ISNVD Abstract Book

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Simple mathematical model for the
hemodynamics of extracranial veins and
effect of posture
Lucas O. Müller 1 *, Gino I. Montecinos 1 , Laura Facchini 1 , Eleuterio F. Toro 1
1 Laboratory of Applied Mathematics, University of Trento, Italy
Purpose
Results
We present a simple mathematical model for the main extracranial cerebral venous
return pathways. The model reproduces physiological patterns of flow distribution
in supine and upright positions. Moreover, we identify which alterations of these
pathways might be associated to abnormal flow distributions recently linked to multiple
sclerosis (MS).
Physiologic behaviour
The model reproduces physiological inversion of cerebral venous return flow distribution
among IJVs and the Vple.
1.5
1.5
Introduction
α = A A 0
1.0
0.5
0.0
0.00 0.05 0.10 0.15
α = A A 0
1.0
0.5
0.0
0.00 0.05 0.10 0.15
1.0
1.0
In supine position cerebral venous return occurs mainly via the internal jugular
veins (IJVs), whereas in upright position the flow is redirected to the venous vertebral
plexus (Vple) due to the collapse of IJVs 1 . A recent work by Monti et al. 5
relates a dysfunction in cerebral venous return to MS patients. The described
anomaly regards an increased flow via IJVs and vertebral veins (VVs) in upright
position with respect to flow through these veins in supine position. Moreover, such
a dysfunction indicates altered venous hemodynamic behaviour, linking these findings
to Chronic Cerebro-Spinal Venous Insufficiency 6 (CSSVI).
Materials and Methods
We consider a simple network of one-dimensional vessels for the main pathways of
cerebral venous return. IJVs are modelled as two collapsible vessels, whereas the
Vple is considered as a rigid vessel. Our model is an improvement of the lumped
parameter model presented by Gisolf et al. 4 .
CBF
CBF
0.5
0.0
0.00 0.05 0.10 0.15
x [m]
CBF
0.5
0.0
0.00 0.05 0.10 0.15
x [m]
Figure 2: Subject in supine (left) and in upright position (right). Low values of α indicate collapse,
while values close to unity or higher indicate vessel distention. In supine position IJVs are distended
an the majority of the CBF passes through them. In upright position IJVs collapse and the flow is
redirected to the Vple, which does not collapse due to its greater wall stiffness.
Increased IJV wall stiffness
Flachenecker et al. 3 found a correlation between MS and cardiovascular autonomic
dysfunction. Such a dysfunction may alter vasoconstriction control. Our model
confirms that changes in the IJVs wall stiffness may invert flow distribution. An
increase from 50 to 2000 Pa was sufficient for inverse flow distribution.
Increased CVP
An increased CVP (from -2 to 10 mmHg) in upright position leads to an inversion in
flow distribution. Moreover, Charkoudian et al. 2 reported that increased CVP may
alter baroreflex control of sympathetic activity in humans. This would result in a
combination of smaller increases of CVP and wall stiffness for determining inverse
flow distribution.
2
1.5
α = A A 0
1.0
α = A A 0
1
0.5
0.0
0.00 0.05 0.10 0.15
0
0.00 0.05 0.10 0.15
1.0
1.0
CBF
0.5
CBF
0.5
LIJV
Vple
RIJV
0.0
0.00 0.05 0.10 0.15
x [m]
0.0
0.00 0.05 0.10 0.15
x [m]
CVP
Figure 1: Simplified extracranial venous network. CBF: cerebral blood flow; CVP: central venous
pressure; Vple: vertebral venous plexus, IJV: internal jugular veins (left and right).
We impose mass conservation and momentum balance for each vessel solving the
following equations ⎧⎨
∂ t A + ∂ x (uA) =0,
2
⎩ ∂ t (uA)+∂ x
(ˆαAu ) + A ρ ∂ (1)
xp = Agsin θ − f .
A(x, t) is the cross-sectional area of the vessel, u(x, t) is the cross-sectional averaged
axial velocity, p(x, t) is the average internal pressure over the cross-section,
g is the acceleration due to gravity, θ is the angle of the vessel with respect to
supine position and f (x, t) is the friction force due to viscous stresses.
A closure is necessary in order to link p(x, t) to A(x, t). This is done via a so called
tube law
[ (A(x, ) m ( ) ] n t) A(x, t)
p(x, t) =p e + K (x)
−
. (2)
A 0 (x) A 0 (x)
p e is the external pressure, m and n are numbers to be specified. A 0 (x) is the
equilibrium cross-sectional area and K (x) is the bending stiffness of the vessel
wall
[ ] 3
E(x) h0 (x)
K (x) =
12(1 − ν 2 . (3)
) R 0 (x)
E(x) is the Young modulus whereas h 0 (x) and R 0 (x) are the equilibrium values for
wall thickness and vessel radius, respectively. Upstream boundary conditions are
obtained by solving a non-linear system arising at the junction of the three vessels,
with a mass source term given by cerebral blood flow (CBF), set to 750 ml min −1 in
supine position and reduced by 10 % in upright position. The downstream boundary
condition is given by fixing the central venous pressure (CVP) value.
We simulate:
- postural changes: variation of θ in (1)
- changes of vessels wall stiffness: by variation of E in (3)
- stenosis: variation of A 0 in (2) along the vessel
- increased CVP: variation of downstream boundary conditions
- blocked Vple
Figure 3: Upright position for increased IJVs wall stiffness (left) and for increased CVP (right). In
both cases, even if there is a reduction of the cross-section area of IJVs, no collapse occurs and
the main portion of the flow still passes via these veins.
IJV Stenosis
No inversion was observed for stenosis in one or both IJVs. In any case, some
considerations should be made:
- IJV stenosis is expected to have a more significant role in supine position, condition
that can not be assessed by our model because of the simplicity of the network
- IJV stenosis might be generally associated to alteration of overall vessel mechanical
properties and might therefore be in any case linked to flow distribution
inversion in upright position.
Blocked Vple
Due to the simplicity of the network the consequences of such a pathology are
obvious. IJVs collapse but the cerebral venous return still occurs entirely via the
IJVs.
Conclusions
Our simplified model reproduces physiological flow distribution of cerebral venous
return in supine and upright positions.
Pathologies that could lead to an inversion of flow distribution are:
- increased IJVs wall stiffness
- increased CVP
- blocked Vple
IJV stenosis does not imply pathological flow inversion but this might be a consequence
of the simplicity of the model.
Current/future work concerns the creation of a closed loop multi-scale model of
the complete cardiovascular network. Such a model will allow a detailed study of
CCSVI 6 implications in venous hemodynamics.
References
[1] Alperin N. et al., JMRI-J MAGN RESON IM 22, 591-596 (2005).
[2] Charkoudian N. et al., AM J PHYSIOL-HEART C 287, H1658-H1662 (2004).
[3] Flachenecker P. et al., J Neurol 246, 578-586 (1999).
[4] Gisolf J. et al., J PHYSIOL 560, 317-327 (2004).
[5] Monti L. et al., PLoS ONE 6 (2011).
[6] Zamboni P. et al., J Neurol Neurosurg Psychiatry 80, 392-399 (2009).
University of Trento *lucas.mueller@ing.unitn.it Via Mesiano 77, 38123 Trento, Italy