Computer Methods in Biomechanics and Biomedical Engineering

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Fractal network model for simulating abdominal and lower extremity blood flow during resting and exercise conditions To link to this article: D OI: 10.1080/10255840601068638 U RL: http://dx.doi.org/10.1080/10255840601068638

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Computer Methods in Biomechanics and Biomedical Engineering, Vol. 10, No. 1, February 2007, 39–51

Fractal network model for simulating abdominal and lower extremity blood flow during resting and exercise conditions Downloaded By: [Ste ele, Brooke] At: 14:25 6 F ebruary 2007

BROOKE N. STEELE†*, METTE S. OLUFSEN‡§ and CHARLES A. TAYLOR{k †Joint Department of Biomedical Engineering, NC State University & UNC-Chapel Hill, Campus Box 7115, Raleigh, NC 27695-7115, USA ‡Department of Mathematics, NC State University, Campus Box 8205, Raleigh, NC 27695-8205, USA {Departments of Mechanical Engineering, Bioengineering, and Surgery, James H. Clark Center, Room E350B, 318 Campus Drive, Stanford, CA 94305-5431, USA (Received 6 August 2006; in final form 26 September 2006)

We present a one-dimensional (1D) fluid dynamic model that can predict blood flow and blood pressure during exercise using data collected at rest. To facilitate accurate prediction of blood flow, we developed an impedance boundary condition using morphologically derived structured trees. Our model was validated by computing blood flow through a model of large arteries extending from the thoracic aorta to the profunda arteries. The computed flow was compared against measured flow in the infrarenal (IR) aorta at rest and during exercise. Phase contrast-magnetic resonance imaging (PC-MRI) data was collected from 11 healthy volunteers at rest and during steady exercise. For each subject, an allometrically-scaled geometry of the large vessels was created. This geometry extends from the thoracic aorta to the femoral arteries and includes the celiac, superior mesenteric, renal, inferior mesenteric, internal iliac and profunda arteries. During rest, flow was simulated using measured supraceliac (SC) flow at the inlet and a uniform set of impedance boundary conditions at the 11 outlets. To simulate exercise, boundary conditions were modified. Inflow data collected during steady exercise was specified at the inlet and the outlet boundaries were adjusted as follows. The geometry of the structured trees used to compute impedance was scaled to simulate the effective change in the crosssectional area of resistance vessels and capillaries due to exercise. The resulting computed flow through the IR aorta was compared to measured flow. This method produces good results with a mean difference between paired data to be 1.1 ^ 7 cm3 s21 at rest and 4.0 ^ 15 cm3 s21 at exercise. While future work will improve on these results, this method provides groundwork with which to predict the flow distributions in a network due to physiologic regulation. Keywords: One-dimensional model; Arterial blood flow; Fractal; Structured tree; Impedance; Exercise

1. Introduction Numerous models have been used to describe the dynamics of blood flow and blood pressure in the cardiovascular system. These models include simple Windkessel models (Pater and van den Berg 1964; Westerhof et al. 1969; Noordergraaf 1978), non-linear one-dimensional (1D) models (Stergiopulos et al. 1992; Olufsen et al. 2000; Wan et al. 2002) and complex three-dimensional (3D) models (Taylor et al. 1996; Cebral et al. 2003). Each class of models is suited to

answer a particular type of blood flow question. For example, the Windkessel can be used to describe the overall dynamics of blood flow in the systemic circulation (Olufsen et al. 2000; Olufsen and Nadim 2004) while spatial models (1D, 2D and 3D models) can describe blood flow and blood pressure through a given geometry. Spatial models span a limited region of interest. The remainder of the circulatory system is represented with a set of boundary conditions that are developed to approximate blood flow and blood pressure outside the modelled domain.

