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J Thorac Cardiovasc Surg 1996;111:502-513
© 1996 Mosby, Inc.
SURGERY FOR CONGENITAL HEART DISEASE |
Supported by grant No. PG/92141 from the British Heart Foundation.
Address for reprints: M. R. de Leval, MD, FRCS, Cardiothoracic Unit, Great Ormond Street Hospital for Children, London WCIN 3JH, United Kingdom.
Abstract
Computational fluid dynamic methods based on a finite-element technique were applied to the study of (1) competition of flows in the inferior and superior venae cavae in total cavopulmonary connection, and (2) competition between flow in the superior vena cava and forward flow from a stenosed pulmonary artery in bidirectional cavopulmonary anastomosis. Models corresponding to various degrees of offsetting and shape of the inferior vena caval anastomosis were simulated to evaluate energy dissipation and flow distribution between the two lungs. A minimal energy loss with optimal flow distribution between the two lungs was obtained by enlarging the inferior vena caval anastomosis toward the right pulmonary artery. This modified technique of total cavopulmonary connection is described. A computational model of the operation was developed in an attempt to understand the mechanisms of postoperative failure. In tight pulmonary artery stenosis (75%), the pulsatile forward flow is primarily directed to the left pulmonary artery, with little influence on superior vena caval pressure and the right pulmonary artery. Pulsatile forward flows corresponding to 15%, 30%, 45%, and 60% of the systemic artery output increased the mean pulmonary artery and superior vena caval pressures by 1, 1.7, 2.4, and 3.6 mm Hg, respectively. Although the modeling studies were not able to determine the cause of postoperative failure, they emphasize the impact of local geometry on flow dynamics. More simulations are required for further investigation of the problem. (J THORAC CARDIOVASC SURG 1996;111:502-13)
Complex flow problems in fields as varied as meteorologic calculation, geophysical fluid dynamics, astrophysical fluid dynamics, flow machinery, and design of aircraft, ships, and automobiles are now studied with numeric methods by means of digital computers. One of the most commonly used methods in computational fluid dynamics (CFD) is the finite-element method (FEM). FEM modeling techniques have been applied to biomedical problems for a number of years. We postulate that CFD technology could usefully be applied to the design and refinement of a number of surgical procedures aimed at mending cardiovascular anomalies.
We tested this hypothesis in two specific clinical situations. (1) One area studied was the flow competition between the superior vena cava (SVC) and inferior vena cava (IVC) in total cavopulmonary connection (TCPC). This is in line with the in vitro modeling and open-channel visualization studies that had led us to recommend the TCPC.
1 (2) The other area studied was flow competition between the SVC flow from a bidirectional cavopulmonary anastomosis (BCPA) and an additional source of pulmonary blood flow (ABF). Clinical recommendations based on the flow simulation studies were made and the preliminary results are reported here. Potential applications of CFD technology to cardiovascular sciences are discussed.
CFD modeling
Governing equations for flows of practical interest are usually so complicated that an analytic solution is unavailable, and it is therefore necessary to seek a computational solution. The mathematic equations (the unsteady Navier-Stokes equations) are replaced by algebraic equations so that a computer can be used to obtain the solution. The process of converting the continuous model into a discrete model of algebraic equations is called discretization. All methods of discretization involve an approximation that approaches the true continuum solution as the number of discrete variables increases. The FEM is a method of discretization where the flow domain is divided into a number of simply shaped regions called finite elements. Within each element, a certain number of points or nodes are defined where the numeric value of the unknown functions, and eventually their derivatives, will have to be determined (Fig. 1). The space discretization consists in setting up a mesh or a grid in which the continuum of space is replaced by a finite number of elements.
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Clinical problem
In basic flow dynamic terms, a TCPC represents two T-junctions with flow velocities in opposite directions from SVC and IVC into the pulmonary arteries. Those abrupt changes in geometry and velocity inevitably generate energy losses. Furthermore, postoperative cineangiograms demonstrate that the SVC flow, which carries approximately one third of the systemic venous return, goes preferentially to the bigger right lung, whereas the IVC flow, which carries approximately two thirds of the systemic venous return, goes mainly to the smaller left lung. This study was aimed toward improving the design of the operation to reduce energy loss and optimize flow distribution between the two lungs.
