ORIGINAL ARTICLE

Dynamics of pulsatile flow in fractal models of vascular branchingnetworks

Anh Bui Æ Ilija D. Sutalo Æ Richard Manasseh ÆKurt Liffman

Received: 1 September 2008 / Accepted: 16 April 2009 / Published online: 26 May 2009

� International Federation for Medical and Biological Engineering 2009

Abstract Efficient regulation of blood flow is critically

important to the normal function of many organs, espe-

cially the brain. To investigate the circulation of blood in

complex, multi-branching vascular networks, a computer

model consisting of a virtual fractal model of the vascu-

lature and a mathematical model describing the transport of

blood has been developed. Although limited by some

constraints, in particular, the use of simplistic, uniformly

distributed model for cerebral vasculature and the omission

of anastomosis, the proposed computer model was found to

provide insights into blood circulation in the cerebral

vascular branching network plus the physiological and

pathological factors which may affect its functionality. The

numerical study conducted on a model of the middle

cerebral artery region signified the important effects of

vessel compliance, blood viscosity variation as a function

of the blood hematocrit, and flow velocity profile on the

distributions of flow and pressure in the vascular network.

Keywords Vascular � Fractal � Pulsatile flow �Branching tree

1 Introduction

The complex network of branching vascular systems can

be described in a physiologically meaningful way by using

a fractal scaling principle. This was first theoretically

substantiated by Murray [21] based on the assumption that

minimal biological work is needed to maintain the blood

flow at a required level in a typical vascular system.

According to the Murray’s law, all the bifurcation seg-

ments are self-similar with the relation between the inlet

and outlet segment radii defined by a power law with the

radius exponent approximately equal to 3:

rnin ¼ rn

out;1 þ rnout;2: ð1Þ

The work by Rossitti and Lofgren [28] confirmed that the

branching of the internal carotid artery system is governed

by the principle of minimal work with the radius exponent

being approximately 3. The recent studies, however,

suggested that the radius exponent can significantly vary

in a large vascular network and the range of this parameter

between 2 and 3 was assumed to be physiologically relevant

[16]. The work by Gafiychuk and Lubashevsky [8] found

that the approximately square radius exponent (n ^ 2)

(corresponding to preservation of the branching area) would

help to minimize wave reflection at the branching points,

which is important for waves traveling through large

systemic arteries (see also [36]). Olufsen et al. [23], using

the principle of minimum work, defined the optimal value

of n for laminar and turbulent flows to be approximately 3

and 2.33, respectively. The theoretical work by West et al.

[36] identified a step change of n from 2 (area-preserving

flow in major blood vessels) to 3 (Poiseuille-type flow in

small blood vessels) at a certain branching level that is just

after a few branchings from the largest systemic arteries

in humans. The latest study by Kaimovitz et al. [12]

investigating porcine coronary arterial trees, however,

showed the possible variation of the radius component

beyond the range 2–3.

A vascular system obeying Murray’s Law with the

branching exponent equal to 3 has a constant and equal

shear stress throughout its network. For such a system, both

A. Bui (&) � I. D. Sutalo � R. Manasseh � K. Liffman

Division of Materials Science and Engineering,

Commonwealth Scientific and Industrial Research Organisation

(CSIRO), PO Box 56, Highett, VIC 3190, Australia

e-mail: Anh.Bui@csiro.au

123

Med Biol Eng Comput (2009) 47:763–772

DOI 10.1007/s11517-009-0492-6

the non-pulsatile and the pulsatile flow resistance, R, can be

described by Poiseuille formula:

R ¼ 8ll=pr4 ð2Þ

with ll and r denoting blood viscosity and vessel radius,

respectively.

The work of Gafiychuk and Lubashevsky [8] suggested

that a vascular network satisfying Murray’s Law is devel-

oped through some adaptation mechanisms which governs

both short- and long-term changes of arterial diameters and

seems to involve the sensitivity to shear stress of the

endothelial cells. Karau et al.’s work [13] indicated that,

for a branching network with n having a heterogenous

distribution about the mean value, the average of the radius

exponent seemed irrelevant to the possible interdepen-

dency of vascular structuring and shear stress distribution.

