arterial stiffness identification of the human carotid artery using the

Ultrasonics xxx (2011) xxx–xxx

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Ultrasonics

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Arterial stiffness identification of the human carotid artery using the stress–strainrelationship in vivo

T. Khamdaeng a, J. Luo b, J. Vappou b, P. Terdtoon a, E.E. Konofagou b,⇑a Department of Mechanical Engineering, Chiang Mai University, Chiang Mai, Thailandb Department of Biomedical Engineering, Columbia University, New York, NY, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 5 July 2011Received in revised form 20 September2011Accepted 20 September 2011Available online xxxx

Keywords:Arterial stiffnessCarotid arteryCollagenElastinStress–strain relationship

0041-624X/$ - see front matter � 2011 Elsevier B.V.doi:10.1016/j.ultras.2011.09.006

⇑ Corresponding author. Address: Columbia Univerical Engineering, 351 Engineering Terrace, Mail coAvenue, New York, NY 10027, USA. Tel.: +1 212 342 0

E-mail address: [email protected] (E.E. Konof

Please cite this article in press as: T. Khamdaenin vivo, Ultrasonics (2011), doi:10.1016/j.ultras.

Arterial stiffness is well accepted as a reliable indicator of arterial disease. Increase in carotid arterial stiff-ness has been associated with carotid arterial disease, e.g., atherosclerotic plaque, thrombosis, stenosis,etc. Several methods for carotid arterial stiffness assessment have been proposed. In this study, in vivononinvasive assessment using applanation tonometry and an ultrasound-based motion estimation tech-nique was applied in seven healthy volunteers (age 28 ± 3.6 years old) to determine pressure and walldisplacement in the left common carotid artery (CCA), respectively. The carotid pressure was obtainedusing a calibration method by assuming that the mean and diastolic blood pressures remained constantthroughout the arterial tree. The regional carotid arterial wall displacement was estimated using a 1Dcross-correlation technique on the ultrasound radio frequency (RF) signals acquired at a frame rate of505–1010 Hz. Young’s moduli were estimated under two different assumptions: (i) a linear elastictwo-parallel spring model and (ii) a two-dimensional, nonlinear, hyperelastic model. The circumferentialstress (rh) and strain (eh) relationship was then established in humans in vivo. A slope change in the cir-cumferential stress–strain curve was observed and defined as the transition point. The Young’s moduli ofthe elastic lamellae (E1), elastin–collagen fibers (E2) and collagen fibers (E3) and the incremental Young’smoduli before (E06eh<eT

h) and after the transition point (EeT

h6eh

) were determined from the first and secondapproach, respectively, to describe the contribution of the complex mechanical interaction of the differ-ent arterial wall constituents. The average moduli E1, E2 and E3 from seven healthy volunteers were foundto be equal to 0.15 ± 0.04, 0.89 ± 0.27 and 0.75 ± 0.29 MPa, respectively. The average moduli EInt

06eh<eTh

andEInt

eTh6eh

of the intact wall (both the tunica adventitia and tunica media layers) were found to be equal to0.16 ± 0.04 MPa and 0.90 ± 0.25 MPa, respectively. The average moduli EMe

06eh<eTh

and EAdeTh6eh

of the tunicaadventitia were found to be equal to 0.18 ± 0.05 MPa and 0.84 ± 0.22 MPa, respectively. The averagemoduli EMe

06eh<eTh

and EMeeTh6eh

of the tunica media were found to be equal to 0.19 ± 0.05 MPa and0.90 ± 0.25 MPa, respectively. The stiffness of the carotid artery increased with strain during the systolicphase. In conclusion, the feasibility of measuring the regional stress–strain relationship and stiffness ofthe normal human carotid artery was demonstrated noninvasively in vivo.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Arterial stiffness has been shown to be an excellent indicator ofcardiovascular morbidity and mortality in a large percentage of thepopulation [1], patients with hypertension [2,3], atherosclerosis [4]and myocardial infarction [5]. Several carotid stiffness indices havebeen proposed based on the pressure–diameter relationship ofarterial distension from the end-diastolic to the end-systolic phase,e.g., arterial distensibility, arterial compliance, Peterson’s elasticmodulus (Ep) and stiffness index (b) [6–17]. They, however, repre-

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sity, Department of Biomed-de 8904, 1210 Amsterdam863; fax: +1 212 342 1648.

agou).

g et al., Arterial stiffness iden2011.09.006

sent a global stiffness measurement of the entire arterial wallbased on a single measurement site. The Young’s modulus can de-pict more subtle changes in the relative proportions of the constit-uents of the arterial wall [18]. The in vivo noninvasive Young’smodulus estimated from the regional stress–strain relationship in-cludes more comprehensive information on the arterial wall prop-erties regarding the effects of the different constituents [19]. Only afew studies have reported on the in vivo Young’s modulus mea-surement of the carotid artery based on the pressure–strain rela-tionship [18,20–22] or from the slope of the stress–strainrelationship [23] at end-diastole and end-systole. Previous studies,however, have not investigated on the complex mechanical inter-action of the arterial wall constituents in humans in vivo.

