# Study of Reversible Platelet Aggregation Model by Nonlinear Dynamics

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{0}denotes the initial concentration of single platelets in the suspension.

_{0}= 0), and at time t = 0 they are assumed to become “sticky”, i.e., the platelets assume an ability to form aggregates. The relationship between the concentration of activator and the “stickiness” of platelets (parameter values) was not investigated previously [2]. Another unconventional feature of Equation (1) is that the mass conservation law does not hold (one aggregate could be formed from two aggregates and vice versa). The addition of another variable, the average size of aggregates, restores the mass conservation law [2]. However, for a description of the aggregation curve with as a low number of parameters as possible, the two variables are sufficient.

_{i}) of the model depend on the concentration of activator (ADP) with a Pearson’s correlation coefficient of 0.9 for the ${k}_{1}$ parameter [2]. However, conventional analysis utilizing nonlinear dynamics for this model was not performed.

_{0}). We demonstrated that in most cases for positive values of parameters and variables only one singular point exists, and its type is always the stable node. For the first and second methods of bifurcation analysis, the possible bifurcations occur only for negative and zero values of parameters for the probabilities of the aggregate dissociation. However, after conducting a five-dimensional analysis, we found bifurcation points in a positive range of parameters. Additionally, we performed an explicit sensitivity analysis to evaluate the impact of parameter values on the aggregation curves.

## 2. Materials and Methods

#### 2.1. Computational Methods

#### 2.2. Reagents

^{®}, Greiner Bio-One GmbH, Kremsmünster, Austria).

#### 2.3. Blood Collection and Aggregometry

## 3. Results

#### 3.1. Transformation of the Model

_{0}denotes the initial platelet concentration, and n denotes the concentration of platelet aggregates; therefore, y is the dimensionless concentration of aggregates and a is the ln of optical density of the solution. The system of Equation (1) with the application of Equation (2) becomes the following system:

_{1}is the probability of new aggregate formation from two single platelets, k

_{2}is the probability of another platelet attachment to an existing aggregate, k

_{3}is the probability of formation of one aggregate from two existing ones, and k

_{−1}and k

_{−3}are the probabilities of an aggregate fragmenting (Figure 1b).

_{0}) and the activation strength. Here, p

_{0}is already incorporated in the values of k

_{1}, k

_{2}, and k

_{3}. However, the dependence on the concentration of ADP is not obvious.

#### 3.2. Bifurcation Analysis of the System

_{i}on the behavior of the system, we have determined the types of singular points for different sets of parameters and looked for the bifurcation points. For all three ADP concentrations, the type of singular point was a stable node (see Figure 3 legend for the eigenvalues of Jacobian). For each set of parameters, the coordinates of the singular point were calculated using the analytical Equation (6). The set of coordinates of the singular point corresponds to the eigenvalues of the Jacobian of Equation (5). For each of the parameter sets, the eigenvalues are negative real numbers; therefore, the type of the singular point is a stable node. The characteristic form of the phase plane for this type of singular point is given in Figure 3. It was surprising because the aggregation curves were reversible in all cases, and the same values of the variable a were achieved for different values of the variable y (Figure 3).

_{−3}= 0 for curve ADP = 5 μM, the type of singular point changed from «stable node» to «stable focus». The corresponding real eigenvalue of the Jacobian remained negative (Figure S1a,b).

_{−3}, which was varied in the range {−1000, 1000}. For most sets of parameters, only singular points of the type “stable node” were found. The type “stable focus” was found for 2.5 μM ADP (Figure 4b) and 10 μM ADP (Figure 4e) model with both k

_{−3}and k

_{1}varied, and for 5 μM the ADP models with k

_{−3}and k

_{−1}(Figure 4c), or k

_{−3}and k

_{3}(Figure 4d) varied. Unstable singular points were also found for 2.5 μM ADP and 10 μM ADP models with both k

_{−3}and k

_{1}varied (Figure S1), the type “saddle” was not found.

_{−3}= 0 for the system for 5 μM ADP, and k

_{−3}< 0 for other concentrations of ADP. For greater negative values of the parameters, a loss of stability of the singular point could occur (Figure S1). The negative values of the parameters do not bear any physiological meaning, because the reverse reactions are described by different equations (Figure 1b). Thus, we can conclude that during the reversible platelet aggregation, the system is outside of the influence area of the steady state.

