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.

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

Development costs for new medicin

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

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

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

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.

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.

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 I.

Cartoon of the blood coagulation pathway II.

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

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).

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)

Cartoon model of the perfusion experiment. Model IV.

Activated Platelet

Va:XaVVIIa XaX Va

II IIa

Unactivated Platelet Activated PlateletIIa

IIa

Cartoon of the blood coagulation pathway. Model V.

Ref.: Julie Refsgaard Lawaetz (master thesis 2010) and Nina Marianne Andersen (PhD thesis 2011).

Enzyme kinetics

PECES Reaction scheme:

Reaction equations:

Note that:

sekckkdtde

121 )(

1k

1k

2k

sekckdtds

11

ckksekdtdc

)( 211 ckdtdp

2

0ece

Enzyme kinetics

Scaling:

Mathematical model:

0ss

0ec

x 01

21

skkk

0

0

s

e01

1

skk

)( xdd

)(

xddx

Quasi steady state approximation:

)/( 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).

Competitive inhibition

PECES 1Reaction scheme:

Inclusion of flow and diffusion:

1k

1k

2k

sekcksvsDts

s 111)(

2CIE 3k

3k

Diffusion constant: sD Convective flow velocity: v

Reaction scheme at the boundary:

BPBP 0BBPB

Binding sites on boundary: B4k

4k

)( 044 bbkbpkdtdb

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

xy

P

Bindings sites on the boundary:

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).

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).

The perfusion chamber

t=1

The perfusion chamber

t=5

The perfusion chamber

t=10

The perfusion chamber

t=300

The perfusion chamber

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

The perfusion chamber

The perfusion chamber

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).

Simulations – localized collagen site

Concentration of T, Ta og IIa• Plottet after 5 minutes, constant dosis faktor rVIIa (NovoSeven)

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)

Concentration of bounded platelets. TaB• Plottet every half minutes, constant dosis faktor rVIIa (NovoSeven)

Simulations – Reduction of cross section area

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).