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Fractal network model for simulating abdominal and
lower extremity blood flow during resting and exercise
conditions
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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
40
B. N. Steele et al.
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
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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).
42
B. N. Steele et al.
2.4 Impedance boundary condition for vascular
networks
2.5 The geometry of the structured tree
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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<
3s0 r 0
;
2Eh
ð11Þ
where s0 is the reference cross-sectional area, Eh=r 0 is
defined in equation (4), and g ¼ cC: The impedance at
v ¼ 0 can be found as:
Zð0; 0Þ ¼ lim
ð12Þ
v!0
where lrr ¼ L=r is the length-to-radius ratio described
below and viscosity, m ¼ 0.049 g cm21 s.
The small arteries that form the vascular bed are described
using a bifurcating self-similar tree characterized by three
parameters as described below (Olufsen 1999). The first
parameter describes the branching relationship across
bifurcations between the radius of the parent vessel r p and
the radii of the daughter vessels r di ; i ¼ 1; 2: There are two
methods for defining this relationship, the area ratio and
the power law. The area ratio, h, is given by:
h¼
r 2d1 þ r 2d2
r 2p
ð13Þ
The power law is defined by:
r kp ¼ r kd1 þ r kd2
ð14Þ
If k ¼ 2, then area will be conserved. Murray (1926)
studied the physiologic organization of the vascular
system and applied the principle of minimum work to
examine the correlation between structure and function of
the arteries. Murray derived that maximum efficiency for
blood flow is attained when the relationship between
flow and vessel radius is of the form q / r 3 ; or k ¼ 3:
However, observations show that k is neither constant nor
organ specific. It has also been suggested that k may vary
depending on the radius of the branches involved. Several
studies show that k varies from 2 to 3 with mean values
given by 2.5, 2.7 and 2.9 (Iberall 1967; Zamir 1999; Karch
et al. 2000). Zamir (1999) introduces the concept of
sub-ranges of vessels based on diameter. In this work, we
adopt this concept and use a three tiered structure with
k ¼ 2:5 for small arteries r , 250 mm, k ¼ 2:7 for
resistance arteries 250 mm , r , 50 mm; and k ¼ 2:9
for vessels r , 50 mm (table 1).
The second parameter, g, is known as the bifurcation
index or the asymmetry index. The asymmetry index
describes the relative relationship between the daughter
vessels:
g¼
r d1
:
r d2
ð15Þ
Assuming that r d1 # r d2 ; g is between 0 and 1. The
asymmetry index varies widely throughout vascular beds
and does not appear to be organ specific (Papageorgiou
et al. 1990; Zamir 1999). We chose to vary the asymmetry
ratio in the tiered system described above with g ¼ 0.4,
0.6 and 0.9 (table 1).
Table 1. Parameters used to describe the structured tree. The tree is divided into three levels as a function of the vessel radius (second column). For each
level, the parameters that describe the power exponent k (third column) and the asymmetry ratio g (last column) are varied.
Level
Small arteries
Resistance vessels
Capillaries
Radius
Power exponent
Asymmetry ratio
250 mm , r
50 mm , r , 250 mm
r , 50 mm
k ¼ 2.50
k ¼ 2.76
k ¼ 2.90
g ¼ 0.4
g ¼ 0.6
g ¼ 0.9
Fractal network model for rest and exercise
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Finally, the length of a given artery (between
bifurcations) can be expressed as a function of the mean
radius of the vessel. Iberall (1967) recommends the use of a
length-to-radius ratio, lr r, of 50. This conclusion was drawn
from analysis of data collected in several studies that
produced a range of estimates. Zamir (1999) suggests that
the mean lr r is 20 with a maximum of 70. Others (Suwa
et al. 1963; Iberall 1967; Zamir 1999) have shown that that
the lr r in the vascular bed is widely varied and that the value
is organ specific. We elect to use lrr as a mechanism to vary
the relative impedance between outlets.
