = 22.5 cm/sec

(Numerical)

Mean Axial Velocity = 15.1 cm/sec(Experimental and Numerical)

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0 20 40 60 80 100 120 140 160Pressure (mmHg)

Flow

Rat

e (m

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Zero flow pressure= 35.4 mmHg

A

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10

Figure 4: Estimation of flow using linear pressure – flow relationship: (A) Extrapolated linear pressure –flow relationship for baseline and elevated pressure; (B) Measured in vivo and computed spatially averagedaxial velocity profiles at cuff location.


For the compliant model, the artery was firststretched by 48% of its initial length [30,31] suchthat the final length is 5.2 cm, the in vivo length.The no – load radius at the inlet and outlet are ad-justed such that the computed di and do is 0.38and 0.36 cm, respectively. Further, the wall thick-ness at no – load is adjusted such that the ratio ofwall thickness (t) to radius (r) is about 0.14 [3].

b) Elevated pressure and flow

For the elevated pressure profiles, the adjustedtime averaged pressure drop (Δp) obtained fromthe baseline case is multiplied by a factor (> 1) toincrease its value to Δpe. The value is increasedtill the computed mean flow rate at the cuff loca-tion is 188.2 ml/min and 289.8 ml/min for po of104.1 and 140.7 mmHg, respectively (Figure 4Band 5A). Thus, the procedure used for these com-putations is as follows:

(1) First, the compliant artery is stretched to its invivo length as described in baseline case.

(2) Then, the Δpe(t) is added to the pressurepo(t) (pressure profiles with mean values of104.1 and 140.7 mmHg) at the outlet to obtainthe corresponding pressure pi(t) at the inlet.

(3) Then, the new pi(t) is specified as normaltraction at the inlet while the elevatedpo(t) isprescribed as normal traction at the outlet.

(4) Multiple computations are performed till val-ues of Δpe(t) and pi(t) satisfy the estimatedflow from the extrapolated linear pressure–flow relationship for corresponding po of104.1 and 140.7 mmHg, respectively.

Using this procedure, the computed uz profiles atthe cuff location for the elevated pressures aresimilar in shape to our in vivo measured pulse(Figure 4B). The no – load inlet radius, outlet ra-dius and wall thickness have been kept the samefor these computations as those for the baselinecase.

3 Mesh and convergence

The finite element method has been used to solvethe blood flow and arterial wall equations simul-taneously [32]. The mesh consists of bilinear,

quadrilateral, axi-symmetric elements. The num-ber of nodes in the blood and arterial wall zone are1515 and 1000, respectively. Mesh independencyhas been checked till the solutions for two differ-ent mesh sizes differed by less than 0.5%. Theconvergence criteria for the fluid and solid degreesof freedom are 10−6 and 10−7, respectively. The2nd order trapezoidal rule is used for time integra-tion of the blood flow and arterial wall mechanicsequations. A fixed time step size of 2.345 × 10−4

sec is used. Each computation is run for 3 cycles.It is observed that the solution remain unchangedafter the first cycle. Thus, only results from thesecond cycle are presented in this paper.

3.1 Velocity profiles and pressure drop

The spatially averaged axial velocity profiles uz(t)are plotted in figure 4B for baseline and elevatedpressure po. The computed uz are 22.5 cm/sec and29.9 cm/sec for poof 104.1 and 140.7 mmHg, re-spectively; a 49% and 98% increase from a mea-sured value of 15.1 cm/sec for the baseline case.As seen in figure 4B, the axial velocity profiles atdifferent t are similar, though they differ in magni-tude for baseline and elevated pressure. The closeagreement between our in vivo [22] and computeddata for po = 72.5 mmHg provides confidence toour estimates for uz at different t for the elevatedpressures. The Womersley number [33] for thebaseline calculation is 3.7 which increases to 4.2for the elevated pressure.

Figures 5A shows the baseline and elevated po

vs. t. These profiles have been used for the com-pliant wall – blood flow computations. The cor-responding −Δp(t) (= pi(t)− po(t)) profiles forthe baseline and elevated pressure at different t areshown in figure 5B. It can be seen that all −Δp(t)are similar in shape but differ in magnitude. Asignificant increase in instantaneous pressure dropis seen with increasing po, flow and the diame-ter of the vessel. The in vivo peak −Δp(t) dur-ing systolic and diastolic forward flow is 4.25 and1.5 mmHg, respectively (Figure 1). Comparingthe baseline case and in vivo measurement, peakΔp during systole is 29% lower, and during dias-tolic forward flow is 31 % lower. If the time aver-aged −Δp’s are compared, then computed value


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−Δp

(mm

Hg)

(Experimental)(Numerical)

(Numerical) (Numerical)

mmHg5.72p~; o =

mmHg1.140p~; o =

mmHg5.72p~; o =

mmHg1.104p~; o =

Figure 5: (A) Measured in vivo pressure profiles: baseline and elevated; (B) Measured in vivo and computedpressure drop between inlet and outlet.


for the baseline case is 0.72 mmHg as comparedto 0.59 mmHg measured in vivo; a difference of18%. Thus, the difference in −Δp is significantlylower than instantaneous −Δp(t). The computed−Δp is 1.06 mmHg and 1.35 mmHg for po=104.1and 140.7 mmHg, respectively.

