Accelerated estimation of pulmonary artery stenosis pressure gradients with distributed lumped parameter modeling vs. 3D CFD with instantaneous adaptive mesh refinement - Experimental validation in swine

Abstract

Branch pulmonary artery stenosis (PAS) commonly occurs in congenital heart disease and the pressure gradient over a stenotic PA lesion is an important marker for re-intervention. Image based computational fluid dynamics (CFD) has shown promise for non-invasively estimating pressure gradients but one limitation of CFD is long simulation times. The goal of this study was to compare accelerated predictions of PAS pressure gradients from 3D CFD with instantaneous adaptive mesh refinement (AMR) versus a recently developed 0D distributed lumped parameter CFD model. Predictions were then experimentally validated using a swine PAS model (n=13). 3D CFD simulations with AMR improved efficiency by 5 times compared to fixed grid CFD simulations. 0D simulations further improved efficiency by 6 times compared to the 3D simulations with AMR. Both 0D and 3D simulations underestimated the pressure gradients measured by catheterization (−1.87±4.20mmHg and −1.78±3.70mmHg respectively). This was partially due to simulations neglecting the effects of a catheter in the stenosis. There was good agreement between 0D and 3D simulations (ICC 0.88 [0.66 – 0.96]) but only moderate agreement between simulations and experimental measurements (0D ICC 0.60 [0.11 – 0.86] and 3D ICC 0.66 [0.21 – 0.88]). Uncertainty assessment indicates that this was likely due to limited medical imaging resolution causing uncertainty in the segmented stenosis diameter in addition to uncertainty in the outlet resistances. This study showed that 0D lumped parameter models and 3D CFD with instantaneous AMR both improve the efficiency of hemodynamic modeling, but uncertainty from medical imaging resolution will limit the accuracy of pressure gradient estimations.

Introduction

Branch pulmonary artery stenosis (PAS) commonly occurs in congenital heart disease^23,30 either as a congenital lesion or as a complication of cardiac surgery. The pressure gradient over a stenotic PA lesion is an important marker for re-intervention as high gradients can cause PA hypertension, pulmonary valve insufficiency, and increase RV afterload ^6,7, all contributing to long term morbidity and mortality for patients with branch PAS ^4,9,23. There are no defined criteria dictating what severity of PAS warrants intervention. Anecdotally less than 30% of pulmonary blood flow to the stenotic lung has been used as a cut-off but there is no evidence supporting this criterion. The lack of criteria to define when an intervention to relieve PAS is warranted underscores the need for new tools to better understand the severity and hemodynamic impact of a stenotic PA lesion. Invasive catheterization is required to measure PAS pressure gradients and non-invasive modified Bernoulli estimates Doppler ultrasound velocimetry perform poorly⁸. 4D Flow MRI can be used to accurately calculate pressure gradients across stenoses non-invasively², but even the highest spatial resolution sequences will have insufficient spatial resolution for severe stenotic lesions or the smallest pediatric patients. Image based computational fluid dynamics (CFD) offers an alternative approach to non-invasively estimate pressure gradients and has shown promise for carotid artery stenosis²⁷ and aortic coarctations^24,29. However, in addition to the pitfalls of attempting to predict physiology from anatomy¹², CFD is limited by long simulation times. One possible efficiency improvement would be to use solution adaptive mesh refinement (AMR) to optimally change the 3D CFD grid throughout the cardiac cycle as flow conditions change from systole to diastole¹⁹. Another way to improve efficiency would be to use a recently developed 0D distributed lumped-parameter fluid dynamics model that has shown good agreement with 3D CFD¹⁵. Single vessel lumped parameter models have been used to estimate pressure gradients for coarctation of the aorta¹¹ but more generalized model that can simulate multiple vessels are needed in PAS as flow to the contralateral lung can substantially alter the stenotic pressure gradient. The goal of this study was to compare predictions of PAS pressure gradients from 3D CFD with AMR to 0D lumped parameter modeling and experimentally validate these predictions using a swine PAS model¹⁸.

