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The Haemodynamics of Cerebral Aneurysms of Uncoiled and Coiled Conditions: A Computational Modelling Method for Investigating the Influence of Newtonian and Non-Newtonian Flow Behaviour Hamad K. Alkhareef*
Abstract—This study investigates the heamodynamics inside the cerebral aneurysm and the influence of the Newtonian and nonNewtonian fluid behaviour on the flow patterns for coiled and uncoiled cases. Computational Fluid Dynamics (CFD) technique was used to simulate a symmetric 2D bifurcation cerebral aneurysm. There were small differences in the velocity magnitude, pressure, and shear stress magnitude between the Newtonian and non-Newtonian fluid behaviour. Although the pressure residual is more stable in the non-Newtonian fluid for both cases; coiled and uncoiled, Newtonian fluid assumption can be used to evaluate the flow characteristics as the percentage error is small. The laminar viscosity of blood is higher in the treated (coiled) aneurysm than the uncoiled case. The increase in local blood viscosity in the regions of low strain rate and low flow, decreases the velocity magnitude of the non-Newtonian fluid. Coil porosity plays a major role in flow velocity and blood vortices reduction. Computing Reynolds number in the abnormal flow condition reduces the inaccuracy in flow patterns obtained. The heamodynamic features are independent of grid resolution. Index Terms—Cerebral aneurysms, CFD, Haemodynamics, Newtonian & Non-Newtonian fluid flow.
I. INTRODUCTION
T
intracerebral aneurysm (also known as cerebral aneurysm) is a weak spot in the blood vessels of the brain which expands, inflates and fills with blood [1]. Consequently, the pressure on the surrounding nerves and tissues will rise. Cerebral aneurysms are usually grow along twisted arteries that are located between the brain and the skull [1]. It is popular in people that have genetic diseases such as tissue disorders and high blood pressure. Due to the high pressure, blood circulation may be formed in the bulging cerebral aneurysms which increases the risk of aneurysms rupture resulting in what is called a hemorrhage; splitting blood into the surrounding brain vessels which may lead to death [1]. The fatality rate after the HE
*Alkhareef, H. MSc student in Mechanical Engineering Department, University College London, London WC1X 9AH, UK (e-mail:
[email protected]).
hemorrhage was estimated at 32% to 67% [9]. Cerebral aneurysms can be treated by two surgical options; either Microvascular clipping or Endovascular embolization [1]. The first option isolates and cuts the blood flow to the aneurysms by placing a small metal clip on the nick of the aneurysm while the second option fills the aneurysm with coils which increases the resistance of the aneurysms to the flow by blocking it as much as possible. Fig. 1 represents the endovascular embolization and the microvascular clipping methods. The dynamics of blood flow is an important factor in diagnosing and treating cerebral aneurysms, for instance, reducing blood circulation. The diagnosis of aneurysms and the success of Endovascular surgery could be determined by considering the Haemodynamic factors such as pressure, velocity and wall shear stress [3]. The study of aneurysms pathogenesis and treatment operations can be done by using Computational Fluid Dynamics (CFD) technique to model and simulate the flow of the blood. Cebral et al. [4], analysed the development of an efficient pipeline containing number of aneurysms using CFD simulation. They found out that the sensitivity of flow patterns is independent of the viscosity of the model, mesh resolution and the mean flow input, where the geometry of the model has the great impact on the computational analysis. 3D data of human anatomy can be constructed with the current advancements in medical imaging [4]- [9]. Previous studies on simulating aneurysms have been carried out by Hassan et al. [6]- [5], Cebral et al. [4] and Matthew et al. [7] using the virtual angiography technique to validate patient-specific image-based CFD models. Computational Fluid Dynamics is a great tool that can be used to deduce the impact of Endovascular surgery on haemodynamics in brain vessels. In addition, modelling and simulating the haemodynamics in the aneurysms can provide treatment planning [7].
