Boundary conditions for hemodynamics: The structured tree revisited

doi:10.1016/j.jcp.2012.04.038

Journal of Computational Physics

Volume 231, Issue 18, 15 July 2012, Pages 6086-6096

https://doi.org/10.1016/j.jcp.2012.04.038 Get rights and content

Abstract

The structured tree boundary condition is a physiologically-based outflow boundary condition used in hemodynamics. We propose an alternative derivation that is considerably simpler than the original one and yields similar, but not identical, results. We analyze the sensitivity of this boundary condition to its parameters and discuss its domain of validity. Several implementation issues are discussed and tested in the case of arterial flow in the Circle of Willis. Additionally, we compare results obtained from the structured tree boundary condition to the Windkessel boundary condition and measured data.

Introduction

The human vascular system is comprised of ten of billions of vessels spanning several orders of magnitude in size. Modeling the entire system with high fidelity approaches from fluid dynamics and elasticity theory is thus not feasible. In many applications, only a specific area of the system is of prime interest. A small number of vessels are then chosen and modeled in detail while the remainder of the vascular network is accounted for through boundary conditions. We focus here on the design, analysis and implementation of conditions at the outlets on the boundary of the computational domain, i.e., in those vessels through which blood flows out of the computational domain into the “un-modeled” part of the vasculature.

Significant efforts are currently being devoted to the construction of appropriate boundary conditions for three-dimensional hemodynamics simulations, see for instance [18] or [30]. These issues are fundamental when dealing with a small number of relatively large vessels. For larger networks of smaller vessels, such as in the simulation of cerebral blood flow and perfusion, it has been recently confirmed that there is usually little to be gained by a full three-dimensional computation versus a much simpler one-dimensional approach [6]. The present paper investigates boundary conditions for such one-dimensional formulations.

A variety of boundary conditions have been proposed for one-dimensional models. One option is to prescribe the pressure at each outlet. Although simple to implement, this method has been reported to give physiologically incorrect results in certain cases [7]. Another option involves using an electrical circuit model analogy where pressure and flowrate are respectively thought of as voltage and current. Examples of this include a pure resistor model [24], as well as the 3-element RCR Windkessel model [1], [3], [21], [23]. Although each of these methods has been used with success in certain cases (see above references), they have the drawback of not being physiologically based. Additionally, they are sensitive to parameters (resistances and capacitance) which can not be directly measured and must be tuned to generate accurate model output, see for instance [3], [10].

The structured tree boundary condition or impedance boundary condition originates from the work in [28] and was developed by Olufsen [15], [16], [17], see also [26]. It remedies some of the undesirable aspects of the above boundary conditions. The vascular trees located downstream from the outlet vessels are assumed to have simple geometric structures – structured trees – subject to certain scaling laws. In addition, a number of drastic simplifying assumptions, including linearization of the governing equations, are made about the fluid dynamics in these trees. This allows for the analytical derivation of a time-dependent expression relating pressure and flowrate at outlets. This boundary condition is physiologically based and can be implemented at moderate computational cost.

The contributions of this paper are as follows. First, we propose in Section 2 a new derivation of the impedance boundary condition. Our approach is much simpler than Olufsen’s. The resulting condition is not equivalent but very similar to that of [15], [16], [17]. The key difference is that we first average the governing equations on cross-sections and then linearize as opposed to linearizing first and then averaging. Second, we perform, in Section 3, analytical and numerical dependence studies of the impedance boundary condition with respect to its defining geometrical parameters. Model output is found to critically depend on scaling law factors and minimum vessel radius in the tree. Third, Section 4 is devoted to issues related to the numerical implementation of the impedance condition. Its definition as a convolution integral raises both numerical and theoretical questions which are, to the authors’ knowledge, discussed here for the first time. We also propose an efficient simplified model allowing for the calculation of the initial flow history. Finally, numerical experiments are discussed in Section 5. Using the example of the Circle of Willis, we compare numerical results obtained with the structured tree boundary condition to numerical results obtained with the Windkessel condition and to data. Conclusions are offered in Section 6.