The properties described above are used in an
asymmetric structured tree originally devised by Olufsen
(1999). The limbs of the structured tree are systematically
ordered to take advantage of pre-computed branches to
minimize the computational cost in arriving at the root
impedance. In the tree, the radii of successive daughter
vessels (rd1 and rd2) were obtained by introduction of
scaling parameters a and b for the radius of the root vessel
(rroot) such that:
r d1 ¼ ar root ;
i
r i; j ¼ a b
r d2 ¼ br root
j2i
and
r root
ð16Þ
for the radius of the ith daughter vessel in the jth
generation of the tree and i ¼ {0; 1; . . . ; j} (see figure 1).
The power law, area and asymmetry ratios are
combined to give:
h¼
¼
r 2d1 þ r 2d2
1þg
¼
r 2p
ð1 þ g k=2 Þ2=k
ðar root Þ2 þ ðbr root Þ2
r 2p
ð17Þ
and
a ¼ ð1 þ g k=2 Þ2ð1=kÞ
and
pffiffiffi
b¼a g
ð18Þ
Olufsen used an asymmetry ratio g ¼ 0:41; a power k ¼
43
2:76 and lr r ¼ 50: The terminal minimum radius
(300 2 100 mm) was used as the mechanism to vary the
relative impedance between outlets.
In our implementation of the structured tree, we
developed a three-tiered model with variables rroot and lrr
to mimic the behaviour of a specific vascular bed. The
tiers in table 1 were developed based on the literature
described above and to provide asymmetry. These
modifications to Olufsen’s structured tree allow the
extension of the structured tree to a minimum radius of
3 mm where pressure is set to be 0.0 mmHg. The extension
of the structured tree to include resistance vessels
facilitates the use of a scaling factor, described below to
model physiologic regulation. Although the apparent
viscosity of blood in microcirculation is a function of both
vessel diameter and hematocrit (Fahraeus-Lindqvist
effect) Pries et al. (1990), we do not include this variation
in viscosity in this impedance model.
2.6 Modeling exercise
During exercise, regional resistance vessels regulate the
blood supply to active muscle and non-essential organs.
To mimic this behaviour, the radii of the “resistance”
vessels ðr , 300 mmÞ were adjusted by a scaling factor,
f, to simulate the increase or decrease in effective crosssectional area of the vascular bed during exercise. If f , 1;
the effective vessel cross-section is decreased and if f . 1;
the effective cross-sectional area is increased. In order to
preserve the structure of the tree, the length and radii of the
vessels and number of generations of the tree are
determined before scaling vessel radii.
2.7 Application with non-invasively measured data
The method described above is validated using flow data
obtained non-invasively from 11 healthy subjects Taylor
et al. (2002). Through-plane flow velocities were acquired
using a Cine PC-MRI sequence on a 0.5 T open magnet
(GE Signa SP, GE Medical Systems, Milwaukee, WI,
USA). To minimize blood flow regulation associated
with digestion, subjects were instructed to fast 2 h before
scanning. Subjects were seated on a custom built magnetic
Table 2. Standard dimensions for idealized model. Columns describe
the large vessels in the idealized model with inlet and outlet radii and
vessel lengths (cm)
Vessel
Figure 1. Structured tree and sub-unit. The root of the tree is the
interface between the modeled domain and boundary condition. Radii of
vessels determined using scaling parameter a ib j2i. In the convention
shown, the leftmost daughter vessel assumes i ¼ j and the right most
daughter vessel assumes i ¼ 0.
Aorta
Celiac
Superior mesenteric
Renal
Inferior mesenteric
Iliac
Internal iliac
Femoral
Profunda
Inlet radius
(cm)
Outlet radius
(cm)
Length
(cm)
0.79
0.33
0.33
0.28
0.20
0.43
0.20
0.40
0.20
0.63
0.30
0.33
0.26
0.18
0.40
0.20
0.30
0.20
16.31
4.14
4.29
3.23
2.14
8.92
3.96
44.01
12.34
44
B. N. Steele et al.
Table 3. Estimated flow distribution expressed as a percentage of CO.
Structured tree boundary conditions were created for each outlet using
the specified length-to-radius ratio. This uniform set of parameters was
used for each subject. Target distribution in extremities estimated from
measured data and physiologic blood pressure. Symmetry was assumed
in right/left-paired vessels (renal, profunda, internal iliac, and femoral).
