Coronary fractional flow reserve measurements of a stenosed side branch: a computational study investigating the influence of the bifurcation angle

Coronary hemodynamics and physiology specific for bifurcation lesions was not well understood. To investigate the influence of the bifurcation angle on the intracoronary hemodynamics of side branch (SB) lesions computational fluid dynamics simulations were performed. A parametric model representing a left anterior descending—first diagonal coronary bifurcation lesion was created according to the literature. Diameters obeyed fractal branching laws. Proximal and distal main branch (DMB) stenoses were both set at 60 %. We varied the distal bifurcation angles (40°, 55°, and 70°), the flow splits to the DMB and SB (55 %:45 %, 65 %:35 %, and 75 %:25 %), and the SB stenoses (40, 60, and 80 %), resulting in 27 simulations. Fractional flow reserve, defined as the ratio between the mean distal stenosis and mean aortic pressure during maximal hyperemia, was calculated for the DMB and SB (FFRSB) for all simulations. The largest differences in FFRSB comparing the largest and smallest bifurcation angles were 0.02 (in cases with 40 % SB stenosis, irrespective of the assumed flow split) and 0.05 (in cases with 60 % SB stenosis, flow split 55 %:45 %). When the SB stenosis was 80 %, the difference in FFRSB between the largest and smallest bifurcation angle was 0.33 (flow split 55 %:45 %). By describing the ΔPSB−QSB relationship using a quadratic curve for cases with 80 % SB stenosis, we found that the curve was steeper (i.e. higher flow resistance) when bifurcation angle increases (ΔP = 0.451*Q + 0.010*Q2 and ΔP = 0.687*Q + 0.017*Q2 for 40° and 70° bifurcation angle, respectively). Our analyses revealed complex hemodynamics in all cases with evident counter-rotating helical flow structures. Larger bifurcation angles resulted in more pronounced helical flow structures (i.e. higher helicity intensity), when 60 or 80 % SB stenoses were present. A good correlation (R2 = 0.80) between the SB pressure drop and helicity intensity was also found. Our analyses showed that, in bifurcation lesions with 60 % MB stenosis and 80 % SB stenosis, SB pressure drop is higher for larger bifurcation angles suggesting higher flow resistance (i.e. curves describing the ΔPSB−QSB relationship being steeper). When the SB stenosis is mild (40 %) or moderate (60 %), SB resistance is minimally influenced by the bifurcation angle, with differences not being clinically meaningful. Our findings also highlighted the complex interplay between anatomy, pressure drops, and blood flow helicity in bifurcations.


Background
Intracoronary hemodynamics can directly be assessed during percutaneous coronary intervention (PCI) using sensor-equipped guide wires, measuring pressure and/or flow [1]. Pressure and/or flow measurements in stenosed arteries have provided us a profound understanding of the coronary physiology [1]. Fractional flow reserve (FFR), defined as the ratio between the mean distal stenosis and mean aortic pressure during maximal hyperemia, has shown to be a valuable tool to assess the functional severity of coronary stenoses in daily clinical practice. Multiple (randomized) trials, including the landmark 'FAME' trial, have shown that FFR-guided PCI improves patient outcomes with respect to relief of angina complaints and the necessity of (repeat) angiography [1][2][3][4]. Combined use of FFR with coronary flow reserve measurements may provide the clinician an even better understanding of the functional severity of a coronary stenosis and its prognosis [5,6]. In contrast, FFR guidance for side branch (SB) lesion PCI did not show clinical benefit compared to angiography-guided SB interventions [7,8]. However, trials on FFR treatment guidance were not specifically designed for bifurcation lesions.
Also after treatment of bifurcation lesions some counter-intuitive FFR measurements have been observed. Treatment of the main branch (MB) in a bifurcation region with a small angle often results in SB compromise, whereas that is less often the case for large bifurcation angles [9]. However, the FFR after treatment was much less compromised for the small angle bifurcations compared to the large bifurcation angles. These contrasting findings imply that coronary hemodynamics and physiology in bifurcations is more complex than in non-bifurcation segments.
Based on the observations described before, we hypothesize that distal bifurcation angle on itself might play a major role in determining SB FFR values. Computational fluid dynamics (CFD) has been demonstrated to be an effective tool to study the hemodynamics of coronary bifurcations, allowing to investigate multiple scenarios characterized by different anatomy and flow conditions [10][11][12][13][14][15][16][17]. Therefore, we performed CFD simulations on a population-based coronary bifurcation model of the left anterior descending (LAD)-first diagonal branch with varying distal bifurcation angles to investigate the influence of the bifurcation angle on the intracoronary hemodynamics, including pressure drops and FFR, of SB lesions.

