Mathematical Models of Blood Coagulation: Developing Anticoagulants

Joana Lilibeth PerdomoAdvisor: Lisette G. de Pillis

Department of Mathematics, 2015-2016 HMC Senior Thesis

IntroductionBlood coagulation is a series of biochemical reactions that take place to form a blood clot. Abnormalities in coagulation, such as under-clotting or over-clotting, can lead to significant blood loss, cardiac arrest, damage to vital organs, or even death. Thus, understanding quantitatively how blood coagulation works can help inform clinical decisions when treating bleeding disorders. Some areas of coagulation of particular interest include coagulation kinetics, the spatial organization of blood coagulation, the effects of blood flow on clotting, and the effects of drugs on the coagulation system. Various mathematical and computational models have been developed in the past 30 years that have addressed some of these topics. For my Senior Thesis, I have worked on an extensive literature review (200+ papers) of mathematical models of blood coagulation. Here, I present three models that have been used to investigate and develop anticoagulants. Anticoagulants, such as warfarin, heparin, and rivaroxaban, are used to correct hypo-coagulation and restore a non-functioning coagulation system.

Blood CoagulationThe formation of a blood clot (thrombus) involves a network of biochemical reactions that result in the production of the key enzyme thrombin (fIIa). Proteins known as clotting factors must be activated, react with other clotting factors, or bind to surfaces for coagulation to take place. To produce fIIa, newly exposed tissue factor (TF) from the subendothelium combines with clotting factors, activated platelets (AP), endothelial surfaces in the protein C pathway (APC) or other cell surfaces to produce fIIa. This converts fibrinogen (fI) to fibrin (fIa), which leads to the formation of a thrombus.4

Figure 1: Blood coagulation takes place in spatially heterogeneous environment, meaning that different reactions occur at different locations. Note that blood flow is essential in regulating blood coagulation because it diffuses active factors throughout the system.4

Anticoagulants

References1. Burghaus R, Coboeken K, Gaub T, et al. Evaluation of the efficacy and safety of rivaroxaban using a

computer model for blood coagulation. PLoS One 2011;6:e17626. 2. Burghaus R, Coboeken K, Gaub T, et al. Computational investigation of potential dosing schedules for a

switch of medication from warfarin to rivaroxaban- an oral, direct Factor Xa inhibitor. Front Physiol 2014;5:1-13.

3. Lazzaro MA, Zaidat OO. Multimodal endovascular reperfusion therapies. Neurology 2012;78:501-506.4. Shibeko AM, Panteleev MA. Untangling the complexity of blood coagulation network: use of computational

modelling in pharmacology and diagnostics. Briefings in Bioinformatics 2015;11:1–11.5. Wajima T, Isbister GK, Duffull SB. A comprehensive model for the humoral coagulation network in humans.

Clin Pharmacol Ther 2009;86:290–8.

Figure 2: Elements of the coagulation cascade with anticoagulant medication interactions. Warfarin inhibits activation of fII, fVII, fIX, fX and proteins C and S (not shown). Heparin binds with antithrombin-III (AT-III) to inhibit fXa and fIIa. Direct fXa and fIIa inhibitors interact with their respective factors. Rivaroxaban is a direct fXa inhibitor.3

Diagnostic Coagulation TestsTo evaluate how well a patient’s blood clots, typically, two diagnostic tests are used:- a prothrombin time (PT) test— measures extrinsic pathway- an activated partial thromboplastin time (aPTT) test— measures

the intrinsic pathway

Model Objective Method Conclusions

Wajima et al. (2009)5

To develop a comprehensive

coagulation model

- Developed a multicompartmental model which included: extrinsic, intrinsic, and common pathways, in vitro coagulation tests (i.e., PT and aPTT), vitamin-K cycle, and AT-III:heparin complex

- Used a system of ordinary differential equations (ODEs)

- First comprehensive model of in vivo coagulation network

- Model accurately predicts time course of coagulation kinetics in clinical situations (i.e., effects of warfarin and heparin on in vitro tests) and drug development

Burghaus et al. (2011)1

To assess the efficacy, safety and dosing of a

drug (rivaroxaban)

- Based on Wajima et al. (2009), but added blood flow and drug action mechanisms

- ODE-based coagulation model

- Rivaroxaban doses up to 20 mg found to be safe and effective

- Model accurately simulated in vitro tests, therapeutic ranges, in vivo coagulation (including flow), and evaluated efficacy and safety limits of drugs

Burghaus et al. (2014)2

To evaluate dose scheduling for a shift from

warfarin to rivaroxaban

- Based on Burghaus et al. (2011), but added a warfarin decay model to describe warfarin treatment (or discontinuation) effects

- Results suggest strong synergy between rivaroxaban and residual warfarin

- Developed specific dosing schedules

Figure 3a: Simulated time course of warfarin plasma concentration of typical dose of 4 mg once daily (OD) and, 3b: INR (4, 10, and 50 mg OD) after warfarin therapy. Note that a 4 mg OD yields INR = 2 at steady state. Warfarin concentrations reached steady state in ~1 week but INR values took ~2 weeks.5

Figure 5: Simulation of patients with INR = 3.5. Pharmacokinetic curves for rivaroxaban 10 and 20 mg OD are black. Blue and red lines are threshold edges for safety and efficacy. Rivaroxaban exposure between the outermost blue and red lines is effective (prevents clotting). Region between solid red and blue lines describes possible safe dose range for patients with high risk of bleeding or clotting.2

Figure 4: Anticoagulant safety evaluation based on INR. Warfarin effect at INR = 3 was an upper limit for safety, so all therapies below this level (not in the red region) are safe because they do not increase the risk of bleeding. Effective therapies (INR >1.5) prevented clotting (not shown).1

Results: Burghaus et al. (2011)

Results: Wajima et al. (2009)

Results: Burghaus et al. (2014)

Conclusions- Mathematical models can be useful in drug development, therapy planning, and diagnostics.

- Most models used in clinical applications are simple, reliable models of in vitro coagulation (with the exception of Burghaus et al. (2011, 2014), which include in vivo blood flow).

An international normalized ratio (INR) value, based on PT tests, was provides a standardized measure of the anticoagulant effect of vitamin-K antagonists (VKAs), like warfarin. Higher INR values means blood takes longer to clot. A safe target range for patients on anticoagulants is an INR between 2.0 and 4.0 (healthy INR is ~1).3