Establishing Hemodynamic Convergence Framework for Coronary Digital Twins Under Realistic Dynamic Heart Rates

Abstract

The advent of digital twins has increased the demand for longer-duration simulations that span multiple physiological states. Digital twins have emerged as powerful tools in cardiovascular modeling, enabling patient-specific simulations of coronary blood flow for non-invasive diagnosis and treatment planning. Although these simulations achieve high fidelity under steady or periodic heart rates, modeling real-world transitions, such as those arising from physical activity, requires careful evaluation of temporal convergence, the stabilization of hemodynamic parameters through the simulation of preceding cardiac cycles, or ‘pre-flows’. In this study, we present a physiologically grounded approach for determining the minimum number of preceding cardiac pre-flows necessary to achieve temporal convergence following abrupt heart rate (HR) changes. Using high-resolution patient-specific 3-dimensional (3D) simulations and inflow waveforms scaled from both synthetic and wearable-derived HR data, we quantify convergence behavior across velocity, pressure gradient, and wall shear stress at both cross-sectional and full-domain levels. Results show that simulating just two pre-flows is sufficient to achieve physiologically stable outputs across high-to-low and low-to-high HR transitions (<2% difference). These findings are further verified using continuous HR data obtained from wearable devices, with low and high HR segments extracted to represent natural extremes, confirming the robustness of the proposed convergence criterion under real-world dynamic inputs (<1% difference). This work establishes a computationally efficient and physiologically consistent criterion for dynamic-state simulations, facilitating the integration of cardiovascular digital twins with real-time sensing technologies.

I. INTRODUCTION

The rise of digital twins in cardiovascular medicine has created new demands for simulations that capture not just static physiological states, but dynamic transitions over time. These state-of-the-art models, designed to reflect patient-specific anatomy and evolving heart rate (HR) conditions, reform how we understand and predict vascular health. However, their clinical potential hinges on an often overlooked technical foundation: temporal convergence. That is, ensuring that the simulated hemodynamics reflect the true physiology rather than the artifacts of the initialization of the model. This requirement becomes especially critical when digital twins incorporate real-world transitions, such as rest to higher HR-scaled inflow conditions, where flow inputs evolve rapidly and boundary conditions change over time.

Although many existing cardiovascular digital twin studies are limited to single heartbeat analyzes under static conditions, typically representative beats entirely in rest or exercise states^1–8, they do not address transitions between these HR conditions. This leaves a gap in the guidance to ensure convergence when sharp physiological changes occur. For applications of longitudinal monitoring, such as predicting vascular disease progression or detecting restenosis in the stent, the ability to track the hemodynamic state of a patient throughout dynamic HR changes spanning naturally occurring low and high HR ranges, such as those captured by wearable devices, is essential. In these scenarios, accurate modeling requires that the simulated flow fields converge reliably after each state change. Identifying the first convergent heartbeat ensures that clinically relevant metrics, such as fractional flow reserve or pressure–flow relationships, are derived from physiologically stable outputs without unnecessary simulation overhead, enabling efficient real-time and remote assessments.

Accurate hemodynamic simulations are critically dependent on temporal convergence, stabilization of flow fields after initial transients have dissipated, so that the computed outputs reflect true physiological behavior rather than numerical artifacts^9–11. This requirement becomes even more essential in coronary digital twins^12,13 that dynamically replicate the physiological state of a patient and aim to replicate dynamic heart rate (HR) conditions, where flow inputs change rapidly in response to activity or stress¹⁴. Without a reliable convergence framework, simulations risk misrepresenting clinically important hemodynamic biomarkers, especially in regions of stenosis or complex geometry. Moreover, as wearable technologies increasingly enable longitudinal tracking, the ability to integrate these data into accurate^15–17, stable simulations becomes essential for the real-world deployment of digital twin systems in cardiovascular medicine^18–22. The objective of this work is to establish a physiologically grounded and computationally efficient criterion for temporal convergence in 3-dimensional (3D) coronary blood flow simulations under dynamic HR transitions. This criterion is fundamental to enabling digital vascular twins to move beyond static snapshots of disease and toward state-of-the-art real-time models capable of tracking cardiovascular physiology through continuous changes.

Vascular digital twins, based on high-fidelity computational fluid dynamics (CFD), have been proven to be valuable for non-invasive diagnosis, risk stratification, and treatment planning by simulating pressure, velocity, and wall shear stress (WSS) in patient-specific coronary anatomies^5–8,23. These models enable patient-specific assessments and in silico treatment planning, offering a powerful complement to invasive procedures^24–27. Despite advances in model fidelity and clinical relevance, most existing digital twin frameworks operate over short temporal windows, often spanning only one or a few cardiac cycles^1–4,28,29. Simulations are typically constructed and validated under static or idealized physiological states, such as resting heart rate and fixed boundary conditions^1,3. Even as CFD solvers have matured to support large-scale 3D flow modeling, their application to dynamic, time-varying conditions remains limited. In particular, it is still uncommon for digital twins to incorporate continuous dynamic HR transitions across physiological ranges, such as abrupt increases or decreases in heart rate, within a single unified simulation.

Recent advances in sensing and computational infrastructure are beginning to make state-varying simulations in the longer term feasible^30,31. However, this shift introduces a new challenge: when simulating a dynamically changing HR profile, how many preceding cardiac cycles must be included to ensure the solution is free from initialization transients and yields physiologically accurate hemodynamic outputs? Most prior work has focused on achieving convergence in periodic or steady-state settings^4,32, but little is known about convergence behavior when modeling large transitions in input conditions, such as an abrupt HR increase or decrease. In addition, convergence is often assessed in single cross sections, which can miss critical spatial variations in complex stenosed anatomies where local flow features are highly sensitive to upstream changes¹⁴.