Materials and Methods

Fourteen domestic male swine were obtained from the University of Wisconsin Swine Research and Teaching Center (2 weeks old, 5±1 kg). Discrete proximal left PA stenosis (LPAS) was surgically created as previously described¹. Five animals had imaging and catheterization at 5 weeks of age. Five animals had imaging and catheterization at 10 weeks of age. Four animals had imaging and catheterization at 20 weeks of age. Further details on the imaging and catheterization techniques can be found in our previous publications^5,18,20 but the imaging methods will be described briefly. Multi-slice computed tomography (MSCT) was performed using a 64-slice CT scanner (GE 750 CT, GE Healthcare, Waukesha, WI) using retrospective cardiac gating. Scan variables included: collimation 40 mm, slice thickness 0.625 mm, 140 kV (peak), tube current 570 mA, acquisition diameter 29 cm and a 512 × 512 reconstruction matrix. Three dimensional rotational angiography (3DRA) was performed on a single plane Artis Z system (Siemens Healthcare. Forchheim, Germany) and utilized a 200° rotational acquisition over 5 s at 60 frames/sec with a 50% diluted contrast injection directly into the main PA (0.8–1 cc/kg, Omnipaque 350) preceding image acquisition by 1 second with simultaneous IVC balloon occlusion. Contrast free phase contrast magnetic resonance angiography (PC-MRA) was performed with a three-dimensional radial under sampled isotropic projection reconstruction sequence (PC-VIPR) [11, 12] on a 3.0 T imaging system (Discovery MR750, GE Healthcare, Waukesha, WI). The Institutional Animal Care and Use Committee of the University of Wisconsin reviewed and approved this protocol.

Model Segmentation

One 20-week old swine was excluded from analysis due to poor image quality. For nine swine pulmonary vascular geometries were segmented from 3D rotational angiography (3DRA: voxel size 0.47mm), and for one the geometry was segmented from multi-slice computed tomography (MSCT: voxel size 0.7mm). For two 5-week old swine and one 10-week old swine geometries were segmented from phase-contrast magnetic resonance angiography (PC-MRA: voxel size 1.25mm) due to artifacts present in datasets from the higher spatial resolution imaging modalities. Geometries were segmented manually using a combination of the software packages Simvascular²⁸, Mimics (Materialize, Leuven), and 3-matic (Materialize, Leuven). All inlet and outlets were trimmed to ensure that these faces were perpendicular to the vessel centerlines. Representative geometries for 5-week, 10-week, and 20-week swine are shown in Figure 1 along with a PC-MRA segmentation to show the limited distal vascular that could be seen with PC-MRA.

Figure 1:

Representative segmentations for 5-week, 10-week, and 20-week old LPAS swine. An additional segmentation is shown for PC-MRA to show the limited distal vasculature that could be seen.

Boundary Condition Estimation

Animal specific MPA flow rates measured from 4D Flow MRI using the flow visualization software Ensight (Ansys, Cannonsburg, PA) were used as inflow boundary conditions assuming a flat velocity profile^13,32. For two 10-week old swine 4D Flow MRI data was not obtained and instead an average MPA flow profile was calculated from the other three 10-week old swine. Outlet boundary conditions were three-element Windkessel models with resistance and compliance distributed by outlet area as previously described³². The left and right lung resistances were assigned using the median left and right pulmonary vascular resistance index (PVRI) from all 14 swine calculated from the hemodynamics catheterization measurements. There was much greater variability in the left PVRI (8.1 IQR [3.1 – 21.9] WU/m²) than in the right PVRI (4.2 IQR [3.5 – 5.1] WU/m²). The median PVRI values were used on the assumption that if CFD were used clinically to estimate PAS pressure gradients, population values for PVR could be used to assign boundary conditions similar to the process used by HeartFlow²⁷. Compliance was then assigned assuming an RC time constant of 0.5 sec as negligible pulse pressures distal the stenosis made direct estimation difficult. All simulations were run for four cardiac cycles.

