Arterial compliance has been shown to be a strong indi-cator of vascular disease, cardiovascular disease, periph-eral vascular occlusive disease, diabetes and renal fail-ure. Change in the ratio of collagen to elastin in theextracelluar matrix of the arterial media is believed to beone of the causes of arterial stiffness (Faury 2001; Bilatoand Crow 1996; Bruel and Oxlund 1996). By measuringmechanical properties of tissue, elasticity imaging couldnoninvasively monitor vascular pathologies developingwithin the vascular wall. Previous attempts at noninva-sive vascular elasticity imaging include arterial wall mo-tion estimation (Bonnefous et al. 1996; Taniwaki et al.2001; Luik et al. 1997; Guerin et al. 2000), intraparietalstrain imaging (Bonnefous et al. 2000) and pulse-wavevelocity measurement (Eriksson et al. 2002; Persson etal. 2001). Arterial compliance measurement was alsoconducted by monitoring internal pulsatile deformationin tissues surrounding the normal brachial artery (Mai
Address correspondence to: Dr. Kang Kim, 2200 BonisteelBlvd., 2117 Carl A. Gerstacker Bldg., Biomedical Engineering Depart-ment, University of Michigan, Ann Arbor, MI 48109-2099 USA.
and Insanna 2002). With some limits, these measure-ments have been correlated with clinical events, includ-ing stroke (Duprez et al. 2001) and claudication symp-toms (Taniwaki et al. 2001) in non-ESRD (end-stagerenal disease) patients and adverse cardiovascular eventsin patients with ESRD (Blacher et al. 1998, 1999), aswell as length of time on dialysis (Luik et al. 1997).
One factor limiting the success of previously usedmethods is that arteries normally distended under phys-iologic pressure produce only small strain. The normalarterial wall, however, is a highly nonlinear elastic me-dium, as illustrated by the solid curve in Fig. 1. Thechange of arterial elasticity due to intraluminal pressurewas reported and analyzed over 40 years ago (Bergel1961). Figure 1 qualitatively captures the essential fea-ture of nonlinear arterial wall compliance. Under physi-ologic loading, the mean arterial pressure produces ahigh effective elastic modulus in the wall. Consequently,the arterial pressure pulse only creates small radial strain(closed bar, Fig. 1). In contrast, if the mean arterialpressure can be lowered to reduce preload, then thearterial pressure pulse can create much larger strain(open bar, Fig. 1). The inherent elastic nonlinearity of the
arterial wall provides an opportunity greatly to expand
762 Ultrasound in Medicine and Biology Volume 30, Number 6, 2004
the strain dynamic range if the mean arterial pressure canbe manipulated with an external force to reduce preload.
By lowering preload, it may be much easier todifferentiate diseased from normal arterial wall. Arterialpathologies are often associated with decreased compli-ance (Hansen et al. 1995; Blacher et al. 1998), as illus-trated qualitatively by the dashed curve in Fig. 1. At lowpreload (open bar, Fig. 1), the difference in radial strainbetween healthy and diseased arteries is much larger thanat high preload (closed bar, Fig. 1). The focus here is todevelop an elasticity imaging procedure exploiting thisrelationship for highly sensitive characterization of arte-rial compliance. In previous reports on arterial elasticityover a wide range of intraluminal pressure (Bank et al.1999; Kaiser et al. 2001), the compliance was inferredfrom geometric changes, such as artery diameter andlumen cross-section, based on a numerical model,Langewouters’ model (Langewouters et al. 1984). Wehypothesize that arterial elasticity can be more accuratelydetermined by measuring localized intramural strain. Inthis paper, the intramural radial normal strain is directlyestimated using a phase-sensitive 2-D speckle-trackingalgorithm (Lubinski et al. 1999). In a clinical setting,larger arterial strains with corresponding higher strainsignal-to-noise ratios (SNRs) are demonstrated usingfreehand deformation to induce transmural pressureequalization. By counteracting the baseline internal arte-rial pressure with offsetting external pressure, the arterialwall preload is effectively reduced. When the preload onarterial wall is lowered by equalizing transmural pres-sure, arterial wall strain and deformation increase witheach cardiac pulsation. Strain and strain-rate measure-
Fig. 1. Arterial elasticity characteristics at different preloads.Solid curve � normal tissue; dashed curve � less compliant
diseased tissue.
ments at maximum pulsation (i.e., minimum preload by
transmural pressure) correspond to the compliance of theartery under the same condition as when blood pressureis taken with a blood pressure cuff. The feasibility of thistechnique is demonstrated using ex vivo and in vivomeasurements and a straightforward elasticity recon-struction algorithm is presented quantitatively to assessthe results.
