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Can the modified Allen's test always detect sufficient collateral flow in the hand? A computational study

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Can the modiﬁed Allen’s test always detect sufﬁcientcollateral ﬂow in the hand? A computational study
J. ALASTRUEY†‡, K. H. PARKER‡, J. PEIRO´† and S. J. SHERWIN†*
†Department of Aeronautics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK ‡Department of Bioengineering, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
(Received 10 January 2006; in ﬁnal form 15 August 2006)
Blood ﬂow in the largest arteries of the arm up to the digital arteries is numerically modelled using theone-dimensional equations of pressure and ﬂow wave propagation in compliant vessels. The model canbe applied to different anatomies of arterial networks and can simulate compression of arteries, theseallowing us to simulate the modiﬁed Allen’s test (MAT) and to assess its suitability for the detection of sufﬁcient collateral ﬂow in the hand if radial blood supply is interrupted. The test measures blood ﬂowin the superﬁcial palmar arch before and during compression of the radial artery. The absence of reversal ﬂow in the palmar arch with the compression indicates insufﬁcient collateral ﬂow and isreferred to as a positive MAT. This study shows that small calibres of the superﬁcial palmar arch andinsufﬁcient compression of the radial artery can lead to false-positive results. Measurement of the dropin digital systolic pressures with compression of the radial artery has proved to be a more sensitive testto predict the presence of sufﬁcient ulnar collateral ﬂow in networks with small calibres of thesuperﬁcial palmar arch. However, this study also shows that digital pressure measurements can fail indetecting enough collateral ﬂow if the radial artery is insufﬁciently compressed.
Keywords
: Modiﬁed Allen’s test; Hand haemodynamics; Collateral ﬂow; One-dimensional model
1. Introduction
Thehandisperfusedbytheulnarartery(UA)andtheradialartery (RA), both of which branchoff of the brachial artery(BA) usually below the elbow (ﬁgure 1, left). The UA andthe RA anastomose across the hand in the form of asuperﬁcial palmar arch (SPA) and a deep palmar arch(DPA) that provide collateral ﬂow pathways to the digitalarteries,the SPA being the mostimportant route. However,arterial anatomical variations are frequent, with anincomplete SPA reported in approximately 30–40%of subjects (Lippert and Pabst 1985, Olave
et al.
1993,Moraes
et al.
2003).
It is of clinical interest to determine the presence of sufﬁcient ulnar collateral ﬂow through the SPA or the DPAto assess whether the RA can be harvested to be used as acoronary artery bypass graft without ischemic compli-cations (insufﬁcient blood supply) to the hand, particularlythe thumb and index ﬁnger (Kamienski and Barnes 1976,Pola
et al.
1996, Serricchio
et al.
1999, Starnes
et al.
1999,Winterer
et al.
2001, Zimmerman
et al.
2001, Broadman
et al.
2002, Zhen
et al.
2002). Furthermore, the RA is alsoa favourable access for haemodialysis if ulnar collateralﬂow is present (Kapoian
et al.
1999). The modiﬁedAllen’s test (MAT) is a common technique to detect thepresence of sufﬁcient ulnar collateral ﬂow. It consists of measuring blood ﬂow in the radial part of the SPA (SPAIII) by Doppler ultrasonography (US) before and duringcompression of the RA at the wrist. The presence of reversal ﬂow (towards the RA) is considered to be anindicator of adequate collateral ﬂow across the palm, andis identiﬁed as a negative MAT.
Several authors have studied the reliability of the MAT.Studies by Pola
et al.
(1996), Serricchio
et al.
(1999) andZimmerman
et al.
(2001) showed that all the follow-uppatients who underwent harvesting of the RA after anegative MAT did not present signs of hand ischemia.However, a study with 129 patients by Starnes
et al.
(1999) showed that the test may unnecessarily excludesome patients for RA graft or RA access for haemodyalisis(false-positive) and may also place a number of patients atrisk of hand ischemia (false-negative). Kamienskiand Barnes (1976) reported a 73% false-positive rate of the MAT.
