Physiological flow simulation in residual human stenoses after coronary angioplasty.

To evaluate the local hemodynamic implications of coronary artery balloon angioplasty, computational fluid dynamics (CFD) was applied in a group of patients previously reported by [Wilson et al. (1988), 77, pp. 873-885] with representative stenosis geometry post-angioplasty and with measured values of coronary flow reserve returning to a normal range (3.6 +/- 0.3). During undisturbed flow in the absence of diagnostic catheter sensors within the lesions, the computed mean pressure drop delta p was only about 1 mmHg at basal flow, and increased moderately to about 8 mmHg for hyperemic flow. Corresponding elevated levels of mean wall shear stress in the midthroat region of the residual stenoses, which are common after angioplasty procedures, increased from about 60 to 290 dynes/cm2 during hyperemia. The computations (Ree approximately equal to 100-400; alpha e = 2.25) indicated that the pulsatile flow field was principally quasi-steady during the cardiac cycle, but there was phase lag in the pressure drop-mean velocity (delta p - u) relation. Time-averaged pressure drop values, delta p, were about 20 percent higher than calculated pressure drop values, delta ps, for steady flow, similar to previous in vitro measurements by Cho et al. (1983). In the throat region, viscous effects were confined to the near-wall region, and entrance effects were evident during the cardiac cycle. Proximal to the lesion, velocity profiles deviated from parabolic shape at lower velocities during the cardiac cycle. The flow field was very complex in the oscillatory separated flow reattachment region in the distal vessel where pressure recovery occurred. These results may also serve as a useful reference against catheter-measured pressure drops and velocity ratios (hemodynamic endpoints) and arteriographic (anatomic) endpoints post-angioplasty. Some comparisons to previous studies of flow through stenoses models are also shown for perspective purposes.

Estimated flow resistance increase in a spiral human coronary artery segment.

Coronary flow estimates were made for a spiral coronary artery segment (identified from a post-mortem replica casting) by using a modified Dean number based on the approximate coil radius of curvature, as suggested earlier. The estimates were found to correlate experimental pressure drop data for helical coiled tubes. Over a physiological range of mean Reynolds numbers from 100 to 400 for blood flow through main coronary arteries, estimates of the flow resistance increase relative to a straight lumen segment ranged from about 20 to 80 percent, and were of similar magnitude to those found in a flow study in a sinuous coronary vessel segment with no spiral.

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 delta p(t)-u(t) relation during the cardiac cycle at the lower hydraulic Reynolds numbers and frequency parameter. Hemodynamic interactions with smaller catheter-based pressure sensors evolving in clinical use require subsequent study since artifactually elevated translesional pressure gradients can occur during measurements with current angioplasty catheters.

Hemodynamic consequences of stenosis remodeling during coronary angioplasty.

Quantitative hemodynamic assessment during various endovascular interventions including balloon angioplasty is lacking. Translesional pressure drops measured by angioplasty catheters can cause flow blockage and thus lead to inaccurate estimates of preintervention and postintervention flow rates. A new analytical model of the flow rate-pressure drop relation across vascular stenoses is utilized that is nonlinear yet relatively simple in principle, easily applicable in vivo, and compatible with the presence of catheters. The model incorporates in vitro experimental evidence, angiographic data on the dimensions and shapes of coronary arterial stenoses before and after balloon angioplasty, reported translesional pressure gradients, and measurements of coronary flow reserve. Reasonable estimates of mean coronary artery flow rates and translesional pressure drops in the absence of angioplasty catheters are obtained. Prior to angioplasty significant flow restriction across a 68% diameter stenosis exists during hyperemic flow conditions. Following successful balloon dilation, increased minimal cross-sectional area (residual 40% diameter stenosis) results in an improved flow rate-pressure drop relation. Despite minimal flow restriction during hyperemic conditions following angioplasty remodeling, residual luminal constriction leads to elevated wall shear stress levels within the entry region of the stenosis. The flow analysis described may be of clinical utility in evaluating the hemodynamic significance of the anatomic severity of stenoses in coronary and peripheral arteries before and after endovascular therapeutic interventions.

Flow measurements in an aortocoronary bypass graft casting.

