Biomechanical analysis of staples for epiphysiodesis

  Citation: Frydrýšek, K.; ˇ Cepica, D.; Halo, T.; Skoupý, O.; Pleva, L.; Madeja, R.; Pometlová, J.; Losertová, M.; Koutecký, J.; Michal, P.; et al. Biomechanical Analysis of Staples for Epiphysiodesis. Appl. Sci. 2022,12, 614. https://doi.org/10.3390/ app12020614 Academic Editor: Claudio Belvedere Received: 9 December 2021 Accepted: 6 January 2022 Published: 9 January 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). applied sciences Article Biomechanical Analysis of Staples for Epiphysiodesis Karel Frydrýšek 1,2,* , Daniel ˇ Cepica 1,2, Tomáš Halo 1,2 , Ondˇrej Skoupý1,2, Leopold Pleva 1,3, Roman Madeja 1,3, Jana Pometlová1,3 , Monika Losertová4,5 , Jan Koutecký5, Pavel Michal 5, Vojtˇech Havlas 6, Šimon Kraus 6,7, Dominik ˇ Durica 7, Kateˇrina Peterek Dˇedková8, Marek Pagáˇc 9, Pavel Krpec 10 and Paweł Osemlak 11 1Institute of Emergency Medicine, Faculty of Medicine, University of Ostrava, Syllabova 19, 703 00 Ostrava-Vítkovice, Czech Republic; [email protected] (D. ˇ C.); [email protected] (T.H.); [email protected] (O.S.); [email protected] (L.P.); r[email protected] (R.M.); [email protected] (J.P.) 2 Department of Applied Mechanics, Faculty of Mechanical Engineering, VSB—Technical University of Ostrava, 17. Listopadu 2172/15, 708 00 Ostrava, Czech Republic 3Trauma Center, 17., University Hospital Ostrava, Listopadu 1790, 708 52 Ostrava-Poruba, Czech Republic 4Faculty of Materials Science and Technology, VSB—Technical University of Ostrava, 17. Listopadu 2172/15, 708 00 Ostrava, Czech Republic; [email protected] 5Medin, a.s, Vlachovicka 619, 592 31 NovéMˇesto na Moravˇe, Czech Republic; [email protected] (J.K.); [email protected] (P.M.) 6Motol University Hospital, V Úvalu 84, 150 06 Prague 5, Czech Republic; [email protected] (V.H.); [email protected] (Š.K.) 7Department of Orthopaedics, Second Faculty of Medicine, Charles University, V Úvalu 84, 150 06 Prague 5, Czech Republic; [email protected] 8 Center of Advanced Innovation Technologies, VSB—Technical University of Ostrava, 17. Listopadu 15/2172, 708 00 Ostrava-Poruba, Czech Republic; katerina.peter[email protected] 9Department of Machining, Faculty of Mechanical Engineering, VSB—Technical University of Ostrava, Assembly and Engineering Metrology, 17. Listopadu 2172/15, 708 00 Ostrava, Czech Republic; [email protected] 10 V-NASS, a.s., Halasova 2938/1a, 703 00 Ostrava-Vítkovice, Czech Republic; [email protected] 11 Pediatric University Hospital Named by Prof. Antoni G˛ebala in Lublin, ul. Prof. A. G˛ebali 6, Department of Pediatric Surgery and Traumatology, Medical University of Lublin, 20-093 Lublin, Poland; [email protected] *Correspondence: kar[email protected]; Tel.: +420-597323495; Fax: +420-596916490 Abstract: Limb asymmetry can, and often does, cause various health problems. Blount bone staples (clips) are used to correct such uneven growth. This article analyzes the performance of a biomechanical staple during bone (tibia) growth arrest. The staples considered in this study were made of 1.4441 stainless steel, the model of tibia consisted of two materials representing corticalis and spongiosis. Hooke’s law was used for modeling materials’ behaviors for finite element analysis (FEA). The maxima of stress and total staple displacement were evaluated using the finite element method and verification of the results, along with the determination of the maximum loading (growing) force that the staples are capable of withstanding, was performed experimentally. The presented method can be used to determine the safety and usability of staples for bone growth arrest. According to our results, the design of Blount staples considered in this paper is safe and suitable for orthopedic treatment. Keywords: biomechanics; orthopedics; Blount staple; FEA; experiment; epiphysiodesis 1. Introduction The growth deformities, one of which is Blount’s disease, are globally among the most common conditions to present in pediatric orthopedic clinics [ 1 , 2 ]. These deformities can, although rarely, be associated also with childhood obesity, i.e., high body mass index (BMI); for further information, see [3]. Appl. Sci. 2022,12, 614. https://doi.org/10.3390/app12020614 https://www.mdpi.com/journal/applsci Appl. Sci. 2022,12, 614 2 of 16 These growth deformities have been, for many years, treated surgically, utilizing the manipulation of natural growth capabilities of the bone; see [ 1 ]. Epiphyseal stapling is one of the most commonly and traditionally used methods for such correction, using inert metal staples (clips) implanted into a specific part of a child’s long bone to temporarily prevent its growth. Hence, epiphyseal stapling (also known as epiphysiodesis, Blount epiphysiodesis, bone growth restriction, or bone growth surgery) involves placing such staples in a way to bridge the growth plate to slow down the growth of the long bone; see [1,4]. This short surgical procedure is performed under general anesthesia. There are no comparable nonsurgical alternatives to epiphyseal stapling. Surgical alternatives include leg shortening (i.e., a surgery during which a section of the bone is cut out and fragments are joined together with a plate), percutaneous epiphysiodesis, see [ 5 ], open epiphysiodesis, and, recently, tension band technique. Epiphyseal stapling was introduced by Walter Blount in 1949, see [ 6 ], and since then, it has become a common procedure of correcting, in particular, angular deformities of the knee (genu varum or genu valgum) in children. The growth of the child’s or adolescent’s bone is associated mainly with