1 Introduction

Metabolic bone disorders such as osteoporosis or bone/muscle disuse cause bone loss which ultimately leads to severe fractures in weight-bearing bones (Dirschl et al. 1997; Maimoun et al. 2006; Lau and Guo 2011; Vico et al. 2000). Several in vivo studies have shown that a dynamic loading on bone may effectively inhibit bone resorption as it encourages new bone formation at the sites of peak strain magnitude (Rubin and Lanyon 1985; Hillam and Skerry 1995; Inman et al. 1999; Rubin et al. 2003; Shackelford 2004). Normal strain is believed to act as a principal stimulus of osteogenesis (Turner et al. 1994; Kotha et al. 2004; De Souza et al. 2005; Sample et al. 2010; Moustafa et al. 2012). However, based on a site-specific comparative analysis between the new bone distribution reported in several in vivo studies (such as Gross et al. 1997; Judex et al. 1997; Mosley et al. 1997; Srinivasan et al. 2002; LaMothe 2004; Zhang 2006; Kuruvilla et al. 2008; Matsumoto et al. 2008; Roberts et al. 2009; Lynch et al. 2011; Sakai et al. 2011; Weatherholt et al. 2013; Wallace et al. 2014; Birkhold et al. 2014) and the corresponding strain distribution, especially the position of neutral axis, reported in the same (Judex et al. 1997; Mosley et al. 1997; Zhang 2006; Matsumoto et al. 2008; Wallace et al. 2014) or other studies (Akhter et al. 1992; Srinivasan and Gross 1999; Srinivasan et al. 2013; Willie et al. 2013; Wagner et al. 2013; Patel et al. 2014; Yang et al. 2014; Razi et al. 2015), we find several instances where new bone formation occurs even in the absence of normal strain, specifically near neutral axis of bending in long bones. For example, an in vivo study on murine tibiae of female C57Bl/6J mice (10 weeks old) subjected to cantilever bending of 0.5 N for 500 loading cycles at 1 Hz led to osteogenesis away from as well as near the neutral axis; thus, the normal strain alone was unable to explain the site-specificity of new bone formation (Srinivasan et al. 2002). In that study, it can also be seen that new bone formation in the case of 0.5 N loading is more than that in 0.25 N loading group, site-specifically near the neutral axis at the endocortical surface, and it was also reported that no significant difference in new bone distribution was noticed between the control and 0.25 N loading groups. Hence, osteogenesis is mostly due to mechanical loading although natural growth is slightly present at endocortical surface (compare Fig. 1c with Fig. 3a). As such, the mid-diaphyseal histological section reported in Srinivasan et al. (2002) represents a typical new bone distribution for cantilever loading case which can be observed in several similar in vivo studies (Srinivasan 2003; LaMothe 2004; Srinivasan et al. 2010). Osteogenesis was similarly enhanced significantly in loaded limb as compared to natural bone growth in intact limb, in most of the above-cited studies. These studies were done on 10- to 16-week-old mice, which are equivalent to 18.4- to 24-year-old humans. This age group has been widely considered suitable in the literature to study bone’s adaptation as the active bone growth decreases by the age of 16 weeks (Beamer et al. 1996; Sheng et al. 1999; Lee et al. 2002; Ferguson et al. 2003; Somerville et al. 2004; Wergedal et al. 2005; Flurkey et al. 2007; Main et al. 2010; Sakai et al. 2011). New bone formation near minimal strain sites has also been observed in fully adult mice (Srinivasan et al. 2002, 2010), growing rats (Mosley et al. 1997), juvenile sheep (Wallace et al. 2014) and adult turkey (Judex et al. 1997).

