Catheter Obstruction Effect on Pulsatile Flow Rate-Pressure Drop During Coronary Angioplasty

Transcription

1 . K. Banerjee Fluent, Inc., Lebanon, NH L. H. Back Jet Propulsion Laboratory, California Institute ot Technology, Pasadena, CA Fellow ASME M. R. Back Assistant Protessor ot Surgery, Division of Vascular Surgery, University of South Florida, College of Medicine, Tampa, FL Y. I. Cho Professor, Mechanical Engineering and Mechanics Department, Drexel University, Philadelphia, PA Catheter Obstruction Effect on Pulsatile Flow Rate-Pressure Drop During Coronary Angioplasty The coupling of computational hemodynamics to measured translesional mean pressure gradients with an angioplasty catheter in human coronary stenoses was evaluated. A narrowed flow cross section with the catheter present effectively introduced a tighter stenosis than the enlarged residual stenoses after balloon angioplasty; thus elevating the pressure gradient and reducing blood flow during the measurements. For resting conditions with the catheter present, flow was believed to be about 40 percent of normal basal flow in the absence of the catheter, and for hyperemia, about 20 percent of elevated flow in the patient group. The computations indicated that the velocity field was viscous dominated and quasi-steady with negligible phase lag in the Ap(t) u(t} relation during the cardiac cycle at the lower hydraulic Reynolds numbers and frequency parameter. Hemodynamic interactions with smaller catheterbased pressure sensors evolving in clinical use require subsequent study since artifactually elevated translesional pressure gradients can occur during measurements with current angioplasty catheters. Introduction Quantitative methods to measure the hemodynamic consequences of various endovascular interventions, including balloon angioplasty, are of clinical importance. Various authors, including Gruentzig et al. (1979), Ganz et al, (1983, 1985), Leimgruber et al. (1985), Anderson et al. (1986), and Redd et al. (1987), have described use of catheters to measure translesional pressure gradients to assess the hemodynamic significance of coronary artery stenoses. Catheters measuring translesional pressure drops during balloon angioplasty procedures can cause flow blockage effects (e.g., Anderson et al., 1986; Wilson et al., 1988). Wilson and Laxson (1993) note that the specific effects of changes in hemodynamic conditions on pressure gradient measurements are not well described in the human coronary circulation. In the present investigation computational hemodynamics is used to acquire information on changes in flow, flow regime, phase relations, and the magnitude of phasic and mean pressure gradients across a coronary artery stenosis in the presence of a translesional catheter. This investigation presents a new pulsed flow rate-pressure drop relation across vascular stenoses for in vivo flow in the presence of catheters and extends the earlier work of Young and Tsai (1973) and many others along with Back et al. (1997) for various stenosis configurations without catheters present. Numerical flow modeling was coupled with angiographic data on the dimensions and shape of stenotic vessel segments after angioplasty to estimate the degree of flow blockage effect with the catheter present by comparison with measured pressure gradients obtained from a patient group undergoing PTC A (Wilson et al., 1988). A significant catheter obstruction effect was predicted from an approximate method for mean flow (Back et al., 1996) using elementary flow analysis (Back, 1994). The present application provides a reference datum for subsequent evaluations of smaller catheter-based pressure sensors evolving in clinical use. Contributed by ttie Bioengineering Division for publication in the JOURNAL or BiOMECHANicAL ENGINEER[NG, Manuscript received by the Bioengineering Division June 20, 1998; revised manuscript received December 14, Associate Technical Editor: S. E. Rittgers. Methods The in vivo data set of Wilson et al. (1988) in a 32 patient group undergoing percutaneous transluminal balloon coronary angioplasty (PTCA) was used. The patients had single-vessel, single-lesion coronary artery disease. Dimensions and shape of the coronary stenosis after angioplasty were obtained from biplanar angiograms (Fig. 1). Figure 1 shows the lumen size relative to a catheter probe. Blood flow direction is from left to right. The centerline of the probe in Fig. 1 is identical to that of the lumen. The redistributed lesion following angioplasty is shown in a dotted shade. After PTCA the average minimal diameter increased to d, = 1.8 mm {A, = 2.5 ± 0.1 mm^) within treated lesions compared to a stenosis diameter before PTCA of rf = 0.95 mm (A, = 0.7 ± 0.1 mm"). Average proximal diameter d^ = 3 mm was relatively unchanged by the procedure producing a residual 40 percent mean diameter stenosis after PTCA. Average diameter distal to the stenosis was equivalent to the upstream diameter (dr '=' d^) and was unchanged by PTCA. Dimensional data on the shape of a similar size lesion are from Back and Denton (1992). Balloon angioplasty both lengthened and widened the narrowest region of the stenosis with resulting /, = 3 mm. Despite axial redistribution of plaque away from the narrowest region, the constriction length 4 = 6 mm and divergence length I,. = 1.5 mm were roughly unchanged by balloon dilation. The constriction segment was approximately conical and the divergence