It is hypothesized that the slope change of the stress–strain rela-tionship is related to the change in material properties resulting

tification of the human carotid artery using the stress–strain relationship

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2 T. Khamdaeng et al. / Ultrasonics xxx (2011) xxx–xxx

from the elastin and collagen contribution in the aortic wall [6,24].The elastic lamellae of the arterial wall are primarily dilated at lowpressure levels. As the lumen pressure increases, the collagen fibersstart elongating [25,26] and reach high tensile lengths. The tunicaadventitia stiffness thereby increases with pressure in order to pre-vent the artery from overstretching or rupture [27].

Higher vascular stiffness is typically found in older subjects be-cause the elastic lamellae decrease in number with age while theconnective tissue and collagen fibers increase [28]. The elasticmodulus and spatial arrangement of the arterial wall constituentsdepend on the level of circumferential stress. The incrementalYoung’s modulus of the carotid artery, as a function of blood pres-sure and circumferential stress, was shown to be higher with ageand hypertension [3]. The change in the carotid artery stiffnesswith age has been shown similar to that of the abdominal aorta[7]. The stiffness of the carotid artery has been shown to be higherthan in the abdominal aorta in young subjects (<20 years old)[12,29] and in children [30]. The difference in stiffness betweenthe carotid artery and the abdominal aorta is due to their differ-ence in elastin and collagen proportionality [10]. The linear rela-tionship between the carotid stiffness and aortic pulse wavevelocity (APWV) was proposed as a rough estimate of the cardio-vascular risk factor [10,31]. In the aforementioned studies, how-ever, insufficient information on the contribution of the arterialwall constituents was shown, which may be useful for the earlydetection of cardiovascular disease.

Two well-known mechanical models, i.e., the Kelvin–Voigt andgeneralized Maxwell models, have been applied to describe thedilation mechanism of the arterial wall during the systolic phaseof the cardiac cycle. Dobrin and Canfield [32] identified the serieselastic component (SEC) representing the elastin, collagen andsmooth muscle cell behavior, in the intact canine carotid arteryvia the Kelvin–Voigt and generalized Maxwell models. They foundthat the generalized Maxwell model represented the carotid arte-rial wall more accurately than the Kelvin–Voigt model. The smoothmuscle cells, which are overlaid and arranged in the circumferen-tial direction, mainly exhibit a viscous behavior. This way, theyregulate the caliber of the blood vessels and result in a time delaybetween the pressure and the dilation waveforms [6]. In order todescribe the constituents’ mechanism of the abdominal aortic wallof mice in vivo, Danpinid et al. [19] modified a generalized Maxwellmodel (i.e., a two-parallel spring model) by compensating for thetime delay between the pressure and the dilation waveforms andthus ignoring the viscous properties. However, in this model, vis-cosity was not expected to affect the stress–strain relationshipbut only the time delay between the pressure and the dilationwaveforms, which was removed for the purposes of this study.Due to the nonlinear and anisotropic characteristics of the artery,other approaches based on the use of strain-energy functions havebeen proposed. In-vivo hyperelastic constitutive equations (stressand strain relations) have been proposed [33,34] to identify mate-rial properties and to calculate wall stress in human arteries basedon in vivo clinical images.

To obtain the stress–strain relationship, the carotid pressureand wall displacement need to be simultaneously measured.Applanation tonometry can provide noninvasive assessment ofthe wall pressure. It provides a continuous measurement of theblood pressure waveform [35] and has been shown to accuratelyprovide peripheral arterial pressure waveforms, e.g., in the femoraland radial arteries. In this paper, measurements in the left CCAwere performed since it is more accessible for imaging while atthe same time more closely correlated with the central aorta con-ditions [18]. The calibration method for the carotid pressure wave-form assumed that the mean and diastolic blood pressuresremained constant throughout the arterial tree [36], i.e., wereequivalent to those of the radial artery [37].

Please cite this article in press as: T. Khamdaeng et al., Arterial stiffness idenin vivo, Ultrasonics (2011), doi:10.1016/j.ultras.2011.09.006

Ultrasound imaging is a well-known noninvasive method for di-rect visualization of vascular function. The arterial wall motion canbe estimated using motion estimation, or speckle tracking, tech-niques on the RF signals. These techniques have been mainly usedon the brachial, femoral and carotid arteries and the abdominalaorta [38]. In this study, a 1D cross-correlation technique was ap-plied noninvasively to estimate and image the local carotid arterialwall displacement in humans in vivo [39].

In this paper, we noninvasively applied the applanation tonom-etry methodology and ultrasound-based estimation motion tech-nique [39] to obtain local, in vivo pressure measurements andwall displacements, respectively, in the left CCA of healthy hu-mans. The in vivo regional stress–strain relationship was proposedto characterize the complex mechanical interaction of the arterialwall constituents and determine the respective Young’s modulifrom (i) a linear elastic two-parallel spring model and (ii) a two-dimensional, nonlinear, hyperelastic model.

The experimental procedure, pressure and diameter relation,and stress and strain calculation derived from (1) the in vivo mea-surements (2) the two-parallel spring model and (3) the two-dimensional hyperelastic model are first provided in Section 2.The Young’s moduli of the elastin and collagen fibers in both thetunica media and tunica adventitia in the left CCA of healthy hu-mans are presented in Section 3, followed by discussion and con-clusions on the methodology and results presented.

