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: s/pathway/pathway.php?pat hway=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:Xa V V VIIa Xa X X Va II IIa Unactivated Platelet Activated Platelet IIa

Reaction schemes, one example. Ref: P.M. Didriksen, Modelling hemostasis - a biosimulation project, internal report, Dept. 252 Biomodelling, Novo Nordisk Factor II (prothrombin): II Factor IIa (thrombin): IIa Prothrombinase complex: Xa_Va_Ta A total of 17 equations. 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. Corresponding model equations in the space Ω. Poiseuille’s flow

Boundary conditions and parameters Boundary condition x=0 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: , Boundary condition x=l: Outflow boundary conditions. Top and bottom boundary condition: No flow crossing.

Numerical results. Time = 0.6 sec. T-IIa IIaT Ta

Numerical results. Time = 5 sec. TIIa T-IIaTa

Numerical results. Time = 10 sec. T-IIa TIIa Ta

Future work: Boundary attachment of Ta Reaction schemes on Corresponding model equations on.

One simple example including diffusion and flow Reaction scheme in the bulk: Reaction scheme on parts of the boundary: Conservation of binding sites at the boundary: Boundary binding sites:

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