Section snippets

Geometry of the structured tree

The concept of structured tree in the arterial network, as described for instance in [16], assumes that vessels end by bifurcating into two daughter vessels. Vessels are taken to be cylindrical. For a parent vessel of radius $r_{p}$ , the radii of the daughter vessels $r_{d_{1}}$ and $r_{d_{2}}$ are determined by two scaling parameters ξ and η $\begin{matrix} r_{p}^{ξ} = r_{d_{1}}^{ξ} + r_{d_{2}}^{ξ}, \\ η = {(r_{d_{2}} / r_{d_{1}})}^{2}, \end{matrix}$ where η is the ratio of the cross-sectional areas of the two daughter vessels. The meaning of ξ is less obvious, although $ξ = 2$ corresponds to

Dependence on minimum radius

In [15], Olufsen performs numerical sensitivity studies with respect to the minimum radius, $r_{\min}$ , and concludes that the structured tree boundary condition is highly sensitive to this parameter. We now provide an analytical explanation of this fact by analyzing the average root impedance, ${\hat{Z}}_{0}^{root} (0)$ . Consider first that instead of terminating according to a minimum radius threshold, the tree terminates after a specific number of generations. An induction argument shows that the average root

Well-definedness of the structured tree boundary condition

The algorithm outlined in Section 2 allows for the calculation of ${\hat{Z}}_{k} = {\hat{P}}_{k} / {\hat{Q}}_{k}$ for each $k \in Z$ at outlet boundaries of the network. However, implementing these values as a boundary condition is not trivial. In [17], P and Q are described as being related by the following convolution $P (t) = \frac{1}{T} \int_{t - T}^{t} Z (τ) Q (t - τ) d τ,$ where Z is the function whose Fourier coefficients are ${\hat{Z}}_{k}$ . In practice, the integral in (17) is replaced by an N-point quadrature rule ([17] suggests a composite trapezoidal rule), and values of

Numerical experiments

We consider the numerical simulation of blood flow in the Circle of Willis, a ring-like structure of arteries in the human brain, see Fig. 4. Vessel radii and length measurements are taken from [3]. Within individual vessels, we use the system (8), (9) for the cross-sectional area, A, and the flowrate, Q. A characteristic study shows that, at standard operating regime, changes in cross-sectional area and flowrate propagate at speed $U \pm s$ , where $U = Q / A$ is the flow velocity and $s > | U |$ . In other

Conclusion

The structured tree boundary condition has the advantage of being physiologically based, unlike most of the other outflow boundary conditions used in hemodynamics. The analytical and numerical sensitivity studies from Section 3 show the solutions to critically depend on the minimal vessel radius $r_{\min}$ which is used in the construction of the impedance. In practice, $r_{\min}$ has to be considered as a parameter to be tuned through calibration rather than measured through clinical experimentation. This

Acknowledgments

The authors are indebted to Vera Novak for the use of the cerebral blood velocity data, as well as to Mette Olufsen and Tim David for helpful discussions and insights. They also acknowledge an anonymous referee for remarks that led to a better presentation of the results.

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¹: Partially supported by the National Science Foundation (NSF) through Grant DMS-0636590 and East Asia and Pacific Summer Institutes (EAPSI) award 1015642.

²: Partially supported by the National Science Foundation (NSF) through Grant DMS-0811150 and through the Statistical and Applied Mathematical Sciences Institute (SAMSI), Grant DMS-0635449.

View full text

Boundary conditions for hemodynamics: The structured tree revisited

Abstract

Introduction

Section snippets

Geometry of the structured tree

Dependence on minimum radius

Well-definedness of the structured tree boundary condition

Numerical experiments

Conclusion

Acknowledgments

J. Biomech.

Biophys. J.

Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi-symmetric vessels

Math. Methods Appl. Sci.

Blood flow in the circle of willis: modeling and calibration

Multiscale Model. Simul. SIAM Interdiscip. J.

The viscosity of blood in narrow capillary tubes

Am. J. Physiol. – Legacy Content

Analysis of a geometrical multiscale blood flow model based on the coupling of ODEs and hyperbolic PDEs

Multiscale Model. Simul.

Simulation of the human intracranial arterial tree

Phil. Trans. R. Soc.

Outflow boundary conditions for arterial networks with multiple outlets

Ann. Biomed. Eng.

Iterative Methods for Optimization