Physiological flow analysis in significant human coronary artery stenoses

Transcription

1 Biorheology 40 (2003) IOS Press Physiological flow analysis in significant human coronary artery stenoses Rupak K. Banerjee a,, Lloyd H. Back b, Martin R. Back c and Young I. Cho d a Department of Mechanical, Industrial and Nuclear Engineering, University of Cincinnati, Cincinnati, OH, USA b Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA c Division of Vascular & Endovascular Surgery, University of South Florida, College of Medicine, Tampa, FL, USA d Mechanical Engineering and Mechanics Department, Drexel University, Philadelphia, PA, USA Received 10 June 2002 Accepted in revised form 19 December 2002 Abstract. To evaluate the local hemodynamics in flow limiting coronary lesions, computational hemodynamics was applied to a group of patients previously reported by Wilson et al. (1988) with representative pre-angioplasty stenosis geometry (minimal lesion size d m = 0.95 mm; 68% mean diameter stenosis) and with measured values of coronary flow reserve (CFR) in the abnormal range (2.3±0.1). The computations were at mean flow rates ( Q) of 50, 75 and 100 ml/min (the limit of our converged calculations). Computed mean pressure drops p were 9 mmhg for basal flow(50 ml/min), 27 mmhg for elevated flow (100 ml/min) and increased to an extrapolated value of 34 mmhg for hyperemic flow (115 ml/min), which led to a distal mean coronary pressure p rh of 55 mmhg, a level known to cause ischemia in the subendocardium (Brown et al., 1984), and consistent with the occurrence of angina in the patients. Relatively high levels of wall shear stress were computed in the narrow throat region and ranged from about 600 to 1500 dyn/cm 2, with periodic (phase shifted) peak systolic values of about 3500 dyn/cm 2. In the distal vessel, the interaction between the separated shear layer wave, convected downstream by the core flow, and the wall shear layer flow, led to the formation of vortical flow cells along the distal vessel wall during the systolic phase where Reynolds numbers Re e(t) were higher. During the phasic vortical mode observed at both basal and elevated mean flow rates, wide variations in distal wall shear stress occurred, distal transmural pressures were depressed below throat levels, and pressure recovery was larger farther along the distal vessel. Along the constriction (convergent) and throat segments of the lesion the pulsatile flow field was principally quasi-steady before flow separation occurred. The flow regimes were complex in the narrow mean flow Reynolds number range Ree = and a frequency parameter of α e = The shear layer flow disturbances diminished in strength due to viscous damping along the distal vessel at these relatively low values of Ree, typical of flow through diseased epicardial coronary vessels. The distal hyperemic flow field was likely to be in an early stage of turbulent flow development during the peak systolic phase. 1. Introduction Endovascular Doppler catheter measurements of coronary flow reserve (CFR) and separate catheter measurements of translesional pressure drops ( p) (referred to as pressure gradients in the medical literature) are often made in conjunction with quantitative angiography to assess the physiological significance of lesions in patients with coronary artery syndromes (e.g., Wilson and Laxon [33], who also * Address for correspondence: R.K. Banerjee, Dept. of Mechanical, Industrial and Nuclear Engineering, 598 Rhodes Hall, PO Box , Cincinnati, OH , USA. Tel.: ; Fax: ; X/03/$ IOS Press. All rights reserved

2 452 R.K. Banerjee et al. / Physiological flow analysis in significant human coronary artery stenoses include a short primer in coronary hemodynamics). Brown et al. [15] have earlier proposed that the fall in distal mean coronary pressure ( p r ) across lesions to a level of about 55 mmhg causes ischemia in distal subendocardium when perfusion is inadequate to match metabolic demand. Efforts to directly measure pressure losses by using catheters spanning lesions introduce a tighter artifactual stenosis, thus elevating the pressure drop and reducing blood flow during measurements. Consequently, the mean pressure drop-flow rate curve ( p Q) is not known for physiological flow nor with a catheter present within lesions of patients for diagnostic purposes. In this paper detailed pulsatile hemodynamic computations and analysis were made in conjunction with clinical data in patients with physiologically significant coronary lesions and angina. The in vivo data set by Wilson et al. [32] for a 32 patient group was used. The flow computations are for physiological conditions without catheters present to alter the flow in the native lesions. Proximal lesion measurements of CFR (with minimal blockage) was 2.3 ± 0.1 in the abnormal range, indicating hyperemic flow limitation in the 68% mean diameter stenoses with average minimal lesion size d m = 0.95 mm before angioplasty. The flow regimes were complex in the narrow mean flow Reynolds number range Re e = with a frequency parameter of α e = In the computations using the finite element method, shear rate dependent non-newtonian viscosity of blood was accounted for with the Carreau model [12] and the normal coronary flow waveform was phase shifted to systolic predominance as measured in significant coronary lesions. The Methods section gives details on the stenoses geometry, the clinical data and measurements by Wilson et al. [32], the coronary flow waveform, and the numerical flow computational method in which we have confidence [12]. Results of the computations at basal and elevated flow conditions give a wealth of numerical data on hemodynamic variables difficult to measure in human coronary lesions: pulsatile velocity profiles proximal and along the composite lesion and distal vessel; phasic wall shear stress distributions; phasic pressure drop distributions and mean pressure drops; and phase relations. The Discussion section contains comparisons of the flow computations (in non-dimensional form) to some previous experimental in vitro pressure data [37] and some flow calculations [29] in similar stenotic models for perspective purposes by using mean flow Reynolds number similarity. 2. Background-flow instabilities The results show the important physiological relation between mean pressure drop, flow rate and distal coronary pressure; relatively high levels of wall shear stress in the narrow throat region; and separated flow shear-layer wave instabilities in the distal vessel compatible with the measured velocity profiles by Kajiya et al. [23] with an externally mounted, gated pulsed Doppler velocimeter distal to severe stenoses in the left anterior descending (LAD) coronary artery of a group of 9 patients before bypass surgery. Their measurements showed the existence of flow separation and recirculation near the vessel wall, broadening of the velocity spectrum along the central axial regions suggesting the occurrence of flow disturbance, and augmentation of systolic velocity with relatively small diastolic velocity component. Flow instabilities in steady water flow through an abrupt circular channel expansion (d m /d r ) = have been observed by visual dye filament studies many years ago [8] and indicated the shear-layer between the central jet and the reverse flow along the wall downstream to behave differently in the various flow regimes over a large Reynolds number (ratio of inertia to viscous forces) range (Re e = ). With increasing flow Reynolds number these regimes changed progressively from an orderly laminar flow (similar to the measurements by Macagno and Hung [25]) to an undulating vortex sheet-like flow

