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Am J Physiol Heart Circ Physiol 289: H392-H397, 2005. First published February 25, 2005; doi:10.1152/ajpheart.00798.2004
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Delineating the guide-wire flow obstruction effect in assessment of fractional flow reserve and coronary flow reserve measurements

Abhijit Sinha Roy,1 Rupak K. Banerjee,1,2 Lloyd H. Back,3 Martin R. Back,4 Saeb Khoury,5 and Ronald W. Millard6

Departments of 1Mechanical Engineering and 2Biomedical Engineering, 5Cardiac Catheterization Laboratory, Department of Cardiology, and 6Department of Pharmacology and Cell Biophysics, University of Cincinnati, Cincinnati, Ohio; 3Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California; and 4Division of Vascular and Endovascular Surgery, University of South Florida, College of Medicine, Tampa, Florida

Submitted 5 August 2004 ; accepted in final form 23 February 2005


   ABSTRACT
TOP
 ABSTRACT
METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 GRANTS
 REFERENCES
 
Hemodynamic analysis was conducted to determine uncertainty in clinical measurements of coronary flow reserve (CFR) and fractional flow reserve (FFR) over pathophysiological conditions in a patient group with coronary artery disease during angioplasty. The vasodilation-distal perfusion pressure (CFR-rh) curve was obtained for 0.35- and 0.46-mm guide wires. Our hypothesis is that a guide wire spanning the lesions elevates the pressure gradient and reduces the flow during hyperemic measurements. Maximal CFR-rh was uniquely determined by the intersection of measured CFR and calculated rh of native and residual epicardial lesions in patients without microvascular disease, during angioplasty. Extrapolation of the linear curve gave a zero-coronary flow mean pressure (zf) of ~20 mmHg and a corresponding rh of 55 mmHg in the native lesions, which coincided with the level that causes ischemia in human hearts. On this linear curve, values of CFR and FFRmyo (pathophysiological condition) and CFRg and FFRmyog (in the presence of the guide wire) were obtained in native and residual lesions. A strong linear correlation was found between CFR and CFRg [CFR = CFRg x 0.689 + 1.271 (R2 = 0.99) for 0.46 mm and CFR = CFRg x 0.757 + 1.004 (R2 = 0.99) for 0.35 mm] and between FFRmyo and FFRmyog [FFRmyo = FFRmyog x 0.737 + 0.263 (R2 = 0.99) for 0.46 mm and FFRmyo = FFRmyog x 0.790 + 0.210 (R2 = 0.99) for 0.35 mm]. This study establishes a strong correlation between CFR and CFRg and between FFRmyo and FFRmyog, which could be used to obtain the true state of occlusion in the coronary artery during angioplasty.

hemodynamics; stenosis; microvascular impairment; pressure drop

THE DIAGNOSTIC COMPETENCE of guide wires for measuring mean transstenotic pressure drop (17, 19, 22, 38) (referred to as pressure gradient in clinical practice) and flow (15, 21, 29, 41) in stenosed coronary arteries has been enhanced with the use of small-diameter guide-wire sensors. Several authors (1, 24, 25, 27, 30, 40) have described the use of catheters to assess the hemodynamic significance of coronary artery stenoses. The use of a single guide wire has been extended to measurement of physiological parameters, such as the fractional flow reserve (FFR) and coronary flow reserve (CFR) (18, 23, 36, 43), for simultaneous evaluation of epicardial coronary disease and distal microvascular disease in clinical settings. However, the introduction of a guide wire causes an obstructive effect, creating an "artifactual" stenosis (1, 3, 5, 42).

Although guide wires can be used, in principle, to measure CFR and FFR, absolute measurements of flow are more difficult (29, 42) because of uncertainties in velocity, transducer positioning and sample volume, determination of accurate luminal area, and in vivo calibration. Current understanding behind mean pressure gradient ({Delta}) measurements with guide wires assumes that in moderately (<70%) blocked arteries the obstruction effect of the guide wire can be neglected, because the ratio of the area of the guide wire to the minimal luminal area of the stenosis is not large. For severely (>90%) blocked arteries, the stenosis itself produces a very large pressure drop, and a guide wire need not be used to detect such severe blockages. Although this is true in the case of severely stenosed arteries, in the case of moderately blocked arteries, this assumption can be erroneous, because the moderate stenosis produces a smaller {Delta}, and introduction of the guide wire produces a larger percent difference in {Delta} above the pathophysiological values. Thus diagnosis of severely blocked arteries may not be a major concern, but errors could be inherently made in diagnosis of moderately stenosed arteries.

