Integrating Uncertainty Quantification into Computational Fluid Dynamics Models of Coronary Arteries Under Steady Flow

Abstract

Computational fluid dynamics simulations are increasingly being integrated into clinical medicine, where they have the potential to support clinicians in disease diagnosis, prognosis, and treatment. However, these models frequently use deterministic approaches, neglecting inherent variability (or uncertainty) in input parameters, thereby undermining model credibility and limiting clinical adoption. Herein, we integrate modern and certifiable uncertainty quantification (UQ) techniques to characterize and quantify the variability in coronary artery wall shear stress (WSS) under steady flow conditions due to intrinsic uncertainty in model-dependent quantities. Univariate probability distributions were fit to hemodynamic parameters (density, pressure, radius, velocity, viscosity), and sampled parameter ensembles were applied to an analytical solution (Poiseuille flow) and a patient-specific coronary artery model. Results from the analytical solution demonstrated that variability in input parameters propagated to uncertainty in WSS values, with uncertainty in velocity accounting for the majority (~81%) of WSS variability. In the patient-specific model, spatial medians in WSS varied by ~50% due to input parameter uncertainties, with viscosity (~59%) and velocity (~40%) emerging as the dominant contributors to WSS variability. Across each use case unary interactions dominated (i.e., first-order Sobol indices accounted for the majority of the variance), contributing to ~93% and ~99% of the total WSS variance in the analytical and patient-specific model, respectively. Collectively, this study establishes an uncertainty-aware framework to strengthen computational biomechanics model credibility, aligning with emerging regulatory guidance and enabling more trustworthy modeling-based decision support in the management of coronary artery disease.

1. INTRODUCTION

There is a growing integration of computational fluid dynamics (CFD) modeling techniques into the clinical setting. Modeling results augment anatomic, physiologic, and functional data to advance patient management and inform diagnosis, prognosis, and therapeutic intervention [1]. For example, patient-specific CFD models in cardiovascular medicine have been applied in the setting of congenital heart disease [2], Kawasaki disease [3], pulmonary hypertension [4], abdominal aortic and cerebral aneurysms [5], and coronary atherosclerosis [6,7]. Unfortunately, most modeling simulations do not account for the intrinsic uncertainties that arise from variability in input parameters (i.e., aleatoric uncertainty) and assumptions made due to limited data (i.e., epistemic uncertainty). In fact, a recent review of in silico clinical trials in cardiology evaluating the effectiveness and feasibility of model-directed interventions reported that <20% of studies accounted for input parameter uncertainty [8]. Accordingly, establishing computational frameworks that are robust to model uncertainty is essential to ensure that model-based predictions are accurate, reliable, and safe before their clinical adoption.

Computational uncertainty quantification (UQ) provides an advanced mathematical framework to characterize the variability in simulations due to intrinsic uncertainty in model-dependent quantities of interest [9]. In contrast to deterministic approaches, whereby model input quantities are fixed, and a single simulation provides model results, UQ in the forward model setting formalizes a statistical model for input uncertainty and uses computational algorithms to deliver mathematical and statistical guarantees of output accuracy. Studies have successfully integrated UQ into blood flow simulations, analyzing generalized and patient-specific geometries and quantifying sensitivities in flow, pressure, and wall shear stress (WSS) [10-14]. Importantly, however, nearly all prior uncertainty-aware hemodynamic simulations have relied on Monte Carlo (MC) methods for parameter sampling, which can be computationally demanding. As uncertainty-aware CFD frameworks advance toward clinical adoption, the associated computational costs of MC approaches remain unmanageable and impractical. Thus, integrating efficient sampling strategies is critical to achieving computationally tractable and clinically applicable uncertainty-aware CFD models.

In this study, we leveraged our established UQ-FE framework, which couples the open-source UQ software UncertainSCI and the FEBio software suite [15] to examine how uncertainty in variables within the Navier-Stokes equations propagates to variability in WSS. The goal of the study was to utilize polynomial chaos expansion (PCE)-based UQ methods to quantify WSS variability derived from analytical or FE approaches under steady flow conditions. Recognizing the clinical interest in understanding the prognostic value of WSS in the natural history of coronary artery disease and plaque vulnerability [16,17], the focus was directed at blood flow in the epicardial coronary arteries.

