Introduction

Bone is an example of an optimized structure. The internal architecture of bone is adapted to external mechanical load exerted during daily physical activity1,2,3,4. However, certain conditions, such as extended space travel5,6, prolonged bed rest7,8, or lower-limb paralysis due to spinal cord injury9, disrupt regular physical activities. Reduced mechanical loading on weight-bearing bones under these conditions results in significant bone loss10. Recent studies show that the microgravity environment in spaceflight is so detriemental that even one year post-mission, lost bone tissue is not fully recovered, potentially leading to osteoporosis11. For example, data from the Russian Mir and the International Space Station reveal that crew members of long-duration spaceflights lose approximately 1.2–1.5% of bone mass per month in their proximal femur12, and a six-month mission results in a 2% loss in cortical density in the tibial diaphysis. However, spaceflight studies are prohibitively expensive. Therefore, partial weight-bearing13,14 and hindlimb unloading (via tail suspension) have been used to simulate such condition15. For example, 14 days of hindlimb unloading causes significant bone loss on the endocortical surface of the femoral diaphysis16. Botox has also been studied as a method to induce muscle dysfunction17, resulting in bone disuse and subsequent bone loss18,19. This method has shown that Botox-induced endo-cortical bone loss is heterogeneous along the tibial length20. Despite these findings, the quantitative relationship between bone loss and mechanical loading deficit remains poorly understood.

It is well-established that osteocytes function as mechanosensory cells21,22 and are intricately embedded within the bone matrix via the lacuno-canalicular network (LCN). Experimental studies suggest that load-induced fluid flow within the lacuno-canalicular system (LCS) is critical for nutrient transport and enhancing osteocyte metabolic activity-essential for bone health23. Additionally, fluid flow induced by loading is recognized as a principal stimulus for bone mechanotransduction23,24,25,26. Research also indicates that this fluid flow through the LCS creates shear stresses on the cell walls27, which activate the mechanosensing osteocytes. Tracer techniques confirmed fluid flow within the LCS28, although experimentally measuring fluid velocity is difficult due to LCN inaccessibility. Only a few studies have successfully measured fluid velocity29, which is why the finite element method is commonly employed to estimate it30,31,32,33,34,35,36. Recent studies have demonstrated that Botox-induced muscle paralysis significantly alters bone morphology37, resulting in decreased fluid velocity around the osteocytes38. This finding motivates the present study, which models bone considering reduced load-induced fluid flow.

When bone is mechanically strained during physiological loading, a pressure gradient develops within the LCN, causing fluid to flow through the canalicular channels (Fig. 1). Due to the viscosity of the fluid, this flow generates shear stress on the cell membrane and leads to energy dissipation. The dissipated energy, as a function of fluid velocity, is assumed to serve as a stimulus for mechanosensitive osteocytes34. Studies indicate that in loaded bone, the resultant fluid velocity is higher at the endocortical surface but negligible at the periosteal surface39 due to the respective surfaces being permeable and impermeable40. Botox-induced calf muscle paralysis leads to loss of cyclic loading on the load-bearing bone, thereby significantly reducing or ceasing fluid flow within the porosities of the LCN (Fig. 1d). Consequently, there is a greater loss of fluid velocity at the endocortical surface, which correlates with the in-vivo pattern of bone loss observed at the endocortical surface following Botox-induced muscle paralysis.

Fig. 1
figure 1

Comparison of fluid flow within the LCN porosity between a physiologically loaded and a disused tibia: (a) axially loaded tibia, (b) schematic of the osteocyte network at a cross-section of interest (black-filled circles represent cortical canals), (c) fluid flow within the LCS of a physiologically loaded tibia (indicated by arrows showing fluid movement), and (d) absence of fluid flow within the LCS in the case of bone disuse.

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This study focuses on the loss of mechanical stimulus—specifically, the reduction in fluid flow-based dissipation energy density due to bone disuse—and introduces a mathematical model to predict cortical bone loss accurately. The developed model was validated using published data on Botox-induced bone loss20. Statistical tests were conducted to assess the model’s accuracy in predicting in vivo results. Given relevant experimental data, the model may be extended to predict bone loss in other disuse scenarios, such as long space travel or prolonged bed rest. The paper is structured as follows: Section "Methods" outlines the adopted methodology, Section "Results" presents the results, Section "Discussions" discusses the findings, and Section "Conclusions" provides the conclusions.

