Computational mechanobiological model combining epiphyseal, apophyseal, and appositional growth and inner bone remodeling of the juvenile femur

Computational mechanobiological model combining epiphyseal, apophyseal, and appositional growth and inner bone remodeling of the juvenile femur | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Computational mechanobiological model combining epiphyseal, apophyseal, and appositional growth and inner bone remodeling of the juvenile femur Andreas Lipphaus, Ralf-Bodo Tröbs, Matthias Klimek, Sascha Selkmann, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7224633/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 11 You are reading this latest preprint version Abstract In silico models for simulating bone growth based on mechanical or non-mechanical epigenetic factors are widely used. In this study, a well-known mechanobiological model, which states that octahedral shear stress accelerates longitudinal bone growth and hydrostatic stress retards it, is applied to a finite element model of the femur of an 8-year-old boy. Proximal and distal epiphyseal plates as well as the growth plate of the greater trochanter, cartilaginous growth at the femoral isthmus, and appositional bone growth are included in the model. Furthermore, changes in the density of the cancellous bone in the metaphyses are modeled based on Wolff's law using compressive stresses as the mechanical stimulus. Muscle forces during a dynamic gait cycle were determined for nine discrete loading cases by optimizing to minimize bending stress. The highest stresses in the femoral shaft were determined as medial compressive stresses with a maximum of -33.2 MPa. Highest internal axial load in the shaft was 985 N during loading response. The simulated bone growth resulted in an increase in femur length of 26 mm and a decrease in femoral neck angle by -0.4°, anteversion angle by -1.7°, articulo-trochanteric distance by 1 mm and lateral distal femur angle by -1.9° per year. The bone remodeling led to an increase in bone density, particularly in the medial proximal metaphysis. The consideration of different growth mechanisms allowed a comprehensive simulation of femoral growth with high agreement with anthropometric data. Possible applications are the simulation of the correction of deformities. biomechanics finite element bone growth endochondral ossification growth plate femoral neck isthmus in silico modeling pediatric orthopedics Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 1. Introduction In children, longitudinal bone growth occurs at the cartilaginous epiphyseal plates by endochondral ossification [ 1 ]. These growth plates are located at the end of the long bones between the metaphysis and the epiphysis and can be divided into horizontal zones. Next to the epiphysis, the resting zone contains stem-like cells and secretes morphogens influencing the alignment of the proliferative chondrocytes and inhibiting their terminal differentiation [ 2 ]. The finite proliferative capacity of resting chondrocytes leads to growth deceleration and finally epiphyseal fusion by growth plate senescence as children age [ 3 ]. The resting zone is followed by the proliferation zone, where chondrocytes rapidly divide, form columns of flattened, disc-shaped chondrocytes, and synthesize new matrix [ 4 ]. Towards the end of the proliferative zone, chondrocytes enlarge and take a more spherical shape forming the hypertrophic zone [ 5 ]. Two main hypotheses of ossification at the end of the hypertrophic zone involve either the apoptosis of hypertrophic chondrocytes followed by invasion of osteoblast precursors by the vascular system or transdifferentiation of chondrocytes to osteoblasts [ 6 ]. Appositional growth increases the thickness or diameter of the bone. In this mechanically stimulated process, osteoblasts deposit new matrix on the periosteal and endosteal surfaces and differentiate into osteocytes [ 7 ]. Bone growth in children is orchestrated by various mechanical [ 8 , 9 ] and non-mechanical factors like inflammation [ 10 ], growth [ 11 ] and sex hormones [ 12 ], and genetics [ 13 ]. Several studies investigated the rate of longitudinal growth in the growth plates as a function of static or dynamic loading. The Hueter-Volkmann law states that epiphyseal growth is increased by tensile stress and retarded by compressive stress acting at the growth plate [ 14 ]. While in some animal models only minor femoral length differences are reported due to different physiological dynamic loadings [ 15 ], a growth retardation of 52–63% is observed in paralyzed embryonic chicks due to a lack of mechanical stimulation [ 16 ]. In animal models, a linear relationship between sustained tensile and compressive stresses and acceleration and deceleration of growth was observed [ 17 ]. This effect is due to reduced numbers of both proliferative and hypertrophic chondrocytes and increases approximately linearly with time [ 18 ]. The effect of cyclic or intermittent loading is still not fully understood: While some studies found no significant bone length differences with different exercises in rats [ 19 ], others reported an increase in the length of radius and ulna in the dominant arm of professional tennis players [ 20 ]. Smith experimentally determined the principal stress directions in the epiphyses and concluded that the stress trajectories often are orthogonal to the direction of bone growth [ 21 ]. A second hypothesis is that due to the low stiffness of the cartilage under load, a deformity occurs and that growth proceeds in the direction of the distortion of the cartilage columns as long as this is too small to lead to tissue damage [ 22 – 24 ]. Pauwels proposed that deformities, especially after fractures, lead to asymmetric growth by uneven loading of the epiphyseal plate and thereby works to correct these deformities [ 25 ]. Carter and Wong proposed a mechanobiological model that epiphyseal growth is inhibited by dynamic hydrostatic compression and accelerated by octahedral shear stress. They defined the osteogenic index as the sum of maximum octahedral shear stress and minimum hydrostatic stress overall load cases and calculated the resulting growth as the product of the osteogenic index and the mechanical growth rate [ 26 ]. Endochondral growth primarily causes functional adaptations through angular adaptations till the fusion of the growth plates. In contrast, bone formation and resorption optimize bone density and volume, especially in childhood, but also in adults. Appositional growth occurring at the periosteum is influenced by local strains [ 27 ]. Higher loading is associated with greater periosteal growth [ 28 ]. Thereby, the bone's diameter increases, and not only the axial strength but in particular the strength of bone in bending and torsion is increased. Indeed, previous finite element simulations suggested that torsional load is the most important stimulus in forming the medullary cavity [ 29 ]. In adults, periosteal formation is seen in regions stressed with more than 20 MPa [ 30 ] or 1500 µε [ 31 ]. Above a threshold of about 4000 µε overload resorption occurs and bone density decreases. In growing bones exercise leads to a higher cortical area, while disuse results in decreased cortical area compared to normal developing bones [ 32 ]. However, due to the lower Young´s modulus, higher deformations occur [ 33 ]. In addition, the threshold of bone formation is lower. A study in adolescent rats has reported significant and prolonged bone formation with strains of 850 µε [ 34 ]. Overall, mechanically controlled growth patterns allow the bone to adapt to its mechanical needs. Thereby, bending stresses can be reduced to achieve lightweight design [ 30 , 35 , 36 ]. Based on finite element models, several computational approaches were established to simulate bone growth due to metabolic and mechanobiological signaling. Early models depicted either the biochemical or the mechanobiological component of growth. Heegard et al. used a pure mechanobiological model to simulate prenatal joint development [ 37 ], bone straightening [ 38 ], and epiphyseal growth of the distal femur [ 39 ]. Based on this, the development of bone deformities resulting from pathologies such as cerebral palsy [ 40 ], cam deformity [ 41 ], and developmental dysplasia of the hip [ 42 – 44 ] has been simulated. A similar approach was developed by Stokes et al. and uses tensile or compressive axial stresses to describe the longitudinal bone growth [ 45 ]. Alonso et al. presented a mechanobiological model simulating epiphyseal growth and remodeling after virtual implantation of staples for hemiepiphysiodesis [ 46 ]. In the latest models, reaction-diffusion-equations of morphogens and mechanobiological approaches were combined to model the embryonic development of synovial joints and the femur [ 47 – 49 ]. Also, changes in bone density of the trabecular bone in the proximal femur of children have been modeled [ 50 , 51 ], resulting in an increase in bone mineral density on the medial side of the femoral neck [ 52 ]. The proximal femur has three growth plates: the growth plate located between the metaphysis and the femoral head, the trochanteric growth plate, and the femoral neck isthmus [ 53 ]. Because chondral growth is faster than periosteal growth, the cartilage of the femoral neck isthmus is necessary to compensate for length growth and shape the femoral metaphysis. The growth of the trochanteric apophysis affects the statics of the hip joint as the hip resultant force is mainly defined by the body weight and the hip abductors attached to the greater trochanter. The force resultant at the greater trochanter is formed by forces through a muscle sling of the vastus lateralis with the opposing gluteus medius and minimus and the iliotibial band [ 54 ], reducing bending in both the femoral shaft and near the hip joint. Growth of the greater trochanter results in a reduction of the femoral neck-shaft angle [ 55 ], and subsequently, epiphysiodesis of the greater trochanter has been described as a treatment for infantile coxa vara [ 56 ]. The morphological changes during growth allow the bone to maintain physiological loading and a constant cartilage load [ 57 ]. Current in silico models either focus on very early phases of bone development or focus on a single growth mechanism. For transition into clinical practice, a more comprehensive model is needed. As the greater trochanter plays an important role in the biomechanics of the hip, it is of interest to include its growth in in silico models of bone development. Therefore, this study aims to develop a mechanobiological model of the growth of the juvenile femur including all growth plates, appositional growth of the femoral shaft, and inner remodeling. Stresses and internal axial loading are calculated during gait and muscle forces are optimized to reduce bending stresses. For validation, growth predicted by the in silico model is compared to several in vivo anthropometric measurements. 2. Results 2.1. Optimization of muscle forces Before the growth simulation was initiated, muscle forces were calculated using a bending minimization approach (Table 1 ). Two peaks in joint reaction forces, the first during loading response and the second during terminal stance (Fig. 1 ) were associated with peaks in muscle forces of the gluteal muscles and the iliopsoas and iliacus muscles respectively. Table 1 Resultant muscle forces in Newton (N) for all nine load cases in an 8-year-old boy using a bending minimization approach. Muscle forces are rounded to full numbers. Forces below 1 N are considered as no muscle activation (-). Additionally, resulting axial loading is given for all load cases. Muscle/Load case 1 2 3 4 5 6 7 8 9 Adductor brevis 4 - - - - - - - 32 Adductor longus 1 - - - - - - 2 34 Adductor magnus 10 52 - - - - - - - Biceps femoris 45 - - - - 18 96 64 37 Gastrocnemius lateralis 11 - - - - 55 125 37 - Gastrocnemius medialis 9 - - - - 123 372 32 - Gemellus - 2 6 5 3 9 9 - 12 Gluteus maximus 11 188 235 152 35 2 - - - Gluteus medius - 131 609 368 282 266 206 109 23 Gluteus minimus - 10 64 27 27 59 60 57 12 Iliacus - - - - - 137 880 524 244 Pectineus - - - - - - - - 5 Piriformis - 14 62 17 8 29 67 12 29 Psoas - - - - - 3 208 64 169 Quadratus femoris 11 4 2 - - - - - - Vastus intermedius - 23 47 28 13 - - - - Vastus lateralis - 100 191 94 16 - - - - Vastus medialis - 45 86 43 8 - - - - Axial loading 316 727 985 575 586 656 924 592 206 Finite element analysis of the physiological superposition of all nine load cases in the initial model of the femur of an 8-year-old shows that 3rd principal stresses are highest in the proximal femur on the medial side (Fig. 2 a, b). In contrast, compression in the distal femur is mostly homogenously distributed. In the shaft, compression is minimal on the anterior (-10.9 MPa) and lateral side (-11.7 MPa), whereas higher values are predicted on the posterior (-21.1 MPa) and medial (-33.2 MPa) side. 1st principal stresses are opposite to the compressive stresses (Fig. 2 c, d). Values are generally lower with tensile stresses of 10 MPa anterior, 21.1 MPa lateral, 6.6 MPa posterior, and 5 MPa medial. Before optimization of muscle forces but including the iliotibial band and using physiological boundary conditions bending stresses are 24 MPa in the sagittal and 24.9 MPa in the frontal plane. After optimization, bending stresses in the frontal plane decrease by 78% while increase in the frontal plane by 9%. Loading results in axial forces between 200 and 985 N at midshaft with minimal shear loading. Cross sections show corresponding stresses in the trabecular bone (Fig. 3 ). In the frontal section, the highest compression is observed in the proximal femur on the medial side, while tension peaks at the lateral side. As in cortical bone, the trabecular bone of the distal femur is more uniformly stressed. Of note, the compression near the distal growth plate is slightly higher anterior and lateral. Other than that, in the sagittal section compressive and tensile stresses are mostly homogenously distributed. 2.2. Simulation of physiological growth The predicted growth direction of the proximal femur is highly dependent on Young´s modulus of the growth plate. We find a value of 10 MPa best matching anthropometric data with lower values resulting in an unphysiologically high decrease in anteversion angle. The growth rate as the sum of the biological baseline and the osteogenic index is calculated for all growth plates (Fig. 4 ). The growth plate of the proximal femur exhibits a centric distribution of the growth rate. In the growth plate of the distal femur, the growth rate shows a homogenous distribution The simulation predicted an increase in both bone length and diameter in accordance with anthropometric data (Table 2 ). The ATD shows a decrease of 0.08 mm per month. The NSA decreased on average by 0.03° per month, the anteversion angle by 0.14°. The LDFA decreased by 0.16° per month. Table 2 Changes in the morphometries per year as predicted by the in silico model compared to anthropometric data Measurement In silico model Anthropometric data Bone length + 26.6 mm + 20.8 mm [ 58 ] Neck-shaft angle − 0.4° -1.1° [ 59 ] Articulo-trochanteric distance − 1 mm 0 mm [ 60 ] Lateral distal femoral angle − 1.9° 0° [ 61 ] Shaft diameter (Anterior-Posterior/Medial-Lateral) 0.9 mm/1.1 mm 1 mm [ 62 ] Anteversion angle − 1.7° -0.7° [ 63 ] A second model without modeling the growth of the greater trochanter reduced the decrease of the NSA to 75%. In our model, closure of the apophyseal plate of the greater trochanter results in a higher NSA, matching in vivo data [ 55 , 56 ]. Compressive stresses are highest at the medial side of the femoral shaft (Fig. 5 , a). The bone apposition rate is 20, 40, 20, and 60 µm/month anterior, posterior, lateral and medial. This results in a decrease of bending stresses in the sagittal plane of 9% while bending in the frontal plane is essentially unchanged. Inner remodeling predicted an increase in bone density on the medial side of the proximal femur given by an increase of Young´s modulus (Fig. 5 , b). These areas correspond to regions exhibiting higher compression. 