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Cor rec tion for Si mul ta neous Cath e ter Ec cen tric ity and Tilt inIntravascular Elastography
H. SHI,1 T. VARGHESE,1, 2 Q. CHEN1 AND G. GIMELLI3
1De part ment of Med i cal Phys ics2De part ment of Bio med i cal En gi neer ing
3Sec tion of Car dio vas cu lar Med i cine, De part ment of Med i cineThe Uni ver sity of Wis con sin-Mad i son
Mad i son, WI [email protected]
Intravascular elastography can pro vide sig nif i cant new in for ma tion about the elas tic prop er ties ofvas cu lar tis sue and plaque, use ful for the di ag no sis of dis ease and ap pro pri ate se lec tion of interventionalmeth ods. Knowl edge of the plaque com po si tion, vul ner a bil ity and its elas tic prop er ties can as sist the cli -ni cian in se lect ing ap pro pri ate interventional tech niques. How ever, sev eral noise sources have to be ad -dressed to ob tain qual ity intravascular elastograms. Mis align ment of the ves sel lu men and theul tra sound beam can pro duce er ro ne ous strain es ti mates in elastography. Er rors in the strain es ti mate are in tro duced due to the ec cen tric ity and tilt of the intravascular trans ducer within the ves sel lu men. Pre vi -ous work in this area has pro vided the o ret i cal ex pres sions for the cor rec tion of ec cen tric ity and tilt er rorswhen they oc cur in de pend ent of each other. How ever, un der most im ag ing con di tions, both ec cen tric ityand tilt er rors are si mul ta neously pres ent. In this pa per, we ex tend the the o ret i cal cor rec tion fac tor by ac -count ing for the in flu ence of both of these er rors oc cur ring si mul ta neously in the po si tion ing of the cath -e ter within the ves sel lu men.
KEY WORDS: Cath e ter; elastography; elas tic ity; intravascular; IVUS; im ag ing; strain; ul tra sound.
NOTATION OF VARIABLES
a: Ec cen tric po si tion of trans ducer on X-axish: Distance be tween point where ul tra sound beam reaches ves sel wall to XY planer: Radius of lu meny: Distance be tween trans ducer and ves sel walla: Strain beam an gle, the an gle be tween ul tra sound beam and ra dial strainb: Angle be tween ul tra sound-cath e ter plane and XY planed: Angle be tween cath e ter and in ter sec tion line of ul tra sound beam-cath e ter plane and XY
planef
0: Man u fac turer built-in tilt an gle, the an gle be tween trans ducer and cath e ter; if f
0 = 0,
then there is no man u fac turer built-in tilt an glef
1: Tilt an gle of cath e ter, the an gle be tween ves sel lu men axis and cath e ter;
f2: Angle be tween ul tra sound beam and in ter sec tion line of ultrasound beam-cath e ter
plane and XY planef
3: An gle be tween in ter sec tion line of ul tra sound beam-cath e ter plane and XY plane, and
the line con nect ing the trans ducer and pro jec tion point where the ul tra sound beam reachesthe ves sel wall on XY plane
q: Az i muthal an gle, de fined as the an gle be tween the X-axis and the in ter sec tion line ofthe ul tra sound beam-cath e ter plane and XY plane
x: Ec cen tric ity, de fined as a/r
ULTRASONIC IMAGING 25, 262- 283 (2003)
262 0161-7346/03 $18.00
Copy right 2003, 2004 by Dynamedia, Inc.
All rights of re pro duc tion in any form re served.
IN TRO DUC TION
Intravascular ul tra sound (IVUS) is cur rently the only method for the clin i cal ex am i na tionof vas cu lar tis sue. How ever, IVUS re veals only the ge om e try of the ves sel wall and the pres -ence/ab sence of plaque. Char ac ter iza tion of the plaque com po si tion in vas cu lar tis sue cansig nif i cantly help in the se lec tion of ap pro pri ate interventional tech niques. Sev eral re -search ers have used ul tra sonic tis sue char ac ter iza tion pa ram e ters, such as ul tra sonic back -scat ter, at ten u a tion and scat terer pa ram e ters, to char ac ter ize atherosclerotic plaque.1-8
How ever, intravascular ul tra sound mainly iden ti fies fi brous and cal ci fied plaques, whilethe com po si tion of lipid and mixed plaques are rarely iden ti fied in most cases.9 In ad di tion,plaque vul ner a bil ity is de ter mined pri mar ily by the me chan i cal (elas tic) prop er ties of theves sel wall and the plaque. Var i ous stud ies have in di cated that the pulsatile pres sure in duced due to blood flow may rup ture the thin cap over ly ing fatty tis sue (or lipid rich le sions) on theplaque, which may lead to sub se quent throm bo sis and myo car dial ischemia.10 Tech niquesthat are able to char ac ter ize the me chan i cal (elas tic) prop er ties of plaque may there fore pro -vide clin i cal in for ma tion that may have a sig nif i cant im pact on pa tient care.11-17
Elastography is a method that has shown prom ise in im ag ing the elas tic prop er ties of tis sue.It is a new im ag ing mo dal ity where elas tic tis sue pa ram e ters re lated to the struc tural or ga ni -za tion of nor mal and patho log i cal tis sues are im aged.11-13 In elastography, radio fre quen cy ul -tra sound frames ac quired be fore and af ter a small amount (about 1%) of com pres sion arecom pared. Dif fer en tial dis place ments in small re gions are de tected us ing Time De lay Es ti -ma tion tech niques. Fi nally the strain is ob tained as the gra di ent of the tis sue dis place mentsor time de lay. The ba sic prin ci ple is that soft tis sue de forms more than stiffer tis sue, with thedif fer ences quan ti fied in im ages of tis sue ax ial-strain, lat eral-strain, Pois son’s ra tio orYoung’s modulus, all termed elastograms.
