Galina Eremina, Alexey Smolin

Institute of Strength Physics and Materials Science of Siberian Branch of Russian Academy of Sciences, 2/4, pr.Akademicheskii, Tomsk, 634055, Russia

Keywords:Knee joint Total knee replacement Shock-wave therapy Computer simulation Method of movable cellular automata

ABSTRACT Degenerative diseases significantly reduce the quality of human life.Non-invasive treatments are used in the initial stages of osteoarthritis (OA).Total knee arthroplasty is used in the late stages of osteoarthritis of the knee joint.Non-invasive methods based on mechanical action are also used for the rehabilitation of a patient after arthroplasty.This paper presents numerical models of the knee joint with degenerative OA changes and arthroplasty.Using these models, a computational study was made of the influence of the intensity of shock-wave exposure on the conditioning for the regeneration of bone and cartilage tissues.Based on the modeling results,it was found that in the knee joint with degenerative OA changes,conditions for the regeneration of cartilage and meniscus tissues were fulfilled under medium and highintensity loading.Under high-intensity loading (up to 0.9 mJ/mm2), the stress level was significantly below the ultimate value required for fracture.At knee arthroplasty, the conditions for bone tissue regeneration around the tibia component are fulfilled only under high-intensity loading.

1.Introduction

Arthritis is the most common musculoskeletal disease in the world.The exact cause of this type of degenerative disease is unknown.Arthritis affects people of different ages, most often it begins to appear in women after 40 years of age [1,2].Osteoarthritis(OA) causes enormous financial damage to the economy of developed countries,and in particular to the healthcare system.Cartilage plates and meniscus perform the function of redistributing physiological loads and contribute to the dissipation of energy received as a result of a dynamic impact on the joint.These structural elements of the knee joint are most prone to degenerative arthritis changes.

Most acutely, this problem affects the elderly, in whom, due to age,degradation processes in the soft and bone tissues of the joints begin to appear.Former athletes are also very prone to degenerative changes.They have a high level of cartilage wear and soft tissue injury (meniscus tear).Analysis of the recent military conflicts shows that low extremities have been the most commonly affected body and often result in complex injuries that require long term follow up and rehabilitation.This,in turn,may also lead to various degradation processes in low extremities.Operations are usually performed for catastrophic degenerative changes in the elements of the lower extremities of the human skeleton.Pharmacotherapy(based on medication, injections, etc.) and nonpharmacologic therapies are used to treat mild to moderate degenerative changes in the human musculoskeletal system [3,4].Nonpharmacologic therapy consists of a set of physical exercises [5]and external physical impact.Shock wave therapy is one of the methods of treatment with the help of external exposure.For therapy, shock wave loading with a pressure of up to 100 MPa and a pulse duration of up to 1 μs is used.This combination of very short time and high pressure leads to a low value of the energy flux density and to the absence of a destructive effect [6,7].

The therapeutic effect of external mechanical action is based on mechanobiological principles.Mechanobiological principles are formulated on the experimental fact that a certain level of pressure(mechanical stress) and deformation leads to the growth and differentiation of a different type of biological tissue [8-10].The process of osteoblast differentiation(regeneration of bone tissue)is promoted by stresses below 0.15 MPa.Chondrocyte differentiation is promoted by compressive stresses in the range from 0.15 to 2 MPa (cartilage tissue regeneration).It was noted that at stresses below 0.003 MPa, chondrogenesis and osteogenesis do not occur,and compressive stresses of the order of 0.7-0.8 MPa are most favorable for the formation of cartilaginous tissue.Optimal for the migration of living cells is the fluid pressure in the pores in the range from 20 to 2 MPa (68 kPa is the most favorable value) [11].Differentiation of cartilage and fibrous tissues contributes to the magnitude of deformation (distortional strain) in the range from 0.05 to 1.1% (0.5% is the most favorable value) [12,13].Features of the physiological state (age, disease, injury) of the patient require an individual selection of parameters for low-energy acoustic(mechanical) exposure.Energy flux density (EFD) is the main characteristic of shock wave therapy(SWT)[14].The load is applied through metal and ceramic plates (applicators) of various shapes with a certain level of energy flux density, which is calculated via the parameters of the plates.Currently, active research is being underway to determine the optimal load parameters that promote the regeneration of bone and cartilage tissue.SWT has been shown to be very effective in the treatment of osteoarthritis of the knee[15,16].In the treatment of knee arthritis,applicators are placed in the meniscus and subchondral of the tibia[17].For the treatment of osteoarthritis,shock wave therapy with a load of low,medium,and high intensity is effective [18].Despite the available results, many important problems of shock wave therapy remain insufficiently studied due to the limited experimental studies of local changes in tissues due to acoustic (mechanical) loading of varying intensity.The use of computational(in-silico)methods can significantly help in elucidating the mechanical foundations of bone and cartilage tissue regeneration under shock wave impact.

