Several approaches have been used in the literature to model blood flow in large vessels in the cardiovascular system. Modeling the pulsatile flow using Fourier analysis 2728 and using the mass and momentum conservation equations coupled with a state equation 6918 seem to be two of the most widely used approaches, while several other techniques, including flow modeling using closed-loop systems mimicking electrical circuits 2930, can also be found. Most physiological parameters (including the temporal variations in the cardiac cycle itself) are more directly applicable to time domain models and we believe that time domain analysis (as opposed to frequency domain analysis) would provide additional, easier-to-interpret information in terms of subject pressure and flow profiling, especially where genesis and progression of ischemic conditions would be studied. On a secondary note, clinicians and most non-technical personnel normally find it difficult to interpret results explained in the frequency domain and would prefer the more intuitive time-domain approach.

In this study, blood is assumed to be Newtonian fluid a common assumption in blood flow analysis in large to medium sized vessels 5916. A typical Reynolds number in the abdominal aorta is around 590 16 and this is well below the critical Reynolds number (which is generally considered to be 2300) above which the transition from laminar to turbulent flow usually occurs 31, therefore laminar flow is assumed throughout the study. Further, blood is considered to be an incompressible, homogeneous fluid with an axisymmetric flow and constant viscosity. Under these assumptions, using a cylindrical coordinate system (r, θ, x) where the x axis is aligned with the local vessel axial direction and assuming a zero velocity in the circumferential direction, the complete 3-dimensional Navier-Stokes equations can be reduced to a set of 1-dimensional flow equations:

∂ R ∂ t + V ∂ R ∂ x + R 2 ∂ V ∂ x = 0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabgkGi2kabdkfasbqaaiabgkGi2kabdsha0baacqGHRaWkcqWGwbGvdaWcaaqaaiabgkGi2kabdkfasbqaaiabgkGi2kabdIha4baacqGHRaWkdaWcaaqaaiabdkfasbqaaiabikdaYaaadaWcaaqaaiabgkGi2kabdAfawbqaaiabgkGi2kabdIha4baacqGH9aqpcqaIWaamaaa@444E@

and

∂ V ∂ t + 2 ( 1 − α ) V R ∂ R ∂ t + α V ∂ V ∂ x + 1 ρ ∂ p ∂ x = 2 υ R [ ∂ v x ∂ r ] R MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabgkGi2kabdAfawbqaaiabgkGi2kabdsha0baacqGHRaWkcqaIYaGmcqGGOaakcqaIXaqmcqGHsisliiGacqWFXoqycqGGPaqkdaWcaaqaaiabdAfawbqaaiabdkfasbaadaWcaaqaaGGaaiab+jGi2kabdkfasbqaaiabgkGi2kabdsha0baacqGHRaWkcqWFXoqycqWGwbGvdaWcaaqaaiabgkGi2kabdAfawbqaaiabgkGi2kabdIha4baacqGHRaWkdaWcaaqaaiabigdaXaqaaiab=f8aYbaadaWcaaqaaiabgkGi2kabdchaWbqaaiabgkGi2kabdIha4baacqGH9aqpdaWcaaqaaiabikdaYiab=v8a1bqaaiabdkfasbaadaWadaqaamaalaaabaGaeyOaIyRaemODay3aaSbaaSqaaiabdIha4bqabaaakeaacqGHciITcqWGYbGCaaaacaGLBbGaayzxaaWaaSbaaSqaaiabdkfasbqabaaaaa@636A@

where p, R, V, ρ and ν represent pressure, inner vessel radius, average velocity, blood density and blood viscosity respectively. The parameter α is used to specify the shape of the axial velocity profile, with α = 1 corresponding to a flat profile.

The right hand side of (2) can be determined by specifying an axial velocity profile in the x direction (v_x) of the form

v x = α 2 − α V [ 1 − ( r R ) 2 − α α − 1 ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG2bGDdaWgaaWcbaGaemiEaGhabeaakiabg2da9maalaaabaacciGae8xSdegabaGaeGOmaiJaeyOeI0Iae8xSdegaaiabdAfawnaadmaabaGaeGymaeJaeyOeI0YaaeWaaeaadaWcaaqaaiabdkhaYbqaaiabdkfasbaaaiaawIcacaGLPaaadaahaaWcbeqaamaalaaabaGaeGOmaiJaeyOeI0Iae8xSdegabaGae8xSdeMaeyOeI0IaeGymaedaaaaaaOGaay5waiaaw2faaaaa@4673@

This form in (3) was deemed suitable by Hunter 32 to give a compromise fit to experimental data obtained at various different points in the cardiac cycle. The form of the axial velocity profile with a value of α = 1.1, V = 200 mm/s and R = 3 mm is shown in Fig 4.