2 The relevance of haemodynamics of non-Newtonian behavior in cerebral aneurysms is demonstrated in different studies. Castro et al. [10] showed the importance of the nonNewtonian blood behaviour when the flow is decelerating and approaches to zero. Furthermore, they found out that the flow is significant for 30% of the complete heartbeat (cardiac cycle). Khanafer et al. [6] studied the non-Newtonian behaviour of blood in aortic vessel using finite element model. Carolyn et al. [8] investigated the effect of non-Newtonian behaviour of cerebral aneurysms. They found out that the non-Newtonian condition has a significant effect on the flow and fluid forces such as shear stress as well as the non-Newtonian fluid was greater than the Newtonian at high blood pressure. Valencia et al. [12] evaluated the impact of non-Newtonian behaviour in two virtual saccular aneurysms and the result was that the nonNewtonian flow is more stable than the Newtonian flow with the same initial conditions. Morales et al. [10] investigated the haemodynamics in coiled cerebral aneurysms for Newtonian and non-Newtonian flow and they found out that the velocity of the Newtonian velocity was overestimated before and after coiling due to the viscous forces which are not considered by the Newtonian model. This paper will investigate the haemodynamics of a 2D symmetric cerebral aneurysm under steady flow conditions. Uncoiled and coiled intracerebral aneurysm with Newtonian and non-Newtonian flow conditions will be simulated using CFD approach. The study will be carried out for two different inlet flow velocities; physiological flow (low inlet velocity) and abnormal flow (high inlet velocity) to evaluate the impact of Newtonian and non-Newtonian flow on the haemodynamics parameters between the various cases. In addition, geometric grid sensitivity test will be carried out for the uncoiled model with Newtonian low inlet flow velocity.
II. METHODS The analysis of the haemodynamics in the cerebral aneurysm using CFD approach consists of four main steps: 1) Vessels construction and placing an ellipse-shaped aneurysm in place 2) Grid generation 3) Numerical solution of the flow 4) Post-processing and visualization Detailed description of each step is illustrated in the next sections. Different methods and approaches can be used in order to model and simulate the heamodynamics in aneurysms as suggested by [7], [3]. In this study, simulation was done by using the commercial software CFD-ACE+ package by the ESI Group, version 2014. A. Model Geometry Basilar bifurcation approach was used to model a 2D cerebral aneurysm. The blood vessels are 10 mm long with a 1.5 mm inlet and 1 mm outlets. The two outlets are branched at an angel of 130o. The aneurysm has a shape of ellipse with a major radius of 4.5 mm and minor radius of 3.5 mm. The coil in the treated case was part of the aneurysm and considered as a porous medium with a porosity of 0.8 to reduce blood recirculation. Fig. 2 shows the geometry of the 2D model of the coiled and uncoiled cerebral aneurysms used in this analysis. In this study, an unstructured triangular meshes were used with approximately 10,000 to 27,000 elements.
Fig. 2. 2D Basilar bifurcation cerebral aneurysms: uncoiled and coiled
B. Flow Module Theory In this simulation, it has been assumed that the flow is laminar and the fluid (blood) is incompressible and isothermal. Therefore, the Navier-stokes and continuity equations based on mathematical statements; mass conservation and momentum conservation laws of physics for flow are [4]: ∇. 𝑉 = 0 (1) 𝜌
Fig. 1. Cerebral aneurysms treatment methods: Endovascular embolization and Microvascular clipping [2]
𝜕𝑉 + 𝑉. 𝛻𝑉 = −𝛻𝑝 + 𝛻. 𝜏 𝜕𝑡
(2)
where V is the fluid velocity, 𝜌 is the density of blood which is usually has the value of (1060 kg/m3), p is the pressure, and τ is the deviatoric stress tensor which can be related to the strainrate by: 𝜏/0 = 𝜇𝛾/0
(3)