Outlet
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Celiac
Superior mesenteric
Renal (2)
Inferior mesenteric
Internal iliac (2)
Profunda (2)
Femoral (2)
Target CO (%)
Length-to-radius ratio
14
12
23
4
4
4
5
26
38
21
24
60
60
80
resonance (MR) compatible cycle ergometer with their
torsos in the field of view of the magnet. To minimize
movement, the subjects were securely strapped to the seat.
PC-MRI images of velocity and cross-section were
captured at the supraceliac (SC) and infrarenal (IR) levels
of the abdominal aorta at rest and during moderate cycling
exercise. The velocities were integrated over the crosssectional area of the vessel to compute the flow rate. The
single cardiac cycle produced by PC-MRI is a composite
of many gated cardiac cycles. Patient specific anatomy
and pressure data were not acquired.
In order to develop subject specific geometric models, an
existing geometry of an idealized abdominal aorta with
major branching vessels (Moore and Ku 1994) was scaled to
match measured SC and IR cross-section areas. The
idealized model includes one inlet vessel and 11 outlet
vessels as shown in figure 2 and in table 2. The scaling was
performed using an allometric scaling law (West et al. 1997):
R ¼ R0 M b
ð19Þ
where R represents the desired radius (cm), R0 is the known
scaling constant (cm), M represents the body mass and b is
the scaling exponent. The measured cross-sectional area of
the aorta for one test subject matched the aorta of the
idealized model. This subject was assumed “ideal” and the
SC radius was used as the scaling constant Y0. This subject’s
body mass was designated M0. In order to balance the units,
M ¼ M i =M 0 where Mi is the test subjects body mass. Using
the measured radius from the PC-MRI slice obtained at the
SC aorta (R) during diastole, the size of the idealized aorta
(R0), and the mass for each subject (M), the scaling exponent,
b, was found to be 0.385 for the measured data. This scaling
exponent was then used to scale the idealized model for all
remaining vessels. The scaled idealized models represent the
large vessels in the computational domain including the
aorta (inlet), celiac, superior and inferior mesenteric, renal,
iliac, internal iliac, femoral and profunda arteries (see figure
2 and table 2).
Next, boundary conditions were determined. The inlet
boundary condition for each subject was specified from
the SC flow waveforms measured using PC-MRI. All
outlet boundary conditions were specified using the
modified structured tree with root radius set to the model
domain boundary radius and neutral tone, f ¼ 1:0:
Initially, visceral outlet boundaries were assigned lrr ¼
20 and leg outlet boundaries were assigned lrr ¼ 70: The
lrr values were adjusted so that the distribution of flow to
the viscera matched distributions reported in the literature
and to maintain a physiologic blood pressure of
approximately 120/70 mmHg (see table 3). For normal,
healthy individuals, it is assumed that approximately
20 –27% of CO flows through the renal arteries (Ganong
1995) and approximately 27% flows through the celiac,
superior mesenteric artery (SMA), and inferior mesenteric
artery (IMA) combined. The resting, fasting mesenteric
flow can be further estimated as 10% of the CO to the
celiac artery, 13% to the SMA, and 4% to the IMA
(Ganong 1995; Perko et al. 1998). Because CO was not
measured, each subject’s blood volume in litres was
estimated as 7% of body mass (self reported) in kg. This
value was used to estimate resting CO by assuming the
entire blood volume is circulated in 1 min (table 4). Using
this target CO, lrr values were determined for all outlets
(table 3). These outlet boundary conditions were used to
perform resting flow analysis for all subjects.
Finally, steady exercise was simulated by modifying the
boundary conditions. The inlet boundary condition for each
subject was specified using the SC flow measured using PCMRI during exercise. The outlet boundary conditions were
modified by specifying a scaling factor, f, as described
above. To determine appropriate factors, a series of
analyses were performed on one data set. Initially,
peripheral beds were dilated by a factor of 5 as described
in Ganong (1995) to increase flow to the active muscles
Table 4. Estimated resting CO based on weight and HR. Estimated SC aortic flow based on target percentage (66%) of estimated CO. Measured SC
flow and error between measured and estimated SC flow.
