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The physical governing equations of fluid flow phenomena, the Navier-Stokes equations, were formulated in the 19th centrury [1,2]. For incompressible flows these reduce to a 4 scalar partial differential equations for the three velocity components (u, v, w) and static pressure (P), shown in Equation 1.
The first equation (1a) is the Continuity Equation governing conservation of mass. In a rigid control volume, incompressible mass flows in and out of it should remain constant. The last three equations (1b-1d) represent Newton’s 2nd law of motion which relates external forces (for a fluid element - pressure, body force, and fluid shear force) to acceleration. These forces are also related to the kinematic strain rates through the Constitutive Equations (For a Newtonian fluid shear stress is related to the velocity gradient or strain rate: ). Analytical solutions to these equations are still limited to a few simple cases and idealized flow regimes . A mathematical solution for the full-set of equations is one of the unsolved Millennium prize problems .
Computational fluid dynamics (CFD) is an engineering analysis technique which provides a numerical solution to the discretized versions of the Navier-Stokes equations. A solution to the Navier-Stokes equation provides the velocity and pressure at every point in a three-dimensional domain of interest, which in turn can be used to calculate various parameters including mechanical loading, drag force, energy loss, or wall-shear stress. The history of the CFD technique is in parallel to the development of computational hardware, originally fueled by the aerospace/mechanical applications. It can be traced back to the 1930’s. Discretization steps require a computational mesh, which divides the domain of interest (in our case, the vessel segment) into millions of small control volumes or elements. The discretized Navier-Stokes equations are then applied and solved for each element individually through sophisticated numerical methods and fluid flow solvers. This is an active and dedicated research area where the state-of-the-art can be grasped by several classical reference books [5-11] and dedicated journals [12-14].
For complex cardiovascular flow fields targeting surgical planning or physiological understanding our relatively recent review is still valid and provides a good bibliography on the limitations of CFD . There are three important aspects of a credible CFD model targeting an initially unknown/unexplored flow field (like patient-specific cardiovascular TCPC flows): (1) Code (Fluid Solver) verification, (2) Model verification (specific to the application) and (3) Experimental validation. Once all these three aspects are proven, for the application of interest, than the CFD model can be applied to a large number of models [16,17] (anatomies in our case) providing important quantitative and functional data. Code verification is to ensure that the numerical solver does not contain any programming errors and it behaves as expected for simple problems where an analytical solution is available for comparison. Model verification is application depended. It involves several auxiliary, time consuming tests to determine the effects of boundary conditions, mesh sensitivity (generally done in three grid sizes, should be local and global), solver parameters, convergence, order of accuracy and phase-errors in transient simulations to the computed flow fields. The last and arguably the definitive step is experimental validation, where the exact same geometry and flow conditions are replicated in the laboratory and detailed velocity and pressure measurements performed. These measurements are then quantitatively compared to the computed parameters. For experimental validation, particle image velocimetry is the most practical technique employed in our group to generate the in vitro velocity fields. For cases requiring fine time-resolved measurements laser Doppler velocimetry or hot-wire anemometry is preferred. The second and third requirements for a credible CFD model also implies that the large number of models should not deviate significantly from each other in terms of topology and flow regimes. For example, one should be cautious in applying a CFD model developed for a Fontan model to a Glenn stage anatomy without any additional experimental validation.
Power Loss Calculations
There are two primary methods employed for calculating the power loss of internal flows: by the control volume method and by the calculation of the dissipation function. The method employed in this study is the control volume method, as it has proved more robust in agreement with in vitro validation. If we are given the control volume illustrated below where the solid lines represent the TCPC region of interest and the broken line the control volume, then the power loss can be calculated by integrating the energy flux over the surface of the control volume.
where is the range energy loss, is the pressure at the volume surface, is fluid density, is the velocity component in the j direction, is the unit normal in the i direction, is the surface differential operator, v is the average velocity at the surface, and Q is the total flow rate at that surface. This has been described more thoroughly in previous work. In practical terms, the total energy for each surface is calculated by summing the inertial and static energy components at each surface. From this, the net power loss can be calculated by simply subtracting the outlet power from the inlet power:
where is the net rate of energy loss in the control volume.