Sev eral in ves ti ga tors have used elastography and elas tic ity im ag ing tech niques to ex am -ine vas cu lar tis sue prop er ties.13-16 Intravascular elastographic im ag ing has to ac count for sev -eral ad di tional noise sources, such as cath e ter mo tion dur ing im ag ing, ec cen tric ity of thecath e ter in the lu men, mo tion of the scat ter ing par ti cles in the lu men, and com pres sion lev elsof the vas cu lar tis sue and plaque to pro duce qual ity strain im ages.18, 19
In clin i cal prac tice, the intravascular cath e ter po si tion is gen er ally both ec cen tric and tilted with re spect to the ves sel lu men. The po si tion of the cath e ter has a sig nif i cant im pact on theprop er ties of the ul tra sound echo sig nal ac quired. Hiro et al20 showed that IVUS im ages varyin in ten sity due to the an gle of the trans ducer rel a tive to the plaque and ves sel wall. Courtney et al21 show that the back scat tered in ten sity de creases as ei ther the dis tance be tween the tis -sue and trans ducer in creases, or as the beam’s an gle of in ci dence on the tis sue in creases.
Mis align ment of the ul tra sound beam and ra dial stress in IVUS elastography pro vide er ro -ne ous strain es ti mates re ferred to as ‘strain pro jec tion ar ti facts.’19 Shapo et al18 has pro posedthe use of a geo met ric cen ter al go rithm to ob tain true dis place ment es ti mates when the trans -ducer is off-cen ter but par al lel to the ves sel lu men, thereby ac count ing for the ec cen tric po si -tion ing of the im ag ing cath e ter for elas tic ity im ag ing. Fur ther, de Korte et al19 havecom puted cor rec tion fac tors for cases where the cath e ter is ei ther ec cen tric (cath e ter par al lelto ves sel lu men but not po si tioned at the cen ter) or tilted (cath e ter cen tered but tilted with re -spect to the ves sel lon gi tu di nal axes). The ex pres sions de rived by de Korte et al19 showedthat the mea sured strain is re lated to the ra dial strain by the beam-strain an gle. The re la tion -ship be tween beam-strain an gle and az i muthal an gle for ec cen tric and tilted po si tion ing ofthe cath e ter was eval u ated in de pend ently. These cor rec tion fac tors can then be ap plied tocor rect the pro jec tion ar ti facts in her ent un der these two spe cific con di tions.
How ever, un der gen eral im ag ing con di tions, the cath e ter will be both tilted and ec cen tricwithin the ves sel. There fore, a more gen eral so lu tion that ac counts for both of these ef fects is
COR REC TION FOR CATH E TER EC CEN TRIC ITY AND TILT IN IVUS ELASTOGRAPHY 263
of in ter est. An other ap proach is to use strain invariants22, 23 (i.e., mea sures of elas tic strainthat are in de pend ent of the choice of a co or di nate sys tem and de for ma tion ge om e try), so thatthe re sult ing stress-strain law has com po nent stresses (modes of stress) that are mu tu ally or -thogo nal. For iso tro pic ma te ri als, the fol low ing three strain invariants can be de fined, i.e.,vol u met ric strain, mag ni tude of dis tor tion and the mode of dis tor tion as con sti tu tive func -tions for ma te rial in their un loaded ref er ence state.
In this pa per, we pres ent a the o ret i cal so lu tion where the cath e ter is both tilted and ec cen -tric within the ves sel, i.e., both of these con di tions oc cur si mul ta neously, un der the as sump -tion that the cath e ter in ter sects the cen ter line of the ves sel along the lon gi tu di nal axes. Thecor rec tion fac tor ob tained in this pa per can there fore be used to cor rect for both of these con -di tions when they oc cur si mul ta neously dur ing elastographic im ag ing.
THE ORY
The the o ret i cal cor rec tion fac tors are de rived un der the as sump tion of a cy lin dri cal ves selwith a cir cu lar cross-sec tion of uni form thick ness com posed of ho mo ge neous iso tro pic ma -te rial. The der i va tion is ob tained us ing a three-di men sional XYZ co or di nate sys tem, wherethe or i gin O de notes the ves sel cen ter, and the Z-axis is along the lon gi tu di nal di rec tion of the ves sel (Fig. 1). We uti lize a sim i lar no ta tion to that used by de Korte et al,19 where the un der -es ti ma tion of ra dial strain in the pres ence of the beam-strain an gle a is de scribed. Thebeam-strain an gle a de notes the an gle be tween the ul tra sound beam and the ra dial strain.
In gen eral, stress and strain are ten sors that con sist of three nor mal and six shear straincom po nents. How ever, for a cir cu lar ves sel with a ho mog e nous iso tro pic wall ma te rial,strain com po nents in the ra dial (er), cir cumfer ential (e
q) and lon gi tu di nal (ez) di rec tions are
suf fi cient to char ac ter ize the ves sel19, 24 un der plane stress con di tions, i.e., the nine strain ten -sor com po nents are re duced to com po nents in the ra dial, cir cumfer ential and lon gi tu di nal di -rec tions. The three strain com po nents can be writ ten as:
264 SHI ET AL
X
Y
Z
OC
P
FIG. 1 Sche matic de pic tion of the ves sel and cath e ter in the ves sel lu men. The cath e ter is in serted along the lon -gi tu di nal z-axis and the cen ter of the ves sel lu men, where PC de notes the cath e ter and C is the trans ducer.