In that work, the shear stress was also found to be non-

uniform in a vascular network with distributed n. In vivo

measurements by Reneman and Hoeks [27] suggested a

non-constant shear stress distribution along the arterial tree.

In contrast to the work by Murray, which considered only

the energy minimization of the blood flow metabolism, the

work by Taber [33] included the metabolism dissipation by

structural components (i.e., blood vessel walls) and found

that the shear is also non-uniform in the optimal geometry

of a vessel segment. The study by Liu and Kassab [17],

extending Taber’s theory for a single vessel segment to a

vascular tree, confirmed the validity of the previously

proposed morphology scaling relations with the total met-

abolic consumption found proportional to the vascular tree

volume.

The flows in vascular branching networks have been

studied in the past [7, 19, 22, 23]. However, most of these

studies considered the networks of small arteries as binary

asymmetric structured trees attached to the terminals of

large arteries [22, 23]. Therefore, the detailed structure of

the small arterial network and the flow inside it were

beyond the scope of these works. In the investigations by

Mayer [19] and Gabrys [7], the flows in different bifurca-

tion levels of detailed asymmetric branching networks—

constructed on the basis of the fractal scaling principle—

were modeled, but the important effect of flow pulsation

(and the arterial vessel compliance) was not treated.

The aim of this work is to develop a computational fluid

dynamics (CFD) model of pulsatile flow in the branching

tree model of the human cerebral vascular system. The

model has been constructed on the basis of a fractal

description of the complex vascular network and a math-

ematical representation of the flow dynamics in the

deformable segments of that network.

Unlike the above-mentioned works investigating the

flows in vascular branching systems, the flow model

developed in this work can predict the time-dependent

distributions of the flow and the pressure in the whole

hierarchy of vascular networks and their alterations in

response to the local structural or geometrical changes and/

or the variations of the boundary conditions. As such, the

proposed model can be used to study the transient blood

circulation in a normal operational vasculature and also in

a disease-affected circulatory system. Computational

investigations have been conducted using the developed

flow model to provide insights into the blood circulation in

the branching vascular systems and the factors which may

affect their normal function.

2 Modeling approach

2.1 Fractal branching tree models of cerebral

vascular systems

Using the fractal scaling concept, a branching tree model,

which has physiologically meaningful geometrical prop-

erties, can be created for any vascular branching system. In

this work, a three-dimensional (3D) branching tree model

of the cerebral vasculature has been constructed using the

Constrained Constructive Optimization method (CCO)

proposed by Schreiner et al. [14, 29, 30]. The vascular

branching tree models generated by the CCO method sat-

isfy bifurcation rules such as fractal scaling and conser-

vation of flow. The CCO method involves geometric and

structural optimizations which minimizes the tree volume

and ensures uniform blood perfusion in the designated

perfusion domain. Good agreement of the vascular

branching tree model generated by the CCO method and

experimental observations was previously reported (see

[14, 29, 30]).

In addition to the fractal scaling rule described in the

introduction, other optimization conditions are also applied

in the CCO method, such as the minimization of the total

volume of blood (or vascular volume) as well as the geo-

metrical and structural restrictions ensuring the physio-

logical boundary of the perfusion volume, the uniform

space-filling of the capillaries, and the non-degeneration of

each segment length [14].

More details about the CCO method and its recent

developments can be found elsewhere (e.g., [14, 29, 30])

and were not reproduced in this work.

A simplified morphology of the human brain vasculature

was considered with the radius exponent assumed to be

constant (n = 2.76) and the arterial networks were

assumed to be distributed uniformly in the brain volume.

The radius exponent of 2.76 was chosen inside the range

suggested by Li [16]. The branching tree models generated

for different parts of the cerebral vasculature, namely

anterior (ACA), middle (MCA) and posterior (PCA)

764 Med Biol Eng Comput (2009) 47:763–772

123

cerebral arteries, are shown in Fig. 1. The number of ter-

minal segments in each cerebral vascular territory has been

chosen proportional to its volumetric flow rate as given in

Table 1 [20]. Pressure at the terminal segments which

supplies their microcirculatory areas was assumed to be

similar to the capillary pressure and set as in the work by

Ursino and Lodi [35].

