The simulation of bone remodelling is highly complex due to the high nonlinear coupled relations between the damage and repair behaviours. In this paper, a new bone remodelling model based on the fatigue damage mechanism is proposed. Then the repair process of the human femur is simulated based on the finite element method. The results show that the new damage model can well simulate the bone damage evolution caused by overloadings. This indicates that the new mathematical model suggested in this paper can simulate the effect of external load on the cortical bone density and the related mechanical properties of the cortical bone.

#### Keywords

- bone remodelling
- bone damage
- human femur
- finite element method

#### MSC 2010

- ???

Bone is a hierarchically structured biomaterial consisting of collagen and hydroxyapatite. These two constituents are coupled to form mineralized collagen fibres at nanoscale. The lamellar structure is made up of fibres. The thick and thin lamellar structures are organized circularly into osteons, which are the main structural unit of the cortical bone. Compared with single collagen or hydroxyapatite, the mechanical properties of bones are significantly superior. Bone has a high level of damage tolerance because of the toughening effect related to its complex structures. But bone can also be weakened by fatigue and finally lead to fracture resulting from daily loading activities. Fortunately, bone has an ability to find such damages and heal them by a metabolic process called remodelling [1,2,3]. The most important feature of bone damage resistance is the self-healing mechanism of bone and its adaption to environmental conditions. This adaption is the process by which bone can adjust its structure and other mechanical properties to adjust with the outer loadings. However, if extremely high cycle-loadings are applied, the damage growth may be beyond the bone's repair capacity and may result in bone fractures. Taylor et al. [4] evaluated the fatigue of bones using an experimental method. The results showed that the fatigue life of bones subjected to repeated loadings without repairing is within only 3 years.

Since the damage repair is performed by remodelling, it is reasonable that micro-damage in bone can stimulate the bone remodelling [5,6,7,8,9,10]. Evidence shows that if the remodelling process of bones was suppressed by drugs or genetic manipulation [11], the damages would not be repaired although bones formed and grew. The accumulations of the damage made bone tissues more vulnerable to fatigue. Besides repairing the fatigue damage, bone remodelling also adapted its structures and mechanical properties to the changing mechanical loadings. Thus, fatigue damage has both mechanical and biological consequences. It not only plays a negative role for bone materials as for the artificial ones, but it also has a positive aspect of strengthening and renewing the bone. Then the bone damage repair process can be seen as in Figure 1.

The damage resistance mechanism of bones has aroused the interest of scientists and engineers for many years. It is difficult to predict the process of bone remodelling and damage healing in a single mathematical framework. Researchers have proposed a number of models to simulate the external factors (electromechanical environment [12,13,14], the electromagnetic field [15, 16], etc.) and internal factors (micro-damage [17,18,19,20,21]) that may play a potential role in the remodelling process. Many researchers believe that damage caused by fatigue is the trigger for bone repairing. Taylor et al. [4] suggested that bone is a mechanical system including micro-crack growth and repair. The relationship between damage evolution and osteocyte behaviour has attracted extensive attention. Results [22] show that micro-damage can sense many unknown physiological signals to initiate the repairing process. This point has received great attention from researchers, and it is believed that the micro-fluid potential and damage of ostrocytes are related to micro-crack. McNamara et al. [23] proposed a mechanical regulation system combining strain and micro-damage stimulation to model the formation and repair of trabecular bones. In this model, the propagation/healing process of trabecular bone micro-damage subjected to mechanical loadings was well studied. The stimulation factors to activate bone healing are stress, strain and the damage evolution rule, which is defined by the Miner's rule [24]. Wang et al. [25] proposed a remodelling model for dental implant models. In this theory, strain is assumed to be the dominant factor in remodelling when the damage is below the critical value. However, if it exceeds the critical threshold, the damage model should be used to simulate the damage evolution.

