- Email: [email protected]
In Vitro Doppler Assessment of Pressure Gradients Across Modified Blalock-Taussig Shunts
The potentially important influence of PA stenoses on the exercise capacity of patients with TOF has also been suggested by a number of past investigations. Wessel et al1 found that patients with differential perfusion of the right and left lung tended to have lower exercise capacity and larger cardiothoracic ratios than patients with normal pulmonary perfusion. Marx et al17 found that pulmonary regurgitation was an important determinant of exercise capacity in patients with TOF, and that the patients with the most severe regurgitation had PA stenoses distal to their right ventricular outflow patches or conduits. Similarly, Rowe et al5 noted that ventilatory abnormalities and pulmonary regurgitation were associated with diminished exercise capacity. We suspect that PA stenoses contributed to both the ventilatory abnormalities and the pulmonary regurgitation observed by these investigators. However, their study did not explore this possibility. ˙ CO2 slope may be It must be noted that the VE/V affected by other factors besides PA stenoses. For instance, recent investigators have noted that the VE/ ˙ CO2 slope is elevated in some patients with chronic V congestive heart failure.18 Although this phenomenon has been found to have important clinical and prognostic implications, in that group, it is clearly unrelated to the presence of PA stenoses. It must also be recognized that PA stenosis is only one of several factors that may influence the exercise function of patients with TOF. However, we believe that the potential importance of this condition has been underappreciated. This study suggests that residual PA stenoses can have a dramatic effect on the cardiorespiratory response to exercise in the patient with TOF, and may have a substantial impact on his/her exercise capacity. An aggressive approach to this problem may therefore be warranted.
1. Wessel HU, Cunningham WJ, Paul MH, Bastanier CK, Muster AJ, Idriss FS.
Exercise performance in tetralogy of Fallot after intracardiac repair. J Thorac Cardiovasc Surg 1980;80:582–593. 2. Jonsson H, Ivert T, Jonasson R, Holmgren A, Bjork VO. Work capacity and central hemodynamics twenty-six years after repair of tetralogy of Fallot. J Thorac Cardiovasc Surg 1995;110:416 – 426. 3. Meijboom F, Szatmari A, Deckers JW, Utens EMWJ, Roelandt JRTC, Boss E, Hess J. Cardiac status and health-related quality of life in the long term after surgical repair of tetralogy of Fallot in infancy and childhood. J Thorac Cardiovasc Surg 1995;110:833– 891. 4. Clark AL, Gatzoulis MA, Reddington AN. Ventilatory responses to exercise in adults after repair of tetralogy of Fallot. Br Heart J 1995;73:445– 449. 5. Rowe SA, Zahka KG, Manolio TA, Horneffer PJ, Kidd L. Lung function and pulmonary regurgitation limit exercise capacity in postoperative tetralogy of Fallot. J Am Coll Cardiol 1991;17:461– 466. 6. James FW, Kaplan S, Schwartz DC, Chou TC, Sandker MJ, Naylor V. Response to exercise in patients after total surgical correction of tetralogy of Fallot. Circulation 1976;54:671– 679. 7. Mocellin R, Bastanier C, Hofacker W, Buhlmeyer K. Exercise performance in children annd adolescents after surgical repair of tetralogy of Fallot. Eur J Cardiol 1976;4:367–374. 8. Grant GP, Garofano RP, Mansell AL, Leopold HB, Gersony WM. Ventilatory response to exercise after intracardiac repair of tetralogy of Fallot. Am Rev Respir Dis 1991;144:833– 836. 9. Strieder DJ, Aziz K, Zaver AG, Fellows KE. Exercise tolerance after repair of tetralogy of Fallot. Ann Thorac Surg 1975;19:397– 405. 10. Wasserman K, Hansen JE, Sue DY, Whipp BJ. Principles of Exercise Testing and Interpretation. Philadelphia: Lea & Febiger, 1987:61– 64. 11. Rhodes J, Geggel RL, Marx GR, Bevilacqua L, Dambach YB, Hijazi ZM. Excessive anaerobic metabolism during exercise following repair of coarctation. Evidence for functionally significant residual arch obstruction. J Pediatr 1997; 131:210 –214. 12. Clark AL, Poole-Wilson PA, Coats AJS. Effects of motivation on indices of exercise capacity in chronic heart failure. Br Heart J 1994;71:162–165. 13. Hansen JE, Sue DY, Wasserman K. Predicted values for clinical exercise testing. Am Rev Respir Dis 1984;129(suppl):S48 –S55. 14. Linehan JH, Dawson CA. Pulmonary vascular resistance. In: Fishman AP, ed. The Pulmonary Circulation. Philadelphia: University of Pennsylvania Press, 1990:41– 45. 15. Chaturvedi RR, Kilner PJ, White PA, Bishop A, Szwarc R, Reddington AN. Increased airway pressure and simulated pulmonary artery stenosis increase pulmonary regurgitation after repair of tetralogy of Fallot. Circulation 1997;95:643– 649. 16. Ilbawi MN, Idriss FS, DeLeon SY, Muster AJ, Gidding SS, Berry TE, Paul MH. Factors that exaggerate the deleterious effects of pulmonary insufficiency on the right ventricle after tetralogy repair. J Thorac Cardiovasc Surg 1987;93:36 – 44. 17. Marx GR, Hicks RW, Allen HD, Goldberg SJ. Noninvasive assessment of hemodynamic responses to exercise in pulmonary regurgitation after operations to correct pulmonary outflow obstruction. Am J Cardiol 1988;61:595– 601. 18. Chua TP, Ponikowski P, Harrington D, Anker SD, Webb-Peploe K, Clark AL, Poole-Wilson PA, Coats AJS. Clinical correlates and prognostic significance of the ventilatory response to exercise in chronic heart failure. J Am Coll Cardiol 1997;29:1585–1590.