segment relatively abrupt. The pressure measurements by Wilson et al. (1988) were made with the angioplasty catheter d, = \.A mm spanning the lesions relative to coronary ostium values measured through the guiding catheter. The relative catheter to vessel sizes proximal and distal was dilde = 1.4/3.0 = 0.47, and in the throat region, d)ld, = 1.4/1.8 = 0.78, after angioplasty. The coronary flow waveform used in the flow simulations (Fig. 2) was obtained in our laboratory from in vitro calibration (Cho et al., 1983) and smoothing the fluctuating Doppler signal. The spatially averaged velocity across the flow t7(f), needed for flow simulations, is similar to that from Doppler catheter measurements in the left anterior descending (LAD) and circumflex (LCX) coronary arteries of patients undergoing PTCA Journal of Biomechanical Engineering Copyright 1999 by ASME JUNE 1999, Vol. 121 / 281

2 Proximal Region DIstai Region lc= 6.0 mm Constriction Region (I) lm = 3.0mm Throat Region (ii) i,= 1.5 mm Divergent Region (III) Fig. 1 Stenoses configuration with dimensions and finite eiement mesh (Sibley et al., 1986). The ratio of peak systolic to peak diastolic velocity was 0.4 with ratios for the right coronary artery (RCA) often larger than for the LAD and LCX. The minimum in the phasic signal also may lie above or below zero in patients. The quasi-steady flow behavior (as will be seen subsequently) implies that the mean pressure drop-flow rate (Ap - Q) relation is rather insensitive to the details of patient flow waveform shape, which in any event, was not measured simultaneously with a Doppler catheter because of the presence of the angioplasty catheter in the lesions measuring pressure gradients (Wilson et al., 1988). Numerical Method. The flow simulations were carried out by numerically solving the following conservation equations for mass and momentum with the catheter assumed to lie concentrically in the lesion, using a Galerkin finite element method (FEM) (Baker, 1983): Ujj = 0 and p dui dt ' ' = o-ijj + Pfi (1) Fig. 2 Coronary flow waveform UlOp-t versus t where ij = 1, 2 for axisymmetric flow, w, is the ith component of the velocity vector, p is density, ay is the stress tensor, and fi is the body force (gravitational effects were negligible). Furthermore, <^ij = -P^ij + rij = njih Jij = «.\/- + Hi (2) Here, p is the pressure, Sy is the Kronecker delta, TJ, is the deviatoric stress tensor, jy is the rate of strain tensor, and blood viscosity, TJ, is a function of shear rate 7,^. The stress vector j, at a point on the boundary of a fluid element is defined by Nomenclature A = flow cross-sectional area de = proximal vessel diameter dh = hydraulic diameter = da di di = catheter diameter d, = minimal lesion (throat) diameter d = mean vessel diameter dr = distal vessel diameter 4 = length of constriction region /, = length of narrowest (throat) region Ir = length of divergence region Hj = normal vector p = pressure Ap = overall pressure drop Q = blood flow rate r = radial distance ri = catheter radius To = mean vessel radius Re,, = proximal mean Reynolds number = Ue(de - di)lv Re, = throat mean Reynolds number = Umid - di)lv Si = stress vector t time Ui = velocity vector u = axial velocity X = axial distance a = frequency parameter = {{de- di)l jij = rate of strain tensor y = shear rate Tj - blood viscosity u = kinematic viscosity = rjl p p = blood density (T;,- = stress tensor T = shear stress T = period of cardiac cycle u) = circular frequency = 27r/T Subscripts e = proximal vessel i - catheter m = narrowest (throat) condition o = vessel p s = peak spatial value p - t = peak temporal value r = distal vessel w = wall condition Superscripts = time average (mean) over cardiac cycle = spatial average across flow 282 / Vol. 121, JUNE 1999 Transactions of the ASi\^E

3 Si = aijhj (3) For a known element and the solution field, the stress component Si on the boundary at the Gaussian integration points is evaluated. Subsequently, the normal and tangential components of stress vectors are obtained after applying the appropriate transformations. For brevity, the matrix equations that discretize these equations are not given here. 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 (Cho and Kensey, 1991). Model constants were obtained by curve-fitting blood viscosity data in the literature. The respective relations for local shear rate y and blood viscosity ri are given as follows: = 4n = Vi[S S mj>] 2i(n-l)/2 V = ^0^ + ivo~ ^o,)[i + (\t) ] where \ (characteristic time) = s, n = , r] = 0.56 poise, i]oc = poise. From these relations the local shear stress, T = 77-y, is calculated. In the proximal vessel, the spatial velocity profile in the annular gap was initially taken to be the analogue Poiseuille flow relation for the axial velocity u (Ward-Smith, 1980) ii = 2< 1 In h 1 - In ^ 1 ' «' In 1 _ (_i where Ue(t) is the spatial average velocity across the flow. Since this profile is for a Newtonian fluid and also not the pulsatile profile, the calculations were initiated a distance proximal to the lesion («1cm) to allow the pulsatile non-newtonian blood velocity profile to develop before the inlet of the stenosis, consistent with the mean flow waveform shape, i.e., M;(0- This procedure produces stable numerical calculations. The no-slip boundary condition M, = 0 is specified on the remodeled plaque wall, which is assumed to be rigid, and on the catheter wall. The outflow boundary condition in the