1. Biomerhmics Vol 21. No Printed m Great Britain
2. pp
107-113.
,X,2, -9290’88
1988.
$3.00 +
00
6’) 1988 Pergamon Journals Ltd.
BIOMECHANICS OF THE ARTERIAL WALL UNDER SIMULATED FLOW CONDITIONS ARTHUR M. BRANT, SANJAY S. SHAH, VICTOR G. J. RODGERS, JEFFREY HOFFMEISTER, IRA M. HERMAN,* ROBERT L. KORMOS and HARVEY S. BOROVETZ~ Department of Surgery, University of Pittsburgh School of Medicine, Pittsburgh, PA 15261, U.S.A. Abstract-A perfusion apparatus is employed to reproduce quantifiable pulsatile hemodynamics within freshly excised canine carotid arteries. From measurements of pulsatile intraluminal and transmural pressure and the dynamic radial motion of the vessel wall, calculations are made of the vascular incremental modulus of elasticity and hoop, axial, and radial wall stresses. The results of this investigation suggest that an increase in transmural pressure from 120/80 to 240/120 mmHg produces a marked elevation in incremental modulus and arterial wall stress. These parameters are reduced when transmural pressure is lowered while maintaining intraluminal pressure at physiologic values.
LISTOFSYMBOLS E h
mc
LID Pl P2 P3 Ri
RO R 01 R a2 R 03
R”
S ;; TP, TP, Txx T00 r t P 0
AR R
Incremental modulus of elasticity Wall thickness Ratio of vessel length to vessel diameter Proximal perfusion pressure (Fig. 1) Distal perfusion pressure (Fig. 1) Adventitial fluid pressure (Fig. 1) Inner radius Outer radius Minimum (diastolic) outer radius [equation (2)] Mean outer radius [equation (2)] Maximum (systolic) outer radius [equation (2)] Reference vessel midwall radius (Fig. . _ 2) Radial stress Transmural pressure Minimum (diastolic) transmural pressure [equation (2)] Maximum (systolic) transmural pressure [equation (2) 1 Longitudinal (axial) stress Circumferential (hoop) stress Radial fluid velocity component Radial displacement component of the vessel wall (Fig. 2) Dynamic fluid viscosity Poisson ratio Ratio of the change in vessel radius to the Perfused radius INTRODUCTION
There is considerable evidence that the rate of passage of macromolecules (lipoproteins) from the arterial lumen into the wall substance is affected by fluid mechanical factors. The pioneering work of Fry (1968, 1977, 1981) demonstrates the importance of such parameters as protein concentration, intimal surface exposure time, shear rate and arterial wall circumferen-
Received January 1986; in revised form March 1987. *Department of Anatomy/Cell Biology, Tufts University School of Medicine, Boston, MA 02111, U.S.A. t Correspondence address: Department of Surgery, University of Pittsburgh, School of Medicine, 952 Scaife Hall, Pittsburgh, PA 15261, U.S.A.
tial stretch on transvascular transport processes and atherogenesis. Based upon Fry’s early demonstrations of the role of fluid shearing stress subsequent investigations have been performed to further elucidate this hypothesis. Research includes measurement of the cell biologic response to fluid shearing stresses (Dewey et al., 198 1; Frangos et al., 1985), correlation of lesion position with measurements of shearing rate in rigid model casts of the human vasculature (Friedman et nl., 1981; LoGerfo et al., 198 I; Zarins et al., 1983), the quantitative determination of the uptake of labelled macromolecules by the arterial wall as a function of shearing rate (Caro and Nerem, 1973; Caro, 1974), and the investigation of the role of blood shear rate at the vessel wall on platelet adhesion, thrombus growth, and platelet and fibrin involvement in thrombogenesis (Baumgartner, 1973; Baumgartner and Sakariassen, 1985: Turitto et d., 1980). Considerably less attention has been directed toward quantitating the relationship between the mechanical properties of the vessel wall and arterial disease. A review of the literature indicates that Bergel (1961) and Peterson et a!. (1960) are among the first to study the viscoelastic properties of intact canine vessels (by measuring the dynamic pressure-radius relationships at frequencies up to 20 Hz). Furthering of current understanding is provided by Pate1 and Fry (1969) who studied elastic symmetry in the middle descending thoracic aorta, abdominal aorta, and left common carotid artery in dogs under physiologic ranges of static loading. Based upon their measurements it is concluded that the canine arterial tree possesses elastic properties that are nearly symmetrical, under conditions of physiologic loading, about the planes perpendicular to principal stresses. The implication is that the canine artery can be modelled as a cylindrical elastic tube, an assumption employed in the present investigation. More recently Gow and colleagues (1974) excised the canine left circumflex coronary artery (LCCA) and 107
