On the relationship between dominance order and $ \theta $-dominance order on multipartitions

Kai Zhou; Kai Zhou

doi:10.3934/math.2025139

AIMS Mathematics

2025, Volume 10, Issue 2: 2998-3012. doi: 10.3934/math.2025139

Previous Article Next Article

Research article

On the relationship between dominance order and $\theta$ -dominance order on multipartitions

Kai Zhou ^,

School of Science, Shandong Jianzhu University, Jinan, 250101, China

Received: 08 October 2024 Revised: 08 February 2025 Accepted: 13 February 2025 Published: 18 February 2025
MSC : 05E10, 06A07, 20C08

Many cellular bases have been constructed for the cyclotomic Hecke algebras of $G(\ell, 1, n)$ . For example, with dominance order on multipartitions, Dipper, James, and Mathas constructed a cellular basis $\{m_{{\mathfrak{s}}{\mathfrak{t}}}\}$ and Hu, Mathas constructed a graded cellular basis $\{\psi_{{\mathfrak{s}}{\mathfrak{t}}}\}$ . With $\theta$ -dominance order on multipartitions, Bowman constructed integral cellular basis $\{c^{\theta}_{{\mathfrak{s}}{\mathfrak{t}}}\}$ . Following Graham and Lehrer's cellular theory, different constructions of cellular basis may determine different parameterizations of simple modules of the cyclotomic Hecke algebras of $G(\ell, 1, n)$ . To study the relationship between these parameterizations, it is necessary to understand the relationship between dominance order and $\theta$ -dominance order on multipartitions. In this paper, we define the weak $\theta$ -dominance order and give a combinatorial description of the neighbors with weak $\theta$ -dominance order. Then we prove weak $\theta$ -dominance order is equivalent to dominance order whenever the loading $\theta$ is strongly separated. As a corollary, we give the relationship between weak $\theta$ -dominance order, $\theta$ -dominance order, and dominance order on multipartitions.

Keywords:

Citation: Kai Zhou. On the relationship between dominance order and $\theta$ -dominance order on multipartitions[J]. AIMS Mathematics, 2025, 10(2): 2998-3012. doi: 10.3934/math.2025139

Related Papers:

[1]	Li Cai, Yu Hao, Pengfei Ma, Guangyu Zhu, Xiaoyu Luo, Hao Gao . Fluid-structure interaction simulation of calcified aortic valve stenosis. Mathematical Biosciences and Engineering, 2022, 19(12): 13172-13192. doi: 10.3934/mbe.2022616
[2]	Annalisa Quaini, Sunčica Čanić, David Paniagua . Numerical characterization of hemodynamics conditions near aortic valve after implantation of left ventricular assist device. Mathematical Biosciences and Engineering, 2011, 8(3): 785-806. doi: 10.3934/mbe.2011.8.785
[3]	Zongchao Liu, Linhui Wu, Junwei Yang, Fangsen Cui, Pei Ho, Liping Wang, Jianghui Dong, Gongfa Chen . Thoracic aorta stent grafts design in terms of biomechanical investigations into flexibility. Mathematical Biosciences and Engineering, 2021, 18(1): 800-816. doi: 10.3934/mbe.2021042
[4]	Li Cai, Qian Zhong, Juan Xu, Yuan Huang, Hao Gao . A lumped parameter model for evaluating coronary artery blood supply capacity. Mathematical Biosciences and Engineering, 2024, 21(4): 5838-5862. doi: 10.3934/mbe.2024258
[5]	EYK Ng, Leonard Jun Cong Looi . Numerical analysis of biothermal-fluids and cardiac thermal pulse of abdominal aortic aneurysm. Mathematical Biosciences and Engineering, 2022, 19(10): 10213-10251. doi: 10.3934/mbe.2022479
[6]	Meiyuan Du, Chi Zhang, Sheng Xie, Fang Pu, Da Zhang, Deyu Li . Investigation on aortic hemodynamics based on physics-informed neural network. Mathematical Biosciences and Engineering, 2023, 20(7): 11545-11567. doi: 10.3934/mbe.2023512
[7]	Lorenzo Civilla, Agnese Sbrollini, Laura Burattini, Micaela Morettini . An integrated lumped-parameter model of the cardiovascular system for the simulation of acute ischemic stroke: description of instantaneous changes in hemodynamics. Mathematical Biosciences and Engineering, 2021, 18(4): 3993-4010. doi: 10.3934/mbe.2021200
[8]	Christopher D. Bertram, Bernard O. Ikhimwin, Charlie Macaskill . Modeling flow in embryonic lymphatic vasculature: what is its role in valve development?. Mathematical Biosciences and Engineering, 2021, 18(2): 1406-1424. doi: 10.3934/mbe.2021073
[9]	Zhenwu Xiang, Qi Mao, Jintao Wang, Yi Tian, Yan Zhang, Wenfeng Wang . Dmbg-Net: Dilated multiresidual boundary guidance network for COVID-19 infection segmentation. Mathematical Biosciences and Engineering, 2023, 20(11): 20135-20154. doi: 10.3934/mbe.2023892
[10]	Eun-jin Kim, Massimo Capoccia . Mechano-electric effect and a heart assist device in the synergistic model of cardiac function. Mathematical Biosciences and Engineering, 2020, 17(5): 5212-5233. doi: 10.3934/mbe.2020282

