Endovascular coil embolization is now widely used to treat cerebral aneurysms (CA) as an alternative to surgical clipping. It involves filling the aneurysmal sac with metallic coils to reduce flow, induce clotting, and promote the formation of a coil/thrombus mass which protects the aneurysm wall from hemodynamic forces and prevents rupture. However, a significant number of aneurysms are incompletely coiled leading to aneurysm regrowth and/or recanalization. Computational models of aneurysm coiling may provide important new insights into the effects of intrasaccular coil and thrombus on aneurysm wall stresses. Porcine blood and platinum coils were used to construct an in vitro coil thrombus mass (CTM) for mechanical testing. A uniaxial compression test was performed with whole blood clots and CTM, with coil packing densities (CPDs) of 10%, 20%, and 30% to obtain compressive stress/strain responses. A fourth-order polynomial mechanical response function was fit to the experimentally obtained stress/strain responses for each CPD in order to represent their mechanical properties for computational simulations. Patient-specific three-dimensional (3D) geometries of three aneurysms with simple geometry and four with complex geometry were reconstructed from digital subtraction angiography (DSA) images. The CPDs were digitally inserted in the aneurysm geometries and finite element modeling was used to determine transmural peak/mean wall stress (MWS) with and without coil packing. Reproducible stress/strain curves were obtained from compression testing of CTM and the polynomial mechanical response function was found to approximate the experimental stress/strain relationship obtained from mechanical testing to a high degree. An exponential increase in the CTM stiffness was observed with increasing CPD. Elevated wall stresses were found throughout the aneurysm dome, neck, and parent artery in simulations of the CAs with no filling. Complete, 100% filling of the aneurysms with whole blood clot and CPDs of 10%, 20%, and 30% significantly reduced MWS in simple and complex geometry aneurysms. Sequential increases in CPD resulted in significantly greater increases in MWS in simple but not complex geometry aneurysms. This study utilizes finite element analysis to demonstrate the reduction of transmural wall stress following coil embolization in patient-specific computational models of CAs. Our results provide a quantitative measure of the degree to which CPD impacts wall stress and suggest that complex aneurysmal geometries may be more resistant to coil embolization treatment. The computational modeling employed in this study serves as a first step in developing a tool to evaluate the patient-specific efficacy of coil embolization in treating CAs.

## Introduction

Coil embolization is a commonly used procedure for the treatment of intracranial aneurysms [14]. This procedure involves placing metallic coils within the aneurysm sac in order to create a coil-thrombus mass (CTM) that provides protection to the aneurysm wall from hemodynamic forces that could cause rupture and hemorrhage. Coil embolization offers a less invasive option for neutralizing the risk of rupture and has been shown to reduce dependency and death at 10 years compared to neurosurgical clipping [5]. Successful placement of coils within the aneurysm depends on aneurysm morphology, size, location, and operator skill [6].

The goal of coil embolization is to achieve an adequate coil packing density (CPD) or the least amount of remaining residual volume as estimated with digital subtraction angiography (DSA) or magnetic resonance angiography. However, due to aneurysm geometry and location, as well as variability in technique, it may not always be possible to fill the aneurysm with a high CPD. A one-year rebleeding rate of 2.4% has been observed following coil embolization treatment in ruptured aneurysms [2]. Furthermore, recurrence, defined by an increase in the size of the residual volume, has been documented after 11–36% of embolization procedures [7,8]. Even rehemorrhage correlates to the degree of aneurysm occlusion after treatment with up to a 17% chance for aneurysms coiled to less than 70% occlusion [9]. Therefore, inadequate coil packing is a major consideration which may impact outcomes.

Cerebral aneurysm (CA) risk of rupture and response to treatment is thought to depend on numerous morphological and other factors, such as aneurysm shape, sphericity, degree of undulation, aspect ratio of neck to dome, size, wall thickness, and location of the aneurysm relative to the parent vessel and within the cerebral circulation [10]. While each of these factors can be individually correlated to rupture risk, mechanical wall stress and wall tensile strength are the actual determinants [10]. That is, when the mechanical stress acting on the wall at any location exceeds the ability of the tissue to withstand that stress, failure—aneurysm rupture—will occur. Since the goal of coil embolization is to mechanically protect the aneurysm wall with a CTM, the presence of a tool that can accurately predict wall stress in patient-specific aneurysms before and after treatment could have tremendous clinical value.

