Simulation and Validation of a Closed-Loop Circuit Model of Thrombin Factor X Interactions

Introduction: Clinical anticoagulation therapy is dosed at a standard dose or according to weight, with adjustments made based on laboratory test targets. Patients are at risk of over- or under-anticoagulation given lengthy return times for many of these tests, with blood clot treatment costing the U.S. billions of dollars to treat annually. In an effort to improve anticoagulation therapeutic intervention, we aim to develop a model of the physiologic coagulative response to anticoagulation administration that reflects real-time dose adjustments. Feedback control is integral to the response of biologic organisms to their environment, and this is evident in the clotting cascade, a nonlinear system which consists of multiple interconnected closed-loop control steps. Coagulation is a convective-diffusion-reaction process with spatial dynamics on the scale of millimeters to centimeters. However, its nested-loop control processes occur at the micron--or less--scale. The purpose of the project was to test the hypothesis that local control processes are conservative. Our approach is to model biologic coagulation using a closed network feedback model. We have first selected a trackable part of the biological network of coagulation to understand. We have shown that this part of the cascade may be simplified into a validated circuit model.

Materials and

Methods: We have created an electrical circuit analog model of 64 coupled differential equations, from chemical rate equations, describing the human coagulation cascades and feedback paths that incorporate rate kinetics of factor activation. We developed a Python script with the ability to convert a large number of differential equations to a Simulation Program with Integrated Circuit Emphasis (SPICE) circuit modeling software compatible form to generate a multi-input circuit model of the differential equations. The equations were then converted to a state space representation and used to derive a linearized approximation about the steady state. Internal feedback loops of the coagulation cascade are simulated over time in SPICE with initial conditions for species concentrations being represented as the charge on a capacitor. A current source is used for each species or combination of species that causes a change of charge over time. The SPICE voltage output represents the species concentration for the biological network. We performed piecewise validation of the model, focusing on the loop highlighted in the Figure 1 which is fundamental to the coagulation dynamics while being both simple and isolated from the rest of the system. We predict changes in species concentration in a normal state of coagulability to be minimal, whereas when initial species concentrations reflect tissue injury, non-monotonic dynamics and stabilization of species at a much larger concentration is expected. We use the law of conservation of mass to validate this model by observing species for concentration changes.

Results, Conclusions, and Discussions: Our model demonstrates changes in each coagulation factor concentration over time given initial species concentrations as adjustable inputs to reflect a starting physiologic state of coagulation. When initial species concentrations reflect a normal coagulation profile, little change in voltage is observed for each species of the isolated loop (Figure 2, orange). When initial species concentrations reflected tissue injury, changes in species concentration over time reflected expected coagulation physiology (Figure 2, blue). Tissue injury is simulated by increasing initial FXa concentration, and over time FXa concentration decreases as it is incorporated into other complexes (Figure 2a). Figure 2b shows that in a normal state there is little change in the concentration of tissue factor pathway inhibitor (TFPI) because its binding partner FXa has low concentration. In tissue injury, TFPI concentration decreases at the same rate as FXa as it binds in a 1:1 ratio. TFPI binds FXa to form a complex (FXa_TFPI), making FXa unavailable to amplify coagulation. Figure 2c shows in a normal state there is no change in concentration because there is little FXa to incorporate. In the injured state, FXa_TFPI concentration decreases in the latter half of the timeframe as FXa_TFPI binds to other complexes. Figure 2d demonstrates a complex of tissue factor (TF), activated factor VII (FVIIa), FXa and TFPI (TF_VIIa_FXa_TFPI). In the normal case the concentration of this complex is small and unchanging as there is little FXa available to incorporate. In the injured state there is an expected increase of TF_VIIa_FXa_TFPI complex over time because of the availability of FXa. Individual species are conserved in either condition. A representative example for one species is shown in Table 1. We observed 3 eigenvectors whose eigenvalues equal 0 which physiologically corresponds to no overall change in concentration for a given species. In conclusion, coagulation physiology may be simplified into a circuit model that allows for real-time physiologic visualization. This will be useful in timely adjustments of anticoagulation therapy and may aid in identification of new therapy targets. Future directions include investigation of other portions of the coagulation cascade and incorporating anti-coagulation therapy into this model.

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