A Novel Simplified Model for Blood Coagulation: A piecewise dynamical model for thrombin with robust predictive capabilities
AA Novel Simplified Model for Blood Coagulation:A piecewise dynamical model for thrombin withrobust predictive capabilities
Jayavel Arumugam ∗ Arun Srinivasa † May 23, 2017
Abstract
Realistic description of patient-specific mechanical properties of clot-ting dynamics presents a major challenge. Available patient-specific datafalls short of robustly characterizing myriads of complex dynamic inter-actions that happen during clotting. We propose a simplified switchingmodel for a key part of the coagulation cascade that describes dynamicsof just four variables. The model correctly predicts prolonged activity ofthrombin, an important enzyme in the clotting process, in certain plasmafactor compositions. The activity sustains beyond the time which is con-ventionally considered to be the end of clotting. This observation alongwith the simplified model is hypothesized as a necessary step towards effec-tively studying patient-specific properties of clotting dynamics in realisticgeometries.
Patient-specific geometry modeling, simulations with more realistic boundaryconditions, multiscale models that combine molecular mechanisms with clinicalmanifestation are some of the open problems discussed in vascular biomechanics[1]. Blood constituents vary drastically in patients and their composition playsa significant role in determining the mechanical properties of the resulting clot[2, 3, 4].Current methods model generation and depletion of various constituents inblood flow using convection reaction diffusion equations [5], ∂ [ Y i ] ∂t + div ([ Y i ] v ) = div ( D i grad [ Y i ]) + G i , i = 1 , ..., N, (1)where [ Y i ] is a constituent, v is velocity, D i are diffusion coefficients, and G i aresource terms due to clotting reaction dynamics. Number of constituents N is in ∗ [email protected], Dept. of Mechanical Eng., Texas A & M University † [email protected], Dept. of Mechanical Eng., Texas A & M University a r X i v : . [ q - b i o . M N ] M a y he order of thousands; typical models consider between 30 to 40 constituents[6, 5]; and clotting reaction models considering hundreds of constituents havebeen proposed [7, 8]. Models accounting for myriads of chemical reactions are considered as a first steptowards understanding the complex phenomena of clotting [5]. Further modelingmany such constituents are useful in a diagnostic setting where often diseases arereflected as abnormalities in the coagulation cascade [9]. In particular reactiondynamics alone, for example, dynamics of thrombin [10, 11], is known to beassociated with many diseases such as acute coronary syndrome [12], acutecerebrovascular disease [13], rheumatoid arthritis [14], etc. Recent studies inthis direction include use of machine learning tools to characterize and use suchabnormalities for disease classification tasks such as identifying patients proneto heart attacks [15, 16].However, simplified models for the coagulation cascade are required so that:1. the model predictions are more readily verifiable using experiments2. augmenting chemical reaction dynamics in other phenomena such as flowand mechanical characterization of clots in patient-specific terms becomefeasible and readily comprehensible.