physis, i.e., a cartilage structure near joints; see [ 1 , 4 ]. Blount staples are also used for pseudoarthrosis treatment [7]. Using staples, the physis (epiphyseal plate) can be relatively simply restrained either on both sides when correcting limb length discrepancy (i.e., “epiphysiodesis”) or only on one side when correcting angular deformities (i.e., “hemiepiphyseodesis”) (see Figures 1and 2 ). Unlike the irreversible method of permanent epiphysiodesis, see [ 8 ], epiphyseal stapling does not destroy the epiphyseal plate and, therefore, allows resumption of the growth once the optimal correction is achieved; see [4]. Appl. Sci. 2022, 12, x FOR PEER REVIEW 2 of 16 These growth deformities have been, for many years, treated surgically, utilizing the manipulation of natural growth capabilities of the bone; see [1]. Epiphyseal stapling is one of the most commonly and traditionally used methods for such correction, using inert metal staples (clips) implanted into a specific part of a child’s long bone to temporarily prevent its growth. Hence, epiphyseal stapling (also known as epiphysiodesis, Blount epiphysiodesis, bone growth restriction, or bone growth surgery) involves placing such staples in a way to bridge the growth plate to slow down the growth of the long bone; see [1,4]. This short surgical procedure is performed under general anesthesia. There are no comparable nonsurgical alternatives to epiphyseal stapling. Surgical alternatives include leg shortening (i.e., a surgery during which a section of the bone is cut out and fragments are joined together with a plate), percutaneous epiphysiodesis, see [5], open epiphysiodesis, and, recently, tension band technique. Epiphyseal stapling was introduced by Walter Blount in 1949, see [6], and since then, it has become a common procedure of correcting, in particular, angular deformities of the knee (genu varum or genu valgum) in children. The growth of the child’s or adolescent’s bone is associated mainly with physis, i.e., a cartilage structure near joints; see [1,4]. Blount staples are also used for pseudoarthrosis treatment [7]. Using staples, the physis (epiphyseal plate) can be relatively simply restrained either on both sides when correcting limb length discrepancy (i.e., “epiphysiodesis”) or only on one side when correcting angular deformities (i.e., “hemiepiphyseodesis”) (see Figures 1 and 2). Unlike the irreversible method of permanent epiphysiodesis, see [8], epiphyseal stapling does not destroy the epiphyseal plate and, therefore, allows resumption of the growth once the optimal correction is achieved; see [4]. Figure 1. Parts of the growing bone (tibia). Figure 1. Parts of the growing bone (tibia). Recently, the tension band technique, introduced by Stevens in 2007 [ 9 ], using nonlocking plates (similar to those used for osteosynthesis) and screws almost in the same position as staples, has gradually become a preferable alternative to stapling. However, Blount’s original method still remains an effective means for the treatment of lower limb deformities in adolescents; see [1,4,10]. According to [ 1 , 4 ], besides accurate diagnosis confirmed by a radiogram of the whole limb (see Figure 2), good timing of the treatment is also very important. The surgical procedure involves a short longitudinal incision through soft tissues over the physis and extraperiosteal implantation of the staple using a special instrument under radiography control [ 11 ] (see Figure 3). The staple must bridge the physis but not penetrate it to prevent its impairment (see Figure 4). The staples should not be restricting the physis for longer than 2 years to prevent permanent growth cessation [ 12 ]. Other complications during treatment, such as damaging the physis by imprecise staple implantation, mechanical failure of the staple (bending, rarely break), or staple migration can occur. The last one is also the most common complication and disadvantage compared with the tension band technique; see [1,4]. Appl. Sci. 2022,12, 614 3 of 16 Appl. Sci. 2022, 12, x FOR PEER REVIEW 2 of 16 These growth deformities have been, for many years, treated surgically, utilizing the manipulation of natural growth capabilities of the bone; see [1]. Epiphyseal stapling is one of the most commonly and traditionally used methods for such correction, using inert metal staples (clips) implanted into a specific part of a child’s long bone to temporarily prevent its growth. Hence, epiphyseal stapling (also known as epiphysiodesis, Blount epiphysiodesis, bone growth restriction, or bone growth surgery) involves placing such staples in a way to bridge the growth plate to slow down the growth of the long bone; see [1,4]. This short surgical procedure is performed under general anesthesia. There are no comparable nonsurgical alternatives to epiphyseal stapling. Surgical alternatives include leg shortening (i.e., a surgery during which a section of the bone is cut out and fragments are joined together with a plate), percutaneous epiphysiodesis, see [5], open epiphysiodesis, and, recently, tension band technique. Epiphyseal stapling was introduced by Walter Blount in 1949, see [6], and since then, it has become a common procedure of correcting, in particular, angular deformities of the knee (genu varum or genu valgum) in children. The growth of the child’s or adolescent’s bone is associated mainly with physis, i.e., a cartilage structure near joints; see [1,4]. Blount staples are also used for pseudoarthrosis treatment [7]. Using staples, the physis (epiphyseal plate) can be relatively simply restrained either on both sides when correcting