VALIDATION OF 1D MODEL OF THE SYSTEMIC ARTERIAL TREE
INCLUDING THE CEREBRAL CIRCULATION
Philippe Reymond (1), Fabrice Merenda (1), Fabienne Perren (2),
Daniel Rüfenacht (3) and Nikos Stergiopulos (1)
(1)
Laboratory of Hemodynamics and Cardiovascular Technology,
Ecole Polytechnique Fédérale de Lausanne EPFL, Switzerland
(2)
Neurology Service, Clinical Neurosciences Department,
Hôpitaux Universitaires de Genève HUG, Switzerland
(3)
Neurointerventional Service, Clinical Neurosciences Department,
Hôpitaux Universitaires de Genève HUG, Switzerland
METHODS
Abstract: Blood flow phenomena play an important
role in cerebrovascular disease. The wall shear stress
acting on the endothelial cells is linked to growth and
possible rupture of the aneurysm wall and seems to
play an important role in atherogenesis and the
stability of plaques. Clinical assessment of
hemodynamical forces within the cerebral
circulation is still difficult because pressure can be
measured only invasively and flow, especially in
small deep intracranial vessels cannot be measured
directly. This renders the modeling of blood flow
within the cerebral circulation an attractive
alternative.
The one-dimensional form of the fluid equations was
applied over each arterial segment. A non-linear
viscoelastic constitutive law for the arterial wall was
considered. The arterial tree dimensions and
properties were taken from the literature and
completed with real patient scans and coupled to a
heart model. To validate model predictions, we
performed non-invasive measurements of pressure
and flow waves. Pressure was measured with
tonometry and cerebral blood flow velocities with
transcranial ultrasound and gated phase contrast
MRI.
The model predictions are in good qualitative
agreement with in-vivo measurements. The results
obtained here allow us to obtain pressure and flow in
central arteries as well as in major arteries of the
brain, validating thus the general applicability of the
model.
The 1D forms of the continuity and momentum
equations are applied over a tapered arterial segment.
The convective non linear term in the longitudinal
momentum equation is computed according to the
unsteady velocity profile from the Witzig-Womersley
theory [4]. The same theory is used to model the intimal
shear stress τw.
τ w (t ) = −
4μ
πri 3
q harm 1
⎧
⎫
⎪
⎪
⎪
⎪
⎪
⎪
3/2
⎪
⎪
⎪
⎪
J
α
i
1
μ
1
⎪
⎪
3/2
i ϖt ⎪
⎪
+ Re ⎨ ∑
e ⎬
q harm α i
3
3/2
⎪ harm πri
⎪
2 J (α i )
J0 αi
⎪
⎪
⎪
⎪
1−
⎪
⎪
⎪
⎪
α i J (α i )
⎪
⎪
⎪
⎪
⎩
⎭
(
(
)
)
3/2
1
3/2
3/2
0
Where qharm(z,t) are the volumetric flow rate harmonics,
J0, J1 are the Bessel functions of first kind and α is the
Womersley number.
A non-linear viscoelastic constitutive law for the
arterial wall is considered, according to Holenstein et al
[5].
The arterial tree is coupled to the heart, which is
modeled using the time varying elastance model. This
allows to take into account parameters like heart rate
HR, contractility and changes in preload.
pLV (t ) = ⎡⎣VLV (t ) −VD ⎤⎦ E *(t )[1 − κQ(t )]
pLV is the left ventricular pressure, E* the isovolumic
elastance and Vd the dead volume of the left ventricle.
All distal vessels and vascular beds are terminated
with three-element windkessel models to account for the
proximal, distal resistance and compliance of the distal
small arteries, arterioles and capillaries.
Main coronary arteries are modeled assuming a
systolic flow impediment dependent on the varying
elastance of the ventricles Krams [6].
INTRODUCTION
The aim of this study is to develop a distributed
model of the entire systemic arterial tree, coupled to a
heart model and including a detailed description of the
cerebral arteries. Distributed models of the arterial tree
have been studied extensively in the past (Avolio [1],
Stergiopulos et al [2], Westerhof et al [3]), however, no
model has been developed so far that offers a
physiologically relevant coupling to the heart and
includes the entire cerebral arterial tree.
1
A
B
Figure 1. Arterial tree of major systemic arteries, based
of the Stergiopulos et al. [2] tree (A) connected to a
detailed cerebral arterial tree added (B).
The arterial tree dimensions and properties were
taken from Stergiopulos [2] and extended to include a
detailed description of the cerebral arterial tree (Fig. 1).
Geometry of the cerebral arterial network was obtained
from real patient scans, whereas their elastic properties
were taken from values reported in the literature [7].
The set of equations with the boundary conditions
described above are solved using an implicit finite
difference scheme to yield pressure and flow waveforms
over the entire arterial tree.
To validate the model predictions for the main
systemic arteries and cerebral circulation, we performed
noninvasive measurements in young volunteers.
Pressure waveforms were measured using applanation
tonometry on carotid and superficial temporal arteries.
Flow rate waveforms were measured in common,
internal carotid and middle cerebral arteries using
transcranial Doppler and in systemic arteries (Asc. ao.,
Thoracic ao., Abdominal Ao., Iliac a., Femoral a. and
Common Carotid a.) using gated phase contrast MRI.
Figure 2: Model predictions (bottom panels) compared
to in vivo measurements of flow and pressure waves
(top panels) at various systemic and cerebral arteries.
Flow is measured with transcranial ultrasound in the
MCA (A) and with MRI in the main systemic arteries
(C, D, E and F). Pressure was measured with
applanation tonometry in the superficial temporal artery
(B).
RESULTS AND DISCUSSION
The model predicts pressure and flow waves which
are in good qualitative agreement with in-vivo
measurements (Fig. 2). The agreement is especially
good for the shape and wave details of the flow wave,
where all features are reproduced in a rather faithful
manner.
The results obtained here allow us to that model
predictions of pressure and flow in central arteries as
well as in major arteries of the brain are in good
agreement with measurements, validating thus the
general applicability of the 1D model to the entire
systemic and cerebral circulation.
2
when the pressure gradient is known," The Journal of
physiology, 127(3), pp. 553-563.
5. Holenstein, R., Niederer, P., and Anliker, M., 1980,
"A viscoelastic model for use in predicting arterial pulse
waves," Journal of biomechanical engineering, 102(4),
pp. 318-325.
6. Krams, R., Sipkema, P., and Westerhof, N., 1989,
"Varying elastance concept may explain coronary
systolic flow impediment," The American journal of
physiology, 257(5 Pt 2), pp. H1471-1479.
7. Hayashi, K., Handa, H., Nagasawa, S., Okumura, A.,
and Moritake, K., 1980, "Stiffness and elastic behavior
of human intracranial and extracranial arteries," Journal
of biomechanics, 13(2), pp. 175-184.
REFERENCES
1. Avolio, A. P., 1980, "Multi-branched model of the
human arterial system," Med Biol Eng Comput, 18(6),
pp. 709-718.
2. Stergiopulos, N., Young, D. F., and Rogge, T. R.,
1992, "Computer simulation of arterial flow with
applications to arterial and aortic stenoses," Journal of
biomechanics, 25(12), pp. 1477-1488.
3. Westerhof, N., Bosman, F., De Vries, C. J., and
Noordergraaf, A., 1969, "Analog studies of the human
systemic arterial tree," Journal of biomechanics, 2(2),
pp. 121-143.
4. Womersley, J. R., 1955, "Method for the calculation
of velocity, rate of flow and viscous drag in arteries
3