Biomechanics and Modeling in Mechanobiology

, Volume 15, Issue 5, pp 1345–1353 | Cite as

Adaptive outflow boundary conditions improve post-operative predictions after repair of peripheral pulmonary artery stenosis

  • Weiguang Yang
  • Jeffrey A. Feinstein
  • Irene E. Vignon-Clementel
Original Paper

Abstract

Peripheral pulmonary artery stenosis (PPS) is a congenital abnormality resulting in pulmonary blood flow disparity and right ventricular hypertension. Despite recent advance in catheter-based interventions, surgical reconstruction is still preferred to treat complex PPS. However optimal surgical strategies remain unclear. It would be of great benefit to be able to predict post-operative hemodynamics to assist with surgical planning toward optimizing outcomes. While image-based computational fluid dynamics has been used in cardiovascular surgical planning, most studies have focused on the impact of local geometric changes on hemodynamic performance. Previous experimental studies suggest morphological changes in the pulmonary arteries not only alter local hemodynamics but also lead to distal pulmonary adaptation. In this proof of concept study, a constant shear stress hypothesis and structured pulmonary trees are used to derive adaptive outflow boundary conditions for post-operative simulations. Patient-specific simulations showed the adaptive outflow boundary conditions by the constant shear stress model to provide better predictions of pulmonary flow distribution than the conventional strategy of maintaining outflow boundary conditions. On average, the relative difference, when compared to the gold standard clinical test, in blood flow distribution to the right lung is reduced from 20 to 4 %. This suggests adaptive outflow boundary conditions should be incorporated into post-operative modeling in patients with complex PPS.

Keywords

Peripheral pulmonary artery stenosis (PPS) Blood flow modeling Outflow boundary condition Adaptation 

1 Introduction

Peripheral pulmonary artery stenosis (PPS) is characterized by multiple and often diffusely distributed narrowed ostia of the branch pulmonary arteries (PA) (Fig. 1). It is frequently associated with Alagille and Williams syndromes (genetic disorders affecting approximately 1 in 10,000 births) and can result in pulmonary flow disparity and right ventricular (RV) hypertension (Krantz et al. 1997; Turnpenny and Ellard 2012; Cunningham et al. 2013). For severe PPS, catheter-based or surgical intervention is required. Catheter-based techniques, in general, have mixed results, and given the risk of vessel dissection and rupture, balloon angioplasty of PPS remains a high-risk procedure. As the stenoses are often distal and complex, surgical repair may be quite difficult though excellent results are possible (Monge et al. 2013). Due in large part to the inherent complexity of the stenoses, and the individuality of each case, no established, uniform, optimal surgical strategy exists. The ability to predict the post-operative hemodynamics remains more of a guessing game than anything else, leading to highly variable and often unpredictable results. With the proper tools, however, standardized approaches based on quantitative data could be developed, analyzed and refined to improve outcomes; computational fluid dynamics (CFD) may serve as one such tool. CFD has been used to study congenital heart diseases since 1990s, and Dubini et al. (1996) performed the first numerical flow simulations for the extracardiac Fontan procedure, the final stage to treat single ventricle defects (de Leval 2005). Their numerical studies led to the adoption of the simulation-derived off-set connection in clinical practice (de Leval 2005). Hemodynamic modeling has since been extended to a variety of surgeries for congenital heart diseases (de Zelicourt et al. 2010; Tang et al. 2010; Vignon-Clementel et al. 2010; LaDisa et al. 2011; Pennati et al. 2013). In these studies, three-dimensional (3D) models are usually created for large vessels where the surgery is performed, and lumped parameter models that represent distal small vessels or the heart are coupled to the 3D model to provide boundary conditions.
Table 1

Pre-operative clinical evaluations. PAP \(=\) pulmonary artery pressure; BSA \(=\) body surface area; MPA \(=\) main pulmonary artery. PAP is shown as systolic/diastolic/mean, and lung perfusion scans are shown as % flow to the right lung/left lung

Patient

Gender

Age (year)

PAP (\(\mathrm{mmHg}\))

Lung perfusion (%)

BSA (\(\mathrm{m}^2\))

MPA flow (\(\mathrm{L}/\mathrm{min}\))