In arthritis, the meniscus is one of the first to change its mechanical characteristics and is also prone to degeneration[19].Twophase poroelastic models are mainly used to study the mechanical behavior of meniscus tissues[20].Cartilage plates perform an antifriction function and help to dissipate mechanical energy in the knee joint [21,22].It is known that the cartilaginous tissue of the knee joint has low permeability; this property of the tissue prevents rapid outflow of interstitial fluid under dynamic load[23,24].Therefore, for a computational study of the conditions of regeneration under dynamic action, two-phase poroelastic models are used[25].Bone tissues are described by both single-phase and twophase poroelastic models.However,poroelastic models are mainly used for the computational study of the conditions of bone tissue regeneration [26].

Arthroplasty is used in the case of absence of positive dynamics in the treatment of OA of the knee joint by non-invasive methods.It is worth noting that arthroplasty has its own disadvantages.Thus,there is tissue resorption (decrease in density) in the pin area,leading to aseptic loosening and a decrease in the service life of the implant [27,28].Bone resorption is observed in both the distal femur and the proximal tibia for arthroplasty using both cement technology and cementless technology [29,30].Experimental data[31]show that there is a decrease in tissue resorption around the pin during drug therapy.However, as shown in Ref.[32]the effectiveness of drug therapy is significantly higher in the distal femur than in the proximal tibia.In Ref.[33]this effect is associated with the distribution of stresses in the distal region of the femur,similar to the distribution in a healthy bone.The lack of therapeutic effect of drug therapy in the proximal tibia increases the need to develop therapeutic approaches to reduce the risk of resorption.The data presented in Ref.[34]indicate that drug therapy is highly effective for one year after surgery,and then significantly decreases.The mineral density of the cancellous tissue around the implant pin decreases in the range from 5 to 20% and similar to osteoporotic changes in the properties of the cancellous tissue are observed in the next two years after knee replacement operation.Such a decrease in density and an increase in porosity leads to a more intensive flow of fluid near the “bone-implant” contact area.According to Ref.[35]the area under the pin is subject to the greatest degenerative changes (the greatest decrease in mineral density in the postoperative period) in the distal tibia.

The aim of this article is to develop a numerical model of shock wave impact on the knee joint affected by degenerative diseases and after arthroplasty.As a modeling method, the method of movable cellular automata was chosen,which is a representative of the computational particle mechanics (discrete elements) and allows explicit describing the initiation and development of damages in heterogeneous materials.

2.Material and methods

2.1.Method of movable cellular automata

To describe the mechanical behavior of biological tissues,herein we used the model of a poroelastic body implemented in the method of movable cellular automata (MCA) [36].It has been established that discrete methods have proven themselves to be very promising for modeling contact loading of different materials at the macro and mesoscale [37,38].In the MCA method, a solid is considered as an ensemble of discrete elements of finite size(cellular automata) that interact with each other according to certain rules,which,within the particle approach and due to manybody interaction forces, describe the deformation behavior of the material as an isotropic elastoplastic body.The motion of the ensemble of elements is governed by the Newton-Euler equations for their translation and rotation.Within the framework of the MCA method, the value of averaged stress tensor in the volume of an automaton is calculated as a superposition of forces that act to the areas of interaction of the automaton with its neighbors [37].It is assumed that stresses are homogeneously distributed in the automaton volume.Knowing the components of the averaged stress tensor allows adapting to MCA different models of plasticity and fracture of classical mechanics of solid.