Fig. 1
figure 1

a Loading configuration of murine tibia and b the idealized geometry of tibial cross section at section a–a\(^\prime \). The neutral axis (N.A.) is shown as a solid straight line. c Natural bone growth on control section (adapted from Fig. 3a of Srinivasan et al. 2002)

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Table 1 Values of the parameters used in the computer simulation
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This work presents a computational study that attempts to investigate and explain new bone formation at minimal strain sites. A number of computational models are already available in the literature which can predict and/or simulate site-specific new bone formation (Huiskes et al. 1987; Brown et al. 1990; Fridez et al. 1998; Levenston et al. 1998; Taylor et al. 2003; Ausk et al. 2006; Martínez and Cerrolaza 2006; Chennimalai Kumar et al. 2009; Florio and Narh 2012). These models assume normal strain as a sole primary stimulus for osteogenesis and hence fall short to simulate those experiments where osteogenic activities are observed around minimal strain magnitude. Lack of spatial correlation between locations of new bone formation and that of elevated normal strain proposes that osteogenesis may be driven by secondary components of induced mechanical environment such as bending-induced shear stresses and canalicular fluid flow, which may be potent stimuli of osteogenesis near minimal strain sites (Thompson 1942; Srinivasan and Gross 2000; Burr et al. 2002). In fact, several in vivo and in vitro studies have explained the role of shear strain (De Margerie et al. 2005; Skedros et al. 2003; Taylor et al. 2009) and fluid shear (Lu et al. 2012; Turner and Pavalko 1998) in new bone formation. Skedros et al. (2003) have shown that sites experiencing shear strain had increased cortical thickness and lacunae density in turkey ulna mid-shaft, namely near the neutral axis. Srinivasan and Gross (2000) have also reported maximal fluid flow at these sites. Thus, shear strain may indirectly influence the fluid flow and thus the new bone formation. Carter et al. (1998) have also suggested that tissue level shear strain may induce distortional strain at cellular level and change cell shape. It disrupts the cell–extracellular matrix interaction and may initiate integrin-mediated bone remodelling activities (Marie 2013). Accordingly, osteogenesis as a function of shear strain versus as a function of fluid shear has been studied here, using a computational model based on algorithms proposed in previous studies (Cowin et al. 1985; Huiskes et al. 1987; Chennimalai Kumar et al. 2009). The model simulates loading-induced osteogenesis noticed in the in vivo study of Srinivasan et al. (2002). In the current approach, bone formation rate is assumed roughly proportional to strain energy densities (SED) due to primary as well as secondary stimuli in excess of their osteogenic thresholds. The osteogenic potentials of these stimuli are individually investigated. Subsequently, the effect of bending shear strain or fluid shear stress is combined with normal strain to predict osteogenesis. The combinations, viz. normal strain with shear strain and normal strain with fluid shear, predict results close to experimental values. This computational study may help to identify a simple and computationally easy relationship between site-specific osteogenesis and components of loading-induced mechanical environment, which may, in turn, prove useful in the development of therapeutic biomechanical interventions (such as physical exercises) for the prevention and treatment of bone loss.

2 Methods

2.1 Estimation of normal and shear strains

In the present study, we model osteogenesis for a case where tibia of a 10-week-old female C57Bl/6J mouse was subjected to a continuous cyclic cantilever loading of 0.5 N for 500 cycles at 1 Hz. This loading has been referred to as ‘high-magnitude regimen’ in Srinivasan et al. (2002). A mid-diaphyseal cross section located at 1.8 mm proximal to the tibia–fibula junction of the murine tibia has been idealized using 120 surface coordinates as shown in Fig. 1a, b. Geometric properties such as centroid, cortical area and area moment of inertia are estimated assuming the cross section is enclosed by linear piecewise curves. The neutral axis is assumed to be passing through the centroid. Its orientation has been adapted from the literature (Prasad et al. 2010; Srinivasan et al. 2013), where it was computed using finite element method (Fig. 1). Tibia is idealized as a linearly elastic, homogeneous, isotropic cantilever beam with Young’s and shear moduli of 20 and 7.69 GPa, respectively, and is subjected to a load of 0.5 N at the free distal end (Fig. 1a). The loading waveform is taken similar to that given in Srinivasan et al. (2002). The beam theory was used to estimate normal and shear strains in the cortex, especially at periosteal and endocortical surfaces (Popov and Balan 1998):

$$\begin{aligned} \varepsilon _i= & {} \frac{M \cdot y_i }{E \cdot I} \end{aligned}$$
(1)
$$\begin{aligned} \gamma _i= & {} \frac{V \cdot Q_i }{G \cdot I \cdot t_i } \end{aligned}$$
(2)

