Am J Physiol Heart Circ Physiol 309: H222–H234, 2015.
First published April 17, 2015; doi:10.1152/ajpheart.00857.2014.
Reducing the number of parameters in 1D arterial blood flow modeling: less
is more for patient-specific simulations
Sally Epstein,1 Marie Willemet,1 Phil J. Chowienczyk,2 and Jordi Alastruey1
1
Division of Imaging Sciences and Biomedical Engineering, St. Thomas’ Hospital, King’s College London, London, United
Kingdom; and 2Department of Clinical Pharmacology, St. Thomas’ Hospital, King’s College London, London, United
Kingdom
Submitted 1 December 2014; accepted in final form 11 April 2015
1D modeling; aortic pulse wave; digital pulse wave; windkessel
model; hypertension
PERSISTENTLY ELEVATED BLOOD PRESSURE (hypertension) is now
identified as the most important single cause of mortality
worldwide. By 2025 it is predicted that 1.56 billion people will
have hypertension (22). The correlation between hypertension
and independent biomarkers has been extensively analyzed
(25, 47). However, understanding how properties of the cardiovascular system affect the aortic blood pressure waveform
and its relation to the pressure waveform in the upper limb
(where blood pressure is routinely measured) has not yet been
fully understood (20). One-dimensional (1D) arterial blood
flow modeling has the potential to improve this understanding,
specially if patient-specific simulations could be achieved.
Several studies have shown the ability of the nonlinear 1D
equations of blood flow in compliant vessels to capture the
Address for reprint requests and other correspondence: J. Alastruey,
Division of Imaging Sciences and Biomedical Engineering, St. Thomas’
Hospital, King’s College London, London, UK (e-mail: jordi.alastruey-arimon
@kcl.ac.uk).
H222
main features of pressure and flow waves in the aorta and other
large arteries (26, 30, 32, 38, 50). 1D models offer good
accuracy with considerable less computational cost than equivalent three-dimensional (3D) models (15, 52). However, a
major drawback of 1D modeling is that the determination of
model input parameters from clinical data is a very challenging
inverse problem (35, 36). In the 1D formulation, the arterial
network is decomposed into arterial segments connected to
each other at nodes. The larger the number of arterial segments
in a given 1D model, the greater the number of model input
parameters that need to be estimated. Thus a method of
minimizing the number of 1D model arterial segments should
help maximize the percentage of input parameters that can be
estimated from clinical measurements and, hence, that can be
patient specific.
Within the field of multibranched 1D modeling, there has yet
to be a consensus reached in terms of the optimum number of
arterial segments necessary to obtain good quality pressure and
flow measurements at the aorta. The number of arterial segments used in 1D modeling has increased in recent years from
55 to over 4 million (7, 11, 30, 33, 38, 42). Obviously, with an
increasing number of arterial segments the more challenging it
becomes to estimate a greater percentage of the total number of
input parameters from available patient-specific data. Running
a distributed 1D model requires geometric and material properties for each arterial segment, a blood inflow waveform, and
outflow boundary conditions for every single terminal vessel.
Thus with each generation of bifurcations in the arterial tree the
number of input parameters that need to be prescribed increases exponentially.
Lumped parameter models of the cardiovascular system are
also commonly used to simulate arterial blood flow. They were
introduced by Otto Frank in 1899 (17) under the guise of a
“hydraulic windkessel,” which consists of a resistor and capacitor in series. The windkessel model is adept at portraying
the exponential decay in diastole, but it is less able to describe
the pressure waveform during the systolic phase of the cardiac
cycle (39). More elaborate combinations of resistors, capacitors,
and impedances have been suggested to improve fitting in systole
(24, 31, 43). However, with the drop in the space dimension, the
windkessel model neither accounts for wave propagation and
reflection nor can simulate the effect of pulse wave velocity
(PWV), which is assumed to be infinite. Wave reflections play
an important role in shaping the pressure waveform and central
PWV has been identified as a highly valuable marker of arterial
stiffness, which is being used in the classification and diagnosis
of hypertension (27). Wave propagation and reflection can be
studied using 1D modeling (51).
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Epstein S, Willemet M, Chowienczyk PJ, Alastruey J. Reducing
the number of parameters in 1D arterial blood flow modeling: less is
more for patient-specific simulations. Am J Physiol Heart Circ
Physiol 309: H222–H234, 2015. First published April 17, 2015;
doi:10.1152/ajpheart.00857.2014.—Patient-specific one-dimensional
(1D) blood flow modeling requires estimating model parameters from
available clinical data, ideally acquired noninvasively. The larger the
number of arterial segments in a distributed 1D model, the greater the
number of input parameters that need to be estimated. We investigated
the effect of a reduction in the number of arterial segments in a given
distributed 1D model on the shape of the simulated pressure and flow
waveforms. This is achieved by systematically lumping peripheral 1D
model branches into windkessel models that preserve the net resistance and total compliance of the original model. We applied our
methodology to a model of the 55 larger systemic arteries in the
human and to an extended 67-artery model that contains the digital
arteries that perfuse the fingers. Results show good agreement in
the shape of the aortic and digital waveforms between the original
55-artery (67-artery) and reduced 21-artery (37-artery) models.
Reducing the number of segments also enables us to investigate the
effect of arterial network topology (and hence reflection sites) on
the shape of waveforms. Results show that wave reflections in the
thoracic aorta and renal arteries play an important role in shaping
the aortic pressure and flow waves and in generating the second
peak of the digital pressure and flow waves. Our novel methodology is important to simplify the computational domain while
maintaining the precision of the numerical predictions and to
assess the effect of wave reflections.
H223
REDUCING THE NUMBER OF 1D MODEL PARAMETERS
The aim of this study is to provide a new methodology for
investigating the minimum number of tapered arterial segments
required to simulate, using nonlinear 1D modeling, the blood
pressure and flow waveforms in the aorta and digital artery in
the hand, where the upper limb pressure waveform is usually
measured noninvasively. This is achieved by lumping peripheral 1D model branches into windkessel models that preserve
the net resistance and total compliance of the original model. In
addition, this methodology enables the analysis of the effect of
network topology and, hence, reflections sites on the shape of
pressure and flow waves. Our novel methodology is tested for
the aorta of a 55-artery model under both normotensive and
hypertensive conditions and for the digital artery of a 67-artery
model under normotensive conditions (Fig. 1).
