Introduction
Measurements of fractional flow reserve (FFR) and/or coronary flow reserve (CFR) are widely used for hemodynamic characterization of coronary stenosis. An FFR value of 0.80, calculated as the mean distal and proximal coronary pressure in hyperemia, is the threshold below which percutaneous coronary intervention (PCI) and stent implantation are recommended [1–4]. Multiple clinical studies demonstrated that FFR is a consistent metric to support decisionmaking in treatment options during heart catheterization [1–4]. However, despite the highly significant difference between the PCI and the medical therapy groups in the primary endpoint [5], the risk assessment potential of FFR on the natural history can be interpreted to be limited. Therefore, the need for a more complex hemodynamic evaluation of coronary lesions is warranted.
Coronary flow reserve is defined as the ratio of a cycleaveraged coronary flow rate in the stenosed artery segment during maximum vasodilation and the resting condition. A threshold of hemodynamic significance is proposed at coronary lesions with CFR < 2.0. In contrast to FFR that addresses the hemodynamic consequences of focal stenosis of the conductance vessels, CFR could be the subject of processes that involve the entire coronary artery system. Thus CFR can also be impaired due to microvascular dysfunction originating from atherosclerotic or nonatherosclerotic processes [6–11]. CFR was an independent factor of mortality in patients with suspected coronary artery disease, providing incremental risk stratification over clinical variables and perfusion imaging [12, 13].
Despite the correlation between CFR and FFR, a relevant discrepancy between these two parameters may occur, reflecting distinct aspects of the coronary pathophysiology [14]. These differences underline the importance of a comprehensive description of stenosis physiology by both the pressure and flow characteristics [15].
The resistance of the microvascular system is the ratio of the distal coronary pressure divided by the distal coronary flow. During maximal vasodilation, the hyperemic (minimal) resistance of the microcirculation can be characterized by the thermodilution method by calculating the index of microcirculatory resistance (IMR): IMR = mean transit time × distal coronary pressure.
While the IMR is influenced significantly by the perfused myocardial mass [7–10], the Doppler flow velocity measurement shows the hyperemic microvascular resistance (HMR) without this limitation but with the laborious technical problems of the Doppler measurement. In cases when both the FFR and the CFR can be determined, the CFR/FFR ratio will provide useful information about microvascular resistance; < 2 value indicates microvascular dysfunction [16].
The microvasculature has become focused on the pathogenesis of coronavirus disease 2019 (COVID19)related cardiovascular involvement. The role of microvascular injury has arisen both in the acute phase and in the postCOVID cardiac syndrome provoked by possible immunemediated mechanisms [17, 18].
Such a diagnostic approach that can provide information on the microvasculature without having to apply further invasive device testing during a routine FFR measurement may serve as good guidance for followup on COVID19related microvascular abnormalities [19].
The present study aims at establishing a clinically applicable method for simultaneous CFR determination during a routine FFR measurement. We have developed a simple threedimensional (3D) angiography and intracoronary pressure measurementbased hemodynamic calculations which do not require the timeconsuming modeling of the flow. In this paper, this simple method was validated by the results of detailed unsteady and steadystate modeling of the flow using a finite volume method that solves the NavierStokes and continuity equations.
Methods
Study population
Nine patients (4 males, 5 females) who underwent invasive pressure wire measurement were selected for this retrospective study from 20 consecutive patients with stable coronary artery disease. Inclusion criteria were single welldefined intermediate stenosis (40–70% diameter stenosis by visual assessment) in a main branch of the coronary system with a diameter > 2 mm, appropriate angiographic views for 3D reconstructions, and good quality hyperemic and resting pressure traces with checking for pressure drift by the pullback of the pressure sensor at the end of the procedure. In addition, patients with an acute coronary syndrome left main coronary artery stenosis, ostial stenosis, previous coronary artery bypass graft surgery, or diffuse coronary artery disease were excluded. Segment detection, 3D reconstruction, and pressure measurement of 1 case are presented in Figure 1.
Detection of more than 2 mmHg drift at the end of the measurement was also an exclusion criterion. Among the 20 screened patients, 5 cases did not have appropriate angiograms for 3D coronary artery reconstruction from two angiographic views (with at least 25° difference), and in 6 cases more than 2 mmHg drift was detected and was a cause for exclusion. The remaining 9 patients were analyzed.
The present study was performed in concordance with the ethical standards laid down in the 1964 Declaration of Helsinki and its later amendments and approved by the Hungarian Office of Health Authorisation and Administrative Procedures (Project identification number: 44270/2013/OTIG). Patient characteristics can be found in Table 1.
Characteristics, Unit 
Data 
General 