0D Distributed Lumped Parameter Model

At first glance, a distributed lumped parameter model appears to be variation of solving the 1D Navier-Stokes equations. However, distributed lumped parameters models are actually derived from integrating the linearized 1D Navier-Stokes equations over the length of a vessel segment¹⁴. This results in a 0D model where pressure and flow rate no longer have spatial dependence along the vessel length, unlike the 1D Navier-Stokes equations. Centerlines for the 0D lumped parameter model were created with Mimics (Materialize, Leuven). 0D simulations were implemented in MATLAB using the recently developed method of Mirramezani and Shadden¹⁵. Mass conservation is enforced at each vascular junction

$$\mathrm{\Sigma }{Q}_{in}=\mathrm{\Sigma }{Q}_{out}$$ (1)

Conservation of momentum is solved for each vascular segment

$$\left(\rho {\int }_0^L\frac{1}{A(x)}dx\right)\frac{\partial Q}{\partial t}+RQ=\mathrm{\Delta }P$$ (2)

Where ρ is fluid density, A(x) is the vessel area that varies axially, L is the segment length, R is a non-linear resistance, Q is flow rate, and ΔP is the pressure gradient. The non-linear resistance was calculated

$$R=\frac{8\mu }{\pi }{\int }_0^L\max \{\gamma ,\zeta \}\frac{1}{r^4}dx+\sum _{i=1}^n\frac{\rho K_t}{2A_{0,i}^2}{\left(\frac{A_{0,i}}{A_{s,i}}-1\right)}^2|Q|$$ (3)

Where the first term is the linear resistance and the second term is the non-linear Bernoulli type resistance. The γ and ζ modify the Poiseuille resistance depending on whether viscous losses due to curvature or unsteadiness are greater. γ is defined

$$\gamma =0.1033\sqrt{K}{\left(\sqrt{1+\frac{1.729}{K}}-\frac{1.315}{sqrt(K)}\right)}^{-\frac{1}{3}}$$ (4)

Where K is the Dean number calculated from the Reynolds number (Re), radius (r), and vessel curvature (a).

$$K=Re\sqrt{\frac{R}{a}}$$ (5)

ζ is calculated from Womersly flow³³

$$\zeta =\left|1+\frac{2}{\text{A}}\frac{J_1(\mathrm{\Lambda })}{J_0(\mathrm{\Lambda })}\right|$$ (6)

Where Λ is calculated from Womersly’s number,

$$\mathrm{\Lambda }=r\sqrt{\frac{\rho }{\mu }}\frac{(i-1)}{\sqrt{2}}$$ (7)

and J is a Bessel function of the first kind.

In the non-linear resistance term to account for sudden expansions, A_s is the local area minimum, A₀ is the average of the maximum areas before and after the stenosis, and K_t is an empirically determined correction factor, 1.52.

The non-linear branching model that Mirramezani and Shadden included was neglected because preliminary simulations found that the branching model accounted for ~50% of the resistance over the stenosis segment which was unphysical as the primary source of resistance should be the stenosis. Results from the original developers of the branching model showed that for a bifurcation with a highly asymmetric flow split the reduced order branching model overestimated pressure losses by a comparable amount to findings from the present study¹⁷. Furthermore, the close proximity of the stenotic lesion to the PA bifurcation in PAS is not consistent with the geometric assumptions made in the control volume analysis used to derive the junction pressure loss model¹⁷ and therefore the junction pressure loss model may not be appropriate to include in simulation of PAS.

Equations 1 and 2 were then discretized to a system of non-linear algebraic equations using a backwards Euler time-stepping scheme and solved in MATLAB using the Levenberg-Marquardt algorithm¹⁶.

3D CFD with AMR

All 3D CFD simulations were performed with Cartesian cut-cell grids using the commercial software CONVERGE v3.0 (Convergent Science Inc., Madison, WI)²⁶. The fluid domain was filled with orthogonal polyhedras with user-specified base grid and refinement criteria. The Cartesian grid was cut at boundary to exactly match the geometry surface and conserve the volume. If a cut-cell volume was less than 30% of the adjacent Cartesian cell, the two cells were paired to form a single cell to improve simulation stability and efficiency. A second-order spatial accuracy centered finite-volume discretization was used and the Rie-Chow algorithm collocates pressure and velocity at the cell center. For pressure-velocity coupling a pressure implicit with splitting of operator (PISO) method was used¹⁰. The PISO equations were solved to a tolerance of 1e-3 with a maximum of nine PISO iterations. Windkessel outlets were implemented as a standard feature in the CONVERGE v3.0 software. Simulations also assumed rigid walls even though the PAs are highly distensible. All 3D CFD simulations were performed with 40 CPUs on the University of Wisconsin- Madison High Performance computing cluster which is tightly networked (56 Gbit/sec Infiniband).