2. METHODS

The computational framework developed to analyze the influence of uncertainty in input hemodynamic parameters is summarized in Figure 1. Briefly, probability distributions were fit to selected hemodynamic m-input parameters (e.g., velocity, viscosity), and samples were drawn from these distributions to generate n-parameter sets (or ensembles), representing independent combinations of the input parameter space. Parameter sets were integrated into an analytical solution for WSS under steady flow of an incompressible, Newtonian fluid in a rigid cylinder (i.e., Poiseuille flow) or a batch-processing scheme to predict WSS in a patient-specific coronary artery CFD model. Finally, statistical and global sensitivity analyses were carried out to quantify the impact of parameter variability on WSS uncertainty.

Fig. 1.

Schematic representation of the implemented computational framework to quantify the impact of hemodynamic parameter variability on uncertainty in WSS values derived from an analytical solution and patient-specific FE model.

This study employed the open-source, Python-based software suite UncertainSCI² [18,19]. UncertainSCI leverages modern PCE techniques to construct emulators, enabling the prediction of model outputs for general parameter sets and probing model uncertainty. The modularity of the software enables broad application across diverse domains (e.g., biomechanics [15], electrocardiography [20], neuromodulation [18]), and it benefits from mathematical and statistical capabilities and guarantees that are not present in other UQ toolkits. The software is non-intrusive, enabling integration with external modeling software without modifying the underlying solver implementation.

2.1. Hemodynamic Parameter Distributions

The distributions for input parameters were derived from population-based or patient-specific data, as reported in Table 1. Patient-specific distributions were determined by evaluating invasive measurements, as discussed in Section 2.3, across 6 cardiac cycles. Normal distributions were assumed for density as well as patient-specific velocity and pressure, while gamma distributions were used for viscosity as well as Poiseuille flow velocity and radius. These assumptions were based on available empirical data, the absence of evidence that alternative distributions provided a better fit, and the need to ensure physically meaningful (i.e., positive) values. Distribution parameters (e.g., mean, standard deviation) were specified within UncertainSCI, and parameter sets were efficiently sampled across the multi-dimensional parameter space via a weighted approximate Fekete points (WAFP) procedure ([21,22]; Fig. 1). Briefly, UncertainSCI uses a two-step procedure to define the parameter sets. First, a random sampling approach is used to identify a candidate set of points in the parameter space, analogous to MC sampling. Second, a subset of these points is selected using a weighted D-optimal design (i.e., WAFP) that approximately maximizes the stability of the polynomial design matrix, yielding a well-conditioned set of sampling locations for estimating PCE coefficients. As this procedure seeks near-optimal sampling sets, it yields accurate estimates of output statistics and sensitivities with substantially fewer model evaluations than, for example, MC approaches, as demonstrated in the Appendix (Fig. App. A1) and in prior studies [23].

Table 1.

Fluid parameter values for analytical and patient-specific model. Data are reported as the mean ± standard deviation. References are provided for population-derived values.

Parameter	Poiseuille Flow	Patient-specific
Viscosity ($\mu$) [mPa·s]	3.48 ± 0.51 (± 14.7%) [47]
Velocity ($\mathit{v}$) [m/s]	0.27 ± 0.16 (± 59.2%) [48]	0.145 ± 0.01 (± 6.9%)
Radius ($\mathit{r}$) [mm]	1.75 ± 0.40 (± 22.9%) [49]	—
Density ($\rho$) [kg/m³]	—	1051.50 ± 10 (± 1.0%) [50]
Pressure ($\mathit{P}$) [mmHg]	—	97.54 ± 5.42 (± 5.6%)

2.2. Approaches to Quantify Coronary Artery Wall Shear Stress

2.2.1. Analytical Solution (Poiseuille Flow)

Recognizing the need to verify software coupling, as well as the pre- and post-processing workflows, the initial implementation of the parameter ensembles utilized an analytical solution to the Navier-Stokes equations. Assuming a steady flow of an incompressible, Newtonian fluid in a rigid cylinder (i.e., Poiseuille flow) [24], WSS is calculated as,

$${\tau }_{wall}=\frac{4\mu Q}{\pi r^3}or\frac{4\mu v}{r},$$ (1)

where $\mu$ is the fluid dynamic viscosity, $Q$ is the flow rate, $v$ the velocity, and $r$ is the cylinder radius. WSS values were determined across parameter sets via a custom Python script and stored for UQ analysis.

