A mathematical model of the metabolic process of atherosclerosis

A mathematical model of the metabolic process of atherosclerosis is constructed.


M a t h e M a t i c a l M o d e l i n g o f b i o c h e M i c a l p r o c e s s M a t h e M a t i c a l M o d e l i n g o f b i o c h e M i c a l p r o c e s s
I n the present work with the help of a mathe matical modeling, we continue the study of a prostacyclinthromboxane system of blood. We will investigate how low density lipoproteins (LDL) influence the dynamics of this metabolic process. In the construction of equations (Eqs.) of our model and the determination of its parameters, we used the re sults obtained by Prof. S.D. Varfolomeev and Prof. A. T. Mevkh. Their book and the fruitful collabo ration with them [13] allow Prof. V.P. Gachok and the author to obtain calculation results similar to the experimental ones in the case where the system is in a stable state of hemostasis [49], which state charac terizes a healthy blood vessel. It is the ideal state, which is attained by synchronization of the systems of thrombosis and antithrombosis. The dynamical stationary equilibrium arises. The desynchroniza tion of these systems results in the appearance of autooscillatory modes in the metabolic process of a prostacyclinthromboxane system. If the stationary kinetics is broken so that the level of thromboxane increases, then the coagulability of blood grows as well, and the appearance of thrombosis becomes possible in the circulatory system. On the contrary, if the level of prostacyclin increases, then the coagu lability of blood decreases, and hemophilia occurs. If the autooscillatory mode arises, then the appearance of a thrombus as a result of increased coagulabil ity on some time interval and its abruption under a decrease of the coagulability in the following time interval are possible. The actions of external and in ternal factors induce various modes in the system.
The metabolic process of coagulability of blood is considered by the author of the article as an open nonlinear system. The study was conducted using methods of nonlinear dynamics.
The kinetic model [49] allowed us to trace the effect of various levels of activities of phospholipases and concentrations of prostacyclin and thromboxane on properties of a biosystem, to determine the role of the arachidonic acid exchange between thrombo cytes and endothelium, to analyze the influence of parallel processes running with the participation of arachidonic acid on kinetics of changes and on sta tionary levels of prostanoids, and to find the struc turalfunctional connections of selforganization in the biosystem.
We will modify the given model by adding four nonlinear Eqs.. The other parameters remain un modified. Within the model, we will study the influ ence of concentration of "bad cholesterol" (LDL) on the metabolism of a hemostasis of blood vessels. The principal reason for its elevated level is high dietary fat content. The excessive content of fat in organism causes formation of atheromatous plaques. They are aggregates of LDL on internal walls of blood ves sels, causing stenosis. The atheromatous plaques grow over time. As a result, blood circulation slows down, which creates a deficit of nutrients in tissues. In this case, the arteries become denser and gradual ly lose their elasticity; i.e., atherosclerosis develops [1017].
Atherosclerosis does not appear instantly, but arises gradually during the whole life. The excess of LDL is accumulated on arterial walls and is chemi cally modified. The modified LDL then stimulate adhesion of the endothelial cells to monocytes and Т-cells. In addition, the endothelial cells secrete chemokines, which entrap Т-cells in a trap of intima. Macrophages and Т-cells produce numerous media tors of inflammation such as cytokines and cell divi sion signaling molecules. In addition, macrophages express waste receptors, which help them to absorb modified LDL. Macrophages absorb LDL, by filling themselves by drops of fat. These foamy macropha ges loaded with fat and Т-cells form fat strips, which are earlier manifestations of atherosclerotic plaques. Molecules participating in inflammation facilitate further growth of a plaque and formation of a fibrous capsule above the lipid core. Thus, the dynamics of the metabolic process of accumulation of cholesterol in the intima of an arterial wall has the autocatalytic character. The unusual accumulation of cholesterol occurs. Its amount depends on aggregated throm bocytes and oxidized lipoproids, which depend, in turn, on the concentration of cholesterol in blood. This can explain the sharp growth of cholesterol plaques and the unexpected appearance of athero sclerosis in a person with medium level of choles terol in blood [10].
During person's life, the metabolic process of the circulatory system permanently adapts to the conditions of nutrition. Respectively, the amount of cholesterol in the intima of blood vessels varies. At a high level of cholesterol, atherosclerotic damage to walls of blood vessels occurs. Inflammation induces propagation of atherosclerosis under imbalanced nutrition, unhealthy life style, and associated path ological conditions. Inflammation of blood vessels and thrombosis result from the permanent shift of hemostasis to a critical state for an organism. Studies of the oscillatory dynamics of the given metabolic process will allow one to investigate the process of selforganization of the metabolic process of hemo stasis of a circulatory system under changes in blood cholesterol level. The presented model can serve as an example of the selforganization in the open dis sipative system of human organism. This will allow one to study the regularity of metabolic processes in human organism from a single physical viewpoint of synergetics.

the mathematical model and methods of its study
The general scheme of the hemostasis with re gard for the entry of "bad cholesterol" into blood is presented in Fig. 1 [6,10]. According to this scheme, we construct the mathematical model of the given metabolic process (1)(12) [6,10]: , , , , . , , .