Coronary bifurcation model
A parametric coronary bifurcation model that represents the LAD with its first diagonal branch was created using the open-source software PyFormex (http://www.nongnu.org/ pyformex/) (Fig. 1a). The model has a proximal main branch (PMB) diameter of 3.30 mm [18]. The diameters of the distal main branch (DMB) and the SB obeyed Finet's law [19] and they were set as 2.77 mm and 2.10 mm, respectively. Three different distal angles (α; 40°, 55°, and 70°) were chosen according to the studies by Onuma et al. [20] and Godino et al. [21]. The PMB to DMB angle (β) was set to 150° [21]. The PMB segment length (from the inlet cross-section to the stenosis starting point) is equal to eight diameters, i.e. 26.4 mm. This value is in accordance with the measurements by Yamamoto et al. for the human proximal LAD segment, in which a length of 26.6 ± 9.3 mm was measured (n = 101) [22]. The bifurcation model is characterized by a stenosis affecting all segments of the bifurcation: the PMB, DMB, and SB. A diameter stenosis of 60 % was chosen for the PMB and DMB, while the SB diameter stenosis varied among the different experiments in the clinical range (i.e. 40, 60, and 80 %) [23]. Consequently, these bifurcation stenoses represent 1,1,0 or 1,1,1 bifurcation lesions according to the Medina classification [24], which assigns a binary value (1, 0) to each of the three portions of the bifurcation (i.e. PMB, DMB, and SB) depending on whether they have more than (1) or less than (0) 50 % lesion. The lesions are eccentric with the plaque located in the inner arc of coronary vessels where low wall shear stress was present as a consequence of the vessel curvature (Fig. 1b). This modelling feature follows the findings by Iwami et al. [25]. The total lesion length was set to 12 mm for both branches, consistent with what was previously found in 1028 patients [26]. In order to take into account the curvature of the bifurcation due to the presence of the heart, the model was placed on a sphere with radius of 56.25 mm [27] which corresponds to a curvature ratio (i.e. vessel radius/radius of curvature) of 0.03. This value was in the range (0.02-0.50) as previously reported for the left coronary tree [28][29][30].
An unstructured tetrahedral mesh was generated in ANSYS ICEM CFD v.15 (ANSYS Inc., Canonsburg, PA, USA) to discretize the bifurcation model. The fluid grid was characterized by smaller elements in the stenosis region and by a prism layer close to the arterial lumen to efficiently resolve the fluid dynamics quantities in the entire fluid domain (Fig. 2). The mesh element number was ~2,400,000 after a mesh independence study, which is briefly described at the end of this section.

Fluid dynamic simulations
Since we aimed to calculate mean pressure values to derive the FFR, steady-state CFD simulations were performed, as done in previous studies [31,32]. The finite volume software ANSYS Fluent v.15 (ANSYS Inc.) was used to carry out the fluid dynamics analyses. A hyperemic state was replicated by imposing a flow-rate of 120 mL/min at the inlet. This value is equal to three times the physiological value at rest (coronary flow reserve of 3) [33] that was obtained by solving the following equation [34]: where q is the flow and d is the diameter of the PMB (diameter of the inlet). Three different flow splits were applied at the bifurcation. First, a physiological (i.e. assuming the absence of stenoses) flow split was calculated following the relation between the diameter ratio of two daughter branches and the flow ratio through the bifurcation branches [34]: where Q SB and Q DMB are the flow values and d SB and d DMB the diameters of the two daughter branches SB and DMB. The calculated flow split was 65 %:35 % for the DMB and SB, respectively. To account for population flow split variability, two additional flow
The blood was modeled as a non-Newtonian fluid using the Carreau model: where μ is the dynamic viscosity, μ 0 and μ ∞ are the viscosity values as the shear rate goes to infinity and zero, respectively, Ṡ is the shear rate, λ is the time constant, and n is the Power-Law index. The following parameter values of the Carreau model were used [35]: μ ∞ = 0.0035 Pa s, μ 0 = 0.25 Pa s, λ = 25 s, and n = 0.25. A blood density of 1060 kg/m 3 was chosen [35]. The flow was assumed to be laminar. Indeed, Reynolds number is ~235 at the inlet for all investigated cases. The bifurcation models with 80 % SB stenosis and flow split 55 %:45 % for the DMB and SB outlets, respectively, represent the extreme scenarios with Reynolds number of ~605 at the SB stenosis. The solver settings and the computing platform specifications that were used to perform the CFD simulations are summarized in Table 1.