In Vitro Doppler Assessment of Pressure Gradients Across Modified Blalock-Taussig Shunts Theresa A. Tacy,
MD,
Kevin K. Whitehead,
he purpose of this investigation was to determine the relation between Doppler-predicted pressure T gradient and directly measured pressure gradient in actual Gore-Tex shunts used in surgical procedures. We hypothesized that application of the simplified Bernoulli equation would result in significant error in pressure gradient prediction due to viscous losses From the Cardiac Dynamics Laboratory, Children’s Hospital of Pittsburgh, University of Pittsburgh, Pittsburgh, Pennsylvania. Dr. Tacy’s address is: Division of Cardiology, Children’s Hospital of Pittsburgh, 3705 Fifth Avenue, Pittsburgh, Pennsylvania 15213. Manuscript received August 21, 1997; revised manuscript received and accepted January 9, 1998. ©1998 by Excerpta Medica, Inc. All rights reserved.
BS,
and Edward G. Cape,
PhD
within the shunt. It was also our goal to explain the pattern of error using the principles of fluid mechanics which may ultimately lead to an accurate technique for use of Doppler velocities in shunt flow to predict the aortopulmonary pressure gradient in patients. •••
We chose an in vitro approach to address our hypothesis so that shunt length and diameter, as well as flow conditions, could be systematically varied. The pulsatile flow model consisted of a compressible bulb that was in series with a model of systemic vasculature (Figure 1). A model of the pulmonary vasculature was placed in parallel with the systemic 0002-9149/98/$19.00 PII S0002-9149(98)00096-4
1219
FIGURE 1. Pulsatile flow model.
electromagnetic flowmeter were interfaced to a Macintosh Quadra 800 through a 16bit analog-to-digital converter for simultaneous acquisition and display of data. This system allowed for continuous display of the aortic and pulmonary artery pressures and flow rates. The pressure and flow waveforms were each sampled at 1,000 Hz, digitized, and stored using LAB Viewt (National Instruments, Corp., Austin, Texas) programmable software. The maximum aortopulmonary pressure gradient was recorded for 3 consecutive pulses and averaged. Continuous-wave Doppler shunt velocity was obtained simultaneously using a Vingmed CFM750 equipped with a 2.0-MHz nonimaging transducer placed on the aortic side (Figure 1). The maximal velocity and pressure gradient (predicted by the simplified Bernoulli FIGURE 2. Relation between pressure gradient predicted by the simplified Berequation) were recorded for 3 beats and noulli equation and the actual gradient for all shunt diameters and lengths. averaged. The line of identity (y 5 x) is shown for comparison. The Blalock-Taussig shunt length varied between 3, 6, and 9 cm. Diameter varied between 4, 5, and 6 mm. A 33% vasculature. The systemic and pulmonary beds were aqueous glycerin solution (3 cP) was used as connected with Gore-Tex shunt material by suturing the working fluid. Cardiac output was held at 3 the shunt material to pledgets attached to the external L/min for all conditions studied, and pulse rate walls. This created 2 systems in parallel, connected was constant at 60 beats/min. The compliance and resistance of both the pulmonary and systemic only by the aortopulmonary shunts. Aortic and pulmonary pressure transducers and an sections were tuned to produce aortopulmo1220 THE AMERICAN JOURNAL OF CARDIOLOGYT
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MAY 15, 1998
Re 5
FIGURE 3. Error in Doppler pressure gradient prediction (6 1 SD) for each shunt diameter.
