1D and Directed Continuum Models for Arterial Flows

Anne Marie Robertson
University of Pittsburgh

It remains computationally unfeasible to model the full unsteady, three-dimensional (3D) equations for blood flow in large sections of the circulatory system. As a result, one-dimensional (1D) and lumped parameter models play an important role in studies of the arterial systems. The use of 1D models has intensified recently, including clinical applications to medical problems such as a thoraco-thoraco aortic bypasses [1] and coronary arterial networks [2].

A variety of 1D models are in use, differentiated by the method of achieving closure of the governing equations. In classical 1D models, the closure approximations are obtained by assuming a particular purely axial flow, typically either a uniform or parabolic profile. We will discuss an alternative approach, in which the Navier-Stokes equations are taken as the “gold standard” for selecting these constants. Parameter identification is performed using analytical and numerical results for the Navier-Stokes equations for chosen benchmark problems. In an earlier work, we used this approach for 1D models of flow in slender bodies with Reynolds number (Re) less than or equal to 200 [3]. These results have been extended to a larger range of Re [4].

In the second part of the talk, attention will be turned to an extension of the classical 1D theory for Navier-Stokes fluids to include generalized Newtonian fluids. Particular emphasis is given to power-law models. The predictive capability of the 1D power-law model is evaluated for the clinically relevant benchmark problem of flow through tapered vessels. Significantly, judicious choice of the closure constants results in a very good agreement between the 1D model and the full 3D (axisymmetric) problem, over a wide range of Reynolds numbers (100-500).

The last part of the talk will be focused on alternatives to classical 1D models. We will discuss applications of a directed continuum theory for viscous fluid flow in pipes (see, e.g. [5],[6]) to arterial systems. While more complex than classical 1D models, director theory has a number of advantages including (i) the theory is hierarchical making it possible to increase the capabilities of the model by including more directors; (ii) the wall shear stress enters independently as a dependent variable; and (iii) there is no need to make somewhat ad hoc approximations about the nonlinear convective terms. Recent results in our group demonstrate the power of these 9-director models for studying arterial flows [3].

[1] BN Steele, J Wan, JP Ku, TJ R Hughes and CA Taylor, In-vivo validation of a one-dimensional finite-element method for predicting blood flow in cardiovascular bypass grafts, IEEE Trans. Biomed. Engrg. 50, p 649–656, 2003.
[2] NP Smith, AJ Pullan, and PJ Hunter, An Anatomically Based Model of Transient Coronary Blood Flow in the Heart, SIAM J Appl. Math., 62, p 990-1018, 2002.
[3] A.M. Robertson and A. Sequeira, A Director Theory Approach for Modeling Blood Flow in the Arterial System: An Alternative to Classical 1D models, M3AS, 15(6), p871-906, 2005.
[4] H Zakaria, AM Robertson, One dimensional models for arterial flow based on parameter identification using benchmark problems, Proceedings of BIO2006, 2006 Summer Bioengineering Conference, June 21-25, Amelia Island Plantation, Amelia Island, Florida, USA.
[5] AE Green and PM Naghdi, A direct theory of viscous fluid flow in pipes I. Basicgeneral developments, Phil. Trans. R. Soc. London A342, p 525–542,1993.
[6] D. Caulk and P. M. Naghdi, Axisymmetric motion of viscous fluid flow inside a slender surface of revolution, J. Appl. Mech. 54, p 190–196, 1987.

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