Computational analysis of aortic haemodynamics in the presence of ascending aortic aneurysm

2.1Image acquisition and segmentation

This work used CT thorax 3D imagery with an ascending aortic aneurysm [16] and the image of a healthy aorta from the SimTK database [17]. The SimVascular software package [18] was used to create a 3D aortic model using sections from medical images. The first step was to determine which parts of the medical image depicted the object of interest, which was achieved through volumetric image generation and threshold setting. Initially, the aortic centre points were postponed (Fig. 1), and the centre line of the flow beam was formed. Next, the left posterior artery, the left carotid artery, and the brachiocephalic artery were added to the baseline. The second step, as shown in Fig. 1, was the segmentation of the vessel, where the contour of the vessel was determined at the central points. In the process, 2D contours were generated at each point on the centreline. The contour was initiated by a dot and displayed in the directions of changing the intensity values to find the location of the most pronounced changes. After segmentation, the modelling tool was used to generate a 3D model of the aorta. At the sites of the aortic arch where the arterial bifurcations were present, surface smoothing and rounding of the arterial junctions were performed, and Meshmixer software was used to regenerate the model [19]. The SimVascular program was used to join all arterial walls into a single body. After model processing, 6 surfaces were obtained: 1 vessel wall and 5 caps (Fig. 2).

Figure 1.

Geometric model construction steps; flow waveform as an inlet boundary condition and RCR model as outlet boundary conditions.

Figure 2.

Geometric models of aortas.

2.2Blood flow modelling

Blood flow was modelled using Navier-Stokes equations for Newtonian fluids. The storage of the mass and impulses of a non-compressed fluid in three dimensions can be expressed as follows:

where ρ is the blood density, vi is the i-th liquid velocity component, vj is the derivative of its time, p is the pressure, and τi⁢j is the part of the viscosity stress tensor, where the index “, j” denotes the corresponding derivatives of spatial coordinates [20].

Appropriate equations were needed to calculate blood viscosity in order to use this system of equations. The simplest model is Newtonian fluid, which has constant viscosity. Blood is described as an uncompressed Newtonian fluid, and flow is laminar. Dynamic blood viscosity μ= 0.04 g/cm⋅s2, while blood density ρ= 1.06 g/cm3.

The inflow boundary conditions are the waveform of the fluid flow that simulates flow from the aortic valve. Because the flow of blood in the human body pulsates, a constant velocity at the inlet does not simulate the actual flow, which should be reported as a variable periodic profile in this case. Each cardiac cycle is a combination of a systolic phase and a diastolic phase. Therefore, this work used a temporal profile describing the flow during systole and diastole. Assuming a heart rate of 60 beats per minute, the duration of each cycle is 1 s; under these conditions, the blood flow Q= 5.5 l/min. A parabolic flow wave profile was used (Fig. 1).

The Windkessel haemodynamic model [21], used to restore a patient’s aortic blood pressure, describes the heart and blood vessels as a closed hydraulic system. The Windkessel model is most used to describe load during the cardiac cycle. This approach links the pressure and blood flow in the aorta and describes the ability of the arteries to withstand deformities as the volume changes. In comparison, SimVascular uses resistance-capacitance-resistance (RCR) outlet conditions [22]. The downstream pressure is expressed through an ordinary differential equation similar to the relation between voltage and current in electric circuits. This type of boundary condition uses a reduced-sequence vascular model based on an analogue of the electrical circuit described above. According to this theory, vascular behaviour is represented by three parameters: proximal resistance Rp, capacitance C and distal resistance Rd (Fig. 1). Since patient-specific data were not available for the present models, common peak systolic (120 mm Hg) and diastolic pressures (80 mm Hg) for a healthy person were used as target values. At time t, the pressure P is related to the flow rate Q according to this relationship (if the simulation starts from t= 0) [23]:

where Rp represents the proximal resistance, C is the volume pressure, and Rd indicates distal resistance, while Q is the flow, t indicates time, τ is a time constant, P represents pressure, and Pd is the downstream pressure.