A

M

2.8

71/16/45

86/14

0.58

2.5

B

M

1.9

72/14/38

80/20

0.48

2.5

Fig. 1

CT angiograms of a representative patient with bilateral PPS (patient B). Stenoses in the RPA and LPA are marked by arrows. The proximal left PA (LPA) is diminutive. In the right PA (RPA), multiple stenoses are located at the origins of lobar branches and followed by aneurysmal dilation

A common assumption employed in previous virtual surgical studies is that the anatomic and physiologic properties of the downstream vasculature do not change significantly after the surgery, and as such, the boundary conditions are kept unchanged (Bove et al. 2003; Yang et al. 2012). Although a closed-loop heart model allows one to model change of incoming flow, no changes are made in boundary condition parameters (Baretta et al. 2011). While this may be generally acceptable for most questions (and models), those for pulmonary artery diseases may yield inaccurate results as experimental studies have shown that changes in blood flow can result in acute vessel adaptation and chronic growth and remodeling. More specifically, arteries respond to flow perturbations in an attempt to maintain the same level of wall shear stress (Kamiya and Togawa 1980; Langille and O’Donnell 1986; Brownlee and Langille 1991; Razavi et al. 2013). Since PA patch plasty for complex PPS significantly alters the pre-operative geometry and pulmonary flow is redistributed, pulmonary adaptation likely occurs in distal branches. Thus maintaining pre-operative outflow boundary conditions in post-operative models of patients with PPS has the potential to yield inaccurate results.

The goal of this study is to develop a mathematical model that incorporates the aforementioned experimental observations (i.e., constant shear stress) into flow simulations and to better predict post-operative hemodynamic properties.

2 Methods

2.1 Model construction and flow simulation

Computed tomography (CT) images for two Alagille patients with PPS were identified from a cardiac image database under an institutional review board approved protocol. Figure 1 demonstrates the classic pulmonary artery stenoses commonly seen in this syndrome. Pre-operative lung perfusion shows greater than 80 % of pulmonary flow going to the RPA. Pre-operative demographics and diagnostic data are summarized in Table 1.

Based on CT images, SimVascular (simvascular.org) was used to create patient-specific models which include the central pulmonary arteries and lobar/segmental branches (Schmidt et al. 2008; Marsden et al. 2009). Up to four generations of bifurcation are modeled in order to include PPS identified in CT images with sufficient extension distal to the PPS. Post-operative models were created by locally enlarging the ostial stenosis in the LPA under the guidance of the surgeon.

A finite element solver that employs a coupled momentum method to account for the arterial compliance is employed to simulate blood flow in the PAs (Figueroa et al. 2006). The wall is modeled by linear elastic equations. The equations for wall deformation are coupled into the fluid governing equations (Figueroa et al. 2006). Inflow waveforms were taken from the study by Tang et al. (2011) and scaled to match patient’s cardiac index and PA pressures. Inflow conditions were assumed to be unchanged immediately after the surgery. To verify the effect of the assumptions for inflow on cardiac output, simulations with 75 and 125 % of the baseline cardiac output were performed, respectively. For outflow boundary conditions, three-element Windkessel models (resistor-capacitor-resistor, RCR) were employed following previous simulations for the PAs (Spilker et al. 2007; Vignon-Clementel et al. 2006; Troianowski et al. 2011). The downstream arteries in the RPA or LPA are parallel circuits, and the total downstream resistance for the arteries beyond the 3D model can be approximated by the following equation,
$$\begin{aligned} \frac{1}{R}=\sum ^{n}_{i=1}\frac{1}{R_{i}}, \end{aligned}$$
(1)
where R is the total downstream resistance for the RPA or LPA, n is the number of outlets of the RPA or LPA in the patient-specific model, and \(R_{i}\) is the resistance of the Windkessel model for the ith outlet in the RPA or LPA. Individual outlet resistance \(R_{i}\) for the RPA or LPA is assumed to be inversely proportional to the outlet area \(A_{i}\) (Troianowski et al. 2011):
$$\begin{aligned} \frac{1}{R_{i}}=\frac{A_{i}}{\sum ^{n}_{j=1}{A_{j}}R}, \end{aligned}$$
(2)
where R is the total downstream resistance for the RPA or LPA and n is the number of outlets of the RPA or LPA. Initial resistance values for the RPA or LPA can be estimated by dividing the PA pressure by the RPA or LPA flow. For each \(R_{i}\), a three-element Windkessel model is obtained by fitting the impedance spectra of a morphometry-based tree with the closest resistance. We refer readers to the study by Spilker et al. (2007) for technical details. Calculating steps from resistance values to Windkessel model parameters were repeated to achieve target PA pressures and pulmonary flow splits in the pre-operative stage. These tuned RCR parameters are referred to as baseline boundary conditions.

