12124 FACTA UNIVERSITATIS Series: Mechanical Engineering Vol. 22, No 3, Special Issue, 2024, pp. 399 - 422 https://doi.org/10.22190/FUME230919051P © 2024 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND Original scientific paper IN SILICO ANALYSIS OF AN ARTICULAR CARTILAGE REGENERATIVE REHABILITATION UNDER CONDITIONS OF MESENCHYMAL STEM CELLS IMPLANTATION AND THEIR MECHANICAL STIMULATION Aleksandr Poliakov, Vladimir Pakhaliuk Sevastopol State University, Sevastopol, Russia ORCID iDs: Aleksandr Poliakov https://orcid.org/0000-0002-2940-8945 Vladimir Pakhaliuk https://orcid.org/0000-0002-9992-1189 Abstract. One of the most important tasks of modern medicine is the development of effective technologies for the treatment of joint diseases caused by damage to the articular cartilage. The results of experimental studies and a number of successful clinical practices indicate that its solution can be found within the framework of a new medical direction - regenerative rehabilitation, which synergistically combines the methods of regenerative and rehabilitation medicine. In particular, regenerative rehabilitation of articular cartilage defects involves the use of cellular technologies, the effectiveness of which is enhanced by mechanical stimulation of chondrogenic cells, which accelerates their proliferation, differentiation, and formation of an extracellular matrix. The simulation results indicate that its outcome depends not only on a set of parameters determined by the state of the tissue in the defect aria, but also on their combination. One of the main goals of this work is to find the best combination of parameter values that are practically achievable in the process of articular cartilage regenerative rehabilitation using cellular technologies and mechanical stimulation of cells. Its solution is based on the study of a regenerative tissue rehabilitation mathematical model, the state parameters set of which is determined by the Sobol-Statnikov method, based on a systematic study of the parameter space uniformly distributed in a multidimensional cube. The practical significance of the results of the work lies in the fact that they can be used to evaluate the effectiveness of mechanical stimulation various methods of articular cartilage defects in the process of regenerative rehabilitation. Key words: Articular cartilage, Osteoarthritis, Stem cell implantation, Mechanical cell stimulation, Regenerative rehabilitation, Mathematical model Received: September 19, 2023 / Accepted December 25, 2023 Corresponding author: Aleksandr Poliakov Sevastopol State University, 299053, Universitetskaya, 33, Sevastopol, Russian Federation E-mail: a.m.poljakov@sevsu.ru https://orcid.org/0000-0002-2940-8945 https://orcid.org/0000-0002-9992-1189 400 A. POLIAKOV, V. PAKHALIUK 1. INTRODUCTION It is known that the current mechanical and biophysical properties of the cellular tissue microenvironment depend on the mechanical impact experienced by cells in the past and largely determine their future fate [1]. In particular, mechanical impacts on mesenchymal stem cells (MSC), which have a high regenerative potential, affect their ability to synthesize elements of the extracellular matrix (ECM), including fibronectin, laminin, collagen and proteoglycans, and form the microenvironment necessary to increase their own viability, proliferation and differentiation [2,3]. Taking into consideration these facts, a hypothesis was formulated according to which, by artificially creating a certain cellular microenvironment, it is possible to control the life cycle of MSCs implanted in the area of articular cartilage defect, stimulate their proliferation and differentiation in the desired direction in order to ensure the regeneration of damaged tissue. This hypothesis is confirmed by the results of a number of studies in silico [4–7], in vitro [8– 10], and in vivo [11–13]. There is an opinion that the regenerative rehabilitation methods of cartilage tissue defects, which have been intensively developed in recent years, have great potential for its successful practical implementation [14]. However, these methods still need to be improved. One of the main issues to be resolved on the way to their widespread introduction into medical practice is how exactly it is necessary to influence MSC implanted in the area of articular cartilage defect in order to achieve the best effect of its restoration. This is an extremely complex task, the solution of which depends on many factors, the influence of which on the achievement of the final results is ambiguous and is characterized by a high degree of uncertainty. At the same time, it is known that the mechanical and biophysical state of articular cartilage is characterized by many measurable parameters that change as a result of mechanical stimulation of the tissue. These include: the rates of diffusion, proliferation, differentiation and death of cells, the effect of growth factors and nutrients on the deposition and degradation of ECM, the degree of reproduction and degradation of growth factors, the rate of consumption of nutrients by cells, etc. The question arises: what are the optimal values of the above, as well as other parameters that ensure a stable, irreversible and final process of tissue regeneration? In addition, no less important (and perhaps even more important) is the question of how to ensure the creation of a cellular microenvironment corresponding to optimal parameters. In other words: how to carry out mechanical stimulation of cells, providing the best conditions for the regeneration of damaged tissue? The answer to this question, if it exists, can be obtained by comparing the results of in vivo experiments with the results of in silico studies performed on the basis of adequate mathematical models for tissue regenerative rehabilitation. It is clear that this is an extensive way, but, unfortunately, currently there are no other ways apparently to find the optimal solution to the multiparametric problem of repairing articular cartilage defects. Known methods of its treatment cannot be considered not only optimal, but also sufficiently reliable, because in most cases they do not lead to positive results. This is largely due to the peculiarities of the cartilage tissue, which is practically devoid of a vascular network, as a result of which the nutrition of its cells is carried out mainly due to the diffusion of nutrients. Therefore, unlike, for example, bone tissue, hyaline articular cartilage after destruction in the vast majority of cases is not able to fully recover. After reparative regeneration, at best, it is transformed into hyaline-like cartilage [15]. At the same time, it should be noted that cases of complete restoration of local articular cartilage defects as a result of MSC cell therapy followed by physiological procedures are known. It can be assumed that such outcomes are possible as a result of achieving an optimal cellular microenvironment in the tissue defect area as a result of In Silico Analysis of an Articular Cartilage Regenerative Rehabilitation under Conditions of ... 