Regional pulse wave velocity and stress in aneurysmal arch-shaped aorta

Abstract

The pulse wave velocity (PWV) has been shown to be associated with the properties of blood vessel and a cardiovascular risk factor such as aneurysm. The global PWV estimation is applied in conventional clinical diagnosis. However, the geometry of blood vessel changes along the wave traveling path and the global PWV estimation may not always detect regional wall changes resulting from cardiovascular diseases. In this study, a fluid structure interaction (FSI) analysis was applied on arch-shaped aortas with and without aneurysm aimed at determining the effects of the number of aneurysm, aneurysm size and the modulus ratio (aneurysm to wall modulus) on the pulse wave propagation and velocity. The characterization for each stage of aneurysmal aorta was simulated by progressively increasing aortic stiffness and aneurysm size. The pulse wave propagations and velocities were estimated from the two-dimensional spatial-temporal plot of the normalized wall displacement based on elastic deformation. The descending forward and arch reflected PWVs of aneurysmal aortic arch models were found up to 9.7% and 122.8%, respectively, deviate from the PWV of non-aneurysmal aortic arch model. The PWV patterns and magnitudes can be used to distinguish the characterization of the normal and aneurysmal aortic walls and shown to be relevant regional markers utilized in clinical diagnosis.

Keywords

Aneurysm heterogeneous aortic wall pressure wave stiffness

1 Introduction

Several cardiovascular diseases occur from the change of the vascular mechanical properties. The responses to stress and strain of the arterial wall, i.e., inflation, contraction and extension result from the blood pressure. Numerous studies have reported that the blood pressure increases with the arterial wall stiffening. There are several in vivo noninvasive estimations to assess the arterial stiffness [1–4]. The conventional method is global measurement of the traveling time delay of radial pulse wave profiles at two different distant locations using tonometry. The distance between the two different locations is divided by the traveling time delay called pulse wave velocity (PWV). The arterial stiffness is therefore related to PWV via the Moens-Korteweg equation. Since the PWV measured represents the average value between two remote sites, this conventional measurement is inaccurate in application such as errors of distance and time delay measurements. In fact, the arterial wall geometry changes between two measurement points and the diseases are not always in the travel line of the pulse wave [5]. Another measurement is regional using ultrasound and MRI-based methods to locally detect the changes in stiffness. Using these imaging techniques, the PWV is not only quantitatively determined but also the pulse wave propagation can be qualitatively visualized. The results of these methods are the 2D regional PWV in longitudinal direction and the aortic wall displacement in radial direction which are related to the arterial stiffness through the mathematical formulas. The regional PWV is therefore properly used as a parameter to detect the early onset or focal of the cardiovascular diseases such as aneurysms. These methods involving assessments of regional aortic stiffness have been proposed and developed to apply in clinical detection such as pulse wave imaging (PWI) using ultrasound-based method [4,6,7]. The PWI has been investigated in phantoms [4] and is applied in normal and aneurysmal aortas, both mouse and human [5,8]. To verify the PWI, the Fluid-Structure Interaction (FSI) finite-element simulation is validated against experimental phantom which present discontinuities of arterial wall composition due to aneurysm [9]. However, the regional variations in composition of diseased arterial wall and the diseased locations along the arterial wall still lead to erroneous estimation of both global and regional PWV. The arterial stiffness based on the regional PWV has mainly been determined in the straight-geometry aorta with accessible ultrasound, not locate behind bone or air-contained organs, such as abdominal aorta. In cases of ascending, aortic arch and descending aorta aneurysms, the PWV-based ultrasound technique cannot be revealed. The effect of geometry, especially at the presence of disease, on propagation speed and patterns of PWV remains to be investigated. Therefore, it is interesting to analyze the stiffening of aortic arch using regional PWV technique in order to investigate the propagation speed and patterns of PWV using arch shape with non-aneurysmal and aneurysmal models.

