The reasons why fractional flow reserve and instantaneous wave-free ratio are similar using wave separation analysis

The iFR is defined as the ratio of distal pressure (P_d) to aortic pressure (P_a) at rest during a wave-free period, as shown in Eq. 1:

Assuming that P_wedge or \({\text{P}}_{back} + {\text{P}}_{static}\) is baseline in the wave-free period, the slopes of P_d and P_a, and \({\text{P}}_{for} \left( {prox} \right)\) and \({\text{P}}_{for} \left( {dist} \right)\) in pre-systolic phase could be the same, as shown in Fig. 1. Thus, reconstructed iFR is redefined as follows in Eq. 2:

Reconstructed iFR is calculated as the average pressure of an entire cycle at rest and is not the same as conventional iFR, which is determined only in the wave-free period. Reconstructed iFR, which is the ratio of P_for at distal and proximal locations averaged over a whole cycle, is assumed to be similar to conventional iFR. This assumption will be shown to be appropriate by in vitro and in vivo experimental results in this study.

The corresponding FFR of the coronary artery (\(FFR_{cor}\)) is overestimated based on central vein pressure (P_v) instead of P_wedge considering collateral flow [4]. \(FFR_{cor}\) is defined as the ratio of distal pressure (P_d) and aortic pressure (P_a) when the effect of P_wedge is subtracted at hyperemia as shown in Eq. 3:

Reconstructed FFR is the ratio of P_for at the distal and proximal locations when P_wedge is assumed to be \({\text{P}}_{back} + {\text{P}}_{static}\) as shown in Eq. 4:

WIA was performed to obtain wave-free periods using representative flow speed (U) and pressure (P) as in Eqs. 5 and 6:

where ρ is the density of blood [1050 kg/m⁻³], and c is wave speed [m/s] calculated using the single-point equation. The wave-free period is defined as the time from WI (-) = 0 to the end of diastole for 5 ms [7].

WSA was performed to obtain P_for and P_back using representative flow (F(t)) and pressure (P(t)) obtained from Eqs. 7 and 8:

where Z_c is characteristic impedance and is defined as an input impedance (Z_i) in the absence of wave reflection. Z_i is defined as resistance or impedance obtained by frequency analysis of representative pressure and blood flow using Fourier analysis [9]. At the same time, the modulus (division) and phase (subtraction) of impedance were automatically calculated. Therefore, the impedance modulus at zero frequency (0-impedance) is mean pressure/mean flow. There are many methods of obtaining Z_c. In general, Z_c is defined as the modulus at the zero crossing point or a point close to zero in phase. The reason for this distinction is that the negative phase is the imaginary component of Fourier analysis. Previous studies have addressed Z_c with a fixed frequency [9‐15]. However, Z_c can be changed depending on the situation [9]. In this study, we used flexible Z_c, which is defined as the average modulus of four harmonics of the fundamental frequency after zero crossing or close to zero in phase less than 10 Hz. The process of calculating the reconstructed FFR and iFR is briefly shown in Fig. 2.

In this study, we designed an in vitro coronary blood circulation system. As shown in Fig. 3, a catheter (Combo Wire XT ®, Volcano Corporation, San Diego, CA, USA) was inserted into a tube to simultaneously measure pressure and flow speed at the proximal and distal sites of stenosis. A pulsatile pump (Model 55–3305, Harvard Apparatus Corp., USA) was used to mimic heart motion. The pump rate was fixed to 60 rotations/min, and the operative phase ratio (OPR; systolic time over a cyclic time) was set to 60%. The tube was filled with 1.5 L of Doppler fluid (Model 707, ATS Laboratories, USA). The viscosity of this fluid (5 cP) is similar to that of human blood. The tube was an IXAK® silicon tube (SL-0710, TOMMYHECO, KOREA) with an internal diameter of 5 mm. To reflect stenotic coronary arteries in the system, stenotic vessels of 48, 71, and 88% (minimum vessel area/maximum vessel area) were created using a three-dimensional printer. The minimal luminal dimensions of each model were 28, 46, and 64%. A Windkessel model was constructed using an air tank to control blood flow, pressure waveforms, and phase differences, which were similar to those observed in the human coronary artery. This approach can eliminate negative pressure and exerts zero flow on the system. Measurements were performed at 20 mm proximal to the stenosis site, and a catheter was inserted 200 mm proximal to the site of stenosis.