where \(\varepsilon _{i}\) and \(\gamma _{i}\) are normal and shear strains, respectively, at any surface coordinate i (i = 1–120); M is the bending moment at the cross section; \(y_{i}\) is perpendicular distance between the surface coordinate and neutral axis; I is the second moment of area; V is the vertical shear force (0.5 N); \(Q_{i}\) is the first moment of the area lying above an axis passing through the point i and parallel to the neutral axis; \(t_{i}\) is the width of the section along an axis passing through point i and parallel to the neutral axis; E and G are Young’s modulus and shear modulus, respectively (Table 1). The initial cortical area and area moment of inertia were \(0.5042\,\hbox {mm}^{2}\) and \(0.06\,\hbox {mm}^{4}\), respectively. Maximum normal strain calculated in the model was approximately \(1311 \, \upmu \upvarepsilon \) which is close to the in vivo strain values, i.e. \(1330 \pm 50\,\upmu \upvarepsilon \).

2.2 Estimation of fluid shear

Osteocyte-level canalicular fluid flow parameters such as pore pressure and fluid shear stress induced due to bending has been estimated based on Weinbaum et al. (1994); Kameo et al. (2009), with an assumption that osteocytes are located close to surface coordinates and sense fluid shear to initiate osteoblastic activity on surfaces. Only steady-state response of fluid flow has been considered as the transient response attains steady state within a few seconds of loading (Srinivasan and Gross 2000).

The in vivo study considered here was subjected to a bending of trapezoidal waveform, which can be closely approximated as a Fourier series of sinusoidal waves (Srinivasan and Gross 2000). For numerical simplicity, a sinusoidal waveform is assumed. The periosteal and endocortical surfaces are assumed to be permeable to the fluid flow. The pore pressure due to a sinusoidal bending moment of frequency \(\omega \) and amplitude M has been calculated as follows (Kameo et al. 2009):

$$\begin{aligned} P_{steady} (y^*,t) = \frac{BM (1+v_u )}{2a^{2}} \times \left\{ y^*\sin \Omega t^*-\hbox {Im}\left[ {\left\{ {\frac{\hbox {i}\Omega H\sinh \sqrt{\hbox {i}\Omega }y^*+3y^*(\sinh \sqrt{\hbox {i}\Omega } -\sqrt{\hbox {i}\Omega }\cosh \sqrt{\hbox {i}\Omega })}{\hbox {i}\Omega H\sinh \sqrt{\hbox {i}\Omega }+3(\sinh \sqrt{\hbox {i}\Omega }-\sqrt{\hbox {i}\Omega }\cosh \sqrt{\hbox {i}\Omega })}} \right\} e^{\mathrm{i}\Omega t}} \right] \right\} \end{aligned}$$
(3)

where \(P_{steady}\) is the steady-state component of pore pressure induced due to bending; B is relative compressibility constant or Skempton coefficient; c is diffusion coefficient; M is the amplitude of bending moment; \(\upsilon _{u}\) is Poisson’s ratio of solid bone matrix under undrained conditions; i is the imaginary unit; a is the thickness of the cortex; \(y^{*}, \varOmega , t^{*}\) and H are non-dimensional parameters and are defined as:

$$\begin{aligned} y^*=\frac{y_i }{a}, \quad t^*=\frac{ct}{a^{2}}, \quad \Omega =\frac{a^{2}\omega }{c}, \quad H=\frac{1-\nu }{\nu _u -\nu } \end{aligned}$$

where t is the time and \(\omega \) is angular frequency of the applied load.