zz
rr
sinˆ
,sinˆ
,cosˆ
eae
eae
eae
×=
×=
×=
2
2
2 (1)
For a ma te rial with a Pois son ra tio of 0.5, and un der con di tions where the cath e ter is po si -tioned ec cen tri cally and tilted to the ves sel wall, we have er = -(e
q+ez), and the mag ni tude of
the mea sured strain e is
where e is the ra dial strain.The geo met ric re la tion ship of cath e ter and co or di nate sys tem is shown in fig ure 2. For a
ves sel with a cir cu lar cross-sec tion of ra dius r, we as sume that the cath e ter crosses the Z-axisand is tilted at an an gle f
1 with re spect to the Z-axis. The trans ducer is po si tioned at point C
on the X-axis with co or di nates of (a,0,0), where a/r de notes the ec cen tric ity of the trans -ducer. Some IVUS trans duc ers are de signed to sub tend an an gle of f
0 with the cath e ter axis
to avoid me chan i cal vi bra tion. For IVUS cath e ter sys tems with out a built-in tilt an gle f0, theabove der i va tion can be sim pli fied by set ting the value of f0 = 0. The cath e ter-beam plane,de noted by PCA, now crosses the XY plane with an gle b along the line CE. The az i muthal an -gle q is de fined as the an gle be tween X-axis and the line CE. When the cath e ter ro tates alongPC, the ul tra sound beam CA would scan a con i cal swath along PC. The ul tra sound pulsereaches the ves sel at point A, where A sat is fies the con di tion that the pro jec tion of A on the XYplane (de fined as B) has a dis tance r from the or i gin point O. Sim i larly, we can also drawAF//BO, and point F (with co or di nates (0,0,h)) on the Z- axis). Note also that AF = r, is thera dius of the ves sel. The an gle a be tween ul tra sound beam CA and ra dial line AF has been
COR REC TION FOR CATH E TER EC CEN TRIC ITY AND TILT IN IVUS ELASTOGRAPHY 265
( ) ( )( )
r
rzr
zrzr
cos
sincossincos
sinsincosˆˆˆ
ea
eaaeeaea
eaeaeaeeee
q
12 2
2222
222
-=
-=×++×=
×+×+×=++= (2)
X
Y
Z
O
P
Q
F
A
E
B
C
f1
b
r
ah
h
r
y
dq
f2
f0
a
f3
FIG. 2 Geo met ric re la tion ship of the cath e ter PC in the XYZ co or di nate sys tem used in this pa per.
de fined as the strain-beam an gle.19 Due to the com plex ity of the re la tion ship be tween thestrain-beam an gle a and az i muthal an gle q, and to im prove the clar ity of the der i va tion, thefol low ing six in ter me di ate equa tions are uti lized to com pute the re la tion ship be tween a andq. The er ror prop a ga tion in tro duced by the vari a tion of the in di vid ual geo met ric pa ram e tersin these for mulas is dis cussed in the next sec tion.
In ter me di ate Equa tion I: De not ing the length of ul tra sound beam CA from trans ducer toves sel wall as y, the an gle be tween BC and CE as f
3, and the an gle be tween AC and CE as f
2
(Fig. 2) we ob tain:
Equa tion (3) is the same as Eq. (A-4) in Ap pen dix A.In ter me di ate Equa tion II: From tri an gle OBC in fig ure 2, when q<p, the ÐOCB =
p-q+f3, while for q>p, ÐOCB = f
3-p+q, so:
In ter me di ate Equa tion III: In tri an gles ABE and ACE, we have AB^BE, AE^EC, so
In ter me di ate Equa tion IV: To find the an gle b be tween cath e ter-beam plane ACP in fig ure 2 and the x-y plane, we ex tend one unit dis tance from point C along the pos i tive x di rec tion topoint F (Fig. 3). From point F we draw pro jec tion lines to the in ter sec tion line at G.
The ACP plane in fig ure 2 is the same as the GDCP plane in fig ure 3. There fore, DG^GC,the an gleÐFCD = p/2-f1. From tri an gle DFC, we have DF^FC, FC = 1, so DF can be writ -ten as ctanf
1. Sim i larly, in tri an gle FGC, we have FG^GC, ÐFCG = q, FC = 1, so FG = sin
q. There fore, in tri an gle DFG, since DF^FG, sin b can be writ ten as:
In ter me di ate Equa tion V: We also re quire a re la tion ship be tween an gle f2 and the az i -
muthal an gle q, as il lus trated in fig ure 4. For con ve nience, we ex tend dis tance a from point C along the CE di rec tion to point H. The an gle f
2 can be writ ten as:
Equa tion (7) is the same as Eq. (B-8) in Ap pen dix B.In ter me di ate Equa tion VI: Fi nally, from tri an gle ACF, we have
266 SHI ET AL
22
2
3
hy
cosyarccos
-
×=
ff
(3)
1
22
1
fq
fb
ctansin
ctansin
+=
(6)
( )
( ) ( )3
222222
222222
2
2
fq --+-+=
--+-=
coshyahyar
OCBcoshyaahyr (4)
bf sinsinyh2
= (5)
[ ] 2012pfqff -+= cossinarccos (7)
acosyRRyha 22222 -+=+ (8)
So far, we have six in de pend ent equa tions, (3)- (8), and six in de pend ent vari ables, h, y, a,b, f
2 and f
3, that we use to ob tain a closed form so lu tion for the beam-strain an gle a.
COR REC TION FOR CATH E TER EC CEN TRIC ITY AND TILT IN IVUS ELASTOGRAPHY 267
a
G
F
D
C
Oq
f1
b
X
Z
p/2-f1
P
FIG. 3 Geo met ric re la tion ship be tween the an gle b (be tween the cath e ter-beam plane ACP and the x-y plane) and az i muthal an gle q.
O
P
H
f1
aX
Z
qC
f0
a
a(2+2cosq+tan -2f1 ) 0.5
a/sinf1
d
a/tanf1
a(2+2cosq)0.5f
2
FIG. 4 Geo met ric re la tion ship be tween f2 and az i muthal an gle q.
From Eqs. (7) and (8), we have
Sub sti tut ing Eqs. (3) and (9) into Eq. (4), we ob tain a so lu tion for the term y as fol lows:
Equa tion (12) is ob tained af ter dis card ing the neg a tive root, be cause y>0, where
From Eqs. (8) and (9), we ob tain:
Solv ing for the strain beam an gle, we ob tain:
where y and f2 are func tions of q, and their ex pres sions are shown in Eqs. (10) and (7), re -spec tively. Sub sti tut ing for y and f2 into Eq. (13), we ob tain an ex pres sion for thebeam-strain an gle a as a func tion of the az i muthal an gle q.
Case I: Eccentric catheter par al lel to ves sel lu men
We now con sider the two spe cial cases of Eq. (13) de rived by deKorte et al.19 To eval u atethe sit u a tion when the cath e ter is ec cen tric but par al lel to the ves sel axis, the tilt an gle f
1 = 0.