Flow visualization and pressure measurements were carried out in a singel valve saphenous vein casting which was made from a saphenous vein segment obtained from a bypass patient at Cedars Sinai Medical Center. Dye was injected to understand the flow around the valve. The dye showed very complex flow patterns around the valve and in the valve sinus, and the cavity formed by a ligated branch. For steady flow, pressure drops across the valve were 0.72, 2.0 and 6.3 mmHg for the physiological flow rates of 45, 84, and 169 ml/min, respectively. Overall pressure drop across the casting (compared to Poiseuille flow for a straight tube) increased with the flow rate, being 130 to 290 percent higher over this flow rate range. In the case of pulsatile flow, pressure drops across the valve were 0.95 and 3.0 mmHg for the flow rates of 47 and 87 ml/min which were 26 and 43 percent higher than those of steady flow. Overall pressure drop was 220 and 360 percent higher for those flow rates compared to Poiseuille flow. The measured spatial pressure distributions along the casting and flow visualization indicated the global nature of the flow field with the accelerated flow through the valve separating and reattaching downstream along the wall in the pressure recovery region. Atherosclerosis may be prone to occur in the lower shear region along the wall beyond the valve tip in the reattachment region, as we have observed in vivo in rabbit experiments.

Flow rate-pressure drop relation in coronary angioplasty: catheter obstruction effect.

Quantitative methods to measure the hemodynamic consequences of various endovascular interventions including balloon angioplasty are limited. Catheters measuring translesional pressure drops during balloon angioplasty procedures can cause flow blockage and thus inaccurate estimates of pre- and post-intervention flow rates. The purpose of this investigation was to examine the influence of the presence and size of an angioplasty catheter on measured mean pressure gradients across human coronary artery stenoses. Analytical flow modeling and in vitro experimental evidence, coupled with angiographic data on the dimensions and shape of stenotic vessel segments before and after angioplasty, indicated significant flow blockage effects with the catheter present.

Flow-pressure drop measurement and calculation in a tapered femoral artery of a dog.

This study describes the in vivo measurement of pressure drop and flow during the cardiac cycle in the femoral artery of a dog, and the computer simulation of the experiment based on the use of the measured flow, vessel dimensions and blood viscosity. In view of the experimental uncertainty in obtaining the accurate velocity profile at the wall region, the velocity pulse at the center was measured and numerical calculations were performed for the center line instantaneous velocity and within the two limits of spatial distribution of inlet flow conditions: uniform and parabolic. Temporal and spatial variations of flow parameters, i.e., velocity profile, shear rate, non-Newtonian viscosity, wall shear stress, and pressure drop were calculated. There existed both positive and negative shear rates during a pulse cycle, i.e., the arterial wall experiences zero shear three times during a cardiac cycle. For the parabolic inlet condition, the taper of the artery not only increased the magnitude of the positive and negative shear rates, but caused a steep gradient in shear rate, a phenomenon which in turn affects wall shear stress and pressure. In contrast, for the uniform inlet condition, the flow through the tapered artery was predominantly the developing type, which resulted in reduction in magnitude of wall shear rate along the axial direction.

Estimated mean flow resistance increase during coronary artery catheterization.

The purpose of this investigation is to examine the influence of the presence and size of the catheter on the measurement of mean pressure drop and, thus, flow resistance in coronary vessels. Relatively large mean translesional pressure gradients have been reported, but they may be due to obstruction effects. To evaluate this hypothesis, analytical flow modeling coupled with in vitro experimental evidence was used to estimate mean flow resistance increases due to the presence of a catheter in a proximal vessel for concentric and eccentric catheter configurations. For an angioplasty catheter, over the relative range of catheter size to coronary vessel size (di/d(o)) from 0.3 to 0.7 (which is currently being used clinically), the flow resistance increased by a large factor of 3-33 for the concentric configuration. For smaller infusion catheters, the flow resistance increase was less, although still appreciable. Very small angioplasty guidewire also leads to sizeable increases in flow resistance. Effects of catheter eccentricity also indicated substantial increases in flow resistance, although the magnitude was less. These initial results might be used by clinicians to obtain rough estimates of actual mean pressure gradients in vivo in relatively straight proximal segments of artery from values measured with catheters. Since catheters are used so widely clinically, these initial results may be useful also for other vessels in the vascular system where the mean flow is describable by the Poiseuille relation. Whereas there is reasonable confidence in the flow modeling methodology, hemodynamic data are needed to evaluate the actual magnitude of the effects of obstruction in vivo.

