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Coronary artery pressure-drop and distensibility (compliance) are two major, seemingly unrelated, parameters in the cardiovascular clinical setting, which are indicative of coronary arteries patency and atherosclerosis severity. While pressure drop is related to flow, and therefore serves as a functional indicator of a stenosis severity, the arterial distensibility is indicative of the arterial stiffness, and hence the arterial wall composition. In the present study, we hypothesized that local pressure drops are dependent on the arterial distensibility, and hence can provide information on both indices. The clinical significance is that a single measurement of pressure drop could potentially provide both functional and bio-mechanical metrics of lesions, and thus assist in real-time decision making prior to stenting. The goal of the current study was to set the basis for understanding this relationship, and define the accuracy and sensitivity required from the pressure measurement system. The investigation was performed using numerical fluid-structure interaction (FSI) simulations, validated experimentally using our high accuracy differential pressure measurement system. Simplified silicone mock coronary arteries with zero to intermediate size stenoses were used, and various combinations of arterial distensibility, diameter, and flow rate were simulated. Results of hyperemic flow cases were also compared to fractional flow reserve (FFR). The results indicate the potential clinical superiority of a high accuracy pressure drop-based parameter over FFR, by: (i) being more lesion-specific, (ii) the possibility to circumvent the FFR dependency on pharmacologically-induced hyperemia, and, (iii) by providing both functional and biomechanical lesion-specific information.
Introduction
Arterial flow and compliance are used in the cardiovascular clinical setting, as indicators of coronary arteries patency and atherosclerosis severity.
Fractional flow reserve (FFR) has recently been established as the gold standard functional indicator for epicardial coronary lesions, utilizing trans-stenotic pressure measurement as a surrogate for flow under maximal hyperemia. This essentially estimates the ischemic severity of a lesion, regardless of its geometrical features, and is mostly useful for intermediate size stenoses (defined by cross-sectional areas of 40-70%), where coronary angiography sensitivity is limited (Tobis et al., 2007). The common clinical practice today is to stent lesions that reduce hyperemic flow and to avoid stenting those that do not reduce hyperemic flow (Pijls and Sels, 2012).
Arterial compliance (or distensibility) has been a primary measure for assessing arterial stiffness as an indicator for the arterial stenosis severity (Ovadia-Blechman et al., 2003), and distinguishing between white or yellow plaques (Takano et al., 2001). Atherosclerosis was correlated with impaired distensibility in comparison to normal healthy arteries (Mokhtari-Dizaji et al., 2006; van Popele et al., 2001), even in sites accompanying occult atherosclerosis, which could not be detected by conventional angiography (Nakatani et al., 1995).
Over the years, arterial pressure drop and distensibility were studied separately, as two independent parameters. Konala et al. have investigated the effect of coronary plaque compliance on pressure drop (Konala et al., 2011b) and FFR (Konala et al., 2011a) for cases of severe stenoses (70-90%) during hyperemic flow using computational fluid-structure interaction (FSI) simulations. These studies, however, presented no information on FFR-relevant pressure drops for intermediate size stenosis. The present study hypothesizes that arterial pressure drop is dependent on the arterial distensibility, and seeks to investigate it through a combination of coronary flow rates and diameters, for zero to intermediate size stenoses. The motivation was the development of the capability to obtain both functional and biomechanical information from a single pressure measurement during cardiac catheterization that will assist in improving the decision making prior to stenting. While this manuscript refers mostly to coronary flow, it is also applicable to the entire arterial system.
Materials and methods
Computational model
Coronary models
Simplified coronary arteries were simulated as straight compliant tubes to match our experimental silicone models (Fig. 1). A 0.835mm radius tube (rc) was set concentrically along the lumen to depict a 5 French fluid-filled double-lumen catheter that was used in the experimental setup. The models were scaled up from a 3mm coronary artery to a baseline diameter (D0) of 5mm to keep the simulations identical to the experimental model, utilize flow and pressure recordings as boundary conditions (B.C.), and subsequently validate the computational results. Scaling up the models in the experimental setup was necessary to allow sufficient measurements with the 5 French catheter within the stenosis models. The numerical models were divided into four sections along the planes of symmetry to save computational time. Fixed extensions were added on both sides of the compliant model to retain similarity to the experimental setup (40cm rigid extension was added at the compliant tube inlet where flow was measured, to allow flow development at the inlet, and 5cm rigid extension at the outlet where pressure was measured).
