Linearity of Pulsatile PressureFlow Relations in the Embryonic Chick Vascular System
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Abstract
The calculation and modeling of vascular input impedance are based on the assumption that pressure and flow are linearly related in the frequency domain. However, this assumption has not been proven for the embryonic circulation. Therefore, we investigated the linearity of pulsatile pressureflow relations in vivo with acute alterations in cycle length. We simultaneously measured dorsal aortic pressure with a servonull system and flow velocity with a 20MHz pulsedDoppler system in stage 24 chick embryos (n=38). Cycle length was acutely altered using thermal probe(s) applied to the sinus venosus. We determined the impedance spectra at several cycle lengths for each embryo and a reference curve from a threeelement Windkessel model with the use of nonlinear curve fitting. We then assessed the scatter of experimental impedance along the reference curve as a measure of linearity in the frequency domain. We found that mean vascular resistance did not change after thermal probe applications (P>.20 for each), indicating that acute alterations in cycle length did not alter peripheral vascular properties. Superpositioned impedance spectra showed minimal scatter along the model impedance from 0 to 6 Hz. Goodness of fit values (R^{2}) were near unity (.94 to .97) and were similar for all interventions (P>.07 for Fisher's z, by F test). Above 6 Hz, both modulus and phase spectra exhibited significant scatter (P<.05, by F test). Experimental impedance spectra tended to have a fluctuation and a phasezero crossover, indicating significant wave reflection in the embryonic circulation. Thus, the embryonic vascular system can be approximated as a linear system from 0 to 6 Hz, the range in which the majority (96.0±0.18%) of hydraulic energy is dissipated.
The arterial circulation develops and expands rapidly, coincident with the structural and functional maturation of the embryonic heart. The growing dorsal aorta is relatively stiff versus the mature aorta.^{1} Collagen type I first appears in the avian dorsal aorta on embryonic day 4,^{2} and elastogenesis first occurs on embronic day 5.^{3} Smooth muscle cell differentiation begins on embryonic day 4,^{4} when the dorsal aortic wall is three or four cell layers thick.^{5} These data support relative stiffness of the early embryonic vascular system.
Hydraulic energy is imparted during systole from the heart and dissipated as blood courses through the body. A portion of this energy is pulsatile and distends blood vessels while the remaining energy produces forward blood flow. Pulsatile energy may be a potential promoter of vascular development. Pulsatile pressureflow relationships are also critical determinants of vascular load that may influence cardiac morphogenesis. The quantitative analysis of pulsatile pressureflow relationships provides insight into the structure and regulation of the cardiovascular system.^{6}
The calculation of vascular impedance by Fourier methods and Windkesseltype lumped parameter models is based on the assumption that pressure and flow are linearly related in the frequency domain. Linearity in the frequency domain can be referred to as “independence of harmonics,” which means that one harmonic is not influenced by the other harmonics. Independence of harmonics ensures a unique relationship between corresponding harmonics of pressure and flow and, therefore, justifies the calculation of vascular impedance and hydraulic energy. This assumption of linearity has been tested in the mature pulmonary and systemic circulation and forms the basis of our understanding of arterial afterload.^{7} ^{8} ^{9} ^{10} ^{11} ^{12}
The direct proof of linearity was obtained in vitro by driving an isolated vascular system with a pure sinusoidal flow input and then determining pulsatile pressureflow relationships with the use of Fourier analysis.^{7} Linearity in vivo was first reported by Bergel and Milnor,^{9} who studied the effects of changing the cycle length for the pulmonary circulation. In response to changes in cycle length, impedance spectra were relatively constant, and the scatter of the harmonics was acceptably small. Thus, they concluded that the pulmonary circulation was “approximately” linear on the basis of the qualitative consideration. This finding, in a highly compliant pulmonary circulation, actually conflicts with the nonlinear properties of arterial pressureflow relations. In a strict sense, if arterial compliance was related to cycle length, there should have been some changes in curvature in the impedance spectrum and some scatter, particularly in the intermediate frequency range. Nevertheless, this method to test linearity in vivo was soon applied to the systemic circulation,^{10} ^{11} ^{12} and a similar conclusion was drawn. Thus, the use of cycle length change has been accepted as a valid, though imperfect, method to test the linearity of pressureflow relations in the frequency domain.^{6}
In the present study, we characterize the hemodynamic factors previously reported to be involved in the regulation of morphogenesis, growth, and function of the embryonic cardiovascular system.^{13} Although arterial impedance has been calculated in the embryonic circulation,^{1} ^{14} the assumption of linearity in the embryonic vascular system has not been proven. Therefore, we tested the linearity of the chick embryonic vasculature by changing cycle length without altering peripheral vascular resistance. We found that the stage 24 embryonic chick vascular system can be approximated as a linear system from 0 to 6 Hz, and this finding supports our analytic paradigm using Fourier analysis and lumped parameter modeling.