This study evaluates how abrupt changes in heart rate affect the convergence of patient-specific coronary flow simulations, with a focus on pressure, velocity, and WSS computed in both localized slices and throughout the vascular domain. The analysis spans synthetic transitions designed to bracket physiological extremes, as well as wearable-derived heart rate sequences that reflect real-world variability. The convergence behavior is examined across anatomical regions and hemodynamic metrics, revealing that flow fields typically stabilize after two cardiac cycles, regardless of the direction of transition or the input source. This result establishes a practical and physiologically justified standard for initializing simulations under dynamic conditions. The findings support the extension of vascular digital twins beyond static evaluations toward continuous, time-aware modeling, enabling broader integration with real-time sensing technologies in both clinical and remote settings. Moreover, by quantitatively characterizing convergence under dynamic HR transitions, this work strengthens the modeling foundations for the longitudinal hemodynamic mapping framework which aims to enable wearable-informed real-time coronary simulations^30,31.

II. METHODS

A. Patient data and power analysis

To evaluate convergence behavior under real-world conditions, this study leveraged a robust dataset of previously collected, de-identified patient cases with confirmed coronary artery disease. No new patient recruitment, experimentation, or interaction was conducted. Instead, the coronary geometries used in this analysis were drawn from a validated dataset originally published in a prior study on non-invasive computation of fractional flow reserve using digital twins⁵. That study included 160 patients with angiographically confirmed disease, and was approved for retrospective analysis of coronary angiograms and associated clinical measurements. As the original data collection involved no direct patient contact, signed informed consent was not required.

For the current analysis, we began with the cohort of 160 previously processed left coronary artery (LCA) geometries. A two-sample t-test was performed to determine the minimum sample size required to detect statistically significant differences in $L_2$ error between simulation groups. An $L_2$ error threshold of 10⁻³ was adopted, consistent with prior CFD studies involving patient-derived geometries that report this limit as an accepted numerical accuracy criterion^1,33–37. The maximum expected error was set to 0.007, corresponding to the largest deviation observed in the cohort. The standard deviation was taken as 0.003, consistent with the findings of Vardhan et al.¹, who reported a mean bias of approximately 0.003 for resting gradients when validating computational and angiographic pressure gradients. These values remained stable within 95% agreement limits, demonstrating that a standard deviation on the order of 0.003 is representative of rigorous coronary CFD comparisons. Using these parameters, the t-test indicated that $N=6$ representative LCA models would be sufficient to achieve statistical power for the specified effect size. Six models were therefore selected from the cohort to capture a broad range of anatomical and physiological variability, including differences in fluid domain volume, Euclidean distances between main branch radii, degrees of stenosis, and distribution across fractional flow reserve (FFR) classification categories (see Supplementary Table SI). Furthermore, continuous HR data from a different individual, spanning one hour and capturing extreme low and high HR segments within the recording, were sourced from an open access FitBit dataset³⁸.

The algorithm described in previous studies was utilized to reconstruct the entire coronary tree model from pairs of coronary angiograms^39,40. The processes of reconstruction of angiograms and recording of clinical measurements of aortic blood pressure, cardiac output, baseline heart rate, and hematocrit for the construction of a personalized digital twin are described in^{5,30,31,33,41}.

B. Coronary Geometry Reconstruction and Computational Flow Modeling

To generate anatomically accurate digital twin models, patient-specific 3D coronary artery geometries were reconstructed from paired coronary angiograms using a previously established and validated reconstruction pipeline^39,40. The reconstruction pipeline generated 2D vessel centerlines with associated diameters, which were then used to automatically compute full 3D coronary tree geometries. This method has been extensively validated and applied in prior work^{5,30,31,33,41}.

Hemodynamic simulations were performed using HARVEY, a massively parallel fluid dynamics solver based on the lattice Boltzmann method (LBM)^42,43. The solver uses a D3Q19 velocity discretization and single-relaxation-time Bhatnagar–Gross–Krook (BGK) collision operator to solve the Navier-Stokes equations. The lattice Boltzmann equation governs the time evolution of the particle distribution function $f_i(\mathit{x},t)$ as:

$$f_i\left(\mathit{x}+{\mathit{c}}_i\mathrm{\Delta }t,t+\mathrm{\Delta }t\right)=fi(\mathit{x},t)+{\mathrm{\Omega }}_i(\mathit{x},t)$$ (1)

where ${\mathit{c}}_i$ denotes the discrete lattice velocities and $\mathrm{\Omega }i$ is the collision term. No-slip boundary conditions were enforced on rigid vessel walls using the halfway bounce-back method, while finite-difference boundary conditions were applied at the inlet and outlets⁴⁴. All simulations were run on the Duke Compute Cluster using 1440 CPU tasks across 36 Intel Xeon E5–2695V4 ’Broadwell’ nodes.

Blood was modeled as an incompressible Newtonian fluid with a density of 1060 kg/m³. The dynamic viscosity values were derived from the measured hematocrit level ($\varphi$) using the empirical relationship⁴⁵:

$$\mu =\frac{{\mu }_0}{1-\varphi }$$ (2)

where ${\mu }_0=1.2\mathrm{c}\mathrm{P}$ is the assumed plasma viscosity^1,33,45. The baseline clinical and geometric characteristics, including hematocrit, viscosity, severity of stenosis, FFR, and mesh resolution are summarized in Supplementary Table SI for the six geometries of the left coronary artery selected for this study.