Numerical studies of three-dimensional arterial flows in reverse curvature geometry: Part I--Peak flow.

A three-dimensional flow simulation at Repeak = 192 and 580 was made in a smooth reverse curvature model that conformed to the gentle "S" shape from a human left femoral artery angiogram. The objective of this numerical investigation was to find the changes in pressure, shear stress, velocity profile, and particle path occurring in the double-curved arterial vessel. Due to the impingement of blood at the outer wall in the first bend region, the wall shear stress approached 40 dyne/cm2--a value over twice as large as in the straight upstream segment. Conversely, at the inner wall in the first bend, a low shear stress region was found where the value of the shear stress was consistently smaller than that in the straight section. The initiation of centrifugal effects caused by the first bend could clearly be seen at Repeak = 580, but due to the close proximity of the reverse curvature segment, the momentum effect due to the second bend overshadowed the centrifugal effect. Hence, only near the end of the second bend did the centrifugal effect due to the second bend result in a double-spiral-secondary motion. In addition, the numerically calculated pressure drop data were in agreement with prior experimental values.

Wall shear stress estimates in coronary artery constrictions.

Wall shear stress estimates from laminar boundary layer theory were found to agree fairly well with the magnitude of shear stress levels along coronary artery constrictions obtained from solutions of the Navier Stokes equations for both steady and pulsatile flow. The relatively simple method can be used for in vivo estimates of wall shear stress in constrictions by using a vessel shape function determined from a coronary angiogram, along with a knowledge of the flow rate.

Flow measurements in a highly curved atherosclerotic coronary artery cast of man.

Flow visualization and wall pressure measurements were made in a polyurethane cast of a cadaver coronary artery with a significant "s" shaped reverse curvature. A sucrose solution was used to simulate the kinematic viscosity of blood, with flow rates in the physiologic range. Flow visualization demonstrated significant secondary flow patterns in the wall vicinity, which increased with increasing Reynolds number. Random dye dispersion was observed at a Reynolds number of about 400, but not at 200. Dye filament patterns in the transition between the first and second curved region were predominantly influenced by the second curved region at lower Reynolds numbers, and by the first curved region at higher Re. Local wall pressure measurements demonstrated a significant centrifugal effect with large radial pressure differences across the casting. Flow resistances for the casting were considerably greater than reference Poiseuille flow values, and increased further with pulsatile flow.

Flow measurements in a model of the mildly curved femoral artery of man.

Steady flow observations in a smooth curved femoral artery model with a gradual bend revealed a flow pattern like that observed in coiled pipes. A double helical type flow was found to develop, with converging streamlines in the wall vicinity from the upper and lower plane of curvature merging asymptotically along the inner curvature in a stable manner. The helical or swirl angle of the labeled fluid particle paths increased with flow rate and thus Dean number. Flow in the wall vicinity was altered by centrifugal effects almost immediately downstream of the transition from the straight to curved segment for steady flow, although the propagation of this effect was observed farther downstream along the inner curvature side. This observed distance along the inner curvature became shorter with increasing Dean number. Pressure measurements for steady flow revealed progressively larger pressure drops with distance along the entrance region of the curved segment relative to that for a straight lumen. The overall pressure drop or flow resistance increased in a nonlinear way with increasing flow rate and thus Dean number. Time average pressure drop measurements across another similar smooth curved femoral model were found to be about the same for simulation of femoral artery pulsatile flow as for steady flow.

In vitro flow measurements in ion sputtered hydrocephalus shunts.

Accurate in vitro measurements of the pressure drop-flow (drainage) rate relationship were made for perforated Teflon microtubules using a fish hook arrangement. These measurements indicated that appropriate drainage rates could be obtained in the physiological range for hydrocephalus shunts. Animal experimentation is required for in vivo evaluation of these microtubules with micron-sized holes that may prevent cellular ingrowth and recurrent obstruction, and thus extend implanted shunt life.

Flow measurements in a human femoral artery model with reverse lumen curvature.

