Influence of Blood Flow on the Coagulation Cascade

Nina Marianne Andersen, Mads Peter Sørensen, Emil Sokoler

Department of Mathematics (MAT), Techn. Univ. of Denmark

Steen Ingwersen and Ole Hvilsted Olsen

Biomodelling and Haemostasis Biochemistry, Novo Nordisk, Denmark

Content:

1) Introduction, blood coagulation.

2) Perfusion experiment, cartoon model and reaction schemes in a fully stirred model.

3) Cartoon model and reaction schemes for a simplified model with diffusion and flow. Platelet activation.

4) Numerical results and relation to other models.

5) Summary and future work plan.

Ref: http://www.ambion.com/tools/pathway/pathway.php?pathway=Blood%20Coagulation%20Cascade

Cartoon of the blood coagulation pathway.

Perfusion experiments and modelling

Perfusion chamber

Top glass lid coated with collagen

Thrombocytes (platelets), red and white blood cells.

Factor X in the fluid phase

X

Factor VIIa in the fluid phase

VIIa

Active thombocytes (Ta) binds to the collagen coated lid. vWF.

Reconstructed blood.

Content: Thrombocytes (T), Erythrocytes.

[T] = 14 nM (70,000 platelets / μ litre blood)

Cartoon model of the perfusion experiment

Activated Platelet

Va:XaVVIIa XaX Va

II IIaIIa

Unactivated PlateletUnactivated Platelet Activated PlateletActivated PlateletIIaIIa

IIaIIa

Reaction schemes, one example.

Ref: P.M. Didriksen, Modelling hemostasis - a biosimulation project, internal report, Dept. 252 Biomodelling, Novo Nordisk

TaIITaII 10Ta

10F

IIaTaXaTaXaTaII 16R

IIaTaVaXaTaVaXaTaII 3S

Factor II (prothrombin): II

Factor IIa (thrombin): IIa

Prothrombinase complex: Xa_Va_Ta

A total of 17 equations.

TaIITaVaXaSTaIITaXaRdtTadII

316

TaIIFTaIITa 1010

11750016 sMR

11103

7 sMS

1110

sF

114300010

sMTa

Reaction rates:

Numerical results.

T

VIIa Ta

IIa

Initial conditions: FVIIa = 50 nM FX = 170 nM T = 14 nM sTa = 0.1*14 nM FII = 0.3 nM

Reaction diffusion model with convection

Reaction scheme for T, Ta and IIa.

IIaTasTaIIaT 6T

7T

Corresponding model equations in the space Ω.

))((27 TayvTaDsTaT

dtdTa

Ta

))((26 TyvTDIIaTT

dtdT

T

))((276 sTayvsTaDIIaTaTIIaTT

dtdsTa

sTa

))((267 IIayvIIaDIIaTTsTaT

dtdIIa

IIa

Poiseuille’s flow

)/1()( Hyayyv

Boundary conditions and parameters

Boundary condition x=0 )(102.1 16 yfnMIIa

Ref.: M. Anand, K. Rajagopal, K.R. Rajagopal. A Model Incorporating some of the Mechanical and Biochemical Factors Underlying Clot Formation and Dissolution in Flowing Blood. Journal of Theoretical Medicine, 5: 183-218, 2003.

)(1014 29 yfnMT

0Ta 0sTa

Boundary condition x=l: Outflow boundary conditions.

Top and bottom boundary condition: No flow crossing.

Numerical results. Time = 0.6 sec.

T-IIa

IIa

T

Ta

Numerical results. Time = 5 sec.

T IIa

T-IIa Ta

Numerical results. Time = 10 sec.

T-IIa

T IIa

Ta

Future work: Boundary attachment of Ta

Reaction schemes on

Corresponding model equations on.

TaBTa TaBT

IIaTaBCTaBII 2

2k 3k

4k 5k

TaBIIkdtdII 4

25 CkdtdIIa

TkCkTaBIIkTakdtdTaB 32542

2542 CkTaBIIk

dtdC

One simple example including diffusion and flow

xy

P

PECES Reaction scheme in the bulk: Reaction scheme on parts of the boundary:

BPBP

0BBPB Conservation of binding sites at the boundary:

Boundary binding sites: B

Including pro-coagulant and anti-coagulant thrombin

Ref.: V.I. Zarnitsina et al, Dynamics of spatially nonuniform patterning in the model of blood coagulation, Chaos 11(1), pp57-70, 2001.

E.A. Ermakova et al, Blood coagulation and propagation of autowaves in flow, Pathophysiology og Haemostasis and Thrombosis, 34, pp135-142, 2005.

Model consisting of 11 PDEs in 2+1 D, including diffusion

Summary and future work

1. Modelling of perfusion experiment for blood coagulation.

2. Reduced PDE model including blood flow and diffusion.

3. Modelling of attachment of activated thrombocytes on collagen coated boundary.

4. Full PDE model.

5. Model of in vivo blood coagulation.

Simpel Model consisting of 3 PDEs, including diffusion