Fluid shear stress developed in canaliculi has been estimated based on the literature (Zeng et al. 1994; Weinbaum et al. 1994) as follows:

$$\begin{aligned} S_i =\frac{b_o }{\lambda }\frac{\partial P_{steady} }{\partial y^{{*}}}\left[ {A_1 I_1 (\lambda /q)-B_1 K_1 (\lambda /q)} \right] \end{aligned}$$
(4)

which implies

$$\begin{aligned} S_i= & {} \frac{b_o }{\lambda }\frac{BM(1+v_u )}{2a^{2}} \times \left\{ \sin \Omega t^*-\hbox {Im}\left[ {\left\{ {\frac{\root 3 \of {\hbox {i}\Omega }H\cos \sqrt{\hbox {i}\Omega }y^{*}+3(\sinh \sqrt{\hbox {i}\Omega }-\sqrt{\hbox {i}\Omega }\cosh \sqrt{\hbox {i}\Omega })}{\hbox {i}\Omega H\sinh \sqrt{\hbox {i}\Omega }+3(\sinh \sqrt{\hbox {i}\Omega }-\sqrt{\hbox {i}\Omega }\cosh \sqrt{\hbox {i}\Omega })}} \right\} e^{i\Omega t}} \right] \right\} \nonumber \\&\times \left[ {A_1 I_1 (\lambda /q)-B_1 K_1 (\lambda /q)} \right] \end{aligned}$$
(5)

where \(S_{i}\) is fluid shear stress at surface coordinate i; \(b_{o}\) is the radius of canaliculus; the resultant fluid flow or shear is oriented perpendicular to the neutral axis of bending towards direction ‘y’, i.e. directed from compression side to tensile side of the section; and

$$\begin{aligned} A_1= & {} \frac{K_O (\lambda )-K_O (\lambda /q)}{I_O (\lambda /q)K_O (\lambda )-I_O (\gamma )K_O (\lambda /q)} \end{aligned}$$
(6)
$$\begin{aligned} B_1= & {} \frac{I_O (\lambda /q)-I_O (\lambda )}{I_O (\lambda /q)K_O (\lambda )-I_O (\lambda )K_O (\lambda /q)} \end{aligned}$$
(7)

where \(K_{o}\) and \(I_{o}\) are the modified Bessel functions of the third and the first kind, respectively; q is a ratio of the radius of canaliculus \((b_{o})\) to the radius of cell process \((a_{o})\); \(\lambda =b_o /\sqrt{k_p}\) is a dimensionless length ratio, where \(k_{p}\) is Darcy law permeability constant.

2.3 Computation of strain energy density (SED) and evolution of surface

The model computes normal and shear SED (\({U}_{\mathrm{i}}^\varepsilon \) and \({U}_{\mathrm{i}}^\gamma \)) along with the fluid shear SED \(({U}_{\mathrm{i}}^s)\) at periosteal and endocortical surfaces as follows:

$$\begin{aligned} U_i^\varepsilon= & {} \frac{1}{2}E.\varepsilon _i^2 \end{aligned}$$
(8)
$$\begin{aligned} U_i^\gamma= & {} \frac{1}{2}G.\gamma _i^2 \end{aligned}$$
(9)
$$\begin{aligned} U_i^s= & {} \frac{S_i^2 }{2G_{osc} } \end{aligned}$$
(10)

where \(G_{osc}\) is the shear modulus of osteocyte. The model uses SED as a stimulus for new bone formation. In the literature (Fritton et al. 2005; Srinivasan et al. 2007; Weinbaum et al. 1994), normal strain of magnitude \(800 \, \upmu \upvarepsilon \) and fluid shear of \(6 \, \hbox {dyne}/\hbox {cm}^{2} (0.6 \hbox { Pa})\) are considered sufficient to initiate bone remodelling activities at cellular level. Accordingly, reference SEDs \((U_{ref})\) have been computed corresponding to these values of osteogenic thresholds of each stimulus and are listed in Table 1. The difference between estimated and reference SED is assumed to govern osteogenesis (Cowin 1993). The growth law is applied to simulate new bone formation and is similar to that proposed by Chennimalai Kumar et al. (2009):

$$\begin{aligned} \frac{\partial c_i}{\partial t}=\left\{ {\begin{array}{ll} C(U_i -U_{ref}), &{} if \, U_i >U_{ref}\\ 0, &{} \text {otherwise} \\ \end{array}} \right. \end{aligned}$$
(11)

where \(\frac{\partial c_i }{\partial t}\) represents the bone formation rate and C is the bone’s surface remodelling rate coefficient.

The following assumptions have been made based on Carter et al. (1998); Burr et al. (2002); Carpenter (2006); Carpenter and Carter (2008); Feldman et al. (2007):

  • The osteogenic response may depend on the nature of normal strain (viz. compressive vs. tensile).