2.2.2. Patient-specific Coronary Artery CFD Model

A representative patient-specific coronary artery geometry, including the left main and left anterior descending coronary arteries, was constructed with established approaches using data acquired in a prospective study [7,16]. Enrollment criteria, patient consent, and clinical data acquisition have been described in detail [16]. In brief, biplane angiography and virtual histology intravascular ultrasound (VH-IVUS) images were acquired, as well as Doppler flow and pressure measurements in the interrogated vessel (Fig. 2a). The 3D lumen geometry was constructed by stacking the VH-IVUS images perpendicular to the catheter centerline and wrapping a surface to the lumen pointcloud (Fig. 2b). The geometry was meshed in SimVascular [25] (global edge size = 0.5 mm, 4 boundary layers, decreasing ratio = 0.8), resulting in a mesh size of ~2 million linear terahedral elements (Fig. 2c). A mesh convergence study showed that increasing mesh densities led to a relative median difference in WSS of 1.2% at increasing mesh densities (Supp. Fig. S1). The volume mesh was imported into FEBio Studio. Patient-specific invasive hemodynamic data were applied as boundary conditions. A steady-state plug velocity profile, derived from the temporal average of velocity data across a cardiac cycle, was applied at the inlet surface and resistance boundary conditions at all outlet surfaces with values derived from area scaling of the total resistance ($R_{tot}$) of the system, which was determined as,

$$R_{tot}=\frac{P}{Q},$$ (2)

where $P$ and $Q$ are the mean pressure (temporal average of cyclic pressure data) and volumetric flow rate, respectively. A no-slip boundary condition was applied at all wall surfaces. The FE simulations were executed using the CFD solver in FEBio [26]. Broyden’s quasi-Newton method, in combination with the mkl_dss linear solver, was employed to accelerate convergence, and the simulation was executed for 1000 time steps with a step size of 0.001 s. Following solution convergence, data were post-processed to determine WSS across the spatial domain (Fig. 1d). Patient-specific CFD simulations were executed on the GRACE and FASTER clusters ain the High Performance Research and Computing at Texas A&M University using a SLURM job batch array, with at least 8 models running in parallel, each allocated 8 CPUs (2.2 – 4.0 GHz) and 64GBs of memory.

Fig. 2.

Illustration of patient-specific computational fluid dynamics framework, including data acquisition, geometry construction, meshing and boundary conditions, and post-processing.

2.3. Uncertainty Quantification Analysis Framework

Extending our previously developed framework [15], we utilized UncertainSCI to examine uncertainty in calculated or FE-predicted WSS values. Following the evaluation of the parameter sample sets and quantification of WSS using either Eq. 1 or a CFD simulation, an emulator was constructed. The emulator is a sum of orthogonal, polynomial functions with a maximal order p, which is user-specified and results in the generation of n-parameter sets via the computationally efficient WAFP procedure (Fig. 1) [21,27]. The PCE emulator serves as a surrogate model, approximating the mapping between input parameter(s) and model output(s), and enabling the model output to be queried without solving the computationally expensive forward problem. Parameter sets served as inputs to either the analytical solution or an FE batch processing scheme that iterated through ensembles using the same FEBio input file (Fig. 1). Calculated or predicted WSS values were retrieved, permitting emulator construction.

