Indhold:

1) Udvikling af medicin og matematisk modellering.

2) Blodkoagulation.

3) Insulinproducerende beta celler.

4) Sammenfatning.

Matematik i biologi og farmaceutisk industri.

Mads Peter Sørensen

DTU Matematik, Kgs. Lyngby

Årskursus i matematik, kemi og fysik, Rosborg Gymnasium, Vejle, 24/10, 2008

1) Nina Marianne Andersen, DTU Matematik og Novo Nordisk

2) Steen Ingwersen, Biomodellering, Novo Nordisk.

3) Ole Hvilsted Olsen, Hæmostasis biokemi, Novo Nordisk.

4) Morten Gram Pedersen, Department of Information Engineering, University of Padova, Italy.

5) Oleg V. Aslanidi, Institute of Cell Biophysics RAS, Pushchino, Moscow, Russia.

6) Oleg A. Mornev, Institute of Theoretical and Experimental Biophysics RAS, Pushchino, Moscow, Russia.

7) Ole Skyggebjerg, Novo Nordisk.

8) Per Arkhammar og Ole Thastrup, BioImage a/s, Søborg.

9) Alwyn C. Scott, DTU Informatik og University of Arizona, Tucson AZ, USA.

10) Peter L. Christiansen, DTU Fysik og DTU Informatik.

11) Knut Conradsen, DTU Informatik

Samarbejdspartnere.

Sponsorer: Modelling, Estimation and Control of Biotechnological Systems (MECOBS). EU Network of Excellence BioSim.

Udviklingsomkostninger for ny medicin.

Ref.: Erik Mosekilde, Ingeniøren 10. oktober, side 9, (2008).

EU Network of Excellence BioSim. http://biosim-network.eu

Udviklingsprocessen for ny medicin.

Ide, hypotese, forskning.

Dyremodeller. Dyreforsøg.

1) Opdagelse.

Udviklingsfase. Dyreforsøg.

Protokol for sikkerhed og effektivitet.

Mekanisme og potentiel giftpåvirkning af organer.

2) Prækliniske forsøg.

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

3) Kliniske forsøg.Regulerende

myndigheder.

Godkendelse af medikamentet.

Marketing autorisation.

Sikker og effektiv medicin.

4) Godkendelse.

Lægemiddelovervågning

5) Kontrol.

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.

Disorders of Coagulation

Hypocoagulation:

Hemophilia A

Hemophilia B

Others

Hypercoagulation:

Cardiovascular diseases:

Arthroscleroses

Emboli and thrombi development

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

Cartoon of the blood coagulation pathway.

Perfusions eksperiment og modellering

Perfusions kammer

Glaslåg coated med collagen

Thrombocyter (blodplader), røde og hvide blod celler.

Faktor X i fluid fase

X

Faktor VIIa I fluid fase

VIIa

Aktive thrombocyter (Ta) binder til et collagen coated låg. vWF.

Rekonstrueret blod.

Indhold: Thrombocyter (T), Erythrocyter.

[T] = 14 nM (70,000 blodplader / μ litre blood)

Enzym kinetic

PECES Reaktions skema:

Reaktions ligningerne:

Bemærk at:

sekckkdtde

121 )(

1k

1k

2k

sekckdtds

11

ckksekdtdc

)( 211 ckdtdp

2

0ece

Enzym kinetic

Skalering:

Matematisk model:

0ss

0ec

x 01

21

skkk

0

0

s

e01

1

skk

)( xdd

)(

xddx

Kvasistationær tilstand:

)/( x

)(

dd

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).

Konkurrerende inhibitor (hæmningsstof)

PECES 1Reaktions skema:

Inklusion af flow og diffusion:

1k

1k

2k

sekcksvsDts

s 111)(

2CIE 3k

3k

Diffusionskonstant: sD Konvektions flow hastighed:

v

Reaktionsskema ved rand:

BPBP 0BBPB

Bindingssites på rand:

B4k

4k

)( 044 bbkbpkdtdb

To dimensionalt eksempel med flow, diffusion og bindingssites på randen

xy

P

Bindingssites på randen:

Cartoon model of the perfusion experiment

Activated Platelet

Va:XaVVIIa XaX Va

II IIaIIa

Unactivated PlateletUnactivated Platelet Activated PlateletActivated PlateletIIaIIa

IIaIIa

Reaction schemes, one example.

Ref: P.M. Didriksen, Modelling hemostasis - a biosimulation project, internal report, Dept. 252 Biomodelling, Novo Nordisk

TaIITaII 10Ta

10F

IIaTaXaTaXaTaII 16R

IIaTaVaXaTaVaXaTaII 3S

Factor II (prothrombin): II

Factor IIa (thrombin): IIa

Prothrombinase complex: Xa_Va_Ta

A total of 17 equations.

TaIITaVaXaSTaIITaXaRdtTadII

316

TaIIFTaIITa 1010

11750016 sMR

11103

7 sMS

1110

sF

114300010

sMTa

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.

IIaTasTaIIaT 6T

7T

Corresponding model equations in the space Ω.

))((27 TayvTaDsTaT

dtdTa

Ta

))((26 TyvTDIIaTT

dtdT

T

))((276 sTayvsTaDIIaTaTIIaTT

dtdsTa

sTa

))((267 IIayvIIaDIIaTTsTaT

dtdIIa

IIa

Poiseuille’s flow

)/1()( Hyayyv

Boundary conditions and parameters

Boundary condition x=0 )(102.1 16 yfnMIIa

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.

)(1014 29 yfnMT

0Ta 0sTa

Boundary condition x=l: Outflow boundary conditions.

Top and bottom boundary condition: No flow crossing.

Numerical results. Time = 0.6 sec.

T-IIa

IIa

T

Ta

Numerical results. Time = 5 sec.