limb length discrepancy (i.e., “epiphysiodesis”) or only on one side when correcting angular deformities (i.e., “hemiepiphyseodesis”) (see Figures 1 and 2). Unlike the irreversible method of permanent epiphysiodesis, see [8], epiphyseal stapling does not destroy the epiphyseal plate and, therefore, allows resumption of the growth once the optimal correction is achieved; see [4]. Figure 1. Parts of the growing bone (tibia). Figure 2. ( a ) Long radiogram of preoperative genu valgum and ( b ) consecutive correction with staples in femur; see [4]. Appl. Sci. 2022, 12, x FOR PEER REVIEW 3 of 16 Figure 2. (a) Long radiogram of preoperative genu valgum and (b) consecutive correction with staples in femur; see [4]. Recently, the tension band technique, introduced by Stevens in 2007 [9], using nonlocking plates (similar to those used for osteosynthesis) and screws almost in the same position as staples, has gradually become a preferable alternative to stapling. However, Blount’s original method still remains an effective means for the treatment of lower limb deformities in adolescents; see [1,4,10]. According to [1,4], besides accurate diagnosis confirmed by a radiogram of the whole limb (see Figure 2), good timing of the treatment is also very important. The surgical procedure involves a short longitudinal incision through soft tissues over the physis and extraperiosteal implantation of the staple using a special instrument under radiography control [11] (see Figure 3). The staple must bridge the physis but not penetrate it to prevent its impairment (see Figure 4). Figure 3. Implantation of staples. Figure 4. Position of staples bridging the physis in femur. The staples should not be restricting the physis for longer than 2 years to prevent permanent growth cessation [12]. Other complications during treatment, such as damaging the physis by imprecise staple implantation, mechanical failure of the staple (bending, rarely break), or staple migration can occur. The last one is also the most common complication and disadvantage compared with the tension band technique; see [1,4]. There is a lack of information regarding the biomechanical aspect of Blount’s staples; hence, one of the goals of our publishing is to fill the gap in this field. There are two main methods for solving biomechanical problems: • Numerical approach—(the main subject of this paper). • Experimental approach (used and described only marginally here). In this paper, the stress and deformation of staples during epiphysiodesis are evaluated by a numerical approach using finite element analysis (FEA). The finite element method (FEM) is a recognized instrument of numerical analysis widely used in Figure 3. Implantation of staples. Appl. Sci. 2022, 12, x FOR PEER REVIEW 3 of 16 Figure 2. (a) Long radiogram of preoperative genu valgum and (b) consecutive correction with staples in femur; see [4]. Recently, the tension band technique, introduced by Stevens in 2007 [9], using nonlocking plates (similar to those used for osteosynthesis) and screws almost in the same position as staples, has gradually become a preferable alternative to stapling. However, Blount’s original method still remains an effective means for the treatment of lower limb deformities in adolescents; see [1,4,10]. According to [1,4], besides accurate diagnosis confirmed by a radiogram of the whole limb (see Figure 2), good timing of the treatment is also very important. The surgical procedure involves a short longitudinal incision through soft tissues over the physis and extraperiosteal implantation of the staple using a special instrument under radiography control [11] (see Figure 3). The staple must bridge the physis but not penetrate it to prevent its impairment (see Figure 4). Figure 3. Implantation of staples. Figure 4. Position of staples bridging the physis in femur. The staples should not be restricting the physis for longer than 2 years to prevent permanent growth cessation [12]. Other complications during treatment, such as damaging the physis by imprecise staple implantation, mechanical failure of the staple (bending, rarely break), or staple migration can occur. The last one is also the most common complication and disadvantage compared with the tension band technique; see [1,4]. There is a lack of information regarding the biomechanical aspect of Blount’s staples; hence, one of the goals of our publishing is to fill the gap in this field. There are two main methods for solving biomechanical problems: • Numerical approach—(the main subject of this paper). • Experimental approach (used and described only marginally here). In this paper, the stress and deformation of staples during epiphysiodesis are evaluated by a numerical approach using finite element analysis (FEA). The finite element method (FEM) is a recognized instrument of numerical analysis widely used in Figure 4. Position of staples bridging the physis in femur. There is a lack of information regarding the biomechanical aspect of Blount’s staples; hence, one of the goals of our publishing is to fill the gap in this field. There are two main methods for solving biomechanical problems: •Numerical approach—(the main subject of this paper). •Experimental approach (used and described only marginally here). In this paper, the stress and deformation of staples during epiphysiodesis are evaluated by a numerical approach using finite element analysis (FEA). The finite element method (FEM) is a recognized instrument of numerical analysis widely used in engineer- Appl. Sci. 2022,12, 614 4 of 16 ing mechanics (see, e.g., [ 13 ]) and biomechanics. It has been previously used for various biomechanical tasks [14–16], including Blount staple applications in epiphysiodesis [17]. To verify the results of the numerical solution, i.e., to assess the usability of staples under the chosen loading (growing) force, and to find the maximum loading (growing) force that the staples can withstand, a simple experimental approach was also used in this paper. Experiments can be used in combination with FEA (as we do here or, e.g., in [ 18 ]), or experiments can serve as a standalone tool for simulation of reality, see, e.g., [19]. Our approach can be further used for another types or modifications of Blount’s staples, plates, and similar implantates. 