Governing equations and FSI formulation
The fluid flow interactions with the compliant tube were modeled using coupled continuity (Eq. (1)) and momentum (Eq. (2)) equations for incompressible, viscous, and laminar flow (Reynolds numbers are detailed in Section 2.1.6) along with the compliant tube wall stress equilibrium equation (Eq. (3)). The fluid was assumed slightly compressible (Eq. (4)) to allow for easier convergence of the models. The arbitrary Lagrangian-Eulerian formulation was used to solve Eqs. (1)-(3) subjected to the B.C. described in Section 2.1.3. [Formula omitted. See PDF] [Formula omitted. See PDF] [Formula omitted. See PDF] [Formula omitted. See PDF] where u is the fluid velocity [m/s], P is the pressure [Pa], µ is the dynamic viscosity of the fluid [Pas], ρ is the fluid density [kg/m3], ρ0 is the density at P = 0Pa, K is the fluid bulk modulus of elasticity [Pa], and σijS is the stress tensor for the compliant tube wall [Pa]. For the stenois models, the k-epsilon turbulence model was adopted since it provided the best matching between the computational and experimental pressure drops.
Boundary conditions
Three consecutive flow/pressure cycles, that were recorded from two matching experiments (normal or hyperemic flow), were set as B.C. Average velocity (u(t)) was set at the inlet, and pressure (Pg(t)) was set at the outlet, at the same locations where they were measured experimentally (Fig. 2). A relatively slow pulsation rate of 0.8Hz (50bpm) was used to allow enhanced stability of the pressure measurements in the experimental setup. It was critical to use recorded data as B.C. since the mechanical properties of the compliant model affect the pressure-flow phase-lag and hence the calculated pressure drops. The baseline pressure in the simulations (Pg(t)) was offset to zero to keep the nominal diameter of the tube (D0) uniform throughout the simulations of each of the five diameters simulated (for varying ΔD cases). A no-slip condition was applied at the catheter wall-fluid interface, and at the compliant tube wall-fluid interface. In addition, the following B.C. were assumed: [Formula omitted. See PDF] [Formula omitted. See PDF] [Formula omitted. See PDF] [Formula omitted. See PDF] [Formula omitted. See PDF] [Formula omitted. See PDF] where dS and dF are the displacements of the solid and fluid domains [m], respectively, and σijF is the stress tensor for the fluid flow [Pa].
Material properties
The working fluid was distilled water-glycerol solution (volumetric ratio of 1:0.56 at room temperature) defined as Newtonian and slightly compressible (Mach number was 0.002 « 0.1). The dynamic viscosity was taken as µ = 3.8mPas, as measured using a glass capillary viscometer (Canon-Fenske, No. T625, size 50, Canon® Instrument Company Inc., PA, USA). The fluid density was estimated as ρ = 1100kg/m3, and the bulk modulus K = 109Pa (Cheng, 2008).
The silicone tube wall (MED-6020, Nusil Silicone Technology, CA, USA) was homogeneous, incompressible, and hyperelastic using the first order Mooney-Rivlin model ( HE model in Rotman et al. (2015)): [Formula omitted. See PDF] where σ is the engineering stress [Pa], λ is the stretch, and C1 = 33,300Pa and C2 = 27,100Pa are the material constants. The material properties were ρ = 1050kg/m3, bulk modulus of B = 1.5GPa, and Poisson ratio of [theta] = 0.49. Varying distensibility (∆D) cases were simulated by adjusting the slope of the silicone stress-strain curve. For the stenoses cases, only the material properties of the plaque region were adjusted (Table S1).
Meshing and numerical model
The models were meshed with 8-node brick elements in both the solid and fluid domains (Table S2). Large deformations and large strains were considered for the solid elements. The coupled FSI equations employing a direct scheme were solved by the finite element code ADINA (ADINA R&D Inc., Watertown, MA, USA). A time discretization of 10ms was selected, since it yielded the best agreement between the calculated and the experimental pressure drops. Mesh convergence check was performed to ensure that the solutions of the pressure drop values in each case differed by no more than 0.005mmHg.
