Indhold: Udvikling af medicin og matematisk modellering. Blodkoagulation.

1. Matematik i biologi og farmaceutisk industri. Årskursus i matematik, kemi og fysik, Rosborg Gymnasium, Vejle, 24/10, 2008 Mads Peter Sørensen DTU Matematik, Kgs. Lyngby Indhold: Udvikling af medicin og matematisk modellering. Blodkoagulation. Insulinproducerende beta celler. Sammenfatning.

2. Samarbejdspartnere. Nina Marianne Andersen, DTU Matematik og Novo Nordisk Steen Ingwersen, Biomodellering, Novo Nordisk. Ole Hvilsted Olsen, Hæmostasis biokemi, Novo Nordisk. Morten Gram Pedersen, Department of Information Engineering, University of Padova, Italy. Oleg V. Aslanidi, Institute of Cell Biophysics RAS, Pushchino, Moscow, Russia. Oleg A. Mornev, Institute of Theoretical and Experimental Biophysics RAS, Pushchino, Moscow, Russia. Ole Skyggebjerg, Novo Nordisk. Per Arkhammar og Ole Thastrup, BioImage a/s, Søborg. Alwyn C. Scott, DTU Informatik og University of Arizona, Tucson AZ, USA. Peter L. Christiansen, DTU Fysik og DTU Informatik. Knut Conradsen, DTU Informatik Sponsorer: Modelling, Estimation and Control of Biotechnological Systems (MECOBS). EU Network of Excellence BioSim.

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

4. Udviklingsprocessen for ny medicin. 1) Opdagelse. 2) Prækliniske forsøg. Ide, hypotese, forskning. Dyremodeller. Dyreforsøg. Udviklingsfase. Dyreforsøg. Protokol for sikkerhed og effektivitet. Mekanisme og potentiel giftpåvirkning af organer.

5. 4) Godkendelse. 3) Kliniske forsøg. Regulerende myndigheder. Godkendelse af medikamentet. Marketing autorisation. Sikker og effektiv medicin. Godkendelse fra regulerende myndigheder. Test på mennesker. Test for sikkerhed og effektivitet. >50% af udviklings tiden. 1 ud af 10-15 medikamenter overlever til fase 3 5) Kontrol. Lægemiddelovervågning

6. Matematisk modellering som et redskab i udviklingen af ny medicin Udviklingsomkostningerne for et nyt medikament ligger typisk mellem 1 og 7 milliarder kr. Udviklingstid: 10 – 15 år. Anvendelse af moderne modellerings og computer simuleringsværktøjer til udvikling af ny medicin. Kompleksitet. Mere rationel og hurtigere udviklings proces med færre økonomiske omkostninger. Forbedret behandling af patienter. Bedre, mere sikker og mere individuel behandling. Reduktion i anvendelse af dyre eksperimenter. Computer model af menneske.

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

9. Perfusions eksperiment og modellering Perfusions kammer Aktive thrombocyter (Ta) binder til et collagen coated låg. vWF. Glaslåg coated med collagen Faktor X i fluid fase X Thrombocyter (blodplader), røde og hvide blod celler. Faktor VIIa I fluid fase VIIa Rekonstrueret blod. Indhold: Thrombocyter (T), Erythrocyter. [T] = 14 nM (70,000 blodplader / μ litre blood)

10. Enzym kinetic Reaktions skema: Reaktions ligningerne: Bemærk at:

11. Enzym kinetic Skalering: Matematisk model: Kvasistationær tilstand: 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).

12. Konkurrerende inhibitor (hæmningsstof) Reaktions skema: Inklusion af flow og diffusion: Diffusionskonstant: Konvektions flow hastighed: Reaktionsskema ved rand: Bindingssites på rand:

13. To dimensionalt eksempel med flow, diffusion og bindingssites på randen Bindingssites på randen:

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

15. Reaction schemes, one example. Factor II (prothrombin): II Factor IIa (thrombin): IIa Prothrombinase complex: Xa_Va_Ta A total of 17 equations. Reaction rates: Ref: P.M. Didriksen, Modelling hemostasis - a biosimulation project, internal report, Dept. 252 Biomodelling, Novo Nordisk

16. Numerical results. Initial conditions: FVIIa = 50 nM FX = 170 nM T = 14 nM sTa = 0.1*14 nM FII = 0.3 nM IIa T VIIa Ta

17. Reaction diffusion model with convection Reaction scheme for T, Ta and IIa. Corresponding model equations in the space Ω. Poiseuille’s flow

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

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

20. Numerical results. Time = 5 sec. T IIa T-IIa Ta

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

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

23. 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.