Multiple studies using computational fluid dynamic (CFD) models have identified wall shear stress as an independent predictor of aneurysm rupture [1113]. However, the more likely mechanism of aneurysm rupture that has been studied to a lesser degree is transmural wall stress, i.e., compressive or tensile forces acting within the wall tissue. Transmural wall stresses obtained from patient-specific, finite element models of abdominal aortic aneurysms have been shown to help predict rupture potential [14,15]. Similar models evaluating transmural wall stress in cerebral aneurysms have shown 70% greater stresses in the neck compared to the dome [16] and higher stresses in the domes of ruptured compared to nonruptured aneurysms [17]. However, to our knowledge, no study has created a computational model to simulate the effect of coil embolization on transmural wall stress in cerebral aneurysms.

In this study, we utilize finite element analysis to evaluate the effect of coil embolization on transmural wall stress in patient-specific aneurysms. We hypothesized that the presence of a CTM reduces peak wall stress (PWS) in the dome of an aneurysm. Further, we hypothesized that the transmural wall stress distribution and peak wall stress depend on aneurysm morphology and CPD after complete formation of CTM. To test these hypotheses, we combined in vitro mechanical testing of a CTM with patient-specific finite element models of aneurysms with a coil embolization treatment to compute the transmural wall stress distribution.

## Materials and Methods

### Reconstruction of Aneurysm Geometry.

Under an Institutional Review Board (IRB) approved protocol (IRB # PRO13080334), digital subtraction angiography scans of seven representative patients undergoing elective coil embolization at the University of Pittsburgh Medical Center were obtained. These seven patients had saccular aneurysms of either a simple geometry (characterized by the presence of only two parent vessels and the absence of daughter aneurysms or vessels originating from the aneurysm dome; n = 3) or a more complex geometry (n = 4). The decision to pursue treatment on each aneurysm was left to the discretion of the attending physician. Given that a vast majority of intracranial aneurysms are saccular [18], we chose to use only saccular aneurysms from various intracranial locations for our study.

The DSA scans for each aneurysm were first imported into Mimics (Materialise, Plymouth, MI), where lumen boundaries of the CA and associated parent vessels were segmented using a level-set thresholding algorithm. The CTM was modeled by placing a boundary in the neck of each aneurysm without occluding the parent vessels. This selection was chosen based on visual inspection of post-treatment DSA images showing coils filling the aneurysm sac without extension into the parent vessels. After isolation, the boundaries were rendered into coarse three-dimensional (3D) volume models of the CA. The 3D models were further smoothed, corrected for rendering flaws, patched in Geomagic (3D Systems, Rock Hill, SC), and exported as aneurysm wall surface geometries.

### Mechanical Properties of Aneurysm Wall.

The aneurysm wall was modeled as a homogenous shell with the mechanical properties of a nonlinear, isotropic, hyperelastic, and incompressible material. To characterize the mechanical properties of the aneurysm wall, we chose the three-parameter Mooney–Rivlin strain energy function (Eq. (1)) based on the biomechanical classification of human CAs previously described by Costalat et al. [19]
$W=C10(I1−3)+C01(I2−3)+C11(I1−3)(I2−3)$
(1)

The coefficients C10 (59.8 kPA), C01 (16.8 kPA), and C11 (5710 kPA) are the material constants characterizing the material properties of an intermediate stiffness CA as reported by Costalat et al. [19], and I1 and I2 are the first and second invariants of the Cauchy-Green strain tensor, respectively. The aforementioned strain energy function was implemented in abaqus finite element analysis software (Version 6.12-3, © Dassault Systèmes Simulia Corp., 2012, Providence, RI) using the UHYPER subroutine. The thickness of the aneurysm wall was set as 0.36 mm and assumed to be uniformly distributed [19].

### Characterizing Mechanical Properties of the Coiling Thrombus Mass.

To our knowledge, the bulk mechanical properties of a CTM have not been previously reported. We therefore performed in vitro confined compression testing on CTM analogs to estimate these properties for use in our finite element modeling.