In this study, CFR and FFR are the values expected in the native, intermediate and residual lesions (pathophysiological condition without guide wire). Similarly, CFRg and FFRg are the values measured with the guide wire. Two guide wires, 0.35 mm (0.014 in.) and 0.46 mm (0.018 in.), are evaluated, and hemodynamic analyses for pathophysiological flow are used to construct the correlations between CFR and CFRg and between FFR and FFRg.

Thus, once CFRg and FFRg are measured with guide wire, the correlations could be used to obtain the physiological (without guide wire) CFR and FFR for a coronary stenosis, i.e., removal of the guide-wire flow obstruction effect from clinical measurements.


   METHODS
TOP
 ABSTRACT
 METHODS
RESULTS
 DISCUSSION
 APPENDIX
 GRANTS
 REFERENCES
 
The in vivo data set of Wilson et al. (45) in a 32-patient group undergoing percutaneous transluminal 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 stenoses before and after intervention were obtained from quantitative biplanar X-ray angiography. Biplanar 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 an equation for an ellipse. This area was converted to mean diameters. The measured (mean ± SD) values of minimal area of stenosis, mean pressure measured proximal to the stenoses at the ostium (a), and CFR by Wilson et al. are summarized in Table 1.


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  		Table 1. Lesion geometry and hemodynamics: pre-, mid-, and post-PTCA

 
PTCA widened and lengthened the narrowest region of the stenoses: 0.95–1.8 mm diameter and 0.75–3 mm long. Mean proximal and distal vessel diameter was 3.0 mm. Despite axial redistribution of the plaque away from the narrowest region, overall stenosis shape and length (constriction length = 6 mm and divergence length = 1.5 mm), were roughly unchanged. Patients with abnormalities that might affect the vasodilator capacity of the arteriolar vasculature were excluded from the study (45). In the flow analysis, the residual composite lesion was assumed to have a smooth, rigid plaque wall and round concentric shape. Plaque geometry was presumed to remain rigid, and the same was assumed for hyperemic conditions. Additional dimensional data on the shape of similar-sized lesions are from Back and Denton (4).

Additionally, an intermediate stenosis with maximal area blockage on the basis of minimal diameter of 80% was used in this study [Table 1, data from Back and Denton (4)], thus including a wide range of lesion sizes, to obtain the correlations. However, this intermediate lesion size was not measured by Wilson et al. (45). Bech et al. (11, 12), Baumgert et al. (10), and several others measured FFRg using 0.35-mm guide wire in lesions of size and reference artery diameter similar to those of the intermediate stenosis used in this study. Their measurements were very close to our analyses.

The coronary velocity waveform (spatially averaged at each time across the cross-sectional area) used in the flow analyses was obtained in our laboratory from in vitro calibration (14) by smoothing the fluctuating Doppler signal and phase shifting the normal pattern for the proximal left anterior descending artery. This result is consistent with Doppler catheter measurements in patients where normal peak diastolic velocity is reduced by significant lesions (9, 28, 41, 42). After intervention, the velocity waveform was phase shifted back to normal diastolic predominance.

Details of the numerical method used to calculate the pulsatile hemodynamics in coronary artery and lesions with and without guide wire have been described by Banerjee et al. (7–9). The Carreau model was used for shear rate-dependent non-Newtonian blood viscosity. Instantaneous pressure differences [{Delta}p(t) = pe – pr] between the stenosis inlet and distal region (including pressure recovery in the separated flow reattachment processes in the distal vessel) were integrated over the cardiac cycle to obtain the mean pressure drop ({Delta}). Distal mean coronary pressures (r = e – {Delta}) were computed with ~= a, where a is mean aortic pressure. Computations were carried out for a range of mean flow rates (), consistent with measurements of CFR and flow limitation. A typical basal physiological value ( = 50 ml/min) for a 3-mm coronary vessel was used (6). Mean proximal Reynolds number (ee = 4/{pi}de{nu}, where de is proximal vessel diameter and {nu} is kinematic viscosity) ranged from 100 to 230 and from 100 to 360 before and after coronary intervention for pathophysiological flow, respectively. The Womersley number for the pathophysiological condition in native, intermediate and residual lesions was 2.25. In the presence of 0.35- and 0.46-mm guide wire, the Womersley number was 1.99 and 1.9, respectively. Cycle time of 0.8 s, density of blood ({rho}) = 1.05 g/cm3, and a kinematic viscosity ({nu}) = 0.035 cm2/s, the value near the asymptote in the shear rate-dependent non-Newtonian Carreau model, were used to calculate the Womersley number and ee.

