Fractional Flow Reserve Evaluated as Metric of Coronary Stenosis — A Mathematical Model Study

Figures

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  Figure 1. Electrical analog of the coronary circulation. The model consists of an aortic pressure PA and a venous pressure PV with two resistances, serially connected between these pressure sources, to represent the stenotic RS and capillary RC fluid resistances. F is the flow through the resistances, while ΔPS and ΔPC are the pressure drops over the resistances RS and RC, respectively. PD is the pressure distal to the stenosis and proximal to the capillaries.

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  Figure 2. (A) Cartesian and polar coordinates. The pressure PA and PD represented in Cartesian coordinates (PA, PD) (see dotted lines) and in polar coordinates (PA(1+FFR2),FFR). Note that the pair of pressure (PA, PD) define exactly one point in the graph, while the single FFR defines a complete line in the graph. So, mathematically speaking, the single FFR without the companion FFRC leaves ambiguity. As a result, all points on a line through the origin share the same value of FFR. (For details see Methods section). (B) Ambiguous change of FFR from point (PA, PD) for three different cases of a pressure decrease: The FFR is unchanged (see notation FFR =), if the change in PD is equal to the change of PA times FFR. Similarly, the FFR decreases (see FFR ↓) and increases (see FFR ↑), if the change in PD is smaller and larger than the change of PA times FFR, respectively (see section Methods for details).

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  Figure 3. Simulation results for the non-linear dependence of: • the FFR with common reference values at 0.75 and 0.8 (upper-left panel), • its companion FFRC (upper-right panel), • the distal-to-stenosis pressure PD (lower-left panel), to the degree of stenosis α (α = 1 is no stenosis, α = 0 means complete occlusion) for different values of R0/RC and PV/PA (see legend). • The relation of FFR as function of PD (lower-right panel), (The legend in upper-left panel applies also to the upper-right and lower-left panel). See section Results for details.

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  Figure 4. Simulation of distal-to-stenosis pressure PD vs. mean aortic pressure PA (Equation 17) with each dot representing a case with PA running from 55 to 115 mmHg for different values of the degree of stenosis α (α running from 1 to 0), and with lines (Equation 7) indicating FFR at 1, 0.75, 0.5, and 0.25.

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  Figure 5. Post-stenotic pressure (PD) vs. mean aortic pressure (PA) during hyperemia with stenosis at different anatomical locations. Data points refer to 22 patients with right coronary artery (RCA) stenosis. The three symbols specify the anatomical location of each stenosis, being distal (d), medial (m), or proximal (p). The black (identity) line corresponds with a fractional flow reserve (FFR) of 1.00 (i.e., no stenosis). Regression line: PD = 0.873 PA – 6.836, R = 0.728, P < 0.001, N = 22. In the section Discussion, this graph is compared with Figure 4 which shows results from the in silico study.

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  Figure 6. Fractional flow reserve (FFR) and the intrinsic companion (FFRC). The interpretation of FFR is not unique, as a large spread of the co-metric FFRC can be discerned. In the past the region between 0.75 and 0.80 was termed the “gray zone.” Data points having FFR > 0.80 (purple line) are often considered to refer to patients who do not require revascularization. However, the situation becomes different if the FFR companion (FFRC) is accepted as a second criterion. When, for example, the green line is taken as an additive cut-off level, then the data located in the red shaded area do not simultaneously fulfill both criteria. Obviously, establishment of combined cut-off levels requires robust evaluation in large patient groups. The yellow curve is calculated for PA = 90 mmHg on the basis of Equation (7), i.e., FFRC=PA1+FFR2.

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  Figure 7. Fractional flow reserve (FFR) in dependence of degree of coronary artery stenosis. For the patients studied (N = 22), the FFR is shown vs. percentage diameter stenosis (squares) as well as vs. the metric α as defined in Equation (8) (triangles). On the abscissa the scales for these metrics run in opposite directions, and therefore all data points are shown twice. Note that FFR sharply drops if the stenosis exceeds 70% diameter reduction. The curve for α (Equation 17) shows a sigmoid pattern. In the section Discussion, this graph is compared with results from the in silico study (Figure 3, upper-left).

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  Figure 8. Fractional flow reserve (FFR) against post-stenotic mean pressure (PD). Data points for cardiac patients (N = 22) with right coronary artery (RCA) stenosis. The blue curve refers to the best fitting regression (R2 = 0.581), predicted on the basis of the theoretically derived formula FFR = PD/(c1+c2PD). The black broken line indicates the traditional cut-off level at 0.80 for FFR. The various data points near this line can be further characterized by specifying the prevailing PD value. The red line reflects a tentative complementary cut-off for the co-variable PD, assuming that a driving PD below 50 mmHg is inadequate for appropriate perfusion. Acceptance of this criterion implies that the patients within the red rectangular area are in jeopardy. Clearly, any suggested combination of cut-off levels requires future robust evaluation in large patient groups. In the section Discussion, this graph is compared with results obtained from the in silico study (Figure 3, lower-right).

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  Figure 9. Survey of relationship between FFR and diameter stenosis. Average values reported in 38 studies on the basis of four measurement techniques: optical coherence tomography (OCT), intravascular ultrasound (IVUS), computed tomography angiography (CTA), and intracoronary angiography (ICA). Most authors employ linear regression or fit a second order polynomial for their study participants. The yellow shaded area refers to values of 0.78 ≤ FFR ≤ 0.82 around the common cut-off point, which narrow range corresponds with (averaged) diameter reduction anywhere between 45 and 55%. Data from Chu et al. (17).