3 R.K. Banerjee et al. / Physiological flow analysis in significant human coronary artery stenoses 453 due to the growth of small amplitude shear-layer waves propagating downstream, and then to a random fluctuating eddying motion of fluid which was turbulent at higher Reynolds numbers. The axial distance between the step and the flow reattachment location (l R /h s where h s is step height) correspondingly increased, reached a maximum, then decreased to a minimum value, and then increased again somewhat to a plateau value typical of turbulent flow over the Re m range from 40 to The authors suggested the need for unsteady methods in the numerical calculation of flows where instabilities may occur. Much other earlier work has been reported in the literature on distal separated flow instabilities (e.g., [1,17,26,36,37]) in various stenosis models. The onset of these instabilities depends on the magnitude of the Reynolds number, and primarily on throat to distal diameter ratio (d m /d r ). Damping can occur in the more distal region due to viscous dissipative processes, as we observe herein and discuss in the text. Burger and Jou [14] have reviewed recent work on flows in stenotic vessels, particularly with regard to separated flow regions, which are difficult to analyze or compute. The effect of spatial flow acceleration in constriction (convergent) and throat segments of arterial stenoses is to suppress fluctuations in the flow (e.g., Back et al. [4] on the laminarization of turbulent boundary layers in convergent sections of rocket type nozzles at throat Reynolds numbers (Re m )aslarge as 1,000,000). This is due to the predominantly spatial favorable pressure gradient ( p/ z) < 0, i.e., decreasing pressures, even during unsteady flow decelerations, ( ū/ t) < 0, during the cardiac cycle. In contrast, the magnitude of the adverse pressure gradient ( p/ z) > 0, i.e., increasing pressures in divergent stenotic segments, can lead to flow separation and distal reattachment. Numerical flow computations have improved significantly in spatial resolution over the years, but require long computational times for pulsatile blood flow through diseased vessels where time varying separated flow and reattachment occur on the back side of stenotic segments and distal vessel as mean velocity, and thus Reynolds number vary widely during the cardiac cycle. Separated flow shear-layer wave instabilities near peak systolic flow further compound the problem of obtaining converged solutions (see Section 3.1). Since we are solving the unsteady Navier Stokes equations for blood flow, the initiation, growth and damping of flow fluctuations can be computed in principle, and these conservation equations have also been used to determine Reynolds stresses in turbulent flows that arise from the momentum term (u i u j )/ x j in large computer flow simulations. The role of shear-layer flow instabilities in distal vessel, and lowered luminal pressures in the throat and distal vessel at elevated blood flow rates has not been correlated with flow limiting processes in significant coronary lesions of patients experiencing ischemia. The flow computations and analysis shed some light on these hemodynamic flow phenomena that were found to occur at relatively low Reynolds numbers for pulsatile blood flow in the patient group before angioplasty. 3. Methods The in vivo data set of Wilson et al. [32] in a 32 patient group undergoing percutaneous transluminal balloon coronary angioplasty (PTCA) was used. The patients had single-vessel, single-lesion coronary artery disease with unstable or stable angina pectoris. Dimensions and shape of the coronary stenosis before angioplasty were obtained from quantitative biplanar X-ray angiography (Fig. 1). Biplane angiography of each lesion in orthogonal projections (60 left anterior oblique and 30 right anterior oblique) resolved vessel widths with cross-sectional area calculated from the equation for an ellipse, which we converted to mean diameters. The average minimal diameter d m = 0.95 mm (A m = 0.7 ± 0.1 mm 2 ) and the average proximal diameter was d e = 3 mm, producing a 68% mean diameter stenosis (90 ± 1%