Number of patients 
9 
Female, n 
5 
Mean age (SD), years (years) 
68.8 (8.2) 
CAD risk factors 

Hypertonia, n (%) 
9 (100) 
Diabetes mellitus, n (%) 
5 (55) 
Hyperlipidemia, n (%) 
2 (22) 
Chronic renal failure, n (%) 
2 (22) 
Cardiovascular conditions 

Aortic valve stenosis, n (%) 
3 (33) 
Previous ACS, n (%) 
2 (22) 
Interrogated coronary artery 

LAD, n (%) 
6 (66) 
LCx, n (%) 
1 (11) 
RCA, n (%) 
3 (33) 
FFR measurement 

FFR (SD), mean value (mean value) 
0.76 (0.14) 
Fractional flow reserve measurement
Fractional flow reserve measurements were performed using a pressuresensor guidewire via a 6 French (F) guiding catheter. First, the target artery was passed with an intracoronary pressure wire (RADI Medical, Uppsala, Sweden). After calibrating the wire, pressures were equalized with the sensor positioned at the level of the catheter tip. The sensor was then advanced through the coronary stenosis and measurements were performed approximately 2 cm distal to the lesion. 150–200 µg intracoronary adenosine was administered, and the ratio of the distal coronary pressure (pd) and the aortic pressure (pa) gave the FFR (Fig. 1B, lefthand side, green line). The resting pressure ratio was measured after the disappearance of the adenosine effect (defined by returning to the same pressure trace as before the adenosine injection) (Fig. 1B, righthand side).
Threedimensional quantitative coronary reconstruction
The invasive coronary angiography was followed by offline 3D angiographic reconstruction using validated software (e.g., QAngio XA Research Edition 1.0, Medis Specials bv, Leiden) [20–23]. During the automatic reconstructions, two angiographic recordings of good quality were used with at least 25° difference in angle. A target vessel segment in 3D, starting from the coronary orifice to the location of the pressure wire sensor, was generated for further analysis. Subsequent to the 3D reconstruction, parameters required for the computations, including the proximal reference crosssectional area, the average stenotic area, the distal reference area, the length of the proximal reference area, the stenotic area, the distal reference area, as well as the minimal lumen area were calculated using the aforementioned software. The average crosssections are determined automatically by the angiographic software based on volume equivalence; i.e., the crosssections are computed by dividing the segment volume by its length.
The software evaluates the shortaxis diameter (Fig. 1A, blue curve) and the reference diameter curve (Fig. 1A, dashed green curve) to find the above measurements. The shortaxis diameter curve is based on the ellipses along the centerline that defines the entire 3D reconstruction. The reference diameter curve is a linear approximation for a hypothetical healthy vessel. At first, the algorithm finds the minimal lumen area (MLA), marked with a red vertical line, this is the smallest crosssectional area in the stenosed segment. Then the software finds the intersection between the shortaxis diameter curve (blue) and the reference diameter curve (dashed green), both proximally and distally from the position of the MLA. These proximal and distal positions (marked with green vertical lines) are the boundaries of the stenosed segment. The Ap (proximal), As (stenosis), and Ad (distal) crosssectional areas are calculated accordingly.
Description of the simple method for the calculation of CFRModel (CFRM)
A flow resistance caused by an obstacle in a flow manifests itself in a pressure drop. As discussed above, the pressure drop, in general, has a linear term (as a function of the velocity or the flow rate), representing viscous friction losses (fQ), and a quadratic term which represents separation losses (sQ2). The method called the “simple model” used here is the reformulation of the similar methods in [14, 15, 24, 25] (Fig. 2):
∆pt = ∆pviscprox + ∆pviscsten + ∆psep + ∆pviscdist = ƒproxQ + ƒstenQ + sQ2 + ƒdistQ