2.2 Constant shear stress model

To predict post-operative pulmonary blood flow distribution, a constant shear stress model is proposed to adapt outflow boundary conditions. The model incorporates the constant shear stress hypothesis into the structured trees (Olufsen 1999; Olufsen et al. 2000, 2012). Based on the studies by Kamiya and Togawa (1980), we hypothesize the distal PAs attempt to maintain the pre-operative shear stress level by vasodilation/vasoconstriction in response to surgically altered pulmonary flow.

The structured tree model consists of bifurcating trees (Olufsen 1999; Olufsen et al. 2000, 2012; Kheyfets et al. 2013). For a parent segment with a radius of \(r_p\), two daughter segments are scaled to have radii of \(\alpha r_p\) and \(\beta r_p\). The recursive bifurcating process is terminated when the minimum radius \(r_{min}\) is reached. For each segment, the length l is related to the radius, \(l=12.4r^{1.1}\) (Olufsen 1999; Olufsen et al. 2000, 2012). Figure 2a illustrates a structured tree and the scaling for daughter segments. If a root radius \(r_{0}\) is given, one can generate a structured tree with parameters \(\alpha \), \(\beta \), l and \(r_{min}\). A connectivity map is created to record the parent and daughter segments for each segment. The resistance for a segment with a radius r is given by \(\frac{8\mu l}{\pi r^4}\), where \(\mu \) is viscosity of the blood. Thus the total resistance of a structured tree can be obtained by summing up the resistance from the most distal segments based on the bifurcation map. The shear stress \(\tau \) in a segment with a radius r and flow rate q is given by \(\tau =\frac{4\mu q}{\pi r^3}\). The flow rate and shear stress in each segment can be calculated if the structured tree and its root flow rate are determined.
Fig. 2

a Illustration of structured trees for the PAs. If a root segment has a radius of r, the size for the daughter segments is scaled by \(\alpha \) and \(\beta \). For each 3D outlet, a structured tree is created such that the total resistance of the structured tree is equal to the resistance of the three-element Windkessel model used for the same 3D outlet in simulations. b Algorithm of structured tree adaptation for post-operative outflow boundary conditions (BCs)

By applying the constant shear stress hypothesis to the structured tree, a segment in the structured tree will be dilated if its flow rate is increased and vice versa. The new segment radius \(r'\) is given by
$$\begin{aligned} r'=\root 3 \of {\frac{q'}{q}}r, \end{aligned}$$
(3)
where \(q'\) is the new flow rate. For a given new flow rate \(Q'\) at the root of the structured tree, the segments in the structured tree are dilated and constricted to obtain an adapted resistance value that maintains the original shear stress in the structured tree.

Figure 2b summarizes the major steps to predict post-operative outflow resistance. First, baseline pre-operative conditions (inflow and three-element Windkessel models) are identified using the conventional methods described in Sect. 2.1. For each 3D outlet, a baseline structured tree is created to match the resistance of the three-element Windkessel model used in the pre-operative simulation. The matching structured tree can be identified by optimizing the root radius, while bifurcation parameters \(\alpha \) and \(\beta \) and minimum radius \(r_{min}\) are taken from Olufsen (1999), Olufsen et al. (2012) and kept unchanged because the lesions are found in the proximal pulmonary arteries only and distal vasculature is not malformed. Herein, \(\alpha =0.9\), \(\beta =0.58\) and \(r_{min}=0.05\) mm are used (Olufsen et al. 2012). Although it is possible to find a structured tree that satisfies the target resistance by optimizing \(\alpha \) and \(\beta \), varying the root segment size minimizes the number of uncertain parameters without significantly changing the tree structure. Then a flow simulation is carried out with the post-operative geometry and pre-operative boundary conditions to obtain the flow changes due to the surgical reconstruction only. By comparing the differences in the outlet mean flow, the structured trees are adapted to restore shear stress using Eq. 3. Since the structure of the tree is unchanged and only segment radius is updated, it is easy to obtain the new set of total resistance for each outlet by adding up the resistance backward again from the most distal segments. Then a new set of Windkessel parameters is generated by fitting morphometry-based trees after the new resistance for each outlet is determined. Lastly, the post-operative flow simulation is performed with adaptive outflow boundary conditions to predict immediate post-operative pulmonary flow distribution.

