Quantitative characterization of postnatal growth trends in proximal pulmonary arteries in rats by phase-contrast magnetic resonance imaging

Malformations of the pulmonary arteries can increase right heart workload and result in morbidity, heart failure, and death. With the increased use of murine models to study these malformations, there is a pressing need for an accurate and noninvasive experimental technique that is capable of characterizing pulmonary arterial hemodynamics in these animals. We describe the growth trends of pulmonary arteries in 13 male Sprague-Dawley rats at 20, 36, 52, 100, and 160 days of age with the introduction of phase-contrast MRI as such a technique. PCMRI results correlated closely with cardiac output measurements by ultrasound echocardiography and with fluorescent microspheres in right-left lung flow split (flow partition). Mean flow, average cross-sectional area, distensibility, and shear rates for the right and left pulmonary arteries (RPA and LPA) were calculated. The RPA was larger and received more flow at all times than the LPA (P < 0.0001). Right-left flow split did not change significantly with age, and arterial distensibility was not significantly different between RPA and LPA, except at 160 days (P < 0.01). Shear rates were much higher for the LPA than the RPA (P < 0.0001) throughout development. The RPA and LPA showed different structure-function relationships but obeyed similar allometric scaling laws, with scaling exponents comparable to those of the main pulmonary artery. This study is the first to quantitatively describe changes in RPA and LPA flows and sizes with development and to apply phase-contrast MRI techniques to pulmonary arteries in rats.     

Multiscale modeling of the cardiovascular system: application to the study of pulmonary and coronary perfusions in the univentricular circulation

The objective of this study is to compare the coronary and pulmonary blood flow dynamics resulting from two configurations of systemic-to-pulmonary artery shunts currently utilized during the Norwood procedure: the central (CS) and modified Blalock Taussig (MBTS) shunts. A lumped parameter model of the neonatal cardiovascular circulation and detailed 3-D models of the shunt based on the finite volume method were constructed. Shunt sizes of 3, 3.5 and 4 mm were considered. A multiscale approach was adopted to prescribe appropriate and realistic boundary conditions for the 3-D models of the Norwood circulation. Results showed that the average shunt flow rate is higher for the CS option than for the MBTS and that pulmonary flow increases with shunt size for both options. Cardiac output is higher for the CS option for all shunt sizes. Flow distribution between the left and the right pulmonary arteries is not completely balanced, although for the CS option the discrepancy is low (50-51% of the pulmonary flow to the right lung) while for the MBTS it is more pronounced with larger shunt sizes (51-54% to the left lung). The CS option favors perfusion to the right lung while the MBTS favors the left. In the CS option, a smaller percentage of aortic flow is distributed to the coronary circulation, while that percentage rises for the MBTS. These findings may have important implications for coronary blood flow and ventricular function.     

The influence of systemic-to-pulmonary arterial shunts and peripheral vasculatures in univentricular circulations: Focus on coronary perfusion and aortic arch hemodynamics through computational multi-domain modeling

Initial palliation for univentricular hearts can be achieved via a systemic-to-pulmonary shunt (SPS). SPS configurations differ depending on the proximal anastomosis location, which might lead to dissimilar coronary and upper body perfusions. Mathematical modeling can be used to explore the local and global hemodynamic effects of the SPSs. In literature there are few patient-specific models of SPS that specifically address the influence of both the local and peripheral vasculature. In this study, multi-domain models of univentricular circulations were developed to investigate local hemodynamics and flow distribution in the presence of two shunt configurations. We also analyzed the relative impact of local and peripheral vascular resistances on coronary perfusion and flows through the upper aortic branches.A two-step approach was followed. First, two patient-specific models were based on clinical data collected from univentricular patients having different shunts and peripheral vasculatures. Each model coupled a three-dimensional representation of SPS, aortic arch (AA) and pulmonary arteries, with a lumped parameter model (LPM) of peripheral vasculature closing the circulatory loop. Then, two additional models of hypothetical subjects were created by coupling each customized LPM with the other patient’s three-dimensional anatomy.Flow rates and pressures predicted by the patient-specific models revealed overall agreement with clinical data. Differences in the local hemodynamics were seen during diastole between the two models. Varying the three-dimensional models, while keeping an identical LPM, led to comparable flow distribution through the AA, suggesting that peripheral vasculatures have a dominant effect on local hemodynamics with respect to the shunt configuration.     