401 its optimal mechanical stimulation. With this in mind, the most important task, the solution of which is necessary for organizing effective processes of articular cartilage defects regenerative rehabilitation, is to determine the optimal parameters of the cellular microenvironment, which must be achieved by mechanical stimulation of cells. Articular cartilage has a very complex structure, represented by solid and liquid phases, and its rigidity is determined to the greatest extent by the fibrous structure of the extracellular matrix and the incompressibility of the interstitial fluid [16]. When loaded, the stiffness of cartilage changes along with a change in the ratio of solid and liquid phases. At the same time, at the initial stage of loading, the degree of hydration of the matrix does not change, since osmotic pressure prevents the release of interstitial fluid. Its extrusion from the matrix begins only when the load acting on the joint will lead to an intra-articular pressure greater than the osmotic pressure. When the load is removed, the reverse process occurs - the absorption of interstitial fluid by cartilage. The viscous component of the synovial fluid, consisting of molecules larger than the pores of the matrix, always remains in the joint space and the low-viscosity component moves from the joint space to the cartilage and back during loading/unloading of the joint. This promotes the transfer of cells and nutrients into the cartilage and the removal of metabolic products from the cartilage into the joint cavity. The state of articular cartilage under conditions of mechanical loading (stimulation) is characterized by a number of parameters, estimates of which can be obtained as a result of in vivo experiments, thanks to modern methods and analysis tools. At the same time, the measured values of the key parameters αi, which have the greatest impact on the course of cellular processes, are within some segments min max i i i    . When modeling the processes for regenerative rehabilitation of articular cartilage, taking into account the personalized characteristics of the tissue, the boundary values of the parameters * min M i i = and ** max M i i = may be deliberately chosen to be larger than min i and max i , respectively, to be able to evaluate as many methods of mechanical stimulation of cells as possible. Obviously, the simulation results will depend on which point Ak=(αk1, αk2,…,αkn) in the parameter space A=(α1, α2,…,αn) will be investigated. Theoretically, it is possible to simulate an infinite number of variants for regenerative rehabilitation processes, if we assume that the parameters αi are continuously distributed on the segments * **[ , ]i i  . If the parameters αi are represented as discrete sets, then a finite number of models can be investigated. But in any case, in order to obtain the most complete information about the processes of regenerative rehabilitation, the parameter space A should be systematically investigated. One of the main objectives of this work is to search for the values of the cellular microenvironment key parameters, which are practically achievable using cellular technologies and mechanical stimulation of cells, under which the best conditions for regenerative rehabilitation of articular cartilage defects are achieved. To solve it, a mathematical model of regenerative tissue rehabilitation is used, the combination for state parameters of which is determined by the Sobol-Statnikov method, based on a systematic study of the parameter space A uniformly distributed in a multidimensional cube [17,18]. 402 A. POLIAKOV, V. PAKHALIUK 2. MATERIALS AND METHODS 2.1. Mathematical Model for Regenerative Rehabilitation of an Articular Cartilage Defect Fig.1 shows the layout of a local articular cartilage defect on the articular surface of the knee joint, as well as its element on an enlarged scale and the associated coordinate system. Fig. 1 Representation scheme of the elementary area for the articular cartilage defect and the associated coordinate system If to assume that the development of the process for articular cartilage regenerative rehabilitation begins with the subchondral bone, and the properties of the tissue or tissue- engineered structure located in the defect area change slightly in planes parallel to the subchondral bone, then, for the purpose of simplification, this problem can be approximately considered as one-dimensional. We will place the origin of the coordinate system on the surface of the subchondral bone in the conditional center of the defect, and direct the x coordinate axis along the outer normal vector ν to the articular surface. We will assume that the articular cartilage regenerative rehabilitation is performed under conditions of implantation in the area of the MSC defect, i.e., based on the ASI - Articular Stem cell Implantation cellular technology that meets the criteria established by the International Society for Cellular Therapy's Mesenchymal Stromal Cell Committee (ISCT MSC) [19]. To solve this work problems, a mathematical model of the “diffusion-reaction” type is used, the advantages of which were confirmed when modeling various types of biological processes. The history of this type model development is briefly presented in [20]. Initially, starting in the mid-1980s, it was used to simulate the dynamics of tumor-induced angiogenesis. In 1990, it was modified to predict epidermal wound healing by taking into account the interaction between In Silico Analysis of an Articular Cartilage Regenerative Rehabilitation under Conditions of ... 403 endothelial cells and a chemoattractant. This took into account the chaotic movement of cells, their proliferation and death, while the chemical substance produced by the cells diffused and underwent natural breakdown. The results of this model were in very good agreement with the results of experimental studies. In 1993, Chaplain and Stuart developed a model for the chemotactic response of endothelial cells to tumor angiogenesis factor [21], which was modified in 1995 by Byrne and Chaplain [22] to account for the development of blood vessels in response to tumor angiogenesis factor. In addition, in 1996, the same authors established an important link between mathematical models of tumor-induced angiogenesis and wound healing angiogenesis [23], which led to the emergence of numerous mathematical models of wound angiogenesis, which were based on the concept of the “diffusion-reaction” model. The most important among them, according to many researchers, are the models developed in 1996 by Pettet, Byrne, McElwain and Chaplain, which inspired many researchers to study wound angiogenesis [24]. In 2001, Bailón-Plaza and Meulen [25] used a “diffusion-reaction” model to study fracture healing processes taking into account the influence of growth factors, the equations of which had the following structure: 1) MSC density = migration + mitosis + differentiation; 2) chondrocytes density = mitosis + differentiation - endochondral replacement; 3) osteoblasts density = mitosis + differentiation – removal; 4) connective/cartilage ECM density = synthesis – degradation; 5) bone ECM density = synthesis – degradation; 6) osteogenic growth factor concentration = diffusion + production – decay; 7) chondrogenic growth factor concentration = diffusion + production – decay. The migration of MSC in the model was simulated based on the experimentally observed behavior of randomly populated cells, considering the effects of ECM density and environmental heterogeneity; to account for cell proliferation, the logistic growth law was used, according to which the rate of cell