It should be noted that there are two singularities with this equation; when α = 1 and when R = 0. When α = 1 it is not physiological and results in a flow profile which is a step function with no flow at the walls and maximum flow just off the walls. The case where R = 0 corresponds to a fully collapsed or occluded vessel. Although rare, this is a condition that can occur physiologically and it can be represented in the model by decoupling a particular vessel segment and replacing the terminals with a no-flow boundary condition.

Further manipulation of equations (1) – (3) gives

∂ V ∂ t + ( 2 α − 1 ) V ∂ V ∂ x + 2 ( α − 1 ) V 2 R ∂ R ∂ x + 1 ρ ∂ p ∂ x = − 2 ν α α − 1 V R 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@66B5@

Equations (1) and (4) provide us with two equations for the three unknowns, P, R and V. A third equation can now be determined by taking the vessel mechanics into account, and in this study we have chosen a pressure-radius relationship of the form

p ( R ) = G 0 [ ( R R 0 ) β − 1 ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGWbaCcqGGOaakcqWGsbGucqGGPaqkcqGH9aqpcqWGhbWrdaWgaaWcbaGaeGimaadabeaakmaadmaabaWaaeWaaeaadaWcaaqaaiabdkfasbqaaiabdkfasnaaBaaaleaacqaIWaamaeqaaaaaaOGaayjkaiaawMcaamaaCaaaleqabaacciGae8NSdigaaOGaeyOeI0IaeGymaedacaGLBbGaayzxaaaaaa@3EFA@

where G₀ and β are constants defining a particular wall behavior and R₀ is the initial unstressed vessel radius. This was the empirical relationship originally deduced by Hunter 32 and then adapted by Smith et al 18 in their work. The chosen form assumes a purely elastic behavior of the arterial wall and closely agrees with the conclusions drawn by Saito et al 33 who concluded from their experiments that in large artery models, viscoelasticity may be neglected and the arterial walls may be considered purely elastic. Similar pressure-radius relationships have been proposed by Sherwin et al 6 and Olufsen et al 9 by assuming a purely elastic wall.

The governing equations cannot be solved analytically and the use of numerical techniques is required. The Two-Step-Lax-Wendroff finite difference method was selected as a suitable explicit scheme as it is second order accurate in both space and time while eliminating large numerical dissipations 1832.

Equations (1), (4) and (5) were then solved numerically using the above finite differencing technique for an N grid point arterial segment to determine the values (P, R and V) at each of the interior grid points (i = 2 to N-1, where i denotes a grid point) while a boundary scheme is required to determine the values at the two ends of the vessel segment. The specified boundary condition was chosen to be pressure, as opposed to velocity or flow pulses chosen by Parker et al 34 in their work, as pressure can be measured in a clinical environment and is also less sensitive to small measurement errors. Radius is simply a function of pressure via the constitutive equation (5) and by considering the characteristic directions along which information propagates in (x,t) space, an expression for velocity at the first and last grid points (i = 1 and i = N respectively) in a single vessel can be derived (refer to 18 for details).

Following the studies of 1832 G₀ was set to 21.2 kPa (158 mmHg) and β was set to 2.0 due to the nature of the arterial walls. A value of α = 1.1 was chosen to define the axial velocity profile. Blood density was assumed to be 1.05 gcm^-3 and viscosity was considered to be 3.2 cm²s^-1 (these parameter values are used for all simulations presented here).

In order to test our numerical scheme and its implementation, we simulated the flow in an approximately 55 mm long tapering section of the abdominal aorta (the chosen location was just below the SMA and slightly above the IMA since we needed a single vessel with no branching) and the initial conditions were set to pi0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGWbaCdaqhaaWcbaGaemyAaKgabaGaeGimaadaaaaa@308B@ = 12.5 kPa, Ri0=R0i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGsbGudaqhaaWcbaGaemyAaKgabaGaeGimaadaaOGaeyypa0JaemOuai1aaSbaaSqaaiabicdaWmaaBaaameaacqWGPbqAaeqaaaWcbeaaaaa@3539@ and Vi0=0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGwbGvdaqhaaWcbaGaemyAaKgabaGaeGimaadaaOGaeyypa0JaeGimaadaaa@3255@ for each grid point i. The initial radius was specified at various locations along the vessel (using the information extracted during the digitizing process) and the variation in radius along each segment between 2 specified locations was assumed to be linear. The pressure at the inlet was raised from 12.6 kPa to 14.6 kPa over 0.2 s (the approximate pressure change in the heart during the ejection phase shown in Fig. 7) and the spatial changes were plotted in Fig. 5.

