Proper tangential vector field for the wall shear stress evaluation in numerical blood flow simulations in human carotis tree

04/08/2022
by   Tristan Probst, et al.
0

In numerical simulations of the cardiovascular system the realistic depiction of the blood vessel is essential. The vessel morphology plays a critical role in the underlying hemodynamical computation, for which 3D geometries are obtained via imaging techniques from patients CTA or MRI scans. To evaluate the patients current risks suffering vessel wall damage or an ischaemic stroke, areas of interest are closely observed. A main indicator for these defects is the stress the blood is exerting on the surrounding vessel tissue. The wall shear stress (WSS) and its cyclic evaluation as the oscillatory shear index (OSI) can be used as an indicator for future medical procedures. As a result of low resolution imaging and craggy stenotic surface areas, the geometry model's mesh is often non smooth on the surface areas. The automatically generated tangential vector fields of such a surface topology are therefore corrupted. Making an interpretation of the orientation dependent WSS not only difficult but partially unreliable. In order to smooth out the erratic surface tangential field we apply a projection of the vessel's center-line tangents of the to the surface, based on a method introduced by Morbiducci et al. The validation of our approach is made by comparing the orientation focused evaluation of the WSS to the generic method with automatically generated tangential vectors. The results of this work focus on the carotid bifurcation area and a distal stenosis area in the internal carotid artery.

READ FULL TEXT VIEW PDF

Authors

page 5

page 7

page 9

page 10

page 11

page 12

05/10/2020

Towards a Mathematical Model for the Solidification and Rupture of Blood in Stenosed Arteries

In this paper, we present a mathematical and numerical model for blood s...
09/29/2021

Accurate quantification of blood flow wall shear stress using simulation-based imaging: a synthetic, comparative study

Simulation-based imaging (SBI) is a blood flow imaging technique that op...
12/08/2019

Post-processing techniques of 4D flow MRI: velocity and wall shear stress

As the original velocity field obtained from four-dimensional (4D) flow ...
09/10/2021

Mesh convolutional neural networks for wall shear stress estimation in 3D artery models

Computational fluid dynamics (CFD) is a valuable tool for personalised, ...
06/10/2019

Efficient Parallel Simulation of Blood Flows in Abdominal Aorta

It is known that the maximum diameter for the rupture-risk assessment of...
03/06/2021

Machine Learning versus Mathematical Model to Estimate the Transverse Shear Stress Distribution in a Rectangular Channel

One of the most important subjects of hydraulic engineering is the relia...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction

The importance of a healthy and functioning cardiovascular system is reflected in the WHO death statistics of 2019. Ischemic stroke was the disease responsible for the highest proportion of deaths across all countries and wealth levels [37]. The cause of ischemic stroke is an arterial vascular disease, which in its most common form, atherosclerosis, is an inflammatory response of the vessel wall to lipid metabolism disturbances and endothelial stress. This leads to the formation of multi-focal plaques and thus to the narrowing and hardening of the arteries and consequently to an insufficient supply of oxygen to the brain [8]

. A special role in atherosclerosis development plays the carotid artery, which is responsible for an estimated 18 - 25 % of thromboembolic strokes

[17]. In the carotid bifurcation the common carotid artery splits into the external and the internal carotid artery. While the former is responsible for supplying blood to the head and upper neck organs, the latter supplies blood to the brain. Both, the death toll of ischemic strokes and the drastic increase in the general prevalence of atherosclerosis, which is related to demographic change and the accompanying burden on health and care system, make it necessary to adequately address the danger posed by atherosclerosis. To provide necessary tools predicting the locations of sites susceptible to atherosclerotic damage, as well as to make recommendations for their optimal treatment, is one of the main goals of modern medicine.

The predictions of atherosclerosis and further cardiovascular risk through numerical simulations have become popular over the last decades. Especially in the field of fluid-structure-interaction there is an inexhaustible range of publications. They span from the incorporation of different mathematical models, such as non-Newtonian fluids [16, 19], different structure models, e.g. shell models and membrane models [5, 6], to exploring new numerical methods, which effectively tackle the multiphysicality with splitting techniques [15, 33, 4], just to name a few.

For the purpose of risk quantification parameters derived from numerical flow data as e.g., the wall-shear-stress (WSS) [1, 2, 22, 34, 40], or oscillatory shear index (OSI) [1, 27, 38], both referred to be correlated to cardiovascular risk, are of interest. It has been reported [21], that apart from regions with high amplitude WSS, areas with low and temporary oscillating WSS promote atherosclerotic processes. The multi-directional behaviour of WSS has also been linked to potential risk zones, see results on transversal WSS and other metrics in [14].

Some of the recent results report on proper visualisation tools for the localisation of cardiovascular risk zones. These are based on the interplay of imaging techniques for exploring the vessel morphology and simulated data as velocity streamlines or WSS, see, e.g., [9, 10] and further citations therein. In those tools the patient-based morphology of the chosen region in the vessel plays a crutial role for reliable risk prediction parameters mentioned above, based on numerical simulations.

