Journal of Biomechanics 44 (2011) 869–876

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Journal of Biomechanics

0021-92

doi:10.1

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www.JBiomech.com

Application of 1D blood flow models of the human arterial network todifferential pressure predictions

David A. Johnson a,b, William C. Rose c, Jonathan W. Edwards d, Ulhas P. Naik a,b,e,f,g, Antony N. Beris a,n

a Department of Chemical Engineering, University of Delaware, Newark, DE 19716, USAb Delaware Biotechnology Institute, University of Delaware, Newark, DE 19716, USAc Department of Health, Nutrition, and Exercise Sciences, University of Delaware, Newark, DE 19716, USAd Eastern Virginia Medical School, Norfolk, VA 23501, USAe Department of Biological Science, University of Delaware, Newark, DE 19716, USAf Department of Chemistry and Biochemistry, University of Delaware, Newark, DE 19716, USAg Cardiovascular Research Center, University of Delaware, Newark, DE 19716, USA

a r t i c l e i n f o

Article history:

Accepted 6 December 2010A new application of 1D models of the human arterial network is proposed. We take advantage of the

sensitivity of the models predictions for the pressure profiles within the main aorta to key model

Keywords:

Blood flow model

Flow impedance

Arterial network

Pressure profiles

Sensitivity

90/$ - see front matter & 2010 Elsevier Ltd. A

016/j.jbiomech.2010.12.003

esponding author. Tel.: +1 302 831 8018; fax

ail address: beris@udel.edu (A.N. Beris).

a b s t r a c t

parameter values. We propose to use the patterns in the predicted differences from a base case as a way to

infer to the most probable changes in the parameter values. We demonstrate this application using an

impedance model that we have recently developed (Johnson, 2010). The input model parameters are all

physiologically related, such as the geometric dimensions of large arteries, various blood properties,

vessel elasticity, etc. and can therefore be patient specific. As a base case, nominal values from the

literature are used. The necessary information to characterize the smaller arteries, arterioles, and

capillaries is taken from a physical scaling model (West, 1999). Model predictions for the effective

impedance of the human arterial system closely agree with experimental data available in the literature.

The predictions for the pressure wave development along the main arteries are also found in qualitative

agreement with previous published results. The model has been further validated against our own

measured pressure data in the carotid and radial arteries, obtained from healthy individuals. Upon

changes in the value of key model parameters, we show that the differences seen in the pressure profiles

correspond to qualitatively different patterns for different parameters. This suggests the possibility of

using the model in interpreting multiple pressure data of healthy/diseased individuals.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

In cardiovascular medicine, it can be difficult to predict thepressure and flow of blood at the innermost arteries. With non-sophisticated, non-invasive techniques, such as pressure cuff mea-surements, it is a challenge to be able to extrapolate to the rest ofthe network, which has been approached recently by severalresearchers (Lowe et al., 2009). The development of more sophis-ticated methods for predicting local blood pressure and flow fromsimple measurements for medical applications has the potential ofelucidating the presence of disease conditions, but this requires thecoupling to blood flow models. The development of a model for theflow of blood in the human circulatory system is a difficult task,thus explaining the large number of publications devoted to thissubject from the original work by (Womersley, 1955a, 1955b, 1957a,1957b; O’Rourke, 1967) to the more recent (Duan and Zamir, 1995;

ll rights reserved.

: +1 302 831 1048.

Fung, 1996, 1998; Zamir, 1998, 2000; West, 1999; Olufsen, 2004;Lorenz et al., 2004; Truskey et al., 2009; Reymond et al., 2009) andreferences therein. An equally large number of papers has beendedicated to blood flow model applications, involving both some ofthe above-mentioned works (Duan and Zamir, 1995; Zamir, 1998,2000; Olufsen, 2004; Lorenz et al., 2004; Reymond et al., 2009) as wellas others (Taylor et al., 1998; Gijsen et al., 2003; Dong et al., 2006;Badia et al., 2008; Nguyen et al., 2008).