FIGURE 4. Error in Doppler pressure gradient prediction (6 1 SD) for each shunt length.
nary pressure gradients of 20, 40, 60, 80, and 100 mm Hg. Linear regression analysis was performed to compare pressure gradients predicted by the simplified Bernoulli equation to the actual pressure gradient. The percent error in pressure gradient prediction was calculated as (DRDoppler 2 DrCatheter) DRCatheter for each shunt. The effect of flow conditions on the degree of error in pressure gradient prediction was assessed by comparing the percent error to the Reynolds number. The Reynolds number is estimated by the equation
rvD , m
where r is the fluid density (1.05 g/ml), v is the peak Doppler velocity within the shunt (cm/s), D is shunt diameter (cm), and m is fluid viscosity (Poise). The fluid viscosity was measured with a Cannon-Fenske viscometer. There was excellent correlation between Bernoullipredicted pressure gradient and actual pressure gradient for all aortopulmonary shunt sizes (r 5 0.97) (Figure 2). However, there was a tendency for underestimation of pressure gradient by the simplified Bernoulli equation, as reflected by the slope of the regression line (y 5 0.8x). The intercept was statistically equal to zero and the slope was significantly different from unity (p 5 0.01). An overall mean error of 21% was observed for these physiologic aortopulmonary pressure gradients. The error of the Doppler-predicted pressure gradient was more closely correlated with shunt diameter (r 5 0.74) than with shunt length (r 5 0.08). Larger errors for smaller diameter shunts were observed, consistent with the deletion of the viscous term in the Bernoulli equation (Figure 3). For the 4-mm diameter shunts, the mean error in pressure gradient prediction was 235 6 18% (the negative sign denotes underestimation), for the 5-mm shunts the error was 225 6 10%, and for the 6-mm shunts the error was 16 6 9%. For shunt lengths of 3 cm, the mean error in pressure gradient prediction was 215 6 16, for the 6-cm shunts the error was 243 6 10%, and for the 9-cm shunts the error was 212 6 20% (Figure 4). The error magnitude was unrelated to the dimensionless term L/D (r 5 0.17). In this investigation the shunt diameter was the most significant geometric determinant of the error in pressure gradient prediction by Doppler velocimetry using the simplified Bernoulli equation. Another means of evaluating determinants of the errors in the pressure gradient prediction is to look at flow conditions within the shunt affected by the shunt geometry, as described by the Reynolds number. The Reynolds number is a dimensionless quantity that represents the ratio of inertial forces to viscous forces for a given flow condition.1 A high Reynolds number describes a flow condition in which inertial forces predominate, whereas a low value describes a flow condition predominated by viscous forces. Therefore, one would expect the pressure gradient in low Reynolds-number flows to be underestimated owing to extensive viscous losses that are not included in the simplified Bernoulli equation. As the Reynolds number increases (in this study, by increasing diameter), underestimation should be reduced, with possible overestimation due to pressure recovery effects. This correlation between discrepancy and Reynolds number was demonstrated in this investigation (p , 0.001) (Figure 5). The purpose of this study was to investigate the relation between Doppler-predicted gradient and directly measured instantaneous pressure gradients across modified Blalock-Taussig shunts. Application BRIEF REPORTS
1221
the simplified Bernoulli equation in viscous flows neglects a potentially important contribution to the pressure decrease. This study, using actual Gore-Tex shunts, showed this contribution to be important. •••
Clinical studies investigating Doppler assessment of aortopulmonary shunts have either focused on determining patency of shunts,2– 4 or were performed in patients with aortopulmonary shunts other than modified Blalock-Taussig shunts.5 In vitro investigations of the accuracy of the simplified Bernoulli equation applied to Doppler velocities in narrow tunnels have noted that tunnel areas .0.25 cm2 (equivalent to a diameter of 5.6 mm) provide good Doppler-to-manometer correspondence.6,7 Teirstein et al6 demonstrated that tunnel areas corresponding to shunt diameters of 5.6 mm produced an average of 15% underestimation of the pressure gradient by the simplified Bernoulli equation. This is highly consistent and fits within FIGURE 5. Relation between percent error in Doppler pressure gradient prediction and flow conditions (represented by the Reynolds number). A significant our results, which showed 225% error for correlation between error and Reynolds number is demonstrated. a 5-mm shunt and a 16% error for a 6-mm shunt. Because modified Blalock-Taussig shunts used in the neonatal patient are of the simplified Bernoulli equation using flow veloc- often #4 mm, one should expect significant underesity in small tubes such as those often used in aorto- timation of pressure gradient due to viscous losses pulmonary shunt procedures should result in a signif- when the simplified Bernoulli equation is applied to icant error in pressure gradient prediction due to sig- Doppler velocities in these patients. In the 6-mm diameter shunts, there was some overnificant viscous losses within the shunt. The Bernoulli estimation of the actual pressure gradient by the simequation, plified Bernoulli equation. This may be explained by 2 the concept of pressure recovery, which has been well described in Doppler investigations of valvar and sub1 p1 2 p2 5 r(v22 2 v12) 1 r valvar stenosis.8 –14 This effect occurs as flow expands 2 distal to the obstruction and kinetic energy is recon-