CFD studies
Anatomic and hemodynamic variables necessary to construct and validate the three-dimensional models of the investigated domains were obtained from angiocardiograms and cardiac catheterization data. The developed three-dimensional geometric models for TCPC and BCPA simulations are fully parametric; that is, all their dimensions are mathematically related to a basic set of about 30 measurements estimated from the angiocardiograms and corrected on the basis of the magnification factor (computed as the ratio between the measured catheter diameter on the angiocardiographic frames and its actual diameter). Anteroposterior and lateral views were used to obtain the measurements and to build the three-dimensional geometries of the reference models. A commercial CFD code (FIDAP; Fluid Dynamics International, Evanston, Ill.) was used to compute the fluid dynamic variables (pressure and velocity components). This code combines the FEM and the Galerkin method of weighted residuals to solve the full Navier-Stokes s. The geometric models were fully parametric. The FEM mesh was made of three-dimensional brick elements containing eight nodes; quadrilateral two-dimensional elements with four nodes were used at the inlet and outlet sections and along the walls. The number of nodes for each model ranged from 15,000 to 33,000. In addition to the three-dimensional computational model, a lumped parameter model of the afterload, representing the two lungs and the left atrium, was developed. A pulmonary arteriolar resistance of 1.62 Wood units (millimeters of mercury per liter per minute) and a left atrial pressure of 4 mm Hg were selected for the simulations reported on here. A FORTRAN program was written to obtain the parametric construction and the mesh of the three-dimensional simulations and to connect the two computational models, with the calculated afterload used as the boundary conditions for the three-dimensional model. The computations were done on an IBM RISC-6000 Model 340 workstation (International Business Machines Corp., Armonk, N.Y.) with 32 Mb RAM. Detailed descriptions of the methods and their limitations have been previously reported 2,3 and are summarized in Appendixes 1 and 2.
The grid of the TCPC model is represented in Fig. 1. Flow distributions between the left and the right lung were qualitatively studied with particle path plots and calculated from velocity vector plots. In a particle path plot, a massless particle is introduced at some point in the flow domain and the path of motion of the particle is tracked on the basis of the computed flow field. Two energetic indexes were calculated 4: the hydraulic dissipated power, Wdiss, which is the difference between the total energy at the inlet and at the outlet of the domain, and the total energy-loss coefficient, Ce, which is the ratio between the hydraulic dissipated power and the inlet kinetic energy (Appendix 2). Two sets of simulations were carried out. In the first series, an IVC anastomosis with the same diameter as the SVC anastomosis was displaced alongside the pulmonary artery while the position of the SVC anastomosis was kept constant. Results are reported as functions of the offset, which represents the distance between the centers of the two caval anastomoses measured toward the right pulmonary artery axis. The offset is negative if the center of the anastomosis of the IVC is displaced toward the reader's right side. The best flow distribution between the right and the left lung, considered to be 55% to the right lung and 45% to the left, was obtained for an offset of the IVC anastomosis of 0.5 to 0.7 cm toward the right pulmonary artery. Unfortunately, this corresponds to the maximum energy dissipation (Fig. 2). In an attempt to improve the balance between flow distribution and energy dissipation, the IVC anastomosis was enlarged and moved alongside the pulmonary artery (Fig. 3). The results of those simulations are summarized in Table I. A minimal loss of energy was obtained by angling the IVC anastomosis toward the right pulmonary artery (model T2.5, 
= 17 degrees, width of IVC anastomosis 2.5 cm), and this corresponds to a flow ratio between the right and the left lung of just above 1 (Fig. 4). The particle path and the vector velocity plots (Fig. 5) showed that the SVC drains preferentially to the left lung whereas the IVC drains preferentially to the right lung, the opposite of the pattern usually observed angiocardiographically after the classical Fontan or TCPC operation. In terms of fluid dynamics, the enlargement of the IVC anastomosis acts as a diffuser or recuperator, which is a device used in hydraulic plants to reduce dissipation of kinetic energy at abrupt changes of cross-section. There is an optimum range of angles for which the decrease of head loss is maximal ( = 14 to 17 degrees, models T2.3 and T2.5).