2. Methods

2.1. Experimental procedure

The study was performed in seven healthy male human subjectsof ages varying between 22 and 32 years (28 ± 3.6 years old, aver-age ± std). The brachial blood pressure of the subjects was firstmeasured using a sphygmomanometer to allow for calibration ofthe radial pressure waveform via a SphygmoCor system (AtCorMedical, Sydney, NSW, Australia). The applanation tonometer (Mil-lar SPT-301 probe; Millar Instruments, Houston, TX) was thenplaced on the subject’s wrist against the radial artery where thestrongest pulse signal was manually detected. The diastolic andsystolic blood pressures of the radial artery were assumed equalto those of the brachial artery [37]. The mean blood pressure wasestimated using numeric integration of the radial pressure wave-form [36]. In order to obtain the carotid pressure waveform, thesubjects were placed in the supine position. The carotid pressurewaveform was obtained with the sensor placed perpendicularlyon the left CCA where the strongest pulse signal was detected.The carotid blood pressure was calibrated by assuming that themean and diastolic blood pressures were equivalent to those ofthe radial artery [36,37].

In order to acquire RF signals, the subjects were placed in a sit-ting position. The ultrasound transducer was placed on the left CCAusing coupling gel. High temporal resolution RF frames of the leftCCA were obtained with a 10-MHz linear array transducer usinga clinical ultrasound system (Sonix TOUCH; Ultrasonix Medical,Burnaby, British Columbia, Canada). The radial plane of the carotidartery was closely aligned with the axial direction of the ultra-sound beams in the longitudinal view. The local incremental walldisplacement along the carotid arterial wall was determined usinga 1D cross-correlation technique [39] between consecutive RFframes acquired at a sampling frequency of 20 MHz, depth of30 mm, width of 38 mm, and a line density depending on the sub-ject from 16 to 32 beams per full sector. Since the frame rate isassociated with the beam density, the frame rate was varied be-tween 505 and 1010 Hz depending on the depth scanned in eachsubject that had to be sufficient to determine the diameter of the



Fig. 1. (A) The carotid pressure variation of the human carotid artery over one cardiac cycle. (B) Envelope-detected B-mode image. (C) The cumulative displacements of thecarotid arterial wall. The red and blue curves indicate the cumulative displacement at the near and far wall, respectively. The black curve indicate the difference of the nearand far wall cumulative displacement. (D) The diameter variation with time.

T. Khamdaeng et al. / Ultrasonics xxx (2011) xxx–xxx 3

lumen in each subject. At least three measurements were per-formed and averaged in each subject case.

2.2. Pressure and diameter relationship

During the systolic phase of the cardiac cycle, the carotid wallwas assumed to exhibit a purely passive elastic behavior. The vis-cosity effects were thus ignored. Uniform pressure acting on the in-ner arterial wall was assumed. The external pressure was assumedto be zero.

The carotid pulse pressure acquired from applanation tonome-try was calibrated to obtain absolute values. The carotid bloodpressure was estimated over the entire cardiac cycle (Fig. 1A)according to the aforementioned assumptions of calibration.

The carotid artery region most perpendicular to the ultrasoundbeam was selected in order to obtain the most accurate displace-ment estimation [19]. Selected points on the envelope-detectedB-mode image (Fig. 1B) were defined as ‘near wall’ (wall nearestto the ultrasound probe (top)) and ‘far wall’ (wall furthest fromthe ultrasound probe) position and they were mapped onto theB-mode image. The radial near and far wall incremental displace-ments along the carotid arterial wall were determined using a 1Dcross-correlation technique [39] and were averaged across the wallthickness. The cumulative displacement was calculated using Eq.(1) (see below). The1 red and blue dots represent near and far wallcumulative displacements towards and away from the ultrasoundtransducer (top), respectively (Fig. 1C). The diameter of the carotid

1 For interpretation of color in Figs. 1,3, and 4 the reader is referred to the webversion of this article.


artery (Fig. 1D) was, then, calculated from the difference betweenthe near and the far wall cumulative displacements added on the ref-erence diameter measured on the first frame as follows:

ucumðiÞ ¼ ucumði� 1Þ þ uincðiÞ; ð1Þ

dðiÞ ¼ dref þ ½ucum;nearðiÞ � ucum;farðiÞ�; ð2Þ

where ucum(i) is the cumulative displacement, uinc(i) is the incre-mental displacement, ucum,near(i) and ucum,far(i) are the near and farwall cumulative displacement, respectively, d(i) is the diameter, dref

is the reference mean diameter, and i is the frame index.However, a delay between the pressure and the dilation wave-

forms during the systolic phase was noted due to the viscoelasticbehavior of the carotid wall. According to the aforementionedassumptions regarding the arterial properties, the minima andmaxima of the carotid pressure and diameter waveforms werealigned and matched to remove the viscosity effect [19], i.e., onlythe dilation of the carotid artery was considered with negligiblevascular vasodilation effects.

2.3. Stress and strain calculation

The carotid arterial wall is well-defined into three layers whichare the tunica intima, tunica media and tunica adventitia respec-tively arranged from the innermost to outermost wall layer, asillustrated in Fig. 2. The mechanical behavior is different for eachwall layer depending on the constituents. The functional compo-nents of the tunica media are elastin lamellae, collagen fibersand smooth muscle cells while that of the tunica adventitia arecollagen fibers and some elastin merged with the surrounding



Fig. 2. Diagram of the elastic carotid arterial wall is well-defined into three layers which are tunica intima, tunica media and tunica adventitia respectively arranged from theinnermost to outermost wall layer (Adapted from Humphrey, 2002).