Q is the volumetric flow rate; ∆pt is the total pressure drop; ∆pvisc(i) are the frictionrelated pressure losses, respectively for the proximal, stenotic, and distal part; and ∆psep is the separationrelated pressure loss term of the idealized stenosis. Thus, fi is the linear coefficient in the viscous pressure loss terms, and s is the quadratic coefficient in the separationrelated pressure loss term. A simple model, shown in Figure 2, consists of three segments: the proximal, the stenotic, and the distal segment. All three parts are straight, cylindrical, rigidwalled tube sections. The length of the stenotic segment, denoted by Ls in the model, is defined in Figure 1 as the segment where the red real diameter curve deviates from the blue reference diameter curve at more than a certain threshold. The steps of the volumetric flow determination are described in the patent of the method: https:/patents.google.com/patent/WO2019175612A2/en.
Computational fluid dynamics calculation for CFR
Computational fluid dynamics (CFD) techniques to study coronary artery disease is a widely used and quickly advancing research tool [26] used in pre [27–30] and poststenttreatment analysis [31]. After 3D reconstruction by QAngio XA, the geometries were imported to MeshLab [20] for further preprocessing to have a smooth surface for the CFD analysis. The software package ANSYS 18.2. (ANSYS Inc. Canonsburg, USA) was used for the 3D CFD analysis [21, 22] using the CFX numerical solver.
The measured aorta and distal pressure profiles were imposed on the inlet and outlet in unsteady simulations, respectively. CFRunsteady (CFRU) was calculated as the ratio of flow rates computed from resting and hyperemic conditions. The unsteady flow results were averaged within one heart cycle, and the CFRsteady (CFRS) was calculated from these data.
Using the ANSYS software, all numerical meshes consisted of about 1.5 to 2 million numerical cells consisting mostly of linear tetrahedral cells and 5 prismatic layers adjacent to the wall for better nearwall resolution (see the inset image in Fig. 3). In the present simulations, the blood was assumed to be an incompressible Newtonian fluid, and the vessel walls to be rigid. The density and viscosity were set to be 1055 kg/m3 and 3.5 mPas, respectively.
The setup of the steadystate simulations was as follows: the spatial discretization scheme was set to high resolution, which uses a blending function between firstorder and secondorder discretization according to the computation demands. The criterion 10−6 was set for the residuals, and the flow rate was monitored for convergence. A cycleaveraged pressure was calculated in the aorta and at the distal point in the coronary artery after the stenosis from the invasive pressure measurements. Since the flow rate is independent of the absolute pressure level, and depends only on the pressure difference, the difference of the twocycleaveraged pressures were imposed as total pressure at the inlet to drive the flow within the vessel section, and zero pa static pressure at the outlet. To obtain the ∆p(Q) relationship and eventually the f and s coefficients, a set of 5 steadystate simulations were carried out with increasing inlet pressure values, including those corresponding to the resting and hyperemic conditions. A parabolic curve was fitted on these 5 (∆p, Q) points, and the coefficients f and s were determined from the curve fitting. The measured aorta and distal pressure profiles were imposed on the inlet and outlet in the unsteady simulations, respectively. CFRunsteady (CFRU) was calculated as the ratio of flow rates computed from the resting and hyperemic conditions.
Because the pressure detection in the aorta using a fluidfilled system, and in the coronary artery, is based on electrical conduction, the time signal of the pressure wire is shifted in time (comes earlier) relative to the aortic pressure trace. This 2–5 ms shift is corrected before being used in the CFD simulations. A secondorder backward Euler method was chosen for the temporal discretization scheme with adaptive time stepping to fulfill the CFL criterion of 1.0. Three cycles were simulated, and only the last cycle was postprocessed to exclude any form of the initial transient. The unsteady flow results were then averaged within that one last heart cycle, and the CFRsteady (CFRS) was calculated from these data.
Results
In Table 2 a summary of the computed CFR values is reported.
calculated based on [16].
Case 
CFRM 
CFRS 
CFRU 
CFRM/FFR 
1 
1.14 
1.14 
1.15 
2.59 
2* 
1.61 
1.52 
1.46 
1.89 
3** 
1.27 
1.32 
1.02 
1.74 
4** 
1.31 
1.30 
1.34 
1.58 
5* 
1.33 
1.35 
1.42 
1.92 
6** 
1.08 
1.08 
1.06 
1.46 
7 
1.84 
1.74 
1.61 
2.16 
8 
2.67 
2.70 
3.27 
3.22 
9 
2.41 
2.11 
1.77 
2.73 
Average (SD) 
1.63 (0.57) 
1.58 (0.52) 
1.57 (0.68) 
2.14 (0.59) 
Correlation analysis 
r 
p 

