Research on topology optimization method of bone plate for rabbit femur

Abstract

The diagnosis and treatment of the special types of the fractures faces an important challenge for the hand and foot surgery. Due to the mismatch between the mechanical properties of traditional solid plate materials and bones, the high stiffness and density pose a risk of fatigue fracture caused by excessive stress levels. In this paper, the topology optimization method of the bone plate for the rabbit femur based on the finite element analysis is present. This method focuses on designing the mechanical structure based on the initial pore layout of the bone plates by searching for the material density distribution within a predetermined domain. By reducing the total volume of the bone plate, the design objective of minimizing the strain energy is implemented to control the mechanical stress and displacement for avoiding the stress concentration and geometric deformation. Finite element method is utilized to construct the relationships between the material density and strain energy, which clearly characterizes the mechanical properties under different volume fractions. The results show that compared with the initial design, the proposed method effectively addresses the issues of the stress concentration, meanwhile obviously reducing the mass of the bone plate and preferably fitting to the bone surface.

Keywords

bone plate topology optimization stress level geometric deformation finite element analysis

1. Introduction

The bone plate, as the core instrument in orthopedic internal fixation surgery, has its biomechanical properties directly affecting the quality of the fracture healing. Currently, the bone plates are primarily made from metal materials and biodegradable materials. But the elastic modulus of metal materials (such as stainless steel, titanium, and their alloys) differs significantly from that of human bone tissue.¹ The mismatch of the high elastic modulus and skeletal biomechanics will have a negative impact on the bones.² The occurrence of the stress concentration phenomena that may lead to the issues such as bone resorption and delayed healing.³ Through the research on the elastic-plastic deformation and damage response of the bone plate under the load conditions, it has been verified that the traditional bone plates undergo slight deformation under load, thereby affecting the performances of the implant.⁴ In addition, the implant is firmly in contact with the surface of the bone at the fracture site, making it easy to form a necrotic area at the contact surface. The impaired blood flow in the necrotic area is detrimental to the remodeling of the fracture site.⁵ It is seen that an ideal bone plate should possess appropriate structural stiffness and reasonable stress transmission characteristics. This places high demands on the selection of the materials, geometric configuration, and pore structure design of the bone plate. Therefore, the development of a lightweight, highly load-bearing, and closely fitting bone plate has become a research focus in the field of orthopedics in recent years.

In the field of orthopedic research, the rabbit femur has attracted the significant attention from researchers due to its high similarity to the human bones.⁶ After measuring the titanium levels in rabbit serum and organs following the implantation of 3D printed and traditionally the manufactured titanium plates, the biocompatibility of the custom 3D printed titanium plates was verified.⁷ Artificial defects were created on the rabbit femur, and plates fixed with Ti-6Al-4V ELI (Ti-64) and Ti-29Nb-13Ta-4.6Zr (TNTZ) were im-planted at the defect site, finding that the low-stiffness titanium alloys facilitate fracture healing.⁸ Surgery on the fractured femur of the rabbit, measuring the thickness of the healing tissue, can assess the optimal time for implant removal and the biomechanical stability of the fracture site.⁹ Obviously, it is evident that the substitutability of the rabbit femur model in the study of the human bone plates cannot be overlooked, and it is urgent to find the advanced design methods for the bone plates of the rabbit femur to promote the fracture healing.