Fig. 3. Envelope-detected B-mode image with five longitudinal locations on thenear and far carotid walls (indicated by different colors) where the stress–strainrelationship was determined.


connective tissue. Regarding its actions, the tunica media and tuni-ca adventitia predominantly adjust the mechanical behavior at thelower and higher pressure levels respectively [25]. The tunica med-ia and tunica adventitia, therefore, constituted the focus of thisstudy in order to determine the stress–strain relationship. Thetwo-layer wall, i.e., including both the tunica media and tunicaadventitia, was defined as the ‘intact wall’.

2.3.1. Stress and strain relation calculated directly from experimentaldata

Laplace’s law was applied to calculate the stress in the circum-ferential direction, the most dominant of the arterial wall deforma-tion. Infinitesimal strain theory was assumed. The circumferentialstress–strain relationship, therefore, was established along the car-otid artery via Eqs. (4) and (5) to characterize the complexmechanical interaction of arterial wall constituents. The mean cir-cumferential stress, rh(t), was given by

rhðtÞ ¼PiðtÞdiðtÞ

2 h; ð3Þ

or in terms of d(t) as follows:

rhðtÞ ¼PiðtÞdðtÞ

2 h� PiðtÞ

2; ð4Þ

where h denotes the carotid arterial wall thickness (intact wallthickness; the addition of the tunica adventitia and tunica mediathickness), Pi(t) and di(t) denote the inner pressure and diameterof the carotid arterial wall, respectively. The carotid arterial wallthickness of 0.48 ± 0.047 and 0.61 ± 0.018 mm (mean ± std) was ap-plied for ages varying between 22–29 and 30–32 years, respectively[40]. In this study, the wall thickness was evaluated on the B-modeimage though manual tracing by a trained expert.

Using the Cauchy strain definition, the mean circumferentialstrain, eh(t), was equal to the ratio of the diameter change to thereference mean diameter, dmin, defined as the minimum diameterover a cardiac cycle, i.e.,

ehðtÞ ¼dðtÞ � dmin

dmin: ð5Þ


Fig. 3 shows the locations of the five selected points along thecarotid wall corresponding to five circumferential stress–strainrelationships as shown in Fig. 4A. In this study, 5–9 points arrangedover the longitudinal locations of the carotid arterial wall were se-lected for each subject in order to calculate the mean and standarddeviation. As mentioned above, different number of points (5–9points) were used depending on the number of points on the car-otid wall in each subject that were most perpendicular to the ultra-sound beam in order to obtain the most accurate displacementestimation.

2.3.2. The two-parallel spring modelThe arterial model was assumed to be an axisymmetric, single-

walled layer cylindrical tube with isotropic, linearly elastic, homo-geneous, incompressible and non-viscous properties. As shown inFig. 4A, the stress–strain relationship of the carotid artery in hu-mans was observed a clear inflection point similar to the abdomi-nal aorta in mice [19]. The Young’s moduli of the elastic lamellaeand elastin–collagen fibers were defined as E1 and E2, respectively,



Fig. 4. (A) The circumferential stress–strain relationship in the same five longitudinal locations as in Fig. 3. (B) The mean stress–strain relationship corresponding to Fig. 4A.The stress–strain relationship was separated into two linear relations by the transition point. (C) The 1st and the 2nd derivative of the circumferential stress and strain ratioover systolic phase of the cardiac cycle. The transition point is defined as the maximum of the 2nd derivative of the circumferential stress and strain ratio (indicated by the redsquare in both (B) and (C)).


from a bilinear fit of the non-linear stress–strain curve (Fig. 4B). Atthe change of the slope, defined as the transition point, it washypothesized that the collagen fibers start engaging and theYoung’s modulus of the vessel wall depends on the elastin–colla-gen fibers modulus instead. E1 and E2, therefore, were assessedusing two linear regression fits separated by the transition point(Fig. 4B). In this study, the transition point was defined as the max-imum of the second derivative of the ratio of the circumferentialstress to the circumferential strain (Fig. 4C). The difference be-tween E1 and E2 was defined as the modulus of the collagen fibers(E3). The relationship between E1, E2 and E3 is thus as follows [19]:

E1 ¼rh � r0

eh; 0 6 eh < eT

h ð6Þ

E1 ¼ E2; eh ¼ eTh ð7Þ

and

E2 ¼rh � E1eT

h

eh � eTh

¼ E1 þ E3; eTh 6 eh ð8Þ

where r0 is the stress at diastolic reference state (the stress at theminimum diameter over a cardiac cycle), eT

h is the strain at the tran-sition point, i.e., where the collagen fibers start engaging. The strainratio of the actively engaged collagen fibers, eT

heh

, was unity at eh ¼ eTh ,

i.e., when the collagen fibers started engaging and underwenttension.