distal vessel does not need to be specified in detail, since its values are effectively determined by extrapolation similar to finite difference schemes. The calculations extended 1.2 cm distal to the lesion to allow for separated flow reattachment processes in the distal vessel with the angioplasty catheter therein to measure pressures near the tip (Wilson et al., 1988). The mixed formulation FEM was used so that both velocity and pressure in the conservation equations are calculated simultaneously. Four nodal quadrilateral elements were used in the flow domain and at the boundary, two nodal elements were used (total number of elements = 1596; number of nodes = 1395). The mesh plot for the stenosis is shown in Fig. 1. For spatial integration of the matrix equations, the number of iteration steps was limited to 30 at each time step with a combination of the successive substitution and quasi-newtonian scheme. The implicit time integration scheme used was the second-order trapezoidal method with a variable time step, which is dependent on the magnitude of temporal inlet velocity and its gradient change. Depending on the velocity pulse shape, the time steps are varied between 1 X 10 * to 1 X 10"' s. The finite-element computer code (FIDAP, 1997) was used to formulate and solve the matrix equations. For this study 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. In comparison to the core elements in the lumen, element sizes near the walls were smaller to achieve accuracy for flow parameters. The aspect ratio of the elements was chosen to be (4) (5) (6) less than 10. For validation of the numerical computation, two separate computer modeling runs were performed at peak diastolic flow with different convergence criteria as follows: The relative velocity error with respect to the previous step and the relative residue error compared to the initial value were set to be 2 and 1 percent, respectively (Banerjee et al., 1997). Furthermore, the overall convergence was confirmed by increasing the total number of meshes by 20 percent over that of the previous run, and the two results were compared to check for accuracy. When the improvement with 20 percent more meshes was less than 1 percent in velocity vectors, wall shear stress, and pressure, the computation was considered to be accurate. The results are from the computation with the least CPU time, i.e., with less than 2 percent for both relative velocity error and relative residue error. The CPU time for each time step was approximately 2.8 s. The calculations were for two consecutive pulse cycles in order to compare them and to obtain accurate results: the results for the second pulse cycle were calculated in continuation of the first one. Numerical data are reported for the second cycle where convergence was obtained. Heart rate was 75 beats/min (period of a heart beat T = 0.8 s) and the density of blood p = 1.05 g/cm^. Axial components of the calculated flow field are reported as a function of radius (r) along the axial (x) direction, and wall shear stress and pressure distributions are shown along the lesion. The appendix contains previous investigations that contribute to the validation of the code. Results Velocity Profiles. Velocity profiles at the midpoint in the throat region are shown in Fig. 3 at various times during the cardiac cycle noted in the flow waveform (Fig. 2). Results are shown for time-averaged (mean) flow rates Q of 10, 30, and 50 ml/min. Corresponding throat mean hydraulic Reynolds numbers are Re, = 19, 57, and 95 with proximal Re,, values being less than Re,,, by the factor The arrow path superimposed on the profiles depict increasing times during the second cycle corresponding to near-peak systolic (t = 0.89 s), peak diastolic (f = 1.20 s), and times t = 1.41 and 1.53 s during the deceleration phase in diastole (Fig. 2). The profiles are remarkably similar during the cardiac cycle with viscous effects extending to the midregion of the gap during the acceleration and deceleration phases. Because of the narrow gap width, velocities are high, particularly during peak diastolic flow at the higher mean flow rates. From the continuity equation, instantaneous spatial mean velocities u, are higher than proximal values u^ by the area ratio AJA,,, = 5.50 and the peak spatial velocities Up-,, in the gap are about a factor of 1.48 u, at peak diastolic flow, in particular. Unsteady inertial effects appear to be secondary in the relatively low Reynolds number viscous-dominated flow field during the cardiac cycle, and this is consistent with the relatively small value of the hydraulic frequency parameter in the throat region a, = In the proximal region, a,, = 1.20, or a factor of 4a,, but still of order unity. The flow field appears to be quasi-steady, and as will be seen subsequently, the instantaneous pressure drop A/:)(t) across the lesion is in phase with the flow waveform velocity u(t). Shear Rate Profiles and Viscosity. Spatial variations in the shear rate y across the gap region at the midpoint in the throat are shown in Fig. 4 with arrows indicating the phasic oscillations near the arterial wall (on the right side). Shear rates y along the catheter wall (left side) are about 7 percent higher than the arterial wall values. The level of the oscillations proportionately increases with mean flow rate as evident in Fig. 4 at 2 = 10, 30, and 50 ml/min. At these relatively high shear Journal of Biomechanical Engineering JUNE 1999, Vol. 121 / 283