#### 3.3. Effects of Parameter Variation

_{1}is the probability of new aggregate formation from two single platelets, and thus with the increase in this parameter, initial aggregation should occur faster, and so the time before the achievement of maximum aggregation should decrease.

_{3}is the probability of forming one aggregate from two existing ones, and so with the increase in the parameter aggregation becoming more intense in the late period when there is a significant amount of aggregate, the height of the steady state thus decreases. The parameters k

_{−1}and k

_{−3}are the probabilities of an aggregate fragmenting, and so with the increase in these parameters, the height of steady state is increased because of more intense disaggregation.

## 4. Discussion

_{−3}= 0 for the set of parameters corresponding to 5 µM ADP (Table 1, Figure 4a). The “stable focus” type of the singular point corresponds to the non-physiological type of aggregation curve. The second mode of bifurcation analysis consisted of the simultaneous variation of two parameters. In most cases, bifurcations in this mode were also not found. Negative values of the bifurcation parameters correspond to the opposite directions of reactions; therefore, in most cases, negative values of constant rates represent an experimental or procedural artifact [25]. However, in the present study, it is notable that bifurcations were found while varying the parameter k-

_{3}which equals k-

_{3}- k

_{-1}of the original model (Equation (1)), thus bifurcations in the negative values have physiological meaning, corresponding to the high probability of the aggregate disintegration into two platelets. We have performed an additional numerical investigation of the five-dimensional parameter space (Figure 5). Although there are no bifurcations close to the points given in Table 1, in distant areas of the parameter space, a bi-stable behavior could be observed (Figure 5b,c).

_{1}and k

_{2}determine the initial aggregation rate (Figure 6), while the parameter k

_{3}does not have an impact on it. This effect is more noticeable for small concentrations of ADP (Figure 6a,d). Therefore, this part of the aggregation curve is mostly dependent on the formation of small aggregates, most probably on the initial activation of integrins [26,27]. This result is consistent with the experimental data on platelet aggregate formation in diluted solutions [28].

_{3}, k

_{−1}, k

_{−2}and k

_{−3}determine the steady state level and thus the stability of aggregates (Figure 6). This finding lies in line with our previous statement [2], and experimental results obtained in previous studies [29], that reversible platelet aggregation is the process of destabilization of large aggregates formed in the first phase. Although in this study the dependence of the model parameters on ADP concentration was not observed, we expect the parameters k

_{1}and k

_{2}to be linearly proportional to the logarithm of reagent concentration as was described earlier by Babakhani et al. [30].

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schemes of experimental (

**a**) and theoretical (

**b**) processes of platelet aggregation in suspension. (

**a**) Conventional light transmission aggregometry. A ray of light after passing through a cuvette with a suspension of platelets enters the photocell. Due to the formation of large local light scattering centers—platelet aggregates—the total light transmission of the suspension increases. The light transmission measured by the aggregometry determines the degree of platelet aggregation. (

**b**) Illustration of a mathematical model of platelet aggregation based on the law of mass action between aggregates (n) and single platelets (p) [2].

**Figure 2.**Fitting of Equation (3) to experimental data on platelet aggregometry in response to ADP. (

**a**–

**c**) Five parameters of Equation (3) were fitted by means of the “Particle Swarm” parameter estimation method to describe experimental data of platelet aggregation in response to ADP at 2.5 μM (

**a**), 5 μM (

**b**), or 10 μM (

**c**). Blue lines denote variable y, black lines denote variable

**a**, red squares denote the logarithm of experimental OD values. Experimental data are typical curves from n = 10.

**Figure 3.**Phase planes for the models, presented in Figure 2. The models were fitted to experimental data of platelet aggregation in response to ADP at 2.5 μM (

**a**), 5 μM (

**b**), or 10 μM (

**c**). Nullclines are given in grey, trajectories from point {a,y} = {1,0} into steady state (same as in Figure 2) are given in black. Steady state parameters (the asterisk(*) denotes the stationary values of the variables): (

**a**) a * = 1.047, y * = 0.327, λ

_{1}= −0.24, λ

_{2}= −0.041; (

**b**) a * = 1.06, y * = 1.06, λ

_{1}= −0.17, λ

_{2}= −0.028; (

**c**) a *= 1.012, y *= 0.78, λ

_{1}= −0.68, λ

_{2}= −0.014.