In ad di tion, we also have
Equa tion (13) now re duces to:
268 SHI ET AL
22
0
0
2
02
2
rak
,coscosan
,cosm
,
-=
=
=
=
fq
f
ff (14)
( ) ( )( )1
222
222 fqqffq ctansinsinsincoscosan ++=
( )22 rak -=
( ) ( )ïþ
ïýü
ïî
ïíì
úû
ùêë
é
+
++-=
1
2
1
2
2
22
222
21
fq
ffqqa
ctansin
ctancossinyararccos
2yr
(13)
1
22
1
2
fq
ff
ctansin
ctansinyh
+=
(9)
( ) ( )1
2
1
2
2
22 fqffq ctansinctancossinm 2 ++= (10)
( ) ( )1
2
1
2
2
22 fqffq ctansinctancossinm 2 ++= (11)
afq
ff cosyrry
tancsin
ctansinya
2222
1
2
1
2
2
222 -+=+
+(12)
( ) ( )[ ]{ }0
2222
21 fqa cosyararccos yr +-= (15)
In the ex pres sion de rived by deKorte et al,19 they also set f0 = 0° (the man u fac turer built-in
tilt an gle f0). Un der these con di tions, Eq. (15) re duces to:
The com plete der i va tion of Eq. (16) is shown in Ap pen dix D.The ex pres sion de rived by de Korte et al,19 un der sim i lar con di tions (f
0 = 0, f
1 = 0), is given
by:
where x = a/r is the ec cen tric ity. Equa tions (16) and (17) have slightly dif fer ent for mats dueto the dif fer ence in the co-or di nate sys tem used to de rive the in di vid ual ex pres sion. How -ever, we dem on strate in Ap pen dix E, that Eqs. (16) and (17) are equiv a lent. In ad di tion, wealso plot the strain beam an gle ver sus the az i muthal an gle, which il lus trates that both of these equa tions are equiv a lent in the fol low ing sec tion.
Case II: Tilted cath e ter at cen ter of ves sel lu men
If we set a = 0, which cor re sponds to the case where the cath e ter is tilted but not ec cen tricwithin the ves sel lu men, Eq. (10) re duces to:
Equa tion (18) is ob tained af ter sub sti tut ing for m from Eq. (11). The gen eral ex pres sionfor the strain beam an gle, Eq. (13), can now be writ ten as:
The ex pres sion in de Korte et al19 for this case is given by:
The fol low ing re la tion ship re lates our ex pres sion with de Korte et al19 ex pres sion, i.e.,
or
Equa tion (22) has been ver i fied us ing the pro gram Mathematica (Wol fram Re search, Inc., Cham paign, IL, USA) and found to be equiv a lent.
COR REC TION FOR CATH E TER EC CEN TRIC ITY AND TILT IN IVUS ELASTOGRAPHY 269
( ) ( )qffffqa p coscossincossinacos01102 -×--= (20)
( ) ( )pqaqa +=deKorteours (21)
( )1
2
1
2
2
22
fq
ffqqa
ctansin
ctancossinarccos
2 +
+=
(19)
( )( )
1
2
2
22
1
2
ffq
fq
ctancossin
ctansinr
m
ry
2
+
+== (18)
( ) ( )( )qpxqa -×= sinarcsin (17)
( ) ( ) úúû
ù
êêë
é
--
---=
qqqqa
cosasinarr
cossinarasinararccos
222
222222 (16)
( )( ) ( )( )pqaqa +=deKorteours
coscos 22 (22)
Both ex pres sions cor re spond ing to real cases, as shown in fig ure 5. Fig ure 5(a) de pictsEq. (19) and fig ure 5(b) de picts Eq. (20). Both con di tions are ac cu rate with the re la tion shipshown in Eqs. (21) and (22). Equa tion (19) is al ways greater than zero while Eq. (20) can beless than zero. The sign dis ap pears, how ever, when the cos2
a term is com puted.In the next sec tion on the o ret i cal re sults, we com pare the o ret i cal plots us ing the gen eral
ex pres sion de rived in this pa per to the two cases de scribed by de Korte et al.19
THE O RET I CAL RE SULTS
Fur ther con fir ma tion of the va lid ity of the gen eral ex pres sion de rived in this pa per is pre -sented by plot ting the beam-strain an gle as a func tion of the az i muthal an gle de rived by us tothe re sults ob tained by the ex pres sions de rived by de Korte et al.19
For the case when the cath e ter is ec cen tric and par al lel to the ves sel axis, we plot Eq. (16)and Eq. (17) in fig ure 6. In the fig ure, the x-axis de notes the az i muthal an gle q, while they-axis is the beam-strain an gle a. We ob tain a se ries of plots by vary ing the ec cen tric ity pa -ram e ter x = a/r from 0.1 to 0.9. The solid lines shown in fig ure 6 are ob tained us ing the spe -cial case of the gen eral ex pres sion de rived in this pa per, i.e., Eq. (16), while the solid cir cleson the curve are ob tained us ing the ex pres sion de rived by de Korte et al19 or Eq. (17). Ob -serve that the solid cir cles lie ex actly on the solid curve, dem on strat ing the close cor re spon -dence be tween the two re sults. In ad di tion, the vari a tion of the strain beam an gle issym met ri cal about the az i muthal an gle of 180°.
In a sim i lar man ner, we com pare our the o ret i cal re sults with ex pres sions de rived by deKorte et al19 for the case where the cath e ter is tilted but at the cen ter of the ves sel lu men, asshown in fig ure 7. We as sume a man u fac turer’s tilt an gle f
0 = 10°, which is a typ i cal value,
and vary the tilt an gle f1 over 10°. The solid lines are ob tained us ing the ex pres sion de rived
in this pa per, mod i fied for the case where the trans ducer is tilted but at the lu men cen ter, de -scribed by Eq. (19). The solid cir cles rep re sent the ab so lute value of Eq. (20) with the vari -able (q+p). Once again, note the close cor re spon dence be tween the two the o ret i cal re sults. Note also that when f
1 = 0°, the strain beam an gle is 10°, which is the man u fac turer’s tilt an -
gle, as shown in fig ure 7.The re sults pre sented in figures 6 and 7 dem on strate that the ex pres sion de rived in this pa -
per, which com bines both the tilt and ec cen tric po si tion of the cath e ter dur ing strain im ag ing,
270 SHI ET AL
FIG. 5 Two sit u a tions of trans ducer cath e ter an gle f0, the re sul tant re la tion ship be tween beam-strain an gle a and
az i muthal an gle q has p an gle shift.
