Original Contribution - Medical Physics DepartmentIn this paper, we present a general theoretical...

Ultrasound in Med. & Biol., Vol. 31, No. 11, pp. 1509–1526, 2005Copyright © 2005 World Federation for Ultrasound in Medicine & Biology

Printed in the USA. All rights reserved0301-5629/05/$–see front matter

doi:10.1016/j.ultrasmedbio.2005.07.004

● Original Contribution

A GENERAL SOLUTION FOR CATHETER POSITION EFFECTS FORSTRAIN ESTIMATION IN INTRAVASCULAR ELASTOGRAPHY

HAIRONG SHI,* QUAN CHEN* and TOMY VARGHESE*†

Departments of *Medical Physics and †Biomedical Engineering, The University of Wisconsin-Madison,Madison, WI, USA

(Received 28 November 2004, revised 8 June 2005, in final form 7 July 2005)

Abstract—Intravascular ultrasound (US) elastography reveals the elastic properties of vascular tissue andplaque. However, misalignment of the US catheter in the vessel lumen can cause incorrect strain estimation inintravascular US elastography caused by strain projection artifacts. In this paper, we present a generaltheoretical solution where the impact of catheter eccentricity, tilt and noncoplanar errors on the strain estimatesare derived. Appropriate corrections to strain estimates can then be applied with prior knowledge of the catheterposition information to reduce the strain projection artifacts. Simulations using a frequency-domain–basedalgorithm that models intravascular US imaging before and after a specified deformation are presented. Thesimulations are used to verify the theoretical derivations for two displacement situations (linear and nonlinear)under intraluminal pressure, with and without stress decay. The linear displacement case demonstrates that thecorrection factor is dependent only on the angle between the US beam and the cross-sectional plane of the vessel.For the nonlinear displacement case, where a l/r stress decay in the displacement is modeled, the correction factorbecomes a more complicated function of the azimuthal angle. (E-mail: [email protected]) © 2005 WorldFederation for Ultrasound in Medicine & Biology.

Key Words: Catheter, Elastography, Elasticity, Elasticity imaging, Intravascular, IVUS, Imaging, Strain, Ultra-
sound.
INTRODUCTION

Atherosclerosis is a disease of the vessel wall that mayoccur anywhere in vessels such as in the aorta, carotid orthe coronary arteries. Intravascular ultrasound (US), orIVUS, provides real-time high-resolution cross-sectionalimages of coronary/peripheral arteries, enabling visual-ization of vessel wall morphology and plaque (Picano etal. 1985; Gussenhoven et al. 1989a, 1989b; Urbani et al.1993; Wilson et al. 1994; Bridal et al. 1997a, 1997b;Lizzi et al. 1997; Moore et al. 1998; Zhang et al. 1998;Watson et al. 2000). Furthermore, calcified and noncal-cified plaque components can be identified. However, therelative sensitivity to identify lipid components remainslow and IVUS primarily reveals the geometry of thevessel wall and the presence/absence of plaque. Charac-terization of the plaque composition in vascular tissuecan significantly help in the identification of vulnerable

Address correspondence to: Dr. Tomy Varghese, Department of
Medical Physics, The University of Wisconsin-Madison, Madison, WI53706 USA. E-mail: [email protected]
1509

plaque, enabling selection of appropriate interventionaltechniques to prevent plaque rupture.

Imaging tissue elastic properties has been rapidlydeveloping into a new imaging modality that providessignificant new information for diagnosis and treatment(Wilson and Robinson 1982; Krouskop et al. 1987;Parker et al. 1990; Yamakoshi et al. 1990; O’Donnell etal. 1991; Ophir et al. 1991; Varghese et al. 2001). Elas-tography has been used to characterize strain-based prop-erties of vascular tissue (de Korte et al. 2000a, 2002). deKorte et al. (1998), (2000b) incorporated the 1-D cross-correlation technique (Ophir et al. 1991) for IVUS elas-tography using different levels of intraluminal pressureto strain the vascular tissue. Local displacement of thetissue is determined using cross-correlation analysis ofgated radiofrequency (RF) signals, with the finite differ-ences between successive displacement estimates ap-plied to determine the local strain. IVUS strain is calcu-lated for the vessel wall and plaque and presented as acomplementary image to the IVUS image. In other in-stances, strains calculated in sections of the vessel wall
are presented as a color-coded ring around the lumen
mailto:[email protected]

1510 Ultrasound in Medicine and Biology Volume 31, Number 11, 2005

vessel-wall boundary. de Korte et al. (2000) have per-formed both in vitro and in vivo validation studies onexcised human coronary and femoral arteries, and Yu-catan minipig models with atherosclerosis (de Korte etal. 2002). Initial catheter position correction studies inIVUS elastography was also performed by de Korte et al.(1999), where they derive an expression that corrects forcatheter eccentricity and tilt occurring independentlywithin the lumen.

In clinical IVUS practice, the intravascular catheteris inserted into the vessel with the transducer probelocated at the tip of the catheter. Unfortunately, thecatheter is not always located at the lumen center and,thus, has a significant impact on the resultant IVUS andstrain image or elastogram. Hiro et al. (1999) have dem-onstrated that US signals from plaque depend on thereflection angle and Courtney et al. (2002) showed thatboth the angle of incidence of the US beam and thedistance from the catheter to the region-of-interest havesubstantial effects on backscattered intensity. Thompsonand Wilson (1996) also studied the impact of variationsin transducer position and sound speed in IVUS.

In IVUS elastography, misalignment of the USbeam and the radial stress introduce errors into strainestimates referred to as “strain projection artifacts”(SPA) (de Korte et al. 1999). Several methods have beenused to correct these artifacts, such as the geometriccenter algorithm (Shapo et al. 1996). Correction factorshave also been derived for cases where the catheter iseither eccentric (catheter parallel to vessel lumen but notpositioned at the center) or tilted (catheter centered buttilted with respect to the vessel longitudinal axes) (deKorte et al. 1999). These expressions show that themeasured strain is related to the radial strain by thebeam-strain angle, with the derived correction factorsapplied to correct the SPA inherent under these twospecific conditions. However, under general imagingconditions, the catheter will be tilted, eccentric and non-coplanar with respect to the vessel axis, with all of theseconditions occurring simultaneously. Therefore, a moregeneral solution that accounts for all of these effects is ofinterest.

