Optimizing Non-invasive Fractional Flow Reserve Estimation with Machine Learning-Enhanced 1D Hemodynamic Modeling

Abstract

Patient-specific computational models exhibit strong concordance with invasively measured fractional flow reserve (FFR)—the clinical gold standard for diagnosing coronary ischemia. However, current modeling techniques frequently rely on computationally intensive assumptions such as pulsatile flow dynamics and often fail to optimally leverage patient-specific clinical data that is routinely available, limiting their practical clinical adoption. In this study, we propose a hybrid coronary angiography-based approach that reduces computational complexity through simplified steady-state flow assumptions, while simultaneously better leveraging available clinical information. Specifically, we integrate physics–based modeling with a machine learning (ML) feedback loop designed to refine and improve FFR predictions. We evaluated this hybrid framework using a retrospective two-center cohort comprising 132 patients with 132 coronary lesions. Our results demonstrate that steady-state models effectively capture essential hemodynamic patterns, closely matching pulsatile model predictions. The ML refinement step enhances diagnostic accuracy, yielding a sensitivity of 83.3%, specificity of 100.0%, positive predictive value of 100.0%, negative predictive value of 88.2%, and overall precision of 92.6%. By effectively combining efficient computational modeling with targeted ML-driven refinements, our approach represents a robust, clinically viable solution for accurate patient-specific FFR estimation.

Introduction

Personalized computational fluid dynamics (CFD) simulations of blood flow offer a powerful, non-invasive approach for assessing cardiovascular function. When combined with machine learning (ML), these models can incorporate richer patient-specific information, including demographics, physiology, and disease history, enabling more accurate and clinically relevant predictions than purely physics–based approaches. One key application is the estimation of fractional flow reserve (FFR), the gold-standard metric for evaluating the functional severity of coronary stenosis and guiding revascularization decisions.

Coronary artery disease remains one of the leading causes of death worldwide, with over 600,000 percutaneous coronary interventions performed annually in the United States alone [1]. Localized narrowing of coronary vessels impairs myocardial perfusion and, if left untreated, can lead to myocardial infarction. While pressure-wire-based FFR measurements provide critical diagnostic insight, they are invasive and underutilized in routine clinical workflows. As a result, there has been a growing interest in developing non-invasive computational methods to estimate FFR using coronary imaging data. Most existing computational FFR frameworks adopt an “open-loop” design, where either steady or pulsatile flow assumptions are imposed based on imaging data (e.g., coronary angiography or computed tomography) and routinely measured physiological data. Integrating simulation outputs with routinely collected clinical data as a “feedback loop” remains underexplored, leaving significant opportunities to refine predictive power and clinical accuracy.

Physics–based CFD estimations have seen commercial success in clinical translatability including companies like HeartFlow, which combine CFD simulations and deep-learning aided segmentation methods to visualize patient-specific geometry and aid in informing optimal treatment plans through FFR prediction. Foundational work in personalized coronary perfusion simulation by Taylor et al. [2], Sankaran et al. [3] and Updegrove et al. [4] have also contributed to these initiatives of FFR prediction. Purely ML-based FFR estimations have also been documented in literature indicating success in data and image-driven models trained to estimate FFR as an alternative to exclusively physics–based modeling. These models have seen auspicious success in demonstrating that ML methods can compute FFR with comparable diagnostic performance to CFD-based FFR[5–10]. The current work aims to combine aspects of these techniques to enhance 1D-FFR estimations with physiologically-informed ML methods.

Current computational FFR models assume either steady inflow conditions [11–15, 63]—which reduce computational cost but may overlook important flow variability—or pulsatile inflow [16–20], where time-varying hemodynamics are captured at the expense of higher computational overhead. It remains unclear, however, whether full pulsatile modeling is always necessary for accurate FFR predictions or if simpler steady inflow conditions suffice [21]. Moreover, while recent studies have examined uncertainty quantification and sensitivity analysis [19, 20], few have incorporated patient-specific clinical data in a feedback loop to refine model predictions. We hypothesize that machine learning can be used to close this loop by learning the residual between simulated and invasively measured FFR values based on patient features and thereby improve diagnostic accuracy without increasing simulation complexity.