The current trend involves sophisticated computational fluiddynamics (CFD) models that can simulate blood flow in three dim-ensions and provide exceptionally detailed results (Taylor et al.,1998; Gijsen et al., 2003; Lorenz et al., 2004; Dong et al., 2006; Badiaet al., 2008; Nguyen et al., 2008). However, the high computationalworkload associated with these simulations limits their applic-ability to a selected subsystem of the overall arterial network. Sim-plified, one-dimensional (1D) in space blood flow network models,such as the one used here developed recently (Spaeth, 2006; Roseet al., 2008; Johnson, 2010; Johnson et al., 2010a), have alreadyfound application in setting up better outlet conditions for detailed3D and time-dependent simulations under in vivo conditions of

D.A. Johnson et al. / Journal of Biomechanics 44 (2011) 869–876870

selected arterial vessel systems (Olufsen, 1999; Pontrelli and Rossoni,2003; Taylor and Draney, 2004; Bessems et al., 2008; Grinberg andKarniadakis, 2008; Formaggia et al., 2009; Johnson et al., 2010b).

The objective of the present work is to predict changes in thepressure profiles in the arterial system in response to variations inkey physiological parameters, such as in the blood flow viscosity,the vessel elasticity, and viscoelasticity. The central outcome is theidentification of characteristic patterns associated with each one ofthe physiological factors varied in the analysis. This provides a newbasis to correlate changes observed in blood flow pressure mea-surements to their underlying physiological, possibly disease-related, causes. As a vehicle for this analysis we used a recentlydeveloped impedance-based 1D model (Johnson, 2010). It is fairlyrepresentative of the recently developed 1D models (Formaggia et al.,2003; Canic et al., 2005; Formaggia et al., 2006; Alastruey, 2006, Azerand Peskin, 2007; Huo and Kassab, 2007; Alastruey et al., 2009;Formaggia et al., 2009; Reymond et al., 2009). Although in these worksthe models are developed in the time domain, for enhanced compu-tational efficiency the 1D model used here is implemented in thefrequency domain (Johnson et al., 2010a), an approach also followedby others (Duan and Zamir, 1992, 1995; Olufsen, 2004).

Fig. 1. Schematic of the proposed use of 1D models interpreting non-invasive

pressure and/or flow data.

Table 1Blood modeling parameters for this model and Olufsen (2004).

Parameters Our model (Johnson et al., 2010a)

Plasma viscosity 0.00135 Pa s (Koenig et al., 1994; Rosenson e

Discharge hematocrit 45 (Pries et al., 1990)

Network viscosity Fahraeus–Lindqvist model (Pries et al., 1990)

Terminal arterial pressure 2000 Pa (�15 mmHg)-gage pressure (Mart

Terminal artery/capillary radius 3 mm (Milnor, 1982; Nichols and O’Rourke, 2

Capillary length to diameter ratio 50 (Milnor, 1982; Nichols and O’Rourke, 200

In Section 2 we describe briefly the model and method ofoperation. In Section 3 we present validation and pressure differ-ence results. In Section 4 we discuss these results. Section 5 containsour conclusions.

2. Methods

The method by which we wish to identify pressure patterns is described here,

and is visually represented in Fig. 1. Pressure and/or flow data can be recorded at one

or multiple locations within the body, using non-invasive techniques, such as

pressure cuff (Lowe et al., 2009), applanation tonometry (Nichols and O’Rourke,

2005; Edwards et al., 2008), or Doppler tomography (Chen et al., 1997). For this

work, blood pressure was measured at the radial, carotid, and femoral arteries by

applanation tonometry (Millar Instruments, Houston, TX). The pressure was

calibrated by diastolic and mean pressure obtained from a cuff on the brachial

artery. All procedures were approved by the University of Delaware Institutional

Review Board, and informed consent was obtained from all subjects. The pulse

waveform can then be fed into a 1D model for further analysis. The objective is,

although not performed here, to repeat the measurements on specific individuals

over time and note the differences. Instead, we use a 1d model to make predictions

on the patterns observed in the pressure profile differences.

Olufsen (2004) Model parameters

t al., 1996) –

–

Constant at 0.0049 Pa s

ini and Nath, 2008; Truskey et al., 2009) 0 Pa (0 mmHg)-gage pressure

005) 400 mm

5) 25

Fig. 2. Model vascular network. The network is the same as that of Olufsen (2004).