**Figure 4.**Bifurcation diagrams. Singular point values of a and y for Equation (4) as a function of parameter ${k}_{-3}$ from {−1, −0.9, −0.8, …, −0.1, −0.01, −0.0001, 0, 0.0001, 0.001, 0.01, 0.1, …, 0.9, 1}. (

**a**) Parameter ${k}_{-3}$ is varied, other parameters are given in Table 1 for ADP = 5 μM; (

**b**) parameter ${k}_{-3}$ is varied, ${k}_{1}=1$, other parameters are given in Table 1 for ADP = 2.5 μM; (

**c**) parameter ${k}_{-3}$ is varied, ${k}_{-1}=0.01$, other parameters are given in Table 1 for ADP = 5 μM; (

**d**) parameter ${k}_{-3}$ is varied, ${k}_{3}=0.0001$, other parameters are given in Table 1 for ADP = 5 μM; (

**e**) parameter ${k}_{-3}$ is varied, ${k}_{1}=1$, other parameters are given in Table 1 for ADP = 10 μM.

**Figure 5.**Numerical investigation of the parameter space for ADP = 2.5 μM. (

**a**) Typical bifurcation diagrams demonstrating an area with 4 singular points (log

_{10}(a) as a function of k

_{−1}and k

_{3}, other parameters were set at 0.49 (k

_{1}), 0.067 (k

_{2}), 0.026 (k

_{−3}). It can be seen that all four singular points exist for almost any combination of parameters. However, the third and fourth points are always “unstable”, while the second is always stable and positive, and the first could be either stable or unstable. (

**b**,

**c**) Bifurcation diagrams for the values of a (

**b**), inset show the value of k

_{3}for which 4 singular points with positive values of a exist) and the real part of the first eigenvalue of Jacobian (

**c**), inset shows a close up view near Re(λ) = 0), k

_{3}is varied, k

_{−1}= 5, k

_{1}= 0.49, k

_{2}= 0.067, k

_{−3}= 0.026. Together, (

**b**,

**c**) demonstrate the existence of two “stable nodes” (first and second) with positive values of a for a subspace of parameter space. At k

_{3}= 0.0025, a “stable node”–“unstable node” bifurcation happens for the second singular point (black lines). (

**d**) Phase plane for the system with the following parameter set: k

_{−1}= 0.05, k

_{1}=4.93, k

_{2}= 0.675, k

_{3}= 0.1, k

_{−3}= 0.026. Nullclines are given in red, trajectories from point {a,y} = {1,0} into steady state are given in blue. Stable singular points are given in blue, unstable singular points are given in red, directions of phase trajectories are shown in grey.

**Figure 6.**The effects of variations in parameter values on the model responses. Time courses for a(t) and y(t) for parameter sets given in Table 1 for the corresponding ADP concentrations with one parameter being varied for each plot. For panels (

**a**–

**c**) ${k}_{1}$ was varied; for (

**d**–

**f**)${k}_{2}$ was varied; for (

**g**–

**i**)${k}_{3}$ was varied.

**Table 1.**Estimated parameter values for Equation (3) for experimental datasets presented in Figure 2.

2.5 μM ^{1} | 5 μM | 10 μM | |
---|---|---|---|

k_{−1} | 0.5027 | 1.73 × 10^{−11} | 0.1995 |

k_{−3} | 0.0262 | 0.02778 | 0.01329 |

k_{1} | 0.004933 | 0.01469 | 0.003955 |

k_{2} | 0.6757 | 0.1617 | 0.86886 |

k_{3} | 0.1042 | 0.02627 | 0.01731 |

^{1}ADP concentrations for the corresponding experimental data.

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**MDPI and ACS Style**

Vasilev, G.A.; Filkova, A.A.; Sveshnikova, A.N. Study of Reversible Platelet Aggregation Model by Nonlinear Dynamics. *Mathematics* **2021**, *9*, 759.
https://doi.org/10.3390/math9070759

**AMA Style**

Vasilev GA, Filkova AA, Sveshnikova AN. Study of Reversible Platelet Aggregation Model by Nonlinear Dynamics. *Mathematics*. 2021; 9(7):759.
https://doi.org/10.3390/math9070759

**Chicago/Turabian Style**

Vasilev, Grigorii A., Aleksandra A. Filkova, and Anastasia N. Sveshnikova. 2021. "Study of Reversible Platelet Aggregation Model by Nonlinear Dynamics" *Mathematics* 9, no. 7: 759.
https://doi.org/10.3390/math9070759