To validate the above results, we also derived a steady state analytical solution using mass conservation. Considering the vessel area S = πR², a steady state expression can be derived from (2) and (3) where all ∂/∂t terms are set to 0.

α V d V d x + 1 ρ ∂ p ∂ x = − 2 π υ α α − 1 V S MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFXoqycqWGwbGvdaWcaaqaaiabdsgaKjabdAfawbqaaiabdsgaKjabdIha4baacqGHRaWkdaWcaaqaaiabigdaXaqaaiab=f8aYbaadaWcaaqaaiabgkGi2kabdchaWbqaaiabgkGi2kabdIha4baacqGH9aqpcqGHsislcqaIYaGmcqWFapaCdaWcaaqaaiab=v8a1jab=f7aHbqaaiab=f7aHjabgkHiTiabigdaXaaadaWcaaqaaiabdAfawbqaaiabdofatbaaaaa@4C36@

Using a constant flow rate Q = VS and considering that initial radius R₀ (hence S₀) to be a function of distance x, equation (6) can be modified to:

[ − α Q 2 S + G 0 β 2 ρ S 0.5 β + 1 S 0 0.5 β ] d S d x − G 0 β 2 ρ S 0 − 0.5 β − 1 S 0.5 β + 2 d S 0 d x = − 2 π υ α α − 1 Q MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWadaqaaiabgkHiTmaalaaabaacciGae8xSdeMaemyuae1aaWbaaSqabeaacqaIYaGmaaaakeaacqWGtbWuaaGaey4kaSYaaSaaaeaacqWGhbWrdaWgaaWcbaGaeGimaadabeaakiab=j7aIbqaaiabikdaYiab=f8aYbaadaWcaaqaaiabdofatnaaCaaaleqabaGaeGimaaJaeiOla4IaeGynauJae8NSdiMaey4kaSIaeGymaedaaaGcbaGaem4uam1aa0baaSqaaiabicdaWaqaaiabicdaWiabc6caUiabiwda1iab=j7aIbaaaaaakiaawUfacaGLDbaadaWcaaqaaiabdsgaKjabdofatbqaaiabdsgaKjabdIha4baacqGHsisldaWcaaqaaiabdEeahnaaBaaaleaacqaIWaamaeqaaOGae8NSdigabaGaeGOmaiJae8xWdihaaiabdofatnaaDaaaleaacqaIWaamaeaacqGHsislcqaIWaamcqGGUaGlcqaI1aqncqWFYoGycqGHsislcqaIXaqmaaGccqWGtbWudaahaaWcbeqaaiabicdaWiabc6caUiabiwda1iab=j7aIjabgUcaRiabikdaYaaakmaalaaabaGaemizaqMaem4uam1aaSbaaSqaaiabicdaWaqabaaakeaacqWGKbazcqWG4baEaaGaeyypa0JaeyOeI0IaeGOmaiJae8hWda3aaSaaaeaacqWFfpqDcqWFXoqyaeaacqWFXoqycqGHsislcqaIXaqmaaGaemyuaefaaa@7BCA@

Now considering the variation of R₀ to be linear with x between two grid points, i.e.,

R₀ = a + bx (8)

where a, b are constants that can be easily determined when radii at two locations along the vessel are known, and:

S 0 = π R 0 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGtbWudaWgaaWcbaGaeGimaadabeaakiabg2da9GGaciab=b8aWjabdkfasnaaDaaaleaacqaIWaamaeaacqaIYaGmaaaaaa@3503@

we get

S₀ = π(a + bx)² (10)