*Corresponding author. Tel: þ 1-919-513-8231. Fax: þ 1-919-515-3814. Email: [email protected] §Tel: þ1-919-515-2678. Fax: þ 1-919-515-3798. Email: [email protected] kTel: þ1-650-725-6128. Fax: þ 1-650-725-9082. Email: [email protected] Computer Methods in Biomechanics and Biomedical Engineering ISSN 1025-5842 print/ISSN 1476-8259 online q 2007 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/10255840601068638

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Downloaded By: [Ste ele, Brooke] At: 14:25 6 F ebruary 2007

To describe boundary conditions for spatial models, researchers often prescribe blood flow or pressure profiles (Taylor et al. 1999a). Although this approach is simple, specifying flow or pressure will influence the fluid dynamics inside the model domain and is only appropriate when profiles and distribution between outlets is known. Often complete boundary profile information is not available, so constant relationships between pressure and flow are used (Wan et al. 2002). Resistance boundary conditions provide a convenient method to specify a boundary relationship without prescribing a pressure or flow waveform. However, pure resistance boundary conditions cannot account for non-proportional variations between pressures and flow as observed in compliant vessels. An alternative to the constant resistance boundary condition is the impedance boundary condition, which is the frequency analogue to resistance. Impedance has long been recognized as an important tool for evaluating the reflections and damping of flow and pressure waves (Taylor 1966; Brown 1996; Nichols and O’Rourke 2005). Impedance boundary conditions are often implemented using simple three-element Windkessel model (Burattini et al. 1994; Manning et al. 2002). While useful, the Windkessel model has two limitations: (1) parameters cannot be specified as a function of model geometry; and (2) Windkessel models cannot account for flow and pressure wave changes including damping or amplification and dispersion that occur in a branched network of compliant blood vessels with spatially varying properties (Olufsen and Nadim 2004). An alternate method not subject to these limitations is to compute the impedance using a fractal network (Taylor 1966; Brown 1996; Olufsen 1999) representing the vascular bed. In this work, the objective is to extend the structured tree model developed by Olufsen (1999) to compute the impedances of vascular beds during rest and exercise. Short-term regulatory mechanisms in the body continuously alter the impedance of vascular beds to control the distribution of blood due to varying demands of organs and tissues. These regulatory mechanisms act on the vascular beds resulting in changes in vascular anatomy such as vasodilation or vasoconstriction and recruitment or closure of capillary beds by the opening and closing of pre-capillary sphincters. Following the onset of leg exercise, heart rate (HR) and cardiac output (CO) are increased and as a result, the aortic flow waveform is changed from tri- to bi-phasic as negative flows are eliminated. Impedance in the leg is decreased due to the dilation (3 – 5 times) of arterioles or recruitment of non-flowing capillaries to meet the metabolic demand of the active muscles. Meanwhile, the vascular beds that supply non-essential organs and inactive muscles reduce flow, using constriction of arterioles or pre-capillary sphincters to direct more of the CO to high-demand locations and maintain blood pressure. These impedance-regulating mechanisms can be incorporated by using geometric alterations in the structured tree impedance boundary.

A number of in vitro and numerical studies have been performed to visualize the changes in flow features in the abdominal aorta during exercise (Pedersen et al. 1993; Moore and Ku 1994; Boutouyrie et al. 1998; Taylor et al. 1999b). In these studies, the goal was to understand current hemodynamic conditions with a prescribed, known outflow condition. This method would not be suitable in determining the change in flow features following a change in the geometry of the modelled region. The ability to simulate both rest and exercise is desired because diagnostic data required for modelling is primarily collected with the patient at rest and symptoms of lower extremity vascular disease are most evident during exercise. One of the most pronounced symptoms of lower extremity vascular disease is claudication, pain in the thigh and buttock during exercise due to diminished capacity to deliver blood to active muscle. Currently, the success rate of relieving claudication is not easily predicted as it is related to the location and extent of disease, the ability of proximal vessels to supply blood to the region, and the capacity of distal beds to accommodate runoff. As a consequence of this difficulty, potential negative outcomes include: (1) patient may be required to undergo a re-do operation to relieve symptoms following an under aggressive treatment; (2) patient may not benefit due to being a poor candidate; or (3) patient may suffer unnecessary complications from overaggressive treatment. Computational modelling for surgical planning in the scenario above, with the ability to model the exercise state based on data collected during rest, may improve the success rate and reduce the risk to patients suffering from claudication. In summary, this paper shows how to model blood flow in large vessels during rest and exercise. We demonstrate the effect of changing inlet HR, CO, and the geometry of the structured tree attached at the outlet. This model is validated against non-invasively recorded phase contrastmagnetic resonance imaging (PC-MRI) flow data for eleven healthy subjects during rest and exercise.

