Enhanced 4D Flow MRI-Based CFD with Adaptive Mesh Refinement for Flow Dynamics Assessment in Coarctation of the Aorta

Abstract

4D Flow MRI is a diagnostic tool that can visualize and quantify patient-specific hemodynamics and help interventionalists optimize treatment strategies for repairing coarctation of the aorta (COA). Despite recent developments in 4D Flow MRI, shortcomings include phase-offset errors, limited spatiotemporal resolution, aliasing, inaccuracies due to slow aneurysmal flows, and distortion of images due to metallic artifact from vascular stents. To address these limitations, we developed a framework utilizing Computational Fluid Dynamics (CFD) with Adaptive Mesh Refinement (AMR) that enhances 4D Flow MRI visualization/quantification. We applied this framework to five pediatric patients with COA, providing in-vivo and in-silico datasets, pre- and post-intervention. These two data sets were compared and showed that CFD flow rates were within 9.6% of 4D Flow MRI, which is within a clinically acceptable range. CFD simulated slow aneurysmal flow, which MRI failed to capture due to high relative velocity encoding $\left(V_{\text{enc}}\right)$. CFD successfully predicted in-stent blood flow, which was not visible in the in-vivo data due to susceptibility artifact. AMR improved spatial resolution by factors of 10¹ to 10³ and temporal resolution four-fold. This computational framework has strong potential to optimize visualization/quantification of aneurysmal and in-stent flows, improve spatiotemporal resolution, and assess hemodynamic efficiency post-COA treatment.

INTRODUCTION

Coarctation of the aorta (COA) is a congenital heart defect characterized by a narrowing in the aortic arch or proximal descending aorta.^1,40 Prevalence ranges from 5 to 8% of all congenital heart defects, making it one of the most common forms of structural heart disease.^7,40,44 COA can induce altered aortic hemodynamics including high pressure gradients across the constriction, hypertension related to obstruction or altered vascular compliance, increased risk of ventricular dysfunction, vascular aneurysm, or even aortic rupture.^1,32 Current American Heart Association (AHA)/American College of Cardiology (ACC) guidelines call for COA repair to improve aortic blood flow dynamics in cases with pressure gradients greater than 20 mmHg across the constriction.⁴⁶ The optimal therapeutic procedure is often not well defined by guidelines and based on patient size, anatomic substrate and preference of the treating surgeons and interventionalists.^12,48 Treatment procedures for COA include surgery, intravascular stent placement, and balloon angioplasty.^44,46 Despite the success and low early mortality of COA repair (approximately 2%),^3,19 1 in 3 patients will have post-interventional secondary complications such as hypertension, re-coarctation, late aneurysm formation, or aortic dissection.^7,33,39,46

Current tools for deciding on the optimal type of COA intervention are limited and often rely on the expertise and preference of the interventionalist. Improved diagnostic imaging could aid clinicians in intervention planning and assessment of treatment success. 4D Flow MRI is a diagnostic imaging technique capable of characterizing time-resolved hemodynamic metrics within a three-dimensional (3D) acquisition region.^22,23 Studies have used 4D Flow MRI to evaluate blood flow information in COA patients.^2,8,14,15,18 4D Flow MRI has high potential to improve understanding of pre- and post-repair hemodynamics, which may help to better identify the optimal repair strategy for patients.^8,22 Despite recent developments, shortcomings of 4D Flow MRI include phase-offset errors,^9,28 limited spatial and temporal resolution,²² aliasing,¹¹ and distortion of images due to metal artifact (stent).¹² Computational Fluid Dynamics (CFD) is capable of complementing and improving 4D Flow MRI to address the stated limitations, providing enhanced patient-specific hemodynamic data that can aid clinical decision-making.

This manuscript presents a patient-specific CFD method based on 4D Flow MRI. Research has shown image-based CFD as a predictive tool that can be used to support clinical decision-making,²⁴ and has been used to study COA.²⁵ The CFD method used in this study applies a novel numerical method called Adaptive Mesh Refinement (AMR) during the simulation. AMR enables higher spatial resolution while maintaining reasonable computational demands.^37,43 We hypothesize that MRI-based CFD with AMR can enhance 4D Flow MRI visualization and quantification to improve the assessment of aortic hemodynamics pre- and post-COA repair. The purpose of this manuscript is to implement and validate this CFD method on a COA patient population and to perform a systematic side-by-side comparison of 4D Flow MRI vs. CFD, with a particular focus on how this computational procedure addresses common shortcomings of 4D Flow MRI.

MATERIALS AND METHODS

Patient Population

Pediatric patients with a history of COA who had at least two standard-of-care cardiothoracic MRI exams, one before and one after surgical or transcatheter intervention (including 4D flow MRI acquisitions between 2011 and 2020), were retrospectively included in this study. Institutional Review Board (IRB) approval with waiver of consent was obtained for this HIPAA-compliant retrospective study. All patients underwent cardiothoracic MRI using 1.5-Tesla MR-systems (Aera, Siemens Healthcare, Erlangen, Germany). Each subject underwent free-breathing, prospective ECG- and respiratory navigator-gated 4D Flow MRI covering the entire thoracic aorta in sagittal oblique orientation. Scan parameters were as follows: spatial resolution = 1.8–2.2 × 1.8–2.2 × 1.9–4.2 mm³, temporal resolution = 35.0–42.4 ms, TR = 4.4–5.3 ms, TE = 2.2–2.7 ms, flip angle = 15° and velocity sensitivity $\left({\mathrm{V}}_{\text{enc}}\right)=150-300\mathrm{c}\mathrm{m}/\mathrm{s}$. All patients also received standard of care imaging including contrast-enhanced magnetic resonance angiography (CE-MRA) using Ablavar, Gadavist or Magnevist; spatial resolution = 0.55–1.79 × 0.55–1.79 × 1.3–1.5mm³, TR = 2.2–2.4 ms, TE = 1.22–1.55 ms, flip angle = 18°). In addition, computed tomography angiography (CTA) was performed in two patients who had stents placed in the aorta as part of the coarctation treatment.