We created three conditions. Figure 3a is the basal condition. There is a combination of forward and backward pressures in the coronary artery. Resistance with stenosis can be used to control the ratio of forward and backward flow. By adjusting the inner diameter of the resistor, the amount of fluid directed to the stenotic phantom can be adjusted. Therefore, the ratio of forward and backward flow can be controlled, which makes it possible to reproduce a human-like automatic control ability of the blood flow. If the inner diameter of the resistor is larger than that of each site of stenosis, a phenomenon occurs where the backward flow is larger than the forward flow. Forward and backward pressures were separated using WSA in this condition. The reservoir was used to control P_static in the blood flow system, which was adjusted by varying the height.

Figure 3b is a condition of forward flow only without backward flow. High P_static and low P_static values are assumed to represent pre-hyperemia and hyperemia conditions, respectively. P_d/P_a values measured in pre-hyperemia and hyperemia conditions were assumed to indicate reconstructed iFR and reconstructed FFR, respectively.

To confirm that this in vitro circulatory system mimics the blood flow of the coronary system, the flow speed of Case 2 (only forward condition) was divided by that of Case 1 (basal condition), and the ratio was compared to the coronary flow reserve.

Figure 3c depicts the P_wedge. In clinical practice, P_wedge is measured at a distal site when the artery is blocked with a balloon. To reflect this in the in vitro system, we blocked the branch point toward the coronary phantom through the valve. The measured pressure in this case was compared with the backward pressure that was calculated using WSA.

The study protocol was approved by the institutional review board of Jeju National University Hospital (2016-07-011).

Coronary angiography and pressure-flow measurements were obtained using standard techniques [16]. Angiographic views were obtained following administration of intracoronary nitrate in all cases (200 or 300 µg). We used 0.014-inch pressure and Doppler sensor-tipped wires (ComboWire XT, Volcano Corporation, San Diego, CA, USA). The distal pressure was set to zero and equalized to the aortic pressure at the coronary ostium before being positioned at least three vessel diameters distal to the site of stenosis. Adenosine was administered for hyperemia by intravenous infusion based on 11 measurements (140 µg/kg/min). When a ComboWire was used, the electrocardiogram, pressures, and flow velocity signals were directly extracted from the digital archive of the device console (ComboMap, Volcano Corporation). Data were analyzed off-line, using a custom software package designed by Labview (National Instruments, Austin, TX, USA). Stenosed vessels were defined as vessels that had an angiographically visible stenosis from 40 to 70% severity, as determined visually by an operating physician at the time of coronary angiography.

Resting indices were calculated at a time of stability, allowing for a return to stable baseline conditions after any preceding injection of contrast or saline. Hyperemic indices were determined during stable hyperemia, excluding cases with ectopy or conduction delay. Representative flow and pressure waves were obtained by an average method using recordings of 5–15 consecutive cycles both at rest and during hyperemia. These procedures were necessary to achieve linearity and time invariance. FFR and iFR were calculated as the ratio of mean P_d to P_a at hyperemia during a whole cycle and at rest during a wave-free period, respectively.

The reconstructed FFR and reconstructed iFR were calculated by the following equation of \(\left( {{\text{P}}_{d} - {\text{P}}_{back} } \right)/\left( {{\text{P}}_{a} - {\text{P}}_{back} } \right)\) at hyperemia and at rest conditions, respectively.

All statistical analyses were performed using the Statistical Package for the Social Sciences (SPSS), version 23.0 software (SPSS Statistics for Macintosh, IBM Corp. Armonk, NY, USA). The values of continuous variables are mean and standard deviation (SD), and categorical variables are expressed as frequency and percentage. The comparison of continuous variables between groups was performed using the independent sample t-test, and the categorical variables were assessed with a chi-square test. The correlation analysis between groups was performed by simple correlation analysis. For each statistic, the significance level was less than 0.05.

A total of 18 cases were analyzed according to stenoses (48, 71, and 88%), and P_static (10 and 30 mmHg) values obtained from mock circulatory experiments. The measured and calculated data are summarized in Table 1.

When the static pressure was 10 mmHg, the distal flow ratio (only forward flow/basal flow) according to stenosis increased to 2.2, 1.5, and 1.2 as the stenosis increased to 49, 71, and 88%, respectively. When the static pressure was 30 mmHg, the distal flow ratios ​were 2.5, 1.6, and 1.2 at stenosis rates of 49, 71, and 88%, respectively.

