## Abstract

The expected blood flow improvement following a coronary intervention is inversely related to the stenotic-to-normal flow ratio Q_{s}/Q_{n}. Since Q_{n} cannot be measured prior to intervention, treatment decisions rely on stenosis-severity indexes, e.g., area stenosis (%AS), hyperemic stenosis resistance (HSR), and fractional flow reserve (FFR), where treatment cut-off levels have been established for each index based on presence of inducible ischemia. Here, we studied the dependence of these indexes-predicted Q_{s}/Q_{n} under physiological perturbations of stenosis features and of hemodynamic and mechanical conditions. Dynamic coronary flow was simulated based on measured coronary morphometric data and a physics-based computational model. Simulations were used to evaluate the relationship between each index level and Q_{s}/Q_{n}. Under each perturbation, an independence measure (IM) was calculated for each index based on the relative change in Q_{s}/Q_{n} associated with each perturbation. The results show that while %AS prediction of Q_{s}/Q_{n} is largely independent (IM > 90%) of physiological changes in heart rate, venous pressure, and lesion length and location on the epicardial tree, HSR is also independent of changes in left ventricle pressure. FFR-predicted Q_{s}/Q_{n} is also independent of changes in aortic pressure, blood hematocrit, and stenotic vessel stiffness. Nevertheless, independence of all indexes is substantially compromised (IM < 70%) under changes in vasculature stiffness. Specifically, a physiological stiffening elevates Q_{s}/Q_{n} value by 21% at the FFR cut-off value (0.75). These findings suggest that the current FFR cut-off value for treatment of stenotic lesions overestimates the benefit of coronary intervention in patients with a stiffer coronary vasculature (e.g., diabetics, hypertensives).

- myocardial ischemia
- flow restoration
- clinical assessment
- hemodynamic effects
- model simulation
- fractional flow reserve
- hyperemic stenosis resistance

the goal of coronary intervention is to treat the lesion in order to restore normal blood flow (Q_{n}) of a stenotic vessel flow (Q_{s}). The Q_{s}/Q_{n} ratio represents the anticipated hyperemic flow restoration postrevascularization. Unfortunately, Q_{s}/Q_{n} cannot be quantified a priori since Q_{n} is unknown prior to intervention. Hence the clinical choice of treatment strategy, whether drug administration, percutaneous coronary interventions (PCI), or coronary artery bypass grafts (CABG), is based on indirect indexes of stenosis severity. Some of the current indexes (e.g., absolute and relative coronary flow reserve, CFR and rCFR, respectively) depend on the state of coronary vasoactivity (autoregulation), which can be quite variable. Hence, the present study focuses on the most common hyperemic (full vasodilation) indexes, such as the relative area of stenotic occlusion, %AS; the hyperemic stenosis resistance, HSR (31); and the pressure-based fractional flow reserve, FFR (35). The relationship between these indexes' values and Q_{s}/Q_{n} level is presently unknown. Furthermore, for a given index value the flow ratio Q_{s}/Q_{n} may be affected by hemodynamic and mechanical determinants of the downstream microcirculatory flow (29, 39), such as heart rate (HR), blood pressure, and contractile state, which may vary during interventional procedures (6). Assessment of Q_{s}/Q_{n} should ideally be independent of changes in hemodynamic loading and myocardial function.

A qualitative analysis based on an analog of two resistors in series (Fig. 1*A*), which represents the stenosed vessel and the distal microvasculature (38), suggests that the indexes' relations with Q_{s}/Q_{n} potentially depend on hemodynamic perturbations. For example, for a fixed value of stenosis geometry (%AS), Q_{s}/Q_{n} is expected to decrease under high aortic blood pressures. Since vascular compliance is nonlinear (5, 11), high inlet pressure increases the microvascular lumen diameters and, therefore, decreases the resistance to a higher extent than that of the stenosed vessel (Fig. 1*B*). Under low resistance of the distal microvasculature, the relative effect of a stenosis on total resistance increases, causing the stenosis to be functionally more severe and its treatment to be more effective in terms of flow restoration. Furthermore, since Q_{s}/Q_{n} is determined by the stenosis-resistance relative to distal vascular resistance, elevated blood pressure is expected to similarly affect the independence of HSR, which attempts to measure the absolute stenosis-resistance. Based on a similar qualitative two-resistor analysis, it was previously argued (39, 41) that FFR exactly represents Q_{s}/Q_{n} only if the downstream microvessels are rigid. When the vessels are compliant (5, 11), however, vascular distensibility, which is not measured in the clinic (14), causes FFR to underestimate Q_{s}/Q_{n}.