Determining the boundary conditions in the SimVascular program requires first calculating the total system resistance. This value can be found by dividing the mean pressure P𝑎𝑣𝑔 by the mean flow Q over one cardiac cycle:

where P𝑎𝑣𝑔 is the mean pressure, dyn/cm2; Q indicates average flow demand during the cardiac cycle, ml/s.

By introducing the total resistance, SimVascular concretely calculates the proximal and distal resistance for each outlet using the outlets cups areas.

SimVascular uses the open-source TetGen meshing method [24]. In the aorta with an aneurysm (Fig. 3), an FE mesh of 146,262 elements was generated. In contrast, the healthy aorta model mesh consisted of 67,572 elements.

2.3Vessel wall parameters

The calculation was performed using different wall parameters. Rigid settings, meaning that the walls were rigid and non-slip, were used to examine blood velocities and pressure. The use of the rigid wall approximate condition reflected the fact that, under normal conditions, didn’t need to use deformable wall settings, because the displacement of the walls does not significantly change the velocity field. Obtaining the value of wall displacements as well as the wall shear stress (WSS) and oscillatory shear index (OSI) involved using the boundary conditions under which the aortic wall was modelled as a linear elastic material that could have had the same or variable modulus of elasticity and thickness along the vessel (Table 1). This approach entailed a simplified fluid-structure interaction (FSI) method using the coupled momentum method (CMM) [25]. This CMM-FSI formulation shows the traditional stabilized finite element formula for the Navier-Stokes equations in the rigid wall region and modifies it to account for the deformation of the fluid domain.

Figure 3.

Generated FE mesh.

Figure 4.

Schematic illustration of wall shear stress of blood flow in the aorta.

Figure 5.

Distribution of blood flow velocities in the aorta with aneurysm.

Figure 6.

Distribution of blood flow velocities in healthy aorta.

Figure 7.

Aortic wall displacements.

WSS and OSI were determined during the calculation. WSS a frictional force caused by blood flowing along the vessel wall (Fig. 4) and can alter the integrity and biomechanical strength of the intercellular matrix of the vessel wall. WSS expresses the force per unit area by which a wall acts on a liquid in the direction of a local tangent plane [26] as follows:

where μ is the dynamic viscosity, u is the flow velocity parallel to the wall, y is the distance to the wall, and τω is the wall shear stress.

The fluctuating shear index is another significant haemodynamic parameter used to assess WSS because they are related. OSI is a dimensionless parameter used to identify WSS fluctuations from flow direction during the cardiac cycle [27]:

where WSS is the instantaneous vector of wall shear stresses, 𝑊𝑆𝑆𝑚𝑒𝑎𝑛 is the mean, cardiac cycle time-averaged vector, TAWSS is the time-averaged WSS and T denotes the waveform period of one heart cycle.

OSI shows the physical deviation of the WSS vector from its predominant axial flow direction along the parallel vector [28].

2.4Simulation

Simulations were performed over 6 cardiac cycles to stabilise flow velocity and pressure fields. The number of time steps per cycle was 1,000 with a fixed time step size of 0.001 s. The simulation results were evaluated during the last cardiac cycle.

4.1Flow rates

In the aorta with an aneurysm, the maximum blood flow velocity is almost 1.6 times lower than the systolic blood flow velocity in the healthy aorta. According to one study [30] where researchers measured flow velocities with an MRI scanner, the findings showed that the mean blood flow Q = 5.5 ± 1.2 l/min in the group of 25–30 subjects, and the maximum velocity in the ascending aorta was 1.18 ± 0. 23 m/s. The maximum velocity obtained in this work in a healthy aorta is higher compared to the experiment described earlier, which may be due to different aortic geometry and modelling with a rigid artery wall. The velocity profiles and contours are similar in the systolic phases for both the healthy aorta and the aorta with aneurysm. In the aorta with aneurysm during diastole, the axial profile of velocity differs significantly from healthy aortic results in that the blood velocity at the centre of the aneurysm slows down due to the increased diameter of the ascending aorta.