108
A. M. BRANT~ al.
attached the vessel to an in vitro apparatus for determination of vascular static and dynamic elastic properties. The latter are derived from pressure-diameter measurements according to the expressions of Bergel (1961). Their results are the first to demonstrate that the LCCA possesses static and dynamic elastic properties consistent with those of others muscular arteries. A novel noncontacting method for analysis of vascular elasticity has previously been described by Fronek et al. (1976). Their method consists of a closedcircuit TV system in conjunction with a video dimension analyzer which tracks continuously changes in longitudinal and circumferential strains as the intraluminal pressure is varied. Static determinations of the incremental modulus of elasticity of rabbit arteries are in reasonable agreement with the data of Bergel(1961), Pate1 and Janicki (1970) and others obtained from dog vessels. A unique advantage of their noncontacting system is that there is no possibility of deformation due to sensor-induced friction of the arterial specimen. The present study has as its principal aim the measurement and calculation of the biomechanical stresses acting upon freshly excised canine carotid arteries perfused under various hemodynamic conditions. A perfusion apparatus (Brant et al., 1986) is employed to simulate quantifiable and repeatable pulsatile hemodynamics in vitro. The effects of individual flow parameters, i.e. intraluminal and transmural pressure, pulse pressure and rate of flow upon arterial wall biomechanics (incremental modulus of elasticity, hoop, axial, and radial wall stresses) are derived from simultaneous dynamic measurements of these parameters and changes in vessel radius. The latter are obtained in a noncontacting fashion using a scanning helium-neon laser micrometer (Brant et al., 1986b).
EXPERIMENTAL
The perfusion apparatus used to acquire the present experimental data is described by Brant et al. (1986a). Briefly this systein simulates the natural (in oioo) environment of an arterial segment, providing pulsatile pressures, pH, PO,, and pCOt, temperature, pulse frequency, and fluid flow through a freshly excised securely tethered canine carotid artery. ’ Figure 1 shows an overview of a typical perfused vessel and associated instrumentation. Canine carotid arteries, excised in accordance with NIH and University guidelines for the care and use of laboratory animals, serve as the test vessels in these experiments. The arteries are carefully secured at both ends in the perfusion apparatus with suture material (Ethicon, Inc., Somerville, NJ; 000 silk) and maintained at their natural length (10 cm) during in vitro perfusion. Temperatures are measured using a thermistor thermometer (Omega Engineering, Inc., Stanford, CT, Model 747 Meter and OL-710PP probes). Volumetric arterial flow rates are determined using electromagnetic flow meters and probes (Carolina Medical Electronics, King, NC, Model 501 Meter and EP3OOA probes). Simultaneous dynamic pressure measurements are acquired using strain gauge pressure catheters (Millar Instruments, Inc., Houston, TX; Model TCB-100 Control Unit and PC-360 probes). The viscosity and density of the perfusate (canine serum) are measured using a Ubbelohde viscometer (Schott America, Yonkers, NY; Ubbelohde Model 24525-03 (ASTM)) and pycnometer (Fisher Scientific Co., Pittsburgh, PA; Model 03-247). Finally, the dynamic variation in the outer diameter of the perfused carotid is followed using a noncontacting scanning laser micrometer (Techmet Company, Dayton, Ohio;
TISSUE HOUSING CHAMBER AND ASSOCIATED IN-VITRO INSTRUMENTATION
Fig. 1. Tissue housing chamber and associated instrumentation. Nomenclature: T = temperature; F = flow rate; P = pressure; 1= proximal; 2 = distal; 3 = adventitial.