Methods

Theory of poroelasticity

The mouse tibia was considered a linear isotropic porous material consisting of a solid matrix and fluid-saturated pores. The constitutive equations for such a material are as follows41.

$$\begin{array}{*{20}c} {\sigma_{ij } = 2G\varepsilon_{ij} + \frac{2G\nu }{{1 - 2\nu }}\varepsilon_{kk } \delta_{ij } - \alpha p\delta_{ij } } \\ \end{array}$$
(1)
$$\begin{array}{*{20}c} {\zeta = \alpha \Delta + \frac{p}{Q}} \\ \end{array}$$
(2)

where \(\sigma_{ij}\) and \(\varepsilon_{ij}\) are stress and strain tensors, respectively (i or j represents the Cartesian coordinates, i.e., i or j = 1,2,3 corresponds to the x, y, z coordinates). \(G\) and \(\nu\) represent the modulus of rigidity and Poisson’s ratio, respectively. \(p\) is the interstitial pore pressure, \(\zeta\) is the increment of the water volume per unit volume of the solid matrix, \(\delta_{ij }\) is the Kronecker delta (i.e., \(\delta_{ij } = 1\) if i = j; \(\delta_{ij } = 0\) otherwise), \(\Delta\) is the dilation that represents the volumetric strain developed in the pore. \({\upalpha },{ }\) and \(\frac{1}{{\text{Q}}}\) are the constants.

Equation (3) describes the fluid flow rate through a porous media following Darcy’s law:

$$\begin{array}{*{20}c} {q_{i } = - \frac{k}{\mu }\frac{\partial p}{{\partial x_{i} }}} \\ \end{array}$$
(3)

where \(q_{i}\) is the fluid mass flow rate, while \(k\) and \(\mu\) represent the isotropic intrinsic permeability and the fluid’s dynamic viscosity, respectively.

The mass balance equation is as follows:

$$\begin{array}{*{20}c} {\frac{\partial \zeta }{{\partial t}} = - \frac{{\partial q_{1} }}{{\partial x_{1} }} - \frac{{\partial q_{2} }}{{\partial x_{2} }} - \frac{{\partial q_{3} }}{{\partial x_{3} }}} \\ \end{array}$$
(4)

The above equations finally lead to the following differential equation that needs to be solved:

$$\begin{array}{*{20}c} {k\nabla^{2} p = \alpha \frac{\partial \Delta }{{\partial t}} + \frac{1 }{Q}\frac{\partial p}{{\partial t}}} \\ \end{array}$$
(5)

Additionally, the linear momentum balance can be written as

$$\begin{array}{*{20}c} {G\nabla^{2} u_{i} + \frac{G}{1 - 2\nu }\frac{\partial \Delta }{{\partial x_{i} }} = \alpha \frac{\partial p}{{\partial x_{i} }}} \\ \end{array}$$
(6)

where \(u_{i}\) is the displacement of the solid matrix. Equations (5) and (6) together have four unknowns to be solved for—three displacement components (viz. \(u_{1}\), \(u_{2}\) and \(u_{3}\)) and the fluid pore pressure p. The problem was solved using the commercial finite element software ABAQUS42, as detailed in section "Finite element analysis".

Cage activity of a mouse

The cage activity data for the mice were taken directly from an earlier study by Hasriadi et al., in which six mice were monitored over a 24-h period, and their activities were recorded43. The typical activities observed in the cage included locomotion, rearing, immobilization, and climbing:

  • Locomotion: Locomotion was the most common activity. During locomotion, the tibia experiences ground reaction forces, which can be resolved into axial compression and transverse shear forces44. It was found that the magnitude of the shear force is minimal compared to the axial component and could therefore be ignored38,45. As a result, the primary force acting on the mouse tibia during locomotion is axial compression.

  • Rearing: Rearing was the second most frequent activity. When rearing, the mouse stands on its hind limbs, subjecting the tibia to pure axial loading.

  • Immobilization: During immobilization, static loading is applied to the tibia. However, it is well established that static loading does not promote new bone formation46.

  • Climbing: Climbing leads to no ground reaction forces on the mouse tibia.

These observations suggest that axial loading is typically the dominant mechanical stimulus during a mouse’s cage activities.

Bone geometry

The tibia of a 16-week-old C57BL6J mouse was scanned at a voxel size of 21 µm, and micro-CT images were obtained44. The raw micro-CT images were processed using appropriate tools, including flipping, rotating, resizing, cropping, and converting to black and white. An appropriate threshold was selected to capture the tibial cross-section. All processing steps were performed using the in-house software “Priffer,” as illustrated in Fig. 2. The processed micro-CT images were then rendered to generate a voxel-based 3D model of the tibia. After convergence analysis, 76,400 brick elements were used for the finite element analysis, as discussed in Section "Finite element analysis".

Fig. 2
figure 2

Construction of a voxel-based tibia model: (a) a set of raw micro-CT images of a 16-week-old mouse tibia, (b) processed micro-CT images: The raw images were processed using necessary tools such as flipping, rotating, resizing, cropping, and converting to black and white. An appropriate threshold was chosen to capture the bone cross-section. (c) Rendered 3D mouse tibia with brick elements (hexahedral elements) for finite element analysis.