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The clinical problem
The BCPA is used as a staged procedure or a definitive palliation in the treatment of hearts with a single ventricular chamber. An ABF arising from either the native pulmonary outflow tract or from a systemic–pulmonary artery shunt is often maintained to provide a better (higher oxygen saturation) and longer palliation. Setting aside concerns about volume load on the ventricular chamber, the preoperative clinical problem is often to know what magnitude of ABF is acceptable while constructing a BCPA. In fluid dynamic terms, the operation creates a competition in the pulmonary arteries between the continuous flow from the SVC and the pulsatile flow from the ABF. CFD was applied to analyze the flow dynamic effects of such a competition and to see whether they correlated with the postoperative morbidity associated with the surgical procedure.
CFD studies
The same computational approach used to study flow competition in the TCPC was used in the study of flow competition in the BCPA. The grid of a BCPA model with forward flow from the main pulmonary artery is represented in Fig. 8. The geometry of the flow domain was obtained from angiocardiograms of hearts with ventriculoarterial concordance, where the main pulmonary artery is "pointing" toward the left lung. A steady flow was imposed on the SVC with a fully developed parabolic velocity profile, whereas a pulsatile flow was introduced into the native pulmonary artery. Waveforms obtained from multiple-catheterization data of pulmonary stenosis of variable severity were used to simulate forward flow. Two sets of simulations were carried out to differentiate the effects of forward flow and flow configuration on the pulmonary arterial branches and the SVC. The flow from the SVC was kept constant (1 L/min) for all the simulations, and a heart rate of 120 beats/min, with a systolic and diastolic time of equal duration, was selected. In the first set of simulations, forward flows of 0.45, 0.90, 1.35, and 1.80 L/min were tested. Forward flows of 1.0 and 1.8 L/min through discrete valvular and discrete supravalvular (banded pulmonary artery) stenoses of different degrees were tested in the second set of simulations. Particle path plots and velocity vector plots were constructed and used to derive flow dynamics. Without stenosis, increments in forward flows altered the distribution of the caval and the forward flow between the two lungs. There is a preferential distribution of the forward flow to the left lung, and consequently a higher peak velocity in the left pulmonary artery (Fig. 9). The overall flow ratio between the two lungs, however, remains constant at 1.1 (Table II). With discrete valvular stenosis, the proportion of flow going to the left lung increases as the stenosis becomes more severe. With a forward flow of 1.8 L/min and a stenosis of 75%, nearly three quarters of the blood goes to the left lung (Table III). This implies that during systole a greater proportion of the forward flow goes to the left lung and in diastole a greater proportion of the caval flow also goes to the left lung. This preferential flow to the left lung is less marked with supravalvular discrete stenosis. The ABF has only a moderate influence on the mean SVC pressure, which increases from 6.4 to 8.6 mm Hg when the forward flow rises from 0.45 to 1.8 L/min.
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Table IV. There was no statistical difference between the two groups. Within the range of preoperative flows and pressures of patients subjected to BCPA and ABF, it was not possible to predict outcome and complications.
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5 Pulmonary vascular resistance is expected to be one of the main determinant factors of postoperative hemodynamics. Pulmonary vascular resistance was not directly measured before operation. A preoperative resistance index (mean pulmonary arterial pressure divided by the ratio of pulmonary to systemic flow) was found to have no correlations with outcomes. This obviously indicates that patients with elevated pulmonary vascular resistance were not subjected to BCPA plus ABF. The reason that some patients have a complicated postoperative course remains unclear and warrants further investigation. Discussion
The science of fluid dynamics started with Newton. The governing equations for Newtonian fluid dynamics, the unsteady Navier-Stokes equations, were formulated more than 150 years ago. The steady improvement in the speed of computers and their memory size since the 1950s has led to the emergence of CFD as a means of modeling real flows. In the cardiovascular system, the goal of CFD studies is to predict flow, pressure, and velocity fields as a function of time and position in a realistic model of the domain under investigation. The technology has been applied to the study of the normal cardiovascular system 6,7 and to the design of prosthetic devices. 8 With few exceptions, such as the simulation of the fluid dynamics of an aorta-coronary bypass, 9 CFD has rarely been applied to the actual design and refinement of interventions on the cardiovascular system.
As far as the methodology is concerned, important assumptions are made to generate appropriate models and to solve the equations that predict flow distribution. We are currently looking at the possibility of directly translating three-dimensional geometries from magnetic resonance imaging to computer-aided design formats transferable to FEM packages. This direct translation would avoid the phase of mesh creating by means of a specially written FORTRAN program.