CFR and rh without guide wire (45) and computed values of CFR and rh for the different lesions with guide wire (0.35 and 0.46 mm) were used to construct the maximal vasodilation-distal perfusion pressure curve, also known as the CFR-rh relation. The maximal CFR-rh curve was plotted by joining the measured (45) and computed values of CFR and rh at hyperemia for the native, intermediate and residual lesions after angioplasty, because blood was supplied to the same distal vasculature, which was originally characterized by marked arteriolar dilation. The CFR-rh relation was then used to construct the correlations between FFR and FFRg and between CFR and CFRg in native, intermediate and residual lesions for the two guide wires. The linear CFR-rh relation was also extrapolated toward its origin to estimate zero-coronary flow mean pressure (zf) for the patient group of Wilson et al. (45).


   RESULTS
TOP
 ABSTRACT
 METHODS
 RESULTS
DISCUSSION
 APPENDIX
 GRANTS
 REFERENCES
 
Banerjee et al. (7–9) published detailed pulsatile analyses of preintervention (with and without 0.46-mm guide wire) and physiological postintervention (without guide wire) conditions from the clinical diagnostic perspective to ascertain changes in {Delta} and fall in distal coronary pressure (r) due to insertion of a guide wire. These results also showed the detailed variation in pressure difference (pe – p) along the lesion and distal vessel associated with momentum changes and larger viscous effects with a guide wire spanning the lesions. New data before and after intervention with 0.35- and 0.46-mm guide wire and an intermediate stenosis with maximal area blockage of 80% with 0.35- and 0.46-mm guide wire and without guide wire are included in this study (Tables 2 and 3).


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  		Table 2. Hyperemic mean pressure gradients and distal mean pressure pre-, mid-, and post-PTCA with guide wire

 

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  		Table 3. Effect of guide wire on hyperemic CFR and FFRmyo vs. pathophysiological CFR and FFRmyo

 
Estimation of zf. Figure 1 shows the maximal CFR-rh relation for the native, intermediate and residual lesions after angioplasty with and without guide wire. Distal mean perfusion pressure for hyperemic flow (rh) increased from 55 mmHg before PTCA without guide wire to ~75 mmHg after PTCA without guide wire. rh of 55 mmHg is indicative of subendocardial ischemia (13). Extrapolation of the nearly linear CFR-rh relation toward its origin gave zf of ~20 mmHg, which is near a measured value (2) of 18 mmHg, where myocardial blood flow ceased in the subendocardial layer of dog hearts that were maximally dilated by adenosine infusion. Bache and Schwartz (2) placed a variable occluder on the proximal left circumflex artery and measured distal rh by cannulation. After removal of the heart, myocardial blood flow (ml·min–1·g tissue–1) measured in transmural layers with radioactive microspheres decreased linearly with rh, even at pressures close to zf. In contrast, flow ceased in the subepicardial layer at a lower zf of 10 mmHg in the dog hearts (2).



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  		Fig. 1. Increase in mean coronary flow rate (/b) vs. distal mean coronary pressure (r) before and after percutaneous transluminal coronary angioplasty (PTCA). Maximum vasodilation (i.e., coronary flow reserve)-distal perfusion pressure relation (CFR-rh) is shown by the nearly linear solid line.

 
Furthermore, Wilson and Laxson (44) observed that human coronary arteries with focal lesions (not more widespread atherosclerosis as observed by angiography) behave similarly to dog normal coronary arteries when occluded with respect to compensatory arteriolar dilation response. Because Wilson et al. (45) used a Doppler crystal embedded on the side of the catheter proximal to the tip, the sample volume was in the gap between the proximal vessel and catheter wall, which is a more stable flow region. These results lend some confidence to our hemodynamic analysis, particularly before angioplasty, where distal flow disturbances were found (7, 9).