division decreases linearly with increasing cell density due to space and nutrient limitations. The size of the cell population, in accordance with experimental data, stabilized around the maximum density, and cell death due to apoptosis was balanced by the mitosis of new cells. In addition, the enhancing and inhibitory effects observed at low and high ECM densities, respectively, were taken into account. The differentiation of mesenchymal cells into osteoblasts or chondrocytes was regulated by the concentrations of osteogenic and chondrogenic growth factors, respectively. In this case, the dependence of the differentiation rate on the concentration of growth factors was determined considering experimental dose-dependent curves. Other processes occurring in the fracture area during its healing were similarly modeled. In 2011, adopting the approach proposed by Bailón-Plaza and Meulen, Lutianov, Naire, Roberts and Kuiper presented a “diffusion-reaction” model to predict articular cartilage regeneration using cell therapy [4], which was studied in a modified form by Campbell, Naire and Kuiper in 2019 [5,6]. This model, which previously studied by Popov, Poliakov and Pakhaliuk under other conditions [7], is also used in this work. Its equations are: 404 A. POLIAKOV, V. PAKHALIUK 01 1 2 3 1 0 [ ] ( ) ( ) ( )S S S S S S S S diffusion deathdifferentiation proliferation С n D C p C H n n p C H C C p C H n n t n n  =   + − − − − −  + , (1) 04 1 2 5 1 0 [ ] ( ) ( ) ( )C C C C S S S C diffusion deathdifferentiation proliferation С n D C p C H n n p C H C C p C H n n t n n  =   + − + − − −  + , (2) 6 7 0 { } ( )n S C diffusion reaction n n D n p C p C t n n  =   + +  + , (3) 8 0 [ ]m C diffusion reaction m n D m p C t n n  =   +  + , (4) 9 11[ ]g S reactiondiffusion g D g p C p g t  =   + −  , (5) 12 13[ ]b C diffusion reaction b D b p C p g t  =   + −  , (6) where i j k x y z     = + +    ; ( , , )x y z i j k grad x y z          = + + =    ; 2 2 2 2 2 2 grad i j k divgrad x y z          = + + =    ; where φ(x,y,z) is a scalar function. A detailed description of the model, represented as system of Eqs. (1-6) and the results of its study are presented in [7]. The numerical values of the parameters used in this case corresponded to the values given in the literature [4–6]. Their brief description is presented in Appendix A. Previously, these values were established experimentally or justified theoretically and used in a number of studies on bone fracture healing, implant osseointegration, and articular cartilage regeneration [26,27]. Boundary conditions of the problem [7]: ▪ at the point x=0: ( ), 0, 0, 0, 0, 0S C S C g b n m C C g b n m D f t D D D D D x x x x x x       − = = = = = =       ; (7) ▪ at the point x=d: 00, 0, , , , 0S C S C g b m C C g b m D D D g D b n N D x x x x x      = = = − = − = =        . (8) In Silico Analysis of an Articular Cartilage Regenerative Rehabilitation under Conditions of ... 405 Initial conditions at t=0 [7]: (0) 0 3( ), 0, 0, 0, ,S S C sC C h x g b C n N m m m= = = = = = + , (9) where (0) SC is the initial MSC density; h(x) is the seeding profile of MSC by defect depth; m3 is the initial ECM density; ms is the scaffold density. 2.2. Methodology for the Systematic Study of a Mathematical Model by the Sobol-Statnikov Method 2.2.1. Research Objective It should be noted that the problem solved in the framework of this work is essentially a multicriteria one. This is due to the fact that it is rather difficult to formulate its global criterion, despite the fact that its qualitative content is interpreted quite unambiguously: complete regeneration of the articular cartilage defect. But, as practice shows, even if it is possible to achieve this result, the ways to achieve it are most likely ambiguous. In addition, there is a possibility that during the regeneration of a tissue defect, undesirable complications may occur or other quality criteria will deteriorate. Another problem is that the global criterion, the qualitative content of which is formulated above, is quite difficult to give a quantitative form. Therefore, its evaluation should be carried out indirectly along with the evaluation of other criteria. For example, the mathematical model represented as system of Eqs. (1-6) makes it possible to evaluate the rate of ECM synthesis and interstitial cartilage growth by increasing the density of chondrocytes. Therefore, ECM density (m) and chondrocytes density (CC) on the articular surface can be taken as quality criteria for the process of an articular cartilage defect regenerative rehabilitation, assuming that their increase promotes the regeneration of cartilage tissue. In addition, MSC, which can differentiate into chondrocytes, play an important role in the processes of articular cartilage regeneration. Therefore, the density of these cells in the defect area or on its surface can also be attributed to the quality criterion of the cartilage tissue regeneration process. Thus, the task is to determine the set of model parameters at which the highest density of ECM (m), chondrocytes (CC) and MSC (CS) on the articular surface in the area of the defect is achieved over a certain period of time. 2.2.2. Parametric Constraints The solution of the problem is sought taking into account the set of variable parameters of the αi model, which, depending on the state of the cellular microenvironment, can take different values belonging to some segments * **[ , ]i i  . Their description is given in Table 1. The minimum * i and maximum ** i values of the parameters are accepted to be smaller or larger, respectively, than in [4–6]. This was done in order to evaluate as many options as possible for potentially possible processes of regenerative rehabilitation, in which the parameters of the cellular environment αi can take extreme values. 406 A. POLIAKOV, V. PAKHALIUK Table 1 Variable parameters of the mathematical model Parameter Functional dependencies and parameter values Parameter dimension Dimensionless parameter values, * ** i i − DS0 – MSC diffusion constant 0 * 12S SD m D (mm3/hour)·(g/mm3) 0.001 – 0.01 DC0 – chondrocyte diffusion constant 0 * 12C CD m D (mm3/hour)·(g/mm3) 0.0005 – 0.0015 Dn – nutrient diffusion coefficient Dn ≈ 4.6 mm2/hour 100 – 300 Dm – ECM diffusion coefficient Dm ≈ 2.5·10-5 mm2/hour 0.001 – 0.01 Dg – FGF-1 diffusion coefficient Dg ≈ 2.0·10-3 mm2/hour 0.5 – 1.5 Db – BMP-2 diffusion coefficient Db ≈ 2.0·10-3 mm2/hour 0.5 – 1.5 p2 – MSC differentiation rate p2 ≈ 3.75·10-3 1/hour 0.5 – 1.5 p3 – MSC death rate p3 ≈ 3.75·10-3 1/hour 0.5 – 1.5 p40 – chondrocyte proliferation constant 0 9 4 4 10p −  g/(hour·mm3) 0.006 – 0.018 p400 – chondrocyte proliferation rate 00 4 4 2 10p −  1/hour 0.006 – 0.018 p5 – chondrocyte death rate p5 ≈ 3.75·10-3 1/hour 0.5 – 1.5 p80 – ECM production constant 0 13 8 3.75 10p −  (g/mm3)/((cells/mm3)·(1/hour)) 0.2 – 1.2 p800 – FGF-1 ECM deposition rate 008 0 1p =  - 0.2 – 1.2 p81 – ECM degradation constant 1 9 8 3.75 10p −  (1/hour)·(1/(cells/mm3)) 0.5 – 1.5 p9 – FGF-1 production constant p9 ≈ 10-17 (g/mm3)/((cells/mm3)·(1/hour)) 22.0 – 28.0 p11 – FGF-1 degradation rate p11 ≈ 5.8·10-2 1/ hour 13.0 – 16.0 p12 – BMP-2 production constant p12 ≈ 10-17 (g/mm3)/((cells/mm3)·(1/hour)) 22.0 – 28.0 p13 – BMP-2 degradation rate p13 ≈ 5.8·10-2 1/hour 13.0 – 16.0 2.2.3. Functional Limitations The functional limitations of the regenerative rehabilitation process are set directly in the equations of the mathematical model. The following effects are taken into account. Proliferation of MSC and chondrocytes can occur as long as the concentration of nutrients n in the defect area is greater than the critical value n1, which is ensured by introducing the In Silico Analysis of an Articular Cartilage Regenerative Rehabilitation under Conditions of ... 