Pulmonary artery hemodynamics with varying degrees of valvular stenosis: an in vitro study

The study was to investigate the effects of varying degrees of valvular stenosis on the hemodynamics of the main (MPA), left (LPA), and right (RPA) pulmonary arteries. Particle flow visualization was used to examine the flow patterns in a series of pulmonary artery models manufactured out of glass. These glass models were made based on the geometry of the porcine pulmonary arteries with dilatation in the MPA and LPA. Also, detailed pressure mappings in the models were conducted using a side-hole catheter. As the valve became stenotic, a jet-like flow was observed in the MPA. A higher degree of valvular stenosis corresponded to a narrower jet. This jet-like flow was noted to deflect away from the centerline and impinge on the roof of the dilated MPA. Additionally, a notable pressure gradient across the deflected jet-like flow in the direction of its radius of curvature was seen. Moreover, secondary flows started to appear in the dilated MPA. This suggested that the change in geometry in the MPA, due to its dilatation, had a marked effect on the pulmonary artery hemodynamics. In the LPA and RPA, the strengths of the secondary flows increased as the valve became more stenotic. The flow patterns observed in the LPA appeared to be more disturbed than in the RPA, due to the poststenotic, dilatation present in the LPA. Pressure recovery along the axial direction in the MPA was observed for all the stenotic valves studied. As the degree of valvular stenosis increased, the transvalvular energy loss increased. Moreover, it was observed that the energy loss decreased progressively as the flow traveled downstream. This tendency was consistent with the phenomenon of pressure recovery observed in the pressure measurement. The study demonstrates the importance of analyzing biological flows from a three-dimensional viewpoint.     

Numerical investigation of regurgitation phenomena in pulmonary arteries of Tetralogy of Fallot patients after repair

Pulmonary regurgitation is a very common phenomenon in pulmonary arteries after repair of patients of Tetralogy of Fallot (TOF) which is the most common complex congenital heart diseases. The aim of this study is to use numerical approaches to simulate flow variations in pulmonary artery after repair of patients of TOF. We analyze the flow patterns in an in-vitro bifurcation pulmonary artery and consider effects of various regurgitation fractions (RF or b/f) in left pulmonary artery (LPA) and right pulmonary artery (RPA). We not only observe the variation of flow patterns, but also analyze the results of b/f and net volumetric flow rates in LPA and RPA. In general, the b/f of LPA is higher than RPA in the measured data provided by phase-contrast magnetic resonance imaging (PC-MRI). We validate the result using numerical approaches to analyze the flow patterns in pulmonary artery in this study. The results will be useful for medical doctors when they perform operations for TOF patients.     

Axial flow velocity patterns in a pulmonary artery model with varying degrees of valvular pulmonic stenosis: pulsatile in vitro studies

With the advent of noninvasive clinical techniques which can measure blood flow velocities (Doppler ultrasound), it is suggested that a fundamental knowledge of the axial flow velocity patterns in the pulmonary artery, and the changes caused by stenosis, may be used to support accurate diagnosis of valvular pulmonic stenosis. The present study was designed to characterize the axial flow velocity patterns in an in vitro model of a human adult pulmonary artery with varying degrees of valvular pulmonic stenosis. A two-dimensional laser Doppler anemometer (LDA) system was used to map the flow fields in the main (MPA), left (LPA), and right (RPA) branches of the pulmonary artery model. The study was conducted in the Georgia Tech. right heart pulse duplicator system. It was observed that the axial flow velocity patterns in the MPA and the LPA change dramatically with increasing degree of valvular stenosis. This indicates that the axial flow velocity patterns in these two branches are strongly influenced by the degree of valvular stenosis. The axial flow velocity patterns in the RPA, however, do not change much with varying degrees of valvular stenosis, indicating that the axial flow fields in the RPA are mainly influenced by the geometry of the bifurcation. It may be concluded therefore, that the changes in the axial flow velocity patterns in the MPA and LPA (rather than in the RPA) could be sensitive and reliable indicators of the severity of the defect.     