The network includes 45 arteries, each characterized by its length and upstream and

downstream radii. The network also includes 23 branching terminal sub-networks

denoted by ‘‘B’’s. (See Johnson et al., 2010a for details.)

D.A. Johnson et al. / Journal of Biomechanics 44 (2011) 869–876 871

The 1D blood flow arterial circulation model used in this work is not unlike the

last generation 1D models as described in Table 1 of Reymond et al. (2009). It is

based on a simple analytical approximation available to describe the time-periodic

pulsatile flow of a viscous fluid within an elastic, partially viscoelastic, wall vessel

(Womersley, 1955a). The geometric information of 45 main arteries, as developed

by Olufsen (2004) and given in Fig. 2, is used to predict the local and sub-network

structure and resistance, using scaling laws (West, 1999) to predict the sub-

networks. Suitable modifications have also been introduced in order to account for

the non-Newtonian character of the blood rheology for the steady-state flow

resistance using the Casson Equation (Truskey et al., 2009), as well as the Fahraeus

(1929) and Fahraeus and Lindqvist (1931) effects using variable effective properties

of the blood (Pries et al., 1990), the viscoelasticity of the arterial vessels (Pedley,

1980), and a full accounting of the wave reflections at the arterial bifurcations

(Zamir, 1998, 2000; Alastruey et al., 2009). For more details regarding this model

and the specific equations used we refer to the supplemental material and for the

computational implementation of the solution to the Ref. (Johnson et al., 2010a).

3. Results

In the pursuit of developing usages of 1D models in medicalapplications, it is important that the model is first validated(Matthys et al., 2007; Reymond et al., 2009).

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Fig. 3. Input impedance of the systemic arterial tree, modulus (a) and phase lag

(b) using the geometry shown in Fig. 2. Comparisons of model predictions (solid blue

lines) against reported experimental measurements by Nichols and O’Rourke (2005)

and Nichols et al. (1977) obtained as an average of five adults (isolated red dots). For

an estimate of the experimental variance, reported data obtained from Nichols and

O’Rourke (2005) a single individual are also supplied (green crosses). (For inter-

pretation of the references to color in this figure, the reader is referred to the web

version of this article.)

3.1. Aortic input impedance predictions and comparison against

literature results

The aortic input (total) flow impedance of all arteries and their sub-networks describes an Ohm’s law-like pressure–flow relationship inthe Fourier frequency domain:

Zn ¼Pn

Qn

, ð1Þ

where Zn, Pn, and Qn represent the impedance, pressure, and flow,respectively, nth order harmonics. The zeroth contribution is real andcorresponds to the steady-state flow resistance. As a result of the ref-lections at the bifurcations and of the viscoelasticity of the vessel wall,a phase lag is introduced between the pressure and flow harmonics.Therefore, following Li (2004), we can also express the nth harmo-nic of the total impedance as a function of its modulus (9Zn9) and

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0 cm Olufsen (2004)0 cm R & W (1956)20 cm Olufsen (2004)20 cm R & W (1956)40 cm Olufsen (2004)40 cm R & W (1956)

Fig. 4. Comparison of simulation predictions for the pressure profile against

experimental data (Fig. 8 of Remington and Wood, 1956) at several locations down

the main arterial trunk (0, 20, 40 cm). The network geometry shown in Fig. 2 was

used as the basis for developing impedances. (a) Predictions correspond to the

current model and for parameter values as described in Johnson et al. (2010a) and

given in the left column of Table 1. (b) Predictions correspond to the model and

parameter values described by Olufsen (2004) and shown in the right column of

Table 1.

D.A. Johnson et al. / Journal of Biomechanics 44 (2011) 869–876872

phase-lag angle (yn).

Zn ¼ Znj jeiyn : ð2Þ

Impedance information is important on its own right, and one ofthe main usages of 1D models is to provide it at any desirablelocation, as it can be used to describe proper boundary conditionsfor detailed in vivo simulations of isolated components of the humanarterial circulation network domain (Spilker et al., 2007; Clipp andSteele, 2009; Johnson et al., 2010b).