2.3.3. The two-dimensional hyperelastic modelUsing the noninvasively acquired pressure–diameter data in vivo,

the approach was proposed by Schulze-Bauer et al. based on thetwo-dimensional Fung’s type model [33]. The arterial wall was as-sumed to be infinitesimally thin with respect to the arterial diame-ter. Radial stress and residual stresses, therefore, were ignored. Thearterial model was assumed to be an axisymmetric, double-walledlayer cylindrical tube with plane strain, anisotropic, nonlinearlyelastic, homogeneous (in each layer), incompressible and non-vis-cous properties. The collagen fiber orientation was assumed to alignonly in the circumferential direction (no dispersion). The hyperelas-tic constitutive equation in the circumferential and axial directionsassociated with the strain-energy function (W) and principalstretches (k) was proposed [27] as follows, respectively:

rmodhh ¼ kh

@W@kh

ð9Þ

and

rmodzz ¼ kz

@W@kz

; ð10Þ

where rmodhh and rmod

zz are the circumferential and axial stresses pre-dicted by the model, and kh and kz are the circumferential and axialstretches defined as kh ¼ DiðAþ d2

i pÞ=diðAþ D2i pÞ and kz = z/Z = 1,

respectively. A = dihp is the cross-sectional wall area. Di and di

respectively denote unloaded and loaded inner diameters, and z



Table 1E1, E2, and E3 of 7 subjects (age 28 ± 3.6 years old).

Subject E1 (MPa) E2 (MPa) E3 (MPa)

1 0.13 0.70 0.572 0.13 1.06 0.933 0.18 0.60 0.424 0.12 1.40 1.285 0.10 0.94 0.846 0.15 0.77 0.627 0.22 0.78 0.56

Average 0.15 ± 0.04 0.89 ± 0.27 0.75 ± 0.29

Fig. 5. The averaged Young’s moduli of E1, E2, and E3 (mean ± std) over sevensubjects.


and Z are the actual and unloaded lengths of the arterial segment,respectively.

The two-dimensional strain-energy function was equal to

W ¼ C2

eQ � 1� �

ð11Þ

with

Q ¼ chhðeGhhÞ

2 þ 2chzeGhhe

Gzz þ czzðeG

zzÞ2; ð12Þ

where C, chh, chz and czz are the constitutive parameters. eGhh and eG

zz

are the circumferential and axial Green–Lagrangian strains definedas eG

hh ¼ 12 ðk

2h � 1Þ and eG

zz ¼ 12 ðk

2z � 1Þ, respectively.

Four material parameters of the model, i.e., Di, C, chh and chz,were optimized using a nonlinear least-squares method by mini-mizing the sum of differences between the circumferential and ax-ial stresses calculated experimentally and from the model definedas

v2 ¼Xn

j¼1

rmodhh � rhh

� �2

j þ ðrmodzz � rzzÞ2j

h i; ð13Þ

where j was the jth of n data points. rhh and rzz are the circumferen-tial and axial stresses calculated directly from the experimentaldata using the equilibrium equation of the force acting on the arte-rial wall with the inflation as follows:

rhh ¼Pid

2i

d2o � d2

i

1þ d2o

d2

!; ð14Þ

rzz ¼pd2

i Pi þ 4F4phðdi þ hÞ ; ð15Þ

where do denotes the outer diameter. F is the external axial forcedetermined explicitly by a known constant ratio of the axial tothe circumferential stress of the intact wall (both the tunica adven-titia and tunica media layers), tunica adventitia and tunica mediadefined as jk ¼ rk

z=rkh , Where index k represents the intact wall

(jInt), tunica adventitia (jAd) and tunica media (jMe).In order to investigate the contribution of the elastic lamellae

and the collagen fibers in the intact wall, tunica adventitia and tu-nica media, j was adopted from existent experimental data of anin vitro human layer-dissected CCA at physiologic loading(Pi = 13.3 kPa) [41]. In this study, the axial stretches of the intactwall, tunica adventitia and tunica media in vivo were assumed tobe unity (0% axial stretch) as the arterial wall was subjected toarterial pressure. The jInt, jAd and jMe at 0% axial stretch was as-sumed to be equal to 0.361, 0.195 and 0.482, respectively [41].The d of the tunica adventitia and tunica media defined asdAd = d + h � hAd and dMe = d � h + hMe, respectively. hAd and hMe de-note the tunica adventitia and tunica media thicknesses. The hAd

and hMe were respectively assumed to be equal to 40% and 60%of the carotid arterial wall thickness (h) [41]. Eqs. (9)–(15) were ap-plied to determine the material parameters and stresses of the in-tact wall, tunica adventitia and tunica media of the carotid arterialwall, which underwent strain before (0 6 eh < eT

h ) and after thetransition point (eT

h 6 eh). A generalized reduced gradient algorithmwas used to determine the best-fit parameters including convexity[27]. The incremental Young’s moduli before and after the transi-tion point of the intact wall, tunica adventitia and tunica mediawere determined using Eq. (17), i.e., differentiated the strain en-ergy function twice with respect to the Cauchy strain, i.e.,

rhh ¼ kh@W@kh¼ @W@ehh

¼ Eehh; ð16Þ

thus


E ¼ @2W@ehh@ehh

: ð17Þ

Since E changed with the amount of deformation, the estimatedE was separately averaged for the stress–strain curves before andafter the transition point, which was respectively defined asE06eh<eT

hand EeT

h6eh

, e.g., EInt06eh<eT

hand EInt

eTh6eh

for the intact wall,EAd

06eh<eTh

and EAdeTh6eh

for the tunica adventitia, and EMe06eh<eT

hand EMe

eTh6eh

for the tunica media.