407 Heaviside function into the model equations of the form 1 1 1 0, ( ) 1, n n H n n n n  − =   . On the contrary, as soon as n becomes less than n1 due to a lack of nutrients, the cells gradually begin to die, which is ensured by the Heaviside function of the form 1 1 1 0, ( ) 1, n n H n n n n  − =   . The process of MSC differentiation into chondrocytes is explained similarly. It occurs as long as the MSC density (CS) is greater than the threshold value CS0 . As soon as CS becomes smaller CS0 , differentiation stops. These effects in the model equations are provided by the Heaviside function of the form 0 0 0 0, ( ) 1, S S S S S S C C H C C C C  − =   , where 0 0max 0min ( ) ab S S SC C C e−= − ; 0maxSC , 0minSC are the maximum/minimum threshold MSC density, respectively, a is the threshold MSC density reduction factor. In addition, the functional limitations that must be taken into consideration in the simulation process can include the initial values of the key parameters of the model given in Table 2 [4]. Table 2 Initial values of the mathematical model key parameters Parameter Functional dependencies and parameter value Parameter dimension Dimensionless parameter value (0) SC – initial MSC density (0) 52.5 10SC   cells/mm3 0.25 (0) CC – initial chondrocytes density (0) 210 %CC − of total cell density cells/mm3 0.0001 m3 – initial ECM density assumed mmax/104 g/mm3 0.0001 ginit – initial FGF-1 concentration ginit ≈ 10-12 g/mm3 0.01 binit – initial BMP-2 concentration binit ≈ 10-12 g/mm3 0.01 It should be noted that in practice these values significantly depend on the state of a particular tissue, as a result of which the processes of regenerative rehabilitation are highly personalized. 2.2.4. Criteria Constraints It is known that after reaching a certain critical value, the reproduction of ECM stops. Therefore, in order to exclude obtaining impossible results of the model study, a criterion constraint of the form m≤mmax=10-4 g/mm3 is introduced. The same applies to the differentiation of MSC into chondrocytes. Those, after reaching the critical density of chondrocytes, their reproduction in the defect area stops, which is taken into consideration in the model by introducing a criterion constraint of the form CC≤CCmax0=106 moles/mm3. Finally, it is assumed that the density of MSC in the defect area should also be limited. As a first approximation, one can take CS≤CSmax0=106 moles/mm3. 408 A. POLIAKOV, V. PAKHALIUK Taking into consideration the results of studies presented in [19-21], these criteria restrictions in dimensionless form are presented as follows: m≤mmax=1; CC≤CCmax0=0.4; CS≤CSmax0=0.6. (10) Constraints in Eq. (10) assume that the maximum density of chondrogenic cells in the Ctotal,max0 articular cartilage defect should not exceed 106 cells/mm3 at zero ECM (m=0) density 1 6 3 ,max 0 max 0 max 0 max 1 ( ) 10 /total S C m C C C cells mm m −   = − +     . In the dimensionless form Ctotal,max0 ≤ 1, therefore, CS + CC ≤ 1. 2.2.5. Selection of Trial Points According to the Sobol-Statnikov method, to select test points Ak = (αk1, αk2,…, αkr), it is advisable to use sequences Q1, Q2,…,Qk,… uniformly distributed in the parameter space with fairly good uniformity characteristics and, if possible, simple algorithms for calculating coordinates [18]. Then, by the Cartesian coordinates of the next point Qk = (qk1, qk2,…, qkr), you can find the generalized coordinates of the point Ak using the following relationship * ** *( ), 1,ki i ki i iq i r   = + − = , (11) where r is the number of mathematical model variable parameters. To determine Qk, LПτ-sequences of points are used, which are based on the concepts of a binary segment, a binary parallelepiped, a multidimensional cube, and a Пτ-grid. Definition 1. Any segment that can be obtained by dividing segment [0,1] into 2m equal parts (m=0,1,2,…) is called binary. All binary segments can be numbered and denoted as ls, s=1,2,… Let k=(k1,k2,…,kr), where some or even all of kj, j=1,2,…,r may coincide. Definition 2. A set of points with coordinates (x1,x2,…,xr) such that , 1,2,...,i kix l i r = is called a binary parallelepiped Пk. Any binary box belongs to the unit dimensional cube Kr. Definition 3 [17]. A grid consisting of N=2ν points of an r-dimensional cube Kr is called a Пτ- grid, if each binary parallelepiped Пk with volume 2 k V N   = has 2τ points of the grid at ν > τ. Definition 4 [17]. A sequence of points P0,P1,…,Pi,… of an r-dimensional cube Kr is called an LПτ-sequence if any of its binary segments containing at least 2τ+1 points is a Пτ- grid. It is proved that all LПτ-sequences are uniformly distributed in Kr [17]. Moreover, all LПτ-sequences are perfectly distributed [28]. The main idea of the Sobol-Statnikov method is that for a systematic study of the parameter space of a model, perfectly distributed LПτ- sequences in Kr can be successfully used. Let i = em…e2e1 be the number of the trial point in the binary system. Then its Cartesian coordinates Qk = (qk1,qk2,…,qkr) can be calculated by the formula (1) (2) ( ) 1 2 ... m ki i i m iq eV e V e V=    , (12) where the symbol (*) denotes the "exclusive OR" operation. In Silico Analysis of an Articular Cartilage Regenerative Rehabilitation under Conditions of ... 409 Eq. (12) can also be calculated in the usual arithmetic way. To do this, considering the given k, first calculate the number m = 1 + [ln k / ln 2], (13) and then for i=1,2,…,r 1 ( ) 1 1 1 2 2{ 2 } 2{ 2 } 2 m m l s s i s ki i l s l q k r− + − − − = =               , (14) where ( )s ir are the numerators of guiding numbers [17]. In formulas (13) and (14), [*] is the integer part of the number *, and {*} is the fractional part of the number indicated as *. Thus, the set of Cartesian coordinates qki, calculated by Eq. (14) makes it possible to determine the set of generalized coordinates of test points given in Eq. (11), uniformly distributed in the parameter space of the model for the process of an articular cartilage defect regenerative rehabilitation, which makes it possible to systematically study this process in silico. 