All the computations show significant phase dif-ference between in vivo and computed −Δp(t).On an average, the phase difference between thein vivo and computed −Δp(t) is 20 degrees forthe baseline case. The phase difference betweencomputed −Δp(t) and measured uz(t) [22] forthe baseline case, which is ∼ 26.6 degree. Thisis consistent with Womersley theory for pulsatileblood flows [33]. The phase difference betweenthe computed −Δp(t) and in vivo −Δp(t) [22]is due to response limitation of diaphragm basedpressure transducer, which was available to theauthors and used for the in vivo experiment. How-ever, the −Δp value is not affected much if timeaveraging is done synchronized with the heartrate. Table 1 summarizes the results of the com-putations, with mean values of di, do, thi and tho.

3.2 Compliance of the femoral artery of dog

Due to variability in artery dimensions, wallthickness and hemodynamic conditions ,compli-ance is a useful parameter to compare experimen-tal data using computational models. Compliance(c) of a vessel is given as follows [34]:

c =(

1r

)(∂ p∂ r

)−1

(16)

Figure 6A-C show the variation of mid-wall radiiat the inlet and outlet for the baseline and elevatedpressures. Many researchers have linearised eq.16 to estimate a simple measure of c for a givenpressure and flow pulse, expressed as percent:

c =[d (max)−d (min)]

d (min)× [p(max)− p(min)]×100 (17)

Here d(max) and d(min) are the maximum andminimum diameter at p(max) and p(min), respec-tively. The values of c at the inlet and outlethave been calculated and compared with in vivodata published for femoral artery of dog [35].

Megerman et al. [35] measured in vivo the com-pliance, using only the outer wall diameter, in dis-sected and undissected femoral artery of mongreldog. For the present computations, table 2 sum-marizes the maximum and minimum outer walldiameters and pressures (from figure 6A-C) at theinlet and outlet. Here, dissection implies that theartery is dissected free from the surrounding tis-sue similar to ex vivo conditions (figure 7). Infigure 7, the computed c decreases from 0.198%diameter change/mmHg for po= 72.5 mmHg to0.145% diameter change/mmHg for po= 140.4mmHg. In comparison with in vivo data [35],the c tends to match at lower pressure (∼70-110mmHg) and is overestimated at higher pressure byour computations (figure 7). The compliance ofdissected artery is lower than undissected artery[35]. This suggests an alteration in the vascularwall mechanics of the artery from the in vivo con-ditions. The strong dependence of c on pressureis evident.

3.3 Stress at the mid-wall radius

Figure 8 shows the cauchy circumferential andlongitudinal stress vs. time for baseline and ele-vated pressures at mid wall radius both at the inletand outlet. The phasic variation in circumferen-tial stress and longitudinal stress (at the inlet) isidentical to the pressure profile. In contrast, thelongitudinal stress at the oulet is inverted in shapefor all the computations. The mean circumfer-ential stress increases from 42 kPa for po= 72.5mmHg to 146 kPa for po = 140.7 mmHg. In con-trast, the mean values of longitudinal stresses areconsiderably lower, which are 28 kPa and 31 kPa,respectively. In the physiological pressure rangeof 100 mmHg, the circumferential stresses shouldbe in the range of 50-100 kPa under in situ pre-stretch conditions [31,36]. The present calcula-tion for po = 104.1 mmHg (mean circumferentialstress value of 92 kPa) using our material modelequation agrees well with the past studies [31,36].

4 Discussion and Conclusion

In this study, the reproducibility of in vivo compli-ance of femoral artery of a dog from ex vivo mea-sured wall properties has been studied. While the


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Figure 6: Pressure vs. mid – wall radius curves at inlet and outlet: (A) po = 72.5 mmHg; (B) po = 104.1mmHg; (C) po = 140.7 mmHg.


Table 1: Summary of mean pressure drop (Δp), mean axial velocity (uz), mean radius at inlet (di) and outlet(do), wall thickness at inlet (ti) and outlet (to) for baseline and elvated pressures.

−Δp (mmHg) uz (cm/sec) di (cm) do (cm) thi (cm) tho (cm)In vivo4 0.59 15.1 0.38 0.36 – –

po = 72.5mmHg (Numerical) 0.72 15.1 0.38 0.36 0.027 0.027po = 104.1mmHg (Numerical) 1.06 22.5 0.41 0.39 0.026 0.026po = 140.7mmHg (Numerical) 1.35 29.9 0.44 0.42 0.024 0.024

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Pressure (mmHg)

Com

plia

nce

(% d

iam

eter

cha

nge/

mm

Hg)

Megerman et al. (1986); UndissectedMegerman et al. (1986); DissectedInletOutlet

70 − 110 mmHg

Figure 7: Compliance vs. pressure evaluated at inlet and outlet. The computed data is compared with invivo data from reference.