Established AMR methods^3,19,26 were implemented based on the sub-grid field (ϕ′) or difference between the actual field (ϕ) and resolved field $(\overline{\varphi })$²¹

$${\varphi }^{\prime }=\varphi -\overline{\varphi }.$$ (8)

The sub-grid field is defined by an infinite series but in practice only the first term is used to approximate the sub-grid field

$${\varphi }^{\prime }\approx -\frac{d{x}_{[k]}^2}{24}\frac{{\partial }^2\overline{\varphi }}{\partial x_k\partial x_k}$$ (9)

In this study the sub-grid field was calculated from velocity. When the sub-grid field (ϕ′) was above a specified level named as the sub-grid scale (SGS), a cell was refined into 8 identical cells recursively until the target SGS is reached. And if the sub-grid field was below 1/5^th of the SGS the embedding was released. Cells were refined up to a maximum of three times (1/8^th the original Δx). Figure 2 shows the effect of different SGS values on the adapted grids for a representative case.

Figure 2:

Results from a representative 20-week old LPAS swine on a plane placed through the MPA, RPA, and LPA. Velocity contours and AMR grids with SGS values of 0.01, 0.02, and 0.05 m/s are shown. Increased grid refinement can be seen during systole compared to diastole. Increasing SGS from 0.01 m/s to 0.05 m/s decreased grid refinement.

Additionally, an adaptive time step algorithm was used to maintain a constant convection Courant-Friedrichs-Lewy (CFL) number $\left(\frac{u\mathrm{\Delta }t}{\mathrm{\Delta }x}\right)$ and diffusion CFL number $\left(\frac{\mu \mathrm{\Delta }t}{\rho \mathrm{\Delta }{x}^2}\right)$ with changing flow conditions during a heartbeat. Simulations were started with a time step of 1e-5 seconds and the time step size was increased by 25% for each time step until the maximum CFL number was reached.

Based on a grid refinement and AMR parameter study¹⁹ the base-grid was chosen to be 2mm (minimum cell size 250μm), the AMR SGS to be 0.02 m/s, and the maximum convection CFL number 5.0. These parameters were found to improve efficiency while maintaining good accuracy compared to highly refined AMR simulations and fixed grid simulations.

All 3D CFD figures were generated with Tecplot 360 (Tecplot, Bellevue, WA).

Parameter Uncertainty Assessment

To assess the effects of uncertainty in parameters on our model results a series of simulations were run with a representative 5-week old case. The majority of these uncertainty simulations were performed with the 0D model for efficiency. The effect of imaging resolution was quantified by changing the minimum lumen diameter of the stenosis by approximately +/− 1 voxel (0.5mm). The effect of outlet resistance was quantified by simulating both 1^st quartile LPVRI with 3^rd quartile RPVRI, and 3^rd quartile LPVRI with 1^st quartile RPVRI, to increase and decrease the amount of flow through the stenosis respectively. The effect of cardiac output (CO) was quantified by simulating +/− 10% changes to CO. Lastly, to the effect of a pressure catheter in the stenosis was quantified by simulating a 4 French (1.3mm OD) catheter through the stenosis with 3D CFD. Grid parameters were the same as above except in the stenosis the minimum cell size was changed to 62.5μm to resolve flow in the small gaps. To best represent the in vivo measurement, the pressure gradient was calculated as the peak systolic MPA pressure from the simulation without the catheter in the stenosis minus the peak systolic LPA pressure from the simulation with a catheter in the stenosis

Statistics

Data are reported as mean ± standard deviation (SD). The variable of interest that was compared between modalities was the peak systolic pressure gradient across the LPAS. The differences between modalities were found to be normally distributed with both a Shapiro-Wilkes test and visual inspection of histogram plots. Differences between modalities were assessed with both Bland-Altman analysis and a 1–1 agreement intra-class correlation coefficient (ICC) model. ICC estimates are presented with the 95% confidence interval and can be interpreted as: 0.0–0.5 poor agreement, 0.5–0.75 moderate agreement, 0.75−−.90 good agreement, and 0.9–1.0 excellent agreement.

Results

Summary data of hemodynamics, pulmonary blood flow distributions, and pulmonary artery dimensions are shown in Table 1. At 5-weeks of age the stenosis would be considered moderate and progressed to severe at 10-week and 20-week ages. For further information of the physiological consequences of LPAS in swine the reader is referred to our previous publication¹⁸.