The aim of our study is a reliable numerical exploration of ischemic stroke risk via quantification of the impact of the fluid flow dynamics in means of endothelial stresses in the carotid artery. The domain of interest is the carotid bifurcation area with its’ separation of the common carotid artery into the ICA - internal carotid artery and the ECA - external carotid artery. The 3D inner flow lumen shape of the carotid vessel tree including stenosed region, see Fig.2, as well the shape of the surrounding (thick) wall tissue, Fig.2, have been reconstructed from the computer tomography angiography (CTA) data set of a clinical patient with the method described in [9]. This carotid vessel tree as well as its arterial wall shape has been imported for fluid and solid-mechanical simulations in the finite-element-based simulation software Comsol Multiphysics. We perform numerical simulations based on the incompressible fluid flow model for both rigid walls, as well as compliant walls. For the latter the fluid-structure interaction (FSI) model describing the interplay of fluid and thick-wall dynamics is considered. The corresponding wall tissue mechanics is modeled by deformation of linear elastic material.

The obtained numerical results are used to evaluate the hemodynamic risk parameters: the wall shear stress (WSS) and the oscillatory shear index (OSI), measuring the temporal change (oscillations) of the wall shear stress direction over one cardiac cycle. Hereby, we consider the longitudinal component of WSS instead of its amplitude. The choice of proper tangential vectors is crucial for the specification of the WSS value. The sensitivity of, for example, the transversal WSS to the orientation of the tangential field has been addressed in [11],[14] [23] and [24].

Regarding the realistic geometries with spatially non smooth surface topology, the mesh-based erratic tangential vector field leads to problematic, spatially non smooth behaviour of longitudinal (or transversal) WSS on the surface of the extracted geometry. We address this problem and improve the evaluation of the longitudinal component of WSS in complex realistic geometries. Our approach is based on choosing tangential vectors obtained by the projection of the center-line of the vessel tree to the vessel surface, previously used by, e.g. Morbiducci et al. [25] or Arzani Shadden [2]. We present the numerical data and wall parameters as the longitudinal WSS and OSI derived for projected as well as automatically generated (mesh-based) tangential vector fields. Further, we compare the values obtained for rigid as well as compliant carotid vessel walls in order to examine the importance of the compliance of wall tissue in considered mathematical model of the carotid artery.

2 Mathematical modeling of the carotis flow

To investigate the individual risk of arterial defects and imminent health issues the individual examination of the carotid geometry can give a more reliable assessment of the patient’s current situation. A CTA scan of the patient is processed to a detailed three-dimensional arterial geometry, as described in [9] and the references therein. This technique was applied on the arterial lumen and expanded on the vessel wall domain as well. The resulting rigid domain of arterial lumen which includes part of the common carotid, its branching into internal and external carotid artery and two sub-branches of the latter, compare Fig. 2, has been considered for numerical simulations. On the other hand, simulations which featured fluid-structure-interaction are supplemented with a vessel domain which also reveals stenotic regions and the distinct wall thickness distribution of the patient throughout the area of interest, see Fig. 2. By cutting the computational domain proximal to the bifurcation area for the flow to fully develop and distal such that the boundary conditions don’t affect the dynamics of the flow, the geometry is defined.

In particular, we denote with the deforming fluid and structural domain in time , respectively and their shared boundary with .

Figure 1: Computational geometry of the inner human carotis lumen for the fluid flow modeling with rigid walls.
Figure 2: Computational geometry for the FSI-modeling of the human carotis including wall tissue, area of interest.

To model the blood flow in a deforming vessel the incompressible Navier-Stokes equations in a moving domain are considered in the arbitrary Lagrangian-Eulerian formulation. They read as

(1)

with the Cauchy stress tensor

and the strain rate tensor . The fluid velocity and pressure are solved in a fluid domain for constant viscosity and density . Here describes the fluid domain deformation velocity with respect to the reference domain and is the total (material) time derivative of the fluid velocity.

The boundary of the fluid domain consists of inflow and outflow boundary as well as the shared fluid-structure interface, . On the fixed inflow boundary a pulsating blood flow-rate was implemented, based on measured data taken from [26] as shown in Fig. 7. The outflow was assumed under an zero normal stress (do-nothing condition), that is

(2)

The mechanics of the arterial wall is modeled by a linear elastic material model for the wall-tissue deformation that reads in the reference configuration,

(3)

Here denotes the deformation, the deformation gradient, and the second Piola-Kirchhoff tensors is denoted with . Outer forces acting on the volume are incorporated in . The elasticity tensor is given with dependency on the Young’s modulus () and the Poisson’s ratio (). The elastic strain tensor is given by the Green-Lagrange strain . Note, that for small deformation gradients it holds .
The surfaces of the domain clipping, i.e. the annular boundaries of the vessel cuts, surrounding the in- and outflow boundaries of the fluid domain, and , are constraint to have deformation. In contrast, the outer surface of the vessel, which would be in contact with surrounding tissue, is able to move freely.
To maintain continuity of velocities and forces at the fluid-structure boundary layer, we enforce the coupling conditions balancing the velocities and the normal stresses of the fluid and the solid material,

(4)

here stand for fluid quantities transformed to the reference fluid-solid layer.
Note that the model considering rigid walls consists only of the fluid sub-problem (1) defined in the inner carotis lumen moreover and . In analogy to the velocity continuity condition in (4) the non-slip condition is prescribed on the vessel wall surface in case of rigid walls.