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

25. Sammenfatning og fremtidig arbejde Modellering af perfusionseksperiment for blod-koagulation. Reduceret PDL model, som inkludere blod flow og diffusion. Modellering af vedhæftning af aktive thrombocyter på collagen coated rand. Fuld PDL model. Model af in vivo blod koagulation.

26. Synthesis and secretion of insulin Transcription Pre-proinsulin Insulin Golgi complex packed in granules Proinsulin Endoplasmatic reticulum B Exocytosis of insulin caused by increased Ca concentration

27. The β-cell Ion channel gates for Ca and K B

28. Mathematical model for single cell dynamics The modified Hodgkin-Huxley model for a single β-cell Ion currents due to the ion-gates Ref.: A. Sherman, (Eds. Othmar et al), Case studies in mathematical modelling, ecology physiology and cell biology, Prentice Hall (1996), pp.199-217. Ref.: Fall, Marland, Wagner, Tyson, Computational Cell Biology, Springer, (2002).

29. Mathematical model for single cell dynamics The gating variables Ref.: Fall, Marland, Wagner, Tyson, Computational Cell Biology, Springer, (2002).

30. Ref.: Fall, Marland, Wagner, Tyson, Computational Cell Biology, Springer, (2002).

31. Dynamics and bifurcations Ref.: E.M. Izhikevich, Neural excitability spiking and bursting, Int. Jour. of Bifurcation and Chaos, p1171 (2000).

32. Dynamics and bifurcations Simple polynomial model Parameters Ref.: J. Keener and J. Sneyd, Mathematical Physiology, Springer, New York, (1998).

33. Sketch of the homoclinic bifurcation

34. Mathematical model for single cell dynamics Topologically equivalent and simplified models. Polynomial model with Gaussian noise term on the gating variable. Voltage across the cell membrane: Gating variable: Slow gate variable: Gaussian gate noise term: where Ref.: M. Panarowski, SIAM J. Appl. Math., 54 pp.814-832, (1994). Ref.: A. Sherman, (Eds. Othmar et al), Case studies in mathematical modelling, ecology physiology and cell biology, Prentice Hall (1996), pp.199-217.

35. The influence of noise on the beta-cell bursting phenomenon. Ref.: M.G. Pedersen and M.P. Sørensen, SIAM J. Appl. Math., 67(2), pp.530-542, (2007).

36. Mathematical model for coupled β-cells Gap junctions between neighbouring cells Coupling to nearest neighbours. Coupling constant: Ref.: A. Sherman, (Eds. Othmar et al), Case studies in mathematical modelling, ecology physiology and cell biology, Prentice Hall (1996), pp.199-217.

37. Coupled β-cells Image analysis experiments of in vitro islets of Langerhans

38. Experiments on Islets of Langerhans

39. The gating variables Calcium current: Potassium current: ATP regulated potassium current: Slow ion current: The gating variables obey.

40. Glycose gradients through Islets of Langerhans Ref.: J.V. Rocheleau, et al, Microfluidic glycose stimulations … , PNAS, vol 101 (35), p12899 (2004).

41. Glycose gradients through Islets of Langerhans. Model. Continuous spiking for: Bursting for: Silence for: Coupling constant: Note that corresponds to

42. Wave blocking Units

43. Glycose gradients through Islets of Langerhans

44. PDE model. Fisher’s equation Continuum limit of Is approximated by the Fisher’s equation where Velocity: Simple kink solution Ref.: O.V. Aslanidi et.al. Biophys. Jour. 80, pp 1195-1209, (2001).

45. Numerical simulations and comparison to analytic result

46. Sammenfatning Støj på ion porte reducerer burst perioden. Blokering af bølgeudbredelse ved rumlig variation af den ATP regulerende Na ion kanal. Koblingen mellem beta celler fører til en forøget excitation af ellers inaktive celler. Ref.: M.G. Pedersen and M.P. Sørensen, SIAM J. Appl. Math., 67(2), pp.530-542, (2007). M.G. Pedersen and M.P. Sørensen, To appear in Jour. of Bio. Phys. Special issue on Complexity in Neurology and Psychiatry, (2008). Bio-kemiske processer er meget komplekse og kræver omfattende modellering. Simple og overskuelige modeller kan give kvalitativ indsigt. Der er lang vej til pålidelige kvantitative modeller. Matematiske modeller forventes dog at kunne bidrage til hurtigere og mere sikker udvikling af medicin med færre dyreforsøg.

47. Studieretningsprojekter for gymnasiet Se: http://www.dtu.dk/Moed_DTU/Studieretningsprojekter.aspx