4 SJ200.0 E ~^150.0 'o : ioo.o < "^ '>J50.0 'o _o ^100.0 TD < S ^150.0 'o ;^100.0 "TO x < ml/mi n -f- - - = 0.89 s = 1.20 s = 1.41 s = 1.53s SOml/min 1 /'.^ / y "", '~^ = 0.89 s ~-t = 1.20s -^t= 1.41s - -t= 1.53 s ^S. \ ml/min - // y^ 1 / / / y^ 1 / ^^-^''^ f ^ ^ L- 1 ^ Radial Direction (cm) t = 0.89s -t = 1.20 s -^t = 1.41 s -»-t = 1.53s \ \ \ \ Fig. 3 Velocity profiles at midpoint of throat region at various times during the cardiac cycle (for Q = 10, 30, and 30 ml/min) rates in the throat, variations in blood viscosity r\ are relatively small, but do increase somewhat at the lower mean flow rate (Fig. 4). Spatial locations where shear rates are small and blood more viscous are in the wall vicinity of flow separation in the divergent portion of the lesion, and in the flow reattachment region along the wall of the distal vessel. In these regions, which are oscillatory spatially during the cardiac cycle, non-newtonian effects are more important, as they are also at small flow rates near the beginning of systole and end of diastole (not shown in Fig. 4). Calculated profiles of «, y, and 77 are not shown herein along the entire lesion during the cardiac cycle for brevity, since primary interest lies in the shear stress variation along the wall (T ) and the pressure distributions along the stenosis. Wall Shear Stress. The variation of shear stress r along the arterial wall is shown in Fig. 5 for peak diastolic flow at mean flow rates Q of 10, 30, and 50 ml/min. In the constriction region where spatial flow acceleration occurs, T increases as the core flow velocities become greater; and a local spatial peak value occurs at the end of the constriction region. In the throat inlet, the flow is no longer subjected to spatial flow acceleration effects, and r decreases and remains at a nearly constant level through the throat region. In the vicinity of the throat exit there is another local peak in r, before it decreases rapidly in the divergent region where spatial flow deceleration occurs. At mean flow rates of 30 and 50 ml/min, flow separation occurred along the divergent portion of the lesion, and the flow reattached along the wall of the distal vessel. At the lower mean flow rate of 10 ml/min, where spatial inertial forces are lower compared to viscous forces, i.e., lower Reynolds number, flow separation did not occur in the divergent region. The local spatial peaks in^t. in the throat region are also relatively smaller at the lower Re for the same reason. The magnitude of T, in the throat region scales on the instantaneous flow rate with the levels from Fig. 5 being 280, 840, and 1400 dynes/cm' at peak g,,., = 18.6, 55.9, and 93.1 ml/ min respectively, corresponding to mean flow rates of 10, 30, and 50 ml/min by using the integrated flow waveform factor QIQp-t = Or conversely, the time-averaged (mean) wau shear stress f levels in the throat region would be about 150, 450, and 750 dynes/cm^ for the range of mean flow rates based on the similarity of the velocity profiles in the throat region during the cardiac cycle. By using the mean translesional pressure gradient measurements in the patient group by Wilson et al. (1988) (see section on Mean Pressure Drop-Flow Rate Relation) estimated values of g are 18 and 35 ml/min for basal and hyperemic flow conditions. Thus levels of f,, in the throat region of the patient group are inferred to have been about 250 and 500 dynes/cm^ for basal and hyperemic flow conditions, respectively. These elevated levels of in vivo wall shear stress are moderately high, but of course act over a short time during which the angioplasty catheter was present to make pressure drop measurements in the patients after PTCA. Moreover, because of the common residual coronary stenosis present after PTCA in the patients, elevated levels of f, expected to occur in the throat region for normal physiologic flow (i.e., no catheter present) require subsequent evaluation. The corresponding physiological Reynolds numbers Re, = 4-Q/ni^dc are 100 and 360 at Q >«50 and 180 ml/min, respectively, for resting and hyperemic conditions. Pressure Distributions. The variation of the pressure distribution p p, along the arterial wall is shown in Fig. 6 for peak diastolic flow at mean flow rates g of 10, 30, and 50 ml/min. The general shape of the pressure variation consists of increasing pressure drops along the constriction section, a linear decrease in the throat region, and a relatively flat variation in the divergent region. In the throat exit region there is a local rise in pressure that becomes more pronounced as the mean flow rate (or Re) increases. The predominant pressure drop contribution occurs across the narrow gap throat region. The magnitude of the pressure drop along the 284 / Vol. 121, JUNE 1999 Transactions of the ASME