3 Derived hemodynamic wall parameters

3.1 Wall shear stress

The wall shear stress (WSS) measures the endothelial stress exerted by blood on the vessel tissue, its unit is given by . To explain the relationship between WSS and zones susceptible to atherosclerosis two main explanatory approaches can be found in the literature. The high shear stress theory identifies sites with prolonged high WSS as risk zones, the low shear stress theory considers also sites with oscillating and low WSS as potentially at risk, the correlation was reported in [21]. For a systematic review of both see [32].

The wall shear stress is defined on interface , or on the rigid vessel wall either as non directional quantity [28], describing the amplitude of the projection of the normal stress to the tangential plane,

(5)

where are unit vectors spanning the tangential plane.
Alternatively, direction-based indicators of WSS are used to measure the stress exerted by the fluid as well, [1, 11, 14, 23, 24]. For cylinder-like or other simple geometrical objects the vector quantities as the rotary (transversal) or longitudinal components of the WSS can be considered. In this study, we evaluate the longitudinal wall shear stress component (longitudinal WSS) for the carotid artery bifurcation vessel tree, defined as

(6)

where is the tangential vector showing in the longitudinal, i.e., the main flow direction.
Due to the non smooth surface of the studied 3D computational geometry, the specification of the proper longitudinal tangent vectors may be problematic. The direction of both tangent vectors spanning the tangential plane, obtained automatically for the surface topology may jump for neighbouring mesh element surfaces and even point in the opposite direction of the flow, as observed on the left picture in Fig. 3.

[scale=.35,,tics=10] Bilder/Abb531  (0,0)(25,0)(25,20)(0,20)[scale=.35,,tics=10] Bilder/Abb532 (0,0)(25,0)(25,20)(0,20)

Figure 3: Tangential vector field obtained in Comsol Multiphysics (left) and corrected tangential field flipped to overall flow direction (right), flow direction: from left to right.

This non continuous spacial behaviour and opposite direction of affect the value of longitudinal WSS (6) substantially. In order to correct the opposite direction of we flip it, i.e. switch its sign according to the angle between and the overall flow vector . For it holds . Thus, we can define the new tangential vector as follows

(7)

Note that the overall flow direction vector has to be specified locally for different sections of the computational domain tree.

3.2 Oscillatory shear index

The oscillatory shear index (OSI) introduced by Ku et al. [21] is a common indicator of disturbed flow. It characterises the temporal oscillations of WSS through its directional change at any point on the surface in the considered time period. The degree of oscillation is expressed by the ratio of the negative mean WSS compared to the mean absolute size of WSS, over the whole time interval. Note that OSI does not express the frequency of the sign change of the WSS.
Two definitions of OSI based on the definition of directional WSS (6) can be found in literature, see e.g., [1, 3, 27, 28, 35, 38, 40] differing in the ratio and absolute ratio of mean WSS, ,

(8)
(9)

The formula (9) defines values of OSI between 0 and 0.5, whereby 0 stands for no directional changes and 0.5 for completely balanced sign changes (oscillations) of WSS. Values in between imply corresponding measure of sign change of WSS. In contrast, in definition (8) negative values of mean WSS are considered as well. This leads to a range of OSI values from 0 to 1. Similarly as in definition (9), the value of 0.5 describes completely balanced sign changes of WSS as well. The OSI value 0 at one point of surface means that the WSS at this point is positive over the entire time interval considered. For the value 1, on the other hand, the WSS is negative over the entire time interval. Values between 0 and 0.5 show predominantly positive, values between 0.5 and 1 describe predominantly negative WSS over the whole time interval. The definition (8) thus allows to locate not only sites of oscillating WSS, but also sites with long-lasting or predominantly negative WSS. Thus, in contrast to (9), (8) provides an index that can represent both indicators of low shear stress theory. In what follows we therefore refer to (8) by mentioning the OSI.

3.3 Projection method for tangential field

Considering the (bi-)directional WSS, the tangential plane’s orthonormal base vectors are crucial for its evaluation, compare (6). For the longitudinal component of the WSS a proper longitudinal tangential vector, denoted here with , along the overall flow is necessary, whereby the orthogonal transversal tangent plays a role in transversal WSS. As depicted above, see Fig. 3, on complex surfaces the automatically rendered tangent vectors do not follow the overall flow direction in some topologically complicated surfaces. In order to overcome this difficulty we apply the approach based on the knowledge of the centerline of the vessel tree and its projection onto the vessel surface, similarly to the method of Morbiducci et al [25], presented in Fig. 5.