MATERIALS AND METHODS
Like any other tissue, arteries exhibit nonlinear elas-ticity (Fung 1993). To demonstrate and quantify arterialnonlinearity with respect to internal loading, a controlledexperimental protocol was designed. The intramuralstrain of an ex vivo bovine artery was measured when theintraluminal pressure was increased with a fluid-filledsyringe pump.
Experimental setupA closed-loop compression system was designed to
pressurize an artery sample while simultaneously scan-ning with US. A programmable commercial syringepump (74900-series, Cole-Parmer Instrument, VernonHills, IL) served as pressure source. An acoustic windowwas designed to hold an arterial sample between inletand outlet ports. The outlet was sealed so intramuralpressure develops while the syringe pump compresses. Apressure gauge was placed between the syringe and ar-tery sample close to the artery to measure intraluminalpressure. The acoustic window was positioned in a watertank with anechoic material at the bottom to suppresspossible reverberation, and the tank was placed under-neath US transducer positioning device. A PC-basedradiofrequency (RF) data-acquisition system is con-nected to a commercial ultrasound scanner (SonolineElegra, Siemens Medical System, Issaquah, WA). A
Fig. 2. Experimental setup for ex vivo arterial elasticitymeasurements.
block diagram is presented in Fig. 2.
Vascular intramural strain imaging ● K. KIM et al. 763
Bovine artery ex vivoA 50-mm long bovine carotid artery segment pre-
served in 30% ethanol (Artegraft, Brunswick, NJ) wasplaced in the middle of the acoustic window connected tothe flow path filled with degassed water. A commercialsyringe pump (74900-series, Cole-Parmer) was pro-grammed to pump water at a fixed rate over a fixedperiod (70.5 mL/min for 13 s) to build intraluminalpressure to 120 mmHg. While the artery distended fromthe resting position, a 12.0-MHz linear US array con-nected to a commercial US scanner (Sonoline Elegra,Siemens) imaged the arterial cross-section at a rate of 22frames per s for 13 s. RF data from every frame in thesequence were captured. The intraluminal pressure overtime was also recorded. Data were subsequently pro-cessed using a phase-sensitive 2-D speckle-tracking al-gorithm to determine displacement and strains (Lubinskiet al. 1999). Correlation-based algorithms were used totrack internal displacements. Frame-to-frame lateral andaxial displacements were estimated from the position ofthe maximum correlation coefficient, where the correla-tion kernel size equaled the speckle spot (0.2 mm) foroptimal strain estimation, and axial displacements wererefined using the phase zero-crossing of the complexcorrelation function. A spatial filter that was twice as bigas the kernel size was used to enhance the SNR withreasonable spatial resolution. Frame-to-frame displace-ment estimates were integrated from and registered to theinitial coordinate system (i.e., Lagrangian presentation).Spatial derivatives of the displacements were computedin one region of the artery to estimate the radial normalstrain (i.e., the radial derivative of the radial displace-ment). In the rest of this paper, the radial normal strainwill be called simply “the strain,” where appropriate.
Assuming incompressibility and plane strain in thecross-section of the artery, the radial normal strain, �r, onthe artery wall can be expressed as a function of intralu-minal pressure, pi, outside pressure, p0, and Young’smodulus �. (Appendix 1):
� � �r �3a2b2
2�b2 � a2�r2 �po � pi
� � , a � r � b, (1)
where a is the lumen radius, b is outer radius of the arteryand r is the strain measurement point. Using this relation,� can be estimated as a function of �. In this experiment,outside pressure po can be ignored because the artery sitsless than 2.5 cm from the surface of the water. Note thatthis model applies only to a purely elastic material. Tosatisfy this condition, intramural pressure was slowlyincreased. In vivo, a very high frame rate was used to
overcome the limit associated with pulsatile deformation.
