Uncertainty quantification in coronary blood flow simulations: Impact of geometry, boundary conditions and blood viscosity
Introduction
Sufficient blood flow in the coronary arteries is essential for perfusing the myocardium and ensuring normal cardiac function. Atherosclerosis in the coronary arteries can obstruct blood flow and result in myocardial ischemia, or low myocardial blood flow particularly during physical activity, and may necessitate treatment with medical therapy, angioplasty and stenting or bypass surgery. The most effective test for assessing the functional significance of coronary artery disease is invasive fractional flow reserve (FFR) which is the ratio of mean pressure downstream of a coronary lesion to the pressure in the aorta under conditions of maximal hyperemia induced through the administration of adenosine to dilate the coronary microcirculation and increase coronary blood flow in a manner mimicking physical activity. Importantly, large prospective randomized-control clinical trials have demonstrated that the use of FFR in clinical decision-making can identify patients that should be treated medically (Tonino et al., 2009) and those patients that benefit from revascularization using stents (De Bruyne et al., 2014). While FFR is the gold-standard for identifying lesions causal of ischemia, it is an invasive method requiring diagnostic cardiac catheterization and is negative in roughly half the patients that receive the test (Min et al., 2012, Koo et al., 2011, Norgaard et al., 2014). As a result, there has been significant motivation to develop a noninvasive test that could better identify patients who can be deferred from invasive diagnostic catheterization and those patients that would most likely benefit from this invasive procedure. Recently, a technique called has emerged for noninvasively predicting FFR using coronary computed tomography angiography (coronary CTA) to inform simulation studies of blood flow performed using computational fluid dynamics (Taylor et al., 2013). has demonstrated high diagnostic accuracy as compared to measured FFR (Min et al., 2012, Koo et al., 2011, Norgaard et al., 2014) and has been shown to significantly reduce unnecessary diagnostic cardiac catheterizations without adverse clinical events (Douglas et al., 2015), to improve the quality of life of patients receiving the test and reduce health care costs (Hlatky et al., 2015).
Patient-specific models of blood flow in arteries include a description of the anatomic region of interest created from image data, the mathematical equations representing the physical laws of blood flow within the region of interest and boundary conditions to define physiologic relationships between variables at the boundaries of the region of interest and the remainder of the circulation. Each of these elements can introduce uncertainty in the simulation and are discussed in turn in the following.
For modeling blood flow in the human coronary arteries, coronary CTA data provides input for the patient-specific anatomic model. Image artifacts, which can depend on imaging hardware, image acquisition protocols and reconstruction techniques and inherent patient characteristics can affect the quality of the image data and the segmentation of the coronary arteries. Owing to the reasons above, the reconstructed geometry from cCTA is an approximation of the true geometry (which is unknown), which has to be accounted for when performing blood flow simulations.
For many patient-specific simulations of blood flow, a Newtonian rheological model is used and a single viscosity value is assumed based on population averages. The effect of variations in blood viscosity from the population-based average will depend on the quantity of interest. For example, viscosity will have a direct effect on shear stress, but may or may not affect computed pressure gradients or fractional flow reserve values.
Inlet and outlet boundary conditions have a profound effect on blood flow simulations in patient-specific models. A robust strategy is to prescribe the flow rate or pressure at the inlet and a lumped-parameter relationship between flow rate and pressure at the outlets of the patient-specific domain. In this work, for simplicity, we use a resistance model relating pressure to flow to model properties of the micro-circulation downstream of the large coronary arteries represented in the image-based model (Taylor et al., 2013). Although resistance values in each artery cannot be directly measured non-invasively, parameter values can be estimated based on form-function relationships applied to an individual patient and population-based physiologic responses. However, the true resistance remains unknown and it is necessary to account for this uncertainty in patient-specific models.
The main goal of this work is to understand the impact of uncertainties in lumen geometry (minimum lumen diameter and lesion length), boundary conditions and blood viscosity on the blood flow and pressures in the coronary artery. We investigate the relative importance of each of these model parameters and calculate the impact of these on , in comparison with measurement variability. Finally, we perform a combined uncertainty quantification analysis where all variables are perturbed simultaneously to determine whether the uncertainty in the parameters is additive.
To accomplish this assessment of solution uncertainty, we use data-driven techniques for calculating the stochastic models. To account for uncertainty in geometry, we utilize data comparing minimum dimensions of the lumen segment from coronary CTA against invasive measurements obtained using optical coherence tomography (OCT). Uncertainty in lesion length is modeled based on variability in modeling stenoses by three different users segmenting the same image data. For modeling uncertainty in resistance values, we selected a cohort of patients whose reconstructed geometry matches invasive measurements obtained from intravasular ultrasound (IVUS) data. We then make the assumption that all of the differences between actual and measured FFR occurs due to error in modeling boundary resistance. This uncertainty model for boundary resistance is computed by fitting an empirical distribution to the observed data. Uncertainty in viscosity is modeled by fitting a distribution to viscosity calculated from measured values of hematocrit obtained in a recent clinical trial.
The differences between this work and prior work (Sankaran et al., 2015, Sankaran et al., 2015) are (i) in the previous work, we used a machine learning surrogate for whereas we use the Navier–Stokes equations directly in this paper, (ii) uncertainties in minimum lumen diameter, lesion length, boundary resistance and viscosity are all included in this paper, whereas only geometric uncertainty was discussed earlier, and (iii) the input uncertainty model for all the parameters is computed using a data-driven approach.
The paper is organized as follows. In the methods section, we describe a data-driven approach for stochastic modeling of the various sources of uncertainties. We also describe the stochastic Navier–Stokes equations. In Section 3, we describe the results obtained on an idealized and patient-specific model, including the computed standard deviation, confidence intervals and probability density functions. In Section 4, we discuss implications of this work, including ranking of parameters based on their importance.
Section snippets
Methods
We first describe the geometry of the system under consideration. Subsequently, we describe the Navier–Stokes equations governing blood flow and physiologic boundary conditions. Then, we describe the stochastic models used in this work for each of the variables, namely minimum lumen diameter (MLD), lesion length, blood viscosity and boundary resistance. Then, we describe the adaptive stochastic collocation algorithm used to solve the stochastic Navier–Stokes equations.
Results
We describe the impact of uncertainty in MLD, lesion length, boundary resistance and viscosity on an idealized stenosis model first. Following this, we describe the results on a patient-specific model, which has a focal lesion in each of the main coronary vessels (left circumflex - LCx, left anterior descending - LAD and right coronary artery - RCA). Uncertainty in pressure, velocity and are quantified.
Discussion
We applied an adaptive stochastic collocation method for analyzing the impact of uncertainty in minimum lumen diameter, lesion length, boundary resistance and blood viscosity on blood flow simulations. We showed these results on both an idealized stenosis model and a patient-specific geometry with lumen narrowing on each of the major coronary arteries (LAD, LCx and RCA).
We observed that the relative importance of uncertainty in minimum lumen diameter exceeds that of the other variables
Conflict of interest statement
All the authors in this manuscript are employed at HeartFlow, Inc., which provides FFRCT as a service to physicians for assessing the functional significance of coronary artery disease.
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