Unfortunately, most previous clinical studies of lesions (6, 28, 40) did not include pre- and post-interventional flow data to evaluate the indexes' relations with Q_{s}/Q_{n}. Previous studies focused primarily on the effects of coronary hemodynamic factors on the measured index levels, rather than on their associated predicted Q_{s}/Q_{n}. Animal studies claimed, for example, that FFR is affected by HR (27). Clinical studies showed FFR to be only slightly affected by HR and contractility (6), to be highly reproducible, to have low intraindividual variability, and to be independent of sex and coronary artery disease (CAD) risk factors such as hypertension and diabetes (33). Pijls et al. (35) found that the relationship between FFR and Q_{s}/Q_{n} is linear below a certain level of stenosis severity and independent of aortic blood pressure. In severe stenoses FFR was found to underestimate Q_{s}/Q_{n} in dogs. In a clinical study (31), the HSR index was found to better predict reversible perfusion deficits than FFR, as determined by SPECT.

In silico studies have been hindered by the complexities of both the coronary network geometry (22–24) and the myocardial-vessel dynamic interaction (1). Hence, previous simulation studies of clinical indexes used lumped network structures and linear coronary pressure-flow relationships [i.e., they do not account for the compliance of the coronary vasculature (38)] and imposed various ad hoc assumptions on the effects of myocardial contraction on the coronary flow (42). Although such approaches facilitate understanding of qualitative behavior under specific empirical conditions, a more realistic distributive analysis is required to gain both quantitative and mechanistic insights into the physical factors that affect the performance of stenosis severity indexes.

Here, we report a biophysical micromechanical model of the relationships of %AS, HSR, and FFR with the flow ratio Q_{s}/Q_{n}. It is hypothesized that these relationships depend on hemodynamic loads and cardiac environment. To explore this hypothesis, the relationship between indexes and flow ratio Q_{s}/Q_{n} under variations of pressure, HR, vessel compliance, and contractility were analyzed based on measured network morphometry of the left anterior descending (LAD) artery, on vascular and myocardial mechanics, and on fundamental physical principles.

## METHODS

#### Indexes definition.

The following standard indexes of stenosis severity (percent stenosis area, %AS; hyperemic stenosis resistance, HSR; and fractional flow reserve, FFR) were defined as: *A*_{s} and *A* denote the stenotic and nonstenotic vessel cross-section area, respectively; P_{d} and P_{p} denote the hyperemic intravascular pressures distal and proximal to the stenotic vessel, respectively; *V*_{d} is the hyperemic average-peak velocity distal to the stenosis (31); and P_{v} is the venous pressure. The commonly used approximation FFR = P_{d}/P_{ao} (P_{ao} denoting aortic blood pressure) was avoided here since it has been shown (38) to induce errors in the representation of Q_{s}/Q_{n} for distal lesions with diffuse CAD. In the clinic, each index has an empirically established cut-off value, used to determine therapeutic decisions: 75% for %AS (3), 0.8 mmHg·s/cm for HSR (31) and 0.75 for FFR (40).

#### Methodological approach.

The relationship between each index (*Eqs. 1a*–*1c*) and both total as well as subendocardial Q_{s}/Q_{n} (Q_{s,total}/Q_{n,total} and Q_{s},_{endo}/Q_{n,endo}, respectively) was evaluated under a variety of hemodynamic and biorheological conditions (Table 1) and under different characteristics of the stenotic lesion (Table 2). Subendocardial Q_{s}/Q_{n} is of particular interest given the propensity of subendocardium to ischemia (14). At a particular index clinical cut-off value, all other indexes were allowed to change in response to the imposed hemodynamic perturbations.

#### Independence definition.

For each severity index (*Eqs. 1a*–*1c*) and for each perturbation listed in Tables 1 and 2, an independence measure (IM) was defined as _{s}/Q_{n}) and min(Q_{s}/Q_{n}) denote the predicted values of Q_{s}/Q_{n} (whether total or subendocardial) correspondingly at the “High” and “Low” values of the perturbed parameter in Table 1, or at the “Variation” and “Reference” values in Table 2. The term ref(Q_{s}/Q_{n}) denotes the Q_{s}/Q_{n} value predicted at the reference state, defined by the “Reference” columns in both Tables 1 and 2. Since *Eq. 2* changes with severity-index value, we report IM only at the index cut-off value, where excessive sensitivity most affects the clinical decision. Moreover, since clinical experience has shown that Q_{s}/Q_{n} varies by several percent between repeated measurements in the same patient under identical index values (6, 35), IM values that are higher than 90% were considered to render the respective index independent of hemodynamic and cardiac conditions for evaluation of postrevascularization flow restoration.

