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

This study enhances the clinical credibility of coronary artery computational fluid dynamics models by integrating uncertainty quantification via polynomial chaos expansion, revealing that velocity and viscosity are the dominant factors influencing wall shear stress variability in analytical and patient-specific scenarios, respectively.

The Gist

Imagine you are a doctor trying to predict how blood flows through a patient's heart arteries. To do this, you use a super-smart computer program (a "digital twin") that simulates the flow. Usually, these programs act like a strict recipe: they take exact numbers for blood speed, thickness, and pressure, run the simulation once, and give you a single answer.

The Problem: The "Perfect World" Trap
The authors of this paper argue that this "perfect world" approach is risky. In reality, nothing is exact. Blood isn't always the same thickness; it might be slightly thicker in one moment and thinner in the next. Blood pressure fluctuates. If your computer model ignores these tiny, natural wiggles and variations, the answer it gives you might look precise but could actually be wrong. It's like trying to predict the weather by only looking at the temperature at exactly 12:00 PM, ignoring that it might rain at 12:05 PM.

The Solution: The "Weather Forecast" Approach
Instead of asking, "What happens if the blood is exactly this thick?", the researchers asked, "What happens if the blood is anywhere between this thick and that thick?"

They built a new system that treats the inputs (like blood speed and thickness) not as fixed numbers, but as a range of possibilities, similar to how a weather forecast gives you a "70% chance of rain" rather than a guarantee. They used a mathematical trick called Polynomial Chaos Expansion. Think of this as building a "smart shortcut" or a digital emulator.

• The Analogy: Imagine you want to know how a car handles on a bumpy road.
  • Old Way: You drive the car on the road 1,000 times, changing the tire pressure slightly each time, and record the results. This takes forever and costs a lot of gas.
  • New Way (This Paper): You drive the car 30 times with different tire pressures. Then, you build a "smart map" (the emulator) based on those 30 drives. This map can instantly predict how the car would handle any tire pressure within that range without you ever having to drive it again.

What They Did
They tested this "smart map" in two ways:

1. The Simple Test: They simulated blood flowing through a perfect, straight, rigid tube (like a garden hose). This is a known math problem, so they could check if their "smart map" was accurate.
2. The Real Test: They used a real patient's heart artery geometry (scanned from medical images) and ran the simulation on a supercomputer.

The Big Discoveries
By using their "smart map," they found out which factors actually matter most when predicting Wall Shear Stress (WSS). WSS is a fancy term for the "friction" or "rubbing" force the blood exerts on the artery walls. High or low friction can be a sign of heart disease.

• In the Simple Tube: The biggest factor causing changes in friction was blood speed. If the speed varied, the friction changed the most.
• In the Real Patient's Artery: The biggest factor was blood thickness (viscosity). Even though speed mattered, the natural variations in how thick the blood was had the biggest impact on the friction results.

They also found that these factors mostly acted alone. It wasn't usually a complex dance where speed and thickness and pressure all changed together to cause a problem. Instead, one factor usually dominated the outcome.

Why This Matters
The paper concludes that by adding this "uncertainty" layer to the computer models, doctors can trust the results more. It stops the models from pretending to be 100% certain when they aren't.

However, the authors are careful to note that this study was a proof-of-concept. They made some simplifications to keep the math manageable:

• They assumed the blood flow was steady (like a river flowing at a constant speed), not pulsing like a heartbeat.
• They assumed the artery walls were rigid (like a hard pipe), not flexible (like a real, squishy artery).
• They treated blood as a simple fluid, ignoring that real blood can get thicker or thinner depending on how fast it flows.

The Bottom Line
This paper doesn't claim to have a new drug or a new surgery. Instead, it built a better calculator. It showed that if you want to use computer models to help diagnose heart disease, you need to account for the fact that real-life numbers wiggle. By using their "smart map" method, they can tell doctors, "Based on the natural variations in your patient's data, the friction on the artery wall is likely this range, not just this single number." This helps make the computer models more honest and reliable for future medical decisions.

Technical Summary

1. Problem Statement

Computational Fluid Dynamics (CFD) models are increasingly used in clinical settings for diagnosing and treating cardiovascular diseases, particularly coronary artery disease (CAD). However, current clinical adoption is hindered because most models rely on deterministic approaches. These approaches treat input parameters (e.g., blood viscosity, velocity, pressure) as fixed values, ignoring inherent aleatoric uncertainty (natural variability) and epistemic uncertainty (lack of data/assumptions).

• The Gap: A recent review indicated that fewer than 20% of in silico clinical trials account for input parameter uncertainty.
• The Consequence: Without quantifying how input variability propagates to output metrics (like Wall Shear Stress, WSS), model predictions lack credibility, reliability, and safety guarantees necessary for clinical decision-making.
• The Challenge: Traditional methods for Uncertainty Quantification (UQ), such as Monte Carlo (MC) simulations, are computationally prohibitive for complex, patient-specific 3D CFD models, making them impractical for clinical workflows.