4 454 R.K. Banerjee et al. / Physiological flow analysis in significant human coronary artery stenoses Fig. 1. Stenosis configuration with dimensions and finite element mesh. area stenosis) before PTCA. Dimensional data on the shape of a similar size lesion are from Back and Denton [5]. Before angioplasty, the length of the constriction region was l c = 6 mm, the narrow throat length was l m = 0.75 mm, and the divergence length l r = 1.5 mm. Average distal diameter is d r d e. Measurements of abnormal Coronary Flow Reserve (CFR) (ratio of time mean Doppler shift frequency between hyperemic and basal flow conditions) by Wilson et al. [32], with a 3F pulsed Doppler ultrasound catheter (d 1.0 mm) with tip positioned proximal to the coronary lesions (causing minimal flow obstruction) was 2.3 ± 0.1 (e.g., [34]) for the patient group before angioplasty. The authors have noted the importance of repeated doses of papaverine to maximize arteriolar bed vasodilation and potential flow augmentation during flow measurements. Accurate quantification of volumetric blood flow rates in stenotic vessels is difficult to determine clinically. Patients with abnormalities that might affect the vasodilator capacity of the distal vasculature were excluded from their study. Stenoses that are clinically significant reduce maximum achievable hyperemic blood flow since compensatory arteriolar dilation maintains resting blood flow (e.g., [24,33]). Mean arterial pressures p a = 89 ± 3 mmhg were measured in the coronary ostium. The large measured mean pressure drop of 56 mmhg at basal flow using the angioplasty catheter was artifactually contributed by flow obstruction of the catheter [6,32]. The coronary flow waveform used in the flow computations (Fig. 2) was obtained in our laboratory from in vitro calibration [18], smoothing the fluctuating Doppler signal, and phase shifting the normal pattern for the proximal LAD. The spatial average velocity across the flow ū(t) needed for flow computations, is similar to that from Doppler catheter measurements in patients where normal peak diastolic flow is reduced by significant lesions (e.g., [28,32]) and observed in the patient group of Kajiya et al. [23]. The ratio of relative average diastolic to peak systolic velocity was 0.4. In Fig. 2, the peak systolic velocity ū p t corresponds to a normalized velocity of 1.0, so that the ratio of mean to peak velocity ū/ū p t is as shown by the dotted line Numerical method In the flow computations, the composite lesion was assumed to have a smooth, rigid plaque wall, and round concentric shape with mean diameter d o. Plaque geometry was presumed to remain the same

5 R.K. Banerjee et al. / Physiological flow analysis in significant human coronary artery stenoses 455 Fig. 2. Phase shifted coronary flow waveform ū/ū p t vs t where S indicates the beginning of systole and D indicates the beginning of diastole. for hyperemic conditions, consistent with studies documenting the failure of flow-dependent dilation mechanisms in atherosclerotic coronary arteries secondary to endothelial dysfunction [20,22,31]. Arterial wall motion associated with pressure pulsation is believed to be relatively small in significant plaque regions because of the decreased wall elasticity. Glagov et al. [22] noted that functionally important coronary lumen stenosis might be delayed until the lesion occupies about 40 percent of the internal elastic lamina area. The arterial wall was considered to be essentially rigid. The flow computations were carried out by solving the mass and momentum equations for pulsatile blood flow using a Galerkin Finite Element Method (FEM) [10]. The Carreau model was used for shear rate dependent non-newtonian viscosity of blood with local shear rate calculated from the velocity gradient through the second invariant of the rate of strain tensor [19]. The conservation equations for mass and momentum for axi-symmetric flow, relations for local shear rate γ and blood viscosity η( γ), and details of the FEM have been previously described [12]. In our reported values of Reynolds number (Re) a kinematic viscosity of ν = cm 2 /s was used, a value near the asymptote in the Carreau model for blood (η poise, as γ ), which gives ν cm 2 /s. The calculations were initiated a distance of 2.5 cm proximal to the lesion by using the Poiseulle flow relation to allow the pulsatile non-newtonian blood velocity profile to develop before the stenosis inlet, consistent with the mean flow waveform shape i.e., ū e (t) (Fig. 2), and extended a distance of 3.0 cm distal to the lesion to allow for separated flow reattachment processes in the distal vessel. Proximal flow development distances were much shorter than 2.5 cm. A stress free boundary condition (b.c.) is used at the outlet. The no slip b.c. (u i = 0) was specified on the plaque wall (assumed to be rigid), and the symmetry condition was used along the axis of the lesion. The companion paper by Banerjee et al. [11] provides discussion on the usual presence of side branches. Reported pressure drops p(t) = p r p e are instantaneous pressure differences between the proximal vessel (z = 0) and distal region, thereby including pressure recovery. The calculations were carried out at time average (mean) flow rates Q of 50 ml/min (typical of basal values in a coronary vessel of 3 mm size [7]) and at 75 and 100 ml/min. The calculations were done for the first pulse cycle and a portion of the second cycle in order to compare them and to obtain accurate results since the computations were initiated near zero velocity. Numerical data are reported for the second cycle where convergence was obtained. Heart rate was 75 beats/min (period of a heartbeat T = 0.8 s) and the density of blood ρ = 1.05 gm/cm 3. Depending on the velocity pulse shape, the calculation time steps are varied between to s. The