4D Flow MRI Data Analysis

All 4D Flow MRI data were analyzed according to a previously described pre-processing workflow, i.e., corrections for Maxwell terms, eddy currents, noise masking, and velocity aliasing.⁵ A sum of squares 3D phase contrast angiogram (PC-MRA) was calculated to depict vessel anatomy. The thoracic aorta was segmented using commercial software Mimics (Materialise NV, Leuven, Belgium). The 3D aorta segmentation was used to mask the 4D Flow MRI data for further hemodynamic analysis.

Patient-Specific Anatomy

Generation of patient-specific 3D computer models were based on CE-MRA or CTA data. CTA was used only in patients who had stents placed as part of the treatment. In the cases without metal artifact (no stent), CE-MRA was used. The 3D angiograms from CE-MRA or CTA were loaded into semiautomatic segmentation software packages Mimics and 3-matic (Materialise NV, Leuven, Belgium). The geometries were manually segmented such that the inlets and outlets were perpendicular to the vessel centerlines. A total of ten patient-specific 3D models were generated for the five patients, both pre- and post-intervention.

Patient-Specific Boundary Conditions

Patient-specific boundary conditions (BC) were applied to the inlet and outlets of the CFD simulation for each patient case. The inlet of the patient anatomy was defined at the ascending aorta, prior to the aortic arch. There were four outlets in each patient case: the brachiocephalic, left common carotid, and left subclavian arteries, and the descending thoracic aorta. The wall of the aorta was assumed to be rigid in the CFD simulations. Figure 1 shows the 3D anatomy of a post-surgical patient that was generated from CE-MRA using the segmentation software packages.

FIGURE 1.

3D model of a patient’s thoracic aorta showing inlet (in red) and outlets (in blue).

A special case of the Dirichlet BC, called mass flow velocity $\mathrm{B}\mathrm{C}$, was imposed on the inlets of the geometries. The imposed inlet velocity was derived from the mass flow rate using the equation

$$\mathit{u}=\frac{1}{\rho A_{\mathrm{i}\mathrm{n}}}\frac{\mathrm{d}m}{\mathrm{d}t}\widehat{\mathit{n}}$$ (1)

where $\mathit{u}$ is the velocity vector at the inlet, $\rho$ is the density of blood, $A_{\text{in}}$ is the cross-sectional area at the inlet, $\frac{dm}{dt}$ is the temporally varying mass flow rate of the blood, and $\widehat{\mathit{n}}$ is the unit normal vector of the surface of the inlet. Blood was assumed to behave as an incompressible, Newtonian fluid with density of 1060 kg/m³ and viscosity of 0.0036 Pa s. The mass flow rate was calculated from 4D Flow MRI data. The software package EnSight (Ansys, Inc., Canonsburg, Pennsylvania, USA) was used to visualize and quantify the 4D Flow MRI data. As shown in Fig. 2, three planes were placed at the approximate location corresponding to each inlet/outlet, and the flow curves (volumetric flow vs. time) were extracted from each plane. The mean of the flow curves from the three planes at each inlet/outlet was calculated to reduce the effect of noise from the in-vivo data.

FIGURE 2.

Visualization of blood flow 4D Flow MRI data. Three planes are placed at the inlet, normal to the cross-section. Flow rate (cm³/s) vs. time (ms) data were extracted from each plane. The mean of the three flow curves was used to impose the inlet BC for the CFD simulation.

A three-element Windkessel model was used for the outlet BCs. The following differential equation corresponds to the three-element model that was used:

$$P=\left(R_{\mathrm{p}}+R_{\mathrm{d}}\right)Q-R_{\mathrm{d}}C\frac{\mathrm{d}P}{\mathrm{d}t}+R_{\mathrm{p}}{R}_{\mathrm{d}}C\frac{\mathrm{d}Q}{\mathrm{d}t}$$ (2)

In the above equation, $P$ is pressure at the outlet, $R_p$ is proximal resistance, $R_d$ is distal resistance, $Q$ is volumetric flow rate, and $C$ is capacitance. For each outlet $i$, three parameters, $\left(R_p^i,R_d^i,C^i\right)$, were calculated based on the following equations from Zambrano et al.⁵⁰

$$R_{\mathrm{T}}=\frac{P_{\text{mean}}}{Q_{\text{mean}}}$$ (3)

$$C_{\mathrm{T}}=\frac{Q_{\mathrm{m}\mathrm{a}\mathrm{x}}-Q_{\mathrm{m}\mathrm{i}\mathrm{n}}}{P_{\mathrm{s}\mathrm{y}\mathrm{s}}-P_{\mathrm{E}\mathrm{D}}}\mathrm{\Delta }t$$ (4)

$$\frac{1}{R_{\mathrm{T}}^i}=\left(\frac{A^i}{A_{\mathrm{T}}}\right)\left(\frac{1}{2R_{\mathrm{T}}}\right)$$ (5)

$$C^i=\left(\frac{C_{\mathrm{T}}}{2}\right)\left(\frac{A^i}{A_{\mathrm{T}}}\right)$$ (6)

$$R_{\mathrm{p}}^i=\frac{\rho c_{\mathrm{E}\mathrm{D}}}{A_{\mathrm{i}\mathrm{n}}}$$ (7)

$$R_{\mathrm{d}}^i=R_{\mathrm{T}}^i-R_{\mathrm{p}}^i$$ (8)