E 1
¡ 3 d v 3 3 d s 1 R ~v ! dt
(1)
relates pressure decrease across a narrowed region of flow to the fluid velocity increase (from v1 to v2) across the narrowing. In the above equation, 3
dv 3 * ds dt 2 1
represents acceleration effects, and R~v3 ! represents viscous losses. If these 2 components are assumed to be negligible, the resulting equation is the simplified Bernoulli equation: DR 5 4v2
2
S D
rv22 5 (assuming v1,,v2). 2
(2)
with pressure in mm Hg and velocity in m/s. Comparing equations (1) and (2) reveals that application of 1222 THE AMERICAN JOURNAL OF CARDIOLOGYT
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verted into potential energy. It is reflected by a significant discrepancy between the Doppler-predicted pressure at the point of maximal velocity and the pressure measured by the catheter downstream of this site. Thus, the accuracy of the Doppler pressure gradient prediction across aortopulmonary shunts may be affected by interacting effects of viscous losses versus effects of pressure recovery. The degree of under- or overestimation can potentially be related to the fluid conditions within the shunt that are embodied in the Reynolds number. Our results demonstrate that underestimation was more pronounced at low Reynolds numbers, and underestimation was virtually eliminated at higher Reynolds numbers. This study demonstrates that estimation of pressure gradients across modified Blalock-Taussig shunts using the simplified Bernoulli equation will underestimate the catheter gradient for shunts #5 mm diameter. This is important because it is common clinical practice to estimate the aortopulmonary pressure gradient from the continuous-wave Doppler and the simplified Bernoulli equation. The relation between DopMAY 15, 1998
pler-predicted and catheter pressure gradients can be related to the flow conditions within the shunt by use of the Reynolds number. Although further studies will be required in vivo, this suggests that noninvasive prediction of pulmonary artery pressures may be possible for patients with a modified aortopulmonary shunt. Shunt diameter is known at implantation, velocity is easily measured by Doppler echocardiography, and blood viscosity and density are known. Therefore, viscous correction factors based on a Reynolds number may potentially be derived for use in conjunction with the simplified Bernoulli equation.15 The simplified Bernoulli equation should only be applied to modified Blalock-Taussig shunt flow with an understanding of its associated limitations, namely that underestimation should be expected for Blalock-Taussig shunts with diameters <5 mm. Doppler estimation of pressure gradients approaches agreement only in extreme cases of large shunts. The pattern of underestimation is controlled by conditions within the shunt which are characterized by the Reynolds number. 1. Fox RW, McDonald AT. Introduction to Fluid Mechanics. 2nd ed. New York: John Wiley, 1978. 2. Allen HD, Sahn DJ, Lange L, Goldberg SJ. Noninvasive assessment of surgical systemic to pulmonary artery shunts by range-gated pulsed Doppler echocardiography. J Pediatr 1979;94:395– 402.
3. Stevenson JG, Kawabori I, Bailey WW. Noninvasive evaluation of Blalock-
Taussig shunts: determination of patency and differentiation from patent ductus arteriosus by Doppler echocardiography. Am Heart J 1983;106:1121–1132. 4. Serwer GA, Armstrong BE, Anderson PA. Noninvasive detection of retrograde descending aortic flow in infants using continuous wave Doppler ultrasonography. J Pediatr 1980;97:394 – 400. 5. Marx GR, Allen HA, Goldberg SJ. Doppler echocardiographic estimation of systolic pulmonary artery pressure in patients with aortic-pulmonary shunts. J Am Coll Cardiol 1986;7:880 – 885. 6. Teirstein PS, Yock PG, Popp RL. The accuracy of Doppler ultrasound measurement of pressure gradients across irregular, dual, and tunnel-like obstructions to blood flow. Circulation 1985;72:577–584. 7. Popp RL, Teplitsky I. Lessons from in vitro models of small, irregular, multiple and tunnel-like stenoses relevant to clinical stenoses of valves and small vessels. J Am Coll Cardiol 1989;13:716 –722. 8. Levine RA, Jimoh A, Cape EG, McMillan S, Yoganathan AP, Weyman AE. Pressure recovery distal to a stenosis: potential cause of gradient ‘‘overestimation’’ by Doppler echocardiography. J Am Coll Cardiol 1989;13:706 –715. 9. Stewart WJ, Schiavone WA, Salcedo EE, Lever HM, Cosgrove DM, Gill CC. Intraoperative Doppler echocardiography in hypertrophic cardiomyopathy: correlations with the obstructive gradient. J Am Coll Cardiol 1987;10:327–335. 10. Valdes-Cruz LM, Yoganathan AP, Tamura T, Tomizuka F, Woo Y, Sahn DJ. Studies in vitro of the relationship between ultrasound and laser Doppler velocimetry and applicability of the simplified Bernoulli relationship. Circulation 1986;73:300 –308. 11. Baumgartner H, Schima H, Tulser G, Kuhn P. Effect of stenosis geometry on the Doppler-catheter gradient relation in vitro: a manifestation of pressure recovery. J Am Coll Cardiol 1993;21:1018 –1025. 12. Voelker W, Reul H, Stelzer T, Schmidt A, Karsch KA. Pressure recovery in aortic stenosis: and in vitro study in a pulsatile flow model. J Am Coll Cardiol 1992;20:1585–1593. 13. Ohlsson J, Wranne B. Non-invasive assessment of valve area in patients with aortic stenosis. J Am Coll Cardiol 1986;7:501–508. 14. Yoganathan AP, Cape EG, Sung H, Williams FP, Jimoh A. Review of hydrodynamic principles for the cardiologist: applications to the study of blood flow and jets by imaging techniques. J Am Coll Cardiol 1988;12:1344 –1353. 15. Cape EG, Valdes-Cruz LM, Yamada I, VanAuker MD, Jones M. In vivo studies of aortic stenosis: role of inertial and viscous forces in Doppler/catheter discrepancies (abstr). J Am Coll Cardiol 1995;(suppl): 245A.