Differentiating (10) with respect to x gives:

d S 0 d x = 2 π ( a + b x ) b MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdsgaKjabdofatnaaBaaaleaacqaIWaamaeqaaaGcbaGaemizaqMaemiEaGhaaiabg2da9iabikdaYGGaciab=b8aWnaabmaabaGaemyyaeMaey4kaSIaemOyaiMaemiEaGhacaGLOaGaayzkaaGaemOyaigaaa@3EAF@

and (8) and (11) can now be substituted into (7) to derive an expression for the variation of S with x for a single vessel segment:

d S d x = π G 0 β ρ b [ π ( a + b x ) 2 ] − 0.5 β − 1 S 0.5 β + 2 ( a + b x ) − 2 π ν α α − 1 Q − α Q 2 S 2 + G 0 β 2 ρ S 0.5 β + 2 [ π ( a + b x ) 2 ] 0.5 β MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdsgaKjabdofatbqaaiabdsgaKjabdIha4baacqGH9aqpdaWcaaqaaGGaciab=b8aWnaalaaabaGaem4raC0aaSbaaSqaaiabicdaWaqabaGccqWFYoGyaeaacqWFbpGCaaGaemOyai2aamWaaeaacqWFapaCdaqadaqaaiabdggaHjabgUcaRiabdkgaIjabdIha4bGaayjkaiaawMcaamaaCaaaleqabaGaeGOmaidaaaGccaGLBbGaayzxaaWaaWbaaSqabeaacqGHsislcqaIWaamcqGGUaGlcqaI1aqncqWFYoGycqGHsislcqaIXaqmaaGccqWGtbWudaahaaWcbeqaaiabicdaWiabc6caUiabiwda1iab=j7aIjabgUcaRiabikdaYaaakmaabmaabaGaemyyaeMaey4kaSIaemOyaiMaemiEaGhacaGLOaGaayzkaaGaeyOeI0IaeGOmaiJae8hWda3aaSaaaeaacqWF9oGBcqWFXoqyaeaacqWFXoqycqGHsislcqaIXaqmaaGaemyuaefabaGaeyOeI0YaaSaaaeaacqWFXoqycqWGrbqudaahaaWcbeqaaiabikdaYaaaaOqaaiabdofatnaaCaaaleqabaGaeGOmaidaaaaakiabgUcaRmaalaaabaGaem4raC0aaSbaaSqaaiabicdaWaqabaGccqWFYoGyaeaacqaIYaGmcqWFbpGCaaWaaSaaaeaacqWGtbWudaahaaWcbeqaaiabicdaWiabc6caUiabiwda1iab=j7aIjabgUcaRiabikdaYaaaaOqaamaadmaabaGae8hWda3aaeWaaeaacqWGHbqycqGHRaWkcqWGIbGycqWG4baEaiaawIcacaGLPaaadaahaaWcbeqaaiabikdaYaaaaOGaay5waiaaw2faamaaCaaaleqabaGaeGimaaJaeiOla4IaeGynauJae8NSdigaaaaaaaaaaa@8E82@

This cannot be solved for x using standard analytical solution techniques and therefore a simple numerical integration scheme was implemented using an explicit Runge-Kutta formula of 4^th/5^th order. Both the analytical solution and the numerical solutions are plotted in Fig. 5, showing excellent agreement.

Following the analysis of Smith et al 18, a bifurcation in the arterial network is approximated using three short elastic tubes which are short enough to assume a constant velocity along them and zero losses due to fluid viscosity. No fluid is assumed to be stored within the junction. The grid points associated with each vessel segment are shown in Fig. 6.

Equations (1) and (5) can be manipulated (for R ≠ 0) to obtain the following expression:

∂ p ∂ t + 1 2 π R ∂ F ∂ x d p d R = 0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabgkGi2kabdchaWbqaaiabgkGi2kabdsha0baacqGHRaWkdaWcaaqaaiabigdaXaqaaiabikdaYGGaciab=b8aWjabdkfasbaadaWcaaqaaiabgkGi2kabdAeagbqaaiabgkGi2kabdIha4baadaWcaaqaaiabdsgaKjabdchaWbqaaiabdsgaKjabdkfasbaacqGH9aqpcqaIWaamaaa@44CD@

Applying the principle of conservation of momentum for tube a yields:

π R a 2 ( p a − p o ) = ∂ ( ρ l a π R a 2 V a ) ∂ t MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFapaCcqWGsbGudaqhaaWcbaGaemyyaegabaGaeGOmaidaaOGaeiikaGIaemiCaa3aaSbaaSqaaiabdggaHbqabaGccqGHsislcqWGWbaCdaWgaaWcbaGaem4Ba8gabeaakiabcMcaPiabg2da9maalaaabaGaeyOaIyRaeiikaGIae8xWdiNaemiBaW2aaSbaaSqaaiabdggaHbqabaGccqWFapaCcqWGsbGudaqhaaWcbaGaemyyaegabaGaeGOmaidaaOGaemOvay1aaSbaaSqaaiabdggaHbqabaGccqGGPaqkaeaacqGHciITcqWG0baDaaaaaa@4E51@