#### Development of In Vitro Coil Thrombus Mass Analogs.

Ethylenediaminetetraacetic acid chelated porcine blood and one metallic aneurysm coil (Target, Stryker Neurovascular) were obtained. Specially designed cuvettes that could be directly mounted onto our custom mechanical compression testing device were used as wells to make four different types of specimens: blood clot without coils and CTM at varying packing densities (CPD: 10%, 20%, and 30%). The surgical coil manufacturer specifications, diameters, and Angiocalc software were used to provide the specific length of coil required to achieve each of the desired CPD for the specified cuvette volume. Ethylenediaminetetraacetic acid chelated porcine blood was freshly reconstituted with calcium chloride and injected into the cuvette from bottom up to avoid entrapment of any air bubbles. The cuvettes were left at room temperature for about 2 h to ensure that thrombus was present in the CTM prior to biomechanical testing.

##### Confined Compression Testing.

The CTM was contained in a cuvette and mounted onto a 5 N load cell (Transducer Techniques, Temecula, CA) in a custom, calibrated micro-indentation system for confined compression testing. The CTM height and inner diameter of the cylindrical cuvette were measured before testing. The previous unpublished work in our laboratory suggests that the viscoelastic effects on the estimation of elastic properties of intraluminal thrombus are minimal [20]; therefore, preconditioning was not applied prior to compression testing. A perforated flat punch was used as an indenter to allow displacement of the plasma during compression of the CTM. A schematic of the confined compression testing is provided in Fig. 1. The force measurement obtained from the load cell prior to the punch tip contacting the CTM was considered to be a baseline, and all subsequent force measurements were normalized to zero. For compression testing, the punch was driven toward the mounted specimen at the rate of 0.01 mm/s. As the punch advanced into the CTM sample, the force measurement provided by the load cell gradually increased until cessation of the indentation. The force–displacement data were synchronously collected at a sampling rate of 50 Hz using LabVIEW and converted into stress/strain curves. Three samples (n = 3) were subjected to compression testing for whole blood clot and each CPD. The slopes of tangential lines at the linear portions of the stress/strain curves were used to provide stiffness values for the samples.

#### Mathematical Model for Characterization of the Coiling Thrombus Mass.

To allow for characterization of the CTM in abaqus, a constitutive strain energy function that adequately modeled the measurements obtained from compression testing was developed. First, compressive force measurements were converted into axial First Piola–Kirchhoff stress (Pa)
$Pa=fAo$
(2)
where f is the measured force and Ao is the cross-sectional area of the sample. Uniaxial compression gives rise to the following deformation gradient:
$F=[λa000λt000λt]$
(3)
where the axial stretch ratio λa is defined as
$λa=L0−dL0$
(4)
where $I$ is the fourth-order unit tensor and C is the right Cauchy–Green tensor. The isochoric component of the deformation gradient $F¯$ is given as
$F¯=J−13F , with J=det(F)$
(5)
We used a volumetric-isochoric additive split for the Helmholtz free energy function or strain energy function ($ψ$) as follows:
$ψ(C)=ψiso(C)+ Ψvol(J)$
(6)
It is convenient to use the second Piola–Kirchhoff stress tensor S when developing a strain energy function due to the thermodynamic relation between $ψ$ and S. The fictitious S can be found from isochoric component of $ψ$ as follows [21]:
$S¯=2∂ψiso(C¯)∂C¯$
(7)
and isochoric component of S can be written as
$Siso=J−23ℙ:S¯$
(8)
Upon visual inspection of the data and using statistical regression, it was determined that the axial component (S11) of S varied with the first invariant of the distortional component of C, $I¯1$. Multiple polynomial orders were tested in abaqus and the fourth order was chosen because it was the lowest one that provided a “stable” curve fit
$ψiso=∑i=14Ci(I¯1−3)i$
(9)
For the volumetric part for the strain energy density function $Ψvol$, we used a commonly used finite element formulation for nearly incompressible materials [2224]
$Ψvol(J)=0.5κ(J−1)2$
(10)

where $κ$ is a penalty parameter enforcing incompressibility condition, and $J=det($F), with F as the deformation gradient. The magnitude of $κ$ was guided by using a ratio of bulk modulus to shear modulus near the default ratio of 20 used in abaqus.

The experimental stress–strain data were smoothed by moving average of 30 samples. These data sets were then imported into abaqus and the built-in curve-fitting function was used to curve-fit a reduced fourth-order polynomial strain energy function to the portion of the finite element model representing the aneurysm dome containing the CTM for whole blood clot and CPDs of 10%, 20%, and 30%.