To assess the quality of the PCE (i.e., convergence), the relative error (${\epsilon }_{\delta }$) was quantified between the emulator approximations and calculated/predicted WSS values, whereby

$${\epsilon }_{\delta }=\frac{1}{N}\sum _{i=1}^N\frac{{\Vert A\widehat{x}-\overrightarrow{b}\Vert }_2}{{\Vert \overrightarrow{b}\Vert }_2},$$ (3)

where $A\widehat{x}$ is the approximation, $\overrightarrow{b}$ is the analytical/FE model output, $N$ is the number of elements or takes the value of 1 for the analytical solution, and ${\Vert \cdot \Vert }_2$ indicates the 2-norm of the vector. For orders $p=\{1,2,\dots 5\}$, the relative error was calculated across 5 independent PCE runs to account for random parameter sampling (i.e., 5 PCE runs were performed for each order). For the patient-specific model, WSS values for each mesh element were utilized to evaluate the relative error, and these values were spatially averaged across runs at a given order. In addition, parameter sets were oversampled (2×) in the Poiseuille flow solution to confirm error stability across sampling rates, ensuring that aliasing error was minimized (see Appendix, Fig. App. A2).

2.4. Statistical and Global Sensitivity Analyses

Statistical measures (e.g., mean, standard deviation) were determined directly from the constructed PCE emulator. Also, Sobol’s indices [28], which quantify the relative contribution of individual parameters or parameter sets to variance in the output (here, WSS), were computed directly from the polynomial emulator coefficients, as described in [18]. The convergence criteria for the PCE emulators were defined as a relative error <0.01% and first-order Sobol index value changes <0.02 (i.e., index stability) between increasing polynomial orders. Data are reported as the mean ± standard deviation for normally distributed variables and as median (Q1, Q3) for non-normally distributed variables.

3. RESULTS

The number of parameter sets generated across the 5 orders and run times for the analytical solution are shown in Table 2. Parameter sets ranged from 14 to 66 across orders, with a total run time of ~47 seconds for 5 runs at order 5 on a multi-core personal computer (Windows 11 server machine; Intel^® Xeon^® w5-3433 CPU, 16 cores at 2.0-4.2 GHz). The relative error for WSS between the PCE emulator and analytical solution demonstrated a reduction in magnitude and variability across runs at increasing order (Fig. 3a). For example, error was 5.38 ± 2.71%, 0.01 ± 0.004%, and 3.91e-5 ± 2.1e-5% for orders 1, 3, and 5, respectively. Moreover, first-order Sobol indices stabilized at higher orders (Fig. 3b), with minimal changes in index values (<0.02) and small standard deviations (~0.003) across orders 3 through 5. Similar trends in relative error and Sobol index convergence were observed at a 2× sampling rate (see Appendix, Fig. App. A2). Although order 2 with 2× sampling achieved lower error and Sobol index stabilization (i.e., the PCE emulator converged), it required more samples than order 3 without oversampling, which also met the convergence criteria. Thus, order 3 without oversampling captured the dominant uncertainty modes while remaining computationally efficient.

Table 2.

Parameter sets and compute time for the Poiseuille flow UQ analysis. Run times are reported for five PCE runs in total at each order, and include parameter set generation, WSS calculation, PCE emulator construction, and statistical and global sensitivity analysis.

Order	Parameter Sets	Run Time (s)
1	14	13.28 ± 1.31
2	20	16.09 ± 1.7
3	30	20.99 ± 1.67
4	46	37.56 ± 3.39
5	66	46.79 ± 5.48

Fig. 3.

PCE emulator and sensitivity indices convergence analysis for the analytical solution – (a) relative error and (b) first-order Sobol indices for each PCE order across 5 runs. (c) Kernel density estimate (KDE) of WSS values for the order 3 emulator. (d) Percent output variance due to unary (single parameter), binary (pairwise), and tertiary (triplewise) interactions for order 3 PCE analysis.

Input parameter uncertainties propagated to yield variability in Poiseuille flow-derived WSS values, as evident by the considerable variation in WSS values across the constructed order 3 emulator (Fig. 3c). Across the 30 parameter ensembles at order 3, median, minimum, and maximum WSS values were 2.56 (1.45, 5.77), 0.26 ($v=0.051\frac{m}{s}$, $\mu =0.0037Pa.s$, $r=0.002m$), and 15.94 Pa ($v=2.09\frac{m}{s}$, $\mu =0.004Pa.s$, $r=0.002m$), respectively. Sensitivity analysis indicated that uncertainty in velocity dominated the variance in WSS. First-order Sobol indices at order 3 were 0.81, 0.13, and 0.05 for velocity, radius, and viscosity, respectively (Fig. 3b). Unary interactions accounted for >93% of the variance in WSS, with binary (i.e., interactions between two parameters) and tertiary interactions accounting for approximately 6.4% and 0.09% of the variance, respectively (Fig. 3d). Across the 3 possible pairwise combinations, the interaction between velocity and radius accounted for 67.4% of variance due to binary interactions; thus, accounting for approximately 4.3% of the total variance (67.4% of 6.4%).