Materials and Methods
Hemodynamic Measurements
Fertilized white Leghorn chicken eggs were incubated to HamburgerHamilton stage 24 (4 days, n=38) of a 46stage (21day) incubation period.^{15} The details of embryo preparation are published elsewhere.^{1} We measured dorsal aortic blood pressure with a servonull pressure system (model 900A, World Precision Instruments). The servonull system measures the resistance of the 2 mol/L NaCl–filled pipette (7 to 10μm tip) and then prevents changes in resistance by generating an opposing pressure to the pressure present at the pipette tip. The actual pressure decay can be approximated by a secondorder system. This manometer has a relatively low frequency response, which varies with the size of the pipette tip. We then performed a “pop test”^{16} to acquire two determinants of system frequency response, the damping coefficient β and the natural frequency ω_{0} (angular frequency). The two parameters were calculated as described elsewhere^{16} and determined with various sizes of the pipette tip. The size of the pipette tip was quantified by the resistance of the pipette at a current of 50 μA. The correlations between the resistance (R, in MΩ) and these two parameters were as follows: β=–0.11·R+0.18 (r=.69, P<.0001) and ω_{0}=−4.4·R+12.7 (r=.72, P<.001), with R ranging from 0.2 to 0.6 MΩ. We measured the resistance of each pipette before the individual experiments and corrected Fourier terms of the pressure in the analysis process.^{16}
Dorsal aortic blood flow velocity was measured simultaneously with a 20MHz pulsedDoppler velocimeter (model 545C, University of Iowa). A 0.5mmdiameter Doppler crystal was positioned at a 45° angle to the aorta using a protractor jig at the same level as the pressure pipette. The angle was corrected in the analysis process. The Doppler velocimeter focus range is 0 to 10 mm, with a sample volume of 1 to 4 mm.^{17} This sample volume includes the entire dorsal aortic cross section (0.41±0.01 mm in diameter, stage 24).^{1} Pressure and velocity analog waveforms were continuously digitized at a sampling rate of 500 Hz and analyzed using a custom virtual instrument (LabVIEW, National Instruments).
We also measured the internal diameter of the dorsal aorta using videomicroscopy. A calibration scale scribed with 50μm divisions was positioned perpendicular to the long axis of the dorsal aorta. At the current level of optical resolution, we noted no change in aortic diameter during the normal and altered cardiac cycles. Therefore, we determined a single value for mean dorsal aortic diameter. Assuming aortic cross section to be circular, instantaneous aortic blood flow (Q) was calculated using the following equation: Q=π·d^{2}·V/4, where d is the mean aortic diameter and V is the instantaneous velocity.
Experimental Protocols
Baseline data were recorded for at least 15 seconds, and intrinsic cycle length was determined. We then changed cycle length using a 1mmdiameter steel probe preheated (hot probe alone group, n=12) or precooled (cold probe alone group, n=11) in a water bath.^{18} ^{19} The probe was applied to the sinus venosus for 3 seconds. Cycle length decreased in response to the hot probe and increased in response to the cold probe. Only embryos that recovered to 90% to 110% of intrinsic cycle length 30 seconds after the intervention were included for analysis. In a separate set of embryos, the alternate probe was applied after recovery to intrinsic cycle length from the first intervention (hotandcold group, n=15). In the hotandcold group, a hot probe was applied first in nine embryos and was followed by a cold probe; the order was reversed in six embryos. Sham (control) embryos were treated by the probe, heated to 37°C to 38°C. All experiments were performed within 180 seconds.