6 456 R.K. Banerjee et al. / Physiological flow analysis in significant human coronary artery stenoses finite element computer method [21] was used to formulate and solve the matrix equations. The mixed formulation FEM was used so that both velocity and pressure in the conservation equations are calculated simultaneously. The mesh plot for the stenosis is shown in Fig. 1. For this simulation a Sun Ultrasparc 2 computer, having 200 MHz, 256 MB RAM, 4 GB disk was used with results downloaded to an IBM-PC computer for plotting. The CPU time for each time step was approximately 2.8 s. Calculation time steps decreased below s and hence, the number of time steps increased significantly with increasing mean flow rate, which precluded obtaining converged solutions at a higher value than 100 ml/min. Consequently, the mean flow rate range spanned a factor of two. Simpler and much faster steady flow calculations at the same mean flow rates were also made for reference purposes. The Appendix in Banerjee et al. [12] contains previous investigations that contribute to the validation of the methodology. 4. Results and discussion Axial velocity components of the computed pulsatile flow field are reported as a function of radius (r) along the axial (z) direction (Figs 3 and 4) and wall shear stress (Figs 5 and 7) and pressure distributions (Figs 6 and 8) are shown along the entire lesion and distal vessel for the reference steady and pulsatile flow. The origin of the axial coordinate (z = 0) is just proximal to the lesion a distance of 0.2 cm (Fig. 1); values of z = 0.5 and cm correspond to the mid-points of the constriction and throat segments, respectively, and z = 1.225, and cm are distal to the lesion. The pulsatile data are denoted by points I, II, III and IV in the coronary flow waveform (Fig. 2) for which values of ū/ū p t are respectively 0.50, 0.99, 0.75 and Point I (t = 1.05 s) is near the beginning of systole (near the end of the QRS complex of the ECG) which puts coronary flow in phase with the arterial pressure pulsatile waveform as observed clinically (Section 3). Point II is near peak systolic flow (t = 0.40, 1.20 s); points III (t = 0.58, 1.38 s) and IV (t = 0.71, 1.51 s) are during the deceleration phase early and later in diastole. Discussion of the results shown in Figs 3 8 is divided by flow regimes into two parts. One is the principally quasi-steady pulsatile mode along the constriction and throat segments before flow separation occurred just downstream of the shoulder of the divergent segment. The other is the vortical pulsatile mode that developed in the separated flow region in the distal vessel as the Reynolds number increased during the systolic phase Pulsatile velocity profiles along lesion The variation of axial velocity profiles during the cardiac cycle at progressive locations along the lesion are shown vertically upward in Figs 3A 3C, 3D 3F and 3G 3I at mean flow rates Q of 50, 75 and 100 ml/min, respectively. Proximal to the lesion, the profiles (if normalized by the centerline velocity u cl ) were close to parabolic in shape at peak systolic (point II) and at point III, but deviated somewhat at points I and IV where values of ū(t) were lower. This trend is evident over the range of mean flow rates, Q = ml/min. Along the constriction region where spatial flow acceleration occurs, the velocity profiles at the midpoint were steeper in the wall region and more uniform in the higher velocity core flow, although the shape of the profiles vary during the cardiac cycle and with the mean flow rate. At elevated flow rate ( Q = 100 ml/min) unsteady effects are evident in the relatively uniform core flow (points II and III). Weaker unsteady effects are also evident in the core flow region at the lower flow rates (points II and III). In the throat region where mean flow velocities were highest along the lesion, the velocity profiles at the midpoint indicate that viscous effects were confined to the near wall region with a uniform

7 R.K. Banerjee et al. / Physiological flow analysis in significant human coronary artery stenoses 457 Fig. 3. Velocity profiles at some locations along the stenosis at various times during the cardiac cycle (for Q = 50, 75 and 100 ml/min).

8 458 R.K. Banerjee et al. / Physiological flow analysis in significant human coronary artery stenoses Fig. 4. Velocityprofiles at some locations in the distalseparated flow region at various timesduring the cardiac cycle (for Q = 50, 75 and 100 ml/min).

9 R.K. Banerjee et al. / Physiological flow analysis in significant human coronary artery stenoses 459 core flow (boundary layer type flow) during the cardiac cycle. The viscous wall shear layer thickness (δ t ) decreased at the higher mean flow rates. At elevated flow rate ( Q = 100 ml/min) the spatial variations in core flow velocity (points II and III) are believed to be associated with propagation of disturbances from the distal vortical separated flow region. Weaker core flow velocity perturbations were also present at the lower mean flow rates ( Q = 50 and 75 ml/min; points II and III) although these are difficult to see in the scale of Fig. 3. Another way to view these profiles that depict the spatial transition of pulsatile viscous blood flow proximal to the lesion to a viscous wall shear layer flow along the constriction and the throat regions, is to follow the instantaneous velocity profiles shown in Figs 3A 3C, 3D 3F, 3G 3I, at particular times I, II, III, IV during the cardiac cycle, thus providing a snapshot picture of the flow computations. The pulsatile velocity profiles in the distal vessel flow separation region (Fig. 4) are subsequently discussed in conjunction with the wall shear stress (Fig. 7) and pressure distributions (Fig. 8) for pulsatile flow. But first we discuss the results of the reference steady flow calculations at flow rates which are the same as the mean flow values. Although these calculations have no physiological counterpart since unsteady accelerations and decelerations ( u i / t) of blood during the cardiac cycle are not accounted for, they nevertheless, show similar characteristics as the pulsatile variables in some regions of the flow field Wall shear stress distributions (reference steady flow) The variation of shear stress τ w along the arterial wall is shown in Fig. 5 as a reference datum for steady flow rates Q of 50, 75, and 100 ml/min. In the constriction region, τ w increased as flow velocities increased; a local spatial peak value occurred at the shoulder of the throat inlet. In the throat region, τ w decreased as the wall shear layer grew in thickness along the throat, and core flow acceleration occurred. Values of τ w decreased sharply at the shoulder of the divergent segment where spatial flow deceleration occurred, and the flow separated from the wall (τ w = 0). The reference flow reattached along the wall of Fig. 5. Arterial wall shear stress τ w distributions along the stenosis and distal vessel for the reference steady flow calculations at Q = 50, 75 and 100 ml/min.

10 460 R.K. Banerjee et al. / Physiological flow analysis in significant human coronary artery stenoses the distal vessel (τ w = 0) at Q = 50 ml/min and τ w increased again in the flow redevelopment region. In the flow separation region, values of τ w < 0, due to the reverse flow velocities ( u r ) in the wall region of the clockwise recirculation zone. In the reverse flow region, minimum values of τ w < 0 also increased with flow rate, and the reattachment location moved downstream beyond the calculation domain. It may be noted that the boundary condition at the end of the distal vessel allows blood to flow in and out to satisfy the mass balance Pressure distributions (reference steady flow) The variation of the pressure distribution p p e along the arterial wall is shown in Fig. 6 as a reference datum for steady flow rates Q of 50, 75 and 100 ml/min. Here, pressure differences are referenced to p e upstream of the lesion inlet, (z = 0). The general shape of the pressure variation consisted of increasing pressure drops along the constriction region, a nearly linear decrease in the throat region to a minimum value of magnitude p 1 and pressure recovery in the divergent region and distal vessel. The predominant pressure drop contribution occurred across the constriction region due to changes in momentum as the flow undergoes spatial acceleration and wall friction. In the throat region, the smaller pressure drop contribution was due to wall friction, and the additional momentum change as the core flow accelerated due to wall shear layer growth. In the divergent region, the initial rise in pressure was associated with flow deceleration, and the plateau region with flow separation. Further pressure recovery occurred in the distal vessel due to viscous processes in the separated flow region, and downstream of flow reattachment in the flow redevelopment region. These calculations indicate a slow pressure recovery process, and are discussed subsequently in terms of a pressure recovery coefficient c pr. Overall pressure drops, p s = p e p r, increased appreciably with flow rate as evident in Fig. 6. Values of p s, shown in Table 1, increased from 8.3 to about 29 mmhg for the two fold increase in Q. Fig. 6. Axial pressure drop p p e along the stenosis and distal vessel for the reference steady flow calculations at Q = 50, 75 and 100 ml/min.