Abstract

1. Introduction

Pathological dilation of the ascending aorta is a common aortic disease, which, if untreated, may progressively weaken aortic wall and lead to the formation of aortic dissection/aneurysm or even sudden rupture ^[1,2]. In clinical practice, the maximum diameter of aorta is usually taken as the main indicator for decision-making ^[3], although there is increasing evidence that the morphotype of aortic valve is an important factor associated with the risk and progression of aortic dilation ^{[4,5,6,7,8,9,10]}. It has been found that the incidence of aortic dilation or aneurysm formation in patients with bicuspid aortic valve (BAV) is 50% to 70% ^[4], and that patients with BAV often have larger aortic root dimensions and a higher progression rate of dilation in comparison with patients with a normal tricuspid aortic valve (TAV) ^[5,6]. Numerous in vivo or in vitro studies have demonstrated that hemodynamic characteristics in aortas with BAV not only differ remarkably from those in aortas with TAV, but also depend strongly on the morphotype of BAV ^{[7,8,9,10,11,12]}. The fusion configuration of valve cusps provides an anatomical basis for the categorization of BAV and explaining the associated characteristics of aortic hemodynamics. For instance, in cases of RL-BAV with fusion of the right and left coronary cusps, the flow jet through the aortic valve is directed toward the right anterior wall of the ascending aorta, leading to elevated wall shear stress (WSS) in this region; whereas in cases of RN-BAV with fusion of the right and noncoronary cusps, the flow jet is directed toward the posterior wall of the ascending aorta, making the distribution of high WSS differ significantly from those observed in aortas with RL-BAV ^[9,10,11,12]. Despite the useful insights from these studies, detailed quantification of hemodynamic parameters using in vivo measurement or in vitro experimental methods remains challenging due to issues related to the limited spatio-temporal resolutions of available measurement techniques.

In this context, computational fluid dynamics (CFD) methods have been widely employed to quantitatively study the characteristics of aortic hemodynamics in the context of various aortic valve abnormalities (e.g., anomaly, stenosis, and regurgitation) ^{[13,14,15,16]}. A major advantage of CFD methods is their high precision in computing near-wall hemodynamic parameters that are considered to be associated closely with the pathophysiology of vascular wall while challenging for in vivo measurement. On the other hand, the fidelity of a CFD model in reproducing in vivo hemodynamics is affected by a variety of factors involved in model setup. For instance, it has been found that imposing idealized flow velocity profiles at the inlets of patient-specific aortic models causes significant deviations of model outputs from real in vivo hemodynamic conditions ^[17,18,19], which highlights the importance of patient-specifically prescribing model boundary conditions. Another important issue is relevant to the assumption on the physical properties of blood flow and the corresponding choice of fluid dynamics modeling method. While laminar flow assumption is usually considered to be valid for blood flows in human arteries and has been widely adopted in computational model-based studies on aortic blood flow ^{[18,19,20,21]}, in vivo measurements have revealed the presence of highly disturbed unstable flows even in the aortas of healthy subjects ^[22,23] and the occurrence of marked flow turbulence (confirmed through Fourier decomposition of point-wise flow velocity waveform) in the ascending aortas of patients with aortic valve diseases ^[22]. In addition, in vitro experiments have shown that RL-and NL-BAVs generate high degrees of flow turbulence in the ascending aorta with normal anatomy ^[12]. Given that lumen enlargement in aortic dilation tends to reduce mean flow velocity ^[7], one may ask whether flow turbulence would occur in dilated aortas as well and how the morphotype and functional state of aortic valve would affect the intensity and spatial distribution of flow disturbance. To quantify flow turbulence with CFD methods, direct numerical simulation (DNS) could be the most reliable approach since it can theoretically resolve all spatial and temporal scales of both laminar and turbulent flows, however, DNS is extremely draconian in terms of the requirements for mesh quality, hardware resource and computation time ^[24,25], and hence is generally impractical to study blood flows in anatomically realistic arteries. Relatively, computational methods based on Reynolds-averaged Navier-Stokes (RANS) models are computationally much cheaper, but are limited by their relatively poor performance in predicting laminar-turbulent transition ^[26,27]. Lying between DNS and RANS methods is the LES method, which has been demonstrated to provide predictions agreeing well with experimental data for flows with Reynolds numbers in the range of arterial blood flow ^[28,29,30], and has been employed to simulate flow turbulence in various pathophysiological scenarios of aortic hemodynamics ^[29,30].

In the present study, a LES-based computational method was employed to simulate blood flow patterns and flow turbulence in two aortas with pathological dilation of the middle ascending aortic segment while differential aortic valve conditions. Computed results for the two aortas were compared to assess the influence of aortic valve disease on aortic hemodynamics, especially the intensity and spatial distribution of flow turbulence. When building the computational models, in vivo flow information derived from phase contrast magnetic resonance imaging (PC-MRI) data was used to prescribe model boundary conditions so as to improve the fidelity of computer models in reproducing patient-specific hemodynamics. In the meantime, four-dimensional magnetic resonance imaging (4D-MRI) data of in vivo aortic blood flow were acquired and used as a reference for validating the results of hemodynamic simulations.

2. Materials and method

2.1. Acquirement of clinical data

Two patients diagnosed to have aortic dilation at the middle ascending segment were enrolled in the study upon ethical approval and the receipt of patient consent. Besides aortic dilation, the patients were found to suffer from various types and severities of aortic valve abnormality. One patient (patient #1) had a RL-BAV with severe stenosis (mean pressure gradient: 43 mm Hg) and moderate regurgitation (regurgitant fraction: 42.86%), while the other patient (patient #2) had a quasi-normal TAV with mild regurgitation (regurgitant fraction: 8.24%). Diagnosis of aortic valve stenosis or regurgitation was based on transthoracic echocardiographic and PC-MRI data. Detailed demographic information of the patients and the basic geometrical and hemodynamic parameters of the aortas and aortic valves are summarized in Table 1. Chest CTA (computed tomography angiography) and ECG-gated PC-MRI examinations were performed in sequence on each patient with a 64-channel multi-detector CT scanner (Philips Healthcare, The Netherlands) and a 3T MR scanner (Philips Healthcare, The Netherlands), respectively. During PC-MRI scanning, oblique sagittal MR slices were aligned with the aortic inlet and selected cross-sections of aortic branch arteries so that the measured flow velocity information can be used to prescribe the boundary conditions of CFD model. In addition, flow velocity information in the entire aorta was also acquired through isotropic spatial sampling of flow field (field of view = 300 × 350 mm², voxel size = 2.5 × 2.5 mm², slice thickness = 2.5 mm) using flow-sensitive 4D-MRI and visualized with the Circle Cardiovascular Imaging software (CVI, Calgary, Alberta, Canada).