In this paper, we present a general theoretical solu-tion along with correction factors when all of theseconditions occur simultaneously during elastographicimaging. The underestimation of radial strain in thepresence of the beam-strain angle (angle between the USbeam and radial strain) � is described. Simulation resultsare used to validate the accuracy of the theoretical cor-rection factors derived in this paper. The simulationresults presented enable the clear visualization of theSPA generated with various catheter positions, alongwith subsequent correction of these artifacts in the strain
image or elastogram.
TheoryThe theoretical correction factors are derived under

the assumption of a cylindrical vessel with a circularcross-section of uniform thickness composed of homo-geneous isotropic material. The derivation is obtainedusing a 3-D xyz coordinate system, where the origin Odenotes the vessel center, and the z-axis is along thelongitudinal direction of the vessel, as illustrated in Fig.1. The theoretical expression for radial displacement,U(r), as a function of radius from the center of the vessel,previously derived by Ryan and Foster (1997), is givenby:

U(r) � A · r �B

r, (1)

where

A ��1 � 2��1 � ��

E·

�pi � p0��Ro

2 � Ri2� (2a)

B ��1 � ��

E·

�pi � p0��Ro

2 � Ri2� · Ro

2Ri2, (2b)

where A and B are constants when the vessel dimensionsand pressure conditions are as specified below, E is thecircumferential elastic modulus of the vessel wall mate-rial, � is the Poisson’s ratio, pi is the mean intraluminalpressure over the applied deformation, p0 is the meanpressure external to the vessel and Ri and Ro are the innerand outer vessel radii at the mean intraluminal pressureof pi, respectively.

The expression for the radial displacement in eqn(1) can be separated into two terms (i.e., a linear dis-placement component and a component accounting forthe l/r stress decay in the displacement). For each radial

Fig. 1. Cylindrical vessel and catheter within the vessel lumen.Catheter P-C is inserted along the longitudinal z-axis, withtransducer C at the x-axis (a, 0, 0); the catheter passes through

the y-z place at (0, b, c) point.

displacement component, we calculate the impact of

Catheter position effects for strain estimation ● H. SHI et al. 1511

catheter position errors and then combine both expres-sions to obtain a composite result. The US beam CApasses through the specified region-of-interest in thelumen wall at point A, which is at a distance r from thevessel center axis (see Fig. A1 in Appendix A). Wedenote FA as the radial line from the center of the vesselaxis to point A and line FA that is perpendicular to thevessel center axis. To study the impact of catheter posi-tion, we first calculate the angle � between the US beamCA and the radial line FA as a function of azimuthalangle. A component of the derivation was presented byShi et al. (2003) and the complete derivation is nowpresented in Appendix A.

The linear displacement caseStrain is determined from the gradient of the mea-

sured local displacements in the vessel wall, as follows:

� ��t1 � �t2

�(3)

where � denotes the strain, and �t1 and �t2 are thedisplacements in the gated windows 1 and 2 separated bya distance �, respectively. The strain projection artifactis first evaluated for the simplest case, where the catheteris eccentric but parallel to the vessel axis. In this situa-tion, as we have shown in Fig. 2, the transducer ispositioned at point C, assuming that the US beam istransmitted along the CD direction and that E and Fdenote the center points of two data segments separated

Fig. 2. Eccentric position of catheter from vessel center. Cath-eter is parallel to the vessel axis.

by � over which the local displacements are computed.

These two points E and F are located at radii r1 and r2

with respect to the center of the lumen. Estimation of thelocal strain uses the gradient of the displacement esti-mates over the two windowed data segments with radii r1

and r2 as shown in Fig. 2. After the expansion of thelumen caused by changes in the intraluminal blood pres-sure, points E and F move to E� and F�, respectively,changing the radii subtended by these points to:

r1 →r1�1 � A� (4a)

r2 →r2�1 � A�. (4b)

The angle between EE� and FF� with CD (US beam) aregiven by 1 and 2, respectively. To calculate the strain,we require the projection of the displacement estimatealong the beam direction CD. The projections of therespective displacements are [r1(1 � A)cos1 � r1

cos1] and [r2(1 � A)cos2 � r2 cos2]. The projectionof the separation (�) along the US beam (CD) is given byr1 cos1 � r2 cos2. Using eqn (3), the estimated straincan be written as follows:

� �

�r1�1 � A�cos1 � r1cos1� � �r2�1 � A�cos2 � r2cos2�r1cos1 � r2cos2

� A. (5)

Equation (5) illustrates that the strain calculated is inde-pendent of the eccentricity of the catheter.

One of the manufacturers of IVUS equipment pro-duces catheters with a built-in tilt angle 0 between thetransducer at the tip and the vessel cross-section, toreduce vibration artifacts in the IVUS image. In thissituation, as shown in Fig. 3, the US beam is tilted at anangle 0 with the vessel cross-sectional plane. Themovement of the scatterers within the data segment canbe projected along two directions, one along the beam-catheter plane A1 and the other perpendicular to thebeam-catheter plane A2, where the following relation-ships hold:

A1 � A cos,

A2 � A sin.(6)

Because A2 is perpendicular to the US beam, it has nocontribution to the calculated strain. Similarly, using thederivation from de Korte et al. (1999), the measureddisplacement is smaller than the real displacement by afactor of cos0 and the distance between the two win-dowed data segments along the US beam is increased bya factor of 1/cos0. In a similar manner to the derivationof eqn (5), the calculated strain in the presence of a built
in tilt angle 0, can be written as:


� � r1A cos1 cos0 � r2A cos2 cos0

r1 cos1

cos0�

r2 cos2

cos0

� Acos20.

(7)

Equation (7) illustrates that the calculated strain is scaledby a factor of cos0

2 when the catheter has a built-in tiltangle of 0.