Simulation plan
In investigating the local fluid pressure as affected by the tube distensibility, diameter and flow rate, it is important first to understand these relationships in a straight tube without stenosis. Therefore, in addition to the scaled-up baseline diameter of 5mm, four additional diameters were simulated for the zero-stenosis cases (4.70, 4.85, 5.15, and 5.30mm), taking into account matching dimensional analysis; peak Reynolds (Re) of ~200 and ~800 (Konala et al., 2011b) for normal flow and hyperemic flow cases, respectively (Table S3). Four small to intermediate cases of stenosis severity were simulated: 14%, 38%, 55%, and 64% (by cross-sectional area), for stenosis diameters (Dstenosis) of 4.63, 3.94, 3.35, and 3.00mm, respectively. The normal flow B.C. was kept uniform for all the stenosed cases (coronary flow reserve (CFR) =1). For the hyperemic flow cases CFR values of 4, 3.85, 3.4, and 2.7 were used for the 14%, 38%, 55%, and 65% stenosis severity, respectively (Gould, 2009; Gould et al., 1974). This corresponds to peak Re in the range of 229-391 and 823-1034 for normal and hyperemic flows, respectively. For every combination of flow and diameter, or flow and stenosis severity, 4-8 cases of distensibility (∆D) were simulated. ∆D was maintained within the human physiological range of 0-8% (Shimazu et al., 1986). Overall, 86 cases were simulated (Table S3).
In investigating the effects of flow rate variability on pressure drops, additional hyperemic flow simulations were performed with the intermediate stenoses models (38%, 55%, and 64%): 60%, 80%, and 120% of the nominal hyperemic flow condition (Re ~800). This was also used for comparison with FFR.
Post-processing of the pressure drop signals included extraction of the mean and max values. The pressure drop products were then analyzed for correlations with the model distensibility, diameter, flow rate, axial location along the model, and stenosis severity.
Validation
Four marginal cases (V1-V4) were validated experimentally using our catheterization simulator (Table 1 and S2 Supplementary material). Even though the proposed FSI models are relatively simple, it was still important to perform validation since the analyzed pressure drops are very small in magnitude. The validation also showed that such measurements are practically feasible. The silicone models were mounted in the flow loop in a sealed water chamber. The silicone models wall motion was recorded using a digital camera (uEye, IDS imaging, Germany) and analyzed in real-time to measure the models' inner diameter and distensibility (Rotman et al., 2015). The working fluid was distilled water-glycerol solution as described above. The same pressure and velocity cycles were then set as B.C. in the matching FSI simulations, except that the pressure cycles were offset to zero (to ensure identical D0). Accordingly, the wall material properties used for the simulations were adjusted (Table S1). Differential pressure was acquired via a 5 French fluid-filled double-lumen catheter, and the acquired signal was corrected from common mode pressure (CMP) distortion to achieve an accuracy of 0.05mmHg as described in Rotman et al. (2015) and (2014). For the validation cases the pressure drops were acquired at locations P1, P4, P7 (Fig. 1). For each of the measurement points, separate matching FSI simulation was performed (for different B.C.). Validation of the FSI simulations was done by qualitative comparison of the pressure drop signals using Bland-Altman plots.
Results
Validation
Four marginal cases, V1-V4 were used to validate the FSI models. These cases cover combination of stenosis severity in the range of 0-64%, and two flow rates: normal or hyperemia. As an example, Fig. 3 depicts the pressure drop comparison for case V4 (64% stenosis, normal flow). It can be seen that the agreement between the experimental and numerical pressure drops was very good. Similarly good agreements were found for the other tested cases (Figs. S2-S5).
Numerical analysis
Analysis of the pressure drop signals (Pd(t)) involved correlating the mean (Pd-mean) and max (Pd-max) products with the following independent parameters: distensibility, diameter, flow rate, axial location along the model, and stenosis severity.