CFRM–CFRS 
0.984 
< 10–5 

CFRM–CFRU 
0.876 
< 10–2 

CFRS–CFRU 
0.940 
< 10–3 
The correlation between the CFR was determined by the simple equations and from the steady flow simulation (CFRM−CFRS) was excellent (r = 0.984, p < 10−5). Table 2 also shows that the correlation between the results of the steady and the cycleaveraged unsteady simulations (CFRS−CFRU) remained good and significant, with a Pearson correlation coefficient of 0.94 and (p < 0.001). Finally, the correlation between the model values and unsteady simulations (CFRM–CFRU) remained pleasant (r = 0.876, p < 10−2).
Figure 4 shows the scatter plots of the comparisons and their corresponding BlandAltman analysis for the simple method, steady and unsteady state simulations. The BlandAltman analysis of the abovedemonstrated methods did not systematically skew the values in the investigated range.
The CFRM/FFR was derived as an analogue to the microvascular reserve characterizing the microvascular state [16]. Values below 2.0 indicate a microvascular dysfunction. All 5 of those patients with CFRM/FFR < 2 had clinical evidence for microvascular disease (Table 2).
The overall pressure loss (∆pt, the sum effect of the friction (fQ) and separation (sQ2) related pressure loss types) of each case, derived by the simple model, is smaller than that calculated by CFD simulation. In Figure 5, flow rate as a function of the pressuredrop (Q(∆p)), the diagram is shown containing 3 cases with decreasing resistances. The solid and dashed curves are defined by the simple model and the CFD simulations, respectively. The lowest resistance (lowest f and s coefficients) was obtained for Case 8 by the simple model (the solid blue steep curve in Fig. 5), rendering the flow rate calculation to be quite sensitive.
/second; CFR — coronary flow reserve; Dp — pressure loss; mmHg — millimeter of mercury.
While the actual numerical values of the quadratic losses are rather underestimated by the simple model because of the fact that vessel tortuosity and lesion eccentricity could be the sources for separationrelated losses but these are not incorporated into the simple model yet, the difference between the CFR values is small, being that the CFR is a ratiotype variable.
The present results for unsteady and steady flow simulations are in line with earlier studies [23, 32], that also relied on using pressure histories.
Discussion
Morris et al. [33] recently published a method for absolute flow calculation using similar pressure input data for fluid dynamic computation by ANSYS software (QCFD), this is the simplified method used herein. However, our simple method only requires an Excel sheet for the flow calculations, and — according to available research, the first one of its kind — furthermore, it is, to a large extent, automated and requires only a couple of minutes to perform the whole procedure.
The full 3D CFD simulations using the programs mentioned above are computationally demanding and need professional preprocessing. The preprocessing and meshing take around half an hour. In contrast, the setup and the computation of the steadystate and unsteady analysis takes around 1 hour and 2 days, respectively, utilizing a 4core 16 GB RAM computer. This time span makes it unrealistic to use this workflow outside the framework of clinical research. On the other hand, the proposed method with a simplified calculation of the CFR requires the most detailed possible data and shows results comparable to that of detailed CFD modeling. Therefore, this simple calculation method seems suitable for clinical applications that offer a more comprehensive evaluation than FFR measurement alone.
Apart from its small sample size and retrospective nature, some further limitations of the present study should be mentioned. First, the blood is a suspension of particles and the blood plasma, and on a microscopic level, it behaves like a nonNewtonian fluid; however, in the present CFD simulations, a Newtonian approach was used. This approximation is widely accepted and used by the hemodynamic research community.
The CFD simulations used “single tube” 3D reconstructions discarding side branches of the main segments. In this way, the same volumetric flow rate in the reconstructed coronary artery was assumed, not considering flow losses through side branches. Others showed that excluding side branches significantly affects calculating flow rates but can still be used to calculate a ratiotype value [34].
Conclusions
It was demonstrated herein, that during a routine FFR investigation, the measured intracoronary pressure data and 3D reconstructed coronary angiography offer a feasible means of calculating the volumetric flow rate and the CFR. The flow calculation needs very little time and can provide online comprehensive flowpressure relation. Validation of the current simple calculation technique has been carried out for coronary lesions of a smallsized patient population in a widely accepted finite volume solver ANSYS CFX.
The CFR and FFR (CFR/FFR) relation will be an indicator of microvascular function analog to microvascular resistance reserve. In the current study, some patients showed markedly low CFR values. Among them, most had stenotic aortic valves. This observation is in line with the earlier data showing that CFR values in patients with severe aorta stenosis are radically reduced despite visually intermediate coronary lesions [35, 36].
According to these values, the clinical decisionmaking on medical therapy or percutaneous/ /surgical revascularization could be guided more precisely. The CFR results from simplified calculations show strong agreement with those from steady flow simulations and thus may be accurate enough for clinical applications. For a comparison of the results a larger size study with established clinical flow investigation, e.g., by intracoronary Doppler wire measurements, is planned.
Acknowledgments
The research reported in this paper was supported by the BME NC TKP2020 grant of NKFIH Hungary. Áron Üveges was supported by the ÚNKP2003 New National Excellence, Program of the Ministry for Innovation and Technology from the source of the National Research, Development, and Innovation Fund.