Comparison by Computerized Numeric Modeling of Energy Losses in Different Fontan Connections
Background Different surgical techniques for creating a Fontan circulation can be used. The option of including an atrium in the circuit, or the technique used for connecting the caval veins to the pulmonary artery in a total cavopulmonary connection, frequently is empirical and is based on personal experience and preference. The hemodynamic and energetic differences between the different circuits are small, and short-term results are comparable. However, small, energetic differences may have significant implications for the long-term follow-up. The finite element method allows a computer-based modeling of the flow dynamics and pressure losses. It permits comparison of different Fontan connections in a single patient with identical geometry and functional conditions.
Methods and Results We compared the atriopulmonary connection with different types of cavopulmonary connections, which differed in the degree of symmetry of implantation of both caval veins into the right pulmonary artery. Based on anatomic models and physiological flow dynamics, three-dimensional geometries and finite element meshes were created with patran; flows were calculated with polyflow (b), and results were visualized with data visualizer.
Conclusions The atriopulmonary connection produces higher energy losses than the cavopulmonary connection (±1 mm Hg at rest). The cavopulmonary connection is more efficient when the connection of the caval veins to the pulmonary artery is asymmetrical.
Different surgical techniques for creating a Fontan circulation can be used. If the right AV valve is not required in the circuit as in tricuspid atresia, the surgeon has the option to include the right atrium in the systemic venous system. The technique used for connecting the caval veins into the pulmonary artery in a total cavopulmonary connection frequently is empirical and is based on personal experience and preference. The hemodynamic and energetic differences between the different circuits are small, and short- and medium-term results are comparable.1 2 3 4 5 6 However, small, energetic differences can have important implications for the long-term follow-up of Fontan patients. To elucidate this question, a refined statistical analysis of long-term follow-up data of several hundreds of comparable patients will be required. To gain more insight into the energetic differences between the different types of surgical circuits, computer simulation of the connections and the calculation of energy losses can be helpful in clarifying the relative advantages and disadvantages of the different connections.
The finite element method7 allows a computer-based modeling of the flow dynamics and pressure losses and permits comparison of different Fontan connections in a single patient with identical geometry and functional conditions. The aim of this study was to compare the atriopulmonary connection (APC) with various geometrically different cavopulmonary connections (CPC) from an energetic point of view. We were especially interested in the systemic venous pressure differences and how these were determined by the geometry of the connections.
Creation of Geometrical Models
First the geometry of the veins, right atrium, and pulmonary arteries was modeled on a computer (Fig 1⇓). Therefore, angiographies were used to determine the geometry and relative sizes of superior and inferior caval vein, right atrium, pulmonary trunk, and proximal pulmonary arteries. Based on these data, a model of an APC and three different models of CPCs were generated. These three circuits differed in the symmetry of the connection of both caval veins to the pulmonary artery: A symmetrical crossing implies that the two veins form a perfect cross with the pulmonary artery; the two other crossings differed in the location of the inferior caval vein vis-à-vis the superior caval vein by shifting of the inferior caval vein to the left.
For modeling of the APC, we chose a spherical representation of the atrium. To ensure the similarity between the two models, we placed the modeled CPC at the exact location as the right atrium in the APC. For the nonpulsatile model with steady flow, the vessels were considered as rigid cylinders; this assumption does not influence the final flow and pressure results. These computer-generated models then were divided into discrete elements (hexahedral elements) to form a “finite element mesh.” This type of modeling is required because the complexity of the geometrical models and equations does not allow calculation of an analytic solution. Therefore, a numerical method was used with approximate results.7
The modeling and the finite mesh creation were performed using patran (PDA Engineering), a classic computer-aided design of software frequently used in the motor and aeronautic industries.
The boundary conditions must be defined to allow numerical calculation of the model’s physics: For the given size of the modeled veins, the inflow at the superior and inferior caval vein was set at 1 and 2 L/min, respectively. This is based on the assumption that in older children and adults, inferior vena caval flow contributes about two thirds and superior caval flow about one third of total venous return. The outflow was determined at the pulmonary artery outlets; the walls of all vessels were considered rigid if steady nonpulsatile flow passed through the system; plane of symmetry in the CPC then was determined. With the patran.msh generator, boundary sets were defined. The boundary conditions were defined with polydata (Polyflow).