The description of the fluid-saturated material in the MCA method is based on the implicit use of such effective characteristics as the volume fraction of interstitial fluid, porosity, permeability,and the ratio of the macroscopic bulk modulus of elasticity to the bulk modulus of the solid skeleton of the material [39].The fluid filtration in the material is governed by Darcy’s law.The mechanical effect of pore fluid on stress and strain of the solid skeleton of the automaton is described using Biot’s linear poroelasticity model,therefore,pore fluid pressure affects only the diagonal components of the stress tensor.Herein, to describe the strength properties of the sold skeleton of the bone tissues,we used the model of elasticbrittle medium with the Drucker-Prager criterion of fracture(with different values for compressive and tensile strengths).

2.2.Models of the knee joint and knee total replacement

Fig.1. Model of the knee joint: (a) General view and main components; (b) Initial and boundary conditions.

We used a geometric model consisting of the epiphysis femur,proximal tibia,cartilage plates,and meniscus to numerically study the mechanical behavior of the knee joint under external acoustic exposure (Fig.1(a)).We neglect the patella to simplify the model and analyze the results obtained, as well as due to absence of the need to observe the processes taking place inside the patella.Moreover, most studies of SWT of the knee joint are carried out with the installation of the SW applicator in the exact location at a distance of 0.5 cm from the joint line and on the tibia(in the area of the medial tibia plateau) with a different angle of knee flexion,which confirms the absence of a practical effect of patella deformation on the mechanical behavior of the knee joint tissues[40,41].The knee joint was placed in a parallelepiped mimicking an articular (synovial) capsule consisting of fibrous tissue.The standard computer-aided design (CAD) models of the corresponding components available on the Internet(https://www.3dcadbrowser.com/3d-model/human-knee-joint) were used as a geometric basis of the developed model.Mesh models were built in stereolithography file format (ASCI STL) based on solid models of the components of the knee joint and the implant.Obtained models were then imported into the preprocessor of the software package that implements the MCA method.The applicator was modeled as a thin square copper plate 20×20 mm2in size and was located in the meniscus region to apply an external load.The applicator is shown as the pink square in Fig.1(b),the letter“V”near it denotes applying the load by setting velocity to the automata of this region,and the arrow indicates the direction of the load.The upper and lower squares shown by thick solid lines in Fig.1(b)denotes positions of automata, which are fixed in space by setting zero to all their velocity componentsVX=VY=VZ= 0.

In this study, we also model the knee joint with arthroplasty(Fig.2(a)).The prosthesis consisted of three parts: femoral and tibial components made of titanium alloy Ti6AlV4, and a plastic spacer made of high-molecular-weight polyethylene (HMWPE)(Fig.2(b)).The area around the pin of the tibial component of the prosthesis was divided into lateral(RO 1),medial(RO 2)and distal(RO 3) zones (Fig.2(c)) according to Ref.[42].

The height of the lateral and medial regions was 4 cm,the height of the distal region was 2 cm in compliance with the literature data[43].According to experimental data [44], within two years after arthroplasty, the resorption zone in the tibia component is characterized by a thickness of about 2 mm and a loss of elastic characteristics of up to 20%.The resorption area in the tibial component was modeled by the region around the contour of a pin with 2 mm thick.The decrease in the elastic characteristics of the bone tissue in the resorption zone was 20%.