The self-similarity of the cat pial vasculature considered

at different scale levels, which indicates its fractal geo-

metrical characteristic, was demonstrated in the work by

Herman et al. [9]. The empirical studies of the rat and

human cerebral vasculatures by Lapi et al. [15] and Cassot

et al. [4] show that the cerebral morphometry obeys Hor-

ton’s Law [10], which, in the stream morphometry, governs

the number as well as other geometric characteristics of the

segments in the successive stream orders. The branching

tree models of the cerebral vasculature generated in this

work (Fig. 1) are also found to have the vessel diameter,

length and number approximately satisfying the Horton’s

Law. The number of segments per vessel order, N, is

compared well against the correlation derived by Schreiner

et al. [31] (see Fig. 2):

NðKstrÞ ¼ exp � lnðNtermÞmaxfKstrg

� ðKstr �maxfKstrgÞ� �

ð3Þ

where Kstr is the segment Strahler order and Nterm is the

number of terminal segments.

The mean flow rate and shear stress of the steady-state

Poiseuille flow in the MCA are calculated and shown in

Fig. 3. For this flow, the shear stress is determined by the

equation: s = 8Ql/(pr3) with Q indicating the flow in the

segment. The simulation results show the significance of

the fractal scaling parameter n on the flow in the branching

tree model, which is in qualitative agreement with the

analysis by Gabrys et al. [7]. The shear stress distribution is

more uniform for n = 3. However, significant fluctuation

around the mean shear stress value is seen near the segment

diameter of 0.5 mm. This statistical deviation from the

constant shear rate could be caused by the optimization

method and/or the averaging procedure which depends on

the number of the branches and their ordering method. The

deviation is expected to reduce with the increase in the

number of the branches and the branching tree size.

Once the branching tree model(s) of the cerebral blood

vessels is/are created, variations of the geometrical or

structural properties of the cerebral arteries caused by

patho-physiological disorders can be taken into account by

modifying the characteristics of the affected blood vessels.

2.2 Pulsatile flow in fractal models of vascular systems

Flow in a branching vascular network is affected by the

network geometry which is defined by the branching pattern

[5]. Distributions of pressure and flow rate in a tree-like

network can easily be defined if the branch deformation

(compliance) is ignored (see [19]). The numerical study of

flows in fractal models by Gabrys et al. [7] ignored blood

vessel compliance in describing the flow and pressure dis-

tributions in fractal models with different bifurcation

exponents. Their work suggested that the effects of flow

non-linearities caused by velocity profile development after

bifurcations in small diameter vessels are small. A more

detailed numerical model of blood flows in a branching tree

system was developed by Olufsen et al. [23]. In that work,

the pulsatile flows in a structured tree consisting of small

arteries and arterioles were investigated. Each tree segment

was considered as a one-dimensional (1D) straight pipe.

Although the blood vessel compliance was taken into

account, the non-linear inertia term in the flow equation was

neglected, so that an analytical solution of the flow could be

obtained. The blood flow in the network of large 1D

deformable arteries has been investigated by Sherwin et al.

[18, 32]. In these works, the wave propagation and reflec-

tion in the human arterial system were studied with the

Fig. 1 An elementary branch

bifurcation and the branching

tree models generated for the

human anterior, middle, and

posterior cerebral vasculatures

(the number of terminal

segments in each brain territory

is reduced by a factor of 10 to

assist the visualisation). The

figure does not include the

major arteries of the circle of

Willis

Med Biol Eng Comput (2009) 47:763–772 765

123

Galerkin numerical techniques, which were used to solve

the 1D mass and momentum conservation equation system.

In this work, a method has been developed to consider

the effect of blood vessel compliance. The blood is

assumed to flow in relatively small vessels, so that the

Womersley number, a, (defined in terms of the Reynolds

and Strouhal numbers as a ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pReSrp

¼ rffiffiffiffiffiffiffiffiffiffiffiffixq=l

pwith

the vessel radius r, the blood density q and viscosity l, and

the circular frequency of oscillation x) is small. For such a

flow, the effects of oscillatory wave propagation and

reflection can be ignored and the pressure–flow relation can

be described by the Poiseuille’s Law. As noted in the book

by Westerhof et al. [37], the above assumption is appro-

priate in most arteries of intermediate size.