Table 1. Baseline information of patients and main geometrical and hemodynamic parameters of the aortas and aortic valves.

	Patient #1	Patient #2
Age	77	61
Sex	Male	Female
Heart rate (bpm)	70	56
Aortic disease	Mid-ascending aortic dilation	Mid-ascending aortic dilation
Morphology of aortic valve	RL-BAV	TAV
Aortic valve stenosis	Severe	None
Aortic valve regurgitation	Moderate	Mild
Mean pressure gradient (mmHg)	43.0	7.0
Regurgitant fraction (%)	42.86	8.24
Maximum diameter of aorta (mm)	45	52
Diameter of aortic inlet (mm)	38	38
Severity of aortic dilation^#	1.18	1.37
Mean Re at aortic inlet	521	922
Peak Re at aortic inlet	3257	4342
Wo at aortic inlet	27.86	24.96
Abbreviations: BAV, bicuspid aortic valve; RL, right-left fusion; TAV, tricuspid aortic valve; Re, Reynolds number; Wo, Womersley number. ^#assessed by maximum-to-aortic-inlet diameter ratio

| Show Table

DownLoad: CSV

2.2. Image-based geometrical model reconstruction and mesh generation

The CTA images (which have a slice spacing of 0.4 mm and a spatial resolution of 512 × 512 pixels) of each patient were segmented in Mimics 15.0 (Materialise, Belgium) to reconstruct a geometrical model of the aorta and its main branch arteries (see Figure 1(a)). The reconstructed geometrical model was subsequently imported into ICEM CFD 16.0 to generate a high-resolution mesh model. Herein, the fluid domain was firstly divided by tetrahedral elements with a minimum size of 0.4 mm, followed by a mesh refinement treatment in which multiple layers of prism elements were mapped along the vascular wall to improve the precision of flow computation in the near-wall regions (Figure 1(b)). The thickness of the initial prism layer was set to 30 μm (which allows the dimensionless wall distance y⁺ to be always below 1.0 under the simulated flow conditions, fulfilling the requirement of mesh size nearest to the wall in LES), and was increased exponentially toward the interior until the thickness of the last layer approached the size of tetrahedral elements in the interior domain. Mesh sensitivity analyses were conducted on the two aortic models by varying the minimum size of tetrahedral elements from 0.5 mm to 0.4 mm and 0.2 mm. The results showed that the differences in computed space-mean wall shear stress and oscillatory shear index in the ascending aortic segment between the median and finest mesh models were less than 1%. Therefore, adopting a minimum tetrahedral element size of 0.4 mm was considered sufficient to yield numerically acceptable results. The mesh models of the two aortas contained 5592372 and 7061034 elements, respectively.

Figure 1. CTA image-based model reconstruction: (a) reconstructed geometrical model for the aorta of patient #1; (b) outside view of the mesh model and longitudinal/cross-sectional views of internal tetrahedral elements and prism layers mapped along the wall.

DownLoad: Full-Size Img PowerPoint

2.3. Modeling methods for blood flow

Aortic blood flow was assumed to be an incompressible fluid with a constant density of 1055 kg/m³ governed by the unsteady mass conservation and Navier-Stokes (N-S) equations. The classical Carreau model was employed to dynamically relate blood viscosity to shear rate so as to account for the non-Newtonian rheology of blood ^[31]. In addition, vascular walls were assumed to be rigid to which the non-slip boundary conditions were imposed.

While the mass conservation and N-S equations can theoretically describe both laminar flow and turbulent flow at an arbitrary point in the flow field, when they are discretized and numerically solved on a mesh model with finite element size, a problem arises as to how sub-grid flow should be represented. In the case of strictly laminar flow, flow variables can generally be assumed to be homogeneous within each individual element ^[32], the assumption may however lead to considerable numerical errors when applied to turbulent flow due to the presence of sub-grid turbulent eddies smaller in size than the element that will induce non-negligible intra-element viscous energy loss. Therefore, we employed the LES method to account for intra-element viscous energy loss associated with potential flow turbulence. In the LES method, flow velocity ( ${u_i}$ ) in an arbitrary element is separated into a grid-resolved component ( ${\bar u_i}$ ) and a sub-grid component ( ${u_i}^\prime$ ) ^[33]. In the fluid domain of a mesh model, the grid-resolved velocity components depict the large-scale flow patterns, while the sub-grid velocity components account for the effects of turbulent eddies whose scales are smaller than the grid size.

${u_i} = {\bar u_i} + u_{i}^\prime.$

(1)

With the above flow velocity separation, the mass conservation and N-S equations can be rewritten as

$\frac{{\partial {{\bar u}_i}}}{{\partial {x_i}}} = 0,$

(2)

$\frac{{\partial {{\bar u}_i}}}{{\partial t}} + {\bar u_j}\frac{{\partial {{\bar u}_i}}}{{\partial {x_j}}} = - \frac{1}{\rho }\frac{{\partial \bar p}}{{\partial {x_i}}} + \frac{\partial }{{\partial {x_j}}}\left( {v\frac{{\partial {{\bar u}_i}}}{{\partial {x_j}}}} \right) + \frac{1}{\rho }\frac{{\partial {\tau _{ij}}}}{{\partial {x_j}}},$

(3)

where ${\tau _{ij}}$ is the sub-grid stress tensor that accounts for the dissipative effect of small scale eddies, and is by definition expressed as

${\tau _{ij}} = \overline {{u_i}{u_j}} - {\bar u_i}{\bar u_{j}}.$

(4)