Bone, cartilage, and meniscus tissues were modeled using the Biot poroelasticity model.But the properties of these tissues at the microscale and at the macroscale are different.At the microscale,cartilage is modeled as a composite of collagen and aggrecan [45].To model the knee joint at the macroscale,the effective poroelastic properties of the corresponding tissues are used.At the macroscale,the value of the effective elastic modulus of cartilage lies in the range of 1-25 MPa, and its permeability is (1-10)·10-18m2(hydraulic permeability (1-10)·10-15m4/(Ns)) [46,47].Elastic and poroelastic properties of the knee joint tissues including ones with degenerative changes [47-49]are shown in Table 1.The synovial fluid in this model has the properties of salt water: bulk modulusKf= 2.4 GPa, density ρf= 1000 kg/m3.

To describe the mechanical behavior of the copper applicator plates, an elastic body model is used with the following parameters: density ρ = 8950 kg/m3, bulk modulusK= 115 GPa shear modulusG= 41.6 GPa[50].

Fig.2. Model of the knee joint replacement: (a) General view and main components of the model; (b) Model of implant; (c) Model section with zone designation around the implant pin; (d) Implant with the resorption zone; (e) Model section with the resorption zone.

Table 1Elastic and poroelastic properties of the knee joint tissues (healthy and with OA).

The results of verification and validation of poroelastic models of knee joint tissues are presented in our works[51-55].The results of verification and validation of the model of the knee joint are presented in Ref.[56].

2.3.Model verification

The main purpose of verification is to assess the correctness and efficiency of the numerical scheme for solving the governing equations of the method.The key component of verification of a numerical model is the analysis of the convergence of the obtained results with increasing the resolution of a discrete model(decreasing the discrete element size in the case of computational particle mechanics).The discrete representation of the model is considered optimal when a further increase in its resolution gives no more than a 5% difference with the available resolution [57].

In this work, the analysis of the result convergence for a threedimensional model of the knee joint was carried out by studying the stiffness of the system and the pattern of equivalent strain distribution at different discretization of the considered geometric model(Fig.2)under its uniaxial compression.Herein,the numbers of discrete elements (automata) in the model sample varied from 123391 to 2356312.The compression of the model samples along the vertical direction was carried out by setting a constant velocity of 0.001 m/s to the upper layer of particles (Fig.1(b)).

The results on the convergence of the stiffness value of the model knee joint showed that the convergence is nonlinear,and the total scatter between the values for the minimum and maximum size of elements does not exceed 2%(Fig.3).A very small difference between the values obtained for the size of elements smaller than 1.3 mm (509757 elements) indicates a good accuracy of the numerical model for determining its integral parameters.

Fig.3. Stiffness of the model knee joint versus the number of discrete elements in the model.

However, the pattern of hydrostatic pressure (Fig.4) indicates that sufficient accuracy in determining the zones of strain concentration is obtained only for the models with the numbers of elements of 992776 and 2356312.Since the calculation times for these models are 13 h and 6 h, respectively, we will take a sample with the numbers of automata equal to 992776, which was the optimal choice for subsequent calculations.

2.4.Model validation

Validation of the model of the entire knee joint was carried out by comparing the simulation results with experimental data from Ref.[58]and other numerical simulations from Ref.[59].For this purpose, a compressive load was applied to the upper layer of the femur by its displacement by 0.3 mm for 1 s in accordance with the reference experiment (Fig.1(b)).

As mentioned above, the tissues of the cartilaginous plate and meniscus are subject to the greatest degenerative changes; therefore, the model was validated by comparing the distribution of contact pressure(equivalent stresses in the contact zone)and fluid pressure in these tissues.The distributions of fluid pressure in the pores for femur cartilage obtained from our calculations(Fig.5(a))are in good qualitative and quantitative agreement with the data presented in the literature [59](see Fig.5(a) on page 285).The distributions of fluid pressure in the pores for meniscus obtained from our calculations (Fig.5(b)) are also in good qualitative and quantitative agreement with the data presented in the literature[59](see Fig.2(a) on page 284).