Data Analysis and Statistics
The cycle length of most embryos recovered to 90% to 110% of its intrinsic cycle length within 50 to 60 beats (20 to 30 seconds) after probe application. Therefore, we chose three consecutive waveforms at baseline and then immediately, 10 seconds, and 20 seconds after the probe application. Cycle length, mean pressure, pulse pressure, mean blood flow, stroke volume, and mean vascular resistance at each time point were determined by averaging the three consecutive cycles. Similarly, the impedance spectrum at each time point was determined by averaging three corresponding harmonics with the use of statistics for circular data.^{20}
The calculation of input impedance spectrum is published elsewhere.^{1} ^{6} Experimental impedance spectra with varying cycle lengths were superpositioned on the same axes to identify the scatter of the harmonics along a reference curve. Since each spectrum has a zeroorder modulus at 0 Hz, we averaged all moduli at 0 Hz to determine the single value at 0 Hz. All harmonics above 12 Hz were discarded because of the limited frequency response of the servonull pressure system. Superpositioned impedance spectra were thus determined for individual embryos. The number of harmonics up to 12 Hz varied among groups, because the impedance spectrum derived from longer cycle lengths has more harmonics than from shorter cycle lengths.
Hemodynamic parameters in the time domain were summarized as mean±SEM and compared with matched sham (control) values by repeated measures ANOVA. The significance of the order of interventions in the hotandcold group was also tested by repeated measures ANOVA. The response to individual probe in the hotandcold group was compared with corresponding probe alone by twoway repeated measures ANOVA. Statistical significance was defined by a value of P<.05. Calculations were performed with SPSS (SPSS Inc) and Statistica (Statsoft) software.
Linearity of Pulsatile PressureFlow Relations
We evaluated the scatter of the harmonics along a reference curve as a quantitative measure of the linearity in the frequency domain. The reference curve was derived from the threeelement Windkessel (3WK) model, which consists of a proximal resistor in series with a parallel arrangement of a distal resistor and a capacitor.^{21} Biological correlates of the proximal and distal resistors and the capacitor are the characteristic impedance (R_{c}), peripheral resistance (R_{p}), and total arterial compliance (C), respectively. Model impedance Z_{m}(jω) of the 3WK model is given in rectangular form by the following:where jω is complex angular frequency. For individual embryos, we optimized three lumped parameters using modified LevenbergMarquardt nonlinear regression. To calculate the regression, the experimental impedance spectrum, Z_{e}(jω), was expressed by rectangular form:where Z_{re}(ω) and Z_{im}(ω) are the real and imaginary components of the harmonics at frequency ω, respectively. The residual sum of squares, SS_{res}, was defined as follows^{22} :where h is the angular frequency at the highest harmonic <12 Hz, subscripts e and m denote experimental and model impedance, respectively, and re and im denote real and imaginary components, respectively. In Equation 3, the quantity in braces is a squared distance between experimental and model impedance plots in the complex plane. Estimated lumped parameters were thus optimized minimizing SS_{res}. For statistical comparison, total sum of squares, SS_{tot}, was also defined as follows:where Z̅_{e̅,̅r̅e̅} and Z̅_{e̅,̅i̅m̅} are means of the real and imaginary components, respectively. Then, the coefficient of determination, R^{2}, as a measure of “goodness of fit” was calculated by (SS_{tot}−SS_{res})/SS_{tot}. We applied Fisher's z transformation and F test to compare the difference in R^{2} among groups with different interventions. In addition, we compared the difference in the scatter of harmonics among three frequency ranges (0 to 4, 4 to 8, and 8 to 12 Hz). SS_{res}/df (where df indicates degrees of freedom) was calculated for individual frequency ranges in the hotandcold group embryos (n=15) and assessed by F test. The harmonic at 0 Hz was eliminated from this analysis.