Analysis of the results
Pressure in the PMB, more precisely at one diameter proximal to the stenosis, was set at the average aortic pressure for humans (100 mmHg) [32]. Pressure drops across the stenosis were calculated from PMB to DMB (ΔP MB = 100 mmHg−pressure DMB) and from PMB to SB (ΔP SB = 100 mmHg−pressure SB). Pressures in the DMB and SB were evaluated at cross-sections of one diameter distal to the stenosis, as indicated in Fig. 1. FFR of the DMB (FFR MB ) was calculated as the ratio of the pressure in the DMB and the pressure in the PMB. FFR of the SB (FFR SB ) was calculated as the ratio of the pressure in the SB and the pressure in the PMB.
The SB pressure drop was plotted against the absolute Q SB for the 9 simulations in which the SB stenosis is 80 %. The relationship between ΔP SB and Q SB was described for the cases with SB stenosis of 80 % as ΔP = AQ + BQ 2 . The first term (A) of these relationship describes the viscous friction losses over the stenosis according to Pouseuille's law while the second term (B) describes the pressure losses caused by convective acceleration along the narrowing according to Bernoulli's law [36]. The goodness of the quadratic fits was evaluated by calculating the root mean squared error (RMSE), which is defined as: where n is number of response values, m the number of fitted coefficients estimated from the response values, y i is the ith value of the variable to be predicted, and ŷ i is the predicted value of y i . Smaller values of RMSE indicate that the observations are closer to the fitted line.
In order to visualize the flow patterns inside the coronary bifurcations, the local normalized helicity was calculated. This quantity has been widely adopted in the cardiovascular field of biomechanical engineering to describe the arrangement of fluid streams into spiral patterns [35,[37][38][39][40][41][42]. Positive and negative local normalized helicity values point out clockwise and counter-clockwise rotating fluid structures along the main flow direction, respectively. Additionally, to quantify the strength of the spiral flow structures that develop in the bifurcation, the helicity intensity was computed, as previously done in several recent numerical studies on coronary arteries, carotid bifurcations, and aortas [37,42,43].

Mesh independence study
To ensure the independence of the results from the mesh size, a mesh independence study was conducted on one representative geometry (i.e. case with stenosis degree of 60 % in the SB, distal angle of 70°, and flow split 55 %:45 % for the DMB and SB outlets, respectively). Three meshes were created, from a coarser to a finer one, by increasing the element number by a factor ~1.5 between each consecutive mesh:  Table 3 Figure 5 shows the ΔP SB plotted against the absolute Q SB for the 9 simulations in which the SB stenosis is 80 %. A good quadratic fit was obtained for cases with different distal angle, as highlighted by the small values of RMSE (0.58, 0.47, and 0.73 mmHg for cases with 40°, 55°, and 70° distal angle, respectively). Both terms in the equation defining the ΔP SB −Q SB relationship (i.e. ΔP = A Q + B Q 2 ) were larger with increasing bifurcation angles, resulting in steeper curves describing the ΔP SB and Q SB relationship, suggesting that the stenosis resistance of 80 % SB stenosis increases in larger bifurcation angles.  Figure 6 shows the complex flow patterns in the bifurcation. In particular, a jet with high velocity is visible in the SB, downstream of the stenosis (Fig. 6a). Recirculations can also be observed in the same region. The vessel curvature generates secondary flows in all segments, with more complex patterns in the SB, as highlighted by the in-plane velocity pathlines at SB selected cross-sections (Fig. 6b). Complex spiral flow patterns with clockwise and counterclockwise rotating fluid structures originate in the stenosed bifurcation region and develop into the two daughter vessels (with 60 % PMB and DMB stenosis and 80 % SB stenosis, see Fig. 6c). These spiral flow patterns were observed in all cases and they were more pronounced in the models with large bifurcation angle and severe SB stenosis, as shown by Fig. 7. In this figure, the helicity intensity versus the distal angle is reported for all cases with different SB stenosis. Helicity intensity is not affected by distal angle when SB stenosis is 40 % (Fig. 7a). On the contrary, helicity intensity increases when distal angle becomes larger for cases with 60 and 80 % SB stenosis, for all flow splits applied at the bifurcation outlets (Fig. 7b, c).