Description of the terms used in Eqs. (3)–(8) are in Table 1. The software CONVERGE CFD (Convergent Science, Inc., Madison, Wisconsin, USA)⁴¹ was used to run all CFD simulations. Once the Windkessel parameters, $\left(R_{\mathrm{p}}^i,R_{\mathrm{d}}^i,C^i\right)$, were inputted into CONVERGE 3.0, the pressure at the outlets varied temporally based on Eq. (2).

TABLE 1.

Definition of the terms used in Eqs. (3)–(8).

$R_{\mathrm{T}}$	Total arterial resistance	$P_{\mathrm{E}\mathrm{D}}$	Diastolic pressure
$C_{\mathrm{T}}$	Total arterial compliance	$Q_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}$	Mean volumetric flow rate
$R_{\mathrm{T}}^i$	Total resistance associated with outlet $i$	$Q_{\mathrm{m}\mathrm{a}\mathrm{x}}$	Maximum volumetric flow rate
$R_{\mathrm{p}}^i$	Proximal resistance associated with outlet $i$	$Q_{\mathrm{m}\mathrm{i}\mathrm{n}}$	Minimum volumetric flow rate
$R_{\mathrm{i}}^d$	Distal resistance associated with outlet $i$	$\mathrm{\Delta }t$	Time interval between maximum and minimum flow
$C^i$	Compliance associated with outlet $i$	$A_{\mathrm{T}}$	Total outlet area
$P_{\text{mean}}$	Mean pressure	$A^i$	Area of outlet $i$
$P_{\text{sys}}$	Systolic pressure	$c_{\mathrm{E}\mathrm{D}}$	Diastolic pulse wave velocity

Due to the lack of patient-specific catheterization data, $P_{\text{sys}}$ and $P_{\text{ED}}$ were assumed to be 120 and 80 mmHg, respectively, and so $P_{\text{mean}}=\frac{1}{3}{P}_{\mathrm{s}\mathrm{y}\mathrm{s}}+\frac{2}{3}{P}_{\mathrm{E}\mathrm{D}}=93.3\mathrm{m}\mathrm{m}\mathrm{H}\mathrm{g}$. The pulse wave velocity, $c_{\mathrm{E}\mathrm{D}}$, was assumed to be 7 m/s based on values from literature.²⁰

Adaptive Mesh Refinement

CFD simulations require discretizing the aorta geometry. Discretization is the process of dividing the entire aorta into numerous computational grid cells. CONVERGE 3.0 employed a finite volume approach to numerically solve the unsteady conservation equations using the Pressure-Implicit with Splitting of Operators (PISO) algorithm.⁴¹ The computational grid cells approximate the infinitesimal fluid element in the continuum fluid domain, and this approximation leads to discretization errors in the numerical results. Reducing the size of each computational grid cell, i.e., increasing the total number of grid cells, reduces the discretization error at the expense of excessive demands of computational resources.⁴³ AMR is a novel numerical method that was implemented in the CFD simulations in this study. AMR dynamically updates the computational mesh grid based on calculated discretization errors.⁴³ The grid is updated locally, instead of globally, allowing a fine mesh only in regions of large velocity gradients and complex flow structures. In regions of smooth laminar flow, the mesh remains coarse. The AMR process we implemented is based on the Sub-Grid Scale (SGS) method.

In the SGS method, the computational grid is refined based on the velocity vector field, $\mathit{u}$. The resolved velocity field is $\stackrel{-}{\mathit{u}}$, and the sub-grid velocity field is ${\mathit{u}}^{\prime }$. The sub-grid velocity field is defined in Eq. (9). The sub-grid field can be expressed as an infinite series⁴ and truncated such that ${\mathit{u}}^{\prime }$ is approximated by only the first (second-order) term, as in Eq. (10). The expansion is for rectangular grid cells, which was used in the CFD simulations. If the absolute value of ${\mathit{u}}^{\prime }$ in a grid cell is above a threshold, that cell is divided until the sub-grid field is below the threshold in all cells.

$${\mathit{u}}^{\prime }=\mathit{u}-\stackrel{-}{\mathit{u}}$$ (9)

$${\mathit{u}}^{\prime }\approx -\frac{d{x}_k^2}{24}\frac{{\partial }^2\stackrel{-}{\mathit{u}}}{\partial x_k\partial x_k}$$ (10)

This form of SGS-based AMR was investigated by our group previously.^36,37 Simulations for the sudden expansion of fluid inside an FDA nozzle showed that setting the minimum velocity sub-grid value, i.e. the threshold for $\left|{\mathit{u}}^{\prime }\right|$ mentioned above, to 1% of the mean throat velocity provided highly accurate results.³⁷ Therefore, the SGS criterion in the COA CFD simulation cases was set to 1% of mean blood velocity at the ascending aorta from in-vivo 4D Flow MRI.