#### Flow analysis.

A previously validated, morphologically based distributive coronary flow analysis (1) was modified for the present study to incorporate the epicardial arteries and stenosis (Fig. 1*C*; details in appendix a). Briefly, the computational platform incorporates the 3D network anatomy, the vascular nonlinear mechanics, the myocardial-vessel dynamic interaction, and the stenosis hemodynamics. The flow analysis was dynamic and under hyperemic conditions (full vasodilation). For each perturbation (Tables 1 and 2), the index values (*Eqs. 1a*–*1c*) and Q_{s} were calculated from the time-averaged predicted results of flow and pressure. Subsequently, a second dynamic flow analysis that assumed no stenosis was used to determine Q_{n}. Hence, the relationship between each index and Q_{s}/Q_{n} was obtained for a wide range of hemodynamic perturbations.

#### Hemodynamic and mechanical perturbations.

The hemodynamic determinants included heart rate (HR), aortic, venous, and left ventricle pressures (Pao, Pv and LVP, respectively), and blood hematocrit (Hct). The effect of HR on the diastolic time fraction (9) was included. Each hemodynamic and mechanical parameter was varied in the range of lowest to highest physiological value (Table 1). In perturbation of one parameter, all other hemodynamic and mechanical conditions were maintained at their reference values (Tables 1, 2). Additionally, the coronary vascular stiffness (STF) and the size of the vascular bed downstream from the stenosis was varied as described below. Finally, the stenosis location, length, and stiffness were also varied (Table 2). The stenosis stiffness was varied independent of the entire coronary vascular stiffness, since unlike the latter it is affected by atherosclerosis (12, 43).

#### Vascular stiffness.

Vessels stiffness was varied by the following scaling of *Eq. A3* (see appendix a) parameters: *D*_{+}, *D*_{0}, and *D*_{−} denote the maximum inflated, 0-pressure and maximum compressed vessel diameters, respectively, prior to scaling; and *D**_{+} and *D**_{−} are the scaled parameters, implying that the stiffness increases with decrease of β. At β = 1, the parameters remain unchanged compared with their measured or predicted values (see appendix a). High and low stiffness values (β = 0.4 and β = 1.6, respectively; Table 1) reflect three standard deviations from the average of measured pressure-diameter curves (11). The scale factor β was used due to lack of data regarding the changes in the material constitutive laws of both vessel wall and myocardium [see *In situ Vessel Compliance* in the supplement available with the online version of this article (the Online Supplement)] in disease states that affect coronary stiffness (e.g., diabetes, hypertension, atherosclerosis).

#### Vascular bed size.

Variability in the size of the vascular bed downstream from the stenotic vessel has been implemented by reducing the lumen diameter of all network vessels by 15%. Preliminary results indicated that such perturbation reduces the model-predicted nonstenotic flow rate, Q_{n}, from 304 to 148 ml/min under the reference hemodynamic conditions (Table 1). Assuming a nonregulated blood supply of 4 ml·min^{−1}·g tissue^{−1} (13) and a LAD that supports 45–55% of the left ventricle (LV) tissue, such flow rates reflect the physiological range of LV mass (15).

#### Flow analysis validation.

The flow analysis predictions were compared with available experimental data (7, 35, 44) (Fig. 2). To permit comparison, Q_{s}/Q_{n} values (which might change between data sets due to changes in hemodynamic conditions) were scaled to span within physiological range of hemodynamic conditions (Tables 1 and 2). This was done by varying vascular stiffness, since preliminary results suggested that stiffness changes (Table 1) have the highest effect on Q_{s}/Q_{n}. Additional validations included comparison with coronary pressures, diameters, and blood flow velocities at different vessels and under various loading conditions [see *Flow Scheme Validation* in the Online Supplement].

## RESULTS

#### Relation between indexes and total Q_{s}/Q_{n}.

Changes in aortic blood pressure (from 90/60 to 140/90 mmHg; Table 1) affect all the index-predicted Q_{s,total}/Q_{n,total} throughout the entire index range (Fig. 3), but to different degrees. The results suggest that the %AS prediction of Q_{s,total}/Q_{n,total} is sensitive to variations in aortic pressure: Q_{s,total}/Q_{n,total} changes by 21% at the cutoff level of %AS = 75% (Fig. 4*A*). Thus, according to the IM independence criterion (IM > 90%), %AS prediction of Q_{s,total}/Q_{n,total} is pressure dependent. Similarly, HSR is also pressure dependent (IM = 81%; Figs. 3*B* and 4*B*). Interestingly, HSR prediction of Q_{s,total}/Q_{n,total} is more sensitive to a reduction than to elevation in aortic pressure, probably due to the sigmoid shape of vascular pressure-diameter relations (Fig. 1*B*).

