This study was approved by the institutional review board of Peking University People’s Hospital and the Second Affiliated Hospital of Zhejiang University School of Medicine. All patients signed an informed consent. 136 coronary heart disease patients with 154 moderate-to-severe epicardial stenosis were retrospectively enrolled (between 2019 and 2021). The patient inclusion criteria were shown in Figure 1. Under the guidance of coronary angiography based on the Azurion 7M20 DSA system, all patients had undergone the FFR catheter surgery measurement with FFR system and Verrata Plus pressure guide wire (Philips Healthcare, Netherland). The period between the CTA examination and the cardiac catheterisation did not exceed 1 week. The Biomechanics Laboratory of Beijing University of Technology analyzed the anonymized data independently.

The coronary CTA images were obtained using of a dual-layer detector CT system (IQon, Philips Healthcare), with a matrix size of 512 × 512 and a slice of 0.625 mm thickness. Segmentation and 3D reconstruction of the coronary artery for each patient were performed using Mimics (Materialise, Leuven, Belgium), with the results being reviewed by two radiologists with 15 years of experience in cardiac CTA. Only arteries with a diameter bigger than or equal to 1 mm were retained in the reconstructed model for further computational fluid dynamics (CFD) analysis (Sankaran et al., 2016).

In a previous study, we proposed a closed-loop geometric multiscale model to compute FFR (Liu et al., 2021) non-invasively. Although the closed-loop model has improved computational accuracy, the clinical application is limited due to its superior computational speed and complexity in determining individualized parameters. In order to fulfill the needs of clinical FFR calculation, a steady-state-based model mimicking the closed-loop geometric multiscale model was proposed in this study. It replaced the transient state with steady-state boundary conditions to reduce the computation time, optimize the geometric multiscale module, and reduce the optimization of individual parameters.

The steady-state geometric multiscale model consists of lumped parameter model (LPM) (0D) and a coronary global three-dimensional (3D) model. LPM uses the circuit structure to simulate the microcirculation network downstream of the coronary artery, the resistance “R” to simulate the resistance of blood flow, and the inductance “L” to simulate the inertia of blood flow (Pietrabissa et al., 1996). The full-scale 3D model of the coronary artery preserves the real structure of the coronary artery instead of only stenotic vessels. The 0D part provides the outlet boundary conditions for the patient’s 3D coronary model. As shown in Figure 2C, the details of the geometrical multiscale model of 0D and 3D coupling are described previously (Zhao et al., 2016; Li et al., 2020; Mao et al., 2020).

The inlet boundary condition of the coronary model is set to the aortic pressure, which could be equivalent to the mean pressure (Wilson et al., 2001) and calculated from the cuff pressure based on systolic blood pressure (SBP), diastolic blood pressure (DBP), and heart rate (HR) (Sharma et al., 2012) as follows: Pa = DBP + [ 1 3 + ( HR ∗ 0.0012 ) ( SBP − DBP ) ] (1)

The outlet boundary condition of the branch of the coronary model is composed of microcirculation resistance. The microcirculatory resistance of the downstream branch of the coronary artery was termed as resistance “R,” which could be estimated as follows: R resting = P Q (2) where P is the aorta pressure, and Q is the flow rate of blood in the target coronary branch while resting. The latter can be estimated using Murray’s Law (Murray, 1926) based on the patient’s cardiac output (Opie, 2003). Since the FFR needs to be calculated in the hyperemia state, according to the assumption (Wilson et al., 1990; Sdringola et al., 2011; Taylor et al., 2013), the R_resting becomes 0.24 of the original in the hyperemia state: R hyperemia = 24 % ∗ R resting (3)

Notably, in the computation model of FFR_SS for the resistance of the outlet of the ascending aorta (Figure 2C) connected to the systemic circulation, the resistance is calculated based on the cardiac output (Opie, 2003): R doa = P a c o ∗ 96 % (4) where Pa is the pressure at the aorta pressure, and CO is the cardiac output.

The model described in this study was similar to the 0D/3D coupling method (Zhao et al., 2016; Liu et al., 2021) of the previous closed-loop model and used specific interface conditions and coupling algorithms to establish the 0D-3D coupling model. The 3D model calculation of the whole coronary artery relies on the fluid calculation software ANSYS, while the calculation of the lumped parameter model relies on the FORTRAN program of the CFX junction box. The data transmission between them was completed by the CFX User CEL Function, and the specific geometric multiscale coupling model solution process is shown in Figure 3.