⭸U
⭸t
⫹U
⭸U
⭸x
⭸t
⫹
⭸x
⫽ 0,
(1A)
67-Artery Model
A
C
␤
Ad
␤⫽
␳A
,
(1B)
共兹A ⫺ 兹Ad兲 ,
(2)
4
3兹
␲Eh,
(3)
with E the elastic modulus and h the thickness of the vessel wall. ␤ Is
related to the PWV, denoted by c, through
c⫽
⭸ 共AU兲
f
where Ad is the cross-sectional area at diastolic pressure (Pd). The
parameter ␤ characterizes the material properties of the vessel through
One-Dimensional Formulation
⭸A
␳ ⭸x
⫹
冑
␤
2␳Ad
A1⁄4 .
(4)
Equations 1 and 2 are solved numerically using a discontinuous
Galerkin scheme with a spectral/hp spatial discretization (4).
55-Artery Model
The baseline 1D arterial network considered in this work consists
of the larger 55 arteries (Fig. 1C, black) with the properties of a
normotensive human. Each arterial segment is described by a set of
55-Artery Model
D
1
400
0.5
200
0
0
Q (ml/s)
Q (ml/s)
400
200
0
0
0.5
0
1
1
0.5
Time (s)
1
P (kPa)
E
P (kPa)
B
0.5
15
10
0
0.5
Time (s)
1
15
10
0
Fig. 1. A: flow waveform prescribed at the aortic root as a reflective boundary condition (black), and simulated flow waveform at the digital artery (gray) for
the 67-artery model. B: simulated pressure waveforms at the aortic root (black) and digital artery (gray) for the 67-artery model. C: schematic representation of
the 55-artery 1-dimensional (1D) model (black). The circled region shows additional anatomical model of the hand included in the 67-artery model (grey),
containing vessels of the superficial palmar arch and digital arteries. D: flow waveform prescribed at the aortic root as a reflective boundary condition (black),
and simulated flow waveform at the thoracic aorta (grey) for the 55-artery model. E: simulated pressure waveforms for the 55-artery model at the aortic root
(black) and thoracic aorta (gray).
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METHODS
In the 1D formulation, each segment is modeled as a deformable
cylindrical tube whose properties can be described by a single axial
coordinate x. Assuming the tube to be impermeable and the fluid to be
incompressible and Newtonian, conservation of mass and momentum
applied to a control volume of the tube yields (16, 40)
1 ⭸P
where t is the time; ␳ ⫽ 1,050 kg/m3 is the density of the blood;
A(x, t) is the luminal cross-sectional area, assumed to be circular; and
U(x, t) and P(x, t) are the axial blood flow velocity and pressure,
respectively, averaged over A. The frictional force per unit length is
given by f(x, t) ⫽ ⫺2(␰ ⫹ 2)␲␮U, with ␰ the polynomial order of the
velocity profile and ␮ ⫽ 4 mPa·s the viscosity of the blood. We
assume a close to plug flow with ␰ ⫽ 9 (41).
To close the system of nonlinear governing Eqs. 1A and 1B we
need a pressure-area relationship, known as tube law. In this model we
assume the arterial wall to be a linear elastic, thin, incompressible and
homogenous solid, which yields the tube law (34)
P ⫽ Pd ⫹
We begin by describing the 1D formulation (see 1D Formulation)
and the 55 and 67-artery models considered in this study (see 55Artery Model and 67-Artery Model). We then introduce our novel
technique for lumping peripheral 1D model branches into windkessel
0D models (see Reducing the Number of Arterial Segments). Lastly,
we describe the error metrics used to analyze our results (see Error
Metrics).
⫽⫺
H224
REDUCING THE NUMBER OF 1D MODEL PARAMETERS
three independent parameters (length, luminal cross-sectional area at
diastolic pressure, and diastolic PWV), as described in detail in Ref.
4. Energy losses are neglected at bifurcations and the initial conditions
are (A,U, P) ⫽ [A0(x), 0, 0] in all segments, with A0 the luminal area
that yields Ad at diastolic pressure, Pd. The flow at the aortic root (Fig.
1D) is prescribed as a reflective boundary condition. The outlet of
each terminal vessel is coupled to a three-element windkessel model,
with two peripheral resistances (R1 and R2), a compliance (C) and an
outflow pressure ^
pout (Fig. 2A). All four parameters are assumed to be
constant, but with different values in each terminal vessel. The
resistance R1 is equal to the characteristic impedance Z0 at the end
point of the vessel, with
Z0 ⫽
␳c
Ad
.
(5)
67-Artery Model
We extended the baseline model by including the larger arteries of
the hand using previously published data (Table A1 in Ref. 1) (see
Fig. 1C, gray). Each hand consists of the superficial palmar arch and
four digital arteries attached to the ulnar and radial arteries of the
baseline model. Figure 1, A and B, shows the flow and pressure
waveforms at the aortic root and digital artery of the 67-artery model.
Pressure and flow in all four digital arteries are almost identical.
Reducing the Number of Arterial Segments
The baseline model consists of the aorta and five generations of
arterial bifurcations (Fig. 1C, black). We reduced the 55 arterial
segments of this model using two procedures. In the first one, each
successive reduction in the number of segments was achieved by
decreasing the number of generations of bifurcations. For example, by
eliminating the fifth generation of bifurcations, the arterial network is
reduced to 53 arteries, since the right interosseous and right lower
Rnew ⫽ R2 ⫹ R1 ⫹ Rv ,
Cnew ⫽
CvR2 ⫹ CvR1 ⫹ CR2 ⫹ RvCv
R2 ⫹ R1 ⫹ Rv
Rv ⫽ 2共␰ ⫹ 2兲␲␮K3 , K3 ⫽
Cv ⫽
K1
␳
, K1 ⫽
兰
l
1
0
A2d
兰
l
Ad
0
c2d
dx,
(7)
dx.
Reducing a single bifurcation
D
l(x),A(x),c(x)
(6B)
Equations 6A, 6B, and 7 can be derived by neglecting nonlinearities
and inertia terms and assuming that pulse wave transit times within a
A
qin
pin
,
where Rv and Cv are, respectively, the resistance and compliance of
the 1D model vessel (Fig. 2B) given by
Reducing a single vessel
Fig. 2. A–C: reduction of a nonlinear 1D
model single vessel attached to a 3-element
windkessel model (R1–C–R2; A) into a single
2-element windkessel model (Cnew–Rnew; C),
via an intermediary stage (B) in which the 1D
vessel is simplified into a 2-element windkessel model (Cv–Rv). The 1D vessel is characterized by a length l, a cross-sectional area
A(x) and a pulse wave velocity c(x). D–F:
reduction of a nonlinear 1D model single
bifurcation coupled to 3-element windkessel
models (R1–C–R2; D) into a nonlinear 1D
model single vessel coupled to a 3-element
windkessel model (Rnew,1–Cnew–Rnew,2; F).