Topology optimization technology is a mathematical method that optimizes the material distribution based on the given loads, constraints, and performance indicators. Currently, there have been several studies optimizing the bone plates based on the topology optimization methods.^10–12 For examples, Murat et al. conducted the experimental tests and data comparisons on the three models of fixed plates through the finite element analysis method (FEM), proposing a new implant model. This model avoids the formation of necrotic areas due to the smaller contact surface with the bone.⁵ Hu et al. developed a topology optimization method for designing the structures using the Triply Periodic Minimal Surface (TPMS), which achieves the optimized mechanical properties of materials without increasing usage.¹⁰ Alkebsi et al. incorporated the design of the permeable femoral implants for the appropriate stiffness, lightweight, and bio-compatibility of the bone.¹¹ Wang et al. provided the material layout of the internal fixation device by the topology optimization method, which generates the desired light-weight design under the given boundary conditions.¹² To address the stress phenomenon caused by the mismatch in stiffness between the implant and the bone, Al-Tamimi et al. utilized the topology optimization to design the 3D fixation plate to reach the goal of the bone stress regulation^13,14 Moreover, a custom-fit bone plate was developed to improve the stress levels for the patient suffering from distal tibia spiral fracture.¹⁵ Additionally, Shams et al. compared the stress and strain of two different bone plates by the FEM, and discovered the custom bone plates to the better fit based on the patient’s femoral anatomical morphology.¹⁶ For bearing the self-weight loads structures, Novotny et al. studied a regularization formulation for the topology optimization imposing any feasible volume constraint.¹⁷ Since the regularization parameter disappears, the standard formulation based on minimizing flexibility under the volume constraints is restored. Dos Santos et al. considered the von Mises stress constraints and self-weight loads for the bone plate problems.¹⁸ Ni et al. utilized the improved P-norm method to aggregate the local stress constraints of all units into a global stress constraint, and the topology optimization model for the stress constraints of the continuum structures considering the self-weight loads was established.¹⁹ Mehboob et al. employed the topology optimization techniques to obtain the bone plates of the three distinct load conditions.²⁰ Park et al. designed a steel plate for a mandibular condyle fracture case for adapting to various patients, where the FEM simulation was performed to assess the performances of the plates.²¹ In order to maximize the total bone density within the recon-structed area during the final stage of bone remodeling, Wu et al. presented a transient topology optimization program for the bone plate design that takes bone remodeling into account.²² Based on this, it is demonstrated that the topology optimization techniques have the potential to achieving the complex rabbit femur bone plate design.

However, it is worth noted that the most existing studies were overlooked the interaction between the bone plate geometry and bone plate interface stress based on the established arrangements. In addition, the material models are often simplified and equivalent to reduce the computational cost, which ignores the anisotropic behavior of the bone plates in the practical applications. Therefore, a topology optimization method of the bone plates considering the typicality of the rabbit femur is proposed. The remaining of this paper is organized as follows: In Section II, the CT scan bone data of the rabbit femur is used to segment and extract the bone features, and a 3D model of the bone is established to reposition the fractured part. The initial design of the bone plate is proposed, and the FEM is employed to generate the mesh and simulate the mechanical properties by the Hypermesh software. In Section III, the optimization problem is introduced and the model of the topology optimization is constructed. By adjusting the material density, the objective of minimizing the strain energy is achieved to control the mechanical stress and displacement. Applying the constraints to the bone plate to ensure the desired volume fraction. Section IV provides a typical numerical example for the bone plate to verify the effectiveness of the proposed method. In Section V, the conclusions are summarized to illustrate the content of this manuscript.

2. Initial design and FEM analysis

2.1. Rabbit femur skeleton reconstruction

In order to characterize the mechanical properties of the complex 3D bone plate model, the CT bone data of the rabbit femur is scanned to display the medical images of the rabbit femur fracture. By the image segmentation and model stitching techniques, the images are handled to segment and extract the bone features, and the 3D digital model of the rabbit femur is reconstructed to reposition the fractured part by the CAD software, which is shown in Figure 1. Normally, a surface contour is firstly created to express the features in the reconstructed model, and then the 3D solid is generated by filling, stitching, and sealing the contour. It is worth noting that the fracture site needs to undergo the simulated restoration, where the function is carried out through setting a benchmark and performing the operations of the translation and rotation. Subsequently, the reconstructed model is im-ported into the Hypermesh software for the initial design of the bone plate and FEM analysis.