In contrast, FFR-predicted Q_{s,total}/Q_{n,total} is independent (IM > 95%) under perturbation of aortic pressure (Figs. 3*C* and 4*C*). Intriguingly, FFR is independent of aortic pressure but at the same time inaccurate: under reference hemodynamic and mechanical conditions (Tables 1 and 2), FFR = 0.75 predicts a Q_{s,total}/Q_{n,total} value of 0.55 (instead of 0.75) (see discussion). This inaccuracy is even higher at lower FFR values (Fig. 3*C*).

The performance of the three indexes under all hemodynamic variations of Table 1 is presented in Fig. 4. The results indicate that FFR IM values are higher than those of both %AS and HSR under the subjected perturbations, and that FFR varies only under changes in the vascular stiffness (IM = 65% under β variation from 0.4 to 1.6). The vascular stiffness effect on FFR prediction of Q_{s,total}/Q_{n,total} is further investigated in Fig. 5. As deduced from theory (39), the predicted FFR and Q_{s}/Q_{n} values tend to be equal only for rigid vasculature (Fig. 5, β → 0) but may differ substantially in compliant vasculature, thus causing an escalating FFR underestimation of Q_{s}/Q_{n} as the coronary vasculature becomes less stiff. Specifically, a physiological increase in vascular stiffness (from β = 1 to β = 0.4) is associated with a Q_{s}/Q_{n} elevation of 21% (from 0.55 to 0.66) for the commonly used FFR cut-off value of 0.75.

#### Indexes independence in predicting subendocardial Q_{s}/Q_{n}.

Subendocardial Q_{s,endo}/Q_{n,endo} values are lower than the respective Q_{s,total}/Q_{n,total} level. For %AS, HSR, and FFR respective cutoff values of 75%, 0.8 mmHg·s/cm and 0.75, the subendocardial flow ratios Q_{s},_{endo}/Q_{n,endo} are 0.51 (vs. the total flow ratio Q_{s,total}/Q_{n,total} level of 0.58), 0.36 (vs. 0.43), and 0.47 (vs. 0.55) for these indexes, respectively. Moreover, the IM values of Q_{s,endo}/Q_{n,endo} under changes in vascular stiffness are smaller than the respective IM values of Q_{s,total}/Q_{n,total} (Fig. 4) by 37%, 64%, and 38%, respectively, under the same parameter perturbations (Table 1).

It was hypothesized that the differences between indexes' predictions of Q_{s,endo}/Q_{n,endo} and Q_{s,total}/Q_{n,total} result, in part, from a higher compliance of subendocardial vessels with the subepicardial compliance (2). To test this hypothesis, flow was analyzed in a network that hypothetically consisted of similar compliance in subendocardial and subepicardial vessels [see *Abolishing Transmural Compliance Differences* in the Online Supplement]. The results show that the above listed total/subendocardial differences in IM values reduced from 37%, 64%, and 38% to 1%, 3%, and 2%, respectively, a result that supports the hypothesis.

#### Performance of indexes under various stenosis characteristics.

Variation in the stiffness of the stenotic vessel induces higher IM value for FFR than for %AS and of HSR (Fig. 6; IM = 99%, 82%, and 84% for FFR, %AS, and HSR, respectively, under changes of the stenosis-vessel stiffness β from 1.0 to 0.4, Table 2). The effect of changes in stenosis length and location on IM is of secondary importance. The results in Fig. 6 also indicate that Q_{s},_{endo}/Q_{n,endo} is affected similarly to Q_{s},_{total}/Q_{n,total} by changes in stenosis characteristics.

## DISCUSSION

The major findings of this study are as follows. *1*) The total flow ratio Q_{s,total}/Q_{n,total} as predicted by the anatomic degree of occlusion (%AS) is independent of applied perturbations in heart rate and venous pressure. The hyperemic stenosis resistance (HSR) total flow ratio prediction is also independent of perturbations in left ventricle pressure. The fractional flow reserve (FFR)-predicted total flow ratio is also independent of perturbations in downstream vascular bed size, aortic pressure, and hematocrit. *2*) The independence of total flow ratio prediction of all three indexes is compromised under variations in the coronary vasculature stiffness. *3*) Both %AS and HSR indexes' predicted total flow ratios Q_{s,total}/Q_{n,total} are independent of perturbations in the lesion length and location on the epicardial tree. FFR-predicted flow ratio is also independent of perturbation in the stenotic-vessel stiffness. *4*). The predicted subendocardial flow ratio Q_{s,endo}/Q_{n,endo} varies more than that of total coronary flow ratio Q_{s,total}/Q_{n,total} for all three indexes.