5 ' C ^i^ Q) ^ \ 10 ml/mi n : : \ \ _ ' ' " * ^ ~ - ~ ^ ^ - - A 1= 0.89 s t= 1.20 s -^t=1.41 s t=1.53s ^^^ 0)0.044 "O_0.043 "^0.042 '^0.041 O ^0.040 > : C c O ~.- *"' y / "'^"^ B A ml/min '. V 1 = 0.89 s t = 1.20s ->-t= 1.41s *-t = 1.53s ' -. t"^^p^;y:r5srif^::nv.;av.v,vrs=?=^ rrrirq-f-r- - ' ' , CO ^ CO "^20000 ca 0)15000 ^ ; 30 ml/min ^rr;it---^^^^:::=!=irr::^^^^-^^ B t = 0.89s f= 1.20s = 1.41 s...( = 1.53 s... ; \ - ' 1.. 0)0.044 CO 'O0.043 CL P.042 '3)0.041 O ^0.040 >0.039 C.$50,038 c o.036 '0.035 ^0.034 : 30 ml/min i,. ' L^^^r^^^^^S^ ' -..' :» ',_ t = 0.89s t= 1.20 s -H-t= 1.41s *-t = 1.53s o) ^0000 ^25000 '^20000 (C 0)15000 ''^ :., 50 ml/min ^ ^ * - ^ «... 1 = 0.89 s t= 1.20s -^t= 1.41s t=1.53s. - * ",,, *.. ' Radial Distance (cm).^..- * ^ ).CO ).042 do.041 (0 > ? ) Z O ml/min Radial Distance (cm) -t = 0.89s t = 1.20s -t = 1.41 s t = 1.53s Fig. 4 Shear rate y and blood viscosity rj, profiles at midpoint of throat region at various timed during the cardiac cycle (for Q = 10, 30, and 50 ml/min) constriction and throat regions nearly scales on the instantaneous flow rate being 10, 30, and 55 mmhg at peak diastolic flow rates in Fig. 6. This variation is similar to the wall shear stress, although momentum change contributions to the pressure drop do become of some importance at the higher mean flow rates. Of note is that the wall frictional pressure drop contribution is due to the shear stress acting on the catheter wall as well as on the arterial wall. Overall Pressure Drop. The instantaneous overall pressure drops Ap = /?r - Pe across the stenosis is shown in Fig. 7 during the cardiac cycle at mean flow rates of 10, 30, and 50 ml/ min. The shape of Ap with time is remarkably similar to the normalized flow waveform velocity U shown in Fig. 2. The largest peak values App_, occur at peak diastolic flow Qp^,, and the secondary peak in Ap occurs at peak systolic flow. There is virtuauy no phase lag in the instantaneous A/7(f) - M(0 Journal of Biomechanical Engineering JUNE 1999, Vol. 121 / 285

6 Axial Distance (cm) I : Constriclior II: Throat III: Divergent -loml/mln - 30 mt/mln - 50 ml/min Fig. 5 Arterial wall shear stress along the stenosis at peak diastolic flow (for Q = 10, 30 and 50 ml/min) relation, i.e., hysteresis effects are negligible, and the flow is basically quasi-steady. Integration of Ap(?) over the cardiac cycle gives the timeaveraged or mean pressure drops A/5 shown in Fig. 7 by the dotted lines. The calculated values of A/J are 5.4, 16.4, and 28.7 mmhg at corresponding mean flow rates Q of 10, 30, and 50 ml/min. Mean Pressure Drop-Flow Rate Relation: Results and Discussion. The overall time-averaged (mean) pressure drops Ap across the residual stenoses are shown in Fig. 8 as a function of time-averaged (mean) flow rates Q by the solid curve fitting the calculated values at 10, 30, and 50 ml/min. The dashed curve shows values from Back et al. (1996) obtained by an approximate method using local mean Poiseuille flow in the constriction and throat regions, and a pressure recovery coefficient in the divergence and distal regions. There is good correspondence between the approximate method, and the detailed solutions of the Navier-Stokes equations for pulsatile flow with shear dependent blood viscosity. The difference between the predictions is a few percent. Since the calculated pulsatile Ap is expected to lie above calculated mean values, the lower values here may be partly due to the asymptote in the Carreau model 77a, = poise compared to a Newtonian viscosity 77 = poise used in the calculations by Back et al. (1996). Since blood viscosity is affected by hematocrit and many other factors (e.g., Cho and Kensey, 1991) either value of 77 is considered adequate for calculation purposes in the range of relatively high mean wall shear rates ly ) in stenotic regions of main coronary arteries. Blood viscosity is nearly constant in the shear rate range larger than about 200 s~'. Results of the present calculations and those of Back et al. (1996) are given in Table 1. The important result of the calculations is that large mean pressure drops Ap occur at nominal mean flow rates Q. To place these results in perspective, for a typical mean resting flow rate g == 50 ml/min for the physiologic case (absence of a catheter) the estimated Ap is of order 1 mmhg (Back et al., 1997). If Q were the same value with the catheter present, Ap would be over a factor of 30 times higher as observed in Fig. 8 where Ap «30 mmhg at g = 50 ml/min. The flow blockage effect was estimated from the mean translesional pressure drops measured by Wilson et al. (1988) in patients with single vessel, single lesion coronary artery disease so that measurement error could not originate from under appreciation of pressure losses across diffusely diseased vessel segments. Wilson et al. (1988) gave a value of 12 ± 1 mmhg for the mean translesional pressure gradient at basal flow measured in the patient group after angioplasty. Using a value of about Ap!^]0 mmhg across the lesions, since the flow resistance in the proximal branched vessels is also increased due to the presence of the catheter therein, the estimated flow rate from Fig. 8 is 2 =i 18 ml/min. The peak mean translesional pressure gradient for hyperemia induced by repeated doses of papaverine to maximize arteriolar bed vasodilatation and flow augmentation, was 26 ± 3 mmhg in the patient group of Wilson et al. (1988). Using a value of about Ap «* 20 mmhg across the lesions. Fig. 8 indicates about a doubling of the mean flow rate (or coronary flow reserve) above basal flow to 2 '^^ 35 ml/ min. However, separate measurements of coronary flow reserve by Wilson et al. (1988) with a 3F pulsed Doppler ultrasound catheter (d = 1.0 mm) with tip positioned proximal to the coronary lesions (with minimal blockage) indicated a value of 3.6 ± 0.3 in the patient group after angioplasty. Clearly, this large increase in flow could not have occurred during the time that the angioplasty catheter spanned the lesions and mean translesional pressure gradients were measured. The catheter significantly