Let us now consider the case where the transducer issimultaneously tilted and eccentric within the vessel lu-men; however, we assume that the catheter and vesselaxis are coplanar, as shown in Fig. 4. The projection ofthe US beam (CD) on the cross-sectional plane is alongC�D�, which is at an angle with the point D� to thelumen center. The expansion of the vessel wall can onceagain be separated into two parts: these are Ar*cosalong the beam projection direction (C�D�) and Ar*sinperpendicular to the beam projection direction. Thestrain calculation is similar to that shown in eqn (7),except that now the angle between Ar*cos and the USbeam is the beam cross-sectional plane angle �. Thecalculated strain can, therefore, be written as:

� � r1A cos1cos� � r2A cos2 cos�

r1 cos1

cos��

r2 cos2

cos�

� A cos2�. (8)

where the value of cos� is given by:

cos� �cos2

cos3, (9)

where 2, 3 are intermediate angles and are defined in

Fig. 3. Catheter with built-in tilt angle, where the catheter isplaced eccentric within the vessel lumen.

Appendix A. From the derivation above, we show that

the correction factor is only related to the angle betweenthe US beam and the cross-sectional plane.

Finally, in the most general case, the catheter andvessel axis are also assumed to be noncoplanar in addi-tion to being eccentric and tilted within the lumen. Asshown in Fig. 4, the transducer is at (a,0,0) and thecatheter passes through the y-z plane at the (0,b,c) point.Note that, in the derivation of eqn (8), we do not limit thecatheter to be coplanar with the lumen axis. Instead, wedenote the catheter position using the angle � (i.e., 2,3), where � is dependent on the catheter coordinates.Therefore, as long as the dependence of the angle � withthe rotation of the US beam is known, the correctionfactor can be easily computed. Thus, eqn (8) is valid forthe noncoplanar case, as long as the dependence of theangle � with the rotating US beam can be accuratelyrepresented by catheter coordinates a, b and c.

The 1/r stress decay displacement caseFor this case, we also start with the simplest situa-

tion, where the catheter is eccentric, but parallel to thevessel axis. After expansion of the vessel, the radii sub-tended by the center points of the two data segments canbe written as:

r1 →r1 �B

r1(10a)

r2 →r2 �B

r2(10b)

where B is defined in eqn (1). To calculate the strain, thedisplacement has to be projected along the beam direc-tion. In a similar manner to that shown in eqn (5), thestrain can be calculated using:

Fig. 4. Catheter placed in the most general situation in thevessel. Catheter is eccentric, tilted and noncoplanar within the

vessel lumen.

) strain


� ��r1 �

B

r1�cos1 � r1cos1�� r2 �

B

r2�cos2 � r2cos2�

r1cos1 � r2cos2.

(11)

Using the relationship r1 sin1 � r2 sin2 � doc andtrigonometric relationships, we can simplify the expres-sion as follows:

� � �Bsin1sin2 cos�1 � 2�

doc2 . (12)

Because the two windowed data segments chosen tocalculate strain are very close, 1 � 2 (i.e., r1 � r2 �r. The local strain estimate can be written as:

� � �B

r2cos�2�. (13)

Similarly, if the catheter has a built-in tilt angle 0,the calculated strain is given by:

� � �B

r2cos�2�cos20. (14)

Fig. 5. Simulation results for linear displacement caseSimulated B-mode image; (b) elastogram before SPA coat a 7-mm radial position before SPA correction; and (e

For the case where the catheter is simultaneously tilted

and eccentric within the lumen, the expression can bewritten as:

� � �B

r2cos20cos�2�cos2�

� �B

r2cos20�2cos2 � 1�cos2�

� �B

r2cos20�2cos2� � cos2��. (15)

The derivation of the correction factor for the generalnoncoplanar case is similar to the linear displacementcase described in the previous subsection. As long asthe dependence of the angle � on the rotating US beamis accurately represented by catheter coordinates a, band c, eqn (15) remains valid for the general nonco-planar case.

SIMULATION METHOD AND RESULTS

Simulation methodA simulation program developed in our laboratory

e catheter is only placed in an eccentric position. (a)n; (c) elastogram after SPA correction; (d) strain profileprofile at a 7-mm radial position after SPA correction.

wherrrectio

by Li and Zagzebski (1999) was used to verify the

) strain


validity of the theoretical expression of the correctionfactor derived in the previous section. The geometry ofthe cylindrical vessel wall with an inner radius of 4 mmand an outer radius of 10 mm is initially established forthe simulation. In the next step, scatterers of a sufficientnumber to generate Rayleigh statistics are randomly dis-tributed within the vessel wall. This initial distribution ofthe scatterers is referred to as the precompression state ofthe simulated phantom. The displacement of the scatter-ers under intraluminal pressure is then modeled usingeqn (2). By setting B � 0 and A � 0, we simulate thelinear displacement and 1/r stress decay displacementcases, respectively. The scatterer distributions before andafter the application of an intraluminal pressure are savedas corresponding pre- and postcompression cases of thesame vessel wall. A simulated US catheter is then placedwithin the vessel lumen, with the position and orientationof the catheter determined by the catheter position pa-rameters a, b and c that are user-defined variables. Thevalues of the catheter coordinates for different simulationconditions are changed by varying the values of the

Fig. 6. Simulation results for the linear displacementSimulated B-mode image; (b) elastogram before SPA coat a 7-mm radial position before SPA correction; and (e

parameters a, b and c. The transmitted US beam that

originates from the catheter propagates to the vessel walland the corresponding backscattered echoes from scat-terers within the vessel wall are received and saved as theradiofrequency (RF) data for pre- and postcompressionstates of the vessel. For simulations where the catheterincorporates a built-in tilt angle, appropriate tilting of thebeam is included in the simulation. Simulation of theIVUS data-acquisition process for generating a cross-sectional image of the vessel is obtained by rotating theUS beam along the axis of the catheter.