#### Q_{s}/Q_{n} as the measure of index independence.

In the present study, coronary blood flow restoration following PCI is regarded as the measure of interest since downstream myocardial tissue suffers from ischemia if it is deprived of blood flow. Changes in the stenosis anatomy (represented by %AS index), resistance (HSR), or pressure drop (FFR) are all secondary (indirect), clinically measurable indexes. The expected flow restoration ratio following PCI is given by

The inverse flow ratio Q_{s}/Q_{n} has been used herein since it is bounded between 0 (high potential of restoration) and 1 (low potential). The study outcome that at a single index level, Q_{s}/Q_{n} is sensitive to hemodynamic variability implies that different patients with an identical index level are likely to be treated equally even though the expected benefit from the intervention (in terms of flow restoration) might be substantially different. Notably, the flow restoration (represented by Q_{s}/Q_{n}) is not determined solely by the stenosis characteristics (e.g., geometry, resistance) but also by the physical characteristics of the downstream microvascular bed (mainly by vascular compliance).

In contrast to FFR, %AS and HSR are not intended to predict Q_{s}/Q_{n}. In fact the correlation between these two indexes and Q_{s}/Q_{n} is negative (Fig. 3, *A* and *B*, respectively). They are used in the clinic, however, to describe the stenosis severity. Since the flow reduction associated with a stenotic lesion (quantified by Q_{s}/Q_{n}) is one determinant of stenosis severity, the ability to predict Q_{s}/Q_{n} based on these indexes is important, especially at the indexes' cut-off values. Accordingly, the present study considered independence of index-predicted Qs/Qn (i.e., whether at a single index value, the index is associated with a single Q_{s}/Q_{n} value under varying hemodynamic and mechanical conditions) rather than accuracy (i.e. whether at any index value the index equals Q_{s}/Q_{n}).

The clinical cut-off value of each index is based on the ability to distinguish if a stenosis is physiologically significant. From a flow restoration point of view, the prediction that Q_{s,total}/Q_{n,total} is similar for both %AS and FFR at their cut-off values (0.58 and 0.55, respectively; Fig. 3) agrees with the notion that flow restoration is an important determinant of clinical stenosis severity. In fact, the recent clinical recommendation to increase FFR cutoff levels to 0.8 reflects a Q_{s}/Q_{n} value of 0.6 (Fig. 3*C*), implying an even better correspondence between %AS and FFR cut-off levels with respect to flow restoration. In contrast, at the HSR cut-off value of 0.8, the associated predicted Qs/Qn value has a lower value of 0.43. The difference from the %AS- and FFR-predicted Qs/Qn may possibly stem from the model prediction that HSR-associated Q_{s}/Q_{n} is highly affected by vascular bed size (Fig. 4*B*). In fact, for a diameter reduction of 25% of all downstream vessels at reference conditions (e.g., LAD flow of 84 instead of 304 ml/min at reference conditions), an HSR value 0.8 mmHg·s/cm predicts a Qs/Qn values of 0.59. At these conditions, however, %AS values of 75% are associated with Qs/Qn values of 0.86.

The predicted dependence of flow ratio on hemodynamic conditions may vary with the level of Qs/Qn rather than with the index value. To evaluate independence under equal Q_{s}/Q_{n} values, we further tested IM values under HSR levels of 0.37 rather than 0.8 (reflecting a Q_{s}/Q_{n} value of 0.6, close to the value at FFR and HSR cut-off values, Fig. 3). The simulations show that at this comparable levels of Qs/Qn, the variability of HSR-predicted Qs/Qn under perturbations of aortic blood pressure (IM = 88%), hematocrit (IM = 86%), and vascular stiffness (IM = 61%) is indeed lower. Although the variability of HSR becomes comparable to the variability of %AS, it is still higher than that of FFR and is outside the acceptable range of independence (*Eq. 2*).

#### FFR-predicted flow restoration is less affected by changes of aortic blood pressure, hematocrit, and downstream vascular bed size.

Low aortic blood pressure (hypotension, Fig. 3), high hematocrit level [polycythemia (13)], and small downstream vascular beds were all found to increase Q_{s,total}/Q_{n,total} at the cut-off values of %AS and HSR, thus compromising the predicted independence of Q_{s,total}/Q_{n,total} (Fig. 4). This poor performance of %AS may account for the observed (30) low dependence of cardiac prognosis in patients with confirmed CAD on the extent of epicardial vessel stenosis. As described above, these effects on Q_{s,total}/Q_{n,total} can be readily understood since both low aortic blood pressure and high hematocrit levels increase the coronary microvascular resistance and thus reduce the effective severity of stenosis.

The effect of a smaller vascular bed deserves further elaboration. For a patient with a small vascular bed, treatment of a stenosis that is regarded severe solely based on its anatomical size (%AS) or resistance (HSR) will result in a poor clinical flow restoration since in addition to the vascular bed being small, the expected flow restoration (*Eq. 4*) is also low.