Inlet flow waveforms were generated by scaling a representative coronary velocity profile⁴⁶ according to patient-specific baseline cardiac output and resting HR ($H{R}_{\text{baseline}}$). The baseline coronary flow rate was estimated using the following:

$$Q_{\text{baseline}}=\alpha Q_{SV}H{R}_{\text{baseline}}$$ (3)

assuming a constant stroke volume $Q_{SV}$, approximated from clinical cardiac (CO) output data, and a literature-based coronary flow fraction, $\alpha$^47,48. To model dynamic conditions, the waveform was scaled to reflect instantaneous HR ($H{R}_{\text{instant}}$), using the corresponding coronary ostial area $A_{\text{ostial}}$ to calculate the maximum inlet velocity:

$$v_{\mathrm{m}\mathrm{a}\mathrm{x}}=\frac{2Q_{\text{baseline}}}{A_{\text{ostial}}}\cdot \frac{H{R}_{\text{instant}}}{H{R}_{\text{baseline}}}$$ (4)

The cardiac period was scaled according to the following:

$${\tau }_{\mathit{cardiac}}=\frac{60}{H{R}_{\mathit{instant}}}$$ (5)

This scaling approach was adopted intentionally to maintain consistent boundary conditions across cases and isolate HR as the primary variable of interest when assessing temporal convergence behavior, particularly for the largest possible shift between two consecutive cardiac cycles. By keeping waveform characteristics such as the systolic-to-diastolic ratio constant, any observed differences in convergence could be attributed solely to heart rate changes. In patient-specific coronary artery simulations, waveforms are often filtered and scaled to patient-specific blood pressures and heart rates at rest and stress to enable controlled comparisons between states⁴⁹. Prior CFD models of coronary flow have similarly scaled resting waveforms by flow reserve factors to represent hyperemia, preserving waveform morphology while varying flow or heart rate to isolate specific effects⁵⁰. Such scaling approaches are widely used in cardiovascular modeling to reduce complexity and focus on primary variables while maintaining a consistent baseline waveform shape for controlled analysis⁵¹.

We employed constant 2-element Windkessel models for the outlet boundary condition, consisting of peripheral resistance ($R_p$) and compliance ($C$). The Windkessel models were applied to the ends of each terminal branch to account for microvascular hemodynamics⁵²:

$$Q=\frac{P}{R_p}+C\frac{dP}{dt}$$ (6)

$C$ was assumed to be constant at 1 × 10⁻¹² m³mmHg⁻¹ based on⁵³. $R_p$ was distributed among the terminal vessels using relationships commonly used for patient-specific coronary simulations^{1,33,41,47,54}:

$$R_i=P_{\mathit{mean}}\frac{\sum _{j=1}^{N_{\mathit{terminal}}}{r}_j^3}{Q_{\mathit{ostial}}{r}_i^3}$$ (7)

where $R_i$ is the peripheral resistance in each terminal vessel, $P_{\mathit{mean}}$ is the mean arterial pressure, $Q_{\mathit{ostial}}$ is ostial flow rate, $r$ is the radius of the average terminal branch, and $N_{\mathit{terminal}}$ is the number of terminal branches. We used a constant ostial flow rate in the resting state to parameterize the outflow conditions.

$P_{\mathit{mean}}$ was computed from measured systolic and diastolic blood pressures (SBP and DBP, respectively):

$$P_{\mathit{mean}}=\frac{\mathrm{S}\mathrm{B}\mathrm{P}+2\mathrm{D}\mathrm{B}\mathrm{P}}{3}.$$ (8)

Patient-level SBP, DBP, CO, dominance, and terminal-branch counts are provided in Supplementary Table S2.

C. Hemodynamic Parameter Analysis

Three key hemodynamic parameters were evaluated in this study across all simulations: pressure distribution, velocity, and WSS. The pressure values computed in the lattice units were converted to physical units using a calibrated transformation by the Boltzmann density field of the lattice and matched to clinical manometric pressure data, following the procedure described in^1,5. To assess the pressure drop across the lesion, we computed the pressure gradient (PG) along the left anterior descending artery (LAD) in multiple cross-sectional slices. PG was defined as the ratio of beat-averaged cross-section-averaged pressure downstream of the stenosis $\left(P_d\right)$ to that at the ostium $\left(P_a\right)$:

$$PG=\frac{P_d}{P_a}$$ (9)

For complete 3D comparisons, the pressure distribution was evaluated point-wise across all fluid nodes in the domain. Velocity was extracted both as pointwise velocity vectors and as cycle-averaged magnitudes at each lattice point. WSS was computed at fluid nodes adjacent to vessel walls using a method adapted from⁵⁵, with the WSS vector ${\tau }_i$ given by:

$${\tau }_i=-\frac{\mu \omega }{c_s^2\rho }{f}_{\alpha }^{neq}{c}_{\alpha j}{n}_j\left(c_{\alpha j}-c_{\alpha k}{n}_i{n}_k\right)$$ (10)

where $f_{\alpha }^{\mathit{neq}}$ is the non-equilibrium distribution function, $\mu$ is the blood viscosity, $\omega$ is the relaxation rate, $\rho$ is the fluid density, $c_{{\alpha }_j}$ is the lattice velocity vector, and $n_j$ is the outward unit normal at the vessel wall. The magnitude of WSS was computed as the norm of ${\tau }_i$. To evaluate differences across simulation conditions, we performed pointwise comparisons of WSS magnitude for each geometry and computed the distribution of absolute differences across all cases.