The clinical implications of the simulation studies of the competitive flow in cavopulmonary anastomoses presented in this article are still preliminary. Flow distribution studies between the two lungs should be validated. This could be assessed by nuclear magnetic resonance velocity mapping by the technique described by Rebergen and associates. 10 In the TCPC, one could conceive ways to obtain an even more favorable flow pattern by placing a second angulation patch on the medial aspect of the SVC–pulmonary artery anastomosis to direct more SVC blood to the left lung. The flow simulation studies are quite convincing in demonstrating that it is not necessary to divide the right pulmonary artery and anastomose the SVC to its proximal end and the IVC to its distal end to obtain a more physiologic distribution of pulmonary blood flows. 11 Such a procedure loses the flexibility of the BCPA cavopulmonary anastomosis, which could become useful if, for example, the pulmonary vascular resistance in one lung were to increase or one of the anastomoses were to become restrictive. In addition the left lung, being deprived of IVC blood, might be at higher risk for development of arteriovenous fistulas (the so-called hepatic factor).
The purpose of this article is not to discuss the pros and cons of ABF in construction of a BCPA. ABF is an optional solution for patients who are likely to become good candidates for a Fontan circulation. It is a more logical solution for those patients who are not optimal candidates or whose anatomy will not be suitable for the Fontan operation. Two recent reports 12,13 have shown that BCPA with ABF had a higher postoperative morbidity rate than BCPA without ABF. This difference must be related to flow dynamic problems. Although we have been unable to define these problems so far, the modeling studies have shown the important effects of local geometry on flow dynamics and flow distribution between the two lungs, and more simulations are needed to attempt to elucidate the problem. The concept of BCPA with forward flow from the native pulmonary artery has also been applied to a subset of patients with a small right ventricle and intact ventricular septum without pulmonary stenosis. 14-16 The first BCPA simulations, which modeled this arrangement, showed that a forward flow of about two thirds of the systemic venous return (IVC flow) interferes minimally with the mean SVC pressure, with less pulsatility in the right pulmonary artery than the left.
There are multiple theoretic applications of CFD to cardiovascular surgery. Pressure gradients and aneurysmal formations after vascular patch enlargements are no doubt related to wall stresses resulting from the geometry and the material properties of the patches. This has been demonstrated in an elegant study with CFD technology on aortic wall stress profile after repair of coarctation of the aorta. 17 Computation of ventricular flow dynamics will open the field for numerous CFD studies in areas such as outflow tract patch enlargement, repair of ventricular aneurysms, repair and replacement of atrioventricular valves, and designs of ventricular-assist devices. Preliminary comparative CFD studies of multiple sequential anastomoses versus simple end-to-side anastomosis in coronary revascularization have demonstrated the superiority of the former procedures (Menicanti L. Personal communication, 1995). Flow dynamic studies of stented arteries could help to predict restenosis and late outcomes, and could contribute to the design of new devices. There is little doubt that obstruction and attrition of extracardiac valved conduits are in great part related to local flow fields determined by the configuration of the anastomosis to the ventricular chamber and the distal vessel, the position of the valve, and the presence or absence of external compression. CFD and FEM can help in the understanding of the mechanics of vascular catastrophes such as atherosclerotic plaque rupture, a common cause of acute myocardial infarction, or shear failure as a cause of acute aortic dissection. 18,19
With regard to the future, one could foresee that with increasingly powerful computers FEM could be used to investigate mechanical stress at a cellular level. The body of knowledge describing the ability of cells to respond to mechanical stresses at a molecular level is growing rapidly. 18,20 The endothelium functions as a mechanically sensitive signals-transduction interface between blood and vascular wall. 21 Changes in shear stress may lead to changes in cell shape, growth, and function, and in a wide variety of gene regulatory events. Are rheology and cellular and molecular biology going to meet in the near future? Such a convergence would open an area of fascinating research.