BRIEF REPORTS
1223
1. Wessel HU, Cunningham WJ, Paul MH, Bastanier CK, Muster AJ, Idriss FS.
Exercise performance in tetralogy of Fallot after intracardiac repair. J Thorac Cardiovasc Surg 1980;80:582–593. 2. Jonsson H, Ivert T, Jonasson R, Holmgren A, Bjork VO. Work capacity and central hemodynamics twenty-six years after repair of tetralogy of Fallot. J Thorac Cardiovasc Surg 1995;110:416 – 426. 3. Meijboom F, Szatmari A, Deckers JW, Utens EMWJ, Roelandt JRTC, Boss E, Hess J. Cardiac status and health-related quality of life in the long term after surgical repair of tetralogy of Fallot in infancy and childhood. J Thorac Cardiovasc Surg 1995;110:833– 891. 4. Clark AL, Gatzoulis MA, Reddington AN. Ventilatory responses to exercise in adults after repair of tetralogy of Fallot. Br Heart J 1995;73:445– 449. 5. Rowe SA, Zahka KG, Manolio TA, Horneffer PJ, Kidd L. Lung function and pulmonary regurgitation limit exercise capacity in postoperative tetralogy of Fallot. J Am Coll Cardiol 1991;17:461– 466. 6. James FW, Kaplan S, Schwartz DC, Chou TC, Sandker MJ, Naylor V. Response to exercise in patients after total surgical correction of tetralogy of Fallot. Circulation 1976;54:671– 679. 7. Mocellin R, Bastanier C, Hofacker W, Buhlmeyer K. Exercise performance in children annd adolescents after surgical repair of tetralogy of Fallot. Eur J Cardiol 1976;4:367–374. 8. Grant GP, Garofano RP, Mansell AL, Leopold HB, Gersony WM. Ventilatory response to exercise after intracardiac repair of tetralogy of Fallot. Am Rev Respir Dis 1991;144:833– 836. 9. Strieder DJ, Aziz K, Zaver AG, Fellows KE. Exercise tolerance after repair of tetralogy of Fallot. Ann Thorac Surg 1975;19:397– 405. 10. Wasserman K, Hansen JE, Sue DY, Whipp BJ. Principles of Exercise Testing and Interpretation. Philadelphia: Lea & Febiger, 1987:61– 64. 11. Rhodes J, Geggel RL, Marx GR, Bevilacqua L, Dambach YB, Hijazi ZM. Excessive anaerobic metabolism during exercise following repair of coarctation. Evidence for functionally significant residual arch obstruction. J Pediatr 1997; 131:210 –214. 12. Clark AL, Poole-Wilson PA, Coats AJS. Effects of motivation on indices of exercise capacity in chronic heart failure. Br Heart J 1994;71:162–165. 13. Hansen JE, Sue DY, Wasserman K. Predicted values for clinical exercise testing. Am Rev Respir Dis 1984;129(suppl):S48 –S55. 14. Linehan JH, Dawson CA. Pulmonary vascular resistance. In: Fishman AP, ed. The Pulmonary Circulation. Philadelphia: University of Pennsylvania Press, 1990:41– 45. 15. Chaturvedi RR, Kilner PJ, White PA, Bishop A, Szwarc R, Reddington AN. Increased airway pressure and simulated pulmonary artery stenosis increase pulmonary regurgitation after repair of tetralogy of Fallot. Circulation 1997;95:643– 649. 16. Ilbawi MN, Idriss FS, DeLeon SY, Muster AJ, Gidding SS, Berry TE, Paul MH. Factors that exaggerate the deleterious effects of pulmonary insufficiency on the right ventricle after tetralogy repair. J Thorac Cardiovasc Surg 1987;93:36 – 44. 17. Marx GR, Hicks RW, Allen HD, Goldberg SJ. Noninvasive assessment of hemodynamic responses to exercise in pulmonary regurgitation after operations to correct pulmonary outflow obstruction. Am J Cardiol 1988;61:595– 601. 18. Chua TP, Ponikowski P, Harrington D, Anker SD, Webb-Peploe K, Clark AL, Poole-Wilson PA, Coats AJS. Clinical correlates and prognostic significance of the ventilatory response to exercise in chronic heart failure. J Am Coll Cardiol 1997;29:1585–1590.