The .msh file then becomes the input file for the polydata session. This file contains the finite mesh elements (nodes), their coordinates, and the indices of the nodes as solicited by a physical condition. Blood was defined in the mathematical model as a newtonian fluid because the viscosity is considered to remain constant for a given velocity. This is realistic in this geometrical model with wide vessels. The viscosity of blood μ was set at 4×10−3 N/m2, with a specific density of 1060 kg/m3. At the end of this session, the names of the input and output files were redefined as a .dat file, containing all information on the mesh, the fluid, the border conditions, and the file management.
The model was then calculated by polyflow version 3.2.0 or 3.2.2. This program was chosen because of the nonlinear Navier-Stokes equation system. polyflow is based on the finite element method. This consists of determining the unknowns, such as blood velocity and pressure, in accurate points of each mesh element called a node. The values in the neighboring areas then are further extrapolated from the calculations obtained in these nodes. Blood velocities and pressures over the whole domain can thus be determined. After about 20 hours of run time for each problem on a CONVEX machine of 25 MFlops, we obtained the result files. As a comparison, an 80486 processor at 66 MHz operates at a speed lower than 1 MFlop. To reduce the CPU calculation time, the vessels of the CPC connections were modeled in the same plane, providing a perfect plane of symmetry. This peculiarity allowed us to use half of a three-dimensional representation that reduced the calculation time and cost by a factor of 8.
The files then were exported in different files: .patr files were patran compatible, the .wave files were used by a color visualization program, wavefront (Wavefront Technologies), and .p3r results file contained the various numerical values of the velocities and pressures at the mesh nodes.
First, the data were visualized during a classic wavefront session with the .wave results files. The geometrical models were first set in a visualization box. The cut plane tool was applied to create a visualization plane inside the box. This plane coincides with the plane of symmetry in the CPC. In the APC, we created a first cut plane including both axes of the caval veins and a second one slightly inclined compared with the first plane through the axis of the pulmonary arteries. The velocities in meters per second and the pressures in pascals were visualized in those domains. The absolute pressure values were calculated by the program in relative values. The plots represent the pressure differences from one point to another.
To compare the exact values of the pressure losses in the four geometrical models, the .patr and .p3r result files were used. The differences were calculated between a point at the inlet of the caval vein and one at the outlet of the pulmonary artery. This difference represented the pressure drop along the connection. patran was used as a postprocessor for this procedure. The exact values of both node indices were extracted. The practical procedure consisted in creating new grids at the exact coordinates of the extraction pressure points. We then looked at the exact index of the closer node from that grid by zooming the screen. patran has a very efficient procedure to extract values from definite nodes of various result files.
polyflow is based on an iterative method with many result files. The Navier-Stokes equation system is highly nonlinear because of the inertia terms. However, multiplied by the volumetric mass, the impact of the inertia terms is proportional to the density. Therefore, the Navier-Stokes equations were applied by progressively increasing the importance of these inertia terms. The calculation was started with a small value for the specific density, which was increased during each iteration to reach the desired value of 1060 kg/m3 of blood. In our program, this value was reached after 20 iterations. The desired nodes are extracted from the .patr files and the corresponding pressure values from the .p3r files. The pressure value of the node at the right pulmonary artery is then subtracted from the pressure value of the node in the superior caval vein. The same procedure is repeated for each geometry.
All the preceding studies were generated with a nonpulsatile atrium in the APC because it was modeled like a rigid sphere. To model pulsatility, a second mesh was created representing the end-systolic phase of the atrium. The deformation was imitated by imposing a shift for each mesh node with a mathematical function. To model the pulsatility, the program passed progressively from the first geometry to the other in a fixed period of time. This wall movement generates blood flow velocity, which becomes a new border condition. The most important problem in modeling pulsatility was the choice of impedance values for the inflow and outflow of the caval veins and pulmonary arteries in order to account for the reflection and attenuation of the pressure waves. No data are available on the compliance of caval veins in a Fontan connection, which are chronically submitted to elevated venous pressures.