The centerline is obtained as the set of centerpoints of the maximally inscribed spheres. We use the 3D Voronoi diagram of the geometry to find and connect the centerpoints as implemented in the vascular modeling toolkit [18]. The method yields robust and detailed results with a resolution of about 3000 points. After getting the 3D curves of the center-line, its tangential vectors are obtained by subtraction of two points of the curve, see Fig. 5.

Figure 4: The Carotis domain with centerline and the normed tangential field of the centerline
Figure 5: Projection method by Morbiducci, adapted from [25]

Afterwards, the centerline tangent vectors are projected to the vessel surface, i.e. into each surface point . This is done in two steps, first is extrapolated to the surface points by geometry tool extrapolate with linear settings in Comsol. Then, the extrapolated centerline tangents are projected into the tangential plane of the carotid artery surface by subtracting the normal component from ,

(10)

where are the normal vectors to the carotid surface. The procedure is illustrated in Fig. 5. Here the centerline tangent is denoted by and the corresponding longitudinal component of the WSS by WSS. The resulting longitudinal tangential field , see Fig. 6, shows at first sight a more uniform alignment compared to the flipped tangential field (Fig. 3). In this manner, longitudinal tangent vectors related to the center-line of the geometry, with proper unidirectional behaviour on the surface, are implemented in Comsol. Both, projected (10) as well as flipped tangents (7) will be used for the evaluation of longitudinal WSS in what follows.

Figure 6: Tangential vector field obtained from the centerline by the projection method.

4 Results

The numerical simulations have been performed with Comsol Multiphysics [7] using MEMS Module to incorporate the interaction of laminar fluid flow with the linear elastic wall material. The software uses the arbitrary Eulerian-Lagrangian formulation, which consists of the Navier-Stokes equations in the Eulerian frame and the solid mechanics equations using the Lagrangian formulation. The interaction was chosen to be bidirectional, such that the fluid loading acted on the structure and the wall velocity is transmissioned to the fluid. We chose a monolithic, fully-coupled solving scheme to provide a robust solution for the dependent variables consisting of the solid deformation , the fluid flow velocity , pressure and the spatial mesh displacement . For the discretization of the fluid velocity and pressure linear and for the solid deformation quadratic finite elements have been chosen. For the time discretization BDF method of order 1 or 2 and an adaptive time-stepping has been applied. The simulation has been performed on a computational mesh consisting of about tetrahedral elements in the solid domain and elements in the fluid domain, about of which are tetrahedral and are prisms acting as two boundary layers. The whole simulation spans over two cardiac cycles, i.e. , and is driven by a pulsatile flow rate presented in Fig. 7, starting with the domain at rest, i.e., zero deformation and zero flow at . The presented results are chosen from the second cycle where the flow is fully developed.

Figure 7: Flow waveform at the inflow cut of common carotid artery with 7.9 ml/s mean flow rate, waveform adapted from [1, 26].
Figure 8: A part of computational mesh for solid walls (left) and for compliant walls FSI simulations (right), chosen sections of interest.

The evaluation of numerical data and derived wall parameters are compared for four model configurations. In two different vessel wall models: rigid elastic vessel walls two different tangential fields: flipped (7) projected tangential field (10) has been used to calculate the WSS and OSI, the overview can be found in Table 1. In every simulation fluid density kg.m and constant viscosity Pa.s was chosen. The wall parameters density kg.m, Young’s modulus MPa and Poisson’s ratio 222The choice of Poisson’s ratio has negligible impact on the vessel wall displacement in our modeling. were used in simulations (c) and (d) with fluid-structure interaction.

Configuration Properties
(a) rigid walls / flipped tangents
(b) rigid walls / projected tangents
(c) elastic walls / flipped tangents
(d) elastic walls / projected tangents
Table 1: Configurations of the model evaluations.

In Fig. 9 the blood velocity streamlines in the stenosed region of the ACI is presented at the time of maximal flow-rate, . The observed vortices are located in the stenotic bulges, but also in the area adjacent to the stenosis. Note, that the appearance of vortices is related to high OSI values, presented in Fig. 14, and may imply progression of lesions along the carotid artery tree, which is a common hemodynamic hypothesis [36]. On the other hand, high and unidirectional velocity streamlines are observed along the inner wall of the ICA and are related to high WSS values, as seen in Fig. 11. The latter fits to the velocity profile observations in [39].

[height=0.29,tics=10] Bilder/streamlines_1_.png stenosis

Rigid walls
Elastic walls
Figure 9: Velocity streamlines in the stenosed region of ACI (stenotic plaque marked with arrows) for both vessel wall models colored by velocity magnitude, , viewpoints from front and back.