3 where 𝜇 is the viscosity and 𝛾 is the strain-rate which can be defined as follow: 𝜕𝑢𝑖 𝜕𝑢𝑗 (4) 𝛾/0 = ( + ) 𝜕𝑥𝑗 𝜕𝑥𝑖 The fluid’s viscosity is assumed to be constant for the Newtonian flow behaviour, 𝜇9:; =0.00357 kg/m-s. However, the nature of blood is non-Newtonian simply because its viscosity varies with the applied strain-rate. This type of fluids is typically called rheological fluids and therefore, we modeled the non-Newtonian flow behaviour according to the rheology power law (blood) approach which has the following mathematical expression: 𝜇=𝜆𝛾
than the Newtonian one. Fig. 3 shows the velocity magnitude of the Newtonian flow and the non-Newtonian flow for both cases. Although the Newtonian flow has a greater maximum pressure than the non-Newtonian case by 1.466%, the residual pressure is more stable in the non-Newtonian flow. In addition, we have observed that the maximum strain rate and the shear stress magnitude for the Newtonian flow is higher than the nonNewtonian case by 34 1/s and 0.81 N/m2 respectively. Furthermore, the total mass flow rate of the two conditions was identical, 4.293x10-1 kg/s. Non-Newtonian Fluid
Newtonian Fluid
(5)
;@A
𝛾 ∝
𝑒𝑥𝑝
−𝑏 𝛾
𝑛 𝛾 = 𝑛B + ∆𝑛. exp − 1 +
𝛾 𝑐
𝑒𝑥𝑝
−𝑑 𝛾
and
where: 𝛾 = the local calculated shear stress 𝜆 = the consistency constant 𝜇B = 0.035 [the limiting (Newtonian) viscosity] ∆𝜇 = 0.25 a = 50 b=3 c = 50 d=4 The inlet fluid velocities in the main vessel were taken as, 0.27 m/s for the physiological flow and 10 m/s for the abnormal flow. Reynolds number was computed for the high velocity flow case by using the standard k-epsilon model approach as follow: (6) 𝑢. 𝐷Q 𝑅O = 𝜈 where 𝐷Q is the hydraulic diameter (diameter of the inlet), u is the flow velocity, and 𝜈 is the kinematic viscosity which is S given by the relationship; . The turbulent kinetic energy (k) T
and turbulent dissipation rate (𝜀) can be calculated from the following equations [ESI Group]: 3 (7) 𝑘 = . (𝑢. 𝑙)Z 2 and 0.1643. 𝑘 A.] (8) 𝜀= 𝑙 where l is the turbulence length and given as; 𝑙 = 0.07. 𝐷Q III. RESULTS AND DISCUSSION A. Newtonian and non-Newtonian flow behaviour in Cerebral Aneurysms We have examined the un-treated (uncoiled) cerebral aneurysm model for the physiological flow inlet velocity under the conditions of Newtonian and non-Newtonian flow behaviour. The flow velocity magnitude was found to be slightly higher in the non-Newtonian flow condition by 0.34%
Coiled
𝜆 𝛾 = 𝜇B + ∆𝜇. exp − 1 +
Uncoiled
where
Fig. 3. Maximum velocity magnitude for coiled and uncoiled aneurysms under Newtonian and non-Newtonian fluid behavior.
For the treated (coiled) aneurysm, the velocity magnitude and the pressure were found to be higher for the Newtonian flow than the non-Newtonian one by 0.96% and 1.44% respectively. This is due to the increase of local viscosity in the regions of low flow. However, the maximum strain rate in the nonNewtonian flow was found as 4407 1/s and 4399 1/s in the Newtonian flow behaviour, resulting in higher shear stress magnitude in the non-Newtonian model. B. The impact of Endovascular on the haemodynamic parameters The physiological flow velocity for the Newtonian fluid was examined before and after coiling the aneurysm. The result shows that the reduction in velocity was found to be 9.72% when the coil was deployed with 0.8 porosity, resulting in less blood circulation zones by 18.3%. However, the pressure was recognized to be high in the uncoiled case at the inlet vessel where the high-pressure areas in the coiled case were at the neck of the aneurysm and the inlet vessel. This because of the increase in pressure amplitude oscillations at the aneurysm wall when its coiled. Predominantly, this high-pressure area at the neck of the dome has an instantaneous protective effect as it subject to a new rapture during the treatment process [3]. When the power low (blood) was applied at the same flow velocity, a deceleration in the flow was observed when the aneurysm was coiled. This reduction in velocity magnitude was
4 Uncoiled
Coiled
Uncoiled
Non-Newtonian
Newtonian
Coiled
Fig. 4. The flow stream and pressure of the Newtonian fluid and-
-Non- Newtonian fluid for the coiled and uncoiled cases.