Use of Finite Element Method (FEM) can provide the insight and visualization of the complicated variations of the biomechanical problems. The simulations using Finite Element Analysis (FEA) are applied to characterize the deformation, stress and strain, and flow dynamics of healthy and diseased arteries which are modeled under static or pulsatile pressure-deformation analysis [10,11]. Due to the complexity of blood flow in arteries, the model of the luminal pressure variations with altered hemodynamics is utilized to realistically simulate the fluid-solid arterial responses. A few numerical simulations have been studied on transient blood flow in human aorta [12–14]. The algorithms of coupled fluid-structure dynamic problems have been developed in 2D and 3D numerical simulations using finite element method [15,16]. The finite element computation of coupled fluid-structure interaction (FSI) simulations have been applied to investigate the effects of wall heterogeneities on the aortic pulse wave propagation speed and pattern [9] and to estimate the stiffness, wall stress, flow dynamics and mass transfer [17–20]. The arterial geometry changes such as arterial branches and arch affect the propagation speed and patterns of PWV[21]. The numerical modeling in aortic arch aneurysm has been developed, it has been shown that the aneurysm occurs in the aortic arch has higher risk of rupture than other locations along the aorta [22]. While the variations of arterial rupture criteria have been studied, it has been shown that the risk of rupture increases with the level of luminal pressure acting on the arterial wall and the inappropriate strain due to the different deformation of each arterial layer [23]. In this study, the simulation of fluid-structure interaction is therefore developed in arch-shaped aorta using finite element method. The finite element models with normal (non-aneurysm) and pathological (aneurysm) conditions are established. The regional propagation speed and patterns of PWV, and wall stress are estimated.

2 Methods

2.1 Numerical model

The pulse wave propagation and velocity of the aortic wall subjected to the luminal pulsatile pressure have been observed by numerical studies. The governing equations were solved by using the finite element method. The governing equations, the mechanical formulations and the boundary conditions under consideration are described below.

2.1.1 Governing equations

The blood flow resulting in the luminal pulsatile pressure was assumed to be laminar, viscous and incompressible. The governing equation for the fluid flow can be described by the Navier–Stokes equation, given by $\begin{eqnarray}\displaystyle {\rho}_{f}\frac{\partial v}{\partial t}+{\rho}_{f}\boldsymbol{v}{\nabla}\boldsymbol{v}=-{\nabla}p+{\mu}{\nabla}^{2}\boldsymbol{v}+\boldsymbol{f} & & \displaystyle\end{eqnarray}$ (1) where 𝜌_f denotes the blood density, v is the velocity vector of the blood flow, p is the luminal pressure, 𝜇 is the dynamic viscosity of blood, f is the body force, and 𝛻 is the gradient operator with respect to the position vector in deformed configuration.

The aortic wall was deformed corresponding to the applied load. The deformations of the aortic wall were solved under the elastic response consideration. The motion equation of continuum was used as the governing equation for the solid deformation, which is $\begin{eqnarray}\displaystyle {\nabla}\boldsymbol{\sigma}+\boldsymbol{G}={\rho}_{s}\boldsymbol{a} & & \displaystyle\end{eqnarray}$ (2) where 𝝈 denotes the Cauchy stress tensor, G is the body force affecting the aortic wall, 𝜌_s is the density of aortic wall, and a is the acceleration of aortic wall.

The conservation of mass was described by the equation of continuity as $\begin{eqnarray}\displaystyle \frac{\partial {\rho}_{s}}{\partial t}+{\rho}_{s}{\nabla}\boldsymbol{v}=0 & & \displaystyle\end{eqnarray}$ (3) where 𝜌_s is the density of aortic wall, t is time, and v is the velocity vector.

2.1.2 Mechanical formulations

The aorta consisted of ascending part, arch, and descending part. Two domains were considered, i.e., aortic wall and lumen. The blood flow results in the luminal pressure acting on the aortic wall. The interaction between blood flow and aortic wall presents the stress and strain (deformation) of the aortic wall. The mathematical formulations were detailed as follow.

2.1.3 Aortic wall

The aortic wall was assumed to be of isotropic material and exhibited purely elastic behavior. The aortic geometry was arch-shaped. The schematic illustration of aortic geometry and the dimensions of aortic wall and aneurysm are shown in Figs 1 and 2(a)–(d). The aortic wall thickness s and luminal diameter d _i were respectively assumed to be 2.2 mm and 12 mm. The longitudinal lengths of ascending and descending parts were 12.5 mm and 300 mm, respectively. The different lengths of aneurysm L _a equal to 25 mm, 37.5 mm, and 50 mm were taken to be either located at 50 mm from the inlet of descending part or at aortic arch. The ratio of length to height of aneurysm was equal to 12. The aortic wall and aneurysm were modeled with wall density 𝜌_s = 1,050 kg/m³, Poison’s ratio 𝜈 = 0.48 and Young’s modulus values E = 2.56–10.24 MPa. Young’s moduli of E = 2.56, 5.12, 7.68 and 10.24 MPa [9] were used to represent the stiffness of aneurysm. Young’s modulus of non-aneurysmal wall was assumed to be equal to E = 2.56 MPa [24]. The modulus ratios of aneurysm to non-aneurysmal wall were therefore equal to MR = 1, 2, 3 and 4, respectively. Young’s moduli and aneurysm lengths at any location of the aortic arch models were summarized in Table 1.