3. Discussion Several computational models of femoral long bone growth have been presented in the literature [ 64 – 69 ]. These models provide valuable insights into the principles of the mechanobiology of bone growth. Furthermore, the development of various pediatric deformities can be explained [ 40 – 44 ]. However, they are limited to certain aspects of growth. Complex deformities, such as resulting from fractures, are corrected by a combination of altered growth of all growth plates and bony remodeling [ 70 ]. This study investigates stresses acting on the juvenile femur during gait using finite element analysis. A well-established mechanobiological model that uses octahedral shear stresses and hydrostatic stresses as the mechanical input values is employed to calculate the endochondral growth of the proximal and distal femur as well as that of the greater trochanter. Appositional growth of the femoral shaft and inner remodeling of the trabecular bone are also included. The calculated stresses depend to a large extent on the boundary conditions. The question of bending stress on bones has long been the subject of controversial debate in biomechanics. For adults, Sverdlova and Witzel [ 71 ] and later Lutz et al. [ 72 ] showed that the femoral shaft is primarily subjected to compressive stresses through muscle activation. This allows weight to be saved, which represents an evolutionary advantage [ 30 ]. On the other hand, Bertram and Biewener [ 73 ] argue that residual bending allows the direction of bending to be predicted. In this study inertia relief, muscle force optimization, and the iliotibial band are combined to model physiological boundary conditions. The stress analyses of the juvenile femur reveal reduced bending stresses in the sagittal plane. Nevertheless, bending in the frontal plane cannot be compensated for entirely. The highest appositional growth is predicted for the medial site, but the decrease in the NSA and the growth of the femoral neck result in a more eccentric hip joint resultant. In this simulation the functional adaption and compensate the additional bending, but is not able to further reduce bending stresses in the frontal plane. It can be assumed that functional adaption and muscles forces mainly reduce bending in the sagittal plane whereas the iliotibial tract is the primary mechanism for reducing bending stresses in the frontal plane. However, with a modulus of elasticity of 12000 MPa for cortical bone, the bones in children and adolescents are significantly more elastic than in adults and its elasticity allows greater bending until material failure. Bones become stiffer as they grow. At the same time, the simulation of bone growth predicts functional changes that reduce bending stresses. The changes in bone morphology during growth can therefore be understood as adaptions to mechanical loading. In the proximal femur, peak compressive stresses are found medially and result in an increase in bone density similar to previous studies [ 52 ]. This shifts the center of mass of the neck of the femur medially towards the resulting axial force. In the diaphysis, internal axial loading is between 200 and 985 N during gait, while shear loading is considerably lower. This is in agreement with comparable studies in adults [ 74 ], whereas the weight-adapted axial load is higher. They have important implications in the healing processes of pediatric femoral shaft fractures, as axial loading can enhance fracture healing, while shear loading decelerates fracture healing [ 75 ]. The values for changes in bone length, diaphyseal diameter, NSA, and anteversion agree well with anthropometric values. However, the decline in the LDFA is not observed during adolescent growth. A possible explanation might be that the ring of LaCroix, which is not modeled, reduces the deformity of the cartilage in the growth plate. The predicted changes are also a mechanism of functional adaptation, as a decline in the NSA is associated with lower loading on the proximal femur [ 76 ]. Growth of the greater trochanter and the femoral neck isthmus could preserve the morphology of the metaphysis during longitudinal growth. The slight increase in ATD not observed in human growth indicates that either the apophyseal plate is to be modeled with a greater thickness or be orientated more horizontally. The highest increase in cortical thickness is predicted on the medial side of the diaphysis. Indeed, the femoral shaft has a higher thickness on the medial compared to the lateral side [ 77 ]. However, the highest cortical bone thickness is observed posterior. Due to the curvature of the femur in activities resulting in bending of the shaft, high compressive stresses can be expected on the posterior side. As the physiological loading of the bone was simplified as gait, higher impact loading like running and jumping are probable causes of bone formation on the concave side. In conclusion, it could thus be shown that the mechanobiological model of bone growth presented here is suitable for reproducing the physiological growth of the femur with some limitations. Other limitations of the model include the assumption of constant muscle strength. However, as the angle of the femur changes, the lever arms of the muscles also change, and thereby the vector of the muscle force. Furthermore, the increase in weight leads to an increase in ground reaction force and a higher level of muscle strength due to muscle build-up. Secondarily, this also results in altered hip and knee joint forces. Further developments of the model should therefore include a stronger coupling of the calculations of bony and muscular growth. As discussed above, the simulated load only includes the gait cycle and therefore does not include the maximum loads that occur during sport and play. However, frequent loads are particularly relevant for the morphological adaptation of the bones to the stresses placed on them and gait is subject to less inter-individual variation than sporting activities. These reasons support the choice of gait as a functional load. Also, the cartilaginous growth at the surface of the femoral head and the greater trochanter was not included and periosteal growth was only considered for the diaphysis. Material models are simplified as linear elastic and isotropic and the geometry is simplified. Yadav et al. [ 69 ] were able to demonstrate that modeling of patient-specific growth plate geometry increased the accuracy of the predictions. Future research using medical images to develop a patient-specific model and compare the in silico results with the respective radiological follow-up may allow a more detailed validation. The model contains a constant biological baseline, which in reality is influenced by hormones and cellular senescence. For clinical decision-making, it is often necessary to consider growth till its completion. Further research should aim to simulate a longer period of growth. Including the effects of sex hormones and cellular senescence on the growth plate might allow for modeling fusion of the growth plates. However, in the current model growth is simulated by updating the location of the nodes. This only allows the simulation of growth over a few months, otherwise element distortions will occur. Manipulating a volume model and remeshing instead of directly manipulating the mesh might be a feasible solution to this problem [ 46 ]. 4. Conclusions Bone growth in children does not only lead to an increase in length and thickness but also to complex three-dimensional changes in shape. The implementation of all growth plates and periosteal growth in computational simulations of bone growth can contribute to a better understanding of the interaction between morphological changes and functional loads. Optimizing muscle strength and functional adaptation together reduce bending stresses in growing bone in this in silico model. Future research should aim to simulate bone growth in complex deformities and might be a tool to predict bone remodeling after fractures. 5. Materials and Methods A finite element mesh of the adolescent right femur, made freely available at SimTK by Kainz et al. [ 64 ] based on MRI data of a typically developed child (8 years old, weight: 20.4 kg, height: 1.24 m), was imported into ANSYS Mechanical version 19.2 (ANSYS Inc, Canonsburg, PA, USA). The modeling approach has previously been shown to accurately predict growth in 18 out of 20 cases [ 78 ]. The mesh has 22560 elements and consists of 3-D 8-node structural solids (SOLID185). The outer shaft diameter was adapted to 15.5 mm matching values for boys of the age of 8 [ 62 ]. The model includes the medullary cavity. The diaphysis is composed of cortical bone, and the meta- and epiphyses are composed of cancellous bone covered by a layer of cortical bone. The proximal and distal growth plates with thicknesses of 0.9 mm and 2.1 mm, respectively, are modeled [ 79 ]. The widths are in a ratio of approximately 30:70 to simulate the different contributions of the two epiphyseal plates to femoral growth [ 80 ] and correspond to the literature values of an 8-year-old male [ 81 , 82 ]. Additionally, a layer of elements representing the resting zone is included in the model. The growth plate of the apophysis of the trochanter major is modeled with an angle of 50° to the body horizontal [ 83 ]. Pozdnikin et al. [ 60 ]. reported an almost constant articulotrochanteric distance measured between the tip of the greater trochanter and the upper point of the femoral head in normal hips of children aged 3 to 17 years. Therefore, the growth plate of the greater trochanter was assumed to have the same thickness as the proximal growth plate to map growth at the same rate. The cartilage of the femoral neck isthmus as well as the articular cartilage is modeled as one layer of proliferative chondrocytes. To reduce computation time and like comparable finite element models [ 37 , 64 ], all materials are assumed to be linearly elastic, isotropic, and homogeneous. The material properties are given in Table 2 . Young´s modulus of the growth plate highly affects the deformation and therefore the growth direction. Values in the literature range, starting with 0.49 MPa for the proliferative and hypertrophic zone and 0.99 MPa for the resting zone [ 84 ]. Therefore, a sensitivity analysis is performed to investigate the influence of the rigidity of the growth plates on growth tendency. Table 2 Initial material properties Material Density in kg/m^3 Young's modulus in MPa Poisson's ratio Cortical bone 1100 [ 85 ] 12000 [ 87 ] 0.3 [ 90 ] Cancellous bone 150 [ 85 ] 345 [ 88 ] 0.25 [ 90 ] Cartilage 1100 [ 86 ] 0.5–10 [ 37 , 64 , 84 ] 0.47 [ 84 ] Bone marrow 1000 [ 86 ] 0.01 [ 89 ] 0.49 [ 89 ] Rigid boundary conditions with fixed bearings in all directions at the condyles lead to artificially high stresses [ 91 ]. Therefore, physiological boundary conditions with a bearing fixed in all directions at the femoral notch, a bearing fixed in the anterior-posterior direction at the lateral epicondyle of the distal femur, and a bearing fixed in the anterior-posterior and medial-lateral directions at the middle of the femoral head were used [ 71 ]. To prevent artificial stresses near the bearings and to take into account the inertia forces generated during movement, inertia relief is used. This method can be used to balance the forces when analyzing individual load steps during the gait and achieve physiological stresses [ 92 ]. The boundary conditions are thereby only used to prevent rigid body movements, but the bearings themselves do not generate a reaction force. Instead, the applied forces are balanced by the inertial forces that the body experiences in the acceleration field. The acceleration field is defined as the standard gravitational force on the earth in the inferior direction. To model bone growth by thermal expansion, the femur was fixed at both condyles in all directions. Using load cases, a dynamic gait cycle can be represented by a finite number of static simulations. These discrete load cases can be combined as a functional load history to calculate the mechanical stimulus [ 26 ]. For this purpose, a physiological superposition of all loading cases is performed by cumulating the highest values of compressive stresses occurring in one of the loading cases [ 36 ]. Nine load cases based on a published multibody simulation were identified and used as initial muscle and joint forces [ 64 ]. In addition, the force of the iliotibial band is included, which reduces bending stresses in the diaphysis. This force is taken into account with 50% of the amount of the hip joint resultant of the respective load case [ 83 ]. Optimization of muscle forces has been shown to reduce bending stresses, and bending minimization can be used as a principle for determining muscle forces [ 71 ]. Therefore, the ANSYS subproblem routine is performed with 30 iterations for each load case. The muscle forces are used as design variables and are allowed to vary within a range of 50 to 200% of the initial value. Forces acting on the hip and knee joint are adopted as fixed values. Bending stresses are calculated in the mediolateral and anteroposterior directions by analyzing the absolute highest principal stresses at four points in the middle of the shaft [ 93 ]: $$\:{\sigma\:}_{bending}=\pm\:\left|\frac{{\sigma\:}_{medial/anterior}-{\sigma\:}_{lateral/posterior}}{2}\right|$$ The sum of both bending stresses at the midshaft is minimized (objective function). Additionally, both are independently used as state variables with a target value below 10 MPa. After importing the mesh and optimizing the muscle forces, growth is iteratively simulated (Fig. 6 ). Epiphyseal growth is modeled as thermal expansion. Proliferation increases the overall volume and therefore accounts for the isotropic growth ratio, while hypertrophy enlarges the columns unidirectional. A combination of biomechanical and histological analyses found a high correlation between the direction of the distortion of the hypertrophic chondrocytes and orientation of the columns of the growth plate, a mechanism by which shear stresses are reduced [ 22 ]. Therefore, for every element representing hypertrophic chondrocytes, the average deformation over all nine load cases is aligned with the z-axis of the element´s local coordinate system. The growth rate is the sum of the biological baseline r b0 and a mechanically stimulated growth rate r m and is modeled as temperature difference. The growth of an element of the growth plate g can be calculated by the local thickness of the respective zone l and both growth rates as $$\:g=l\bullet\:({r}_{b0}+{r}_{m})$$ Proliferation and production of extracellular matrix in the proliferative zone account for about 27%, and hypertrophy and production of extracellular matrix in the hypertrophic zone account for 73% [ 94 ]. Both growth mechanisms are modeled by the choice of thermal expansion coefficients. r b0 and r m are calculated based on a model developed by Wong and Carter [ 65 ], Stevens et al. [ 66 ], and Yadav et al. [ 67 – 69 ]. Briefly, based on a reported growth rate of the proximal femur of 9 mm/year [ 80 ], an assumed percentage of nonmechanically stimulated growth of 66% [ 16 , 67 ], and a thickness of the proximal growth plate of 0.9 mm, r b0 is given as 0.6 mm/month. r m is given by the sum of the maximum octahedral shear stress and the minimum hydrostatic stress obtained over all nine loading cases, where the coefficients a and b describe the respective influence of octahedral shear stress and hydrostatic stress on r m [ 66 ]: $$\:{r}_{m}=a\bullet\:max\left[\frac{\sqrt{{\left({\sigma\:}_{1}-{\sigma\:}_{2}\right)}^{2}+{\left({\sigma\:}_{2}-{\sigma\:}_{3}\right)}^{2}+{\left({\sigma\:}_{3}-{\sigma\:}_{1}\right)}^{2}}}{3}\right]+b\bullet\:min\left[\frac{{\sigma\:}_{1}+{\sigma\:}_{2}+{\sigma\:}_{3}}{3}\right]$$ The ratio of b/a was set at 0.5 in agreement with previous studies [ 40 ] and calibrated to correspond to 33% of total growth. Changes in bone density in cancellous bone and thereby changes in Young´s modulus can be described by a differential equation developed by Li et al. [ 95 ]. It considers bone resorption due to disuse, bone formation due to physiological overload in a dose-dependent manner, and bone resorption due to microdamage and pathological overload. 2 MPa is used as thresholds for bone formation [ 30 ]. Pathological overload is assumed for loading four times higher [ 31 , 34 ]. Furthermore, it is assumed that the curve is symmetrical. This can be used to solve a system of equations for 3 support points. The maximum bone formation rate per bone volume in trabecular bone is 4% per month [ 96 ]. The Young´s modulus of a specific element representing cancellous bone after one month of simulated loading E i+1 can therefore be expressed as function of the volume at the beginning of the iteration E i $$\:{E}_{i+1}={E}_{i}+\left(0.2916\bullet\:{e}^{0.7702{\sigma\:}_{3}-0.0772{{\sigma\:}_{3}}^{2}}-1\right)\bullet\:0.04\:MPa$$ Physiological loading of cortical bone is around ten times higher than in cancellous bone [ 30 ]. In contrast to cancellous bone and the endosteal surface, animal models found no periosteal resorption even without any loading [ 97 ]. With a maximal apposition rate of 0.06 mm/month [ 9 ], the mechanically stimulated apposition rate per month AR is given by $$\:AR=\left({e}^{0.0346574{\sigma\:}_{3}-0.000433217{{\sigma\:}_{3}}^{2}}-1\right)\bullet\:0.06\:mm/month$$ Principal stresses are analyzed by superposition of the minimal and maximal principal stresses, respectively. To calculate internal axial loading, a surface perpendicular to the femoral shaft axis is created. Normal stresses on this surface are averaged and multiplied with the cross-sectional area to obtain axial forces. Values are converted to percent body weight. After simulation of bone growth, the following values were measured and compared to reference values for validation (Fig. 7 ): Femur length: The distance between the center of the femoral head and the intercondylar femoral fossa is used here, as these two points are also used to determine the mechanical femoral axis [97]. Neck-shaft angle (NSA): The projected NSA angle is defined in the frontal plane as the angle between the femoral neck axis and the femoral shaft axis. To determine this, a circle is drawn around the center of the femoral head that intersects the femoral neck medially and laterally. The femoral neck axis can then be constructed by drawing a line through the middle of this line and the center of the head. The femoral shaft axis is the line between the centers of two transverse diameters below the lesser trochanter [98]. Articulo-trochanteric distance (ATD): The articulo-trochanteric distance is calculated as the distance between two parallel lines orthograde to the femoral shaft axis that are tangential to the maximum points of the femoral head and the greater trochanter [99]. Lateral distal femoral angle (LDFA): The mechanical lateral distal femoral angle is the lateral angle between the mechanical femoral axis and the knee line as a tangent to the most distal points of both knee condyles. There is also the anatomical lateral distal femoral angle between the femoral shaft axis and the knee line, but this is less commonly used and is therefore not used in this study [100]. Antetorsion angle (AT): The antetorsion angle describes the rotation of the femoral neck in relation to the shaft. It is measured in the axial plane between the projection of the femoral neck axis and a line tangential to the distal condyles [101]. Anteroposterior and mediolateral diameter: Since the femoral shaft is not cylindrical and shows an eccentric growth behavior, the diameter is determined at the level of the middle of the femur in the axial section between the most anterior and posterior as well as medial and lateral points. Declarations Ethics approval and consent to participate: Not applicable Consent for publication: Not applicable Availability of data and materials: The data presented in this study are available on request from the corresponding author. Competing interests: The authors declare that they have no competing interests. Funding: This research was funded by Deutsche Forschungsgesellschaft (DFG, German Research Foundation), grant number 445465815. The content is solely the responsibility of the authors and does not necessarily represent the official views of the DFG. Authors' contributions: AL conceptualized the project, developed the model, performed the finite element analysis and wrote the manuscript. RB analyzed, interpreted, and validated the modeling results. MK and SS assisted in computational modeling and were major contributors in writing the manuscript. UW supervised the project and was a major contributor in conceptualization of the project and interpreting the results. All authors read and approved the final manuscript. Acknowledgements: The authors are grateful to Hans Kainz and colleagues for making the mesh of the juvenile femur used in this study publicly available under the MIT Use Agreement at SimTK: https://simtk.org/projects/normal-load/. The authors also want to thank Beate Bender, Product Development, and the Institute of Product and Service Engineering, Ruhr-University Bochum, for the provision of computational equipment. References Emons J, Chagin AS, Sävendahl L, Karperien M, Wit JM. Mechanisms of growth plate maturation and epiphyseal fusion. Horm Res Paediatr. 