The model predictions for the modulus and phase lag of theaortic input impedance are shown in Fig. 3. Nichols and O’Rourke(2005), and Nichols et al. (1977), as well as others, experimentallymeasured the modulus of impedance and phase angle at the aortaand their data are also shown in Fig. 3. In Fig. 3a we see that, as far asthe modulus is concerned, we have very good agreement betweenthe predictions and the experimental data: we only slightly over-predict the steady-state value (by less than 8%) whereas all theother values are within the same range 30–170 dynes s/cm. Ourdata show exactly the same trough-peak behavior, with the onlydifference being that those are seen to occur at smaller frequencies:2 and 4 Hz instead of 4 and 7, respectively. Regarding the phase lag,shown in Fig. 3b, we see that our predictions almost exactly matchthe experimental data at low frequencies, below 3 Hz. Even above3 Hz we see the same qualitative behavior, the phase angle reaching a

Fig. 5. Pressure profiles for various inputs to the 1D impedance model. Pressure was meas

model, using the geometry shown in Fig. 2 and the parameters in the left column of Table 1

the first 10 harmonics (+), was fed into the model to predict the radial artery profile. (

prediction used the first 10 harmonics of carotid pressure as input to the model. (c) Mea

10 harmonics (+) was fed into the model to predict the carotid artery profile. (d) Measure

used the first 10 harmonics of radial pressure as input to the model.

peak before eventually going down and eventually settle at almost0 rad. However, above around 3 Hz there is a significant quantitativevariance: the predicted peak value is much smaller (around 0.1 rad)and the peak occurs earlier (around 4 Hz) than in the data (around0.8 rad and 6 Hz, respectively). Nevertheless, when examining adifferent set of data (green stars in Fig. 3b) we see that althoughthe same qualitative behavior is observed, there is an even highervariation observed between the two experimental data beyond 3 Hzindicating that the results in this region may be a sensitive function ofperson-specific information.

3.2. Comparison of pressure predictions against the literature results

Aortic pressure waveforms were obtained from Remington andWood (1956) and implemented as input into the model. In Fig. 4,a comparison is made against the data of Remington and Wood(1956) at the locations of 20 and 40 cm from the aortic entrance. InFig. 4(a) we use model parameter values chosen through our ownmodel development (Johnson, 2010), whereas in Fig. 4(b) we usethe values implemented by Olufsen (2004), as given in Table 1. As itcan be seen, the model used here performs equally well predictingalmost exactly (0.22 and 0.25 s) the same primary peak times (0.21and 0.24 s) as in the data and exactly the same as in the older model.On other measures, our predictions get even closer: for example,

ured using applanation tonometry on the carotid and radial arteries to input into the

. (a) Measured carotid artery pressure (filled circles) and pressure reconstructed from

b) Measured (dashed) and predicted (continuous) radial artery pressure. Pressure

sured radial artery pressure (filled circles) and pressure reconstructed from the first

d (dashed) and predicted (continuous) carotid artery pressures. Pressure prediction

D.A. Johnson et al. / Journal of Biomechanics 44 (2011) 869–876 873

the relative peak heights errors were 0.88% and 2.81% for our theoryas compared to 2.63% and 4.68% for Olufsen’s, for the 20 and 40 cmcurves, respectively. Moreover, calculating the area under thecurves, we see that the new model gives a relative error of 2.59%and 3.24% as opposed to 2.88% and 4.12%, for the 20 and 40 cmcases, respectively. In the work of Olufsen (2004), there were modelassumptions made towards important parameters—see Table 1. Inthe development in our model, we chose to match more closelywith the physiology of the capillaries. Fig. 4 shows that with theparameters chosen in this model, the agreement with the experi-mental data, though not perfect, is better.

3.3. Experimental validation using applanation tonometry

Additional experimental work has been performed whereapplanation tonometry has been applied upon the carotid, radial,

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Fig. 6. Effect of several model parameters on pressure predictions in the aorta using the

(b) flow predictions for the base case, using normal values for hematocrit (45%), plasma v

et al. (2010a). (c) Pressure profiles for a 20% increase in hematocrit from base case. (d) P

profiles for a 100% increase in stiffness in all vessels from base case. (f) Pressure profile

and femoral arteries of young healthy volunteers for further vali-dation of the model. Several pressure measurements were madeof the carotid and radial arteries. Fig. 5 shows the comparison ofmeasured and predicted pressures. In Fig. 5b the predicted radialartery pressure is compared against the measured one after fittingthe carotid pressure (Fig. 5(a)). In the lower panels (Fig. 5(c,d)), wedo the reverse.