11 R.K. Banerjee et al. / Physiological flow analysis in significant human coronary artery stenoses 461 Table 1 Results of the physiologic flow analysis before angioplasty Q Proximal Throat Throat Mid-throat Pressure drop (mmhg) Distal Pressure p/ p s (ml/min) Ree Rem dynamic τ wm Pulsatile Steady Local coronary drop pressure (dyn/cm 2 ) flow flow Poiseuille mean coeff. (1/2)ρ ū 2 m p p s p pressure c p (mmhg) [9] p r (mmhg) Pulsatile flow field before separation The pulsatile flow calculations of wall shear stress distributions during the cardiac cycle are shown in Fig. 7A, 7B, 7C at corresponding mean flow rates Q of 50, 75 and 100 ml/min. By using the time integrated waveform factor ū/ū p t = Q/Q p t = shown in Fig. 2, the corresponding peak systolic flow rates Q p t = 93, 140 and 186 ml/min, and peak Reynolds numbers Re e = 188, 282, and 376. Along the constriction and throat segments there were wide variations in the magnitude of oscillatory values of τ w at time points I, II, III and IV during the cardiac cycle. The relative level of these oscillations increased with the mean flow rate. For example, at the mid-point of the throat region, values of τ wm near peak systolic flow (point II) increased from 1170 to 2840 dyn/cm 2 for the two-fold increase in mean flow rate in the numerical calculations. Elevation in phasic peak values τ wp were higher at the shoulder of the throat inlet (z 0.8 cm) as seen in Fig. 7C where values reached about 4400 dyn/cm 2 at an elevated flow rate Q = 100 ml/min. These elevated wall shear stress levels in the throat region are discussed later in connection with Fig. 11 where values of τ w (t) are shown as a function of flow rate Q(t) in a composite plot over the mean flow rate range at various times during the cardiac cycle. The pulsatile distributions of τ w shown in Fig. 7 along the constriction and throat segments are similar in shape as the reference values for steady flow (Fig. 5). Although there are some differences in detail, the pulsatile flow field appears to be principally quasi-steady in nature before flow separation. In contrast, the complex variations in τ w in the flow separation region are described in the following section. The pulsatile flow pressure distributions during the cardiac cycle are shown in Fig. 8A, 8B, 8C at corresponding mean flow rates Q of 50, 75 and 100 ml/min. As with the wall shear stress distributions along the constriction and throat segments, there were wide variations in magnitude of these oscillating distributions at time points I, II, III, IV during the cardiac cycle. The relative level of these oscillations increased with mean flow rate. At peak systolic flow, the magnitude of the pressure drop to the minimal throat value p 1 was about 31 mmhg for basal flow and increased to about 108 mmhg at elevated flow ( Q = 100 ml/min). Application of the Bernoulli equation for one-dimensional inviscid flow at peak systolic flow gives pressure drops across the throat regions ( p 1 ) of 19 and 75 mmhg at corresponding peak systolic flows rates Q p t [( ū/ t) = 0] of 93 and 186 ml/min. Thus, contribution of wall frictional pressure drops and more precise determination of net momentum changes in the computations considerably increase p 1. The shape of the pulsatile flow pressure distributions retains the features of the reference values for steady flow (Fig. 6) along the lesion.

12 462 R.K. Banerjee et al. / Physiological flow analysis in significant human coronary artery stenoses Fig. 7. Arterial wall shear stress τ w distributions along the stenosis and distal vessel at various times during the cardiac cycle (for Q = 50, 75 and 100 ml/min).