### Boundary Conditions and Finite Element Mesh.

A no-slip or “tie” interaction condition was applied at the interface between the CTM and the aneurysm wall. The ends of the parent vessel were fixed in space (i.e., no translation or rotation). A systolic pressure of 120 mmHg was applied uniformly everywhere on the lumenal surface that would be in contact with nonclotted blood (i.e., inner surface of aneurysm wall, CTM, and parent vessel).

The geometries of the cerebral aneurysm wall surface and CTM solid were meshed in abaqus CAE as a structured mesh and were determined to be mesh independent using the Abaqus mesh inspection tool. The finite element mesh of the aneurysm wall was composed of linear, finite membrane strain, reduced-integration, quadrilateral shell elements (S4R), which are robust and suitable for a wide range of applications. The solid geometry of the CTM was meshed using linear, continuum, hybrid ten-node tetrahedral elements (C3D10H), which are suitable for the hyperelastic materials (that was derived from the confined compressive testing). abaqus standard using implicit integration was used to compute each solution for each finite element mesh geometry.

### Stress Analysis.

abaqus was used to calculate the von Mises stress, which is frequently used to describe the stress field of materials under multi-axial loading conditions [25]. We used Simpson's five-point integration through which the wall thickness and the wall stress at the midsurface are reported. We define the PWS for a given aneurysm model as the 99th percentile von Mises stress acting anywhere on that model [26]. Finite element simulations were run for five conditions for each CA model: no filling, filling with whole blood clot, and filling with CPDs of 10%, 20%, and 30%. The simulations of filling with whole blood clot were both to test a previously unexamined aneurysm filling condition (though, to our knowledge, there is no intervention in clinical practice that fills the aneurysm with clot without the use of coils) and to serve as a control against the CPD10–30% cases.

### Statistical Methods.

ANOVA was utilized to compare stiffness measurements obtained from compression testing of the in vitro CTM with various CPDs. The nonparametric Friedman test was used to compare the mean wall stress (MWS) from finite element simulations between the five conditions of no filling, filling with whole blood clot, and filling with CPDs of 10%, 20%, and 30%.

## Results

### Stiffness Measurements of Coil Thrombus Mass Analogs.

Stress/strain curves obtained from compression testing of CTMs are shown in Fig. 2. The stress–strain relationship was nonlinear, with stiffness increasing as compressive deformation (i.e., negative strain) was applied. An exponential increase of stiffness was observed with increasing CPDs (Table 1). Mean stiffness measurements for blood clots without coil were 3.22 ± 1.47 N/mm2, 10% CPD were 7.31 ± 0.23 N/mm2, 20% CPD were 27.4 ± 13.6 N/mm2, and 30% CPD were 196.0 ± 86.3 N/mm2 (Table 2). Statistical analysis showed a significant difference in stiffness between the 20% and 30% CPD samples compared to blood clot with no coil, and also between the 10% and 30% CPD samples when compared against each other (Table 1).

### Curve Fitting of Coiling Thrombus Mass Stress/Strain Data.

Our mathematical model (Eq. (9)) approximates the experimental stress/strain relationship obtained from mechanical testing to a high degree represented in Fig. 3, where CPD 20% had an R2 value of 0.997 and the R2 values for CPD 10% and 30% were greater than 0.980 (data not shown). The CTM material properties obtained from the regression analysis against experimental data for whole blood clot and the three CPDs are provided in Table 2.

### Mean Wall Stress in Aneurysm Following Coil Embolization Simulation.

Figure 4 displays the finite element-estimated wall stress distribution with no filling, filling with whole blood clot only (no coil), and filling with CPDs of 10%, 20%, and 30% for all seven aneurysms, grouped as simple or complex geometries. Elevated wall stresses were found throughout the dome, neck, and parent artery in simulations of the aneurysm with no filling. Complete, 100% filling of the aneurysms (i.e., without any residual volume) with whole blood clot and CPDs of 10%, 20%, and 30% eliminated regions of high wall stress from the dome and neck regions in simple geometry aneurysms. Complete filling with whole blood clot and CPDs of 10%, 20%, and 30% eliminated high stress regions from the aneurysm neck in two of the four models and in the dome in all four models of the complex geometry aneurysms. The MWS was significantly lower (p < 0.001) with all filling conditions compared to without filling in both simple and complex geometry aneurysms as shown in Fig. 5. In the simple geometry aneurysms, sequential increases in CPD from no coil through 30% CPD resulted in statistically significant (p < 0.05) decreases of MWS. While a similar trend was noted, there were no significant differences in MWS with increasing CPD in complex geometry aneurysms.