3. Results

Fig. 4A shows an example of the circumferential stress–strainrelationship along five longitudinal locations, indicated in differentcolors on the envelope-detected B-mode images of the human car-otid artery as shown in Fig. 3. Fig. 4B shows the mean circumferen-tial stress–strain relationship of Fig. 4A. The stress–strainrelationship shows nonlinearity and a transition point.

E1, E2 and E3 were calculated from the two-parallel spring mod-el using Eqs. (6)–(8). The Pearson product moment correlationcoefficients (r) of the two linear fits were calculated. Good correla-tion for both fits found with r2 = 0.967 and 0.996 before and afterthe transition point, respectively. The three Young’s moduli in se-ven subjects, averaged across all longitudinal locations of the car-otid artery, are shown in Table 1. Fig. 5 shows the averagedYoung’s modulus of E1, E2 and E3 over all subjects. The E1, E2 andE3 were found to be equal to 0.15 ± 0.04, 0.89 ± 0.27 and0.75 ± 0.29 MPa, respectively.

The stress–strain relationship of the intact wall (Fig. 6A), tunicaadventitia and tunica media at constant 0% axial stretch before andafter the transition point were estimated using the two-dimen-sional hyperelastic model with optimized material parameters(Fig. 6C and D). Good correlation was found with r2 = 0.961,0.965 and 0.964 in the case of the intact wall, tunica adventitia



-0.02 0 0.02 0.04 0.06 0.08 0.140

50

60

70

80

90

100

110

Circumferential strain

Circ

umfe

rent

ial s

tress

(kPa

)

before transition pointafter transition point

C D

B

-0.2 0 0.2 0.4 0.6 0.8 1 1.248

50

52

54

56

58

60

Normalized circumferential strain (-)

Circ

umfe

rent

ial s

tress

(kPa

)

empirical data (before transition point)model at 10% axial stretch of Adventitiamodel at 10% axial stretch of Media

-0.2 0 0.2 0.4 0.6 0.8 1 1.250

60

70

80

90

100

110

Normalized circumferential strain (-)

Circ

umfe

rent

ial s

tress

(kPa

)

empirical data (after transition point)model at 10% axial stretch of Adventitiamodel at 10% axial stretch of Media

A

Fig. 6. (A) The circumferential stress–strain relationship of the human carotid artery averaged along longitudinal locations. The solid red line indicates the model at 0% axialstretch of the intact wall. (B) The modified two-parallel spring model depicted E06eh<eT

hand EeT

h6eh

of the tunica adventitia and tunica media. (C) and (D) The stress–strainrelations in the tunica adventitia and tunica media before and after the transition point were fitted using the two-dimensional hyperelastic model, respectively. The solid blueand magenta lines indicate the model at 0% axial stretch of the tunica adventitia and tunica media, respectively.

Table 2E06eh<eT

hand EeT

h6eh

(mean ± std) of seven subjects (age 28 ± 3.6 years old) determinedat 0% axial stretch of (A) the intact wall and (B) the tunica adventitia and tunicamedia.

A

Subject EInt06eh<eT

h(MPa) EInt

eTh6eh

(MPa)

1 0.17 ± 0.00 0.70 ± 0.092 0.15 ± 0.00 1.07 ± 0.053 0.22 ± 0.01 0.63 ± 0.034 0.13 ± 0.01 1.37 ± 0.245 0.11 ± 0.00 0.92 ± 0.096 0.17 ± 0.00 0.80 ± 0.067 0.19 ± 0.00 0.80 ± 0.03

Average 0.16 ± 0.04 0.90 ± 0.25

B

Subject EAd06eh<eT

h(MPa) EMe

06eh<eTh

(MPa) EAdeTh6eh

(MPa) EMeeTh6eh

(MPa)

1 0.17 ± 0.00 0.23 ± 0.01 0.66 ± 0.09 0.70 ± 0.102 0.14 ± 0.00 0.16 ± 0.00 0.90 ± 0.04 1.06 ± 0.043 0.26 ± 0.02 0.27 ± 0.02 0.63 ± 0.04 0.64 ± 0.034 0.15 ± 0.01 0.16 ± 0.01 1.28 ± 0.24 1.38 ± 0.265 0.12 ± 0.00 0.12 ± 0.00 0.80 ± 0.08 0.92 ± 0.096 0.19 ± 0.00 0.19 ± 0.00 0.80 ± 0.07 0.80 ± 0.057 0.20 ± 0.00 0.20 ± 0.00 0.80 ± 0.05 0.83 ± 0.03

Average 0.18 ± 0.05 0.19 ± 0.05 0.84 ± 0.22 0.90 ± 0.25


and tunica media before the transition point, and r2 = 0.993, 0.991and 0.991 in those after the transition point. Using Eq. (17), theE06eh<eT

hand EeT

h6eh

were determined and average from seven sub-jects for the intact wall, tunica adventitia and tunica media, as


shown in Table 2A and 2B. The average EInt06eh<eT

hand EInt

eTh6eh

of the in-tact wall were respectively found to be equal to 0.16 ± 0.04 MPaand 0.90 ± 0.25 MPa. The average EAd

06eh<eTh

and EAdeTh6eh

of the tunicaadventitia were respectively equal to 0.18 ± 0.05 MPa and0.84 ± 0.22 MPa. The average EMe

06eh<eTh

and EMeeTh6eh

of the tunica mediawere respectively equal to 0.19 ± 0.05 MPa and 0.90 ± 0.25 MPa.The resulting E06eh<eT

hand EeT

h6eh

were related to the elastin lamellaeand collagen fibers contribution, respectively, based on the two-parallel spring model [19] as depicted in Fig. 6B.