4.2Pressure

The pressures obtained in this work show the normal pressure distribution during systole and diastole phases. The pressures are provided to verify that the modeling parameters (proximal and distal resistances) are selected correctly under the model outflow boundary conditions. In this study, it is not meaningful to compare pressure changes in a healthy aorta and in the aorta with an aneurysm, as the mean cardiac cycle pressure of 100 mm Hg was included in the calculation of boundary conditions (Eq. (4)).

4.3Displacement

The results obtained show that using the same boundary conditions, the value of wall displacements is 45% lower in a healthy aorta compared to an aorta with an aneurysm. This finding means that in the dilated aorta, the amplitude of wall movements increases during the cardiac cycle. In general, the greatest dilation of vascular walls occurs in the systolic phase, when the rate of blood inflow reaches its maximum value. In a previous study that examined aortic haemodynamic parameters in the presence of dissection, wall displacement before and after correction surgery were investigated. The study showed that in the case of arterial dissection, large wall displacements of 11 mm occurred due to arterial swelling and the resulting stresses. After aortic lumen correction, the displacements decreased to 0.5 mm [31]. In summary, displacements are also an important parameter indicating certain abnormalities and pathologies. Nevertheless, no studies to date have demonstrated any specific effects of displacements on aneurysm progression.

4.4WSS and OSI

In the presence of ascending aortic aneurysms, large areas of the low shear stress zone have been detected, as well as a corresponding increase in internal deformation of the aortic wall at that site. This case is characterised by a change in flow and a sudden drop in wall shear stresses in the aorta. In a study that examined blood flows in the aorta using MRI technology, the mean value of shear stresses in normal aortas was 1.5 ± 0.3 Pa, and OSI was 0.325 ± 0.025. In the same study, the results for four patients with aneurysms showed higher mean WSS values during the cardiac cycle: 4.29 ± 1.11 Pa and lower, OSI 0.2 ± 0.05. Furthermore, the diagrams of the study results revealed that the OSI was evenly distributed in the normal aortas and indicated zones of high and low values in the pathological arteries [32].

Other researchers examined 224 aortas without pathologies [33] using four-dimensional flow MRI technology to determine wall shear stresses in a population of normal thoracic aortas. The mean peak systolic WSS was 1.79 ± 0.71 Pa in the aortic arch and was significantly higher, at 2.23 ± 1.04 Pa, in the descending aorta. The systolic WSS distribution obtained in this work in the healthy aorta similarly demonstrated an increase in shear stresses in the descending aorta. Furthermore, in the ascending aorta, aortic arch and descending aorta, Simao et al. argued that higher or lower WSS values could lead to inner aortic layer dysfunction, which could lead to progression of atherosclerosis. Extremely high WSS values were associated with vascular structure changes responsible for aneurysm initiation and progression, with low WSS regions correlating with atherosclerosis progression [34].

In contrast to the studies described above, in this work, the same initial parameters were used for different aorta geometries, which shows how much the blood flow velocities and WSS/OSI distribution can differ in a healthy aorta and aorta with aneurysm under the same conditions.

4.5Limitations

The personalised geometry models analysed in this study use non-individualised parameters of blood flow and boundary conditions. The second limitation of this analysis is the small number of models. During the analysis, the blood was simulated as a Newtonian fluid, and the flow was assumed to be laminar. We encountered mesh problems because the SimVascular software uses a Tetgen mesh type and has limitations of mesh enhancement functions. This led to some distortion of the results such as high OSI values in the branches of aortic arch and distorted flow distribution within the aneurysm velocity profile. Future studies should be based on numerical modelling of fluid-structure interactions while considering the exact property of elasticity of the aortic wall and surrounding tissues from patient images. Although FSI analysis is more complex and requires more time and resources for modelling than simple CFD structural analysis or a coupled momentum method, the solutions obtained in FSI studies are more accurate. A patient-specific flow rate profile and turbulent flow modelling should be used to accurately assess blood flow and vascular stress and the associated risks of wall pathologies.