3. RESULTS OF THE STUDY For a systematic study of the influence the parameters of the mathematical model on the course of the processes for an articular cartilage defect regenerative rehabilitation, according to Eq. (13) and Eq. (14), the Cartesian coordinates LПτ-sequences of points uniformly distributed in an 18-dimensional cube were calculated, which were used to calculate by the Eq. (11) generalized coordinates of sample points uniformly distributed in the 18-dimensional parameter space, taking into consideration the parametric constraints given in Table 1. As an illustration, Fig.2 shows the distribution of sample points Ak = (αk1, αk2,…, αkr) in the 3-dimensional parameter space. Fig. 2 Illustration of the test points Ak={αk1, αk2, αk3} distribution in a 3-dimensional parameter space: a) 10 test points (k=1..10); b) 100 trial points (k=1..100) In this work, the study of the mathematical model represented by the system of Eqs. (1- 6) was carried out taking into consideration the generalized coordinates of 56 test points uniformly distributed in the 18-dimensional parameter space. The numerical values of the 410 A. POLIAKOV, V. PAKHALIUK generalized coordinates of the test points are presented in Appendix B; the results of the mathematical model study are in Appendix C. The first stage of the mathematical model study was carried out taking into consideration the time of the regenerative rehabilitation process of about 110 days (in a dimensionless form t=10). At the same time, at 6 test points (Nos. 20, 22, 25, 27, 36, 56), unstable solutions of the system of Eqs. (1-6) were obtained, and therefore they were excluded from consideration. The analysis of the results for the mathematical model study was carried out under the assumption that all test points are independent and uniformly distributed in the 18- dimensional parameter space. In principle, another 56 sample points evenly distributed in this space could be chosen for the study. Formally, this would have no effect on the results of the numerical experiment. To confirm this fact, in Fig. 3 shows different sets of 10 sample points in 3D parameter space. Their generalized coordinates were determined using LПτ-sequences, which means that from a mathematical point of view they are uniformly distributed in the parameter space, which, in general, is also confirmed visually. Therefore, we can assume that the test points used to study the mathematical model were randomly selected, which means that the results of the study are also random. Fig. 4 shows the distributions of the quality criteria values {m, CC, CS} in 3-dimensional and 2-dimensional criteria spaces, and Table 3 presents the correlation coefficients of these criteria. The results of the mathematical model study indicate that there is a strong negative relationship between the ECM density and the MSC density in the tissue defect area, which is explained by the essence of the tissue regeneration process. It is known that MSC under certain conditions can differentiate into chondroblasts, which, in turn, provide the secretion of ECM, producing type II collagen and sulfated glycosaminoglycans associated with non- collagen proteins - proteoglycans. Ultimately, chondroblasts clog in the matrix cavities (lacunae) and gradually turn into mature cells with lower synthetic activity - chondrocytes. Thus, MSC produced during proliferation differentiate into chondroblasts, which contribute to the secretion of ECM and increase its density. At the same time, the MSC density decreases. Fig. 3 Illustration of the test points Ak={αk1, αk2, αk3} distribution in a 3-dimensional parameter space: а) 10 test points (k = 11…20); b) 10 test points (k = 31…40) In Silico Analysis of an Articular Cartilage Regenerative Rehabilitation under Conditions of ... 411 Table 3 Quality criteria correlation coefficients {m, CC, CS} m CC CS m 1 -0.4186 -0.9977 CC -0.4186 1 0.4341 CS -0.9977 0.4341 1 At the same time, a moderate negative relationship is observed between the ECM density and the density of chondrocytes, indicating that the dynamics of changes in these parameters are interrelated, but expressed not brightly. That is, there is a tendency for a decrease in the density of ECM with an increase in the density of chondrocytes and vice versa. Fig. 4 Distribution of quality criteria values in criterion spaces calculated at 50 trial points: a) 3-dimensional space of criteria {m, CC, CS}; b) 2-dimensional space of criteria {m, CS}; c) 2-dimensional space of criteria {m, CC }; d) 2-dimensional space of criteria {CC, CS}. Regression lines are shown in blue. And, finally, from the data of Table 3 and the nature of the location of points in the 2- dimensional criterion space {CS, CC}, shown in Fig. 4d, it follows that there is a moderate positive relationship between the density of MSC and the density of chondrocytes, which 412 A. POLIAKOV, V. PAKHALIUK is also explained by the essence of the regenerative process: an increase in the density of MSC in the defect area contributes to an increase in the density of chondrocytes and, consequently, the regeneration of cartilage tissue. Table 4 presents the values of the correlation coefficients between the quality criteria of the problem under study and the mathematical model variable parameters. It is easy to see that p80 – ECM production constant parameter, which determines the ECM synthesis rate, has the most significant impact on the quality criteria values 0 18 8 8p p p m= − . (15) Table 4 Correlation coefficients between quality criteria {m, CC, CS} and the mathematical model variable parameters m CC CS DS0 -0.1146 0.1763 0.1311 DC0 -0.0127 -0.0283 0.0113 Dn 0.1612 0.0400 -0.1738 Dm 0.0132 0.1602 -0.0145 Dg 0.1795 -0.1186 -0.1786 Db -0.1130 -0.3195 0.1078 p2 0.1132 0.2691 -0.1351 p3 0.1091 -0.1554 -0.0998 p40 0.0661 -0.0452 -0.0773 p400 0.1151 0.0201 -0.1216 p5 -0.0966 -0.0354 0.0881 p80 0.8771 -0.5113 -0.8724 p800 -0.0936 0.0031 -0.0916 p81 -0.1924 0.3921 0.1985 p9 0.0274 -0.1558 -0.0415 p11 -0.1151 0.0277 -0.1076 p12 -0.1548 0.2757 0.1574 p13 0.0187 0.1269 -0.0072 This parameter does not depend on the density of the ECM, but has a direct impact on its dynamics. In addition, as follows from Table 3, it has a moderate effect on the change in the density of chondrocytes and a strong effect on the change in the density of MSC. Another parameter: p81 – ECM degradation constant, has a moderate effect on the change in the density of chondrocytes. It follows from Eq. (15) that its increase contributes to a decrease in the ECM synthesis rate with an increase in the ECM density. The results of the mathematical model study indicate that other parameters that are important in terms of the implementation of a sustainable process of articular cartilage regeneration have an insignificant effect on its dynamics. At the same time, some of their combination can lead to a violation of the process stability or its atypical development. By typical is meant a process of regenerative rehabilitation in which an increase in matrix density is constantly observed in the area of the articular cartilage defect, which ultimately ends in the formation of native tissue characterized by a physiologically limited density. Atypical processes are multidirectional during regenerative rehabilitation and, as a rule, do not lead to the formation of tissue with physiological density. In addition, processes that cannot be In Silico Analysis of an Articular Cartilage Regenerative Rehabilitation under Conditions of ... 413 implemented within the framework of the mathematical model under study should be considered atypical. That is, in such cases, solving the system of Eqs. (1- 6) is impossible. An illustration of a typical increase in ECM density and the corresponding pattern of changes in nutrient density in space and time are shown in Fig. 5, and an example of an atypical process is shown in Fig. 6. Fig. 5 Illustration of changes in state variables of a mathematical model corresponding to a typical regenerative rehabilitation process (test point No. 51): a) change in ECM density (m); b) change in nutrient concentration (n) Fig. 6 Illustration of changes in state variables of a mathematical model corresponding to a typical regenerative rehabilitation process (test point No. 56): a) change in ECM density (m); b) change in nutrient concentration (n) In typical cases, the growth of ECM density from the very beginning of the process is purposeful. At the same time, the process of nutrient consumption quickly becomes established. In atypical cases, at the initial stage, the process of ECM density growth is multidirectional, and the process of nutrient consumption only after a certain period of time tends to take a stable form. In some cases, the process of an articular cartilage defect regenerative rehabilitation becomes unstable and in most of them cannot be assessed. 