2.2. Details of the bone plate and FEM analysis

In order to design the bone plate of the rabbit femur, the FEM analysis is utilized to characterize the mechanical performance details of the bone plate. Considering the trans-verse fracture of the mid femur in a rabbit, where the femur is composed of the external cortical bone and internal bone trabeculae. Based on the 3D digital model of the recon-structed rabbit femur, the initial design of the fitted bone plate is customized to show the geometry configuration and basic dimensions with a length of 58 mm, width of 12 mm, and thickness of 2 mm, as shown in Figure 2(a), where the fracture wound of the femur is targeted by the red circle in this figure. The bone plate is configured to adhere to the uneven bone surfaces and fixed with eight interface screws to avoid the issues of the stress concentration. The material properties of the rabbit femur and bone plate are shown in Table 1. Among them, the material of the bone plate is steel. $ρ$ is the density, $E$ is the elastic modulus, and $ν$ is the Poisson’s ratio.

Table 1.

The material properties of the rabbit femur and bone plate.

	Material	$ρ$ (g/cm³)	$E$ (GPa)	$ν$
Rabbit femur	Biomaterial	1.15	15	0.30
Bone plate	Steel	7.8	210	0.36

The simulation analysis is implemented to express the mechanical and displacement responses of the bone plate in the rabbit femur using the Hypermesh software. In order to reduce the complexity of simulation, a simplified material model is constructed to approximate the anisotropic behavior of the bone plate. Figure 2(b) indicates the mesh generation details and the display of local grids. Owing to the coupling model of the rabbit femur and bone plate, the tetrahedral size elements are employed to ensure the computational accuracy, where the screw holes of the plate are applied to the hexahedral mesh with a size of

1 e^{- 3}

. In order to control the error of the grids on the simulation results, the mesh independence verification for the bone plate is given in Table 2. Compared with the Mesh I and II, the maximum stress of Mesh III changed by 18.9% and 7%, respectively. The maximum stress of Mesh IV showed only a slight adjustment about 1.1% than that of Mesh III. In addition to the comparison of the maximum stress, the minimum element size, worst Jacobian ratio, and computation time are characterized, and Mesh III is selected to calculate the mechanical performances. The grid numbers of the rabbit femur and bone plate are 137586 and 25282, respectively.

Table 2.

Independence verification of the grids.

	Mesh I	Mesh II	Mesh III	Mesh IV
Number of grids	4588	9221	25282	224098
Minimum element size (mm)	7e-4	5e-4	3e-4	1e-4
Worst Jacobian ratio	>0.7	>0.7	>0.7	>0.7
Maximum stress (MPa)	787.1	686.4	637.7	630.5
Computation time (s)	3.02	3.91	12.36	62.63

In Figure 2(c) and (d), the loading and boundary conditions are imposed to simulate the state of the bone plate, which include the fixed support at the proximal femur, the equivalent loadings at the eight screws on the bone plate ( $F_{1 i}, F_{2 i}, i = 1, \dots, 4$ ), and the horizontal and torsional loadings along the z-axis direction ( $F_{z}$ ). Among them, the interface screw layout is set at the eight positions and applied with the equal forces to improve the stability of the bone plate. All the loadings are set to 10 N based on the reaction force exerted on the rabbit femur to the ground after the surgery.²² One side of the femur is fixedly supported, meanwhile the other end is subjected to a force from the z-direction. The binding constraints are applied to contact the bone plate and the femur surface through the interface screws. Thus, the mechanical stress of the bone plate is implemented to consider the quasi-static elastic behavior by the FEM analysis, which is calculated as follows:

σ^{e} = D ε^{e} = DB α^{e}

(1)

where

σ^{e}

denotes the element stress of the bone plate, and it is given by the strain field

ε^{e}

of the physical constitutive relationship.

D

denotes the elastic matrix related to the material properties (

E

ν

B

is the derivative of the shape function, which connecting the displacement and strain.

α^{e}

is the node displacement vector. Based on the principle of virtual work, the stiffness matrix

K

of the material is calculated as:

K = \int_{V} B^{T} DB d V

(2)

where

K

and

F

are integrated into the equilibrium equation

K u = F

, the reaction force

F

is equivalent to the nodal force vector of the element as the boundary condition. The unknown displacement vector

u

are obtained. Based on this, the stress and displacement performances of the bone plate can be calculated in the subsequent simulation.