Parameter sets for the patient-specific CFD model ranged from 15 (order 1) to 136 (order 5) and required ~4 to 30 days to run the simulations, respectively (Table 3). As observed in the analytical solution results, the emulator exhibited a reduction in relative error with increasing polynomial order that can be appreciated spatially and quantitatively (Fig. 4a). Error reduced at each element across the computational domain, with average values of 4.27% ± 1.1 %, 0.02 ± 0.002%, and 8e-4 ± 2e-4% at orders 1, 3, and 5, respectively. Furthermore, Sobol indices stabilized at higher orders, again observed spatially and quantitatively (Fig. 4b). Average first-order Sobol indices changed by <0.0008 at order 3 and higher with standard deviations of ~1e-4 across the 5 PCE runs (i.e., order 3 met the defined convergence criteria).

Table 3.

Patient-specific model parameter sets and compute time across PCE orders. Five PCE runs were performed across each order.

Order	Parameter Sets	CFD Simulation Run Time (hh:mm:ss)
1	15	90:41:43 ± 2:54:30
2	25	108:24:24 ± 28:21:39
3	45	240:12:31 ± 6:53:56
4	80	399:18:34 ± 64:26:25
5	136	627:58:38 ± 5:55:00

Fig. 4.

PCE emulator and sensitivity indices convergence analysis in the patient-specific CFD model – (a) relative error and (b) first-order Sobol indices for each PCE order across 5 runs. Element-wise data are shown in the fringe plots and summarized in the graphs. For conciseness, only first-order Sobol indices across the computational domain are presented for viscosity, but the others also demonstrated stability as illustrated in the bar graphs.

Variations in WSS values were observed across the 45 patient-specific CFD models at order 3. Representative models with the highest, average, and lowest mean WSS values are shown in Fig. 5a. Although WSS magnitudes differed at the same spatial locations across the models, the relative spatial distributions within each model were consistent (i.e., areas of elevated and reduced WSS were preserved). The highest, average, and lowest models exhibited median WSS values of 1.33 Pa (0.93, 2.18), 0.91 Pa (0.63, 1.52), and 0.47 Pa (0.31, 0.79), respectively (Fig. 5b). The observed consistency in the relative WSS distributions in space is complemented by the shape of the violin plots. As outlet flow divisions are governed by the pressure (i.e., resistance, Eq. 2), variability in volumetric flow rate was quantified at each branching vessel (Fig. 5c). Flow rates in the left circumflex (LCx), first diagonal (D1), first septal (S1), second septal (S2), second diagonal (D2), and distal left anterior descending (LAD) arteries were 69.19 mL/min (63.84, 74.98), 54.47 mL/min (50.24, 59.01), 18.02 mL/min (16.61, 19.52), 5.42 mL/min (4.99, 5.86), 4.65 mL/min (4.28,5.02), and 34.75 mL/min (32.00,37.60), respectively.

Fig. 5.

Representative WSS (a) fringe plots and (b) violin plots for models exhibiting the highest, average, and lowest mean values at order 3. Violin plots display data between the 10^th and 90^th percentiles for visual clarity. (c) Volumetric flow rate uncertainty for the coronary branches, identified in 5a, due to variability in outlet pressure (i.e., resistance, Eq. 2).

At the converged PCE order of 3 for the CFD models, sensitivity analysis revealed that variability in viscosity and velocity had the greatest impact on WSS variability. Unlike the analytical solution, first-order Sobol indices varied spatially (Fig. 6). The Sobol indices for pressure and density were relatively homogeneous throughout the model; however, those for viscosity and velocity were higher in contrary regions across the model. For example, the Sobol indices for viscosity were high in the inner wall of a branching vessel, whereas velocity values were low (Fig. 6).

Fig. 6.

Spatial variation in normalized first-order Sobol indices across the patient-specific models at the converged order 3.