D. Evaluation

metrics and tolerance To assess the convergence of hemodynamic waveforms, we employed two complementary evaluation metrics: the percent difference and difference in relative $L_2$ norm. Percent difference provides a relative measure of deviation between two time-varying waveforms and is defined as:

$$\mathrm{Percent}\mathrm{difference}=\frac{\sum \left|{\overrightarrow{x}}_1-{\overrightarrow{x}}_2\right|}{\sum \left|{\overrightarrow{x}}_2\right|}\times 100$$ (11)

Here, ${\overrightarrow{x}}_1$ and ${\overrightarrow{x}}_2$ are vectors representing the evaluated and reference waveforms, respectively, over 40 time points of a cardiac cycle at a given cross-sectional slice. This metric is well suited for slice-based comparisons with relatively few spatial points, where cumulative absolute differences provide an interpretable measure of localized hemodynamic variation^31,32. The relative $L_2$ norm quantifies spatial deviations between two solutions across the full 3D domain and is computed as:

$$\mathrm{Relative}{L}_2\mathrm{norm}=\frac{{‖{\overrightarrow{y}}_1-{\overrightarrow{y}}_2‖}_2}{\sqrt{\sum {\overrightarrow{y}}_2^2}}$$ (12)

Here, ${\overrightarrow{y}}_1$ and ${\overrightarrow{y}}_2$ are vectors containing values of pressure, velocity, and WSS at each fluid point in the domain at a specific cardiac timepoint. This difference in relative $L_2$ norm captures the energy-normalized Euclidean distance between the reference and ground-truth waveforms and is suitable for time-dependent convergence involving millions of fluid and shell points^9,56. In this study, the relative $L_2$ difference was evaluated at three physiologically important cardiac phases: ventricular end-diastole (T_ED), corresponding to the completion of ventricular filling; end-systole (T_ES), marking the end of ventricular ejection; and end of the cardiac cycle (T_End), representing a full beat-to-beat transition. Both the domain-averaged and maximum (98th percentile) values of the relative $L_2$ difference were examined to capture overall and peak deviations. A 5% threshold is commonly used in cardiovascular CFD studies as a clinically relevant standard for convergence^57–59. To support stricter convergence assessment without significantly increasing computational cost, we adopted a 2% threshold^60,61 for percent difference and an equivalent ${\mathrm{L}}_2$ norm of 0.02⁶² which provides a practical balance between physiological accuracy and simulation efficiency for dynamic-state modeling.

E. Methodological framework for evaluating convergence under dynamic heart rate transitions

To assess the temporal convergence of simulated hemodynamics in dynamic conditions, we developed a systematic framework leveraging both real-world and synthetic heart rate (HR) transitions. For each heartbeat or region of interest (ROI) within a wearable-derived HR sequence, a pulsatile inlet velocity waveform was generated by scaling a reference waveform from the literature, as described in Section IIB. To analyze the hemodynamic behavior in the stenosed LAD artery, three axial cross-sections were selected downstream of the ostium (used as the reference point for PG calculations): (1) immediately post-stenosis, (2) post-stenotic recovery region, and (3) a distal, near-outlet locations (Figure 1A). These locations were chosen for consistency across all patient-specific geometries and represent regions of high clinical and hemodynamic relevance..

FIG. 1.

Overview of the methodological framework for assessing temporal convergence in coronary simulations under heart rate (HR) transitions. (A) Inlet waveforms are scaled from literature using wearable-derived heart rate data and applied to patient-specific coronary geometries. All patients have stenosed left anterior descending artery (LAD) where three arterial slice locations were specified distal to the 1. ostium (reference point for PG) slice namely - 2. immediately after stenosis, 3. post-stenosis, and 4. near-outlet. (B) For each heartbeat of interest, simulations with varying pre-flows ($N_{pre}$) are compared to a ground-truth (GT) ($N_{pre}=5$) on the heartbeat or region of interest (ROI) selected from any real-world HR sequence to assess convergence. (C) Two extreme HR transitions: 200→50 bpm and 50→200 bpm – are modeled to bracket dynamic fluctuations in HR sequence.

Given a continuous HR time series, the primary objective was to determine the number of preceding cardiac cycles (denoted as $N_{pre}$) required for the simulated hemodynamics to converge within the region of interest. For each selected ROI, simulations were repeated using increasing values of $N_{pre}=\mathrm{1,2},3,\dots$, and compared against a ground-truth (GT) simulation using $N_{pre}=5$ (Figure 1B)¹. While convergence may occur in fewer than five cycles for some metrics, we selected $N_{\mathrm{p}\mathrm{r}\mathrm{e}}=5$ as a conservative standard to ensure robust GT across all anatomical models and hemodynamic variables^32,63,64. This choice is consistent with our prior workflow, where five pre-flow cycles were used to define convergence in patient-specific CFD simulations^1,30,31. All three hemodynamics parameters were analyzed on the region of interest (ROI) between the reference and GT waveforms.

To ensure generalizability across a wide physiological spectrum, we modeled two extreme HR transitions: high-to-low (200 to 50 bpm) and low-to-high (50 to 200 bpm), representing the full range of resting and exertional HRs observed in population-scale studies^65,66 (Figure 1C). These scenarios bracket real-world variability and allow robust evaluation of temporal convergence in dynamic HR conditions.

F. Approach for assessing convergence under HR transitions

To systematically evaluate convergence behavior in dynamically evolving HR conditions, we developed a simulation approach to address two key questions: (i) how many cardiac cycles upstream of a rapid HR shift influence the ROI? (ii) for each such scenario, how many preceding cardiac cycles (pre-flows, $N_{pre}$) are required to achieve convergence.

For each HR transition scenario of high-to-low and low-to-high – we systematically varied the timing of the HR shift relative to the ROI. The shift timing was defined as the number of beats before the ROI at which the HR change occurred. For example, a shift occurring one beat before the ROI is expressed as:

$$\mathrm{S}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}\mathrm{T}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{g}:\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}=\mathrm{R}\mathrm{O}\mathrm{I}-n$$ (13)

This process is depicted in Figure 2, and the low-to-high case in Figure 3, each illustrating the shift construction and waveform comparison approach used to assess convergence.

FIG. 2.