2. Materials and Methods The bone growth occurs in the epiphyseal plate, where a new bone mass is created. Considering this fact, the simulation of the growing process turned out to be difficult. For this reason, we came up with a solution where the artificial bone (tibia) is cut in two at the position of the epiphyseal plate; furthermore, we assume that the bone grows predominantly in the direction of the bone axis (i.e., only oppositional growth is considered), which means that growth can be simulated by pulling the two bone segments away from each other. Models of the staples, both 3D CAD and physical, were provided by MEDIN, a.s.; see Figure 5and [20]. Appl. Sci. 2022, 12, x FOR PEER REVIEW 4 of 16 engineering mechanics (see, e.g., [13]) and biomechanics. It has been previously used for various biomechanical tasks [14–16], including Blount staple applications in epiphysiodesis [17]. To verify the results of the numerical solution, i.e., to assess the usability of staples under the chosen loading (growing) force, and to find the maximum loading (growing) force that the staples can withstand, a simple experimental approach was also used in this paper. Experiments can be used in combination with FEA (as we do here or, e.g., in [18]), or experiments can serve as a standalone tool for simulation of reality, see, e.g., [19]. Our approach can be further used for another types or modifications of Blount’s staples, plates, and similar implantates. 2. Materials and Methods The bone growth occurs in the epiphyseal plate, where a new bone mass is created. Considering this fact, the simulation of the growing process turned out to be difficult. For this reason, we came up with a solution where the artificial bone (tibia) is cut in two at the position of the epiphyseal plate; furthermore, we assume that the bone grows predominantly in the direction of the bone axis (i.e., only oppositional growth is considered), which means that growth can be simulated by pulling the two bone segments away from each other. Models of the staples, both 3D CAD and physical, were provided by MEDIN, a.s.; see Figure 5 and [20]. Figure 5. Physical model and 3D CAD model with main dimensions (mm), supplied by MEDIN, a.s. 2.1. Finite Element Analysis The numerical analysis is performed using the Ansys Workbench 2020 R2 sw; see [21]. Homogenous and isotropic material models are assumed to be good approximations of reality. Staples are made of biocompatible stainless steel 1.4441 (AISI 316L), see [4], and the artificial bone model consists of corticalis and spongiosis (i.e., the cortical and spongy parts); mechanical properties were taken from [22], where Young’s modulus for spongiosis was reported to range between 0.1 and 0.5 GPa and for corticalis between 12 and 18 GPa. From this, values closer to the upper limit were chosen; this can, for example, illustrate obesity (i.e., stronger bones to accommodate for higher body mass). Used material models are presented in Table 1. Table 1. Material models of bone and stainless steel. Material Young’s Modulus (GPa) Poisson’s Ratio (1) Yield Strength (MPa) Ultimate Strength (MPa) 1.4441 183 0.33 690 800 Corticalis 16.1 0.3 Spongiosis 0.4 0.3 Figure 5. Physical model and 3D CAD model with main dimensions (mm), supplied by MEDIN, a.s. 2.1. Finite Element Analysis The numerical analysis is performed using the Ansys Workbench 2020 R2 sw; see [ 21 ]. Homogenous and isotropic material models are assumed to be good approximations of reality. Staples are made of biocompatible stainless steel 1.4441 (AISI 316L), see [ 4 ], and the artificial bone model consists of corticalis and spongiosis (i.e., the cortical and spongy parts); mechanical properties were taken from [22], where Young’s modulus for spongiosis was reported to range between 0.1 and 0.5 GPa and for corticalis between 12 and 18 GPa. From this, values closer to the upper limit were chosen; this can, for example, illustrate obesity (i.e., stronger bones to accommodate for higher body mass). Used material models are presented in Table 1. Table 1. Material models of bone and stainless steel. Material Young’s Modulus (GPa) Poisson’s Ratio (1) Yield Strength (MPa) Ultimate Strength (MPa) 1.4441 183 0.33 690 800 Corticalis 16.1 0.3 Spongiosis 0.4 0.3 Appl. Sci. 2022,12, 614 5 of 16 Research by Halo et al. [ 4 ] focused on a simpler bone material model, considering the cortical part as the only material of the bone. In the current paper, however, we improved the bone material model by dividing it into corticalis and spongiosis parts. 