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Porosity and permeability of cortical bone

The cortical bone has porosities at different length scales, such as vascular and lacuno-canalicular porosities, which influence bone permeability. It is widely accepted in the literature that the load-induced fluid flow inside the LCN porosity plays an important role in bone mechanotransduction. Accordingly, only LCN porosity has been considered, i.e., \(3.79 \times 10^{ - 21}\) \({\text{m}}^{{2{ }}}\) 47, as also presented in Table 1.

Table 1 Poroelastic material properties used in the simulations (adapted from Kumar et al.34 except for the permeability value from Zhou et al.47).
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Loading and boundary conditions

The physiological loading of a caged mouse is dominated by axial loading (Fig. 3a). To mimic the physiological condition, a Haversine loading waveform with a frequency of 1 Hz (Fig. 3c) was considered in accordance with the literature38. The axial component (1.3 N) of the ground reaction force exerted during physiological loading conditions was obtained from the literature44. The proximal end of the tibia was fixed, and the load was applied at the distal end, as shown in Fig. 3a. In accordance with the literature, the periosteal and endocortical surfaces were assumed to be completely impermeable and permeable, respectively38,39,40 implemented through zero-flow and zero-pressure boundary conditions (Fig. 3b).

Fig. 3
figure 3

Loading and boundary conditions (a) The proximal end of the tibia was fixed, and the distal end was axially loaded with 1.3 N, (b) The periosteal and endocortical surfaces of the bone were prescribed zero-fluid-flow and zero-pressure (\(p=0\)) conditions, respectively. (c) To mimic the physiological loading, a Haversine loading waveform with a frequency of 1 Hz was assumed.

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Finite element analysis

The finite element analysis was conducted using the commercial software ABAQUS (Dassault Systèmes SE). To simulate fully or partially saturated fluid flow through porous media, the soil element type C3D8RP (continuous, three-dimensional, eight-noded, reduced integration pore-pressure hexahedral element) was assigned to the brick element of the tibial bone, as available in ABAQUS42. These elements had four degrees of freedom per node—three for displacement and one for pore pressure. The bone material was modeled as a linear isotropic poroelastic material, with parameters taken directly from Kumar et al.34 and listed in Table 1.

A coupled pore fluid diffusion-stress analysis was performed in ABAQUS to capture the fluid flow. The fluid velocity was computed at the integration point during the applied loading. The strain induced by physiological loading (idealized as axial loading) and the corresponding fluid flow velocity at three sections—distal, mid-section, and proximal—along the tibial length (as studied by Ausk et al.20) are presented in Fig. 4. The obtained strain patterns were also validated against the existing literature48.

Fig. 4
figure 4

Induced strain and fluid velocity for axially loaded tibia: (a) Longitudinal strain contour of a mouse tibia; (b) Longitudinal strain (EE33) pattern at the three sections, i.e., distal, mid-section, and proximal (as studied by Ausk et al.20); (c) Corresponding fluid velocity (mm/s) along the full tibia; (d) Fluid velocity (mm/s) at the three sections, viz., distal, mid, and proximal.

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Prismatic bar analysis

A voxel-based 3D model of a mouse tibia exhibits jagged edges at the periosteal and endocortical surfaces, which may affect fluid flow predictions. To mitigate these surface irregularities and allow for smoother geometries, prismatic bars of the distal, midshaft, and proximal sections were created. Each prismatic bar (length: 3 mm) had a cross-section representative of a 16-week-old C57BL6J young adult mouse tibia, as described in Section "Finite element analysis" and shown in Fig. 5. This approach also provides a computationally efficient alternative for poroelastic fluid flow analysis. Nevertheless, the non-use of a smooth full-bone model is acknowledged as a limitation, and its incorporation is identified as a direction for future work.

Fig. 5
figure 5

The bone (a) is idealized as a prismatic bar (b) of length 3 mm, with a cross-section similar to that of the (b) “proximal section,” (c) “mid-section” and (d) “distal section.” (e) loading condition of a prismatic bar: one end was fixed, and the other end was loaded such that it induced strain similar to that of the full tibia.

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The prismatic bars shown in Fig. 6a were meshed with 10,650 hexahedral elements after performing a convergence analysis. These bars were modeled as a linear poroelastic material, with material properties identical to those of the whole tibia, as outlined in Table 1. One end of each bar was fixed, while the other end was subjected to loading in all three directions (x, y, and z), as shown in Fig. 5e, to generate strain similar to that in the full tibia. The resulting strain patterns and fluid velocities are shown in Fig. 6b,c. A comparison of the induced strain and corresponding fluid velocity between the voxel-based 3D tibia model and the prismatic bar analysis is provided in Table 2.

Fig. 6
figure 6

The bone is idealized as (a) a prismatic bar of length 3 mm, with a cross-section similar to that of the “proximal section,” “mid-section,” and “distal section” of the original tibia studied in Section "Finite element analysis"; (b) induced strain at the three sections; (c) corresponding fluid velocity (mm/s).

