The Blood Coagulation Cascade in a Perfusion Experiment: Example from the Pharmaceutical Industry

Abstract

In the last Chap. 6, we shall illustrate the use of mathematical modelling in the pharmaceutical industry by an example from the development of a blood coagulation treatment with a coagulation factor. More specifically we derive a mathematical model for a blood coagulation cascade set up in a perfusion experiment conducted at the pharmaceutical company Novo Nordisk A/S in Denmark. We investigate the influence of blood flow and diffusion on the blood coagulation pathway by deriving a model consisting of a system of partial differential equations taking into account the spatial distribution of the biochemical species. The validity of the model is established via positivity criteria proved in Chap. 3. The model is solved using a finite element code in order to illustrate the influence of diffusion and convection on the coagulation cascade with dynamic boundary condition modelling adhesion of blood platelets to a collagen coated surface.

Appendices

Appendix A

The system consists of recombinant factor VIIa (rFVIIa), factor X (FX), factor V (FV), prothrombin (FII) and thrombocytes (T) or platelets. Most of these reactions take place on activated thrombocytes (Ta) and therefore the mathematical model consists of reactions for the binding of coagulation factors to activated thrombocytes. It is postulated [81] that initially some platelets adhere to the surface of the thrombocytes and become activated and expose procoagulant phospholipids. Subsequently rFVIIa binds to these phosphatidylserine-exposing platelets, on which it activates FX independently of tissue factor TF. The resulting FXa combines with factor V released from platelets α-granules, and the resulting prothrombinase complex converts prothrombin to thrombin. This thrombin activates platelets and the activated platelets adhere to the collagen surface. A list of the species entering the biochemical reactions are given in Table 6.1.

Table 6.1 List of the notation for chemical factors and species

Below we list the complete biochemical reactions modelling a perfusion experiment as the one in [81].

Appendix B

The reaction schemes in Appendix A result in the below system of partial differential equations, including diffusion and advection terms with flow velocity v. In comparison to Eq. (3.1) we have u=(u ¹,u ²,…,u ¹⁷) where the components are given by u ¹=[X], u ²=[Xa], u ³=[Xa_Ta], u ⁴=[X_Ta], u ⁵=[II], u ⁶=[IIa], u ⁷=[II_Ta], u ⁸=[VIIa_Ta], u ⁹=[VIIa], u ¹⁰=[T], u ¹¹=[Ta], u ¹²=[V], u ¹³=[Va], u ¹⁴=[V_Ta], u ¹⁵=[Va_Ta], u ¹⁶=[Xa_Va_Ta], u ¹⁷=[V_Xa_Ta]. The diffusion matrix in Eq. (3.1) is diagonal with components D ₁=D _X, D ₂=D _Xa and similarly for the other diagonal elements.

The reaction rates used are listed in Tables 6.2 and 6.3. For the parameter a in the flow velocity profile we have chosen the value a=64⋅1/s. It is assumed that the coagulation factors and platelets have the same diffusion constant D ₁=D ₂=⋯=D ₁₇=6.47⋅10⁻¹¹ m²/s (see Ref. [6]).

Table 6.2 MG10 is decreased in comparison to Bungay to avoid saturation of binding sites

Table 6.3 Reaction rates