Subject
1
2
3
4
5
6
7
8
9
10
11
Weight (kg)
HR
Estimated CO (l min21)
Estimated SC (l min21)
Measured SC (l min21)
Error (%)
63.64
56.82
70.45
60.00
81.82
75.91
63.64
54.55
54.55
63.64
83.18
72
75
80
80
70
75
80
80
65
63
70
4.45
4.14
5.48
4.67
5.57
5.54
4.95
4.24
3.45
3.90
5.70
2.94
2.73
3.62
3.08
3.68
3.65
3.27
2.80
2.28
2.57
3.76
3.28
3.30
2.86
3.46
3.32
3.42
2.25
2.96
2.40
2.60
2.87
10
20
214
11
211
27
245
5
5
1
231
Fractal network model for rest and exercise
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Figure 2. Computational domain for the larger arteries. This model
represents an idealized aorta and major branching vessels. Branching
vessels include celiac, SMA, IMA, renal, internal iliac, profunda and
femoral arteries. Note that the femoral arteries have been truncated in this
image. Dimension for each of these vessels for a standard subject is
shown in table 2.
in the legs. After a cursory study, it was found that f ¼ 5:0
is the maximum effective dilation factor and alone did not
create a large enough pressure drop to draw an appropriate
amount of flow away from the viscera (as measured with
45
PC-MRI) through the IR aorta. To decrease flow through
the visceral vessels, a constriction factor was applied
uniformly to all of the viscera beds in increments of 0.1
until the appropriate mean flow was computed in the IR
aorta. This method provided a reasonable result, however,
other combinations of parameters may provide similar
results. Using this method, a constricting factor of f ¼ 0:7
and a dilating factor of f ¼ 5:0; applied to the viscera and
lower extremity outlets respectively, were found to provide
the appropriate flow redistribution, matching the measured
IR aortic flow and approximating physiological pressures.
This scheme was then applied to all data sets.
3. Results
3.1 Rest
Pressure, flow and cross-sectional area were computed in
the large arteries of the abdomen and legs for 11 healthy
Figure 3. Flow comparisons for healthy young adults at rest for one cardiac cycle. Graphs show comparison between measured (thin line) and
computed (bold line) IR aortic flow.
46
B. N. Steele et al.
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body weight for subjects 3, 5, 7 and 11 (table 4). The
boundary condition scheme was not altered for these
subjects. Although this reduced level of flow would
undoubtedly result in a physiologic change in distribution,
i.e. less flow to non-essential organs and resting muscle,
there does not seem to be a large difference between
measured and predicted infra renal flow.
In figure 6, computed inlet pressures are shown with the
corresponding measured inlet flow waveforms. As
described previously, measured SC inlet flow is lower than
estimated for several subjects and the boundary condition
scheme was not altered for these subjects. Consequently, the
computed pressures are lower than expected in subjects
with lower than expected measured flow.
3.2 Exercise
Figure 4. Scatter plot comparison between predicted and measured flow
(cm3 s21) waveforms. The identity line is shown. A correlation
coefficient of 0.94 was computed.
subjects. For each subject, the model geometry (see figure 2)
was based on an idealized model scaled for each subject
using body weight and an allometric scaling law. Inlet
boundary conditions were obtained from non-invasively
measured data and outlet boundary conditions were
specified using structured trees with parameters based on
approximations from literature (see table 3). PC-MRI was
used to acquire non-invasive flow measurements for each
subject at the IR aorta. Paired waveforms of computed and
measured IR flow for each subject are shown in figure 3.
A scatter plot of all paired data points is shown in figure 4.
While overall means between paired measured and
computed data are within 1.1 cm3 s21, the computed
waveforms tend to overestimate the amplitude of the
waveforms with a standard deviation of 7.00 cm3 s21. Mean
flow distribution through each outlet is shown in figure 5.