2.1.1. CAD and FEM Model The used CAD model obtained from a 3D scan and the bone model used in the experiment are not 100% identical; nevertheless, they are sufficiently similar to allow experimental verification of the calculation results; see Figure 6. The model of the whole bone is not necessary for our purposes and, for this reason, only the proximal part of the tibia was used in this calculation. This proximal part was then “cut” in two at the site of the epiphyseal plate. The staples were virtually placed in the bone in the way they usually are during epiphysiodesis, i.e., in the general area bridging the physis. Appl. Sci. 2022, 12, x FOR PEER REVIEW 5 of 16 Research by Halo et al. [4] focused on a simpler bone material model, considering the cortical part as the only material of the bone. In the current paper, however, we improved the bone material model by dividing it into corticalis and spongiosis parts. 2.1.1. CAD and FEM Model The used CAD model obtained from a 3D scan and the bone model used in the experiment are not 100% identical; nevertheless, they are sufficiently similar to allow experimental verification of the calculation results; see Figure 6. The model of the whole bone is not necessary for our purposes and, for this reason, only the proximal part of the tibia was used in this calculation. This proximal part was then “cut” in two at the site of the epiphyseal plate. The staples were virtually placed in the bone in the way they usually are during epiphysiodesis, i.e., in the general area bridging the physis. Figure 6. 3D CAD model of the bone and staples with main dimensions (mm). The CAD model of staple provided by MEDIN a.s. contains notches, which are not suitable for FEA. For this reason, sharp edges were rounded; see Figure 7. However, these sharp edges are important for properly inserting the staple in a bone. Figure 7. 3D CAD model of the staple with rounded sharp edges (mm). Figure 6. 3D CAD model of the bone and staples with main dimensions (mm). The CAD model of staple provided by MEDIN a.s. contains notches, which are not suitable for FEA. For this reason, sharp edges were rounded; see Figure 7. However, these sharp edges are important for properly inserting the staple in a bone. Appl. Sci. 2022, 12, x FOR PEER REVIEW 5 of 16 Research by Halo et al. [4] focused on a simpler bone material model, considering the cortical part as the only material of the bone. In the current paper, however, we improved the bone material model by dividing it into corticalis and spongiosis parts. 2.1.1. CAD and FEM Model The used CAD model obtained from a 3D scan and the bone model used in the experiment are not 100% identical; nevertheless, they are sufficiently similar to allow experimental verification of the calculation results; see Figure 6. The model of the whole bone is not necessary for our purposes and, for this reason, only the proximal part of the tibia was used in this calculation. This proximal part was then “cut” in two at the site of the epiphyseal plate. The staples were virtually placed in the bone in the way they usually are during epiphysiodesis, i.e., in the general area bridging the physis. Figure 6. 3D CAD model of the bone and staples with main dimensions (mm). The CAD model of staple provided by MEDIN a.s. contains notches, which are not suitable for FEA. For this reason, sharp edges were rounded; see Figure 7. However, these sharp edges are important for properly inserting the staple in a bone. Figure 7. 3D CAD model of the staple with rounded sharp edges (mm). Figure 7. 3D CAD model of the staple with rounded sharp edges (mm). Appl. Sci. 2022,12, 614 6 of 16 The radius size of 0.25 mm, according to Figure 7, is quite small, because this part of the staple is relatively thin, and using bigger radius size (e.g., 0.5 mm) would result in near complete removal of this part. In this paper, we used only the bone model and staples for FEA. The reason for this acceptable simplification lies in the fact that the limb growth is primarily determined by the bone (or, more accurately, epiphyseal plate). The bone is intact (i.e., without fracture); therefore, the influence of the muscles, ligaments, menisci, and synovia on bone growth is negligible compared to the load on the bone. Muscles, ligaments, menisci, and synovia could play a small role in restricting the staple migration, but this effect is not noticeable in our study and hence is considered negligible. Thus, muscle and other tissues and fluids were omitted in this paper. The influence of anatomical parts in cavitas articularis (i.e., mentioned muscles, ligaments, menisci, and synovia) might play significant role in ambulation of patients with Blount staples, see [23]. The transformation of the CAD model into the FEM model is presented in Figure 8. Considering the complexity of the bone shape, the tibia was discretized by tetrahedral elements (SOLID187 in Ansys sw) with a global maximum size of 2.5 mm. The element size in holes for staples was locally refined to mirror the element size of the staples. In addition, a refinement to 0.5 mm was performed in a small circular area in the immediate vicinity of holes for staples. The global element size is relatively large as we are focusing on and evaluating only the staple response, not that of the bone. The corticalis and spongiosis FE meshes are continuously connected by nodes and elements sharing faces (i.e., conformal mesh achieved by shared topology function in Ansys SpaceClaim sw), see [21]. Appl. Sci. 2022, 12, x FOR PEER REVIEW 6 of 16 The radius size of 0.25 mm, according to Figure 7, is quite small, because this part of the staple is relatively thin, and using bigger radius size (e.g., 0.5 mm) would result in near complete removal of this part. In this paper, we used only the bone model and staples for FEA. The reason for this acceptable simplification lies in the fact that the limb growth is primarily determined by the bone (or, more accurately, epiphyseal plate). The