Multiscale modelling in biofluidynamics: application to reconstructive paediatric cardiac surgery

Multiscale computing is a challenging area even in biomechanics. Application of such a methodology to quantitatively compare postoperative hemodynamics in congenital heart diseases is very promising. In the treatment of hypoplastic left heart syndrome, which is a congenital heart disease where the left ventricle is missing or very small, the necessity to feed the pulmonary and systemic circulations is obtained with an interposition shunt. Two main options are available and differ from the sites of anastomoses: (i) the systemic-to-pulmonary conduit (Blalock-Taussig shunt known as the Norwood Operation (NO)) connecting the innominate artery (NO-BT) or the aorta (NO-CS) to the right pulmonary artery and (ii) the right ventricle to pulmonary artery shunt (known as Sano operation (SO)). The proposition that the SO is superior to the NO remains controversial. 3-D computer models of the NO (NO-BT and NO-CS) and SO were developed and investigated using the finite volume method. Conduits of 3, 3.5 and 4 mm were used in the NO models, whereas conduits of 4, 5 and 6 mm were used in the SO model. The hydraulic nets (lumped resistances, compliances, inertances and elastances) which represent the systemic, coronary and pulmonary circulations and the heart were identical in the two models. A multiscale approach was adopted to couple the 3-D models with the circulation net. Computer simulation results were compared with post-operative catheterization data. Results showed that (i) there is a good correlation between predicted and observed data: higher aortic diastolic pressure, decreased pulmonary arterial pressure, lower pulmonary-to-systemic flow ratio and higher coronary perfusion pressure in SO; (ii) there is a minimal regurgitant flow in the SO conduit. The close correlation between predicted and observed clinical data supports the use of mathematical modelling, with a mandatory multiscale approach, in the design and assessment of surgical procedures.     

Secondary flow velocity patterns in a pulmonary artery model with varying degrees of valvular pulmonic stenosis: pulsatile in vitro studies

The objective of this study was to characterize in detail the secondary flow velocity patterns in an in vitro model of a human (adult) pulmonary artery with varying degrees of valvular pulmonic stenosis. A two-dimensional laser Doppler anemometer (LDA) system was used to map the flow fields in the main (MPA), left (LPA), and right (RPA) branches of the pulmonary artery model. The study was conducted in the Georgia Tech right heart pulse duplicator system. A pair of counter-rotating secondary flows were observed in each daughter branch in which the fluid moved outwardly along the side walls and then circled back inwardly toward the center of the vessel. For the case of the "normal" valve, the two counter-rotating secondary flows were symmetric about the centerline. The strength of secondary flows in the RPA was much stronger than in the LPA. However, as the pulmonic valve became more stenotic, the two counter-rotating secondary flows in both the LPA and RPA were no longer symmetric. In addition, the strength of secondary flows in both daughter branches increased with increasing degree of valvular stenosis. The increment in the LPA was, however, greater than in the RPA. The study demonstrates the importance of analyzing complex biological flows from a three-dimensional viewpoint.     