The carotid and radial profiles show very distinct systolic anddicrotic peaks, which were reproduced well in the model using aten-harmonic Fourier decomposition as Figs. 5a and c show. Whencarotid pressure is used as an input, the corresponding predictionfor the radial pressure (Fig. 5(b)) shows systolic and dicrotic peaksof heights 137 and 76, and at times 0.28 and 0.62, respectively, veryclose to the measured values, 140 and 75 for the height and 0.26and 0.62 for the times. Similarly, when the radial pressure is used asinput, the predicted carotid pressure (Fig. 5(d)) shows systolic anddicrotic peaks of heights 124 and 76, and at times 0.16 and 0.50,

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geometry of Fig. 2 and the parameters in the left column of Table 1. (a) Pressure and

iscosity (1.35 cP) and the elasticity and viscoelasticity models described in Johnson

ressure profiles for a 10% increase in plasma viscosity from base case. (e) Pressure

s for a 100% increase in viscoelasticity in all vessels from base case.

D.A. Johnson et al. / Journal of Biomechanics 44 (2011) 869–876874

respectively, very close to the measured values, 116 and 79 for theheight and 0.19 and 0.47 for the times. The predicted and measuredprofiles overlap quite well indicating the absence of a systematicdeviation.

3.4. Sensitivity analysis to key model parameters

Further analysis was made to investigate the sensitivity of pre-ssure predictions to cardiovascular properties, which vary betweenindividuals age and disease state (Nichols et al., 1993). For example,with increased age or atherosclerosis, the vessels become stiff (i.e.less elastic), or in diseases, such as type 2 diabetes, plasma viscositycan rise in a high sugar diet. When hemopoetin is administeredto fight anemia this results in elevated hematocrit levels, whichincrease the whole blood viscosity. Furthermore, the levels of varia-tion considered have been selected to be well within typical varia-tions as seen from the literature. Therefore, the properties consideredinclude the hematocrit (20% increase; Tefferi, 2003), the plasmaviscosity (20% increase; Houston et al., 1949), the elasticity/rigidity ofthe vessel wall (100% increase; Learoyd and Taylor, 1966), and theviscoelasticity of the vessel wall (100% increase; Learoyd and Taylor,1966). In Fig. 6 we report the predictions on the progression ofpressure profiles along the aorta as a function of distance from theheart corresponding to different conditions, keeping the same inputflow profile (shown in Fig. 6b) as in the base case (shown in Fig. 6a).Note that the negative flow seen in Fig. 6b is primarily a model artifact

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Fig. 7. Effect of several model parameters on predicted aortic pressures. Differences b

parameters in the left column of Table 1) and altered parameter cases are shown. (a)

differences for a 10% increase in plasma viscosity from baseline. (c) Pressure differenc

differences for a 100% increase in viscoelasticity in all vessels from baseline.

caused by the lack of an aortic valve, which cannot be accommoda-ted in the linear (frequency-domain) model. Fig. 6c and d showthat the main effect of increases in hematocrit and plasma viscosityis a seemingly uniform vertical (y)-shift in the pressure, due to anincrease in Poiseuille resistance, while Fig. 6e shows that increases invascular stiffness cause significant enhancement to the main pressurepeak.