In Fig.10 the movement of the carotid in regions close to its bifurcation is presented, whereby the clipped ends of the vessel are cut out. The arrows at the common fluid-solid interface (inner wall) and at the outer vessel wall show the dominance of the lateral vessel wall movement compared to the inflating effects. On the one hand, this is caused by the boundary condition at the outer wall, allowing free movement without any constraining effects of outer stresses by surrounding tissue, on the other hand, by the length of the whole computational domain considered. Indeed, the effects of clipping the bottom and top part of the vessel geometry, whose length is identical to the fixed geometry presented in Fig. 2, are weakened in the middle of the computational domain. This leads to the fluid-driven lateral motion.

Figure 10: The deformation vectors at the inner and outer surface of the thick vessel tissue shows the translational vessel movement in the zoomed area prior and subsequent to the carotis bifurcation. The surface is coloured by the deformation amplitude in [cm].

4.1 Longitudinal WSS

The longitudinal WSS is evaluated on the carotid surface as well as along chosen surface curves.


(a)
(b)
(c)
(d)
Figure 11: WSS distribution at t=1.1s for all configurations, frontal viewpoint.

In Fig. 11 the surface distributions of WSS for the four configurations at the time of highest flow-rate are presented. All plots (a)-(d) show high positive WSS up to 35 in the bifurcation and in the sinusoidal constrictions of the stenotic bulge of the ACI. The center of the bifurcation and the regions around the stenotic bulges of the ACI are regions of low and negative WSS. In configuration (a), local point-wise extreme values of WSS appear using the flipped tangent vectors on rigid surface. These are smoothed out in (c) on the deformed surface with elastic walls. Beside these very local phenomena in (a), wider areas of negative extreme values occur close to the separation point of the bifurcation in configurations (b) and (c), compared to (d). The occurrence of these extreme values is associated with the alignment of chosen tangential fields, which differ for the considered model configurations at the bifurcation point. We demonstrate this coherence for automatically rendered flipped (c) and projected tangents (d) on compliant walls in Fig. 12.

(c)
(d)
Figure 12: WSS distribution used flipped (c) and projected (d) tangent fields, evaluation with compliant walls, viewpoint from back (different viewpoint as in Fig.11).

Indeed, a clear side separation of the flipped tangent vectors loosing their longitudinal alignment even before the bifurcation can be observed in plot (c), explaining the discontinuity and the appearance of negative WSS values up to -20 in this area. On the other hand, the longitudinal continuance of projected tangents until the separation point is apparent in plot (d). Obviously, configurations with flipped tangent vectors lead to a spurious longitudinal WSS evaluation on the carotid surface close to their bifurcation point, whereas the tangent field (d), which is projected from the centerline, seems to appropriately map the main flow and its separation in this problematic area.

To get more detailed comparisons, the WSS along the chosen surface curves is shown in Fig. 13 for configurations (b), (c), (d).

Intersection curve (1)
Intersection curve (1)
Longitudinal intersection curve (2)
Surface intersection curves, frontal viewpoint
Figure 13: Circumferential (1) and longitudinal (2) intersection curves for the evaluation of WSS. The arc length of (1) starts at the respective green point and follow a clockwise direction. Below: comparison of longitudinal WSS for configurations (b)-(d) along intersection curves (1),(2).
Surface intersection curves, frontal viewpoint

The WSS values in configurations (b) and (d) are almost identical and differ at most by along the circumferential curve (1). On the longitudinal line (2) we observe a very good agreement of (b) and (d). The situation is slightly different in case (c) with flipped tangents, where local deflections of WSS values can be observed. These are caused by the previously discussed alignment of differently flipped tangent vectors presented in Fig. 12. The negative deflections of WSS along the longitudinal curve (2) are not visible in Fig. 12 due to the different viewpoint, neverthless they can be identified with local blue areas on the outer wall of ACI in plot (c) of Fig. 11.

Generally, we can conclude that the choice of the tangential field has a considerable effect on the longitudinal WSS evaluation in the bifurcation region of the carotid artery surface, with the projected tangent vectors being the most suitable for this analysis. After passing the problematic area of the bifurcation the WSS results for projected and flipped tangent fields are comparable at maximal systolic flow, whereas certain inaccuracies occur using automatic flipped tangent vectors. This is because of the misalignment in longitudinal direction on complex and uneven surfaces. In addition, effects of the wall movement in FSI models, comparing to rigid wall models are present around the bifurcation point, compare e.g., plots (b) and (d) in Fig. 11. In the next section we present the temporal change of WSS during the whole cardiac cycle and compare it for compliant and rigid wall models.

4.2 Oscillatory shear index

The OSI index (8) for longitudinal WSS (6) is presented and compared for the fixed wall model using projected tangents (b) and for the FSI models using flipped (c) and projected tangents (d) in Fig. 14.

(b)
(c)
(d)
Figure 14: Oscillatory shear index evaluated for the second cardiac cycle (), red areas show the occurrence of long lasting negative WSS.