2.3 Validation

A post-operative lung perfusion scan is routinely performed to evaluate the results of PPS repair and determine a new baseline perfusion distribution before the patient is discharged (1–2 weeks after the surgery). We compare simulation-based predictions with these post-operative lung perfusion scans.
Fig. 3

Time-averaged pressure distribution in the pre- and post-operative models. The pre-operative baseline boundary conditions are tuned to match the clinical measurements. Post-operative results include baseline boundary conditions and adaptive outflow boundary conditions. PPS is marked by arrows

Fig. 4

Volume rendered velocity magnitude in the pre- and post-operative models. The baseline boundary conditions are tuned to match the pre-operative clinical measurements. Adaptive outflow boundary conditions given by the constant shear stress model are applied to the post-operative model to predict the post-operative flow fields

3 Results

Figure 3 shows the pressure distribution for the pre- and post-operative stages. Post-operative models with baseline boundary conditions show the impact of obstruction relief only. The size of the proximal LPA in patient A was initially 15 % of the RPA diameter. Thus it is not surprising that the ostial stenosis is the leading factor that causes RV hypertension. As expected, the overall PA pressure is effectively reduced after the hypoplastic LPA is reconstructed. In patient B, a significant pressure drop is also found in the the right upper lobe in the pre-operative stage because the right upper lobe is more stenotic than other lobes with stenoses followed by aneurysmal dilation. Although the surgical reconstruction is performed on the LPA only, the pressure difference between the right upper lobe and other lobes in the RPA is reduced as flow is redistributed away from the RPA. We observed lower PA pressures in the models with adaptive outflow boundary conditions due to reduced resistance of the structured trees for the LPA.

In a second step, post-operative simulations are carried out with the adaptive boundary conditions. Figure 4 shows a comparison of velocity fields in the pre- and post-operative stages. In the pre-operative model, flow with high velocity is evident in the stenotic right lung segments, but low velocity is seen in LPA branches due to the lack of perfusion. In the post-operative stage, velocity magnitude in the main LPA is reduced due to surgical reconstruction, while improved perfusion in the LPA results in an expected increased velocity in branch PAs as the branches are not enlarged. In patient B, the flow to the right upper lobe is reduced by 16 % in the post-operative stage, but the velocity magnitude is still relatively high due to an unrepaired stenosis.
Table 2

Immediate post-operative lung perfusion scans (LPS) and simulation-derived pulmonary flow distribution (RPA/LPA). Simulations are performed on virtual post-operative models with baseline and adaptive outflow boundary conditions (BCs). Prediction errors are relative to the measured flow distribution to the RPA

Patient

Post-op LPS

Baseline BCs

Adaptive BCs

Errors

A

53 %/47 %

65 %/35 %

52 %/48 %

23 % \(\rightarrow \) 2 %

B

63 %/37 %

74 %/26 %

67 %/33 %

18 % \(\rightarrow \) 6 %

Fig. 5

Resistance for ostial LPA stenosis and distal PAs in the pre- and post-operative stages. The post-operative downstream resistance is predicted by the adaptive boundary conditions. The resistance contributed by PPS in the pre- and post-operative stages is evaluated under the baseline boundary conditions

Table 2 shows post-operative pulmonary flow distribution from the post-operative lung perfusion scans and baseline and adaptive outflow boundary conditions. The conventional strategy that maintains pre-operative outflow boundary conditions yields larger errors than the adaptive outflow boundary conditions derived by the constant shear stress model.