Kinematics of aortic wall was considered in cylindrical coordinate system. The deformation was determined from the transformation of the material particle point X (R, 𝛩, Z) in reference configuration 𝛺_o to x (r, 𝜃, z) in deformed configuration 𝛺. The deformation gradient F is given by $\begin{eqnarray}\displaystyle \boldsymbol{F}=\frac{\partial \boldsymbol{x}}{\partial \boldsymbol{X}}. & & \displaystyle\end{eqnarray}$ (4)

The rotation-independent deformation tensor was applied to measure the shape change in the deformed body. The Eulerian-Almansi finite strain tensor 𝜺 referenced to the deformed configuration can be written as follow. $\begin{eqnarray}\displaystyle \boldsymbol{\varepsilon}=\frac{1}{2}(\boldsymbol{I}-\boldsymbol{B}^{-1}) & & \displaystyle\end{eqnarray}$ (5) where I denotes identity tensor, B ⁻¹ denotes inverse of the left Cauchy-Green deformation tensor and is given in term of the deformation gradient F which is $\begin{eqnarray}\displaystyle \boldsymbol{B}^{-1}=\boldsymbol{F}^{-T}\boldsymbol{F}^{-1} & & \displaystyle\end{eqnarray}$ (6) where superscript T is transpose of matrix.

The stress measure of the force acting on area element in the deformed configuration is known as the Cauchy stress tensor 𝝈, which can be written as the derivative of strain energy function 𝜓 with respect to the Eulerian-Almansi finite strain tensor 𝜺, i.e., $\begin{eqnarray}\displaystyle \boldsymbol{\sigma}=\frac{\partial {\psi}}{\partial \boldsymbol{\varepsilon}} & & \displaystyle\end{eqnarray}$ (7)

The Hooke’s law relationship between stress and strain, i.e., 𝝈 = E𝜺, was used to describe the response of aortic wall. Therefore, the strain energy function was ${\psi}=\frac{1}{2}$ 𝝈𝜺.

For isotropic material, the Hooke’s law can be expressed in terms of Young’s modulus E and Poisson’s ratio 𝜈 by $\begin{eqnarray}\displaystyle \boldsymbol{\varepsilon}=\frac{\left(1+{\nu}\right)}{E}\boldsymbol{\sigma}-\frac{{\nu}}{E}tr(\boldsymbol{\sigma})\boldsymbol{I} & & \displaystyle\end{eqnarray}$ (8) where tr is trace of the matrix.

The Cauchy stress tensor can be furthermore transformed onto the von Mises stress 𝝈_v by following equation. $\begin{eqnarray}\displaystyle \boldsymbol{\sigma}_{\boldsymbol{v}}=\sqrt{3J_{2}} & & \displaystyle\end{eqnarray}$ (9) where J ₂ is the second deviatoric stress invariant, which is $\begin{eqnarray}\displaystyle J_{2}=\frac{1}{2}\left[tr(\boldsymbol{\sigma}^{2})-\frac{1}{3}tr(\boldsymbol{\sigma})^{2}\right]\!. & & \displaystyle\end{eqnarray}$ (10)

2.1.4 Lumen

The luminal pressure profile was assigned at the tube inlet. The pulsatile pressure was represented as an axisymmetric flow varied along the longitudinal direction. The pressure at the tube outlet and extravascular pressure were assumed to be null. The displacements of the inlet and outlet ends were fixed in three dimensions. The fluid structure interface was defined as frictionless. The blood flow was modeled as Newtonian. The fluid density 𝜌_f = 1,000 kg/m³, reference speed of sound c _o = 1,483 m/s and fluid viscosity 𝜇 = 0.0001 Ns/m² were applied in the model. Since the flow was assumed to be Newtonian, axisymmetric flow and not affected by the gravitational force, the Navier–Stokes equation can be described as $\begin{eqnarray}\displaystyle \frac{\partial p}{\partial r}=0 & & \displaystyle \nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle \frac{\partial p}{\partial {\theta}}=0 & & \displaystyle\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle {\rho}_{f}\frac{\partial u_{z}}{\partial t}+u_{z}\frac{\partial u_{z}}{\partial z}=-\frac{\partial p}{\partial z}+{\mu}\frac{\partial ^{2}u_{z}}{\partial r^{2}}+{\mu}\frac{\partial ^{2}u_{z}}{\partial z^{2}} & & \displaystyle \nonumber\end{eqnarray}$ where u _z denotes the velocity component in longitudinal direction, p is the luminal pressure, t is time, r, 𝜃 and z are the radial, circumferential and longitudinal positions in deformed configuration, respectively.