Steady flow velocity measurements in a pulmonary artery model with varying degrees of pulmonic stenosis

Velocity and flow visualization studies were conducted in an adult size pulmonary artery model with varying degrees of valvular stenosis, using a two dimensional laser Doppler anemometer system. Velocity measurements in the main, left and right branches of the pulmonary artery revealed that as the degree of pulmonic stenosis increased, the jet type flow created by the valve hit the distal wall of the LPA farther downstream from the junction of the bifurcation. This in turn led to higher levels of turbulent and disturbed flow, and larger secondary flow motion in the LPA compared to the RPA. The high levels of turbulence measured in the main and left pulmonary arteries with the stenotic valves, could lead to the clinically observed phenomenon of post stenotic dilatation in the MPA extending into the LPA.     

Blood flow conditions in the proximal pulmonary arteries and vena cavae: healthy children during upright cycling exercise

Diagnostic testing in patients with congenital heart disease is usually performed supine and at rest, conditions not representative of their typical hemodynamics. Upright exercise measurements of blood flow may prove valuable in the assessment of these patients, but data in normal subjects are first required. With the use of a 0.5-T open magnet, a magnetic resonance-compatible exercise cycle, and cine phase-contrast techniques, time-dependent blood flow velocities were measured in the right (RPA), left (LPA), and main (MPA) pulmonary arteries and superior (SVC) and inferior (IVC) vena cavae of 10 healthy 10- to 14-yr-old subjects. Measurements were made at seated rest and during upright cycling exercise (150% resting heart rate). Mean blood flow (l/min) and reverse flow index were computed from the velocity data. With exercise, RPA and LPA mean flow increased 2.0 +/- 0.5 to 3.7 +/- 0.7 (P < 0.05) and 1.6 +/- 0.4 to 2.9 +/- 0.8 (P < 0.05), respectively. Pulmonary reverse flow index (rest vs. exercise) decreased with exercise as follows: MPA: 0.014 +/- 0.012 vs. 0.006 +/- 0.006 [P = not significant (NS)], RPA: 0.005 +/- 0.004 vs. 0.000 +/- 0.000 (P < 0.05), and LPA: 0.041 +/- 0.019 vs. 0.014 +/- 0.016 (P < 0.05). SVC and IVC flow increased from 1.5 +/- 0.2 to 1.9 +/- 0.6 (P = NS) and 1.6 +/- 0.4 to 4.9 +/- 1.3 (P < 0.05), respectively. A 56/44% RPA/LPA flow distribution at both rest and during exercise suggests blood flow distribution is dominated by distal pulmonary resistance. Reverse flow in the MPA appears to originate solely from the LPA while the RPA is in relative isolation. During seated rest, the SVC-to-IVC venous return ratio is 50/50%. With light/moderate cycling exercise, IVC flow increases by threefold, whereas SVC remains essentially constant.     

In vitro steady-flow analysis of systemic-to-pulmonary shunt haemodynamics.

A modified Blalock-Taussig shunt is a connection created between the systemic and pulmonary arterial circulations to improve pulmonary perfusion in children with congenital heart diseases. Survival of these patients is critically dependent on blood flow distribution between the pulmonary and systemic circulations which in turn depends upon the flow resistance of the shunt. Previously, we investigated the pressure-flow relationship in rigid shunts with a computational approach. to estimate the pulmonary blood flow rate on the basis of the in vivo measured pressure drop. The present study aims at evaluating, in vitro how the anastomotic distensibility and restrictions due to suture presence affect the shunt pressure-flow relationship. Two actual Gore-Tex shunts (3 and 4 mm diameters) were sutured to compliant conduits by a surgeon and tested at different steady flow rates (0.25-11 min(-1)) and pulmonary pressures (3-34 mmHg). Corresponding computational models were also created to investigate the role of the anastomotic restrictions due to sutures. In vitro experiments showed that pulmonary artery pressure affects the pressure-flow relationship of the anastomoses. particularly at the distal site. However, this occurrence scarcely influences the total shunt pressure drop. Comparisons between in vitro and computational models without anastomotic restrictions show that the latter underestimates the in vitro pressure drops at any flow rate. The addition of the anastomotic restrictions (31 and 47% of the original area of 3 and 4 mm shunts, respectively) to the computational models reduces the gap, especially at high shunt flow rate and high pulmonary pressure.     