The input substances in a blood vessel are ara chidonic acid s and molecules of fat F, which are supplied into blood from the intestinal tract. The output agents of the system are aggregated thrombo cytes t x * and oxidized lipoproteins L * , which are ac cumulated on internal walls of arteries. In the model we utilize the law of mass action and the kinetics of enzyme catalysis. The Eqs. involve the balance of masses of the intermediate products of reactions on separate stages of the metabolic process.
Eqs. (1) and (3) describe, respectively, changes in the concentrations of arachidonic acid a t and a p in thrombocytes and in endothelial cells of the vessel. These processes are affected by activity of the corre sponding phospholipases. The coefficients k 5 and k 2 characterize, respectively, the rate of these proces ses. The accumulated arachidonic acid is transformed then by prostaglandinHsynthase of thrombocytes E 1 and prostaglandinHsynthase of prostacyclins E 2 . Respectively, thromboxanes t x (2) and prostacyclins P (4) are formed. The rate of these enzymatic pro cesses is determined by the coefficients k 7 and k 10 . The coefficients k t and k p charac terize the exchange by arachidonic acid between thromboxanes and en dothelial cells. Eqs. (5) and (6) describe, respectively, changes of the concentrations of enzymes E 1 and E 2 . The coefficients k 13 and k 15 determine the intensity of biosynthesis in thromboxanes and endothelial cells. The inactivation of these enzymes is guided by the corresponding terms with the coefficients k 7 and k 10 in Eqs. (2) and (4). Eqs. (1)(6) satisfy completely the balance of masses in enzyme catalysis. Eq. (7) de scribes changes in the concentration of the control ling component, cyclic adenosine monophosphate (cAMP) r. Its presence in Eqs. (1) and (3) creates the negative feedback that affects the level of activity of phospholipases of thrombocytes a t and endothe lial cells a p . Equation (8) describes aggregation of thrombocytes t x * and their dissipation under effect of prostacyclin. Its concentration depends also on blood LDL level. Eqs. (9)(12) characterize the metabolic process of formation of LDL in blood and the accu mulation of plaques on walls of the vessel. Fat mole cules F (9) are supplied by blood to arteries from liver and small intestine. Eqs. (9)(10) describe the process of creation of "bad cholesterol" L from fat. Its deposition on walls of blood vessels in the form of oxidized lipoproteids L * (plaques) is described by Eqs. (10) (11). In the metabolic process, the positive feedback controlled by enzyme E 3 is formed (12). The accumulation of cholesterol in arteries and the growth of plaques cause thrombophilia. In this case, the lumen of an artery becomes narrower, i.e., steno sis develops. The above Eqs. involve also the dissipa tion of the corresponding substances at the expense of other metabolic processes and the flow of blood in an artery.
The given system is an open nonequilibrium one. Its study was carried out with the use of the theory of nonlinear differential Eqs. [18,19] and the methods of mathematical modeling used earlier in [2027].

results of studies
The investigation of the mathematical model (1)(12) has shown that, in addition to the ideal stationary modes of the metabolic process of the thrombosisantithrombosis system [49], the model includes also autooscillatory modes. Depending on the rate of supply of molecules of fat F 0 to blood, the level of "bad cholesterol" L varies. The metabolic process of hemostasis becomes unstable. The study of autooscillatory modes will enable us to compre hend the dynamics of the metabolic process and to reveal the structuralfunctional connections in this system.
In Fig. 2, we show a phaseparametric dia gram of the system for L(t) at a variation of F 0 in the corresponding intervals. In order to construct the phaseparametric diagrams, we have used the cut ting method. In the phase space of a trajectory of the system, we place a cutting plane for the value of concentration of cyclic adenosine monophosphate (cAMP) r = 1.77. If the trajectory crosses this plane in some direction, we mark the value of chosen vari able on the phaseparametric diagram (L(t) in this case). Such choice is explained by the symmetry of oscillations of the given component relative to such point in multiply calculated earlier modes. For every value of L, we mark the intersection of the trajectory and this plane after the trajectory falls into the at tractor. If a multifold periodic limiting cycle arises, we will see a number of points on the plane, which coincide in the period. If a deterministic chaos ari ses, the points of intersection are located chaotically. It is convenient to consider the phasepara metric diagram from right to left. As F 0 decreases (Fig. 2, a), we observe the successive transition to autoperiodic modes with higher multiplicity due to a cascade of bifurcations with the doubling of a period, until the chaotic mode is finally established due to the intermittence. Such scenario is shown in Fig. 3, a-f. The phase portraits of regular attractors transfer to a strange attractor. As the given parame ter decreases further, the chaotic modes hold in the system. In the interval F 0 ∈ (0.010007, 0.010012) (Fig. 2, a, b), chaos is destroyed, and the periodic ity window arises. In this window, the transition from autoperio dic modes to chaotic ones occurs also as a result of the cascade of bifurcations with the doubling of a period by Feigenbaum's scenario. The transition finishes analogously to the previous scenario. Due to the intermittence, chaos is formed. The analogy of these scenarios indicates the fractal nature of the given cascades of bifurcations.