Computer Methods in Biomechanics and Biomedical Engineering
ISSN 1025-5842 print/ISSN 1476-8259 online
q
2006 Taylor & Francishttp://www.tandf.co.uk/journalsDOI: 10.1080/10255840600985477
*Corresponding author. Email: s.sherwin@imperial.ac.uk
Computer Methods in Biomechanics and Biomedical Engineering
,Vol. 9, No. 6, December 2006, 353–361
In this work, we apply a previously published (Sherwin
et al.
2003) non-linear, one-dimensional (1-D) model of pressure and ﬂow pulse wave propagation in compliantvessels to simulate the MAT in the three most prevalentarterial anatomies formed by the largest arteries in the arm(ﬁgure 1, right). We have a network without either acomplete SPA or DPA that we call
Model 0
, a network with a complete SPA that we call
Model 1
and a network with both a complete SPA and DPA that we call
Model 2
.We focus our study on analysing if small calibres of themiddle SPA (SPA II) and insufﬁcient compression of theRA are responsible for false-positive or false-negativeresults of the MAT. When a false result occurs, we analyseif the alternative technique used by Starnes
et al.
(1999)can assess the presence of sufﬁcient ulnar collateral ﬂowbetter. This technique consists of measuring the change inthe systolic pressure of the digital arteries III and IV withthe compression of the RA. Pressures are measured with adigital pressure cuff placed on the proximal phalanx. Adecrease in digital systolic pressure of 40mmHg or less isconsidered to be an indicator of sufﬁcient collateral ﬂow.
2. Methodology
2.1 Mathematical model
2.1.1 Governing equations
. The mathematical model isbased on the non-linear, 1-D equations of pressure andﬂow wave propagation in compliant vessels. Thegoverning system of equations results from conservationof mass and momentum applied to a 1-D impermeable anddeformable tubular control volume of incompressible andNewtonian ﬂuid. It takes the form (Sherwin
et al.
2003,Alastruey 2006):
›
A
›
t
þ
›
ð
AU
Þ
›
x
¼
0
;
ð
1
Þ
›
U
›
t
þ
U
›
U
›
x
¼
2
1
r
›
p
›
x
þ
f
r
A
;
ð
2
Þ
where
x
is the axial coordinate along the vessel,
t
is thetime,
A
ð
x
;
t
Þ
is the cross-sectional area of the vessel,
U
ð
x
;
t
Þ
is the average axial velocity,
p
ð
x
;
t
Þ
is the average internalpressure over the cross-section,
r
is the density of theblood taken here to be 1050kg m
2
3
, and
f
ð
x
;
t
Þ
is thefriction force per unit length, which is modelled as
f
¼
2
22
p
m
U
according to Smith
et al.
(2001), with
m
theviscosity of the blood taken here to be 4.5mPa s.
The system of equations is completed with a pressure-area relation previously used in Olufsen (1999),Sherwin
et al.
(2003) and Alastruey (2006). It assumes athin, homogeneous and elastic arterial wall and it takes theform:
p
¼
p
0
þ
b
A
0
ð
ﬃﬃﬃ
A
p
2
ﬃﬃﬃﬃﬃ
A
0
p
Þ
;
b
¼
ﬃﬃﬃﬃ
p
p
hE
ð
1
2
s
2
Þ
;
ð
3
Þ
where
A
0
and
h
are the sectional area and wall thickness atthe reference state
ð
p
0
;
U
0
Þ
(with
p
0
and
U
0
assumed to bezero),
E
is the Young’s modulus, and
s
is the Poisson’sratio, typically taken to be
s
¼
1
=
2 since biological tissueis practically incompressible. The parameter
b
is related to
Figure 1. Anatomy of the largest arteries in the arm (left). Schematic of the three anatomically different models studied (right). They contain thebrachial artery (BA), the ulnar artery (UA), the radial artery (RA), the superﬁcial palmar arch (SPA I, SPA II and SPA III), the deep palmar arch (DPA I,DPA II and DPA III), and the digital arteries (DIG I, DIG II, DIG III and DIG IV). The thumb is perfused by DIG IV.
J. Alastruey
et al.354
the speed of pulse wave propagation,
c
, through (Sherwin
et al.