Human artery in vivoTwo subjects were tested under a study protocol
approved by our local investigational review board. Thefirst was a 43-y-old healthy male volunteer and thesecond was a 48-y-old man with ESRD secondary todiabetes mellitus, on hemodialysis and with a history ofperipheral vascular occlusive disease, with prior ampu-tation right below the knee.
A 7.2-MHz linear array was used with continuousfreehand compression performed on the surface of theright upper arm close to the brachial artery. While im-aging the cross-section of the brachial artery at a rate of107 frames per s and collecting US data frame by frame,surface compression was performed by the investigators.The applied external force produces internal pressurecomparable to that generated in measuring a subject’sblood pressure. The compression was increased untilbrachial artery pressure exceeded diastolic pressure, asevidenced by viewing B-scan images. An artery pulsatesthe most when applied external pressure hits diastolicpressure and the artery collapses when the applied pres-sure exceeds the systolic pressure. This was confirmedwith pressure readings when the artery was compressedby a blood pressure cuff. Pulse pressure of each subjectwas recorded by measuring blood pressure before andafter the US scan. Collected RF data were processedoff-line in the same way as described above. In particu-lar, a correlation kernel size of 0.2 mm was also used forin vivo measurement with the 7.2-MHz array.
Elastic modulus reconstructionFor an isolated artery, eqn (1) can be used to recon-
struct the arterial elastic modulus over the entire strainrange (Fig. 3). In vivo, an artery is connected to sur-rounding tissue with finite elastic modulus. This meansthat the reconstructed modulus from eqn (1) will not beaccurate if the finite elasticity of surrounding tissue isignored. To understand how surrounding tissue can in-fluence elastic modulus reconstruction, a simple model isanalyzed in Appendix 2, in which surrounding tissue is tobe viewed as a continuous medium with elastic modulus�2 and the artery wall is considered to be homogeneouswith an elastic modulus �1. The radial strain for thismodel can be written as:
� � �r � � �3a2b2pi
2�b2 � a2�r2����1 �a2
�b2 � a2��2� . (2)
To reconstruct the modulus, this equation must be in-verted,
a2 �3a2b2 pi
��1 ��b2 � a2�
�2� � �2�b2 � a2�r2� � �� . (3)
764 Ultrasound in Medicine and Biology Volume 30, Number 6, 2004
As a first step in solving this equation, a, b and r must beestimated from B-scan images. In this study, we com-puted the constants by hand but, in real-time clinicaloperation, it is very feasible to design automatic lumen-detecting algorithms to define both intimal and adventialboundaries (i.e., a and b). To determine a, b and r, anaveraging procedure was used in this study because theartery was not perfectly circular over the entire pressure-equalization procedure. This averaging should not intro-duce significant error into the estimated modulus as longas a and b change at the same rate for reasonably smalldeformations. Assuming a, b and r can be determinedwithin an error as large as half of the speckle size (0.1mm), the estimation error in elastic modulus, left side ofeqn (3), will be within 20% for the artery sizes in thisstudy.
Given a and b and the coordinates of the strainmeasurement position, the radius r can also be capturedautomatically. Consequently, eqn (3) can be written as:
�1 � K2�2 � K1��p
��� , (4)
where
K1 ��3a2b2
2�b2 � a2�r2, (5)
K2 �a2
2 2 (6)
Fig. 3. Bovine carotid artery B-scans at different intraluminalpressures. From the top left, pressure increases clockwise. Topthree frames � low-preload region; bottom three � high-preload region. Each picture is separated by the same timeinterval. Determining � from the arterial diameter is very hardin the high-preload region because the arterial diameter hardly
changes for a given pressure differential.