Comparing the results in Fig. 14, almost no difference in OSI evaluation for configurations (b) and (d) with projected tangents can be observed. The only region of difference worth mentioning spreads out in the bifurcation area, where the centerline tangents have been projected on different surfaces, obviously due to the wall deformation in case (d). In contrast to (b) and (d), configuration (c) shows many small-scale and point-like extreme values, which may be a consequence of the discontinuity and deflections of WSS arising from the erratic alignment of the flipped tangents in some surface regions discussed above.

Concerning the hemodynamical interpretation of the OSI results presented in Fig. 14, conspicuous regions of maxima indicating long-lasting negative WSS inside and oscillating WSS at their edges are observed in the bifurcation zone as well as prior and after the sinusoidal stenotic occlusion of the ICA in all model configurations. Further punctual abnormalities are present, e.g., after the stenotic occlusion on the left side of the ACI wall. In consistency to the streamlines presented in Fig. 9, red OSI regions are related to vortices adjacent to the stenotic bulges of ACI or prior to the bifurcation. According to the hemodynamic hypothesis, those maxima regions may be a indicator of the pathological progression of mechanical damage of the artery wall.

5 Conclusion

In this contribution a computational study of fluid dynamics has been performed in human carotis artery. The effects of the fluid stresses on the arterial wall has been quantified using established hemodynamic risk factors: wall shear stress and oscillatory shear index. Following the low shear theory we focused on exploration of low and negative longitudinal component of the WSS allowing to track the reverse flow and its oscillatory behaviour. Model configurations with both rigid as well as compliant artery wall using fluid-structure interaction model have been used and hemodynamic parameters have been evaluated respectively. The presented results demonstrate the strong dependency of the orientation-based longitudinal WSS and of the derived OSI-index on the proper construction of tangential vectors and address this problem on topologically complex surfaces obtained from patient CTA scans. For the studied carotis artery tree we applied the projection of center-line tangents to the inner arterial surface, and compared the results of derived wall parameters for projected as well as generic mesh-based tangents. Our results confirm, that the projected tangential field retain the longitudinal alignment on the craggy surface and map the flow separation in the bifurcation area much better then the automatically generated tangent vectors. Applying the tangential projection method reliable numerical results for longitudinal WSS allowing hemodynamic predictions have been obtained.

For further work, the range of prediction tools can be extended by considering further multi-directional WSS parameters, where the choice of the tangential field is of particular importance. According to [14], time-averaged WSS (TAWSS; [2, 20]) and relative residence time (RRT; [12, 13, 35]) are strong predictive clinical markers for disease development, even in early stages of atherosclerosis. Moreover, transversal WSS ([23, 30, 31]) and cross flow index (CFI, [23]) shown a predictive value when complex fluid flow appears in later phases of atherosclerosis. Taking those parameters into account, a more differentiated prediction of atherosclerotic development can be achieved.

Acknowledgements

This work was funded by the BMBF joint Project 05M20UNA-MLgSA.