To facilitate numerical solution, ${\tau _{ij}}$ is often related to the large-scale strain rate via an assumed turbulent eddy viscosity (TEV) ^[33]

${\tau _{ij}} - \frac{{{\delta _{ij}}}}{3}{\tau _{kk}} = - 2{\nu _t}{\bar S_{ij}},$

(5)

where ${\delta _{ij}}$ is the Kronecker symbol, ${\nu _t}$ is the turbulent eddy viscosity, and ${\bar S_{ij}}$ is the strain rate tensor of the grid-resolved velocity field defined as

${\bar S_{ij}} = \frac{1}{2}\left( {\frac{{\partial {{\bar u}_i}}}{{\partial {x_j}}} + \frac{{\partial {{\bar u}_j}}}{{\partial {x_i}}}} \right).$

(6)

${\nu _t}$ was estimated with the wall-adapted local eddy-viscosity (WALE) LES model ^[34]

${\nu _t} = {\left( {{C_\omega }\Delta } \right)^2}\frac{{{{(S_{ij}^dS_{ij}^d)}^{3/2}}}}{{{{({{\bar S}_{ij}}{{\bar S}_{ij}})}^{5/2}} + {{(S_{ij}^dS_{ij}^d)}^{5/4}}}}$

(7)

Herein, ${C_\omega }$ is a coefficient fixed at 0.5 for homogeneous isotropic turbulence, $\Delta$ is the cube root of the element volume, and $S_{ij}^d$ is the traceless symmetric part of the square of the velocity gradient tensor

$S_{ij}^d = \frac{1}{2}\left( {\bar g_{ij}^2 + \bar g_{ji}^2} \right) - \frac{1}{3}{\delta _{ij}}\bar g_{kk}^2, {\rm{with}}\;\bar g_{ij}^2 = {\bar g_{ik}}{\bar g_{kj, }}{\bar g_{ik}} = \frac{{\partial {{\bar u}_i}}}{{\partial {x_j}}}.$

(8)

2.4. Patient-specific prescription of model boundary conditions based on PC-MRI data

Physiologically, the spatial distribution of flow velocity at the aortic inlet is highly complex and prone to change with time due to the dynamic hemodynamic interactions among the left ventricle, aortic valve and aorta. It has been extensively demonstrated that imposing an idealized (e.g., plug, parabolic, or Womersley) flow velocity profile at the aortic inlet may cause the model predictions to deviate significantly from in vivo hemodynamics, especially in the ascending aortic region ^[19,20]. In this study, we extracted the time-varying velocity vector information (including the trans-plane normal velocity component and in-plane tangential velocity component) at the aortic inlet based on the PC-MRI data acquired from each patient and projected the extracted velocity data to the mesh model using an in-house code written in MATLAB according to the methodology proposed in a previous study ^[35] (see Figure 2(b) for the volumetric flow rate waveform and flow velocity distributions at the aortic inlet of patient #1). In this way, the inflow conditions of each aorta were fully patient-specifically prescribed. The inlet boundary conditions prescribed for the aortas of the two patients are compared in Figure 3 with respect to volumetric flow rate waveform, trans-plane mean flow velocity waveform, trans-plane maximum flow velocity waveform, and spatial distribution of flow velocity at peak systole. It is evident that flow conditions at the aortic inlet differ remarkably between the two patients in terms of both flow waveform and in-plane flow velocity distribution, demonstrating the necessity of patient-specific boundary condition prescription. As for the prescription of outflow boundary conditions, we imposed the volumetric flow rate waveforms derived from the PC-MRI data measured in each patient to the outlets of the aortic branch arteries (i.e., BCA, brachiocephalic artery; LCCA, left common carotid artery; LSA, left subclavian artery) (see Figure 2(a) & (c) for the locations of the outlets and PC-MRI-measured flow waveforms in patient #1, note that the flow waveforms measured in patient #2 were not shown in order to save space). At the outlet of the descending (DAo) aorta of each patient, although patient-specifically measured flow waveform with PC-MRI was available as well, we prescribed the Neumann type free outflow boundary condition instead of imposing the measured flow waveform directly, mainly because: 1) the Neumann type outflow boundary condition can play a role of absorbing potential numerical errors in the fluid domain when Dirchlet boundary conditions are imposed at the inlet and other three outlets, and 2) the rigid wall assumption renders our model unable to tolerate the phase lag between inflow waveform and the sum of outflow waveforms caused by the deformation of aortic wall under in vivo conditions. The latter is evidenced by the considerable differences between the model-predicted and measured flow waveforms at the DAo outlets of patient #1 and patient #2 (see Figure 4).

Figure 2. Setup of boundary conditions for the aortic model of patient #1: (a) locations of the inlet (denoted by the red plane) and outlets (denoted by the blue planes) in the model; (b) PC-MRI data-derived volumetric flow waveform (upper panel) and the spatial distributions of the normal and tangential flow velocity components at the model inlet at three time moments in a cardiac cycle (i.e., T1, T2 and T3) (lower panel); (c) PC-MRI data-derived volumetric flow rate waveforms at the outlets of the three branch arteries (i.e., BCA, LCCA, LSA). Abbreviations: AAo, ascending aorta; DAo, descending aorta; BCA, brachiocephalic artery; LCCA, left common carotid artery; LSA, left subclavian artery; BC, boundary condition; A, anterior; P, posterior; R, right; L, left.

DownLoad: Full-Size Img PowerPoint

Figure 3. Comparisons of the PC-MRI data-derived inlet boundary conditions between the aortas of the two patients in terms of trans-plane volumetric flow rate waveform (a), mean flow velocity waveform (b), maximum flow velocity waveform (c), and spatial distribution of trans-plane velocity at peak systole (denoted by the filled red circles in panel (c)) (d).

DownLoad: Full-Size Img PowerPoint

Figure 4. Comparisons between model-predicted and measured (with PC-MRI) flow waveforms at the outlet of descending aorta in patient #1 (a) and patient #2 (b).