All velocities were initialized to zero and the initial mesh consisted of 1 × 1 × 1 mm³ cubic elements. The zero-velocity initial condition with the course initial mesh would produce physiologically inaccurate results. To solve this issue, we simulated five cardiac cycles for every case. We allowed the velocities (and corresponding computational mesh) to converge to more physiologically accurate values in the first four cardiac cycles. The results were derived from the fifth cardiac cycle.

Time-Step Control

We implemented an adaptive time-step procedure based on the Courant-Friedrichs-Lewy (CFL) condition. The CFL condition is defined as

$$\mathrm{C}\mathrm{F}\mathrm{L}=\mathrm{\Delta }t\left(\frac{u}{\mathrm{\Delta }x}+\frac{v}{\mathrm{\Delta }y}+\frac{w}{\mathrm{\Delta }z}\right)\le {\mathrm{C}\mathrm{F}\mathrm{L}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$$ (11)

where CFL is the Courant number, $\mathrm{\Delta }t$ is the time-step size, $(u,v,w)$ are the $(x,y,z)$-components of velocity respectively, $(\mathrm{\Delta }x,\mathrm{\Delta }y,\mathrm{\Delta }z)$ are the spatial step sizes in the $(x,y,z)$-directions respectively, and ${\mathrm{C}\mathrm{F}\mathrm{L}}_{\text{max}}$ is the maximum Courant number that ensures numerical stability. Since our solver was an explicit method, we used ${\mathrm{C}\mathrm{F}\mathrm{L}}_{\mathrm{m}\mathrm{a}\mathrm{x}}=1$. The upper and lower bounds of the time-step size were set at 0.01 and 10⁻⁷ s, respectively. AMR varied the spatial step sizes, and therefore, the time-step size varied based on Eq. (11) within the mentioned bounds. As a result, our time-step size varied during the CFD simulation such that the numerical method remained stable, and the runtime was minimized.

Turbulence Modeling

Studies have shown turbulence in the blood flow inside the ascending aorta in normal subjects and subjects with stenoses.^10,35,45 Some of our COA cases also involve severe stenoses which would produce high-velocity jets, high acceleration, and turbulent flow. Researchers have used Large Eddy Simulation (LES) when studying turbulent blood flow.^30,34,35 In this technique, larger energy-carrying turbulent scales are resolved, and smaller energy-dissipating scales are modeled. We employed the One-Equation Viscosity Model formulated by Yoshizawa and Horiuti,⁴⁹ and Menon et al.²⁶ This enabled us to estimate the turbulent kinetic energy and dissipation rates inside the high-velocity jet in the stenotic cases.

Figure 3 shows a schematic of the framework of the 4D Flow MRI-based CFD method described above.

FIGURE 3.

Framework of the MRI-based CFD method. (a) CE-MRA/CTA data was loaded to the segmentation software. (b) The 3D anatomy with the inlets and outlets was generated. (c) 4D Flow MRI data was used to quantify blood flow dynamics by placing planes in locations of interest, i.e. inlets and outlets. (d) The in-vivo data provided the BCs for the CFD simulations. (e) CFD simulations were executed for each patient case and provided enhanced flow information compared to 4D Flow MRI alone. The plane shows regions of varying mesh refinement demonstrating the successful implementation of AMR.

Validation Methodology

Since the volumetric flow rate at the inlet (ascending aorta), and outlets (brachiocephalic, left common carotid, left subclavian arteries, and descending thoracic aorta) were extracted from the in-vivo data, the volumetric flow rates at the same locations were calculated from the CFD results. Instead of comparing the mean flow rates at the outlets over one cardiac cycle, the flow rates at each outlet were normalized by the total outflow, i.e., the sum of the mean flow rate at all four outlets. We are calling this quantity the fractional volumetric flow rate at outlet $i$, denoted as $q^i$, and is defined as

$$q^i=\frac{Q^i}{\sum _j Q^j}\times 100\mathrm{\%}$$ (12)

where $Q^i$ is the mean volumetric flow rate at outlet $i$ over one cardiac cycle and ${\sum }_j Q^j$ is the total outflow (sum of flow rates in brachiocephalic, left common carotid, and left subclavian arteries, and the descending thoracic aorta). $\mathrm{\Delta }q$ is defined as $\underset{i}{\text{max}}\left|q_{\text{MRI}}^i-q_{\text{CFD}}^i\right|$ where $i$ is brachiocephalic artery (BCA), left common carotid artery (LCC), left subclavian artery (LS), or descending thoracic aorta (Desc).

CFD numerically solves the Navier-Stokes equations, which are derived from the conservation of mass and momentum. Due to the continuity condition, CFD ensures that the inlet and total outlet flows are equal, i.e., mass is conserved. Conversely, 4D Flow MRI imposes no such continuity conditions. Flow metrics from 4D Flow MRI are solely computed from the acquired velocity information. This results in 4D Flow MRI providing flow rates that do not necessarily satisfy mass conservation, requiring normalization for a fair comparison. Furthermore, there are numerous intercostal and subcostal arteries that branch from the descending thoracic aorta which are not visible in 4D Flow MRI due to its limited spatial resolution or suboptimal ${\mathrm{V}}_{\mathrm{e}\mathrm{n}\mathrm{c}}$ setting. This contributes to the total outflow not matching the inflow in the in-vivo data. Therefore, we decided to normalize the flow rates by the total flow out of the four outlets in the manner shown in Eq. (12).