Literature

References

  • [1] Antiga, L.:Patient-specific Modeling of Geometry and Blood Flow in Large Arteries. PhD thesis, Politecnico di Milano, 2002
  • [2] Arzani, A., Shadden S.C.: Characterizations and correlations of wall shear stress in aneurysmal flow, Journal of Biomechanical Engineering, 138, 0145031-01450310, 2016. DOI: 10.1115/1.4032056
  • [3] Blagojević, M., Nicolić, A., Zivković, M., Zivković, M., Stanković, G.,Pavlović, A.: Role of oscillatory shear index in predicting the occurrence and development of plaque, Journal of the Serbian Society of Computational Mechanics, 7(2), 29-37, 2013.
  • [4] Bukač, M et al.: A modular, operator-splitting scheme for fluid–structure interaction problems with thick structures, International journal for numerical methods in fluids 74.8, 577–604, 2014
  • [5] Čanić, S., Tambača, J., Guidoboni, G., Mikelić, A., Hartley, C.J., Rosenstrauch, D.: Modeling viscoelastic of arterial walls and their interaction with pulsatile blood flow, SIAM J. Appl. Math. 67(1), 164–193, 2006
  • [6] Ciarlet, P.G.: Mathematical Elasticity, Volume III: Theory of Shells, North-Holland, Amsterdam, 2000
  • [7] Comsol Multiphysics. Reference Manual,
    https://doc.comsol.com/5.5/doc/com.comsol.help.comsol/COMSOL-ReferenceManual.pdf, November 2021
  • [8] Debus, E.S., Torsello, G., Schmitz-Rixen, G., Flessenkämper, I., Storck, M., Wehk. H., Grundmann, R.T.: Ursachen und Risikofaktoren der Arteriosklerose, Gefässchirurgie, 18, 2413-2419, 2013. DOI: 10.1007/s00772-013-1233-6
  • [9] Eulzer, P., Meuschke, M., Klinger, C. M., Lawonn, K.: Visualizing carotid blood flow simulations for stroke prevention, Computer Graphics Forum, 40 435–446, 2021, DOI: 10.1111/cgf.14319
  • [10] Eulzer, P., Richter, K., Meuschke M., Hundertmark A., Lawonn K.: Automatic Cutting and Flattening of Carotid Artery Geometries, in Proceedings of Eurographics Workshop on Visual Computing for Biology and Medicine 2021, (Editors: S. Oeltze-Jafra, N. N. Smit, and B. Sommer)
  • [11] Gallo, D., Steinman, D.A., Morbiducci, U.: Insights into the co-localization of magnitude-based versus direction-based indicators of disturbed shear at the carotid bifurcation, Journal of Biomechanics, 49(2), 2413-2419, 2016. DOI: 10.1016/j.jbiomech.2016.02.010
  • [12] Gorring, N., Kark, L., Simmons, A., Barber, T.: Determining possible thrombus sites in an extracorporeal device, using computational fluid dynamics-derived relative residence time. Computer Methods in Biomechanics and Biomedical Engineering, 18(6), 628-634, 2014. DOI: 10.1080/10255842.2013.826655
  • [13] Hashemi, J., Patel B., Chatzizisisi, Y.S., Kassab, G.S.: Study of Coronary Atherosclerosis Using Blood Residence Time Front. Physiol., 12, 625420, 2021. DOI: 10.3389/fphys.2021.625420
  • [14] Hoogendoorn, A., Hartman, E. MJ., Kok, A., De Nisco, G.: Multidirectional wall shear stress promotes advanced coronary plaque development - comparing five shear stress metrics, Cardiovascular Research, 116(6), 1136-1146, 2019. DOI: 10.1093/cvr/cvz212
  • [15] Hundertmark, A., Lukáčová, M.: Numerical study of shear-dependent non-Newtonian fluids in compliant vessels, Computers and Mathematics with Applications, 60, 572-59, 2010.
  • [16] Hundertmark, A., Lukáčová, M., Rusnáková, G.: Fluid-structure interaction for shear-dependent non-Newtonian fluids, In: Kaplický P. (Ed.) Topics in Mathematical Modeling and Analysis Vol.7, 109–158, 2012
  • [17] Iannuzzi, A., Rubba, P., Gentile, M., Mallardo, V., Calcaterra, I., Bresciani, A., Covetti, G., Cuomo, G., Merone, P., Di Lorenzo, A., Alfieri, R., Aliberti, E., Giallauria, F., Di Minno, M., Iannuzzo, G.: Carotid Atherosclerosis, Ultrasound and Lipoproteins. Biomedicines, 9(5), 521, 2021. DOI: 10.3390/biomedicines9050521
  • [18] Izzo, R., Steinman, D., Manini, S., Antiga L.: The Vascular Modeling Toolkit: A Python Library for the Analysis of Tubular Structures in Medical Images The Open Journal 25(3), 2018.
  • [19] Janela, J., Moura, A., Sequeira, A.: A 3D non-Newtonian fluid-structure interaction model for blood flow in arteries, J. Comput. Appl. Math. 234, 2783–2791, 2010
  • [20] John, L., Pustêjovská, P., Steinbach, O.: On the influence of the wall shear stress vector form on hemodynamic indicators, Computing and Visualization in Science, 18, 113-122. 2017. DOI: 10.1007/s00791-017-0277-7
  • [21] Ku, D. N., Giddens, D. P., Zarins, C. K., . Glagov, S.: Pulsatile flow and atherosclerosis in the human carotid bifurcation, positive correlation between plaque location and low oscillating shear stress. Arteriosclerosis, 5, 293–302, 1985.
  • [22]

    Lawton, M., Higashida, R., Smith, W. S., Young, W. L., Saloner, D., Boussel, L., Rayz, V., McCulloch, C., Martin, A., Acevedo-Bolton, G.,: Aneurysm growth occurs at region of low wall shear stress: Patient-specific correlation of hemodynamics and growth in a longitudinal study,