The simulation program developed by Li and Zag-zebski (1999) is based in the frequency domain, whereecho signals at each frequency component within aneffective frequency band width range of 0 � 50 MHz aresimulated. In the final stage of the simulation, we select20 MHz as the center frequency of the transducer, with a50% band width. The simulated echo signal spectra overthe frequency range from 10 MHz to 30 MHz aresummed up, followed by a Fourier transform operation toconvert the frequency-domain signal to time-domain sig-nals to obtain RF echo signals. The sampling rate for the

here catheter is only placed in an tilted position. (a)n; (c) elastogram after SPA correction; (d) strain profileprofile at a 7-mm radial position after SPA correction.

case wrrectio

simulation was set to 400 MHz and 360 A-line data were

correc


acquired for each rotation of the catheter about its axis toform the IVUS image. The simulated displacement of thescatterers at the inner radius is 0.5% of the inner radius(i.e., the intraluminal compression is 0.5% at inner radiuslocation). The displacement of scatterers at other radiilocations follows eqn (2). For each catheter position, wesimulate 25 independent realizations of the pre- andpostcompression echo signals. The corresponding elas-tograms are estimated over these 25 independent realiza-tions and then averaged to obtain a statistical mean andSD of the strain estimates at each pixel position in theelastogram of the vessel. The scatterer positions for eachof the independent realizations are randomly distributed,but the scatterer concentration and effective scatterer sizeare maintained the same for all of the simulations. Theeffective scatterer size is 50 �m in diameter and thescatterer concentration is 1010/m3.

The US simulation program is written in C�� andruns in a Linux environment. After the completion ofeach independent simulation, the strain images or elas-

Fig. 7. Simulation results for the linear displacement cas(a) Simulated B-mode image; (b) elastogram before SPprofile at a 7-mm radial position before SPA correc

SPA

tograms in polar coordinates are calculated using MAT-

LAB (MathWorks, Inc., Natick, MA, USA). The vesselwall boundaries in the elastograms are calculated usingeqn (A-9) from Appendix A, and the strain estimates thatlie within the inner boundary or lumen and outside theouter boundary (outside the vessel wall) are automati-cally set to zero. Finally, a graphic user interface pro-gram in MATLAB is used to reconstruct the correspond-ing elastogram in the Cartesian coordinate system.

Simulation resultsSimulation results mimicking the two cases dis-

cussed in the theoretical section, namely the linear dis-placement model and the stress decay of the displace-ment, are presented in the following sections.

The linear displacement case. Simulation results inthis section present the corresponding IVUS B-modeimage and the elastograms before and after correction ofthe SPA for the known catheter coordinates. These re-sults provide a comparison of the improvement in the

re catheter is both tilted and eccentric within the lumen.rection; (c) elastogram after SPA correction; (d) strainnd (e) strain profile at a 7-mm radial position aftertion.

e wheA cor

tion; a

strain estimates after correction for the SPA. Each figure

positio


also provides a comparison of the theoretical and simu-lation results obtained at the center of the vessel wall at7 mm (note that the lumen has a radius of 4 mm with theouter radius of the vessel set to 10 mm in the simulation).

Figure 5a shows the B-mode image from the sim-ulated vessel when the catheter is placed at an eccentricposition 3 mm from the vessel center, but parallel to thevessel axis. In this case, the simulated catheter has abuilt-in tilt angle of 30°; Fig. 5b shows the correspondingelastogram with the SPA, and Fig. 5c shows the cor-rected elastogram, with a significant reduction in theSPA. Both the theoretical and simulated strain profiles ofthe vessel wall before correction of the SPA at a radiusof 7 mm are shown in Fig. 5d, where the close corre-spondence between the theoretical and simulation resultsare shown. Note that the eccentricity of the catheter doesnot contribute to SPA, as shown in the elastograms.Figure 5e shows the corrected strain profile of the vesselwall after correction of the SPA at the 7-mm radius.

Generally, the manufacturer’s built-in tilt angle is

Fig. 8. Simulation results for linear displacement case wh(i.e., the most general situation within the lumen). (a) Sim(c) elastogram after SPA correction; (d) strain profile at

profile at a 7-mm radial

around 10°. Equation (7) shows that, for the eccentric

situation, the calculated strain deviates by a factor ofcos20 from the true value, where 0 is the built-in tiltangle. For catheters with a built-in tilt angle 0 � 10°,the calculated strain is cos210, or approximately 0.97 ofthe true value. In simulations or experiments, it is diffi-cult to separate the strain component because of thebuilt-in tilt angle from the true strain values caused byelectronic and other noise artifacts, because the reductionin the true strain is very small. The use of a built-in tiltangle of 30° enables us to verify eqn (7), because itintroduces a larger deviation of the true strain valuesover the errors caused by noise. We, therefore, simulatea manufacturer’s built-in tilt angle of 30° as opposed tothe smaller value of 10°. Figure 5b and d denote theelastogram and strain profile at 7 mm, before correctionof SPA. The variation in the strain value matches thetheoretical prediction. As shown in Fig. 5c and e, thecorrected elastogram is uniform and the strain valuesreach the true applied strain of 0.5% that was simulated.The results shown in Fig. 5c and d confirm the accuracy

heter is simultaneously tilted, eccentric and noncoplanard B-mode image; (b) elastogram before SPA correction;m radial position before SPA correction; and (e) strainn after SPA correction.

ere catulatea 7-m

of eqn (7) in computing the SPA correction. For catheters

g radi


without a built-in tilt angle, we set 0 � 0 degrees andthe expression shown in eqn (7) still holds.

Figure 6a illustrates the B-mode image where thecatheter is tilted at an angle of 30° with the vessel axisand placed at the center of the vessel. A built-in tilt anglewas not simulated for this condition. Because the cath-eter is tilted, the B-mode image is no longer circular, butellipsoidal, in shape as seen in Fig. 6a. Figure 6b illus-trates the corresponding elastogram with SPA caused bythe tilting of the catheter, and Fig. 6c shows the correctedelastogram after reduction of SPA artifacts. In a similarmanner as in the previous results, Fig. 6d shows thesimulated strain profile and the theoretical strain profilebefore correction at a radius of 7 mm within the vesselwall. Because of the tilted position of the catheter, theacquired strain value at a specified radius is no longeruniform, although a uniform stress was applied. Notethat the estimated strain follows eqn (8), which incorpo-rates SPA caused by the tilted position of the catheter.Reduction of the SPA is easily achieved using the theo-retical result obtained with eqn (8) to correct the esti-

Fig. 9. Simulation results for nonlinear 1/r stress decay dand along vessel axis. (a) B-mode image from one

alon

mated strain with SPA in Fig. 6b to obtain the corrected

elastogram in Fig. 6c. Figure 6e shows the correctedstrain profile at a radius of 7 mm.