2011,75:383-91 Abad V, Meyers JL, Weise M, Gafni RI, Barnes KM, Nilsson O, Bacher JD, Baron J. The Role of the Resting Zone in Growth Plate Chondrogenesis. Endocrinology. 2002,143:1851-57 Nilsson O, Mitchum RD, Schrier L, Ferns SP, Barnes KM, Troendle JF, Baron J. Growth plate senescence is associated with loss of DNA methylation. J Endocrinol. 2005,186:241-9 Prein C, Warmbold N, Farkas Z, Schieker M, Aszodi A, Clausen-Schaumann H. Structural and mechanical properties of the proliferative zone of the developing murine growth plate cartilage assessed by atomic force microscopy. Matrix Biol. 2016,50:1-15 Buckwalter JA, Mower D, Ungar R, Schaeffer J, Ginsberg B. Morphometric analysis of chondrocyte hypertrophy. J Bone Joint Surg Am. 1986,68:243-55 Aghajanian P, Mohan S. The art of building bone: emerging role of chondrocyte-to-osteoblast transdifferentiation in endochondral ossification. Bone Res. 2018,6:19 Watanabe-Takano H, Ochi H, Chiba A, Matsuo A, Kanai Y, Fukuhara S, Ito N, Sako K, Miyazaki T, Tainaka K, Harada I, Sato S, Sawada Y, Minamino N, Takeda S, Ueda HR, Yasoda A, Mochizuki N. Mechanical load regulates bone growth via periosteal Osteocrin. Cell Rep. 2021,36:109380 Robling AG, Turner CH. Mechanical Signaling for Bone Modeling and Remodeling. Crit Rev Eukaryot Gene Expr. 2009,19:319-38. Rauch F. Bone growth in length and width: the Yin and Yang of bone stability. J Musculoskelet Neuronal Interact. 2005,5:194-201 De Luca F. Impaired Growth Plate Chondrogenesis in Children with Chronic Illnesses. Pediatr Res. 2006,59:625-9 Ranke MB. Short and Long-Term Effects of Growth Hormone in Children and Adolescents With GH Deficiency. Front Endocrinol. 2021,12:720419 Perry RJ, Farquharson C, Ahmed SF. The role of sex steroids in controlling pubertal growth. Clin Endocrinol. 2008, 68, 4-15 Nilsson O. Genetic and epigenetic regulation of childhood growth. Nat Rev Endocrinol. 2018,14, 70-2 Hueter C. Anatomische Studien an den Extremitätengelenken Neugeborener und Erwachsener. Archiv f. pathol. Anat. 1863,26:484–519 Niehoff A, Kersting UG, Zaucke F, Morlock MM, Brüggemann GP. Adaptation of mechanical, morphological, and biochemical properties of the rat growth plate to dose-dependent voluntary exercise. Bone. 2004,35: 899-908 Hall BK, Herring SW. Paralysis and growth of the musculoskeletal system in the embryonic chick. J Morphol. 1990,206:45-56 Stokes IA, Aronsson DD, Dimock AN, Cortright V, Beck S. Endochondral growth in growth plates of three species at two anatomical locations modulated by mechanical compression and tension. J Orthop Res. 2006,24:1327-34 Stokes IAF, Spence H, Aronsson DD, Kilmer N. Mechanical Modulation of Vertebral Body Growth: Implications for Scoliosis Progression. Spine. 1996,21:1162-7 Snyder A, Zierath JR, Hawley JA, Sleeper MD, Craig BW. The effects of exercise mode, swimming vs. running, upon bone growth in the rapidly growing female rat. Mech Ageing Dev. 1992,66:59-69 Krahl H, Michaelis U, Pieper HG, Quack G, Montag M. Stimulation of bone growth through sports. A radiologic investigation of the upper extremities in professional tennis players. Am J Sports Med. 1994,22:751-7 Smith JW. The relationship of epiphysial plates to stress in some bones of the lower limb. J Anat. 1962,96:58-78 Alonso G, Yawny A, Bertolino G. How do bones grow? A mathematical description of the mechanobiological behavior of the epiphyseal plate. Biomech Model Mechanobiol. 2022,21:1585-601 Hunziker E. Mechanism of longitudinal bone growth and its regulation by growth plate chondrocites. Microsc Res Tech. 1994,28:505-19 Arkin AM, Katz JF. The effects of pressure on epiphyseal growth; the mechanism of plasticity of growing bone. J Bone Joint Surg Am. 1956,38:1056-76 Pauwels F. Gesammelte Abhandlungen zur funktionellen Anatomie des Bewegungsapparates. Springer: Berlin/Heidelberg/New York, Germany, 1965 Carter DR, Wong M. The role of mechanical loading histories in the development of diarthrodial joints. J Orthop Res. 1988,6:804-16 Feik SA, Storey E, Ellender G. Stress induced periosteal changes. Br J Exp Pathol. 1987,68:803-13 Gross TS, Edwards JL, McLeod KJ, Rubin CT. Strain gradients correlate with sites of periosteal bone formation. J Bone Miner Res. 1997,12:982-8 Lipphaus A, Witzel U. Biomechanical Study of the Development of Long Bones: Finite Element Structure Synthesis of the Human Second Proximal Phalanx Under Growth Conditions. Anat Rec. 2019,302:1389-98 Witzel U. Eine neue Methode zur virtuellen Schädelsynthese am Beispiel Camarasaurus. Hallesches Jahrb Für Geowiss. 2007,23:73-8 Frost HM. Bone's mechanostat: a 2003 update. Anat Rec A Discov Mol Cell Evol Biol. 2003,275:1081-101 Biewener AA, Bertram JE. Structural response of growing bone to exercise and disuse. J Appl Physiol. 1994,76:946-55 Willie BM, Birkhold AI, Razi H, Thiele T, Aido M, Kruck B, Schill A, Checa S, Main RP, Duda GN. Diminished response to in vivo mechanical loading in trabecular and not cortical bone in adulthood of female C57Bl/6 mice coincides with a reduction in deformation to load. Bone. 2013,55:335-46 Mustafy T, Londono I, Moldovan F, Villemure I. Isolated Cyclic Loading During Adolescence Improves Tibial Bone Microstructure and Strength at Adulthood. JBMR Plus. 2020,4:e10349 McCabe K, Henderson K, Pantinople J, Richards HL, Milne N. Curvature reduces bending strains in the quokka femur. PeerJ. 2017,5:e3100 Witzel U, Preuschoft H. Finite-element model construction for the virtual synthesis of the skulls in vertebrates: case study of Diplodocus. Anat Rec A Discov Mol Cell Evol Biol. 2005,283:391-401 Heegaard JH, Beaupré GS, Carter DR. Mechanically modulated cartilage growth may regulate joint surface morphogenesis. J Orthop Res. 1999,17:509-17 Carpenter RD, Carter DR. Computational simulation of spontaneous bone straightening in growing children. Biomech Model Mechanobiol. 2010,9:317-28 Shefelbine SJ, Tardieu C, Carter DR. Development of the femoral bicondylar angle in hominid bipedalism. Bone. 2002,30:765-70 Shefelbine SJ, Carter DR. Mechanobiological predictions of femoral anteversion in cerebral palsy. Ann Biomed Eng. 2004,32:297-305 Sadeghian SM, Lewis CL, Shefelbine SJ. Predicting growth plate orientation with altered hip loading: potential cause of cam morphology. Biomech Model Mechanobiol. 2020,19:701-12 Shefelbine SJ, Carter DR. Mechanobiological predictions of growth front morphology in developmental hip dysplasia. J Orthop Res. 2004,22:346-52 Giorgi M, Carriero A, Shefelbine SJ, Nowlan NC. Effects of normal and abnormal loading conditions on morphogenesis of the prenatal hip joint: application to hip dysplasia. J Biomech. 2015;48:3390-7 Lipphaus A, Tröbs RB, Wegener-Panzer A, Witzel U. Biomechanische Simulation der Entstehung eines Genu valgum bei Hüftdysplasie. In Proceedings of the Deutscher Kongress für Orthopädie und Unfallchirurgie, Berlin, Germany, 28 10 2021 Stokes IAF. Analysis and simulation of progressive adolescent scoliosis by biomechanical growth modulation. Eur Spine J. 2007,16:1621-28 Alonso MG, Bertolino G, Yawny A. Mechanobiological based long bone growth model for the design of limb deformities correction devices. J Biomech. 2020,109:109905 Carrera-Pinzón AF, Márquez-Flórez K, Kraft RH, Ramtani S, Garzón-Alvarado DA. Computational model of a synovial joint morphogenesis. Biomech Model Mechanobiol. 2020,19:1389-1402 Vaca-González JJ, Moncayo-Donoso M, Guevara JM, Hata Y, Shefelbine SJ, Garzón-Alvarado DA. Mechanobiological modeling of endochondral ossification: an experimental and computational analysis. Biomech Model Mechanobiol. 2018,17:853-75 Sadeghian SM, Shapiro FD, Shefelbine SJ. Computational model of endochondral ossification: Simulating growth of a long bone. Bone. 2021,153:116132 Terryn PJ, Fernandes PR, João F, Veloso A, Baptista F. Modeling the musculoskeletal loading in bone remodeling at the hip of a child. Comput Methods Programs Biomed. 2021,210:106365 Quexada-Rodríguez D, Trabelsi O, Hobatho MC, Ramtani S, Garzón-Alvarado D. Influence of growth plate morphology on bone trabecular groups, a framework computational approach. Bone. 2023,171:116742 Beaupré GS, Orr TE, Carter DR. An approach for time-dependent bone modeling and remodeling-application: a preliminary remodeling simulation. J Orthop Res. 1990,8:662-70 Lee MC, Eberson CP. Growth and development of the child's hip. Orthop Clin North Am. 2006,37:119-32 Heimkes, B. Die großen Apophysen. Orthopäde. 2016:45:206-12 Struijs PA, Oostra RJ, van Rijn RR, Besselaar PP. Abnormal growth of the proximal femur due to apophyseal-epiphyseal coalescence resulting in coxa valga--a report of two cases in adolescents. Acta Orthop. 2011,82:507-9 Langenskiöld A, Salenius P. Epiphyseodesis of the greater trochanter. Acta Orthop Scand. 1967,38:199-219 Kainz H, Koller W, Bastir M, Svehlik M, Schwartz MH. The shape and size of the femur adapts during growth to maintain a constant cartilage load. Gait Posture. 2024,113:105-7 Anderson M, Green WT, Messner MB. Growth and predictions of growth in the lower extremities. J Bone Joint Surg Am. 1963,45-A:1-14 Van Geel P, Cools W, Laumen A. Evolution of head-shaft angle and neck-shaft angle in childhood. Acta Orthop Belg. 2021,87:435-41 Pozdnikin IY, Baskov VE, Barsukov DB, Bortulev PI, Krasnov AI. Relative overgrowth of the greater trochanter and trochanteric-pelvic impingement syndrome in children: causes and X-ray anatomical characteristics. Pediatric Traumatol Orthop Reconstr Surg. 2019,7:15-24 Sabharwal S, Zhao C, Edgar M. Lower limb alignment in children: reference values based on a full-length standing radiograph. J Pediatr Orthop. 2008,28:740-6 Bradney M, Karlsson MK, Duan Y, Stuckey S, Bass S, Seeman E. Heterogeneity in the growth of the axial and appendicular skeleton in boys: implications for the pathogenesis of bone fragility in men. J Bone Miner Res. 2000,15:1871-8 Upadhyay SS, Burwell RG, Moulton A, Small PG, Wallace WA. Femoral anteversion in healthy children. Application of a new method using ultrasound. J Anat. 1990,169:49-61 Kainz H, Killen BA, Wesseling M, Perez-Boerema F, Pitto L, Garcia Aznar JM, Shefelbine S, Jonkers I. A multi-scale modelling framework combining musculoskeletal rigid-body simulations with adaptive finite element analyses, to evaluate the impact of femoral geometry on hip joint contact forces and femoral bone growth. PLoS One. 2020,15:e0235966 Wong M, Carter DR. A theoretical model of endochondral ossification and bone architectural construction in long bone ontogeny. Anat Embryol. 1990,181:523-32 Stevens SS, Beaupré GS, Carter DR. Computer model of endochondral growth and ossification in long bones: biological and mechanobiological influences. J Orthop Res. 1999,17:646-53 Yadav P, Shefelbine SJ, Pontén E, Gutierrez-Farewik EM. Influence of muscle groups' activation on proximal femoral growth tendency. Biomech Model Mechanobiol. 2017,16:1869-83 Yadav P, Shefelbine SJ, Gutierrez-Farewik EM. Effect of growth plate geometry and growth direction on prediction of proximal femoral morphology. J Biomech. 2016,49:1613-19 Yadav P, Peña Fernández M, Gutierrez-Farewik EM. Influence of loading direction due to physical activity on proximal femoral growth tendency, Med Eng Phys. 2021,90:83-91 Naik P. Remodelling in Children's Fractures and Limits of Acceptability. Indian J Orthop. 2021,55:549-559 Sverdlova NS, Witzel U. Principles of determination and verification of muscle forces in the human musculoskeletal system: Muscle forces to minimise bending stress. J Biomech. 2010,43:387-96 Lutz F, Mastel R, Runge M, Stief F, Schmidt A, Meurer A, Witte H. Calculation of muscle forces during normal gait under consideration of femoral bending moments. Med Eng Phys. 2016,38:1008-15 Bertram JE, Biewener AA. Bone curvature: sacrificing strength for load predictability? J Theor Biol. 1988,131:75-92 Duda GN, Schneider E, Chao EY. Internal forces and moments in the femur during walking. J Biomech. 1997,30:933-41 Augat P, Burger J, Schorlemmer S, Henke T, Peraus M, Claes L. Shear movement at the fracture site delays healing in a diaphyseal fracture model. J Orthop Res. 2003,21:1011-7 Kainz H, Mindler GT, Kranzl A. Influence of femoral anteversion angle and neck-shaft angle on muscle forces and joint loading during walking. PLoS One. 2023,18:e0291458 Someya K, Mochizuki T, Hokari S, Tanifuji O, Katsumi R, Koga H, Takahashi Y, Kobayashi K, Morise Y, Sakamoto M, Koga Y, Endo N. Age- and sex-related characteristics in cortical thickness of femoral diaphysis for young and elderly subjects. J Bone Miner Metab. 2020,38:533-43 Koller W, Svehlik M, Wallnöfer E, Kranzl A, Mindler G, Baca A, Kainz H. Femoral bone growth predictions based on personalized multi-scale simulations: validation and sensitivity analysis of a mechanobiological model. Biomech Model Mechanobiol. 2025. 24:879-94 Lipphaus A, Wegener-Panzer A, Tröbs, RB, Witzel U. Computational modelling of proximal and distal epiphyseal and appositional growth of the femur in children. In Proceedings of XXVIII Congress of theInternational Society of Biomechanics (ISB): 25-29 June 2021; Digital congress:A287. Pritchett JW. Longitudinal growth and growth-plate activity in the lower extremity. Clin Orthop Relat Res. 1992,275:274-9 Abiodun AA, Falana BA, Olatunbosun A. Baseline Thickness of Proximal Femoral Epiphysis in the Diagnosis of Hip Diseases in Infants and Children in Nigeria. Int J Surg Res. 2014,3:22-26. Kim A, Dombi E, Solomon J, Fox E, Balis FM, Widemann BC. Automated volumetric growth plate measurement using magnetic resonance imaging for monitoring skeletal toxicity in children treated on investigational drug trials. Clin Cancer Res. 2011,17:5982-90 Heimkes B, Posel P, Plitz W, Zimmer, M. Die altersabhängige Kräfteverteilung am koxalen Femurende des normal wachsenden Kindes. Z Orthop. 1997,135:17-23 Gao J, Roan E, Williams JL. Regional variations in growth plate chondrocyte deformation as predicted by three-dimensional multi-scale simulations. PLoS One. 2015,10:e0124862 Rauch F, Schoenau E. Changes in bone density during childhood and adolescence: an approach based on bone's biological organization. J Bone Miner Res. 2001,16:597-604 Hasgall PA, Di Gennaro F, Baumgartner C, Neufeld E, Lloyd B, Gosselin MC, Payne D, Klingenböck A, Kuster N, “IT’IS Database for thermal and electromagnetic parameters of biological tissues,” Version 4.1, Feb 22, 2022, DOI: 10.13099/VIP21000-04-1. itis.swiss/database Öhman C, Baleani M, Pani C, Taddei F, Alberghini M, Viceconti M, Manfrini M. Compressive behaviour of child and adult cortical bone. Bone. 2011,49:769-76 Piszczatowski S. Material aspects of growth plate modelling using Carter's and Stokes's approaches. Acta Bioeng Biomech. 2011;13:3-14 Jansen LE, Birch NP, Schiffman JD, Crosby AJ, Peyton SR. Mechanics of intact bone marrow. J Mech Behav Biomed Mater. 2015,50:299-307. Lai YS, Chen WC, Huang CH, Cheng CK, Chan KK, Chang TK. The effect of graft strength on knee laxity and graft in-situ forces after posterior cruciate ligament reconstruction. PLoS One. 2015,10:e0127293 Speirs AD, Heller MO, Duda GN, Taylor WR. Physiologically based boundary conditions in finite element modelling. J Biomech. 2007,40:2318-23 Heyland M, Trepczynski A, Duda GN, Zehn M, Schaser KD, Märdian S. Selecting boundary conditions in physiological strain analysis of the femur: Balanced loads, inertia relief method and follower load. Med Eng Phys. 2015,37:1180-5 Main RP, Biewener AA. Ontogenetic patterns of limb loading, in vivo bone strains and growth in the goat radius. J Exp Biol. 2004,207:2577-88 Wilsman NJ, Farnum CE, Leiferman EM, Fry M, Barreto C. Differential growth by growth plates as a function of multiple parameters of chondrocytic kinetics. J Orthop Res. 1996,14:927-36. Li J, Li H, Shi L, Fok AS, Ucer C, Devlin H, Horner K, Silikas N. A mathematical model for simulating the bone remodeling process under mechanical stimulus. Dent Mater. 2007,23:1073-8 Glorieux FH, Travers R, Taylor A, Bowen JR, Rauch F, Norman M, Parfitt AM. Normative data for iliac bone histomorphometry in growing children. Bone. 2000,26:103-9 Li B, Sankaran JS, Judex S. Trabecular and Cortical Bone of Growing C3H Mice Is Highly Responsive to the Removal of Weightbearing. PLoS One. 2016,11:e0156222 Guenoun B, Zadegan F, Aim F, Hannouche D, Nizard R. Reliability of a new method for lower-extremity measurements based on stereoradiographic three-dimensional reconstruction. Orthop Traumatol Surg Res. 2012,98:506-13 Fischer CS, Kühn JP, Völzke H, Ittermann T, Gümbel D, Kasch R, Haralambiev L, Laqua R, Hinz P, Lange J. The neck-shaft angle: an update on reference values and associated factors. Acta Orthop. 