�b � a �
are geometric factors computed from B-scan images, �pis pulse pressure and �� is intercardiac strain (i.e.,change in strain from systole to diastole). Note that theconstant K1 is the same as the proportionality factor ineqn (1). Consequently, the effective modulus computedby inverting eqn (1) is:
�effective � K1��p
��� � �1 � K2�2. (7)
RESULTS
Bovine artery ex vivoThe accumulated displacement and axial normal
strain within the artery wall were referenced to a framewhere the artery was at rest. Representative imageframes over the full pressure range are presented in Fig.3. From the resting position of the bovine artery at 0mmHg internal pressure, the syringe pump pressurizedthe lumen to 120 mmHg. The pressure is plotted in Fig.4 vs. axial normal strain at the arterial bottom wall,approximating the radial strain. The strain initially in-creases rapidly with pressure and then hardly changes.Because it is difficult to distinguish diseased fromhealthy arteries with small strain under physiologic pres-sure, it is important to bring the artery into the low-preload region to better determine its elasticity through
Fig. 4. Pressure-strain curve for a bovine carotid artery. Lumenradius � 3.5 mm; adventitial surface radius � 4.25 mm. Thestrain at different points across the wall (between arrows)shows similar nonlinear behavior. The ratio of elastic modulusat high preload to that at low preload ranges from 6 to 12 acrossthe wall. The strain in the plot denotes the representative radialnormal strain in artery wall, as indicated by the dot in the
inserted cross-sectional image.
strain measurement.
Vascular intramural strain imaging ● K. KIM et al. 765
Human artery in vivoThe accumulated radial displacement of the brachial
artery wall was estimated relative to the original frame,as illustrated in Fig. 5, where results are compared fromtwo points in the wall of the healthy subject separated by0.2 mm. Using frame-to-frame images of the correlationcoefficient produced by speckle tracking procedure, thelumen surface could be easily identified. Consequently,these points were clearly within the intima-media com-plex. The overall slope of the average displacementsignals corresponded to the compression by the trans-ducer. Cyclic displacement represents deformation bythe pulse pressure. The point on the bottom wall closer tothe transducer displaced more than the point furtheraway from the transducer at systole and both points cameback to the resting position at diastole. This shows thatthe artery wall compresses maximally at systole andreturns to the resting position at diastole.
Based on the displacement information depicted inFig. 5, radial normal strain and strain rate were esti-mated. When the applied pressure matched the internalbaseline diastolic pressure of 80 mmHg (i.e., low pre-load), strains increased by a factor of 10 with peak strainsof 20% over a cardiac cycle (top plot in Fig. 6). Inaddition, the strain rate computed from the accumulateddisplacements increased over several cardiac cycles. Thepeak strain rate under physiologic conditions rangedfrom 0.1 Hz during diastole to �0.2 Hz during systole.After arterial pressure equalization (i.e., low preload),
Fig. 5. Accumulated radial displacement of two points on theartery wall (the two arrows in the inserted cross-sectionalimage of a healthy volunteer). Dashed line � the accumulatedradial displacement of the point at white arrow; solid line � thepoint at black arrow. All lines are fit using cubic spline smooth-
ing within an error of � 0.01 mm.
the peak strain rate increased to 1.0 Hz during diastole
and �2.5 Hz during systole, similar to the increase inpeak strains (bottom plot in Fig. 6).
The radial normal strain and strain rate of the dis-eased subject are compared with those of the healthyover approximately one cardiac cycle in Fig. 7. The firstpanel illustrates differences at high preload (i.e., physi-ologic condition), and the second at an intraluminalpressure that is nearly equalized by the applied force(i.e., low preload). Over all, the artery of the diseasedsubject produces smaller strain and strain rate than theartery of the healthy one. The blood pressure of thediseased subject was 160/85 mmHg and that of thehealthy subject measured 114/78 mmHg. Noting that thepulse pressure (�p � 75 mmHg) of the diseased subjectis twice as big as that of the healthy subject (�p � 36mmHg), the strain and strain rate contrast will be dou-bled after normalization by the pulse pressure. Theseresults demonstrate that differences in elastic propertiesbetween the two subjects become more pronouncedwhen measured in the low-preload region using pressureequalization.