Figure 7 shows the case where the catheter is si-multaneously tilted and eccentric within the vessel lu-men. The catheter is placed at an eccentric position 3 mmfrom the central axis of the lumen and tilted at an angleof 45°. The built-in tilt angle was not simulated for thiscase. Figure 7a presents the simulated B-mode image,which is similar to that shown in Fig. 6a, where theB-mode image is no longer circular because of the cath-eter position; Fig. 7b denotes the corresponding esti-mated elastogram that contains SPA. Note that we canclearly observe the bright and shadowed areas on theelastogram introduced caused by SPA. The correctedelastogram with the reduced SPA using the theoreticalcorrection in Fig. 7c is fairly uniform. The solid line anderror bars in Fig. 7d denote the simulated strain profileand the standard error at a radius of 7 mm before SPAcorrection; the dotted line is the theoretical prediction ofthe strain estimates at the 7-mm radial location. Thetheoretical prediction corresponds very closely to the

ement case. Catheter is placed at center of vessel lumention; (b) elastogram obtained; and (c) strain profileus.

isplacsimula

simulated profile, as shown in Fig. 7d. Finally, Fig. 7e

fore SPofile a


shows the corresponding strain profiles after correctionfor the SPA, where the simulated strain profile is nowquite uniform.

Fig. 10. Simulation results for nonlinear 1/r stress decaposition within lumen. (a) Simulated B-mode image; (bcorrection; (d) strain profile at a 7-mm radial position be

after SPA correction; and (f) strain pr

Figure 8 presents an example of the most general

case, where the catheter is also noncoplanar with thevessel axis, in addition to being simultaneously tilted andeccentric within the vessel lumen. In this simulation, we

lacement case. Catheter is placed only in an eccentricgram before SPA correction; (c) elastogram after SPAA correction; (e) strain profile at a 7-mm radial positionlong the radius after SPA correction.

y disp) elasto

set a � 3 mm, b � 2 mm and c � �3 mm. In a similar


manner as in the previous simulation, the built-in tiltangle was set to zero. Figure 8a represents the B-modeimage and Fig. 8b and c are the corresponding elasto-grams before and after SPA correction, respectively. TheSPA corrected elastogram in Fig. 8c is uniform with astrain value of 0.5%, the applied value of the compres-sion in the simulation. Figure 8d denotes the strain pro-file at radius of 7 mm location before SPA correction andFig. 8e shows the corresponding strain profile at the samelocation after SPA correction. Observe the close corre-spondence between the strain variations in the theory andsimulations showing the strain estimates caused by SPAin Fig. 8d.

The 1/r stress decay displacement case. In a similarmanner as in the previous subsection, we also simulatetissue displacement with the 1/r decay in the appliedstress with depth. The intraluminal tissue displacement is0.02 mm at the inner wall or lumen (equivalent to a 0.5%compression of the inner radius) and tissue displacementat other radii undergo a stress decay that follows the l/rrule. The simplest situation, again, is the case where thecatheter is at the lumen center and parallel to the vesselaxis. Figure 9a is the B-mode image obtained from arealization of the simulation. Figure 9b shows the stressdecay elastogram, where the brightness of the strainestimates in the elastogram decays with depth. Plots ofthe strain profile with depth over radii of 4 to 10 mm areshown in Fig. 9c. From the strain profile, observe that thestrain value decays after a 1/r2 rule, except for thedeviation around the vessel edges, which are artifactscaused by the fact that parts of the windowed datasegment lie outside the vessel wall, thereby introducingerrors in the strain estimates.

Figure 10 shows the case where the catheter isplaced at an eccentric position 3 mm away from thevessel center, but parallel to the vessel axis. The B-modeimage is shown in Fig. 10a, and Fig. 10b is the corre-sponding elastogram before SPA correction. We canclearly visualize the brightness changes along the azi-muthal angle direction because of the SPA. Figure 10cshows the SPA-corrected elastogram, where, except forthe two bright zones in the region closer to the inner wallor lumen, the strain distribution is very similar to theelastogram depicted in Fig. 9a. The existence of thesetwo bright zones in the elastogram can be explainedusing Fig. 11 and eqn (15). In eqn (15), the strain valuebecause of the eccentric position of the catheter is pro-portional to (2cos2 � 1) � cos2 of the true strainvalue, where the angle can be as large as 45° at certainregions, as shown in Fig. 11. So, the resultant strainestimate would be close to zero; however, this is veryunlikely because of noise artifacts in the elastogram.
Nevertheless, these strain estimates would be quite
small. Now, when SPA are corrected for in the elasto-gram, strain estimates are divided by the correction fac-tor obtained using eqn (10), which amplifies the noise inthese regions of the corrected elastogram.

Figure 10d shows the strain profile at a radial loca-tion of 7 mm before SPA correction, where we observea close correspondence between the simulation and the-oretical prediction. Figure 10e shows the correspondingstrain profile after correction and Fig. 10f shows thestrain profile along the radii from 4 mm to 10 mm.Observe that the strain profile follows the 1/r2 decaycurve well, except near the lumen or initial portion of thevessel wall, which is because of the amplified noiseartifacts.

Figure 12 shows the situation where the catheter isat the center of the vessel, but tilted at 30° from thevessel axis. Figure 12a denotes the B-mode image andFig. 12b is the elastogram before SPA correction. Be-cause of the catheter position, the B-mode image andelastogram are not circular, but elliptical, as previouslydescribed. Figure 12c denotes the elastogram after SPAcorrection. Note that Fig. 12b and c are similar becausethe variance of the SPA is quite small. We can alsoclearly observe this aspect in Fig. 12d, which shows thestrain profile before correction at a 7-mm radial location.The figure indicates a strain variation along the azi-muthal angle and the theoretical curve is able to predictthe simulation results obtained. Finally, Fig. 12e showsthe strain profile after correction at the same location.

