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Content: 1)Development of medicine and mathematical modelling as an aid. 2)Blood coagulation. 3)Lumped models. 4)Enzyme kinetics. Inhibition and cooperative effects. 5)Influence of flow and diffusion. 6)Computational methods. The finite element method. Mathematics in Biology and Pharmaceutical Industry Mads Peter Sørensen DTU Mathematics, Technical University of Denmark, Kongens Lyngby, Denmark Industrial Mathematics course, University of Warsaw, Poland, May 2011.

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Collaborators 1)Nina Marianne Andersen, DTU Mathematics and Novo Nordisk 2)Ole Steen Ingwersen, Biomodelling, Novo Nordisk. 3)Hvilsted Olsen, Haemostasis biochemistry, Novo Nordisk. 4)Julie Refsgaard Lawaetz 5)Tudor Gramada, DTU Mathematics 6)Kristian Rye Jensen, DTU Mathematics

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

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Development process for new medicine Ideas, hypothesis, research. Animal models. Animal experiments. 1) Discovery. Development phase. Animal experiments. Protocol for safety and effektiveness. Function mechanisms and potential poissiones of organs. 2) Pre-clinical tests

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Aproval from the regulating authorities. Tests on humans. Tests for safety and effectiveness. >50% of the development time. 1 out of 10-15 medicaments survives untill phase 3 3) Clinical tests. Regulating authorities. Approval of drug. Marketing authorization. Safe and effective medicin. 4) Approval. Medicament supervision 5) Control.

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Mathematical modelling as a tool for development of new drugs Development costs 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 development processes at reduced financial costs. Improved treatment of patients. Better, more safe and a more individual treatment. Reduction of applications of animal experiments. Computer model of humans.

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

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

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

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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).

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

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

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

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Enzyme kinetics Reaction scheme: Reaction equations: Note that:

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Enzyme kinetics 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).

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

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Two dimensional example with flow, diffusion and binding sites on the boundary Bindings sites on the boundary:

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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).

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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).

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The perfusion chamber t=1

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The perfusion chamber t=5

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The perfusion chamber t=10

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The perfusion chamber t=300

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

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The perfusion chamber

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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).

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Simulations – localized collagen site Concentration of T, Ta og IIa Plottet after 5 minutes, constant dosis faktor rVIIa (NovoSeven)

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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)

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

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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).

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