DownLoad: Full-Size Img PowerPoint

2.5. Numerical methods

The governing equations of blood flow were solved along with the boundary conditions using a commercial CFD package (ANSYS CFX 16). Second-order schemes were employed for both spatial discretization and time integration. The numerical time step was fixed at 0.1 ms. At each time step, convergence of numerical solution was judged when the root mean-square residuals of the mass and momentum conservation equations all went below 10^-5. Moreover, to eliminate uncertain errors introduced by the estimation of initial conditions that were not known in advance, each set of numerical simulation was continuously run for several cardiac cycles. Our numerical tests showed that the space-mean value of model-simulated time-averaged wall shear stresses in each aorta changed by less than 0.2% from the fourth to the fifth cardiac cycle, indicating that the numerical solution is close to periodic from the fourth cardiac cycle. Accordingly, the numerical results obtained in the fifth cardiac cycle were analyzed and reported. Numerical simulations were run on a Dell workstation equipped with an Intel Xeon (2.9GHz, 32 CPUs). The total computation times for the two aortic models were 58 and 65 hours, respectively.

2.6. Data analysis

The simulated time-varying WSS vector ( $\overrightarrow {{\tau _w}}$ ) was analyzed over a cardiac cycle (T) to derive the time-averaged wall shear stress (TAWSS) and oscillatory shear index (OSI) that have been extensively proved to be typical hemodynamic parameters associated with arterial disease ^[19,36,37].

${\rm{TAWSS}} = \frac{1}{T} \int_0^T \left| {\overrightarrow {{\tau _w}} } \right|dt, {\rm{OSI}} = \frac{1}{2}\left( {1 - \frac{{\left| {\mathop \smallint \nolimits_0^T \overrightarrow {{\tau _w}} dt} \right|}}{{\mathop \smallint \nolimits_0^T \left| {\overrightarrow {{\tau _w}} } \right|dt}}} \right).$

(9)

TAWSS and OSI were further spatially averaged in the ascending or descending aortic segment to facilitate quantitative comparisons of these parameters between the two aortas.

${\rm{SA}}-{\rm{TAWSS}} = \frac{1}{{{S_{{\rm{Ao}}}}}}\mathop \sum\nolimits_{i = 1}^n {\rm{TAWS}}{{\rm{S}}_i} \cdot \Delta {S_i}, {\rm{SA}}-{\rm{OSI}} = \frac{1}{{{S_{{\rm{Ao}}}}}}\mathop \sum\nolimits_{i = 1}^n {\rm{OS}}{{\rm{I}}_i} \cdot \Delta {S_i}.$

(10)

Here, S_Ao is the total wall area of the aortic segment, i is the index of a mesh surface on the aortic wall that contains a total number of n mesh surfaces, and ∆S_i is the area of the ith mesh surface, with the corresponding TAWSS and OSI being denoted by TAWSS_i and OSI_i, respectively.

Similarly, turbulent eddy viscosity (TEV), which reflects the intensity of flow turbulence in each mesh element, was averaged over the space of ascending or descending aortic segment.

${\rm{SA}}-{\rm{TEV}} = \frac{1}{{{V_{{\rm{Ao}}}}}}\mathop \sum\nolimits_{i = 1}^n {\rm{TE}}{{\rm{V}}_i} \cdot \Delta {V_i},$

(11)

where, V_Ao is the total volume of the aortic segment, i is the index of a mesh element with its volume and TEV being denoted by ∆V_i and TEV_i, respectively.

3. Results

Figure 5 shows the model-simulated flow streamlines at three time moments in the aortas of patient # 1 and patient #2 and their comparisons with in vivo 4D-MRI visualizations (videos of the simulated and measured flow streamlines in the aorta of patient #1 are available in the supplementary material). The presence of high-speed jet flows impinging on the anterior wall of the ascending aortic segment was captured by both model prediction and 4D-MRI measurement, which demonstrates the ability of our computational models to reasonably reproduce the main features of in vivo hemodynamics. Nevertheless, when the details of model-simulated and measured flow fields were compared, considerable discrepancies were observed. For instance, the model-predicted flow structures in the ascending aorta were more complex and flow velocities in the descending aorta were higher (especially at T2) in comparison with those presented by the 4D-MRI data. The discrepancies are likely to arise from the inherent limitations of the modeling method and in vivo measurement: 1) the adoption of rigid wall assumption renders our models unable to account for wall deformation and the variations in phase angle and shape of flow waveform along the aorta (see Figure 4) that both considerably affect intra-aortic flow patterns; and 2) the limited spatio-temporal resolutions of 4D-MRI measurement may compromise the visualization of detailed flow patterns, especially in the ascending aortic region where flow field is highly complex and unstable. With regard to the comparison of flow patterns in the two aortas, although high-speed jet flows impinging on the anterior wall of the ascending aortic segment were detected in both aortas, the area subject to flow impingement was much larger and flow disturbance was stronger in the aorta of patient #1.

Figure 5. Comparisons of 4D-MRI-visualized aortic flow streamlines at three representative time moments (i.e., T1 (early systole), T2 (peak systole) and T3 (late systole)) with those obtained by numerical simulations for patient #1 (a) and patient #2 (b). The curved arrows indicate the main path of the impingement flows directed from the aortic inlet toward the descending aorta through the anterior region of the ascending aorta.