Distribution of Hydraulic Energy Over the Frequency Range
In the mature vascular system, the ratio of pulsatile energy to total hydraulic energy is considered to be a reverse index of energy efficiency.^{23} Of note, the pulsatile ratio is higher in the embryonic vascular system than in the mature circulation,^{14} indicating that pulsatile energy likely plays a significant role in the maturation process. To assess physiological relevance of linearity, distribution of the hydraulic energy over the frequency range was also calculated as pulsatile energy, W_{puls}(α), distributed from >0 Hz to α Hz (note that 0 Hz is not included)^{23} :where f is the angular frequency at the first harmonic, and Q(jω), Z(jω), and θ(ω) are the modulus of flow, the modulus of impedance, and the phase of impedance, respectively. The highest value of α was set to 12 Hz so that “total pulsatile” energy was defined by W_{puls}(12). Steady state energy, W(0) (because it is energy at 0 Hz), was calculated from the product of mean pressure and mean flow. Then, hydraulic energy, W(α), distributed from ≥0 Hz to α Hz (note that 0 Hz is included) was defined by W(0)+W_{puls}(α). Hence, total hydraulic energy was defined by W(12). Then, percentage ratios of W_{puls}(α) to total pulsatile energy W_{puls}(12) and W(α) to total hydraulic energy W(12) were calculated for individual embryos. The calculations were performed for the baseline data in the hotandcold group (n=15).
Results
Hemodynamic Parameters in the Time Domain
Representative waveforms of aortic pressure and flow during acute alterations in cycle length are displayed in Fig 1⇓. In response to hot probe alone, mean and peak pressures and mean flow did not change, whereas pulse pressure and stroke volume decreased (Table 1⇓). In response to cold probe alone, mean and peak pressures, mean flow, and stroke volume decreased, whereas pulse pressure did not change. Of note, mean resistance did not change in response to either intervention (P>.20 for each by repeated measures ANOVA). The order of the intervention in the hotandcold group was not significant (P>.34 for each hemodynamic parameter by repeated measures ANOVA). The response of hot probe in the hotandcold group was similar to the hot probe alone (P>.21 for each hemodynamic parameter by twoway repeated measures ANOVA). The response of cold probe in the hotandcold group was similar to the cold probe alone (P>.11 for each hemodynamic parameter by twoway repeated measures ANOVA).
Impedance Spectra With Varying Cycle Lengths
When we varied cycle length by a single intervention, experimental impedance spectra were consistent, and most of the harmonics up to 6 or 7 Hz fell on the same curve (Fig 2⇓). Model impedance spectra with optimized lumped parameters are also superpositioned in the figures. Experimental impedance spectra in response to sequential intervention (hotandcold group) and fitted model impedance are shown in Fig 3⇓. A larger number of harmonics were obtained with the sequential intervention. In all groups, we noted minimal scatter of the impedance data along the reference curve from 0 to 6 Hz, although there was larger scatter in the higher frequency range. The experimental impedance phase spectrum had a zero crossover between 6 and 8 Hz. In addition, difference between experimental and model impedance in the phase spectrum was larger in the higher frequency range. Table 2⇓ summarizes optimized lumped parameters and the parameters for “goodness of fit.” The overall coefficient of determination (R^{2}) was near unity. The parameters for “goodness of fit” were similar among groups regardless of the type of interventions (P>.07 for Fisher's z, by F test). Fig 4⇓ shows comparison among three different frequency ranges in terms of the scatter along the model impedance. F test revealed larger scatter in the higher frequency range (8 to 12 Hz) compared with lower frequency ranges.
Distribution of Hydraulic Energy in the Frequency Range
Percentage ratios of W_{puls}(α) to total pulsatile energy, W_{puls}(12), increased from 4.2±1.66% at 2 Hz to 68.5±1.13% at 4 Hz, 85.0±0.76% at 6 Hz, 93.7±0.35% at 8 Hz, and finally 97.4±0.23% at 10 Hz. Percentage ratios of W(α) to total hydraulic energy, W(12), increased from 73.1±0.56% at 0 Hz to 74.3±0.59% at 2 Hz, 91.5±0.3% at 4 Hz, 96.0±0.18% at 6 Hz, 98.3±0.1% at 8 Hz, and finally 99.3±0.07% at 10 Hz.
Discussion
In the linearity test in vitro, a single harmonic sinusoidal pump was connected surgically to isolated and perfused mature systemic vascular beds.^{7} Generated pressure and flow at the input of the isolated vessels had a single harmonic with the same frequency regardless of the changes in source frequency, which indicated an “independence of harmonics” in the mature vascular system. The in vitro test thus provided a direct proof of linearity, permitting the use of Fourier methods in the analysis of pulsatile pressureflow relationships. For the embryonic vascular system, however, a similar test in vitro is technically difficult with the present measurement system. Therefore, we pursued an in vivo test in the presence of ventricular/vascular interactions.
