Cardiovascular Engineering and Technology, Vol. 10, No. 1, March 2019 (Ó 2018) pp. 32–45 https://doi.org/10.1007/s13239-018-00383-1 A New Method for Simulating Embolic Coils as Heterogeneous Porous Media 1,2 1 2,3 2,4 HOOMAN YADOLLAHI-FARSANI , MARCUS HERRMANN, DAVID FRAKES, and BRIAN CHONG 1School for Engineering of Matter, Transport and Energy, Arizona State University, Tempe, AZ, USA; 2School of Biological and Health Systems Engineering, Arizona State University, 501 E Tyler Mall, BLDG ECG RM#334, Tempe, AZ 85287, USA; 3School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ, USA; and 4Mayo Clinic Hospital, Phoenix, AZ, USA (Received 21 June 2018; accepted 3 October 2018; published online 18 October 2018) Associate Editors Dr. Ajit P. Yoganathan & Dr. Matthew J. Gounis oversaw the review of this article. Abstract greater accuracy was enabled by considering heterogeneity Purpose—To gain insight into the influence of coils on compared to the homogenous approach. aneurysmal hemodynamics, computational fluid dynamics (CFD) can be used. Conventional methods of modeling coils Keywords—Brain aneurysm, CFD, Porous medium, Hetero- consider the explicit geometry of the deployed devices within geneous, Embolic coils. the aneurysm and discretize the fluid domain. However, the complex geometry of a coil mass leads to cumbersome domain discretization along with a significant number of mesh elements. These problems have motivated a homoge- INTRODUCTION neous porous medium coil model, whereby the explicit geometry of the coils is greatly simplified, and relevant Endovascular coiling is frequently employed to treat homogeneous porous medium parameters are approximated. unruptured brain aneurysms. One goal of the treat- Unfortunately, since the coils are not distributed uniformly ment is to reduce blood flow into the aneurysm and in the aneurysm, the homogeneity assumption is no longer 14 valid. thereby promote thrombosis. Researchers have used Methods—In this paper, a novel heterogeneous porous computational fluid dynamics (CFD) to better medium approach is introduced. To verify the model, we understand the efficacy of coiling and the treatment’s performed CFD simulations to calculate the pressure drop specific effects on aneurysmal hemodynamic.3,7,8 caused by actual deployed coils in a straight cylinder. Next, we considered three different anatomical aneurysm geome- Unfortunately, the complexity of deployed embolic tries virtually treated with coils and studied the hemody- coils is such that considerable time and effort is namics using the presented heterogeneous porous medium required to generate high quality meshes of the devices. model. Additionally, small coil diameters demand a high Results—We show that the blood kinetic energy predicted by number of boundary layer mesh elements, which re- the heterogeneous model is in strong agreement with the 2 conventional approach. The homogeneity assumption, on the sults in high computational cost. These issues have other hand, significantly over-predicts the blood kinetic motivated researchers to look for less complex and energy within the aneurysmal sac. costly ways of performing CFD on coiled aneurysms. Conclusions—These results indicate that the benefits of the Homogeneous porous medium theory is often applied porous medium assumption can be retained if a heteroge- to address the aforementioned modeling challenges.29 neous approach is applied. Implementation of the presented method led to a substantial reduction in the total number of Using this approach, endovascular coils are considered mesh elements compared to the conventional method, and as homogeneous porous media to simplify the explicit geometries of the devices. In other words, local char- acteristics of the coil mass geometry do not influence local porosity and permeability—instead, both are held constant throughout the fluid domain of the coil mass. Address correspondence to Hooman Yadollahi-Farsani, School Several works have implemented the homogeneity of Biological and Health Systems Engineering, Arizona State assumption. For example, in Refs. 18 and 23, authors University, 501 E Tyler Mall, BLDG ECG RM#334, Tempe, considered coils as a homogeneous porous domain. AZ 85287, USA. Electronic mail: [email protected] 32 1869-408X/19/0300-0032/0 Ó 2018 Biomedical Engineering Society A New Method for Simulating Embolic Coils 33 They observed that increasing the number of coils re- geometries of three different anatomical aneurysms duced the porosity, thereby decreasing intra-aneurys- were first constructed; embolic coils were then de- mal blood flow velocity. However, they mentioned that ployed virtually in each aneurysm using a finite ele- uneven distribution of coils within the aneurysmal sac ment method (FEM). CFD simulations were invalidates the homogeneity assumption. A formula to performed on the explicit coil geometries, as well as calculate the minimum coil length needed to arrest corresponding homogeneous and heterogeneous por- blood flow in an aneurysm by applying the