In this work, we present an ML-refined, 1D coronary angiography FFR framework that establishes a “feedback loop” by merging initial simulation outputs with routinely measured patient characteristics, thereby systematically improving FFR predictions (Fig. 1). Using a retrospective two-center cohort of 132 patients (132 lesions), we systematically evaluate the necessity of pulsatile flow modeling, benchmark steady versus pulsatile simulations against invasive FFR measurements, and demonstrate that ML refinement can recover clinical accuracy even with reduced model complexity. By integrating physics–based simulation with patient-informed ML correction, our approach provides a scalable and clinically practical alternative for achieving near–accurate non-invasive FFR estimations.

Fig. 1.

Overview of the machine learning–enhanced, patient-specific modeling framework. Clinical data (such as angiograms and routinely collected physiological measurements) inform the geometry and boundary conditions of the 1D blood flow model. The outputs of this model, together with anatomical information, then train an ML model to refine FFR predictions against clinical ground truth

Methods

Patient Data

This study did not involve direct patient experimentation, interaction, or the use of human tissue samples. The research protocol was approved by the Institutional Review Boards of the Duke University Medical Center and of the Massachusetts General Hospital Brigham Healthcare Center and was carried out according to the relevant guidelines and regulations under the approved protocols. Because there was no interaction or intervention between patients and the study was not prospective, the IRB did not require the signed informed consent of individual patients. Coronary angiograms and clinical measurements were obtained from 132 patients, all of whom had angiographically confirmed coronary artery disease. Each angiogram included at least four standard orthogonal views of the left coronary circulation. Physiological measurements collected during coronary angiography, such as aortic blood pressure, cardiac output, heart rate, and hematocrit, were used to inform patient-specific hemodynamic models. FFR measurements based on wires were performed by administering intravenous adenosine (140 mcg/kg/min for 120 s) to induce hyperemia. A coronary guidewire pressure sensor (Volcano Corporation, San Diego, CA) was placed distal to the coronary stenosis to compute invasive FFR. An FFR value of 0.80 or less was considered ischemic, whereas an FFR above 0.80 was considered non-ischemic.

Patient-Specific Coronary Geometry Reconstruction

To obtain 1D patient-specific models, 3D reconstructions of the coronary tree were generated from pairs of coronary angiograms using a well-established algorithm described previously [22, 23]. Previous works from our lab show the pipeline for model creation and obtaining hemodynamic data from the model [24, 25]. As a first step, the 2D vessel centerlines and their corresponding cross-sectional diameters were calculated from the angiographic images. A fully automated process then converted these 2D centerlines into 3D coronary tree models. Although information required for 1D modeling could be extracted from planar images, this reconstructive method was previously clinically validated and was therefore selected [22]. An expert interventional cardiologist verified the topology and anatomy of each reconstruction using ImageJ v1.52k (NIH, Bethesda, MD, USA) through an iterative feedback loop. This loop included the measurement of critical dimensions of the coronary tree by the interventional cardiologist, such as bifurcation angles, ostial diameter and segment lengths on both the reconstruction and patient planar imaging to ensure agreement in the cardiologist’s expert opinion. All identifiable main and side branches larger than 1 mm in diameter were incorporated into the 3D model. This reconstruction algorithm has been widely applied to coronary arteries in multiple studies [18–20, 24, 26].

The 1D geometries were then derived from the 3D reconstructions by using the centerlines of the vessels and their corresponding hydraulic diameters. The centerline data were extracted with Mimics (Materialise, Leuven, BE), and vessel lengths were calculated at a resolution of 100 μm. These 1D centerlines were validated by comparing the minimal luminal diameter in the 1D model against the ground truths measured by the physician of the angiograms. The validated 1D geometries were then used as inputs for patient-specific CFD simulations.

1D Computational Fluid Dynamics Solver

1D hemodynamic simulations were performed with our internal reduced-order blood flow simulator [28]. The governing equations for the 1D models are as follows:

$$\frac{\partial A}{\partial t}+\frac{\partial Q}{\partial x}=0$$ (1)

$$\frac{\partial Q}{\partial t}+\frac{\partial }{\partial x}(\alpha \frac{Q^2}{A})+\frac{A}{\rho }\frac{\partial P}{\partial x}=-C_f\frac{Q}{A}$$ (2)

where A is the arterial cross-sectional area, P is pressure, Q is flow rate, $\rho$ is blood density, $\alpha$ controls the velocity profile and $C_f=22\pi \mu$ is a frictional term. $\alpha$ and C_f were empirically estimated from previous work [29, 30]. To close the loop, P and A are related by the following constitutive equation:

$$P=P_{\mathit{ext}}+\beta (\sqrt{A}-\sqrt{A_0}),\beta =\frac{\sqrt{\pi }hE}{\left(1-v^2\right)A_0}$$ (3)

where P_ext is the external pressure exerted on the arteries and A₀ is the undeformed cross sectional area when $P=P_{\mathit{ext}}.\beta$ is an arterial stiffness term and is a function of wall thickness (h = 0.945 mm) [31], A₀, elastic modulus (E = 1.41 MPa) [32], and Poisson’s ratio (ν = 0.5) [33]. Conservation of mass (Eq. 1), conservation of momentum (Eq. 2), and pressure-area constitutive relationship (Eq. 3) were numerically solved using a MacCormack finite difference scheme.