Spatially averaged first-order Sobol indices were approximately 0.59, <1e-3, 0.40, and <1e-4 for viscosity, density, velocity, and pressure, respectively (Fig. 7a). Unary interactions were dominant (~99.5%), however, higher-order interactions were present (Fig. 7b). Interactions between two (binary) and three (tertiary) parameters accounted for approximately 0.51% and 0.01%, respectively, of variance in WSS. For the binary interactions, the interaction between velocity and viscosity accounted for ~90% of that variance in WSS (i.e., the interaction between velocity and viscosity accounted for ~0.46% of the total variance in WSS). The interaction between viscosity, velocity, and pressure accounted for ~65% of the variance in WSS due to tertiary interaction.

Fig. 7.

Sensitivity analysis at order 3 PCE analysis – (a) normalized first-order Sobol indices for viscosity ($\mu$), density ($\rho$), velocity ($v$), and pressure ($P$), and (b) percent output variance due to unary (single parameter), binary (pairwise combinations), and tertiary (triplewise combinations) interactions.

4. DISCUSSION

In this study, we extended our previously developed UQ-FE framework, which coupled UncertainSCI with the FEBio software suite, and quantified how uncertainty in hemodynamic parameters influences variability in WSS under steady flow conditions derived from an analytical solution and a patient-specific CFD model. The accuracy of the PCE surrogate models was confirmed through convergence testing, demonstrating that a third-order polynomial expansion (i.e., PCE order 3) without parameter set oversampling was sufficiently accurate and provided reliable statistical estimates and stable sensitivity indices. Examining flow in a non-distending cylinder (i.e., Poiseuille flow), we demonstrated that WSS is most sensitive to uncertainties in viscosity and radius, both derived from population data, whereas in a patient-specific left anterior descending coronary artery, values were most sensitive to viscosity (population-based data) and velocity (patient-specific data). Furthermore, the sensitivity analysis highlighted that unary interactions (first-order) were dominant, accounting for >93% of variance in WSS values, indicating limited interaction between input parameters.

UQ has been broadly applied in nearly all biomechanics domains, with applications, for example, in evaluating surgical implants [29], joint kinematics [30], soft tissue mechanics in the brain [31], and myocytes transduction [32]. For blood flow simulations, studies have successfully integrated UQ to quantify the confidence in predicted hemodynamic measures across the vascular tree. Early studies examined parameterized geometries (abdominal aorta aneurysm, carotid bifurcation, stenosed coronary artery) and utilized stochastic collocation sampling methods, enabling efficient exploration of variability in geometry, boundary conditions, and material properties to quantify uncertainties in WSS and fractional flow reserve (FFR) [14,33]. Results clearly demonstrated the importance of incorporating uncertainties into hemodynamic simulations (e.g., FFR was most sensitive to minimum lumen diameter [14]), motivating continued investigation and highlighting the need for mathematically rigorous analyses when assessing the impact of model input uncertainty. Investigations expanded on these initial findings and integrated UQ strategies in patient-specific frameworks, including studies on the coronary circulation [34,35], coronary artery bypass grafts [36], and abdominal aorta [37,38]. Even when evaluating the same vascular territory, it is not straightforward to compare results across studies, largely due to how data are sampled and uncertainty is characterized. However, recognizing that numerous sources of uncertainty, coupled with the high computational costs of 3D patient-specific modeling, necessitate efficient approaches to achieve tractable results. Integrating efficient sampling methods [23], surrogate models, and dimensionality-reduction techniques [10] all hold promise for reducing computational burden while preserving the fidelity of UQ in hemodynamic simulations. For example, herein UncertainSCI utilized a WAFP sampling approach and PCE techniques, both of which are computationally efficient for characterizing variability in simulations due to uncertainty in model-dependent quantities. Collectively, these studies and others underscore the growing importance of UQ in hemodynamic simulations and highlight the need for continued development to fully realize its potential in promoting confidence in patient-specific models in the clinical setting.