Assuming that the distension of each blood vessel seg-

ment is driven by pressure, P, at the segment inlet, the

following flow balance can be derived for a straight blood

vessel segment as follows:

Qin ¼ Qout þ Qdeform ¼Pin � Pout

Rþ C

dPin

dtð4Þ

where C and R designate the vessel compliance and

resistance, respectively, and Q is the volume flow rate.

Table 1 Parameters of the cerebral vascular models

Cerebral

territory

Flow rate

mL/s

Perfusion

volume cm3Pterm, Pa

(mmHg)

Resistance 9 109,

Pa s/m3Number of term

segments

Total number

of segments

ACA 1.39 140 3,333 (25) 6.906 1,390 2,779

MCA 2.78 280 3,333 (25) 3.453 2,780 5,559

PCA 2.08 209.5 3,333 (25) 4.615 2,080 4,159

Fig. 2 The number of vessels and the average vessel diameter at different vessel Shahler order numbers. The analytical results were obtained

with Eq. 3

Fig. 3 Predicted mean flow rate and shear stress in the vessels of different sizes in the MCA under a steady-state flow condition

766 Med Biol Eng Comput (2009) 47:763–772

123

The vessel resistance is calculated using the Poiseuille

formula shown in Eq. 2. Assuming that the blood vessel

wall is a purely elastic material the vessel compliance is

obtained from the balance of the forces acting on the vessel

wall (blood pressure), and the tensile stress inside the

deformable wall, as follows [23]:

C ¼ dV

dP’ 3pr3

0l

2Ehð5Þ

with V the volume, r0 the diastolic vessel radius, h the

vessel thickness, and E the vessel Young’s modulus in the

circumferential direction (which is assumed to be

1.6 9 106 Pa for the MCA [1]). Following the works by

Olufsen et al. [22, 23], the blood vessel elastic modulus can

also be made dependent on the diastolic vessel radius as:

Eh

r0

¼ k1 expðk2r0Þ þ k3 ð6Þ

where k1, k2, k3 are the material constants [k1 = 2 9 106

kg/(s2m), k2 = -2.253 9 103 m-1, and k3 = 8.65 9 104

kg/(s2m)] which were obtained from data fitting [22].

At a branching point of an elementary bifurcation shown

in Fig. 1, the following flow balance equation can be

obtained:

ðC2 þ C3ÞdP0

dt¼ � 1

R1

þ 1

R2

þ 1

R3

� �P0 þ

P1

R1

þ P2

R2

þ P3

R3

:

ð7Þ

Equation 7 is applicable to the arteries and arterioles,

which are muscular vessels and able to dilate and contract

to regulate the blood flow. In the smaller capillaries, which

are responsible for microcirculation, the effect of vessel

compliance is negligible and the transient term in Eq. 7 can

be omitted. In this case, the differential equation 7 will

reduce to a linear algebraic equation.

An arbitrarily complex branching tree system can be

constructed from the above-described elementary branch

bifurcation models. A mixed system of differential alge-

braic equations (DAEs) and algebraic equations similar to

Eq. 7 can be derived to describe the inter-dependency of

pressures at the branching points (nodes) of the tree model.

This system of mixed differential and algebraic equations

can be solved using the MATLAB ODE software package.

The flow passing through each node can be determined

from the pressure drop and the resistance of the upstream

segment as defined in Eq. 4. The pressure distributions at

the root of the tree and the ends of the arterioles (terminal

segments) are to be prescribed as the boundary conditions.

In the cerebral blood circulation, anastomoses, i.e.,

blood vessels which connect the elements of one branching

tree or of different branching trees, are important features

of the brain vasculature and play an important role in blood

flow regulation and stroke development [3]. However, the

presence of anastomoses may alter the flow balance

equation 7 and they were not considered in this work.

3 Simulation results

The pulsatile blood flow in the arterial network has been

numerically studied in the works of Olufsen et al. [23],

Sherwin et al. [18, 32], and Huo and Kassab [11]. The

emphasis in those works was placed on the flow in the large

blood vessels. The small arteries were mostly treated in a

semi-analytical manner by employing the root impedance

defined for the whole structured tree of the small arteries.