DownLoad: Full-Size Img PowerPoint

Figure 6 shows the simulated flow velocity distributions in a transverse plane of the ascending segments of the two aortas at three time moments (i.e., T1 in the acceleration phase, T2 at peak systole, and T3 in the deceleration phase). The corresponding flow velocity distributions in a longitudinal cutting plane are presented in Figure 7. Basically, flows in both aortas were disturbed, asymmetric and time-varying. However, the spatial distribution of velocity and its variation with time differed remarkably between the aortas. For instance, in the ascending aorta of patient #1, high-speed flow jets were observed in the vicinity of both the anterior and right walls, whereas they appeared only in the anterior area of the ascending aorta of patient #2. To further explore the differences in three-dimensional flow patterns and the intensity and distribution of flow turbulence between the two aortas, intra-aortic vortex structures were visualized in form of turbulent eddy viscosity-colored iso-surfaces of Lambda 2 (at a level value of 0.05) (see Figure 8). Overall, complex vortex structures and high turbulent eddy viscosity were present mainly in the ascending aortic segment. For both aortas, the complexity of vortex structure and turbulent eddy viscosity changed with time, and were remarkably enhanced in the deceleration phase. When the vortex structure and turbulent eddy viscosity in the deceleration phase (i.e., late systole at T3) were compared between the two aortas, vortex breakdown accompanied by increased turbulent eddy viscosity in the ascending aortic segment was much more pronounced in patient #1 than in patient #2. Accordingly, the space-averaged turbulence eddy viscosity (SA-TEV) in the ascending aorta of patient #1 was 176% higher than that in patient #2 (see Figure 9). In contrary, the SA-TEV in the descending aortic segment was lower in patient #1 than in patient #2, although the low intensity of flow turbulence (indicated by the small value of TEV) in the descending aortic segment reduces the significance of the comparison.

Figure 6. Simulated flow fields (illustrated in form of filled flow velocity contours in a transverse cutting plane at three time moments (i.e., T1, T2 and T3)) in the aortas of patient #1 (left panel) and patient #2 (right panel).

DownLoad: Full-Size Img PowerPoint

Figure 7. Simulated flow fields (illustrated in form of filled flow velocity contours in a longitudinal cutting plane at three time moments (i.e., T1, T2 and T3)) in the aortas of patient #1 (left panel) and patient #2 (right panel).

DownLoad: Full-Size Img PowerPoint

Figure 8. Vortex structures represented by iso-surfaces of Lambda-2 at a level value of 0.05 colored by the turbulent eddy viscosity at three time moments (i.e., T1, T2 and T3) in the aortas of patient #1 (upper panel) and patient #2 (lower panel).

DownLoad: Full-Size Img PowerPoint

Figure 9. Comparisons of the simulated SA-TEV in the ascending and descending aortic segments (denoted by blue and yellow in the geometrical models) at three time moments (i.e., T1, T2 and T3) between patient #1 and patient #2.

DownLoad: Full-Size Img PowerPoint

Figure 10 shows the model-predicted distributions of TAWSS and OSI in the two aortas. Regional high TAWSS in the ascending segment was predicted for both aortas, but with its degree and spatial distribution differing considerably between the aortas. Relatively, TAWSS in the descending aortic segment was more evenly distributed. As for the distribution of OSI, considerable difference between the two aortas was observed. For instance, focal high OSI appeared in the entire aorta of patient #1 while only in the ascending segment of the aorta of patient #2. To further quantitatively compare the values of TAWSS and OSI in the two aortas, the space-averaged TAWSS and OSI (i.e., SA-TAWSS, SA-OSI) in the ascending and descending segments of the two aortas are plotted in Figure 11. It is evident that SA-TAWSS in the ascending aortic segment was much larger in patient #1, whereas SA-TAWSS in the descending aortic segment was comparable between the two patients. If a local TAWSS of at least two times of the mean TAWSS over the entire aorta is set as the threshold of high TAWSS, the total area of ascending aortic wall exposed to high TAWSS was 26.1 cm² in patient #1, which was much larger than that (4.6 cm²) in patient #2. The comparison of SA-OSI between the two aortas however revealed a phenomenon different from that of SA-TAWSS. For instance, ascending aortic SA-OSI was comparable between the two patients, whereas descending aortic SA-OSI was evidently higher in patient #1 than in patient #2.

Figure 10. Simulated distributions of TAWSS and OSI in the aortas of patient #1(left panel) and patient #2 (right panel).

DownLoad: Full-Size Img PowerPoint

Figure 11. Comparisons of the simulated SA-TAWSS (a) and SA-OSI (b) in the ascending and descending aortic segments (denoted by blue and yellow respectively in the geometrical models) between patient #1 and patient #2.

DownLoad: Full-Size Img PowerPoint

4. Discussion

In the present study, blood flows in two aortas with pathological dilation of the ascending segment while differential aortic valve conditions were numerically simulated and compared. The presence of high-speed eccentric jet flows and regional high WSS was predicted in the ascending aortic segment of both aortas. In comparison with the aorta with quasi-normal TAV, the aorta with RL-BAV had stronger and more widely distributed jet flows in the ascending aortic segment that caused a marked elevation of WSS at both the anterior and right walls. While most of these model-predicted hemodynamic features and their differences between the two aortas have been reported in previous studies on similar subtypes of aortic dilation and aortic valve disease ^[9,10,11], the adoption of the LES method for flow modeling in the study allowed us to further explore the intensity and spatial distribution of flow turbulence and their association with the morphotype and functional state of aortic valve.