3.Results and analysis

Different authors[60]report on the range from 0.05 mJ/mm2to 3 mJ/mm2for intensity values of shock wave (SW) impact on the knee joint, which has a regenerative effect.In this study, acoustic effect was studied in the range from 0.120 to 0.9 mJ/mm2and frequencyf= 1.5 MHz [61].We analyzed patterns of distribution of hydrostatic pressure,distortional strain,and fluid pressure in pores in accordance with mechanobiological principles [13].

3.1.Shock-wave impact on the model of knee joint with OA

In the initial stages,osteoarthritis is characterized by a change in the properties of cartilage and meniscus tissues.In the last stages,local areas of cartilage tissue abrasion appear which requires surgical treatment[62].It worth noting that the geometric parameters of cartilage and its contact area can change significantly due to generative processes such as OA, which, in turn, change the distribution of mechanical stresses[63].But herewith we restrict our study to considering only the corresponding changes in the physical and mechanical properties of the tissues, since they are the most important.For example, OA of the cartilage of the knee joint can lead to the degradation of the tissues of the meniscus [48].Degenerative changes in the elastic and poroelastic properties of meniscus tissues and cartilage plates can reach 50% [64,65].

Fig.4. Distributions of hydrostatic pressure (negative is compression) in the samples with different numbers of elements: (a) 123391; (b) 295679; (c) 509757; (d) 992776; (e)2356312.

Fig.5. Results of the model validation as distributions of fluid pressure in: (a) the femoral cartilage pores; (b) the meniscus.

Based on the above-mentioned,we investigated the mechanical behavior of the knee joint taking into account the initial stage of osteoarthritic changes in the joint tissues (Table 1).In this case,changes in the poroelastic properties of cartilage and meniscus tissues are taken into account, and degenerative changes in bone tissues that appear at later stages are not taken into account.

When analyzing the patterns of distribution of hydrostatic pressure obtained in the calculations, it was found that loading with an energy flux density of 0.12 mJ/mm2resulted in conditions favorable for osteogenesis and chondrogenesis (hydrostatic pressure above 3 kPa,but compressive stresses more 0.2 MPa promoted at chondrocyte differentiation and of the order of 0.7-0.8 MPa are most favorable for the formation of cartilage tissue)(Fig.6(a)).The pattern of distribution of fluid pressure within pores (Fig.8(a))indicates fulfillment of the conditions for the transfer of biological cells(the values of this parameter should be in the range from 40 to 2 MPa).However, the distribution pattern and the value of the amplitude of distortional strain (Fig.7(a)) indicate that there is no possibility of cartilage tissue regeneration over a large area.The necessary minimum level of distortions is observed in small regions(points) near the applicator.

The positive effect of low and middle intensity exposure is noted in Refs.[66-68].The authors of works[69,70]note that an increase in the intensity of loading contributes to an increase in the therapeutic effect.

Fig.6. Fields of hydrostatic pressure (negative is compression) in the model knee joint with OA at different EFD of SW impact (mJ/mm2): (a) 0.12; (b) 0.25; (c) 0.33; (d) 0.9.

Fig.7. Fields of distortional strain in the model knee joint with OA at different EFD of SW impact (mJ/mm2): (a) 0.12; (b) 0.25; (c) 0.33; (d) 0.9.

Under shock-wave loading with an energy flux density of 0.25 mJ/mm2,small areas begin to form in the cartilage plates with conditions for cartilage tissue regeneration [12,13]: the value of compressive stress exceeds 0.15 MPa(Fig.6(b)),and the distortional strain ranges from 0.05 to 1%(Fig.7(b))at pore fluid pressure values of more than 20 kPa(Fig.8(b)).Compressive stresses of maximum amplitude (0.2-1.5 MPa) are concentrated in the area of cartilaginous plates and meniscus near the loaded surface.The effectiveness in the treatment of OA by shock wave exposure of this amplitude was noted in the works of many authors[71,72].Under loading with an energy flux density higher than 0.3 mJ/mm2, vast regions are formed in the cartilage plates (Fig.5(c); Figs.6(c), Fig.7(c)) that correspond to the conditions for cartilage tissue regeneration.