Elastic modulus reconstructionThe strain (top plot in Fig. 6) for the healthy subject
referenced to the arterial geometry at high preload (underphysiologic pressure, first frame in N frame sequence) isconverted into the strain referenced to the arterial geom-etry at low preload (after pressure equalization, Nthframe in N frame sequence), to conform to the normalgeometry used to compute nonlinear elastic parameters.That is, the reference frame must be converted to the one
Fig. 6. Radial strain and strain rate of a healthy volunteer withcontinuous external compression.
closest to the undistended arterial geometry to present
766 Ultrasound in Medicine and Biology Volume 30, Number 6, 2004
Lagrangian strain for nonlinear analysis, similar to thatpresented in Fig. 4 for the isolated artery. This meansframe-to-frame displacements must be accumulated“backwards” from the last frame to the first, yielding thefinal simple expression relating the strain relative to thefirst frame (�i
o for the ith frame) to the strain relative tothe Nth frame (�i for the ith frame):
�i ��N�i�1
o � �No
1 � �No , (8)
where �N0 is the maximum strain at the pressure equal-
ization frame (Fig. 8).
Fig. 7. Strain rate over one cardiac cycle. (a) Under physiologicpressure, healthy (solid) and diseased (dashed) arteries are hardto distinguish. (b) After pressure equalization, diseased(dashed) artery can be easily differentiated from healthy (solid)
one.
Intercardiac strain of varying amplitude developed
over each cardiac cycle �� and corresponding meanstrain �� was calculated from Fig. 8. With intimal andadventitial boundaries, a and b, and the coordinates ofthe strain measurement position, r, estimated from B-scan images,
K1��p
��� (9)
from eqn (4) is plotted on a semilog scale vs. mean strainat each cardiac cycle, �� (Fig. 9). The pulse pressuremeasured before and after the experiment, �p, was as-sumed to be constant over cardiac cycles. The recon-structed elastic moduli were fit to a straight line, assum-ing a purely exponential model (i.e., � � �0e��), where� is a dimensionless constant describing the degree ofnonlinearity. If the elastic modulus of the surroundingmuscle, �2, can be considered smaller than the arterialelastic modulus, the intercept �0 will determine the un-distended (i.e., zero preload) in vivo arterial elastic mod-ulus. Otherwise, the elastic modulus of surrounding mus-cle must be measured correctly to reconstruct the arterialelastic modulus. If the ratio of elastic moduli between thewall and surrounding muscle remains high over preload,then the slope of this curve (i.e., nonlinear coefficient �)will be correctly reconstructed, independent of the elasticmodulus of the surrounding medium.
For the healthy subject, the intercept �0 ranges from14.7 kPa to 16.5 kPa (mean � one SD of logarithmic fit)and the slope � is 2.9 with a SD of 0.1. The interceptranges from 153.2 kPa to 193.7 kPa (mean � one SD oflogarithmic fit) and the slope � is 4.0 with SD of 0.6 for
Fig. 8. Radial normal strain referenced to the arterial geometryat maximum pulsation (i.e., minimum preload). Closed circles
� mean strain at each cardiac cycle, �.
the subject with known vascular disease. The undis-
Vascular intramural strain imaging ● K. KIM et al. 767
tended elastic modulus of the diseased subject is over 10times that of the healthy subject. Note also that the ratioof the elastic modulus of the diseased artery over thehealthy artery will only increase when accounting for theoverestimation by K2�2 from surrounding tissue, as-sumed to be small compared to �1. Because the sur-rounding tissue modulus may be comparable betweensubjects, overestimation for the diseased subject is muchless pronounced than that for the healthy subject. Inaddition, the nonlinear coefficient � is significantlylarger for the diseased subject.
DISCUSSION AND SUMMARY
The intramural strain range in peripheral arteriesproduced by the pressure pulse can be significantly ex-tended by simply applying pressure comparable to asubject’s blood pressure. Intramural strain can be moni-tored directly with high precision using phase-sensitiveultrasonic speckle-tracking algorithms developed forelasticity imaging. By combining pressure equalizationwith phase-sensitive speckle tracking, new diagnosticinformation may be gathered about the nonlinear elasticproperties of the arterial wall. As illustrated in Fig. 4, theelastic modulus at low preload (� � 39 kPa at 40%preload, representing an intraluminal pressure of about 2kPa) is over an order of magnitude smaller than that atphysiologic preload (� � 454 kPa at 60% preload,
from eqn (4): Open circles � reconstructed moduli; solid line� the fit for the healthy subject; open squares � reconstructedmoduli; dashed line � the fit for the subject with known
vascular disease.
representing an intraluminal pressure of about 10 kPa).