2.2 Finite element method

The time dependent pressure and velocity variables can be approximated by the form expansion, given by $\begin{eqnarray}\displaystyle P(x)=\mathop{\sum }_{n=1}^{N}{\phi}_{n}\left(x\right)P_{n}=\boldsymbol{\Phi}^{T}\mathbf{P} & & \displaystyle\end{eqnarray}$ (12) $\begin{eqnarray}\displaystyle u(x)=\mathop{\sum }_{n=1}^{N}{\psi}_{n}\left(x\right)u_{n}=\boldsymbol{\Psi}^{T}\boldsymbol{u} & & \displaystyle\end{eqnarray}$ (13) where 𝜱 and 𝜳 denote the shape function vectors of pressure and velocity, respectively. P is the nodal pressure and u is the nodal velocity vector.

The weighted residual method was used to establish the finite element equations [15]. The weak forms of governing equations were developed over the element domain 𝛺_e. By substituting the shape function forms of variables into the weak forms, the finite element equations were obtained. The finite element equations and boundary conditions were applied to estimate the deformations of the aortic wall. The equations of discretized finite elements for each element were solved and interconnected to be the system by the nodal points.

The input parameters for aortic wall and lumen domains were summarized in Table 2 and the numerical procedure is illustrated in Fig. 3.

2.3 Fluid-structure interaction

The finite element method was applied to solve the mathematical formulations concerning the mechanical models of the lumen and aortic wall. The three-dimensional hexagonal elements were used to mesh the fluid and solid domains (Fig. 4(a) and (b)). The non-aneurysmal aortic arch model contained 107,110 elements. The numbers of elements and nodes of the aneurysmal aortic arch models were changed between 135,240 to 226,936 elements. The calculations were performed and the coupled models were simulated to visualize the responses of the aortic wall to the pulsatile blood flow. The wall stress was estimated under Von Mises criterion. The nodal points along the longitudinal length of the tube were selected to determine the radial displacement of the aortic wall with respect to time.

2.4 Pulse wave velocity (PWV) and propagation determination

The wave propagations were characterized by mapping on a two-dimensional plane with the x-axis of time and the y-axis of longitudinal location along the tube. The wave peaks of the wall displacement at each time increment were then tracked. The pulse wave velocities were determined by fitting the wave peaks, which were the slopes of the longitudinal location and time relation.

3 Results

The pulse wave velocity (PWV) including its propagation and wall stress of the arch-shaped aortic wall were assessed in the non-aneurysmal and aneurysmal aortic arch models. The aneurysmal aortic arch model can be separated into the homogeneous and non-homogeneous models, which denoted the wall containing the same and different values of the Young’s modulus, respectively. For the non-homogeneous aneurysmal aortic arch model, the aneurysm was assumed to be stiffer than the aortic wall with non-aneurysm.

Figures 5–9 illustrate the 2D spatial-temporal plot of the normalized wall displacement of the non-aneurysmal (E = 2.56 MPa) and aneurysmal aortic walls with different aneurysm numbers, aneurysm lengths, modulus ratios and aneurysm locations. The void circles are wave peaks. The wave peaks on the relationship between the distance and time were fitted using linear regression model to yield the PWV and its propagation.

Figure 5 shows the 2D spatial-temporal plot of the normalized wall displacement of the non-aneurysmal (E = 2.56 MPa) (Fig. 5(a)) and aneurysmal aortic walls with representative conditions of the modulus ratio, aneurysm length, and aneurysm numbers (Fig. 5(b)–(d)), respectively. The forward wave on the ascending aortic wall was traveled toward the arch and separated into two portions, which were the reflected and forward waves unless the standing wave can be observed at the arch aneurysm. When the forward wave on the descending aortic wall was traveled toward the aneurysm, the wave was reflected at the inlet aneurysm site and then continued forward traveling when time increased.