bone is intact (i.e., without fracture); therefore, the influence of the muscles, ligaments, menisci, and synovia on bone growth is negligible compared to the load on the bone. Muscles, ligaments, menisci, and synovia could play a small role in restricting the staple migration, but this effect is not noticeable in our study and hence is considered negligible. Thus, muscle and other tissues and fluids were omitted in this paper. The influence of anatomical parts in cavitas articularis (i.e., mentioned muscles, ligaments, menisci, and synovia) might play significant role in ambulation of patients with Blount staples, see [23]. The transformation of the CAD model into the FEM model is presented in Figure 8. Considering the complexity of the bone shape, the tibia was discretized by tetrahedral elements (SOLID187 in Ansys sw) with a global maximum size of 2.5 mm. The element size in holes for staples was locally refined to mirror the element size of the staples. In addition, a refinement to 0.5 mm was performed in a small circular area in the immediate vicinity of holes for staples. The global element size is relatively large as we are focusing on and evaluating only the staple response, not that of the bone. The corticalis and spongiosis FE meshes are continuously connected by nodes and elements sharing faces (i.e., conformal mesh achieved by shared topology function in Ansys SpaceClaim sw), see [21]. Figure 8. FEM model of the bone and staples. Staples were discretized by a hex-dominant mesh (SOLID186 + some SOLID187 elements) with a global maximum element size of 0.5 mm. The element size was locally refined on radii and in the immediate vicinity of these areas; see Figure 9. Figure 8. FEM model of the bone and staples. Staples were discretized by a hex-dominant mesh (SOLID186 + some SOLID187 elements) with a global maximum element size of 0.5 mm. The element size was locally refined on radii and in the immediate vicinity of these areas; see Figure 9. The presented mesh in its final form was used for the final calculation and result evaluation. The sensitivity analysis started with a coarse mesh, and after each computation, a new, refined, mesh with half the element size of the previous mesh was created. This process was repeated until results for two different meshes were close (within a 1% margin of error); in this way, the mesh sensitivity analysis was performed. Additional information about FE mesh regarding the number of elements and nodes is presented in Table 2. Appl. Sci. 2022,12, 614 7 of 16 Appl. Sci. 2022, 12, x FOR PEER REVIEW 6 of 16 The radius size of 0.25 mm, according to Figure 7, is quite small, because this part of the staple is relatively thin, and using bigger radius size (e.g., 0.5 mm) would result in near complete removal of this part. In this paper, we used only the bone model and staples for FEA. The reason for this acceptable simplification lies in the fact that the limb growth is primarily determined by the bone (or, more accurately, epiphyseal plate). The bone is intact (i.e., without fracture); therefore, the influence of the muscles, ligaments, menisci, and synovia on bone growth is negligible compared to the load on the bone. Muscles, ligaments, menisci, and synovia could play a small role in restricting the staple migration, but this effect is not noticeable in our study and hence is considered negligible. Thus, muscle and other tissues and fluids were omitted in this paper. The influence of anatomical parts in cavitas articularis (i.e., mentioned muscles, ligaments, menisci, and synovia) might play significant role in ambulation of patients with Blount staples, see [23]. The transformation of the CAD model into the FEM model is presented in Figure 8. Considering the complexity of the bone shape, the tibia was discretized by tetrahedral elements (SOLID187 in Ansys sw) with a global maximum size of 2.5 mm. The element size in holes for staples was locally refined to mirror the element size of the staples. In addition, a refinement to 0.5 mm was performed in a small circular area in the immediate vicinity of holes for staples. The global element size is relatively large as we are focusing on and evaluating only the staple response, not that of the bone. The corticalis and spongiosis FE meshes are continuously connected by nodes and elements sharing faces (i.e., conformal mesh achieved by shared topology function in Ansys SpaceClaim sw), see [21]. Figure 8. FEM model of the bone and staples. Staples were discretized by a hex-dominant mesh (SOLID186 + some SOLID187 elements) with a global maximum element size of 0.5 mm. The element size was locally refined on radii and in the immediate vicinity of these areas; see Figure 9. Figure 9. Refined FE mesh: ( a ) Detail of the mesh in/around the hole for the staple; ( b ) Mesh of the staple; (c) Element size on staple (mm). Table 2. Number of FE elements and nodes. Part Number of FE Elements Number of FE Nodes Tibia Epiphysis 66,083 102,706 Metaphysis-Diaphysis 80,183 126,033 Staple Medial 132,292 438,008 Lateral 132,581 442,394 Total 411,139 1,109,141 2.1.2. Boundary Conditions The global coordinate system was oriented so that the Z-axis is parallel to the bone axis. As mentioned above, we assumed that the bone grows predominantly in the direction of the bone axis (i.e., oppositional growth) and, therefore, the alignment of the axes allows simple loading of the bone segments in the epiphyseal plate in the Zdirection. We assumed that for the growth to occur, the loading force Fz needs to match the weight of the person. As the bone growth happens in the early stages