2. Methodology

The authors developed a computationally efficient UQ framework coupling the open-source software UncertainSCI with the FEBio finite element solver. The study utilized Polynomial Chaos Expansion (PCE) to create surrogate models (emulators) that approximate the relationship between inputs and outputs without solving the full CFD equations for every sample.

Key Methodological Steps:

• Input Parameterization: Probability distributions were assigned to hemodynamic inputs based on population data and patient-specific measurements:
  • Analytical Model (Poiseuille Flow): Viscosity ($\mu$), Velocity ($v$), and Radius ($r$).
  • Patient-Specific Model: Viscosity, Velocity, Density ($\rho$), and Pressure ($P$).
  • Distributions included Normal (for density, patient velocity/pressure) and Gamma (for viscosity, Poiseuille velocity/radius) to ensure physical positivity.
• Sampling Strategy: Instead of random sampling, the study used Weighted Approximate Fekete Points (WAFP) to efficiently sample the multi-dimensional parameter space.
• Model Execution:
  1. Analytical Solution: Solved the Navier-Stokes equations for steady flow in a rigid cylinder to validate the framework.
  2. Patient-Specific CFD: Reconstructed a Left Main Coronary Artery geometry from biplane angiography and VH-IVUS. Simulations were run under steady-state conditions with plug velocity inlet and resistance outlet boundary conditions.
• Surrogate Construction: PCE emulators were built using polynomial orders up to 5. The emulator maps input parameters to the output Quantity of Interest (QoI): Wall Shear Stress (WSS).
• Convergence & Analysis:
  • Convergence Criteria: Relative error ($\varepsilon_\delta$) < 0.1% and Sobol index stability (<0.01 change) between orders.
  • Sensitivity Analysis: Sobol indices were computed directly from the emulator coefficients to quantify the contribution of individual parameters (first-order) and their interactions (higher-order) to the output variance.

3. Key Contributions

• Efficient UQ Framework: Demonstrated that PCE-based UQ coupled with WAFP sampling can achieve high accuracy with significantly fewer simulations than Monte Carlo methods, making UQ tractable for patient-specific CFD.
• Dual-Model Validation: Validated the framework first on an analytical solution (Poiseuille flow) and then applied it to a complex, patient-specific coronary artery model.
• Quantification of Interaction Effects: Provided a detailed breakdown of how much of the WSS variability is caused by single parameters versus interactions between parameters (e.g., velocity-viscosity coupling).
• Clinical Insight: Highlighted the critical need for patient-specific viscosity measurements, as literature-based viscosity values were a primary source of uncertainty in the patient-specific model.

4. Key Results

A. Analytical Model (Poiseuille Flow):

• Convergence: The PCE emulator converged at Order 3 with minimal computational cost (~21 seconds for 30 parameter sets).
• Dominant Factor: Velocity was the primary driver of WSS variability, contributing ~79% (Sobol index ~0.81) of the total variance.
• Interactions: Unary interactions (single parameters) dominated, accounting for ~93.2% of the variance. Binary (pairwise) and tertiary interactions were negligible (<7%).

B. Patient-Specific Model:

• Convergence: Order 3 was sufficient for convergence, though computationally intensive (~240 hours for 45 parameter sets on a cluster).
• Dominant Factors: Unlike the analytical model, Viscosity became the dominant driver (~59%), followed by Velocity (~40%).
  • Note: Viscosity was derived from population data (high uncertainty), while velocity was patient-specific (low uncertainty), yet both drove variability.
• Spatial Variability: Sobol indices varied spatially. For instance, viscosity sensitivity was high in the inner walls of branching vessels, while velocity sensitivity was lower in those same regions.
• Interactions: Unary interactions accounted for ~99% of the variance. Binary interactions (specifically between velocity and viscosity) contributed ~0.46% of total variance.

C. Output Variability:

• In the patient-specific model, the median WSS varied significantly across the ensemble:
  • Highest mean WSS model: 1.33 Pa.
  • Average model: 0.91 Pa.
  • Lowest mean WSS model: 0.47 Pa.
• Despite magnitude differences, the spatial distribution patterns of high and low WSS remained consistent across models.

5. Significance and Implications

• Clinical Credibility: This study proves that ignoring input uncertainty leads to potentially misleading deterministic predictions. By quantifying uncertainty, clinicians can assess the "confidence interval" of a model's prediction, which is vital for risk stratification and therapeutic planning.
• Data Acquisition Priorities: The results suggest that blood viscosity is a critical, yet often overlooked, source of uncertainty in patient-specific modeling. The authors advocate for the development of point-of-care devices to measure patient-specific viscosity to reduce epistemic uncertainty.
• Scalability: The framework demonstrates that efficient sampling (WAFP) and surrogate modeling (PCE) can overcome the computational barriers of Monte Carlo methods, paving the way for routine UQ in clinical CFD.
• Future Directions: The study identifies limitations (steady flow, rigid walls, Newtonian fluid) and sets a foundation for future work involving unsteady flows, fluid-structure interaction (FSI), and non-Newtonian blood models.

In conclusion, this paper provides a rigorous mathematical and computational pathway to transition CFD from a deterministic research tool to a robust, uncertainty-aware clinical decision support system for coronary artery disease.