In Fig. 4, a,b, as an example, we show the pro jections of the strange attractor 2 x for F 0 = 0.01 in the planes (L, F) and (E 1 , a t ). The obtained strange at tractor is formed due to the funnel effect. An element of the phase volume of such attractor is stretched in some direction and contracts in other directions, by preserving its stability. Therefore, the mixing of tra jectories happens in narrow contracted regions of the phase space of a funnel, and the deterministic chaos arises.
For the given strange attractor in Fig. 5 a, b, c, we constructed the projection of the intersection with the plane r = 1.77 and the Poincaré image. The inter section plane was chosen so that the phase trajectory r(t) crosses it the maximal number of times, as the given component decreases, without any touching of the intersection plane by the phase curve.
The obtained intersection points and the Poin caré image do not possess a geometric selfsimilari ty. The number of points permanently increases with the duration of a numerical integration of the sys tem. This demonstrates the chaoticity of the attractor and the impossibility of some reduction of the given complicated kinetic scheme of metabolic processes to a onedimensional discrete approximation of the system under study.
In Fig. 6, we compare the kinetics of some com ponents of the system in the periodic (1) and chaotic (2) modes.
Changes of the concentrations of fat and "bad cholesterol" in the chaotic metabolic mode of athe rosclerosis are shown in Fig. 7. Such nonuniform change of LDL affects the thrombosis-antithrombo sis system, by destroying a steady hemostasis of an artery. The balance between the amounts of choles terol deposited in a blood vessel and that taken out with blood is violated. At certain point the amount of deposited cholesterol becomes larger. This favors the formation of plaques in a blood vessel. Thus, the appearance of atherosclerosis depends on the selforganization of the metabolic process in the thrombosisantithrombosis system. Under the self organization, blood vessels adapt to the conditions of nutrition. If a desynchronization of these processes occurs, the risk of atherosclerosis development be comes more significant.
For the unique identification of the type of the obtained attractors and for the determination of their stability for various values of parameter F 0 , we calculated the complete spectra of Lyapunov in dices and their sum: . The calculation was carried out by Benettin's algorithm with the orthogonali zation of the vectors of perturbations by the GramSchmidt method [18].
Using the Pesin theorem [28] and the values of Lyapunov indices, we calculated also the KSentro py (KolmogorovSinai entropy) h and the Lyapunov index "predictability horizon" t min [29]. The Lyapu nov dimension D Fr of the fractality of strange attrac tors was found by the KaplanYorke formula [30,31]: . Below, as an example for comparison, we pre sent some results of calculations of the mentioned indices.
For F 0 = 0.01, the strange attractor 1·2 x arises. For F 0 = 0.010015, the strange attractor 1·2 x arises. λ 1 -λ 16  These results show the variety of geometric structures of the obtained attractors and the predicta bility of the metabolic process depending on the con centration of molecules of fat in blood and on the level of "bad cholesterol".
Calculating successively various strange attrac tors, we can find some regularity in the hierarchy of their chaotic behavior. Respectively, the variation of the given indices changes a geometric view of attrac tors of the system.
Autooscilations in the metabolic process of he mostasis of a blood vessel arise due to the interac tion between thrombosis and antithrombosis systems of blood, which is regulated by the level of cyclic adenosine monophosphate. The presence of "bad cholesterol" in blood causes desynchronization of these systems and the appearance of chaotic modes in the metbolism of a hemostasis. LDL affects the binding of thrombocytes and deposits on the walls of blood vessels. This leads to the autocatalysis of cholesterol in blood.
Thus, the hemostasis under a change of the amount of cholesterol in blood characterizes the adap tation of the metabolic process of a blood vessel to these changes, by preserving its functionality in this case.
We have constructed a mathematical model of the process of atherosclerosis of a blood vessel. The mathematical model describes the metabolic process of the thrombosisantithrombosis system based on the prostacyclinthromboxane system of blood. We have studied how molecules of LDL affect the im balance of this system. The autooscillatory modes determined with this model indicate a complicated internal dynamics of formation of the selforgani zation in a blood vessel, i.e. that of the homeostasis.
We have studied the dependence of autooscillatory modes on the concentration of fat in blood. Moreo ver, we determined the chaotic modes of strange at tractors. During such modes, the imbalance between the amount of "bad cholesterol" deposited in a blood vessel and its removal from the system happens. This provokes the formation of plaques in an artery. It is shown that affects the binding of thrombocytes and deposits on walls of blood vessels. This causes the autocatalysis of cholesterol in blood and the increase of its level. The mathematical study of the obtained modes is performed. The phaseparametric diagram, kinetic curves, projection of phase portraits, and Poincaré crosssections and images are constructed. The Lyapunov indices, divergencies, "predictability horizons," and Lyapunov dimensions of the fractali ty of strange attractors are calculated. These indices characterize the stability and structure of calculated attractors.
The obtained results clarify the metabolic pro cess of hemostasis and to find the structural-func tional connections affecting the appearance of atherosclerosis of blood vessels.
The work is supported by the project N 0113U001093 of the National Academy of Scien ces of Ukraine.