109
Biomechanics of the arterial wall Table 1. Hemodynamic simulations in oitro. Nomenclature:
“STRESSED”
Pl, proximal perfusion pressure; Q, mean rate of flow
Normotension Increased TP Decreased TP Hypertension High flow-hypertension
Pl (mmHg)
& (cc/min)
120180 115/85 115/85 240/120 240/120
150 150 150 150 300
AND “REFERENCE” OF ARTERIAL WALL
POSITION
I
accuracy Model No. 50-03) whose inherent (0.0125 mm), ease of operation, and repeatability (standard error of measurement * 0.25 % for a vessel of nominal diameter = 5 mm) make it especially well suited for the present biomechanics application (Brant et al., 1986b). Table 1 documents the various flow conditions which are simulated in the present study. Advantage is taken of the fact that individual hemodynamic parameters can be varied on a one-by-one basis and the biomechanical response of the arterial wall studied in detail. For example, the ‘pulsatility’ of a perfused vessel is altered by either increasing the pulse pressure or varying the adventitial fluid pressure (P3 in Fig. 1) in the tissue housing chamber of Fig. 1. This latter maneuver did not influence the perfusate pressures (Pl, P2 of Fig. 1) but rather modified the calculated transmural pressure, TP. defined as: TP = (Pl + P2)/2 - P3.
(1)
MODEL OF VESSEL WALL RHEOLOGY Incremental
modulus of elasticity
The design properties of the perfusion apparatus allow for controlled hemodynamic parameters to be applied to excised canine vessels. The experimentally derived data of pulsating radius, wall thickness, pressure drop and transmural pressure serve as input to model descriptions of the biomechanics of pulsatile vessel wall motion. Following the classical work of Pate1 and Fry (1969), the present calculations of elasticity modulus are based upon the assumptions of cylindrical geometry and an elastic vessel wall. However, several experimental modifications are incorporated in the present analysis. The reference midwall radius R, (Fig. 2) is defined as the minimum value over the cycle. At rest, this value would correspond to the situation in which net transmural wall force is zero. However, in a physiologic system this situation never occurs. Accordingly an alternative definition is employed based upon the work of Bergel (1961). He noted that although the modulus of elasticity, which is calculated based on the value of R,, changes significantly from the unstressed state (i.e. atmospheric pressure) to the stressed state (i.e. the radius associated with systolic pressure), the elasticity modulus varies less so during a
/
I
” Reference ” Positmn of Small Element
Fig. 2. Stressed and reference position of arterial wall. R. = reference radius (diastole); RU+ q = stressed radius; q = change in radius during pulsatile perfusion as measured with the scanning laser micrometer. Adapted from Kuchar and Ostrach (1966).
physiologic pressure pulse. Hence the use of an incremental modulus, Einc for calculation of longitudinal and hoop stresses is appropriate: TP3 - TP, 2(1 -a’) R;,R,+ Einc =
Ro2 - RO,
R2,-R;
’
(21
Here P refers to the transmural pressure, D = Poisson’s ratio, R = radius, and the subscripts i, o, 1, 2, 3 represent inside, outside, and the minimum (diastolic), mean value, and maximum (systolic) value of the parameter over the cycle. Incremental
arterial wall stresses
The model of arterial wall stress used in the present analysis is adapted from the work of Kuchar and Ostrach (1966). The assumptions upon which their model is based are: the blood vessel wall is elastic, with constant properties; the vessel is axisymmetric, semiinfinite in length and straight with circular crosssection; the radial displacement of the vessel wall during its motion is taken to be small in comparison to the mean radius; and there is no longitudinal displacement or motion of the tube wall. Given these assumptions, the general form for the equations for wall stresses shown in Fig. 3 becomes:
SF, =[TP-2gj-, oEinc
TX, = (1 -c?)
4°C
Tee = (1 -a’)
(3) 1
R, A+ R,
Ei”ch’V 12R5 (1 - 0’)
(51
where: r = radial coordinate, q = radial displacement component of the vessel wall as shown in Fig. 2, B = Poisson ratio (assumed = 0.5), h = wall thickness.
110
A. M. BRANT et STRESSES ACTING ON SMALL ELEMENT OF ARTERIAL WALL
al.
HYPERTENSION
HIGH FLOW
PROdlAL
PR;SS”R;
PRESSURE DROP
Tee Fig. 3. Definition sketch showing stresses acting on a small element of the arterial wall. S,, = radial stress; Toe = circumferential stress; TX, = axial stress. Adapted from Kuchar and Ostrach (1966).
