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Nucleation of Fibrin Filaments in Intensive Blood Flow

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Mathematical modeling of
blood coagulation processes
in intensive blood flow
conditions
A.S. Rukhlenko1
K.E. Zlobina2
G.Th. Guria1,2
– Moscow Institute of Physics and Technology
2 – National Research Center for Hematology Research
1
Moscow
2014
Intravascular thrombus formation
General view to the interplay of processes
Change of rheological
properties of the
media
Chemical kinetics of cascade
reactions and processes of
polymerization
Change of vessel wall
permeability and of
concentration of activators
Vessel wall and surrounding
tissue
Vessel wall
elasticity
Convection
and diffusion
Wall shear stress and
transmural pressure
Hemodynamics and rheology
Intravascular clot growth at low Reynolds number
[A.L. Chulichkov et al., 2000]
Intravascular clot growth in slow blood blow
(Re<<1)
Results of numerical simulations
[A.L. Chulichkov et al., 2000]
Intravascular clot growth in slow blood flow
(Re<<1)
Results of numerical calculations
[A.P. Guzevatikh et al., 2000]
Structure of diagram of liquid state of blood
stability (Re<5*10 -3)
• Retarding of blood flow
promotes activation of
blood coagulation
Thrombus formation
starts
"II"
• The bigger stenosis the
more likely thrombosis to
occur
"I"
Liquid state of blood
is stable
Vessel with 50% stenosis,
[A.P. Guzevatykh et al., 2000]
where Ој denotes the rate of activator
substances infiltration into the vessel
Formation of polydisperse microthrombi
•
It was shown previously in our lab [Uzlova et
al., 2008], that in intensive flow (Re ~ 100) a
number of stages of microthrombi
formation and growth precedes formation of
large thrombi
•
Clouds of microthrombi may be detected by
ultrasound techniques
•
Theoretical investigation of formation of
friable clots without sharp phase border is
the subject of present work
[Uzlova S., Guria K., Guria G., 2008; Uzlova S.G., Guria K.G.,
Shevelev A.A. et al., 2008]
Influence of blood flow on vessel wall
permeability
•
Experimental in vitro [Warboys et al., 2010, Jo et al., 1991, McIntire et al., 1995,
Sill et al., 1995], and in vivo (а также ex vivo) [Kim et al., 2005, Lever et al.,
1992, Williams, 1999, 2003] investigations give evidence for the fact that rise of
wall shear stress may lead to reversible growth of endothelium
permeability up to orders of magnitude for some substances [McIntire
et al., 1995].
• Rise of wall shear stress may also lead to irreversible
permeability growth e.g. due to the rupture of fibrous cap of
atherosclerotic plaque. Comparative researches show [Gertz and
Roberts, 1990, Fukumoto et al., 2008, Lovett and Rothwell, 2003, Slager et al.,
2005] that atherosclerotic plaque rupture happens predominantly in the
high wall shear stress zones.
Pattern of thrombus formation processes
development in intensive flows
qualitatively differs
from that in slow flows
Re << 10 -2
Re >> 10
Only solid thrombi are formed
Microthrombi and friable clots
are also formed
Blood flow doesn’t alter vessel wall
permeability
Blood flow may drastically alter
vessel wall permeability
As a rule flow topology is trivial
As a rule recirculation zones are
formed behind stenoses
Blood coagulation in high-Reynolds flows
• Wall shear stress in intensive flow in
stenosed vessel may drastically
change the permeability of the
vessel
• This situation refers to one of the
most dangerous disease –
atherosclerosis and subsequent
intravascular thrombosis
• We analyzed simplified case of 2D
vessel geometry (length Lx=7.5 cm,
width Ly=1 cm)
Blood flow description
• αp
reflects the influence of the fibrin polymers network
on blood flow (Darcy law)
Blood flow description
• αp
reflects the influence of the fibrin polymers network
on blood flow (Darcy law)
Fibrin polymerization kinetics
• Association and fragmentation processes:
• Master equations:
• Statistical moments:
• Average number of monomers in polymer molecules:
[Guria, Herrero, Zlobina, 2009]
Fibrin polymerization kinetics
• Kinetics of polymerization in terms of moments:
• Cutoff assumption:
[Guria, Herrero, Zlobina, 2009]