2020,91:53-7 Yoon PW, Shin YH, Yoo JJ, Yoon KS, Kim HJ. Progression of a fracture site impaction as a prognostic indicator of impacted femoral neck fracture treated with multiple pinning. Clin Orthop Surg. 2012,4:66-71 Marques Luís N, Varatojo R. Radiological assessment of lower limb alignment. EFORT Open Rev. 2021,6:487-94 Scorcelletti M, Reeves ND, Rittweger J, Ireland A. Femoral anteversion: significance and measurement. J Anat. 2020,237:811-26 Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 19 Nov, 2025 Reviews received at journal 18 Nov, 2025 Reviewers agreed at journal 28 Oct, 2025 Reviews received at journal 30 Aug, 2025 Reviewers agreed at journal 27 Aug, 2025 Reviewers agreed at journal 08 Aug, 2025 Reviewers agreed at journal 05 Aug, 2025 Reviewers invited by journal 05 Aug, 2025 Editor assigned by journal 04 Aug, 2025 Submission checks completed at journal 04 Aug, 2025 First submitted to journal 27 Jul, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7224633","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":495256962,"identity":"2d148446-69fc-42f9-8c16-3a0adb5f8ac7","order_by":0,"name":"Andreas Lipphaus","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA90lEQVRIiWNgGAWjYHACAzDJx8zAwAxi8DMwsIERQS1sMC2SDURrYYBqMThAQAv/7OZtHz7U1DGwsfMee1xQc9je+PYZswcMZTY4tUjcOVY8c8axw0CH8aUbAxmJ287lmBswnEvDbc2NHGNmHjagY5h5zKR52A4nmJ3hMZNgbDuMU4c8SMuff3VQLf+ADusBa/mP2+8gLYxtzBAtvG2HGTfwgLUcwKnF8EZaMWNv32EesF94+9ITZ5xhKzdIOJeMU4vcjeTNDD++1cnx85899pjnm7U9fw/ztgcfyuxwex8KeIAIKS4SCGqA6MIf46NgFIyCUTByAQBGv0a5QltDnQAAAABJRU5ErkJggg==","orcid":"","institution":"Ruhr University Bochum","correspondingAuthor":true,"prefix":"","firstName":"Andreas","middleName":"","lastName":"Lipphaus","suffix":""},{"id":495256963,"identity":"623d8aae-d863-4d0d-a698-ef5834723c8c","order_by":1,"name":"Ralf-Bodo Tröbs","email":"","orcid":"","institution":"St. Vinzenz Krankenhaus Paderborn","correspondingAuthor":false,"prefix":"","firstName":"Ralf-Bodo","middleName":"","lastName":"Tröbs","suffix":""},{"id":495256964,"identity":"ba622524-954d-48be-aa10-ccc198437966","order_by":2,"name":"Matthias Klimek","email":"","orcid":"","institution":"University Hospital Münster","correspondingAuthor":false,"prefix":"","firstName":"Matthias","middleName":"","lastName":"Klimek","suffix":""},{"id":495256965,"identity":"4c3e70fb-6d27-49f3-90e8-915587aa3a2a","order_by":3,"name":"Sascha Selkmann","email":"","orcid":"","institution":"Ruhr University Bochum","correspondingAuthor":false,"prefix":"","firstName":"Sascha","middleName":"","lastName":"Selkmann","suffix":""},{"id":495256966,"identity":"e50274e9-40c8-49e9-8cc2-d4903260f952","order_by":4,"name":"Ulrich Witzel","email":"","orcid":"","institution":"Ruhr University Bochum","correspondingAuthor":false,"prefix":"","firstName":"Ulrich","middleName":"","lastName":"Witzel","suffix":""}],"badges":[],"createdAt":"2025-07-27 07:08:29","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7224633/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7224633/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":88349349,"identity":"d4d676cb-5ee5-4c16-b5a7-7095cd5a37a5","added_by":"auto","created_at":"2025-08-05 14:03:39","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":63186,"visible":true,"origin":"","legend":"\u003cp\u003eCourse of the axial load for all nine load cases. The respective phases of the gait cycle are shown accordingly.\u003c/p\u003e","description":"","filename":"floatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-7224633/v1/9331f2437b4754d58e4d4f26.png"},{"id":88349350,"identity":"c4268e2c-02c6-41fc-8904-3c584e0df2b6","added_by":"auto","created_at":"2025-08-05 14:03:39","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":105334,"visible":true,"origin":"","legend":"\u003cp\u003e3rd principal stresses in frontal (a) and sagittal (b) view highlighting compressive stresses and 1st principal stresses in frontal (c) and sagittal (d) view highlighting tensile stresses in cortical bone.\u003c/p\u003e","description":"","filename":"floatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-7224633/v1/c03267f6d497c31a119e0aab.png"},{"id":88349351,"identity":"9a531a55-1fe1-485d-bcd0-08743f89d651","added_by":"auto","created_at":"2025-08-05 14:03:39","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":110225,"visible":true,"origin":"","legend":"\u003cp\u003e3rd principal stresses in frontal (a) and sagittal (b) sections highlighting compressive stresses and 1st principal stresses in frontal (c) and sagittal (d) sections highlighting tensile stresses in trabecular bone.\u003c/p\u003e","description":"","filename":"floatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-7224633/v1/774c64af6ca1126762659a7b.png"},{"id":88349353,"identity":"e362862c-de9d-4f5f-b970-f9ccd87c3d7d","added_by":"auto","created_at":"2025-08-05 14:03:40","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":130020,"visible":true,"origin":"","legend":"\u003cp\u003eTop view of the proximal hypertrophic (a) and proliferative (b) zone as well as the distal proximal hypertrophic (c) and proliferative (d) zone showing the growth rate in mm/month.\u003c/p\u003e","description":"","filename":"floatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-7224633/v1/fed5969e98ee5f25f7646840.png"},{"id":88349354,"identity":"4a03f2e6-15b0-4f59-912b-4c584e9a8a2b","added_by":"auto","created_at":"2025-08-05 14:03:40","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":58830,"visible":true,"origin":"","legend":"\u003cp\u003eCompressive stresses in transversal section of the femoral shaft (a). Increased density due to inner remodeling in the proximal femur. Darker areas correspond to an increased bone mineral density (b).\u003c/p\u003e","description":"","filename":"floatimage5.png","url":"https://assets-eu.researchsquare.com/files/rs-7224633/v1/aa274ff05586fe8c6a2bc312.png"},{"id":88349359,"identity":"5e9cafcc-4ef2-409a-b7b0-3cce52b32d8b","added_by":"auto","created_at":"2025-08-05 14:03:40","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":117995,"visible":true,"origin":"","legend":"\u003cp\u003eFlow diagram of the mechanobiological model: first, the finite element mesh is established in ANSYS and bending-minimized muscle forces are calculated. Stresses for all load cases are physiologically superimposed and hydrostatic and octahedral shear stresses at the growth plates and 3rd principal stresses in the bone are analyzed. The mechanical stimulus is calculated. The working plane of each growth plate element is aligned with the average deformation during gait. The density of cancellous bone is updated according to equation 3 based on 3rd principal stresses. Appositional periosteal growth is modeled by moving the external nodes on a line to the midpoint of the respective transverse plane. Forces are removed and growth rate is applied as temperature, and the model geometry is updated after thermal expansion. Nodes are moved to their final location and the thickness of the growth plates is reset to initial values to model ossification. For the next iteration, forces are applied again to the modified mesh.\u003c/p\u003e","description":"","filename":"floatimage6.png","url":"https://assets-eu.researchsquare.com/files/rs-7224633/v1/2edac69f9e5c29dba40467c5.png"},{"id":88349368,"identity":"21186808-9534-4005-84ab-082bbb32ae64","added_by":"auto","created_at":"2025-08-05 14:03:40","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":55805,"visible":true,"origin":"","legend":"\u003cp\u003eAnthropometric measurements of the femur used in the present study. (a) Length between the center of the femoral head and the femoral notch. (b) Neck-shaft angle (NSA). (c) Articulo-trochanteric distance (ATD). (d) Lateral distal femoral angle (LDFA). (e) Anteversion angle.\u003c/p\u003e","description":"","filename":"floatimage7.png","url":"https://assets-eu.researchsquare.com/files/rs-7224633/v1/9580593fb82c554603625404.png"},{"id":88352050,"identity":"b2f7cfe4-77ae-45d0-89db-abf7b02e81d5","added_by":"auto","created_at":"2025-08-05 14:27:41","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1385681,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7224633/v1/46886d07-61da-4802-b06a-d1b84d82a86a.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Computational mechanobiological model combining epiphyseal, apophyseal, and appositional growth and inner bone remodeling of the juvenile femur","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eIn children, longitudinal bone growth occurs at the cartilaginous epiphyseal plates by endochondral ossification [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. These growth plates are located at the end of the long bones between the metaphysis and the epiphysis and can be divided into horizontal zones. Next to the epiphysis, the resting zone contains stem-like cells and secretes morphogens influencing the alignment of the proliferative chondrocytes and inhibiting their terminal differentiation [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. The finite proliferative capacity of resting chondrocytes leads to growth deceleration and finally epiphyseal fusion by growth plate senescence as children age [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. The resting zone is followed by the proliferation zone, where chondrocytes rapidly divide, form columns of flattened, disc-shaped chondrocytes, and synthesize new matrix [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. Towards the end of the proliferative zone, chondrocytes enlarge and take a more spherical shape forming the hypertrophic zone [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. Two main hypotheses of ossification at the end of the hypertrophic zone involve either the apoptosis of hypertrophic chondrocytes followed by invasion of osteoblast precursors by the vascular system or transdifferentiation of chondrocytes to osteoblasts [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e]. Appositional growth increases the thickness or diameter of the bone. In this mechanically stimulated process, osteoblasts deposit new matrix on the periosteal and endosteal surfaces and differentiate into osteocytes [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eBone growth in children is orchestrated by various mechanical [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e] and non-mechanical factors like inflammation [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e], growth [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e] and sex hormones [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e], and genetics [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]. Several studies investigated the rate of longitudinal growth in the growth plates as a function of static or dynamic loading. The Hueter-Volkmann law states that epiphyseal growth is increased by tensile stress and retarded by compressive stress acting at the growth plate [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]. While in some animal models only minor femoral length differences are reported due to different physiological dynamic loadings [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e], a growth retardation of 52\u0026ndash;63% is observed in paralyzed embryonic chicks due to a lack of mechanical stimulation [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. In animal models, a linear relationship between sustained tensile and compressive stresses and acceleration and deceleration of growth was observed [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e]. This effect is due to reduced numbers of both proliferative and hypertrophic chondrocytes and increases approximately linearly with time [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e]. The effect of cyclic or intermittent loading is still not fully understood: While some studies found no significant bone length differences with different exercises in rats [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e], others reported an increase in the length of radius and ulna in the dominant arm of professional tennis players [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e]. Smith experimentally determined the principal stress directions in the epiphyses and concluded that the stress trajectories often are orthogonal to the direction of bone growth [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e]. A second hypothesis is that due to the low stiffness of the cartilage under load, a deformity occurs and that growth proceeds in the direction of the distortion of the cartilage columns as long as this is too small to lead to tissue damage [\u003cspan additionalcitationids=\"CR23\" citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e]. Pauwels proposed that deformities, especially after fractures, lead to asymmetric growth by uneven loading of the epiphyseal plate and thereby works to correct these deformities [\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e]. Carter and Wong proposed a mechanobiological model that epiphyseal growth is inhibited by dynamic hydrostatic compression and accelerated by octahedral shear stress. They defined the osteogenic index as the sum of maximum octahedral shear stress and minimum hydrostatic stress overall load cases and calculated the resulting growth as the product of the osteogenic index and the mechanical growth rate [\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eEndochondral growth primarily causes functional adaptations through angular adaptations till the fusion of the growth plates. In contrast, bone formation and resorption optimize bone density and volume, especially in childhood, but also in adults. Appositional growth occurring at the periosteum is influenced by local strains [\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e]. Higher loading is associated with greater periosteal growth [\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e]. Thereby, the bone's diameter increases, and not only the axial strength but in particular the strength of bone in bending and torsion is increased. Indeed, previous finite element simulations suggested that torsional load is the most important stimulus in forming the medullary cavity [\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e]. In adults, periosteal formation is seen in regions stressed with more than 20 MPa [\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e] or 1500 \u0026micro;ε [\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e]. Above a threshold of about 4000 \u0026micro;ε overload resorption occurs and bone density decreases. In growing bones exercise leads to a higher cortical area, while disuse results in decreased cortical area compared to normal developing bones [\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e]. However, due to the lower Young\u0026acute;s modulus, higher deformations occur [\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e]. In addition, the threshold of bone formation is lower. A study in adolescent rats has reported significant and prolonged bone formation with strains of 850 \u0026micro;ε [\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e]. Overall, mechanically controlled growth patterns allow the bone to adapt to its mechanical needs. Thereby, bending stresses can be reduced to achieve lightweight design [\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e, \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e, \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eBased on finite element models, several computational approaches were established to simulate bone growth due to metabolic and mechanobiological signaling. Early models depicted either the biochemical or the mechanobiological component of growth. Heegard et al. used a pure mechanobiological model to simulate prenatal joint development [\u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e], bone straightening [\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e], and epiphyseal growth of the distal femur [\u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e39\u003c/span\u003e]. Based on this, the development of bone deformities resulting from pathologies such as cerebral palsy [\u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e], cam deformity [\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e41\u003c/span\u003e], and developmental dysplasia of the hip [\u003cspan additionalcitationids=\"CR43\" citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e44\u003c/span\u003e] has been simulated. A similar approach was developed by Stokes et al. and uses tensile or compressive axial stresses to describe the longitudinal bone growth [\u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e45\u003c/span\u003e]. Alonso et al. presented a mechanobiological model simulating epiphyseal growth and remodeling after virtual implantation of staples for hemiepiphysiodesis [\u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e46\u003c/span\u003e]. In the latest models, reaction-diffusion-equations of morphogens and mechanobiological approaches were combined to model the embryonic development of synovial joints and the femur [\u003cspan additionalcitationids=\"CR48\" citationid=\"CR47\" class=\"CitationRef\"\u003e47\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e49\u003c/span\u003e]. Also, changes in bone density of the trabecular bone in the proximal femur of children have been modeled [\u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e, \u003cspan citationid=\"CR51\" class=\"CitationRef\"\u003e51\u003c/span\u003e], resulting in an increase in bone mineral density on the medial side of the femoral neck [\u003cspan citationid=\"CR52\" class=\"CitationRef\"\u003e52\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eThe proximal femur has three growth plates: the growth plate located between the metaphysis and the femoral head, the trochanteric growth plate, and the femoral neck isthmus [\u003cspan citationid=\"CR53\" class=\"CitationRef\"\u003e53\u003c/span\u003e]. Because chondral growth is faster than periosteal growth, the cartilage of the femoral neck isthmus is necessary to compensate for length growth and shape the femoral metaphysis. The growth of the trochanteric apophysis affects the statics of the hip joint as the hip resultant force is mainly defined by the body weight and the hip abductors attached to the greater trochanter. The force resultant at the greater trochanter is formed by forces through a muscle sling of the vastus lateralis with the opposing gluteus medius and minimus and the iliotibial band [\u003cspan citationid=\"CR54\" class=\"CitationRef\"\u003e54\u003c/span\u003e], reducing bending in both the femoral shaft and near the hip joint. Growth of the greater trochanter results in a reduction of the femoral neck-shaft angle [\u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e55\u003c/span\u003e], and subsequently, epiphysiodesis of the greater trochanter has been described as a treatment for infantile coxa vara [\u003cspan citationid=\"CR56\" class=\"CitationRef\"\u003e56\u003c/span\u003e]. The morphological changes during growth allow the bone to maintain physiological loading and a constant cartilage load [\u003cspan citationid=\"CR57\" class=\"CitationRef\"\u003e57\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eCurrent in silico models either focus on very early phases of bone development or focus on a single growth mechanism. For transition into clinical practice, a more comprehensive model is needed. As the greater trochanter plays an important role in the biomechanics of the hip, it is of interest to include its growth in in silico models of bone development. Therefore, this study aims to develop a mechanobiological model of the growth of the juvenile femur including all growth plates, appositional growth of the femoral shaft, and inner remodeling. Stresses and internal axial loading are calculated during gait and muscle forces are optimized to reduce bending stresses. For validation, growth predicted by the in silico model is compared to several in vivo anthropometric measurements.