Figure 13 shows the case where the catheter issimultaneously eccentric (3 mm from the vessel center)and tilted at 45°. Figure 13a represents the B-modeimage and Fig. 13b is the elastogram before SPA cor-rection. Again, because of the position of the catheter,

Fig. 11. The reason for the bright artifacts caused by noiseamplification that appear close to the lumen in Fig. 10c.

the B-mode image is no longer circular, but ellipsoidal,

r SPA


in shape. The elastogram in Fig. 13b is quite dark be-cause of the SPA artifacts, whereas, in Fig. 13c, theelastogram after correction, the whole image is brightand the 1/r2 trend is obvious. We can also clearly visu-alize the two brighter regions closer to the inner wallbecause of the amplification of noise artifacts. Figure 13dshows the strain profile before correction at a 7-mmradial position, which indicates a large strain variationalong the azimuthal angle, with the theoretical curveproviding a good prediction of the simulation result aswell. Figure 13e shows the strain profile after correctionof the SPA at the same location.

Finally, Fig. 14 provides an example of the mostgeneral case (i.e., where the catheter is also noncoplanarwith the vessel axis in addition to being simultaneouslyeccentric and tilted). For this simulation, we set a � 3mm, b � 2 mm and c � �3 mm. Figure 14a again is theB-mode image and Fig. 14b shows the elastogram beforeSPA correction. In a similar manner as shown in previ-

Fig. 12. Simulation results for nonlinear 1/r stress decaywithin the lumen. (a) Simulated B-mode image; (b) ecorrection; (d) strain profile at a 7-mm radial position b

position afte

ous simulation examples; the B-mode image and elasto-

gram are not circular in shape. Figure 14c is the elasto-gram after correction of SPA artifacts. In Figure 14d,once again we can clearly observe the variation in thesimulated strain profile predicted by the theoreticalcurve. Figure 14e shows the strain profile after correctionat the same location.

DISCUSSION

Knowledge of plaque composition, especially lip-id-rich and mixed (fibrous, lipid, calcified), and plaquevulnerability through its elastic properties can signif-icantly assist the surgeon in selecting appropriate in-terventional techniques. In this study, we derive ageneral solution for catheter position correction forintravascular strain estimation. This general solutioncovers the entire gamut of possible catheter positionswithin the vessel lumen (i.e., eccentric, tilted andnoncoplanar). As shown in the paper, these situations

cement case. Catheter is placed only in a tilted positionram before SPA correction; (c) elastogram after SPASPA correction; and (e) strain profile at a 7-mm radialcorrection.

displalastogefore

can exist independent of each other or could exist

r SPA


simultaneously. However, the general solution pre-sented provides a solution for all possible catheterpositions, either occurring independently of each otheror simultaneously. The correction for the SPA is onlydependent on the catheter position in the vessel, whichhas been verified using simulations. The results ob-tained at various conditions all match the theoreticalpredictions. Previous theoretical expressions for SPAsolutions by de Korte et al. (1999) deal only with caseswhere the catheter position is either tilted or eccentric,with neither condition occurring simultaneously. Inaddition, they derived separate expressions to dealwith each condition. In this paper, we present a moregeneral situation (i.e., the tilted, eccentric and nonco-planar situations that may occur either independentlyof each other or simultaneously), and the generalsolution is obtained using a single expression.

Generally because of the flexible nature of the cath-eter, it is difficult for the clinician to accurately positionthe catheter within the vessel lumen. The cross-sectional

Fig. 13. Simulation results for nonlinear 1/r stress decatilted within the lumen. (a) Simulated B-mode image; (bcorrection; (d) strain profile at a 7-mm radial position b

position afte

US image from the IVUS clinical scanner can help the

clinician visualize the plaque; however, it is difficult todetermine the catheter position parameters (a, b and c)from the IVUS image itself. Other imaging modalities(such as x-ray angiography) or other independent track-ing techniques have to be used to determine the catheterposition.

The blood vessel model used in this paper as-sumes the vessel to be an infinite length circularcylinder, where the center of the inner wall and outervessel wall are the same. However, under practicalconditions for in vivo imaging, this is usually not thecase. The noncircular shape of the vessel will intro-duce errors in the theoretical expression derived in thispaper, under the conditions where the expansion of thevessel wall tissue under intraluminal pressure may notbe only along the radial direction, introducing errors inboth the strain estimation algorithm and the theoreticalcorrection factor and, even with a fixed catheter posi-tion, the catheter position parameters (a, b and c) willvary for different locations within the vessel because

acement case. Catheter is simultaneously eccentric andogram before SPA correction; (c) elastogram after SPASPA correction; and (e) strain profile at a 7-mm radialcorrection.

y displ) elastefore

of its curvature. This artifact could be overcome by

7-mm


acquiring several adjacent points on the vessel wall todetermine an estimate of the equivalent local radius ofthe vessel.

The general solution presented in this paper is anexpression that includes the azimuthal angle , where isdefined as the angle between the x-axis and the intersec-tion line of the US beam-catheter plane and the x-yplane. This definition of the azimuthal angle is, therefore,around the transducer C, instead of the usual definition ofthe angle around the vessel center O, as shown in Fig. A1in Appendix A. This scheme is convenient for our sim-ulation results, because the beam originates from thetransducer and sweeps in 360° arcs around the trans-ducer. In the theoretical expression and simulation re-sults, we can easily set the initial � 0 position at thex-axis and increase the values counter-clockwise fromthe z-axis. However, under practical conditions, the sit-uation may be more complicated because it is difficult topinpoint the exact azimuthal direction of the beam withour experimental data-acquisition system on a single RF

Fig. 14. Simulation results for nonlinear 1/r stress decay(i.e., simultaneously eccentric, tilted and noncoplanar wbefore SPA correction; (c) elastogram after SPA correc

correction; and (e) strain profile at a

data frame. In the meantime, the catheter could vibrate

and rotate off its previous center of mass during differentsweep cycles (Thompson and Wilson 1996) used togenerate the real-time IVUS images. All of the above-described conditions make SPA correction more chal-lenging.

Because many factors, such as the actual trans-ducer dimensions, catheter vibration effects, align-ment of pre- and postcompression RF data, inhomo-geneity of tissue etc., affect the theoretical correctionexpression in clinical practice, with some of thesebeing more critical than the possible errors in thecatheter position parameters (a, b and c), we did notderive the error propagation formula because of theuncertainties in the catheter-position parameters inthis paper. The mathematical derivation is possible;however, the final expression obtained is fairly com-plicated. Because we derive the theoretical closed-form solutions for the correction factors in this paper,errors caused by incorrect estimations of the parame-ters a, b and c can be evaluated by varying the param-

cement case. Catheter is placed in the most general casee lumen). (a) Simulated B-mode image; (b) elastogramd) strain profile at a 7-mm radial position before SPAradial position after SPA correction.

displaithin thtion; (

eters a, b and c in the corresponding theoretical cor-


rection expressions and obtaining the difference be-tween the theoretical correction values. Interestedreaders are referred to Shi et al. (2003), where we havedemonstrated the error-propagation approach dis-cussed in this paragraph.