Figure 5 compares the downstream resistance as calculated by Eq. 1 with the resistance due to PPS in the pre- and post-operative stages. The post-operative downstream resistance is predicted by the constant shear stress model. The resistance due to PPS in the pre- and post-operative stages is obtained by evaluating the pressure drop and flow rate across the stenosis under the same boundary conditions. Enlarging the stenotic area reduces the corresponding resistances to a negligible level. In patient A (Fig. 5), the downstream LPA resistance is four times greater than the downstream RPA resistance. The LPA ostial stenosis resistance is nearly four times higher than the LPA downstream resistance. By relieving the ostial stenosis in the LPA, the difference between the LPA and RPA has been greatly reduced. In patient B, the LPA downstream resistance is three times higher than the RPA. But the LPA ostial stenosis is less severe than for patient A. Its resistance is 60 % lower than the LPA downstream resistance. As in patient A, the surgery has eliminated the negative impact of the ostial stenosis, although less drastically.

By calculating the shear stress for each segment in the structured trees, mean shear stress and standard deviations (STDs) for the distal RPA and LPA under different conditions are shown in Fig. 6. The shear stress in the LPA trees is elevated when the baseline boundary conditions are used in the post-operative models. In the RPA trees, the reverse happens. In patient A, untreated focal stenoses at the origin of the right middle and lower lobe PAs resulted in uneven flow distribution in the RPA. Therefore the shear stress in structured trees for patient A’s RPA varies largely.
Fig. 6

Mean wall shear stress and standard deviations in the structured trees that represent the distal RPA and LPA. For each patient, three conditions were simulated: (1) pre-operative, (2) post-operative with baseline boundary conditions and (3) post-operative with adaptive boundary conditions

Additional simulations were performed to evaluate the impact of inflow uncertainty on the pulmonary flow split. The flow split varied by only 1 % with a change of \(\pm 25~\%\) in the inflow rate.

4 Discussion

In this study, a constant shear stress model was employed to predict downstream pulmonary vascular resistance in response to surgically altered pulmonary flow. We have demonstrated that the adaptive outflow boundary conditions outperform the constant boundary conditions in estimating the in vivo measurements for post-operative pulmonary flow distribution. This is the first hemodynamic modeling for PPS with respect to surgical reconstruction and pulmonary artery adaptations.

Previous simulations associated with surgeries done on the pulmonary arteries usually assumed that the resistance created by the 3D model is insignificant when compared to the distal pulmonary vascular resistance, and that the outflow boundary conditions could be maintained for post-operative simulations. Compared to those PA stenoses simulated in the cavopulmonary connections (Marsden et al. 2010; Yang et al. 2012; Schiavazzi et al. 2015), the patients with PPS show a more complex structure with a larger pressure drop across the LPA stenosis. Figure 5 shows the resistance created by PPS is comparable to the total downstream pulmonary vascular resistance. Repairing severe PPS can significantly alter the total resistance, and consequently the pulmonary flow is redistributed.

In addition to the morphological changes by the surgical reconstruction, the discrepancies between the in vivo measurements and results predicted by the conventional strategy indicate that the distal PAs, which are not included in the 3D model, also play a role in redistributing pulmonary flow as they can adapt to the changes from the upstream. Kamiya and Togawa (1980) demonstrated that surgically altered flow perfusion could lead to changes in vessel size, yet shear stress was maintained. Razavi et al. (2013) found that banding the LPA in rats resulted in significant changes in vessel caliber and pulmonary vascular resistance such that normal shear stress was restored. Therefore surgical reconstruction for PPS created an initial change in pulmonary flow distribution; then, mechanobiological responses were triggered by the altered pulmonary perfusion making the downstream resistance different from the baseline. In other words, the resulting post-operative pulmonary flow distribution was determined by the obstruction relief in the proximal PAs combined with the vasoactive responses in the distal PAs. Thus, the use of constant outflow boundary conditions ignored the contribution from the latter factor. In the adaptive outflow boundary conditions, the post-operative outflow resistance is modeled as a function of the flow perturbations solely due to the surgical construction. Since the LPA perfusion was increased by 21 % in patient A by enlarging the diminutive LPA, the constant shear stress model reduced the resistance for the distal LPA by 60 % achieving satisfactory agreement with the in vivo measurement. Although a similar result was obtained in patient B, the drop in resistance for the distal LPA was moderate due to 6 % increase in the LPA perfusion. This is consistent with the lower PPS severity of this patient, as assessed by the relative contribution of the ostial stenosis resistance to the total resistance (Fig. 5).