\u003c/p\u003e"},{"header":"2. Results","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\u003ch2\u003e2.1. Optimization of muscle forces\u003c/h2\u003e\u003cp\u003eBefore the growth simulation was initiated, muscle forces were calculated using a bending minimization approach (Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). Two peaks in joint reaction forces, the first during loading response and the second during terminal stance (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) were associated with peaks in muscle forces of the gluteal muscles and the iliopsoas and iliacus muscles respectively.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eResultant muscle forces in Newton (N) for all nine load cases in an 8-year-old boy using a bending minimization approach. Muscle forces are rounded to full numbers. Forces below 1 N are considered as no muscle activation (-). Additionally, resulting axial loading is given for all load cases.\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"10\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c10\" colnum=\"10\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eMuscle/Load case\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003e1\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003e2\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003e3\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e\u003cp\u003e4\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e\u003cp\u003e5\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c7\"\u003e\u003cp\u003e6\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c8\"\u003e\u003cp\u003e7\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c9\"\u003e\u003cp\u003e8\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c10\"\u003e\u003cp\u003e9\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eAdductor brevis\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e4\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c10\"\u003e\u003cp\u003e32\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eAdductor longus\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e1\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c10\"\u003e\u003cp\u003e34\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eAdductor magnus\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e10\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e52\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c10\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eBiceps femoris\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e45\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e18\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e96\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e64\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c10\"\u003e\u003cp\u003e37\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eGastrocnemius lateralis\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e11\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e55\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e125\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e37\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c10\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eGastrocnemius medialis\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e123\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e372\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e32\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c10\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eGemellus\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e6\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e5\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c10\"\u003e\u003cp\u003e12\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eGluteus maximus\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e11\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e188\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e235\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e152\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e35\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c10\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eGluteus medius\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e131\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e609\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e368\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e282\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e266\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e206\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e109\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c10\"\u003e\u003cp\u003e23\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eGluteus minimus\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e10\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e64\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e27\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e27\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e59\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e60\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e57\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c10\"\u003e\u003cp\u003e12\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eIliacus\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e137\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e880\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e524\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c10\"\u003e\u003cp\u003e244\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003ePectineus\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c10\"\u003e\u003cp\u003e5\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003ePiriformis\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e14\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e62\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e17\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e8\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e29\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e67\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e12\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c10\"\u003e\u003cp\u003e29\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003ePsoas\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e208\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e64\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c10\"\u003e\u003cp\u003e169\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eQuadratus femoris\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e11\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e4\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c10\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eVastus intermedius\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e23\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e47\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e28\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e13\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c10\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eVastus lateralis\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e100\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e191\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e94\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e16\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c10\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eVastus medialis\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e45\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e86\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e43\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e8\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c10\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eAxial loading\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e316\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e727\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e985\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e575\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e586\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e656\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e924\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e592\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c10\"\u003e\u003cp\u003e206\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eFinite element analysis of the physiological superposition of all nine load cases in the initial model of the femur of an 8-year-old shows that 3rd principal stresses are highest in the proximal femur on the medial side (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003ea, b). In contrast, compression in the distal femur is mostly homogenously distributed. In the shaft, compression is minimal on the anterior (-10.9 MPa) and lateral side (-11.7 MPa), whereas higher values are predicted on the posterior (-21.1 MPa) and medial (-33.2 MPa) side. 1st principal stresses are opposite to the compressive stresses (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003ec, d). Values are generally lower with tensile stresses of 10 MPa anterior, 21.1 MPa lateral, 6.6 MPa posterior, and 5 MPa medial. Before optimization of muscle forces but including the iliotibial band and using physiological boundary conditions bending stresses are 24 MPa in the sagittal and 24.9 MPa in the frontal plane. After optimization, bending stresses in the frontal plane decrease by 78% while increase in the frontal plane by 9%. Loading results in axial forces between 200 and 985 N at midshaft with minimal shear loading.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eCross sections show corresponding stresses in the trabecular bone (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e). In the frontal section, the highest compression is observed in the proximal femur on the medial side, while tension peaks at the lateral side. As in cortical bone, the trabecular bone of the distal femur is more uniformly stressed. Of note, the compression near the distal growth plate is slightly higher anterior and lateral. Other than that, in the sagittal section compressive and tensile stresses are mostly homogenously distributed.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\u003ch2\u003e2.2. Simulation of physiological growth\u003c/h2\u003e\u003cp\u003eThe predicted growth direction of the proximal femur is highly dependent on Young\u0026acute;s modulus of the growth plate. We find a value of 10 MPa best matching anthropometric data with lower values resulting in an unphysiologically high decrease in anteversion angle.\u003c/p\u003e\u003cp\u003eThe growth rate as the sum of the biological baseline and the osteogenic index is calculated for all growth plates (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e). The growth plate of the proximal femur exhibits a centric distribution of the growth rate. In the growth plate of the distal femur, the growth rate shows a homogenous distribution\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe simulation predicted an increase in both bone length and diameter in accordance with anthropometric data (Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e2\u003c/span\u003e). The ATD shows a decrease of 0.08 mm per month. The NSA decreased on average by 0.03\u0026deg; per month, the anteversion angle by 0.14\u0026deg;. The LDFA decreased by 0.16\u0026deg; per month.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eChanges in the morphometries per year as predicted by the in silico model compared to anthropometric data\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"3\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eMeasurement\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eIn silico model\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eAnthropometric data\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eBone length\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e+\u0026thinsp;26.6 mm\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e+\u0026thinsp;20.8 mm [\u003cspan citationid=\"CR58\" class=\"CitationRef\"\u003e58\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eNeck-shaft angle\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e\u0026minus;\u0026thinsp;0.4\u0026deg;\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e-1.1\u0026deg; [\u003cspan citationid=\"CR59\" class=\"CitationRef\"\u003e59\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eArticulo-trochanteric distance\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e\u0026minus;\u0026thinsp;1 mm\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e0 mm [\u003cspan citationid=\"CR60\" class=\"CitationRef\"\u003e60\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eLateral distal femoral angle\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e\u0026minus;\u0026thinsp;1.9\u0026deg;\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e0\u0026deg; [\u003cspan citationid=\"CR61\" class=\"CitationRef\"\u003e61\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eShaft diameter (Anterior-Posterior/Medial-Lateral)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e0.9 mm/1.1 mm\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e1 mm [\u003cspan citationid=\"CR62\" class=\"CitationRef\"\u003e62\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eAnteversion angle\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e\u0026minus;\u0026thinsp;1.7\u0026deg;\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e-0.7\u0026deg; [\u003cspan citationid=\"CR63\" class=\"CitationRef\"\u003e63\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eA second model without modeling the growth of the greater trochanter reduced the decrease of the NSA to 75%. In our model, closure of the apophyseal plate of the greater trochanter results in a higher NSA, matching in vivo data [\u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e55\u003c/span\u003e, \u003cspan citationid=\"CR56\" class=\"CitationRef\"\u003e56\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eCompressive stresses are highest at the medial side of the femoral shaft (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e, a). The bone apposition rate is 20, 40, 20, and 60 \u0026micro;m/month anterior, posterior, lateral and medial. This results in a decrease of bending stresses in the sagittal plane of 9% while bending in the frontal plane is essentially unchanged.\u003c/p\u003e\u003cp\u003eInner remodeling predicted an increase in bone density on the medial side of the proximal femur given by an increase of Young\u0026acute;s modulus (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e, b). These areas correspond to regions exhibiting higher compression.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e"},{"header":"3. Discussion","content":"\u003cp\u003eSeveral computational models of femoral long bone growth have been presented in the literature [\u003cspan additionalcitationids=\"CR65 CR66 CR67 CR68\" citationid=\"CR64\" class=\"CitationRef\"\u003e64\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR69\" class=\"CitationRef\"\u003e69\u003c/span\u003e]. These models provide valuable insights into the principles of the mechanobiology of bone growth. Furthermore, the development of various pediatric deformities can be explained [\u003cspan additionalcitationids=\"CR41 CR42 CR43\" citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e44\u003c/span\u003e]. However, they are limited to certain aspects of growth. Complex deformities, such as resulting from fractures, are corrected by a combination of altered growth of all growth plates and bony remodeling [\u003cspan citationid=\"CR70\" class=\"CitationRef\"\u003e70\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eThis study investigates stresses acting on the juvenile femur during gait using finite element analysis. A well-established mechanobiological model that uses octahedral shear stresses and hydrostatic stresses as the mechanical input values is employed to calculate the endochondral growth of the proximal and distal femur as well as that of the greater trochanter. Appositional growth of the femoral shaft and inner remodeling of the trabecular bone are also included.