Blood coagulation kinetics description
[Guria, Herrero, Zlobina, 2009; Guria, Herrero, Zlobina, 2010]
Blood coagulation kinetics description
[Guria, Herrero, Zlobina, 2009; Guria, Herrero, Zlobina, 2010]
Blood coagulation kinetics description
[Guria, Herrero, Zlobina, 2009; Guria, Herrero, Zlobina, 2010]
Infiltration of procoagulant substances into
the blood flow
• It is supposed that procoagulant
substances are able to infiltrate into
the blood flow from surrounding
tissue (variable u)
• It is supposed that vessel wall
permeability for the procoagulant
substances Ој depends on wall
shear stress Оіsh in picewise-linear
manner
Coefficients of polymers mass transfer
• Diffusion coefficient:
• Motility coefficient:
• where:
•
denotes an average number of fibrin monomers in averageweighted polymer clot
• Condition
refers to semi-diluted solution
[de Gennes, 1979 // Scaling concepts in polymer physics]
Under-threshold activation
• The primary activator is infiltrated into the blood flow from the
surrounding vessel tissue through the zone of high wall shear
stress
• As a result of activation of blood coagulation system clouds of
microthrombi are formed in the blood
• Mainly they are accumulated in recirculation zone
Under-threshold activation
• The primary activator is infiltrated into the blood flow from the
surrounding vessel tissue through the zone of high wall shear
stress
• As a result of activation of blood coagulation system clouds of
microthrombi are formed in the blood
• Mainly they are accumulated in recirculation zone
Presumable centres of initiation
of clot growth
• An areas with local maximum of microthrombi concentrations
seem to be the most probable places for initiation of clot
growth processes
Results of numerical simulation
Fibrin filament growth
Scenario в„–1 (Re = 130,
)
Results of numerical simulation
Two centers of fibrin filament growth
Scenario в„–2 (Re = 130,
)
Results of numerical simulation
Stationary flattering fibrin filament formation
Scenario в„–3 (Re = 200,
)
Fluttering fibrin filament
Results of experiments
Uzlova S., Guria K., Guria G. Acoustic determination of early stages of intravascular blood coagulation //
Philos Trans R Soc A. — 2008. — Vol. 366. — P. 3649–3661
Fibrin fibres
Flotating friable fibrin structures
Fibrin clots
Parametric plane relevant to
activation of clot growth (Re > 10)
Parametric plane relevant to
activation of clot growth (Re<5*10 -3)
Thrombus formation
starts
"II"
"I"
Liquid state of blood
is stable
[Guzevatykh et al., 2000]
Parametric plane relevant to
activation of clot growth
Thrombus
formation starts
"II"
"I"
Liquid state of
blood is stable
The influence of stenosis shape
•
•
The influence of stenosis shape on activation
threshold
Scaling power law в„– 1
• It was found that at the fixed Reynolds number the lag time
of clot growth depends on the value of parameter
in
following way:
where (
) corresponds to the distance of
representative point at the parametric plane to the border of
liquid state stability
Scaling power law в„– 2
• It was found that at the fixed values of
(
)
the lag time of clot growth depends on the value of Reynolds
number in a following way:
where (Re - Recrit) corresponds to the distance of
representative point at the parametric plane to the border of
liquid state stability
Probable biomedical significance of the
presented results
1. Activation of blood coagulation may happen both due to blood
flow intensification (e.g. as a result of blood pressure rise)
and due to its retarding (e.g. due to blood pressure drop).
2. The most thrombogeneous are medium sized atherosclerotic
plaques
3. Detection of fibre-like structures in medical practice (e.g. by means
of ultrasound techniques) has to be considered as early
predictor of subsequent thrombosis.
Relevant publications
• A.S. Rukhlenko, K.E. Zlobina, G.Th. Guria. Hydrodynamical activation of
blood coagulation in stenosed vessels. Theoretical analysis // Computer
Research and Modeling, 2012, vol. 4, no. 1, pp.155–184
• A.S. Rukhlenko, O.A. Dudchenko, K.E. Zlobina, G.Th. Guria. Threshold
activation of blood coagulation as a result of elevated wall shear stress //
Proceedings of MIPT. — 2012. — V. 4, N 2.