414 A. POLIAKOV, V. PAKHALIUK 4. CONCLUSION In this paper, a systematic analysis of the regenerative rehabilitation processes of a local articular cartilage defect was performed using a mathematical model represented by a system of non-linear differential equations in partial derivatives (1-6) of the "diffusion- reaction" type. This model is quite plausible and makes it possible to judge the dynamics of the process for given parameters values, which are determined mainly on the basis of the experimental studies results in vitro. At the same time, it is known that many parameters of the cellular environment depend on the mechanical stimulation of cells, which largely determines their future fate. Taking into consideration this fact, the concept of regenerative rehabilitation of articular cartilage defects was developed, based on the simultaneous use of cellular and rehabilitation technologies. However, for its successful practical implementation, it is necessary to find answers to a number of questions, including: ▪ what should be the parameters values of the cellular environment in vivo, at which a stable, irreversible and high-quality regeneration process of damaged tissue is achieved? ▪ how should mechanical stimulation of cells implanted in the defect area be organized in order to ensure the condition of the damaged tissue with parameters that ensure a stable, irreversible and high-quality regeneration process? The results of the mathematical model study performed in this work allow us to note that there are parameters of the cellular microenvironment of the tissue, in which the regenerative rehabilitation process of an articular cartilage defect is stable, irreversible and can result in complete regeneration of the damaged tissue. The problem of achieving the values of these parameters under in vivo conditions remains topical. One way to find its solution is to compare the parameters of the cellular microenvironment observed after physiotherapeutic procedures with the optimal parameters determined as a result of a mathematical model systematic study. If with the help of the available physiotherapeutic methods and means it will not be possible to ensure the compliance of the parameters, then there will be a need to develop new approaches to the mechanical stimulation of tissues and cells. It is necessary to note a number of limitations adopted in this study. 1. First of all, this concerns the rationale for choosing a one-dimensional model for studying, generally speaking, the spatial process of regenerative rehabilitation. This was done primarily to simplify solutions to the system of nonlinear partial differential equations representing the model. It should be noted that even in the one-dimensional case, obtaining a set of such solutions for each set of state parameters - a trial point - is a non-trivial task. Due to the impossibility obtaining analytical solutions, in this work, to study the model at 56 test points, uniformly distributed (in the sense of I.M. Sobol) in an 18-dimensional parameter space, has been used the finite element method, implemented in Matlab using the built-in pdepe function on supercomputer at the Afalina Shared Use Center of Sevastopol State University. 2. In the mathematical model, the size of the defect was neglected and the change in the parameters for the tissue stress-strain state was not taken into account, which did not allow assessing the value of the chondrogenic index and the probability of achieving the desired result - the formation of native hyaline cartilage at the end of the regenerative rehabilitation process for the defect. 3. The mathematical model did not consider the possibility of MSC differentiation into bone and fibrous tissue cells, because of which the information obtained from the results of its study was incomplete and could potentially contribute to the formulation of incorrect conclusions. In Silico Analysis of an Articular Cartilage Regenerative Rehabilitation under Conditions of ... 415 4. The mathematical model did not take into consideration the possibility of using technologies for implanting autologous chondrocytes or their combinations with MSC into the defect area, which limited the obtaining more complete results achieved by methods of regenerative tissue rehabilitation. 5. When determining the density of the formed ECM, the biodegradation for the tissue- engineered structure scaffold populated with cells and implanted into the defect area at the initial stage was not taken into account. 6. The boundaries for the mathematical model variable parameters have been chosen, in general, arbitrarily, considering only a limited number of experimental studies in which they were assessed. However, the results obtained in this study indicate a high potential for articular cartilage regeneration under conditions for parallel use of cellular and rehabilitation technologies, at least in the initial stages of osteoarthritis. In the context of this work, the use of neural networks and machine learning methods to predict and control the processes of regenerative tissue rehabilitation is of interest. In further studies devoted to this topic, the mathematical model for regenerative rehabilitation will be refined, which will make it possible to obtain more accurate information about the course of this process in vivo and plan more reliable protocols for the treatment of osteoarthritis. Acknowledgement: The paper is a part of the research done at the Laboratory of Biomechanics at Sevastopol State University without external funding. Authors are grateful to the management of the “Afalina” Shared Use Center at Sevastopol State University for the opportunity to perform numerical experiments on a hybrid supercomputer cluster (x86/CUDA). REFERENCES 1. Patel, D.M., Shah, J., Srivastava, A.S., 2013, Therapeutic potential of mesenchymal stem cells in regenerative medicine, Stem Cells Int, 2013, 496218. 2. Salinas, E.Y., Hu, J.C., Athanasiou, K., 2018, A Guide for Using Mechanical Stimulation to Enhance Tissue- Engineered Articular Cartilage Properties, Tissue Eng. Part B: Rev, 24, pp. 345-358. 3. Lutianov, M., Naire, S., Roberts, S., Kuiper, J.H., 2011, A mathematical model of cartilage regeneration after cell therapy, J Theor Biol, 289, pp. 136-150. 4. Campbell, K., Naire, S., Kuiper, J.H., 2019, A mathematical model of cartilage regeneration after chondrocyte and stem cell implantation - I: the effects of growth factors, J Tissue Eng, 10, 2041731419827791. 5. Campbell, K., Naire, S.; Kuiper, J.H., 2019, A mathematical model of cartilage regeneration after chondrocyte and stem cell implantation - II: the effects of co-implantation, J Tissue Eng, 10, 2041731419827792. 