This implies that the artery in the physiologic regionproduces only about 1/10 of the strain in the low-preload(i.e., after pressure equalization) region for the samepressure differential. As demonstrated in vivo, the radialstrain and strain rate increased 10-fold in a healthy arterywhen mean arterial pressure was reduced from physio-logic levels. By applying external force opposite to thatgenerated by the mean arterial pressure, higher strainsmay enable easier differentiation of diseased arteriesfrom healthy ones. By equalizing the baseline arterialpressure to approximate the diastolic pressure, the pre-load on the arterial wall decreases to zero, resulting inmaximal strain. Clearly, the elastic properties of thearterial wall can be better characterized with intramuralstrain measurements extending over a large preloadrange. The deformation in a diseased artery, however,changed comparatively little as the pressure was equal-ized. Consequently, a diseased artery was easily differ-entiated from a healthy one simply by observing radialnormal strain and strain rate at low preload. These verypreliminary ex- and in vivo results suggest that evensmall changes in arterial stiffness accompanying vascu-lar disease may be sensitively monitored with elasticityimaging.
In addition to qualitative assessment of vascularcompliance, the nonlinear elastic modulus of the vascularwall can be quantitatively estimated by using a simplereconstruction procedure. If surrounding tissue can bemodeled as a continuous medium with elastic modulus�2, the elastic modulus �1 of the artery wall can bereconstructed using eqn (7). Within an offset propor-tional to �2, it is possible to reconstruct the arterialelastic modulus as a function of mean arterial strain from
the ratio ��P
��� at the following different levels of so-
phistication:
● assume a fixed geometric scale factor K1 for all sub-jects;
● automatically determine a and r from B-scan imagesusing lumen boundary detection algorithms, assuminga fixed wall thickness for all subjects to compute K1;and
● automatically determine a, b and r from B-scan im-ages to compute K1.
This procedure overestimates the arterial elasticmodulus by as much as K2�2. This should not pose apractical problem at high preloads, but it could be asource of error at low preloads (i.e., pressure equalized)where K2�2 may not be much smaller than �1. The offsetmay only be a practical issue for low arterial moduli,where it can limit the detection of subtle changes in
arterial compliance such as that in healthy or near normal
768 Ultrasound in Medicine and Biology Volume 30, Number 6, 2004
arteries. More studies are required to evaluate this po-tential source of measurement error.
Both ex vivo and in vivo measurements presented inthis study, as well as a large body of previous literature(Bergel 1961; Fung 1993; Bruel and Oxlund 1996; Kai-ser et al. 2001, suggest that the nonlinear change inarterial elastic modulus with preload can be modeled asan exponential function. Consequently, a simple linearleast squares fit to the natural log of the estimated elasticmodulus as a function of preload can fully characterizethe vessel wall’s nonlinear mechanical properties. Amajor advantage of this fit procedure is that only a fewpoints are needed over a limited range of preloads toestimate the elastic modulus of the undistended artery.This may be very valuable in applications such as as-sessment of carotid compliance, where it may be difficultto equalize the pressure all the way to the diastolic limit.Again, the elastic properties of the background mediumwill influence the fit, but they should not significantlyalter the results, except in the small preload limit ofhighly compliant arteries. In any event, both the intercept(i.e., the elastic modulus of the undistended artery) andthe nonlinear parameter (i.e., the slope of the fit) canassess vascular compliance. To estimate these parame-ters with optimal accuracy, all image frames from highpreload to low must be used. One limitation of thepresent study was that all image frames for the diseasedsubject were not of high enough quality to contribute tothis accumulation. Consequently, we computed high pre-cision intramural strain only over several nonconsecutivecardiac cycles in the image sequence (three data points inFig. 9). An absolute geometric reference was establishedbetween isolated segments by tracking changes in arte-rial wall thickness over the entire sequence using B-scanimages. These lower precision measurements only pro-vided the geometric reference for high precision intra-mural strain measurements. Nevertheless, the results pre-sented in Fig. 9 show that good results can be obtainedfrom intramural measurements, even when all imageframes are not of sufficient quality for high-precisionphase-sensitive speckle tracking. Because of the largenonlinear parameter in arterial tissue compared withmost soft tissues, the slope should not be greatly affectedby the surrounding medium. This hypothesis, as well asthe predictive value of each parameter, will be tested incontrolled ex vivo studies.