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Table 2 Strains and fluid velocities at three distinct sections along the length of a mouse tibia (All strain values are presented in microstrain while fluid velocities are measured in mm/s).
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Botox-induced disuse activity

Botox-induced paralysis of the calf muscle group impairs the muscle’s ability to apply force, leading to a reduction in cyclic loading on the mouse tibia. Cyclic loading is a fundamental mechanical stimulus46 that facilitates fluid movement within the lacunar-canalicular system. In the referenced experiment, Botox was injected into the calf region of the mouse tibia20, which likely impaired plantar flexion of the ankle. After Botox administration, mice may exhibit two different locomotor adaptations:

  • Sitting on the injected limb: In this case, the limb bear a static load corresponding to the mouse’s body weight. Previous studies have shown that static loading does not induce fluid flow within the LCN46.

  • Dragging the injected limb: If the mouse drags the affected limb, cyclic loading on the tibia is substantially diminished, further reducing the stimulus for fluid flow.

Based on these considerations, we assume that Botox application results in negligible cyclic loading of the tibia and, consequently, minimal fluid movement within the LCN. We acknowledge the limitations associated with this assumption.

Computation of fluid flow-based dissipation energy density

Load-induced viscous fluid flow inside LCN porosities exerts shear stress on the osteocyte membrane, resulting in energy dissipation. The dissipated energy influences the mechanosensory osteocyte, activating further biochemical signalling pathways. According to Kumar et al.34, the dissipation energy density (\(DED)\), which is a function of fluid velocity, can be calculated as:

$$\begin{array}{*{20}c} {DED = - n_{p} {\varvec{v}}^{fl} .\nabla p = \frac{1}{2 }\left( {n_{p} {\varvec{v}}^{fl} } \right).\left( {\frac{k}{\mu }} \right)^{ - 1} .\left( {n_{p} {\varvec{v}}^{fl} } \right)} \\ \end{array}$$
(7)

where \(n_{p}\) is the LCN porosity, \({\varvec{v}}^{fl}\) is the fluid velocity.

Zone of influence

Osteocytes form an interconnected network ( i.e.,LCN) through which biochemical signals are transmitted. These signals act as messengers for communication with other cells, such as osteoblasts, bone-lining cells, and osteoclasts. Such communication has been described in the literature by different mechanisms, including diffusion49 and the zone of influence32,34,50. According to the zone of influence concept, we capture the contribution of each osteocyte within a defined zone. Osteocytes located farther from the node of interest (where a total stimulus is being calculated), depicted in red in Fig. 7a, contribute less than those closer to the node of interest.

Fig. 7
figure 7

(a) The non-local behavior of osteocytes is captured through a spherical zone of influence. The red node represents the point of interest. (b) In vivo cortical bone loss at the distal section of a tibia20; the endocortical bone surface was divided into 360 points. For clarity, points are shown at every 10 degrees.

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This non-local behavior of osteocytes was modeled using a weighting function. We used an exponential weighting function \(\left( {\left| x \right|} \right) = e^{{\left( { - \frac{5\left| x \right|}{R}} \right)}}\), consistent with the literature34,50, where |x| represents the distance between the node of interest and the position of an osteocyte, while R denotes the radius of the spherical zone. To include the contributions of all osteocytes, we set R = 155 µm, i.e., the radius of the spherical zone, equal to the mean cortical thickness of the three sections: “distal,” “mid,” and “proximal”. This is in accordance with earlier work in which optimization was used to determine the optimal radius39.

Considering the zone of influence (ZOI), the dissipation energy density was calculated using Eq. (8) at each ith node of interest. The total dissipation energy density is given by \(\phi = DED*N*d\), where N is the number of cycles and d is the number of days.

$$\begin{array}{*{20}c} {DED_{i} = \frac{{\mathop \smallint \nolimits_{V}^{ } \left( {\mathop \smallint \nolimits_{0}^{T} \frac{1}{2 }\left( {n_{p} {\varvec{v}}^{fl} } \right).\left( {\frac{k}{\mu }} \right)^{ - 1} .\left( {n_{p} {\varvec{v}}^{fl} } \right)dt} \right)w\left( {\left| x \right|} \right)dV}}{{\mathop \smallint \nolimits_{V}^{ } w\left( {\left| x \right|} \right)dV}}} \\ \end{array}$$
(8)

Mineral resorption rate (MRR)

Exogenous loading-induced fluid flow inside the LCS has been widely accepted in the literature as being correlated with bone adaptation. Recently, Sanjay et al.50 presented a robust formulation showing that the MAR (mineral apposition rate), i.e., the rate of change of new bone thickness, is directly proportional to the square-root of total dissipation energy density minus its threshold or reference value. That is,

$$\begin{array}{*{20}c} {MAR = A\left( {\sqrt {\phi_{c} } - \sqrt {\phi_{phy} } } \right),} \\ \end{array}$$
(9)

where A is the remodeling rate, \(\phi_{c}\) is the current dissipation energy density, and \(\phi_{phy}\) represents the dissipation energy density required to maintain bone homeostasis under physiological loading conditions.