0 200 I E 10.0 0.0
I z
Ostrach (1966) have assessed the relative magnitude of each term in equations (3) through (5) by employing representative numerical data for the aorta of a dog as taken from the literature. They conclude that the fluid contribution to the radiaf wall stress S,, [equation (3)] is small compared to the magnitude of the pressure term. This simplification will be applied to the present calculations of S,,. Kuchar
and
APPLICATION
OF EXPERIMENTAL
BIOMECHANICS
DATA TO
1 FiOW
500
z
400
E
300
5
200
ti
100
z
1.98
w
1.93
ii I
1.88
2
1.83
z
I.78
RiiTE
CALCULATIONS
2 0.287
Equations (2) through (5) are used to calculate the incremental modulus and incremental arterial wall stresses at the vessel midpoint during pulsatile perfusion according to the protocol outlined below: (i) the reference radius, R,, wall thickness, h, and radial displacement, 9, are identified from measurements made using the scanning laser micrometer (Brant et al., 1987a); (ii) the transmural pressure difference for a perfused carotid is derived from measurements of Pl, P2 and P3 shown in Fig. 1 and equation (1); (iii) the incremental modulus of elasticity, E,,,, is calculated from equation (2); (iv) S,,, TX,, and T,, are calculated according to equations (3)-(5). RESULTS
Each canine carotid is perfused under one set of hemodynamic conditions (Table 1) for continuous periods between 2 and 24 h. Hemodynamic and wall motion data are acquired at the beginning and end of each experiment and at 8 hour intervals where appropriate. Figure 4 displays typical dynamic tracings of proximal perfusion pressure, pressure drop, flow rate, and arterial geometry for a vessel perfused under conditions of high flow-hypertension. Similar panels are obtained for the other flow situations of Table 1 (Brant
p m 0.279 z ? ; 0.271 L
0
I
I
1
I
i
3
4
J 5
TIME IN SECONDS
Fig. 4. Dynamic tracings of intraluminal proximal pressure (Pl of Fig. l), pressure drop (Pl-P2), flow rate (Fl), inside radius, and wall thickness for one artery perfused under the conditions of high flow-hypertension (Table I).
et ai., 1987b). None of these parameters appears to be influenced by the duration of perfusion. These tracings serve as input to equations (2) through (5) for calculation Of Einc, S,,, TX,, and To.+ Table 2 summarizes the results for incremental modulus. Results are listed as the mean f the standard error of the mean for the number of vessels (in parentheses) studied. An increase in Ei,,is observed for the hemodynamics of hypertension vs normotension and increased TP vs decreased TP. Once values for Ei, are known, calculations of longitudinal (axial) wall stress, TX,, circumferential ‘(hoop) stress, &,, and radial stress, S,,, are made. A typical tracing for TX,, T,, and S,, as a function of time during one cardiac cycle is presented in Fig. 5. The
Biomechanics of the arterial wall Table 2. Incremental modulus of elasticity, Ei,, and group averaged peak circumferential (T,,), axial (TX,),and radial (S,,) stress calculated for the various hemodynamic conditions of Table l*
Yz:f
Normotension Increased TP Decreased TP Hypertension High-flow hypertension
(:: = 3)
551 1222 4+2 14+2 17+3
16&-3 19 * 5 7kl 44*4 56+9
3225 37*11 15+3 88&7 112+ 17
17kO.2 25 * 0.4 9+0.6 31+0.3 30 * 0.3
*Mean + SEM. t x lo6 dynes/cm*. $ x IO“ dynes/cm’.