The underlying conditions for the observed changes in thepressure profiles can be hinted at most sensitively using the differ-ences in the model predictions from the base case. Those differ-ences are shown in Fig. 7 corresponding to the profiles described inFig. 6. Fig. 7 reveals distinctive qualitative differences in the patternof the predicted changes. When either the hematocrit levels or theplasma viscosity is increased (Fig. 7a and b) we see a y-shift uniformchange in the pressure predictions, with a secondary variationproportional to the second harmonic (two peaks and troughs).In contrast, a change in the vessel elasticity causes a fairly welldefined, triangular in shape, main peak (Fig. 7c). A change in visco-elasticity results in an almost null on the average change inthe pressure with a minor contribution proportional to the firstharmonic (one peak and one trough). These are very characteristicchanges, specific to specific parameters whereas invariant to theabsolute magnitude of the changes (as also documented fromadditional results, obtained for different levels of parametervariations, shown in the Supplemental Material). The very char-acteristic specificity between the pattern and the cause can be usedto discriminate between 3 of the 4 factors examined here.

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etween pressure in the baseline (being the model geometry shown Fig. 2 and the

Pressure differences for a 20% increase in hematocrit from baseline. (b) Pressure

es for a 100% increase in vessel stiffness in all vessels from baseline. (d) Pressure

D.A. Johnson et al. / Journal of Biomechanics 44 (2011) 869–876 875

4. Discussion

There has been recently a surge of interest in models and methodsaimed to infer the arterial wall elasticity/viscoelasticity from blood flowmeasurements (Shirwany and Zou, 2010) as changes in those valueshave been associated to aging and/or various diseases (Mackenzie et al.,2002; Shirwany and Zou, 2010). Another avenue is followed here,identifying distinctive patterns in the changes in the blood pressure atselected points across the arterial network as a result to changes incritical parameters. It is really the robustness of these differentialpattern predictions to the details of the base case and the absolutemagnitude of the changes that especially motivates the present work.

The most important results are provided in Fig. 7. We see there thatthe difference in the predicted pressure profiles along the aortic rootexhibits distinct patterns characteristic to the varied parameter.Sensitivity analyses of 1D models have been also performed in thepast (Alastruey, 2006). However, here we present the results differ-ently, focusing on the patterns. The fact that we see a dominance of theconstant term plus the second harmonic for the effects of hematocritand plasma viscosity changes, a triangular main peak for the effects ofthe vessel elasticity and proportionality to the first harmonic, forchanges in the viscoelasticity, is very significant. It means that (a) theobserved results can be used to distinguish with high confidence fromend data (pressure measurements) the principal causes behind them(three from the four) (b) that those results are robust, not dependent onthe magnitude of the events and (c) to a first-order effect can also beused when multiple causes are operating. This has the potential toallow 1D models to be used for (a) direct personalization of keyparameters because observed differences between predictions andexperiments suggest changes to specific parameters, and (b) inferenceof physiological factors responsible for changes in experimental data.Even in the cases of increased hematocrit and plasma viscosity thatprovide indistinguishable results, a very simple hematocrit analysis caneasily resolve the issue. If the analysis points to alterations in plasmaviscosity or hematocrit (which have similar effects on pressureprofiles), a simple blood test can resolve the ambiguity.

5. Conclusions

A new use for a one-dimensional blood flow model has beenpresented based on an analysis of patterns in the differences seenbetween pressure profiles obtained under normal (base case) anddiseased (test cases) conditions. The approach has been illustratedusing a recently developed model using only physical parametersthat can be independently evaluated or can be readily estimatedfrom the literature, if desired, to represent average, physiological,results. With parameters that were not optimized or fitted, weshowed first that the predicted results for pressure showed goodagreement with experimental results obtained from the literatureor in our own laboratories. The demonstrated sensitivity of the modelto key parameter variations can lead to potential applications fordetection of pathological conditions. We believe that 1D models canfind usage for their own right, in, for example, helping physicians inassessing changes observed between routine clinical blood pressuremeasurements as to their possible physiological origin.

Conflict of interest statement

None declared.

Acknowledgements

DAJ acknowledges the generous support of the National ScienceFoundation’s Integrative Graduate Education and Research Traineeship

(NSF-IGERT) Program at the University of Delaware and the NASADelaware Space Grant Consortium Fellowship Program (NASA-DESGC). JWE acknowledges the support received from the Departmentof the Health, Nutrition and Exercise Sciences of the Universityof Delaware. WCR acknowledges support from the University ofDelaware Research Foundation.

Appendix A. Supplementary Information

Supplementary data associated with this article can be found inthe online version at doi:10.1016/j.jbiomech.2010.12.003.

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