The above models were divided into tetrahedral meshes by the ANSYS ICEM CFD software. It was assumed that the vascular wall was rigid and impermeable without slippage, the blood material property was adiabatic and comprised of an incompressible viscous Newtonian fluid, and its flow was unsteady laminar flow. The density of blood flow was set to 1,050 kg/m^3, and the viscosity of blood was set to 0.0035 Pa s (Deplano et al., 2001).

The specific steps of the steady-state geometric multiscale model include four processes: 1) Based on the coronary CTA image, a patient’s accurate personalized epicardial coronary 3D model was established; 2) According to the segment 3D model, the boundary conditions of the inlet (mean pressure) and outlet (microcirculation resistance) were calculated, respectively; 3) The change in the coronary microcirculation resistance in the maximum hyperemia state was quantified, and 0D and 3D coupling was calculated; 4) The Navier–Stokes (N-S) equation of the intracoronary fluid was solved using the ANSYS software, the flow velocity and pressure in each coronary artery were obtained under the hyperemia state, and the FFR_SS was calculated (Figure 2).

The specific steps of closed-loop geometric multi-scale calculation of FFR_CT include five processes: 1) Based on the patient’s coronary artery CTA image, the patient’s accurate and personalized 3D model of the epicardial coronary artery and heart model is constructed. 2) According to the constructed three-dimensional model and based on the allometric scaling law, the branch flow of the coronary arteries and the coronary microcirculation resistance in the resting state (assuming that there is no stenosis) are determined. 3) A closed-loop 0D centralized parameter model is constructed to personalize the physiological parameters of the patients. 4) The change in the coronary microcirculation resistance under maximum hyperemia is quantified, and zero-dimensional and three-dimensional coupling calculations are performed. 5) The control equation (N-S) of the fluid in the coronary artery is solved using a calculation software to obtain the flow velocity and pressure in each blood vessel of the coronary artery under congestion, and FFRCT is calculated. The detailed of FFR_CT calculation steps refer to previous studies (Liu et al., 2021).

In this study, we calculated the FFR of 134 cases based on the steady-state and closed-loop geometric multiscale model and compared the computational accuracy of FFR_SS with clinical FFR and FFR_CT, respectively.

Data are summarized by descriptive statistics. Pearson correlation and linear regression analysis were performed to examine the relationship between FFR and FFR_SS and FFR_CT, respectively. Agreement between the methods was assessed by Bland-Altman plots with corresponding 95% limits of agreement. The optimal cut-off values for FFR_SS was computed based on maximizing the sum of sensitivity plus specificity. The sensitivity, specificity, accuracy, and area under the ROC curves (AUC) with 95% CI classification metrics were computed. Throughout this study, a p-value threshold of 0.05 was considered to infer statistically significant findings.

A total of 154 vessels in 136 patients (57% male, median age:60 years) were analyzed with stenosis severity of coronary lesions evaluated by CCTA and ICA ranging from 40 to 80% luminal narrowing. Invasive FFR interrogation assessed the presence of hemodynamically significant stenosis (FFR ≤0.80) in 154 vessels (28.57%, 44/154) of 136 patients. The clinical and demographics characteristics of the patients’ population are summarized in Table 1.

The medians (interquartile range) of the FFR, FFR_CT and FFR_SS in this study were 0.81 (0.33–0.99), 0.85 (0.34–0.98), and 0.84 (0.26–0.99), respectively. A scatter plot between FFR and FFR_SS is shown in Figure 4A, demonstrating moderate overall linear correlation between the 2 measures, with an r of 0.68 (95% confidence interval [CI]:0.21–0.39) (p < 0.001). Similarly, the correlation of FFR_SS and FFR_CT demonstrated an r of 0.75 (95%CI: 0.67–0.72) (p < 0.001) (Figure 4B).

Bland-Altman plots for FFR_SS are illustrated in Figure 4C. On average, FFR_SS exceeded FFR by 0.03 (95% CI: −0.043 to −0.009). Most of the points in the figure are distributed within the 95% confidence interval, indicating that there is good agreement between FFR and FFR_SS. Similarly, Bland-Altman plots for FFR_SS and FFR_CT are illustrated in Figure 4D. FR_CT exceeded FFR_SS by 0.01 (95% CI: −0.001 to −0.029).

The formula for calculating relative error: Relative   error = C a l c u l a t e   F F R − C l i n i c a l   F F R C l i n i c a l   F F R (5)

The relative error between FFR_SS and FFR is 0.11. The relative error between FFR_SS and FFR_CT is 0.067.