13 R.K. Banerjee et al. / Physiological flow analysis in significant human coronary artery stenoses Pulsatile flow separation region The flow separation location (τ w = 0) just downstream of the shoulder of the divergent segment (origin of the separated shear layer), did not vary much during the cardiac cycle nor with the mean flow rate as indicated by the wall shear stress distributions shown in Fig. 7. The primary clockwise recirculation zone at each time during the cardiac cycle was foreshortened at its terminal point (τ w = 0) and was within the calculation domain. The wall shear stress is a sensitive indicator of the direction of flow velocities (u r ) adjacent to the wall, and from vorticity dynamics, the zeroes of τ w gives the axial spacing ( z) of alternating cellular sub-zones along the wall. To illustrate the nature of the flow instability, we focus on the basal flow field near peak systolic flow (time point II) in Fig. 7A. In this case the primary recirculation zone terminal (or tail) point was at z c1 = 3.25 cm. Following along the τ w contour (back toward the lesion) indicates that the wall shear layer was spacially accelerated by the impressed favorable pressure gradient (see Fig. 8A(II)) i.e., decreasing pressure, so that the magnitude of τ w increased. Small perturbations in τ w grew in amplitude along this contour. The axial spacing of these perturbations, z w (along with others observed in Fig. 7 at different times during the cardiac cycle), was associated with the core flow shear layer wave with wavelength, λ w = z w 0.6 cm (2 vessel diameters d r ) and wave speed, c = λ w f where f = 1/T is the frequency of a heartbeat. The wave speed was small, being only 0.75 cm/s and the wave was convected downstream by the core flow, diminishing in strength due to viscous damping. Further along this τ w contour (toward the lesion) the magnitude of the τ w oscillation grew rapidly and the spacially decelerated wall shear layer separated from the wall (τ w = 0) presumably due to the adverse pressure gradient in this vicinity (Fig. 8A(II)) i.e., increasing pressure. This location z c21 = 1.30 cm where τ w changes sign to a positive value (+τ w ) demarks the tail point of a counterclockwise circulation cell. In this vortical cell τ w > 0 due to forward flow velocities (u r ) in the wall shear layer; τ w reached a peak value near the cell mid-point, and τ w = 0againatz c23 = 1.10 cm, giving a cell width z c2 = = 0.20 cm. This vortical cell formed along the wall underneath the primary zone. Near peak systolic flow (time point II) the axial velocity profile at z = cm in Fig. 4A for basal flow shows this strong counter-clockwise vortical cell (centered nearby at z c2 = 1.20 cm) that developed beneath the crest of the undulating separated shear layer wave that was constrained by the bounding condition along the vessel axis. The axial mid-point of the cell was located at about 3.2 step heights [h s = 0.5(d e d m )] distal to the throat exit (or 3.4 throat diameters (d m )). The instantaneous Reynolds number Re e (t) at time point II for the basal flow ( Q = 50 ml/min) was 186; the throat Reynolds number Re m (t) was 587, thus indicating the significant ratio of inertia to viscous forces prior to flow separation. This vortical cell was also present, roughly in the same location, (at time point III for basal flow at Re e (t) = 141 as observed in Figs 4A and 6A, but was weaker in magnitude. At the lower Re e (t) = 47 and 94, (time points IV and I, respectively) the flow reverted to the primary single cell recirculation mode. The pulsatile flow pressure distributions during the cardiac cycle at basal flow (Fig. 8A) show the influence of this vortical cell in lowering the distal pressure from initial plateau values. Near peak systolic flow (II) there was a second minimum in p 2 slightly below the value p 1, at the throat exit. The velocity profiles shown in Fig. 4B at z = cm are in the pressure recovery region, while at z = cm (Fig. 4C) pressure recovery was nearly complete, although the velocity profiles were still redeveloping. At an elevated mean flow rate of 100 ml/min (the limit of our computational ability in the simulation of hyperemic response) Reynolds numbers were twice those for basal flow, and the undulating separated

14 464 R.K. Banerjee et al. / Physiological flow analysis in significant human coronary artery stenoses Fig. 8. Axial pressure drop p p e along the stenosis and distal vessel at various times during the cardiac cycle (for Q = 50, 75 and 100 ml/min).

15 R.K. Banerjee et al. / Physiological flow analysis in significant human coronary artery stenoses 465 flow field switched to a higher mode with numerous sub-zones as indicated by the wall shear stress distributions (Fig. 7C) and velocity profiles (Figs 4G, 4H, and 4I). The number of zeros of τ w increased from four to twelve near peak systolic flow (time point II) with various counter-clockwise vortical cells τ w > 0. The two counter-clockwise cells farthest downstream from the lesion formed underneath the primary zone due to the stronger core flow shear layer wave. The principal counter-clockwise vortical cell centered at z = 1.20 cm during basal flow, had a weak clockwise sub zone τ w < 0 beneath it. The velocity profile in Fig. 4G(II) at z = cm shows this weak wall cell next to the strong principal counter-clockwise cell (observed in Fig. 4A(II) during basal flow). Another counter-clockwise cell formed in the divergent segment of the lesion. The sharp downward spike in τ w (at z 1.46 cm) to 1440 dyn/cm 2 was about 1/3 of the peak wall shear stress at the shoulder of the throat inlet, and a factor of 37 higher than the proximal vessel value τ we. The pulsatile flow pressure distributions during the cardiac cycle at an elevated mean flow rate of 100 ml/min (Fig. 8C) indicates the significant effect of the vortical cells, particularly near peak systolic flow (time point II). The pressure dropped by about 30 mmhg below the throat p 1 value to p 2 of 140 mmhg in the distal vessel (z 1.45 cm) about 1.9 vessel diameters (d r ) distal to the throat exit. In the multi-vortical mode, net pressure recovery was enhanced, amounting to about 50 mmhg, so that the net pressure loss p across the lesion and distal vessel was 60 mmhg near peak systolic flow. More detail on the flow patterns is seen in Figs 4, 7 and 8 over the mean flow rate range at the various times during the cardiac cycle, particularly in the flow separation region, showing the evolution, growth, and viscous decay of the flow disturbances along the distal vessel. The terminal reattachment locations from the throat exit (in terms of throat step heights) were 23.2, 27.7 and 26.2, respectively, at Q = 50, 75 and 100 ml/min and happen to be in the same range as maximum values of about 25 step heights observed by Back and Roschke [8] at lower Reynolds numbers (Re m 300) for steady water flow through a smaller abrupt circular channel expansion [(d m /d r ) = 0.385]. The separated shear layer wave length λ w in the core flow in their larger diameter tube (d r = 2.48 cm) with step height (h s = 0.76 cm) was about 2.1 tube diameters (d r ), similar to the geometric scaling ratio observed in the smaller coronary distal vessel Phasic and mean flow pressure relations The instantaneous overall pressure drops p = p r p e across the lesion and distal vessel including pressure recovery is shown in Fig. 9 during the cardiac cycle at mean flow rates of 50, 75 and 100 ml/min. The shape of p with time is similar to the phase shifted waveform velocity ū shown in Fig. 2, indicating small phase lag in the p(t) ū(t) relation. The phasic oscillations in p(t) seen in Fig. 9 during the systolic phase are associated with the spatially damped residue of the separated shear layer flow instabilities. At mean flow rates of 50, 75 and 100 ml/min, the peak systolic pressure drops p p were about 20, 40 and 60 mmhg, respectively. Time average (mean) pressure drops p by integration over the cardiac p(t)dt) shown by the horizontal dashed lines in Fig. 9, were 9.15, 16.8 and 26.6 mmhg at the corresponding mean flow rates Q of 50, 75, and 100 ml/min. In the usual way, the absolute magnitude of pressure drops or losses, p e p r, are obviously p in our nomenclature. Global results of the computations are represented in Fig. 10, which shows the mean p Q relation, and fall in distal mean coronary pressure, p r = p e p (with p e p a, since mean flow resistances of proximal vessels were negligible compared to the large values across the stenoses, which increased with flow rate) vs Q. Mean pressure drops p were about 9 mmhg for basal flow, and flow limiting values 34 mmhg (obtained by extrapolation on a log log plot of p vs. Q), consistent with the measurements cycle ( p = (1/T) T 0