Modeling of the Norwood circulation: effects of shunt size, vascular resistances, and heart rate

Hypoplastic left heart syndrome is the most common lethal cardiac malformation of the newborn. Its treatment, apart from heart transplantation, is the Norwood operation. The initial procedure for this staged repair consists of reconstructing a circulation where a single outlet from the heart provides systemic perfusion and an interpositioning shunt contributes blood flow to the lungs. To better understand this unique physiology, a computational model of the Norwood circulation was constructed on the basis of compartmental analysis. Influences of shunt diameter, systemic and pulmonary vascular resistance, and heart rate on the cardiovascular dynamics and oxygenation were studied. Simulations showed that 1) larger shunts diverted an increased proportion of cardiac output to the lungs, away from systemic perfusion, resulting in poorer O2 delivery, 2) systemic vascular resistance exerted more effect on hemodynamics than pulmonary vascular resistance, 3) systemic arterial oxygenation was minimally influenced by heart rate changes, 4) there was a better correlation between venous O2 saturation and O2 delivery than between arterial O2 saturation and O2 delivery, and 5) a pulmonary-to-systemic blood flow ratio of 1 resulted in optimal O2 delivery in all physiological states and shunt sizes.     

Bilateral coronary fistulae to pulmonic valve in presence of severe three-vessel coronary artery disease

A 62-year-old man was admitted to the coronary care unit due to anginal pain and palpitations--coronary angiography revealed three-vessel coronary artery disease. The unexpected finding was the presence of coronary to pulmonary artery fistulae bilaterally, from both the proximal RCA and the proximal LAD. Right heart catheterization revealed normal right ventricular and pulmonary artery pressure and absence of hemodynamically significant left to right shunt. The patient underwent a triple coronary bypass including the closure of bilateral fistulae, which were draining into the left sinus of the pulmonary valve. One month after the operation he was in good health and had no complaints. Bilateral coronary artery fistulae is a rare anomaly diagnosed in 0.002-0.0013% of adult coronary angiograms. (Int J Cardiovasc Intervent 1999; 2: 249-251).     

Personalized stent design for congenital heart defects using pulsatile blood flow simulations

Stent size selection and placement are among the most challenging tasks in the treatment of pulmonary artery stenosis in congenital heart defects (CHD). Patient-specific 3D model from CT or MR improves the understanding of the patient’s anatomy and information about the hemodynamics aid in patient risk assessment and treatment planning. This work presents a new approach for personalized stent design in pulmonary artery interventions combining personalized patient geometry and hemodynamic simulations. First, the stent position is initialized using a geometric approach. Second, the stent and artery expansion, including the foreshortening behavior of the stent is simulated. Two stent designs are considered, a regular stent and a Y-stent for bifurcations. Computational fluid dynamics (CFD) simulations of the blood flow in the initial and expanded artery models are performed using patient-specific boundary conditions in form of a pulsatile inflow waveform, 3-element Windkessel outflow conditions, and deformable vessel walls. The simulations have been applied to 16 patient cases with a large variability of anatomies. Finally, the simulations have been clinically validated using retrospective imaging from angiography and pressure measurements. The simulated pressure, volume flow and flow velocity values were on the same order of magnitude as the reference values obtained from clinical measurements, and the simulated stent placement showed a positive impact on the hemodynamic values. Simulation of geometric changes combined with CFD simulations offers the possibility to optimize stent type, size, and position by evaluating different configurations before the intervention, and eventually allow to test customized stent geometries and new deployment techniques in CHD.     