Appendix: Discussion
Dr. Aldo R. Castaneda (Genolier, Switzerland)
We are once more indebted to Dr. de Leval and his bioengineering colleagues for exposing us first to up-to-date technology in CFM and FEM and second to the application of these powerful investigative tools to the practice of cardiovascular surgery. I think that the simple but effective modification of adding a right-sided augmentation patch plasty to the cardiac segment of the SVC–right pulmonary artery anastomosis, an outgrowth of the effort of achieving the best possible balance between flow distribution and energy dissipation, exemplifies this effort. In addition, this modification should make the so-called unidirectional cardiopulmonary connection proposed by Laks unnecessary, retaining the hemodynamic advantages and much lower risk of formation of arteriovenous fistulas.
I was surprised that this methodology did not identify energy dissipation of IVC blood passing the SVC–right atrial junction. We know that the area ratio between the IVC–right atrial junction and SVC–right atrial junction is about 2:1 because the orifice size or area size of the IVC–right atrial junction is about twice as large as that of the SVC.
It appears to me that the augmentation plasty goes through the SVC–right atrial junction. Could you comment on that? Clearly the concept of allowing pulsatile forward pulmonary flow, either through a stenotic pulmonary valve or through a systemic artery–pulmonary artery shunt, is an option, particularly for patients who are not ideal candidates for a Fontan operation.
I am sure that we will hear much more from your group after you have been able to unravel the remaining problems of various pulmonary vascular geometries and also bilateral pulmonary parenchymal flow distribution.
Dr. de Leval
It is true that the patch goes down, across the narrowest part of the pathway, but the more I look at flow simulations the more I believe that maintaining streamlining is more important than the actual diameter. I think a smooth, progressive reduction of diameter of a pathway is less important or deleterious than the loss of streamlining.
Dr. Serafin Y. DeLeon (Maywood, Ill.)
I was just wondering whether you have already done some preliminary studies in patients in terms of the total cavopulmonary anastomosis in which you have straddled a little bit to the right pulmonary artery, whether you have correlated this with lung scan studies and demonstrated that the flow really goes where you intend it to go and determined the clinical implications of your model.
Dr. de Leval
It is an important question. Obviously, all these results have to be validated, and we are in the process of validating them by using magnetic resonance imaging to look at flow distribution to the two lungs.
Dr. Richard A. Jonas (Boston, Mass.)
Do you have any thoughts on differences in hemodynamics that might result from the use of different materials? For example, there seems to be a lot of enthusiasm right now for prosthetic extracardiac conduits. There also has been a suggestion of using a pedicled pericardial graft that is placed along the external surface of the right atrium, with the implication being that pulsatility is induced by this. Have you looked at any of these other alternatives?
Dr. de Leval
We have not compared the flow dynamics through different materials used to construct the IVC pathway. I am aware of the recent enthusiasm for extracardiac conduits to channel the IVC blood to the pulmonary artery but cannot comment on the advantages of, for example, a pericardial graft sewn on the outside of the heart.
Appendixes
Appendix 1
The basic equations of fluid mechanics consist of the equation of continuity, which expresses the law of conservation of mass, the equation of motion, which expresses the law of conservation of momentum, the equation of balance of energy, and the constitutive equation and boundary conditions. If there is no heat input into the system and no heat source in the region, the equation for balance of energy is the same as that for balance of mechanical energy and work; it leads to no new independent equation. Let us consider an incompressible Newtonian viscous fluid. Let x, y, and z be rectangular Cartesian coordinates. Let the velocity components along the x-, y-, and z-axis directions be denoted by u, v, and w, respectively. Let p denote pressure. Let µ and 
denote the coefficient of viscosity (constant for a Newtonian fluid) and the density of the fluid (constant for an incompressible fluid), respectively.
The condition of incompressibility reduces the equation of continuity to the following equation:
The equation of motion is a vectorial equation. For an incompressible Newtonian viscous fluid, the (scalar) correspondents in the x-, y-, z-axis directions are named Navier-Stokes equations. They are written as follows:
In equations A1.2 to A1.4, each term has a physical meaning. Let us consider equation A1.2 (x-axis direction) and examine its terms: (
u/t) represents the "transient" inertia force per unit volume of fluid; that is, the result of the change of u velocity component in the generic point P(x,y,z) in time. (uu/x + vu/y + wu/z) represents the "convective" inertia force per unit volume of fluid; that is, the result of the change of u velocity component in the neighborhood of the generic point P(x,y,z) in the x-, y-, z-axis directions. p/x represents the force per unit volume caused by the presence of a pressure gradient in the x-axis direction. µ(2u/x2 + 2u/y2 + 2u/z2 represents the force per unit volume due to the viscous friction. X represents the component of the body forces (e.g., gravity force) per unit volume of fluid in the x-axis directions.