16 466 R.K. Banerjee et al. / Physiological flow analysis in significant human coronary artery stenoses Fig. 9. Overall pressure drop p across the stenosis and distal vessel during the cardiac cycle (for Q = 50, 75 and 100 ml/min). Fig. 10. Time average (mean) overall pressure drop flow rate relation, p Q, for physiologic flow before angioplasty. The fall of distal mean coronary pressure p r is shown on the right y-axis. of abnormal CFR of 2.3. Distal mean coronary pressures p rh for hyperemic flow Q h 115 ml/min, were in the range of 55 mmhg known to cause ischemia in the subendocardium [15]. The relative steepness of the p Q relation (Fig. 10) implies that variations of Q h in the patients were only moderately sensitive to variations of critical values of p h across the lesions. Table 1 gives the mean flow results of the physiological flow computations before angioplasty.

17 R.K. Banerjee et al. / Physiological flow analysis in significant human coronary artery stenoses 467 Fig. 11. Variation of wall shear stress τ w(t) in the throat region with flow rate Q(t) over the mean flow rate range ( Q = 50, 75 and 100 ml/min) at time points (I, II, III and IV) during the cardiac cycle Elevated wall shear stress-throat region The variation of τ w (t) in the throat region is shown in Fig. 11 as a function of Q(t) on a log log basis over the mean flow rate range Q = 50, 75 and 100 ml/min at time points (I, II, III and IV) during the cardiac cycle. The upper values are peak values (τ wp ) at the shoulder of the throat inlet, and the lower values (τ wm ) are at the mid-point of the throat segment. The solid points are the steady flow (Q = 50, 75 and 100 ml/min) reference values (τ ws ) at these two locations, which fall on the same curves as the pulsatile computations, thus indicating the quasi-steady nature of the flow field. Estimates of the magnitude of peak values (τ wp ) at the end of the constriction segment at the throat inlet are shown by the bold curve in Fig. 11 from the laminar boundary layer similarity analysis by Back [2], Back and Crawford [3] for conical constrictions, applied herein on a quasi-steady basis. This simple approximate method, amenable to hand calculation for which τ wp Q 3/2, agrees reasonably well with computed levels of τ wp, although there is a weaker dependence of τ wp Q n with, n 1.3, indicated by the near linearity of the computed values on a log log basis. For perspective purposes, the proximal level of wall shear stress (τ we ) ranged from about 4 to 40 dyn/cm 2 over the mean flow rate range at the time points shown, so that relative increases in τ wp /τ we were large, being a factor of 60 to 110 higher. In Fig. 11, the results are presented as physiological variables τ w (t) andq(t), in dimensional form, rather than by normalizing τ w by τ we as often done in reporting computational studies. Along the throat segment, τ w (t) decreased due to wall shear layer growth, and spatial core flow acceleration. The computations indicate a relatively high level of τ w in the throat segment which ranges from about 600 to 1500 dyn/cm 2 over the mean flow rate range of ml/min, with excursions to about 3500 dyn/cm 2 near peak systolic velocity at elevated flow rate. The significance of such elevated wall shear stress levels in the patients before angioplasty was not known, but do suggest potential biologic