Furthermore, it was necessary to include explicit pressure drop terms because energy flow does not only occur in the longitudinal direction of flow in a stenosis. We coupled the 1D model with an explicit pressure drop term:

$$\mathrm{\Delta }{P}_s=\frac{\mu K_v}{2\pi r_u^3}Q+\frac{\rho K_t}{2A_u^2}{\left(\frac{A_u}{A_s}-1\right)}^2\left|Q\right|Q+\frac{\rho K_u{L}_s}{A_u}\frac{\partial Q}{\partial t},$$ (4)

where ΔP_s is the pressure drop in a stenosis, r_u is the radius of an unstenosed artery, A_u is cross-sectional area of an unstenosed artery, A_s is cross-sectional area of a stenosed artery, and L_s is the length of the stenosis. The stenosis model (Eq. 4) was coupled to the 1D governing equations through the continuity of total pressure.

The continuity of total pressure equations relate the pressure gradient from parent to daughter arteries as shown in Eqs. 5 and 6.

$$P_p+\frac{1}{2}\rho {\left(\frac{Q_p}{A_p}\right)}^2=P_{d1}+\frac{1}{2}\rho {\left(\frac{Q_{d1}}{A_{d1}}\right)}^2$$ (5)

$$P_p+\frac{1}{2}\rho {\left(\frac{Q_p}{A_p}\right)}^2=P_{d2}+\frac{1}{2}\rho {\left(\frac{Q_{d2}}{A_{d2}}\right)}^2$$ (6)

To modify these equations to include the coupled stenosis equation a similar structure is imposed accounting for a ΔP_s term, shown in Eq. 7.

$$P_u+\frac{1}{2}\rho {\left(\frac{Q}{A_u}\right)}^2=P_s+\frac{1}{2}\rho {\left(\frac{Q}{A_s}\right)}^2+\mathrm{\Delta }{P}_s$$ (7)

Where the pressure above and below the stenosis are accounted for in addition to the stenosed pressure term ΔP_s.

K_v, K_t, and K_u are viscous, “turbulent” and inertial coefficients, respectively. The “turbulence” term accounts for non-linear effects of converging or diverging flow patterns. These coefficients were parameterized as $K_v=32\left(0.83L_s+1.64D_s\right)\times {\left(A_u/A_s\right)}^2/D_u,K_t=1.52$, and K_u = 1.2 based on [28]. D_u and D_s are non-stenosed and stenosed arterial diameters, respectively. A_s was taken as the minimal luminal diameter. r_u was estimated from the degree of stenosis labeled by the physician and the minimal luminal radius (r_s) using the following relationship:

$$\mathit{Stenosis\; degree}=\left(1-\frac{r_s}{r_u}\right)\times 100\%$$ (8)

Modeling Assumptions

Coronary circulation was modeled as an incompressible Newtonian fluid with a density of 1060 kg/m³. Dynamic viscosity was calculated from clinically measured hematocrit based on the following expression [34]:

$$\mu =\frac{{\mu }_0}{1-\varphi },$$ (9)

where μ is the dynamic viscosity, ${\mu }_0$ is the dynamic viscosity of the plasma, and $\varphi$ is hematocrit. We assumed a plasma dynamic viscosity of 1.2 cP [18, 24, 34].

Boundary Condition Tuning for Personalized Hemodynamic Simulations

The boundary conditions were adjusted to a patient-specific level. Pulsatile or steady-flow conditions were used at the inlet boundary condition. The left and right coronary waveforms were derived from [35] and scaled to a level per patient using clinically measured cardiac output, heart rate, and flow dominance at rest. Hyperemia was estimated by scaling up the flow rate based on empirical observations, where the left and right circulations were scaled by 4x and 3x, respectively [36].

For steady-state simulations, the inlet flow rate was taken as the mean flow rate over the cardiac cycle. The assumed mean flow representation throughout the cardiac cycle was in lieu of diastolic and systolic oscillation. For each patient, both the pulsatile and steady-state simulation outlet boundary conditions, described below, remained consistent and were prescribed per patient.