In contrast, FFR is less affected by these perturbations since its relationship with Q_{s,total}/Q_{n,total} does not depend on the absolute value of the native coronary resistance, but rather on the relative change in coronary stenosis resistance, as described above.

#### Effect of vascular stiffness.

The study results suggest that none of the index-predicted flow restorations are independent of variations in the coronary vascular stiffness. Even for the FFR index (which was shown to be more independent than the others), the results (Fig. 5) imply that under a given value of FFR, Q_{s,total}/Q_{n,total} is higher in patients with higher vessel stiffness (e.g., diabetics, hypertensives, smokers). Hence, the maximal flow restoration due to a coronary intervention is expected to be lower in patients with higher vascular stiffness. This result is of clinical significance since it provides mechanistic insights into the low post-interventional improvement observed in patients with diabetes mellitus (20). This result can be readily understood since treatment reduces the stenosis resistance to flow, which leads to an elevation of the pressure downstream from the stenosis (P_{d}) and an FFR increase. In healthy compliant vessels, this Pd elevation increases the vessels' diameters downstream from the stenosis and thereby increases Qn. In a stiffer vasculature, a similar Pd (FFR) elevation is not expected to increase the vessel diameters as much as in compliant vessels. Hence, the flow restoration due to stenosis treatment is expected to be lower compared with a more compliant microvasculature. Physically, vessel resistance depends highly on diameter (*Eqs. A3*), and hence, a pressure-independent resistance can only occur in rigid vessels. Since passive coronary vessels are always compliant [although less so in diabetics (5, 11)], FFR is shown to underestimate Q_{s,total}/Q_{n,total} (Figs. 2 and 5). A lower flow restoration in a stiffer coronary vasculature supports the view (45) that stratification of the coronary patient, which includes consideration of comorbidities known to affect vascular stiffness, may enhance the a priori prediction of an interventional success. In an apparent contradiction, two previous clinical studies found no significant differences between diabetic and nondiabetic patients with respect to post-interventional flow restoration (8, 37). However, in one of these studies (8), the patient population had high pretreatment FFR values (0.87 ± 0.06). Our results (Fig. 2) suggest that under such a high level of FFR, the predicted effect of stiffness on Q_{s,total}/Q_{n,total} is considerably smaller compared with the cutoff level FFR = 0.75 considered here (IM = 95% instead of 65%), thus providing a possible explanation for the observed similarity between diabetic and nondiabetic patients in that study (8). In another study (37), pretreatment FFR levels were similar not only between diabetic and nondiabetic patients, but also between patients with substantially different degree of stenosis (40–70%), which questions the reliability of the data. These results underscore the importance of well-controlled clinical trials to shed light on this multifactorial clinical issue. The present study suggests that a lower cut-off value should be used for patients at risk for higher coronary stiffness (diabetics, smokers, etc.).

#### Indexes representation of Q_{s,endo}/Q_{n,endo.}

Under all conditions tested, Q_{s,total}/Q_{n,total} values at the clinical cut-off levels are generally higher than the respective Q_{s,endo}/Q_{n,endo} values (Fig. 4, bars vs. rectangles). This result implies that the expected improvement in subendocardial flow due to a coronary intervention is higher than in the total coronary flow. Although this result has not been directly validated in the clinic, it agrees with the known higher vulnerability of subendocardium to ischemia (13). Stenosis compromises subendocardial flow to a higher extent than subepicardial flow (2), and hence, treatment of stenosis should improve subendocardial flow even more than total flow.

#### Study limitations.

The present study focused on the effect of physiological loading conditions on the expected blood flow improvement following a coronary intervention (*Eq. 4*). In clinical practice, the observed flow restoration may be lower due to incomplete lesion removal or due to pathological high stiffness of the downstream vasculature, which has been shown herein to increase Q_{s}/Q_{n} and thus reduce flow restoration. Moreover, the association between index value and inducible ischemia is beyond the scope of the present study, since the reversibility of the ischemic process in myocardial cells depends on parameters that are not considered in the present study (e.g. duration of coronary syndrome).

The pressure drop over a stenosis (*Eqs. A2b*) was assumed to have a commonly used (10, 38, 42) empirical form (46). The validation of this relation in vivo was established by flow conditions in obstructed arteries somewhat larger than the coronary vessels. Yet, the study conclusions regarding the index-predicted flow independences are likely to remain unaffected even if using different stenosis flow formulations (e.g., Ref. 17). This expectation is based on the following: *1*) in the analysis, the empirical constants of the stenosis model (*Eq. A2b*) were widely perturbated (see appendix a) but did not qualitatively affect the conclusions; and *2*) the study predictions are formulated based on physiology and basic physical principles that are stenosis-model independent.