Table 1:

Longitudinal progression of untreated LPAS

	5-week	10-week	20-week
Weight (kg)	6.7±2.4	32±7	57±5
Proximal LPA Diameter (mm)	1.7±0.4	2.2±0.8	1.4±0.2
Distal LPA Diameter (mm)	6.5±0.7	10.3±1.1	6.9±3.8
HR (BPM)	120±16	92±4	83±5
CI (L/min/m2)	2.0±1.7	4.2±0.3	3.2±0.2
L Lung Perfusion (%)	18±8	6±4	7±5
Mean RA Pressure (mmHg)	6±3	8±3	10±1
RV Systolic Pressure (mmHg)	29±7	29±2	38±4
MPA Pressure (sys / dia, mmHg)	27±5 / 13±4	29±4 / 14±1	38±4 / 17±2
RPA Pressure (sys / dia, mmHg)	25±4 / 10±3	28±3 / 14±1	37±3 / 18±3
LPA Pressure (sys / dia, mmHg)	13±6 / 13±6	13±3 / 13±3	15±6* / 13±5
Stenosis/Stent Pressure Gradient (mmHg)	14±7	16±3	23±4
PCWP (mmHg)	8±5	10±1	11±3

Inter-Modality Differences

Results from Bland-Altman analysis and ICC are presented in Table 2 and Figure 3. Both 0D and 3D simulations on average under predicted the peak systolic LPAS pressure gradient measured by right heart catheterization (RHC) by 4–5%. There was high variability between both 0D and 3D simulations with RHC measurements and only moderate agreement by ICC. From Bland-Altman plots (Figure 3) both 0D and 3D simulations tended to over predict smaller LPAS gradients and under predict higher LPAS gradients. 3D simulations were marginally more consistent with RHC as they had a 95% confidence interval from Bland-Altman analysis that was 2mmHg lower than 0D simulations (14.8 vs 16.8 mmHg respectively) and slightly higher ICC (0.66 [0.21 – 0.88] vs 0.60 [0.11 – 0.86] respectively). There was negligible bias between 0D and 3D simulations (−0.09±1.95 mmHg) and good agreement from ICC (0.88 [0.66 – 0.96]).

Table 2:

Inter-modality difference comparisons of peak systolic LPAS pressure gradient

Comparison	Bias		ICC [95% CI]
	mmHg	%
0D-RHC	−1.87±4.20	−4.7±36.7	0.60 [0.11 – 0.86]
3D-RHC	−1.78±3.70	−4.2±30.6	0.66 [0.21 – 0.88]
0D-3D	−0.09±1.95	−1.0±14.4	0.88 [0.66 – 0.96]

Figure 3:

Top row: Bland-Altman plots comparing RHC, 0D simulations, and 3D simulations. Solid line represents the mean bias and the dashed lines are ± 2 standard deviations. Bottom row: 1–1 agreement plots with the solid line representing perfect 1–1 agreement. White marker color represents 5-weeks age, grey 10-weeks age, and black 20-weeks age. Circles represent high resolution segmentation and patient specific inflow, squares represent PC-MRA segmentation and patient specific inflow, triangles represent high resolution segmentation and a generic, averaged inflow

The cases of best and worst agreement between 0D and 3D simulations were both 20-week old LPAS swine and are shown in Figure 4 at peak systole. For 0D simulations pressures are shown along the vessel centerlines, linearly interpolated between vascular junctions. For the best-agreement case, the difference in predicted systolic LPAS pressure gradient was 0.23 mmHg. For the worst-agreement case the difference in predicted systolic LPAS pressure gradient was 3.14 mmHg. For both cases, a progressive decrease in pressure from the proximal to distal RPA is seen in the 3D simulation but not the 0D simulation. This could be due to the peak pressure gradient occurring at different slightly different cardiac phases between the 3D and 0D simulations or due to neglecting energy losses from branching in our 0D simulations.

Figure 4:

Comparison of pressure distributions for 0D and 3D simulation methods. Top row is the case with the best 0D-3D agreement (0.23 mmHg difference), Bottom row is the case with the worst 0D-3D agreement (3.14 mmHg difference).