To better understand this process, it is necessary to use the damage theory under the condition of small strain in elasticity to study its mechanical response. From a biomechanics point of view, it is of great importance to put the theoretical framework in which a hypothetical statement can be made, considering the damage in the process of remodelling of bone. Mathematical modelling proved to be a helpful and useful way in the biomedical area for simulation of the complex and nonlinear biomedical process [26,27,28]. The purpose of this paper is to develop a complete remodelling-damaging model to simulate bone adaption to mechanical loadings. The relationship between the apparent density of bone and its effective anisotropy was established using the experimental data of other researchers. Then the new model coupling the remodelling is used to simulate a two-dimensional human femur remodelling and predict the density and damage change in this adaptive process.

In this paper, the remodelling model [29, 30] was used to simulate the bone remodelling and damage growth/repair. In this model, the apparent density was used to evaluate the mechanical properties of bone tissues and was considered as a single scalar. The changing rate of bone density is
_{v}_{0} is the tissue density, which is a constant;

Then the damage can be included in the model. First, we define _{f}_{f}^{i}^{i}^{i}^{−3}), respectively [MullenderM. G.HuiskesR.

The damage threshold _{crit}^{10} is obtained when the damage reaches 3,500

If damage _{crit}^{i}^{i}^{i}^{−3})^{−1}. At a given location ^{i}^{i}^{i} −U_{ref}

The changing rate of density is
^{ref}^{j}^{ref}_{1} = 3.87; if the strain1,000^{ref}_{1} = 0; If the strain 2,000^{ref}_{1} = 198.

If damage accumulation _{crit}

Thus
_{a}_{max} are the proportion coefficients and the maximal bone density. ^{−3})^{2} MPa^{−1}day^{−1} and _{max} = 1.80gcm^{−3}. ^{i}^{i}^{i}

A human femur bone model was used to predict the change in bone density and damage under different loadings by finite element method (FEM) (Figure 2) to show how the micro-damage-based mechanism works in the clinical practice.

As is shown in Figure 2, a four-node quadrilateral plane strain element is used to generate mesh. It is assumed that bones are compressible isotropic materials. Therefore, Young's modulus, permeability and Poisson's ratio are preliminarily defined, all by mineral density. These parameters are all changing with time.

Bone tissue is affected by repeated loads during daily activities, and the peak value of cyclic loads is dominant in the remodelling process [MullenderMGHuiskesR

Based on the mechanical law, a remodelling and damage model of human proximal femur is established, which takes into account the strain energy and stress of each element under usual stimulation. In the initial model, bone remodelling was considered and the density distribution was uniform. The biomechanical stimulation of each unit was calculated and the bone mineral density (BMD) of each unit was updated. After 1,000 iterative steps, we can clearly see that two high-density cortical layers appear around the model and a low-density area corresponds to the medullary cavity. The density distribution of the femoral head is very complicated. Two high-density areas and two low-density ones can be seen in and around the femoral head. Then the damage mechanism is introduced into the simulation model. The damage region is taken as the initial, and continuous simulation is carried out using the controlled boundary conditions. The initial damage is a constant higher than the damage threshold to activate the damage repair mechanism.

In addition, we divided the bone remodelling process into three stages: absorption period (20 days), recovery (10 days) and formation (90 days) [33]. During each cycle, the FEM was used to calculate and obtain the stresses and strains in the femoral model.

The model of bone remodelling is implemented using the finite element software ANSYS, and each element is regarded as a separate model. Bone remodelling algorithms can be summarized as follows. First, assume the initial properties of the material and apply the boundary condition. The stresses and strains can be obtained according to the formula. Then the mechanical signal of each sensor can be calculated in bones. It will be detected by the tissues and these signals are sent to initiate the bone remodeling. According to the magnitude of these signals, the change in cell density is simulated and the new density can then be calculated. For the element whose density has changed, a new elastic modulus is determined in the next step.

To demonstrate how the above models work in simulating the damage and remodelling process of bone, several representative examples are presented under different loadings in what follows. First, we consider a normal bone remodelling subject to normal loadings. Then a pin is inserted into a broken bone to simulate the clinical case when the bone suffers from “stress shielding.”

In this case, all the material parameters comply with the above model and the loading condition is as shown in Figure 3 and Table 1.