Table 3 Pressure drop across the stenosis from proximal to distal main branch (ΔP MB ), fractional flow reserve in the main branch (FFR MB ), pressure drop across the stenosis from proximal main branch to side branch (ΔP SB ), and fractional flow reserve in the side branch (FFR SB ) for all investigated cases
Finally, to investigate the relation between the SB pressure drop and the complex flow patterns that characterize the stenosed bifurcation models, the SB pressure drop of each investigated case was plotted against the corresponding helicity intensity value (Fig. 8). A good linear correlation (R 2 = 0.80) between the two quantities was found.

Discussion
In the present study, we evaluated the influence of bifurcation angle and SB stenosis on pressure drop and corresponding FFR. The main finding of the study is that in models of bifurcation lesions with 60 % MB stenosis and 80 % SB stenosis, the SB pressure drop is higher suggesting higher flow resistance (i.e. curves describing the ΔP SB −Q SB relationship being steeper) when the distal bifurcation angle is larger. However, when the SB stenosis is mild (40 %), the SB resistance is minimally influenced by the bifurcation angle.
In interventional cardiology, FFR has become a feasible invasive measurement to assess potential myocardial ischemia under high work load by calculating the ratio between the pressure distal to the coronary artery stenosis and the aortic pressure under hyperemic conditions. Despite the widespread acceptance of FFR, a deeper comprehension of its physiological basis and diagnostic features is needed to better understand the meaning of the FFR values measured in each patient, in particular when bifurcation lesions are treated [36]. CFD simulations can provide useful information by systematically calculating pressure drops and FFR values in coronary bifurcation models under different scenarios. In this study we evaluated the influence of bifurcation angle and SB stenosis on pressure drops and corresponding FFR. By plotting the ΔP SB values against the absolute Q SB values (Fig. 5), we could evaluate the ΔP SB −Q SB relationships. We showed that the curves describing these ΔP SB −Q SB relationships become steeper in larger bifurcation angles when SB stenosis is severe. As a consequence, we also found that FFR SB is significantly influenced by the bifurcation angle in case of severe SB stenosis.
Our CFD analyses highlighted also the complex interplay between hemodynamics and vessel geometry. Indeed, the geometric features of coronary bifurcations dictate the local hemodynamic environment, which influences the process of atherosclerotic plaque initiation and progression [44]. While previous numerical studies on coronary bifurcations focused on the relation between geometric features such as vessel tortuosity and bifurcation angle with wall shear stress descriptors [12,15,44,45], in this work we investigated the impact of bifurcation angle and degree of stenosis on pressure drop (and FFR) and on the bulk flow, helicity under hyperemic conditions. The results of our study revealed complex hemodynamics in all investigated bifurcations with marked secondary flows and recirculation areas in the SB. Counter-rotating helical flow structures were evident in the bifurcation region and in the branches (Fig. 6). These hemodynamic patterns were  caused by the combined effect of the curvature of the bifurcation, the presence of the stenosis, and also the bifurcation angle. In particular, larger bifurcation angles resulted in more pronounced helical flow structures (i.e. higher helicity intensity, Fig. 7) when SB stenoses of 60 or 80 % were present. Additionally, a good correlation between the pressure drop in the SB and helicity intensity was found (Fig. 8), suggesting that marked helical flow structures caused by the specific geometric features of the vessel result in higher pressure drops, reflecting higher resistance. This result is in agreement with the pressure-flow relationships (Fig. 5) and the previous CFD findings obtained for patientspecific coronary segments under resting conditions [42].
Currently, FFR measurements are applied to examine the functional severity of a stenosis in order to decide for PCI with promising results compared to the classical angiography [1]. Furthermore, FFR measurements are also used to judge treatment result. During treatment of a bifurcation lesion by stenting of the MB, SB compromise is often observed when the angle between the MB and the SB is small. Interestingly, a poor correlation between ostial SB narrowing due to PCI of the MB and FFR measurements, was observed [46]. These findings together with the findings of the current study suggest that the bifurcation angle also plays an important role to predict SB flow compromise after MB stenting due to higher SB flow resistances when the bifurcation angle is larger. However, this remains speculative and future flow simulation studies are needed to investigate the influence of the bifurcation angle on the hemodynamic impact of the SB after MB stenting. The complex interplay between the bifurcation angle, the degree of stenosis, and the hemodynamics can render pressure drop unreliable for examination of SB perfusion.