Input parameter values and associated distributions were obtained from the literature and supplemented with patient-specific data for the CFD model where available (Table 1). Although the analytical solution parameter specifications utilized population-based data derived from the literature, the results provided general insights into the coupled UQ-FE framework for coronary hemodynamics and WSS under uncertainty. This example was included primarily to demonstrate the UQ framework using a simple analytical solution and was not intended to represent physiological variability in coronary WSS. In contrast, the patient-specific analysis provided insight into how uncertainty propagates through the clinically relevant model. Findings from this analysis highlighted the need for additional data collection to advance model confidence. For example, uncertainty in viscosity (literature-derived) and velocity (measured) accounted for ~99% of the variability in WSS in the patient-specific model (combined unary and their binary interactions). Unlike routine anticoagulation monitoring in the catheterization lab (e.g., activated clotting time), blood viscosity is rarely quantified. Importantly, however, viscosity values vary with factors such as hematocrit, sex, disease severity, and even altitude [39]. The presented data indicate the importance of quantifying patient-specific viscosities, which, with point-of-care devices [40], could perhaps reduce uncertainty in predicted hemodynamic metrics. Unlike viscosity, velocity data were derived from invasive measurements; however, data were collected in the left main coronary artery across only 6 cardiac cycles. Although velocity variability was low, additional data could reduce measurement variability and uncertainty in model predictions, thereby allowing parameter distributions to be more tightly constrained to physiologically plausible ranges. Collectively, more comprehensive personalized data acquisition and reporting would promote model credibility, and the presented framework is sufficiently modular to incorporate future advances in medical technologies aimed at quantifying hemodynamics metrics with greater fidelity.

Interpreting sensitivity indices requires considering factors beyond the magnitude of the input uncertainties. Although one might assume that input parameters with greater variability would yield larger Sobol indices, this is not necessarily the case. Sobol indices quantify how much each uncertain input contributes to the variability of the model output and therefore reflect the magnitude of the input uncertainty, the functional dependence of the model on each parameter, and parameter interactions. While this effect was less apparent in the simplified Poiseuille flow case, where Sobol index magnitudes reflected the relative input variability, the same was not true for the CFD model, where relative sensitivities were shaped by nonlinear governing equations and coupled parameter effects. Consequently, sensitivities cannot be inferred from input variability alone and require explicit UQ analysis to properly characterize.

Clinical studies have demonstrated that hemodynamic forces are markers of plaque development, progression, and vulnerability. In addition to the established association between low and oscillatory WSS and atherosclerosis development [41], longitudinal clinical studies have demonstrated the role of hemodynamics in coronary atherosclerosis progression and heightened vulnerability [16,17]. Unfortunately, UQ-certified computational models have not been linked to clinical data or patient outcomes, providing limited confidence in model predictions. The lack of translation of UQ techniques to guide clinical decision-making motivates efforts to couple uncertainty-aware computational models with clinical observations to establish confidence in model-based predictions. In conjunction with the presented results, which reveal strong sensitivity of WSS values to uncertain inputs, one could reasonably question the credibility of the conclusions drawn in these clinical associations. However, it is important to emphasize that these foundational clinical studies have been central in establishing the associations that motivate continued investigation. Accordingly, incorporating UQ into these analyses is essential to distinguish robust relationships from those that may simply reflect instability in the underlying model inputs. Such rigor is fundamental to establishing trustworthy and safe modeling frameworks necessary for the clinical adoption of patient-specific computational models.

Although this study makes a valuable contribution to integrating UQ into hemodynamic simulations, the limitations must be acknowledged. Importantly, these assumptions were introduced mainly to reduce computational costs, particularly when running hundreds of patient-specific models, and to avoid confounding the interpretation of results in this initial investigation while establishing and verifying the coupled UQ-FE framework. First, velocity was assumed to be steady. This assumption represents a first step toward developing the framework and interpreting results, with future efforts aimed at capturing variability in time-dependent quantities (e.g., unsteady flows) through, for example, principal component analysis. Incorporating pulsatile flow will substantially increase parameter dimensionality and computational demand, requiring additional preprocessing to capture velocity waveform variability, statistically characterize model coefficients, enforce physiologic parameter constraints, and further optimize the FE model to accommodate the expanded computational demands. Second, Poiseuille flow assumes the cylinder is rigid, and the patient-specific model likewise assumes a rigid wall. Although solutions for steady flow of a viscous fluid in a thin-walled elastic cylinder exist and fluid-structure interaction modeling techniques are well established [42,43], integrating material stiffness introduces additional sources of uncertainty that increase study complexity and computational cost. Third, the fluid was assumed to be Newtonian. While constitutive equations characterizing the non-Newtonian viscosity of blood are well developed [44], incorporating shear rate-dependent viscosity introduces additional uncertain parameters whose population-level statistical distributions are not well established, thereby requiring further assumptions. Finally, radius was considered an uncertain parameter in the analytical solution, but the geometry was constant in the patient-specific model. In contrast to material property and boundary condition uncertainty, including geometric uncertainty is non-trivial, particularly due to the need for a non-intrusive, automated model construction and discretization pipeline [45,46].