For the 2D spatial-temporal plot of the normalized wall displacement of the non-aneurysmal aortic walls with E = 2.56 MPa (Fig. 5(a)), the forward waves were found at the ascending and descending parts of the aortic wall. The pulse wave velocities of the forward waves on the ascending and descending parts were found to be equal to 11.29 and 13.53 m/s, respectively. At the aortic arch, the wave was minor traveled backward with the reflected wave velocity of 0.99 m/s.

The pulse wave velocities of the aortic wall containing one aneurysm (at descending part) with the length of aneurysm L _a = 25 mm and modulus ratio MR = 2 were tracked from the 2D spatial-temporal plot of the normalized wall displacement (Fig. 5(b)). The reflected wave of the aneurysmal aortic wall was higher in velocity than that of the non-aneurysmal aortic wall, which was found to be equal to 1.46 m/s. The forward wave velocity of the aneurysmal wall region was found to be equal to 22.40 m/s.

Figure 5(c) shows the normalized wall displacement when the aneurysm length L _a and modulus ratio MR were increased up to 50 mm and 4, respectively. It was found that, the wave was standing and strongly reflected at the arch with reflected wave velocity of 19.04 m/s. The fluctuation of reflected wave pattern was also found to increase with the aneurysm length. The higher forward wave velocity of the aneurysmal wall region was found up to 38.94 m/s. Furthermore, the standing wave distinctly increased with the aneurysm number and was found at the inlet and outlet arch aneurysm sites (Fig. 5(d)).

The effect of the aneurysm number on the PWV is shown in Fig. 6. Figure 6(b) and (c) illustrate the aortic wall containing one aneurysm at descending part and aortic arch, respectively. The results showed that the reflected wave velocity at arch increased when the aneurysm occurred at the aortic arch. The aneurysm at aortic arch made the higher about 6 times in reflected wave velocity than that at descending part. The increasing aneurysm number also increased the reflected wave velocity at arch (Fig. 6(d)).

The effect of the aneurysm length on the PWV is shown in Fig. 7. The aortic wall containing one aneurysm (at descending part) was simulated with the length of aneurysm L _a = 25, 37.5, and 50 mm (Fig. 7(b), (c), and (d)), respectively. The reflected wave velocity at arch and its fluctuation increased with the aneurysm length. The standing wave was also found to increase with the aneurysm length.

The effect of the modulus ratio on the PWV is demonstrated in Figs 8 and 9. For the aortic wall containing one aneurysm at descending part (Fig. 8), the modulus ratio MR was shown to have a non-significant effect on the velocities of the forward and reflected waves and theirs pattern. However, the forward wave velocity of the aneurysmal wall region was found to increase with the modulus ratio MR. For the aortic wall containing one aneurysm at aortic arch (Fig. 9), the standing waves were clearly observed at the inlet and outlet arch aneurysm sites when modulus ratio MR increased. The velocities of the forward and reflected waves also did not change with the increasing of MR.

The pulse wave velocities of the reflected wave at arch and forward waves on the ascending and descending aortic walls obtained from the linear fits of the 2D spatial-temporal plots are shown in Table 3.

The stress and derivative of regional stress was investigated on the non-aneurysmal and aneurysmal aortic arch models. Figures 10–14 illustrate the stress and derivative of regional stress corresponding to the simulated conditions in Figs 5–9. The derivative of stress with respect to longitudinal position in deformed configuration was plotted to determine the magnitude of regional stress variation along the aortic wall. The aneurysm located at descending part provided the higher in magnitude of regional stress variation than that located at aortic arch (Fig. 11). The aneurysm length was shown to have a non-significant effect on the derivative of regional stress (Fig. 12). The increasing modulus ratio resulted in the increasing magnitude of regional stress variation (Figs 13 and 14) and was found to have more effect on the stress wave when the aneurysm located at descending part.