Under disuse conditions, the fluid velocity is zero, making \(\sqrt {\phi_{c} }\) to be zero. Hence, the expression for MAR simplifies to:

$$\begin{array}{*{20}c} {MAR = - A\left( {\sqrt {\phi_{phy} } } \right)} \\ \end{array}$$
(10)

Here, the negative sign indicates bone loss; therefore, the mineral resorption rate (MRR) can be written as:

$$\begin{array}{*{20}c} {MRR_{model} = A\left( {\sqrt {\phi_{phy} } } \right)} \\ \end{array}$$
(11)

Thus, \(MRR_{model}\) is directly proportional to the square-root of the loss in total dissipation energy density due to bone disuse. The parameter A is a constant that needs to be determined. The MRR (\(\upmu {\text{m}}^{3} {/}\upmu {\text{m}}^{2} {\text{/day}}\)) is defined as the total loss of bone volume per unit of bone surface area per unit time. For computational convenience, the endocortical bone surface was divided into 360 segments, and then MRR was calculated at each ith point (i.e., the common point of two consecutive segments) as

$$\begin{array}{*{20}c} {MRR_{model}^{i} = A\left( {\sqrt {\phi_{i} } } \right).} \\ \end{array}$$
(12)

These points are illustrated in Fig. 7b as an example. The constant A was determined by comparing experimental bone loss data from the literature. The experimental endocortical bone loss \(MRR_{exp}^{i}\) for each of the 360 points was computed from Fig. 7b using an in-house MATLAB code. The optimization problem was formulated as the minimization of squared error:

$$\begin{array}{*{20}c} {\min f\left( A \right) = \mathop \sum \limits_{i = 1}^{i = 360} \left( {MRR_{exp}^{i} - MRR_{model}^{i} } \right)^{2} } \\ \end{array}$$
(13)

The Levenberg–Marquardt algorithm51 available in MATLAB was used to fit the model in Eq. (12) to the experimental data, i.e., \({MRR}_{exp}^{i}\) and to find the optimal value of the constant A. The calculated optimal parameter A (6.556 \(\frac{{\text{mm}}^{2}}{\sqrt{N} \text{day}}\)) was then used to predict bone loss at the two remaining sections, viz., the mid and proximal sections. A statistical analysis (as detailed in Section "Statistical Analysis") was performed to compare the model predictions with the experimental values.

Statistical analysis

The endocortical bone loss area per unit time was computed by integrating the MRR over the circumference to compare the model predictions with the experimental values. The bone resorption rate per unit bone surface (BRR/BS) was obtained by dividing the loss area per unit time by the total perimeter of the surface under consideration (i.e., the endocortical surface). A one-sample, two-tailed Student’s t-test was used to compare the BRR/BS values. A circular goodness-of-fit test, namely Watson’s U2 test52, was used to compare the endocortical distribution of bone loss predicted by the model with the in-vivo results.

Results

Distal section

The developed model was first used to predict cortical bone loss at the distal section, as shown in Fig. 8a. The BRR/BS values have been compared in Fig. 8d through a bar chart. The site-specific bone loss distribution predicted by the model is shown in Fig. 8c, whereas the experimental bone loss is shown in Fig. 8b. Statistical analysis showed that the model-predicted BRR/BS (1.64 \(\upmu {\text{m}}^{3} {/}\upmu {\text{m}}^{2} {\text{/day}}\)) was not significantly different from the in vivo BRR/BS(1.15 ± 0.67 \(\upmu {\text{m}}^{3} {/}\upmu {\text{m}}^{2} {\text{/day}}\), p = 0.47, Student’s t-test). The predicted bone loss distribution was also not significantly different from the in vivo distribution (p = 0.33, Watson’s U2 test).

Fig. 8
figure 8

Comparison of the model and experimental cortical bone loss distribution at the three sections of a mouse tibia: (a) The three cross-sections—namely “distal,” “mid,” and “proximal”—along the length of a mouse tibia; (b) In vivo site-specific bone loss20 (shown in red) at the corresponding sections; (c) Bone loss predicted by the model; and (d) Bar chart comparing the model BRR/BS to the in vivo value (error bars represent the standard error).