Zero-stenosis cases
An example for the effect of distensibility (∆D) on the pressure drop (Pd(t)) can be seen in Fig. 4a (D0=5.0mm, normal flow, ∆D=0.7-5.6%, at P4). The peak and average pressure drops are inversely related to the tube distensibility, with excellent approximation to a second order polynomial (R2 > 0.99). Fig. 4b and c summarizes all the zero-stenosis simulation cases. The inverse relationship between Pd-max and ∆D, and Pd-mean and ∆D was maintained for all the simulatated cases. For Pd-max the axial location effect was much more significant in the normal flow cases than in the hyperemia simulations, as the plots of Pd-max(∆D) for the different diameter cases are seen to be overlapping. Under hyperemic flow there was no evidence for overlapping of Pd-max between the cases with different diameters for the physiologic range of ∆D (~0-7%). For Pd-mean, however, the effect of axial location seems negligible, as no overlapping was evidenced both in the normal and hyperemic flow conditions. Nevertheless the slopes for Pd-mean(∆D) are much flatter than for Pd-max(∆D), especially for the normal flow cases (change of up to ~ 0.04and ~0.16 mmHg over the entire range of ∆D for normal flow and hyperemia, respectively), which practically makes Pd-mean useless as an indicator for ∆D. For both the normal and hyperemic flow cases the diameter (D0) had a negative effect on the pressure drop products with second-order polynomial behavior (R2 > 0.99). For the various diameters tested, the sensitivity of Pd-max to a 1% change in ∆D was measured in the range of 0.04-0.07mmHg and 0.08-0.16mmHg for normal and hyperemic flows, respectively. For Pd-mean, the sensitivity to a 1% change in disensibility was measured in the range of 0.002-0.006mmHg for the normal flow, and 0.009-0.02mmHg for the hyperemic flow.
Small to intermediate stenosis cases
Examples for the peak pressure drop profiles over the stenoses (axial locations P1-P7) are shown in Fig. 5, where Pd-max was plotted for each of the simulated stenoses (14%, 38%, 55%, and 64%), for both normal and hyperemic flow, and for several distensibility cases of the stenosed region. The effect of the stenosis distensibility on Pd-max appears to be much more pronounced for the more severe stenosis cases.
The following results are based on the analysis of Pd-max measurements at P4 (center of the lesion). For every stenosis case, a second order polynomial fitting was applied to Pd-max(∆D) (R2 > 0.99). Using this fitting, Pd-max was plotted as a function of the stenosis severity, including the results from the zero-stenosis cases at P4 (Fig. 6). These plots of Pd-max(% stenosis) were found to be best fitted by using a second order exponential function y=e(ax2+bx+c) (R2 > 0.99). The average sensitivity of Pd-max to a 1% change in ∆D under normal flow conditions was 0.04, 0.10, 0.27, and 0.60mmHg for stenoses severity of 14%, 38%, 55%, and 64%, respectively. For hyperemic flow the average sensitivity was 0.20, 0.49, 1.23, and 2.19mmHg for stenoses severities of 14%, 38%, 55%, and 64%, respectively. This means that sensing ∆D variability in intermediate size stenoses is within the capabilities of the proposed high accuracy differential pressure measurement method, for both normal and hyperemic flows. With conventional pressure-wire probes, however, this capability is marginally feasible, and is applicable to hyperemic flow only.
Since functional assessment of coronary stenoses is of great importance in the catheterization laboratory, we seeked to understand how local pressure drops would be affected by variabilities in the blood flow rate. Alterations in the hyperemic flow rate were simulated for the intermediate size stenoses cases, and the maximal and mean pressure drop products were acquired at the center of the stenosis (P4) and plotted as a function of the normalized hyperemic flow (Fig. 7). The plots of the maximal and the average pressure drops show similar trends, and for brevity reasons only the plots of Pd-max(flow) are presented. This relationship can be approximated as linear, with R2 > 0.99. The results indicate that the pressure drop was positively dependent on the flow rate, and very sensitive to changes thereof. The lowest sensitivity (smallest slope) was found in the plot of Pd-mean(flow) for the 38% stenosis, where every 10% change in the flow rate resulted in a pressure changes of about 0.3mmHg. The largest impact of Pd-max(flow) was indicated for the 65% stenosis, where every 10% change in the flow rate resulted in a pressure change of about 4.2mmHg.
Pressure drop is directly related to the flow rate, and therefore reduced flow rates yield smaller pressure drops. Interestingly, Fig. 7 depicts an inverse relationship for the pressure drop and flow with FFR, where, independently of the stenosis severity, the FFR values were becoming larger with reduced flow rates. As indicated by Fig. 7, the ‘normal’ hyperemic flow rate (normalized value 1) for the 65% stenosis yielded a marginal FFR value of 0.83 (slightly above the upper FFR threshold of 0.80) However, when the flow rate is increased by 20%, which should obviously improve coronary perfusion, the FFR value was reduced even further, to 0.80. Fig. 8 depicts plots of Pd-max with FFR. These plots show that the pressure drop and FFR relationship is practically independent of the stenosis severity, as measurements from all the stenoses cases were aligned linearly (R2 > 0.98), and that both variables are mainly affected by the flow rate.
