In Vitro Assessment of Mathematically-Derived Fractional Flow Reserve in Coronary Lesions With More Than Two Sequential Stenoses

Abstract: Background. Myocardial fractional flow reserve (FFR) is a reliable index in coronary intervention. A simple FFR measurement does not predict the true functional severity of an individual stenosis in multiple sequential coronary stenoses because of complex interaction between the stenoses. Application of the theoretical equations to predict the true FFR of individual stenosis is limited in a tandem lesion. Two novel equations applicable to a multiple sequential coronary stenoses are mathematically derived. One predicts the true FFR of each stenosis (equation A), and the other predicts the true FFR after releasing a given stenosis (equation B). The present study aimed to validate the two derived equations in an in vitro model of coronary circulation. Methods and Results. Predictive FFR was compared with true FFR in an in vitro model of three sequential stenoses using linear regression analysis. The difference between apparent FFR and true FFR was compared with the difference between predictive FFR and true FFR. The legitimacy of equation A was first assessed. A close correlation was found between predictive FFR and true FFR (r² = 0.92). The difference between predictive FFR and true FFR was significantly lower compared to the difference between apparent FFR and true FFR (0.18 ± 0.10 vs 0.05 ± 0.05; P<.001). The legitimacy of equation B was also assessed, and a close correlation was found (r² = 0.97). The difference was significantly lower when we applied equation B (0.13 ± 0.06 vs 0.04 ± 0.02; P<.001). Conclusions. Equations A and B strongly predict the true value of FFR in the experimental model of coronary circulation.

J INVASIVE CARDIOL 2013;25(12):642-649

Key words: coronary stenosis, prediction

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Myocardial fractional flow reserve (FFR) is a reliable index to determine the functional significance of coronary stenosis.^1-3 However, a simple FFR measurement does not predict the true functional severity of an individual stenosis in multiple sequential coronary stenoses because of the complex fluid dynamic interaction between the stenoses. Theoretical equations to predict the true FFR of individual stenosis in a tandem lesion were proposed by De Bruyne and colleagues, and the legitimacy of the equations was confirmed in animal model experiments and clinical study.^4,5 However, application of the equation has been limited in a tandem lesion thus far. Coronary lesions with more than two significant stenoses are not rare in everyday clinical practice. We mathematically derived two novel equations applicable to multiple sequential coronary stenoses. Equation A predicts the true FFR of each stenosis, and Equation B predicts the true FFR after releasing a given stenosis in multiple sequential stenoses. The present study aimed to validate the two derived equations in an in vitro model of coronary circulation.

Methods 

Mathematical model. When a mathematical model of multiple sequential stenoses is constructed, stenosis X is defined as a given stenosis within the multiple stenoses (Figure 1A). P_a, P_prox, P_dist, P_d, and P_w indicate mean coronary pressure proximal to the first stenosis, mean coronary pressure proximal to stenosis X, mean coronary pressure distal to stenosis X, mean coronary pressure distal to the most distal lesion, and mean coronary wedge pressure, respectively. Note that P_a is considered constant when measuring P_w in the present study. P_w needs to be adjusted according to the change of P_a. The pressure gradient across stenosis X is expressed as ΔP and the FFR gradient is expressed as ΔFFR. All the parameters are given under maximum hyperemia. The resistance of the microvasculature is minimal and the central venous pressure (P_v) is close to zero. The FFR of entire stenoses (FFR_all) is calculated by dividing P_d by P_a. 

The apparent FFR of stenosis X (FFR(X)_app) is calculated by dividing P_dist by P_prox as described in the previous literature.⁵ 

The apparent FFR of entire stenoses after releasing stenosis X (FFR(X–)_app) is calculated by adding ΔFFR and FFR_all.

The hemodynamic significance of each stenosis is influenced by the presence of other stenoses. The predicted FFR of stenosis X (FFR(X)_pred) can be mathematically derived using the baseline pressure data as described in Appendix 1. 

The predicted FFR of entire stenoses after releasing stenosis X (FFR(X–)_pred) can also be mathematically derived. The derivation of the FFR(X–)_pred is described in Appendix 1. 

After releasing all stenoses except for stenosis X, the true FFR of stenosis X (FFR(X)_true) is calculated by dividing P´_dist by P´_prox (Figure 1B). In this status, P´_a is equal to P´_prox, and P´_dist is equal to P´_d.

After releasing stenosis X, the true FFR of entire stenoses (FFR(X–)_true) is calculated by dividing P˝_d by P˝_a (Figure 1C).

The apparent, predicted, and true values of FFR are compared in the present study. 