Two-element Windkessel models, consisting of peripheral resistance (R_p) and compliance (C), were employed at the outlets to account for the effect of microvascular hemodynamics [37]:

$$Q=\frac{P}{R_p}+C\frac{dP}{dt},$$ (10)

where C was assumed to be 9 μ cm⁴ s² g⁻¹ based on patient averages from [28]. R_p was distributed among the terminal branches using resistance-radius relationships commonly used for coronary simulations [18–20, 24, 27, 38]:

$$R_i=P_{\mathit{mean}}\cdot \frac{{\sum }_{j=1}^{N_{\mathit{terminal}}}{r}_j^3}{Q_{\mathit{ostial}}{r}_i^3}$$ (11)

where R_i is the peripheral resistance in each terminal vessel, P_mean is the mean arterial pressure, Q_ostial is the flow rate in the ostium, r is the average radius of the terminal vessel and N_terminal is the number of terminal vessels. The mean arterial pressure and inlet flow rate in the resting state were obtained from clinical measurements at the time of angiography and the terminal branch radii were calculated through the centerlines of the vessel. The ostial flow rate was determined as a fraction of cardiac output by flow dominance [18, 24]. Terminal resistances were reduced by a factor of 0.22x [18, 24, 36, 39] from the resting state to estimate hyperemia. Although patient-specific peripheral resistances were calculated, a patient average of compliance was used as a representative value based on prior work that, through a global sensitivity analysis, determined that compliance variability had negligible impact on FFR prediction [20].

Pulsatile State and Steady State Simulations

The first step—prior to any comparison with clinically measured ground truths—was to assess whether pulsatility was essential for computing 1D FFR. If steady state simulations alone could capture the necessary hemodynamic results, computational costs would be reduced, making these models more accessible in practice. In a cohort of 132 patients (with one lesion per patient), we compared pulsatile and steady-state 1D FFR predictions. Convergence details for simulations in the pulsatile state can be found in our previous works [24, 40].

Analysis of Hemodynamics

This work focuses on computing FFR, which was computed as follows:

$$FFR=\frac{P_d}{P_a}$$ (12)

where P_d is the distal location labeled by interventional cardiologists and P_a is the proximal location taken at the ostium. In the case of pulsatile simulations, pressure was averaged over the cardiac cycle at the two locations.

Machine Learning-Driven Refinement of 1D FFR Predictions

Following the analysis of pulsatile versus steady-state flow assumptions, we introduced an ML refinement step to improve the accuracy of 1D simulated FFR values. This approach integrates patient-specific data from electronic health records with simulation output to predict and correct residual error, defined as the difference between simulated and clinically measured FFR. A custom weighted loss function was implemented in XGBoost to reinforce accuracy in the FFR “gray zone” (Fig. 9 in Supplementary section). Cases that were in this predefined range (0.75–0.85) received a weight of α while other cases retained a weight of 1. This model predicted the error between predicted and invasive FFR, allowing simulation results to be corrected post-hoc into the final refined FFR prediction.