Construction of subcases for the high-to-low heart rate transition (200 to 50 bpm). Regions of interest (ROI) are selected after the shift at beats ROI–1, ROI–2, ROI-3, …, ROI-n, and each ROI is compared with varying numbers of pre-flows ($N_{\mathit{pre}}=\mathrm{1,2},..$) to a ground-truth simulation ($N_{\mathit{pre}}=5$) to evaluate convergence.

FIG. 3.

Setup of subcases for the low-to-high heart rate transition (50 to 200 bpm). Each region of interest (ROI) beat is chosen post-transition (shift = ROI-1, ROI–2, ROI–3,…., ROI-n) and simulated with an increasing number of pre-flows ($N_{\mathit{pre}}=\mathrm{1,2},..$). Convergence is assessed relative to a ground-truth waveform using $N_{\mathit{pre}}=5$.

For each value of $n$, we compared the corresponding ROI waveforms (e.g., n = 1 vs n = 2) to identify the temporal extent to which upstream HR shifts continue to affect downstream hemodynamics. This approach established the number of meaningful subcases required to capture the influence of HR transitions.

Within each identified subcase, we then varied the number of pre-flows ($N_{pre}=1,2,3,\dots$) and compared the resulting hemodynamic waveforms to a GT simulation with $N_{pre}=5$. This analysis allowed us to determine the minimum number of pre-flows necessary to eliminate transient effects and achieve physiologically accurate hemodynamic predictions at the ROI.

III. RESULTS AND DISCUSSION

A. Heart rate transitions influence hemodynamics for up to two cardiac cycles

We evaluated the impact of upstream HR transitions on hemodynamics by comparing ROIs’s where shifts occur between one and four cardiac cycles (n = 1 to 4) prior to ROI. As shown in Figures 4 and 5, HR changes occurring within two beats prior to the ROI significantly influenced PG, velocity, and WSS, whereas shifts three or more beats upstream had negligible effects. This trend was consistent for both high-to-low and low-to-high transitions, supporting a two-beat temporal window for convergence assessment.

FIG. 4.

Percent difference in hemodynamic metrics across LAD cross-sectional slices under dynamic heart rate transitions in post-shift regions. (A) High-to-low HR transitions with comparisons between successive regions of interest (ROIs) where shift occurs at: (a) ROI–1 vs ROI–2, (b) ROI–2 vs ROI–3, and (c) ROI–3 vs ROI–4. (B) Comparison of ROI’s for low-to-high HR transitions with shifts at (d) ROI–1 vs ROI–2, (e) ROI–2 vs ROI–3, and (f) ROI–3 vs ROI–4. Percent differences in pressure gradient (PG), velocity, and wall shear stress (WSS) are computed at key LAD locations: post-stenosis, recovery region, and near-outlet. Both transitions had 5 pre-flows in pre-shift region for stable comparison. Convergence is evaluated using a 2% threshold and panels with red shading indicate comparisons exceeding the threshold, while green indicates convergence within the accepted tolerance

FIG. 5.

Relative ${\mathrm{L}}_2$ difference analysis averaged over the full 3D coronary domain in post-shift regions at three cardiac timepoints: end-diastole (T_ED), end-systole(T_ES), and end of cycle (T_End). (A) High-to-low HR transitions, comparing successive ROI’s with shifts at (a) ROI–1 vs ROI–2, (b) ROI–2 vs ROI–3, (c) ROI–2 vs ROI–3. (B) Low-to-high HR transitions with shifts (ROI-n) at (d) ROI–1 vs ROI–2, (e) ROI–2 vs ROI–3, (f) ROI–2 vs ROI–3. Convergence is assessed using an ${\mathrm{L}}_2$ threshold of 0.02 for pressure, velocity, and wall shear stress (WSS) and both transitions had 5 pre-flows in pre-shift region for stable comparison. Red-shaded panels indicate differences exceeding tolerance; green panels indicate converged results

1. Percent difference in cross-sectional hemodynamics is highest immediately after transition

Figure 4 presents the percent differences in average PG, velocity and WSS at three locations along the LAD stenosis, post stenosis, and near the outlet, comparing ROIs generated from HR shifts occuring $n$ beats before the ROI. For both cases A and B, with 5 cardiac cycles are there prior to an extreme shift, we observe that when the HR shift occurs one beat before ROI (n = 1), the percent difference relative to n = 2 exceeds the 2% threshold at multiple locations. This trend is especially pronounced in the velocity and the WSS. WSS is highly sensitive to transient flow characteristics and is known to reflect shear-driven endothelial responses^30,67,68. Notably, as the HR shift moves further away from the ROI (i.e., n = 2 vs n = 3 and n = 3 vs n = 4), percent differences drop below the 2% threshold in all measured locations. This finding supports the conclusion that HR changes that occur within two beats prior to ROI (that is, n ≤ 2) can significantly influence local hemodynamics, while shifts further upstream (n>2) have minimal impact, effectively decoupling ROI from earlier dynamic transitions.

2. ${\mathit{L}}_2$ domain difference shows flow stabilization by two beats after HR shift

Relative ${\mathrm{L}}_2$ difference (Figure 5) was averaged over full 3D domain at three cardiac timepoints – T_ED, T_ES, and T_End, comparing ROI’s with n vs n + 1 for n = 1 to 3, where shift = ROI - n. For both transition cases, ${\mathrm{L}}_2$ difference values remain below the 0.02 threshold for ROI’s with n= 2 vs n= 3 and n= 3 vs n = 4 across all timepoints. However, in the ROI’s with n = 1 vs n = 2 comparisons, T_ED exhibits noticeable deviations above the threshold for multiple parameters, while T_ES and T_End remain more stable.

Together with the slice-wise results, 3D domain analysis confirms that an upstream HR shift as early as two beats before the ROI (n = 2) is sufficient for achieving convergence across both localized and whole-domain hemodynamic metrics. This observation underscores the importance of pairing domain-level analysis with localized assessments to fully capture convergence behavior under dynamic conditions. Similar findings were observed when maximum (98th percentile) ${\mathrm{L}}_2$ differences were computed (shown in Supplementary Figure S1).