2003, Alastruey 2006):
c
2
¼
b
2
r
A
0
A
1
=
2
:
ð
4
Þ
2.1.2 Treatment of boundary conditions andbifurcations
. The hyperbolic system of partialdifferential equations (1)–(3) is solved in each arterialsegment of the networks shown in ﬁgure 1 (right) with thefollowing boundary conditions. At the proximal end of theBA we prescribe the ﬂow time history shown in ﬁgure 2(right), measured with Doppler US at the left subclavianartery of a healthy young adult. Boundary conditions of the arterial segments joining at junctions are prescribed byenforcing conservation of mass and continuity of the totalpressure
p
þ ð
1
=
2
Þ
r
U
2
. The distal end of each digitalartery is coupled to a three-element lumped parametermodel (ﬁgure 3), consisting of two resistances,
R
1
and
R
2
,and a compliance,
C
, to account for both the resistive andthe compliant effects of vessels and microcirculationbeyond the distal boundary. It is governed by thedifferential equation
p
1D
þ
R
2
C
d
p
1D
d
t
¼
p
v
þ ð
R
1
þ
R
2
Þ
Q
1D
þ
R
1
R
2
C
d
Q
1D
d
t
;
ð
5
Þ
where
p
1D
and
Q
1D
are the pressure and volume ﬂow rateat the distal end of the 1-D digital artery, and
p
v
representsthe pressure at the entrance of the venous system, taken tobe zero.
At any point in the network we can simulatea compression leading to total occlusion by enforcingthe condition
U
¼
0 at the compression point, and acompression leading to partial occlusion by decreasing thecalibre of the artery locally (Section 3.3).
2.1.3 Numerical scheme
. Equations (1) and (2) aresolved for area and velocity after substitution of equation(3) into equation (2) to express the term
›
p
=
›
x
as
›
A
=
›
x
(Sherwin
et al.
2003, Alastruey 2006). A discontinuousGalerkin scheme with a spectral/
hp
spatial discretisationand a second-order Adams–Bashforth time-integrationscheme is used. Details on this algorithm can be found inSherwin
et al.
(2003) and Karniadakis and Sherwin(2003).
Equation (5) is solved using a ﬁrst-order timediscretisation, taking into account that
p
1D
and
Q
1D
canbe related to
A
1
D
and
U
1
D
(their corresponding
A
and
U
values at the outlet of each 1-D digital artery) throughequation (3) and
Q
1D
¼
A
1D
U
1D
. This lumped parametermodel is coupled to the 1-D digital arteries by solving aRiemann problem at their interface.
The accuracy of our 1-D formulation and the three-element lumped parameter model has been successfullyvalidated in Alastruey (2006).
2.2 Physiological data
The physiological parameters used in each arterialnetwork are given in the tables of Appendix A. They arereferred to as the control case data. We aim at representingthe arterial geometry and elasticity of healthy young adultarms. In the absence of detailed knowledge of theseproperties, the lengths,
l
, and initial radii,
R
0
, of the BA,UA and RA are based on data published in Stergiopulos
Figure 2.
In vivo
pressure (left), measured with planar tonometry, and velocity (right), measured with Doppler US, over one cardiac cycle at theproximal part of the left subclavian artery.
Figure 3. Electrical circuit analogous to the three-element lumpedparameter model that simulates the resistance and compliance of vesselsand microcirculation beyond the 1-D digital arteries.
Computational assessment of the modiﬁed Allen’s test
355
et al.
(1992). The remaining lengths are estimated from ananatomical atlas (Martini 1995), whereas the remaining
R
0
’s are assumed to have an area ratio at bifurcations(ratio of the total cross-sectional area of the daughtervessels to that of the parent vessels) close to 1.2. Thisvalue yields minimal wave reﬂection at a bifurcation forforward travelling waves (Gosling
et al.
1971, Greenwaldand Newman 1982, Papageorgiou and Jones 1987). Thewall thickness of each artery,
h
, is assumed to be 10% of
R
0
. We determine
E
in each vessel by enforcing therelation
ð
A
0
1
=
c
0
1
Þ ¼ ð
A
0
2
=
c
0
2
Þ þ ð
A
0
3
=
c
0
3
Þ
at each junc-tion, where
A
0
1
and
c
0
1
are, respectively, the initial areaand wave speed of the parent artery, and
A
0
2
,
c
0
2
,
A
0
3
and
c
0
3
are the corresponding parameters in the two daughtervessels. This is equivalent, under the assumption of smallperturbations, to a well-matched bifurcation for forwardtravelling waves (Sherwin
et al.