Flows and Velocities
On the basis of the modeled geometry and the border conditions, the mean calculated blood velocity in the caval veins was about 0.25 m/s, which is a realistic value. This suggests that the model functions as a good approach to the real situation in a Fontan circuit.
The flow maps (Fig 2A⇓) clearly show a nonuniform flow velocity across the connections. In the APC, flow is laminar in the veins, in the pulmonary arteries, and in the central flow inside the right atrium. Significant whirling with nearly no flow occurs at the borders of the atrium. At the bifurcation of the pulmonary trunk, a slight flow acceleration with recirculations could be observed.
In the symmetrical CPC (Fig 2B⇑) with the perfect cross, complete mixing of blood flow from the inferior and superior caval vein is present. Flow is nearly laminar, with some flow acceleration proximal in the pulmonary arteries. When the inferior caval vein is slightly shifted to the left, flow separation occurs with the direction of inferior caval flow to the left pulmonary artery. This flow separation is even more pronounced when the inferior caval vein is further shifted to the left. In the last two types of CPC connections, flow remains laminar throughout the largest part of the connection with a more marked flow acceleration and flow recirculations just distally of the crossing at the vessel wall opposing the blood inflow.
The pressure curves were obtained by polyflow, which uses the Navier-Stokes equation system with a progressive increase of the inertia terms. Therefore, at each iteration, the specific density of the modeled fluid was increased. For the CPC connections, a final density of 1060 kg/m3 was reached. The APC calculations were prematurely terminated because the nonlinear inertia terms showed a steeper increase as a result of the important recirculations in that model. Those recirculations cause a significant increase in the pressure loss curves, even at the lower specific density values. The nonpulsatile atriopulmonary connection clearly induces a higher pressure loss (about 1 mm Hg) across the connection when compared with the three different types of CPC models. When the different types of CPC connections are compared, the least pressure loss is noticed in the connection with the largest shift of the inferior caval vein, while the largest pressure drop is detected in the connection with the small displacement. The pressure drop in the perfect cross-model is intermediate. However, the pressure gradient differences between the different CPC connections are very small (about 0.2 mm Hg) (Figs 3⇓ and 4⇓).
It should be noted that the pressure losses on the graphs represent the difference between the calculated absolute pressure at the pulmonary artery outlet and at the caval vein inlet. When interpreted in energetic terms, two different components can be distinguished: an irreversible dynamic energy loss related to friction and the difference in kinetic energy of the fluid at the inlet and outlet of the system. We calculated the relative importance of both terms, and this showed that the energy loss due to friction is the most important component (data not shown). Moreover, as a parabolic velocity profile was imposed on the model, the kinetic energy term is comparable in the different models. This implies that comparison of the different models based on calculation of absolute pressure drop will give the same results when compared with calculations of dynamic pressure losses.
The data we have on the interposition of a pulsating atrium into the connection are still very limited because we have no controlled compliance data available, which are necessary for the various inflow and outflow impedance settings. In vivo measurements are currently in progress that would provide us with the data required for an accurate calculation. However, the limited calculations we could make using a few assumptions on the border conditions indicate that even the smallest degree of pulsatility increases the pressure loss across the APC. This causes a pressure gradient exceeding any pressure loss in a nonpulsatile type of connection (either APC or CPC).
In this study, the energy losses in different types of Fontan circulations were studied with the use of computer modeling. This approach allows the study of different types of connections in a single patient, in which factors extrinsic to the operative circuit are well controlled. From these computer-generated flow data, it can be concluded that a CPC connection causes lower energy losses compared with the nonpulsatile APC. Moreover, when different types of CPC connections are compared, it was calculated that the connection with the largest degree of displacement of the inflow of the inferior caval vein resulted in the best hemodynamic result. For a given cardiac output and pulmonary vascular resistance, a CPC with maximal flow separation of superior and inferior caval venous inflow will function at the lowest systemic venous pressure. Because most short- and long-term problems are related to systemic venous congestion, this study suggests the CPC with maximal displacement to be the better option when a Fontan circulation is considered. It should be stressed, however, that the differences between the different CPC connections are very small. Those small pressure differences could be clinically insignificant, although we believe that one should try to prevent energy losses as much as possible when designing the optimal technique. This is especially important when considering the response to exercise because any energy loss increases with increasing cardiac output. Indeed, increasing the input of the system corresponded with increased energy losses across the different CPC connections (data not shown).