In this study, idealized, population-based bifurcation models were used. Although the geometric dimensions, including the curvature of the heart, were taken from the literature, coronary flow may behave differently in true human coronary anatomy. Currently, it is possible to perform CFD simulations on three-dimensional patient-specific human coronary anatomies reconstructed from computed tomography or quantitative coronary angiography and/or intravascular imaging [35,42,[47][48][49]. Although local hemodynamics (e.g. secondary flows and wall shear stress) cannot be measured in vivo in coronary arteries, pressure and flow (velocity) measurements can be done and used as boundary conditions for the CFD models. However, the advantage of the use of populationbased over patient-specific models is that it is possible to vary one specific anatomic component, such as the bifurcation angle, while keeping other variables constant. Consequently, the direct influence of that specific anatomic component on the local hemodynamics can be investigated. Additionally, as demonstrated in a previous numerical study [11], hemodynamic results in idealized bifurcation geometries are consistent in location and magnitude with those of the patient-specific anatomies that the idealized models represent.
In daily clinical practice, FFR values are in general not obtained in bifurcations in which the PMB is involved since interpretation of the FFR value will be hampered by the impossibility to distinguish relative contribution of the proximal and distal stenosis to the pressure drop. However, by displaying the curves describing the ΔP SB −Q SB relationships of the 80 % SB stenosis for each bifurcation angle separately, we were able to describe the SB stenosis resistances, which were clearly influenced by SB angle when SB stenosis was 80 %. It is likely that such resistances play a role in the flow distribution to the SB and DMB and thus play a role in SB compromise, although future studies are needed to further investigate this.
Moreover, we imposed flow under maximal hyperemia as inlet boundary condition. Hereby we assumed the microvasculature being healthy with the distal resistances (i.e. microvascular resistances) being negligibly low. However, impaired microvascular function would impact the microvascular resistance resulting in a different assumed hyperemic flow and thus would have an impact on the calculated FFR values. Furthermore, the stenosis at the bifurcation would also result in a reduced flow, especially when the stenosis becomes more severe. Therefore, the absolute FFR values we have obtained under hyperemic conditions are higher than what would be expected in vivo. However, by showing the ΔP SB −Q SB relationships using the imposed flow assumptions, we were still able to draw conclusions on the influence of bifurcation angle on the SB stenosis resistance.
We assumed three different flow splits, one representing the natural flow split in case there would have been no stenosis, one with relative more flow diverted to the SB, and one with relative less flow diverted towards to SB. Although physiological (i.e. in case of no stenosis) flow split ratios at coronary bifurcations are well described in the literature [34,50], less is known about the flow split under pathological circumstances (i.e. with different stenosis degrees). In the reality, the flow split depends on the distal resistances, which are related to the patient-specific condition of the myocardium. Future studies using lumped parameter models (LPM), quantifying the entire coronary circulation (including the microvasculature) based on a hydraulic-electrical analogue, can be used to estimate the flow split under different circumstances. Such model could also take into account the influence of collateral flow. The specific ΔP SB −Q SB relationships found under the different circumstances (SB and DMB diameter stenosis, bifurcation angles, etc.) can be included in such LPM models to automatically calculate the flow splits [51]. Furthermore, the investigation of the coronary branch steal phenomenon [52] and its influence on FFR SB values by using these LPM models would be of particular interest.
We assumed laminar flow conditions for all our calculations. However, in the most extreme scenarios (i.e. bifurcation models with 80 % SB stenosis and flow split 55 %:45 % for the DMB and SB outlets, respectively) Reynolds number was ~605 at the SB stenosis. This value is at the borderline in the range between 500 and 1000, for which flow instabilities were observed in non-realistic axisymmetric stenosed vessels [53,54]. In a more realistic geometry, namely a carotid bifurcation, transitional flow was only observed close to the stenosis for higher local Reynolds numbers (peak Reynolds number of ~1200 at the stenosis) [55]. Thus, in our study the flow was assumed to be laminar in all cases for comparative purposes and to simplify numerics, as previously done in [13]. This assumption is conservative because it might result in a slightly underestimation of the pressure drops for the extreme cases, leading to the calculation of slightly higher FFR values.
Finally, it should be highlighted that the models are rigid and fixed. Although these limitations might have effects on near-wall hemodynamics quantities like wall shear stress, the pressure values are minimally affected by the wall movement, as shown in a recent study [56].