Efficiency of 0D and 3D simulations

The efficiency of the 0D and 3D simulations are reported in Table 3. The 0D simulations took on average 20 minutes to complete while the 3D simulations took on average 2 hours to complete, approximately 6 times slower than the 0D simulation. The 3D simulations were performed across 40 cores on a cluster so the actual computational cost of the 3D simulations was on average 82 hours.

Table 3:

0D and 3D Simulation Efficiency

Efficiency Metric	Time (h:m:s)
0D Simulation Wall / CPU Time	0:20:18 ± 0:20:16
3D Simulation Wall Time	2:03:39 ± 1:02:33
3D Simulation CPU Time	82:26:08 ± 41:41:43

For a representative 20-week old LPAS swine, accelerated 3D simulations with instantaneous AMR and adaptive time stepping was compared with both traditional 3D CFD with a fixed grid (Figure 5) and fixed time step were compared as well as a constant AMR grid and fixed time step similar to previous AMR techniques used in cardiovascular CFD. The constant AMR grid was created by only allowing cells to be refined during the simulation but not allowing them to be coarsened. The cell count for the AMR simulation varied greatly between systole when flow is most challenging to resolve and diastole when there is little flow (Figure 5). The is representative of all cases as on average the maximum cell count during systole was 2.15±0.32 times greater than the mean cell count. The instantaneous AMR behavior resulted in lower wall time per time step than the fixed grid simulation and lower wall time per time step during diastole than the constant AMR simulation (Figure 5). The fixed grid and constant AMR simulations had a constant efficiency while for the instantaneous AMR simulation efficiency varied between systole and diastole. Fixed grid, constant AMR, and instantaneous AMR simulations predicted nearly identical LPAS pressure gradients (11.71, 11.77, and 11.74 mmHg respectively) The fixed grid simulation took 11.4 hours to run, the constant AMR simulation took 6.5 hours to run, and the AMR simulation took 2.8 hours to run. The instantaneous AMR simulation was 4.8 times faster than the fixed grid simulation and 2.3 times faster than the constant AMR simulation.

Figure 5:

A. fixed grid that AMR simulations were compared with (see Figure 2 for AMR grids), B. Cell count variation of fixed grid, constant AMR, and instantaneous AMR grid over 4 heart beats, C. Simulation wall time per time step for fixed grid, constant AMR, and instantaneous AMR simulations. Wall time is plotted with a moving average filter to aid visualization. Increased variability in the final heartbeat corresponds with 3D output files being written more frequently.

Parameter Uncertainty

Results from a parameter uncertainty study with a 5-week case are shown in Figure 6. For 0D simulations, changing the minimum lumen diameter by approximately +/− 1 voxel (0.5mm) changed the LPAS pressure gradient by −3.45 mmHg and 4.77 mmHg respectively. Altering LPVR and RPVR from the median values to 1^st or 3^rd quartile values changed the LPAS pressure gradient by −3.96 mmHg for decrease LPA flow and by 3.87 mmHg for increased LPA flow. Changing CO by +/− 10% of its measured value changed the LPAS pressure gradient by 1.23 and −1.19 mmHg respectively. For 3D simulations, including a 4F (1.33mm OD) catheter in the simulation increased the LPAS pressure gradient by 2.0 mmHg compared to a simulation without the catheter.

Figure 6:

Results from a parameter uncertainty study. MLD indicated changing the minimum lumen diameter by approximately +/− 1 voxel size (0.5mm). PVR indicates changing the left and right PVR from median to 1^st and 3^rd quartile values respectively to increase LPA flow or from median to 3rd^t and 1^st quartile values respectively to decrease LPA flow. CO indicates changing cardiac output by +/− 10%. Catheter indicates the effect of including a 4F catheter in the stenosis on the calculated pressure gradient.

Discussion

This study evaluated the feasibility of both 0D and 3D CFD simulations to accelerate non-invasive predict peak systolic pressure gradients in branch PAS. These predictions were then experimentally validated with invasive catheterization measurements in a swine model of left PAS. 3D CFD simulations with AMR improved efficiency by 5 times compared to fixed grid CFD simulations. 0D simulations further improved efficiency by 6 times compared to the 3D simulations with AMR. Overall, there was good agreement between 0D and 3D simulations but only moderate agreement between simulations and experimental measurements. This was likely due to uncertainty in our geometries from medical imaging and uncertainty in the outlet resistance values.