2003, Alastruey 2006).
At the distal end of each digital artery, we assume
R
1
tobe equal to the characteristic impedance of the terminalbranch,
Z
0
¼ ð
r
c
0
=
A
0
Þ
with
c
0
¼
c
ð
A
0
Þ
(4), since it is thevalue that minimizes the total impedance of the
RCR
lumped parameter model (Rainest
et al.
1974). Eachperipheral resistance
R
p
¼
R
1
þ
R
2
is obtained bydistributing the resistance of the entire network,
R
N
, sothat the mean outﬂow rates over one cardiac cycle in thedigital arteries are proportional to the corresponding initialcross-sectional areas. The value of
R
N
is calculated as theratio between the mean pressure shown in ﬁgure 2 (left)and the mean ﬂow rate measured
in vivo
at the proximalpart of the left subclavian artery. The mean ﬂow rate isdetermined by multiplying the velocity shown in ﬁgure 2(right) by the cross-sectional area calculated fromthe pressure in ﬁgure 2 (left) through equation (3). Thevalue
R
N
¼
1
:
41
£
10
10
Pasm
2
3
(106mmHgsml
2
1
) isobtained.
The total compliance of each network,
C
N
, isdetermined by ﬁtting an exponential function during thelast third of diastole in the
in vivo
pressure of ﬁgure 2(left). According to Wang
et al.
(2003), the time constantof the exponential decay is
R
N
C
N
, which leads to
C
N
¼
2
:
02
£
10
2
10
m
3
Pa
2
1
(0.027mlmmHg
2
1
). The totalperipheral compliance is the difference between
C
N
andthe compliance of each artery in the network, calculatedthrough
A
0
l
=
r
c
20
(Milisˇic´ and Quarteroni 2004). The totalperipheral compliance is distributed among the digitalarteries in proportion to their initial cross-sectional areas.
3. Results and discussion
In Section 3.1 we simulate the MAT in the models of ﬁgure 1 (right) with the data shown in Appendix A, and weanalyse the effect of harvesting the RA on digital outﬂowrates. We also study the effect of changing the location of the compression along the RA on the reversal ﬂow peak measured by the MAT. Next, we show how a reduction inthe calibre of the middle SPA (Section 3.2) and aninsufﬁcient compression of the RA (Section 3.3) affect theresult of the MAT.For each simulation we can calculate pressure and ﬂowtime histories at any site in the arterial network. Weinitially assume
A
ð
x
;
0
Þ ¼
A
0
and
U
ð
x
;
0
Þ ¼
0
everywherein the network, and we run each simulation for sufﬁcientcardiac cycles until the waveforms become time-periodic.This typically takes around 10 cycles.
3.1 Control case
Figure 4 compares the velocity in the middle point of theradial part of the SPA (SPA III) of each model before (left)and during (right) compression of the RA 15cm proximalto the brachial bifurcation (approximately at the level of the standard pressure point in the wrist). Differencesbetween the three anatomical variations are minimalbefore compression, with ﬂow coming from the RA andmoving towards the central segment of the SPA (SPA II),except for a small time interval at the end of systole. If acompression leading to total occlusion is simulated, ﬂowbecomes zero in
Model 0
, whereas ﬂow reversesremarkably in
Models 1
and
2
, because it comes fromthe UA through the SPA. These results are in reasonableagreement with the
in vivo
Doppler US measurementspublished by Zimmerman
et al.
(2001). The simulationsalso indicate that the reversal ﬂow peak increases in
Models 1
and
2
as the compression point in the RA ismoved away from the hand (ﬁgure 5). Hence, the peak of
Figure 4. Velocity time histories in the middle point of the SPA III of each model before (left) and during (right) compression of the RA at the wrist.
J. Alastruey
et al.356
the ﬂow signal can be better detected by the MAT if compression is applied as far as possible from the hand.