3. WSS deviations localize near stenosis and curvature

To resolve fine-grained hemodynamic differences induced by HR transitions, we examined absolute WSS differences in points throughout the coronary wall (Figure 6). In the high-to-low case (panel A), the n = 1 vs n = 2 comparison reveals widespread WSS deviations, with localized peaks exceeding 1.7 × 10⁻³ Pa near the stenotic region. However, the n = 2 vs n = 3 comparison shows a marked attenuation in spatial difference, with most differences falling below 1.1 × 10⁻⁴ Pa.

FIG. 6.

3D point-to-point differences in WSS across cardiac cycles for a representative patient geometry at end-diastole timepoint. (A) In the high-to-low HR transition, WSS differences decrease markedly from (a) n = 1 vs. n = 2 to (b) n = 2 vs. n = 3 when shift occurs at ROI-n. (B) Similarly, in the low-to-high HR transition, WSS differences diminish substantially from (c) n = 1 vs. n = 2 to (d) n = 2 vs. n = 3 when shift occurs at ROI-n. These spatial comparisons highlight the rapid stabilization of WSS following HR shifts, supporting the use of two pre-flows for achieving physiologically consistent simulations. Red and green shaded areas indicate converged and unconverged beats, respectively

A similar trend is observed in the low-to-high case (panel B), where the WSS differences for n= 1 vs. n= 2 reach 3.3 × 10⁻⁴ Pa, particularly near bifurcations and curved segments. These differences drop significantly in the comparison n = 2 vs. n = 3, with maximal deviations below 5.1 × 10⁻⁵ Pa.

These spatial patterns reinforce the conclusion that HR shifts occurring more than two beats upstream produce minimal residual influence on WSS. In particular, the peak differences in WSS align with areas of high curvature and stenosis, where flow instabilities are most sensitive to upstream perturbations. This finding further supports the two-beat stabilization window to ensure convergence in wall-dependent biomarkers essential for clinical relevance.

B. Two pre-flows are sufficient to ensure convergence at ROI across hemodynamic metrics

Temporal convergence was affected not only by the position of the HR shift, but also by the number of cardiac cycles simulated before. We analyzed within two pre-flows ($N_{pre}=2$), the parameters converged across two subcases (n=1 and 2) for each of the two dynamic shift scenarios.

1. 1–2 pre-flows ensure convergence at LAD cross-sections depending on transition type

Table I demonstrates for high-to-low transitions, both ROI beats with shifts at n = 1 and n = 2 achieved convergence with $N_{pre}=1$ across all parameters, with percent differences remaining below 2%. This behavior is consistent across all three slice locations for these two ROIs highlights the robustness of early convergence in decelerated HR conditions. On the other hand, for low-to-high transitions, velocity and WSS required $N_{pre}=2$ to drop below the threshold. The elevated sensitivity to shifts occurring towards higher heart rate transition reflects the heightened inertia and transient development in flow structures. The trend of decreasing percent differences with increasing number of pre-flows for all cases have been showed in Supplementary Figure S2.

TABLE I.

Required N_pre for convergence across slice locations (a= post-stenotic, b= recovery region, and c= near outlet) and HR transitions at 2% tolerance.

Location	High-to-Low shift						Low-to-High shift
	ROI-1			ROI-2			ROI-1			ROI-2
	PG	V	WSS	PG	V	WSS	PG	V	WSS	PG	V	WSS
a	<1	1	1	<1	1	1	<1	2	2	<1	2	2
b	<1	1	1	<1	1	1	<1	2	2	<1	1	1
c	<1	1	1	<1	1	1	<1	1	2	<1	1	1

PG = Pressure Gradient, V = Velocity, WSS = Wall Shear Stress.

2. Whole-domain relative ${\mathit{L}}_2$ difference confirms need for two pre-flows in delayed ROIs

The relative ${\mathrm{L}}_2$ difference trends in Figure 7 reinforce the slice-wise results. For both types of transition, the ROI beats immediately after HR shift (n = 1) converged with $N_{pre}=1$ in the T_ED, T_ES, and T_End phases. However, for ROI beats located two cycles after the shift (n = 2), convergence in WSS and velocity required $N_{pre}=2$. Notably, the relative ${\mathrm{L}}_2$ dropped from above the 0.02 threshold at $N_{pre}=1$ to well below it at $N_{pre}=2$, suggesting the necessity of additional upstream flow cycles to fully resolve transient flow evolution in those delayed cases. In several cases, the difference in relative ${\mathrm{L}}_2$ norm exceeded 0.02 for WSS at $Npre=1$ (e.g., T_ES phase in panel D and F), highlighting the persistence of upstream transient effects. At $N_{pre}=2$, all differences consistently dropped below the threshold in all phases and metrics. This consistency reinforces the need for two pre-flows when simulating hemodynamic states not immediately influenced by the HR shift, where residual transients persist over multiple cycles. Comparable trends were also observed when evaluating the maximum (98th percentile) relative ${\mathrm{L}}_2$ differences for this analysis, as presented in Supplementary Figure S3.

FIG. 7.

Influence of number of pre-flows ($N_{\mathrm{p}\mathrm{r}\mathrm{e}}$) on temporal convergence averaged over full 3D domain. (A,B) High-to-low subcases with HR shifts at ROI-1 and ROI-2. (C,D) Low-to-high subcases with HR shifts at ROI-1 and ROI-2. For each case, results are shown for $N_{\mathrm{p}\mathrm{r}\mathrm{e}}=1$ (a, c, e, g) and $N_{\mathrm{p}\mathrm{r}\mathrm{e}}=2$ (b, d, f, h) across three cardiac timepoints: end-diastole (T_ED), end-systole(T_ES), and end of cycle (T_End) with respect to the ground-truth of $N_{pre}=5$. Relative ${\mathrm{L}}_2$ differences are averaged across all patients, highlighting the improved convergence achieved with two pre-flows under both transition types. Purple and orange shaded panels indicate shifts at ROI-1 and ROI-2, respectively.