\u003c/p\u003e\u003cp\u003eThe calculated stresses depend to a large extent on the boundary conditions. The question of bending stress on bones has long been the subject of controversial debate in biomechanics. For adults, Sverdlova and Witzel [\u003cspan citationid=\"CR71\" class=\"CitationRef\"\u003e71\u003c/span\u003e] and later Lutz et al. [\u003cspan citationid=\"CR72\" class=\"CitationRef\"\u003e72\u003c/span\u003e] showed that the femoral shaft is primarily subjected to compressive stresses through muscle activation. This allows weight to be saved, which represents an evolutionary advantage [\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e]. On the other hand, Bertram and Biewener [\u003cspan citationid=\"CR73\" class=\"CitationRef\"\u003e73\u003c/span\u003e] argue that residual bending allows the direction of bending to be predicted. In this study inertia relief, muscle force optimization, and the iliotibial band are combined to model physiological boundary conditions.\u003c/p\u003e\u003cp\u003eThe stress analyses of the juvenile femur reveal reduced bending stresses in the sagittal plane. Nevertheless, bending in the frontal plane cannot be compensated for entirely. The highest appositional growth is predicted for the medial site, but the decrease in the NSA and the growth of the femoral neck result in a more eccentric hip joint resultant. In this simulation the functional adaption and compensate the additional bending, but is not able to further reduce bending stresses in the frontal plane. It can be assumed that functional adaption and muscles forces mainly reduce bending in the sagittal plane whereas the iliotibial tract is the primary mechanism for reducing bending stresses in the frontal plane.\u003c/p\u003e\u003cp\u003eHowever, with a modulus of elasticity of 12000 MPa for cortical bone, the bones in children and adolescents are significantly more elastic than in adults and its elasticity allows greater bending until material failure. Bones become stiffer as they grow. At the same time, the simulation of bone growth predicts functional changes that reduce bending stresses. The changes in bone morphology during growth can therefore be understood as adaptions to mechanical loading.\u003c/p\u003e\u003cp\u003eIn the proximal femur, peak compressive stresses are found medially and result in an increase in bone density similar to previous studies [\u003cspan citationid=\"CR52\" class=\"CitationRef\"\u003e52\u003c/span\u003e]. This shifts the center of mass of the neck of the femur medially towards the resulting axial force. In the diaphysis, internal axial loading is between 200 and 985 N during gait, while shear loading is considerably lower. This is in agreement with comparable studies in adults [\u003cspan citationid=\"CR74\" class=\"CitationRef\"\u003e74\u003c/span\u003e], whereas the weight-adapted axial load is higher. They have important implications in the healing processes of pediatric femoral shaft fractures, as axial loading can enhance fracture healing, while shear loading decelerates fracture healing [\u003cspan citationid=\"CR75\" class=\"CitationRef\"\u003e75\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eThe values for changes in bone length, diaphyseal diameter, NSA, and anteversion agree well with anthropometric values. However, the decline in the LDFA is not observed during adolescent growth. A possible explanation might be that the ring of LaCroix, which is not modeled, reduces the deformity of the cartilage in the growth plate. The predicted changes are also a mechanism of functional adaptation, as a decline in the NSA is associated with lower loading on the proximal femur [\u003cspan citationid=\"CR76\" class=\"CitationRef\"\u003e76\u003c/span\u003e]. Growth of the greater trochanter and the femoral neck isthmus could preserve the morphology of the metaphysis during longitudinal growth. The slight increase in ATD not observed in human growth indicates that either the apophyseal plate is to be modeled with a greater thickness or be orientated more horizontally. The highest increase in cortical thickness is predicted on the medial side of the diaphysis. Indeed, the femoral shaft has a higher thickness on the medial compared to the lateral side [\u003cspan citationid=\"CR77\" class=\"CitationRef\"\u003e77\u003c/span\u003e]. However, the highest cortical bone thickness is observed posterior. Due to the curvature of the femur in activities resulting in bending of the shaft, high compressive stresses can be expected on the posterior side. As the physiological loading of the bone was simplified as gait, higher impact loading like running and jumping are probable causes of bone formation on the concave side. In conclusion, it could thus be shown that the mechanobiological model of bone growth presented here is suitable for reproducing the physiological growth of the femur with some limitations.\u003c/p\u003e\u003cp\u003eOther limitations of the model include the assumption of constant muscle strength. However, as the angle of the femur changes, the lever arms of the muscles also change, and thereby the vector of the muscle force. Furthermore, the increase in weight leads to an increase in ground reaction force and a higher level of muscle strength due to muscle build-up. Secondarily, this also results in altered hip and knee joint forces. Further developments of the model should therefore include a stronger coupling of the calculations of bony and muscular growth. As discussed above, the simulated load only includes the gait cycle and therefore does not include the maximum loads that occur during sport and play. However, frequent loads are particularly relevant for the morphological adaptation of the bones to the stresses placed on them and gait is subject to less inter-individual variation than sporting activities. These reasons support the choice of gait as a functional load. Also, the cartilaginous growth at the surface of the femoral head and the greater trochanter was not included and periosteal growth was only considered for the diaphysis. Material models are simplified as linear elastic and isotropic and the geometry is simplified. Yadav et al. [\u003cspan citationid=\"CR69\" class=\"CitationRef\"\u003e69\u003c/span\u003e] were able to demonstrate that modeling of patient-specific growth plate geometry increased the accuracy of the predictions. Future research using medical images to develop a patient-specific model and compare the in silico results with the respective radiological follow-up may allow a more detailed validation.\u003c/p\u003e\u003cp\u003eThe model contains a constant biological baseline, which in reality is influenced by hormones and cellular senescence. For clinical decision-making, it is often necessary to consider growth till its completion. Further research should aim to simulate a longer period of growth. Including the effects of sex hormones and cellular senescence on the growth plate might allow for modeling fusion of the growth plates. However, in the current model growth is simulated by updating the location of the nodes. This only allows the simulation of growth over a few months, otherwise element distortions will occur. Manipulating a volume model and remeshing instead of directly manipulating the mesh might be a feasible solution to this problem [\u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e46\u003c/span\u003e].\u003c/p\u003e"},{"header":"4. Conclusions","content":"\u003cp\u003eBone growth in children does not only lead to an increase in length and thickness but also to complex three-dimensional changes in shape. The implementation of all growth plates and periosteal growth in computational simulations of bone growth can contribute to a better understanding of the interaction between morphological changes and functional loads. Optimizing muscle strength and functional adaptation together reduce bending stresses in growing bone in this in silico model. Future research should aim to simulate bone growth in complex deformities and might be a tool to predict bone remodeling after fractures.\u003c/p\u003e"},{"header":"5. Materials and Methods","content":"\u003cp\u003eA finite element mesh of the adolescent right femur, made freely available at SimTK by Kainz et al. [\u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e64\u003c/span\u003e] based on MRI data of a typically developed child (8 years old, weight: 20.4 kg, height: 1.24 m), was imported into ANSYS Mechanical version 19.2 (ANSYS Inc, Canonsburg, PA, USA). The modeling approach has previously been shown to accurately predict growth in 18 out of 20 cases [\u003cspan citationid=\"CR78\" class=\"CitationRef\"\u003e78\u003c/span\u003e]. The mesh has 22560 elements and consists of 3-D 8-node structural solids (SOLID185). The outer shaft diameter was adapted to 15.5 mm matching values for boys of the age of 8 [\u003cspan citationid=\"CR62\" class=\"CitationRef\"\u003e62\u003c/span\u003e]. The model includes the medullary cavity. The diaphysis is composed of cortical bone, and the meta- and epiphyses are composed of cancellous bone covered by a layer of cortical bone. The proximal and distal growth plates with thicknesses of 0.9 mm and 2.1 mm, respectively, are modeled [\u003cspan citationid=\"CR79\" class=\"CitationRef\"\u003e79\u003c/span\u003e]. The widths are in a ratio of approximately 30:70 to simulate the different contributions of the two epiphyseal plates to femoral growth [\u003cspan citationid=\"CR80\" class=\"CitationRef\"\u003e80\u003c/span\u003e] and correspond to the literature values of an 8-year-old male [\u003cspan citationid=\"CR81\" class=\"CitationRef\"\u003e81\u003c/span\u003e, \u003cspan citationid=\"CR82\" class=\"CitationRef\"\u003e82\u003c/span\u003e]. Additionally, a layer of elements representing the resting zone is included in the model. The growth plate of the apophysis of the trochanter major is modeled with an angle of 50\u0026deg; to the body horizontal [\u003cspan citationid=\"CR83\" class=\"CitationRef\"\u003e83\u003c/span\u003e]. Pozdnikin et al. [\u003cspan citationid=\"CR60\" class=\"CitationRef\"\u003e60\u003c/span\u003e]. reported an almost constant articulotrochanteric distance measured between the tip of the greater trochanter and the upper point of the femoral head in normal hips of children aged 3 to 17 years. Therefore, the growth plate of the greater trochanter was assumed to have the same thickness as the proximal growth plate to map growth at the same rate. The cartilage of the femoral neck isthmus as well as the articular cartilage is modeled as one layer of proliferative chondrocytes. To reduce computation time and like comparable finite element models [\u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e, \u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e64\u003c/span\u003e], all materials are assumed to be linearly elastic, isotropic, and homogeneous. The material properties are given in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e2\u003c/span\u003e. Young\u0026acute;s modulus of the growth plate highly affects the deformation and therefore the growth direction. Values in the literature range, starting with 0.49 MPa for the proliferative and hypertrophic zone and 0.99 MPa for the resting zone [\u003cspan citationid=\"CR84\" class=\"CitationRef\"\u003e84\u003c/span\u003e]. Therefore, a sensitivity analysis is performed to investigate the influence of the rigidity of the growth plates on growth tendency.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eInitial material properties\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"4\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eMaterial\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eDensity in kg/m^3\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eYoung's modulus in MPa\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003ePoisson's ratio\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eCortical bone\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e1100 [\u003cspan citationid=\"CR85\" class=\"CitationRef\"\u003e85\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e12000 [\u003cspan citationid=\"CR87\" class=\"CitationRef\"\u003e87\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.3 [\u003cspan citationid=\"CR90\" class=\"CitationRef\"\u003e90\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eCancellous bone\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e150 [\u003cspan citationid=\"CR85\" class=\"CitationRef\"\u003e85\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e345 [\u003cspan citationid=\"CR88\" class=\"CitationRef\"\u003e88\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.25 [\u003cspan citationid=\"CR90\" class=\"CitationRef\"\u003e90\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eCartilage\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e1100 [\u003cspan citationid=\"CR86\" class=\"CitationRef\"\u003e86\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e0.5\u0026ndash;10 [\u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e, \u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e64\u003c/span\u003e, \u003cspan citationid=\"CR84\" class=\"CitationRef\"\u003e84\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.47 [\u003cspan citationid=\"CR84\" class=\"CitationRef\"\u003e84\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eBone marrow\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e1000 [\u003cspan citationid=\"CR86\" class=\"CitationRef\"\u003e86\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e0.01 [\u003cspan citationid=\"CR89\" class=\"CitationRef\"\u003e89\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.49 [\u003cspan citationid=\"CR89\" class=\"CitationRef\"\u003e89\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eRigid boundary conditions with fixed bearings in all directions at the condyles lead to artificially high stresses [\u003cspan citationid=\"CR91\" class=\"CitationRef\"\u003e91\u003c/span\u003e]. Therefore, physiological boundary conditions with a bearing fixed in all directions at the femoral notch, a bearing fixed in the anterior-posterior direction at the lateral epicondyle of the distal femur, and a bearing fixed in the anterior-posterior and medial-lateral directions at the middle of the femoral head were used [\u003cspan citationid=\"CR71\" class=\"CitationRef\"\u003e71\u003c/span\u003e]. To prevent artificial stresses near the bearings and to take into account the inertia forces generated during movement, inertia relief is used. This method can be used to balance the forces when analyzing individual load steps during the gait and achieve physiological stresses [\u003cspan citationid=\"CR92\" class=\"CitationRef\"\u003e92\u003c/span\u003e]. The boundary conditions are thereby only used to prevent rigid body movements, but the bearings themselves do not generate a reaction force. Instead, the applied forces are balanced by the inertial forces that the body experiences in the acceleration field. The acceleration field is defined as the standard gravitational force on the earth in the inferior direction. To model bone growth by thermal expansion, the femur was fixed at both condyles in all directions.\u003c/p\u003e\u003cp\u003eUsing load cases, a dynamic gait cycle can be represented by a finite number of static simulations. These discrete load cases can be combined as a functional load history to calculate the mechanical stimulus [\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e]. For this purpose, a physiological superposition of all loading cases is performed by cumulating the highest values of compressive stresses occurring in one of the loading cases [\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e]. Nine load cases based on a published multibody simulation were identified and used as initial muscle and joint forces [\u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e64\u003c/span\u003e]. In addition, the force of the iliotibial band is included, which reduces bending stresses in the diaphysis. This force is taken into account with 50% of the amount of the hip joint resultant of the respective load case [\u003cspan citationid=\"CR83\" class=\"CitationRef\"\u003e83\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eOptimization of muscle forces has been shown to reduce bending stresses, and bending minimization can be used as a principle for determining muscle forces [\u003cspan citationid=\"CR71\" class=\"CitationRef\"\u003e71\u003c/span\u003e]. Therefore, the ANSYS subproblem routine is performed with 30 iterations for each load case. The muscle forces are used as design variables and are allowed to vary within a range of 50 to 200% of the initial value. Forces acting on the hip and knee joint are adopted as fixed values. Bending stresses are calculated in the mediolateral and anteroposterior directions by analyzing the absolute highest principal stresses at four points in the middle of the shaft [\u003cspan citationid=\"CR93\" class=\"CitationRef\"\u003e93\u003c/span\u003e]:\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$\\:{\\sigma\\:}_{bending}=\\pm\\:\\left|\\frac{{\\sigma\\:}_{medial/anterior}-{\\sigma\\:}_{lateral/posterior}}{2}\\right|$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe sum of both bending stresses at the midshaft is minimized (objective function). Additionally, both are independently used as state variables with a target value below 10 MPa. After importing the mesh and optimizing the muscle forces, growth is iteratively simulated (Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e).