homoge- ous domains. neous porous medium assumption was derived in Ref. 19. In their paper, the homogeneous porous medium assumption was applied to overcome the difficulties MATERIALS AND METHODS associated with finding the permeability and porosity of a heterogeneous porous domain. Researchers in Governing Equations Ref. 24 attempted to assess the capability of the The effect of a porous medium on the flow field was homogeneous porous medium assumption in compar- modeled by an added source term in the Navier–Stokes ison to the conventional approach of using explicit coil equation. The term was borrowed from the original geometries. Specifically, they performed CFD simula- Darcy’s law relating pressure drop to fluid velocity via tions on explicit coil geometries and compared the re- permeability.11,12,17,31,32 Permeability and porosity sults to those from simulations applying the were considered to be position-dependent 3D maps homogeneous porous medium assumption. They con- and were used as input for the Navier–Stokes equa- cluded that despite the easier implementation of the tions. homogeneous porous medium approach, it failed to capture the main flow features. Their results showed a @u uu l/u q þr: ¼Àrp þ qg þ lr2u À ð1Þ significant deviation from the conventional approach. @t / K In a recent work,22 the hemodynamics in physical 3 models of two patient-specific aneurysms treated with In the equation above, q (kg/m ) is fluid density, u coils were compared to those in a version of the same (m/s) is superficial velocity, t (s) is time, / is porosity, p 2 2 aneurysm embolized with a homogeneous porous (kg/m s ) is pressure, g (m/s ) is gravity, l (kg/ms) is 2 medium. Authors showed that the homogeneity dynamic viscosity, and K (m ) is permeability. Perme- assumption considerably over-predicted blood flow ability was then calculated using the Carman–Kozeny 6 into the aneurysm. Thus, despite the benefits associ- relation. Defining porosity (/) and specific area (A) ated with reduced computational cost and easier per unit bulk volume as meshing, the non-uniform, complex geometries of coils Device volume degrade the homogeneity assumption. In other words, / ¼ ð2Þ Total volume changes in porosity throughout the domain require a more sophisticated model that defines a position-de- Internal surface area of the porous region pendent porosity map. A ¼ ð3Þ Total volume Experiments and numerical analyses performed in Ref. 27 quantified void spaces within the aneurysmal permeability is found sac after coil deployment, and showed that the coil /3 mass is not uniformly distributed. A Weibull distri- K ¼ ð4Þ cA2 bution model was proposed to represent intra- aneurysmal pore distribution. The authors mentioned where c is the shape factor. This factor is dependent on that the model could be used in numerical studies of the cross section of the coils and was considered to be 2 intra-aneurysmal hemodynamics, but did not apply it in this study.6 themselves. As an alternative to numerical prediction of the permeability of a coil mass, experiments using a Map Generation Procedure falling-head permeameter28 and fluorescence microscopy10 can also be used. Stereo-lithography (STL) representations of the In this paper, we present a novel method that con- deployed coil surfaces were used. This allowed us to siders coils as a heterogeneous porous medium. take advantage of the triangular surface mesh to gen- Specifically, the method applies the porous medium erate the porosity and permeability maps. The assumption but also considers non-uniform changes in bounding box around each device was broken into a domain porosity. First, we verified the capabilities of uniform lattice of hexahedra. Within each hexahedral, the proposed method in predicting pressure drop the device portion was closed by tessellating the open caused by coils placed in a straight cylinder. Next, areas cut by the hexahedral faces. Once the device 34 YADOLLAHI-FARSANI et al. 3 portion within the hexahedral was watertight (Fig. 1), volume, q (kg/m ) is fluid density, and uphysical (m/s) is its area was found by adding the surface areas of all the mesh cell’s physical velocity: triangles forming that device portion including the ! X closure parts. The divergence theorem was then q 1 2 1 K.E. ¼ 8i uphysical i : ð6Þ applied to calculate volume from surface area, con- 2 8 i sidering F~ a vector field built from a triangle’s corner and origin: The physical velocity is a more realistic and accurate Z I representation of velocity in porous region and is 33 X ¼ rÁF~ðxÞdX ¼ F~Á ndS^ ; ð5Þ specified as : u X S u ¼ : ð7Þ physical / where X is the volume, S is the surface, and n^ is the triangle’s surface normal. The generated map was then used to find the porosity and permeability for each Verification Model node in CFD mesh by means of tri-linear interpola- tion. This algorithm is explained further in ‘‘Appendix To verify the porous medium approach suggested A’’. here, the common methodology which was first sug- gested by Darcy in 18565,30 was used here.