# Dynamic Modeling of the Human Coagulation Cascade Using Reduced Order Effective Kinetic Models

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Formulation of Reduced Order Coagulation Models

#### 2.2. Identification of Model Parameters Using Particle Swarm Optimization

_{2}pH 7.4 (HBS/Ca

^{2+}) for 30 min at 37°C. The relipidated TF was incubated with 10 pmol/L factor VIIa for 20 min to allow the formation of FVIIa-TF. Factors V, VIII and thrombomodulin (Tm) (when protein C activation is required) were added to FVIIa-TF complex. Thrombin generation was then initiated by adding equal volumes of this mixture with a mixture containing prothrombin, factor IX and factor X, TFPI, AT-III and protein C (added when required), protein S (added when required) and factor XI (added when required). In the training data, 5 pmol/L FVIIa-TF was used along with 200 µmol/L of phospholipid vesicles (PCPS) to initiate thrombin generation. All other the coagulation factors and inhibitors i.e., factors X, IX, V, and VIII, prothrombin, TFPI and AT-III, protein C and protein S (when applicable) were at their mean plasma concentrations.

_{D}) for each data group. The delay parameter was constant within a data set, but allowed to vary across training data sets. Introduction of the delay parameter allowed the model to simulate multiple training data sets using a single ensemble of model parameters. Taken together, the model identification results suggested that our hybrid approach could reproduce a panel of thrombin generation data sets in the neighborhood of physiological factor and inhibitor concentrations. However, it was unclear whether the reduced order model could predict new data, without updating the model parameters.

#### 2.3. Validation of the Reduced Order Coagulation Model

_{D}, was fixed within each validation data set but allowed to vary between data sets. The reduced order model predicted the thrombin generation profile for ratios of prothrombin and ATIII in the absence of the protein C pathway (Figure 5). Simulations near the physiological range (fII,ATIII) = (100%, 100%) or (125%, 75%) tracked the measured thrombin values (Figure 5B,C). On the other hand, predictions for factor levels outside of the physiological range (fII,ATIII) = (50%, 150%) or (150%, 50%), while qualitatively consistent with measured thrombin values, did show significant deviation from the measurements (Figure 5A,D ). Likewise, simulations of thrombin generation in normal versus hemophilia (missing both fVIII and fIX) were consistent with measured thrombin values (Figure 6). We modeled the dependence of thrombin amplification on factor levels using a product rule $(\mathcal{Z}=fV\times fX\times fVIII\times fIX)$, which was then was integrated using a min integration rule into the control variable governing amplification. Thus, in the absence of fVIII or fIX, the amplification control variable evaluated to zero, and the only thrombin produced was from initiation (Figure 6B). However, the decay of the thrombin signal was underpredicted in the normal case (Figure 6A), while the activated thrombin level was overpredicted in hemophilia simulations, although thrombin generation was far less than normal (Figure 6B). Taken together, the reduced order model performed well in the physiological range of factors, even with unmodeled components such as platelet activation in the hemophilia data set.

^{∗∗}> t

^{∗}). Further increases in trigger strength resulted in decreased thrombin peak time and increased maximum thrombin values (Figure 7A, 50× trigger). Thus, for large trigger values (200× trigger), the hemophilic thrombin profile approximated normal coagulation, where peak thrombin was achieved shortly after administration and 95% of the thrombin was gone by 20 min after initiation. We performed flux analysis to understand how the reduced order coagulation model balanced initiation, amplification and termination of thrombin generation for normal and hemophilic coagulation. Analysis of the reaction flux through the reduced order network for thrombin generation in normal, hemophilia and rFVIIa-treated hemophilia identified three distinct operational modes (Figure 8). We calculated the flux through four lumped reactions, initiation, amplification, thrombin-induced APC generation and total thrombin inhibition (including both APC and ATIII action). Directly after the addition of a trigger (e.g., TF/FVIIa or rFVIIa), the lumped initiation flux was the largest for all three cases. However, within a few minutes enough thrombin was generated by the initiation mechanism to induce the amplification stage. During amplification, thrombin catalyzes its own formation and inhibition by generating activated protein C (APC), a potent inhibitor of the coagulation cascade. For normal coagulation, amplification and thrombin inhibition were the dominate reactions by 6 min after initiation (Figure 8, left). After 10 min, the dominate reaction had shifted to thrombin inhibition (both ATIII and APC action). In hemophilia (missing both fVIII and fIX), the amplification reaction did not occur, and thrombin was produced only by initiation (Figure 8, center). Initiation was quickly inhibited by APC, and the thrombin level stabilized (eventually decaying at longer times because of ATIII activity). Lastly, when 50× trigger was used to induce thrombin formation in hemophilia (absence of fVIII/fIX), initiation mechanisms dominated for up to 6 min following initiation (Figure 8, right). Similar to hemophilia alone, no amplification occurred in the 50× trigger+hemophilia case, and the rate of thrombin generation was extinguished by the combined action of ATIII and APC. Taken together, the hybrid modeling approach captured the transition between the modes of thrombin generation, as well as the role that inhibitors play in attenuating the thrombin generation rate. Thus, the transfer function approach encoded the inhibitory logic of this cascade in the absence of specific mechanism.