RESULTS

Study Cohort

Table 2 presents the demographic information of the patients included in the study. The final cohort included 5 patients (mean age 10 ± 4 years, 4 males) with a median follow-up duration of 5 ± 3 years. Four patients had native COA, and one patient (Patient 4) underwent reintervention for recurrent COA. Table 2 summarizes the patient demographics and 4D Flow MRI scan parameters as well.

TABLE 2.

Patient demographic information and 4D Flow MRI scan parameters (spatial and temporal resolution, V_enc).

Patient	Case	Gender	Age	Intervention procedure	Spatial Resolution (mm)	Temporal Resolution (ms)	Venc (cm/s)
1	Pre	F	6	Resection of mycotic aneurysm and placement of a 16mm gel weave interposition graft	2.73 × 2.00 × 2.20	38.4	150
	Post		8		2.50 × 1.75 × 2.00	40.8	150
2	Pre	M	13	Stent placement, 4010 XL delivered with 22 mm stent	3.93 × 2.50 × 2.00	35.0	150, 300*
	Post		16		1.88 × 1.81 × 1.80	41.1	150
3	Pre	M	7	Resection with extended end-to-end anastomosis and patent ductus arteriosus ligation	2.73 × 2.00 × 2.30	38.4	150
	Post		15		2.92 × 2.19 × 2.20	40.7	150
4	Pre**	M	10	Previously had stent placement, reintervention with 16 mm gel weave end-to-end interposition graft	1.88 × 1.56 × 1.90	40.8	180
	Post		15		4.00 × 1.88 × 2.20	41.1	150
5	Pre	M	14	Ebstein anomaly and dysplastic mitral valve. Repair with interposition graft	4.22 × 1.88 × 2.00	40.8	150
	Post		20		2.90 × 2.13 × 2.20	40.1	180

^*Patient 2 pre-intervention 4D Flow MRI scan was dual-V_enc.

^**Patient 4 underwent two COA interventions, so Pre denotes before second intervention.

In-Vivo 4D Flow MRI Results

The 4D Flow MRI data is visualized in Fig. 4. Patient 1’s mycotic aneurysm was not completely captured because the aneurysmal velocity was too low compared to the ${\mathrm{V}}_{\mathrm{e}\mathrm{n}\mathrm{c}}$ setting. Patient 2 and 4’s in-stent blood flow was also not captured by 4D Flow MRI due to artifact from the stent.

FIGURE 4.

Velocity data from in-vivo 4D Flow MRI at peak systole. Red boxes highlight the limitations of 4D Flow MRI: incomplete aneurysmal blood flow (Patient 1) and missing in-stent blood flow (Patient 2 and 4).

CFD Simulation Results

Ten COA case CFD simulations were successfully executed following the above-mentioned framework without turbulence modeling. Additionally, two CFD simulations with turbulence modeling were executed for Patients 3 and 5 pre-intervention. These two cases involved severe constrictions in the descending aorta, producing high-velocity turbulent jets. CFD results were analyzed using the post-processing software Tecplot 360 EX (Tecplot, Inc., Bellevue, Washington, USA). The presented CFD method is validated by direct comparison with the in-vivo 4D Flow MRI data. Figure 5 visualizes the CFD results for all ten patient cases. CFD successfully predicted Patient 1’s aneurysmal flow, and Patient 2 and 4’s in-stent hemodynamics.

FIGURE 5.

Velocity data at peak-systole from CFD simulation results without turbulence modeling.

Table 3 presents the average number of elements that were used in each CFD simulation. AMR varied the number of elements in the mesh throughout the simulation, and so the average number of elements are reported.

TABLE 3.

Average number of total elements in each CFD simulation.

Number of elements	Patient 1	Patient 2	Patient 3	Patient 4	Patient 5
Pre-intervention	4.76 × 105	9.32 × 105	7.42 × 105	6.67 × 105	7.83 × 105
Post-intervention	5.41 × 105	7.12 × 105	6.30 × 105	6.04 × 105	5.86 × 105

Validation of Outlet Volumetric Flow Rates

Table 4 presents the fractional volumetric flow rates for all five patients and $\mathrm{\Delta }q$. The methods section explains how these quantities were calculated.

TABLE 4.

Fractional volumetric flow rates in percent for all five patients from 4D Flow MRI data and CFD results.

Patient	Case	4D Flow MRI				CFD				Δq
		q BCA	q LCC	q LS	q Desc	q BCA	q LCC	q LS	q Desc
1	Pre	38.8	12.3	14.2	34.8	41.3	11.5	16.4	30.8	3.9
	Post	15.7	9.2	13.1	62.0	27.4	9.2	11.4	52.0	11.8
2	Pre	13.6	14.8	9.3	65.3	25.0	8.8	13.7	52.5	12.8
	Post	15.0	8.2	10.2	66.6	25.1	7.5	12.8	54.7	11.9
3	Pre	33.1	10.4	17.7	38.8	24.0	6.8	22.1	47.2	9.1
	Post	10.7	3.3	31.5	61.6	14.3	4.4	27.7	53.6	7.9
4	Pre	41.7	6.0	31.7	20.0	30.5	7.4	21.1	41.0	21.0*
	Post	8.9	5.3	7.4	76.2	17.9	4.7	13.9	63.5	12.7
5	Pre	43.6	7.4	26.0	35.9	36.2	8.9	23.4	31.5	7.4
	Post	15.7	5.5	21.0	57.9	24.9	6.0	16.4	52.7	9.2

^*Patient 4 pre-intervention reported severe aliasing. The last column is the largest difference in fractional volumetric flow rate among the four outlets.