Flow visualization and wall pressure measurements were made in a smooth reverse curvature model that conformed to the gentle "s" shape of a left femoral artery angiogram of a patient in a clinical trial. Observed lesion localization at the inner (lesser) curvatures appeared to be associated with secondary flows in the wall vicinity directed toward the inner curvatures that tended to reverse direction in the flow entering the reverse curvature region. Moderate flow resistance increases of about 20 percent above the Poiseuille flow relation were found at the higher physiological Reynolds numbers Re above about 600-700 and thus Dean numbers for steady flow. For pulsatile flow simulation, flow resistances did not increase up to the largest Re of 470 tested. Apparently, the large variations in velocity during the cardiac cycle disrupted the stronger secondary flow patterns observed at the higher Reynolds numbers for steady flow.

Flow measurements in an atherosclerotic curved, tapered femoral artery model of man.

Flow visualization and pressure measurements were made for physiological conditions in a model derived from a femoral angiogram of a patient with lesion localization on the inner curvature wall and with vessel taper. Effects of curvature and taper were evaluated separately in other curved, tapered, smooth and straight, tapered, smooth models. Double helical secondary flow patterns were modified by plaque on the inner wall, and flow separations were observed between plaques at higher flow rates and Reynolds numbers. Pressure drop data for the plaque simulation model were similar in trend with Reynolds number as for the smooth model, but flow resistances were 25 to 40 percent higher. Significant pressure drops were measured due to the mild taper which could be estimated from momentum considerations, and smaller increased pressure drops were found due to curvature effects at the higher Dean numbers. Flow resistances for in vivo pulsatile flow simulation were about 10 percent higher than for steady flow for the plaque model, whereas no differences were observed for the smooth model.

Fluid particle motion and Lagrangian velocities for pulsatile flow through a femoral artery branch model.

A flow visualization study using selective dye injection and frame by frame analysis of a movie provided qualitative and quantitative data on the motion of marked fluid particles in a 60 degree artery branch model for simulation of physiological femoral artery flow. Physical flow features observed included jetting of the branch flow into the main lumen during the brief reverse flow period, flow separation along the main lumen wall during the near zero flow phase of diastole when the core flow was in the downstream direction, and inference of flow separation conditions along the wall opposite the branch later in systole at higher branch flow ratios. There were many similarities between dye particle motions in pulsatile flow and the comparative steady flow observations.

Phasic and spatial pressure measurements in a femoral artery branch model for pulsatile flow.

Phasic and spatial time-averaged pressure distributions were measured in a 60-deg femoral artery branch model over a large range of branch flow ratios and at physiological Reynolds numbers of about 120 and 700. The results obtained with an in-vivo like flow wave form indicated spatial adverse time average pressure gradients in the branch vicinity which increased in magnitude with branch flow ratio, and the importance of the larger inertial effects at the higher Reynolds numbers. Pressure losses in the branch entrance region were relatively large, and corresponding flow resistances may limit branch flow, particularly at higher Reynolds numbers. The effect of branch flow was to reduce the pressure loss in the main lumen.

Measurement and prediction of flow through a replica segment of a mildly atherosclerotic coronary artery of man.

Pressure distributions were measured along a hollow vascular axisymmetric replica of a segment of the left circumflex coronary artery of man with mildly atherosclerotic diffuse disease. A large range of physiological Reynolds numbers from about 60 to 500, including hyperemic response, was spanned in the flow investigation using a fluid simulating blood kinematic viscosity. Predicted pressure distributions from the numerical solution of the Navier-Stokes equations were similar in trend and magnitude to the measurements. Large variations in the predicted velocity profiles occurred along the lumen. The influence of the smaller scale multiple flow obstacles along the wall (lesion variations) led to sharp spikes in the predicted wall shear stresses. Reynolds number similarity was discussed, and estimates of what time averaged in vivo pressure drop and shear stress might be were given for a vessel segment.

Fluid dynamic study in a femoral artery branch casting of man with upstream main lumen curvature for steady flow.

An in-vitro, steady flow investigation was conducted in a hollow, transparent vascular replica of the profunda femoris branch of man for a range of physiological flow conditions. The replica casting tested was obtained from a human cadaver and indicated some plaque formation along the main lumen and branch. The flow visualization observations and measured pressure distributions indicated the highly three-dimensional flow characteristics with arterial curvature and branching, and the important role of centrifugal effects in fluid transport mechanisms.