In the present work, the pulsatile flow was calculated for

the whole branching tree starting from the root segments

down to the terminal segments. In the numerical modeling

below, the branching tree model of the MCA (see Table 1)

(generated with the branching radius exponent in Eq. 1

equal to 2.76) was considered, with the pulsatile pressure at

the tree root described by a double-sin wave as:

Proot ¼ 97þ 8:75 � sin½ð2t� 0:5Þp� � 6:25 � sinð4ptÞ: ð8Þ

The above pulsatile pressure imitates the cerebrospinal

fluid pressure waveforms proposed by Ursino [34]. The

maximum and minimum pressures are 110 and 84 mmHg,

respectively, with the pulse period equal to 1 s.

Pressure at the terminal arterioles was assumed to be

constant and equal to the human cerebral capillary pressure

(Pterm = 25 mmHg) [34]. The fluid and vessel structural

characteristics were specified on the case basis as described

below.

3.1 Effect of the vessel mechanical properties

on the flow

The change of blood vessel mechanical properties resulting

from aging, trauma, or diseases such as atherosclerosis or

aneurysm, involves stiffening or weakening of the vessel

wall, which may lead to abnormal vessel deformation and

shape. The effect of blood vessel compliance variation,

which can stimulate disease-affected circulatory systems,

can be investigated using the model developed in this work

(Table 2).

When vessel compliance is very small, Eq. 7 becomes

very stiff. As a result, the numerical method used to solve

this equation may become unstable unless a very small

time step is used. To remedy this problem, a MATLAB

stiff ODE solver, i.e., ode15s, was used in conjunction with

analytical evaluation of the Jacobian matrix qf/qP (f is the

right-hand side of Eq. 7). By default, the solvers in the

MATLAB ODE suite that are intended for stiff problems

approximate Jacobian matrices numerically. However, by

analytical evaluation of this matrix the reliability and

Med Biol Eng Comput (2009) 47:763–772 767

123

efficiency of integration can be improved. The MATLAB

ode15s is a variable-order multistep solver, which is suit-

able for stiff and/or differential-algebraic problems. For

each solution substep, the system of equations is solved

until the error at every solution point satisfies:

errorj j � max RelTol yij j;AbsTolð Þ; with yi being the ith

component of the solution vector y. In the simulations,

RelTol and AbsTol were set at 1 9 10-6 and 1 9 10-12,

respectively.

The results of computer simulations conducted on the

MCA fractal model are shown in Figs. 4 and 5. In the

simulations, the blood viscosity was assumed to be con-

stant and equal to 0.0036 Pa.s. Initially, the pressure in the

whole network was set at the terminal pressure, i.e., 25

mmHg. As can be seen from the figures, amplitudes of

pressure and flow oscillations in small arteries increase

with the decrease of the vessel compliance. In addition,

lowering vessel compliance appears to significantly reduce

transient time and peak flow rate (see Fig. 4), which are the

major factors affecting the transport and residence time of

materials in a branching tree network. The reduction of the

magnitude of pulsatile pressure in highly compliant blood

vessels can partly be explained by the loss of the flow

energy which is absorbed by the blood vessel wall

deformation.

A simulation, in which the compliance of all terminal

segments was assumed to be zero, was also conducted to

clarify the effect of mixed flexible and inflexible blood

vessels on flow and pressure distributions. The simulation

indicated that a solution could be obtained for the mixed

system of differential and algebraic equations as described

in Sect. 2.2. The computational prediction is identical to

the result shown in Fig. 5

3.2 Effect of the variable blood viscosity on the flow

For blood vessels smaller than 0.3 mm, the size of red

blood cells (6–8 lm) may have a significant effect on the

blood flow rheology [25] due to the segregation and sub-

sequent non-uniform distribution of the red blood cells

across the blood vessel cross-section. Due to the higher

concentration of the blood cells near the vessel centre, the

average red cell velocity is higher than the bulk blood

velocity. Consequently, a lower concentration of the blood

cells (and hematocrit level) is effectuated which is termed

the Fahraeus effect. The red blood vessel segregation in the

small blood vessels also leads to a rheology change and

dependency of the blood flow resistance on the vessel

geometry; this is designated as the Fahraeus–Lindqvist

effect. The reduction of the flow resistance in the small

arteries (arterioles) caused by the Fahraeus–Lindqvist

effect would significantly bring down the transmural blood

pressure and lower the force required to pump the blood

through the large number of the arterioles to adequately

perfuse the tissues. The Fahraeus and Fahraeus–Lindqvist

effects have been experimentally investigated by many

authors [24, 25] and also predicted in direct numerical

simulations by Bagchi [2].