X
Y
Z
P
C
X
Y
Z
P
C
(a) (b)
ac cu rately de picts the vari a tion in the strain beam an gle for each con di tion in de pend ent ofthe other. The re la tion ship be tween the strain beam an gle and the az i muthal an gle is morecom pli cated when the tilted and ec cen tric con di tions are pres ent si mul ta neously. In the fol -low ing fig ures and dis cus sion, we pres ent three cases where the com bined so lu tion is used to de pict the the o ret i cal vari a tion in the strain beam an gle.
Fig ure 8 pres ents plots of the beam-strain an gle ver sus the az i muthal an gle in the com -bined tilted and ec cen tric con di tion for f
0 = 10°, x = a/r = 0.1, and f
1 = 10°, as shown in fig ure
COR REC TION FOR CATH E TER EC CEN TRIC ITY AND TILT IN IVUS ELASTOGRAPHY 271
FIG. 6 Re la tion ship be tween the beam-strain an gle a and az i muthal an gle q when the cath e ter is po si tioned ec -cen tri cally within the ves sel lu men. Both man u fac turer tilt an gle (f
0 = 0) and tilt an gle (f
1 = 0) pa ram e ters are set to
zero. The solid lines dem on strate our re sults (Eq. (16)) while the solid cir cles rep re sent deKorte et al19 re sults de -noted by Eq. (17). Note close cor re spon dence be tween the re sults.
FIG. 7 Re la tion ship be tween the beam-strain an gle a and az i muthal an gle q when the cath e ter is tilted but at thecen ter of the ves sel lu men. The solid lines dem on strate our re sults (Eq. (19)) while the solid cir cles rep re sentdeKorte et al19 re sults de noted by Eq. (20). Note close cor re spon dence be tween the re sults.
7. Note that in this case, the trans ducer is placed in a slightly ec cen tric po si tion within theves sel lu men. Com par ing fig ure 8 with fig ure 7, we ob serve that the beam-strain an gle is nolon ger sym met ri cal along the man u fac turer’s tilt an gle of 10°. In ad di tion, note that the shiftin the strain beam an gle and sharp vari a tions in the curve for the f
1 = -10° tilt an gle. If the ec -
cen tric ity were x = – 0.1, then the shift would be re versed and the sharp vari a tion would oc -cur for the curve with f
1= 10°.
Now con sider the case where the ec cen tric ity of the trans ducer is larger, i.e., x = 0.9. Un -der these con di tions, even small tilt an gles can cause large strain pro jec tion ar ti facts. Fig ure9 shows the com bined ef fect where f
0 = 10° and vary ing f
1 over 10° with step size of 2°, the
dot ted lines de note neg a tive f1. We clearly see that the strain beam an gle a changes rap idly
when com pared to fig ure 8, and the vari a tion in a is sig nif i cantly dif fer ent when com pared to fig ure 8, with a smaller value of the ec cen tric ity fac tor.
Fi nally, fig ure 10 pres ents plots of the beam-strain an gle a vari a tions at dif fer ent ec cen tric po si tions (x rang ing from 0.1 to 0.9) within the cath e ter lu men in the pres ence of a tilt an glef
1 = 10° and man u fac turer tilt an gle f
0 = 10°. Note that in con trast to fig ure 6, where the strain
beam an gle be comes zero at an az i muthal an gle of 180°, we ob serve a shift in the strain beaman gle to around 5°, with the strain beam an gle no lon ger sym met ri cal around the 180° point.
The re sults pre sented in this sec tion il lus trate the strain es ti ma tion er rors that will be in tro -duced into an elastogram with im proper po si tion ing of the cath e ter. How ever, if the cath e terpo si tion is known within the ves sel lu men, we can prop erly cor rect the er ror in the strain im -age and elim i nate the strain pro jec tion ar ti facts.
ER ROR PROP A GA TION
Er rors in the mea sure ment the ec cen tric ity and tilt an gle are in ev i ta ble. Thus, the prop a ga -tion of these er rors in Eq. (13) will also af fect the strain-beam an gle a. A the o ret i cal der i va -tion of the er ror prop a ga tion for the strain-beam an gle a is pos si ble; how ever, the ex pres sionsgen er ated are com pli cated. In this pa per, we nu mer i cally com pute the prop a ga tion of er rors
272 SHI ET AL
FIG. 8 Gen eral re la tion ship be tween beam-strain an gle (a) and az i muthal an gle (q) when the cath e ter is si mul ta -neously tilted and ec cen tric within the lu men for a range of tilt an gles (f
1 = ±10°). Ec cen tric ity pa ram e ter value x =
0.1 and man u fac turer tilt an gle f0 = 10°.
in the the o ret i cal der i va tion. Since we al ready have closed form ex pres sions, i.e., Eq. (13),er rors in tro duced due to in cor rect es ti ma tions of the ec cen tric ity and/or tilt an gle can be eval -u ated.
Ob serve from fig ures 6 and 10 that the strain-beam an gle curves orig i nate at 0° at an az i -muthal an gle of 0° and reach a max i mum at a 90° az i muthal an gle for dif fer ent ec cen tric po -si tions of the cath e ter. This im plies that any prop a gated er rors in the strain-beam an glewould be max i mized at 90° for dif fer ent ec cen tric po si tions of the cath e ter. In ad di tion, inthe pres ence of both ec cen tric ity and tilt er rors in the cath e ter po si tion, we ob serve an in -
COR REC TION FOR CATH E TER EC CEN TRIC ITY AND TILT IN IVUS ELASTOGRAPHY 273
FIG. 10 Gen eral re la tion ship be tween beam-strain an gle (a) and az i muthal an gle (q) when the trans ducer is si -mul ta neously tilted and ec cen tric within the lu men, plot ted for a range of ec cen tric po si tions of the trans ducer (x =0.1 to 0.9). Man u fac turer tilt an gle f
0 = 10° and cath e ter tilt an gle f
1 = 5°.