As the energy flux density amplitude increases to 0.9 mJ/mm2,the hydrostatic pressure amplitude increases, but the distribution pattern remains the same (Fig.6(d)).In the region of maximum stresses, the values of distortional strain reach 0.6% (Fig.7(d)) and fluid pressure reaches 1 MPa(Fig.8(d)).In the bone tissues adjacent to the cartilaginous plate, stresses up to 0.2 MPa are observed,the fluid pressure within the pores is about 25-30 kPa.This pattern of the distribution of pressure and deformation indicates the fulfillment of conditions for the regeneration of cartilage and meniscal tissues,as well as bone tissues near the loaded surface according to Ref.[73].

In works [74,75]the authors note that high intensity loading(higher than 0.5 mJ/mm2) can lead to cartilage destruction.The compressive strength of cartilage tissue can vary from 10 to 50 MPa depending on the loading rate according to experimental data[76-78].Our results do not confirm this:the obtained distribution of hydrostatic pressure indicates the absence of conditions for the destruction of cartilaginous tissues (the maximum amplitude of compressive stresses does not exceed 1.7 MPa).

The simulation results indicate localization of the shock wave action in the region of the loading plate.With an increase in the amplitude of the impact,the volume in which conditions for tissue regeneration are fulfilled increases.In addition, conditions for chondrogenesis are observed in the tissues of the meniscus,which is also consistent with experimental data[79].Thus,the simulation results obtained herein are consistent with experimental data on the effectiveness of SW therapy with an energy flux density of more than 0.25 mJ/mm2for the joints affected by osteoarthritis [80,81].

3.2.Shock-wave impact on a model of total knee replacement

It should be noted that there are no exact studies on the effect of acoustic exposure after knee arthroplasty.Most studies were performed in combination with other types of therapy [82].

The parameter of distortional strain is included in the condition for regeneration of only fibrous and cartilage tissues.Therefore,when studying the conditions for the regeneration of biological tissues under acoustic exposure in the case of the model knee joint with a prosthesis,the distribution of distortional strain will not be analyzed.Another important distinction of the model knee joint with prosthesis is a resorption zone,which is identified around the prosthesis pin (Fig.2(d) and Fig.2(e)).

3.2.1.Shock-wave impact on a model of the knee joint without resorption

The model of total knee replacement (TKR) subjected to shock wave loading in the medial and lateral zones along the supporting pin of the tibial component, small local areas with compressive stresses with an amplitude in the range from 0.005 to 0.12 MPa,which correspond to possible osteogenesis,are observed(Fig.9(a)).

With an increase in the intensity of exposure,the amplitude and area of compressive stresses increase (Fig.9(b) - (Fig.9(d)).

On the pattern of distribution of fluid pressure within the pores(Fig.10) the required level (greater than 30 kPa) for starting the processes of bone tissue regeneration is reached in the specified regions.

Fig.8. Fields of pore fluid pressure in the model of knee joint with OA at different EFD of SW impact (mJ/mm2): (a) 0.12; (b) 0.25; (c) 0.33; (d) 0.9.

Fig.9. Fields of hydrostatic pressure(negative is compression)in the model of knee joint replacement at different EFD of SW impact(mJ/mm2):(a)0.12;(b)0.25;(c)0.33;(d)0.9.

Fig.10. Fields of pore fluid pressure in the model of TKR at different EFD of SW impact (mJ/mm2): (a) 0.12; (b) 0.25; (c) 0.33; (d) 0.9.

Under high-intensity loading,the area of these zones increases,and the maximum amplitudes of compressive stresses corresponding to the conditions of cancellous tissue regeneration also increase(Fig.9(b) - (Fig.9(d)).