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Mid-section

Mid-section cortical bone loss was also predicted using the same resorption rate A. The predicted and in vivo BRR/BS values were compared in Fig. 8d. The predicted BRR/BS (1.13 \(\upmu {\text{m}}^{3} {/}\upmu {\text{m}}^{2} {\text{/day}}\)) was found to be not significantly different from the experimental value (1.43 ± 0.5675 \(\upmu {\text{m}}^{3} {/}\upmu {\text{m}}^{2} {\text{/day}}\), p = 0.61, Student’s t-test). The endocortical bone loss distribution predicted by the model and observed in the in vivo experiment is shown in Fig. 8b,c, respectively. The distribution predicted by the model was not significantly different from the in vivo distribution (p = 0.99, Watson’s U2 test).

Proximal section

Bone loss at the proximal section of the same tibial bone was also predicted using the same resorption coefficient, A. The predicted BRR/BS (0.87 \(\upmu {\text{m}}^{3} {/}\upmu {\text{m}}^{2} {\text{/day}}\)) was found to be not significantly different from the corresponding experimental value of 1.23 ± 0.85 (p = 0.68, Student’s t-test) as compared in Fig. 8d. However, the site-specific cortical bone loss distribution predicted by the model (Fig. 8c) was found to be significantly different from that of the experimental case (Fig. 8b) (p = 0.0003, Watson’s U2 test). Thus, the model was unable to accurately predict the experimental bone loss distribution for this case.

Proximal section incorporating muscle attachments

The results so far were obtained based on axial loading of the tibia, simplifying the real scenario in which different muscles act at different points of time during various cage activities. This simplification was able to predict bone resorption for both distal and mid-diaphyseal sections, as detailed in Sections "Distal section" and "Mid-section". Since the model was unable to accurately predict bone loss at the proximal section, the effect of muscle activity needs to be incorporated. The literature indicates that four significant muscles—viz., flexor digitorum longus (FDL), tibialis posterior (TP), extensor hallucis longus (EHL), and tibialis anterior (TA)—are attached53 close to the proximal section under consideration, as schematically shown in Fig. 9a. Since no such muscle attachments are present near the distal and mid-section, the proximal section may need to be analyzed differently and more accurately. This is also consistent with Saint–Venant’s Principle54.

Fig. 9
figure 9

(a) Proximal section of the tibia with relevant muscles, (b) TA muscle attachment, where the black dot indicates the attachment point, (c) Line diagram showing the TA muscle attachments in the whole tibia. (d) Strain distribution at the proximal section under consideration. (e) Corresponding fluid velocity (mm/s).

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Incorporation of muscle activity is complex, however, as various cage activities and involve different muscle forces, which are currently unknown in the existing literature. In light of this, a simplified analysis was carried out as an example.

These muscles, which are attached close to the proximal section of the tibia, contribute to dorsi flexion (TA, EHL) and plantar flexion (TP, FDL) of the ankle during walking and thus apply load to the bone. The maximum load is carried by the dorsiflexion muscle55 during a gait cycle. The dorsi flexor group includes the tibialis anterior (TA) and extensor hallucis longus (EHL). Among these, TA is the more prominent due to its higher physiological cross-sectional area (PCSA); hence, only TA was considered in this study.

The attachment point of the TA muscle at the proximal section (Fig. 9b) and the corresponding muscle force (2.4 N) were obtained from the literature56. The TA muscle force was applied along the line of action of the muscle (Fig. 9c), keeping the material properties and boundary conditions the same as in the earlier analysis. The corresponding strain and fluid flow patterns obtained by the finite element analysis are shown in Fig. 9d,e, respectively.

The experimental and model-predicted bone loss distributions are shown in Fig. 10c,d, respectively. A prismatic bar of the same proximal section was also created and loaded to induce strain similar to that in the whole tibia, as shown in Fig. 10a, and the corresponding fluid velocity is shown in Fig. 10b. The BRR/BS values have also been compared in Fig. 10e. The predicted BRR/BS (2.07 \(\upmu {\text{m}}^{3} {/}\upmu {\text{m}}^{2} {\text{/day}}\)) was not significantly different from the experimental BRR/BS of 1.23 ± 0.85 (p = 0.34, Student’s t-test).

Fig. 10
figure 10

Proximal section with attached muscle idealized as a prismatic bar: (a) Induced strain at the considered proximal section; (b) Fluid velocity at the corresponding section; (c) In-vivo site-specific cortical bone loss20; (d) Site-specific bone loss predicted by the model; (e) comparison of model BRR/BS to the in-vivo values.

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The bone loss distribution was also found to be not significantly different (p = 0.09, Watson U2 test). Note that this is only an illustrative example highlighting the importance of incorporating muscle attachments in the current study. However, due to the lack of muscle force data for all cage activity conditions, a detailed analysis incorporating muscle attachments has been designated as future work.

Sensitivity analysis

While developing a mathematical model, it is crucial to analyze how different parameters affect the final output. A sensitivity analysis is necessary to demonstrate how parameter variations influence the result. In this study, we employed a local sensitivity analysis, also referred to as a one-at-a-time sensitivity measure57. This approach involves varying the model parameter by a set amount and observing its impact on the final output.