4. Discussion

In this paper, the in vivo stress–strain relationship was deter-mined in order to characterize the complex mechanical interactionof constituents of the carotid wall. To establish the stress–strainrelationship in vivo, the arterial pressure and wall displacementmeasurements were respectively acquired and estimated in vivo.In addition, this study was performed in healthy humans; thereforenoninvasive assessment was warranted. The noninvasive pressuremeasurements also avoided the effects of local flow turbulence andpressure encountered in invasive (direct) procedures through cath-eterization in the small arterial lumen. The invasive pressureassessments can affect the elasticity measurements of the arterialwall [4]. The applanation tonometry has been shown to have highaccuracy when peripheral arterial pressure waveforms from super-ficial vessels were acquired [42]. However, it can be affected bymovements from either the subject or user during the measure-ment. This noninvasive assessment, therefore, required the steadyacquisition by the trained experimentalist and short duration of




each measurement to acquire an accurate pressure waveform. Thecarotid pressure waveform was calibrated using the assumptionsof constant mean and diastolic blood pressures throughout thearterial tree. The mean and diastolic blood pressures of the carotidartery were equal to those of the radial and brachial arteries. To ob-tain the brachial pressure, the auscultation method [12,43,44] wasperformed by a physician. This method is generally used by physi-cians to acquire systolic and diastolic brachial blood pressure bymonitoring a pressure gauge or mercury sphygmomanometer to-gether with a pressure cuff and stethoscope. They tend to providelower and higher values than the true intra-arterial pressure forsystolic and diastolic blood pressures, respectively [45]. Underesti-mation of the pulse pressure, therefore, may occur.

For the purposes of this study, the stress calculation using La-place’s law, however, did not use the pulse pressure. The stress isdependent on the pressure and the increasing transmural pressureproduced the arterial wall dilation during the systolic phase over acardiac cycle. The transmural pressure is equal to the differencebetween the existing intravascular and extravascular pressureswhereas the latter pressure cannot be measured in vivo. The sur-rounding tissue provides constraints on the vessels and may alsoaffect the passive mechanics of the arterial wall. A previous study[46] has reported the surrounding tissue effects to reduce the arte-rial wall stress and strain in the circumferential direction by 70%and 20%, respectively. A limitation of this study is thus the extra-vascular pressure and surrounding tissue which have been ignoredin the models.

The strain was calculated in the circumferential direction basedon the diameter waveform obtained using a 1D cross-correlationtechnique [39]. Due to the most dominant deformation in the cir-cumferential direction, only the circumferential strain was consid-ered in the two-parallel spring model and assumed to be isotropic.The approach based on the two-dimensional hyperelastic model,however, can characterize nonlinear anisotropic material re-sponses. Knowledge on the external axial forces was requiredand was adopted from pre-existent data since they could not bemeasured in vivo. The arterial wall viscous parameters were ig-nored because the higher nonlinear deformation under physiologicloading was not clearly visualized in vivo. As indicated in Sections 1and 2, the minima and maxima of the carotid pressure and diam-eter waveforms were aligned and matched to eliminate the viscouseffects. Only the dilation of the carotid artery was considered. Thecarotid artery was assumed to react passively with negligible vaso-dilation effects from the smooth muscle cells. The mechanicalproperties of the carotid artery were mainly influenced by the con-tribution of the elastin and collagen fibers, which reacted as thepredominant elastic response [6]. Therefore, the models used pro-vided the contribution by the elastin and collagen constituents inboth the tunica adventitia and tunica media using the stress–strainrelationship and our ultrasound-based method. The models, how-ever, do not account for the angular dispersion or 3D geometryof the collagen fibers [47], i.e., the collagen fibers in the tunicamedia are mainly oriented in the circumferential direction whilethat in the tunica adventitia are more dispersed. This negligencewould affect the estimated stress–strain relationship and also theYoung’s modulus, especially in the tunica adventitia but not the tu-nica media. The strain estimated from the models, furthermore,may be overestimated due to two limitations. First, as previouslyindicated, the extravascular pressure and surrounding tissue wereignored. Second, the carotid arterial wall was assumed to dilatesymmetrically in the radial direction.

Two separate models [19,33] were employed in this study toestimate the Young’s moduli of the carotid arteries. The first modelwas used for the Young’s moduli of the elastin lamellae (E1), elas-tin–collagen fibers (E2) and collagen fibers (E3). Since the passivebehavior of the carotid artery was only considered here, its dilation


was assumed to occur without vasodilation. The carotid artery wasmainly constituted of elastin and collagen fibers, which providedpredominantly elastic response [6]. A modified generalized Max-well model, i.e., a two-parallel spring model [19] was thus applied.Furthermore, the stress–strain relationship of the carotid arteryestablished noninvasively using our ultrasound-based methodcharacterizes the nonlinear elastic behavior with a clear inflectionpoint defined as the transition strain. Due to the clear inflectionpoint, the stress–strain relationship can be separated into two lin-ear relationships, i.e., the two-parallel spring models before andafter the transition point. The transition point split the stress–strain relation into two curves and was defined as the maximumof the second derivative of the circumferential stress and strain ra-tio. The maximum of the second derivative of the circumferentialstress and strain ratio (Fig. 4C) shows the maximum change ofstress as strain increases to approximately 23%. The second wasused for the incremental Young’s moduli of the intact wall, tunicaadventitia and tunica media to characterize the complex mechan-ical interaction between the arterial wall constituents.