FIG. 9 Gen eral re la tion ship be tween beam-strain an gle (a) and az i muthal an gle (q) when the cath e ter is si mul ta -neously tilted and ec cen tric within the lu men for a range of tilt an gles (f
1 = ±10°). Ec cen tric ity pa ram e ter value x =
0.9 and man u fac turer tilt an gle f0 = 10°.
creased bias at a 180° az i muthal an gle in fig ure 10. We will there fore pres ent the er ror prop a -ga tion plots for these two az i muthal an gles. Fig ure 11 il lus trates the prop a gated strain- beaman gle er rors for ec cen tric ity mea sure ment er rors of Dx = ±0.1, and a tilt an gle f1 = 0° over arange of ec cen tric cath e ter po si tions. Here, the curve de notes the the o ret i cal strain-beam an -gle for dif fer ent ec cen tric ity val ues at az i muthal an gles q = 90° and 180°. The errorbar iscom puted for strain-beam an gle value cor re spond ing to ec cen tric ity x±Dx. Note for thesolid curve at an az i muthal an gle q =90° where a larger ec cen tric ity in tro duces largerstrain-beam an gles, sim i lar in cre ments of Dx=0.1 will in tro duce in creased prop a ga tion er -rors at larger ec cen tric i ties of the cath e ter. In ad di tion, for the same ec cen tric cath e ter po si -tion, ad di tional er rors that in crease the ec cen tric po si tion of the cath e ter (Dx = 0.1) in tro ducesig nif i cantly larger prop a ga tion er rors than for the case where the er rors re duce the over allec cen tric ity, i.e., for Dx = -0.1. Thus, the er ror bars ob tained are not sym met ric for in creased er rors in the ec cen tric ity in this case. How ever, for the case at an az i muthal an gle q =180°,de picted as the dashed curve, we ob serve no er ror bars due to mea sure ment er rors in the ec -cen tric ity. This is due to the ge om e try at 180°, where the ul tra sound beam and ra dius of theves sel are in the same plane, so the trans la tion of the cath e ter will not in tro duce anystrain-beam an gle changes. The strain-beam an gle is at a con stant value of 10° (the man u -fac turer built-in tilt an gle; oth er wise, the strain beam an gle would be fo = 0°).
In a sim i lar man ner, changes in the er ror prop a ga tion in the pres ence of a tilted cath e ter iseval u ated in fig ure 12, where f1 is set at 30° over the range of ec cen tric cath e ter po si tions.Note that the solid curve for q = 90° is iden ti cal to the case in fig ure 11, with small vari a tionsin the er ror bars. On the other hand, the dashed curve ob tained at q = 180° il lus trates the in -creased bias to a strain beam an gle of 40° (which equals f1 + fo = 30° (tilt an gle) + 10° (man -u fac turer tilt an gle) = 40°).
Er ror prop a ga tion in the pres ence of tilt er rors are shown in fig ures 13 and 14 for the sit u a -tion with the cath e ter at the lu men cen ter (Fig. 13) and when the cath e ter is ec cen tric withinthe lu men (Fig. 14), re spec tively. Fig ures 8 and 9 show that the prop a gated strain-beam an -gle er ror reaches a max i mum at a 180° (or 0°) az i muthal an gle for the dif fer ent tilt an gles. Inad di tion, we also pres ent the re sults at q = 90°, where the curves cross over at a strain beam
274 SHI ET AL
FIG. 11 Prop a gated strain-beam an gle er ror for ec cen tric ity mea sure ment er rors Dx = ±0.1 over a range of cath e -ter ec cen tric po si tions x = 0.1 to 0.9 for a tilt an gle f
1 = 0° and man u fac turer tilt an gle f
0 = 10°. The solid line is com -
puted at an az i muthal an gle q = 90° while the dashed line is ob tained for q =180°.
an gle of 10° (the man u fac turer tilt an gle) when the cath e ter is at the lu men cen ter. Fig ure 13pres ents the prop a gated strain-beam an gle er rors from er rors in the mea sure ment of the tiltan gle of Df1 = ±5°, at az i muthal an gles of q = 180° and 90°, with the cath e ter po si tioned at the lu men cen ter x = 0, over tilt an gles from 5° to 55°. Er ror bars are com puted for strain-beaman gle val ues cor re spond ing to tilt an gle vari a tions of f1±Df1. At an az i muthal an gle of 180°,the mea sure ment er ror of ±Df1
in tro duces sym met ric er ror bars as the strain-beam an gle er -rors in crease lin early with in creased tilt an gles, shown on fig ure 12 as the solid curve. Thedashed curve shows the strain-beam an gle vs. tilt an gle at az i muthal an gle q = 90°. Due to
COR REC TION FOR CATH E TER EC CEN TRIC ITY AND TILT IN IVUS ELASTOGRAPHY 275
FIG. 12 Prop a gated strain-beam an gle er ror for ec cen tric ity mea sure ment er rors Dx = ±0.1 over a range of cath e -ter ec cen tric po si tions x = 0.1~0.9 for a tilt an gle f
1 = 30° and man u fac turer tilt an gle f
0 = 10°. The solid line is com -
puted at an az i muthal an gle q = 90° while the dashed line is ob tained for q = 180°.
FIG. 13 Prop a gated strain-beam an gle er rors for tilt an gle mea sure ment er rors of Df1 = ±5° vs. tilt an gle with the
cath e ter at the cen ter of the lu men (x = 0) and man u fac turer tilt an gle f0 = 10°. The solid line is com puted at an az i -
muthal an gle q = 180° while the dashed line is ob tained for q = 90°.
the tilted cath e ter po si tion, the ul tra sound beam is not ex actly at 10° for an az i muthal an gle of 90°; in stead, the strain-beam an gle a starts from 10° at f1 = 0° and de clines with fur ther in -creases in the tilt an gle. How ever, this de crease is slow be cause the tilt an gle in tro duces verysmall changes on the beam di rec tion at q = 90°, lead ing to very small er rors caused by mea -sure ment er rors of the tilt an gle (Fig. 13).
Fi nally, fig ure 14 il lus trates the prop a gated strain-beam an gle er ror where the cath e ter isnow po si tioned ec cen tri cally within the lu men at x = 0.5. All other pa ram e ters are main -tained the same as in fig ure 13. Note that the solid curve for an az i muthal an gle 180° is al -most iden ti cal to the case shown in fig ure 13, while the dashed curve ob tained at q = 90°shows a bias about the strain beam an gle of 30° due to the ec cen tric po si tion of the cath e ter.Also, for the same rea son as in fig ure 13, the dashed line de creases slowly due to mea sure -ment er rors in the tilt of the cath e ter. The er ror prop a ga tion re sults shown in this sec tiondem on strate that Eq. (13) holds in the pres ence of large ec cen tric ity and tilt and can be ap -plied to com pute the strain-beam an gle and thus the strain cor rec tion nec es sary to re ducestrain pro jec tion ar ti facts.