3. Pointwise WSS differences drop sharply with second pre-flow

Figures 8 and 9 provide pointwise maps of WSS differences for n = 2 ROIs. In the high-to-low case (Figure 8), WSS differences at $N_{pre}=1$ ranged from 5.4 to 16.8 Pa, with high gradients observed near the bifurcation and along the lesion boundary. With $N_{pre}=2$, the differences reduced dramatically, falling within 0.092 to 3.9 × 10⁻⁴ Pa, especially at end-diastolic and end-systolic phases.

FIG. 8.

Point-to-point differences in WSS magnitude for a representative patient geometry under a high-to-low HR transition. Results compare simulations using (A) $N_{pre}=1$ and (B) $N_{\mathrm{p}\mathrm{r}\mathrm{e}}=2$ against the ground truth ($N_{\mathrm{p}\mathrm{r}\mathrm{e}}=5$). Contour maps illustrate spatial differences at three cardiac timepoints: end-diastole:(T_ED) (a,d), end-systole: (T_ES) (b,e), and end-of-cycle: (T_End) (c,f). The reduction in WSS discrepancies with $N_{\mathrm{p}\mathrm{r}\mathrm{e}}=2$ demonstrates improved convergence following HR transitions.

FIG. 9.

Point-to-point differences in WSS magnitude for a representative patient geometry under a low-to-high HR transition. Simulations with (A) $N_{\mathrm{p}\mathrm{r}\mathrm{e}}=1$ and (B) $N_{\mathrm{p}\mathrm{r}\mathrm{e}}=2$ are compared to the ground truth ($N_{\mathrm{p}\mathrm{r}\mathrm{e}}=5$). Contour maps show WSS differences at three cardiac timepoints: end-diastole:(T_ED) (a,d), end-systole: (T_ES) (b,e), and end-of-cycle: (T_End) (c,f). The reduction in WSS discrepancies with $N_{\mathrm{p}\mathrm{r}\mathrm{e}}=2$ indicates improved spatial convergence following accelerated heart rate transitions.

Similarly, for the low-to-high case (Figure 9), regions of high WSS variation at $N_{pre}=1$, extending from mid-vessel to distal segments showed absolute differences up to 16.8 Pa. At $N_{pre}=2$, spatial differences fell below 1.0 Pa in most regions, and below 0.4 Pa in distal LAD. These spatial reductions confirm that two pre-flows are sufficient for achieving physiological convergence across the vascular wall, even under abrupt HR transition conditions. Importantly, this behavior was phase-consistent, confirming that the observed convergence is not only spatial but also temporally uniform across the cardiac cycle. These reductions in near-wall stress support the selection of $N_{pre}=2$ as a physiologically justified standard for minimizing transient effects and ensuring convergence.

This analysis highlights that while one pre-flow may suffice for immediate post-transition states, a second pre-flow becomes necessary to resolve hemodynamic stability in slightly delayed regions of interest. This threshold ensures both global and localized convergence and provides a practical guideline for improving efficiency in dynamic cardiovascular digital twin simulations.

C. Real-world HR data confirms generalizability of two pre-flow $N_{pre}=2$ condition

Having established that $N_{pre}=2$ ensures convergence across various synthetic HR shift scenarios, we sought to verify this conclusion using real-world HR data as shown in Figure 10A and Figure 10B. From a continuous one-hour sequence of HR data, two distinct 10-beat segments corresponding to the lowest and highest recorded HR values were extracted (Figure 10(a) and (b)). Specifically, the wearable-derived HR data included a resting segment with a global minimum HR of 48 bpm (Figure 10(c)) and a high-HR segment with a peak HR of 192 bpm (Figure 10(d)). These selections represent natural extrema in HR variability and allowed us to test $N_{pre}=2$ under representative edge-case dynamic inputs on one of the representative LCA geometries.

FIG. 10.

Verification of two pre-flows ($N_{\mathrm{p}\mathrm{r}\mathrm{e}}=2$) using real-world HR sequences during (A) low-HR and (B) high-HR segments. Continuous HR data were analyzed over one-hour periods, with (a) low-HR and (b) high-HR segments identified. Ten consecutive heartbeats with extreme HRs were selected: (c) minimum HR of 48 bpm during the low-HR segment and (d) maximum HR of 192 bpm during the high-HR segment. Temporal convergence was confirmed with $N_{\mathrm{p}\mathrm{r}\mathrm{e}}=2$ for PG, velocity, and WSS at the stenotic LAD location in a representative patient geometry for both (e) low-HR and (f) high-HR conditions.

Figures 10 (e) and (f) show the HR traces and the corresponding percentage difference values for each beat in the two selected HR segments. For the low-HR segment (panel e), HRs remained relatively stable with slight variability between beats. For all 10 beats tested at $N_{pre}=2$, the percent difference in average PG, velocity, and WSS at the stenosed LAD slice remained below 2%. Similarly, for the high-HR segment (panel f), where HR varied more dramatically between successive beats, all beats achieved convergence with $N_{pre}=2$ in all hemodynamic parameters at the stenotic location.

The percent difference values remained consistently below 2% for both HR segments, indicating whole-domain convergence. The fact that percent differences remained below the threshold despite modeling only heart rate–driven fluctuations, derived from wearable input data, demonstrates the robustness of using two pre-flows for convergence even under highly dynamic, real-life HR conditions. Importantly, convergence was observed consistently across all 10 heartbeats in each case, without requiring a beat-specific tuning of $N_{pre}$. This result suggests that $N_{pre}=2$ may serve as a generally applicable convergence guideline for dynamic digital twin simulations incorporating real-time wearable HR data.