of life (childhood, adolescence), we have chosen our bone to come from an adolescent standing on one leg with a chosen 100 kg body weight equivalent to Fz = 980.7 N. In [ 24 ], the growing force was determined to be approximately 500 N, i.e., our force Fz was overestimated to err on the side of safety. The force boundary condition is illustrated in Figure 10. The distal end of the cut tibia is fully fixed (i.e., prescribed displacements are u x = u y = u z = 0); in the proximal part; there is a partial fixation (i.e., prescribed displacements u x = u y = 0) allowing for a movement in the Z direction. Deformation boundary conditions are shown in Figure 11. Apart from the force and deformation boundary conditions, frictional contacts between the staples and the bone must be considered. The Coulomb friction coefficient between stainless steel and bone ranges from approximately 0.25 to 0.7, according to [ 25 ]. The friction coefficient is highly dependent, among other things, on the surface quality of both bone and steel, hardness of bone, etc. In this analysis, the friction coefficient was set to 0.2 (estimated by an educated guess), giving the possibility for staples to migrate out of bone. The friction coefficient used in this study is lower than in [ 25 ], taking into account the body fluids and tissues reducing the friction. Nevertheless, even with the low friction coefficient used in our study, the displacement in contact areas was very small, so friction does not have a major effect on stress distribution. Appl. Sci. 2022,12, 614 8 of 16 Appl. Sci. 2022, 12, x FOR PEER REVIEW 7 of 16 Figure 9. Refined FE mesh: (a) Detail of the mesh in/around the hole for the staple; (b) Mesh of the staple; (c) Element size on staple (mm). The presented mesh in its final form was used for the final calculation and result evaluation. The sensitivity analysis started with a coarse mesh, and after each computation, a new, refined, mesh with half the element size of the previous mesh was created. This process was repeated until results for two different meshes were close (within a 1% margin of error); in this way, the mesh sensitivity analysis was performed. Additional information about FE mesh regarding the number of elements and nodes is presented in Table 2. Table 2. Number of FE elements and nodes. Part Number of FE Elements Number of FE Nodes Tibia Epiphysis 66083 102706 Metaphysis-Diaphysis 80183 126033 Staple Medial 132292 438008 Lateral 132581 442394 Total 411139 1109141 2.1.2. Boundary Conditions The global coordinate system was oriented so that the Z-axis is parallel to the bone axis. As mentioned above, we assumed that the bone grows predominantly in the direction of the bone axis (i.e., oppositional growth) and, therefore, the alignment of the axes allows simple loading of the bone segments in the epiphyseal plate in the Z direction. We assumed that for the growth to occur, the loading force Fz needs to match the weight of the person. As the bone growth happens in the early stages of life (childhood, adolescence), we have chosen our bone to come from an adolescent standing on one leg with a chosen 100 kg body weight equivalent to Fz = 980.7 N. In [24], the growing force was determined to be approximately 500 N, i.e., our force Fz was overestimated to err on the side of safety. The force boundary condition is illustrated in Figure 10. Figure 10. Force boundary condition—forces Fz (equivalent to 100 kg) acting on the epiphyseal plate. Appl. Sci. 2022, 12, x FOR PEER REVIEW 8 of 16 Figure 10. Force boundary condition—forces Fz (equivalent to 100 kg) acting on the epiphyseal plate. The distal end of the cut tibia is fully fixed (i.e., prescribed displacements are ux = uy = uz = 0); in the proximal part; there is a partial fixation (i.e., prescribed displacements ux = uy = 0) allowing for a movement in the Z direction. Deformation boundary conditions are shown in Figure 11. Figure 11. Deformation boundary conditions. Apart from the force and deformation boundary conditions, frictional contacts between the staples and the bone must be considered. The Coulomb friction coefficient between stainless steel and bone ranges from approximately 0.25 to 0.7, according to [25]. The friction coefficient is highly dependent, among other things, on the surface quality of both bone and steel, hardness of bone, etc. In this analysis, the friction coefficient was set to 0.2 (estimated by an educated guess), giving the possibility for staples to migrate out of bone. The friction coefficient used in this study is lower than in [25], taking into account the body fluids and tissues reducing the friction. Nevertheless, even with the low friction coefficient used in our study, the displacement in contact areas was very small, so friction does not have a major effect on stress distribution. 