## Discussion

Intracranial aneurysms are pathologic outpouchings in the walls of the cerebral arteries and their rupture is the most common cause of nontraumatic subarachnoid hemorrhage resulting in devastating morbidity or mortality if left untreated [18]. Hemodynamic factors have been implicated in aneurysm pathogenesis and rupture by influencing underlying molecular mechanisms in endothelial and smooth muscle cells, causing inflammation and extracellular matrix degradation, eventually leading to dysplastic vascular tissue [27]. While coil embolization has become a common form of treating cerebral aneurysms, it is unclear how much coiling is required for a given patient-specific aneurysm morphology, and what is the appropriate compromise between reducing wall stresses/risk of rupture on the aneurysm sac with the increased expense of coil embolization. This study combined the mechanical properties of a CTM through in vitro experimentation and a computational analysis of patient-specific aneurysms to compute transmural wall stress before and after coiling at various CPDs. To our knowledge, this is the first study simulating transmural wall stress in intracranial aneurysms with a CTM of varying packing densities. Among other implications of this work, the observations reported here may help neurointerventionalists to determine the amount of coiling needed to neutralize the risk of rupture. It is feasible that computational modeling such as that reported here could be used in a case-by-case basis to personalize treatment.

Numerous reports have highlighted wall shear stress as the major predictor of aneurysm rupture [1113]. These studies use CFD models to simulate flow in reconstructed 3D aneurysm geometries in order to obtain shear stress, which are thought to stimulate molecular mediators resulting in remodeling of the vessel wall, potentially leading to compromised tissue integrity [13,28]. However, divergent findings exist in the literature regarding whether high or low wall shear stress increases risk of cerebral aneurysm rupture [11,13,18,29]. Additionally, such CFD models have also correlated the presence of complex flow patterns (characterized by multiple vortex structures within the aneurysm sac that changes over the cardiac cycle), large jet sizes (compared to the aneurysm neck), and the presence of aneurysm wall displacement with increased risk of rupture [28]. Other reports describing CFD modeling of coil embolized aneurysms have demonstrated a decrease in fluid velocity and reduction in intraluminal pressure at the aneurysm dome after simulating 20% CPD [3032]. Shear stress and intraluminal pressure are not the only hemodynamic forces that act on the aneurysm wall; however, transmural wall stress is another possible contributor which has not been as widely studied as shear stresses in cerebral aneurysms.

The results of our finite element modeling described in this study suggest that the effect of coiling on transmural wall stress in aneurysms is dependent on both morphology and CPD (Figs. 6 and 7). Our simulations of aneurysms without coiling show PWS frequently occurring at the neck of both the simple and complex geometry aneurysms (Fig. 4), consistent with the findings reported by Cornejo et al. [16]. The presence of blood clot without any coil significantly reduced MWS and PWS in the domes of both simple and complex geometry aneurysms. For the isolated aneurysm neck and sac, we report the MWS and PWS for both simple and complex geometries in Table 3.

Increasing amounts of coil further induced small reductions in the MWS and PWS for both simple and complex geometries that were statistically significant between no-fill case and blood clot/CPD10–30, between blood clot and CPD20/CPD30 for simple geometries, and between blood clot and CPD10–30 for complex geometries (Figs. 5 and 7). Therefore, our results suggest that 100% filling of the aneurysm dome with blood clot of 0% CPD decreases the MWS and PWS in the domes of all aneurysms with additional CPD making slight improvements in both simple geometry and complex geometry aneurysms. Based on our findings, the reduction of wall stresses due to increased CPD may suggest that the expense of inserting additional coils to reduce wall stresses on the aneurysm sac is not warranted for simple aneurysms. For example, a 6 mm diameter spherical aneurysm using Stryker Target Neurovascular coils would require 22.5 cm, 45 cm, and 67 cm length coils to achieve 10%, 20%, and 30% CPD (as estimated by Angiocalc). Gandhoke et al. [32] reported that for aneurysms less than 10 mm with a total coil length less than 50 cm, the patient cost was between $3000 and$6338, and for a coil length greater than 50 cm, the patient cost was between $5500 and$11,982. We report that there is no statistical difference of MWS and PWS between various CPDs that were computed, therefore achieving a high CPD in clinical practice may not warrant the need for additional coils, reducing operational costs for the patient. 100% filling with either blood clot alone or any of the CPDs reduced the wall stress at the neck of all the simple geometry aneurysms but only two of the four complex geometry aneurysm cases (Figs. 4 and 6). It was observed that PWS for complex cerebral aneurysms 2 and 3 relocated from the aneurysm sac to the neck region (Figs. 4 and 6) although the global trend for PWS in complex aneurysm reduced with blood clot and increasing CPD (Fig. 7).