4 Discussion

The aorta plays an important role in delivering the blood containing oxygen and nutrients to all the cells in the body. The aneurysm presented at aortic wall has been reported to be the leading cause of death worldwide. To understand the sophisticated mechanics of aneurysmal aortic wall, the simulation was used as the powerful tool to visualize the phenomenon. Fluid structure interaction (FSI) simulation based on finite element method was performed to solve the numerical models. The results from these numerical models have been validated experimentally and numerically in other aneurysmal aortic wall models. It should be noted that, the results have been carefully interpreted under the model assumptions. The mechanical parameters, i.e. wall deformation, strain, and stress, under pressure variations were investigated on the aneurysmal aortic models. The non-homogeneity geometry, size, distribution, and wall properties have been reported to affect the mechanical parameters [2,9,11,25,26]. In this study, the pulse wave propagation and velocity of arch-shaped aneurysmal aortic models were characterized under different conditions of the different aneurysm size, number of aneurysm, and modulus ratio (aneurysm to aortic wall modulus). The regional wave pattern, velocity and the presence of the aneurysm on the aortic wall can be captured on the 2D spatial-temporal plot of the normalized wall displacement versus time. The main wave patterns, i.e. forward, reflected, and standing waves, were found as same as previous studies on both non-aneurysm and aneurysm models. The feasibility studies of pulse wave propagation and velocity were served as a guide for experimental phantom and pulse wave imaging in vivo [3,17]. In this study, the descending forward and arch reflected PWVs of aneurysmal aortic arch models were found up to 9.7% and 122.8%, respectively, different from the PWV of non-aneurysmal aortic arch model. However, it should be again noted that, the values of PWV were different due to the assumptions of the wall properties, geometry, and boundary of aorta and the geometry, size, and distribution of aneurysm. Higher PWV were always found at the aneurysm wall boundary due to its higher wall stiffness. The increase in wall stiffness increased the PWV. The other multi-layered and sophisticated aortic arch models were previously simulated using numerical methods and resulted in the different outputs [27–29]. In this study, since the vessel branches at the arch and distal abdominal aorta were not included, the reflected wave at the junctions was not found. The main reflected wave was found to be occurred due to the arch of the aorta. The characteristics of regional aortic wall can be therefore determined by the wave patterns and amplitudes. The wave reflection affected the coupled fluid and structural dynamics such as pressure and flow waveforms. The pressure and flow waveforms in the arterial system were not equal result from the wave reflection [21]. The reflected wave also affected the ejection dynamics of the left ventricle. The pressure ejected from the left ventricle was reduced by the reflected wave. The progressive stiffness altered the pressure by increasing not only the velocity magnitudes of forward wave but also the reflected wave, causing the pressure boost of the left ventricle in late systole [30]. The ventricular load therefore increased with the increasing the velocity magnitudes of reflected wave.

The higher pressure gradient inside the aneurysm has been reported to be a cause of higher magnitude of wall deformation [26]. In this study, the stress was found to be dramatically increased, approximately up to 30% higher, at the connection boundaries of aortic wall and aneurysm regions. The change of modulus ratio caused the change in regional stress variation. This discontinuity of the stress can be indicated the relevant lesion such as the onset and progression of aneurysm [31]. Therefore, the increase in heterogeneity of a longitudinal aortic wall also increased the risk of aortic rupture, in addition to the high pressure level resulting from the wall stiffening [23]. The studies of aneurysmal aortic models for rupture and risk predictions have been developed (see, e.g., [32] and references therein). Several researchers have proposed the alternative noninvasive techniques used for diagnosis of atherosclerotic and aneurysmal arteries [33–35]. The effect of the wall viscosity on the arterial wall motion can be found in their reports, i.e., that the viscoelastic deterioration of arterial wall related the atherosclerosis and aneurysm. The viscosity of the aortic wall is limited by the assumptions used in this study. However, the advantages of this study are that the PWV and propagation estimated from the numerical models of non-aneurysmal and aneurysmal aortic walls were investigated, incorporating the consideration of the effects of the number of aneurysm, aneurysm size and the modulus ratio (aneurysm to wall modulus). The influences of arch-shaped geometry of aorta and location of aneurysm were furthermore included in this study. The magnitudes of the stress and derivative of stress along the arch-shaped aortic walls were additionally determined to describe the attributes of the aneurysm. However, the numerical models must be further developed to assess the more realistic solid-fluid mechanics in order to apply as a practical tool in clinical diagnosis.

5 Conclusion

The non-aneurysmal and aneurysmal aortic arch models with boundary conditions under consideration were successfully established to estimate the pulse wave velocity propagation and velocity using the fluid structure interaction (FSI) simulation. The 2D spatial-temporal plot of the normalized wall displacement provided clearly information on the regional pulse wave propagation. The effects of the number of aneurysm, aneurysm size and the modulus ratio (aneurysm to wall modulus) on the pulse wave propagation and velocity were examined. The changes of the number of aneurysm, aneurysm size and the modulus ratio were quantitatively and qualitatively identified by the velocity and traveling of the pulse wave, respectively. The magnitudes of stress and derivative of regional stress in the aneurysmal aortic walls were determined based on the pressure waveform result from the luminal blood flow. In addition, the arch-shaped geometry of aorta was included in this study and found to be the one of the relevant parameters in the pulsatile aortic characterization.