\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eEpiphyseal growth is modeled as thermal expansion. Proliferation increases the overall volume and therefore accounts for the isotropic growth ratio, while hypertrophy enlarges the columns unidirectional. A combination of biomechanical and histological analyses found a high correlation between the direction of the distortion of the hypertrophic chondrocytes and orientation of the columns of the growth plate, a mechanism by which shear stresses are reduced [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e]. Therefore, for every element representing hypertrophic chondrocytes, the average deformation over all nine load cases is aligned with the z-axis of the element\u0026acute;s local coordinate system. The growth rate is the sum of the biological baseline r\u003csub\u003eb0\u003c/sub\u003e and a mechanically stimulated growth rate r\u003csub\u003em\u003c/sub\u003e and is modeled as temperature difference. The growth of an element of the growth plate g can be calculated by the local thickness of the respective zone l and both growth rates as\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$$\\:g=l\\bullet\\:({r}_{b0}+{r}_{m})$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eProliferation and production of extracellular matrix in the proliferative zone account for about 27%, and hypertrophy and production of extracellular matrix in the hypertrophic zone account for 73% [\u003cspan citationid=\"CR94\" class=\"CitationRef\"\u003e94\u003c/span\u003e]. Both growth mechanisms are modeled by the choice of thermal expansion coefficients. r\u003csub\u003eb0\u003c/sub\u003e and r\u003csub\u003em\u003c/sub\u003e are calculated based on a model developed by Wong and Carter [\u003cspan citationid=\"CR65\" class=\"CitationRef\"\u003e65\u003c/span\u003e], Stevens et al. [\u003cspan citationid=\"CR66\" class=\"CitationRef\"\u003e66\u003c/span\u003e], and Yadav et al. [\u003cspan additionalcitationids=\"CR68\" citationid=\"CR67\" class=\"CitationRef\"\u003e67\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR69\" class=\"CitationRef\"\u003e69\u003c/span\u003e]. Briefly, based on a reported growth rate of the proximal femur of 9 mm/year [\u003cspan citationid=\"CR80\" class=\"CitationRef\"\u003e80\u003c/span\u003e], an assumed percentage of nonmechanically stimulated growth of 66% [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan citationid=\"CR67\" class=\"CitationRef\"\u003e67\u003c/span\u003e], and a thickness of the proximal growth plate of 0.9 mm, r\u003csub\u003eb0\u003c/sub\u003e is given as 0.6 mm/month. r\u003csub\u003em\u003c/sub\u003e is given by the sum of the maximum octahedral shear stress and the minimum hydrostatic stress obtained over all nine loading cases, where the coefficients a and b describe the respective influence of octahedral shear stress and hydrostatic stress on r\u003csub\u003em\u003c/sub\u003e [\u003cspan citationid=\"CR66\" class=\"CitationRef\"\u003e66\u003c/span\u003e]:\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e\n$$\\:{r}_{m}=a\\bullet\\:max\\left[\\frac{\\sqrt{{\\left({\\sigma\\:}_{1}-{\\sigma\\:}_{2}\\right)}^{2}+{\\left({\\sigma\\:}_{2}-{\\sigma\\:}_{3}\\right)}^{2}+{\\left({\\sigma\\:}_{3}-{\\sigma\\:}_{1}\\right)}^{2}}}{3}\\right]+b\\bullet\\:min\\left[\\frac{{\\sigma\\:}_{1}+{\\sigma\\:}_{2}+{\\sigma\\:}_{3}}{3}\\right]$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe ratio of b/a was set at 0.5 in agreement with previous studies [\u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e] and calibrated to correspond to 33% of total growth.\u003c/p\u003e\u003cp\u003eChanges in bone density in cancellous bone and thereby changes in Young\u0026acute;s modulus can be described by a differential equation developed by Li et al. [\u003cspan citationid=\"CR95\" class=\"CitationRef\"\u003e95\u003c/span\u003e]. It considers bone resorption due to disuse, bone formation due to physiological overload in a dose-dependent manner, and bone resorption due to microdamage and pathological overload. 2 MPa is used as thresholds for bone formation [\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e]. Pathological overload is assumed for loading four times higher [\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e, \u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e]. Furthermore, it is assumed that the curve is symmetrical. This can be used to solve a system of equations for 3 support points. The maximum bone formation rate per bone volume in trabecular bone is 4% per month [\u003cspan citationid=\"CR97\" class=\"CitationRef\"\u003e96\u003c/span\u003e]. The Young\u0026acute;s modulus of a specific element representing cancellous bone after one month of simulated loading E\u003csub\u003ei+1\u003c/sub\u003e can therefore be expressed as function of the volume at the beginning of the iteration E\u003csub\u003ei\u003c/sub\u003e\u003cdiv id=\"Equd\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equd\" name=\"EquationSource\"\u003e\n$$\\:{E}_{i+1}={E}_{i}+\\left(0.2916\\bullet\\:{e}^{0.7702{\\sigma\\:}_{3}-0.0772{{\\sigma\\:}_{3}}^{2}}-1\\right)\\bullet\\:0.04\\:MPa$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ePhysiological loading of cortical bone is around ten times higher than in cancellous bone [\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e]. In contrast to cancellous bone and the endosteal surface, animal models found no periosteal resorption even without any loading [\u003cspan citationid=\"CR98\" class=\"CitationRef\"\u003e97\u003c/span\u003e]. With a maximal apposition rate of 0.06 mm/month [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e], the mechanically stimulated apposition rate per month AR is given by\u003cdiv id=\"Eque\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Eque\" name=\"EquationSource\"\u003e\n$$\\:AR=\\left({e}^{0.0346574{\\sigma\\:}_{3}-0.000433217{{\\sigma\\:}_{3}}^{2}}-1\\right)\\bullet\\:0.06\\:mm/month$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ePrincipal stresses are analyzed by superposition of the minimal and maximal principal stresses, respectively. To calculate internal axial loading, a surface perpendicular to the femoral shaft axis is created. Normal stresses on this surface are averaged and multiplied with the cross-sectional area to obtain axial forces. Values are converted to percent body weight. After simulation of bone growth, the following values were measured and compared to reference values for validation (Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e):\u003c/p\u003e\u003cul\u003e\n \u003cli\u003eFemur length: The distance between the center of the femoral head and the intercondylar femoral fossa is used here, as these two points are also used to determine the mechanical femoral axis [97].\u003c/li\u003e\n \u003cli\u003eNeck-shaft angle (NSA): The projected NSA angle is defined in the frontal plane as the angle between the femoral neck axis and the femoral shaft axis. To determine this, a circle is drawn around the center of the femoral head that intersects the femoral neck medially and laterally. The femoral neck axis can then be constructed by drawing a line through the middle of this line and the center of the head. The femoral shaft axis is the line between the centers of two transverse diameters below the lesser trochanter [98].\u003c/li\u003e\n \u003cli\u003eArticulo-trochanteric distance (ATD): The articulo-trochanteric distance is calculated as the distance between two parallel lines orthograde to the femoral shaft axis that are tangential to the maximum points of the femoral head and the greater trochanter [99].\u003c/li\u003e\n \u003cli\u003eLateral distal femoral angle (LDFA): The mechanical lateral distal femoral angle is the lateral angle between the mechanical femoral axis and the knee line as a tangent to the most distal points of both knee condyles. There is also the anatomical lateral distal femoral angle between the femoral shaft axis and the knee line, but this is less commonly used and is therefore not used in this study [100].\u003c/li\u003e\n \u003cli\u003eAntetorsion angle (AT): The antetorsion angle describes the rotation of the femoral neck in relation to the shaft. It is measured in the axial plane between the projection of the femoral neck axis and a line tangential to the distal condyles [101].\u003c/li\u003e\n \u003cli\u003eAnteroposterior and mediolateral diameter: Since the femoral shaft is not cylindrical and shows an eccentric growth behavior, the diameter is determined at the level of the middle of the femur in the axial section between the most anterior and posterior as well as medial and lateral points.\u003c/li\u003e\n\u003c/ul\u003e"},{"header":"Declarations","content":"\u003cp\u003eEthics approval and consent to participate: Not applicable\u003c/p\u003e\n\u003cp\u003eConsent for publication: Not applicable\u003c/p\u003e\n\u003cp\u003eAvailability of data and materials: The data presented in this study are available on request from the corresponding author.\u003c/p\u003e\n\u003cp\u003eCompeting interests: The authors declare that they have no competing interests.\u003c/p\u003e\n\u003cp\u003eFunding: This research was funded by Deutsche Forschungsgesellschaft (DFG, German Research Foundation), grant number 445465815. The content is solely the responsibility of the authors and does not necessarily represent the official views of the DFG.\u003c/p\u003e\n\u003cp\u003eAuthors\u0026apos; contributions: AL conceptualized the project, developed the model, performed the finite element analysis and wrote the manuscript. RB analyzed, interpreted, and validated the modeling results. MK and SS assisted in computational modeling and were major contributors in writing the manuscript. UW supervised the project and was a major contributor in conceptualization of the project and interpreting the results. All authors read and approved the final manuscript.\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;Acknowledgements: The authors are grateful to Hans Kainz and colleagues for making the mesh of the juvenile femur used in this study publicly available under the MIT Use Agreement at SimTK: https://simtk.org/projects/normal-load/. The authors also want to thank Beate Bender, Product Development, and the Institute of Product and Service Engineering, Ruhr-University Bochum, for the provision of computational equipment.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eEmons J, Chagin AS, S\u0026auml;vendahl L, Karperien M, Wit JM. Mechanisms of growth plate maturation and epiphyseal fusion. Horm Res Paediatr. 2011,75:383-91\u003c/li\u003e\n\u003cli\u003eAbad V, Meyers JL, Weise M, Gafni RI, Barnes KM, Nilsson O, Bacher JD, Baron J. The Role of the Resting Zone in Growth Plate Chondrogenesis. Endocrinology. 2002,143:1851-57\u003c/li\u003e\n\u003cli\u003eNilsson O, Mitchum RD, Schrier L, Ferns SP, Barnes KM, Troendle JF, Baron J. Growth plate senescence is associated with loss of DNA methylation. J Endocrinol. 2005,186:241-9\u003c/li\u003e\n\u003cli\u003ePrein C, Warmbold N, Farkas Z, Schieker M, Aszodi A, Clausen-Schaumann H. Structural and mechanical properties of the proliferative zone of the developing murine growth plate cartilage assessed by atomic force microscopy. Matrix Biol. 2016,50:1-15\u003c/li\u003e\n\u003cli\u003eBuckwalter JA, Mower D, Ungar R, Schaeffer J, Ginsberg B. Morphometric analysis of chondrocyte hypertrophy. J Bone Joint Surg Am. 1986,68:243-55\u003c/li\u003e\n\u003cli\u003eAghajanian P, Mohan S. The art of building bone: emerging role of chondrocyte-to-osteoblast transdifferentiation in endochondral ossification. Bone Res. 2018,6:19\u003c/li\u003e\n\u003cli\u003eWatanabe-Takano H, Ochi H, Chiba A, Matsuo A, Kanai Y, Fukuhara S, Ito N, Sako K, Miyazaki T, Tainaka K, Harada I, Sato S, Sawada Y, Minamino N, Takeda S, Ueda HR, Yasoda A, Mochizuki N. Mechanical load regulates bone growth via periosteal Osteocrin. Cell Rep. 2021,36:109380\u003c/li\u003e\n\u003cli\u003eRobling AG, Turner CH. Mechanical Signaling for Bone Modeling and Remodeling. Crit Rev Eukaryot Gene Expr. 2009,19:319-38.\u003c/li\u003e\n\u003cli\u003eRauch F. Bone growth in length and width: the Yin and Yang of bone stability. J Musculoskelet Neuronal Interact. 2005,5:194-201\u003c/li\u003e\n\u003cli\u003eDe Luca F. Impaired Growth Plate Chondrogenesis in Children with Chronic Illnesses. Pediatr Res. 2006,59:625-9\u003c/li\u003e\n\u003cli\u003eRanke MB. Short and Long-Term Effects of Growth Hormone in Children and Adolescents With GH Deficiency. Front Endocrinol. 2021,12:720419\u003c/li\u003e\n\u003cli\u003ePerry RJ, Farquharson C, Ahmed SF. The role of sex steroids in controlling pubertal growth. Clin Endocrinol. 2008, 68, 4-15\u003c/li\u003e\n\u003cli\u003eNilsson O. Genetic and epigenetic regulation of childhood growth. Nat Rev Endocrinol. 2018,14, 70-2\u003c/li\u003e\n\u003cli\u003eHueter C. Anatomische Studien an den Extremit\u0026auml;tengelenken Neugeborener und Erwachsener. Archiv f. pathol. Anat. 1863,26:484\u0026ndash;519\u003c/li\u003e\n\u003cli\u003eNiehoff A, Kersting UG, Zaucke F, Morlock MM, Br\u0026uuml;ggemann GP. Adaptation of mechanical, morphological, and biochemical properties of the rat growth plate to dose-dependent voluntary exercise. Bone. 2004,35: 899-908\u003c/li\u003e\n\u003cli\u003eHall BK, Herring SW. Paralysis and growth of the musculoskeletal system in the embryonic chick. J Morphol. 1990,206:45-56\u003c/li\u003e\n\u003cli\u003eStokes IA, Aronsson DD, Dimock AN, Cortright V, Beck S. Endochondral growth in growth plates of three species at two anatomical locations modulated by mechanical compression and tension. J Orthop Res. 2006,24:1327-34\u003c/li\u003e\n\u003cli\u003eStokes IAF, Spence H, Aronsson DD, Kilmer N. Mechanical Modulation of Vertebral Body Growth: Implications for Scoliosis Progression. Spine. 1996,21:1162-7\u003c/li\u003e\n\u003cli\u003eSnyder A, Zierath JR, Hawley JA, Sleeper MD, Craig BW. The effects of exercise mode, swimming vs. running, upon bone growth in the rapidly growing female rat. Mech Ageing Dev. 1992,66:59-69\u003c/li\u003e\n\u003cli\u003eKrahl H, Michaelis U, Pieper HG, Quack G, Montag M. Stimulation of bone growth through sports. A radiologic investigation of the upper extremities in professional tennis players. Am J Sports Med. 1994,22:751-7\u003c/li\u003e\n\u003cli\u003eSmith JW. The relationship of epiphysial plates to stress in some bones of the lower limb. J Anat. 1962,96:58-78\u003c/li\u003e\n\u003cli\u003eAlonso G, Yawny A, Bertolino G. How do bones grow? A mathematical description of the mechanobiological behavior of the epiphyseal plate. Biomech Model Mechanobiol. 2022,21:1585-601\u003c/li\u003e\n\u003cli\u003eHunziker E. Mechanism of longitudinal bone growth and its regulation by growth plate chondrocites. Microsc Res Tech. 1994,28:505-19\u003c/li\u003e\n\u003cli\u003eArkin AM, Katz JF. The effects of pressure on epiphyseal growth; the mechanism of plasticity of growing bone. J Bone Joint Surg Am. 1956,38:1056-76\u003c/li\u003e\n\u003cli\u003ePauwels F. Gesammelte Abhandlungen zur funktionellen Anatomie des Bewegungsapparates. Springer: Berlin/Heidelberg/New York, Germany, 1965\u003c/li\u003e\n\u003cli\u003eCarter DR, Wong M. The role of mechanical loading histories in the development of diarthrodial joints. J Orthop Res. 1988,6:804-16\u003c/li\u003e\n\u003cli\u003eFeik SA, Storey E, Ellender G. Stress induced periosteal changes. Br J Exp Pathol. 1987,68:803-13\u003c/li\u003e\n\u003cli\u003eGross TS, Edwards JL, McLeod KJ, Rubin CT. Strain gradients correlate with sites of periosteal bone formation. J Bone Miner Res. 1997,12:982-8\u003c/li\u003e\n\u003cli\u003eLipphaus A, Witzel U. Biomechanical Study of the Development of Long Bones: Finite Element Structure Synthesis of the Human Second Proximal Phalanx Under Growth Conditions. Anat Rec. 2019,302:1389-98\u003c/li\u003e\n\u003cli\u003eWitzel U. Eine neue Methode zur virtuellen Sch\u0026auml;delsynthese am Beispiel Camarasaurus. Hallesches Jahrb F\u0026uuml;r Geowiss. 2007,23:73-8\u003c/li\u003e\n\u003cli\u003eFrost HM. Bone\u0026apos;s mechanostat: a 2003 update. Anat Rec A Discov Mol Cell Evol Biol. 2003,275:1081-101\u003c/li\u003e\n\u003cli\u003eBiewener AA, Bertram JE. Structural response of growing bone to exercise and disuse. J Appl Physiol. 1994,76:946-55\u003c/li\u003e\n\u003cli\u003eWillie BM, Birkhold AI, Razi H, Thiele T, Aido M, Kruck B, Schill A, Checa S, Main RP, Duda GN. Diminished response to in vivo mechanical loading in trabecular and not cortical bone in adulthood of female C57Bl/6 mice coincides with a reduction in deformation to load. Bone. 2013,55:335-46\u003c/li\u003e\n\u003cli\u003eMustafy T, Londono I, Moldovan F, Villemure I. Isolated Cyclic Loading During Adolescence Improves Tibial Bone Microstructure and Strength at Adulthood. JBMR Plus. 2020,4:e10349\u003c/li\u003e\n\u003cli\u003eMcCabe K, Henderson K, Pantinople J, Richards HL, Milne N. Curvature reduces bending strains in the quokka femur. PeerJ. 2017,5:e3100\u003c/li\u003e\n\u003cli\u003eWitzel U, Preuschoft H. Finite-element model construction for the virtual synthesis of the skulls in vertebrates: case study of Diplodocus. Anat Rec A Discov Mol Cell Evol Biol. 2005,283:391-401\u003c/li\u003e\n\u003cli\u003eHeegaard JH, Beaupr\u0026eacute; GS, Carter DR. Mechanically modulated cartilage growth may regulate joint surface morphogenesis. J Orthop Res. 1999,17:509-17\u003c/li\u003e\n\u003cli\u003eCarpenter RD, Carter DR. Computational simulation of spontaneous bone straightening in growing children. Biomech Model Mechanobiol. 2010,9:317-28\u003c/li\u003e\n\u003cli\u003eShefelbine SJ, Tardieu C, Carter DR. Development of the femoral bicondylar angle in hominid bipedalism. Bone. 2002,30:765-70\u003c/li\u003e\n\u003cli\u003eShefelbine SJ, Carter DR. Mechanobiological predictions of femoral anteversion in cerebral palsy. Ann Biomed Eng. 2004,32:297-305\u003c/li\u003e\n\u003cli\u003eSadeghian SM, Lewis CL, Shefelbine SJ. Predicting growth plate orientation with altered hip loading: potential cause of cam morphology. Biomech Model Mechanobiol. 