Discussion
Atherosclerosis is known to be associated with impaired distensibility in comparison to normal healthy arteries (Mokhtari-Dizaji et al., 2006; van Popele et al., 2001), even in sites accompanying occult atherosclerosis, which cannot be detected by conventional angiography (Nakatani et al., 1995). Unstable coronary syndromes in patients have been previously correlated to increased coronary artery distensibility in comparison to patients with stable angina (Jeremias et al., 2000). Furthermore, coronary artery distensibility was found to be longitudinally more heterogeneous in acute coronary syndrome-related than unrelated plaques, especially between the lesion and the immediate proximal site (Sasaki et al., 2012), or between the stenosis and a distal reference coronary segment (White et al., 2009). The present study thoroughly investigated the relationship between pressure drop and distensibility. The effect of stenosis severity was best fitted by a second order exponential function (R2 > 0.99) for a wide range of stenosis severities (0-64%). Plots of the pressure drop peaks and average were assembled for varying combinations of nominal diameters, flow rates, and stenosis severities. These plots could be used to extract the tube distensibility assuming the tube diameter and pressure drop are known, and to better understand the sensitivity required (defined here as pressure change per 1% of ΔD) of any pressure measurement method used. It might be concluded that for zero stenosis tube sections, measurements at maximal hyperemia are necessary for eliminating the effect of the axial location from the measurement. This is essential if one wants to be able to calculate the arterial distensibility from the differential pressure measurement. For measurements in stenosed regions, a normal flow condition was sufficient for identifying varying distensibility owing to the improved differential pressure measurement accuracy. Just for comparison, FFR, which was the gold standard measurement for functional assessment of coronary stenosis, is completely dependent on pharmacologically-induced maximal hyperemia.
Functional assessment of the present stenosis models was obtained using local pressure drops for varying hyperemia flow cases, and the results were compared to the FFR measurements. An inverse relationship was found between the pressure drop and FFR; increased flow causes higher pressure drop, hence, the higher the flow the lower the FFR. According to Fig. 7, changes in the stenosis resistance (degree of stenosis) have a stronger effect on FFR than variation in the flow rate. Beyond the human variability in hyperemic coronary flow (Pijls et al., 2000) and coronary artery diameter, these inverse relationships of resistance and flow with FFR might partly explain its wide threshold (0.75-0.80). Not surprisingly, the pressure drop products (max and mean) appeared to be linearly related to FFR, as both parameters are strongly related to pressure differences. Moreover, this relationship appeared to be independent of the degree of the stenosis, and solely dependent on the flow rate, which is the essence of a functional indicator.
The main limitation of this study was the performance on simplified models of straight tubes, while real arteries are different. Nevertheless, it was necessary to thoroughly understand the pressure drop-distensibility relationship in such models before considering more complicated geometries. Hence this study set the ground for future research.
Based on these findings it might be concluded that high accuracy pressure measurements possess the potential to improve the functional assessment of coronary lesions. Not only do these local measurements appear to be more sensitive to flow than FFR, they can also eliminate some of the factors governing the FFR variability (e.g., flow rate) since the measurements are sufficiently sensitive to extract information from non-constricted regions as well. Moreover, functional assessment of a lesion based on the local pressure drop is not dependent on local blood pressure, but only on the absolute pressure drop. In addition to functional assessment of stenoses, it was shown that the fluid pressure drop contains information on the tube distensibility, or compliance. It was also shown that by using very accurate differential pressure measurements, the pressure drop can be measured over smaller segments of the compliant tube (e.g., 1-3cm), making the local measurement lesion specific. By withdrawing the catheter along the artery, the compliance map of the arterial wall, as well as the functional assessment of the blood flow, could be calculated.
Acknowledgements
The study was partly supported by Joseph Drown Foundation(grant number 411342) and Herbert Berman Fund(grant number 4561).
Supporting information
Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jbiomech.2016.11.026.
Supplementary material
.
| Validation case | Stenosis [% area] | ∆D[%] | Flow condition |
| V1 | 0 | 6.6% | Hyperemia |
| V2 | 38 | 3.8%a | Normal flow |
| V3 | 38 | 4.1%a | Hyperemia |
| V4 | 64 | 3.6%a | Normal flow |
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