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Wajima et al., (2009). A comprehensive model for the humoral coagulation network in human.

July 2011, model of the month by Michael Schubert
Original models: BIOMD0000000338, BIOMD0000000339, BIOMD0000000340

Blood coagulation is an essential biological process, responsible for general hemostasis and wound healing. The central reaction is the generation of thrombin from its precursor, prothrombin that converts soluble fibrinogen monomers into a tight mesh of cross-linked fibrin. This process is mediated and controlled by a multitude of distinct coagulation-promoting and anticoagulative factors, ensuring an individual's viability. Consequently, the activation of this system at a threshold that is either too low or too high is associated with a variety of pathological conditions. For instance, the occurrence of blood clotting where it could lead to thrombosis, which in turn can be the cause of an embolism and thereby cardiac infarction or stroke. There are many associated risk factors, e.g. artherosclerosis, smoking, age, or a general unhealthy diet. On the other hand, the opposite is also true. When blood is not clotting when it should, this leads to bleeding conditions such as hemophilia.

Wajima et al. [1, BIOMD0000000338, BIOMD0000000339, BIOMD0000000340], have developed one of the most complete mathematical models so far for describing the coagulation process including both intrinsic and extrinsic pathways. It has been applied to successfully describe the concentrations of coagulation factors and their counterparts over time for both in vitro blood coagulation tests as well as the pharmacodynamic (and, to a lesser extend, pharmacokinetic) effects of the anticoagulants enoxaparin, heparin, and warfarin in vivo. Special consideration has been given to the closely intertwined warfarin and vitamin K status, and the reciprocal effects of one substance to the other.

Figure 1

Figure 1: BIOMD0000000339. Factor concentrations as fraction of their initial inactivated factor concentrations (left) and integral of fibrin (right) during 90 seconds the PT test. Blood clotting occurs after 11.4 seconds in this simulation and is marked by a dashed line in the fibrin integral plot.

For measuring the activity of the extrinsic coagulation pathway, the most widely used clinical test is the Prothrombin Time (PT) Test. Therein, tissue factor (TF) is added to a diluted blood sample, that binds to and activates factor VII, which in turn mediates the assembly of the prothrombinase complex (activated factors X and V). As the name suggests, this complex is the main responsible enzyme for converting prothrombin (factor II) to its activated form. The time it usually takes thrombin to convert enough fibrinogen (factor I) to fibrin in order to form a mesh is around 11 to 16 seconds, which is referred to as the standard prothrombin time and corresponds to an International Normalized Ratio (INR) of 0.8 to 1.2. The required AUC of fibrin concentration after addition of 300 nM TF to a 5x diluted standard blood sample in the model was assumed to be 1500 nM.s and occurs after 11.08 seconds (cf. figure 1).

To also account for the contribution of the intrinsic or contact activation pathway to the coagulation process, the PT test is often used in conjunction with the Activated Partial Thromboplastin (aPTT) Test. It is divided into a preincubation and a second stage. In the first, a Ca2+ chelator and a contact activator is added to the sample. The latter mediates activation of an initial amount of the Hagemann factor (XII), which converts Prekallikrein to Kallikrein in a positive feedback loop. Active factor XII can then activate factor XI once the sample is recalcified in the second stage, ultimately leading to thrombin generation again. The preincubation stage was modelled in way that 300 nM contact activator was added to the system and the rate constant of factor XI activation was set to 0 prior to recalcification, while not taking Ca2+ into account explicitly. A simulated time course of this process is shown in figure 2, the required fibrin concentration being the same as for the PT test and reached after 34.4 seconds (normal times between 25 and 39 seconds).

Figure 2

Figure 2: BIOMD0000000338. Factor concentrations as fraction of their initial inactivated factor concentrations (left) and integral of fibrin (right) during 90 seconds the aPTT test. Simulation of the preincubation phase is not shown. Blood clotting occurs after 34.4 seconds in this simulation and is marked by a dashed line in the fibrin integral plot.

Figure 3

Figure 3: BIOMD0000000340. Simulated warfarin plasma concentration after administration of 4 mg daily over the course of 20 days (left) and its effect on the plasma concentrations of vitamin K-related compounds (middle) and factors (right). The latter two are represented as fraction of their untreated steady state concentrations.

Using the above tests as a baseline, the authors examined the effect of warfarin and vitamin K in more detail by modelling the vitamin K cycle with a peripheral compartment and using a simple first-order absorption and degradation model of the drug. The time courses of bioavailable warfarin were described for a standard dose of 4 mg and an excessive dose of 50 mg per day. The former are shown in figure 3 with their respective effect on the serum concentrations of vitamin K-related factors and compounds. Additionally, a patient's INR was modelled by running PT test simulations with different initial conditions throughout exposure to the anticogulant.

Further simulations were carried out to describe the effects of Taipan snake-bite venom, intravenous heparin infusion, and subcutaneous enoxaparin application, and their effects on the INR. These were modelled as compartmentalised prothrombinase analogue, by increasing the potency of antithrombin III, and a two-compartment model thereof, respectively. Still, and while a huge progress has been made in the attempt to simulate the coagulation network in its entirety, the model is for instance not taking into account the in vivo effect of platelet activation on factor activity, leaving some more room for further model extension.

Bibliographic References

  1. Wajima T, Isbister GK, Duffull SB. A comprehensive model for the humoral coagulation network in humans. Clin. Pharmacol. Ther. Sep;86(3):290-8, 2009. [CiteXplore]