changes the lesion mean pressure drop-flow rate relation from that for physiologic flow. Thus, measured mean translesional pressure gradients are severely elevated above the physiologic case at the same mean flow rate. In terms of flow obstruction by the catheter, it is estimated for the patient group of Wilson et al. (1988) that for resting conditions flow was reduced by the factor 18/50 or about 40 percent of normal basal flow, and for hyperemia, 35/3.6(50), or about 20 percent of elevated flow. The flow obstruction effect of the catheter is believed to be greater for hyperemic conditions in the patient group of Wilson et al. (1988) because of maximal flow limitation. Since distal arteriolar reserve is exhausted in the subendocardium when pressure distal to a lesion falls below ~55 mmhg (Brown et al., 1984), translesional pressure gradients of ~25-35 mmhg for mean arterial pressures of mmhg will become flow limiting either in the absence or in the presence of catheters. The Wilson et al. (1988) data fit within these criteria for limiting or choked flow. For basal flow when the arteriolar bed flow reserve can be augmented, the flow is also reduced due to the increased flow resistance with the catheter present, but to a lesser relative extent than for hyperemic conditions. Clearly, significant hemodynamic changes occur during catheter pressure measurements compared to physiologic flow because of the narrower flow cross section that effectively introduces a tighter stenosis than the enlarged residual stenoses after balloon angioplasty. Some other remarks about the clinical relevance of the calculations are in order. Plaque geometry was presumed to remain the same for hyperemic conditions, consistent with studies docu- Axiai Distance (cm) Fig. 6 Axial pressure drop p ~ p, along the stenosis at peal< diastolic flow (for Q = 10, 30, and 50 ml/min) 286 / Vol. 121, JUNE 1999 Transactions of the ASME

7 First Cycle Second Cycle 1 \V,,»»«ft<5L ^ xy A-Bi»^ y ^ y g ^Pavg - - // 1 \l "V J^ 1\ 5.4 mm Hg 16.4 mm Hg mm Hg ml/min -A-30 ml/min ml/min Time (s) Fig. 7 Overall pressure drop Ap across the stenosis during tlie cardiac cycle (for O = 10, 30 and 50 ml/min) menting the failure of flow-dependent dilation mechanisms in atherosclerotic coronary arteries secondary to endothelial dysfunction (Drexler et al., 1989; Vita et al., 1989). Arterial wall motion associated with pressure pulsation is believed to be relatively small in plaque regions because of the decreased wall elasticity. Hence, the arterial wall was considered to be essentially rigid. The preceding calculations of wall shear stress and pressure drop are conservative since plaque irregularities and wall roughness caused by local wall dissections and intimal flaps that may occur after PTCA procedures will further reduce the mean flow rate at the measured mean pressure gradient. Cho et al. (1983) found that frictional pressure drops were about 30 percent larger owing to local disturbances over plaque irregularities during the cardiac cycle in castings of atherosclerotic human coronary arteries at comparable Reynolds numbers as for the present higher Re calculations, but with the catheter present E E, Q Wilson et. al. (1988), Bacl^ et. al. (1996); w/cattieter * Present; w/ catfieter Conversely, since the catheter position may be eccentric within the vessel lumen, there would be some compensatory effect in not reducing the mean flow rate quite as much (Back, 1994; Back et al., 1996) as in a concentric position. Concluding Remarks The purpose of this coronary angioplasty catheter investigation was to carry out numerical calculations using the Navier- Stokes equations for pulsatile blood flow including shear-ratedependent blood viscosity for stenosis configurations obtained from angiographic data in patients. Measurements of mean translesional pressure gradients with the angioplasty catheter were used to infer in vivo mean flow rates after PTCA from the flow calculations, and thus estimate flow obstruction effects due to the presence of the catheter. In this clinical application the coronary lesions were assumed to have a smooth wall, round shape of mean diameter do, and that the catheter was positioned concentrically within the vessel lumen as it spanned the lesion. This allowed the use of an axisymmetric model with a flow waveform shape similar to in vivo coronary artery measurements in patients undergoing PTCA. The CPU time was a few hours per mean flow rate case. In the second cardiac cycle of the computations convergence was obtained. In general, the present analysis provides a reference datum for further investigations of stenotic vessel size and shape, longer length occlusions, arterial wall roughness, noncircular vessel lumen (e.g., ellipticity), lesion eccentricity and curvature, and catheter size (diameter) and positioning.' Study of some of these effects would require a three-dimensional computation, and would be quite complex. Two analyses appear most important to perform in future studies. One is concerned with the hemodynamic interaction with smaller catheter- based pressure sensors evolving in clinical use for diagnostic purposes. The other is for the study of lesion Flow Rate (ml/min) Fig. 8 Time-averaged (mean) overall pressure drop Ap-flow rate Q relation after angioplasty ' One reviewer of this paper has also expressed the need for the generation of data in stenotic animal models or in in vitro stenosis models for a wide range of flow rates and catheter sizes. Some initial in vitro A/? data were reported by DeBruyne et al. (1993) for steady flow of a sahne solution through various blunt hollow plug stenosis models with a small catheter present in the flow. Journal of Biomechanical Engineering JUNE 1999, Vol. 121 / 287