In the intermediary stage (E), the daughter
vessels 1 and 2 and their outlet windkessel
models are transformed into two 2-element
windkessel models (CT–RT). qin(t): inflow;
qout(t): outflow; pin(t): inflow pressure; pout:
outflow pressure.
(6A)
R1
R2
qout
qin
R1,2
C1
Vessel-0
pout
C
R1,1
Vessel-1
R2,1
Vessel-2
pout,1
R2,2
C2
B qin
pin
Rv
qout
R1
pout
C
Cv
R2
qout
R1,T
E
qin
pout
R2,T
C2,T
C
Rnew
qin
pin
Cnew
qout
pout
F
qin
Vessel-0
AJP-Heart Circ Physiol • doi:10.1152/ajpheart.00857.2014 • www.ajpheart.org
Rnew,1
Cnew
qout,1
pout,1
C1,T
Vessel-0
Rnew,2
qout,1
qout,2
pout,2
qout,new
pout,new
qout,2
pout,2
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Figure 1, D and E, shows the flow and pressure waveforms at the
aortic root and thoracic aorta of the baseline model. We also considered a hypertensive model obtained by increasing arterial stiffness in
all vessels of the baseline model and peripheral resistances in all
windkessel models. Both PWV (which is related to the stiffness
parameter ␤ by Eq. 4) and the total peripheral resistance in each
terminal windkessel (R1 ⫹ R2) are increased by a factor of 1.5.
ulnar arteries (the only vessels at the 5th generation of bifurcations)
are lumped into a single three-element windkessel model coupled to
the end point of the right upper ulnar.
In the second procedure, the number of arteries of the model
containing the aorta up to the first generation of bifurcations was
reduced by successively lumping each pair of the most peripheral
vessels into a three-element windkessel model. For example, the
model up to the first generation of bifurcations (with 21 vessels) was
reduced to 19 vessels by combining the left and right common iliac
arteries into a three-element windkessel model coupled to the lower
abdominal aorta.
The 67-artery model (Fig. 1C, grey) was reduced to a model
consisting of only the arterial segments of the upper right limb and
ascending aorta, which is similar to the model of Karamanoglu et al.
(21). Similarly to the 55-artery model, we reduced the number of
arteries by first successively decreasing the number of generations of
bifurcations and then systematically reducing the number of aortic
segments. Thus the 67-artery model was reduced up to the 19-artery
model shown in Fig. 6F. This reduction enables the evaluation of the
waveform at the digital artery (a peripheral measuring site) throughout
the systematic reduction of arteries.
Reducing a 1D model terminal branch into an equivalent 0D
model. We can lump a 1D model vessel coupled to a three-element
windkessel model (with resistances R1, R2 and compliance C, Fig. 2A)
into an equivalent two-element windkessel model (Fig. 2C) with
resistance (Rnew) and compliance (Cnew). These are calculated as
REDUCING THE NUMBER OF 1D MODEL PARAMETERS
vessel are much smaller than the duration of the cardiac cycle. Further
details are given in the APPENDIX.
Reducing a 1D model single bifurcation into an equivalent 0D
model. We consider an arterial bifurcation with the parent vessel
denoted by the index 0 and the daughter vessels by the indexes 1 and
2 (Fig. 2D). Firstly, we apply the methodology described in Reducing
a 1D model terminal branch into an equivalent 0D model to transform
each daughter vessel into two parallel RC-windkessel models, with
elements R1,T, C1,T, and R2,T, C2,T for daughter vessels 1 and 2,
respectively (Fig. 2E). We can then combine these two 0D models
into a single three-element windkessel model, with resistances Rnew,1
and Rnew,2 and compliance Cnew (Fig. 2F). The total peripheral
resistance of the new 0D model is calculated as
Rnew,1 ⫹ Rnew,2 ⫽
1
1
,
1
(8)
R2,T
where Rnew,1 is set to be equal to Z0 (Eq. 5) of the parent vessel to
minimize wave reflections (2). The total peripheral compliance of the
new three-element windkessel is calculated as
Cnew ⫽ C1,T ⫹ C2,T .
(9)
Error Metrics
The pressure and flow waveforms generated by the reduced models
were compared with those generated by the complete 55- or 67models using the following relative error metrics:
僆 P,avg ⫽
僆 P,sys ⫽
1
Nt
兺
Nt i⫽1
冏
PRi ⫺ PCi
PCi
冏
max i共 PRi 兲 ⫺ max i共 PCi 兲
僆 P,dias ⫽
max i共 PCi 兲
min i共 PRi 兲 ⫺ min i共 PCi 兲
共 兲
min i PCi
,
僆Q,avg ⫽
, 僆Q,sys ⫽
1
Nt
兺
冏
QRi ⫺ QCi
Nt i⫽1 max j共QCj 兲
冏
,
max i共QRi 兲 ⫺ max i共QCi 兲
max i共QCi 兲
,
min i共QRi 兲 ⫺ min i共QCi 兲
max i共QCi 兲
,
, 僆Q,dias ⫽
(10)
with Nt the number of time points where the comparison is made. For
R
each time point i, PC
i and Pi are the pressures at the same spatial
location of the complete (C) and reduced (R) model, respectively. The
R
corresponding flows are QC
i and Qi . The metrics 僆P,avg and 僆Q,avg are
the average relative errors in pressure and the flow, respectively, over
one cardiac cycle; 僆P,sys and 僆Q,sys are the errors for systolic pressure
and flow; and 僆P,dias and 僆Q,dias are the errors for diastolic pressure
and flow. To avoid division by small values of the flow, we normalized the flow errors by the maximum value of the flow over the
cardiac cycle, maxj(QC
j ).
RESULTS
We applied our method for reducing the number of 1D
model arterial segments described in Reducing the Number of
Arterial Segments to the normotensive (see Normotensive 55Artery Model) and hypertensive (see Hypertensive 55-Artery
Model) 55-artery models, as well as the normotensive 67-artery
model (see Normotensive 67-Artery Model). In all cases we
analyzed the effect of decreasing the number of arterial segments on the pressure at the aortic root. For the 55-artery
model we also investigated the pressure and flow at the
midpoint of the thoracic aorta. For the 67-artery model we also
studied the pressure and flow waveforms at the digital artery in
the hand. The aortic-root flow is not studied for any model,
since it is enforced as the inflow boundary condition.
Normotensive 55-Artery Model
Figure 3 compares the aortic waveforms simulated by six
reduced models (dashed black) with those produced by the
baseline model (solid grey) under normotensive conditions.
Average, systolic and diastolic errors are provided in each plot
relative to the baseline model. For each reduced model in Fig.