The average of $\mathrm{\Delta }q$ was found to be 9.6%. Additionally, the total outflow from MRI was on average 8.7% less than the inflow. These results agree with previously conducted consistency studies for validation of 4D Flow MRI.⁴²

Flow Inside Mycotic Aneurysm

The mycotic aneurysm in Patient 1 involves a narrowing just after the aortic arch followed by a sudden expansion in the aneurysm sac. In terms of flow dynamics, the blood velocity experiences an acceleration in the constriction, followed by a sudden deceleration along with recirculation and vortices in the aneurysm. This complex flow structure results in complications in the MR-signal acquisition. The high velocity range requires a high $V_{\mathrm{e}\mathrm{n}\mathrm{c}}$ to avoid aliasing. However, the high $V_{\mathrm{e}\mathrm{n}\mathrm{c}}$ setting also results in the low-velocity aneurysmal flow being contaminated and indistinguishable from noise.⁶ Figure 4 clearly shows that the single-$V_{\mathrm{e}\mathrm{n}\mathrm{c}}$ 4D Flow MRI procedure failed to capture the blood flow inside Patient 1’s mycotic aneurysm.

In contrast, our CFD method successfully simulated the blood flow inside the mycotic aneurysm. Since CE-MRA was used to define the anatomy, the more anatomically accurate geometry was inputted to the CFD setup. Figure 6 shows the visualization and quantification of hemodynamics in the aneurysm. Six parallel planes that cover the length of the aneurysm sac were placed and the velocity maps were extracted at systole and diastole. The mean and peak velocities at each plane and timepoint are reported. Highly refined mesh can be noticed in areas of large velocity gradients, such as the vessel wall and edge of jet. This local mesh refinement ensures high spatial resolution and improved flow visualization. The voxel size in our CFD simulations varied between 1 mm³ and (1/8)³ mm³, compared to 12.5 mm³ for the 4D Flow MRI data. In analogy, inside one 4D Flow MRI voxel, CFD can pack between 12 and 6400 computational grid cells. The average time step size of our CFD simulation was 10 ms, whereas the time resolution of 4D Flow MRI was approximately 40 ms.

FIGURE 6.

Visualization and quantification of hemodynamics inside Patient 1’s mycotic aneurysm. CFD completed 4D Flow MRI outcomes by providing mean and peak planar velocities in the aneurysm.

Figure 7 shows the velocity contours from in-vivo 4D Flow MRI at the approximate location as planes in Fig. 6. The mean and peak velocities are also presented. The locations of the planes were difficult to match because the mycotic aneurysm is not fully visible in the 4D Flow MRI data. This is likely the reason for the differences in peak velocity. In most planes, the mean velocity from 4D Flow MRI is higher than the mean velocity from CFD. This is because CFD calculates the slow aneurysmal flow whereas 4D Flow MRI does not detect that flow.

FIGURE 7.

Visualization and quantification of hemodynamics inside Patient 1’s mycotic aneurysm from 4D Flow MRI. 4D Flow MRI did not detect flow inside the whole aneurysm due to insufficient ${\mathbf{V}}_{\mathbf{e}\mathbf{n}\mathbf{c}}$.

Flow Inside Stents

Patients 2 and 4 underwent COA repair by placing a stent. An AHA scientific statement on safety of MRI with cardiovascular devices recognized certain aortic stent grafts reported severe susceptibility artifact making the evaluation of the in-stent lumen problematic.²¹ In these cases, the stent is not visible due to the distortion of MR-signal. Additionally, the artifact caused interference distal to the stent in Patient 2 and severe aliasing in Patient 4. Running CFD simulations allowed us to quantify and visualize the blood flow dynamics in the tendered region.

Four parallel planes that spanned the length of each stent were placed and the planar velocity contour maps are shown in Fig. 8. The mean and peak velocities at the planes are also reported. This CFD method enabled comprehensive quantification of the flow dynamics inside the stent. This region was not visible in PC-MRA.

FIGURE 8.

Visualization and quantification of hemodynamics inside Patient 2 and 4’s stent at peak systole. The in-stent blood flow was not visible in 4D Flow MRI.

Turbulence Model Results

Two CFD simulations with LES turbulence model were successfully executed for the pre-intervention cases for Patients 3 and 5. These two patients were selected for turbulence modeling due to their severe stenoses that would cause high-velocity turbulent jets at the constrictions. We calculated the fractional volumetric flow rates from these simulations and compared them with the results without turbulence modeling in Table 5. It must be noted that the calculated flow rates are spatiotemporally averaged given the fact that the in-vivo 4D Flow MRI data used for comparison is in nature spatiotemporally averaged. For this reason, the turbulence model did not significantly affect the results in this context.

TABLE 5.

Fractional volumetric flow rates in percent for Patients 3 and 5 pre-intervention from CFD simulations with and without turbulence modeling.