8 Table 1 Comparison of calculated mean pressure drops with Poiseullle flow theory (Back et al., 1996) Avg. Flow Rate Q' (ml/min) (Re)m Constriction Pressure Drop {mm Hg) Throat Pressure Drop (mm Hg) Divergent Pressure Rise (mm Hg) Overall Pressure Drop (mm Hg) (Backet.al.1996) Overall Pressure Drop (mm Hg) Calculated Percentage Difference , % ,3 0, % % hemodynamics for physiologic flow conditions without the catheter present. The development of a small optical sensor on a 0.46-mm-dia guide wire holds promise for reducing flow blockage effects in measuring mean transstenotic pressure gradients. Emanuelsson et al. (1993) reported on the use of this sensor to measure the mean Ap across coronary stenoses in a 30 patient group undergoing PTCA. DeBruyne et al. (1993) also reported on the evaluation of a small (d = 0.38 mm) fluid-filled guide wire for pressure monitoring. For critical minimal area coronary lesions of size ~ I mm, the ratio of guide wire size to minimal lesion size would be about The presence of smaller catheters requires investigation from a computational hemodynamic point of view both before and after PTCA. The hemodynamic alterations imparted by smaller pulsed Doppler ultrasound guide wire systems (d = 0.46 mm) with a piezoelectric transducer at its tip to measure velocities distal to lesions (Doucette et al, 1992; Segal et al., 1992) are also of interest. The situation without the catheter present (physiologic case) requires evaluation for the stenoses geometries both before and after PTCA, in conjunction with the measurements of coronary flow reserve, which improved from 2.3 ± 0.1 to 3.6 ± 0.3 in the patient group of Wilson et al. (1988) in the procedure. Since mean flow rates and Reynolds numbers would be higher (than with the catheter present) inertial effects and momentum changes would be larger, but viscous flow resistances would be much smaller because of the enlarged flow cross-sectional areas. The flow would still be expected to be laminar so that the present code with appropriate modifications could be used to estimate the hemodynamics. Boundary layer phenomena may play a role in blood flow during the cardiac cycle in the spatial flow acceleration region along the constriction portion of coronary lesions. Consequently, in the throat region entrance effects may also be important. These two investigations will be carried out and reported subsequently for pulsatile flow of blood. Computational hemodynamics may eventually serve a useful role in clinical practice to aid assessment of appropriate hemodynamic end points following endovascular interventions in coronary and peripheral arterial lesions. References Anderson, H. V., Roubin, G. S., Leimgruber, P. P., Cox, W. R., Douglas, J. S., Jr., King, S. B., and Gruentzig, A. R., 1986, "Measurement of Transstenotic Pressure Gradient During Percutaneous Transluminal Coronary Angioplasty," Circulation, Vol. 73, pp Back, L. H., and Denton, T. A., 1992, "Some Arterial Wall Shear Stress Estimates in Coronary Angiopla.sty," Advances in Bloengineering, ASME BED-VoI. 22, pp Back, L. H., 1994, "Estimated Mean Flow Resistance Increase During Coronary Artery Catheterization," Journal of Biomechanics, Vol. 27, pp Back, L. H., Kwack, E. Y., and Back, M. R., 1996, "Flow Rate-Pressure Drop Relation in Coronary Angioplasty: Catheter Obstruction Effect," ASME Journal of Biomechanical Engineering, Vol. 118, pp Back, M. R., White, R. A., Kwack, E. Y., and Back, L. H., 1997, "Hemodynamic Consequences of Stenosis Remodeling During Coronary Angioplasty," Angiology, Vol. 48, No. 2, pp Baker, A. J., 1983, Finite Element Computational Fluid Mechanics, Hemisphere Publ. Co., Chap. 4, pp Banerjee, R. K., 1992, "A Study of Pulsatile Flow With Non-Newtonian Viscosity of Blood in Large Arteries," Ph.D. Dissertation, Drexel University, Philadelphia, PA. Banerjee, R. K., Cho, Y. I., and Kensey, K. R., 1997, "A Study of Local Hemodynamics in a 90 degree Branch Vessel With Extreme Pulsatile Flows," Int. J. Computational Fluid Dynamics, Vol. 9, pp Brown R. G., Bolson E. L., and Dodge, H. T., 1984, "Dynamic Mechanisms in Human Coronary Stenosis," Circulation, Vol. 70, pp Cho, Y. I., Back, L. H., Crawford, D. W., and Cuffel, R. F., 1983, "Experimental Study of Pulsatile and Steady Flow Through a Smooth Tube and an Atherosclerotic Coronary Artery Casting of Man," J. Biomechanics, Vol. 16, pp Cho, Y. I., and Kensey, K, R., 1991, "Effects of the non-newtonian Viscosity of Blood on Flows in a Diseased Arterial Ve.ssel: Part 1, Steady Flows," Biorheology. Vol. 28, pp DeBruyne, B., Pijls, N. H. J., Paulus, W. J., Vantrimpont, P. J., Sys, S. U., and Heyndrickx, G. R., 1993, "Transstenotic Coronary Pressure Gradient Measurement in Humans: In Vitro and In Vivo Evaluation of a New Pressure Monitoring Angioplasty Guide Wire," / Am. Coll. Cardiol., Vol. 22, pp Doucette, J. W., Corl, P. D., Payne, H. M., Flynn, A. E., Goto, M. N., Nassi, M., and Segal, J., 1992, "Validation of a Doppler Guide Wire for Intravascular Measurement of Coronary Artery Flow Velocity," Circulation, Vol. 85, pp Drexler, H., Zeiher, A. M., WoUschlager, H., Meinertz, T., Just, H., and