The diagnostic performance of FFR_SS and FFR_CT are assessed using clinically measured invasive FFR as the diagnostic criteria. The Youden index was 0.61 and the optimal cut-off was 0.80 for FFR_SS. The overall sensitivity, specificity, PPV, and NPV for FFR_SS ≤ 0.80 versus FFR ≤ 0.80 was 68.18% (95% CI: 52.4–81.4), 93.64% (95% CI: 87.3–97.4), 82.9%, and 91.1%, respectively, with an overall diagnostic accuracy of 88.3%. Similarly, the overall sensitivity, specificity, PPV, and NPV for FFR_CT ≤0.80 versus FFR ≤0.80 was 68.1% (95% CI: 52.4–81.4), 94.5% (95% CI: 88.5–98.0), 82.0%, and 89.5%, respectively, with an overall diagnostic accuracy of 87.6%.

According to the ROC receiver characteristic curve, the area under the curve of FFR_SS and FFR_CT are AUC = 85.7% (95%CI: [0.79–0.90]), AUC = 81.8% (95%CI: [0.74–0.87]), respectively. It suggesting that a good diagnostic performance is achieved of FFR_SS as shown in Figure 5.

The FFR_SS analysis of 6 representative patients was based on each narrowed vessel. We also list the clinical FFR, FFR_CT and FFR_SS of representative patients. The hemodynamic differences between FFR_CT and FFR_SS were compared based on clinically measured FFR (Figure 6). The stenosis of 6 representative patients stenosis was located in the anterior descending artery (LAD), and the degree of stenosis was 40–80%. It can be seen from the figure that the hemodynamic distribution calculated by FFR_SS is basically consistent with that of FFR_CT.

In the previous closed-loop geometric multiscale model (Liu et al., 2021), the physiological parameters of the patient had to be optimized to simulate the individualized physiological state, using transient periodic inlet boundary conditions. The FFR_SS model adjusts the transient boundary condition of the inlet to a steady-state and replaces the transient pressure with the average pressure, which markedly improves the calculation efficiency. Then, the transient and steady-state pressure waveforms of the coronary arteries calculated by the geometric multiscale model were compared (Figure 7). FFR_SS can replace FFR_CT because the steady-state pressure and the transient average pressure are the same, resulting in the same calculation of FFR.

In addition, the FFR_SS model saves the tedious process of optimizing the cardiac parameters of individual patients and replaces it with the average pressure, thereby improving the calculation efficiency and the clinical applicability of the model. The calculation time of FFR_CT is usually 8–9 h, while the calculation time for FFR_SS is only 20 min. Unlike other models that only consider stenotic vessels to calculate FFR (Itu et al., 2012; Itu et al., 2016; Zreik et al., 2018), the calculation model described in this study retains the complete coronary model, which can view the complete hemodynamic state of the coronary artery.

The inlet of the FFR_SS model adopts the mean pressure calculated based on the “physiological formula,” and the outlet adopts the microcirculation resistance model as the closed-loop geometric multiscale model. In a previous study, we presented a numerical investigation of the effects of the computational model’s inlet and outlet boundary conditions on computed CT-FFR. The mean pressure calculated by the “physiological formula” differed from the real aortic pressure wave (Tosello et al., 2021). However, the calculation model was not sensitive to the boundary conditions of the inlet pressure, i.e., the true aortic pressure could be replaced by the mean pressure calculated by the “physiological formula.” The findings revealed that distal boundary conditions (hyperemic vasodilation response of coronary micro-vessels) have a significant impact on FFR. Thus, improving the calculation accuracy of distal microcirculation resistance is the key to further improving the calculation of FFR_SS.

The calculation results show that compared to the closed-loop geometric multiscale model, the improved calculation model does not reduce the accuracy of calculating FFR. The accuracy of traditional FFR_CT is 84.3% (Taylor et al., 2013), and that of FFR_CT based on the closed-loop geometric multiscale model is 87.3% (Liu et al., 2021), which was similar to the 88.3% computational accuracy of FFR_SS proposed in this study. Compared to the closed-loop geometric multiscale model, the FFR_SS improves the computation speed, retaining the computation time within half an hour. Itu et al. proposed a non-invasive FFR calculation based on a neural network with an accuracy rate of 88.3% (Itu et al., 2016). Fredrik et al. (Fossan et al., 2021) proposed a non-invasive rapid calculation method of FFR based on an enhanced neural network, while the standard deviation of repeated FFR measurements was 0.018. However, the premise of improving the calculation speed in the above two studies was that only the coronary arteries in the stenotic segment are retained, and the other coronary arteries are ignored. The advantage of this study is that while improving the calculation speed, it retains the complete model and displays the hemodynamic positions of all coronary arteries, facilitating the diagnosis of myocardial ischemia.