Theoretically, the proximal portion (i.e., the ascending aortic segment) of aorta not only has complex inflow conditions due to the anatomical connection to the left ventricle via the aortic valve, but also has larger diameter and higher curvature/tortuosity in comparison with the distal portion ^[38], which together would render blood flow in the ascending aorta more complex and unstable than in the descending aorta, and especially more vulnerable to the influence of aortic valve function. It has been extensively demonstrated that the morphotype and functional state of aortic valve significantly affect flow patterns in the ascending aorta ^{[9,10,11,12,13,14]}, and that dilation of the ascending aorta per se augments local flow disturbance by inducing larger flow helices and long-lasting retrograde flow ^[7]. With regard to the two aortas investigated in the present study, in comparison with the aorta (of patient #2) with quasi-normal TAV, the aorta (of patient #1) with RL-BAV had remarkably higher intensity and wider distribution of flow turbulence, stronger flow disturbance, and larger area of wall subject to elevated WSS in the ascending aortic segment. Given the fact that the mean Reynolds number at the aortic inlet (521 vs. 922) and the severity of aortic dilation (maximum-to-aortic-inlet diameter ratio: 1.18 vs.1.37) were both lower in patient #1 than in patient #2 (see Table 1), the different aortic valve conditions may be a main causative factor for the observed differential hemodynamic characteristics and flow turbulence in the ascending aortas of the two patients. The presence of a severely stenotic and moderately regurgitant RL-BAV in patient #1 caused the distribution of high-speed flows at the aortic inlet to be more concentrated and asymmetric (see Figure 3(d)), which not only resulted in the enlargement of wall regions subject to high WSS, but also promoted the formation/breakdown of unstable vortices to induce laminar-turbulent transition and associated increase in turbulent eddy viscosity (TEV) in the ascending aorta. Relatively, hemodynamic characteristics in the descending aortic segment were overall less influenced by aortic valve conditions, as demonstrated by the low turbulent eddy viscosity (see Figure 9) and comparable space-averaged wall shear stress between the two aortas (see Figure 11), although OSI in patient #1 was considerably higher than in patient #2 due primarily to the increased diastolic retrograde flow (see Figure 3(a)) caused by aortic valve regurgitation in patient #1.

Both abnormally high WSS and flow disturbance have been suggested to exert negative influence on endothelial function and wall remodeling ^[39,40], therefore one may speculate that regionally elevated WSS and increased flow disturbance/turbulence resulting from aortic valve anomaly or disease may be involved in the initiation and progression of aortopathy. For instance, adverse alterations in extracellular matrix protein expression in the aortic wall exposed to elevated WSS have been observed in patients with BAV ^[41,42]. Exposure of normal aortic tissue to BAV-induced abnormal aortic WSS was found to result in increased MMP-2 and MMP-9 expressions and MMP-2 activity, indicating the potential role of BAV hemodynamic stresses in mediating aortic medial degradation ^[43]. On the other hand, a recent longitudinal study showed that no clinically relevant anatomical remodeling of the ascending aorta was observed during 3-year follow-up in BAV patients compared against healthy volunteers in spite of the regionally elevated aortic WSS associated with BAV, which suggests that WSS alterations may contribute to the onset of aortopathy, but WSS-driven anatomical remodeling may be a very slow process ^[44]. In addition, the congenital nature of BAV and aortic dilation implies that the roles of genetics are indisputable ^[10]. In a sense, the cause-and-effect relationship between hemodynamic factors and aortic dilation seems to be highly complicated and remains inconclusive given the limited evidence available in current literature. To better understand the relationship, large-scale long-term longitudinal studies would be necessary.

The present study is subject to certain limitations. Firstly, only two aortas with dilation of the ascending segment and specific aortic valve conditions were investigated in the study. Given the extensively demonstrated high sensitivities of aortic hemodynamics to aortic morphology and BAV morphotype ^{[9,10,11,12,13,14]}, the presented findings might not be fully applicable to other aortas with different anatomical features and/or aortic valve abnormalities, although the general finding regarding the increased ascending aortic flow complexity and flow turbulence associated with aortic valve disease should hold. Secondly, while the LES modeling method has been widely proved as a practical approach to compensating for the inherent limitation of laminar flow assumption-based method in accounting for flow turbulence ^[29,30], the assumptions introduced in the modeling of sub-grid turbulent eddies and the limited spatio-temporal resolution of computational model render the LES-based numerical results unable to fully represent all scales of turbulent eddies. To overcome the limitation, DNS may be an alternative ^[24,25], but would incur extremely high computational cost. Thirdly, aortic walls were assumed to be rigid in the study, which is physiologically non-rigorous. Previous studies have demonstrated that ignoring wall deformation in the simulation of aortic hemodynamics leads to considerable overestimate of WSS ^[45,46]. In addition, the adoption of rigid-wall assumption makes computer models inherently lack the ability to account for physiological variations in phase angle and shape of flow waveform along the aorta resulting from pulse wave propagation, reflection and superimposition in the aorta ^[47], which may considerably compromise the fidelity of hemodynamic simulation in the distal portion (e.g., descending aortic segment) of aorta when measured flow waveforms are imposed to upstream aortic inlet and branch arteries. Our study did show that the model-predicted and measured flow waveforms at the outlet of the descending aorta differed significantly, and accordingly the discrepancies between the model-predicted and 4D-MRI-measured flow streamlines seemed to be larger in the descending aortic segment than in the ascending segment. Although how wall deformation would affect the occurrence and intensity of flow turbulence in the aorta remains incompletely understood, modeling methods capable of accounting for wall deformation and its interaction with blood flow are no doubt necessary to further improve the fidelity of hemodynamic simulation, especially when a large portion of aorta is involved.

5. Conclusion

Blood flows in two patient-specific aortas with dilation of the ascending segment have been numerically simulated with the LES method to address how the morphotype and functional state of aortic valve would influence the characteristics of aortic hemodynamics. It was found that the presence of a severely stenotic and moderately regurgitant BAV resulted in marked enlargement of wall regions exposed to elevated WSS and enhancement of flow turbulence in the ascending aortic segment. Relatively, the aortic valve disease had much smaller influence on flow patterns in the descending aortic segment, although OSI was considerably elevated due to the increased diastolic retrograde flow caused by valve regurgitation. These findings suggest that aortic valve disease has profound influence on flow patterns in the ascending aorta and further studies would be required to address how hemodynamic alterations associated with aortic valve disease would affect the progression of aortopathy.

Acknowledgments

The study was supported by the National Natural Science Foundation of China (Grant no. 11972231, 11832003), the China Postdoctoral Science Foundation (Grant no. 2018M640385), and the SJTU Medical-Engineering Cross-cutting Research Project (Grant no. YG2015MS53, YG2017MS45).