Following the work by Pries et al. [25, 26], the blood

viscosity is made a function of the vessel diameter and

hematocrit as follows,

l ¼ lp 1þ g�0:45 � 1� �

� ð1� HdÞh � 1

ð1� 0:45Þh � 1� D

D� 1:1

� �2" #

� D

D� 1:1

� �2

; ð9Þ

where lp is the plasma viscosity, Hd is the discharge

hematocrit (i.e., the volume fraction of the red blood cells

in the blood), g0.45 is the relative effective blood viscosity

at Hd = 45%, which is given by

g�0:45 ¼ 6 � expð�0:085DÞ þ 3:2� 2:44

� expð�0:06D0:645Þ;

and h is defined as,

h ¼ ½0:8þ expð�0:075DÞ� � 1

1þ 10�11D12� 1

� �

þ 1

1þ 10�11D12

with D indicating the blood vessel diameter (in lm).

The above dependence of blood viscosity on hematocrit

level and vessel diameter features a gradual (up to 30%)

reduction of blood viscosity when the vessel diameter D

decreases to 43 lm and then rapid increase of blood vis-

cosity when D reduces further.

In the simulation with the results shown in Fig. 6, the

blood viscosity was described by Eq. 9. The plasma vis-

cosity was chosen so that the blood viscosity for vessels

of 1 mm in diameter or larger was approximately equal to

0.0036 Pa.s. The discharge hematocrit level was assumed

to be 45%. Compared to the corresponding case shown in

Fig. 5, this simulation, which takes into account the

variability of blood viscosity in blood vessels of different

sizes, indicates an increase of the flow rate in the whole

branching network and a decrease of pressure in the small

Table 2 Flow at simulation nodes

Node Flow ratio Bifurcation level Strahler order

N1 (root) 1 1 7

N2 0.38 7 6

N3 0.077 15 5

N4 0.024 24 4

N5 0.01 31 3

768 Med Biol Eng Comput (2009) 47:763–772

123

arteries. Since the smallest segment diameter of the

considered branching tree model was about 56 lm, both

of the above predictions can be explained by the overall

reduction of the resistance of the whole network with a

greater decrease in resistance occurring in the smaller

segments.

3.3 Effect of non-parabolic velocity profile on the flow

The assumption of Poiseuille flow leads to a parabolic

velocity profile in the tree segments. However, the flow in

the small and microscopic vasculatures can significantly

deviate from the parabolic profile due to a number of

factors, such as disturbance of the flow at the branching

points, flow pulsatility or the hematocrit level of the flow

(see Popel et al. [24], Gabrys et al. [7], and Bagchi [2]).

Following the work by Gabrys et al. [7], the non-parabolic

shape of the flow velocity can be taken into account by

using the Forchheimer equation as follows,

DP ¼ Pin � Pout ¼ RQþ BQ2: ð10Þ

In this equation, the second term in the left-hand side

describes the additional pressure drop caused by the non-

parabolic flow profile with factor B defined by

B ¼ bqpr4

ð11Þ

and b is approximately equal to 0.639 [7].

Equation 10 is a quadratic equation in the flow rate Q

which can be solved analytically. Using the flow rate-

pressure dependence which is the solution of Eq. 10, the

following flow balance equation can be obtained in place of

Eq. 7:

Fig. 4 Pulsatile pressure at the nodes at different tree levels (see Table 2)—vessel compliance equals 5 9 10-12 m3 Pa-1 (left) and 5 9 10-13

m3 Pa-1 (right), respectively

Fig. 5 Pulsatile pressure and flow at the nodes at different tree levels—size-dependent vessel compliance

Med Biol Eng Comput (2009) 47:763–772 769

123

C2 þ C3ð ÞdP0

dt¼ g1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ P1 � P0

f1

r� 1

� �

� g2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ P0 � P2

f2

r� 1

� �

� g3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ P0 � P3

f3

s� 1

!ð12Þ

with g and f expressed by

g ¼ 1

2

R

B¼ 4ll

bqand f ¼ Bg2 ¼ 1

2gR:

The system of non-linear differential equations created

by applying (12) to each branching points can also be

solved using the MATLAB ODE subroutines.