Input features were grouped into six categories: (1) simulation outputs, (2) vessel geometry from coronary angiograms, (3) clinical measurements, (4) medications, (5) demographic information, and (6) comorbidity history, yielding a total of 29 features per patient (Table 4 in Supplementary section). Feature selection was guided by physiological relevance to FFR in order to avoid spurious associations and preserve interpretability. Although a broad set of demographic, clinical, geometric, and simulation-derived variables was initially explored, only three features were ultimately selected for model training: steady-state 1D FFR, the pressure ratio between the minimal luminal diameter (MLD) and the distal vessel location derived from 1D simulations, and the MLD itself. The steady-state 1D FFR provides a physics–based approximation of invasive FFR under hyperemic conditions and has been shown to agree with clinical measurements in prior work [41]. The pressure ratio at the MLD directly encodes stenosis-induced pressure loss and downstream recovery, reflecting the fundamental pressure-based definition of FFR. Finally, MLD represents a primary geometric determinant of coronary flow resistance, as pressure drop across a stenosis scales strongly with luminal radius and is not fully captured by percent stenosis alone. Additional simulation-derived hemodynamic variables, geometric descriptors, and selected clinical or historical factors were evaluated to assess whether they provided incremental predictive value by modulating coronary pressure–flow relationships (i.e., via microvascular resistance, blood viscosity, or vasodilatory response). However, feature selection consistently prioritized simulation-based and geometric variables, ensuring that final model predictions were driven by mechanistic determinants of FFR rather than indirect comorbid associations. Furthermore clinical data included in this work is based on risk factors associated with coronary artery disease, which correlate to disease severity, indicating potential to improve predictive FFR accuracy (Table 4 in Supplementary section). Beginning in the history category, prior myocardial infarction, congestive heart failure, peripheral artery disease, tobacco use, prior aneurysm and previous percutaneous intervention in the vessel of interest or other vessel are all comorbidities or risk factors pertaining to the progression of CAD [42–45]. In terms of demographic features, both sex and weight, specifically obesity, are factors to consider in assessing risk of CAD development and progression [46, 47]. Sex differences play a critical role in coronary atherosclerosis as men face higher prevalence of CAD and present with CAD at a statistically younger age, indicating that this information would be important to consider [47]. The drugs represented in the medication category are those commonly prescribed to patients diagnosed with CAD and common comorbidities of the disease, that are not well managed with first-line therapies, like lifestyle modifications and diet management, or those recommended to receive combination therapy. Due to their potential to be strongly correlated with disease progression, we included these features for further exploration. This categorical grouping was for exploratory purposes only and is based in what clinical information is available from patient charts and how this clinical data is represented in the charts in an effort to make physiologically grounded feature organization. This organization allowed us to explore interpretable categories that either dominated feature selection or did not have a high mutual information score, as discussed below. Please see Table 4 in Supplementary section for a complete list of features and their categorical binning. Prior to training, the dataset was randomly split into 80% for training and 20% for testing. Within the training set, missing values were inputted using mean substitution, and all continuous variables were standardized to zero mean and unit variance. Feature selection was performed using mutual information regression; the three features with the highest scores were retained for model training to reduce dimensionality and mitigate overfitting, irrespective of which category they fell in..

To address the increased diagnostic uncertainty near the ischemic threshold (FFR: 0.75–0.85), we implemented a custom objective function with stepwise weighting. Clinical FFR values within this “gray zone” were assigned a higher penalty, controlled by a tunable hyperparameter α, to prioritize accurate predictions where clinical decision-making is most sensitive, as shown in equation (13)

$$\sum _i{w}_i\ast {({\mathit{Predicted}}_i-{\mathit{TrueError}}_i)}^2$$ (13)

where $w_i=\alpha$ when FFR is between 0.75 and 0.85 and otherwise 1.

This approach ensured that residuals near the treatment threshold received greater emphasis during training. We employed the XGBoost framework for model development, based on its prior success in structured medical prediction tasks and its demonstrated utility in related 1D FFR modeling efforts [48].

To efficiently search the hyperparameter space, we employed a Bayesian optimization algorithm, specifically the Tree-structured Parzen Estimator [49], over 15 trials. The explored hyperparameters are summarized in Table 1. The training set was further split into five folds for cross-validation, and predictions on the validation fold were used to compute the mean squared error as the optimization metric.

Table 1.

Hyperparameter grid search

Hyperparameter	Range
Number of estimators	[350, 450]
Alpha	[4, 7]
Maximum depth	[7, 9]
Learning rate	[0.01, 0.03]

Once the best set of hyperparameters was identified, a final XGBoost model was trained on the entire training set. The performance of the model was then evaluated on the 20% test set held out from the start. The test results were compared with both steady state 1D FFR and clinical FFR measurements.

Statistical Analysis

The Kolmogorov-Smirnov test was first used to confirm that the data follows the central limit theorem. Comparison between FFR models and clinically measured FFR was performed using ordinary least-squares correlation, concordance correlation, and Bland–Altman analysis. Diagnostic performance was evaluated using scores from confusion matrices and areas under the receiver operating characteristics curve.

Results and Discussion

Steady State 1D FFR Recapitulates Pulsatile State Results

The ordinary least squares correlation for pulsatile state and steady state simulations was near-perfect (r = 0.96, p < 0.001), with a bias and imprecision of 0.03 ± 0.03. The concordance between the two also showcased a high correlation of 0.93 (Fig. 2). These results suggest that steady state simulations closely replicate pulsatile flow outcomes, showing near-perfect correlation and agreement.

Fig. 2.