transmural pressure to 5 175 mmHg at the same fixed perfusion pressure and rate of flow fails to produce points which comprise each tracing the peak value for significant variations vs the normotension case. Because S,, is a boundary condition in the governing each stress component is identified. These latter values are derived for each vessel and averaged to yield ‘group equations of the vessel model, its value is directly computed from the hemodynamic tracings. Thus, average’ peak wall stresses. Table 2 presents the results of such calculations for hoop and axial stress. The the data of Table 2 is directly derived from the corfinding demonstrated here that TX, - 0.5 x T,, foIIows responding tracings of Fig. 4. One notes that the directly from equations (4) and (5) when assuming the largest radial wall stress occurs when the hemodyPoisson ratio = 0.5. That is, the second term in namic simulation is either hypertension or increased transmural pressure. On the other hand, a 60-70x equation (5) is small in comparison to the dominant first term. One also notes the expected increase in TBB reduction in S,, from these values is seen for the and TX,as perfusion pressure is elevated from 120/80 situation of decreased transmural pressure. to 240 mmHg/l20 mmHg. A further rise occurs when flow rate is increased to 300 cc/min. The effect of Assessment of experimental error The calculated biomechanical parameters of intervariations in P, (Fig. I) is also presented. As mean transmural pressure is decreased to -- 40 mmHg at a est, Eino S,,, Tee, and TX, are derived from the fixed perfusion pressure (115/85 mmHg) and flow rate experimental data of hemodynamics (pressure/flow) (150 cc/min), a faff in arterial wall stress vs that and vessel wail geometry {radius/wail thickness). Since calculated for the situation of normotension is pre- each of these latter measurements contains experimendieted. On the other hand, an increase in mean tal error, it is useful to assess the magnitude of the propagation of error associated with the derived biomechanical parameters. Accordingly the accuracies NORMOTENSION of several such parameters are given in Table 3. These I s LoNGlrUolNni STRESS values for “/0error are based upon measurements made 150during a simulation of normotension and employ formulae for calculation of error propagation found in Melissinos (1966). hemodynamic simulation (Table 1) for this particular figure is normotension. From the individual 64 data
Table 3. Assessment of the accuracies of selected biomechanical parameters. The nomenclature follows that defined
in the text. These calculations are based upon the experimental measurements for a typical simulation of normotension (Table I) Parameters
2 4 .6 TIME IN SECONDS
a
10
Fig. 5. Composite tracing of arterial wall stresses showing longitudinal, circumferential, and radial stresses over one complete cycle (1 s).
Pl TP Ri 1 Einc
Txx T08 SI.
Y0error 0.71 1.2 0.33 7.0 9.4 12 12 1.2
A. M. BRANTet al.
112
DISCUSSION
Whatever the biologic understanding may be regarding thrombosis and/or atherogenesis, biomechanical forces acting on the arterial wall appear to play an important, albeit poorly understood role. Stehbens (1979) has suggested that not only the localization but also the severity of atherosclerosis is governed by hemodynamic factors. Pathologic observations which he cites in support of this hypothesis include: (i) hypertension in any system augments the severity of the disease; (ii) augmented severity occurs where pulse and systolic pressures are high; and (iii) severity of the disease is directly related to the calibre of the vessel. Naumann and Schmid-Schonbein (1983) postulate that increased pulsatile transmural pressure over time can lead to a deformation of the arterial wall and, in consequence, to an aneurysm. Using a vascular casting technique Cornhill et al. (1980) note that endothelial cells in rabbit aortae become rounder and less elongated with increasing pressure. Biomechanical forces might also tend to displace the endothelium from the underlying substratum or strain the interendothelial junctions, thereby providing a mechanism for increased uptake of macromolecules by the arterial wall. Evidence to support such a hypothesis is provided by Chien (1983) who studied in situ the effects of changes in arterial pressure on macromolecular transport (‘251-albumin) in canine common carotid arteries. He reports that an increase in arterial pressure leads to an enhancement of 1251-albumin uptake. The present study provides relevant experimental evidence which underscores the importance of individual hemodynamic parameters on the development and magnitude of arterial wall stress. A novel feature of this work is the simulation in vitro of various hemodynamic conditions (Table 1) using a pulsatile perfusion apparatus. We have verified that this perfusion system is capable of reproducing hemodynamic parameters for continuous perfusion periods of 24 h. One set of results which we compare with literature data is the incremental modulus of elasticity. Such measurements have been made for uniform arteries by Bergel (1961) and Gow et al. (1974) who report values for Einc between 1 x lo6 and 1 x lo7 dynes/cm’, in close approximation to the present findings. Bergel also notes that Einc increases as transmural pressure is elevated, a trend also demonstrated in Table 2. Upon review of the data shown in Fig. 5 and Tables 2 and 3, it is apparent that the scatter associated with the calculations of S,, is less than that for Einc, T,, and 7&. One explanation for this finding can be gleaned from the error analysis of Table 3. Another possibility arises from the fact that S,, is solely dependent upon the operation of the pulsatile perfusion apparatus while the other parameters include a biologic component, i.e. they are calculated from measurements of the radial wall motion of living tissue. In an attempt to reduce the importance of animal-