Although the study by Gabrys et al. [7] suggested the

non-parabolic velocity profile correction in the steady-state

flows had little significance, our simulation results for

pulsatile flow (Fig. 7) indicate considerable reduction of the

overall flow rate in response to the additional pressure drop

caused by the non-parabolic flow. A more complicated

relationship between flow rate and pressure drop is also

seen to substantially lower pressure at all levels of the

branching tree model. The decreases of the flow and

pressure are indications of significant increase of resistance

to the flow in the whole branching tree system, which

results from the additional pressure drop term in Eq. 10.

4 Discussion and conclusion

In this work, a model of the cerebral vascular network

structure and pulsatile flows within the brain has been

developed to investigate the transient pulsatile flows in the

fractal network of the cerebral vasculature. Unlike previous

work studying the flow in the vascular branching networks,

this work took into consideration the important effects of

blood vessel compliance and the variation of the blood

rheology, as well as the possible change of flow velocity

profile in small vasculature.

Based on the fractal scaling principle and by employing

the CCO method, the branching tree models for different

territories of the human cerebral vasculature have been

constructed. These models satisfied the fractal scaling law

as well as several other constraint and optimization criteria,

such as minimal vasculature volume and minimal network

resistance.

A mathematical model of pulsatile flows in a complex

dichotomous branching tree network has also been devel-

oped and was used to identify and study important factors,

which may affect the blood transport in the asymmetric

network of blood vessels of different sizes. The develop-

ment of the flow model was based on the assumption that,

in the small vasculature, the effects of oscillatory pressure

propagation and reflection can be ignored and, therefore,

that there is no phase shift between blood pressure and flow

rate fluctuations. While this model can not be employed to

simulate the flow in large blood vessels, it can be coupled

with a 3D CFD model to provide predictions for the whole

ranges of blood vessel sizes. Using pressures prescribed at

the root and the terminal segments of the vascular tree as

boundary conditions, the flow model was numerically

solved using the MATLAB ODE library with optimal and

robust solution procedures identified for different branch-

ing tree characteristics and flow conditions.

The numerical study conducted on a branching model of

the MCA region signified the important effects of vessel

compliance, flow viscosity variation as a function of the

blood hematocrit, and flow velocity profile on the distri-

butions of flow and pressure in the vascular network. In

particular, the increase of blood vessel compliance was

Fig. 6 Pulsatile pressure and flow at the nodes at different tree levels—effect of variable blood viscosity

770 Med Biol Eng Comput (2009) 47:763–772

123

predicted to reduce flow and pressure oscillations in the

branching tree, while the reduced blood viscosity due to the

Fahraeus–Lindqvist effect was seen to significantly boost

the flow in the whole vascular network. The effect of

additional pressure drop and the associated flattening of

velocity profile was shown to substantially diminish both

the flow and pressure in the branching vascular model

considered.

The combined models describing the cerebral vascular

branching networks and the pulsatile flow inside are cur-

rently restricted by several shortcomings which limit its

usefulness in modeling real-life problems. In particular, the

fractal models of cerebral vascular territories were con-

strained by the assumption about uniform distribution of

the terminal points in the brain volume, but there was not

any condition imposed on the branch distribution except

for the branch non-intersection requirement. In reality,

morphology of the cerebral vasculature is more heteroge-

neous with the distribution of blood vessels greatly

depending on their size (for instance, there is a clustering

of small arteries on the pial surface with the blood supply

to the brain substance provided by penetrating arterioles

[6]). While this shortcoming can be remedied by the

imposition of additional constraints on the branch distri-

bution during the vascular tree creation, the limitation

caused by neglecting anastomosis is harder to overcome, as

it requires the modification of the system of mass balance

equations. Currently, the models are undergoing further

development and will be used to study the transport of

materials in the complex network of cerebral vasculature.

The effect of anastomosis, which may have a great impact

on the redistribution of pressure and flow in the vascular

network, is also a subject worthy to be considered in future

work.

Acknowledgment The authors acknowledge the constructive com-

ments made by the reviewers.

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