DIS CUS SION AND CON CLU SION
Strain es ti mates ob tained in intravascular elastography are a func tion of the strain beaman gle as dem on strated in this pa per. Knowl edge of the cath e ter po si tion within the lu mencan be used to cor rect for the strain pro jec tion ar ti facts. Pre vi ous the o ret i cal ex pres sionseval u ated and cor rected for the ef fect of ei ther tilt or ec cen tric ity of the cath e ter in de pend -ently. We de rive a the o ret i cal ex pres sion that pro vides for cor rec tion of the strain pro jec tionar ti facts when the cath e ter is si mul ta neously tilted and ec cen tric within the cath e ter lu men.
The strain beam an gle has been de fined as the an gle a be tween ul tra sound beam and thera dial strain.19 If the strain-beam an gle were ex actly the same as the tilt an gle of the cath e ter,the strain-beam an gle could then be eas ily eval u ated us ing sim ple geo met ric re la tion ships.How ever, for si mul ta neous ec cen tric ity and tilt in the cath e ter po si tion, the strain-beam an -gle is no lon ger de scribed by a sim ple geo met ric re la tion ship due to the cou pling of the tiltand ec cen tric ity of the trans ducer.
The Z-axis in our der i va tion is along the lon gi tu di nal axes of the ves sel, while the X andY-axis is along the ra dial cross-sec tion of the ves sel, with the or i gin O at the cen ter of the ves -sel. For the closed form ex pres sions de rived in this pa per, we ini tially as sume that the trans -ducer is along the pos i tive X-axis, and thus the tilt an gle f1 is pos i tive. How ever, thetrans ducer can also be placed along the neg a tive X-axis, where for this case the tilt an gle f1
will be neg a tive. The ex pres sion de rived in this pa per can be ap plied un der those con di tionsalso.
Er ror prop a ga tion re sults pre sented in this pa per dem on strate that the er rors as so ci atedwith mea sure ment of the ec cen tric ity and tilt of the cath e ter vary sig nif i cantly with the az i -muthal an gle. For az i muthal an gles where the prop a ga tion of er rors are large (q = 90° for ec -cen tric ity er rors and q =180° for tilt er rors), small mea sure ment er rors in the ec cen tric ity ortilt of the cath e ter can lead to large er rors in the strain beam an gle for large ec cen tric i ties inthe cath e ter. In ad di tion, we ob serve that er rors that in crease the over all ec cen tric ity have alarger im pact than er rors that re duce the over all ec cen tric ity of the cath e ter.
Un der gen eral im ag ing con di tions, it is dif fi cult to con trol the po si tion of the cath e ter dur -ing car diac catheterization. Minimization of er rors as so ci ated with cath e ter po si tion there -fore re quires an other de vice or im ag ing mo dal ity that could pro vide in for ma tion re gard ingthe cath e ter po si tion.
276 SHI ET AL
AC KNOWL EDGE MENTS
The work was sup ported in part from a grant to the Uni ver sity of Wis con sin Med i calSchool un der the Howard Hughes Med i cal In sti tute Re search Re sources Pro gram for Med i -cal Schools, and start-up funds pro vided to Dr. Varghese by the De part ment of Med i calPhys ics, Med i cal School and Grad u ate School at the Uni ver sity of Wis con sin-Mad i son.
Ap pen dix A
For the tri an gu lar prism ABCE in fig ure 2, we have AB^BC, BE^EC, and AE^EC. De not -ing AC = y and AB = h and since AB^BC, we have:
For tri an gle BEC in fig ure 2, we have
In a sim i lar man ner, for tri an gle AEC in fig ure 2, we ob tain:
COR REC TION FOR CATH E TER EC CEN TRIC ITY AND TILT IN IVUS ELASTOGRAPHY 277
FIG. 14 Prop a gated strain-beam an gle er rors for tilt an gle mea sure ment er rors of Df1 = ±5° vs. tilt an gle with the
cath e ter po si tioned ec cen tri cally within the lu men (x = 0.5) and man u fac turer tilt an gle f0 = 10°. The solid line is
com puted at an az i muthal an gle q = 180° while the dashed line is ob tained for q = 90°.