  • The osteogenic response may be different for different components of mechanical milieu (viz. normal strain vs. shear strain vs. fluid flow-induced shear stress).

  • The osteogenic response of periosteal and endocortical surfaces may differ from each other according to their mechano-biological environment as endocortical surface has been reported to be more mechano-responsive than periosteal surface due to additional recruitment of osteoblasts from bone marrow osteoprogenitors (Birkhold et al. 2016; Turner et al. 1998).

Based on the above assumptions, different thresholds were considered for different surfaces (periosteal vs. endocortical) and different components of mechanical environment (normal strain vs. shear strain vs. fluid shear stress). We have incorporated the natural growth in the model based on new bone distribution noticed in 0.25 N loading group (Fig. 1c) which has been treated as control due to unavailability of original control’s site-specific data and similarity in bone formation rate to control group. The growth law is modified accordingly as follows:

Normal strain as the stimulus for periosteal and endocortical surfaces, respectively:

$$\begin{aligned} \frac{\partial c_i^{\varepsilon _p } }{\partial t}=\left\{ {\begin{array}{ll} C_i^{\varepsilon _p} (U_i^{\varepsilon _p} -U_{ref}^{\varepsilon _p} ) +\Delta _i, &{}\quad if \, U_i^{\varepsilon _p } >U_{ref}^{\varepsilon _p } \\ \Delta _i, &{} \quad \text {otherwise} \\ \end{array}} \right. \end{aligned}$$
(12)

and

$$\begin{aligned} \frac{\partial c_i^{\varepsilon _c } }{\partial t}=\left\{ {\begin{array}{ll} C_i^{\varepsilon _c } (U_i^{\varepsilon _c } -U_{ref}^{\varepsilon _c }) +\Delta _i, &{}\quad if \, U_i^{\varepsilon _c } >U_{ref}^{\varepsilon _c} \\ \Delta _i, &{}\quad \text {otherwise} \\ \end{array}} \right. \end{aligned}$$
(13)

Shear strain as the stimulus for periosteal and endocortical surfaces, respectively:

$$\begin{aligned} \frac{\partial c_i^{\gamma _p } }{\partial t}=\left\{ {\begin{array}{ll} C_i^{\gamma _p } (U_i^{\gamma _p } -U_{ref}^{\gamma _p } ) + \Delta _i, &{}\quad if \, U_i^{\gamma _p } >U_{ref}^{\gamma _p } \\ \Delta _i, &{}\quad \text {otherwise} \\ \end{array}} \right. \end{aligned}$$
(14)

and

$$\begin{aligned} \frac{\partial c_i^{\gamma _c } }{\partial t}=\left\{ {\begin{array}{ll} C_i^{\gamma _c } (U_i^{\gamma _c } -U_{ref}^{\gamma _c } ) +\Delta _i, &{}\quad if \, U_i^{\gamma _c } >U_{ref}^{\gamma _c } \\ \Delta _i, &{}\quad \text {otherwise} \\ \end{array}} \right. \end{aligned}$$
(15)

Fluid shear stress as the stimulus for periosteal and endocortical surfaces, respectively:

$$\begin{aligned} \frac{\partial c_i^{s_p } }{\partial t}=\left\{ {\begin{array}{ll} C_i^{s_p } (U_i^{s_p } -U_{ref}^{s_p } ) +\Delta _i, &{}\quad if \, U_i^{s_p } >U_{ref}^{s_p } \\ \Delta _i, &{}\quad \text {otherwise} \\ \end{array}} \right. \end{aligned}$$
(16)

and

$$\begin{aligned} \frac{\partial c_i^{s_c } }{\partial t}=\left\{ {\begin{array}{ll} C_i^{s_c } (U_i^{s_c } -U_{ref}^{s_c } ) +\Delta _i , &{}\quad if \, U_i^{s_c } >U_{ref}^{s_c } \\ \Delta _i, &{}\quad \text {otherwise} \\ \end{array}} \right. \end{aligned}$$
(17)