The chemical reaction kinetics particularly poses many challenges. Models de-scribing kinetics using M reactions are of the form, d [ Y i ] dt = G i = M (cid:88) j =1 S ij r j ([ Y ]; k j ) ,s.t. [ Y i ](0) = [ Y i ] , S [ Y ] = S [ Y ] , [ Y i ] ≥ , (2)where S ij are stoichiometric coefficients ( i = 1 , ..., N, j = 1 , ..., M ), r j are re-action rates, k j are parameters of the reaction rates, and [ Y i ] are the initialplasma composition.Concentration of proteins involved in coagulation and rates of reactions varyby orders of magnitude (see Figure 1). Essentially, picomoles of trigger resultsin the formation of hundreds of nanomoles of certain enzymes. This in turnresults in macroscopic formation of clots. Typically stochastic methods [17] areused to properly account for low concentrations of species [18, 19].The nonlinear chemical kinetics is modeled using reaction rates that typi-cally has quadratic terms or Hill-like terms [20] . The models are very stiff and2igure 1: Concentration of different species during clotting in the extrinsicpathway. The concentration of various species varies by orders of magnitude.This poses stiffness issues in numerical schemes demanding small time steps andexcessive conditioning.solution trajectories are unstable in many directions [21]. The rates involve neg-ative feedback loop or cycles in the reaction cascade [22]. This poses challengesin numerically coupling chemical reactions with flow simulations.Moreover, there is uncertainty in the parameters of the reaction kineticsmodels. It is hard to measure most of these protein factors. Many rate parame-ters are inferred indirectly rather than being directly measured. Models used tosimulate clot in small two dimensional regions ( ∼ µ m) consider the dynamicsof many reactants [23]. These are inappropriate for simulations in realistic 3dimensional flow conditions in arteries, say, in order to study atheroscelerosis orthrombosis [23]. Reduced dimensional simplified chemical kinetics models willhelp to advance the quality of patient specific simulations in realistic geometries. Papadopoulos et al. [23] suggested a phenomenological model for thrombindynamics. Based on the mechanism of thrombin dynamics, they propose a sim-plified thrombin dynamics model using four reactions. The reactions includedynamics of thrombin, prothrombin, platelets, and activated platelets. Usingthe assumption of fast platelet activation, they derive analytical expression forthrombin generation. These are similar to thrombin generation functions pre-scribed by Hemker et al. [24]. The model essentially fits patient specific throm-bin generation profiles and the effect of the plasma factor composition and3nhibitors on the dynamics of thrombin were not emphasized.This motivated Sagar et al. [25] to come up with a dynamical model forthrombin generation using a hybrid strategy. The strategy combines differentialequations and several logical rules to model thrombin dynamics. They designtheir approach to model systems where mechanistic insights are poor and ex-perimental interrogation is difficult. This results in reduced order model thathas rates of the product of Hill-like terms and transfer functions that act as thelogical rules, i.e., r i = k i x η i i k i x η i i min (cid:32) k j x η j j k j x η j j , k m x η m m k m x η m m (cid:33) (3)where r is the reaction rate, k and η are parameters, and x pertains to pro-tein concentration or activity. Though the model shows good performance forthrombin dynamics, the transparency in the mechanistic models such as [24]and [23] is lost, i.e., the functionality of the model parameters and effect oftheir changes is not evident. We seek a middle ground between the two simpli-fied models where we find a dynamical model that makes use of the mechanisticknowledge of blood coagulation and is able to account for changes in plasmafactor composition. We suggest a simple phenomenological model for thrombin dynamics based onchemical kinetics:1. We model the stoichiometry and dynamics of certain important and easilymeasurable chemical species. The model is based on the classically viewedinitiation, propagation, and termination of thrombin dynamics. Hence thechemistry involved in the simulations offer physiological insight.2. We model the initiation phase and the propagation/termination phaseof thrombin dynamics separately using a switching criteria for the rateparameters. The switch separates the slow initiation phase from the latterwhich is orders of magnitude faster.3. The model we propose shows varied responses. The functionality of theparameters are evident and different aspects of thrombin dynamics areeasily alterable.4. A good model should be able to capture the necessary rich behavior of thephenomena as well as generalize well in order to predict important qualita-tive and quantitative responses. The model accounts for the physiologicaleffect of antithrombin and is able to predict certain important changesin thrombin dynamics due to changes in prothrombin and antithrombinconcentration. 4. The model is dynamical in the sense that rates for the model species couldbe calculated given the current values of the model variables. In otherwords, we essentially model G i for the important species in equation 1 sothat the model could be extended to account for transport.The simplified model proposed here is expected to make patient-specificmechanical characterization of clots in realistic geometries feasible.Figure 2: A schematic of the extrinsic pathway. We propose a simplified modelfor thrombin dynamics. We note that events that occur after thrombin genera-tion result in changes of mechanical properties. Figure 2 shows a schematic of the key elements of the extrinsic pathway involvedduring clotting. We consider the extrinsic pathway because hemostasis occursdue to tissue factor initiation. Further, we simply simplify thrombin dynamics.Given that flow properties affect and are affected by fibrin formation, such asimple thrombin dynamics model factors out the two phenomena. Reactions inthe intrinsic pathway could be readily accounted for due to the phenomenologicalaspect of initiation in the proposed model. For simulations of the full model,we use the extrinsic pathway developed by Hockin et al. [6]. A schematic of themodel used is shown in Figure 3.