3. Topology optimization model and process

The principle of the topology optimization focuses on the most suitable allocation of the materials within the design domain by constructing the mapping relationships of the material density and the objectives.^23–25 According to the above-mentioned FEM analysis, the mechanical performances of the mechanical stress and displacement are characterized to facilitate the design of the anisotropic bone plate structures for the rabbit femur. Herein, the rabbit femur and the positions of the interface screws in the bone plate are non-design and preserve regions of the topology techniques, while the other regions of the plate belong to the design domain. The implementation of the topology optimization is to divide the initial design of the bone plate into several elements, and achieve the desired material density layout by determining the presence or absence of each element. The material density of each element is set to $ρ_{i}$ . The problem description of the topology optimization is reasonably defined as: Through designing the material density of each element to minimize the compliance of the bone plate under the constraint of the volume fraction, and the optimization formulation is given as:

{\begin{cases} f i n d : ρ_{e} \\ \min : Φ = F^{T} \cdot u (ρ_{e}) \\ s . t . K (ρ_{e}^{p}) u = F \\ \begin{array}{l} \sum_{e = 1}^{N} v_{e} ρ_{e} \leq V_{f} \\ ρ_{e_{i, j}} \geq ρ_{e_{i, j + 1}} \end{array} \\ ρ_{\min} < ρ_{0} \leq ρ_{e} \leq ρ_{\max} \end{cases}

(3)

where

Φ

is the design objective, also known as the compliance.

F

is the column vector of the applied force, and

u

is the unknown displacement vector.

K

denotes the stiffness matrix of the material, and

p

denotes the penalty factor,

p = 3

. The governing equation

K u = F

is calculated by the FEM analysis in the Hypermesh software.

N

is the number of the partition units, and

V

is the total volume within the design domain.

V_{f}

is the volume fraction ratio, and

ρ_{0}

denotes the initial density of the bone plate design, which expresses the interpolation of the minimum density for avoiding the singularity problems.

Based on the constructed topology optimization model, the optimization problem can be successfully solved by the method of moving asymptotes (MMA). The process of the proposed method for the bone plate design is listed, which is taken in Figure 3, specifically:

(1) Start the rabbit femur skeleton reconstruction by the CT data, and generate the initial design of the bone plate.

(2) Mesh the bone plate structure, set the mechanical boundary conditions to perform the FEM analysis.

(3) Pre-select the optimization conditions including the maximum of iteration number $M$ , the convergence criterion $T_{c}$ , etc. Set the parameters of $N$ , $V_{f}$ , and $ρ_{0}$ .

(4) Calculate the $Φ$ in the Eq. (1), and updated the next generation $ρ_{i}^{n e w}$

(5) If $ρ_{i}^{n e w}$ has converged, the optimization procedure stops; If not, it goes to (2)

(6) Obtain the optimal material density by the evaluation criteria, and achieve the required mechanical performances of the stress and displacement.

4. Results and discussion

In this section, a typical numerical example is provided to verify the effectiveness of the proposed topology optimization method for the bone plate. The initial design and FEM analysis process of the bone plate for the rabbit femur are exhaustively illustrated in Section 2. In order to explore the optimal material distribution of the bone plate, all holes of the initial design are filled with the plate material to achieve the restoration. The optimization information includes: $M = 200$ , $N = 381233$ , and $ρ_{0} = 7.85 e^{- 9}$ . The bounds of the variables are $(0, 1]$ . $T_{c}$ denotes the change of the design variables in each iteration step is not greater than 10^-6. $V_{f}$ is 0.3 ∼ 0.7, and the penalty factor is set to 3. The mechanical characteristics are calculated by the Hypermesh software, and the constructed optimization model is solved by the MMA based on the sensitivity information of the design objective related to the material density. The optimization is computed using a 3.60 GHz i7-7700K CPU and 8 GB RAM.