The linearity test in vivo depends on the evidence that vascular properties, particularly peripheral vascular properties, do not change in response to alterations in cycle length.^{9} In the present study, this evidence was confirmed, since mean vascular resistance was unchanged in all groups. As shown in Figs 2 and 3⇑⇑, our data indicate that the vascular system is linear from 0 to 6 or 7 Hz by the small variance around the reference curve. However, because of some scatter in the impedance spectrum with increasing frequency, we propose that the in vivo embryonic vascular system should be considered quasilinear or conditional linear.
As described by previous investigators,^{8} ^{24} ^{25} “nonlinearity” of pulsatile pressureflow relations has been shown in vascular systems. Several factors, including flow turbulence in the vessels, shear forces due to the viscous properties of blood, and nonlinear pressurevolume relations of the arterial wall, have been inferred as sources of nonlinearity. Among these, Stergiopulos et al^{24} used a completely controlled computer model and found that the scatter in the impedance spectrum was mainly due to the elastic nonlinearity of the arterial wall. In fact, as Stergiopulos et al pointed out, even the impedance spectrum shown to be linear in vivo^{9} ^{10} displays scatter, particularly in the intermediate to higher frequency ranges. In our results, particularly some embryos in the cold probe group, scatter already starts around 5 Hz, as seen in Fig 2⇑. Although statistical difference between the hot probe group and the cold probe group in terms of scatter was not demonstrated, this observation may reflect changes in pressurevolume relations of the arterial wall that are possibly due to decreases in mean pressure, particularly seen in the cold probe group. Recognizing that any instantaneous changes in cycle length, pressure, and flow can alter pressurevolume relations of the arterial wall, linearity is valid only when the focus of the analysis is a first approximation. In addition to these physiological factors, measurement errors are the most likely to cause the scatter at higher frequencies, especially with the low frequency response of our manometer system.
Universality and Limitation of the 3WK Model
It is of interest that the same 3WK model, developed for the elastic adult aorta, could accurately describe the impedance characteristics of the embryonic vascular system with an aortic diameter <0.5 mm and 1/70 of the adult mean pressure. Since the pressure waveforms exhibit diastolic pressure, Windkessel capacitance and discharge characteristics of the artery, which deliver steady flow to the periphery from intermittent cardiac output, are already present in the early embryonic circulation. These functional similarities between mature and embryonic circulations may exist because the embryonic vascular system already possesses qualitative characteristics similar to those of the mature vascular system, whereas the difference in scale discriminates between the two systems. Provided that the vascular system is composed of larger proximal vessels, branching vessels, and smaller distal vessels, numerous studies have shown that the pulsatile arterial pressureflow relationships can generally be characterized by the three parameters R_{c}, R_{p}, and C (from Equation 1) at least within the low to mid frequency ranges.^{26} This may apply to the embryonic circulation, since the structure of the embryonic vascular system itself is similar in terms of the basic components. In addition, although speculative, material properties of the embryonic arterial wall are likely matched relative to the pressure range to achieve the Windkessel function of the artery. Thus, the developmental changes in in vitro material properties of the embryonic arterial wall deserve further investigations.
Although the 3WK model provided a good fit to the experimental impedance spectrum in the present study, further refinements can be made. As shown in Fig 3⇑, the superpositioned impedance phase tended to have a zero crossover at 6 or 7 Hz, and the modulus tended to have a minimum at 5 or 6 Hz. These tendencies, summarized as fluctuation of the impedance spectrum, indicate significant wave reflection phenomena in the actual arterial system^{6} that cannot be reproduced by the 3WK model. The 3WK model as a reference curve was suitable for the present study because of its simplicity. However, simple electrical analog models assume an infinite pulsewave velocity and therefore do not account for wave propagation/reflection phenomena. Ideally, to account for reflected waves, distributed models or asymmetrical Ttube models are required.^{27} Still, refinement of the electrical analog model might be helpful because of its simplicity of calculations and biological relevance of those lumped parameters. One possibility is proximal addition of an inductance term to the 3WK model. This type of fourelement model can reproduce phasezero crossover and at least one minimum of the modulus. Actually, parallel addition of an inductance term improved R^{2} from .966±.006 to .976±.007 (Fig 5⇓) in the preliminary study. However, quantitative comparisons among various types of analog models and consideration of overparameterization are required to appropriately characterize the embryonic vascular system. Moreover, biological relevance of the inductance term should be examined if the inductance is necessary and sufficient.