Together, these findings provide strong verification for our earlier results and support the feasibility of using $N_{pre}=2$ as a standard for convergence assessment in both synthetic and wearable-informed digital twin models.

IV. LIMITATIONS AND FUTURE WORK

This study employed constant outlet boundary conditions to isolate the effects of upstream HR transitions on temporal convergence. Although this simplification allowed for standardized comparisons, it does not reflect physiologic variability in coronary microcirculation or downstream autoregulatory effects. Previous work has shown that distal resistance variation significantly affects coronary perfusion under exercise and pharmacological stress^69–71.

Future work should incorporate time-varying or patient-specific outlet models to capture autoregulatory behavior and enhance the realism of dynamic-state simulations. Incorporating these physiological responses will enable more accurate predictions of coronary hemodynamics during activity-induced transitions and further enhance the clinical relevance of coronary digital twins.

V. CONCLUSION

This study presents a systematic and physiologically grounded investigation into temporal convergence in patient-specific coronary simulations under dynamic HR conditions. We addressed two fundamental questions: (1) how far upstream an HR transition can influence hemodynamics in a downstream ROI, and (2) how many preceding cardiac cycles ($N_{pre}$) are required to achieve convergence in key hemodynamic outputs. To evaluate worst-case scenarios, we modeled extreme HR transitions – from high to low (200 to 50 bpm) and low to high (50 to 200 bpm), using high-resolution 3D CFD simulations across anatomically diverse, patient-specific coronary geometries. Our findings show that abrupt changes in HR can affect downstream hemodynamics for up to two cardiac cycles. Simulating two pre-flows ($N_{pre}=2$) was consistently sufficient to ensure convergence of pressure distribution, velocity, and WSS, both in cross-sectional slices and across the full 3D vascular domain. Notably, velocity and WSS were more sensitive to transient upstream conditions, especially near regions of stenosis and high curvature, while pressure distributions stabilized more quickly.

These trends were maintained in all synthetic HR transition cases and further verified using real-world HR sequences derived from wearable devices capturing naturally occurring low and high HR segments. Across all conditions, a two-beat initialization window provided physiologically stable and computationally consistent outputs without requiring manual tuning of simulation parameters.

Together, this work establishes a robust, verified, and generalizable framework to ensure temporal convergence in dynamic-state cardiovascular digital twins. This capability is particularly important for longitudinal disease surveillance, where digital twins are continuously updated using real-time sensing data. By ensuring convergence during transitions across physiological states, the proposed framework supports an accurate assessment of hemodynamic changes over time, improving the reliability of digital twin–based predictions in scenarios such as monitoring in-stent restenosis progression or evaluating treatment efficacy during stress tests. By defining $N_{pre}=2$ as a physiologically justified default, the study lays the foundation for a wearable-informed real-time modeling of coronary blood flow, allowing a greater adoption of digital twin technologies in clinical and remote monitoring applications.

Supplementary Material

See the supplementary material for Supplementary Table S1: Patient-specific clinical and geometric information; Supplementary Table S2:Patient-specific inputs for Windkessel parameterization; Supplementary Fig S1: Maximum (98th percentile) relative ${\mathrm{L}}_2$ difference analysis across full 3D coronary domain at three cardiac timepoints; Supplementary Fig S2: Determining number of pre-flows ($N_{\mathrm{p}\mathrm{r}\mathrm{e}}$) required for temporal convergence in high-to-low and low-to-high HR transitions, based on patient-averaged data; Supplementary Fig S3: Influence of number of pre-flows ($N_{\mathrm{p}\mathrm{r}\mathrm{e}}$) on temporal convergence with maximum (98th percentile) ${\mathrm{L}}_2$ difference taken over full 3D domain.

NOMENCLATURE

BGK

Bhatnagar–Gross–Krook

CFD

Computational fluid dynamics

FFR

Fractional flow reserve

HR

Heart rate

LAD

Left anterior descending artery

LBM

Lattice Boltzmann

LCA

Left coronary artery

PG

Pressure gradient

ROI

Region of interest

T_ED

Ventricular end-diastole time point

T_End

End of cardiac cycle time point

T_ES

Ventricular end-systole time point

WSS

Wall shear stress

$A_{\mathrm{ostial}}$

Coronary ostial area

$\alpha$

Coronary flow fraction

${\mathit{c}}_{\alpha }$

Discrete lattice velocity vector

$c_s$

Lattice speed of sound

$C$

Compliance

$\mathit{n}$

Outward wall unit normal

$\mathit{\tau }$

Wall shear stress vector

${\tau }_{\mathrm{cardiac}}$

Cardiac period

$f_{\alpha }^{\mathrm{neq}}$

Non-equilibrium LBM particle distribution

$H{R}_{\mathrm{baseline}}$

Baseline (resting) heart rate

$H{R}_{\mathrm{instant}}$

Instantaneous (dynamic) heart rate

$\mu$

Dynamic viscosity of blood

${\mu }_0$

Plasma viscosity n: Beat number in upstream shift

$n$

Beat number in upstream shift

$N_{\mathrm{p}\mathrm{r}\mathrm{e}}$

Number of pre-flow cycles

$P_a$, $P_d$

Proximal (ostial/aortic) and distal pressures

$\varphi$

Hematocrit

$Q$

Volumetric flow rate

$Q_{\mathrm{baseline}}$

Baseline coronary flow rate

$Q_{SV}$

Stroke volume

$R_p$

Peripheral (outlet) resistance

$\rho$

Density of blood

$v_{\max }$

Maximum inlet velocity

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.