Variations in each hemodynamic or mechanical determinant of flow were carried out separately to provide insight into the underlying mechanisms. In the clinical arena, however, several hemodynamic parameters may deviate simultaneously from their reference levels. The present flow analysis can readily be used to calculate the expected Q_{s}/Q_{n} under any set of parameters. For example, elevation of both the HR and P_{ao} from their low to their high levels (Table 1) results in IM values of 85, 85, and 96% under %AS = 75%, HSR = 0.8 mmHg·s/cm, and FFR = 0.75, respectively for Q_{s,total}/Q_{n,total}; and 81, 81, and 94% for Q_{s,endo}/Q_{n,endo}. In contrast, an inverse perturbation of elevating HR and lowering P_{ao} induces IM values of 74, 75, and 95% for Q_{s,total}/Q_{n,total}; and 68, 68, and 93% for Q_{s,endo}/Q_{n,endo}.

The effects of variations in cardiac contractility were not considered here since contractility is vaguely defined and difficult to quantify. It could perhaps be represented by the parameter α in *Eq. A4*, which, together with LVP, determines the external pressure on the coronary vessel. Hence, the effects of variation in α are likely to be similar to those of the LVP, which were presented and discussed above.

The morphometry of the simulated coronary network represents the mean of the measured data. The significance of interindividual variability in the morphometry should be addressed in future studies based on large-scale stochastic reconstructions of the entire coronary networks (18).

The present study considered a localized CAD. In diffuse disease, the entire coronary vasculature is likely to become both narrower and stiffer. A narrower vasculature increases coronary resistance. As shown above for elevated hematocrit (which increase coronary resistance as well), FFR prediction of Q_{s}/Q_{n} is only mildly affected. In contrast, the predicted level of Q_{s}/Q_{n} is expected to increase under an increased resistance for %AS and HSR. The effects of a stiffer vasculature were described above.

IM values highly depend on the perturbation range of hemodynamic and mechanical conditions, as given in Tables 1 and 2. Accordingly, the perturbation limits were chosen based on well-established physiological ranges where possible (19). The physiological basis of vascular stiffness (β) limits, however, was more empirical. Nevertheless, the appropriateness of the chosen stiffness limits is supported by the results in Fig. 2*B*, showing that the model predictions under stiffness perturbation are within the boundaries of independently measured data (7, 35, 44). Finally, all severity indexes have been compared under the same perturbation limits, making the comparison between indexes valid regardless of limits value.

#### Clinical significance.

The results (Fig. 4) point to the higher independence (higher IM) of FFR-predicted flow restoration compared with the other stenosis indexes, which confirms the clinical utility of FFR. Additionally, the results (Fig. 5) suggest that use of a lower FFR cut-off value in individuals with suspected high vascular stiffness (e.g., diabetic patients) would result in a more favorable outcome of coronary interventions.

#### Summary and conclusions.

The present study analyzed the ability of various stenosis indexes to consistently predict posttreatment flow restoration under variations in hemodynamic and mechanical conditions. The results suggest that *1*) pressure-based index FFR is more independent of hemodynamic variability in predicting intervention-related flow restoration than the anatomic degree of vessel occlusion (%AS) and the hyperemic stenosis resistance (HSR); and *2*) FFR performance may be compromised by interpatient variability of coronary vascular stiffness. The results may be of clinical relevance as they provide both mechanistic and quantitative insight into the various factors relating index values (measured prior to a coronary intervention) to the expected intervention-induced flow restoration.

## Appendix A

### Flow Analysis Scheme

#### Network anatomy.

The network was reconstructed based on porcine morphometric data (4, 21, 22, 24–26) of the left anterior descending (LAD) tree. It consists of an epicardial arterial tree [vessel diameters 3.0 down to 0.7 mm, orders 11 to 9, respectively (26)] that perfuses several intramural networks [diameters <0.7 mm, orders 8 to −8 of arterioles and venules, respectively (22, 24, 26)]. The latter consist of arterioles, capillaries, and venules (1). The simulated network perfuses four representative myocardial layers across the heart wall that span from subepicardium to subendocardium (Fig. 1*C*).

The epicardial arterial tree, assumed to be symmetric, was reconstructed based on the mean measured statistical data of the coronary tree (20). Five generations of epicardial arteries emerge from the 3.0-mm diameter LAD prior to feeding the intramural networks. Computational results confirmed that the flow analysis predictions agree well with experimental data (34) of the LAD pressure-flow relations [see *Flow Scheme Validation* in the Online Supplement]. The diameters of the epicardial arteries were taken to decrease at each bifurcation according to Murray's law (32), but with an exponent of 2.2 (16, 21). The lengths of epicardial vessels were based on measured element-lengths for each vessel order (26) and the longitudinal position of intra-element bifurcations (25).