Conflict of interest

The authors declare there is no conflict of interest.

References

[1]	J. J. Graham, G. I. Lehrer, Cellular algebras, Invent. Math., 123 (1996), 1–34. https://doi.org/10.1007/BF01232365 doi: 10.1007/BF01232365
[2]	S. K $\ddot{\text{o}}$ nig, C. C. Xi, On the structure of cellular algebras, in Algebras and Modules II, 1998,365–386. Available from: https://api.semanticscholar.org/CorpusID: 15944731.
[3]	S. Ariki, K. Koike, A Hecke algebra of $({\mathbb{Z}}/r{\mathbb{Z}})\wr{\mathfrak {S}}_n$ and construction of its irreducible representations, Adv. Math., 106 (1994), 216–243. https://doi.org/10.1006/aima.1994.1057 doi: 10.1006/aima.1994.1057
[4]	M. Broue, G. Malle, Zyklotomische heckealgebren, Ast $\acute{\text{e}}$ risque, 212 (1993), 119–189. Available from: http://www.numdam.org/item/AST_1993_212_119_0.pdf.
[5]	R. Dipper, G. James, A. Mathas, Cyclotomic q–Schur algebras, Math. Z., 229 (1998), 385–416. https://doi.org/10.1007/PL00004665 doi: 10.1007/PL00004665
[6]	S. Ariki, On the classification of simple modules for cyclotomic Hecke algebras of type $G(m, 1, n)$ and Kleshchev multipartitions, Osaka J. Math., 38 (2001), 827–837. Available from: https://core.ac.uk/download/pdf/35270702.pdf.
[7]	J. Brundan, A. Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, Invent. math., 178 (2009), 451–484. https://doi.org/10.1007/s00222-009-0204-8 doi: 10.1007/s00222-009-0204-8
[8]	J. Hu, A. Mathas, Graded cellular bases for the cyclotomic Khovanov–Lauda–Rouquier algebras of type $A$ , Adv. Math., 225 (2010), 598–642. https://doi.org/10.1016/j.aim.2010.03.002 doi: 10.1016/j.aim.2010.03.002
[9]	C. Bowman, The many graded cellular bases of Hecke algebras, arXiv preprint, 2017, In press. https://doi.org/10.48550/arXiv.1702.06579
[10]	G. James, A. Kerber, The representation theory of the symmetric group, Addison-Wesley Publishing Co., 1981.
[11]	R. Rouquier, $q$ -Schur algebras and complex reflection groups, Monsc. Math. J., 8 (2008), 119–158. Available from: https://www.math.ucla.edu/rouquier/papers/qschur.pdf.

This article has been cited by:

1.	Milan Toma, Shelly Singh-Gryzbon, Elisabeth Frankini, Zhenglun (Alan) Wei, Ajit P. Yoganathan, Clinical Impact of Computational Heart Valve Models, 2022, 15, 1996-1944, 3302, 10.3390/ma15093302
2.	Molly Cherry, Zinedine Khatir, Amirul Khan, Malenka Bissell, The impact of 4D-Flow MRI spatial resolution on patient-specific CFD simulations of the thoracic aorta, 2022, 12, 2045-2322, 10.1038/s41598-022-19347-6
3.	Jean-François Deux, Lindsey A. Crowe, Léon Genecand, Anne-Lise Hachulla, Carl G. Glessgen, Stéphane Noble, Maurice Beghetti, Jin Ning, Daniel Giese, Frédéric Lador, Jean-Paul Vallée, Correlation between Pulmonary Artery Pressure and Vortex Duration Determined by 4D Flow MRI in Main Pulmonary Artery in Patients with Suspicion of Chronic Thromboembolic Pulmonary Hypertension (CTEPH), 2022, 11, 2077-0383, 5237, 10.3390/jcm11175237
4.	Jixin Hou, Xuanyu Li, Zhaojun Li, Lekang Yin, Xin Chen, Fuyou Liang, An In Vivo Data-Based Computational Study on Sitting-Induced Hemodynamic Changes in the External Iliac Artery, 2022, 144, 0148-0731, 10.1115/1.4052292
5.	Antonio Martínez, Martijn Hoeijmakers, Leonardo Geronzi, Valery Morgenthaler, Jacques Tomasi, Michel Rochette, Marco E. Biancolini, Effect of turbulence and viscosity models on wall shear stress derived biomarkers for aorta simulations, 2023, 167, 00104825, 107603, 10.1016/j.compbiomed.2023.107603
6.	Alessandro Mariotti, Simona Celi, Maria Nicole Antonuccio, Maria Vittoria Salvetti, Impact of the Spatial Velocity Inlet Distribution on the Hemodynamics of the Thoracic Aorta, 2023, 14, 1869-408X, 713, 10.1007/s13239-023-00682-2

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(422) PDF downloads(30) Cited by(0)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

AIMS Mathematics

On the relationship between dominance order and $\theta$ -dominance order on multipartitions

Related Papers:

Abstract

1. Introduction

2. Materials and method

2.1. Acquirement of clinical data

2.2. Image-based geometrical model reconstruction and mesh generation

2.3. Modeling methods for blood flow

2.4. Patient-specific prescription of model boundary conditions based on PC-MRI data

2.5. Numerical methods

2.6. Data analysis

3. Results

4. Discussion

5. Conclusion

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Catalog

AIMS Mathematics

On the relationship between dominance order and θ \theta -dominance order on multipartitions

Related Papers:

Abstract

1. Introduction

2. Materials and method

2.1. Acquirement of clinical data

2.2. Image-based geometrical model reconstruction and mesh generation

2.3. Modeling methods for blood flow

2.4. Patient-specific prescription of model boundary conditions based on PC-MRI data

2.5. Numerical methods

2.6. Data analysis

3. Results

4. Discussion

5. Conclusion

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

On the relationship between dominance order and $\theta$ -dominance order on multipartitions