Regardless of the aforementioned limitations, these moduliqualitatively represent the stiffness of the carotid arterial wall con-stituents. E2 was the highest modulus (0.89 ± 0.27 MPa). It wasapproximately onefold and sixfold higher than E3

(0.75 ± 0.29 MPa) and E1 (0.15 ± 0.04 MPa), respectively. The colla-gen fibers were thus found to be significantly stiffer than the elas-tic lamellae (P < 0.05) in agreement with previous reports [25,26].Regarding the quantities of these Young’s moduli, the relation of E1

and E2 to E06eh<eTh

and EeTh6eh

was noted. E1 and E2 were in agreementwith EInt

06eh<eTh

and EInteTh6eh

of the intact wall (EInt06eh<eT

h= 0.16 ± 0.04 MPa

and EInteTh6eh

= 0.90 ± 0.25 MPa), respectively. It was thus hypothe-sized that at small strains (prior to the transition point) the arterialwall contribution was dominated by the elastic lamellae. At thetransition point, collagen fibers were assumed to start engagingand undergo tension in a sharp change of the Young’s modulus. Be-yond the transition point, the arterial stiffness thus increased asthe strain increased due to the dominant collagen fiber contribu-tion, i.e., the collagen fibers reached their straightened lengthsand protected the vessel from overstretching or rupture. The tunicamedia was not significantly stiffer than the tunica adventitia(P > 0.05) both before (EMe

06eh<eTh

= 0.19 ± 0.05 MPa and EAd06eh<eT

h=

0.18 ± 0.05 MPa) and after the transition point (EMeeTh6eh

= 0.90 ±0.25 MPa and EAd

eTh6eh

= 0.84 ± 0.22 MPa).

Previous studies in healthy subjects have reported that the stiff-ness of the carotid artery was higher than that of the abdominalaorta [5,7,8,12]. The difference in stiffness between the carotidand the abdominal aorta is mainly due to the difference in elastinand collagen constituency [10]. Because of this constituency differ-ence, the ratio of E1:E2:E3 was found to be 1:6:5 and 1:3:2 in thehuman carotid artery (this study) and murine abdominal aorta(previous study) [19], respectively. Previous studies in the ascend-ing and descending porcine aorta [26] have reported that theYoung’s moduli, determined from uniaxial testing, of the tunicamedia were approximately fourfold higher than that of the tunicaadventitia. In this study, the EMe

06eh<eTh

and EMeeTh6eh

of the tunica mediawas found to be not significantly higher than those of the tunicaadventitia (P > 0.05) since the tunica media deformation under-went small strain with in vivo physiologic pressure while the angu-lar dispersion of collagen fibers was ignored in the model. As aresult of the pressure and the angular dispersion, the EMe

06eh<eTh

andEMe

eTh6eh

of the tunica media increase with deformation when it is sub-jected to higher pressure, i.e., in case of pathologic condition or un-der in vitro testing. Moreover, the EAd

06eh<eTh

and EAdeTh6eh

of the tunicaadventitia also decrease when the angular dispersion of collagen fi-bers are considered. In arterial histology, the tunica adventitia hasmore dispersion in the collagen fibers than the tunica media




(almost oriented in the circumferential direction or withoutdispersion) and this structure arrangement provides the lowerstrength in the tunica adventitia. Furthermore, the material re-sponse in the tunica adventitia, i.e., pressure–radius relationship,has been reported to be highly sensitive to the angular dispersionof the collagen fibers [47].

5. Conclusion

The in vivo regional stress–strain relationship in the healthy hu-man carotid arterial wall was established noninvasively. A transi-tion point of the stress–strain relationship was detectedrepresenting the change in the contribution of the elastin and col-lagen fiber during the systolic phase of the cardiac cycle. The car-otid wall constituents were characterized by two separatemodels that yielded the Young’s moduli of the elastic lamellae(E1), elastic–collagen fibers (E2) and collagen fibers (E3) or theincremental Young’s moduli of the intact wall (EInt

06eh<eTh

and EInteTh6eh

)which is composed of the incremental Young’s moduli of the tunicaadventitia (EAd

06eh<eTh

and EAdeTh6eh

) and tunica media (EMe06eh<eT

hand EMe

eTh6eh

).The tunica media was found not to be significantly stiffer than thetunica adventitia (P > 0.05), while the collagen fibers were found tobe five times higher in stiffness than the elastic lamellae.

Acknowledgments

This study was supported in part by the Royal Golden JubileePh.D. Program (RGJ) under The Thailand Research Fund (TRF), Con-tract Number PHD/0243/2548 and the National Institutes of Health(NIH R01HL098830). The authors are grateful to Kazue Okajima,M.D., Ph.D., Columbia University Medical Center, for acquiringthe tonometry data. The authors also wish to thank Phrut Sak-ulchangsatjatai, Ph.D., Chiang Mai University and Danial Shahmirz-adi, Ph.D., Columbia University, for all helpful discussions.

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