The presented computational method may potentially be extended to aid in decision making regarding intervention. With recent advancements in neurovascular imaging, many asymptomatic aneurysms are being discovered. As a result, physicians are forced to postulate cerebral aneurysm rupture potential based on prior experience or lessons from randomized controlled trials and make the challenging decision of whether to pursue invasive procedures or surveillance, which could potentially withhold necessary treatment. The computational mechanical modeling described here may lead a more precise tool for personalizing the approach to each patient.

There are some limitations of this study that should be kept in mind. We only simulated coil embolization in seven aneurysms. Additionally, homogeneous material properties were used throughout the models—i.e., for the aneurysm sac and parent vessels—which may have impacted the wall stress results. A wider variety of aneurysms would allow for stronger conclusions regarding the effect of morphology and coil embolization on transmural stress. Finally, our modeling does not simulate flow dynamics or take shear stress into account. A computational approach providing a combined assessment of both shear stress and transmural stress is expected to provide a more holistic view of aneurysm hemodynamics.

## Conclusion

This study illustrates the efficacy of coil embolization and the degree to which coil density and aneurysm morphology affect transmural stress at the aneurysm neck and dome. The findings suggest that complex geometry aneurysms may be more resistant to coil embolization treatment and higher CPDs are unlikely to translate in the elimination of transmural stress as it was reported that the peak wall stress relocated in two of the four complex geometry aneurysms from the dome to the neck. Our mechanical modeling serves as a first step toward developing a computational tool to aid physicians in assessing cerebral aneurysms before and after treatment.

## Acknowledgment

The authors would like to thank Riley Burton for technical assistance, Dr. Spandan Maiti for technical assistance with the mathematical model.

## Funding Data

• University of Pittsburgh Department of Neurosurgery (Funder ID: 10.13039/100007921).

• Swanson School of Engineering (Funder ID: 10.13039/100007125).

## Senior Author's Tribute to Dr. Fung

Like so many others, as the “founding father of biomechanics,” Dr. Fung had a profound impact on the focus of my research interests and my professional development at a formative stage of my career. When I wandered into the University of Pittsburgh School of Medicine's library as a senior mechanical engineering major in 1986 to explore applications of mechanical engineering to the health sciences, I found and skimmed through Dr. Fung's book “Biomechanics: Mechanical Properties of Living Tissues” (Springer, 1981) and fell in love. It opened up the world of biomechanics to me for the first time and I knew at that moment that was what I wanted to do with my career as a mechanical engineer. I read every paper by Dr. Fung and his students that I could get my hands on (which was not quite as easy then as it is now), which firmly guided my pursuit of a Ph.D. in mechanical engineering with a focus on vascular biomechanics.

A few years later, I was thrilled when I learned that Dr. Fung would be visiting Pittsburgh to receive an honor from the Engineering Society of Southwestern PA and would take the occasion to give a lecture. But there was one problem. I had recently had an accident in which I broke both of my wrists and my arms were in casts, making note-taking practically impossible. I did not want to miss the opportunity to soak in Dr. Fung's in-person teaching, so I purchased a microcassette tape recorder and took it with me to his lecture. To this day, two of my prized possessions are those sets of microcassettes and a copy of Dr. Fung's aforementioned Biomechanics book, personally signed by the author. And one of my fondest professional memories is seeing Dr. Fung some time later at a conference and him remembering me from the unusual circumstance of being in twin arm casts during his lecture in Pittsburgh.

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