Equations A1.1 to A1.4 can be solved only if appropriate boundary conditions are specified. Boundary conditions are typically expressed as velocity or pressure distributions on some portion of the boundary that delimits the fluid domain. To characterize an incompressible viscous flow, two dimensionless parameters are often used. They are the Reynolds number (Re) and the Womersley number (Wo). They are defined as follows:
where p and µ are density and viscosity of the fluid and D, V, f are the diameter of the vessel, the mean velocity of the flow, and the frequency of pulsation, respectively. In a steady flow, the transient inertial force vanishes; the first term in equations A1.2 to A1.4 is therefore dropped and Wo is 0.
At low Re, a steady flow is laminar and transition from laminar to turbulent flow occurs when the critical Re is reached. In a pulsatile flow, this threshold also depends on the rate of change of velocity and therefore on the Wo. For full details, the reader is urged to see basic fluid dynamics textbooks (for instance Fung 22).
Appendix 2
Three-dimensional FEM models
This article reports results from two distinct classes of computer simulations:
Vessel diameters in TCPC models are as follows: SVC diameter (DSVC), 1.2 cm; IVC diameter (DIVC), 1.5 cm; left pulmonary artery diameter (DLPA), 1.0 cm; and right pulmonary artery diameter (DRPA), 1.0 cm. In BCPA models, diameters are as follows: DSVC, 1.5 cm; pulmonary artery diameter (DPA) 1.5 cm; DLPA, 1.2 cm; and DRPA, 1.2 cm.
General assumptions adopted for the CFD study described here are as follows: (1) terms X, Y, Z (body forces per unit volume) discarded in Navier-Stokes equations, (2) incompressible homogeneous Newtonian fluid ( = 1.06 x 103 kg/m3, µ = 3 x 10-3 kg · m-1 · sec-1), (3) rigid and impermeable vessel walls, (4) all vessel axes assumed to lie in the same plane (geometric symmetry plane).
Assumption 3 is quite unusual in hemodynamic studies related to arterial districts because flow pulsatility cannot usually be discarded. It is a fairly correct hypothesis in this study of the TCPC, however, because the right atrium and ventricle are totally bypassed and both the incoming flows are from venous districts. Changes in caval flows and pressures caused by both respiratory rhythm and geometry-dependent fluid dynamic effects are discarded as well. As a consequence, the vessel diameters can be assumed to remain constant and the hypothesis of rigid walls is reasonable. Such an assumption appears less reasonable for the study of the BCPA. In fact, nonlinear and viscoelastic properties of pulmonary arterial walls have effects on the details of local flow pattern, but shear stresses, for example, are not significantly altered. 23 If the pressure gradient is low, as encountered in these vessels, the local fluid dynamics will be slightly influenced, and the results obtained with a rigid model are expected not to differ greatly from the real ones.
Assumption 4 allowed us to represent only half of the model, with an important reduction in the storage and memory requirements for computations. The geometry of the developed three-dimensional models is clearly the result of a compromise between the requirement of describing accurately the highly complex anatomy of the pulmonary bifurcation and caval anastomoses and the necessity of using a modular geometric structure that can be automatically created or adapted through a custom-written FORTRAN computer program. We believe that this limitation does not affect the evaluation of the different hemodynamic designs from a comparative point of view. Regarding the boundary conditions for the models, they were adopted as follows:
Lumped-parameter model
The lumped-parameter model is a mechanical model of the pulmonary circulation that represents the two lungs and the left atrium in terms of lumped components. To simplify computations, only resistance components appear in this model; both elastic compliances and inertial effects are absent. Left atrial pressure (PLA) is assumed to be constant and to equal the mean pressure measured during cardiac catheterization. The left atrium is thus assumed to behave like a constant-pressure reservoir. Cardiac catheterizations have also supplied the value of global pulmonary resistance for the model, equally split into two parallel resistances (RLung) corresponding to the two lungs, assuming that both lungs offer the same resistance. Inputs for this model are the volumetric flow rates QLPA = QLUPA+QLLPA and QRPA = QRUPA+QRLPA (where Q is flow and LPA is left pulmonary artery, LUPA is left upper pulmonary artery, LLPA is left lower pulmonary artery, RPA is right pulmonary artery, RUPA is right upper pulmonary artery, and RLPA is right lower pulmonary artery) computed with the three-dimensional model. It yields the pressure values PLPA and PRPA in the two pulmonary branches corresponding to the assigned flow rates:
Pi = PLA + RLungQi (i = LPA, RPA) (A2.2)
These values are then used as new boundary conditions of the FEM three-dimensional model in the next FIDAP run.