Here, we varied the model parameter—remodeling rate (A)—by a given amount and noted the BRR value at the sections under consideration, as depicted in Fig. 11. The results indicated that BRR values change linearly with the remodeling rate. Consequently, we identified the remodeling rate (A) as a proxy for the rate of osteocyte apoptosis in the absence of fluid flow, which leads to osteoclast recruitment and, thus, bone loss. This finding aligns with existing literature, which identifies a similar parameter associated with osteogenesis in response to increased mechanical loading50,58.

Fig. 11
figure 11

The plot depicts the BRR as a function of remodeling rate (A) in four different sections, viz., distal, mid, proximal, and proximal with muscle.

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Discussions

The present work studied cortical bone tissue loss due to the bone disuse and developed a fluid flow-based mathematical model that utilize dissipation energy density as the stimulus. The finite element method was used to simulate fluid flow inside the porosity of the LCN, considering bone as a poroelastic material. The model was fitted to existing bone loss data from the literature.

The developed disuse model showed the linear relationship between MRR and \(\sqrt{\phi }\) (i.e., the square-root of the loss in dissipation energy density), which aligns with earlier work by Sugiyama et al.59. Notably, dissipation energy density is proportional to the square of strain; hence, MRR being proportional to \(\sqrt{\phi }\) is analogous to the bone formation rate (BFR) being proportional to strain60. A study by Singh et al.39 has also demonstrated a similar trend- i.e., the rate of new bone formation is directly proportional to \(\sqrt{\phi }\). Bone formation occurs when the mechanical stimulus exceeds a threshold value.

It is widely accepted that osteocytes sense the mechanical stimulus for mechanotransduction. Mechanosensory osteocytes require nutrients for survival, which are delivered via load-induced fluid flow inside the LCS. However, in the case of bone disuse, there is no significant fluid flow, which may hinder nutrients transport within the LCS. This condition may also result in hypoxia within the LCS61. Such an environment is detrimental for osteocyte viability and may lead to osteocytes apoptosis62. Dying osteocytes damage the LCN, which becomes a beacon for osteoclast recruitment63,64. The active osteocytes near apoptotic ones produce receptor activator of nuclear factor kappa-beta ligand (RANKL), which binds to the receptor activator of nuclear factor kappa-beta (RANK) on the surface of osteoclast precursor cells. This interaction initiates osteoclastogenesis, as illustrated in Fig. 12.

Fig. 12
figure 12

Bone disuse leads to osteoclastogenesis: (a) Loss of fluid flow inside the LCS causes apoptosis of osteocytes; (b) Active osteocytes near the apoptotic osteocytes release RANKL; (c) RANKL binds to RANK present on osteoclast precursor cells, which leads to (d) Osteoclastogenesis and (e) Bone resorption. (Some parts of this illustration have been created using the Servier medical Art, SMART; https://smart.servier.com/).

Full size image

This theory is supported by in vivo studies available in the literature. For example, Emerton et al. showed that ovariectomy in mice caused osteocyte apoptosis and subsequent bone loss via the formation of active osteoclasts65. Another study, in which bone disuse was induced using a tail suspension model, presented similar findings—bone tissue loss coincided with the site of apoptotic osteocytes66. In a related study, disuse caused by hindlimb unloading (via tail suspension) damaged the LCN and led to osteocyte apoptosis, ultimately resulting in subsequent bone loss16.

Although bone disuse was achieved through different methods, it ultimately resulted in a loss of mechanical stimulus—specifically, a reduction in fluid flow inside the LCS. This led to osteocyte apoptosis and initiated osteoclastogenesis.

In line with the above studies, the present work has established that the site of maximum bone resorption coincides with the region of maximum stimulus loss, i.e., the greatest loss of dissipation energy density. The developed model successfully predicted cortical bone loss at the distal and mid-section of the mouse tibia. However, it failed to accurately predict the bone loss distribution at the proximal section. As discussed in Section "Proximal section incorporating muscle attachments", this may be attributed to the different anatomical arrangement (e.g., muscle attachments) of the proximal section compared to the distal and mid-diaphyseal sites67.

It is noteworthy that cortical bone loss at the endocortical surface was primarily concentrated near the surface, which aligns with earlier findings by Berli et al.68,69, who emphasized the role of active surface area in bone remodeling. Our model supports and complements this perspective by integrating fluid flow-driven mechanical stimulus with spatially weighted surface activity.

Exogenous loading-induced bone formation in young-adult mice occurs at both surfaces, viz., the periosteal and endocortical surfaces70,71,72,73,74. Numerous mathematical models have been developed to predict bone formation, but most focus exclusively on the periosteal surface only26,32,49. However, very few models in the literature predict bone formation at both surfaces50.