Factors affecting perfusion distribution in canine oleic acid pulmonary edema

Factors affecting perfusion distribution in oleic acid pulmonary edema were examined in 28 anesthetized open-chest dogs. Sixteen had unilobar oleic acid edema produced by left lower lobe pulmonary artery infusion of 0.03 ml/kg of oleic acid, and 12 had the same amount of edema produced by left lower lobe endobronchial instillation of hypotonic plasma. Lobar perfusion (determined from flow probes) and lobar shunt (determined from mixed venous and lobar venous blood) were measured at base line, 1.5 h after edema, and 10 min after 10 cmH2O positive end-expiratory pressure (PEEP). Fourteen dogs (8 oleic acid, 6 plasma) received sodium nitroprusside (11.72 +/- 7.10 micrograms X kg-1 X min-1). Total and lobar shunts increased to the same extent in all animals. Lobar perfusion decreased by 49.8 +/- 4.8% without nitroprusside and 34.0 +/- 3.6% with nitroprusside in the oleic acid group, corresponding values being 40.3 +/- 0.8% and 26.4 +/- 1.7% in the hypotonic plasma group. PEEP returned perfusion and shunt to base line. In oleic acid edema, most of the decreased perfusion results from mechanical effects of the edema, a smaller fraction results from other vascular effects of the oleic acid, and approximately 30% is reversible by nitroprusside. PEEP normalizes the perfusion distribution.     

Numerical modeling of hemodynamics scenarios of patient-specific coronary artery bypass grafts

A fast computational framework is devised to the study of several configurations of patient-specific coronary artery bypass grafts. This is especially useful to perform a sensitivity analysis of the hemodynamics for different flow conditions occurring in native coronary arteries and bypass grafts, the investigation of the progression of the coronary artery disease and the choice of the most appropriate surgical procedure. A complete pipeline, from the acquisition of patient-specific medical images to fast parameterized computational simulations, is proposed. Complex surgical configurations employed in the clinical practice, such as Y-grafts and sequential grafts, are studied. A virtual surgery platform based on model reduction of unsteady Navier–Stokes equations for blood dynamics is proposed to carry out sensitivity analyses in a very rapid and reliable way. A specialized geometrical parameterization is employed to compare the effect of stenosis and anastomosis variation on the outcome of the surgery in several relevant cases.     

Redistribution in left ventricular regional flow following acute right ventricular pressure overload

The left ventricular dysfunction following acute pulmowary hypertension remains unexplained. We wondered if acute pulmonary hypertension could alter the transmural flow distribution within the left ventricular myocardium, independent of coronary flow and perfusion pressure. We used a canine preparation in which the left coronary system was perfused at constant flow and induced a two- to three-fold increase in pulmonary artery pressure by banding the pulmonary artery. Regional myocardial blood flow of the left coronary system was measured using radioactive microspheres, injected into the left coronary system before and after 10-30 min of banding of the pulmonary artery. The left ventricular subendocardial:epicardial ratio fell by 12 and 31% (p less than 0.05) of control value, 10 and 30 min, respectively, after banding of the pulmonary artery, the total flow to the left coronary system being kept constant. Left atrial mean pressure increased from 2.9 +/- 2.4 to 3.6 +/- 1.9 and 6.0 +/- 2.1 (p less than 0.05) following banding. The mechanism of the redistribution of coronary flow may relate to inappropriate vasodilation of the right septal myocardium with consequent relative left ventricular subendocardial hypoperfusion which might aggravate left ventricular ischemia in the presence of hypotension and hypoxia.     