Bonzel, T., 1989, "Flow Dependent Coronary Artery Dilation in Humans," Circulation, Vol. 80, pp Emanuelsson, H., Lamm, C, Dohnal, M., and Serruys, P. W., 1993, "High Fidelity Translesional Pressure Gradients During PTCA - Correlation With Quantitative Coronary Angiography," J. Am. Coll. Cardiol., Vol. 21, pp. 340A. FIDAP Manual, 1997, Fluent Inc., 10 Cavandish Court, Lebanon, NH 03766, USA. Ganz, P., Harrington, D. P., Gaspar, J., and Barry, W. H., 1983, "Phasic Pressure Gradients Across Coronary and Renal Artery Stenoses in Humans," American Heart J., Vol. 106, pp Ganz, P., Abben, R., Friedman, P. L., Garnic, J. D., Barry, W. H., and Levin, D. C, 1985, "Usefulness of Transstenotic Coronary Pressure Gradient Measurements During Diagnostic Catheterization," American J. Cardiology, Vol. 55, pp Gruentzig, A. R., Senning, A., and Siegenthaler, W. E., 1979, "Nonoperative Dilation of Coronary Artery Stenosis: Percutaneous Transluminal Coronary Angioplasty," New England J. Medicine, Vol. 301, pp Leimgruber, P. P., Roubin, G. S., Anderson, H. V., Bredlau, C. E., Whitworth, H. B., Douglas, J. S., Jr., King, S. B, III, and Gruentzig, A. R., 1985, "Influence of Intimal Dissection on Restenosis after Successful Coronary Angioplasty," Circulation, Vol. 72, pp Redd, D. C. B., Roubin, G. S., Leimgruber, P. P., Abi-Mansour, P., Douglas, J. S., Jr., and King, S. B., Ill, 1987, "The Transstenotic Pressure Gradient Trend as a Predictor of Acute Complications After Percutaneous Transluminal Coronary Angioplasty," Circulation, Vol. 76, pp Segal, J., Kern, M. J., Scott, N. A., King, S. B., Ill, Doucette, J. W., Heuser, R. R., Oflh, E., and Siegel, R., 1992, "Alterations of Phasic Coronary Artery Flow Velocity in Humans During Percutaneous Coronary Angioplasty,'' / Am. Coll. Cardiol., Vol. 20, pp Sibley, D. H., Millar, H. D., Hartley, C. J., and Whitlow, P. L., 1986, "Subselective Measurement of Coronary Blood Flow Velocity Using a Steerable Doppler Catheter," /. Am. Coll. Cardiol., Vol. 8, pp Vita, J. A., Treasure, C. B., Ganz, P., Cox, D. A., Fish, R. D., and Selwyn, A. P., 1989, "Control of Shear Stress in the Epicardial Coronary Arteries of Humans: Impairment by Atherosclerosis," J. Am. Coll. Cardiol., Vol. 14, pp Ward-Smith, A. J., 1980, Internal Fluid Flow, Oxford University Press, Oxford, pp Wilson, R. F., Johnson, M. R., Marcus, M. L., Aylward, P. E. G., Skorton, D. J., Collins, S., and White, C. W., 1988, "The Effect of Coronary Angioplasty on Coronary Flow Reserve," Circulation, Vol. 77, pp Wilson, R. F., and Laxson, D. D., 1993, " Caveat Emptor, A Clinician's Guide to Assessing the Physiologic Significance of Arterial Stenoses," Catheterization and Cardiovas. Diagn., Vol. 29, pp Young, D. F., and Tsai, F. Y., 1973, "Flow Characteristics in Models of Arterial Stenosis 1 Steady Flow," J. Biomechanics, Vol. 6, pp / Vol. 121, JUNE 1999 Transactions of the ASME

9 APPENDIX This section includes some comments on the validation of the numerical code: An earlier version of the present numerical code was applied by Banerjee (1992), to measured in vitro pressure drop across a blunt hollow plug stenosis model (de mm; tfm/jj, = 0.50 ; /,/«?,, = 3.0 ) for steady flow of water at RCj = 75, without a catheter in the flow. The calculated Ap including distal pressure recovery was within 5 percent of the measured Ap values, thus giving some confidence in the numerical method. Cho and Kensey (1991) also found good agreement within a few percent between their calculated Ap values from an earlier version of the numerical code, and the steady flow Ap measurements of Cho et al. (1983), in a casting of an atherosclerotic human coronary artery with a moderate stenosis (without a catheter present) over a wide range of Reynolds numbers (flow rates) Re^ = using a 33 percent sugarwater solution with viscosity similar to that of blood at higher shear rates. More recently, we have applied the present code to calculate pulsatile, shear-rate-dependent blood flow through a segment of the normal tapered femoral artery of a dog. This numerical study was in conjunction with in vivo Doppler flow cuff data and pressure drop measurements via ligated, small, side branches, which, in turn, did not disturb or change the main lumen flow, and angiographic data.^ In this case, the 10 percent difference between calculated and measured time-averaged (mean) pressure drops (A/5 = -0.6 mmhg) was relatively small compared to the measured Ap oscillations ( 4.5 to 2.7 mmhg) during the cardiac cycle. The pulsatile waveform was triphasic with peak flow during systole, unlike the usual waveform pattern in normal coronary vessels. The mean flow rate Q = 102 ml/min, and the mean flow Reynolds number Re^ = 160 in the mean diameter 3.7 mm vessel segment of length 52 mm. The frequency parameter a = 3.7. These comparisons to experimental data lend some credence to the calculation method, and give reasonable accuracies that may be expected in flow simulations, such as, for the presence of a catheter in a residual coronary lesion considered herein. ^ To be published in a chapter entitled, "Computational Fluid Dynamic Modeling Techniques Using Finite Element Methods to Predict Arterial Blood Flow," Banerjee, R. K., Back, L. H., and Cho, Y. I., in the Gordon and Breach International Series in Engineering, Technology and Applied Science, volume on Biomechanic Systems, Techniques and Applications. Journal of Biomechanical Engineering JUNE 1999, Vol. 121 / 289