Content: Development of medicine and mathematical modelling as an aid. Blood c oagulation.

1. Mathematics in Biology and Pharmaceutical Industry Industrial Mathematics course, University of Warsaw, Poland, May 2011. Mads Peter Sørensen DTU Mathematics, Technical University of Denmark, KongensLyngby, Denmark Content: Development of medicine and mathematical modelling as an aid. Blood coagulation. Lumped models. Enzyme kinetics. Inhibition and cooperative effects. Influence of flow and diffusion. Computational methods. The finite element method.

2. Collaborators Nina Marianne Andersen, DTU Mathematics and Novo Nordisk Ole Steen Ingwersen, Biomodelling, Novo Nordisk. Hvilsted Olsen, Haemostasis biochemistry, Novo Nordisk. Julie RefsgaardLawaetz Tudor Gramada, DTU Mathematics Kristian Rye Jensen, DTU Mathematics

3. Developmentcosts for new medicin Ref.: Erik Mosekilde, Ingeniøren 10. oktober, side 9, (2008). EU Network of Excellence BioSim. http://biosim-network.eu

4. Developmentprocess for new medicine 1) Discovery. 2) Pre-clinical tests Ideas, hypothesis, research. Animal models. Animal experiments. Developmentphase. Animal experiments. Protocolfor safety and effektiveness. Functionmechanisms and potential poissiones of organs.

5. 4) Approval. 3) Clinical tests. Regulatingauthorities. Approval of drug. Marketing authorization. Safe and effective medicin. Aproval from the regulatingauthorities. Tests onhumans. Tests for safety and effectiveness. >50% of the development time. 1 out of 10-15 medicamentssurvivesuntillphase3 5) Control. Medicament supervision

6. Mathematicalmodelling as a tool for development of new drugs Developmentcosts for a new drug is typically 200,000 USD up to 1 billion USD. Development time: 10 – 15 years. Application of modelling and computer simulation tools for the development of new medicine. Complexity. More rational and faster developmentprocesses at reducedfinancialcosts. Improvedtreatment of patients. Better, more safe and a more individualtreatment. Reduction of applications of animalexperiments. Computer model of humans.

7. Disorders of Coagulation Hypercoagulation: Cardiovascular diseases: Arthroscleroses Emboli and thrombi development • Hypocoagulation: • Hemophilia A • Hemophilia B • Others

8. Cartoon of the blood coagulation pathway I. Ref: http://www.ambion.com/tools/pathway/pathway.php?pathway=Blood%20Coagulation%20Cascade

9. Cartoon of the blood coagulation pathway II. Ref.: J. Keener and J. Sneyd, Mathematical Physiology, Springer, (1998).

10. Cartoon of the blood coagulation pathway III. Ref.: J. Müller, et al., Tolerance and threshold in extrensic coagulation system, Mathematical Biosciences 211, pp. 226-254, (2008).

11. Perfusion experiment and modelling Perfusion chamber Active thrombocytes(Ta) binds to a collagen coated lid. vWF. Glass lid coated with collagen Factor X in the fluid phase X Thrombocytes(blood platelets), red an white blood cells. Factor VIIain the fluid phase VIIa Reconstructed blood. Content: Thrombocytes(T), Erythrocytes. [T] = 14 nM (70,000 blood platelets / μlitreblood)

12. Cartoon model of the perfusion experiment. Model IV. UnactivatedPlatelet ActivatedPlatelet IIa IIa II IIa Va:Xa VIIa X Xa V Va Activated Platelet

13. Cartoon of the blood coagulation pathway. Model V. Ref.: Julie Refsgaard Lawaetz (master thesis 2010) and Nina Marianne Andersen (PhD thesis 2011).

14. Enzymekinetics Reaction scheme: Reaction equations: Note that:

15. Enzymekinetics Scaling: Mathematical model: Quasi steady state approximation: Ref.: J. Keener and J. Sneyd, Mathematical Physiology, Springer, New York, (1998). M.G. Pedersen, A.M. Bersani and E. Bersani, Jour. of Math. Chem. 43(4), pp1318-1344, (2008).

16. Competitive inhibition Reaction scheme: Inclusion of flow and diffusion: Diffusion constant: Convective flow velocity: Reaction scheme at the boundary: Binding sites on boundary:

17. Two dimensional examplewith flow, diffusion and binding sites on the boundary Bindings sites on the boundary:

18. Thrombin generation and platelet deposition enhanced by rFVIIa Ref.: Lismann et al., Journal of Thrombosis and Haemostasis 3, pp 742-751 (2005). M.G. Pedersen, A.M. Bersani and E. Bersani, Jour. of Math. Chem. 43(4), pp1318-1344, (2008).

19. The perfusion chamber Ref.: N.M. Andersen et al. Modelling of the Blood Coagulation Cascade in an in Vitro Flow System. Int. Jour. of Biomathematics and Biostatistics, 1(1), pp 1.7, (2010).

20. The perfusion chamber t=1

21. The perfusion chamber t=5

22. The perfusion chamber t=10

23. The perfusion chamber t=300

24. The perfusion chamber Ref.: M. Efendiev, N.M. Andersen et al. Submitted to Complexus, advances in mathematical sciences and applications.

25. The perfusion chamber

26. The perfusion chamber

27. Simulations of the perfusion chamber experiment Concentration of activated platelets, Ta • Plottet after 5 minutes for four dosis of factor rVIIa (NovoSeven) Ref.: Julie Refsgaard Lawaetz, Mathematical Modelling of the Blood Coagulation Cascade under Flow Conditions. Master thesis. (2010).

28. Simulations– localized collagen site Concentration of T, Ta og IIa • Plottet after5 minutes, constant dosis faktor rVIIa (NovoSeven)

29. Simulations– Reduction of cross section area Concentration af T, Ta, flow velocity v and presure p • Plottet after 5 minutes, constant dosis faktor rVIIa (NovoSeven)

30. Simulations– Reduction of cross section area Concentration of bounded platelets. TaB • Plottet every half minutes, constant dosis faktor rVIIa (NovoSeven)

31. Surface coverage of activated platelets as function of rFVIIa.Blue bars: experimental results. Red bars: Simulations. Ref.: Nina Marianne Andersen, In Silico Models of Blood Coagulation. PhD Thesis. (2010).