Pulmonary artery relative area change is inversely related to ex vivo measured arterial elastic modulus in the canine model of acute pulmonary embolization

A low relative area change (RAC) of the proximal pulmonary artery (PA) over the cardiac cycle is a good predictor of mortality from right ventricular failure in patients with pulmonary hypertension (PH). The relationship between RAC and local mechanical properties of arteries, which are known to stiffen in acute and chronic PH, is not clear, however. In this study, we estimated elastic moduli of three PAs (MPA, LPA and RPA: main, left and right PAs) at the physiological state using mechanical testing data and correlated these estimated elastic moduli to RAC measured in vivo with both phase-contrast magnetic resonance imaging (PC-MRI) and M-mode echocardiography (on RPA only). We did so using data from a canine model of acute PH due to embolization to assess the sensitivity of RAC to changes in elastic modulus in the absence of chronic PH-induced arterial remodeling. We found that elastic modulus increased with embolization-induced PH, presumably a consequence of increased collagen engagement, which corresponds well to decreased RAC. Furthermore, RAC was inversely related to elastic modulus. Finally, we found MRI and echocardiography yielded comparable estimates of RAC. We conclude that RAC of proximal PAs can be obtained from either MRI or echocardiography and a change in RAC indicates a change in elastic modulus of proximal PAs detectable even in the absence of chronic PH-induced arterial remodeling. The correlation between RAC and elastic modulus of proximal PAs may be useful for prognoses and to monitor the effects of therapeutic interventions in patients with PH.     

Modified heart-lung preparation for the evaluation of systolic and diastolic coronary flow in rats

A modified heart-lung preparation of the rat, which permits measuring systolic and diastolic coronary flow separately and enables coronary compliance to be evaluated, is described. The systemic circulation was substituted by a shunt circuit, and the elastic properties of the arterial tree were mimicked by a rubber balloon. Systolic and diastolic coronary flow was evaluated from the pulmonary and aortic flow signal. Integrated phasic pulmonary flow represented right ventricular stroke volume. Integrated phasic systolic aortic flow represented left ventricular stroke volume minus that volume flowing into the coronary arteries during systole, because the aortic flow probe had to be inserted distal to the origin of the coronary vessels. Because right and left ventricular stroke volume was identical under steady-state conditions, the difference between systolic pulmonary and systolic aortic flow resulted in systolic coronary flow. Diastolic coronary flow was measured by means of the retrograde flow through the aortic flow probe. Coronary compliance was calculated according to Frank's windkessel model from coronary resistance and from central diastolic aortic pressure, which decayed exponentially after switching out the rubber balloon and the shunt circuit. It could be shown that the proportion of systolic to diastolic coronary flow depends on coronary compliance.     

Ventricular septal defect with bidirectional shunting: Mathematical considerations

Previously proposed formulae for the quantitative estimation of bidirectional shunts across ventricular septal defects require determination of the oxygen contents of mixed venous, pulmonary artery, pulmonary venous, and aortic blood. Because these formulae do not take into account the mixing of oxygenated with unoxygenated blood within the ventricles, their use must result in underestimation of shunt flows in each direction. A mathematical model for a ventricular defect is examined, in which it is assumed that mixing of blood occurs in each of six sites in the venae cavae or right atrium, right ventricle, pulmonary artery, left atrium, left ventricle, and aorta. A total of fourteen streams of blood can flow from one to another of these mixing sites. As long as complete mixing occurs in the six specified mixing sites, any degree of mixing or non-mixing of the various streams is permitted. From the equations characterizing the model, formulae are derived in which the shunt flow in each direction is expressed in terms of the oxygen contents in the six mixing sites and the fractions of blood which enter the shunt from either side without prior mixing in a ventricular mixing site. The previously reported formulae, which apply when no ventricular mixing is allowed to occur, lead to theoretical minimum values for the shunt flows in each direction. At the opposite extreme where all the shunting blood is required to mix in a ventricle before entering the shunt, formulae for maximum possible shunt flows are also obtained. The absolute values for the left-to-right and right-to-left shunt flows, which must lie somewhere between the theoretical maximum and minimum values, cannot be computed from blood gas data alone. This work was supported in part by grant HE-07563 from the National Heart Institute of the National Institutes of Health and grants-in-aid from the American and North Carolina Heart Associations and the Life Insurance Medical Research Fund. Work completed during tenure as U.S.P.H.S. post-doctoral fellow.