18 468 R.K. Banerjee et al. / Physiological flow analysis in significant human coronary artery stenoses effects on blood platelets and red cells (e.g., see Table 3 of [5]) based on Couette viscometer data. The studies by Wurzinger and Schmid-Schonbein [35] allowed cellular damage assessment for short exposure times typical of blood flow along the wall of arterial constrictions. At exposure times of the order of 0.1 sec, they observed irreversible damage to platelets at shear stress levels on the order of 500 dyn/cm 2 and a threshold level for red cell lysis (1%) at about 2300 dyn/cm 2. In this regard it should be noted that cellular blood components in the viscous shear layer in the near wall vicinity move at very small fractions of the core flow velocity. In a previous study [11] it was shown that after angioplasty, wall shear stress levels in the enlarged and elongated throat segments were significantly lower in the residual stenoses Pressure drop and pressure recovery The magnitude of axial pressure drops p 1 (t), p 2 (t), and p(t) are shown in Fig. 12 as a function of Q(t) on a log log basis over the mean flow rate range Q = 50, 75 and 100 ml/min at time points (I, II, III and IV) during the cardiac cycle. The value of p 1 at the throat exit lies on a nearly linear curve on a log log basis, p 1 Q n with n The solid points, representing the reference values p 1s for steady flow (Q = 50, 75 and 100 ml/min) fall on the same curve as the pulsatile computations, thus indicating the quasi-steady nature of the pressure field. At flow rates Q(t) above about 90 ml/min the separated flow instabilities increased the pressure drop ( p 2 ) to values above p 1 and along this curve p 2 Q n with n 2.1 in the vortical pulsatile mode. The overall pressure drops p(t) indicate the variability and magnitude of the pressure recovery process in the separated flow region. In the usual way, pressure recovery is defined as p r = p r p 1 = p 1 p, the difference between the p 1 and p values in Fig. 12. To place these results in perspective, values of p r are scaled on the instantaneous dynamic pressure in the throat region 0.5ρū 2 m, to give a pressure recovery coefficient (c pr ). c pr = p r 0.5ρū 2 m = kc pr. (1) Here, c pr, is the often-cited high Reynolds number limit (Re ) across a sudden expansion of area ratio (κ r = A m /A r ) and ignoring wall friction (τ w 0), cf. [6], where the flow momentum decreases and viscous dissipation (turbulence at high Re) occurs increasing the internal energy of the fluid at the expense of pressure energy. c pr = 2κ r (1 κ r ). (2) The factor k in Eq. (1) was evaluated from the flow computations, and is shown in particular at peak systolic flow in Table 2. Table 2 Peak systolic non-dimensional pressure data Q Q p Re ep (1/2)ρū 2 mp c p c p1 c p2 p r/ p 1 k p 1/ p 1 (ml/min) (ml/min) (mmhg)

19 R.K. Banerjee et al. / Physiological flow analysis in significant human coronary artery stenoses 469 Fig. 12. Variation of the magnitude of axial pressure drops p 1(t), p 2(t), p(t) with flow rate Q(t) over the mean flow rate range ( Q = 50, 75 and 100 ml/min) at time points (I, II, III and IV) during the cardiac cycle. The Bernoulli equation at peak systolic flow ( ū/ t = 0) gives the pressure drop p 1 to the throat region with constriction area ratio (κ e = A m /A e ) for one-dimensional inviscid flow due to the increase in flow momentum as c p1 = p 1 0.5ρū 2 = 1 κ 2 e. (3) m

20 470 R.K. Banerjee et al. / Physiological flow analysis in significant human coronary artery stenoses The overall pressure drop ( p ) due to the net change in flow momentum is simply c p = p 0.5ρū 2 m = ( 1 κ 2) 2κ(1 κ) = (1 κ) 2, κ e = κ r = κ, (4) which gives the Borda-Carnot fluid head loss, h L = p /ρ. These asymptotic relations (Eqs (2) (4)) were used in an elementary way to determine the magnitude of wall friction in the computations, which also more precisely determine flow momentum changes, include unsteady effects, and compute pressure recovery. The lower dashed and solid curves in Fig. 12, show values of p 1 and p relative to computed values p 1 and p, respectively. In the asymptotic limit, values for the lesion with κ = are c pr = 0.180, c p1 = 0.990, and c p = The computed pressure data in non-dimensional form are summarized in Table 2 at peak systolic flow over the mean flow rate range. Pressure drop coefficients given here by c p = p 0.5ρū 2 m = c p1 c pr (5) generally decreased with increasing Re e (or Re m ) across stenoses (and distal vessel) since the variation of p with Q is less than Q 2, the asymptotic high Re limit. As seen in Table 2, both c p and c p1 follow this trend, as expected, but not c p2 which increased at elevated flow. The pressure drop to the throat ( p 1 ) was about a factor of 1.5 higher than the Bernoulli value ( p 1 ), and the pressure recovery ( p r ) was about 0.4 of p 1, giving a value of k as large as about 3.3. The overall pressure drop coefficient (c p ) was of order unity at peak systolic flow, which implies that the pressure head loss, p/ρ was about one throat velocity head (ū 2 m /2). At elevated flow rate, the value of c p at peak systolic flow of 0.80 was coincidentally near that given by Eq. (4) (c p = 0.81) but obviously for the wrong reasons since there was much larger pressure drop to the throat and larger pressure recovery than given by asymptotic theory as seen in Fig. 12. The throat dynamic pressure (Table 2) was substantial at elevated flow rate and peak systolic flow, being about 75 mmhg. In the hemodynamic computations, the minimal level of luminal pressure for peak systolic flow, p mp = p ap p mp was not known in the patients for basal flow, nor at hyperemic flow. Here p ap is the peak arterial pressure in the pulsatile waveform, and p mp is either p 1p or p 2p. The phase shift of the coronary flow waveform to systolic predominance favors higher coronary artery pressures during peak flow. In the patient group reported by Wilson et al. [32], measurements of p a (t) in the coronary ostium (through the guiding catheter) were damped by the presence of the Doppler catheter for basal flow, thus giving a mean value p a p e. During the hyperemic flow measurements, the guiding catheter was withdrawn from the coronary ostium of the patients to not obstruct maximal hyperemic blood flow. The results shown in Fig. 12 for p 2p suggest that luminal pressures distal to the lesion may have fallen below threshold ischemic subendocardial values periodically during the short time of the hyperemic flow measurements before the angioplasty procedures. Wilson and co-workers [33] usually fix heart rate at 100 beats/min during coronary flow reserve measurements. Cardiovascular changes during mild exercise in humans show a marked increase in the pulse pressure, p as p ad, mainly due to more rapid ventricular ejection of blood. Vander et al. [30] show increases of about 50% in peak systolic pressure, while mean arterial pressures usually increase much less 15%. For exercise conditions in the patients (before angioplasty) elevated levels of peak systolic pressure (p ap ) may have been of the same order as the computed peak values ( p mp ), with minimal transmural pressures (p mp ) on the order of zero. Periodic