3, A–E, the number of segments is shown in the first column
and the model topology is illustrated in the second. Diastolic,
systolic, and average relative errors in aortic pressure are ⬍3%
for any reduced model, even for the single-vessel model shown
in Fig. 3E. Relative errors in the flow waveform at the thoracic
aorta are greater than in pressure but remain smaller than 20%
(9% for the average relative error).
Considerable discrepancies between reduced and baseline
models appear if 15 or less segments are included. These
models, which do not include the superior mesenteric and renal
arteries, lead to the following features of the aortic waveform
not being captured as well as in the baseline model (Fig. 3, C
and D): the magnitude of the dicrotic notch and the time of
peak systole at the root, and the magnitude of systolic pressure,
the time and magnitude of the peak flow, and the oscillatory
flow in diastole at the thoracic aorta. However, all reduced
models capture well the time and magnitude of the feet of
baseline pressure and flow waves. Moreover, they produce
pressure and flow waveforms in the second half of diastole that
are similar to those simulated by the complete 55-artery model.
Figure 3F shows the aortic-root pressure waveform calculated by a two-element windkessel model with a resistance and
compliance equal to, respectively, the net peripheral resistance
and total compliance of the baseline model. This 0D model is
able to capture precisely the decay in aortic pressure in diastole
but is unable to describe well pressure in systole.
Figure 4, A–C, shows systolic, diastolic and average relative
errors in the pressure at the aortic root with the number of
arterial segments. Errors remain ⬍1% for models containing
53 to 21 segments; the latter consists of the aorta and first
generation of bifurcations only (see its topology in Fig. 3B).
Further reductions in the number of segments yield a greater
increase in the rate at which errors raise. Qualitatively similar
progressions in relative errors are obtained for the flow and
pressure at the thoracic aorta.
Hypertensive 55-Artery Model
Figure 5 compares the aortic pressure and flow waveforms
produced by six reduced models (dashed black) with those
simulated by the complete 55-artery model (solid grey) under
hypertensive conditions. The topology of the reduced models
and the format of the figure are the same as for the normotensive study above. Pulse and mean pressures are greater under
hypertensive conditions: the pulse pressure is approximately
doubled whereas the mean pressure increases by ⬃40%. The
amplitude and mean value of the flow change little with
simulated hypertension.
Average relative errors in both pressure and the flow remain
⬍2% for models containing 53 to 21 segments, in agreement with
the normotensive results (Fig. 4, A and B). However, eliminating
segments of the aorta and first generation of bifurcations from the
21-artery model produces smaller errors under hypertensive conditions. Hypertensive reduced models are able to capture well all
features of the aortic pressure waveform that normotensive re-
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R1,T
⫹
H225
H226
REDUCING THE NUMBER OF 1D MODEL PARAMETERS
Segment
Number
Arterial
Network
Aortic Root
P (kPa)
avg %:0.01
sys %:0.01
dias %:0.01
15
A
Thoracic Aorta
P (kPa)
10
0.5
1
avg %:0.53
sys %:−0.29
dias %:0.44
15
0.5
1
avg %:0.29
sys %:0.37
dias %:0.38
0
0.5
1
avg %:2.67
sys %:−0.51
dias %:1.25
200
21
10
0.5
1
avg %:0.88
sys %:−0.55
dias %:−0.12
15
0
0
0.5
1
avg %:0.55
sys %:1.92
dias %:−0.18
15
0
0.5
1
avg %:5.12
sys %:−6.76
dias %:11.72
200
15
10
0
10
0.5
1
avg %:1.49
sys %:1.07
dias %:−0.66
15
0
0
0.5
1
avg %:1.19
sys %:2.92
dias %:−0.81
15
0
0.5
1
avg %:8.13
sys %:−19.26
dias %:12.12
200
11
10
0
10
0.5
1
0
0
0.5
1
0
0.5
1
avg %:1.90
sys %:2.24
dias %:−1.76
15
1
10
0
Rnew
F
0
0.5
1
avg %:3.45
sys %:1.08
dias %:−0.85
15
Cnew
10
0
0.5
Time (s)
1
Time (s)
Time (s)
Fig. 3. A–F: pressure (P) and flow (Q) waveforms at the aortic root and midpoint of the thoracic aorta of the normotensive 55-artery model (solid grey) and several
reduced models (dashed black), 1 for each row. The number of segments and the arterial topology for each reduced model are given in the 1st 2 columns. The
aortic-root pressure waveform calculated by a 2-element windkessel model of the whole systemic circulation is shown in F. Average (avg), systolic (sys) and
diastolic (dias) relative errors are indicated in the top right corner of each plot.
duced models were unable to reproduce (compare Figs. 3, A–E,
and 5, A–E). Discrepancies in aortic flow waveforms are similar
under both normotensive and hypertensive conditions (compare
last column in Figs. 3, A–D, and 5, A–D). Moreover, similar
evolutions of systolic and diastolic errors are observed under both
normotensive and hypertensive conditions (Fig. 4).
In agreement with the normotensive results, the aorticroot pressure waveform calculated by the two-element
windkessel model of the whole systemic circulation is able
to describe well the decay in pressure during diastole but is
unable to produce most of the features of the pressure wave
in systole (Fig. 5F).
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0
E
0
0
15
10
D
avg %:0.03
sys %:0.00
dias %:−0.03
200
53
0
C
avg %:0.01
sys %:0.01
dias %:0.01
15
10
B
Thoracic Aorta
Q (ml/s)
REDUCING THE NUMBER OF 1D MODEL PARAMETERS
Normotensive
H227
Hyper tensive
D
Average Error %
A
3
3
0
0
−3
−3
60
40
20
0
B
60
40
20
0
3
3
0
0
−3
−3
60
40
20
0
40
20
0
40
20
0
F
Diastolic Error %
C
60
Fig. 4. Evolution of the average (A and D), systolic (B
and E), and diastolic (C and F) relative errors in the
pressure waveform at the aortic root of the 55-artery
model with the number of arterial segments, under
normotensive (A–C) and hypertensive (D–F) conditions. For each plot, the vertical dashed line corresponds to the 19-artery model: arterial networks on its
left (ⱖ19 vessels) include a full aorta, whereas those on
its right (⬍19 vessels) only include portions of it.
3
3
0
0
−3
−3
60
40
20
Vessel #
0
60
Vessel #
Normotensive 67-Artery Model
DISCUSSION
Figure 6 compares the aortic pressure and flow waveforms
produced by six reduced models (dashed black) with those
simulated by the complete 67-artery model (solid grey). Pulse
and mean pressures differ from those of the baseline 55-artery
model by ⫺9 and 13% at the aortic root (Fig. 1). This is due to
the total resistance and compliance being altered by the extended arterial network in the hands. With segment reductions,
average systolic and diastolic relative errors of central aortic
and digital arterial pressure remain ⬍3% for models containing
65 to 19 segments. Relative errors in the flow at the digital
artery remain ⬍5%.