The intramural network reconstruction has been previously reported (1). Briefly, it consists of intramural arterioles with orders ranging from 8 to 5 (26), intramural veins with orders ranging from −8 to −5 (24), and distributive microvascular networks that include smaller arterioles, venules, and capillaries (22, 24, 26). The four representative distributive networks were assumed to be evenly distributed across the myocardial wall from endocardium to epicardium at myocardial relative transmural depths of MRD = 0.125, MRD = 0.375, MRD = 0.625 and MRD = 0.875. Here, to correspond to physiological LAD flow rate, capillary lumen diameters were reduced by 10% (within 1 SD) from their mean measured (22) values.

#### Flow equations.

Dynamic flow was formulated based on the equations of mass conservation in each vessel and in each network bifurcation, namely: _{in}^{i}, Q_{out}^{i} are instantaneous flows in and out of the *i*th vessel, respectively, and dV^{i}/d*t* is the instantaneous vessel rate of volume change, such that ^{j} is the instantaneous flow into the bifurcation from each of the three vessels.

The instantaneous longitudinal pressure gradient (P_{in} − P_{out}) over each intramural (<0.7 mm lumen diameter) vessel is assumed to be governed by viscous effects and thus follows the Hagen-Poiseuille equation: *L* is the vessel length as previously measured (22, 24, 26), Q is the vessel flow, and *D* is the diameter, which depends on the pressure (see *Eq. A3* below).

In contrast, the longitudinal pressure gradient over larger (epicardial) vessels is affected by both kinetic and viscous components (17). Here we use the well-established relation (46): *D* denotes the vessel's nonstenosed (normal) lumen diameter, and *D*s is the lumen diameter of the stenosed vessel. For nonstenosed epicardial vessels, *Eq. A2b* was applied with *D*s values equal to *D*. In *Eq. A2b*, the pressure drop was induced by the following three separate mechanisms: blood viscosity, flow pulsatility, and the inertial pressure losses at the exit of a stenosis (46), where the latter is applicable only in stenotic epicardial arteries. The magnitude of each mechanism is determined by the respective coefficients *k*_{v}, *k*_{i} and *k*_{e}, which depend on the stenosis geometry (see *Hemodynamic Lesion Characteristics* in Online Supplement). The coefficients reference values where *k*_{i} = 1.52, *k*_{e} = 1.23, and *k*_{v} as calculated in accordance with Ref. 46 [see *Eq. S7* in the Online Supplements]. In the sensitivity analysis, each coefficient was varied between 0 and 10 times the reference value. Blood density ρ was assumed to be equal to 1,060 kg/m^{3}.

#### Myocardial vessel mechanics.

Coronary vessels with surrounding myocardial tissue have diameter-pressure relations that are sigmoidal in shape (1, 11), namely: _{IV} and P_{EV}, respectively). *D, D*_{+}, *D*_{0} and *D*_{−} denote the loaded, maximum inflated, 0-pressure, and maximum compressed vessel diameters, respectively, and ΔP_{1/2} is the transvascular pressure corresponding to a diameter equal to the average of *D*_{+} and *D*_{−}. In in situ coronary arteries, *D*_{−} was measured (11) as nonzero due to the vessel tethering by the adjacent myocardium.

The parameters values were obtained for large (>0.7 mm in diameter) epicardial vessels from a curve fit of measured data (11). Such data are unavailable for smaller vessels, and thus the parameters were calculated for each intramural vessel using a previously developed and validated (1) micromechanical model of vessel-in-myocardium [see *In-situ Vessel Compliance* in the *O*nline Supplement].

#### Effects of contraction on coronary flow.

It was recently shown (1) that only an extravascular pressure that combines the left ventricular pressure (LVP)-induced interstitial pressure, and the shortening-induced intramyocyte pressure results in prediction that are in with the large body of experimental data. Accordingly, the extravascular pressure on a vessel was expressed as: *Flow Simulation* in the Online Supplement], respectively, and α is a scale factor (Table 1) that relates myocyte contraction to extravascular pressure (1). The last term in *Eq. A4* is assumed to vanish for epicardial vessels since they are not surrounded by the myocardium. This is enforced using sign (MRD), which equals 0 for epicardial vessels and 1 for all other vessels.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

## AUTHOR CONTRIBUTIONS

Author contributions: D.A., G.S.K., and Y.L. conception and design of research; D.A. performed experiments; D.A. analyzed data; D.A. and Y.L. interpreted results of experiments; D.A. prepared figures; D.A. drafted manuscript; D.A., G.S.K., and Y.L. edited and revised manuscript; D.A., G.S.K., and Y.L. approved final version of manuscript.

- Copyright © 2013 the American Physiological Society