Energy-loss indexes (for TCPC simulations only)
For a quantitative evaluation of the results in terms of the best hemodynamic design for the TCPC, two energetic indexes (related to the whole three-dimensional fluid dynamic domain) have been adopted. These are the hydraulic dissipated power (diss) and the total energy-loss coefficient (Ce, dimensionless coefficient). Total energy includes both kinetic and pressure energy. Their meanings are as follows:
diss =in -out
where in is rate at which total energy is carried by the fluid entering the model and out is rate at which total energy is carried by the fluid leaving the model.
and their mathematic expressions are as follows:
where Pi and Vi (i is LUPA, RUPA, LLPA, RLPA) represent the mean pressure and velocity respectively on each outlet section and PIVC, PSVC, QIVC, and QSVC represent the mean pressure and velocity, respectively, on each inlet section.
In the definition of the loss coefficient Ce, the kinetic energy rate entering the three-dimensional model (inlet sections of the two venae cavae) has been used to obtain a dimensionless coefficient. It is important to note that this amount of energy rate remains constant in all the TCPC simulations because both vessel diameters and inlet flow rates are left unchanged. This allows quantitative comparison of the different geometric designs for the TCPC.
Footnotes
From the Great Ormond Street Hospital for Children, NHS Trust,a London, and the Royal Brompton National Heart and Lung Institute,b and the Dipartimento di Bioingengneria, Politecnico di Milano,c Milan, Italy. 
Read at the Seventy-Fifth Annual Meeting of The American Association for Thoracic Surgery, Boston, Mass., April 23-26, 1995.
References
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G. Baslaim, A. Hussain, A. Kouatli, and A. Jamjoom Bovine valved xenograft conduits in the extracardiac Fontan procedure J. Thorac. Cardiovasc. Surg., August 1, 2003; 126(2): 586 - 588. [Full Text] [PDF]
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M. R. de Leval Beyond Flatland J. Thorac. Cardiovasc. Surg., January 1, 2003; 125(1): 12 - 19. [Full Text] [PDF]
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 S. Fratz, J. Hess, M. Schwaiger, S. Martinoff, and H. C. Stern More Accurate Quantification of Pulmonary Blood Flow by Magnetic Resonance Imaging Than by Lung Perfusion Scintigraphy in Patients With Fontan Circulation Circulation, September 17, 2002; 106(12): 1510 - 1513. [Abstract] [Full Text] [PDF]
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 E M Pedersen, E V Stenbog, T Frund, K Houlind, O Kromann, K E Sorensen, K Emmertsen, and V E Hjortdal Flow during exercise in the total cavopulmonary connection measured by magnetic resonance velocity mapping Heart, June 1, 2002; 87(6): 554 - 558. [Abstract] [Full Text] [PDF]
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 J. M. Chu, Q. Y. Wu, and W. M. Wang Pulmonary Blood Distribution After Total Cavopulmonary Connection Asian Cardiovasc Thorac Ann, December 1, 2001; 9(4): 282 - 285. [Abstract] [Full Text] [PDF]
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C. Mavroudis, C. L. Backer, B. J. Deal, C. Johnsrude, and J. Strasburger Total cavopulmonary conversion and maze procedure for patients with failure of the Fontan operation J. Thorac. Cardiovasc. Surg., November 1, 2001; 122(5): 863 - 871. [Abstract] [Full Text] [PDF]
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T. Tang, I.-S. Chiu, H.-C. Chen, K.-Y. Cheng, and S.-J. Chen Comparison of pulmonary arterial flow phenomena in spiral and Lecompte models by computational fluid dynamics J. Thorac. Cardiovasc. Surg., September 1, 2001; 122(3): 529 - 534. [Abstract] [Full Text] [PDF]
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