Reduced 67-artery models with ⬍33 vessels do not capture
the following features at the central aorta: magnitude of the
dicrotic notch and the time and magnitude of peak systole (Fig.
6, D–F). At the digital artery these reduced models no longer
capture the time of arrival of the second pressure peak, the
magnitude of the first pressure peak, and the oscillatory flow in
diastole. Figure 7 shows average, systolic, and diastolic relative errors in the pressure at the aortic root (A) and digital
artery (B) with the number of arterial segments. The largest
divergence between the complete and reduced models is seen
if less than 33 vessels are included; i.e., if aortic segments are
lumped into windkessel models.
In this study we have presented a novel methodology for
investigating the minimum number of arterial segments required to simulate precisely blood pressure and flow waveforms at multiple arterial sites using nonlinear 1D modeling.
By systematically lumping terminal 1D model segments into
0D windkessel models that preserve the net resistance and total
compliance of the original model (Fig. 2), we can study the
effect on central and peripheral waveforms of reducing the
number of arterial segments and, hence, input parameters. We
have applied our methodology to investigate the effect of
network topology (and hence reflection sites) on the shape of
pressure and flow waves in the aorta and digital artery in the
hand. To carry out this study we have used a nonlinear 1D
model of blood flow in the 55 larger systemic arteries of the
human, under normotensive and hypertensive conditions, and
an extended 67-artery model that includes the larger arteries in
the hand, under normotensive conditions (Fig. 1).
We have focused on analyzing 1) aortic hemodynamics,
because central (aortic) pressure is an important determinant of
cardiovascular function; and 2) peripheral hemodynamics in
the digital arteries, because the pressure and volume pulses at
the digital artery can be measured continuously and noninvasively in the clinic; e.g., using the Finapres device (18) and
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Systolic Error %
E
H228
REDUCING THE NUMBER OF 1D MODEL PARAMETERS
Segment
Number
Arterial
Network
Aortic Root
P (kPa)
avg %:0.03
sys %:−0.05
dias %:0.03
20
A
Thoracic Aorta
P (kPa)
Thoracic Aorta
Q (ml/s)
avg %:0.03
sys %:−0.06
dias %:0.03
20
avg %:0.04
sys %:0.01
dias %:0.01
200
53
15
15
10
0
10
0
0
0.5
avg %:0.96
sys %:−1.44
dias %:1.48
20
B
1
0.5
1
avg %:0.94
sys %:−1.24
dias %:1.52
20
0
0.5
1
avg %:1.86
sys %:−3.73
dias %:3.62
200
21
15
10
0
10
0
0
0.5
avg %:0.78
sys %:−0.38
dias %:1.12
20
C
1
0.5
1
avg %:0.68
sys %:−0.70
dias %:1.15
20
0
0.5
1
avg %:3.06
sys %:−9.53
dias %:3.66
200
15
15
15
10
0
10
0
0
0.5
avg %:0.88
sys %:2.01
dias %:0.03
20
D
1
0.5
1
avg %:0.51
sys %:1.17
dias %:0.00
20
0
0.5
1
avg %:5.59
sys %:−21.69
dias %:3.94
200
11
15
15
10
0
10
0
0
0.5
0.5
1
0
0.5
1
avg %:0.69
sys %:−0.38
dias %:0.75
20
E
1
1
15
10
0
Rnew
F
0
Cnew
0.5
1
avg %:4.57
sys %:−3.90
dias %:−0.99
20
15
10
0
0.5
Time (s)
1
Time (s)
Time (s)
Fig. 5. A–F: pressure (P) and flow (Q) waveforms at the aortic root and midpoint of the thoracic aorta of the hypertensive 55-artery model (solid grey) and several
reduced models (dashed black), 1 for each row. The number of segments and the arterial topology for each reduced model are given in the 1st 2 columns. The
aortic-root pressure waveform calculated by a 2-element windkessel model of the whole systemic circulation is shown in F. Average (avg), systolic (sys), and
diastolic (dias) relative errors are indicated in the top right corner of each plot.
photoplethysmography (5, 8), respectively. Our simulated aortic and digital pressure and flow waveforms contain the main
features observed in vivo (Fig. 1).
Optimizing Arterial Network Topology
The 55-artery model consists of the aorta and other large
arteries up to the fifth generation of bifurcations (Fig. 1) and is
determined by 227 independent parameters. The 67-artery
model consists of the aorta and other large arteries up to the
sixth generation of bifurcations (which include vessels of the
palmar arch and digital arteries Fig. 1) and is defined by 271
independent parameters. Each arterial segment is described by
three independent parameters (length, luminal cross-sectional
area at diastolic pressure, and diastolic PWV), while two
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15
H229
REDUCING THE NUMBER OF 1D MODEL PARAMETERS
Segment
Number
Arterial
Network
Aortic Root
P (kPa)
avg %:0.01
sys %:0.00
dias %:0.01
15
A
Digital Artery
P (kPa)
Digital Artery
Q (ml/s)
avg %:0.01
sys %:0.00
dias %:0.00
15
65
avg %:0.04
sys %:−0.02
dias %:−0.12
0.4
0.2
0
10
0
0.5
10
0
0.5
avg %:0.58
sys %:0.37
dias %:0.62
15
1
avg %:0.74
sys %:0.41
dias %:0.39
0.5
1
avg %:1.28
sys %:−0.22
dias %:2.42
0.4
15
37
0
0.2
0
10
0
0.5
10
0
0.5
avg %:0.59
sys %:0.41
dias %:0.65
15
C
1
1
avg %:0.75
sys %:0.45
dias %:0.45
0.5
1
avg %:1.25
sys %:−0.18
dias %:2.57
0.4
15
33
0
0.2
0
10
0
0.5
10
0
0.5
1
avg %:1.09
sys %:0.20
dias %:−0.06
avg %:0.77
sys %:0.19
dias %:0.17
15
D
1
15
31
0
0.5
1
avg %:2.45
sys %:−0.10
dias %:4.13
0.4
0.2
0
10
0
0.5
10
0
0.5
avg %:1.04
sys %:0.82
dias %:−0.20
15
E
1
1
avg %:1.46
sys %:−0.01
dias %:−0.46
15
29
0
0.5
1
avg %:3.35
sys %:−0.17
dias %:4.05
0.4
0.2
0
10
0
0.5
10
0
0.5
avg %:1.61
sys %:2.90
dias %:−0.66
15
F
1
1
avg %:1.89
sys %:2.62
dias %:−0.86
15
19
0
0.5
1
avg %:3.70
sys %:3.25
dias %:−2.75
0.4
0.2
0
10
0
0.5
Time (s)
1
10
0
0.5
1
0
Time (s)
0.5
1
Time (s)
Fig. 6. A–F: pressure (P) and flow (Q) waveforms at the aortic root and midpoint of the digital artery of the 67-artery model (solid grey) and several reduced
models (dashed black), one for each row. The number of segments and the arterial topology for each reduced model are given in the 1st 2 columns. Average
(avg), systolic (sys), and diastolic (dias) relative errors are indicated in the top right corner of each plot.
additional parameters are required to model each peripheral
arterial segment (peripheral resistance and compliance). The
remaining independent parameters are the density and viscosity
of blood, the shape of the velocity profile, the diastolic pressure, the inflow boundary condition and the outflow pressure at
terminal windkessel models.
