Numerical Simulation of the Subclavian Aneurysm ...

Numerical Simulation of the Subclavian Aneurysm Blood Flow Alin A. Dobre1, Student Member, IEEE, Alexandru M. Morega1,2, Senior Member, IEEE, University POLITEHNICA of Bucharest, Faculty of Electrical Engineering, Bucharest, ROMANIA 2 “Gheorghe Mihoc – Caius Iacob” Institute of Statistical Mathematics and Applied Mathematics, Romanian Academy [email protected], [email protected] 1

Abstract-This paper reports an analysis of blood flow patterns that occur in a saccular subclavian aneurysm, by numerical, Galerkin finite element simulation. Investigations using image based reconstructed and CAD edited computational domains were also conducted on realistic blood vessels. We assume resistive type of vessels and use inlet pressure conditions rather than inlet velocity profiles to solve for the difficulties related to unknown velocity profiles usable as inlet BC. The results – velocity, pressure and stress fields – unveil insights in the flow dynamics within the aneurysm build-up and the flow-vessel interactions. Keywords: saccular aneurysm, blood flow, CT reconstruction, finite element, numerical simulation.

I.

INTRODUCTION

The aneurysm is an abnormal regional balloon-like deformation of the arteries, due to the weakening of blood vessels walls (Fig. 1). The origin of these pathological formations is not yet fully understood: some aneurysms are congenital while other occur there where the arteries walls are affected by either high blood pressure action, high levels of cholesterol or atherosclerotic disease [1]-[3].

(a) (b) (c) Fig. 1. Different types of aneurysms: (a) saccular; (b) fusiform, and highest rupture risk region (c) [4]

In this study we investigate the blood flow pattern in a saccular aneurysm emerging out of the left subclavian artery, and no concern is devoted to the dynamics of the aneurysm built-up. The computational domain is built by reconstruction techniques out of high resolution Computed Tomography (CT) image datasets with the aim of obtaining a more realistic, patient specific model. The aneurysm is then removed from the arteries, using CAD and image processing techniques [5], to investigate the blood flow in normal physiological conditions of the blood vessels. A comparative study between the aneurysm affected vessel and the normal vessel blood flow may then be conducted. The mathematical model of the pulsatile, laminar, incompressible arterial blood flow made of the momentum balance and mass conservation laws, is numerically solved

for by Galerkin FEM technique as implemented in Comsol Multiphysics, [6]. The fluid (blood) is assumed Newtonian, the arteries walls are rigid, and no mass transfer occurs between the blood domain and the surrounding tissue [7-8]. Numerical simulations conducted on the “normal” arteries show regions prone to aneurysm formation due to the high force per area exerted by the flow. As expected, the saccular subclavian aneurysm occurs in this high-risk region. For the aneurysm affected vessel flow, the numerical simulation results highlight regions of the aneurysm where rupture is more likely to occur. Aneurysm rupture often causes death so better understanding its blood flow patterns is a useful additional tool in investigation and treatment, which usually comprises of artery bypass or endovascular embolization. II.

THE COMPUTATIONAL DOMAIN RECONSTRUCTION

To investigate more realistic hemodynamic flows we used computational domains built out of medical image 3D solid masks of the vessels. The data source is a set of 2D high resolution Digital Imaging and Communication in Medicine (DICOM) images acquired via a CT scanner from a patient diagnosed with left subclavian saccular aneurysm (Fig. 2). This approach has the advantage of removing errors produced by the massive idealizations of the computational domain, while preserving a realistic geometry for the arteries.

(a)

(b)

(c) Fig. 2. View of two randomly selected slices from the data set source [(a), (b)], and the segmented aneurysm by the affected arteries (c).

The blood vessels mask reconstruction begins with the optimization of the threshold filter [5] parameters, which singles out the regions of interest from the image data source corresponding to certain desired levels of grey. Thus, a first

gross version of the arteries mask was obtained. The artifacts were removed using bilateral and gradient anisotropic diffusion noise filters [5]. Several adjustments are needed to finish the ready-to-workwith 3D solid model of the arteries: gaussian and binarisation filters are used to smooth out the mask surface; the segmented mask space continuity and morphological consistency are provided by erode, cavity fill and floodfill tools. To compare the aneurysm flow study with the normal (physiological) blood flow, the geometry of the same, normal blood vessels is needed. Using the aneurysm solid model (Fig. 3a), CAD techniques are then applied to remove the saccular aneurysm (Fig. 3b). The 3D solid vessels models – with and without aneurysm – were discretized and used as computational domains in the numerical study of the hemodynamic flows.

laminar [7-8]. The hemodynamic model is made of momentum balance (Navier-Stokes)

[

(

)]

⎡ ∂u ⎤ T ρ⎢ + ( u ⋅∇) u ⎥ = −∇ − pI + η ∇u + (∇u) , ⎣ ∂t ⎦

(1)

mass conservation law

∇ ⋅ u = 0,

(2)

where u [m/s] is the velocity field, p [Pa] is the pressure, ρ [kg/m3] is the mass density (1000 [kg/m3]), η [Pa·s] is the dynamic viscosity (0.005 [Pa⋅s]), and I is the unity matrix. Judging by size (diameters), the subclavian arterial network is part of the resistance-type blood vessel category (Fig. 4) [9]. Because the vessels here have relatively large cross sections, they oppose little resistance to the flow of blood therefore the pressure drop is small. In what follows we conjecture that the pressures at the inlet and outlets vary synchronously. In this way we assume the pressure drop on the arterial segment of interest. Other approaches might value differently the contribution of the distal arterial network by using, for instance, a lumped parameter model of the distal arterial network to provide some values for the outlets pressures [10,11]. We think that this view is prone to too many simplifying assumptions (types of vessels, rheology of blood in arterioles and capillaries, etc.) about the distal arterial tree to be useful here. The boundary conditions (BCs) that close the model are then no-slip conditions for the vessel walls, and uniform pressure conditions for inlet and outlets (Fig. 4).

