ISNVD Abstract Book

A model of filtration and solute
transport across the
Blood-Brain Barrier
Laura Facchini Alberto Bellin Eleuterio F. Toro
Mathematics
Civil and Environmental Engineering Civil and Environmental Engineering
University of Trento (Italy)
University of Trento (Italy)
University of Trento (Italy)
laura.facchini@unitn.it alberto.bellin@unitn.it toro@ing.unitn.it
PURPOSE
Altered venous hemodynamics affects the transport
properties of vessel walls [1], creating microbleed,
perivenular iron stores and hypoxia.
In the present study, we propose a simple
mathematical model for water and solute transport
across vessel wall.
With the help of suitable analytical and numerical
solutions of this model, we investigate
the effect of a blood pressure increase on both
water flow and molecule transport across the
Blood-Brain Barrier (BBB).
1 INTRODUCTION
In a typical (peripheral) microvessel, the lumen
side of the endothelial cells composing the vessel
wall is covered by the glycocalyx, a thin
membrane which is believed to exert a sieving
effect on macromolecules.
Figure 1. Structure of a peripheral blood vessel [modified from
http://www.hubrecht.eu/research/dekoning/research.html].
Since the BBB has the role of protecting the
Central Nervous System (CNS) from the neurotoxic
substances contained in the blood, while
nourishing the surrounding brain tissue, it must
be highly selective. Indeed, the clefts separating
adjacent endothelial cells of the BBB are
partially closed by the tight junctions. Furthermore,
the microvessels of the CNS are protected
externally by the basement membrane
and by the astrocyte feet and the pericytes.
Figure 2. BBB anatomical structure.
Under normal conditions, while macromolecules
cross the BBB through transcellular
pathways [2], water and small hydrophilic solutes
follow paracellular pathways through the
clefts separating adjacent endothelial cells [3].
Figure 3. Paracellular and transcellular transport.
The breakdown of the BBB with the associated
increase of vessel permeability has been observed
in traumatic head injury [4], Alzheimer’s
disease [5] and it has recently been hypothesised
in Multiple Sclerosis ([6], [7]).
2 MATERIALS and METHODS
From the conservation of water and solute mass
in the steady-state case (following [8]) and after
simplifications, we derive the following nonlinear
system of ordinary differential equations
{ (p ′ + rp ′′ ) − σ(π ′ + rπ ′′ )=0,
Aπ(π ′ + rπ ′′ )+rπ ′ (Bπ ′ + Cp ′ (1)
)=0.
Here
• r is the distance from the vessel axis,
• p = p(r) is the hydrostatic pressure,
• π = π(r) is the osmotic pressure,
• σ is the reflection coefficient of the wall,
• A = l p σ 2 −l d , B = l p σ−l d and C = l p (σ−1)
are physiological parameters,
• l p is a permeability coefficient,
• l d is a diffusion coefficient.
Then we calculate
P (r) =p(r) − σπ(r), (2)
q v (r) =−l p [p ′ (r) − σπ ′ (r)], (3)
q s (r) = π(r)
ϕRT [Bπ′ (r)+Cp ′ (r)] , (4)
where
• P (r) is the net pressure,
• q v (r) and q s (r) are the volume and solute
fluxes per unit length of the vessel,
• ϕ is the Stokes radius,
• R is the gas constant,
• T is the absolute temperature.
Now, using the model, we investigate the effect
of an increase of blood pressure p c on both water
flow and molecule transport across the BBB.
Anatomical parameters are obtained from published
studies on electron microscopy observations
of animal brain or mesenteric vessels.
3 RESULT
Figures 4 to 6 show our results in graphical
form.
Figure 4. Volume flux per unit length of the vessel with respect
to the blood pressure p c, with typical venular values of σ and l p
(solid curve), and altered values of σ and l p (dashed curve) simulating
the glycocalyx/BBB breakdown.
Figure 5. Solute flux per unit length of the vessel with respect
to the blood pressure p c, with typical venular values of σ and l p
(solid curve), and altered values of σ and l p (dashed curve) simulating
the glycocalyx/BBB breakdown.
Figure 6. Osmotic (π), hydrostatic (p) and net (P ) pressures
across the vessel wall, with typical values of venular blood pressure
p c (solid curves) and altered values (discontinuous curves),
in both cases of lower and higher than normal blood pressure.
4 CONCLUSIONS
• Blood pressure increase (hypertension)
causes an increase in the volume and solute
fluxes per unit length across the vessel wall.
• The glycocalyx (and thus BBB) breakdown
gives rise to an increase in both fluxes.
• We have depicted the osmotic, hydrostatic and
net pressures across the vessel wall.
• These are very preliminary results and there
is some work on more sophisticated model.
References
[1] Singh A. V. & Zamboni P., J. Cereb. Blood Flow Metab. 29(12), 1867-1878 (2009).
[2] Pardridge W. M., Molecular Biotechnology 30(1), 57-69 (2005).
[3] Hawkins B. T. & Davis T. P., Pharmacological Reviews 57(2), 173-185 (2005).
[4] Fukuda K. et al., Journal of Neurotrauma 12(3), 315-324 (1995).
[5] Farkas E. & Luiten P. G. M., Progress in Neurobiology 64(6), 575-611 (2001).
[6] Zamboni P. et al., J. Vas. Med. 50(5), 1348-1358 (2009).
[7] Tucker T. W., Med. Hypotheses 77(6), 1074-1078 (2011).
[8] Katchalsky A. & Curran P.F., Harvard Univ. Press Cambridge MA (1965).