Increased pressure drop and reduced hyperemic flow due to the presence of guide wire. Tables 1 and 2 give the mean pressure drop ({Delta}h) and distal mean pressure (rh) during hyperemia in native, intermediate and residual lesions with and without guide wire. The ratio of guide-wire diameter to throat diameter for different stenoses is given in Table 2. With 0.35- and 0.46-mm guide wire, there was ~34% {[(9.9 – 7.4)/7.4] x 100%} and 43% {[(10.6 – 7.4)/7.4] x 100%} overestimation in {Delta}h for the postintervention condition, respectively (Tables 1 and 2); hyperemic flow was reduced from 180 ml/min to 172.5 and 170 ml/min, respectively (maximum of ~6%). For the intermediate stenosis, 0.35- and 0.46-mm guide wire caused 32% and 41% overestimation of {Delta}h, respectively; hyperemic flow was reduced from 165 ml/min to 149.7 and 145.6 ml/min, respectively (maximum of ~12%). For the severely blocked condition, i.e., before intervention, 0.35- and 0.46-mm guide wire caused 26% and 35% overestimation of {Delta}h, respectively; hyperemic flow was reduced considerably from 115 ml/min to 86 and 75 ml/min, respectively (maximum of ~35%). Although diagnosis of severely blocked arteries is easier and additional pressure drop due to the guide wire does not affect the diagnosis, the additional pressure drop due to the guide wire could have an important role in clinical evaluation of moderate and intermediate stenoses.

FFRmyo-FFRmyog and CFR-CFRg correlations. Myocardial FFR (FFRmyo) is defined as the myocardial blood flow distal to an epicardial stenosis during maximal vasodilation (hyperemic flow) relative to that without stenosis and accounts for collateral flow (34, 35, 37, 38)

where v is central venous pressure. Similarly, absolute CFR is defined as the ratio of hyperemic to resting coronary (or myocardial) blood flow in stenosed artery and was first proposed by Gould (26). Because Wilson et al. (45) measured coronary flow using a 3-Fr Doppler catheter proximal to the lesion, linear extrapolation of the CFR-rh curve (Fig. 1) provides a zero-flow mean pressure at which coronary flow would become zero, although myocardial blood flow might not be zero because of the presence of collateral flow (26, 34). In humans, it is generally understood that pressure at which myocardial flow is zero (which is measured by inducing long diastoles) is only a few millimeters of Hg above central venous pressure (34). Pressure at which myocardial blood flow is zero would give a true measure of combined epicardial and distal microvascular resistance, as myocardial blood flow accounts for coronary and collateral flow (26, 34). FFRmyog = 0.75 is assumed to accurately distinguish stenosis whether or not it is associated with inducible ischemia (34, 35, 37). The threshold limit of FFRmyog = 0.75 is a measured value with guide wire. However, this measured value of 0.75 must be attributed to FFRmyog and not FFRmyo. In other words, the presence of guide wire will produce an error in the measured values of FFRmyog that is different from FFRmyo, the pathophysiological value without guide wire.

Table 3 provides the values of CFR, CFRg, FFRmyo, and FFRmyog for the different stenosis configurations. Figures 2 and 3 show the correlation plots between CFR and CFRg and between FFR and FFRg with their linear regression lines. In Fig. 2, CFR was related to CFRg by the following equations: CFR = CFRg x 0.689 + 1.271 (R2 = 0.99) for 0.46-mm guide wire and CFR = CFRg x 0.757 + 1.004 (R2 = 0.99) for 35-mm guide wire. With v ~ 0, as used in present clinical practice, FFRmyo and FFRmyog correlated equally well (Fig. 3): FFRmyo = FFRmyog x 0.737 + 0.263 (R2 = 0.99) for 0.46-mm guide wire and FFRmyo = FFRmyog x 0.790 + 0.210 (R2 = 0.99) for 0.35-mm guide wire, which gave FFRmyo = 0.8 for a measured FFRmyog = 0.75. The study showed that the correlations for FFR and CFR for 0.35-mm guide wire are closer to the ideal (expected) relation, i.e., CFR = CFRg and FFR = FFRg, because of less flow obstruction from 0.35-mm than from 0.46-mm guide wire.