    Stroke, 39, 2997-3002, 2008, DOI: 10.1161/STROKEAHA.108.521617
  • [23] Mohamied, Y., Sherwin, S.J., Weinberg, P.D.: Understanding the fluid mechanics behind transverse wall shear stress, Journal of Biomechanics, 50(4), 102-109, 2016. DOI: 10.1016/j.jbiomech.2016.11.035
  • [24] Mohamied, Y., Rowland, E.M., Bailey, E.L. Sherwin, S.J., Schwartz, M. A., Weinberg, P.D.: Change of Direction in the Biomechanics of Atherosclerosis, Annals of Biomedical Engineering, 43, 16-25, 2014. DOI: 10.1007/s10439-014-1095-4
  • [25] Morbiducci, U., Gallo, D., Cristofanelli, S., Ponzini, R., Deriu, M. A., Rizza, G., Steinmann, D. A.: A rational approach to defining principal axes of multidirectional wall shear stress in realistic vascular geometries, with application to the study of the influence of helical flow on wall shear stress directionality in aorta, Journal of Biomechanics, 48(6), 899-906, 2015. DOI: 10.1016/j.jbiomech.2015.02.027
  • [26] Perktold, K., Rappitsch, G.: Computed simulation of local blood flow and vessel mechanics in a compliant carotid artery bifurcation model, Journal of Biomechanics, 28(7), 845–856, 1995.
  • [27] Quarteroni, A., Gianluigi, R.: Optimal control and shape optimization of aorto-coronaric bypass anastomoses, Math. Models Methods Appl. Sci., 13 (12), 1801-1823, 2003.
  • [28] Quarteroni, A., Formaggia, A: Mathematical modelling and numerical simulation of the cardiovascular system. In Ciarlet, P.G. Lions, J.L. (Hrsg.), Handbook of Numerical Analysis, 12 (3-127), 2004, Amsterdam: Elsevier.
  • [29] Quarteroni, A., Dede’, L., Manzoni, A., Vergara, C.: Mathematical Modelling of the Human Cardiovascular System: Data, Numerical Approximation, Clinical Applications, Cambridge Monographs on Applied and Computational Mathematics, 2019.
  • [30] Pedrigi R.M., Poulsen C.B., Mehta VV, Ramsing Holm N., Pareek N., Post A.L., Kilic I.D., Banya W.A.S., Dall’Ara G., Mattesini A., Bjørklund M.M., Andersen N.P., Grøndal A.K., Petretto E., Foin N., Davies J.E., Mario C., Di Fog Bentzon J., Erik Bøtker H., Falk E., Krams R., de Silva R. Inducing persistent flow disturbances accelerates atherogenesis and promotes thin cap fibroatheroma development in D374Y-PCSK9 hypercholesterolemic minipigs.Circulation, 132,1003–1012, 2015. DOI: 10.1161/CIRCULATIONAHA.115.016270
  • [31] Peiffer V., Sherwin S.J., Weinberg P.D.: Computation in the rabbit aorta of a new metric—the transverse wall shear stress—to quantify the multidirectional character of disturbed blood flow. Journal of Biomechanics, 46, 2651–2658, 2013. DOI: 10.1016/j.jbiomech.2013.08.003
  • [32] Pfeiffer, P., Sherwin, S.J., Weinberg, P.D.: Does low and oscillatory wall shear stress correlate spatially with early atherosclerosis? A systematic review, Cardiovascular Research, 99, 242-250, 2013. DOI: 10.1093/cvr/cvt044
  • [33] Rusnákova G., Lukáčová, M., Hundertmark, A.: Kinematic splitting algorithm for fluid-structure interaction in hemodynamics, Computer Methods in Applied Mechanics and Engineering 265, 83-106, 2013
  • [34] Shojima, M., Oshima, M., Takagi, K., Torii, R., Hayakawa, M., Katada, K., Morita, A., Kirino, T.: Magnitude and Role of Wall Shear Stress on Cerebral Aneurysm Computational Fluid Dynamic Study of 20 Middle Cerebral Artery Aneurysms Stroke, 35 (11), 2500–2505, 2004, DOI: 10.1161/01.STR.0000144648.89172.0f
  • [35] Soulis, S. V., Giannoglou, G., Fytanidis, D.K.: Relative residence time and oscillatory shear index of non-Newtonian flow models in aorta, International Workshop on Biomedical Engineering, Biomedical Engineering, 10, 2013. DOI: 10.1109/IWBE.2011.6079011
  • [36] Spanos, K., Petrocheilou, G., Karathanos, C., Labropoulos, N., Mikhailidis, D., Giannoukas, A.: Carotid Bifurcation Geometry and Atherosclerosis Angiology, 68(9), 757-764, 2017. DOI: 10.1177/0003319716678741
  • [37] WHO, The top 10 cause of death,
    https://www.who.int/news-room/fact-sheets/detail/the-top-10-causes-of-death, December 2020
  • [38] Xiang, J., Sabareesh K. Natarajan, S. K., Tremmel, M., Ma, D., Mocco, J., Hopkins L. N., Siddiqui, A. H., Levy, E. I., Meng H.,: Hemodynamic-Morphologic Discriminants for Intracranial Aneurysm Rupture, Stroke, 42(1), 144–152, 2011, DOI: 10.1161/STROKEAHA.110.592923
  • [39] Zarins, C.K., Giddens, D.P., Bharadvaj, B. K., Sottiurai, V.S., Mabon, R.F., Glagov, S.: Carotid Bifurcation Atherosclerosis: Quantitative Correlation of Plaque Localization with Flow Velocity Profiles and Wall Shear Stress Circulation Research, 53(4), 502-514, 1983. DOI: 10.1161/01.RES.53.4.502
  • [40] Taylor, C. A., Hughes, T. J. R., Zarins, C. K.: Finite element modeling of three-dimensional pulsatile flow in the abdominal aorta: relevance to atherosclerosis. Annals of Biomedical Engineering, 26, 975–987, 1998. DOI: 10.1114/1.140