The \(P_{d} /P_{a}\) ratio for Case 1 decreased in the order of stenosis (48, 71, and 88%) at each P_static (10 mmHg, 0.81, 0.64, 0.46; 30 mmHg, 0.91, 0.88, 0.60, respectively). The distal flow ratio in high P_static was higher than that in low P_static.

The \(P_{d} /P_{a}\) ratio for Case 2 decreased in the order of stenosis (48, 71, and 88%) at each P_static (10 mm Hg, 0.90, 0.84, and 0.5; 30 mmHg, 1.00, 0.96, and 0.67, respectively). The change in \(P_{d} /P_{a}\) between Case 1 and Case 2 was larger for low P_static than for high P_static, as shown in Fig. 4.

The waveforms and magnitude of the observed P_wedge and \({\text{P}}_{back} + {\text{P}}_{static}\) were very similar (Fig. 5). This trend was also observed in other cases. P_wedge always contains static pressure. Correlation with the \({\text{P}}_{back} + {\text{P}}_{static}\) and P_wedge was high (r = 0.990, p < 0.0001, Fig. 6), and the slope was 1.0612.

Nine patients in whom we were able to simultaneously measure blood flow and blood pressure in the proximal and distal regions were compared with pre- and post-hyperemia values of distal forward pressure P_for and distal backward pressure P_back using Z_c in 11 coronary blood vessels. The results are summarized in Table 2.

The correlations between conventional FFR and reconstructed FFR and between conventional iFR and reconstructed iFR were positive (r = 0.992, p < 0.001 and 0.930, p < 0.001, respectively; Fig. 7).

In this study, P_back was characterized as undergoing rapid decline and forming baseline observed during pre-systole either with or without hyperemia. This finding is similar to the characteristics of P_wedge [5]. After forming the baseline of P_back, the slope of P_for was similar to the slope of coronary pressure. The period of forming the baseline of P_back was similar to the wave-free period. Eventually, the amplitude of P_for was smaller than the amplitude of coronary pressure (Fig. 1). During the wave-free period, P_a, P_d, and P_for could have the same slope because P_back for ms the baseline. The ratio between the lines with the same slope may be different, but the value in that interval is constant. iFR is defined as P_d/P_a in the wave-free interval. Therefore, iFR may be related to P_for(distal)/P_for(proximal) during the wave-free period. Furthermore, as the amplitude of P_for without P_back is low, the mean P_for of the whole cycle and the mean P_for of the wave-free period may be similar as a factor of ratio. As a result, in this study, reconstructed iFR was defined as P_for(distal)/P_for(proximal) in Eq. 1. The reconstructed and conventional iFRs showed a good correlation based on in vivo results.

During hyperemia, the theoretical FFR of the coronary artery \((FFR_{cor} )\) is \((P_{d} - P_{w} )/(P_{a} - P_{w} )\), while the FFR of the myocardium \((FFR_{myo} )\) is \((P_{d} - P_{v} )/(P_{a} - P_{v} )\), where \(P_{v}\) represents the mean central venous pressure [4]. The FFR is the ratio between mean values. A mean value is decreased when both the peak and the baseline are lowered. In this study, hyperemia mainly reduced the baseline of pressure (Fig. 1). Moreover, P_back was not zero but still decreased during hyperemia, and P_for was constant under the Windkessel effect.

The difference between \(FFR_{cor}\) and \(FFR_{myo}\) is described by collateral flow [4]. P_wedge is closely related to the collateral flow [19]. In addition, hyperemia theoretically reflects the offset of P_wedge and \(P_{v}\) in the conventional FFRs [4]. However, the values of the P_wedge or \(P_{v}\) would not be practically removed in hyperemia.