22 hyBC -= (A-1)
BC
ECcos =3f
(A-2)
2
2
f
f
cosACEC
AC
ECcos
=
= (A-3)
Sub sti tut ing for EC from Eq. (A-3) into (A-2), we ob tain an ex pres sion for the an gle f3 as
fol lows:
Ap pen dix B
For the right-an gled tri an gle COP in fig ure 4, we have OP = a/tanf1 and CP = OP =a/sinf1. There fore, for the tri an gle HCO, we have:
Sim i larly in right-an gled tri an gle HOP,
In ad di tion, points P, C, A, Q, and H are in the same cath e ter-beam plane, ÐQCA = p/2-f0,ÐACH = f2, ÐHCP = d. We have the fol low ing an gu lar re la tion ship ÐQCA+ÐACH+ÐHCP= p, lead ing to:
In tri an gle CHP,
Solv ing for cos d, we ob tain
where
From Eqs. (B-4) and (B-7), we get
278 SHI ET AL
( ) ( )qqp cosacosaaOH +=--= 1222 2222(B-1)
qcosaOH 22 += (B-2)
( )1
22222 22 fq tanacosaOPOHHP ++=+= (B-3)
( )022
ffpd p ---= (B-4)
( ) dq fffcosacosa sin
asin
atan 1
2
12
22
12
12 222 -+=++ (B-5)
[ ]qfpdqfd cossinarccoscossincos11
-=Þ-= (B-6)
[ ]qfpd cossinarccos1
-= (B-7)
[ ]0212
fqff p +-= cossinarccos (B-8)
22
2
3
22223
hy
cosyarccos
hy
ycos
BC
ACcoscos
-=
-==
ff
fff (A-4)
Ap pen dix C
Sub sti tut ing Eq. (3) into Eq. (4) af ter ex pand ing for cos (q-f3) in Eq. (4), we ob tain:
Sub sti tut ing for h from Eq. (9) into Eq. (C-1), we ob tain:
Sim pli fy ing, we ob tain:
Gath er ing the terms as so ci ated with fac tors of y, we ob tain a qua dratic equa tion as fol -lows:
Equa tion (C-4) can be writ ten as:
where
COR REC TION FOR CATH E TER EC CEN TRIC ITY AND TILT IN IVUS ELASTOGRAPHY 279
÷÷
ø
ö
çç
è
æ
++=
1
22
2
2
22
fq
qffq
ctansin
sinsincoscosan
úú
û
ù
êê
ë
é
-
-+
--+-+=
22
2
2
22
22
2222222 2hy
hsinysin
hy
ycoscoshyahyar
fq
fq
(C-1)
÷÷÷÷÷
ø
ö
ççççç
è
æ
÷÷ø
öççè
æ
+-+
++
-+
=
1
22
1
2
2
22
2
1
22
1
2
2
22
22
2
12
2
fq
ffq
fqfq
ff
ctansin
ctansinysina
coscosayctansin
ctansinyya
r
(C-2)
1
22
2
2
2
1
22
1
2
2
2
2222 22
fq
qffq
fq
ff
ctansin
sinsinaycoscosay
ctansin
ctansinyyar
+++
+-+=
(C-3)
( ) 02 22
1
22
2
2
2
1
22
1
2
2
22
2 =-+÷÷
ø
ö
çç
è
æ
++×+
+
+ra
ctansin
sinsincoscosay
ctansin
ctancossiny
fq
qffq
fq
ffq (C-4)
02 =++ knymy (C-5)
1
2
1
2
2
22
fq
ffq
ctansin
ctancossinm
2 +
+=
(C-6)
( )22 rak -=
Solv ing for Eq. (C-5) in terms of y we ob tain:
Equa tion (C-7) is ob tained af ter dis card ing the neg a tive root, be cause y>0.
Ap pen dix D
If f1 = 0, we ob tain from Eq. (5),
Equa tion (11) now re duces to:
If we set f0 = 0, f
1 = 0, then from Eq. (D-1), f
2 = 0 and Eq. (D-2) re duces to
Sub sti tut ing Eq. (D-3) into Eq. (10), we ob tain:
Equa tion (13) can be writ ten as fol lows, by ex pand ing the term in both the nu mer a tor andde nom i na tor:
280 SHI ET AL
[ ]0202202
0 fffqf ppp =-+=-+= cossinarccos o (D-1)
0
2
2
2
1
2
1
2
2
22
1
2
fffqf
ffqfcoscos
cossinsin
coscossinsinm
2
1
2==
+
+=
(D-2)
( )22 rak -=
1=m (D-3)
qq cosasinarm
mknny --=
-+-= 222
2
2
4 (D-4)
qcosan 2=
( )0
1
2
1
22
1
22
2
222 fq
ffq
fqffq coscosa
cossinsin
sinsinsincoscosan =
÷÷
ø
ö
çç
è
æ
++=
( )22 rak -=
( ) ( )ïþ
ïýü
ïî
ïíì
úû
ùêë
é
+
++-=
1
2
1
2
1
2
2
22
1
2222
21
fqf
ffqfqa
cossinsin
coscossinsinyararccos
2yr
(D-5)
m
mknny
2
42 -+-=
(C-7)
Again if we set f0 = 0°, f
1 = 0° and f
2 = 0° (from Eq. (D-1)), we ob tain:
Sub sti tut ing for y from Eq. (D-4), we ob tain:
Sim pli fy ing, the strain beam an gle, when the cath e ter is ec cen tric but par al lel to the ves selaxis, is given by:
which is Eq. (16) in the main body of the manu script.
Ap pen dix E
The ex pres sions for the strain beam an gle a is shown in Eqs. (16) and (17), re spec tively.To prove that both of these ex pres sions are equiv a lent, we use the fol low ing iden tity. Gen er -ally, if we have ex pres sions of the form a = arccos A (Eq. (16)) and a = arcsin B (Eq. (17)),then if A2+B2=1, the ex pres sions for the strain beam an gle are equiv a lent. So, if we de fine:
Sub sti tut ing for A and B and eval u at ing the func tion A2+B2 ,we have:
Where x = a/r. Sub sti tut ing for x and ex pand ing Eq. (E-2), we ob tain:
COR REC TION FOR CATH E TER EC CEN TRIC ITY AND TILT IN IVUS ELASTOGRAPHY 281
( ) ( ) úúû
ù
êêë
é
--
---=
qqqqa
cosasinarr
cossinarasinararccos
222
222222 (D-8)
( ) ( )qpxqq
qqq-=
--
---= sinB,
cosasinarr
cossinarasinarA
222
222222 (E-1)
( ) qxqxqx
qqxxqx22
2
222
222222
22
1
11sin
cossin
cossinsinBA +
úú
û
ù
êê
ë
é
--
---=+
(E-2)
( ) ( ) ( )( ) ïþ
ïýü
ïî
ïíì
--
--+-=
qqqa
cosasinarr
cosasinarararccos
222
222222
2
(D-7)
( ) ( )[ ]{ }222
21 yararccos yr +-=qa (D-6)
( )
( )qqxxqxqx
qqxxqxqxqx
qqqxxqqxx
qxqqxxqx
cossincossin
cossincossinsin
cossinsincossin
sincossinsin
BA222222
22222222
2223222
22222244
22
121
121
1212
211
--+-
÷÷÷÷
ø
ö
çççç
è
æ
--+-+
-+--
--++
=+
(E-3)
Sim pli fy ing the ex pres sion in the nu mer a tor we ob tain:
or A2+B2=1, since the nu mer a tor is the same as the de nom i na tor, thus prov ing that Eq. (16)de rived in this pa per and Eq. (17) ob tained from deKorte et al19 are equiv a lent.
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282 SHI ET AL
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COR REC TION FOR CATH E TER EC CEN TRIC ITY AND TILT IN IVUS ELASTOGRAPHY 283