Patient case	Turbulence model?	4D flow MRI				CFD				Δq
		q BCA	qLCC	qLS	qDesc	q BCA	qLCC	qLS	qDesc
3 Pre	No	33.1	10.4	17.7	38.8	24.0	6.8	22.1	47.2	9.1
	Yes					24.8	6.8	21.7	46.7	8.3
5 Pre	No	43.6	7.4	26.0	35.9	36.2	8.9	23.4	31.5	7.4
	Yes					39.1	9.2	23.9	27.7	8.1

The last column is the largest difference in fractional volumetric flow rate among the four outlets.

The turbulence model simulations calculated fractional volumetric flow rates that were within 10% of those found in in-vivo 4D Flow MRI. Additionally, the difference in fractional volumetric flow rates with and without turbulence modeling was less than 1%. Moreover, the simulation runtimes without turbulence model were about 22 h, and with turbulence model the runtimes exceeded 60 h. Adding a turbulence model increased the simulation runtime three-fold. Since volumetric flow rates at the brachiocephalic, left common carotid, and left subclavian arteries, and the descending thoracic aorta are the metrics for comparing CFD against in-vivo measurements, turbulence modeling was not necessary for all patient cases.

Turbulence modeling enabled us to calculate the turbulent kinetic energy and dissipation rate. Table 6 presents these parameters for the average of the simulated region (thoracic aorta) as well as at the stenosis. The turbulent kinetic energy and dissipation rate is higher at systole than diastole because the higher flow increases the fluctuating component of velocity.

TABLE 6.

Turbulent kinetic energy and dissipation rate at systole and diastole for Patients 3 and 5 pre-intervention.

Patient case	Average Turbulent Kinetic Energy (m2/s2)	Average Dissipation Rate (m2/s3)	Stenotic Turbulent Kinetic Energy (m2/s2)	Stenotic Dissipation Rate (m2/s3)
Systole
3 Pre	7.21 × 10−3	5.09	0.231	893
5 Pre	7.89 × 10−3	5.07	0.439	2980
Diastole
3 Pre	5.39 × 10−4	2.44 × 10−2	7.03 × 10−4	8.10 × 10−2
5 Pre	4.63 × 10−4	2.41 × 10−2	0.550	2.48 × 10−3

Improvement due to AMR

To demonstrate the effects of AMR on the CFD results, we ran two simulations without AMR. Pre-interventional cases for Patient 1 and 3 were chosen. The uniform base-grid consisted of 1 × 1 × 1 mm³ cubic elements. Table 7 shows the fractional volumetric from rates from simulations with and without AMR for comparison. AMR improved the accuracy in the fractional volumetric flow rates by about 5.8%. Moreover, when the Patient 3 pre-intervention case was simulated without AMR, the maximum difference in fractional flow rates between in-vivo and in-silico data was found to be 14.7%, which is slightly above the accepted accuracy range for 4D Flow MRI.

TABLE 7.

Fractional volumetric flow rates in percent for Patients 1 and 3 pre-intervention from CFD simulations with and without AMR.

Patient Case	AMR?	4D Flow MRI				CFD				Δq
		q BCA	qLCC	qLS	qDesc	q BCA	qLCC	qLS	qDesc
1 Pre	Yes	38.8	12.3	14.2	34.8	41.3	11.5	16.4	30.8	3.9
	No					32.6	10.0	12.8	44.6	9.8
3 Pre	Yes	33.1	10.4	17.7	38.8	24.0	6.8	22.1	47.2	9.1
	No					19.8	8.2	18.5	53.5	14.7

The last column is the largest difference in fractional volumetric flow rate among the four outlets.

DISCUSSION

4D Flow MRI is an important non-invasive clinical diagnostic tool due to its numerous advantages, such as reasonable scan times of typically 5–15 min, and its ability to provide comprehensive 3D blood flow information.²² Hemodynamic information from 4D Flow MRI enables clinicians to make informed decisions for the treatment strategy and analysis of treatment success post-COA repair. 4D Flow MRI data is often suboptimal or incomplete due to limited spatiotemporal resolution,²² aliasing,¹¹ and distortion of images due to metal artifact (stent).¹² Image-based CFD is a powerful tool that can be used to support clinical decision-making,²⁴ and has been applied to congenital heart disease, particularly COA.²⁵ Computational Fluid Dynamics (CFD) has high potential to complement 4D Flow MRI and address the stated limitations, providing enhanced patient-specific hemodynamic data.

This study presented a framework for 4D Flow MRI-based CFD using AMR. Our computational method was implemented on five patients with COA, pre- and post-repair (total of ten cases). The patient population comprising various forms of COA and treatment strategies supported the robustness of this method. The benefits of AMR in terms of CFD simulation times were shown in previous studies conducted by our group.^36,37 Successful execution of our CFD method produced a set of in-silico data to compare with in-vivo 4D Flow MRI data.