2020,19:701-12\u003c/li\u003e\n\u003cli\u003eShefelbine SJ, Carter DR. Mechanobiological predictions of growth front morphology in developmental hip dysplasia. J Orthop Res. 2004,22:346-52\u003c/li\u003e\n\u003cli\u003eGiorgi M, Carriero A, Shefelbine SJ, Nowlan NC. Effects of normal and abnormal loading conditions on morphogenesis of the prenatal hip joint: application to hip dysplasia. J Biomech. 2015;48:3390-7\u003c/li\u003e\n\u003cli\u003eLipphaus A, Tr\u0026ouml;bs RB, Wegener-Panzer A, Witzel U. Biomechanische Simulation der Entstehung eines Genu valgum bei H\u0026uuml;ftdysplasie. In Proceedings of the Deutscher Kongress f\u0026uuml;r Orthop\u0026auml;die und Unfallchirurgie, Berlin, Germany, 28 10 2021\u003c/li\u003e\n\u003cli\u003eStokes IAF. Analysis and simulation of progressive adolescent scoliosis by biomechanical growth modulation. Eur Spine J. 2007,16:1621-28\u003c/li\u003e\n\u003cli\u003eAlonso MG, Bertolino G, Yawny A. Mechanobiological based long bone growth model for the design of limb deformities correction devices. J Biomech. 2020,109:109905\u003c/li\u003e\n\u003cli\u003eCarrera-Pinz\u0026oacute;n AF, M\u0026aacute;rquez-Fl\u0026oacute;rez K, Kraft RH, Ramtani S, Garz\u0026oacute;n-Alvarado DA. Computational model of a synovial joint morphogenesis. Biomech Model Mechanobiol. 2020,19:1389-1402\u003c/li\u003e\n\u003cli\u003eVaca-Gonz\u0026aacute;lez JJ, Moncayo-Donoso M, Guevara JM, Hata Y, Shefelbine SJ, Garz\u0026oacute;n-Alvarado DA. Mechanobiological modeling of endochondral ossification: an experimental and computational analysis. Biomech Model Mechanobiol. 2018,17:853-75\u003c/li\u003e\n\u003cli\u003eSadeghian SM, Shapiro FD, Shefelbine SJ. Computational model of endochondral ossification: Simulating growth of a long bone. Bone. 2021,153:116132\u003c/li\u003e\n\u003cli\u003eTerryn PJ, Fernandes PR, Jo\u0026atilde;o F, Veloso A, Baptista F. Modeling the musculoskeletal loading in bone remodeling at the hip of a child. Comput Methods Programs Biomed. 2021,210:106365\u003c/li\u003e\n\u003cli\u003eQuexada-Rodr\u0026iacute;guez D, Trabelsi O, Hobatho MC, Ramtani S, Garz\u0026oacute;n-Alvarado D. Influence of growth plate morphology on bone trabecular groups, a framework computational approach. Bone. 2023,171:116742\u003c/li\u003e\n\u003cli\u003eBeaupr\u0026eacute; GS, Orr TE, Carter DR. An approach for time-dependent bone modeling and remodeling-application: a preliminary remodeling simulation. J Orthop Res. 1990,8:662-70\u003c/li\u003e\n\u003cli\u003eLee MC, Eberson CP. Growth and development of the child\u0026apos;s hip. Orthop Clin North Am. 2006,37:119-32\u003c/li\u003e\n\u003cli\u003eHeimkes, B. Die gro\u0026szlig;en Apophysen. Orthop\u0026auml;de. 2016:45:206-12\u003c/li\u003e\n\u003cli\u003eStruijs PA, Oostra RJ, van Rijn RR, Besselaar PP. Abnormal growth of the proximal femur due to apophyseal-epiphyseal coalescence resulting in coxa valga--a report of two cases in adolescents. Acta Orthop. 2011,82:507-9\u003c/li\u003e\n\u003cli\u003eLangenski\u0026ouml;ld A, Salenius P. Epiphyseodesis of the greater trochanter. Acta Orthop Scand. 1967,38:199-219\u003c/li\u003e\n\u003cli\u003eKainz H, Koller W, Bastir M, Svehlik M, Schwartz MH. The shape and size of the femur adapts during growth to maintain a constant cartilage load. Gait Posture. 2024,113:105-7\u003c/li\u003e\n\u003cli\u003eAnderson M, Green WT, Messner MB. Growth and predictions of growth in the lower extremities. J Bone Joint Surg Am. 1963,45-A:1-14\u003c/li\u003e\n\u003cli\u003eVan Geel P, Cools W, Laumen A. Evolution of head-shaft angle and neck-shaft angle in childhood. Acta Orthop Belg. 2021,87:435-41\u003c/li\u003e\n\u003cli\u003ePozdnikin IY, Baskov VE, Barsukov DB, Bortulev PI, Krasnov AI. Relative overgrowth of the greater trochanter and trochanteric-pelvic impingement syndrome in children: causes and X-ray anatomical characteristics. Pediatric Traumatol Orthop Reconstr Surg. 2019,7:15-24\u003c/li\u003e\n\u003cli\u003eSabharwal S, Zhao C, Edgar M. Lower limb alignment in children: reference values based on a full-length standing radiograph. J Pediatr Orthop. 2008,28:740-6\u003c/li\u003e\n\u003cli\u003eBradney M, Karlsson MK, Duan Y, Stuckey S, Bass S, Seeman E. Heterogeneity in the growth of the axial and appendicular skeleton in boys: implications for the pathogenesis of bone fragility in men. J Bone Miner Res. 2000,15:1871-8\u003c/li\u003e\n\u003cli\u003eUpadhyay SS, Burwell RG, Moulton A, Small PG, Wallace WA. Femoral anteversion in healthy children. Application of a new method using ultrasound. J Anat. 1990,169:49-61\u003c/li\u003e\n\u003cli\u003eKainz H, Killen BA, Wesseling M, Perez-Boerema F, Pitto L, Garcia Aznar JM, Shefelbine S, Jonkers I. A multi-scale modelling framework combining musculoskeletal rigid-body simulations with adaptive finite element analyses, to evaluate the impact of femoral geometry on hip joint contact forces and femoral bone growth. PLoS One. 2020,15:e0235966\u003c/li\u003e\n\u003cli\u003eWong M, Carter DR. A theoretical model of endochondral ossification and bone architectural construction in long bone ontogeny. Anat Embryol. 1990,181:523-32\u003c/li\u003e\n\u003cli\u003eStevens SS, Beaupr\u0026eacute; GS, Carter DR. Computer model of endochondral growth and ossification in long bones: biological and mechanobiological influences. J Orthop Res. 1999,17:646-53\u003c/li\u003e\n\u003cli\u003eYadav P, Shefelbine SJ, Pont\u0026eacute;n E, Gutierrez-Farewik EM. Influence of muscle groups\u0026apos; activation on proximal femoral growth tendency. Biomech Model Mechanobiol. 2017,16:1869-83\u003c/li\u003e\n\u003cli\u003eYadav P, Shefelbine SJ, Gutierrez-Farewik EM. Effect of growth plate geometry and growth direction on prediction of proximal femoral morphology. J Biomech. 2016,49:1613-19\u003c/li\u003e\n\u003cli\u003eYadav P, Pe\u0026ntilde;a Fern\u0026aacute;ndez M, Gutierrez-Farewik EM. Influence of loading direction due to physical activity on proximal femoral growth tendency, Med Eng Phys. 2021,90:83-91\u003c/li\u003e\n\u003cli\u003eNaik P. Remodelling in Children\u0026apos;s Fractures and Limits of Acceptability. Indian J Orthop. 2021,55:549-559\u003c/li\u003e\n\u003cli\u003eSverdlova NS, Witzel U. Principles of determination and verification of muscle forces in the human musculoskeletal system: Muscle forces to minimise bending stress. J Biomech. 2010,43:387-96\u003c/li\u003e\n\u003cli\u003eLutz F, Mastel R, Runge M, Stief F, Schmidt A, Meurer A, Witte H. Calculation of muscle forces during normal gait under consideration of femoral bending moments. Med Eng Phys. 2016,38:1008-15\u003c/li\u003e\n\u003cli\u003eBertram JE, Biewener AA. Bone curvature: sacrificing strength for load predictability? J Theor Biol. 1988,131:75-92\u003c/li\u003e\n\u003cli\u003eDuda GN, Schneider E, Chao EY. Internal forces and moments in the femur during walking. J Biomech. 1997,30:933-41\u003c/li\u003e\n\u003cli\u003eAugat P, Burger J, Schorlemmer S, Henke T, Peraus M, Claes L. Shear movement at the fracture site delays healing in a diaphyseal fracture model. J Orthop Res. 2003,21:1011-7\u003c/li\u003e\n\u003cli\u003eKainz H, Mindler GT, Kranzl A. Influence of femoral anteversion angle and neck-shaft angle on muscle forces and joint loading during walking. PLoS One. 2023,18:e0291458\u003c/li\u003e\n\u003cli\u003eSomeya K, Mochizuki T, Hokari S, Tanifuji O, Katsumi R, Koga H, Takahashi Y, Kobayashi K, Morise Y, Sakamoto M, Koga Y, Endo N. Age- and sex-related characteristics in cortical thickness of femoral diaphysis for young and elderly subjects. J Bone Miner Metab. 2020,38:533-43\u003c/li\u003e\n\u003cli\u003eKoller W, Svehlik M, Walln\u0026ouml;fer E, Kranzl A, Mindler G, Baca A, Kainz H. Femoral bone growth predictions based on personalized multi-scale simulations: validation and sensitivity analysis of a mechanobiological model. Biomech Model Mechanobiol. 2025. 24:879-94\u003c/li\u003e\n\u003cli\u003eLipphaus A, Wegener-Panzer A, Tr\u0026ouml;bs, RB, Witzel U. Computational modelling of proximal and distal epiphyseal and appositional growth of the femur in children. In Proceedings of XXVIII Congress of theInternational Society of Biomechanics (ISB): 25-29 June 2021; Digital congress:A287.\u003c/li\u003e\n\u003cli\u003ePritchett JW. Longitudinal growth and growth-plate activity in the lower extremity. Clin Orthop Relat Res. 1992,275:274-9\u003c/li\u003e\n\u003cli\u003eAbiodun AA, Falana BA, Olatunbosun A. Baseline Thickness of Proximal Femoral Epiphysis in the Diagnosis of Hip Diseases in Infants and Children in Nigeria. Int J Surg Res. 2014,3:22-26.\u003c/li\u003e\n\u003cli\u003eKim A, Dombi E, Solomon J, Fox E, Balis FM, Widemann BC. Automated volumetric growth plate measurement using magnetic resonance imaging for monitoring skeletal toxicity in children treated on investigational drug trials. Clin Cancer Res. 2011,17:5982-90\u003c/li\u003e\n\u003cli\u003eHeimkes B, Posel P, Plitz W, Zimmer, M. Die altersabh\u0026auml;ngige Kr\u0026auml;fteverteilung am koxalen Femurende des normal wachsenden Kindes. Z Orthop. 1997,135:17-23\u003c/li\u003e\n\u003cli\u003eGao J, Roan E, Williams JL. Regional variations in growth plate chondrocyte deformation as predicted by three-dimensional multi-scale simulations. PLoS One. 2015,10:e0124862\u003c/li\u003e\n\u003cli\u003eRauch F, Schoenau E. Changes in bone density during childhood and adolescence: an approach based on bone\u0026apos;s biological organization. J Bone Miner Res. 2001,16:597-604\u003c/li\u003e\n\u003cli\u003eHasgall PA, Di Gennaro F, Baumgartner C, Neufeld E, Lloyd B, Gosselin MC, Payne D, Klingenb\u0026ouml;ck A, Kuster N, \u0026ldquo;IT\u0026rsquo;IS Database for thermal and electromagnetic parameters of biological tissues,\u0026rdquo; Version 4.1, Feb 22, 2022, DOI: 10.13099/VIP21000-04-1. itis.swiss/database\u003c/li\u003e\n\u003cli\u003e\u0026Ouml;hman C, Baleani M, Pani C, Taddei F, Alberghini M, Viceconti M, Manfrini M. Compressive behaviour of child and adult cortical bone. Bone. 2011,49:769-76\u003c/li\u003e\n\u003cli\u003ePiszczatowski S. Material aspects of growth plate modelling using Carter\u0026apos;s and Stokes\u0026apos;s approaches. Acta Bioeng Biomech. 2011;13:3-14\u003c/li\u003e\n\u003cli\u003eJansen LE, Birch NP, Schiffman JD, Crosby AJ, Peyton SR. Mechanics of intact bone marrow. J Mech Behav Biomed Mater. 2015,50:299-307.\u003c/li\u003e\n\u003cli\u003eLai YS, Chen WC, Huang CH, Cheng CK, Chan KK, Chang TK. The effect of graft strength on knee laxity and graft in-situ forces after posterior cruciate ligament reconstruction. PLoS One. 2015,10:e0127293\u003c/li\u003e\n\u003cli\u003eSpeirs AD, Heller MO, Duda GN, Taylor WR. Physiologically based boundary conditions in finite element modelling. J Biomech. 2007,40:2318-23\u003c/li\u003e\n\u003cli\u003eHeyland M, Trepczynski A, Duda GN, Zehn M, Schaser KD, M\u0026auml;rdian S. Selecting boundary conditions in physiological strain analysis of the femur: Balanced loads, inertia relief method and follower load. Med Eng Phys. 2015,37:1180-5\u003c/li\u003e\n\u003cli\u003eMain RP, Biewener AA. Ontogenetic patterns of limb loading, in vivo bone strains and growth in the goat radius. J Exp Biol. 2004,207:2577-88\u003c/li\u003e\n\u003cli\u003eWilsman NJ, Farnum CE, Leiferman EM, Fry M, Barreto C. Differential growth by growth plates as a function of multiple parameters of chondrocytic kinetics. J Orthop Res. 1996,14:927-36.\u003c/li\u003e\n\u003cli\u003eLi J, Li H, Shi L, Fok AS, Ucer C, Devlin H, Horner K, Silikas N. A mathematical model for simulating the bone remodeling process under mechanical stimulus. Dent Mater. 2007,23:1073-8\u003c/li\u003e\n\u003cli\u003eGlorieux FH, Travers R, Taylor A, Bowen JR, Rauch F, Norman M, Parfitt AM. Normative data for iliac bone histomorphometry in growing children. Bone. 2000,26:103-9\u003c/li\u003e\n\u003cli\u003eLi B, Sankaran JS, Judex S. Trabecular and Cortical Bone of Growing C3H Mice Is Highly Responsive to the Removal of Weightbearing. PLoS One. 2016,11:e0156222\u003c/li\u003e\n\u003cli\u003eGuenoun B, Zadegan F, Aim F, Hannouche D, Nizard R. Reliability of a new method for lower-extremity measurements based on stereoradiographic three-dimensional reconstruction. Orthop Traumatol Surg Res. 2012,98:506-13\u003c/li\u003e\n\u003cli\u003eFischer CS, K\u0026uuml;hn JP, V\u0026ouml;lzke H, Ittermann T, G\u0026uuml;mbel D, Kasch R, Haralambiev L, Laqua R, Hinz P, Lange J. The neck-shaft angle: an update on reference values and associated factors. Acta Orthop. 2020,91:53-7\u003c/li\u003e\n\u003cli\u003eYoon PW, Shin YH, Yoo JJ, Yoon KS, Kim HJ. Progression of a fracture site impaction as a prognostic indicator of impacted femoral neck fracture treated with multiple pinning. Clin Orthop Surg. 2012,4:66-71\u003c/li\u003e\n\u003cli\u003eMarques Lu\u0026iacute;s N, Varatojo R. Radiological assessment of lower limb alignment. EFORT Open Rev. 2021,6:487-94\u003c/li\u003e\n\u003cli\u003eScorcelletti M, Reeves ND, Rittweger J, Ireland A. Femoral anteversion: significance and measurement. J Anat. 2020,237:811-26\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"biomedical-engineering-online","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"bmeo","sideBox":"Learn more about [BioMedical Engineering OnLine](http://biomedical-engineering-online.biomedcentral.com/)","snPcode":"12938","submissionUrl":"https://submission.nature.com/new-submission/12938/3","title":"BioMedical Engineering OnLine","twitterHandle":"@BioMedCentral","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"BMC/SO AJ","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"biomechanics, finite element, bone growth, endochondral ossification, growth plate, femoral neck isthmus, in silico modeling, pediatric orthopedics","lastPublishedDoi":"10.21203/rs.3.rs-7224633/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7224633/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn silico models for simulating bone growth based on mechanical or non-mechanical epigenetic factors are widely used. In this study, a well-known mechanobiological model, which states that octahedral shear stress accelerates longitudinal bone growth and hydrostatic stress retards it, is applied to a finite element model of the femur of an 8-year-old boy. Proximal and distal epiphyseal plates as well as the growth plate of the greater trochanter, cartilaginous growth at the femoral isthmus, and appositional bone growth are included in the model. Furthermore, changes in the density of the cancellous bone in the metaphyses are modeled based on Wolff's law using compressive stresses as the mechanical stimulus. Muscle forces during a dynamic gait cycle were determined for nine discrete loading cases by optimizing to minimize bending stress. The highest stresses in the femoral shaft were determined as medial compressive stresses with a maximum of -33.2 MPa. Highest internal axial load in the shaft was 985 N during loading response. The simulated bone growth resulted in an increase in femur length of 26 mm and a decrease in femoral neck angle by -0.4\u0026deg;, anteversion angle by -1.7\u0026deg;, articulo-trochanteric distance by 1 mm and lateral distal femur angle by -1.9\u0026deg; per year. The bone remodeling led to an increase in bone density, particularly in the medial proximal metaphysis. The consideration of different growth mechanisms allowed a comprehensive simulation of femoral growth with high agreement with anthropometric data. Possible applications are the simulation of the correction of deformities.\u003c/p\u003e","manuscriptTitle":"Computational mechanobiological model combining epiphyseal, apophyseal, and appositional growth and inner bone remodeling of the juvenile femur","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-08-05 14:03:35","doi":"10.21203/rs.3.rs-7224633/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-11-19T12:33:30+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-11-18T06:14:32+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"52926864314038045246212970155484580034","date":"2025-10-28T13:58:31+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-08-30T15:53:20+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"266850176347781959959932758371644585017","date":"2025-08-28T00:10:03+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"282283829078344395868403714347488595872","date":"2025-08-08T17:19:56+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"228964930827693116062494187827756353592","date":"2025-08-05T17:06:41+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-08-05T17:02:47+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-08-04T09:03:38+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-08-04T09:02:41+00:00","index":"","fulltext":""},{"type":"submitted","content":"BioMedical Engineering OnLine","date":"2025-07-27T07:03:40+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"biomedical-engineering-online","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"bmeo","sideBox":"Learn more about [BioMedical Engineering OnLine](http://biomedical-engineering-online.biomedcentral.com/)","snPcode":"12938","submissionUrl":"https://submission.nature.com/new-submission/12938/3","title":"BioMedical Engineering OnLine","twitterHandle":"@BioMedCentral","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"BMC/SO AJ","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"38ab8546-df79-4143-8a35-57b1c36b8624","owner":[],"postedDate":"August 5th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-04-23T15:39:23+00:00","versionOfRecord":[],"versionCreatedAt":"2025-08-05 14:03:35","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7224633","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7224633","identity":"rs-7224633","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}