2. Methods 2.1 Governing equations Axisymmetric 1D equations for blood flow and pressure can be derived by appropriately integrating the 3D Navier– Stokes equations over the vessel cross-section and neglecting in-plane components of velocity (Hughes and Lubliner 1973; Hughes 1974). This model is used to describe large vessels in which the blood flow is considered Newtonian, the fluid is considered incompressible and the vessel walls are assumed to be impermeable. We further assume that the velocity profile across the diameter of the vessels is parabolic (Wan et al. 2002). The resulting partial differential equations for conservation of mass (1) and balance of momentum (2)

Fractal network model for rest and exercise

are given by:

41

pressure and flow of the form:

›s ›q þ ¼0 ›t ›z ! " ›q › 4 q 2 s ›p q ›2 q þ þ ¼ 28pn þ n 2 : s ›t ›z 3 s r ›z ›z

ð1Þ ð2Þ


The primary variables are cross-sectional area sðz; tÞ (cm2), volumetric flow rate, qðz; tÞ (cm3 s21), and pressure, pðz; tÞ (dynes s21 cm22); z (cm) is the axial location along the arteries and t (s) is time. The density of the fluid is given by r ¼ 1.06 g cm23, the kinematic viscosity is given by n ¼ 0.046 cm2 s21. 2.2 Constitutive equation The above system has three variables, but only two equations. Hence, to complete the system of equations, a constitutive relationship is needed. In this paper, we have used a model that describes pressure p as an elastic function of the cross-sectional area s. This equation, derived by Olufsen (1999), is given by: sffiffiffiffiffiffiffiffiffiffi ! 4 Eh s0 ðzÞ pðsðz; tÞ; z; tÞ ¼ p0 þ 12 3 r 0 ðzÞ sðz; tÞ

ð3Þ

where p0 is the unstressed pressure, E (g s22 cm21) is Young’s modulus, h (cm) is the thickness of the arterial wall, r0(z) (cm) is the radius of the unstressed vessel at location z, and s0 (cm2) is the cross-sectional area of the unstressed vessel. Young’s modulus times the wall thickness over the radius is defined by: Eh ¼ k1 e k2 r0 ðzÞ þ k3 ; r 0 ðzÞ

ð4Þ

where k1 ¼ 2·107 g s22 cm21, k2 ¼ 2 22.53 cm21, and k3 ¼ 8.65·105 g s22 cm21 are constants obtained from Olufsen (1999). This elastic model is only an approximation and hence, it does not reflect the viscoelastic nature of arteries. 2.3 Initial and boundary conditions Initially, the cross-sectional area is prescribed from model geometry, and the initial flow is set to zero. Since the above system of equations is hyperbolic, one boundary condition must be specified at each end for all vessels. There are three types of vessel endings: inlets, outlets and bifurcations. At the inlet, we specify a flow waveform qð0; tÞ from data, and at the outlets, we use an expression for impedance obtained by solving the linearized version of the Navier –Stokes equations in the structured tree using an approach first described by Womersley (1955) and Taylor (1966). The impedance is computed as a function of frequency, v (s21). It provides a relation between

Pðz; vÞ ¼ Zðz; vÞQðz; vÞ , Qðz; vÞ ¼ Pðz; vÞYðz; vÞ;

where Yðz; vÞ ¼ 1=Zðz; vÞ:

ð5Þ

Pðz; vÞ; Qðz; vÞ; Zðz; vÞ and Yðz; vÞ are frequency dependent pressure, flow, impedance and admittance, respectively. Since these expressions are applied as outlet conditions, for each outflow vessel they are calculated at z ¼ L: For each outlet, the relation between variables expressed in the time domain and their counterparts in the frequency domain is found using the Fourier transform. Hence, time dependent quantities can be obtained using the convolution theorem, i.e.:

qðz; tÞ ¼

1 T

ð T=2

2T=2

yðz; t 2 tÞpðz; tÞ dt;

ð6Þ

where z ¼ L and y is admittance in the time domain. In our implementation, we evaluate the flow waveform by computing the flow at discrete time points using the form:

qðL; nÞ ¼

N21 X

yðL; jÞpðL; n 2 jÞ;

j¼0

ð7Þ

where N is the number of time steps per cardiac cycle and L is the length of the given vessel. Finally, bifurcation conditions are introduced to link properties of a parent vessel xp and daughter vessels xdi, i ¼ 1; 2. For each bifurcation three relations must be obtained, an outlet condition for the parent vessel and an inlet condition for each daughter vessel. One equation is obtained by ensuring that flow is conserved, and two other relations are obtained by assuming continuity of pressure: qp ðL; tÞ ¼ qd1 ð0; tÞ þ qd2 ð0; tÞ;

pp ðL; tÞ ¼ pd1 ð0; tÞ ¼ pd2 ð0; tÞ:

ð8Þ

Pressure losses associated with the formation of vortices downstream from the junctions are accommodated for by including a minor loss term applied in the proximal region of the junction vessels. For a detailed description, see Steele et al. (2003). To solve the system of equations (1) – (3) combined with the inlet condition, outlet conditions (7), and bifurcation conditions (8), we employ a space-time finite element method that include Galerkin least squares stabilization in space and a discontinuous Galerkin method in time. We use a modified Newton – Raphson technique to solve the resultant nonlinear equations for each time step (Wan et al. 2002).

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2.4 Impedance boundary condition for vascular networks

2.5 The geometry of the structured tree


Vascular impedance is the resistance to blood flow through a vascular network. During steady state (i.e. at rest or during steady exercise), impedance can be computed from the structured trees that represent vascular beds and used as an outlet boundary condition. The vascular impedance at the root of the fractal tree is obtained in a recursive manner starting from the terminal branches where pressure is assumed to be 0 mmHg (figure 2). Along each vessel in the structured tree, impedance is computed from linear, axisymmetric, 1D equations for conservation of mass and momentum (Olufsen et al. 2000). Linearized equations are appropriate for use in arteries with diameter smaller than 2 mm where viscosity dominates (Olufsen and Nadim 2004) and the nonlinear advection effects can be neglected as a first approximation (Womersley 1957; Atabek and Lew 1966; Pedley 1980). The details of this computation are given in Olufsen et al. (2000). Briefly, the input impedance is computed at the beginning of each vessel z ¼ 0 as a function of the impedance at the end of a vessel z ¼ L : Zð0; vÞ ¼

ig 21 sin ðvL=cÞ þ ZðL; vÞcosðvL=cÞ cosðvL=cÞ þ igZðL; vÞsinðvL=cÞ

ð9Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L is vessel length, c ¼ s0 ð1 2 F J Þ=ðrCÞ is the wavepropagation velocity, where: Zð0; 0Þ ¼ lim Zð0; vÞ ¼ v!0

8mlrr 2 J 1 ðw0 Þ þ ZðL; 0ÞF J ¼ wm J 0 ðw0 Þ pr30 ð10Þ

J0(x) and J1(x) are the zero’th and first order Bessel functions with w20 ¼ i 3 w and w 2 ¼ r 20 v=y : The compliance C is approximated as: C