Side-by-side comparison of in-vivo and in-silico data showed that fractional volumetric flow rates from CFD agreed well with that from 4D Flow MRI (Table 4). Single-$V_{\mathrm{e}\mathrm{n}\mathrm{c}}$ 4D Flow MRI failed to capture the slow aneurysmal flow in Patient 1. Despite multi-$V_{\mathrm{e}\mathrm{n}\mathrm{c}}$ acquisition methods solving the stated problem, such imaging modalities are yet to be implemented in the clinical setting in large scale. Metal artifacts from the vascular stents placed in Patient 2 and 4 distorted the MR-signal inside and distal to the tendered region (Fig. 4). The CFD method visualized and quantified the hemodynamics in regions that were incomplete from 4D Flow MRI, particularly Patient 1’s aneurysm (Fig. 6), and Patient 2 and 4’s stent (Fig. 8). Implementing AMR in our CFD simulations ensured that the CFD results have on the order of 10¹ to 10³ better spatial resolution compared 4D Flow MRI. Additionally, our CFD procedure had approximately four times better temporal resolution compared 4D Flow MRI. Severe aliasing was reported in Patient 4’s pre-intervention case, making the 4D Flow MRI data unreliable. Aliasing is the phenomenon when different signals are indistinguishable when they are sampled resulting in distortion and artifacts. CFD is not susceptible to aliasing because it numerically solves the conservation equations, it does not rely on signal processing. To summarize, the 4D Flow MRI-based CFD method presented here provides several advantages compared to 4D Flow MRI alone: higher spatial and temporal resolution, visualization of blood flow in regions of very low or high velocity not visible with the selected $V_{\mathrm{e}\mathrm{n}\mathrm{c}}$ setting, and prediction of blood flow inside and distal to the stent.

Previous studies have used image-based CFD to simulate cardiovascular flows. Successful clinical trials and subsequent 2014 FDA approval of a simulation service by HeartFlow, Inc. is one of the best indicators of clinical acceptance of this tool.^27,31,47 The imaging modalities that are primarily used for image-based CFD are CTA or CE-MRA. In recent years, we have seen an increased interest in using PC-MRA-based cardiovascular flow simulations. Studies by Goubergrits et al.,¹⁷ and Miyazaki et al.³⁰ coupled 4D Flow MRI with CFD to simulate aortic blood flow in patients with COA. In addition to 4D Flow MRI, Goubergrits et al. used catheterization to measure aortic pressure drops. The in-vivo pressure drop measurements were subsequently compared with pressure drops calculated from CFD. Our methods solely used non-invasive scans, and as a result, our in-vivo vs. in-silico data compared volumetric flow rates only. Our CFD procedure differed from Miyazaki et al. where the latter used two turbulence models: an LES using the Smagorinsky-Lilly model, and a Re-Normalization Group (RNG) k-e model. Miyazaki et al. found that CFD with turbulence modeling yielded stronger correlation with MRI data. We used an LES model and found no significant difference in our results when a turbulence model is included. The image-based cardiovascular flow dynamics community is yet to adopt AMR in their CFD simulations in large scale. Our group has studied AMR applied to hemodynamics.^36,37 The major novelty in this study is the successful expansion of this useful numerical method to blood flow in COA.

There are several limitations in our CFD method that we must acknowledge. The following assumptions were made for our Windkessel BCs: blood pressure values were assumed to be 120 mmHg systolic and 80 mmHg diastolic, the pulse wave velocity values were taken from literature and not measured in-vivo, and Eqs. (3)–(8) assume the resistances and compliances depend only on the relative cross-sectional areas of the arteries. The blood pressure values were assumed under normal conditions because we did not have catheterization data available. In-vivo pulse wave velocity data was also unavailable which required us to use values from literature.²⁰ The patient-specific components inputted into Eqs. (3)–(8) were the arterial cross-sectional areas. We acknowledge that the assumptions in the Windkessel BCs likely contributed to the differences between in-vivo and in-silico results.

Limitations of this CFD method also include the rigid wall assumption and high simulation times. Our CFD procedure assumed the aorta wall was rigid. As a result, we failed to capture the motion (radial expansion and contraction) of the arterial wall. Fluid-structure Interactions (FSI) is a coupled computational technique that simultaneously simulates the mechanical deformation of a solid structure and surrounding/internal fluid flow. FSI can represent vessel wall behavior and wave propagation and provide important clinical metrics of vessel distensibility.^13,16,38 The CFD simulations in this study were executed on a high-performance computing cluster (160 CPU cores) and the average runtime was about 17.5 h. A recent study presented a reduced-order model that significantly reduces computational costs while providing results that agree with CFD simulations.²⁹ Future goals include implementing FSI and reduced-order modeling strategies to our COA models.

The presented CFD method has future clinical applications. In patient cases where 4D Flow MRI fails to capture aneurysmal or in-stent hemodynamics, or higher spatiotemporal resolution is required, this computational procedure serves as a tool that would provide enhanced outcomes. Furthermore, our methods are capable of reconstructing post-intervention aorta geometries which can be used to predict hemodynamics before intervention is performed. This in turn provides interventionalists additional information that would aid decision-making and analysis of treatment success.

ABBREVIATIONS

COA

Coarctation of the aorta

CFD

Computational fluid dynamics

AMR

Adaptive mesh refinement

${\mathrm{V}}_{\text{enc}}$

Velocity encoding

PC-MRA

Phase-contrast magnetic resonance angiography

CE-MRA

Contrast-enhanced magnetic resonance angiography

CTA

Computed tomography angiography

BC

Boundary condition

$q^{\text{BCA}}$

Fractional volumetric flow rate in brachiocephalic artery

$q^{\text{LCC}}$

Fractional volumetric flow rate in left common carotid artery

$q^{\text{LS}}$

Fractional volumetric flow rate in left subclavian artery

$q^{\text{Desc}}$

Fractional volumetric flow rate in descending thoracic aorta

$\mathrm{\Delta }q$

Maximum absolute difference in fractional volumetric flow rates from 4D flow MRI and CFD among outlets