The shear stress in the distal RPA and LPA represented by structured trees changes accordingly with flow changes. In Fig. 6, the rise in shear stress from the pre-operative stage to the post-operative stage with baseline boundary conditions stems from increased LPA flow and fixed segment size. The shear stress in the distal LPA and RPA is reduced and increased, respectively, after tree segments are adapted by the constant shear stress model. Because pulmonary flow and resistance are coupled, adapting the downstream resistance results in further changes in pulmonary flow and shear stress. Thus, the shear stress does not return to the initial values in the post-operative model under adaptive outflow boundary conditions indicating that the immediate post-operative pulmonary flow distribution does not reach homeostasis. Based on previous findings, we speculate that acute vasoreactivity resulted in a drop in the distal resistance and improved the perfusion in the LPA during the immediate post-operative period, and the interim growth of the LPA in patients, which took several months, was able to maintain this improved pulmonary flow distribution and further increased perfusion in the LPA.

The main contribution of this study is to model the post-operative pulmonary flow accounting for the distal vasoreactivity in response to altered flow. The method used in this study is easy to implement and effective to model patients undergoing PPS repair. However the post-operative predictions in this study are still representative of immediate/short-term results. The structured trees are adapted once to derive post-operative outflow boundary conditions. To predict post-operative pulmonary flow distribution in multiple stages, one may couple the constant shear stress model into simulations to form a feedback control loop. The major challenge is to identify the stopping criteria or conditions of equilibrium. Because obstruction relief results in increased perfusion that decreases downstream resistance to maintain constant shear stress. Then decreased resistance further skews the pulmonary flow distribution. Pulmonary flow will monotonically increase in one lung with obstruction relief and decrease in the other lung ultimately leading to a non-physiologic flow split if a loop of simulations is performed using the constant shear stress model without proper stopping criteria. In addition growth-related issues need to be taken into account for intermediate and long-term predictions. Wall shear stress and pressure have been found to mediate the vascular growth and remodeling (Humphrey 2008). Recently, mechanical stimulus was computed by flow simulations and provided for growth models to determine a grown configuration (Figueroa et al. 2009; Lindsey et al. 2014). Growth models could be coupled to flow simulations in order to better represent the growth of the PAs.

5 Limitations

Although only two patients are included in this proof of concept study, both simulations and previous experimental studies suggest that the use of constant outflow boundary condition does not suffice for modeling blood flow in the PA undergoing surgical reconstruction. Ongoing studies with increased patient numbers may add additional validation.

Since patient-specific inflow waveforms and wall properties are not available, boundary conditions and wall stiffness data were tuned to match the available patient-specific mean flow and pulse pressure measurements. Instantaneous velocity and pressure might contain larger errors than the averaged quantities. Post-operative 3D imaging was not performed; thus, the virtual surgery geometry could not be validated. In a future study, it would be interesting to obtain post-operative 3D images to identify the impact of geometric discrepancies on predictions.

One-dimensional structured tree is employed to derive resistance values that are used in 3D–0D coupling simulations. Pulsatile flow and wave propagation effects in the structured tree are not taken into account. A direct 3D–1D coupling could be used in future studies to provide additional hemodynamic information.

6 Conclusions

The use of constant outflow boundary conditions leads to inaccurate predictions for modeling PPS repair. We have described a constant shear stress model, which can be used to systematically estimate the post-operative outflow boundary conditions and improve the conventional modeling strategy. A longitudinal study is warranted to confirm these findings

Notes

Acknowledgments

This study is supported by the France-Stanford center for interdisciplinary studies and the Vera Moulton Wall Center for Pulmonary Vascular Disease. We would like to acknowledge the assistance of Ana Ortiz for model construction and Dr. Frank Hanley and Dr. Frandics Chan for their expertise on cardiothoracic surgery and imaging. We would also like to thank Prof. Alison Marsden for her helpful suggestions and support.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Weiguang Yang
    • 1
  • Jeffrey A. Feinstein
    • 1
  • Irene E. Vignon-Clementel
    • 2
    • 3
  1. 1.Department of Pediatrics, School of MedicineStanford UniversityStanfordUSA
  2. 2.INRIA Paris-RocquencourtLe ChesnayFrance
  3. 3.Sorbonne Universités UPMC Univ. Paris 6ParisFrance

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