• Rukhlenko A. S., Zlobina K. E., Guria G. T. Threshold activation of blood
coagulation cascade in intensive flow and formation of fibre-like fibrin
polymer networks // Proceedings of the International Conference
“Instabilities and Control of Excitable Networks: From macro- to nanosystems”. Moscow: MAKS Press, 2012. Pp. 113–125.
• Г.Т. Гурия. Как теоретическая физика трактует свертывание крови? //
Наука, 2011, № 9, с. 51-57
Authors are grateful to following persons
• Academician A.I. Vorobjev
• O.A. Dudchenko
• S.G. Uzlova, K.G. Guria
• I.A. Romanets, A.R. Gagarina, D.A. Ivlev, O.A. Starikovskaya
• The work was partially supported by ISTC grant #3744
Many thanks!
Blood Coagulation Cascade Graph
[Guria G.Th., 2002; Uzlova et al. 2008 // Phil Trans Royal Soc A]
Coagulation cascade – threshold
• Blood coagulation
cascade is activated
in threshold manner
• Activation of the BCC
could be achieved
parametrically or
dynamically
• Blood is metastable
under normal
physiological conditions
[Г.Т. Гурия. 2011 // Наука., № 9, с. 51-57]
Problems of В«pretending to completenessВ»
of mathematical description
пЃ¬
There is a number of recently developed models operating with large number of
variables and parameters (i.e. much more than 10):
[Anand et al., 2003, 2005, 2008, Ataullakhanov and Panteleev, 2005, Shibeko et al., 2010,
Jones and Mann; Leiderman and Fogelson, 2011, Hockin et al., 2002, Butenas et al.,
2004; etc...]
пЃ¬
The weak point of this approach is a large amount of uncertainty in constant
rates values
пЃ¬
It was shown in [Wagenvoord, Hemker, Hemker, 2006; Hemker, Kerdelo, Kremers, 2012]
that up-do-date level of experiment-based data does not let to construct verifiable
mathematical models of coagulation with large number of variables and
parameters
пЃ¬
That’s way in present work we have limited ourselves to the use of
qualitative (i.e. phenomenological) mathematical models of
blood coagulation cascade
Gelation criterion
• It was assumed that fibrin gel was formed when the fibrin
solution became half-diluted [de Gennes, 1979]
• This means that neighboring polymer chains start to
interweave each other
• We assume that formation of polymer chains happens on
chemical impurities (presumably phospholipids) with
concentration n
• Assuming that fibrin polymer chains behave as ideal chains
(Rchain ~ l0(Nw K)0.5) we obtain the В«gelationВ» criterion:
Filtration resistance of fibrin gel
• It was assumed that if Nw < Nwpol fibrin polymer chains do not
alter blood flow
• The filtration resistance of porous media is known to depend
on the mesh size:
• The mesh size in half-diluted solution was estimated as (by
[de Gennes, 1979]):
Diffusion and convection of polymer chains
• To describe fibrin chains diffusion in the system the following
asymptotic expression was used:
• It gives well-known dependencies of diffusion on Nw when:
• To take into account decrease of convective mass transfer
due to chains interweaving following expression was used:
Computational mesh example
Zone 1 – located nearby the proximal end of
separation line
Zone 2 – center of recirculation zone
Zone 3 – located nearby the distal end of
separation line
The local maximum of microthrombi concentration in zone
3 was not resolved by numerical calculations
Presumable nucleation centres - summary
•The local maximum of activator (thrombin) concentration
in zone 1 is much more than ~3 orders of magnitude higher
than in zone 2
•The same is true for second momentum (M2) of fibrin
distribution
This means that zone 1 should be treated as primary
nucleation zone in the system considered
Clot growth in non-convective conditions
• In non-convective conditions the clot grows with constant speed
• Such behavior is caused by thrombin autowave
F. I. Ataullakhanov and G. T. Guria. Spatial aspects of human blood clotting dynamics I. Hypothesis. Biophysics, 39(1):89–96, 1994
Ovanesov M.V. – PhD thesis, 2002
Runyon M., Kastrup C., Johnson-Kerner B. et al. Effects of Shear Rate on Propagation of Blood Clotting Determined Using Microfluidics and Numerical
Simulations // JACS. 2008. Vol. 130. Pp. 3458–3464.
Modelling of atherosclerotic plaque growth
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