(a)

Fig. 4. Blood vessels classification and specific pressure levels – after [9]. (b) Fig. 3. Computational domains made of approximately 62,000 Lagrange tetrahedral elements – (a) arteries with aneurysm; (b) normal arteries.

Accuracy tests led to high resolution, grid-independent numerical solutions for the flows (Fig. 3). III.

THE MATHEMATICAL MODEL

We assume that the blood is Newtonian, with constant properties and its flow is pulsatile, incompressible, and

This approach – pressure for the inlet rather than a velocity profile – is justified by the fact that a velocity profile would be a less realistic, applicable option. For instance, a uniform profile is inadequate while a nonuniform profile (parabolic, Hagen-Poiseuille, is the most common) has no meaning in this case considering that the vessels’ geometry results out of image reconstruction. Further more, the available data we use provides for (average) pressure information (Fig. 5).

incompressible flow condition (2), the total stress becomes

[

σ = − pI + η ∇u + (∇u)

T

].

(4)

The flow exhibits a whirl-like pattern within the aneurysm (Fig. 6a). Although the velocity values are not as high as those that occur in the main artery, the Oy component of the total stress σ – in this layout, almost orthogonal to the wall in the top part of the aneurysm – outlines an area of high risk of rupture at the top of the saccular aneurysm where the wall, supposedly, becomes thinner (Fig. 6b). This result is confirmed by common observations in surgical practice.

Fig. 5. Boundary conditions for the flow – time-dependent pressure conditions are set for inlet and outlets.

However, in simulating the flow we started from a stationary regime, by setting a uniform velocity profile at the inlet – 0.4 [m/s]. This value was determined by using the ultrasound method on the subject that provided the CT image sets [12]. Zero pressure boundary conditions were set at the outlets. The inlet pressure that results provides for the pressure drop between the inlet and the outlets of the arterial network (approximately 1,500 [Pa]). Next, we simulate the pulsatile flow provided as imposed by the inlet and outlet pressures pin(t) = 11,500·pi(t) [Pa], pout(t) = 10,000·pi(t) [Pa], respectively. Here pi(t) = 1+ K sin(t+3/2) and K = 0.1 sets the pulsation amplitude level. Although, in general, of importance in the study of the blood flow the structural interaction with the vessel walls is here neglected, due to very low deformation that occurs for this type of large arteries [8]. Also other effects, such as the action of the wall shear stress that would result in degrading the composition of the arterial wall, are not considered – the aneurysm is a given structure here. The dynamics of its emergence and formation are not discussed in this study. IV.

(a)

NUMERICAL SIMULATION RESULTS AND DISCUSSION

The mathematical model – equations (1), (2), and the BCs – was solved by FEM [6]. We used second order Lagrange elements and the P1-P2 pressure model. The algebraic system of equations was solved with the PARDISO parallel solver. First, the aneurysm flow was investigated. Then, the normal vessel flow was studied, and the two cases are analysed. The velocity field and the total stress distribution are quantities of interest obtained by numerical simulation. The total stress tensor is defined, generally, as [6]

[

σ = − pI + η ∇u + (∇u)

T

] − ⎛⎜⎝ 23 η − κ⎞⎟⎠ (∇u)I ,

(3)

where κ [Pa·s] is the dilatational viscosity. By the

(b) Fig. 6. Flow in subclavia with aneurysm – (a) velocity field through streamlines (color is proportional to the velocity amplitude); (b) total stress in Oy direction, [N/m2].

The normal artery flow shows off that the aneurysm emerged in a high-risk region of the main arterial wall (the intense red area in Fig. 7b, prone to aneurysm formation). The mechanisms that may explain the formation of the aneurysm, from the normal vessel (Fig. 7) to the vessel that exhibits the aneurysm (Fig. 6) make the object of future studies.

subclavian arterial wall was investigated using the finite element numerical modeling and simulation. The 3D models for the normal vessel and the vessel affected by an aneurysm are built out of medical image sets. They are realistic, patient specific reconstructed geometries being used as computational domains. The assumption of the resistive type of vessels and the usage of inlet pressure conditions (provided they are available) rather than inlet velocity profiles solve for the difficulties related to maybe more realistic, but unfortunately unknown, velocity profiles usable as inlet BC and the usage of convenient but less accurate methods such as lumped models for the contribution of the (entire) distal arterial network. Numerical simulations conducted on normal vessels confirm an increased probability of aneurysm formation in regions of high pressure and total stress values exerted upon the arterial walls. We aim to pursue the actual analysis by reconstructing the blood vessel walls, to investigate the stress-strain behavior, and to correlate it with the rupture risk probability values. ACKNOWLEDGMENT

(a)

The work was conducted in the Laboratory for Electrical Engineering in Medicine (IEM) – Multiphysics Models, the BIOINGTEH platform, at UPB. The first author acknowledges the POSDRU 88/1.5/S/61178 grant; the second author acknowledges the support offered by the CNCSIS PCCE-55/2008 grant. REFERENCES [1]

(b) Fig. 7. Flow in normal subclavia – (a) velocity field by streamlines (color is proportional to the velocity amplitude); (b) total stress in Oy direction, [N/m2].

V.

CONCLUSIONS

The blood flow pattern in a saccular aneurysm by the left

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