Correlation and agreement of pulsatile and steady state 1D FFR. (Left) Scatter plot of pulsatile vs. steady state FFR for 132 vessels, with dashed lines denoting the 0.80 ischemic threshold and a solid red line representing perfect correlation. (Right) Bland–Altman plot showing the mean difference between pulsatile and steady state FFR, where black dashed lines indicate the upper and lower limits of agreement and the solid red line shows the mean difference

In terms of runtime, pulsatile simulations took an average of 118 seconds versus 53 seconds for steady state simulations, while the respective medians were 90 seconds and 29 seconds (Fig. 3). Although the reduction from 90 to 30 seconds may not be transformative on its own, it offers meaningful efficiency gains, particularly for larger patient cohorts or potential clinical deployment. Shorter runtimes translate to less computational infrastructure and fewer resources, thereby lowering costs and improving scalability. Additionally, adopting steady state assumptions minimizes the degrees of freedom in the model and curtails uncertainty propagation [19, 20].

Fig. 3.

Violin plots illustrating the distribution of time-to-solution for pulsatile and steady state simulations on an M4 Max MacBook Pro

Validation of Steady State 1D FFR Against Clinical Measurements

To establish the validity of the simplified steady-state model, we compared simulated 1D FFR values with invasive FFR measurements obtained via pressure guidewires. Anatomical correspondence between proximal and distal measurement locations in the simulation and clinical data was confirmed. The ordinary least squares correlation was 0.70 (p < 0.001) with bias and imprecision of 0.02 ± 0.08 (Fig. 4). This result indicates a slight bias to underestimate FFR values, corresponding to a conservative overestimation of lesion severity). The concordance between the set was 0.63. Although not a common practice of reporting 1D steady state FFR values vs the clinically obtained FFR values, concordance value seems to agree with some similar literature. [50]

Fig. 4.

Correlation and agreement of steady state 1D FFR and clinically measured FFR. (Left) Scatter plot of 1D steady state FFR vs. clinical FFR for 132 vessels, with dashed lines denoting the 0.80 ischemic threshold and a solid red line representing perfect correlation. (Right) Bland–Altman plot showing the mean difference between 1D steady state FFR and clinical FFR, where black dashed lines indicate the upper and lower limits of agreement and the solid red line shows the mean difference

Beyond assessing correlation as a continuous metric, we evaluated diagnostic performance relative to the clinical 0.80 ischemic threshold. The sensitivity was 79.6%, specificity was 89.8%, positive predictive value was 79.6%, negative predictive value was 89.8%, and the accuracy was 86.4% (Table 2). Additionally, we also estimated an AUC of 0.90 as can be seen from the receiver operating characteristic curve in Fig. 5. These results are comparable with those in existing literature [51–61]. These results confirm that the steady-state 1D approach provides clinically useful diagnostic performance while substantially reducing computational cost. They also establish a baseline for evaluating the added value of machine learning refinement in the subsequent analysis.

Table 2.

Diagnostic performance of steady state 1D FFR compared to clinical FFR with an ischemic cut-off at 0.80.

Metric	Value [%]	95% Confidence interval [%]
Sensitivity	79.6	64.7–90.2
Specificity	89.8	81.5–95.2
Positive predictive value	79.6	67.3–88.0
Negative predictive value	89.8	83.0–94.0
Accuracy	86.4	79.3–91.7

Fig. 5.

Receiver operating characteristic curve comparing steady-state 1D FFR predictions to clinically measured FFR as the gold standard. The baseline (AUC = 0.50) is shown for reference, while the steady-state predictions achieve an AUC of 0.90

Improving 1D FFR Predictions Using Machine Learning Refinement

After establishing the baseline validation of steady state 1D FFR against clinical ground truths, our final step was to combine 1D simulation results with routinely collected clinical data to train an ML model for refining FFR predictions. The dimensional reduction from 3D to 1D modeling inevitably results in the loss of spatial granularity and potential accuracy. The same patient cohort presented in this work has previously been investigated using 3D simulations for FFR prediction, which demonstrated similar correlation with the clinical gold standard (r = 0.76) as the ML-augmented 1D predictions presented here (r = 0.74), but at a substantially higher computational cost [17]. Even with advanced parallelization frameworks on a high-performance computing system, the longest 3D simulation required 12.8 wallclock hours on 10,000 CPU cores (Lawrence Livermore National Laboratory Lassen supercomputer), whereas the 1D simulations ran locally on an M4 Max MacBook Pro [17]. The trade off of spatial granularity for computational expense should be carefully considered and both frameworks are represented in literature [62]. With this consideration, mutual information regression (Fig. 6) was used to quantify the predictive relevance of each input feature. The three most informative variables were: (1) steady-state 1D FFR, (2) the ratio of pressures at the MLD to those at the distal location, and (3) the MLD itself. Derived features exhibited the greatest mutual information, followed by anatomical and clinical measurements, while medications, demographics, and co-morbidity history contributed minimally. To maintain the simplicity and interpretability of the model, only the top three features were selected for training the XGBoost model. To show the difference in performance of the model when trained by 3 features as opposed to all of the 29 features, refer Fig. 8 in Supplementary section to view the confusion matrix.