In vitro experiments. An originally developed coronary circulation model was employed to simulate multiple sequential coronary stenoses within the same coronary artery (Figure 2). The in vitro model consists of a pump, a systemic and coronary circulation, and constrictors. The pump produces a pulsatile flow at 60 rpm. Distilled water was used as the perfusate. The systemic and coronary circulation is made of silicone rubber tubes that mimic the human arterial system. The inner diameter of the coronary artery is 4 mm and the inner diameter of the aorta is 12 mm. The pressure and flow in the coronary artery are adjustable by a valve placed in the aorta. The coronary artery bifurcates to a main and a side branch. The main and side branches merge together peripherally. Three constrictors are placed in the main coronary artery to create multiple sequential stenoses at intervals of 40 mm. Another constrictor placed in the side branch was used to adjust Pw of the main coronary artery with a range of 0 to 20 mm Hg. FFR measurements were conducted using a 0.014-inch pressure wire (St Jude Medical). After creating 3 sequential stenoses with variable degrees, a pressure wire was advanced to a position distal to the most distal stenosis. P_w was obtained during a temporary occlusion of the distal part of the main coronary artery. After releasing the temporary occlusion, the pressure wire was slowly pulled back from the distal coronary artery to the ostium of the coronary artery, thereby recording P_d, P_dist, P_prox, and P_a. The pressure wire was completely pulled back into the guiding catheter to verify that no drift had occurred each time. The pressure wire was advanced again to the position distal to the most distal artery. FFR(X)_true was obtained when all the stenoses except for stenosis X were released. FFR(X–)_true was obtained when stenosis X was completely released. The middle stenosis was defined as stenosis X in the present study. A pressure drop of 5 to 8 mm Hg inevitably occurred without making any stenoses at the position just distal to the most distal stenosis in this model. 

Statistical analysis. Linear regression analysis was conducted between the true value of FFR as the independent variables and the predictive and apparent value of FFR as the dependent variables. A Bland-Altman plot was given for each comparison. The difference between FFR_app and FFR_true was compared with the difference between FFR_pred and FFR_true using a paired t-test. Continuous variables were presented as mean ± standard deviation. A two-sided P-value of <.05 was considered statistically significant. 

Results

Validation of Equation A. A total of 30 different 3 sequential stenoses were created. First, the baseline pressure data including P_a, P_d, P_w, P_dist, and P_prox were obtained. FFR(X)_app and FFR(X)_pred were calculated using the above mentioned equations. Then, the proximal and distal stenoses were removed. The pressure data including P´_a and P´_d were obtained and FFR(X)_true was calculated. A representative case is presented in Figure 3. 

FFR(X)_pred was compared with FFR(X)_true. A close correlation (r² = 0.92) was found between FFR(X)_pred and FFR(X)_true (Figure 4A). The corresponding Bland-Altman plot shows a small overestimation (Figure 4B). FFR(X)_app was also compared with FFR(X)_true (Figure 4C). Although a moderate correlation (r² = 0.60) was found, a relatively large scatter was observed in the plot of the relationship between FFR(X)_app and FFR(X)_true. The corresponding Bland-Altman plot shows that FFR(X)_app systematically overestimates FFR(X)_true (Figure 4D). The difference between FFR(X)_app and FFR(X)_true was significantly higher compared with the difference between FFR(X)_pred and FFR(X)_true (0.18 ± 0.10 vs 0.05 ± 0.05; P<.001 in paired t-test). 

Validation of Equation B. A total of 30 different 3 sequential stenoses were created. The experiment was conducted using a similar method. Briefly, after obtaining the baseline pressure data, FFR(X–)_app and FFR(X–)_pred were calculated. Then, the middle stenosis was removed and FFR(X–)_true was obtained. A representative case in the validation of equation B is presented in Figure 5. 

A close correlation (r² = 0.97) was found between FFR(X–)_pred and FFR(X–)_true (Figure 6A). The corresponding Bland-Altman plot shows a small overestimation (Figure 6B). A moderate correlation (r² = 0.77) was found between FFR(X–)_pred and FFR(X–)_true. However, a relatively large scatter was observed in the plot of the relationship between FFR(X–)_app and FFR(X–)_true (Figure 6C). The corresponding Bland-Altman plot shows that FFR(X–)_app systematically overestimates FFR(X–)_true (Figure 6D). The difference between FFR(X–)_app and FFR(X–)_true was significantly higher compared with the difference between FFR(X–)_pred and FFR(X–)_true (0.13 ± 0.06 vs 0.04 ± 0.02; P<.001 in a paired t-test).