3.2.2.Shock-wave impact on the model of knee joint replacement taking into account the bone resorption

Shock-wave loading of the model of the knee joint taking into account the zone of cancellous bone tissue resorption after arthroplasty leads to the distribution patterns of hydrostatic pressure and pore fluid pressure that differ considerably if compare with the model without resorption.Namely, there is a shift of the local regions of the maximum of the compressive stress towards the cortical shell of the tibia at the distance of the resorption zone width (Fig.11).The area of the zones corresponding to the conditions of tissue regeneration is significantly reduced compared to the model without resorption at the same energy flux density.In addition, the regeneration conditions in the resorption zone are achieved only with medium- and high-intensity SW exposure(Fig.12).

The results obtained indicate the possibility of therapeutic effect of SW exposure on the bone with a prosthesis; similar data were also obtained in works [83,84]which authors believe that shock wave therapy can help in pain-relieving and improving the patient's motor activity after knee arthroplasty.

Fig.11. Fields of hydrostatic pressure in the model of TKR with zone of tibia resorption at different EFD of SW impact (mJ/mm2): (a) 0.12; (b) 0.25; (c) 0.33; (d) 0.9.

Fig.12. Fields of pore fluid pressure in the model of TKR with resorption at different EFD of SW impact (mJ/mm2): (a) 0.12; (b) 0.25; (c) 0.33; (d) 0.9.

Thus, the results of our study show that SWT can be used to reduce the risk of resorption and reduce osteoporotic manifestations in cancellous tissue.At the same time, a hypothesis is currently being developed about the antibacterial effect of SWT in the implant zone, which is produced by the generation of bone tissues(osteoblasts)[81].In addition,SW therapy has proven itself to increase the rate of healing of fractures with stabilizing implants[85,86].

4.Conclusions

Existing experimental data indicate that the external shock wave impact has a regenerative effect.This article presents threedimensional numerical models of the knee joint affected with degenerative changes and subjected to total arthroplasty.Using computer simulation, we studied the therapeutic effect of shockwave exposure in the wide range of its intensity on the knee joint tissues.

According to the results of modeling the shock-wave effect on the knee joint affected with osteoarthritic changes, it was found that there are local (point) regions with conditions for cartilage tissue regeneration under the medium-intensity impact(0.125 mJ/mm2).Larger regions with conditions satisfying the processes of chondrogenesis are observed in the tissues of the cartilage plates and meniscus under high-intensity loading in the range from 0.25 to 0.9 mJ/mm2.It is worth noting the fact that under high-intensity loading (0.9 mJ/mm2) of the model joint, no stress is observed exceeding the tensile strength for cartilage tissue because work[69]indicates that loading with an intensity above 0.5 mJ/mm2can lead to cartilage damage in small mammals.Our study on the model sample corresponding to the human knee joint does not confirm this fact.

Based on the results of modeling the shock-wave effect on the knee joint replacement it was found that under moderate-intensive loading, small local regions) of compressive stresses are observed along the pin of the tibial component of the prosthesis.There are regions of localization of compressive stresses in the RO I and RO II zones with an increase in the intensity of exposure along the“boneimplant” interface.It was found that the zone with conditions for bone tissue regeneration, the maxima of the compressive stress pattern,are shifted from the pin by the thickness of the resorption zone when modeling the knee joint replacement taking into account the tibia bone resorption.However,the pattern of pore fluid pressure distribution remains the same.Our data are consistent with studies of bone tissue regeneration around implants during fracture healing and at arthroplasty.

For the first time,based on the results of computer simulation,it has been shown that shock wave impact on the knee joint affected with OA can result in conditions for the regeneration of tissues of the cartilage plates and meniscus.In the case of total knee replacement SW exposure contributes to the fulfillment of conditions for cancellous bone tissue regeneration in the region of the tibial component.The results of modeling the SW impact on the knee joint are in good agreement with the available experimental data from the literature.Therefore, the presented model can be successfully used to develop recommendations for determining the range of therapeutic SW exposure, taking into account the personalized characteristics of the patient.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The investigation has been carried out at financial support of the Russian Foundation for Basic Research, grant No.20-08-00818(simulation results) and the Government research assignment for ISPMS SB RAS, project FWRW-2021-009 (in-house software development).