Fig. 6.

Mutual information scores for the training set (n = 105 vessels), computed across 29 features encompassing simulation outputs, angiographic geometry, clinical measurements, medications, demographics, and comorbidity history. The top three features selected for downstream ML training were the steady-state 1D FFR, the ratio of pressure at the minimal luminal diameter (MLD) to the distal location from the 1D simulation, and the MLD itself

An extensive hyperparameter grid search was performed and the best-performing model was used to predict error in a test dataset of 27 patients. This predicted error was then added to the original steady state 1D FFR to yield the ML-refined FFR. In the test set, the original ordinary least squares correlation with clinical FFR was 0.60, with concordance of 0.55 and Bland–Altman bias and imprecision of 0.03 ± 0.18 and a percent discrepancy of 9.7%. After ML refinement, the ordinary least squares correlation increased to 0.74, with concordance of 0.68 and bias and imprecision of 0.00 ± 0.12 and a percent discrepancy of 5.7% (Fig. 7). Therefore, the bias was effectively reduced to zero while narrowing the variance in differences between clinical FFR and predicted FFR.

Fig. 7.

Correlation and agreement of steady state 1D FFR, ML-refined steady state 1D FFR, and clinically measured FFR for the test dataset. (Left) Scatter plot comparing steady state FFR, ML-refined FFR, and clinical FFR values across 27 vessels, with dashed lines marking the 0.80 ischemic threshold and a solid red line representing perfect correlation. (Right) Bland–Altman plot showing the mean differences between steady state and ML-refined FFR with clinical FFR, where black dashed lines indicate the upper and lower limits of agreement and the solid red line represents the mean difference

Regarding diagnostic performance, the original sensitivity was 83.3%, specificity was 73.3%, positive predictive value was 71.4%, negative predictive value was 84.6%, and accuracy was 77.8%. Following ML refinement, sensitivity remained at 83.3%, but positive predictive value rose to 100.0%, specificity to 100.0%, negative predictive value to 88.2%, and overall accuracy to 92.6%. All metrics except sensitivity, therefore, improved. Importantly, while overall performance gains with ML refinement were incremental, the improvements were most evident in the clinically relevant ’grey zone’ (FFR 0.75–0.85). In this range, ML refinement reduced both under- and over-predictions, with a greater proportion of cases aligning with invasive FFR. This suggests that the proposed framework may offer added value in precisely the subset of patients where clinical decision-making is most challenging. We also note that this patient cohort has been previously analyzed with 3D FFR modeling [17, 24]. In those studies, 3D simulations compared to invasive FFR achieved a sensitivity of 79.6%, which increased to 83.3% with 1D modeling, and further improved with ML feedback refinement (accuracy 92.6%, specificity 100.0%, PPV 100.0%). Although 3D modeling provides greater spatial resolution, the 1D+ML framework demonstrated comparable or superior predictive performance at a fraction of the computational cost.

Established medical companies offer a framework for non-invasive prediction of fractional flow reserve (FFR) measurement. HeartFlow’s^® FDA-approved FFR_CT, computed tomography-based fractional flow reserve, is one such example that demonstrated superior diagnostic performance compared to imaging-based diagnostic routines [63, 64]. However, a key component of FFR_CT is that it relies heavily on CT imaging, which is not part of the routine cardiac catheterization laboratory workflow that primarily uses angiograms. Additionally, computed tomography images often exclude important side-branch features that play an essential role in defining blood flow in the geometry [17, 24, 65–67]. This limitation can be overcome by using angiography-based imaging [11, 17, 68]. Cathworks FFRangio^® is a computational FFR predictive technology that uses angiograms, with high diagnostic accuracy [69, 70]. FFR calculations require information on the distal and proximal locations of the stenoses. This generates a limitation in multi-stenosed 0D models when lesion-specific information is crucial, thereby creating a need for a model that preserves spatial information. 1D models are less computationally expensive and are often the choice in FFR studies to balance the spatial granularity concerns and computational expense seen between 0D and 3D models. Additionally, previous works have incorporated clinical inputs into machine learning frameworks to enhance the computational prediction of FFR values [71, 72]. Similarly, this study aims to develop a time-efficient, yet informative model informed by an ML feedback loop to enhance FFR predictions.