#### 2.4. Global Sensitivity Analysis of the Reduced Order Coagulation Model

## 3. Discussion

## 4. Materials and Methods

#### 4.1. Formulation and Solution of the Model Equations

_{i}) in our reduced order coagulation model:

_{ij}> 0, species i is produced by reaction j. Conversely, if σ

_{ij}< 0, species i is consumed by reaction j, while σ

_{ij}= 0 indicates species i is not connected with reaction j. The system material balances were subject to the initial conditions

**x**(t

_{o}) =

**x**

_{o}, which were specified by the experimental setup.

_{j}) that could depend upon many regulatory transfer functions:

_{i}denotes the enzyme abundance which catalyzes reaction j, and K

_{js}denotes the saturation constant for species s in reaction j. The product in Equation (3) was carried out over the set of reactants for reaction j (denoted as $m\overline{j}$). The control term v

_{j}depended upon the combination of factors which influenced the activity of enzyme i. For each enzyme, we used a rule-based approach to select from competing control factors (Figure 2). If an enzyme was activated by m metabolites, we modeled this activation as:

_{ij}(Z) ≤ 1 was a regulatory transfer function that calculated the influence of metabolite i on the activity of enzyme j. Conversely, if enzyme activity was inhibited by m metabolites, we modeled this inhibition as:

^{+}and j

^{−}denoted the sets of activating and inhibitory factors for enzyme j. If a process has no modifying factors, we set v

_{j}= 1. There are many possible functional forms for 0 ≤ f

_{ij}(Z) ≤ 1. However, in this study, each individual regulatory transfer function took the form:

_{j}denotes the abundance of the j factor (e.g., metabolite abundance), and k

_{ij}and η are control parameters. k

_{ij}was the species gain parameter, while η was a cooperativity parameter (similar to a Hill coefficient). Applying the general framework to the reduced coagulation network resulted in five ordinary differential equations:

**x**= (fII, FIIa, PC, APC, ATIII)

^{T}. The terms r*v* in the balance equations denote corrected kinetic expressions for initiation, amplification and inhibition processes. The rate of initiation ${\overline{r}}_{init}$ was modeled as:

_{init}, K

_{init,fII}are the rate and saturation constants governing initiation, respectively. The rate of initiation was modified by v

_{init}, the control parameter governing initiation. Initiation was sensitive to the level of trigger (activator) and TFPI (inhibitor):

_{amp}, K

_{amp,fII}denote the rate and saturation constants governing amplification, respectively. The amplification control term, which modified amplification rate, was modeled as a combination of multiple inhibition terms and one activation term:

_{amp}= fV x fX x fVIII x fIX. Although ${f}_{amp}^{+}$ (Z

_{amp}) is an activating term, we included it in the min integration rule; the factors in Z

_{amp}were essential for amplification (if any of these factors was missing the amplification reaction would not occur). Thus, the factors in Z

_{amp}were required components, a classification that we implemented by the min selection rule. The rate activated protein C formation was given by:

_{apc,}

_{formation}and K

_{formation,PC}denote the rate and saturation constants governing activated protein C formation, respectively and TM denotes the thrombomodulin abundance. We modeled the control term which governed APC formation as a single thrombin-dependent activation term:

_{inh,ATIII}was taken to be unity. The model equations were encoded using the Python programming language and solved using the ODEINT routine of the SciPy module [66]. The model files can be downloaded from http://www.varnerlab.org.