Discussion

The present study revealed that of the two mathematically derived equations, one predicts the true FFR of each stenosis in multiple sequential stenoses (Equation A), and the other predicts the true FFR after releasing a given stenosis in multiple sequential stenoses (Equation B). Both strongly predict the true value of FFR in the experimental model of coronary circulation. Both equations are considered a general form of previously stated equations for tandem lesions.^4,5 

With the advent of the drug-eluting stent, more and more complex lesions including multiple sequential stenoses or diffuse coronary lesions are currently treated with coronary interventions.^6-8 Coronary lesions with more than two stenoses within the same coronary artery are frequent in daily clinical practice. A simple FFR measurement does not predict the functional severity of an individual stenosis in multiple sequential stenoses. However, application of the theoretical equations to predict the true FFR of individual stenosis is limited in a tandem lesion. Another approach to evaluate the functional severity of each stenosis is the employment of ΔP or ΔFFR.^9-11 The most severely narrowed lesion is determined by which of the lesions produces the biggest ΔP or ΔFFR in this approach. After stenting the most severe lesion, the pull-back recording of FFR is repeated, and it is decided whether and where a second stent should be placed. Stenting and reassessment of FFR is repeated until the entire FFR becomes of sufficient value. This approach is theoretically correct because the more the lesion becomes stenotic, the greater ΔP or ΔFFR becomes across the lesion, as long as there are no side branches in the sequential lesion and the blood flow is constant. This approach is also practical, since it omits the necessity of recording Pw, which requires balloon occlusion of the coronary artery. However, one should note that the true FFR of entire stenoses after releasing a stenosis is always smaller than the apparent FFR value obtained by simply adding the baseline FFR and ΔFFR across the target lesion.

Both equation A and equation B in the present study are difficult to apply in daily clinical practice, since both equations require wedge pressure measurement during the maximum hyperemia. However, both equations are important when considering the theoretical background of the FFR assessment in a sequential lesion. Equation A suggests that the true FFR of individual stenosis is always smaller than the apparent FFR value. The severity of individual stenosis is always underestimated in multiple sequential stenoses. Equation B suggests that the true FFR of entire stenoses after releasing a given stenosis is always smaller than the apparent FFR value obtained by a simple addition of the baseline FFR plus ΔFFR across the target lesion. This is because the flow through the entire lesion increases after releasing a stenosis, which results in more pressure drop across the stenoses. Figure 7 represents a case with 3 sequential stenoses in the left anterior descending (LAD) coronary artery. The case clearly shows that after an intervention to the most severe stenosis, the ΔFFR across the other stenosis markedly increased. The postintervention FFR was smaller than the FFR calculated by a simple addition of the baseline FFR plus ΔFFR across the stenosis. The case clearly indicates that Equation A and Equation B provide a better understanding of the FFR assessment in sequential stenoses. 

Study limitations. Several important limitations should be described before drawing any conclusions. First, the study was conducted in an in vitro model of coronary circulation, which lacks several important characteristics of complex human coronary circulation. In the present study, distilled water was used as the perfusate. The viscosity of water is lower than blood, which might influence the study results. More importantly, the experimental model did not have the interposition of a side branch between stenoses. The interposition of a side branch between stenoses may change the mutual hemodynamic influence of one stenosis on the other stenoses. In vivo assessments with an appropriate model are required to assess the true legitimacy of the derived equations. Second, a pressure drop of 5 to 8 mm Hg inevitably occurred in this model, which is different from in vivo coronary circulation. This is considered the primary reason that the predictive values of FFR are always overestimated in the present study. 

Conclusion

Two novel equations applicable to multiple sequential stenoses are mathematically derived. One predicts the true FFR of each stenosis in multiple sequential stenoses. The other predicts the true FFR after releasing a given stenosis in multiple sequential stenoses. Both equations strongly predict the true value of FFR in the experimental model of coronary circulation. 

Acknowledgments. The author would like to express the utmost gratitude and respect to Drs Pijls and De Bruyne for their great contribution in the development of functional assessments of coronary circulation. The present study largely depends on their previous achievements. Special thanks also go to Ms Rika Sugiura, Ms Hiromi Yoshida, and Ms Yoko Shimizu for their secretarial support. 

References

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From the Department of Cardiovascular Medicine, Graduate School of Medicine, Kyoto University, Kyoto, Japan, Department of Cardiovascular Medicine, Gifu Heart Center, Gifu, Japan.

Funding: Devices donated by St Jude Medical.

Disclosure: The authors have completed and returned the ICMJE Form for Disclosure of Potential Conflicts of Interest. The authors report no conflicts of interest regarding the content herein.

Manuscript submitted April 21, 2013, provisional acceptance given June 26, 2013, final version accepted August 14, 2013.

Address for correspondence: Naritatsu Saito, MD, Department of Cardiovascular Medicine, Graduate School of Medicine, Kyoto University, 54 Shogoin Kawahara-cho, Sakyo-ku, Kyoto 606-8507, Japan. Email: naritatu@kuhp.kyoto-u.ac.jp