Expressions similar to (13) and (14) can be written for tubes b and c. Expanding these equations using a central difference representation about (k+1/2) time step and further manipulation (refer to 18 for derivation details) of the resulting difference equations give:

p a 1 k + 1 − p b 1 k + 1 − 2 Δ t ( L a F a 1 k + 1 + L b F b 1 k + 1 ) = − 2 Δ t ( L a F a 1 k + L b F b 1 k ) + p b k − p a k MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@7742@

and

p a 1 k + 1 − p c 1 k + 1 − 2 Δ t ( L a F a 1 k + 1 + L c F c 1 k + 1 ) = − 2 Δ t ( L a F a 1 k + L c F c 1 k ) + p c k − p a k MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@774E@

Equations (15) and (16) along with conservation of mass given by:

F a 1 k + 1 − F b 1 k + 1 − F c 1 k + 1 = 0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGgbGrdaqhaaWcbaGaemyyaeMaeGymaedabaGaem4AaSMaey4kaSIaeGymaedaaOGaeyOeI0IaemOray0aa0baaSqaaiabdkgaIjabigdaXaqaaiabdUgaRjabgUcaRiabigdaXaaakiabgkHiTiabdAeagnaaDaaaleaacqWGJbWycqaIXaqmaeaacqWGRbWAcqGHRaWkcqaIXaqmaaGccqGH9aqpcqaIWaamaaa@44A8@

form a system of three nonlinear equations which are then solved using a Newton-Rhapson iterative scheme which attempts to simultaneously satisfy Equations (15) – (17).

Flow was simulated for the aorto-iliac bifurcation and the resulting numerical values satisfied the conservation of mass constraint with a 0.02% error (see Table 2).

The two characteristic paths along which information propagates in (x,t) space for the governing equations are given by 18:

d x d t = α V ± [ α ( α − 1 ) V 2 + β G 0 R β 2 ρ R 0 β ] 0.5 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdsgaKjabdIha4bqaaiabdsgaKjabdsha0baacqGH9aqpiiGacqWFXoqycqWGwbGvcqGHXcqSdaWadaqaaiab=f7aHjabcIcaOiab=f7aHjabgkHiTiabigdaXiabcMcaPiabdAfawnaaCaaaleqabaGaeGOmaidaaOGaey4kaSYaaSaaaeaacqWFYoGycqWGhbWrdaWgaaWcbaGaeGimaadabeaakiabdkfasnaaCaaaleqabaGae8NSdigaaaGcbaGaeGOmaiJae8xWdiNaemOuai1aa0baaSqaaiabicdaWaqaaiab=j7aIbaaaaaakiaawUfacaGLDbaadaahaaWcbeqaaiabicdaWiabc6caUiabiwda1aaakiabcYcaSaaa@556F@

or

d x d t = α V ± [ α ( α − 1 ) V 2 + c 2 ] 0.5 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdsgaKjabdIha4bqaaiabdsgaKjabdsha0baacqGH9aqpiiGacqWFXoqycqWGwbGvcqGHXcqSdaWadaqaaiab=f7aHjabcIcaOiab=f7aHjabgkHiTiabigdaXiabcMcaPiabdAfawnaaCaaaleqabaGaeGOmaidaaOGaey4kaSIaem4yam2aaWbaaSqabeaacqaIYaGmaaaakiaawUfacaGLDbaadaahaaWcbeqaaiabicdaWiabc6caUiabiwda1aaaaaa@497B@

where c is the wave speed at zero mean flow.

For the numerical scheme to be stable, the numerical velocity (ΔxΔt MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabfs5aejabdIha4bqaaiabfs5aejabdsha0baaaaa@3272@) of the finite difference scheme has to be greater than the wave speed of the equations, or else errors will be introduced which will ultimately grow and make the solutions unstable. That is:

Δ x Δ t > | d x d t | MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabfs5aejabdIha4bqaaiabfs5aejabdsha0baacqGH+aGpcqGG8baFdaWcaaqaaiabdsgaKjabdIha4bqaaiabdsgaKjabdsha0baacqGG8baFaaa@3C16@

Substituting (18) into (19) we get:

Δ x Δ t > | α V ± [ α ( α − 1 ) V 2 + c 2 ]