#### 4.2. Estimation of Model Parameters From Experimental Data

_{j}(τ, k) denotes the simulated value for species j at time τ, and ω

_{j}(τ) denotes the experimental measurement variance for species j at time τ. The outer summation is with respect to time, while the inner summation is with respect to state. We minimized the model residual using Particle swarm optimization (PSO) [67]. PSO uses a swarming metaheuristic to explore parameter spaces. A strength of PSO is its ability to find the global minimum, even in the presence of potentially many local minima, by communicating the local error landscape experienced by each particle collectively to the swarm. Thus, PSO acts both as a local and a global search algorithm. For each iteration, particles in the swarm compute their local error by evaluating the model equations using their specific parameter vector realization. From each of these local points, a globally best error is identified. Both the local and global error are then used to update the parameter estimates of each particle using the rules:

_{1}, θ

_{2}, θ

_{3}) are adjustable parameters, $\mathcal{L}$

_{i}denotes the local best solution found by particle i, and G denotes the best solution found over the entire population of particles. The quantities r

_{1}and r

_{2}denote uniform random vectors with the same dimension as the number of unknown model parameters (Κ x 1). In thus study, we used (θ

_{1}, θ

_{2}, θ

_{3}) = (1.0,0.05564,0.02886). The quality of parameter estimates was measured using goodness of fit (model residual). The particle swarm optimization routine was implemented in the Python programming language. All plots were made using the Matplotlib module of Python [68].

#### 4.3. Global Sensitivity Analysis of Model Performance

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Schematic of the connectivity of the reduced order coagulation model. A trigger compound, e.g., TF/FVIIa initiates thrombin production (FIIa) from prothrombin (fII). Once activated, thrombin catalyzes its own activation (amplification step), as well as its own inhibition via the conversion of protein C to activated protein C (APC). APC and tissue factor pathway inhibitor (TFPI) inhibit initiation and amplification, while antithrombin III (ATIII) directly inhibits thrombin. All inhibition steps and trigger-induced initiation were modeled using a rule-based approach. Likewise, the dependence of amplification on other coagulation factors was also modeled using a rule-based approach. The abundance of the highlighted species (in the dashed boxes) was governed by an ordinary differential equation. All other species were assumed to be constant.

**Figure 2.**Schematic of rule-based effective control laws. Traditional enzyme kinetic expressions, e.g., Michaelis-Menten or multiple saturation kinetics are multiplied by an enzyme activity control variable 0 ≤ υ

_{j}≤ 1. Control variables are functions of many possible regulatory factors encoded by arbitrary transfer functions of the form $0\le {f}_{j}(\mathcal{Z})\le 1$. At each simulation time step, the υ

_{j}variables are calculated by evaluating integration rules such as the max or min of the set of transfer functions f

_{1},…, f

_{n}influencing the activity of enzyme E

_{j}.

**Figure 3.**Reduced order coagulation model training simulations. Reduced order coagulation model parameters were estimated using particle swarm optimization (PSO) without the protein C pathway as a function of prothrombin. Solid lines denote the simulated mean value of the thrombin profile for N = 20 independent particles, points denote experimental data. The shaded region denotes the 99% confidence estimate of the mean simulated thrombin value (uncertainty in the model simulation). (

**A–C**) depict training data and results for 150%, 100% and 50% of physiological prothrombin levels in the absence of protein C pathway. The experimental training data was reproduced from the study of Butenas et al. [40]. Thrombin generation in these experiments was initiated using 5 pmol/L FVIIa-TF in the presence of 200 µmol/L of phospholipid vesicles (PCPS). As depicted in (

**A–C**) the prothrombin levels were at 150%, 100% and 50% of their physiological concentration in the absence of protein C pathway. All factors and control proteins in these experiments were at their physiological concentration unless otherwise denoted.

**Figure 4.**Reduced order coagulation model training simulations. Reduced order coagulation model parameters were estimated using particle swarm optimization (PSO) with the protein C pathway as a function of prothrombin. Only APC pathway parameters were allowed to vary in these simulations keeping the parameters estimated without protein C pathways constant. Solid lines denote the simulated mean value of the thrombin profile for N = 20 independent particles, points denote experimental data. The shaded region denotes the 99% confidence estimate of the mean simulated thrombin value (uncertainty in the model simulation). (

**A–C**) depict training data and results for 150%, 100% and 50% of physiological prothrombin levels in the presence of the protein C pathway. The experimental training data was reproduced from the study of Butenas et al. [40,41]. Thrombin generation in these experiments was initiated using 5 pmol/L FVIIa-TF in the presence of 200 µmol/L of phospholipid vesicles (PCPS). As depicted in (

**A–C**) the prothrombin levels were at 150%, 100% and 50% of their physiological concentration in the presence of protein C pathway. All factors and control proteins in these experiments were at their physiological concentration unless otherwise denoted.