In Vitro Doppler Assessment of Pressure Gradients Across Modified Blalock-Taussig Shunts Theresa A. Tacy,
MD,
Kevin K. Whitehead,
he purpose of this investigation was to determine the relation between Doppler-predicted pressure T gradient and directly measured pressure gradient in actual Gore-Tex shunts used in surgical procedures. We hypothesized that application of the simplified Bernoulli equation would result in significant error in pressure gradient prediction due to viscous losses From the Cardiac Dynamics Laboratory, Children’s Hospital of Pittsburgh, University of Pittsburgh, Pittsburgh, Pennsylvania. Dr. Tacy’s address is: Division of Cardiology, Children’s Hospital of Pittsburgh, 3705 Fifth Avenue, Pittsburgh, Pennsylvania 15213. Manuscript received August 21, 1997; revised manuscript received and accepted January 9, 1998. ©1998 by Excerpta Medica, Inc. All rights reserved.
BS,
and Edward G. Cape,
PhD
within the shunt. It was also our goal to explain the pattern of error using the principles of fluid mechanics which may ultimately lead to an accurate technique for use of Doppler velocities in shunt flow to predict the aortopulmonary pressure gradient in patients. •••
We chose an in vitro approach to address our hypothesis so that shunt length and diameter, as well as flow conditions, could be systematically varied. The pulsatile flow model consisted of a compressible bulb that was in series with a model of systemic vasculature (Figure 1). A model of the pulmonary vasculature was placed in parallel with the systemic 0002-9149/98/$19.00 PII S0002-9149(98)00096-4
1219
FIGURE 1. Pulsatile flow model.
electromagnetic flowmeter were interfaced to a Macintosh Quadra 800 through a 16bit analog-to-digital converter for simultaneous acquisition and display of data. This system allowed for continuous display of the aortic and pulmonary artery pressures and flow rates. The pressure and flow waveforms were each sampled at 1,000 Hz, digitized, and stored using LAB Viewt (National Instruments, Corp., Austin, Texas) programmable software. The maximum aortopulmonary pressure gradient was recorded for 3 consecutive pulses and averaged. Continuous-wave Doppler shunt velocity was obtained simultaneously using a Vingmed CFM750 equipped with a 2.0-MHz nonimaging transducer placed on the aortic side (Figure 1). The maximal velocity and pressure gradient (predicted by the simplified Bernoulli FIGURE 2. Relation between pressure gradient predicted by the simplified Berequation) were recorded for 3 beats and noulli equation and the actual gradient for all shunt diameters and lengths. averaged. The line of identity (y 5 x) is shown for comparison. The Blalock-Taussig shunt length varied between 3, 6, and 9 cm. Diameter varied between 4, 5, and 6 mm. A 33% vasculature. The systemic and pulmonary beds were aqueous glycerin solution (3 cP) was used as connected with Gore-Tex shunt material by suturing the working fluid. Cardiac output was held at 3 the shunt material to pledgets attached to the external L/min for all conditions studied, and pulse rate walls. This created 2 systems in parallel, connected was constant at 60 beats/min. The compliance and resistance of both the pulmonary and systemic only by the aortopulmonary shunts. Aortic and pulmonary pressure transducers and an sections were tuned to produce aortopulmo1220 THE AMERICAN JOURNAL OF CARDIOLOGYT
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MAY 15, 1998
Re 5
FIGURE 3. Error in Doppler pressure gradient prediction (6 1 SD) for each shunt diameter.
FIGURE 4. Error in Doppler pressure gradient prediction (6 1 SD) for each shunt length.