The FFR is based on the assumption that resistances both with and without stenosis are the same. Without collateral flow, this assumption implies that FFRmyo progressively overestimates the FFR using flow with increasing stenosis severity [4]. Thus, an attempt has been made to overcome this mismatch in reconstructing the FFR using zero flow pressure (\(P_{zf}\)). The formula is as follows: \({\text{FFR}} = (P_{d} - P_{zf} )/(P_{a} - P_{zf} )\). FFR using \(P_{zf}\) was in good agreement with the FFR using flow [20]. Because of the diastolic characteristics of the coronary arteries, \(P_{zf}\) is independent of contraction and auto-regulation, showing conductance of the vessels and pure resistance [21‐23]. However, P_wedge is generally smaller than \(P_{zf}\) due to the non-linearity of the pressure-flow relationship and existence of cardiac contraction either with or without collateral flow [23‐25]. Conceptually, P_back by WSA was similar to \(P_{zf}\) in this study. This means that both FFR and iFR could be trans-stenotic ΔP_for, which can be expressed using the same formula, although their methods are different (Eqs. 1 and 2).

In order to replace the FFR using flow with FFR using pressure, hyperemia is required to offset P_wedge and P_v [4]. As mentioned above, the reconstructed iFR was calculated by subtracting P_back at rest, which is assumed to be P_wedge. Theoretically, P_for can be determined by the stroke volume, which is related with inflow, resistance, compliance, and volume capacity, because the Windkessel effect is observed and systolic resistance by subtracting P_for is absent [8]. It is similar to systemic circulation. When administered for hyperemia, adenosine is reported to have little effect on the stroke volume or ejection fraction [26]. There is no significant change in blood volume without bleeding. Therefore, the difference in P_for with or without hyperemia is mainly dependent on resistance. The change of resistance according to the situation from rest to hyperemia could be the change of P_static or P_v. Thus, the difference between iFR and FFR is likely to be the difference of P_for in relation to P_static or P_v rather than P_wedge or P_back.

As myocardium oxygen consumption (MVO₂) increases due to enlargement of micro-vessels, resistance is reduced, and flow is increased. This trend is mainly regulated by the adenosine and nitric oxide (NO) metabolites in the myocardium. In the presence of significant stenosis, the role of adenosine may be activated in micro-vessels, so the reactivity of hyperemia by adenosine may be lowered. In other words, resistance due to pharmacological hyperemia may be smaller in significant stenosis than in nonsignificant stenosis [27, 28].

This is the first paper to prove that iFR and FFR are theoretically related using WSA up to our knowledge. The incidence of clinically appropriate hyperemia is not well known. In fact, it is difficult to verify hyperemia even with constant drug increases or drug changes. Thus, nonsignificant changes of P_for during hyperemia may be explained by the limitations of the assumption of constant resistance either with or without stenosis in FFR, and pharmacological hyperemia with inappropriate offsets of P_wedge and P_v. Nevertheless, this study assumes that P_for is the primary factor for determining iFR and FFR using pressure. This assumption was confirmed by in vivo and in vitro results. Theoretically, the wave free period for iFR was made by WIA. The slope of P_for by WSA in the wave free period was similar to that of P_a and P_d in this period. Therefore, it will be possible to create a new algorithm of the wave free period for iFR.

In this paper, we tried to reflect the characteristics of various coronary arteries such as blood flow and pressure waveforms, in the human body. There are many differences in blood flow and pressure waveforms in human coronary arteries. However, this variation did not pose a problem because we used the average values for pressure and blood flow.

It cannot be said that P_back reflects P_wedge experimentally. The constituent waves from WSA are the estimated values [10]. Moreover, the purpose of this study was to prove that iFR and FFR share the same formula. Therefore, the most important factors are morphological pattern and phase; acquiring accurate values was not the main goal. Accordingly, several trials of WSA were performed considering different Z_c values. The results from various trials of WSA showed a similar pattern.

Z_c increased during hyperemia [14]. However, Z_c decreased in this study. Although this result cannot be explained, it is inferred that there are differences in the species or drugs used for hyperemia. To verify this hypothesis, additional experiments for Z_c will be needed.

The combo wire we used could measure pressure and flow at the catheter tip where Pd was measured. However, when measuring Pa, the flow rate was not measured. Measurements in the proximal region were also not performed. So we could not calculate the proximal forward pressure using WSA. In a future study, the pressure and flow of the proximal site will be measured to check the separated pressure of the proximal and distal sites.

When measuring pressure and flow, we extracted one averaged cycle from more than 5 cycles based on the ECG signal. Groups with atrial fibrillation and other arrhythmia were excluded from the analysis. The process of finding Z_c for WSA may vary from person to person. Therefore, we made a program through Labview software to minimize the error of observers. Three investigators calculated the WSA using the software, and there was little error although the accuracy was not quantified.