By subsequently trimming the 55-artery model of generations of bifurcations, the number of arterial segments is reduced to 21, if only arterial vessels up to the first generation of
bifurcations are simulated using 1D modeling. As a result, the
227 input parameters of the initial model are reduced to 91.
Such a considerable reduction in the number of parameters
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B
1
H230
REDUCING THE NUMBER OF 1D MODEL PARAMETERS
of 1D model parameters may still be challenging, even if using
the suggested lumping approach.
Our results also contribute towards understanding the role of
pulse wave reflections in shaping the aortic (see Effect of Wave
Reflections on Aortic Pressure and Flow Waveforms) and
digital (see Effect of Wave Reflections on Digital Pressure and
Flow Waveforms) pressure and flow waves.
A
Error %
3
0
−3
60
40
20
B
0
−3
60
40
20
Vessel #
Fig. 7. Evolution of the average (䊐), systolic (⫹), and diastolic () relative
errors in the pressure waveform at the aortic root (A) and digital artery (B) with
the number of arterial segments in the 67-artery model. Vertical dashed lines
correspond to the 35-artery model: arterial networks displayed on the left and
on the line include a full aorta while those on the right only include portions
of it.
yields aortic pressure and flow waveforms that contain all the
features of the 55-artery model with errors ⬍4% relative to the
initial model (Figs. 3B and 5B). Reducing the 67-artery model
(271 input parameters) to the first generation of bifurcations
(37 arteries, 171 parameters), with all segments of the right
upper limb preserved, results in relative errors ⬍1% in pressure at the central and peripheral measuring sites and ⬍3% in
the flow at the peripheral measuring site, with all errors relative
to the initial model (Fig. 6B). These results suggest that our
original 55-artery (67-artery) models are overparameterized for
simulating aortic (and digital) waveforms.
The focus of previous studies using 1D modeling (7, 11, 30,
33, 38, 42) was not confined to the simulation of aortic and
upper-limb pulse waveforms. However, according to our study,
1D modeling of these pulse waveforms may not require the
large number of arterial segments considered in those studies.
Instead, our results favor the use of reduced models that
consider only the arteries of interest, e.g., the model of Karamanoglu et al. (21) for the ascending aorta and upper-limb
vessels and the model of Willemet et al. (50) for the lower-limb
vessels. This result has important implications for patientspecific 1D modeling. A reduction in the number of 1D model
arterial segments decreases the number of input parameters that
need to be prescribed, which should facilitate the estimation of
a greater percentage of the total number of parameters from
clinical measurements. Lowering the number of parameters,
therefore, should increase the degree of patient-specific modeling that can be achieved to simulate central and upper-limb
hemodynamics. However, it is important to note that the degree
of applicability of 1D models for patient-specific modeling
remains at a pilot stage. As such, the task of clinical estimation
The excellent agreement between the 55-artery model and
the reduced 21-artery model containing segments up to the first
generation of bifurcations (Figs. 3B and 5B) suggests that the
effect of multiple reflection sites downstream vessels of the
first generation of bifurcations can be lumped into single
reflection sites at the end of these vessels. Further reduction in
the number of arterial segments by lumping aortic segments
considerably increases relative errors once the superior mesenteric and renal arteries are removed (Fig. 3, C–E). These
observations are supported by in vivo studies showing that
reflected waves generated artificially by total occlusions in the
aorta distal to the renal arteries have no discernible influence
on the pressure and flow waveforms at the ascending aorta (23,
45). It is also in agreement with the outcome of numerical
studies that used different tools of pulse wave analysis (3, 21).
Our numerical results are also consistent with the horizon
effect hypothesis (13), which states that aortic reflected waves
are an amalgam of individual reflections from multiple reflection sites, with the nearer site contribution at full strength and
sites further away having their contributions attenuated.
Further reductions in the number of 1D model segments of
the 55-artery model by lumping aortic segments and arteries
that branch off the aorta into 0D models, starting from the iliac
bifurcation, yields relative errors in aortic pressure ⬍5% (Fig.
4). Relative errors in aortic-root pressure remain ⬍5% even if
all the 55 segments of the baseline model are integrated into
Frank’s (17) two-element windkessel model of the entire arterial system (Figs. 3F and 5F). Removing aortic segments in the
55-artery model leads to discrepancies in the shape of aortic
pressure and flow waveforms, in systole and early diastole, but
does not in approximately the last half of diastole. This result
suggests that the diastolic part of the pressure waveform along
the aorta can be described by a lumped parameter model with
only four independent parameters (the inflow from the left
ventricle, the total compliance of the systemic circulation, the
net resistance at the aortic root, and the outflow pressure),
which corroborates in vivo (46, 48) and numerical (51, 52)
studies showing that aortic pressure becomes space independent with increasing time in diastole.
Lumping aortic segments of the 55-artery normotensive
model yields reduced models that are unable to capture the
main features of aortic pressure and flow waves in systole (Fig.
3, C–F). Under hypertensive conditions, however, models with
a reduced number of aortic segments maintain all features of
the aortic-root pressure waveform with relative errors ⬍2%
(Fig. 5, C–E). This result suggests that individual distal reflection sites affect less the shape of the central pressure wave with
hypertension. The following mechanism underlies this result.
Greater PWVs in the hypertensive model make more valid the
assumption of our reduction methodology of pulse wave transit
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Error %
3
Effect of Wave Reflections on Aortic Pressure and Flow
Waveforms
H231
REDUCING THE NUMBER OF 1D MODEL PARAMETERS
times within each vessel being much smaller than the duration
of the cardiac cycle (see Remark 1 in APPENDIX). Our additional
assumption of negligible inertia effects is also satisfied better in
the hypertensive model, since increasing PWV (and hence
decreasing arterial compliance) decreases the weight of the
flow acceleration term in the conservation of momentum Eq.