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  		Fig. 2. Relation between CFR and CFR with guide wire (CFRg).

 


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  		Fig. 3. Relation between myocardial fractional flow reserve (FFRmyo) and FFR with guide wire (FFRmyog), with mean venous pressure (v) ~ 0.

 
Stenosis resistance (s) and distal microvascular resistance (d) are shown in Table 4. s was calculated as {Delta}h/h, where h is mean flow during hyperemia. d was calculated as (rh – zf)/h, as zf ~ 20 mmHg provides an effective backpressure to coronary flow, such that as rh -> zf, the coronary flow would become zero. Table 4 shows that the effect of the guide wires (0.35 and 0.46 mm) on d was negligible (slope ~= 0) and nearly constant as the percent area of stenosis decreased from pre- to post-PTCA. This is consistent with the definition of FFRmyo. In contrast, stenosis resistance increases with increasing percent area stenosis.


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  		Table 4. Stenosis resistance and microvascular resistance distal to the stenosis in hyperemia pre-, mid-, and post-PTCA with and without guide wire

 

   DISCUSSION
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
APPENDIX
 GRANTS
 REFERENCES
 
This study provides the corrections that need to be applied to measured values of CFRg and FFRmyog to obtain the true physiological values of CFR and FFRmyo. Although uncertainties are associated with measurements of CFR with guide wires in diagnostic procedures, as these measurements are made distal to the stenosis, these corrections could be useful in measuring CFR and FFR simultaneously. This could provide useful information about the status of the epicardial stenosis and distal microcirculation during the procedure. Thus CFR measurement proximal to the stenosis will be more accurate, as this is the region of more stable flow. Furthermore, the heterogeneity of stenoses could produce variations in the slope of the correlations. However, the correlations were obtained on the basis of the shape of the stenosis and the size measured invasively by Wilson et al. (45) in a group of patients and have shown a direct and stronger relation to the minimal stenosis area than to the overall length and shape of the stenosis. Further clinical evaluation with a larger patient group will be needed to confirm the dominant effect of minimal stenosis area on these correlations.

In in vitro experiments, using steady flow, De Bruyne et al. (17) and Porenta et al. (39) studied the flow obstruction effect of guide wires on flow and pressure measurements. Banerjee et al. (7–9) showed that, in the event of pulsatile flow, as in in vivo measurements, the pressure drop was as much as 5–10% lower before PTCA and 15–20% higher after PTCA for the same mean flow rate compared with steady-flow experiments with comparable flow rates. This implies that the steady-flow in vitro experiments provided inaccurate data for the flow obstruction effect of the guide wires compared with the present pulsatile hemodynamic analyses using physiological flow pulse, as in normal and diseased human coronary arteries. Thus the FFRmyo-FFRmyog and CFR-CFRg correlations presented in this study improve the guide-wire measurements. Although several studies have focused on the relation between CFR measured with guide wire and CFR measured with a noninvasive flow probe, e.g., a Doppler cuff, no similar study has been done to distinguish the effect of guide wire on FFRmyo, inasmuch as pressure can be measured invasively only with guide wires. Thus this study can form the basis of any future study on diagnostics of focal, as well as diffuse, lesions, using guide wires, for CFR-CFRg and FFRmyo-FFRmyog correlations.

Di Mario et al. (20) and De Bruyne et al. (16) measured zf of ~15 mmHg using diastolic instantaneous coronary pressure-flow relation in humans, which accounts for only coronary flow. Therefore, actual zero-flow pressure is overestimated. During coronary artery occlusion, Pijls and De Bruyne (34) calculated maximum recruitable collateral flow to be as much as 18–36% of normal maximum myocardial blood flow in humans. Nanto et al. (31, 32) measured myocardial zero-flow pressure of 14 ± 7 and 21 ± 7 mmHg using long diastoles in humans, although they did not achieve near-zero flow during the measurements. These measurements show considerable variability and indicate that the myocardial zero-flow pressure could be close to the central venous pressure. Bache and Schwartz (2) measured a zero-flow pressure of 18 ± 2.3 mmHg in the subendocardium and 10 ± 2.1 mmHg in the subepicardium in dog hearts (measured distal to a stenosis induced in the left circumflex coronary artery); thus the extrapolated zf of ~20 mmHg in this study is closer to measured zero-flow pressure in the subendocardium. The higher value of zf of ~20 mmHg in this study compared with zf of ~15 mmHg measured by Di Mario et al. and De Bruyne et al. (which are still higher than central venous pressure) can be attributed to use of mean pressure and flow values to construct the CFR-rh curve compared with use of the diastolic instantaneous hyperemic pressure-flow relation (33).