6. Popov, V.L., Poliakov, A.M., Pakhaliuk, V.I., 2022, One-dimensional biological model of synovial joints regenerative rehabilitation in osteoarthritis, Facta Universitatis-Series Mechanical Engineering, 20(20), pp. 421-444. 7. Li, K., Zhang, C., Qiu, L., Gao, L., Zhang, X., 2017, Advances in Application of Mechanical Stimuli in Bioreactors for Cartilage Tissue Engineering, Tissue Eng. Part B: Rev, 23, pp. 399-411. 8. Gamez, C., Schneider-Wald, B., Schuette, A., Mack, M., Hauk, L., Khan, A.U.M., Gretz, N., Stoffel, M., Bieback, K., Schwarz, M,L., 2020, Bioreactor for mobilization of mesenchymal stem/stromal cells into scaffolds under mechanical stimulation: Preliminary results, PloS One, 15(1), e0227553. 9. Ravalli, S., Szychlinska, M.A., Lauretta, J., Musumeci, G., 2020, New Insights on Mechanical Stimulation of Mesenchymal Stem Cells for Cartilage Regeneration, Appl Sci, 10, 2927. 10. Saadat, E., Lan, H., Majumdar, S., 2006, Long-term cyclical in vivo loading increases cartilage proteoglycan content in a spatially specific manner: an infrared microspectroscopic imaging and polarized light microscopy study, Arthritis Res Ther, 8(5), R147. 416 A. POLIAKOV, V. PAKHALIUK 11. King, K.B., Opel, C.F., Rempel, D.M., 2005, Cyclical articular joint loading leads to cartilage thinning and osteopontin production in a novel in vivo rabbit model of repetitive finger flexion, Osteoarthritis and cartilage, 13(11), pp. 971-978. 12. Reinold, M.M., Wilk, K.E., Macrina, L.C., Dugas, J.R., Cain, E.L., 2006, Current concepts in the rehabilitation following articular cartilage repair procedures in the knee, The J of Ortho and Sports Phys Ther, 36(10), pp. 774-794. 13. Rose, L.F., Wolf, E.J., Brindle, T., Cernich, A., Dean, W.K., Dearth, C.L., Grimm, M., Kusiak, A., Nitkin, R., Potter, K., Randolph, B.J., Wang, F., Yamaguchi, D., 2018, The convergence of regenerative medicine and rehabilitation: federal perspectives, NPJ Regen Med, 3, 19. 14. Ezhov, M.Y., Ezhov I.Y., Kashko A.K., Kayumov, A.Y., Zykin, A.A., Gerasimov, S.A., 2015, Unresolved issues of regeneration of cartilage and bone tissue (review and analytical article), Successes of Mod Nat Sci, 5, pp. 126-131. 15. 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In Silico Analysis of an Articular Cartilage Regenerative Rehabilitation under Conditions of ... 417 APPENDIX A Parameters of the mathematical model for regenerative rehabilitation articular cartilage defect Parameter Functional dependencies and parameter value Parameter dimension Dimensionless parameter value CS – MSC density CS=CS(x,t) cells/mm3 - CC – chondrocytes density CC=CC(x,t) cells/mm3 - m – ECM density m=m(x,t) g/mm3 - n – nutrient concentration n=n(x,t) moles/mm3 - g – FGF-1 concentration g=g(x,t) g/mm3 - b – BMP-2 concentration b=b(x,t) g/mm3 - DS – MSC diffusion coefficient 0 2 2 1 S S m D D m m = + mm2/hour - D* S – maximum MSC diffusion coefficient * 4 33.6 (10 10 )SD − −   mm2/hour - DC – chondrocytes diffusion coefficient 0 2 2 1 C C m D D m m = + mm2/hour - D* C – maximum chondrocytes diffusion coefficient * 4 33.6 (10 10 )CD − −   mm2/hour - p1 – MSC proliferation rate max 1 1 S m S C p A C   = −      cells/hour - 01 2 2 2 m m A p m m = + max max 0 max 1S S m C C m   = −    - - p10 – MSC proliferation constant 0 * 6 1 2 12 4 10p m p −= =  g/(mm3·hour) 12 p* 1 – maximum MSC proliferation rate * 1 0.2p = cells/hour - m1 – reference ECM density m1=10-5 g/mm3 0.1 m2 – reference ECM density m2=10-5 g/mm3 0.1 p4 – chondrocyte proliferation rate max 4 1 C m C C p B C   = −      1/hour - 0 004 42 2 2 0 m m g B p p m m g g = + + + max max 0 max 1C C m C C m   = −    - - 418 A. POLIAKOV, V. PAKHALIUK g0 – FGF-1 reference concentration g0=10-10 g/mm3 - p* 4 – maximum chondrocyte proliferation rate * 4 4 2 10p −=  1/hour - n1 – critical nutrient concentration n1=9.5·10-12 moles/mm3 0.1 n0 – threshold nutrient concentration n0=2.3·10-11 moles/mm3 0.24 – 0.81 CS0 – MSC threshold density 0 0 0 0max min min( ) ab S S S SC C C e C−= − + cells/mm3 - CS0 max – maximum threshold MSC density 0 ,max 0 max 2 total S C C = cells/mm3 0.35 CS0 min – minimum threshold MSC density 0 0min max0.9S SC C= cells/mm3 0.315 p6 – nutrient uptake constant by MSC p6=1.5·10-14 moles/(NC·hour) 10000 p7 – nutrient uptake constant by chondrocytes p7=1.5·10-14 moles/(cells·hour) 10000 p8 – ECM synthesis rate 0 18 8 8p p p m= − (g/mm3)/((cells/mm3)· (1/hour)) - a – threshold MSC density reduction factor a =1010 1/(g/mm3) 100 γ – FGF-1 flux coefficient γ=10-2 mm/hour 0.01 χ – BMP-2 flux coefficient χ=10-2 mm/hour 1 b0 – BMP-2 reference concentration b0=10-10 g/mm3 - N0 – initial nutrient concentration N0 = (2.85 – 9.5)·10-11 moles/mm3 - In Silico Analysis of an Articular Cartilage Regenerative Rehabilitation under Conditions of ... 419 APPENDIX B General Coordinates of Test points 1 2 3 4 5 6 7 8 DS0 0.0055 0.0034 0.0078 0.0021 0.0066 0.0044 0.0089 0.0016 DC0 0.001 0.0013 0.0008 0.0011 0.0006 0.0009 0.0014 0.0014 Dn 200.0 150.0 250.0 275.0 175.0 225.0 125.0 237.5 Dm 0.0055 0.0078 0.0033 0.0089 0.0044 0.0021 0.0066 0.0038 Dg 1.0 0.75 1.25 1.125 0.625 1.375 0.875 0.6875 Db 1.0 1.25 0.75 0.625 1.125 1.375 0.875 0.5625 p2 1.0 0.75 1.25 0.875 1.375 0.625 1.125 0.9375 p3 1.0 1.25 0.75 0.875 1.375 1.125 0.625 1.0625 p40 0.012 0.0150 0.009 0.0165 0.0105 0.0075 0.0135 0.0158 p400 0.012 0.009 0.015 0.0135 0.0075 0.0165 0.0105 0.0143 p5 1.0 1.25 0.75 1.125 0.625 0.875 1.375 0.5625 p80 0.7 0.45 0.95 1.075 0.575 0.825 0.325 0.3875 p800 0.7 0.95 0.4500 1.075 0.575 0.325 0.825 0.6375 p81 1.0 0.75 1.2500 0.625 1.125 0.875 1.375 1.0625 p9 25.0 26.5 23.5 24.25 27.25 25.75 22.75 23.875 p11 14.5 13.75 15.25 14.125 15.625 13.375 14.875 15.4375 p12 25.0 23.5 26.5 27.25 24.25 25.75 22.75 26.875 p13 14.5 15.25 13.75 14.875 13.375 14.125 15.625 15.0625 9 10 11 12 13 14 15 16 DS0 0.0061 0.0038 0.0083 0.0027 0.0072 0.0049 0.0094 0.0013 DC0 0.0009 0.0007 0.0012 0.0008 0.0013 0.0011 0.0006 0.0010 Dn 137.5 287.5 187.5 162.5 262.5 112.5 212.5 181.25 Dm 0.0083 0.0061 0.0016 0.0072 0.0027 0.0049 0.0094 0.0030 Dg 1.1875 0.9375 1.4375 1.0625 0.5625 1.3125 0.8125 0.9688 Db 1.0625 1.3125 0.8125 0.6875 1.1875 1.4375 0.9375 0.7813 p2 1.4375 0.6875 1.1875 0.5625 1.0625 0.8125 1.3125 1.4688 p3 0.5625 0.8125 1.3125 1.4375 0.9375 0.6875 1.1875 0.7813 p40 0.0098 0.0068 0.0128 0.0083 0.0143 0.0173 0.0113 0.0071 p400 0.0083 0.0173 0.0113 0.0068 0.0128 0.0098 0.0158 0.0161 p5 1.0625 1.3125 0.8125 1.1875 0.6875 0.9375 1.4375 0.9688 p80 0.8875 0.6375 1.1375 1.0125 0.5125 0.7625 0.2625 1.1063 p800 1.1375 0.8875 0.3875 0.7625 0.2625 0.5125 1.0125 0.8563 p81 0.5625 1.3125 0.8125 1.1875 0.6875 1.4375 0.9375 1.2188 p9 26.875 25.375 22.375 23.125 26.125 27.625 24.625 25.5625 p11 13.9375 14.6875 13.1875 15.0625 13.5625 15.8125 14.3125 14.0313 p12 23.875 25.375 22.375 23.125 26.125 24.625 27.625 26.6875 p13 13.5625 14.3125 15.8125 13.1875 14.6875 15.4375 13.9375 13.6563 17 18 19 20 21 22 23 24 DS0 0.0058 0.0035 0.0080 0.0024 0.0069 0.0047 0.0092 0.0018 DC0 0.0005 0.0008 0.0013 0.0007 0.0012 0.0015 0.0009 0.0010 Dn 281.25 131.25 231.25 206.25 106.25 256.25 156.25 268.75 Dm 0.0075 0.0097 0.0052 0.0086 0.0041 0.0018 0.0063 0.0047 Dg 1.4688 0.7188 1.2188 1.3438 0.8438 1.0938 0.5938 0.7813 Db 1.2813 1.0313 0.5313 0.9063 1.4063 1.1563 0.6563 0.8438 420 A. POLIAKOV, V. PAKHALIUK p2 0.9688 1.2188 0.7188 1.0938 0.5938 1.3438 0.8438 1.0313 p3 1.2813 1.0313 0.5313 0.6563 1.1563 1.4063 0.9063 1.3438 p40 0.0131 0.0161 0.0101 0.0176 0.0116 0.0086 0.0146 0.0154 p400 0.0101 