Initially, tissue strain induced by increased mechanical loading was considered the primary stimulus for predicting bone formation. However, the strain experienced by whole bone under physiological loading conditions is only about 0.2%75, which is insufficient to elicit a response from mechanosensory osteocytes. In such cases, fluid flow within the porosity of the LCN has been explored as a mechanism to amplify the mechanical strain necessary to initiate the intracellular chemical response76,77.

The developed model for increased mechanical loading suggests that different stimuli drive bone formation at the periosteal and endocortical surfaces. Accordingly, bone disuse should lead to stimulus loss and, consequently, bone loss at both surfaces. However, experimental studies have shown that in the absence of physiological loading, bone loss occurs only at the endocortical surface20,78.

Two distinct cell types—osteoblast and osteoclast—are responsible for bone formation and resorption, respectively. It is possible that, under unloading conditions, these two cell types behave differently at the periosteal and endocortical surfaces. Osteoclasts, the bone-resorbing cells, may be more active at the endocortical surface during disuse due to easier access to the bone marrow.

Cortical bone loss due to aging has also been observed to be confined primarily to the endocortical surface73. Skeletal aging reduces periosteal deposition; however, resorption is not typically observed at the outer surface. Previous studies have shown that aging significantly decreases periosteum thickness79,80. This reduction in periosteum thickness, in turn, lowers pore pressure, thereby decreasing new bone formation at the periosteal surface, as suggested by Singh et al.39.

The fluid flow-based model, being a robust framework for bone adaptation, inherently captures the effects of strain rate and loading frequency. These characteristics may be key to explaining skeletal aging-related bone loss, as bone responds more strongly to dynamic, high-strain loading than to static or low-frequency loads.

A study by Gross and Rubin on the turkey’s avian wing bone (viz, radius bone)78 reported that bone loss was uniform along its length and endocortical perimeter. In contrast, Ausk et al.20 found that bone loss varied along the length and endocortical perimeter of the mouse tibia. This discrepancy may be attributed to structural differences (such as curvature) and variations in loading conditions between the two bones—particularly torsional loading, which may be more prominent in avian bones, as discussed by Novitskaya et al.81. Further research is required to model the turkey’s experimental results, and this has also been identified as a future work.

The developed model predicted bone loss distribution due to Botox-induced muscle paralysis only. Incorporating other disuse conditions—such as the tail-suspension model, prolonged bed rest, long-duration space travel, and spinal cord injury—would enhance the robustness of the model. However, essential experimental data, such as cortical bone loss sites and BRR/BS value under these conditions, are currently unavailable in the literature.

Beyond this, the present model has a few limitations, as outlined below:

  1. 1.

    The current mathematical model was unable to accurately predict cortical bone loss in the proximal section. The underlying reasons and detailed explanations are provided in Section "Proximal section incorporating muscle attachments", “Proximal Section Incorporating Muscle Attachments.” However, a more comprehensive gait analysis of the mouse is needed to accurately compute muscle forces during various cage activities.

  2. 2.

    The present model has been developed for mouse tibia. Extending it to the human tibia presents challenges due to its more complex internal architecture. In particular, incorporating the Haversian system into the model would require significant additional development.

Apart from the above, the present model has been developed with several simplifications, as discussed below.

Material properties: The bone has been considered as a linear isotropic poroelastic material. However, in reality, the bone is highly anisotropic. Cortical bone also has different porosities at different length scales. In the present study, only LCN porosities have been considered. Incorporating all relevant porosities may enhance the accuracy of the results.

Biological cues: The present model does not account for bone loss due to natural aging or the changes in the bone’s internal architecture associated with aging82,83. Incorporating the effects of aging has also been identified as future work. Additionally, the current model does not consider the role of different mechanosensors21 such as integrins, tethering fibers, and the pericellular matrix.

Conclusions

This work presents a mathematical model for cortical bone loss under disuse conditions. To the authors’ knowledge, no such model exists in the literature, making this the first of its kind(although models for increased mechanical loading are available). The model takes the loss of mechanical stimulus due to bone disuse as input and predicts the cortical bone loss at the site of interest. The theory of poroelasticity was used to compute the fluid flow pattern in the LCN, and the corresponding finite element problem was solved using ABAQUS software. The developed mathematical model successfully predicted in vivo bone loss at two murine tibial sites (distal and mid-sections), for which data are available in the literature. However, it could not accurately predict the bone loss distribution at the proximal section, possibly due to the influence of muscle attachment near this region. A more detailed analysis incorporating muscle attachments on the tibia may be needed and has been identified as future work.

The current model can also be further improved by incorporating or predicting other disuse conditions, such as extended bed rest, long-duration space travel, and spinal cord injury. However, relevant experimental data for these conditions are currently unavailable in the literature.