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Cau

chy

Stre

ss (d

ynes

/cm

2 )

Longitudinal Stress at InletLongitudinal Stress at Outlet

28 kPa

28.4 kPa

31kPa

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0.4690 0.5628 0.6566 0.7504 0.8442 0.9380Time (sec)

Cau

chy

Stre

ss (d

ynes

/cm

2 )Circumferential Stress at Inlet Circumferential Stress at Outlet

146 kPa

92 kPa

42 kPa

A

B

Figure 8: (A) Circumferential stress vs. time for baseline and elevated pressures at inlet and outlet; (B)Longitudinal stress vs. time for baseline and elevated pressures at inlet and outlet. Mean stress values at theinlet are shown by horizontal lines.


Table 2: Calculated compliance of the femoral artery at different pressures along with maximum and mini-mum outer wall diameter and pressure.

cOuter wall d(cm), Outer wall d(cm),

(% diameter change/mmHg) pi(mmHg) at inlet po(mmHg) at outletMin. Max. Min. Max.

po = 72.5 mmHg ∼ 0.198 0.420, 51.3 0.456, 94.5 0.402, 51.3 0.432, 88.7po = 104.1 mmHg ∼ 0.168 0.448, 82.3 0.502, 154 0.427, 82.5 0.471, 144.2po = 140.7 mmHg ∼ 0.145 0.478, 121.3 0.531, 196.7 0.456, 121.8 0.498, 184.9

computations predict radial dilations and stressdistributions similar to in vivo values, the com-pliance of the vessel differs from its in vivo value.The use of linear equation (eq. 17) to calculatecompliance is justified since Megerman et al. [35]have also used the same equation. Our estimatesfor c and in vivo measurements for femoral arteryof dog match well in the pressure range of 70mmHg to 110 mmHg but are higher for pressuresabove 110 mmHg.

Studies where both in vivo hemodynamics andcompliance have been measured followed bymeasurement of elastic properties of artery underex vivo conditions are extremely rare, if not non-existent. Further, experimental protocols tend todiffer among different studies, i.e., measurementmade from zero – stress state, no – load state,pre – stressed condition in which the vessel isstretch first to in vivo length. Therefore, there isa need of standardization in ex vivo experimentaltechniques for measurement of arterial wall prop-erties. To link this study to the onset and pro-gression of atherosclerosis is difficult because on-set and progression of atherosclerosis is attributedto a combination of several biophysical and bio-chemical parameters. If abnormal changes inphysiological vascular dynamics are to be linkedto diseases, then more studies combining vesselmechanics and blood flow dynamics need to beperformed.

In this study, the effect of residual stresses is notconsidered since such information was not re-ported by Attinger [23]. A recent fluid-structureinteraction study on removal of tethering suggeststhat dissection of surrounding tissue from the ar-terial wall (as in ex vivo measurements) producesnative changes in the elastic properties of the

artery itself [9]. The present analysis supports theconclusions made by Zhang and colleagues [9]and provides direct comparison between in vivoand ex vivo elasticity data. Cox et al. [37] havereported that the viscoelastic contribution to thedynamic elastic modulus of the femoral artery ofdog is less than 2%. Thus, viscoelasticity shouldhave negligible impact on the calculated compli-ance in this study. Studies where in vivo hemody-namics, compliance and ex vivo elastic propertieshave been measured are rare, if not non – exis-tent. Further, experimental protocols tend to dif-fer among studies [2,38]. Therefore, standardizedex vivo experimental techniques for measurementof arterial wall properties are needed. Presently,there are few layer specific models available thatare able to account for directional orientation ofelastin and collagen fibres [e.g., 16]. Data per-taining to material constants for regressed layer– specific model equations for animals are lim-ited. These future studies should compare bothpulsatile hemodynamics and vascular propertiesmeasured ex vivo for normal and diseased arter-ies to in vivo compliance of the arteries.

Acknowledgement: This work is supported byAmerican Heart Association: National-ScientificDevelopment Grant (AHA National-SDG grantreference number: 0335270N).

Nomenclature

u velocity in cm/secp pressure in mmHgd diameter in cmr radius in cmt time in secc compliance in % diameter change/mmHg


λ stretch ratioσ stress in dynes/cm2

n number of data pointsth wall thickness

Superscripts

− spatially averaged∼ time averagedS solid (arterial wall)F fluid (blood flow)c Cauchyl Lagrange· time derivative∗ experimental data

Subscripts

z axialr radialθ circumferentialα ,β co-ordinate directionsi inleto outlet

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