Loading applied on the bone

1 | 4 | 1 |

Figure 4 gives the change in mechanical property of the human femur during the remodelling and healing process. This phenomenon can be explained as follows. When the resorption is in progress for the first 20 days, the sensor cells can detect the damage and send out corresponding stimulation signals. Beyond the critical value, the damage remodelling will be dominant. In addition, after the absorption activity occurs, it often diffuses in the nearby area. The bone density in the model can be reduced to the minimum. Then a reversal period follows from day 20 to 50. It is assumed that the absorption and formation rates are the same. Finally, the formation stage starts from the 50th to the 100th day. With the deposition of new bone, the mechanical properties and microstructure of bone gradually returned to the original state. As is shown in Figure 3, our simulation result is very close to the real bone. The distribution of bone mass in the bone is similar to that of real ones.

In order to describe the damage rate represented by elastic modulus, damage _{j}_{j}

In this case, we will change the loadings to investigate the influence of those loadings on the bone remodelling process as shown in Tables 2 and 3 and Figure 6.

Loading condition for the first case.

1 | 2,317 | 145° | 703 | 45° |

Loading condition for the second case.

1 | 1,700 | 145° | 703 | 45° | 613 | 145° |

The results are shown in Figure 7. As is shown, the implant was tightly in contact with the bone when it was inserted into the tissue. But due to the stress shield effect, the loadings on the inner surface of the bone are not enough to support the normal condition of the bone. Then the inner bone began to absorb and led to a risk of loosening and failure of the structure. This is because the elastic modulus of the pin is much bigger than that of the bone. Most loadings are carried by the implant. The bone will fall into a condition called disuse. This is the main reason which causes the failure of a bone implant.

However, this condition can be improved by changing its loading condition. We can see in Figure 7(a) that the absorption mainly occurred on the left side. Then we adjust the forces on the right side. The results show that the absorption resulting from the disuse is less than that in the first case.

Under the condition of ensuring the support strength of a prosthesis, choosing a material with a smaller elastic modulus can also reduce the bone stress shielding effect. Scanell et al. [63] considered that the higher the elastic modulus, the greater the stress shielding and the more the bone absorption. Sumner et al. [64] found that the material stiffness has great influence on stress shielding. In order to study the influence of stiffness on the service life of a prosthesis, three kinds of prostheses with different elastic modulus are selected in this paper. The specific parameters are

In this paper, a remodelling model of human femur is established, and the damage-induced remodelling is simulated to illustrate the whole process of bone repair. In the bone remodelling model, both the mechanical stimulation and the damage sensors are considered to simulate the remodelling and damage repair process of the human femur. The numerical results obtained by simulation are very similar to the actual situation. Therefore, finite element simulation enables us to make a helpful and effective investigation on the bone damage and remodelling, which has become a very valuable reference for the clinical treatment [34]. The overall healing behaviour of the human proximal femur can be well simulated with the help of this model.

The main purpose of this paper is to study and simulate the remodelling of bone tissue with damage. In order to simplify the calculation, the isotropic elastic model was used to simplify the calculation. However, in reality, bone is a kind of living tissue with complex physiological functions and structures. It is of great importance to consider the anisotropic and heterogeneous nature of bone materials. This should be an important goal of biomechanics. On the other hand, finite element simulation needs a large number of biological experimental data as verification to support the model. However, due to the complexity of these experiments in vivo, more relevant biological or medical experiments are obviously needed to be carried out in the future. From the mechanical setting of daily load to the periodic tracking of damage repair model, targeted experiments are needed to be supplemented. Other drawbacks can be seen in our paper. For example, before triggering the repair process, the degree of damage must exceed a critical value. In the repair process, we did not consider the biological and metabolic effects, and the change in interface conditions. In the future, these models need to be further improved obviously, including some of their effects. Despite these limitations, as we have shown, this model can be used in the clinical setting, such as looking for suitable osteogenic physical activity, monitoring the long-term risk of stress fractures, prediction of bone tissue distribution after orthopaedic surgery and in some other diseases related to bone remodelling.

#### Loading applied on the bone

1 | 4 | 1 |

#### Loading condition for the first case.

1 | 2,317 | 145° | 703 | 45° |

#### Loading condition for the second case.

1 | 1,700 | 145° | 703 | 45° | 613 | 145° |

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