In the derivation of the theoretical correctionfactor, we separate eqn (1) into two parts and derivethe correction factor for both the linear displacementand l/r stress decay in the displacement. For generalPoisson’s ratio material, the final correction factor isthe weighted summation of the correction factor forthe linear-displacement case and the displacement l/rdecay case, as shown in eqn (1). For incompressiblematerials with Poisson’s ratio � 0.5, the linear dis-placement part disappears and only the displacementl/r decay case decides the correction factor.

During the calculation of elastograms from thesimulated RF data, we noticed that the window lengthselection also affects the noise characteristics of theelastogram. For the linear displacement case, we use a0.03-mm window length and, for the l/r stress decaydisplacement case, we evaluated several windowlengths and observed that the combination of a0.05-mm window length and up-sampling of the RFdata to a sampling rate of 400 MHz provides the bestimage quality. Because, in the linear-displacementcase, the displacement increases at the larger radiilocations, smaller window lengths sometimes intro-duce decorrelation at larger radii. On the contrary, forthe l/r stress-decay displacement case, displacementdecreases dramatically with an increase in the radii,where the smaller window lengths perform better.

The boundaries of all the elastograms were cal-culated using eqn (A-9) in Appendix A, where yvalues at the inner radius denote the inner boundaryand the y values at the outer radius denote the outerboundary. The resultant boundaries are smooth and theentire computation is performed automatically, whichalso implies the accuracy of our theoretical derivation.

Figures 10c, 13c and 14c also show the artifactsintroduced because the correction factors are close tozero. Generally, cases where the theoretical correctionfactors become close to zero occur in locations closeto the inner wall of the vessel, which is also generallythe plaque location. Removal or reduction of theseartifacts is difficult. One solution would be the use ofa small regularization factor. However, these artifactsoccur under specific circumstances, such as with spe-cific combinations of inner radius and catheter posi-tion parameters (a, b and c). This implies that, forIVUS elastography plaque characterization, one canadjust the catheter position inside the vessel (i.e., by
moving the catheter away from the plaque in such
cases) to avoid correction factors close to zero atregions that contain plaque.

CONCLUSION

We have derived a closed-form theoretical expres-sion that provides for the correction of the SPA when thecatheter is simultaneously tilted, eccentric and noncopla-nar within the vessel lumen. A frequency-domain simu-lation program is used to verify and validate the theoret-ical results for different practical imaging conditions.The simulation results obtained demonstrate a close cor-respondence to the theoretical derivation.

Acknowledgements—This work was supported in part from a grant tothe University of Wisconsin Medical School under the Howard HughesMedical Institute Research Resources Program for Medical Schools.

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NOTATION OF VARIABLES

a � The eccentric position of the transducer on x-axis

b � The y-coordinate of the point where catheterpasses through the y-z plane

c � The z-coordinate of the point that catheterpasses through the y-z plane

h � The distance between the point where USbeam reaches the vessel wall to the x-y plane

r � The radius of the vessel lumen

y � The distance between the transducer and vesselwall

� � Strain beam angle, the angle between US beamand radial strain

� � The angle between the US-catheter plane and
the x-y plane
� � The angle between the catheter and theintersection line of the US beam-catheter planeand x-y plane

0 � The manufacturer’s built-in tilt angle, the anglebetween transducer and catheter; if 0 � 0,then the manufacturer’s built-in tilt angle iszero

2 � The angle between the US beam andintersection line of the US beam-catheter planeand x-y plane

3 � The angle between the intersection line of theUS beam-catheter plane and x-y plane, the linethat connects the transducer and the projectionof the point where US beam reaches the vesselwall on the x-y plane

� � The angle between the US beam and the x-yplane

� The azimuthal angle, defined as the anglebetween the x-axis and the intersection of theUS beam-catheter plane and the x-y plane

� The angle between the projection line of theUS beam on w-y plane and the radial line thatpasses through the same point on the vesselwall as the projection

� � Eccentricity, defined as a/r

APPENDIX A

The geometric relationship between the catheter and coordinatesystem is shown in Fig. A1. For a vessel with a circular cross-sectionof radius r, we assume that the transducer is positioned at point C onthe x-axis with coordinates of (a,0,0), where a/r denotes the eccentric-ity of the transducer. The catheter crosses the y-z plane at point P(0,b,c). IVUS transducers are typically designed to subtend an angle of0 with the vessel wall to avoid mechanical vibration. The catheter-

Fig. A1. Geometric relationship between the catheter and co-ordinate system. The z-axis denotes the axis of the vessel, thex-y plane is the cross-sectional plane of the vessel, and P-Crepresents the catheter. This shows the variables used in deri-

vation of the closed form expression.


beam plane denoted by PCA now crosses the x-y plane with angle �along the line CE. The azimuthal angle is defined as the anglebetween the x-axis and the line CE. When the catheter rotates along PC,the US beam CA scans a conical swath along PC. The US pulse reachesthe vessel at point A, where A satisfies the condition that the projectionof A on the x-y plane (defined as B) has a distance r from the originpoint O. Similarly, we can also draw AF//BO and point F (withcoordinates (0,0,h) on the z-axis). Note also that AF � r, the radius ofthe vessel. The angle � between US beam CA and radial line AF is thestrain-beam angle. The relationships derived in the following six in-termediate equations are necessary to determine the geometric relation-ship between � and .

Intermediate equation IWe denote the propagation path of the US beam CA from the

transducer to the vessel wall as y, the angle between BC and CE as 3

and the angle between AC and CE as 2, as shown in Fig. A1, weobtain:

3 � arccosy · cos2

y2 � h2 . (A-1)

Equation (A-1) is the same as eqn (B-4) in Appendix B.

