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We describe the development of an optimization algorithm for determining the effects of different properties of implanted biomaterials on bone growth, based on the finite element method and bone self-optimization theory. The rate of osteogenesis and the bone density distribution of the implanted biomaterials were quantitatively analyzed. Using the proposed algorithm, a femur with implanted biodegradable biomaterials was simulated, and the osteogenic effects of different materials were measured. Simulation experiments mainly considered variations in the elastic modulus (20

Biological systems have mechanisms to automatically adapt to environmental changes in order to maintain their required functions under varying conditions and stresses [

Some of the earliest theoretical frameworks for bone adaptation included proposals for the elastic formulation of apparent bone density by an exponential penalization factor [

However, bones are not necessarily always healthy and may have defects or fractures. The rapid development of biomedicine has provided access to new methods such as implantation to support defective bones, stimulate cell growth, and release ions to help generate better bone structure. Moreover, the optimal features of bone materials, including compatibility, reactivity, and degradability, have been studied extensively for use in the clinical setting [

For new bone, the initial mechanical strength is very important. If it is too strong, the concentration of stress on the new bone may be disadvantageous toward osteogenesis [

The two existing research approaches for achieving this goal are experimental biomechanics and FEM analysis. Although constructing an experimental animal model has advantages such as intuitiveness and subjectivity [

In this study, we created a theoretical FEM model of bone remodeling with implanted biomaterials. Our bone growth optimization process takes into consideration Young’s modulus and the degradation rate of the bone material, achieving a simulation of osteoblasts with material and bone density distributions. The simulation results for the metaphyseal bone of the rat left femur and micro-CT images from rats with experimental femur defects are compared. The results validate the effectiveness of the method in modeling bone structure and overall shape optimization. In addition, this method provides theoretical guidance to the matching problem between bones and implant biomaterials.

In this study, a combination of the bone self-adaptive optimization theory and material degradation rules was used to simulate the proximal femur with defects, as detailed below.

A two-dimensional finite element model of the proximal femur was used. The model was obtained from the preprocessing of the femur of a normal adult male using CT, Photoshop 5.0, and ANSYS 10.0 software. With the ANSYS meshing tools, the model was divided into 3,689 nodes and 1,168 elements whose mesh element size was defined as 0.25 cm^{2} as shown in Figure ^{3} kg·m^{−3}) [

Load results for the proximal femur model.

Joint reaction force/N | Angle/° | Rotor rally/N | Angle/° | |
---|---|---|---|---|

1 | 1700 | 25 | 576 | 35 |

2 | 2271 | 66 | 703 | 62 |

3 | 1049 | 15 | 248 | 8 |

Note: the angle is the angle between the direction of force and the horizontal direction.

Finite element mesh for the proximal femur model.

The ranges of implant material properties used in this study were previously defined [

The optimization of the objective function is presented as follows [

The constraint equations are ^{−3} as the maximum density of the cortical bone during the optimization process. Through solving the optimization model, the density value of the next time was predicted to be

Degradation of the material occurs slowly at earlier time points and more rapidly as time passes [

The apparent densities of the defect elements and other part elements, respectively, follow the rules below:

The next apparent density was calculated as follows:

Seven 10-week-old female Sprague-Dawley rats with body weights of 204 ± 4 g were used for the micro-CT imaging study. The experimental protocol was approved by the Institutional Animal Committee. A defect (3 mm in diameter and 3 mm in depth) was drilled on the lateral side of the left femur. The biomaterial CSC (calcium sulfate cement), which has a degradation rate of 4–8 weeks in vivo for the rat model [^{3} cages.

Micro-CT images were reconstructed and the biggest horizontal truncation area of the refilled defect was calculated from the refilled defect volume by after-tracing the CSC edge visualized by the new bone formation, using the system software and a threshold of 110 in gray scale.

The simulation was designed on the basis of the animal experiments. The model, obtained using CT, Photoshop 5.0, and ANSYS 10.0 software, was based experimentally on the metaphyseal bone of the proximal left femur in rats. The implant material properties used were as follows: elastic moduli of 7000 and 2000 MPa and Poisson’s ratio of 0.3 for two parts of the cortical bone; an elastic modulus of 900 MPa and Poisson’s ratio of 0.3 for the cancellous bone; and an elastic modulus of 800 MPa and Poisson’s ratio of 0.3 for the implant material. The joint force was assumed as 0.65 N, which is about one-third of an average rat body weight (204 g) [

The initial proximal femur bone density distribution was obtained, as shown in Figure

Density distribution for the proximal femur and defective area.

The iterations were terminated after the density no longer exhibited significant change (value < 0.001) [

Apparent osteogenic density changes with

The impact of the average elastic moduli for different implant materials on osteogenesis is shown in [

Because optimal osteogenesis occurred at an elastic modulus of 1000 MPa, we used a material with this elastic modulus value to assess the effects of three degradation periods. The osteogenesis simulation results using implantation materials with different degradation periods at an elastic modulus of 1000 MPa are shown in Figure

Contrast simulation results for osteogenesis using implant materials with different degradation periods.

The micro-CT images of the animal experiments and the simulation images showing the CSC degradation and new bone formation at ^{2} for the control and the simulation, respectively. At day 17, the refilled defect areas are reduced in both the control (6.31 ± 0.43 mm^{2}) and the simulation (6.70 mm^{2}) and yet further reduced on day 27 (3.99 ± 0.35 and 4.01 mm^{2}, respectively). The simulation results are well fitted to the experimental data.

Micro-CT images from the animal experiments and simulation results showing the CSC degradation and new bone formation at

Based on the theories and methods of Beaupré et al. [

The computer simulation provided a quantitative analysis method for resolving two basic but complementary problems between an implant material and bone growth: (

For simulation purposes, the degradation rate of the implant material was assumed to be constant; this may be achieved experimentally by using porous materials. However, due to the use of composite materials and the irregular shape of an implant, degradation will not be uniform across the bulk of the implant material. As a result, the next step will be to perform a prospective study based on the stimulation results, considering properties such as the dimensions, proportions, and structure of the material. Assuming that the material is isotropic simplifies the quantitative analyses of material degradation and bone growth; however, the anisotropic properties of the material will need to be considered later. Based on these studies, a more accurate and reasonable degradation analysis can be established by introducing more parameters into the degradation distribution function which will better fit the experimental results.

Application of the bone remodeling theory aided by computationally iterated simulation allows for the calculation of bone density, osteogenesis, and the development of defective bones. This may be helpful for planning during surgery and for the prediction of postoperative bone growth. Although these bone remodeling simulations are still rudimentary, bone remodeling theories are expected to become more refined as our understanding of 3D modeling and complex loading is improved. The applications of these data will be broad, particularly after improvement of the numerical simulation technology.

The authors of this paper declare no conflict of interests.

This work was jointly supported by the National Natural Science Foundation of China (Grant nos. 61304123 and 81320108018), the Specialized Research Fund for the Doctoral Program of Higher Education of China (no. 20133201120022), Jiangsu Province, two postdoctoral research funding schemes (nos. 1202046C and 2013M541722), and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.