Footnotes

Acknowledgements

This research was financially supported by the Office of the Thailand Research Fund (grant no. TRG5680075), and Maejo University.

Conflict of interest

None to report.

Nomenclature

References

Khamdaeng

Luo

Vappou

Terdtoon

and Konofagou

E.E.

, Arterial stiffness identification of the human carotid artery using the stress–strain relationship in vivo, Ultrasonics 52 (2012), 402–411.

Shahmirzadi

and Konofagou

E.E.

, Quantification of arterial wall inhomogeneity size, distribution, and modulus contrast using FSI numerical pulse wave propagation, Artery Research 8 (2014), 57–65.

Shahmirzadi

Narayanan

R.X.

Qaqish

W.W.

and Konofagou

E.E.

, Mapping the longitudinal wall stiffness heterogeneities within intact canine aortas using Pulse Wave Imaging (PWI) ex vivo, Journal of Biomechanics 46 (2013), 1866–1874.

Vappou

Luo

and Konofagou

E.E.

, Pulse wave imaging for noninvasive and quantitative measurement of arterial stiffness in vivo, American Journal of Hypertension 23 (2010), 393–398.

Luo

Fujikura

Tyrie

L.S.

Tilson

M.D.

and Konofagou

E.E.

, Pulse wave imaging of normal and aneurysmal abdominal aortas in vivo, IEEE Transactions on Medical Imaging 28 (2009), 477–486.

Luo

R.X.

and Konofagou

E.E.

, Pulse wave imaging of the human carotid artery: An in vivo feasibility study, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 59(1) (2012), 174–181.

Vappou

Luo

Okajima

Di Tullio

and Konofagou

, Aortic pulse wave velocity measured by pulse wave imaging (PWI): A comparison with applanation tonometry, Artery Research 5 (2011), 65–71.

Luo

Lee

W.N.

Wang

and Konofagou

E.E.

, Pulse wave imaging of human abdominal aortas in vivo, IEEE Ultrason. Symp. Proc. 2008 859–862.

Shahmirzadi

and Konofagou

E.E.

, Detection of aortic wall inclusions using regional pulse wave propagation and velocity in silico, Artery Research 6 (2012), 114–123.

10.

Park

D.E.

Richards

M.S.

Rubin

J.M.

Hamilton

Kruger

G.H.

and Weitzel

W.F.

, Arterial elasticity imaging: Comparison of finite-element analysis models with high-resolution ultrasound speckle tracking, Cardiovascular Ultrasound 8 (2010), 1–10.

11.

Scotti

C.M.

Jimenez

Muluk

S.C.

and Finol

E.A.

, Wall stress and flow dynamics in abdominal aortic aneurysms: Finite element analysis vs. fluid-structure interaction, Computer Methods in Biomechanics and Biomedical Engineering 11 (2008), 301–322.

12.

Qiao

A.K.

W.Y.

and Liu

Y.J.

, Study on hemodynamics in patient-specific thoracic aortc aneurysm, Theoretical & Applied Mechanics Letters 1 (2011), 014001-1-4.

13.

Suh

G.Y.

Les

A.S.

Tenforde

A.S.

Shadden

S.C.

Spilker

R.L.

Yeung

J.J.

, Hemodynamic changes quantified in abdominal aortic aneurysms with increasing exercise intensity using MR excercise imaging and image-based computational fluid dynamics, Annals of Biomedical Engineering 39 (2011), 2186–2202.

14.

Tse

K.M.

Chiu

Lee

H.P.

and Ho

, Investigation of hemodynamics in the development of dissecting aneurysm within patient-specific dissecting aneurismal aortas using computational fluid dynamics (CFD) simulations, Journal of Biomechanics 44 (2011), 827–836.

15.

Mitra

and Sinhamahapatra

K.P.

, 2D simulation of fluid-structure interaction using finite element method, Finite Elements in Analysis and Design 45 (2008), 52–59.

16.

Teixeira

P.R.F.

and Awruch

A.M.

, Numerical simulation of fluid–structure interaction using the finite element method, Computers & Fluids 34 (2005), 249–273.

17.

Shahmirzadi

R.X.

and Konofagou

E.E.