We exploit the following ideas for model simplification:5igure 3: A schematic of the extrinsic pathway model for thrombin dynamicsused in this work. There are essentially three elements in this network, namely,i) thrombin initiation; ii) thrombin propagation; and iii) thrombin inhibition.
Dynamical Parametrization:
Patient specificity in blood coagulation isusually modeled using variations in the initial conditions for the differentialequations. This leads to a high-dimensional state space and specification ofdynamics in that space of many constituents as described earlier. However,we simplify the state of the system using just four important species and thenmodel the dynamics of those four constituents. The effect of other constituentsare parametrized in the rates describing the dynamics of the former, i.e., theeffect of patient specificity of thrombin dynamics is described using variations inthe rate parameters of the simplified model. We show that training the modelfor a specific choice of initial condition in the original state space (physiologicalmean) is able to predict qualitative responses of changes in prothrombin andantithrombin concentration in the reduced state space.
Switching:
We separate the parameters in the initiation and propagation/terminationphase of thrombin dynamics using a switching model. Such switching modelsare commonly used to model complex nonlinear systems [26]. Switching modelsare known to result in simplified models for even chaotic dynamics. For ex-ample, piecewise affine models are known to model chaotic behavior of manynonlinear systems well [27]. The proposed model switches from initiation topropagation/termination phase based on the concentration of thrombin. Suchthreshold-based responses have been used in models describing initiation [23]and platelet activation [5] in the coagulation cascade. The hybrid (switching)model by Makin and Narayanan [28] intends to develop a comprehensive modelrather than a simplified model. 6igure 4: A schematic of the simplified model proposed in this study. Thereis fuel prothrombin, the bursting enzyme thrombin, the inhibitor antithrombin,and the by-product thrombin-antithrombin. K S is a rate constant that modelsinitiation due to injury. K P is the rate of thrombin propagation. K I is the rateof thrombin inhibition. We use the traditional simplification of the thrombin generation cascade and de-scribe kinetics for prothrombin, thrombin, antithrombin, and thrombin-antithrombinusing the following set of reactions:1.
Thrombin Initiation : Tissue factor activates prothrombin to form throm-bin. II K S −−−−−−−−−−−−→ Initiation IIa2.
Thrombin propagation : Given that sufficient amount of thrombin (2nM) is activated, clotting propagates via a different set of reactions.II K P −−−−−−−−−−−−→ Propagation IIa3.