Our data confirm the results of the in vitro experiments of de Leval et al8 in which the better hydrodynamic design of the CPC connection was demonstrated. Moreover, it may be a partial explanation for the trend that patients with a CPC have superior hemodynamics during submaximal exercise when compared with the APC patients.9
The differences between the different CPC models, with the best hydrodynamic profile in case of maximal offset of the caval flows, may have important clinical implications. A symmetric crossing has some theoretical advantages: In this connection, complete venous mixing occurs with the direction of superior and inferior caval venous blood to both lungs. This may be important for the distribution of a putative hepatic factor involved in the prevention of fistulae formation. Moreover, the surgery is theoretically easier, with a shorter bypass time, if the proximal superior caval vein end is used to direct the inferior caval flow to the pulmonary artery. However, in practice it is very difficult to create a perfect crossing three dimensionally, and a slight displacement of the two caval veins causes energy losses. Therefore, we believe that it is safer to promote the use of the connection with a larger displacement, which is hemodynamically more favorable.
In the calculations of the CPC connections, a simplification in the symmetry of the model is present because the superior and inferior caval vein are modeled in the same plane. This pecularity allowed a simplification of the calculations as the plane of symmetry permitted to calculate half of a three-dimensional model. It should be noted, however, that this simplification diverges from the actual three-dimensional CPC connection, in which inferior caval flow enters posterior to the superior caval flow. It is intuitive to assume that an eccentric inflow in the geometry will produce an increase in the flow recirculations with increased energy dissipation. Thus, use of the simplified model may actually underestimate the real energy losses in the complete three-dimensional model. Because the calculation time is proportional to the square of a characteristic mesh size multiplied by the number of variables, use of a complete three-dimensional model would increase calculation time by a factor of 8. Financial limitations restrained us from calculating such a complete three-dimensional model. An additional simplification is present in the APC model because the atrium does not really resemble the in vivo structure, with its multiple pectineal muscles and the atrial appendage. We believe, however, that this simplification does not critically influence the flow patterns within the nonpulsatile atrium. In the nonpulsatile model almost no flow, with a lot of whirling close to the borders of the atrium, was present. This is similar to the angiographic observations in vivo in which a pulsatile atrium is present. Because of the large number of simplifications in the pulsatile model, our data are unreliable, and a more sophisticated computer model of atrial contraction is required.
It should be stressed further that this study is only a strict mathematical approach to a very complex clinical problem in which some modeling and abstraction are implied. A patient cannot be reduced to a collection of abstract tubing on a screen. In a clinical setting, the problems are much more complex. So, although the energy profile is clearly not favorable, there are some clear advantages of an APC: Pulsatility may decrease pulmonary vascular resistance, which is an important determinant of a functioning Fontan connection.10 It is often claimed that the formation of pulmonary fistulae is enhanced by the absence of pulsatility (although the present evidence is not entirely convincing),11 12 13 14 so that an APC may yield better results in the long term by avoiding this problem. If, on the other hand, one can expect limited or no atrial pulsatility such as in atrial flutter, fibrillation, or hypotrophic right atrium (as in double-inlet left ventricle), there is clearly no theoretical advantage of an APC. These considerations all indicate that more than just the hemodynamic energy losses need to be taken into account when deciding about the superiority of one of the techniques. We believe, however, that for the long-term prognosis, the hemodynamic findings will be very important.
From this study it can be concluded that a CPC connection induces lower energy losses in a Fontan connection when compared with a nonpulsatile APC connection. Moreover, a CPC with a good flow separation between inferior and superior caval vein results in the lowest energy losses.
Dr Mertens is a research assistant for the Belgian National Fund for Scientific Research.
- Copyright © 1995 by American Heart Association
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