One limitation of the dataset selected in this study is its retrospective nature. Although the scope of the current work was to establish baseline scientific viability, the authors consider prospective validation, as well as inclusion of data from a broader set of centers, would provide critical additional insight into the robustness and generalizability of the framework, and can be considered in future works.

These results demonstrate that augmenting steady state 1D FFR predictions with an ML-based error refinement markedly improves alignment with clinically measured FFR. The reduction in bias to near zero underscores the model’s ability to eliminate systematic deviations, while the decrease in imprecision highlights more consistent predictions across the patient cohort. Clinically, these improvements translate into greater specificity and accuracy, suggesting that there were fewer false positives and could potentially reduce unnecessary interventions. Importantly, the ability to maintain sensitivity ensures that ischemic cases continue to be reliably identified while also increasing confidence in identifying FFR-positive cases as the positive predictive value increased to 100.0%. In general, incorporating simulation data with routinely collected clinical features paves the way for more accurate, cost-effective, and scalable FFR assessments (Table 3).

Table 3.

Diagnostic performance of the test dataset comparing 1D steady state FFR with ML-FFR trained using both the top 3 features as well as all features against ground truth FFR values

Metric	1D Steady state FFR [%]	ML-FFR (3 features) [%]	ML-FFR (all features) [%]
Sensitivity	83.3 (51.6–97.9)	83.3 (51.6–97.9)	75.0
Specificity	73.3 (44.9–92.2)	100.0 (78.2–100.0)	100.0
Positive Predictive Value	71.4 (51.0–85.7)	100.0 (69.1–100.0)	100.0
Negative Predictive Value	84.6 (60.0–95.3)	88.2 (67.9–96.4)	83.3
Accuracy	77.8 (57.7–91.4)	92.6 (75.7–99.1)	88.9

Conclusion

In this study, we introduced and validated a coronary angiography-based approach designed to minimize computational overhead and improve clinical accuracy of computing non-invasive FFR through ML refinement. Specifically, we minimized computational demands by adopting steady-state flow simulations and integrated an ML feedback loop to refine FFR predictions. Using a retrospective two-center cohort of 132 patients with 132 coronary vessels, we demonstrated that steady-state models closely match the accuracy of pulsatile models while reducing computational overhead by approximately half. By further incorporating ML refinement, we enhanced diagnostic accuracy validated against invasive FFR measurements, the clinical gold standard.

Next, we investigated whether additional patient-level data could enhance these baseline predictions. A mutual information analysis on the training set revealed that three features—the 1D steady-flow FFR value, the pressure ratio at the minimal luminal diameter to the distal location, and the minimal luminal diameter itself—had the highest mutual information scores. Other groups of features, such as medications, demographics, and other clinical variables contributed far less, suggesting that local anatomical and hemodynamic descriptors were paramount and correlated to 1D FFR error.

Leveraging these insights, we trained an XGBoost model with a custom loss function to prioritize accuracy in the diagnostically uncertain FFR “gray zone.” This refinement step significantly improved both continuous and categorical agreement with invasive FFR: bias was reduced to zero, imprecision narrowed from 0.03 ± 0.18 to 0.00 ± 0.12, and the diagnostic accuracy improved from 77.8% to 92.6% . Specificity increased from 73.3% to 100%, substantially reducing false positives.

Overall, our results highlight the power of combining reduced-order CFD with targeted machine learning to enable fast, scalable, and near–accurate non-invasive FFR assessment. This hybrid framework, focused on interpretable, high-yield features, offers a promising pathway to improve diagnostic precision and clinical decision-making. Although this work focused on focal lesions, the framework is poised to extend to more complex scenarios such as bifurcation or serial lesions. Additionally, now that we have confirmed the use of clinical inputs in an ML feedback loop to improve non-invasive FFR predictions, future studies can also investigate more complex ML models along with a bigger dataset to further improve the diagnostic performance of our ML framework. While we did see an improvement in the concordance value with the introduction of the ML-refined methodologies, a bigger dataset can further help push these values to a more satisfactory result. Together, these results mark a significant step towards improved diagnostic performance and clinical decision-making in cardiology.

Supplementary Material

The online version contains supplementary material available at https://doi.org/10.1007/s13239-026-00836-y.

Data Availability

Anonymized, de-identified data is available upon reasonable request. If interested, please contact Amanda Randles (amanda.randles@duke.edu).