Mathematical modeling of blood coagulation processes

in intensive blood flow conditions

A.S. Rukhlenko1

K.E. Zlobina2

G.Th. Guria1,2

1 – Moscow Institute of Physics and Technology2 – National Research Center for Hematology Research

Moscow2014

Hemodynamics and rheology

Chemical kinetics of cascade reactions and processes of

polymerization

Vessel wall and surrounding tissue

Change of vessel wall permeability and of concentration of activators

Change of rheological properties of the media

Intravascular thrombus formationGeneral view to the interplay of processes

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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

• The bigger stenosis the more likely thrombosis to occur

Vessel with 50% stenosis,

where μ denotes the rate of activator substances infiltration into the vessel

"I"

Liquid state of blood is stable

Thrombus formation starts

"II"

[A.P. Guzevatykh et al., 2000]

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 differsfrom that in slow flows

Re >> 10

Microthrombi and friable clots are also formed

Blood flow may drastically alter vessel wall permeability

As a rule recirculation zones are formed behind stenoses

Re << 10 -2

Only solid thrombi are formed

Blood flow doesn’t alter vessel wall permeability

As a rule flow topology is trivial

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 μ depends on wall

shear stress γsh in picewise-linear

manner

Coefficients of polymers mass transfer

• Diffusion coefficient:

• Motility coefficient:

• where:

• denotes an average number of fibrin monomers in average-weighted 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 simulationFibrin filament growth

Scenario №1 (Re = 130, )

Results of numerical simulationTwo centers of fibrin filament growthScenario №2 (Re = 130, )

Results of numerical simulationStationary flattering fibrin filament formation

Scenario №3 (Re = 200, )

Fluttering fibrin filamentResults 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)

"I"

Liquid state of blood is stable

Thrombus formation starts

"II"

[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 № 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 № 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 poly mer networks // Proceedings of the International Conference “Instabilities and Control of Excitable Networks: From macro- to nano-systems”. 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 «pretending to completeness» of mathematical description

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...]

The weak point of this approach is a large amount of uncertainty in constant rates values

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

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 (R

chain ~ l

0(N

w K)0.5) we obtain the «gelation» criterion:

Filtration resistance of fibrin gel

• It was assumed that if Nw < N

wpol 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