2.2. Experiment The experiment was conducted to support the FEA and clinical applications, i.e., to determine the maximum loading force Fz of the bone that the staples can withstand and to partially confirm the findings of the numerical analysis. However, as the experiments are not the main goal of this article (the main goal is FEA), they were performed only once on an anatomical artificial bone [26] and once using certified bone foam blocks [27] and will be only briefly described (see Section 4: Results of the Experiment). One of the mentioned experiments was performed on artificial bones of the SAWBONES brand. The bones are made of composite material mimicking the properties of a real human bone, i.e., they are suitable for experimental purposes. For the use of composite bone models in experimental testing, see, e.g., [28,29]. The experiment was based on the same principles and assumptions as those used in the presented FEA. The full body of the artificial tibia was cut to obtain only the proximal part, which was subsequently split into two segments at the site of the epiphyseal plate. Both bone segments were mechanically adjusted to allow for the use of a jig. The jig consisted of a screw with a washer and nut attached to the upper bone segment (epiphysis) and of a self-tapping screw holding the lower bone segment (metaphysis-diaphysis). Staples were inserted into the bone segments (in a similar location as in FEA), bridging the epiphyseal plate. Bone segments were then pulled away from each other using the jig. Figure 11. Deformation boundary conditions. 2.2. Experiment The experiment was conducted to support the FEA and clinical applications, i.e., to determine the maximum loading force Fz of the bone that the staples can withstand and to partially confirm the findings of the numerical analysis. However, as the experiments are not the main goal of this article (the main goal is FEA), they were performed only once on an anatomical artificial bone [ 26 ] and once using certified bone foam blocks [ 27 ] and will be only briefly described (see Section 4: Results of the Experiment). One of the mentioned experiments was performed on artificial bones of the SAWBONES brand. The bones are made of composite material mimicking the properties of a real human bone, i.e., they are suitable for experimental purposes. For the use of composite bone models in experimental testing, see, e.g., [28,29]. Appl. Sci. 2022,12, 614 9 of 16 The experiment was based on the same principles and assumptions as those used in the presented FEA. The full body of the artificial tibia was cut to obtain only the proximal part, which was subsequently split into two segments at the site of the epiphyseal plate. Both bone segments were mechanically adjusted to allow for the use of a jig. The jig consisted of a screw with a washer and nut attached to the upper bone segment (epiphysis) and of a self-tapping screw holding the lower bone segment (metaphysis-diaphysis). Staples were inserted into the bone segments (in a similar location as in FEA), bridging the epiphyseal plate. Bone segments were then pulled away from each other using the jig. Figure 12a shows a schematic drawing of the experiment. Figure 12b shows the actual experiment. Appl. Sci. 2022, 12, x FOR PEER REVIEW 9 of 16 Figure 12a shows a schematic drawing of the experiment. Figure 12b shows the actual experiment. Figure 12. (a) A schematic drawing of the experiment (dimensions in mm); (b) Actual implementation of the experiment. In the schematic drawing, see Figure 11a, dimension “A” details the distance of the epiphyseal plate from the top of the bone and the dimension “MAX. 90” is related to the limits of the used testing machine. The upper bone segment in Figure 11b was wrapped in duct tape to facilitate manipulation before and during the experiment. Used equipment: • Model of tibia—SAWBONES, Tibia, 4th Gen., Composite, 17 PCF Solid Foam Core; see [27]. • Staples—provided by MEDIN, a.s.; see [20]. • Jig—M12 screw, M12 nut, washer (inner diameter 12 mm), ST12 self-tapping screw, all provided by MEDIN, a.s. • Universal testing machine—TESTOMETRIC M500-50CT; see [30]. The experiment was conducted using deformation-controlled loading with a constant rate of jaw separation set to 10 mm/min. 3. Results of FEA FEA was performed as described in Section 2: Materials and Methods. The distribution of equivalent stress (von Mises) in the staples was determined from the simulation of the bone growth restriction. The maximum stress occurs in the staple radius, see Figures 13 and 14. Figure 12. ( a ) A schematic drawing of the experiment (dimensions in mm); ( b ) Actual implementation of the experiment. In the schematic drawing, see Figure 11a, dimension “A” details the distance of the epiphyseal plate from the top of the bone and the dimension “MAX. 90” is related to the limits of the used testing machine. The upper bone segment in Figure 11b was wrapped in duct tape to facilitate manipulation before and during the experiment. Used equipment: • Model of tibia—SAWBONES, Tibia, 4th Gen., Composite, 17 PCF Solid Foam Core; see [27]. •Staples—provided by MEDIN, a.s.; see [20]. • Jig—M12 screw, M12 nut, washer (inner diameter 12 mm), ST12 self-tapping screw, all provided by MEDIN, a.s. •Universal testing machine—TESTOMETRIC M500-50CT; see [30]. The experiment was conducted using deformation-controlled loading with a constant rate of jaw separation set to 10 mm/min. 3. Results of FEA FEA was performed as described in Section 2: Materials and Methods. The distribution of equivalent stress (von Mises) in the staples was determined from the simulation of the bone growth restriction. The maximum stress occurs in the staple radius, see Figures 13 and 14. The total displacement of staples is presented in Figure 15. In our case, the highest total displacement is at the top end of the staples and the maximum displacement is higher up in the medial staple than in the lateral one. Based on the detected deformation, we can measure the distance between the bone segments from the epiphyseal plate to obtain a rough estimation of how much the bone could grow with the staples applied. Figure 16 shows the average maximum possible growth distance between both bone segments. The acquired FEA results are summarized in Table 3. Appl. Sci. 2022,12, 614 16 of 16 33. 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