Dias nummer 1

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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.

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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.
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Udviklingsomkostninger for ny medicin.
Ref. Erik Mosekilde, Ingeniøren 10. oktober,
side 9, (2008).
EU Network of Excellence BioSim.
http//biosim-network.eu
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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.
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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. gt50 af udviklings tiden. 1 ud af
10-15 medikamenter overlever til fase 3
5) Kontrol.
Lægemiddelovervågning
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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.

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Disorders of Coagulation
Hypercoagulation Cardiovascular
diseases Arthroscleroses Emboli and thrombi
development

• Hypocoagulation
• Hemophilia A
• Hemophilia B
• Others

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Cartoon of the blood coagulation pathway.
Ref http//www.ambion.com/tools/pathway/pathway.p
hp?pathwayBlood20Coagulation20Cascade
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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)
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Enzym kinetic
Reaktions skema
Reaktions ligningerne
Bemærk at
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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).
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Konkurrerende inhibitor (hæmningsstof)
Reaktions skema
Inklusion af flow og diffusion
Diffusionskonstant
Konvektions flow hastighed
Reaktionsskema ved rand
Bindingssites på rand
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To dimensionalt eksempel med flow, diffusion og
bindingssites på randen
Bindingssites på randen
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Cartoon model of the perfusion experiment
Unactivated Platelet
Activated Platelet
IIa
IIa
II
IIa
VaXa
V
VIIa
Xa
X
Va
Activated Platelet
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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
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Numerical results.
Initial conditions FVIIa 50 nM FX 170 nM T
14 nM sTa 0.114 nM FII 0.3 nM
IIa
T
VIIa
Ta
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Reaction diffusion model with convection
Reaction scheme for T, Ta and IIa.
Corresponding model equations in the space O.
Poiseuilles flow
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Boundary conditions and parameters
Boundary condition x0
Boundary condition xl 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.
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Numerical results.
Time 0.6 sec.
IIa
T
T-IIa
Ta
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Numerical results.
Time 5 sec.
T
IIa
T-IIa
Ta
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Numerical results.
Time 10 sec.
T
IIa
T-IIa
Ta
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Future work Boundary attachment of Ta
Reaction schemes on
Corresponding model equations on.
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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.
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Model consisting of 11 PDEs in 21 D, including
diffusion
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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.

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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
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The ß-cell
Ion channel gates for Ca and K
B
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Mathematical model for single cell dynamics
The modified Hodgkin-Huxley model for a single
ß-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).
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Mathematical model for single cell dynamics
The gating variables
Ref. Fall, Marland, Wagner, Tyson, Computational
Cell Biology, Springer, (2002).
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Ref. Fall, Marland, Wagner, Tyson, Computational
Cell Biology, Springer, (2002).
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Dynamics and bifurcations
Ref. E.M. Izhikevich, Neural excitability
spiking and bursting, Int. Jour. of Bifurcation
and Chaos, p1171 (2000).
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Dynamics and bifurcations
Simple polynomial model
Parameters
Ref. J. Keener and J. Sneyd, Mathematical
Physiology, Springer, New York, (1998).
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Sketch of the homoclinic bifurcation
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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.
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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).
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Mathematical model for coupled ß-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.
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Coupled ß-cells
Image analysis experiments of in vitro islets of
Langerhans
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Experiments on Islets of Langerhans
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The gating variables
Calcium current
Potassium current
ATP regulated potassium current
Slow ion current
The gating variables obey.
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Glycose gradients through Islets of Langerhans
Ref. J.V. Rocheleau, et al, Microfluidic glycose
stimulations , PNAS, vol 101 (35), p12899
(2004).
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Glycose gradients through Islets of Langerhans.
Model.
Continuous spiking for
Bursting for
Silence for
Coupling constant
Note that
corresponds to
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Wave blocking
Units
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Glycose gradients through Islets of Langerhans
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PDE model. Fishers equation
Continuum limit of
Is approximated by the Fishers equation
where
Velocity
Simple kink solution
Ref. O.V. Aslanidi et.al. Biophys. Jour. 80, pp
1195-1209, (2001).
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Numerical simulations and comparison to analytic
result
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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.

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