Pantely et al. (33) measured mean zero-flow pressure of 12.1 ± 3.1 mmHg, which was higher than late diastolic zero-flow pressure (7.0 ± 2.2 mmHg), in swine (which have negligible collateral flow). The mean zero-flow pressure was higher than the diastolic zero-flow pressure, because it included the systolic period (when the pressure is high and the flow is low relative to diastole). The late diastolic zero-flow pressure measurements were close to the central venous pressure, inasmuch as measurements were possible at near zero flow, which showed the curvilinear effect at low flow (33). Thus the zf of ~20 mmHg estimated for the 32-patient group (45) in this study should be reasonable, inasmuch as it falls within the range of measured and published values in humans, although data actually obtained from humans is limited. Another issue that concerns zf estimation is its transmural variation in the heart. To estimate zf in different layers of the heart, the percent distribution of total myocardial blood flow to different layers should be measured. Thus our estimate does not reflect the minimal zf for the myocardial segment with the lowest zf. Although the 32-patient group of Wilson et al. (45) had normal microvascular function, it remains to be seen whether the presence of microvascular impairment could produce different correlations between FFR-FFRg and CFR-CFRg with epicardial stenoses and could be the focus of future studies on guide-wire effect and microcirculation.

In conclusion, this study establishes a strong correlation between CFR and CFRg and between FFR and FFRg, which could be used to obtain the true state of occlusion in the coronary artery during angioplasty.


   APPENDIX
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
GRANTS
 REFERENCES
 
For the computations, a basal flow () of 50 ml/min was chosen. Wilson et al. (45) reported CFR of 2.3 and 3.6 in pre- and post-PTCA lesions, respectively, in their 32-patient group. Thus hyperemic pathophysiological = 115 (2.3 x 50) and 180 (3.6 x 50) ml/min was estimated for pre- and post-PTCA stenoses, respectively. With use of hyperemic , mean pressure drop ({Delta}) of 34 and 7.4 mmHg and distal mean coronary pressure (rh) of 55 and 75.2 mmHg were computed before and after PTCA lesion, respectively. These data for before and after PTCA were used to construct the linear hyperemic CFR-rh (CFR = a x rh + b, where a and b are regressed constants; rh = a – {Delta}) curve, initially. Similarly, {Delta} for different starting from 50 ml/min was computed to construct the {Delta}- relation for pre-, mid-, and post-PTCA stenoses with and without guide wire. These relations were regressed with a quadratic function of the form {Delta} = A2 + B, where A and B are constants. Once the quadratic curves were obtained, the points of intersection of linear CFR-rh and {Delta} = A2 + B were obtained (Fig. 1). These intersection points were the hyperemic {Delta} and for the different lesions with and without guide wire. From the hyperemic {Delta} and , FFRmyo (without guide wire) and FFRmyog (with guide wire), along with CFR and CFRg, were calculated. Then correlations between FFR and FFRg and between CFR and CFRg were obtained by regression.

Furthermore, the linear CFR-rh was extrapolated to estimate zf, as in earlier studies, e.g., De Bruyne et al. (16) and Di Mario et al. (20). Although linear extrapolation does not give a true measure of myocardial zero-flow pressure because of curvilinearity at low pressure and flow, it provides an estimate of effective backpressure to coronary flow.


   GRANTS
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 GRANTS
REFERENCES
 
This work is supported by American Heart Association National Scientific Development Grant 0335270N.


   FOOTNOTES
 

Address for reprint requests and other correspondence: R. K. Banerjee, Dept. of Mechanical, Industrial and Nuclear Engineering, 688 Rhodes Hall, PO Box 210072, Cincinnati, OH 45221-0072 (E-mail: rupak.banerjee{at}uc.edu)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.


   REFERENCES
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 GRANTS
 REFERENCES
 

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