External force measurement will provide additionalinformation about the elastic modulus of the surroundingmuscle further to refine the reconstruction procedure. Acommercial blood pressure cuff has been modified tohave an acoustic window, through which an US scan canbe performed. A manometer attached to the cuff moni-tors the external force (Fig. 10). Force will be recorded
and input to a modified reconstruction algorithm ac-
counting for the finite elastic modulus of surroundingtissue. This procedure will also be tested in controlled exvivo studies.
Reconstruction procedures presented in this paperfocused on the static elastic properties of the arterialwall. It is well known that the arterial wall is a viscoelas-tic medium (Hardung 1962). Consequently, additionalinformation can be obtained by comparing strain ratemeasurements with the arterial pressure pulse to derivetime constants related to viscoelastic parameters. We willalso explore this possibility in future ex vivo experi-ments.
It should also be recognized that the method ofgenerating pressure equalization is an important consid-eration in translating this diagnostic procedure into auseful clinical tool. Compressing the vessel with an UStransducer may affect the physiological conditions of thevessel and the surrounding tissue, thereby creating a newpulse-wave reflection site in the artery. Alternatively,compressing the artery with a blood pressure cuff mayhelp to overcome this adverse effect. A blood pressurecuff with a window for the transducer is being explored.This device introduces some undesirable transducer mo-tion and artery motion within the imaged field duringinflation and data collection that requires further study.The optimal way to reduce preload will be an important
Fig. 10. Scanning of an upper arm (a) without and (b) with ablood pressure cuff. While imaging the cross-section of thebrachial artery and collecting US data frame-by-frame, surfacecompression was performed. The applied external pressureover the range of physiologic pressure is comparable to thepressure that would be generated in measuring a subject’sblood pressure. Both (a) and (b) provide enough pressureequalization to bring the artery into the low preload region.External force can be monitored by a manometer attached to
the cuff in (b).
consideration in effectively translating this method of
Vascular intramural strain imaging ● K. KIM et al. 769
study into the clinical setting. These and other technicalissues regarding different pressure equalization tech-niques are being explored.
Assessing arterial elasticity may have many impor-tant clinical applications. This method allows localizedassessment of vascular elasticity that may reflect thedegree of both local and general vascular disease. It maybe useful in preoperative assessment for certain vascularsurgery procedure, because the elastic properties of thevessel may reflect the capacity of the artery to remodel,influencing clinical outcomes. For example, in surgicallycreating an arterial-venous anastomosis in hemodialysisfistula creation, the artery dilates to create a manifoldincrease in volume flow through the fistula to accommo-date hemodialysis (Konner et al. 2003). Inelastic dis-eased arteries, so prevalent in end-stage renal disease,may greatly influence the outcome of the procedure(Konner et al. 2003). Assessing the elasticity of arteriespreoperatively may favorably influence site selection,prevent the development of peripheral ischemia and im-prove clinical outcomes. The ease of collecting datareliably, such as with a modified blood pressure cuff(Fig. 10), will be important in assessing the utility of thismethod in the clinical setting. Further clinical study is inprogress to assess the validity of this technique in strat-ifying health and disease states.
Acknowledgments—This work was supported in part by NIH (grantsHL-47401, HL-67647, HL-68658) and a grant from the Renal ResearchInstitute.
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APPENDIX 1
Consider an elastic circular cylinder subjected to hydrostaticpressure. The cylinder is long enough, compared with its cross-sec-tional area, to assume a plane strain state. Assuming isotropy in theaxial direction and homogeneity in the radial direction, only normalstress exists. Considering axial symmetry, radial displacement dependsonly on the distance r from the center of the cylinder (Timoshenko1970):
ur�r� � �A
r� Br, (A1)
where A and B are constants related to the material characteristics andgeometry. The corresponding radial and tangential strains are:
�r�r� ��ur
�r�
A
r2 � B, (A2)
���r� �ur
r� �
A
r2 � B (A3)
Stress-strain relations for this problem are:
�r�r� �1
r � �� � z� (A4)
�
770 Ultrasound in Medicine and Biology Volume 30, Number 6, 2004
���r� �1
�� � �r � z� (A5)
�z�r� �1
�z � �r � �� (A6)
where i is the ith component of the stress tensor and is Poisson’sratio.