Gross distributions between the viscera and legs were
determined by comparing SC and IR flow and were
measured to be 30 ^ 6% and computed to be 28 ^ 2%. The
measured SC inlet flow was less than estimate based on
The computational models are modified to reflect the
exercise state by adjusting the inlet boundary conditions to
match measured data, with increased abdominal aortic
flow rate and a shorter cardiac cycle. The outlet boundary
conditions are modified to reflect constriction (f ¼ 0:7) of
the visceral beds and dilation (f ¼ 5:0) of lower extremity
vascular beds. Paired waveforms of computed and
measured IR flow during lower extremity exercise are
shown in figure 7. Figure 8 shows a scatter plot of all paired
data points. The mean difference between paired data
is 4.07 cm3 s21 with a standard deviation of 15 cm3 s21.
While overall agreement is good, the computed waveform
for subject 11 is underestimated. The computed mean
outlet flow distribution during exercise is shown in figure 9.
Gross distributions between the viscera and legs were
determined by comparing SC and IR flow and were found
to be 79 ^ 7% during exercise and computed to be
82 ^ 1%. In figure 10, the computed inlet pressures are
shown with the corresponding measured inlet flow
waveforms.
4. Discussion
In this paper, we have described a method for implementing impedance boundary conditions in a 1D fluid dynamic
model using scaleable structured trees to represent
Figure 5. Computed mean flow values at outlet boundaries at rest as determined by uniform outlet boundary conditions. Sum of flow indicates mean
measured superceliac flow for each subject. Paired arteries (renal, femoral, internal iliac and profunda) are summed.
Fractal network model for rest and exercise
47
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Figure 6. Measured inlet flow boundary condition (cm3 s21) (thin line) and computed inlet pressures (mmHg) (bold line) at rest for one cardiac cycle.
downstream vascular beds. The structured trees were
scaled to simulate changes in the effective cross-sectional
area of the resistance vessels during exercise. Using this
method, we were able to predict changes in blood flow
waveform and distribution. This model was validated by
comparing computed flow waveforms to non-invasively
measured data from 11 healthy subjects at rest and during
moderate cycling exercise. Scatter plots and flow waveforms demonstrate good agreement between computed
and measured data. This study lays the groundwork for
modelling the redistribution of flow due to changes in
peripheral impedance resulting from physiologic
regulation.
4.1 Model limitations and future work
As with any model study, this investigation was subject to
errors associated with experimental methods and modelling assumptions. The experimental method used to
measure blood flow, PC-MRI, may have contributed to
observed difference between computed and measured
values. Although PC-MRI is considered the “gold
standard” for non-invasive measurement of blood flow,
errors associated with this technique are estimated to be
as much as 10– 20% (Taylor et al. 2002). This uncertainty
makes it difficult to quantify the error associated with the
computational studies. The MR sequence used for this
study involved constructing the recorded cardiac cycle
from multiple cycles acquired over approximately 30 s.
Beat-to-beat changes in HR and short-term regulation over
the recording timeframe are not considered. The inherent
error between the MRI flow measurements and assumptions are demonstrated in table 4. The estimation of flow
does not consider the gender or ratio of body fat to lean
muscle. Both of these factors will affect resting and
exercise physiology.
Modelling errors may result from assumptions made
while describing the structured trees, creating the
anatomic geometries, and specifying model parameters.
First, when specifying the boundary conditions, subjects
48
B. N. Steele et al.
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Figure 7. Flow comparisons for healthy young adults during exercise for one cardiac cycle. Graphs show comparison between measured (thin line) and
computed (bold line) IR aortic flow. Resting impedance boundary conditions were modified using a uniform exercise factor: f ¼ 0:7 for all viscera and
f ¼ 5:0 for all lower extremity structured trees.
were assigned a flow distribution (table 3). Distribution
is difficult, if not impossible, to quantify as short-term
regulatory effects may change flow through parallel
networks on a beat-to-beat basis. Gross distributions
between the viscera and legs were quantified by
comparing SC and IR flow. During the exercise protocol,
subjects were asked to exercise to a level of approximately
1.5 times their resting HR. However, change in HR does
not correlate to change in CO. In the study, subjects with
the same increase in HR (1.5 £ resting) experienced
increases in CO ranging from 2.0 –2.8 times resting.