nary pressure gradients of 20, 40, 60, 80, and 100 mm Hg. Linear regression analysis was performed to compare pressure gradients predicted by the simplified Bernoulli equation to the actual pressure gradient. The percent error in pressure gradient prediction was calculated as (DRDoppler 2 DrCatheter) DRCatheter for each shunt. The effect of flow conditions on the degree of error in pressure gradient prediction was assessed by comparing the percent error to the Reynolds number. The Reynolds number is estimated by the equation
rvD , m
where r is the fluid density (1.05 g/ml), v is the peak Doppler velocity within the shunt (cm/s), D is shunt diameter (cm), and m is fluid viscosity (Poise). The fluid viscosity was measured with a Cannon-Fenske viscometer. There was excellent correlation between Bernoullipredicted pressure gradient and actual pressure gradient for all aortopulmonary shunt sizes (r 5 0.97) (Figure 2). However, there was a tendency for underestimation of pressure gradient by the simplified Bernoulli equation, as reflected by the slope of the regression line (y 5 0.8x). The intercept was statistically equal to zero and the slope was significantly different from unity (p 5 0.01). An overall mean error of 21% was observed for these physiologic aortopulmonary pressure gradients. The error of the Doppler-predicted pressure gradient was more closely correlated with shunt diameter (r 5 0.74) than with shunt length (r 5 0.08). Larger errors for smaller diameter shunts were observed, consistent with the deletion of the viscous term in the Bernoulli equation (Figure 3). For the 4-mm diameter shunts, the mean error in pressure gradient prediction was 235 6 18% (the negative sign denotes underestimation), for the 5-mm shunts the error was 225 6 10%, and for the 6-mm shunts the error was 16 6 9%. For shunt lengths of 3 cm, the mean error in pressure gradient prediction was 215 6 16, for the 6-cm shunts the error was 243 6 10%, and for the 9-cm shunts the error was 212 6 20% (Figure 4). The error magnitude was unrelated to the dimensionless term L/D (r 5 0.17). In this investigation the shunt diameter was the most significant geometric determinant of the error in pressure gradient prediction by Doppler velocimetry using the simplified Bernoulli equation. Another means of evaluating determinants of the errors in the pressure gradient prediction is to look at flow conditions within the shunt affected by the shunt geometry, as described by the Reynolds number. The Reynolds number is a dimensionless quantity that represents the ratio of inertial forces to viscous forces for a given flow condition.1 A high Reynolds number describes a flow condition in which inertial forces predominate, whereas a low value describes a flow condition predominated by viscous forces. Therefore, one would expect the pressure gradient in low Reynolds-number flows to be underestimated owing to extensive viscous losses that are not included in the simplified Bernoulli equation. As the Reynolds number increases (in this study, by increasing diameter), underestimation should be reduced, with possible overestimation due to pressure recovery effects. This correlation between discrepancy and Reynolds number was demonstrated in this investigation (p , 0.001) (Figure 5). The purpose of this study was to investigate the relation between Doppler-predicted gradient and directly measured instantaneous pressure gradients across modified Blalock-Taussig shunts. Application BRIEF REPORTS
1221
the simplified Bernoulli equation in viscous flows neglects a potentially important contribution to the pressure decrease. This study, using actual Gore-Tex shunts, showed this contribution to be important. •••
Clinical studies investigating Doppler assessment of aortopulmonary shunts have either focused on determining patency of shunts,2– 4 or were performed in patients with aortopulmonary shunts other than modified Blalock-Taussig shunts.5 In vitro investigations of the accuracy of the simplified Bernoulli equation applied to Doppler velocities in narrow tunnels have noted that tunnel areas .0.25 cm2 (equivalent to a diameter of 5.6 mm) provide good Doppler-to-manometer correspondence.6,7 Teirstein et al6 demonstrated that tunnel areas corresponding to shunt diameters of 5.6 mm produced an average of 15% underestimation of the pressure gradient by the simplified Bernoulli equation. This is highly consistent and fits within FIGURE 5. Relation between percent error in Doppler pressure gradient prediction and flow conditions (represented by the Reynolds number). A significant our results, which showed 225% error for correlation between error and Reynolds number is demonstrated. a 5-mm shunt and a 16% error for a 6-mm shunt. Because modified Blalock-Taussig shunts used in the neonatal patient are of the simplified Bernoulli equation using flow veloc- often #4 mm, one should expect significant underesity in small tubes such as those often used in aorto- timation of pressure gradient due to viscous losses pulmonary shunt procedures should result in a signif- when the simplified Bernoulli equation is applied to icant error in pressure gradient prediction due to sig- Doppler velocities in these patients. In the 6-mm diameter shunts, there was some overnificant viscous losses within the shunt. The Bernoulli estimation of the actual pressure gradient by the simequation, plified Bernoulli equation. This may be explained by 2 the concept of pressure recovery, which has been well described in Doppler investigations of valvar and sub1 p1 2 p2 5 r(v22 2 v12) 1 r valvar stenosis.8 –14 This effect occurs as flow expands 2 distal to the obstruction and kinetic energy is recon-
E 1
¡ 3 d v 3 3 d s 1 R ~v ! dt
(1)