In conclusion, this study presents a framework for characterizing forward model uncertainty in analytic and CFD simulations of coronary artery hemodynamics. We report that uncertainties in hemodynamic parameters propagate through the governing equations, leading to appreciable variability in WSS values. These findings highlight the need for improved data acquisition to reduce epistemic uncertainties and support future extensions to more complex hemodynamic conditions. Ultimately, this work demonstrates the critical importance of UQ in evaluating the credibility of patient-specific simulations and provides a foundation for developing computational tools that more reliably support clinical decision-making and mechanically-driven mechanistic investigations.

Supplementary Material

Supp Material

Fig. Supp. S1 WSS fringe plots and distributions across mesh densities of ~0.5M, ~1M, ~2M, ~3M, and ~4M elements. Data are reported as median (Q1, Q3).

APPENDIX. Weighted Approximate Fekete Points (WAFP)-Based PCE Versus Monte Carlo (MC) Sampling Approaches

To demonstrate the computational efficiency of the WAFP-based PCE relative to MC sampling, an illustrative example was performed using the Poiseuille flow case. For the WAFP sampling strategy, data from the converged PCE order p = 3, corresponding to 30 sampled parameter sets, were utilized (Fig. 3). The MC sampling procedure was implemented using the Python-based UQpy toolbox. Sample set sizes of $N=\{1,000;10,000;100,000;400,000;1,000,000\}$ were evaluated, each repeated 5 times as in the WAFP procedure, and sensitivity convergence was assessed using the same criterion (first-order Sobol index value changes <0.01; see Section 2.4). Results demonstrated that MC convergence was achieved at a sample set size of $N=400,000$. Comparing summary statistics between the WAFP-based PCE and MC approaches at the converged PCE order and MC sample size revealed nearly identical estimates of the median (WAFP: 1.91 Pa, MC: 1.91 Pa) and first and third quartiles (WAFP: 1.19, 2.95 Pa, MC: 1.18, 2.97 Pa; Fig. App. A1a); however, the WAFP procedure required orders of magnitude fewer model evaluations (WAFP: 30, MC: 400,000). In addition, first-order Sobol indices demonstrated strong agreement between methods (viscosity: WAFP: 0.05, MC: 0.05; velocity: WAFP: 0.81, MC: 0.81; radius: WAFP: 0.13, MC: 0.13; Fig. App. A1b).

Fig. App. A1.

Comparison of (a) WSS distributions and (b) sensitivity for Poiseuille flow obtained using the WAFP-based PCE and MC sampling approaches. PCE converged at order $p=3$ (30 parameter sets) and MC sampling at a parameter sample set size of $N=400,000$.

Parameter Set Oversampling

To ensure stable estimation of the PCE coefficients and assess convergence with respect to polynomial order ($p$), parameter oversampling was evaluated for the Poiseuille solution of WSS. Specifically, 2× oversampling was examined, with the number of parameter sets doubled at each order. That is, parameter sets of 28, 40, 60, 92, and 132 were evaluated at orders 1, 2, 3, 4, and 5, respectively (cf. Table 2). Results confirm similar trends in relative error and Sobol index convergence as compared with the baseline sampling strategy (i.e., no oversampling; Fig. App. A2).

Fig. App. A2.

Evaluation of PCE emulator and sensitivity indices convergence for the analytical solution - (a) relative error and (b) first-order Sobol indices at 2x oversampling.