Figure 4 demonstrates the optimization results of the bone plates with different $V_{f}$ . It is seen that since $V_{f}$ decreases, the material distribution of the bone plate designs transitions from the dense to the graded structure, forming the adaptive loading path. The larger the porosity of the bone plate by the topology optimization design, the more significant the migration of femoral cells, which is conducive to the fracture healing. In Figure 5, the von Mises stresses of the bone plates at different $V_{f}$ are provided to show the mechanical performances by the proposed method. Table 3 shows the comparison of the initial mass, optimized mass, and percentage reduction on the stress levels. It can be observed that the peak stresses of the optimization results appear the trend of first decreasing and then increasing as the proportion of $V_{f}$ gradually declines. Herein, the optimization result of Figure 4(c) is chose to achieve the bone plate design, which the mass of the plate by the proposed method is significantly reduced by 45% comparing to the initial design. The geometry configuration and basic dimensions of the postprocessing bone plate for the rabbit femur based on the proposed method, which is illustrated in Figure 6.

Table 3.

Comparison of the initial mass, optimized mass, and percentage reduction on the stress levels.

$V_{f}$	Residual material/mass (kg)	Peak stress level (MPa)	Displacement (mm)
0.3	0.001401	199.4	7.127
0.35	0.001634	195.7	6.926
0.4	0.001868	188.6	6.833
0.45	0.002101	186.2	6.797
0.5	0.002335	182.2	6.774
0.55	0.002568	177.4	6.764
0.6	0.002801	195.9	6.786
0.65	0.003035	237.3	6.824
0.7	0.003268	269.5	6.892
1(Initial material/mass)	0.004669	637.7	6.989

In order to comprehensively evaluate the mechanical properties of the rabbit femur and bone plate, the stresses of the initial design and the proposed method are analyzed to compare the ability of resisting external loadings, which is shown in Figure 7. In Figure 7(a), the peak stresses of the bone plates for the initial design and the proposed method reach 637.7 MPa and 177.4 MPa, respectively. In Figure 7(b), the peak stress comparisons of the rabbit femur are 75.92 MPa and 36.19 MPa, respectively. Compared with the initial design, the peak stress of the bone plate design by the proposed method is significantly reduced. The peak stress of the rabbit femur is also effectively controlled by half. In Figure 8, the displacements of the initial design and the pro-posed method are calculated to characterize the situation of the local deformation. It is seen that whether the initial design and the proposed method, the changes of the maximum displacements are not significant. The above-mentioned results demonstrate that the proposed design by the topology optimization method effectively suppresses the issues of the stress concentration without experiencing significant geometric deformation, while the mass of the bone plate is significantly reduced.

5. Conclusions

In this paper, the topology optimization method of the bone plate for the rabbit femur is proposed, where the FEM is implemented to obtain the mechanical properties of the stresses and displacements. Through designing the material density distribution of the bone plate, the mechanical structure of the rabbit femur is generated under the predetermined screw layout. The objective of minimizing the strain energy is achieved to control the mechanical stress and displacement for avoiding the stress concentration and geometric deformation. The findings validate that the proposed method significantly suppresses the peak stress of the bone plate without the significant geometric deformation, meanwhile significantly reducing the mass of the bone plate by 45 % and tightly adhering to the bone surface. In the future, the proposed method has the great potential to the lightweight and the stress distribution improvement around the actual large segment weight-bearing bone models.

Footnotes

ORCID iD

Tiehui Zhang

Author contributions

Conceptualization, Y.S. and W.C.; methodology, Y.S.; software, Y.S., W.C. and H.L.; validation, Y.S., W.C. and T.Z.; investigation, W.C.; resources, W.C. and T.Z.; data curation, Y.S., W.C. and T.Z.; writing—original draft preparation, Y.S. and W.C.; writing—review and editing, Y.S., W.C. and H.L.; supervision, Y.S. and H.L.; project administration, Y.S.; funding acquisition, Y.S. All authors have read and agreed to the published version of the manuscript.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by the Provincial Key Specialties Internal Research Grant Program of Dalian Municipal Central Hospital (Grant Nos. 2023sj021).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

The data used to support the findings of this study are included within the article.*

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