to-animal variability in the interpretation of the present results, each animal serves as its own control in these in vitro perfusions. By that it is meant that if we choose to study one carotid artery under conditions of increased (or decreased) TP (Table l), then the contralateral vessel is perfused under the control normotensive simulation. A similar protocol is followed for the hypertensive simulations. Finally, several sets of experiments are conducted in which one vessel experienced normotensive hemodynamics and the other, hypertension. We employ the mathematical model of Kuchar and Ostrach (1966) for calculation of wall stress along the radial, axial, and circumferential axes. The assumptions upon which their model is based are listed in a previous section of this paper. As relates to the present in vitro experiments, the assumption of a straight vessel with circular cross-section is reasonable given the orientation of the perfused canine carotid in the tissue housing chamber of Fig. 1. It is also assumed that the vessel is semi-infinite in length (a first approximation since the vessel length is 10 cm and the inner diameter 24 mm resulting in L/D > 20). Pressure-wave reflections are neglected in analyzing wall motion owing to the significant wave length of the pressure-flow wave (- 10 m) vs the length of perfused vessel (10 cm). Since the carotid artery is tethered at its proximal and distal ends in the tissue housing chamber, longitudinal vessel motion is small. The argument that the artery deforms isovolumetrically (i.e. 0 = 0.5) during pulsatile perfusion and undergoes small radial perturbations is supported in part by the present perfusion data (Fig. 4) in which the ratio of the change in radius relative to the mean perfusion radius, AR/R, is approximately 5 ‘A. Carew et al. (1968) have previously conducted a more detailed study and also conclude that for small strain extension is isovolumetric. Some insight into the biomechanics of the perfused carotid can be gleaned from the similarities and differences in the data of wall stress for the simulations of hypertension and increased TP (Table 1). For these experiments the average TP = 17&180 mm Hg. Thus the agreement in S,, for these two cases (Table 2) is to be expected. On the other hand, a significantly higher value for T,, and TX,is seen for the hypertensive simulation. This disagreement is not due to differences in values of Einc for these two cases. Rather the absence of a rise in T,, and TX, for the hemodynamics of increased TP is a function of the dynamic vessel geometry during pulsatile perfusion. We recently report that the mean vessel calibre is essentially the same for the simulations of hypertension and increased TP (Brant et al., 1987b). What is different is the decreased dynamic wall motion for the latter simulation. Referring again to equations (4) and (5), this reduction in the radial motion of the vessel wall, q, yields for approximately equal values of E.1nc and R,,a lower computed value for TX,and l&. The data of Table 2 also suggest the importance of
Biomechanics
of the arterial
flow rate in the development of arterial stress. Increased rates of perfusion are known to accompany changes in physiological status of myocardial performance, systemic resistance, etc. In this work we choose to increase the mean arterial perfusion rate from the published value of 150cc/min for canine carotids (Caro and Nerem, 1973) to 300 cc/min while maintaining a mean perfusion pressure of 180 mmHg (Table 1). Peak axial and hoop stresses are elevated over the values for the hypertension-normal flow situation. Finally, we believe that the present results are also consistent with the aforementioned hemodynamic hypotheses of atherogenesis proposed by Stehbens (1979) and Naumann and Schmidt-Schonbein (1983). As a companion phase of the present work, we correlate the uptake of “‘C-4 cholesterol by the arterial wall and alterations in the vascular endothelial cytoskeleton (Herman et al., 1987) with the various hemodynamic cases of Table 1. We identify that the cases of increased TP, decreased TP, and high Aowhypertension produce profound changes in the form and distribution of the cytoskeletal elements and in the integrity of the arterial intima. Interestingly the 2.4 fold rise in Tee and TX, listed in Table 2 for the case of hypertension vs normotension is accompanied by a 1.4 fold elevation in cholesterol uptake which is further exacerbated when the rate of flow is doubled. Acknowledgements-The authors wish to thank Henry T. Bahnson, for his continued encouragement and support of our research. The manuscript was capably typed by Ms. Kathleen Haupr. This research was supported in part by grant-in-aid awards to HSB from the Whitaker Foundation and the National Heart, Lung and Blood Institute (HL34739). REFERENCES
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