**Figure 5.**Reduced order coagulation model predictions versus experimental data for normal coagulation. The reduced order coagulation model parameter estimates were tested against data not used during model training. Simulations of different levels of prothrombin and ATIII were compared with experimental data in the absence of the protein C pathway. Solid lines denote the simulated mean value of the thrombin profile for N = 20 independent particles, points denote experimental data. The shaded region denotes the 99% confidence estimate of the mean simulated thrombin value (uncertainty in the model simulation). (

**A–D**) prediction results for (FII,ATIII): (50%, 150%), (100%, 100%), (125%, 75%) and (150%,50%) of physiological prothrombin and ATIII levels in the absence of the protein C pathway. The experimental validation data was reproduced from the study of Butenas et al. [40]. Thrombin generation in these experiments was initiated using 5 pmol/L FVIIa-TF in the presence of 200 µmol/L of phospholipid vesicles (PCPS). As depicted in (

**A–D**) the prothrombin and ATIII levels were at (50%, 150%), (100%, 100%), (125%, 75%) and (150%, 50%) of their physiological concentrations in the absence of protein C pathway. All factors and control proteins were at their physiological concentration unless otherswise denoted.

**Figure 6.**Reduced order coagulation model predictions versus experimental data with and without coagulation factors VIII (fVIII) and IX (FIX). The reduced order coagulation model parameter estimates were tested against data not used during model training. Simulations of normal thrombin formation with ATIII and the protein C pathway were compared with thrombin formation in the absence of fVIII and fIX. Solid lines denote the simulated mean value of the thrombin profile for N = 20 independent particles, points denote experimental data. The shaded region denotes the 99% confidence estimate of the mean simulated thrombin value (uncertainty in the model simulation). (

**A,B**) prediction results for normal thrombin generation and thrombin generation in hemophilia. All factors and control proteins were at their physiological concentration unless others noted. Coagulation was initiated with 0.2 nmol/L FVIIa. The experimental validation data was reproduced from the study of Allen et al. [42].

**Figure 7.**Reduced order coagulation model predictions of rFVIIa administration. (A) Simulations of thrombin formation in the presence of ATIII and the protein C pathway were conducted for a range of trigger values (1x–200x nominal) in the absence of fVIII and fIX; (B) Comparison of thrombin generation for normal versus hemophilia for 10x nominal trigger. Solid lines denote the simulated mean value of the thrombin profile for N = 20 independent particles. The peak thrombin time for normal coagulation (t

^{∗}) is less than rFVIIa induced coagulation in hemophilia (t

^{∗∗}), while the peak thrombin value was greater in normal coagulation. The shaded region denotes the 99% confidence estimate of the mean thrombin value (uncertainty in the model simulation). All factors and control proteins were at their physiological concentration unless others noted.

**Figure 8.**Reaction flux distribution as a function of time for thrombin generation under normal (left), hemophilia (center) and rFVIIa treated hemophilia (right). Reaction flux was calculated for each particle at T = 0, 4, 6, 8, 10, 12, 14 min after the initiation of coagulation. Reaction fluxes were calculated for each particle in the parameter ensemble (N = 20). Blue colors denote low flux values while red colors denote high flux values.

**Figure 9.**Global sensitivity analysis of the reduced order coagulation model with respect to the model parameters. (

**A**) Sensitivity analysis of the thrombin peak time for different prothrombin levels (150%, 100% and 50% of the physiological value) as a function of activated protein C; (

**B**) Sensitivity analysis of the thrombin exposure for different prothrombin levels (150%, 100% and 50% of the physiological value) as a function of activated protein C. Points denote the mean total sensitivity value, while the area around each point denotes the uncertainty in the sensitivity value. The gray dashed line denotes the 45° degree diagonal, if sensitivity values are equal for different conditions they will lie on the diagonal. Sensitivity values significantly above or below the diagonal indicate differentially important model parameters. The radius of the shaded region around each total sensitivity value was the maximum uncertainty in that value estimated by the Sobol method.

© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/)

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**MDPI and ACS Style**

Sagar, A.; Varner, J.D.
Dynamic Modeling of the Human Coagulation Cascade Using Reduced Order Effective Kinetic Models. *Processes* **2015**, *3*, 178-203.
https://doi.org/10.3390/pr3010178

**AMA Style**

Sagar A, Varner JD.
Dynamic Modeling of the Human Coagulation Cascade Using Reduced Order Effective Kinetic Models. *Processes*. 2015; 3(1):178-203.
https://doi.org/10.3390/pr3010178

**Chicago/Turabian Style**

Sagar, Adithya, and Jeffrey D. Varner.
2015. "Dynamic Modeling of the Human Coagulation Cascade Using Reduced Order Effective Kinetic Models" *Processes* 3, no. 1: 178-203.
https://doi.org/10.3390/pr3010178