Thrombin inhibition : Finally normal hemostasis requires that throm-bin generation is controlled.IIa + AT K I −−−−−−−−−−−−→ Termination IIaAT7able 1: Switching Condition[IIa] < ≥ K S k s > K I k i > k i > K P k p > ddt [II] = − K S − K P [II][IIa] ddt [IIa] = K S + K P [II][IIa] − K I [IIa][AT] ddt [AT] = − K I [IIa][AT] ddt [IIa-AT] = K I [IIa][AT] . (4)where we model initiation using the rate constant K S , propagation using therate constant K P , and inhibition using the rate constant K I . In order to bestoichiometrically consistent, our [IIa] is the sum of both forms of thrombinin the full 34 variable model (the variables used in our model are drawn incontinuos lines in the full model simulation as seen in Figure 5). [IIa-AT] in ourmodel is the sum of the two antithrombin complex formed due to inhibition .Among the four species modeled, methods to measure or infer thrombin,ATIII, and thrombin antithrombin [29, 30, 31, 32] exist. The rates in equation4 are such that they satisfy a stoichiometric constraint, i.e.,[II] + [IIa] + [IIa-AT] = c [AT] + [IIa-AT] = c (5) c and c are constants defined using the initial conditions. These dependencerelations are crucial while designing experiments to properly observe the sys-tem. For example, calibrated automated thrombogram assay is inadequate tomake proper measurements with deficient ATIII [33] due to fluorogenic substratedepletion. In the results section, we will discuss the possibility of sustainedthrombin activity when ATIII is deficient leading to the depletion.We propose the switching rules in Table 1 that changes the response of themodel during initiation and propagation/termination. Essentially, thrombinpropagation occurs if [IIa] crosses a threshold. For normal clotting, rate ofpropagation is expected to be orders of magnitude higher than that of rate ofinitiation. We also use two different inhibition rate constant k i and k i so thatrate of inhibition could be separately tuned during the two phases. there are other antithrombin complexes formed in the full model but they are 3 orders ofmagnitude smaller than [IIa-AT] and we neglect them. Results and Discussion
In the simulations of the full model, clotting was initiated with 5 pM and theplasma factor composition was set to physiological mean values [6]. We used thedata from the full model to fit parameters for the reduced model. We note thatthrombin dynamics of the full model has been corroborated with experimentaldata [6]. Particle swarm optimization [34] was used for parameter estimationand we obtained one set of parameters for the physiological mean composition.Figure 6: Comparison of all the species modeled in the simplified model. Thesimplified model captures the reponse of the corresponding variable in the fullmodel very well. There is a slight mismatch in the maximum amount of throm-bin generation. This could be improved by also choosing the clot propagationthreshold ([IIa] = 2 nM) better.We used the sum of squared differences of the normalized concentrationprofiles between the common species in the full model and the reduced model10s the objective function, u = 1 M M (cid:88) m =1 4 (cid:88) i =1 (cid:32) C reducedi ( m ; k ) − C fulli ( m ) C constanti (cid:33) (6)where m denotes time points and i denotes model species. Mean concentration ofprothrombin and antithrombin were used as normalization constants C constanti .Comparison of the reduced and full model simulation for the physiological meaninitial composition is shown in Figure 6. We show the effect of parameters by changing them one at a time. Clot timedepends exponentially on K S (seen in Figure 7). Rate constant k i also controlsthe clot time Figure 8. For certain combinations of K S and k i it takes morethan 1200 seconds for clot initiation. Both the parameters together offer morecontrol over dynamics of clot initiation.Figure 7: Controlling thrombin initiation using K S . There is an nonlineardependence of clot time on this parameter due to K i .Figures 9 and 10 show the effect of changes in the rate constants K P and k i respectively. The parameters offer a wide range of thrombin generation rates.Similar to the initiation, there are certain values of K P (for a given value of k i )and vice versa where thrombin generation is too low. These two parameterstogether offer control over simulating a wide range of thrombin propagation. Finally, we check the qualitative response of the model predictions for varia-tions in initial prothrombin and antithrombin concentrations. As seen in Figure11, higher values of prothrombin are able to predict more thrombin generation.This has been observed in experiments [35]. Thrombin rates during terminationcould be improved using better training data and using different reaction rates.11igure 8: Controlling thrombin initiation using k i . This parameter along with K S , allows for modeling a wide range of clot times and dynamics during initia-tion.Figure 9: Controlling thrombin propagation using K P . As expected, variationsin the propagation rate constant is able to capture a wide range of thrombingeneration rates.Figure 10: Controlling thrombin termination using k i . This parameter hasmore effect on the termination phase of thrombin generation.12imilarly, lower values of antithrombin are able to predict higher thrombin gen-eration.Figure 11: Prediction on prothrombin variation. Thrombin activity sustains inplasma factor composition that has excessive