Differences Between Embryonic and Mature Circulations
The ratio R_{p}/R_{c} has been reported to be consistent in the mature systemic circulation and range from 16.7 to 23.9 in most mammals.^{26} Therefore, the impedance spectrum normalized by R_{c} represents a fundamental shape of the spectrum for most mammals. Since R_{c} is one of the two determinants (R_{c} and C) of the pulsatile load in the arterial system, R_{p}/R_{c} is inversely correlated to pulsatile ratio of the hydraulic energy in the arterial system. As calculated from the data in Table 2⇑, R_{p}/R_{c} is ≈5.5 to 6.8 in the embryonic circulation. The smaller R_{p}/R_{c} in the embryonic circulation is consistent with a higher pulsatile ratio of the hydraulic energy and may particularly indicate stiffness of the proximal artery. In addition, greater pulse pressure relative to pressure range indicates that the entire embryonic arterial system is less compliant than the mature circulation. Crossspecies comparison of arterial compliance is possible when the time constant (τ) of the diastolic pressure decay, a product of R_{p} and C, is normalized by a duration of the diastolic time (T_{d}).^{28} Smaller animals with shorter τ have shorter T_{d} values to prevent diastolic pressure from decaying to insufficient coronary perfusion pressure, whereas larger animals with longer τ have longer T_{d} values. Therefore, τ/T_{d} is constant among mature animals and ranges from 3.1 to 6.2, and the optimal heart rate for the species is thus set depending on the τ value. The embryonic τ/T_{d} is 1.7 to 1.8, calculated from data previously published^{1} and in Table 2⇑. Again, the smaller τ/T_{d} in the early embryonic circulation supports a less compliant arterial system in the embryo. It is of note that this less compliant system with a relatively low diastolic pressure precedes coronary arterial connections, which are established on embryonic day 7 or 8.^{29} Developmental changes in pulsatility of the pressure waves, arterial compliance, and pulsatile hydraulic energy should be correlated to the development in coronary circulation.
Study Implications
Despite the technical limitations mentioned above, the linearity of embryonic pressureflow relations proven here from 0 to 6 Hz in the frequency domain is important new information, given the dramatic differences in scale, histology, and regulation between developing and mature circulations. Even restricted to 6 Hz, a linear system analysis provides valuable hemodynamic information, because most hydraulic energy, 85.0% of the pulsatile and 96.0% of the total energy, is distributed within this frequency range. Based on this linear relationship, it is justified to superimpose impedance spectra with various cycle lengths in order to improve frequency resolution (maximize the number of harmonics). This improvement in the frequency resolution allows a more accurate assessment of arterial compliance and characteristic impedance, especially given the limitation of the manometer's frequency response. Quantitative assessment shown here for the linear/nonlinear issue may provide further significant information because it enables quantitative comparisons among different species, developmental stages, perturbed hemodynamic conditions, and various types of analog models.
Thus, despite the dramatic differences in scale between the embryonic and mature arterial vasculatures, the stage 24 embryonic chick vascular system can be approximated as a linear system from 0 to 6 Hz. Using these techniques, we should be able to investigate wave propagation/reflection phenomena, which are likely to be physiologically important in the rapidly expanding embryonic vasculature.
Acknowledgments
This study was supported by a National Institutes of Health SCOR grant in Pediatric Cardiovascular Diseases (P50–HL51498). We thank Edward B. Clark, MD, for his critical review of this manuscript.
Footnotes

Reprint requests to Bradley B. Keller, MD, University of Rochester, Department of Pediatrics, 601 Elmwood Ave, Box 631, Rochester, NY 14642. Email [email protected]
 Received February 1, 1996.
 Accepted June 11, 1996.
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 Linearity of Pulsatile PressureFlow Relations in the Embryonic Chick Vascular SystemMasaaki Yoshigi and Bradley B. KellerCirculation Research. 1996;79:864870, originally published October 1, 1996https://doi.org/10.1161/01.RES.79.4.864
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