0.0131 0.0071 0.0116 0.0176 0.0086 0.0146 0.0109 p5 1.4688 1.2188 0.7188 1.3438 0.8438 0.5938 1.0938 0.9063 p80 0.6063 0.8563 0.3563 0.2313 0.7313 0.4813 0.9813 1.0438 p800 0.3563 0.6063 1.1063 0.4813 0.9813 0.7313 0.2313 1.0438 p81 0.7188 1.4688 0.9688 1.0938 0.5938 1.3438 0.8438 0.6563 p9 22.5625 24.0625 27.0625 27.8125 24.8125 23.3125 26.3125 26.6875 p11 15.5313 13.2813 14.7813 13.6563 15.1563 14.4063 15.9063 14.5938 p12 23.6875 25.1875 22.1875 22.9375 25.9375 24.4375 27.4375 22.5625 p13 15.1563 15.9063 14.4063 14.7813 13.2813 14.0313 15.5313 14.5938 25 26 27 28 29 30 31 32 DS0 0.0063 0.0041 0.0086 0.0030 0.0075 0.0052 0.0097 0.0011 DC0 0.0015 0.0012 0.00072 0.0013 0.0008 0.00059 0.0011 0.0013 Dn 168.75 218.75 118.75 143.75 243.75 193.75 293.75 290.625 Dm 0.0092 0.0069 0.0024 0.0058 0.0013 0.0035 0.00803 0.0070 Dg 1.2813 0.5313 1.0313 1.4063 0.9063 1.1563 0.6563 1.2969 Db 1.3438 1.0938 0.5938 0.9688 1.4688 1.2188 0.7188 1.4219 p2 0.5313 1.2813 0.7813 1.4063 0.9063 1.1563 0.6563 1.2344 p3 0.8438 0.5938 1.0938 1.2188 0.7188 0.9688 1.4688 1.3906 p40 0.0094 0.0064 0.0124 0.0079 0.0139 0.0169 0.0109 0.0126 p400 0.0169 0.0079 0.0139 0.0154 0.0094 0.0124 0.0064 0.0159 p5 1.4063 1.1563 0.6563 1.2813 0.7813 0.5313 1.0313 0.7969 p80 0.5438 0.7938 0.2938 0.4188 0.9188 0.6688 1.1688 0.9969 p800 0.5438 0.2938 0.7938 0.4188 0.9188 1.1688 0.6688 1.1531 p81 1.1563 0.9063 1.4063 0.5313 1.0313 0.7813 1.2813 1.0781 p9 23.6875 22.1875 25.1875 25.9375 22.9375 24.4375 27.4375 25.0938 p11 13.0938 15.3438 13.8437 15.7188 14.2188 14.9688 13.4688 13.3281 p12 25.5625 24.0625 27.0625 27.8125 24.8125 26.3125 23.3125 22.4688 p13 13.0938 13.8438 15.3438 13.4688 14.9688 15.7189 14.2188 13.5156 33 34 35 36 37 38 39 40 DS0 0.0056 0.0034 0.0079 0.0023 0.0068 0.0045 0.0090 0.0017 DC0 0.0008 0.00054 0.0010 0.0009 0.0014 0.0012 0.00067 0.00073 Dn 190.625 240.625 140.625 115.625 215.625 165.625 265.625 153.125 Dm 0.0025 0.0048 0.0093 0.0037 0.0082 0.0059 0.0014 0.0099 Dg 0.7969 1.0469 0.5469 0.9219 1.4219 0.6719 1.1719 1.4844 Db 0.9219 0.6719 1.1719 1.2969 0.7969 0.5469 1.0469 1.4844 p2 0.7344 1.4844 0.9844 1.3594 0.8594 1.1094 0.6094 1.2969 p3 0.8906 0.6406 1.1406 1.0156 0.5156 0.7656 1.2656 0.9531 p40 0.0066 0.0096 0.0156 0.0111 0.0171 0.0141 0.0081 0.0103 p400 0.0099 0.0129 0.0069 0.0114 0.0174 0.0084 0.0144 0.0107 p5 1.2969 1.0469 0.5469 1.4219 0.9219 0.6719 1.1719 0.8594 p80 0.4969 0.7469 0.2469 0.3719 0.8719 0.6219 1.1219 1.1844 p800 0.6531 0.4031 0.9031 0.2781 0.7781 1.0281 0.5281 0.7156 p81 0.5781 1.3281 0.8281 1.2031 0.7031 1.4531 0.9531 0.5156 p9 22.0938 23.5938 26.5937 27.3438 24.3438 22.8438 25.8438 26.9688 In Silico Analysis of an Articular Cartilage Regenerative Rehabilitation under Conditions of ... 421 p11 14.8281 14.0781 15.5781 14.4531 15.9531 13.7031 15.2031 15.3906 p12 25.4688 23.9688 26.9688 27.7188 24.7188 26.2188 23.2188 26.5938 p13 15.0156 15.7656 14.2656 14.6406 13.1406 13.8906 15.3906 14.8281 41 42 43 44 45 46 47 48 DS0 0.0062 0.00395 0.0085 0.0028 0.0073 0.00508 0.0096 0.0014 DC0 0.0012 0.00148 0.00098 0.0011 0.0006 0.00086 0.00136 0.00077 Dn 253.125 103.125 203.125 228.125 128.125 278.125 178.125 221.875 Dm 0.0054 0.0031 0.0076 0.00198 0.0065 0.0087 0.0042 0.0062 Dg 0.9844 1.2344 0.7344 0.8594 1.3594 0.6094 1.1094 1.2031 Db 0.9844 0.7344 1.2344 1.3594 0.8594 0.6094 1.1094 1.1406 p2 0.7969 1.0469 0.5469 1.1719 0.6719 1.4219 0.9219 0.7656 p3 1.4531 1.2031 0.7031 0.5781 1.0781 1.3281 0.8281 1.1719 p40 0.0163 0.01331 0.0073 0.0148 0.0088 0.0118 0.0178 0.0129 p400 0.0167 0.0077 0.0137 0.0152 0.0092 0.0122 0.0062 0.0066 p5 1.3594 1.1094 0.6094 1.4844 0.9844 0.7344 1.2344 0.7031 p80 0.6844 0.9344 0.4344 0.3094 0.8094 0.5594 1.0594 0.3406 p800 0.2156 0.4656 0.9656 0.5906 1.0906 0.8406 0.3406 0.5594 p81 1.0156 0.7656 1.2656 0.6406 1.1406 0.8906 1.3906 0.6719 p9 23.9688 22.4688 25.4688 26.2188 23.2188 24.7188 27.7188 22.6563 p11 13.8906 14.6406 13.1406 15.0156 13.5156 15.7656 14.2656 13.7969 p12 23.5938 25.0938 22.0938 22.8438 25.8438 24.3438 27.3438 27.1563 p13 13.3281 14.0781 15.5781 13.7031 15.2031 15.9531 14.4531 13.2344 49 50 51 52 53 54 55 56 DS0 0.0059 0.00367 0.00817 0.00255 0.00705 0.0048 0.0093 0.00198 DC0 0.00127 0.00102 0.00052 0.00139 0.00089 0.00064 0.00114 0.0012 Dn 121.875 271.875 171.875 196.875 296.875 146.875 246.875 134.375 Dm 0.0017 0.00395 0.00845 0.00508 0.0096 0.0073 0.0028 0.0079 Dg 0.7031 1.4531 0.9531 0.5781 1.0781 0.8281 1.3281 1.0156 Db 0.6406 0.8906 1.3906 1.0156 0.5156 0.7656 1.2656 1.2031 p2 1.2656 0.5156 1.0156 0.6406 1.1406 0.8906 1.3906 0.7031 p3 0.67188 0.9219 1.4219 1.2969 0.7969 0.5469 1.0469 0.7344 p40 0.0069 0.0099 0.0159 0.0114 0.0174 0.0144 0.0084 0.0092 p400 0.0126 0.00956 0.0156 0.01406 0.00806 0.0170 0.01106 0.0148 p5 1.2031 1.4531 0.9531 1.0781 0.5781 0.8281 1.3281 0.6406 p80 0.8406 0.5906 1.0906 0.9656 0.4656 0.7156 0.2156 0.2781 p800 1.0594 0.8094 0.3094 0.9344 0.4344 0.6844 1.1844 0.3719 p81 1.1719 0.9219 1.4219 0.5469 1.04688 0.7969 1.2969 1.2344 p9 25.6563 27.1563 24.1563 24.9063 27.9063 26.4063 23.4063 23.7813 p11 15.2969 13.0469 14.5469 13.4219 14.9219 14.1719 15.6719 14.7344 p12 24.1563 25.6563 22.6563 23.4063 26.4063 24.9063 27.9063 22.2813 p13 14.7344 15.4844 13.9844 15.1094 13.6094 14.3594 15.8594 14.9219 422 A. POLIAKOV, V. PAKHALIUK Appendix C Results of the mathematical model investigation 1 2 3 4 5 6 7 8 m 0.5982 0.215 0.7321 1.0776 0.3499 0.5466 0.1400 0.3152 CC 1.0376 0.7795 0.8062 0.3101 0.4375 0.7856 0.9931 1.316 CS 0.2779 0.476 0.2032 0.0148 0.3888 0.3077 0.5108 0.4146 FGF1 3.6621 3.0659 4.0465 4.4377 1.5657 6.9173 3.1281 1.8207 BMP2 3.8201 2.3315 5.3087 1.7329 6.9741 4.6249 2.2303 3.577 9 10 11 12 13 14 15 16 m 0.9373 0.4199 1.0681 0.5450 0.6240 0.3824 0.2195 0.8316 CC 0.7614 1.0803 0.5413 0.7406 1.1207 0.9292 1.6943 0.8042 CS 0.0979 0.3612 0.0273 0.3076 0.2650 0.3829 0.4629 0.1486 FGF1 5.0074 4.5112 4.8063 3.1708 2.2640 5.9213 2.4906 3.0667 BMP2 5.3259 3.8617 2.6619 1.4368 7.4866 5.185 3.4894 3.9660 17 18 19 20 21 22 23 24 m 0.6515 0.5012 0.2882 - 0.2686 - 0.8423 1.0990 CC 0.9789 1.3741 1.1998 - 0.5790 - 0.6557 0.2010 CS 0.2493 0.3152 0.4348 - 0.4516 - 0.1506 0.0019 FGF1 3.8033 2.4157 7.9171 - 3.1099 - 1.9360 1.9905 BMP2 2.7256 2.8415 2.8215 - 4.8565 - 2.0348 2.3157 25 26 27 28 29 30 31 32 m - 0.7490 - 0.3722 0.7919 0.6564 0.8198 0.8320 CC - 0.9692 - 1.3878 0.7556 0.9994 0.6076 0.6377 CS - 0.1949 - 0.3794 0.1730 0.2488 0.1567 0.1472 FGF1 - 1.3102 - 3.3675 3.8432 3.6281 3.0314 3.9435 BMP2 - 4.9400 - 3.8044 7.3184 8.3667 1.9080 6.3305 33 34 35 36 37 38 39 40 m 0.2013 0.5419 0.0856 - 0.8856 0.3978 0.8963 1.0964 CC 0.6829 1.3217 0.7386 - 0.6010 1.4700 0.5758 0.1069 CS 0.4827 0.3055 0.5416 - 0.1229 0.3653 0.1193 0.0023 FGF1 2.9481 3.5707 1.6303 - 6.5989 2.4895 4.3165 4.1863 BMP2 2.2565 2.6339 7.6867 - 5.0906 3.8176 1.9505 6.6589 41 42 43 44 45 46 47 48 m 0.5745 0.7012 0.2586 0.0167 0.4678 0.5642 0.7082 0.1572 CC 0.9367 0.8136 0.9458 0.1744 0.9135 1.3071 0.7062 0.7271 CS 0.2904 0.2307 0.4538 0.5183 0.3407 0.2943 0.2187 0.5069 FGF1 3.4220 3.4029 5.5431 3.4370 6.0843 1.2618 5.6291 4.4711 BMP2 2.4714 2.1959 5.4392 4.1516 2.3561 2.6471 4.0417 5.9224 49 50 51 52 53 54 55 56 m 0.6195 0.3160 0.6739 0.9191 0.4149 0.5855 0.1199 - CC 1.070 0.6936 0.7177 0.5920 1.6777 0.9843 1.3427 - CS 0.2705 0.4232 0.2367 0.1075 0.3547 0.2877 0.5180 - FGF1 2.1485 9.9384 2.8559 1.9884 4.6092 4.4418 3.3682 - BMP2 2.1416 1.6862 4.2940 2.0489 5.0816 3.9575 4.0476 -