prothrombin. Similar predictionis made in Figure 12 based on variations in initial antithrombin concentration.One of the most important predictions of this model is that thrombin termi-nation appears to halt at non-zero values (Figure 12). Such sustained activity isalso in observed in experiments when there is too much prothrombin comparedto antithrombin [35]. Inhibitors such as activated protein C may need to mod-eled in order to account for oscillations observed in such sustained activity. Inthis model, the reaction essentially runs out of the inhibitor [AT] when initiatedwith a certain plasma factor composition. In such a scenario, other phenomenalike diffusion and convection will control the extent of clotting. For example,when there is less inflow of antithrombin concentration, as in the case of stasis,we would expect more clotting due to the presence of excess active thrombin.The significance of antithrombin deficiency in simulations accounting for diffu-sion has already been reported [36]. For a given value of initial prothrombin, andgiven that thrombin generation proceeds normally due to sufficient rates, thesimplified model quantifies the amount of antithrombin required for thrombingeneration to terminate.Further, active thrombin could be propagated downstream and could po-tentially cause clotting elsewhere. This observation and model prediction onsustained activity of thrombin are hypothesized to play a necessary role towardseffectively studying clotting in realistic geometries. The simplified model proposed here, when coupled with fluid flow and transportmodels, is expected to make patient-specific mechanical characterization of clotsin realistic geometries feasible. The parameters of the model need to be alteredin order to better simulate thrombin dynamics on a wider range of plasma factorcompositions. Figure 15 shows simulations from parameters identified for acute13igure 12: Prediction on antithrombin variation. This is the most importantand significant prediction of the model. This has been observed in thrombin gen-eration experiments [35]. Moreover, this phenomena could be blind to markerslike TAT (IIa-AT) that infer thrombin activity.Figure 13: Simplified model fit for different cases. By changing the parame-ters, the model allows for a wide range of thrombin dynamics such as in acutecoronary syndromes and hemophilia A.14oronary syndrome population mean composition [12] and a hemophilia patient[11] (plasma factor composition of patient C in Figure 3. (B)). The hemophiliapatient simulations suggest the form of rate functions as well as the thresholdvalue (2 nM [IIa]) could be modeled better.Figure 14: Comparison of thrombin generation rates in time. The better thedata aligns with the x = y line, the better is the simplification. The effect oftime lag is better emphasized in this plot compared to Figure 6.For the mean plasma composition, Figure 15 shows the full model thrombinrates for the simplified model at different times during clotting plotted againstthe corresponding time points of the full model. The reaction rates could beimproved to better align the simulation data with the x = y line. We alsonote that the objective used for simplification in equation 6 does not explicitlypenalize deviations in rates. Figure 15 shows thrombin rates for the simplifiedmodel at different points in the state space along a particular trajectory ofthe full model plotted against the corresponding rates of the full model. Thealignment is better in this case compared to figure 15. This suggests that thesimplified model takes a slightly different path compared to the full model.We are currently working towards modifying the dynamics and the objectiveso that the rates could be better simulated. We are also currently testing theperformance of the model while accounting for the effects of diffusion.Augmenting the simplified model with fibrin dynamics would allow patient-and event-specific characterization of thromboelastography experiments [37, 38].This would be useful to better characterize clotting during surgeries [39]. Thiswork could be further improved by coupling with simplified platelet aggrega-tion models. To conclude, the drastically simplified model proposed here is anovel and a fertile step and subsequent progress has potential applications suchas virtual flow-diverter treatment planning [40], predicting thrombotic risk ac-counting for flow [41], factors like injuries in tandem [42] and stenosed arteries[43]. 15igure 15: Comparison of thrombin generation rates along the mean physiolog-ical trajectory of the full model. The model predictions are better compared tothose in Figure 15. We proposed a simplified model for thrombin dynamics based on the stoichiome-try of certain important chemical species. The model, using switching of param-eters based on a threshold, describes dynamics of thrombin akin to jump startinga car. The simplified model fits corresponding aspects of the full model well anddifferent features of thrombin dynamics are easily alterable. The model is ableto predict certain important changes in thrombin dynamics due to changes inprothrombin and antithrombin concentration. This prediction using the sim-plified model with the potential for clinical manifestation in hypercoagulablediseases is hypothesized as a necessary step in patient-specific simulations ofclotting in realistic geometries.
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