Calculation of the Equal Vascular Lung Resistance Operating Point
In a patient, the pulmonary flow-split is imposed by the resistance encountered by the blood as it goes through the lungs. In order to assess the quality of the lung perfusion associated with each TCPC design a simple lump parameter model was implemented to incorporate the pulmonary vascular resistance (PVR). Given the small size of the capillaries, the linear Darcy’s model holds for the capillary lung flows:
where and are the left and right lung resistances, and are the flow rates through the LPA and the RPA, and, and are the pressures in the LPA, RPA and in the pulmonary venous return. Subtracting (4) from (3), we obtain the following relation:
The difference in pressure between the two pulmonary artery branches, , is known from the experimental measurements as a function of flow split,. Equation 5 can thus be solved with respect to the pulmonary flow split. The solution provides the amount of blood going to the left and right lung when the TCPC under study is coupled with the two PVR values and .  In this study we have focused on the physiological operating point of equal lung vascular resistance (EVLR) and the effects of exercise on this operating point. This is done by setinng both and to 2.0 Wood units.
 Navier, C. L. M. H., 1822: M´emoire sur les lois du mouvement des fluides. M´em. Acad. Sci. Inst. France, Vol. 6, 389–440.
 Stokes, G. G., 1845: On the theories of the internal friction of fluids in motion. Trans. Cambridge Philos. Soc., Vol. 8.
 Johnson RW, The Handbook of Fluid Dynamics, see Chapter 9: Incompressible Laminar Viscous Flows by I. Gursul.
 http://www.claymath.org/millennium/Navier-Stokes_Equations/ (Last accessed March 27, 2007 12:10am)
 J. H. Ferziger, and M. Peric, Computational Methods for Fluid Dynamics, 2001, Springer-Verlag.
 Date, A.N., Introduction to Computational Fluid Dynamics, 2005, Cambridge University Press, New York.
 Lomax, H., Pulliam T.H. and Zingg, D.W., Fundamentals of Computational Fluid Dynamics, 2001, Springer-Verlag, New York.
 Patankar, Suhas V., Numerical Heat Transfer and Fluid Flow, 1980, Hemisphere, Washington D.C.
 Anderson JD, Computational Fluid Dynamics: The Basics with Applications, 1995, McGraw Hill.
 Hoffmann KA, Chiang ST, Computational Fluid Dynamics Vol 1, 1998, Engineering Education System.
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 Pekkan K., Zelicourt D., Ge L., Sotiropoulos F., Frakes D., Fogel M., Yoganathan A. P., Flow Physics Driven CFD Modeling of Complex Anatomical Flows. -A TCPC Case Study, Annals of Biomedical Engineering, Vol. 33, Issue 3, March 2005.
 Cebral JR, Castro MA, Burgess JE, Pergolizzi RS, Sheridan MJ, Putman CM. Characterization of cerebral aneurysms for assessing risk of rupture by using patient-specific computational hemodynamics models. AJNR Am J Neuroradiol. 2005 Nov-Dec;26(10):2550-9
 Cebral JR, Castro MA, Appanaboyina S, Putman CM, Millan D, Frangi AF. Efficient pipeline for image-based patient-specific analysis of cerebral aneurysm hemodynamics: technique and sensitivity. IEEE Trans Med Imaging. 2005 Apr;24(4):457-67
 Liu Y, Pekkan K, Jones SC, Yoganathan AP. The effects of different mesh generation methods on computational fluid dynamic analysis and power loss assessment in total cavopulmonary connection. Journal of Biomechanical Engineering. 2004;126(5):594-603.
 de Zelicourt DA, Pekkan K, Wills L et al. In vitro flow analysis of a patient-specific intraatrial total cavopulmonary connection. Ann Thorac Surg. 2005;79(6):2094-102.
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