Both our 0D and 3D simulations underestimated the measured pressure gradient by 1.8–1.9 mmHg on average. This systematic underestimation is likely the result of neglecting the presences of the pressure catheter. For one simulated case, including the pressure catheter increased the estimated pressure gradient by 2.0 mmHg. More problematic though, is that both 0D and 3D simulations only had moderate agreement with catheterization measurements. Unlike for coronary artery stenosis, non-invasive measurements of blood flow through stenotic pulmonary arteries can be performed with either PC-MRI or nuclear perfusion scans. It is possible that improvements to the simulation precision could be made by utilizing a fitting algorithm to that would optimize outlet resistances to match patient-specific pulmonary blood flow distributions³¹. However, the uncertainty from limited medical imaging resolution would remain an obstacle. The resolution of clinical imaging modalities remains a primary limitation in attempts to non-invasively estimate pressure gradients from mathematical models¹². In non-stenotic situations the uncertainty from clinical imaging resolution may be less important.

This study presents the first experimental validation of the 0D fluid dynamics modeling framework proposed by Mirramezani and Shadden¹⁵. They reported average simulation times of approximately 5 minutes compared to our 20 minutes, the primary reason for which is likely due to our implementation in MATLAB compared to their implementation in python. Another reason contributing to our longer simulation times could be that the present study only simulated pulmonary models which have many more branch segments and junctions compared to other vascular territories.

The good agreement between 0D and 3D simulations for predicting LPAS pressure gradients indicates that for simulations where pressures and flow distributions are the primary outcomes, 0D simulation are sufficient. There are many research applications however, where quantifying wall shear stress, blood damage, or transport phenomena is the goal and 3D CFD simulations are still required. For instance, in this study a 0D model could not have been used to quantify the effect of having a catheter placed in the stenosis. In settings where 3D simulations are required, 0D simulations could be useful for parameter tuning and uncertainty quantification. AMR could then be utilized to improve the efficiency of the higher resolution 3D CFD simulations. We believe the AMR method used in this work is an improvement over previous cardiovascular AMR methods^22,25 because the grid can vary over time as flow conditions change during the cardiac cycle¹⁹.

The primary limitations of this study is that both the 0D and 3D simulations neglected the distensibility of the PA walls. Using instantaneous AMR with 3D fluid-structure interaction (FSI) simulations is promising as the majority of degrees of freedom to be solved for are in the fluid domain and when using a strongly coupled solver the fluid domain must be solved for multiple iterations each time step. The 0D model could be improved to include FSI. Mirramezani and Shadden proposed developing a “quasi” 1D model to accomplish this¹⁵, but we propose that a simpler approach would be to add a linear compliance term to the conservation of momentum (Equation 2) and retain the 0D nature of the model. Further limitations of this study include that the surgically created PAS was simpler than the complex lesions often seen in patients, number of swine was relatively small, complete imaging datasets were not obtained for every swine, and this study only included male swine.

In conclusion, this study compared the feasibility of both 0D and 3D CFD simulations to accelerate non-invasive predict peak systolic pressure gradients in branch PAS. These predictions were then experimentally validated with invasive catheterization measurements in a swine model of left PAS. 3D CFD simulations with AMR improved efficiency by 5 times compared to fixed grid CFD simulations. 0D simulations further improved efficiency by 6 times compared to the 3D simulations with AMR. Overall, there was good agreement between 0D and 3D simulations (ICC 0.88 [0.66 – 0.96]) but only moderate agreement between simulations and experimental measurements (0D ICC 0.60 [0.11 – 0.86] and 3D ICC 0.66 [0.21 – 0.88]). Uncertainty assessment indicates that this was likely due to limited medical imaging resolution and uncertainty in the segmented stenosis diameter.

Figure 7:

Velocity contours, streamlines and grids at peak systole are shown on plane through the MPA, LPA, and RPA for simulations of the same case without a catheter (top row) and with a 4F catheter through the stenosis (bottom row). Including the catheter forces flow in the stenosis to squeeze through small gaps around the catheter. This had the effect of increasing the estimated LPAS pressure gradient by 2.02 mmHg.