Similarly, distribution between the viscera and legs during
exercise was non-uniform with distributions ranging from
66% (subject 10) to 92% (subject 11). Clearly,
cardiovascular regulation in response to exercise is not
uniform, with potential differences based on a number of
factors including athletic fitness, lean muscle mass, and
level of effort. The variations in CO were minimized by
using measured SC flow as the inlet boundary conditions
for rest and exercise. The overall variations in distribution
Figure 8. Comparison between predicted and measured flow (cm3 s21)
waveform. The identity line is shown. A correlation coefficient of 0.90
was computed.
Fractal network model for rest and exercise
49
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Figure 9. Predicted flow distribution for all outlets during exercise. Sum of flow indicates the mean measured SC flow. Uniform scaling of impedance
boundary conditions used to simulate physiologic changes due to lower extremity exercise: f ¼ 0:7 for all viscera and f ¼ 5:0 for all lower extremity
structured trees.
between viscera and legs were minimal, with good
agreement between measured and computed for 8 of 11
subjects.
Second, because full geometric data sets were not
available for the subjects, this study used idealized
geometry (see figure 2 and table 2) that was scaled to the
subject’s body mass using an allometric scaling law. These
scaled geometries may not have accurately reflected true
subject geometry.
A third source of modelling error is found in the
geometry of the structured trees and scaling factors used to
approximate physiological response to exercise. Variations
Figure 10. Measured inlet flow q (cm3 s21) (thin line) and corresponding computed inlet pressures p (mmHg) (bold line) during exercise for one cardiac
cycle. Uniform scaling of structured tree impedance boundary conditions used to simulate physiologic changes due to lower extremity exercise.
50
B. N. Steele et al.
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in visceral tone due to chemical, neural, or metabolic factors
were not characterized individually, but if quantified, could
be incorporated into the model through scaling factors.
While the factors used were adequate in determining the
redistribution of flow from rest to exercise, the simulation
overestimates the pressure during exercise. Studies run with
alternate exercise factors and alternate resistance vessel
radii values did not significantly alter exercise pressures.
This is mainly because the dilation factor 5.0 produced the
maximal decrease in impedance in the lower limbs.
Additional impedance was required to further direct the
desired flow away from the viscera outlets. The structured
trees follow a simple bifurcating scheme and are generically
defined from literature values. It is clear that structured trees
must be tailored to more accurately reflect the organs they
are perfusing and allow for a greater range of impedance
regulation. One option is to adopt a methodology that would
use a diameter-defined Strahler (Jiang et al. 1994) system
with several higher-order branches emanating from a single
lower-order vessel. In order to determine if any scheme
accurately models a vascular bed, additional data regarding
blood flow in resting and active muscle is required. Such
data has been described in recent work using positron
emission tomography and could be used in further studies to
tune the vascular beds (Kalliokoski et al. 2000; Mizuno
et al. 2003).
Fourth, the purely elastic constitutive model used to
describe vessel response to pressure is not ideal. The
computation tend to overestimate flow during systole and
underestimate flow during diastole, indicating that the
compliance values for the resting case may be too low. It is
possible to reduce the amplitude of the computed resting
IR flow by modifying the compliance, but this change
results in a drastic underestimation of exercise amplitude.
A combination of structured tree modification as
described above as well as a more sophisticated
constitutive model is needed to more accurately predict
blood pressure and flow wave propagation at rest as well
as during an altered physiologic state.
Finally, the physiologic changes accounted for in this
study were limited to a single, steady level of exercise and
were generated using a simple vasodilation/vasoconstriction
scheme. Regulation mechanisms are more sophisticated,
with pre-capillary sphincters that lead to the recruitment of
collapsed capillary beds. It may be assumed that alternate
levels of exercise, fitness (Wilmore et al. 1980; Wright et al.
2002), age, gender (Fu and Levine 2005), and disease may
elicit different physiologic responses. By determining these
relationships, it may be possible to predict the steady-state
regulatory responses of vascular regions from diagnostic
patient data collected at rest.
Acknowledgements
This work was done as part of the thesis work of Dr Steele
under the guidance of Dr Taylor at Stanford University,
and was supported in part by the National Science
Foundation under Grant No. 0205741, the Whitaker
Foundation and the Ayers Foundation.
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