T IIa

T-IIa Ta

Numerical results. Time = 10 sec.

T-IIa

T IIa

Ta

Future work: Boundary attachment of Ta

Reaction schemes on

Corresponding model equations on.

TaBTa TaBT

IIaTaBCTaBII 2

2k 3k

4k 5k

TaBIIkdtdII 4

25 CkdtdIIa

TkCkTaBIIkTakdt

dTaB 32542

2542 CkTaBIIk

dtdC

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.

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

Sammenfatning og fremtidig arbejde

1. Modellering af perfusionseksperiment for blod-koagulation.

2. Reduceret PDL model, som inkludere blod flow og diffusion.

3. Modellering af vedhæftning af aktive thrombocyter på collagen coated rand.

4. Fuld PDL model.

5. Model af in vivo blod koagulation.

Synthesis and secretion of insulin

B

Transcription Pre-proinsulin

ProinsulinEndoplasmatic reticulum

InsulinGolgi complex

packed in granules

Exocytosis of insulincaused by increased Ca concentration

The β-cell

Ion channel gates for Ca and K

B

Mathematical model for single cell dynamics The modified Hodgkin-Huxley model for a single β-cell

CRACCaKATPKsKCa IIIIIIdtdv

C )()(

Ion currents due to the ion-gates

))(( CaCaCa vtvmgI

))()(( KKK vtvtngI

))()(( sss vtvtsgI

))(()()( KATPKATPK vtvgI

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).

Mathematical model for single cell dynamics

The gating variables

n

tnndtdn

)(

s

tssdtds

)(

x

x

s

vvx

exp1

1

OC k

k

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

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

Dynamics and bifurcations

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

Dynamics and bifurcations

Simple polynomial model

zIxxydtdx 23 3

yxdtdy 251

zxxsrdtdz )( 1

2/)51(1 x

Parameters

001.0r 4s

Ref.: J. Keener and J. Sneyd, Mathematical Physiology, Springer, New York, (1998).

Sketch of the homoclinic bifurcation

crzz crzz

crzz

Mathematical model for single cell dynamics Topologically equivalent and simplified models. Polynomial model

with Gaussian noise term on the gating variable.

zwufdtdu )( )()( twug

dtdw ))(( zuh

dtdz

Ref.: M. Panarowski, SIAM J. Appl. Math., 54 pp.814-832, (1994).

Voltage across the cell membrane: )(tuu Gating variable: )(tww

Gaussian gate noise term: )(t where 0)( t

)()0()( tt

Slow gate variable: )(tzz

Ref.: A. Sherman, (Eds. Othmar et al), Case studies in mathematical modelling, ecology physiology and cell biology, Prentice Hall (1996), pp.199-217.

The influence of noise on the beta-cell bursting phenomenon.

0

1.0

3.0

Ref.: M.G. Pedersen and M.P. Sørensen, SIAM J. Appl. Math., 67(2), pp.530-542, (2007).

Mathematical model for coupled β-cells

j

jiijATPKsKCai vvgIIII

dt

dvC )()(

Coupling to nearest neighbours.

Coupling constant: ijg

Gap junctions between neighbouring cells

Ref.: A. Sherman, (Eds. Othmar et al), Case studies in mathematical modelling, ecology physiology and cell biology, Prentice Hall (1996), pp.199-217.

Coupled β-cells

Image analysis experiments of in vitro islets of Langerhans

Experiments on Islets of Langerhans

ijg

The gating variables

))(( CaiiCaCa vvvmgI

The gating variables obey.

Calcium current:

Potassium current: ))(( KiKK vvtngI

)()()( KiATPKATPK vvgI ATP regulated potassium current:

Slow ion current: ))(( Kiss vvtsgI

n

nvndtdn

)(

s

svsdtds

)(

)/)(exp(11

)(xxi

i svvvxx

snmx ,,

Glycose gradients through Islets of Langerhans

Ref.: J.V. Rocheleau, et al, Microfluidic glycose stimulations … , PNAS, vol 101 (35), p12899 (2004).

Coupling constant:

Glycose gradients through Islets of Langerhans. Model.

pSipSg ATPK 1)1(120)( Ni ,...,2,1

Continuous spiking for: pSg ATPK 90)(

Bursting for: pSgpS ATPK 16290 )(

Silence for: )(162 ATPKgpS

Note that 43i corresponds to pSg ATPK 162)(

Wave blocking

Units tkt tphys uku uphys

msgck Cat 3.5/

mVsk mu 12

Glycose gradients through Islets of Langerhans

pSgij 50

PDE model. Fisher’s equation

Continuum limit of

)2(),( 11 iiiciii vvvgsvF

dt

dv

Is approximated by the Fisher’s equation xxt uaufu );(

where )1)(();( uauuauf

2exp1

1),(

00 vtxx

txuSimple kink solution 2/)21( av

Velocity:

Ref.: O.V. Aslanidi et.al. Biophys. Jour. 80, pp 1195-1209, (2001).

Numerical simulations and comparison to analytic result

Sammenfatning 1) Støj på ion porte reducerer burst perioden.

2) Blokering af bølgeudbredelse ved rumlig variation af den ATP regulerende Na ion kanal.

3) Koblingen mellem beta celler fører til en forøget excitation af ellers inaktive celler.

1) Bio-kemiske processer er meget komplekse og kræver omfattende modellering.

2) Simple og overskuelige modeller kan give kvalitativ indsigt.

3) Der er lang vej til pålidelige kvantitative modeller.

4) Matematiske modeller forventes dog at kunne bidrage til hurtigere og mere sikker udvikling af medicin med færre dyreforsøg.

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).

Studieretningsprojekter for gymnasiet

Se:

http://www.dtu.dk/Moed_DTU/Studieretningsprojekter.aspx