# Prediction of Cortical Bone Thickness Variations in the Tibial Diaphysis of Running Rats

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## Abstract

**:**

## 1. Introduction

## 2. Experiments

- Sedentary control group: the rats were 26.7 (±1.1) cm long and weighed 492.1 (±34.6) g. They were left to their everyday activity (eating and walking), without any specific running activity, for 8 weeks. At the end of this program, the mean cortical bone thickness of the tibial diaphysis was 957 (±110) µm. This value was normalized to 1 to facilitate comparisons between groups. The mechanical load (body weight) for the bone remodeling was assumed to be constant over time.
- Continuous running group: the rats were 26.4 (±0.7) cm long and weighed 486.4 (±31.8) g. This group was subjected to 8 weeks of running for 45 min per day at an intensity of 70% of the maximal aerobic speed (MAS) (Figure 1a). At the end of the exercise program, the mean cortical bone thickness of the tibial diaphysis was 708 µm (±65 µm).
- Intermittent running group: the rats were 26.5 (±0.6) cm long and weighed 475.1 (±30.3) g. This group was subjected to an interval-training running activity for 42 min per day for 8 weeks. This protocol consisted in 7 repetitions of blocks of 3 min at 50% of the MAS, followed by 2 min at 100% of the MAS and 1 min of passive rest (Figure 1b). At the end of the exercise program, the mean cortical bone thickness of the tibial diaphysis was 1024 (±112) µm.

#### 2.1. Post-Mortem Micro-Computed Tomography (µCT) and Computation

#### 2.2. Bone Histology

#### 2.3. Occupation Rate of Osteocyte Lacunae

^{2}area, on the same 3 slides used for histomorphometry, for each location (anterior, posterior, lateral, and medial). The osteocyte lacunar density was expressed in osteocyte lacunae/mm2 and the ORL (%) was calculated as: the number of lacunae with an osteocyte / (the number of lacunae with an osteocyte + the number of empty osteocyte lacunae) × 100. The results (see Section 4.4) obtained from each location were added and then averaged from the three slides to obtain a mean ORL per rat.

#### 2.4. TRAP Histochemistry

#### 2.5. Statistical Analysis

## 3. Theoretical Model

#### 3.1. Theory

^{−3}·s

^{−1}(this unit will need to be translated into biological unit at a later stage with enlarged experimental data available) as they start increasing to reach a maximum value that is dependent on the cell density at a given position within the bone. Using this approach, a four parameter model can be proposed as follows:

_{1}, k

_{2}, A

_{1}, and A

_{2}are the cell activity parameters leading the kinetics of bone density change of formed/degraded bone. ${\rho}_{bone}^{ini}$ is the initial bone density in the homeostasis condition. W represents the elastic mechanical energy developed within the structure W

_{1}and W

_{3}represents the two energy thresholds for which: (i) under W

_{1}, the structure is not overloaded, (ii) above W

_{3}, the structure is mechanically overloaded, and (iii) in between W

_{1}and W

_{3}, uncertain conditions are present where we define W

_{2}to be the value for which the sum of osteoblastic and osteoclastic cells is zero, with an unstable equilibrium. Once the cell activities (four parameters) identified from the experiments and the theoretical model are defined, the bone density can be calculated, together with its corresponding Young’s modulus. It is a well-known fact that the determination of the Young’s modulus of bone from experimental data is well scattered through patients (animals, humans, type of bones, age, etc.). Therefore, we used a known relation, $E={E}_{0}\xb7{\rho}_{bone}^{2}$ (Figure 1 and Figure 2 from [52]), that can be used to fit the experimental data available in the literature, where E

_{0}is the cortical bone Young’s modulus. This can be adapted at a later stage with more precise experimental data.

- -
- In the trabecular bone (center part), cells are located in the bone marrow and are ready to be biologically activated. However, they are not very active, as they are far away from the cortical-trabecular interface and the mechanical support of the bone.
- -
- Around the cortical-trabecular interface (on each side) is where the cell activity is at its maximum for the bone remodeling to occur.
- -
- In the cortical bone, mainly osteocytes are present to sense the mechanical load, without osteoblasts and with a minority of osteoclasts; hence almost no bone remodeling occurs.

#### 3.2. Application

^{®}software with a half-length 2D axisymmetric model (Figure 2b). The boundary conditions of the model (mechanical load and material parameters) were changed on the same model geometry and bone density evolution was then extracted.

_{0}= 20.3 GPa and Poisson ratio ν = 0.3 [53]. The applied mechanical load was provided by the rats’ body weight (350 g on average) and bone density was assumed with an idealized distribution (Figure 3) through the cortical-trabecular interface. It was also assumed to be a closed system with no external input.

#### 3.3. Parameter Identification

- -
- W
_{0}: Homeostasis, where no bone density variation occurs within the sedentary control group. It was evaluated directly from the rat body weight and bone geometry at the beginning of the experiments using the standard mechanics of elasticity. - -
- W
_{1}: Energy for the maximum mineral bone density increase, assumed to correspond to the bone density increase after 8 weeks of exercise in the intermittent running group. It was evaluated from the experimental data. This energy was extrapolated from the maximum cortical bone thickness observed experimentally in the intermittent running scenario. - -
- W
_{3}: The energy level corresponding to the maximum resorption rate (depending on cell availability), whatever the extra-increase in the mechanical energy. It was assumed to be the maximum degradation rate observed in the continuous running group after 8 weeks of running with the minimum cortical bone thickness. - -
- W
_{2}: The threshold energy level at which bone density will start to decrease if the energy level increases. It corresponds to the linear interpolation between W1 and W3, where osteoblast and osteoclast activity are considered to be equal.

^{−3}. As it was easy to extract experimentally the average volume of bone being formed or degraded in the two running scenarios. It was, therefore, easy to extract their corresponding weights. Hence, as the cell activity unit is defined in mg·mJ

^{−1}·mm

^{−3}·56 d

^{−1}for k

_{i}and mg·mm

^{−3}·56 d

^{−1}for A

_{i}, it is therefore straightforward to define what will be the corresponding cell activities for each energy level and time in the model from the mechanical energy (mJ) and exercise duration (56 days).

## 4. Results and Discussion

#### 4.1. Numerical Model Parameter Identification

#### 4.2. Theoretical Model Results

_{1}and W

_{3}, of the intermittent and continuous running groups.

_{4}) leads to incorrect results. Hence, the curvilinear form of the cell activity as a function of bone density is highly important for bone remodeling kinetics. In other words, the intensity of the cell activity not only depends on the cell density at a given place and time but also on the mechanical environment (either through mechanical load or their structural basement) for the biological response. This is a well-known fact for cell activity response in the literature [59,60].

_{1}and W

_{3}. In Figure 5b, the polynomial interpolation between the n parameters is presented for different levels of the developed elastic mechanical energy. Both Figure 5a,b present a simplified cell response as a function of bone density and mechanical energy. Of course, the real biological behavior is expected to be more complex, but this simplified approach provides adequate results compared to the experimental results and seems to be in agreement with the known cell behaviors involved in bone remodeling.

#### 4.3. Bone Density Prediction

#### 4.3.1. Intermittent Running Scenario

^{−3}·56d

^{−1}) than the osteoblast formation (around +100 mg·mm

^{−3}·56d

^{−1}). The mistake to avoid here is to link this directly to the average bone density resorption. It only means that the maximum cell activity, which depends on both the geometrical location (hence the intensity of cell activity, as defined in Figure 5a) and the bone density, does not necessarily occur at the place where the bone density variation is at its kinetic maximum. Therefore, the average bone density variation may have different kinetics than the average cell activity. In Figure 7, we show the same running scenario and cell activities but at the three geometrical points located around the cortical-trabecular interface, as defined in Figure 2c. For the radius 0.45 mm, located in the trabecular zone, near-zero cell activity was found. However, maximum cell activity was observed at the mid-radius of 0.5 mm and decreased towards the cortical zone at 0.55 mm.

^{3}is around 3 g.

#### 4.3.2. Continuous Running Scenario

_{x_norm}) responded instantaneously to the applied mechanical load. It increased quickly, as a function of time, up to about 10 to 15 days and then decreased slowly after bone remodeling occurred. For the continuous running scenario, this lead to a very quick bone density variation in a few days (see ${\rho}_{bone}^{norm}$ on Figure 11b).

^{-5}(“slow” variation), and (ii) based on a cell intensity response delay and time, set to 1e

^{-8}(“low” variation). With these two coefficients, the corresponding cellular responses are showed all together in Figure 11a. The cellular responses were defined with the osteoblasts a

_{ob_slow}and a

_{ob_low}, and osteoclasts with a

_{oc_slow}and a

_{oc_low}. The time response and intensity of the cell activity were delayed in time, the and sensitivity of the biological response (peaks in the curves) also seemed lower. We also note that the time delay not only slows down the cellular response but it also slightly changes the evolution of the curve as a function of time. This is due to the fact that bone remodeling occurs locally as a function of the mechanical strain energy field, which is inhomogeneous within the structure. Hence, it impacts the cell activity scenario, although in a small amount, but it is still visible, which inevitably impacts the bone remodeling scenario over long periods of time.

#### 4.4. Correlation between Cell Scale Experimental Results and Bone Scale Numerical Results

^{2}and 8000 µm

^{2}was observed in the continuous running group, while this surface represented between 200 µm

^{2}and 1200 µm

^{2}in the intermittent running group (Figure 16). Although some bone resorption occurred in the intermittent group, it was less pronounced than in the continuous running group and probably not high enough to compensate for the bone formation. In addition, when looking at the distribution of the areas of resorption activity, the maximum resorption in the continuous running group occurred in the intermediate zone (between the cortical and trabecular regions), followed by trabecular bone, and, lastly, the cortical bone. In the intermittent running group, the maximum resorption occurred in the cortical region (still much lower than in the continuous group), followed by intermediate zone, and, finally, the trabecular zone, which was close to zero.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Training Protocols. Moderate Continuous running (

**a**) and Interval running (

**b**). MAS corresponds to the maximal aerobic speed.

**Figure 2.**Developed axisymmetric model of bone, assumed to be cylindrical and under simple compression: (

**a**) schematic of the model, (

**b**) geometry, boundary conditionsand initial bone density distribution (going from cortical bone in red to trabecular bone in blue), (

**c**) localization of numerical results extraction along a mid-height radius line and three points through the cortical-trabecular interface with same color bone density distribution.

**Figure 3.**Idealized bone density distribution as a function of radius within the bone. The µCT measurements extracted from rats are also displayed. To keep continuity and transition between cortical and trabecular bone, the bone density variation as a function of radius was approximated with: ${\rho}_{bone}\left(R\right)=\left(\left(atan\left(12000\ast \left(R-0.0005\right)\right)\right)/3.4\right)+0.56$.

**Figure 4.**Calculated cell activity driving the bone density change. The evaluated parameters for the proposed mode k

_{1}, k

_{2}, A

_{1}and A

_{2}, are presented in Table 3.

**Figure 5.**Parameters used for the definition of the cell activity intensity: (

**a**) the coefficient of cell intensity as a function of the bone density for the construction and resorption phases at the two levels of energy, W

_{1}and W

_{3}and (

**b**) the power n coefficient of ${\left(\alpha -{\rho}_{bone}\right)}^{n}$ that defines the construction-resorption rates in the different running scenarios.

**Figure 6.**Global osteoblast and osteoclast activities over the entire model for the intermittent running scenario.

**Figure 7.**Osteoblast and osteoclast activities at the three locations through the cortical-trabecular interface. Measurements of the radii were 0.45 mm, 0.50 mmand 0.55 mm.

**Figure 8.**Bone density evolution: (

**a**) initial (blue) and final (red), after 8 weeks running, distributions, as a function of radius; (

**b**) The global evolution of bone density over the entire volume as a function of time.

**Figure 9.**Evolution of osteoblast and osteoclast activities at the beginning (

**b**) and the end (

**c**) of the analysis for the intermittent running scenario compared to the initial and final “activation zone” (

**a**).

**Figure 10.**Initial (

**b**) and final (

**c**) (after 56 days running) bone density distributions with a corresponding finite element mesh (

**a**) in the case of the intermittent running scenario.

**Figure 11.**Global cell activity (

**a**) and the corresponding bone density variation (

**b**) for the continuous running scenario at different cell activity scenarios. The osteoblast (a

_{ob_norm}) and osteoclast (a

_{oc_norm}) activities correspond to the one defined in the above initial theoretical model.

**Figure 12.**Initial (

**b**) and final (

**d**) (after 56 days running) bone density distributions, (

**c**) cell activity a

_{ob}+a

_{oc}at 1, 5, 10and 20 days of running, with the corresponding finite element mesh (

**a**) in the case of the continuous running scenario.

**Figure 13.**Initial and final average bone density distribution as a function of the radius through the bone. A comparison of the numerical predictions with the experimental results for the intermittent and continuous running scenarios. The dotted cross on the degradation curve corresponds to the median value of the considered cortical bone thickness after resorption.

**Figure 14.**Occupation rate of osteocyte lacunae after 8 weeks of running in the cortical part of tibia. (

**a**–

**c**) show examples of cortical bone sections for the three groups with blue arrows for full osteocyte lacunae and black arrows for empty osteocyte lacunae. (

**A**,

**B**) generalize the results over all data sets. Sed: Sedentary control group, CR: continuous running group, IR: intermittent running group. Top panels (

**a**) Sed; (

**b**) CR, (

**c**) IR. The blue arrows show a full lacuna (which contains an osteocyte), the black arrows show an empty lacuna. Bottom panel: (

**A**) summarizes the number of full and empty lacunae in the three groupsand (

**B**) includes the occupancy rate of the lacunae (%). Values are expressed as means of full and empty lacunae for (

**A**) and as means +/− SD for B). *: p < 0.05.

**Figure 15.**Osteoclast activity after 8 weeks of running in the cortical part of tibia. Sed: Sedentary control group, CR: continuous running group, IR: intermittent running group. Top panel (

**a**) Sed; (

**b**) CR, (

**c**) IR. (

**d**) The black arrows show the osteoclast TRAP activity (revealed in red). The results of the total resorption surface are given as means +/− SD (in µm

^{2}). *: p < 0.05.

**Table 1.**Experimental results obtained from the rat tests regarding the variations of the cortical tibial bone thickness measured in the first proximal half of the diaphysis, for the three groups.

Sedentary Control | Continuous Running | Intermittent Running | |
---|---|---|---|

Cortical tibial thickness (µm) | 957 (±110) Normalized = 1 | 708 (±65) Normalized = 0.74 | 1024 (±112) Normalized = 1.07 |

Loading type | Constant load (body weight) | 1 week for 25 min/day, 8 weeks for 45 min/day, Oxygen = 70% MAS | 1 week for 25 min/day continuous, 8 weeks intermittent for 42 min/day, Oxygen = 50%, 100% MASand rest |

**Table 2.**Experimental measure of the bone mineral density per the concentration of hydroxyapatite in the bone structure for the different experimental conditions (at the tibial proximal diaphysis).

Sedentary Control | Continuous Running | Intermittent Running | |
---|---|---|---|

BMD (g HA/cm^{3}) | 113.11 (± 4.12) | 109.16 (± 4.31) * | 106.13 (± 4.22) * |

**Table 3.**Parameters for the proposed theoretical model. Energies were determined from only the experimental data (rat weight, size of bone, test conditions, etc.). The k

_{i}and A

_{i}parameters were calculated from the experimental data after the completion of the running tests.

Parameters Determined from Experimental Data (×10^{−4} mJ) | Parameters Calculated (k _{i} = mg·mJ^{−1}·mm^{−3}·56d^{−1}; A_{i} = mg·mm^{−3}·56d^{−1}) | ||||||
---|---|---|---|---|---|---|---|

W_{0} | W_{1} | W_{2} | W_{3} | k_{1} | k_{2} | A_{1} | A_{2} |

3.1847 | 3.7 | 3.72 | 3.78 | 386.64 × 10^{5} | −375 × 10^{5} | 1992.36 | −2232.38 |

**Table 4.**Tuned α and n parameters of the relation ${\left(\alpha -{\rho}_{bone}\right)}^{n}$ for the optimum reconstruction/degradation rate of bone density, as compared to the experimental results.

Osteoblast Activity Coefficient at W_{1} | Osteoclast Activity Coefficient at W_{1} | Osteoblast Activity Coefficient at W_{3} | Osteoclast Activity Coefficient at W_{3} | Cellular Activity Power at W_{1} | Cellular Activity Power at W_{3} |
---|---|---|---|---|---|

α = 1.0009 | α = 1.0 | α = 1.0007 | α = 1.0 | n = 13.6 | n = 5 |

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## Share and Cite

**MDPI and ACS Style**

George, D.; Pallu, S.; Bourzac, C.; Wazzani, R.; Allena, R.; Rémond, Y.; Portier, H.
Prediction of Cortical Bone Thickness Variations in the Tibial Diaphysis of Running Rats. *Life* **2022**, *12*, 233.
https://doi.org/10.3390/life12020233

**AMA Style**

George D, Pallu S, Bourzac C, Wazzani R, Allena R, Rémond Y, Portier H.
Prediction of Cortical Bone Thickness Variations in the Tibial Diaphysis of Running Rats. *Life*. 2022; 12(2):233.
https://doi.org/10.3390/life12020233

**Chicago/Turabian Style**

George, Daniel, Stéphane Pallu, Céline Bourzac, Rkia Wazzani, Rachele Allena, Yves Rémond, and Hugues Portier.
2022. "Prediction of Cortical Bone Thickness Variations in the Tibial Diaphysis of Running Rats" *Life* 12, no. 2: 233.
https://doi.org/10.3390/life12020233