, Pulse wave propagation in straight-geometry vessels for stiffness estimation: Theory, simulations, phantoms and in vitro findings, Journal of Biomedical Engineering 134 (2012), 114502-1-6.

18.

and Kleinstreuer

, Blood flow and structure interactions in a stented abdominal aortic aneurysm model, Medical Engineering & Physics 27 (2005), 369–382.

19.

Karimi

Navidbakhsh

Razaghi

and Haghpanahi

, A computational fluid-structure interaction model for plaque vulnerability assessment in atherosclerotic human coronary arteries, Journal of Applied Physics 115 (2014), 144702-1-8.

20.

Valencia

and Villanueva

, Unsteady flow and mass transfer in models of stenotic arteries considering fluid-structure interaction, International Communications in Heat and Mass Transfer 33 (2006), 966–975.

21.

Fung

Y.C.

, Biomechanics: Circulation, Springer-Verlag, New York, 1997.

22.

Matsuzawa

Gao

Qiao

Ohta

and Okada

, Numerical simulation in aortic arch aneurysm, in: Etiology, Pathogenesis and Pathophysiology of Aortic Aneurysms and Aneurysm Rupture Grundmann

(ed.), InTech, 2011, pp. 207–223.

23.

Khamdaengyodtai

Vafai

Sakulchangsatjatai

and Terdtoon

, Effects of pressure on arterial failure, Journal of Biomechanics 45 (2012), 2577–2588.

24.

Humphrey

J.D.

, Cardiovascular Solid Mechanics: Cells, Tissues, and Organs, Springer-Verlag, New York, 2002.

25.

Khamdaeng

Panyoyai

Wonsiriamnuay

and Terdtoon

, Pulse wave propagation and velocity in aneurysmal aota using FSI model, Research & Knowledge 2 (2016), 67–73.

26.

Lin

and Shahmirzadi

, FSI simulations of pulse wave propagation in human abdominal aortic aneurysm: The effects of sac geometry and stiffness, Biomedical Engineering and Computational Biology 7 (2016), 25–36.

27.

Gao

Qiao

and Matsuzawa

, Numerical simulation in aortic arch aneurysm, in: Etiology, Pathogenesis and Pathophysiology of Aortic Aneurysms and Aneurysm Rupture Grundmann

(ed.), InTech, 2011, pp. 207–222.

28.

Gao

Ueda

and Okada

, Fluid structure interaction simulation in three-layered aortic aneurysm model under pulsatile flow: Comparison of wrapping and stenting, Journal of Biomechanics 46 (2013), 1335–1342.

29.

Nardi

and Avrahami

, Approaches for treatment of aortic arch aneurysm, a numerical study, Journal of Biomechanics , in press.

30.

Nichols

W.W.

and O’Rourke

M.F.

, McDonald’s Blood Flow in Arteries: Theoretical, Experimental and Clinical Principles, 5th edn, Hodder Arnold, London, 2005.

31.

Danpinid

Terdtoon

Sakulchangsatjatai

Vappou

and Konofagou

E.E.

, Stress-strain analysis of a longitudinal heterogeneous arterial wall, in: Analysis and Design of Biological Materials and Structures Ochsner

da Silva

L.F.M.

and Altenbach

(eds), Springer-Verlag, Berlin, Heidelberg, 2012, pp. 19–32.

32.

Humphrey

J.D.

and Holzapfel

G.A.

, Mechanics, mechanobiology, and modeling of human abdominal aorta and aneurysms, Journal of Biomechanics 45 (2012), 805–814.

33.

Taniguchi

Hosaka

Miyahara

Hoshina

Okamoto

Shigematus

, Viscoelastic deterioration of the carotid artery vascular wall is a possible predictor of coronary artery disease, Journal of Atherosclerosis and Thrombosis 22 (2015), 415–423.

34.

Yokobori Jr.

A.T.

Ohmi

Monma

Tomono

Inoue

Owa

, Correlation between the characteristics of Acceleration and Visco elasticity of artery wall under Pulsatile Flow conditions (Physical Meaning of I* as a Parameter of Progressive Behaviors of Atherosclerosis and Arteriosclerosis), Bio-Medical Materials and Engineering, An International Journal 23 (2013), 75–91.

35.

Yokobori Jr.

A.T.

Owa

Ichiki

Satoh

Ohtomo

Satoh

, The analysis and diagnosis of unstable behavior of the blood vessel wall with an aneurysm based on noise science, Journal of Atherosclerosis and Thrombosis 13 (2006), 163–174.