1B relative to the pressure gradient and viscous terms (51).
Effect of Wave Reflections on Digital Pressure and Flow
Waveforms
Perspectives
We believe that our methodology could offer valuable insight into optimizing the number of model input parameters
that are needed to simulate pressure and flow waveforms in a
given region of the systemic circulation other than the aorta
and digital artery. This will be useful to 1) identify which part
of the arterial tree is responsible for shaping a particular
pressure waveform and 2) lessen the number of parameters that
have to be estimated or taken from the literature to simulate
that waveform using 1D modeling. Other studies have considered truncated arterial networks, e.g., to study hemodynamics
in the leg or arm, and have shown that arterial pressure and
flow waveforms can be efficiently modeled without including
the aorta in the computational domain (26, 37, 50). Adaptations
in the inlet boundary condition may be necessary though (49).
We have not proven that our reduction method produces the
optimal result. Other methods for lumping 1D model segments
may lead to more complex outflow 0D models (e.g., those
described in Refs. 19, 24, 32, 43) that yield smaller relative
errors in pressure and the flow. In future work, we will extend
our reduction methodology to account for inertial effects (e.g.,
Concluding Remarks
We have presented a new method for lumping peripheral 1D
model vessels into 0D windkessel models that enables us to
analyze the effect of network topology on pulse waveforms and
determine the optimal number of arterial segments (and hence
input parameters) required to simulate efficiently blood pressure and flow waveforms at the arterial sites that are relevant
for a particular problem. This is important to simplify the
computational domain while maintaining the precision of the
numerical predictions.
APPENDIX
We describe our novel methodology to lump a single nonlinear 1D
model vessel of length l coupled with a linear three-element windkessel model at the outlet (Fig. 2A) into a linear two-element windkessel model (Fig. 2C). The vessel has a pressure and flow pin(t) and
qin(t) at the inlet and pout(t) and qout(t) at the outlet. The outlet forms
the inlet boundary conditions of the windkessel model, which has an
^
outlet flow q^
out(t) and constant pressure pout.
The methodology requires linearization of Eqs. 1 and 2 and
transformation into 0D equations. In previous derivations of the linear
0D formulation from the nonlinear 1D equations, it was assumed
constant cross-sectional area and material properties along the length
of the arterial segment or an average cross-sectional area (2, 28, 31).
In the following analysis we will not make such an assumption and
will allow area and material properties to vary with the position along
the vessel.
We consider the following linearized version of the 1D equations (4),
Ad ⭸ p
␳c2d
⭸q
⭸t
⫹
⭸t
Ad ⭸ p
␳ ⭸x
⫹
⫽
⭸q
⭸x
⫽ 0,
⫺2共␰ ⫹ 2兲␲␮q
␳Ad
(11A)
,
(11B)
where we have used Q ⫽ UA and linearized about the diastolic
conditions (A, P, Q) ⫽ (Ad ⫹ a, Pd ⫹ p, q), where a(x, t), p(x, t), and
q(x, t) are the perturbation variables for area, pressure and flow rate.
The physiological conditions in the arterial system are only weakly
nonlinear; thus many characteristics can be captured by the linearized
system (14). To continue we introduce the following theorem.
Theorem 1
Mean Value Theorem (MVT): suppose f: [b1, b2] ¡ ᑬ is continuous and suppose g ⱖ 0 is bounded and continuous on [b1, b2], then
there is ␨ 僆[b1, b2] such that
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In the 67-artery model we systematically lumped aortic
segments and branches off the aorta into 0D models while
preserving all arterial segments of the upper right limb. For any
reduced models, we obtained relative errors in pressure and the
flow at the aortic root and digital artery ⬍4.5% (Figs. 6, A–F
and 7). The model with the least number of segments (19
segments) contains only the ascending aorta and upper-rightlimb vessels and has a topology similar to that used by
Karamanoglu et al. (21).
Results at the aortic root of the reduced 37-artery model
(Fig. 6B) confirm our conclusion from the previous section that
the effect of multiple reflection sites downstream vessels of the
first generation of bifurcations can be lumped into single
reflection sites at the end of these vessels.
Pressure and flow waveforms at the digital artery have two
peaks, as observed in vivo. The second peak is progressively
flattened as aortic segments are lumped (Fig. 6, B–F). The
greater change in shape occurs if the renal arteries are lumped
into 0D models (Fig. 6, C–E). This result suggests that wave
propagation in the thoracic aorta and renal arteries plays an
important role in generating the second peak of the digital
pressure and flow waves. This observation corroborates the
conclusions from Refs. 9, 10 and is also consistent with the
hypothesis in Refs. 12, 29: the second peak of the digital
volume pulse (which is closely related to the pressure pulse) is
due to pressure waves travelling from the left ventricle to the
finger and later smaller reflections from internal mismatching,
mainly in the lower body.
using inductance terms) and more sophisticated tube laws [e.g.,
as proposed by Valdez Jasso et al. (44)], which may be more
appropriate for hypertensive models (6).
It may not be possible to have a patient-specific model with
all input parameters calculated from clinical measurements, but
we need tools like the one presented here to eliminate unnecessary parameters and advance the usage of patient-specific
simulations of arterial pressure and flow waveforms.
Given the excellent agreement between 1D and 3D pressure
and flow waves in the aorta, as reported by Xiao et al. (52)
under normal conditions, we believe that our methodology
could be adapted to also determine the size of the computational domain in 3D blood flow modeling.
H232
REDUCING THE NUMBER OF 1D MODEL PARAMETERS
兰
b2
f 共x兲g共x兲dx ⫽ f 共␨兲
b1
兰
b2
b1
g共x兲dx,
(12)
and
兰
b2
f 共x兲g共x兲dx
b1
f 共 ␨兲 ⫽
兰
b2
b1
g共x兲dx
.