relates pressure decrease across a narrowed region of flow to the fluid velocity increase (from v1 to v2) across the narrowing. In the above equation, 3
dv 3 * ds dt 2 1
represents acceleration effects, and R~v3 ! represents viscous losses. If these 2 components are assumed to be negligible, the resulting equation is the simplified Bernoulli equation: DR 5 4v2
2
S D
rv22 5 (assuming v1,,v2). 2
(2)
with pressure in mm Hg and velocity in m/s. Comparing equations (1) and (2) reveals that application of 1222 THE AMERICAN JOURNAL OF CARDIOLOGYT
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verted into potential energy. It is reflected by a significant discrepancy between the Doppler-predicted pressure at the point of maximal velocity and the pressure measured by the catheter downstream of this site. Thus, the accuracy of the Doppler pressure gradient prediction across aortopulmonary shunts may be affected by interacting effects of viscous losses versus effects of pressure recovery. The degree of under- or overestimation can potentially be related to the fluid conditions within the shunt that are embodied in the Reynolds number. Our results demonstrate that underestimation was more pronounced at low Reynolds numbers, and underestimation was virtually eliminated at higher Reynolds numbers. This study demonstrates that estimation of pressure gradients across modified Blalock-Taussig shunts using the simplified Bernoulli equation will underestimate the catheter gradient for shunts #5 mm diameter. This is important because it is common clinical practice to estimate the aortopulmonary pressure gradient from the continuous-wave Doppler and the simplified Bernoulli equation. The relation between DopMAY 15, 1998
pler-predicted and catheter pressure gradients can be related to the flow conditions within the shunt by use of the Reynolds number. Although further studies will be required in vivo, this suggests that noninvasive prediction of pulmonary artery pressures may be possible for patients with a modified aortopulmonary shunt. Shunt diameter is known at implantation, velocity is easily measured by Doppler echocardiography, and blood viscosity and density are known. Therefore, viscous correction factors based on a Reynolds number may potentially be derived for use in conjunction with the simplified Bernoulli equation.15 The simplified Bernoulli equation should only be applied to modified Blalock-Taussig shunt flow with an understanding of its associated limitations, namely that underestimation should be expected for Blalock-Taussig shunts with diameters <5 mm. Doppler estimation of pressure gradients approaches agreement only in extreme cases of large shunts. The pattern of underestimation is controlled by conditions within the shunt which are characterized by the Reynolds number. 1. Fox RW, McDonald AT. Introduction to Fluid Mechanics. 2nd ed. New York: John Wiley, 1978. 2. Allen HD, Sahn DJ, Lange L, Goldberg SJ. Noninvasive assessment of surgical systemic to pulmonary artery shunts by range-gated pulsed Doppler echocardiography. J Pediatr 1979;94:395– 402.
3. Stevenson JG, Kawabori I, Bailey WW. Noninvasive evaluation of Blalock-
Taussig shunts: determination of patency and differentiation from patent ductus arteriosus by Doppler echocardiography. Am Heart J 1983;106:1121–1132. 4. Serwer GA, Armstrong BE, Anderson PA. Noninvasive detection of retrograde descending aortic flow in infants using continuous wave Doppler ultrasonography. J Pediatr 1980;97:394 – 400. 5. Marx GR, Allen HA, Goldberg SJ. Doppler echocardiographic estimation of systolic pulmonary artery pressure in patients with aortic-pulmonary shunts. J Am Coll Cardiol 1986;7:880 – 885. 6. Teirstein PS, Yock PG, Popp RL. The accuracy of Doppler ultrasound measurement of pressure gradients across irregular, dual, and tunnel-like obstructions to blood flow. Circulation 1985;72:577–584. 7. Popp RL, Teplitsky I. Lessons from in vitro models of small, irregular, multiple and tunnel-like stenoses relevant to clinical stenoses of valves and small vessels. J Am Coll Cardiol 1989;13:716 –722. 8. Levine RA, Jimoh A, Cape EG, McMillan S, Yoganathan AP, Weyman AE. Pressure recovery distal to a stenosis: potential cause of gradient ‘‘overestimation’’ by Doppler echocardiography. J Am Coll Cardiol 1989;13:706 –715. 9. Stewart WJ, Schiavone WA, Salcedo EE, Lever HM, Cosgrove DM, Gill CC. Intraoperative Doppler echocardiography in hypertrophic cardiomyopathy: correlations with the obstructive gradient. J Am Coll Cardiol 1987;10:327–335. 10. Valdes-Cruz LM, Yoganathan AP, Tamura T, Tomizuka F, Woo Y, Sahn DJ. Studies in vitro of the relationship between ultrasound and laser Doppler velocimetry and applicability of the simplified Bernoulli relationship. Circulation 1986;73:300 –308. 11. Baumgartner H, Schima H, Tulser G, Kuhn P. Effect of stenosis geometry on the Doppler-catheter gradient relation in vitro: a manifestation of pressure recovery. J Am Coll Cardiol 1993;21:1018 –1025. 12. Voelker W, Reul H, Stelzer T, Schmidt A, Karsch KA. Pressure recovery in aortic stenosis: and in vitro study in a pulsatile flow model. J Am Coll Cardiol 1992;20:1585–1593. 13. Ohlsson J, Wranne B. Non-invasive assessment of valve area in patients with aortic stenosis. J Am Coll Cardiol 1986;7:501–508. 14. Yoganathan AP, Cape EG, Sung H, Williams FP, Jimoh A. Review of hydrodynamic principles for the cardiologist: applications to the study of blood flow and jets by imaging techniques. J Am Coll Cardiol 1988;12:1344 –1353. 15. Cape EG, Valdes-Cruz LM, Yamada I, VanAuker MD, Jones M. In vivo studies of aortic stenosis: role of inertial and viscous forces in Doppler/catheter discrepancies (abstr). J Am Coll Cardiol 1995;(suppl): 245A.
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