Because �z � 0 for a plane strain case, eqn (A6) becomes:
z � �r � ��. (A7)
With this expression, eqns (A5) and (A6) can be rewritten as:
�r�r� �1
��1 � ��1 � �r � � (A8)
���r� �1
��1 � ��1 � �� � r (A9)
Combining eqns (A8) and (A9) to eliminate � yields:
�1 � ��r � �� �1
��1 � ��1 � 2�r. (A10)
Substituting eqns (A2) and (A3) and applying boundary conditions atinner (r � � Pi, at r � a, where pi is the intraluminal pressure) andouter (r � �po, at r � b, where po is the external pressure) surfaceslead to:
�1 � 2�A
a2 � B �� Pi
��1 � ��1 � 2� (A11)
�1 � 2�A
b2 � B �� po
��1 � ��1 � 2�. (A12)
Combining eqns (A11) and (A12) determines the unknown coefficients,
Fig. A1. Cross-section of an elastic circular cylinder subjectedto internal pressure pi and external pressure p0.
A and B. The radial strain of interest in this study can be expressed as:
�r �� po � pi�a2b2�1 � �
��b2 � a2�r2 ��b2po � a2pi��1 � ��1 � 2�
��b2 � a2�. (A13)
Assuming incompressibility, � 0.5, eqn (A13) reduces to:
�r �3� po � pi�a2b2
2��b2 � a2�r2 , a � r � b. (A14)
APPENDIX 2
Consider the same cylinder as in Appendix 1, surrounded by ahomogeneous material that has different elastic modulus. Employingthe same assumptions as in Appendix 1, displacements and resultingstrains in media I and II can be expressed as follows. Media I:
uI,r�r� � �A
r� Br (B1)
�I,r�r� ��ur
�r�
A
r2 � B (B2)
�I,��r� �ur
r� �
A
r2 � B, (B3)
and, media II:
uII,r�r� � �C
r� Dr (B4)
�II,r�r� ��ur
�r�
C
r2 � D (B5)
�I,��r� �ur
r� �
C
r2 � D. (B6)
Note that D � 0 to satisfy the boundary condition at infinity (i.e., nodisplacement). Using the same procedure as in Appendix 1, radial
Fig. B1. Cross-section of an elastic circular cylinder sur-rounded by a homogeneous medium. The cylinder is subjected
to internal pressure pi
stresses in medium I and II can be expressed as follows Medium I:
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I,r�r� ��1
�1 � ��1 � 2��1 � ��I,r � �I,� (B7)
and, medium II:
II,r�r� ��2
�1 � ��1 � 2��1 � ��II,r � �II,�. (B8)
where �1 is the arterial elastic modulus and �2 is the modulus of thesurrounding medium. Applying a boundary condition at the innersurface (I,r � �pi, at r � a), and two boundary conditions at the outersurface (II,r � �p0, II,r � I,r, at r � b) yields:
�1 � 2�A
a2 � B �� pi
�1�1 � ��1 � 2� (B9)
�A
b� bB � �
C
b(B10)
�1 � 2��2
b2 C ��1 � 2��1
b2 A � �1B. (B11)
Combining eqns (B9), (B10) and (B11) determines A, B and C:
A �pi�1 � �a2b2�1 � �1 � 2��2
�1���1 � �2�a2 � �1 � �1 � 2��2b2�(B12)
B �� pi�1 � ��1 � 2�a2��1 � �2�
�1���1 � �2�a2 � �1 � �1 � 2��2b2�(B13)
C �2pi�1 � ��1 � �a2b2�1
�1���1 � �2�a2 � �1 � �1 � 2��2b2�. (B14)
Assuming incompressibility, � 0.5, radial strain in medium I can bereduced to:
