1. Introduction
Bone is a dynamic tissue subject to continuous remodeling orchestrated by specialized cellular units known as basic multicellular units (BMUs) in a sequence of well-coordinated metabolic processes [
1,
2]. This delicate balance is overseen by osteoclasts, osteoblasts, and various other cell types, such as osteocytes, bone lining cells, osteal macrophages, and vascular endothelial cells, all functioning within the microenvironment of the BMU [
3,
4]. These processes can be broadly categorized into two main functions: resorption and formation. It is the precise interplay between these functions that ensures the bone’s microstructure aligns with its functional demands [
2,
5].
While, in young individuals, bone turnover through remodeling is advantageous, various natural and pathological factors can disrupt this delicate balance, resulting in net bone alterations that can, in certain cases, be detrimental. One of the most well-studied factors in this regard is menopause, which holds significant implications for women experiencing this physiological transition [
6]. The onset of menopause, marked by the cessation of menstrual cycles due to declining ovarian follicular activity, leads to lowered serum estrogen levels and disrupts the usual bone remodeling process [
2,
7]. This disruption primarily impacts two factors crucial to Basic Multicellular Units (BMUs): the activation frequency, closely related with the bone turnover, and the focal bone balance. The cellular intricacies of bone turnover and focal bone balance have been explored in earlier studies [
8,
9]. As a consequence of the loss of hormonal anti-catabolic action, the frequency of BMU activation increases from premenopausal levels and the remodeling balance tends toward net resorption, resulting in an excessive loss of bone density [
10,
11]. Also, age-related effects such as decreased intestinal calcium absorption [
12], vitamin D deficiency [
13], and impaired synthesis of active 1,25-dihydroxyvitamin D3 by the kidneys [
14] contribute to accelerated bone resorption [
15,
16]. Conversely, retirement frequently leads to a decrease in the mechanical stimulation of bones. In addition, due to the absence of down-regulation in estrogen-mediated sclerostin production by osteocytes, there is also a reduction in osteoid formation [
7,
10]. The resulting imbalance between resorption and formation alters bone quantity and quality. As a result, the material performance, i.e., strength and fragility, are affected [
17]. Understanding the influence of menopause process and its different stages on bone health can help individuals attend this significant life transition.
The mathematical modeling of bone remodeling in peri- and postmenopausal stages allows for a more detailed description of the process of bone density loss during these stages and the development of osteoporosis [
18]. In this context, studies have utilized phenomenological models based on the work by Hernández et al. [
19]. Such models employ variables averaged over the bone volume to evaluate elastic parameters based on bone volume fraction and mineralization content. These variables undergo changes corresponding to the density of activated BMUs at a given time. The activation process is mediated by the stimulus level, a scalar variable proportional to deformations at each point, thereby establishing a feedback system [
2,
20].
The impact of maximum bone density achieved throughout life, age of menopausal onset, and the rate of age-related bone loss on the development of postmenopausal osteoporosis (PMO) has been characterized [
21]. Temporal and permanent dynamic changes in model parameters, including BMU activation rate and the degree of imbalance between resorption and formation, have been calibrated against experimental values, both individually and in combination, to reflect the evolution of peri- and postmenopausal BMD [
22]. The work by Hernández et al. [
22] is limited to 9.5 years, including premenopause and menopause, during whichconclusions were drawn by aligning the computational results with experimental data. While excellent agreement between these results was achieved, the effects of model parameters could be influenced by the BMD decreasing after menopause, an extension for which the experimental results they considered do not account. In a later work, Hernándezd et al. [
21] proposes a model in which the focal balance has a sustained and constant modification after menopause, showing that this generates a constant loss of BMD that is attributed to ageing. However, although his conclusions are highly valuable for clarifying the factors that lead to osteoporosis, this model was not tested against experimental data throughout the complete simulation period. Recently, Martínez Reina et al. [
23] proposed a model of bone remodelling incorporating the effect of bone mineralization, micro-damage, and mechanobiological feedback. They investigated a variety of treatment scenarios, with an emphasis on the combined effects of mechanical loading (including overuse and disuse) together with denosumab treatment in PMO. Their findings include that late treatment for PMO patients shows higher local failure risk compared to early treatment. They showed that an increment of exercise is followed by a decrease in the risk of failure. Conversely, a decrease in mechanical loading reduces the effectiveness of the treatment. and this could seriously compromise bone integrity. The timing of denosumab treatment initiation plays a critical role. A delayed start may potentially increase the risk of failures, especially in cases where there is a significant increase in load [
23].
Previous computational works predict BMD evolution in a representative volume of a human bone cross-section, neglecting the potential effects of macroscopic geometry and stress distribution at the organ level. It is, however, important to investigate whether a whole bone has the same loss, considering that different parts of its geometry are subjected to different loads and contain different degrees of mineralization as starting conditions. Therefore, a more comprehensive approach is necessary to determine whether the phenomena observed in previous studies are specific to particular regions or affect the entire bone uniformly. In this sense, ref. [
24] conducted dual-energy X-ray absorptiometry (DXA) measurements of BMD at posteroanterior lumbar spine and proximal femur in female subjects of age 20–89 years from different regions of China. Results for the total femur showed a steady decline in BMD after the age of 40 years, reaching an almost 36% reduction at 80 years compared to the maximum value reached during the period of 30–35 years. Indeed, Bonicelli et al. [
25] show that maturity is reached at 35 years exactly, and afterwards bone deteriorates at the bone tissue level. A computational model capable of simulating this behaviour for realistic geometries and loads would provide a valuable tool for predicting the evolution of bone mass in women in clinical situations of interest. Recent studies have pursued this goal to obtain a better understanding of the resorption process [
8] and achieved validation in three-dimensional models [
9], albeit without incorporating menopausal effects. Another aspect not considered in conjunction with these effects is the self-accommodation of bone tissue to the stimuli it undergoes [
26]. The objective of the present work is to analyze these effects in a three-dimensional model of the femur under realistic loading conditions, with a particular focus on the biological implications of the model results.
3. Results and Discussion
In this text, the results and discussions are integrated into a cohesive section, facilitating a clear and coherent narrative. This approach prevents unnecessary repetition and strengthens the connection between findings and arguments, thereby simplifying the interpretation of outcomes.
The simulations cover up to 19 years of bone evolution (4 years prior and 15 years after the last menstrual period) allowing us to observe long-term trends. The goal is to establish a model that can effectively simulate the temporal evolution of bone mineral density in the entire femur model for women experiencing menopause, considering their engagement in normal or modified activities, and exploring combined situations involving both menopause and ageing. In this regard, there are two valuable experimental studies that we will use as benchmarks for comparing the outcomes of the current model. In the first work, Recker et al. [
40] reported BMD measurements of the spine and femoral neck from almost five years before to almost five years after the last menses (9.5 years of measurements), collecting data from 75 women over 46 years old who had premenopausal estradiol and gonadotropin levels and regular menses. The results of the measurements are presented as relative BMD values with respect to the initial value before menopause (
). This work was used by Hernández et al. [
22] to compare their computational results with experiments. For a longer period of measurements, Ahlborg et al. [
41] evaluated bone loss in 156 women in the peri- and postmenopausal period, ranging in age from 48 to 64 years. All women were premenopausal at the beginning of the study. The areal bone mineral density (
) of the forearm was measured 1 cm and 6 cm proximal to the styloid process of the ulna every second year by single-photon absorptiometry. The average BMD value of the left and right forearms was used. For the present work, BMD values from the measurements by Ahlborg et al. [
41] are converted to relative values compared to the initial ones, like those from Recker et al. [
40], in order to include and compare trends of all experimental and computational results in each figure. The experimental data mentioned above are shown in
Figure 4a.
Our analysis will first focus on fitting the model parameters ( and long-term trends for and ) to the experimental data, highlighting the effect of turnover frequency () and osteoblast/osteoclast balance (focal balance, ) variations on the tissue remodelling process. Then, we will examine the effects of overload and a sedentary lifestyle on evolution when compared to normal load.
3.1. Parameter Adjustment
According to the measurements by Recker et al. [
40] and Ahlborg et al. [
41], the relative BMD evolution (see
Figure 4a) exhibits a significant decline during the perimenopausal period, followed by a sustained decrease due to ageing. This behavior aligns with the trends predicted by Hernández et al. [
22] for a representative volume of trabecular bone when considering the simultaneous and permanent changes in the parameters that model the effects of bone turnover frequency (
) and focal balance (
). However, it is not entirely clear whether these effects alone can explain the mentioned changes and their respective influences on bone mass and mineral content, which can contribute to clarifying clinical interventions and/or training plans aimed at minimizing pathological effects. To analyze the individual effects of each parameter on bone changes, three scenarios are proposed: (1) variations only on
, (2) variations only on
, and (3) simultaneous variations in both parameters. A fourth theoretical scenario in which both parameters are unchanged is presented for comparison purposes. In the first three cases, the obtained curves correspond to those parameter values providing the best fit to the experimental data throughout the entire range by minimizing the mean squared error. The fitted parameter values are shown in
Table 3.
Figure 4b shows a good fit of these three cases during the perimenopausal period, capturing the evolution of
. In the postmenopausal period, however, although the decreasing trend is captured, there are important differences that will be discussed later. Indeed, certain aspects concerning the dynamics of each phenomenon reveal their relative biological plausibility. This leads us to explore fundamental aspects of the BMU complex’s mechanisms in order to gain new insights and identify clinically relevant factors. These aspects are discussed in more detail below.
The model calibration was carried out through a process of successive approximations. In the initial stage, simulations were performed using exploratory values for the parameters. From the obtained results, the curve of normalized bone mineral density over time was interpolated for intermediate parameter values. Subsequently, the mean squared error resulting from this interpolation was minimized, allowing for an approximation of the optimal parameters. Using these values as a starting point, additional simulations were conducted iteratively, aiming to progressively approach the optimal parameters in each iteration cycle.
3.2. Analysis in the Volume Fraction Space (): The Effects Solely Caused by Variations in Are Biologically Unlikely
In the case of changes solely driven by
, where the focal balance remains unchanged, a tissue with a higher proportion of osteoid but not a significant change in bone volume would be expected.
Figure 5 shows that the average volume fraction (
) throughout the entire femur, representing the average amount of bone, does not undergo significant changes, despite the decrease in
observed in
Figure 4b. These findings carry implications for predictions during the postmenopausal period, indicating that changes solely attributed to
predict smaller losses in comparison to the other two scenarios.
The reason for this is that increased BMU activity would lead them to replace more internally located, and therefore more mineralized, bone segments [
8] because the new and superficial ones do not have enough time to mineralize and be replaced again. Despite this, as the overall amount of bone remains relatively stable, there are newly formed bone segments that mineralize at a slower rate, leading to slightly higher BMD values at the end of the simulation period compared to the other scenarios. If this phenomenon was actually possible in reality, it would result in bone with increased osteoid content and decreased mineral content but with a mostly constant amount of bone, which is unlikely to occur as menopause and ageing are strongly associated with significant bone loss [
42]. Furthermore,
Table 4 illustrates the percentage increase in porosity measurements of the tibia and radius bones extracted from work by Bjørnerem et al. [
42], compared to the average values of the simulations. It is evident that, in the case of changes solely driven by
, the predicted bone volume loss by the model is very small, while experimental measurements show significant changes (
change on total cortical porosity for tibia at peri- to postmenopause transition vs.
on simulation results). Thus, it is highly unlikely for BMD reduction to occur solely due to increased bone turnover, as its effect does not align with the data indicating increased porosity. Indeed, a long-term recovery in the volume fraction can be observed (see
Figure 5). This is because the decrease in the average elastic modulus (due to a reduction in the ash fraction) is sufficient for the mechanical stimulus to enter the bone formation zone until the accommodation process leads it to the dead zone. Since this mechanism remains unaltered, the amount of bone remains almost stable. It is clear that the factors that affect focal balance (changes in
) appear to play a significant role in the process of BMD and bone mass loss. However, the predictions of increased porosity solely due to variations in this parameter result in much higher average losses compared to the maximum observed during each period (
change in total cortical porosity for tibia at peri- to postmenopause transition vs.
on simulation results). It is worth noting that the experimental data used to adjust the parameters and those presented in
Table 4 [
42] are from different bones. Although bone tissue behavior is generally independent of its location, it is highly dependent on its initial state and everyday use. Therefore, these results should be considered in terms of general trends and approximate value ranges. In this regard, the combined effect of both parameters yields porosity percentage increase values that are more consistent with the experimental data in terms of bone loss percentage. While the curve obtained by varying only
can excellently fit the BMD evolution, disregarding the fact that it shows significantly higher values of bone mass loss (porosity) compared to the experimental data, there are factors related to bone density evolution that would indicate that this case may not be feasible. These factors will be discussed below.
3.3. Analysis in the Density Space: Resorption Solely Caused by Increased Has Improbable Effects in Practice
In this case, we will examine the results of the mentioned cases by comparing the model data with the experimental measurements of the
—
relationship obtained from the work by Zioupos et al. [
30], which were previously used to validate the results of the previous version of the current model [
8,
9]. The individual scenarios of (1) increased bone turnover and (2) increased focal balance have opposing effects. When resorption is driven by
, the boomerang-like curve shifts to the right (
Figure 6), indicating that, in general, bone fragments are more mineralized (resulting from increased
). This phenomenon arises due to net resorption, wherein the less mineralized superficial bone fragments gradually diminish over time [
8]. Conversely, the more internal fragments, characterized by higher mineral content per unit volume, are eventually the last to be resorbed. Consequently, this leads to a more fragile tissue [
43]. However, it should be noted that the most significant effect in this case is the drastic reduction in cortical bone, according to the division of previous studies [
8], causing the cortical zone to virtually disappear throughout the bone (see
Figure 6). This phenomenon can be attributed to net resorption, which increases the porosity of the cortical zone, thereby expanding its free surface area (see
Figure 7) and enhancing the activity of the BMUs. To the best of the authors’ knowledge, it is highly improbable for the compact bone to completely disappear in reality. On the other hand, when resorption is driven by
, the curve shifts dramatically to the left, indicating a higher proportion of osteoid content and a lower mineral content (resulting from a reduction in material density), which further supports the previously discussed results in the volume fraction space. Together, the results suggest that adjusting the BMD curves by independently varying each parameter (
and
) leads to unrealistic outcomes. Therefore, a consistent model would be expected to simultaneously involve changes in
and
in order to encompass both hormonal and ageing-related remodeling changes.
3.4. Simultaneous Changes in fbio and fbb Lead to Both Realistic Bone Loss over Time and Mineralization Distribution
In the previous sections, we showed that a combined change in the parameters
and
is necessary for the prediction of BMD evolution to be biologically plausible. In this case,
Figure 4b demonstrates good agreement between the numerical results and experimental data for normal activities, i.e., maintaining pre-menopausal loads. It can be observed that tissue losses fall within reasonable values compared to the data in
Table 4. Simulation results showed changes in porosity at different stages: 1.89%, 4.01%, and 3.14% from pre- to perimenopause, peri- to postmenopause, and postmenopause, respectively. These values are similar to experimental measurements of changes in compact bone porosity in tibia, which were 1.81% ± 0.19%, 3.64% ± 0.34%, and 2.00% ± 0.15% for comparable stages. The density ratio (
Figure 6) exhibits two aspects: on the one hand, there is an increase in mineral content per unit volume due to net resorption (effects of
), which leads to a more fragile tissue [
43]. On the other hand, a reduction in tissue is also noticeable in the cortical zone, although to a lesser extent than when
alone varies. It is interesting to note that the areas of maximum BMD loss are found in the most structural part of the bone.
Figure 7 shows the areas of greatest BMD loss in red and the areas of least BMD loss in blue. The losses are computed by comparing the final state of the bone (19 years after menopause beginning) with the initial bone, prior to the onset of menopause. The complete femur model allows for visualization of the maximum losses concentrated in the areas of compact tissue, precisely the tissue that supports the maximum loads of flexion and torsion. It is evident that the structural integrity of the femur, particularly in the diaphysis, deteriorates, which can increase the risk of fractures in older individuals. The BMU activity is enhanced due to the amount of free surface growth (increase of porosity), which enters into negative feedback for the cortical zone and adds to the increased loss due to the changes in
and
. Conversely, although there is a reduction in BMD in the more spongy and internal areas of the bone, they are already at one extreme of the available surface/porosity curve, so there is no growth in activity beyond that implied by menopause and ageing factors. The relationship between free bone surface and porosity harms the more compact areas and, therefore, their structural integrity and mechanical capacity once osteoporosis is reached. This is one of the reasons why some inevitable losses in the ability to bear loads occur and the reason why special attention should be paid to bone growth and strengthening in the years leading up to menopause, as was pointed out by Hernández et al. [
21]. However, once menopause is reached, a question arises regarding the natural improvement or deterioration in bone health over time. In line with this objective, we will now delve into the potential impact of increasing or decreasing physical activity to examine how modifying loads can affect long-term bone health once prevention is no longer an option.
3.5. Exercise during the Perimenopause Stage Delays the Onset of Osteoporosis
To assess the evolution of BMD under modified loads, the osteoporosis threshold was defined in terms of
. Osteoporosis is diagnosed when the BMD of a specific bone region falls below 2.5 standard deviations from the healthy value of young women [
29]. The osteoporosis threshold was calculated using total femur BMD female data from Looker et al. [
44] and the following equation:
In this equation, BMDs represents the healthy value, which is the average between 20- and 29-year-old women (
),
is the standard deviation between them, and
is the average value between the women of ages from 40 to 59 (
). From the tables provided by Looker et al. [
44], the osteoporosis threshold was calculated for the total femur (see
Table 5). Depending on the population, this threshold has some variations—0.720 for non-Hispanic white (
,
), 0.698 for non-Hispanic black (
,
), and 0.727 for Mexican white (
,
), but all are inside the interval of 0.70–0.73, and these little variations do not affect the conclusion. A value of 0.72 will be used to compare different load scenarios.
By using the model parameters for combined changes, in this section we will compare the evolution for a normal load and two opposite situations: a moderate increase of 30% in load to simulate sustained physical activity (overload), and a reduction of the same percentage to simulate a trend towards sedentary behavior.
In the first scenario, represented by
Figure 8, it is evident that, regardless of the time of the initiation of physical activity, there is a consistent increase in
compared to normal activity. Furthermore, this increase is sustained even after menopause, as long as activity remains. In all cases, there is a temporary boost in
while the bone re-adapts to the new situation, followed by a subsequent evolution of
along a curve that closely parallels the original curve. On the other hand, a decrease in activity leads to a pronounced drop of
, stabilizing the slope over time and resulting in a net bone change over time that is similar to the normal case. This behaviour is closely related to the parameter
that accounts for the tissue’s adaptive capacity to external stimuli. The
value used in this study demonstrated highly accurate predictions of bone mass evolution in response to disuse in a previous work [
9].
These findings have an important implication: it is never too late to engage in physical activity, as this has the potential to mitigate the deterioration in the material composition of bone tissue. This effect is particularly relevant before and during menopause, emphasising the importance of maintaining consistent and long-term engagement in physical activity. On the contrary, sedentary behaviour, which is commonly observed in older women, accelerates the decline in BMD, posing a significant risk of early osteoporosis, which increases the likelihood of fractures. It is worth noting that the effects of overload/underload regimes are not exactly mirrored, as the effects of disuse are more harmful than their overuse counterpart benefits. In fact, our results suggest that a moderate decrease in mechanical stimulation can precipitate the onset of osteoporosis by up to 6 years when compared to bones that receive adequate stimulation. Our findings also highlight the potential for additional improvement by implementing higher levels of controlled stimulation, which were not within the scope of this study.
3.6. Limitations
While our model holds significant promise for future research, it is important to acknowledge certain limitations. First, it is based on a continuum approach, which is not suitable for investigating the effects of trabecular thinning. Additionally, it cannot predict the impact of connectivity loss due to trabecular removal, which can affect strain distribution and, consequently, local remodelling. Furthermore, the model does not currently account for the effect of microdamage, which limits its ability to predict the maximum loads that can be applied, before or after the onset of osteoporosis, without causing local failures. This, however, can and will be incorporated in future versions of the model.
4. Conclusions
This work introduces a bone remodelling model governed by mechanical loads, incorporating the effects of osteoporosis and ageing through constant variation functions that weigh the activity of BMUs (frequency of one turnover) and focal balance. In contrast to previous works, this model assumes only constant (irreversible) variations in these factors, which persist and continue to vary even after menopause to account for ageing. These functions are adjusted so that the model’s outcomes under normal loads align with the experimental data.
According to the results of this work, the effects of increased bone turnover (frequency of BMU activation, ) and net removal (reduced focal balance, ) must be simultaneously present, with a constant variation over time. These effects lead to a consistent and inevitable loss of BMD and bone mass with age, resulting in age-related osteoporosis that depends on a mechanical stimulation of the bone, among other factors. It should be noted that while does not directly affect mass, its increase enhances the frequency of BMU activation with reduced (focal balance). Controlling will impact the control of both factors simultaneously, while controlling only would reduce bone mass losses to a greater extent. These effects can be achieved by planning training programs that gradually increase physical activity. The transient effect of medications can easily be incorporated into the model by adjusting the shape of the curves. The calibration of the model for these scenarios is beyond the scope of this paper.
By emphasizing the significance of maintaining an active lifestyle, especially during menopause and as individuals age, they can proactively safeguard their bone health. The study’s crucial findings demonstrate that engaging in moderate physical activities can delay the onset of osteoporosis by up to 6 years when compared to a sedentary lifestyle, thus reducing the likelihood of developing severe osteoporosis in old age. Moreover, the potential for a greater reduction exists if physical activity programs are implemented to increase activity and further delay or even prevent the onset of osteoporosis.
Overall, the model exhibited an excellent ability to fit the experimental data in previous studies, and an exceptional ability to adapt to the BMD evolution in this study. Once the model was aligned with experimental measurements, it showed excellent predictive capabilities for the average bone density evolution under both normal and pathological conditions. In this sense, the results of this study are promising not only for investigations into the interplay between hormonal factors and BMU activity, but also for predicting the potential effects on the natural prevention of osteoporosis through the personalized planning of physical activities. Subsequent studies with our model will explore the impact of targeted hormonal treatments in conjunction with planned training.