found to be 10.9% with 8% of pressure decrease. In addition, the flow circulation is found to be ironed out by 18.4% in the coiled model resulting in less strain rate which reduces the chance of aneurysms rapture. Furthermore, the laminar viscosity in the treated aneurysm was found to be higher by 0.5033 kg.m-1. s-1 than the untreated one. Fig. 4 summaries the treated and untreated cases under the Newtonian and nonNewtonian fluid behaviour. Moreover, we investigated the reduction in velocity magnitude for different porosity values, 0.3 to 0.9, and the result shows that the flow reduces its velocity by 7.82% as the porous medium increases. Fig. 5 represents the time taken for the blood flowing inside the brain vessels for the uncoiled and coiled aneurysms. The blood flows faster in the treated case which reduces the risk of coagulation.
C. The influence of Reynolds number We have examined the impact of high inlet flow velocity on the uncoiled cerebral aneurysms under the Newtonian flow condition. Two cases will be illustrated, the influence of high inlet flow velocity on the haemodynamic parameters with and without introducing Reynolds number. Fig. 6 shows the variation in the flow velocity magnitude before and after computing Reynolds number of 4454. We observed that the haemodynamic parameters have low values when introducing Re number in the simulations and excluding Reynolds number from the simulation provides inaccurate solution for flow transition. However, this result is not accurate enough to relay on as the nature of blood is non-Newtonian. Convergence difficulties were found when simulating the abnormal flow velocity (10 m/s) for the non-Newtonian fluid behaviour as well as when coiling the aneurysm. This can be solved in the case of resolving the fluid dynamics around the cross section of the actual coil [3]. Furthermore, symmetry breaking bifurcation was realized when computing Re number as illustrated in fig.6. This phenomenon occurs due to the sudden expansion of the aneurysm which leads to the transition from a symmetric to asymmetric flow.
Uncoiled
Coiled
Fig. 5. The flow rate (in seconds) inside the uncoiled and coiled aneurysm.
D. Grid sensitivity test This test was carried out to demonstrate the independence of grid resolution on the flow characteristics obtained previously. The difference in the velocity magnitude of the flow applied for various meshes was found to be very small, which has no significant effect on the overall flow pattern. Consequently, the important haemodynamic parameters can be captured with either high or small number of elements grid. Table 1 shows the results of the uncoiled cerebral aneurysm at the normal flow velocity of 0.27 m/s under the Newtonian fluid behaviour for different unstructured triangular meshes.
5 [5]
Table 1 Sensitivity test for various meshes for the uncoiled case [6]
Case 1 2 3 4 5 6
Uncoiled # elements 12603 17901 21194 29920 37774 50980
Velocity Magnitude (m/s) 0.4355 0.4392 0.4382 0.4382 0.4433 0.4494
[7]
[8] [9]
[10]
[11] [12]
(a)
(b)
Fig. 6. The velocity magnitude for the abnormal flow condition (10 m/s); (a) Without Reynolds number (b) Compute Reynolds number
IV. CONCLUSION In this study, we have examined the effect of Newtonian and non-Newtonian fluids on the haemodynamics in two cases of cerebral aneurysms; uncoiled and coiled. CFD approach was used to simulate a 2D cerebral aneurysm. The Newtonian fluid behaviour was found to be acceptable for studying the flow patterns of blood as it shows a small percentage error comparing with the non-Newtonian one. When coiling the aneurysm, porous medium plays a major role in reducing flow velocity and blood circulation. Computing Reynolds number produces stable flow as shown in fig. 6. Sensitivity test was carried out for different number of elements and the independence of grid was achieved. CFD method has the potential to diagnose the risk of rapture in the cerebral aneurysms and could be a useful tool in the medical field to investigate the haemodynamics of human vessels. REFERENCES [1] [2] [3] [4]
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