Intermediate equation IIFrom triangle OBC in Fig. 2, when � �, the �OCB � � �

� 3 and, for �, �OCB � 3 � � � , so:

r2 � y2 � h2 � a2 � 2ay2 � h2 cos�OCB� (A-2a)

r2 � a2 � � y2 � h2� � 2ay2 � h2 cos� � 3�. (A-2b)

Intermediate equation IIIFor the triangles ABE and ACE, we have AB�BE, AE�EC, so:

h � y sin2 sin�. (A-3)

Intermediate equation IVTo find the angle � between the catheter-beam plane ACP in Fig.

A1 and the x-y plane, we extend PC to point D in Fig. A2, with PC �CD. We then project point D on the x-y plane at J. We can prove thattriangle CJD is exactly the same as triangle CQP. From J, we then drawa perpendicular line with respect to line CE and the intersection pointis G. If the angle JGD is �, then:

tan� � c

b. (A-4)

Fig. A2. Geometric relationship between catheter and coordi-nate system used in the derivation of intermediate equation IV.

a2 � b2 · sin� � arctan a �

Intermediate equation VWe also require a relationship between the angle 2 and the

azimuthal angle , as illustrated in Fig. A3. For convenience, we extendthe line CE to point H and set CH � a; the angle 2 can be written as:

2 � 0 ��

2� arccos

b sin � a cos

a2 � b2 � c2 . (A-5)

Equation (A-5) is the same as the eqn (C-4) in Appendix C.

Intermediate equation VIAnd, finally, from triangle ACF, we have:

a2 � h2 � y2 � r2 � 2yr cos�. (A-6)

So far, we have six independent eqns (A-1)–(A-6) and six independentvariables h, y, �, �, 2 and 3 that are used to obtain a closed formsolution for the beam-strain angle �.

From eqns (A-3) and (A-4), we have:

h � y sin2

c

c2 � �a2 � b2�sin2� � arctanb

a�. (A-7)

Substituting eqns (A-1) and (A-7) into eqn (A-2), we obtain:

y2�1 �c2sin22


a�� y · 2a�cos cos2

� sin sin2

a2 � b2 sin2� � arctanb

a�c2 � �a2 � b2�sin2� � arctan

b

a�� a2 � r2� � 0.

(A-8)

We solve this equation for y and obtain:

y ��n � n2 � 4mk

2m. (A-9)

where

m � �1 �c2 sin2 2


a�� (A-10a)

Fig. A3. Geometric relationship between catheter and coordi-
nate system used in the derivation of intermediate equation V.


n � 2a�cos cos2 � sin sin2

a2 � b2sin� � arctanb

a�c2 � �a2 � b2�sin2� � arctan

b

a��(A-10b)

k � �a2 � r2� (A-10c)

From eqns (A-6) and (A-7), we obtain:

�� arccos 1

2yr��r2 � a2�

� y2�1 �c2 sin2 2


a��. (A-11)

where y and 2 are given by eqns (A-9) and (A-5), respectively, whichare functions of . Substituting into the expression above, we get anexpression for the beam-strain angle � as a function of the azimuthalangle .

We now consider a special case of eqn (A-11). To evaluate thesituation when the catheter is coplanar with the vessel axis, we set b �0 and define the tilt angle as:

1 � arctana

c. (A-12)

Then, eqn (A-4) can be simplified to:

tan� �c tan1

sin . (A-13)

Similarly, eqn (A-5) can be simplified to:

2 � 0 ��

2� arccos��sin1 cos � � 0 �

�

2� arccos�sin1 cos �.

(A-14)

In a similar manner, eqn (A-10) can be simplified to:

m � �1 �c2 sin22

c2 � a2 sin2 �� sin2 � cos22ctan21� ⁄ �sin2

� ctan21�. (A-15a)

n � 2a�cos cos2 � sin2 sin2

a

c2 � a2 sin2 �� 2a�cos cos2

� �sin2 sin2 � ⁄ �sin2 � ctan21�� (A-15b)

k � �a2 � r2� (A-15c)

In addition, eqn (A-11) is reduced to:

�� arccos� 1

2yr��r2 � a2� � y2�1 �c2 sin2 2

c2 � a2sin2 �� arccos� 1

2yr��r2 � a2� � y2�sin2 � cos2 2ctan21

sin2 � ctan21��. (A-16)

The eqns (A-14) to (A-16) have the same notation as in our previouswork (Shi et al. 2003) and, therefore, the solution is similar. Theseequations can also be simplified to obtain the de Korte et al. (1999)solution as shown in Shi et al. (2003).

APPENDIX B

For the triangular prism ABCE in Fig. A1, we have AB�BC,
BE�EC and AE�EC. Denoting AC � y and AB � h and because
AB�BC, we have:

BC � y2 � h2. (B-1)

For triangle BEC, in Fig. A1, we have:

cos3 �EC

BC. (B-2)

In a similar manner for triangle AEC in Fig. A1, we obtain:

cos2 �EC

AC. (B-3a)

EC � AC cos2. (B-3b)

Substituting for EC from eqn (B-3) into eqn (B-2), we obtain anexpression for angle 3 as follows:

cos3 � cos2

AC

BC� cos2

y

y2 � h2 (B-4a)

3 � arccosy cos2

y2 � h2 . (B-4b)

APPENDIX C

In Fig A3, OP � b2 � c2, CP � a2 � b2 � c2 and OH �a2 � 2cos if we define HQ � w and HP � u and the angle COHis � then, because triangle HCO is symmetrical, we can calculate:

cos� �

1

2a2 � 2cos

a� 2

2 1 � cos . (C-1)

We also obtain:

w2 � a2�2 � 2cos � � b2 � 2ab2 � 2cos cos��

2� ��

� a2�2 � 2cos � � b2 � 2ab�sin �. (C-2)

And we can find u2 � c2 � w2. In triangle CHP, we have � � � � 2

� (�/2 � 0) and:

u2 � a2 � �a2 � b2 � c2� � 2aa2 � b2 � c2 cos�

)cos� �b sin � a cos

a2 � b2 � c2 .(C-3)

And, thus, we obtain:

2 � 0 ��

2� arccos

b sin � a cos

a2 � b2 � c2 . (C-4)

Original Contribution - Medical Physics DepartmentIn this paper, we present a general theoretical...

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