To assess the potential risk of hand ischemia we analysethe effect of removing the RA from
Models 1
and
2
onblood supply to the ﬁngers. Figure 6 shows ﬂow rate overone cardiac cycle at the outlets of the digital arteries of
Models 1
(top) and
2
(bottom), when the RA is present(solid lines) and when it has been harvested (dashed lines).The results in all the digital arteries before and afterharvesting do not differ signiﬁcantly. Mean ﬂow rateschange less than 1% in all the digital arteries of
Model 1
,and less than 7% in all the digital arteries of
Model 2
,which suggests that changes in digital mean ﬂow rates aresmall enough for the peripheral arteries to compensate forthem with vasoconstriction (increase in peripheralresistance) in the digital arteries I and II and vasodilation(reduction in peripheral resistance) in the digital arteriesIII and IV. We also notice that the presence of the DPA in
Model 2
does not introduce signiﬁcant changes in thedigital ﬂow rates predicted in
Model 1
.
Figure 6 also compares the collateral ﬂow rates throughthe SPA II of
Models 1
and
2
. If the RA is absent, the meancollateral ﬂow towards the digital arteries III and IV isapproximately nine times higher in
Model 1
, and seventimes higher in
Model 2
. Mean ﬂow through the DPA II of
Models 2
does not increase signiﬁcantly if the RA isremoved, which suggests that collateral ﬂow takes placemainly through the SPA. Finally, we note that some ulnarcollateral ﬂow towards the digital arteries III and IV isobserved in both models when the RA is present.
3.2 Effect of reducing the calibre of the SPA
This section studies the effect of reducing the calibre of the central SPA (SPA II) of
Models 1
and
2
on the peak of the reversal ﬂow measured at the SPA III during occlusionof the RA, and on the mean outﬂows in the digital arteriesIII and IV after harvesting the RA. We reduce the calibreof the SPA II in the control case, referred to as 100%, insteps of 10% until it becomes zero.
Figure 7 (left) shows that
Model 1
presents a higherdecrease in the peak of the reversal ﬂow measured bythe MAT as we decrease the calibre of the SPA II thanthe corresponding decreases in the mean digital outﬂowsafter harvesting the RA. If the calibre of the SPA II is 30%the value of the control case, the peak reversal ﬂow isreduced up to 80% and might be too small to be detectedby Doppler US, resulting in a positive MAT. However, themean outﬂow rates in the digital arteries III and IV arereduced by less than 20% when the RA is absent.Although the MAT predicts insufﬁcient collateral ﬂow,various compensatory mechanisms, such as vasodilatationof the digital arteries III and IV and progressive increase inthe calibre of the SPA and the UA once the RA isharvested (Broadman
et al.
2002, Zhen
et al.
2002), arelikely to increase collateral circulation to avoid handischemia. Therefore, the MAT is likely to produce a false-positive result. A calibre of the SPA II of 30% the value of the control case is approximately a third of the calibre of the digital artery IV. For calibres smaller than 30%, thereductions in mean digital outﬂows are too high and likelyto lead to hand ischemia.
Figure 7 (right) shows that the DPA of
Model 2
ensuressufﬁcient collateral ﬂow to perfuse the digital arteries IIIand IV even if the SPA II is removed, in which case themean outﬂow rates of the digital arteries III and IVdecrease by less than 15%. However, the reversal ﬂowpeak decreases until it becomes almost zero when thecalibre of the SPA II is between 50 and 40% the value of the control case, which is smaller than the calibre of theDPA II. For calibres of the SPA II smaller than 40%,the ﬂow peak changes direction, since blood ﬂows to thedigital arteries III and IV mainly through the DPA.According to these results, the MAT clearly leads to afalse-positive result for a wide range of calibres of theSPA II in
Model 2
.
Ifwemeasurethedecreaseinthesystolic pressureofthedigital arteries III and IVof
Model 1
with the compressionof the RA (as proposed by Starnes
et al.
(1999)), weobtainless than 35mmHg when the calibre of the SPA II is30% the calibre of the control case. For calibres smallerthan 25%, we obtain decreases in digital systolic pressureshigher than 40mmHg. When this test is applied to
Model2
,thedropindigitalsystolicpressureissmallerthan
Figure5. Effectofmovingthe locationofcompressionalongthe RAon theﬂowmeasured in themiddlepointofthe SPAIII.Both
Models1
(left)and
2
(right) are depicted. Compression is applied at the wrist, 5cm distal (
2
5cm) and proximal (
þ
5cm) to the hand.
Computational assessment of the modiﬁed Allen’s test
357

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