(13)
K1 ⫽
兰
l
0
Ad
冉
l
0
0
冉
1 ⭸q
Ad ⭸ t
Ç
c
⭸x
Ç
b
Ç
a
1⭸p
⫹
⭸q
⫹
␳⭸x
Ç
d
冊
冊
dx ⫽ ⫺
dx ⫽ 0,
Ad ⭸ p
l
共b兲 兰0
l
⭸q
⭸x
␳c2d
⭸t
dx ⫽
兰
l
0
2共␰ ⫹ 2兲␲␮q
dx,
␳A2d
Ç
e
(14B)
dt
兰
l
Ad
0
c2d
Ⲑ
1⫹
Ⲑ
共d兲 兰0
l
1 ⭸p
␳ ⭸x
共 e兲 ⫺
dx ⫽
1 ⭸q
Ad ⭸ t
dx ⫽
pout ⫺ pin
␳
2共␰ ⫹ 2兲␲␮
␳
兰
l
q
0
A2d
dq共␨2, t兲
dt
兰
l
1
0
Ad
2共␰ ⫹ 2兲␲␮
␳
兰
l
1
0
A2d
Ⲑ
⫹ qout ⫺ qin ⫽ 0,
K2
dt
Ç
c
with
⫹
pout ⫺ pin
␳
Ç
d
2共␰ ⫹ 2兲␲␮
␳
R1
(16)
(17A)
(17B)
Rv ⫽ 2共␰ ⫹ 2兲␲␮K3 ,
(18)
R2
qout ⫹ CR1
冉
1⫹
R1
R2
⫹
Rv
R2
冊
qin ⫺
dqout
⫽C
dt
Ⲑ
(15A)
K3qout ,
Ç
e
(15B)
dpout
dt
pout ⫺ ^
pout
⫹
R2
冉冉 冊
冉
冊共
pin ⫺ ^
pout
R2
,
(19)
⫺ Cv 1 ⫹
R1
R2
⫹C⫹
R2
dpin
dt
CR1 ⫹ CRv兲 ⫽ 0.
dt
冊
RvCv dpin
dt
(20)
We make the assumption that the last term in Eq. 20 is of negligible
size compared with the other terms. This is deemed reasonable since
冉
A2d
Ç
b
⫽⫺
dx.
pout ⫺ pin ⫽ ⫺Rvqout ,
d qin ⫺ Cv
Pulse waves propagate rapidly within the cardiovascular system
(with velocities of about 4 –15 m/s), while arterial lengths can vary
between few millimeters up to ⬃10 cm. As a result, pulse wave transit
times within a vessel are small compared with the duration of the
cardiac cycle. Thus, at any given time the space averaged values will
be close to the pointwise ones and p(␨1, t) ⫽ pin(t), q(␨2, t) ⫽ qout and
q(␨3, t) ⫽ qout are reasonable (28).
Equations 14A and 14B become
dqout
A2d
where output values of pressure and flow from the windkessel model
are denoted as ^
pout and q^
out(t), respectively. By substituting qout(t)
from Eq. 17A and pout(t) from Eqs. 17B into Eq. 19 we obtain
dx.
Remark 1
Ç
a
1
⫹ qout ⫺ qin ⫽ 0,
d qin ⫺ Cv
q 共 ␨ 3, t 兲
Ⲑ
␳ dt
dt
⫹
Since q(x, t) and ⭸q共x,t兲 ⭸t are continuous and 1 Ad and 1
are bounded and continuous with Ad(x) ⬎ 0 under physiological
conditions, we can apply the MVT to (c) and (e) with ␨2 and
␨3 僆[0, l] (28).
K1 dpin
dpin
dx;
, with pout ⫽ p共l, t兲 and pin ⫽ p共0, t兲;
dx ⫽ ⫺
l
0
where Rv is the integrated resistance of the 1D model vessel (Fig. 2B).
We have shown that a 1D arterial segment can be reduced to a
two-element windkessel model with resistance Rv and compliance
Cv. To close the problem we must also consider the following
equation for the three-element windkessel model prescribed at the
boundary (Fig. 2A),
冉 冊
Since p(x, t) is continuous and Ad c2d is bounded and continuous, with
Ad(x) ⬎ 0 and cd(x) ⬎ 0 under physiological conditions, we can apply
the MVT to (a), with ␨1僆[0, l].
We now evaluate each term of Eq. 14B:
l
Cv
dx;
dx ⫽ qout ⫺ qin, with qout ⫽ q共l, t兲 and qin ⫽ q共0, t兲 .
共c兲 兰0
兰
with
1 dp共␨1, t兲
␳
Ad
K3 ⫽
dx,
Ⲑ
(14A)
where we have divided the last equation by Ad(x). We first evaluate
both terms of Eq. 14A:
共a兲 兰0
1
l
0
dpin
dt
dt
冊
⫽
dqout
dt
,
(21)
according to Eq. 17A, and consistent with our previous assumption
of the inertia term K2 dqout dt being negligible. Each arterial
segment acts as a buffer, due to its compliance; hence the rate of
change of flow at the outlet dqout dt is small compared with the
other terms in Eq. 20.
Rearranging Eq. 20 we get
Ⲑ
qin ⫽
pin ⫺ ^
pout
R2 ⫹ R1 ⫹ Rv
⫹
Ⲑ
共CvR2 ⫹ CvR1 ⫹ CR2 ⫹ RvCv兲 dpin
. (22)
dt
共 R 2 ⫹ R 1 ⫹ R v兲
By comparing Eq. 22 to the two-element windkessel equation (17, 48)
we obtain
qin ⫽
pin ⫺ ^
pout
Rnew
⫹ Cnew
dpin
dt
,
(23)
where ^
pout is the value of pressure at which the outflow stops (most
often approximated by the mean venous pressure), and Eqs. 6A and 6B
for Rnew and Cnew. These denote the resistance and compliance of
the two-element windkessel (Fig. 2C) that describes blood flow in
the original 1D model vessel coupled to a three-element windkessel (Fig. 2A).
AJP-Heart Circ Physiol • doi:10.1152/ajpheart.00857.2014 • www.ajpheart.org
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兰
l
Ad ⭸ p
␳c2d ⭸ t
兰
The term Cv ⫽ K1 ␳ is the integrated arterial compliance of the 1D
model vessel (Fig. 2B). We assume that the fluid inertia term
K2 dqout dt (Eq. 15B) is negligible, since peripheral inertia has a
minor effect on flow waveforms (2). Thus Eqs. 15A and 15B become
Integrating the linearized 1D Eqs. 11A and 11B along the x-axis
yields
兰
K2 ⫽
dx,
c2d
REDUCING THE NUMBER OF 1D MODEL PARAMETERS
GRANTS
We gratefully acknowledge the support of the Centre of Excellence in
Medical Engineering [funded by the Wellcome Trust and Engineering and
Physical Sciences Research Council (EPSRC) under Grant No. WT 088641/
Z/09/Z] and the National Institute for Health Research (NIHR) Biomedical
Research Centre at Guys and St Thomas’ NHS Foundation Trust in partnership
with King’s College London. M. Willemet and J. Alastruey also received
support from EPSRC Project Grant EP/K031546/1. J. Alastruey also acknowledges the support of a British Heart Foundation Intermediate Basic Science
Research Fellowship (FS/09/030/27812).
DISCLOSURES
No conflicts of interest, financial or otherwise, are declared by the author(s).
AUTHOR CONTRIBUTIONS
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