where parameters with superscripts \(\varepsilon _{p}\) and \(\varepsilon _{c}\) correspond to normal strain at periosteal and endocortical surfaces, respectively; similarly, \(\gamma _{\mathrm{p}}\) and \(\gamma _{\mathrm{c}}\) represent shear strains, whereas \(s_{p}\) and \(s_{c}\) correspond to the fluid shear stress at the respective surfaces. \(\Delta _i\) is the parameter representing natural growth rate at surface coordinate ‘i’, which is zero for the periosteal surface since growth is nearly absent in the referred in vivo study (Srinivasan et al. 2002), whereas for the endocortical surface, it has been estimated at each surface coordinate ‘i’ by dividing new bone thickness on control section (Fig. 1c) by total loading time period. A biologically consistent value of remodelling rate coefficient, C, has been calculated based on the mineral apposition rate (MAR) found from earlier in vivo studies by curve-fitting the corresponding bone growth law to the experimental results (Cowin et al. 1985; Srinivasan et al. 2002; Chennimalai Kumar et al. 2009). The baseline values of reference SED \((U_{ref})\) for different components and surfaces are decided based on osteogenic fluid shear and physiological strains (Weinbaum et al. 1994; Srinivasan et al. 2007; Prasad et al. 2010; Verbruggen et al. 2012), which are given in Table 1.

If at any instance, the SED \((U_{i})\) at surface coordinates ‘i’ reaches value above threshold, the coordinate shifts in the normal direction to the bone surface to incorporate osteogenesis. The model updates the geometry of the cross section according to the following time stepping scheme:

$$\begin{aligned} X_i \left( {t+dt} \right)= & {} X_i \left( t \right) +\frac{\partial c_i }{\partial t}dt \cos \theta \end{aligned}$$
(18)
$$\begin{aligned} Y_i \left( {t+dt} \right)= & {} Y_i \left( t \right) +\frac{\partial c_i }{\partial t}dt \sin \theta \end{aligned}$$
(19)

where \(X_{i} (t+dt)\) and \(Y_{i} ({t+dt})\) are the new coordinates at time \(t+{dt}\), while \(X_{i}(t)\) and \(Y_{i}(t)\) are the old coordinates at time t and dt is the time step size (increment). \(\theta \) is the angle that the unit vector perpendicular to the surface at the point in consideration makes with X-direction. The algorithm used for the present study is shown in Fig. 2. The model uses a time step of 1 second and iterates for a total time period of 500 s, which corresponds to number of loading cycles.

Fig. 2
figure 2

Fundamental algorithm used for the simulation of osteogenesis

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Table 2 Combinations of mechanical stimuli used to test the model
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3 Results

The model simulates osteogenesis considering three different components of mechanical environment: normal strain, shear strain and fluid flow-induced shear stress, individually as well as in their combinations, viz. normal strain with shear strain or with fluid shear (Table 2). To measure site-specific osteogenesis, the circumference of the cross section is divided with 36 radial lines, each \(10^{\mathrm{o}}\) apart from the neighbouring lines and passing through the centroid of the cross section. These lines intersect periosteal and endocortical circumferences at the points numbered in counterclockwise direction as 1–36 at periosteal surface and as \(1^{\prime }\) to \(36^{\prime }\) at endocortical surface (see Fig. 3a). The new bone thickness thus measured has been compared with experimental new bone thickness obtained from histological cross section (Srinivasan et al. 2002, 0.5 N loading case), along these radial lines. Simulation results are described in the following subsections.

Fig. 3
figure 3

Computational versus experimental results: a in vivo new bone formation (green) in murine tibia (adapted from Fig. 3b of Srinivasan et al. 2002); and computational prediction of osteogenesis (red) due to: b normal strain only, c tensile normal strain only and d shear strain only. The cross section has been divided into 36 sectors in order to compare site-specific osteogenesis. The overlap of experimental and computational osteogenesis is shown in red-green mixture

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Fig. 4
figure 4

Computational prediction of osteogenesis, when: a normal and shear strains are combined, b tensile normal strain and shear strain are combined, c fluid shear is considered alone, d fluid shear is considered along with tensile normal strain

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3.1 Normal strain

Osteogenic potential of normal strain has been investigated by considering (i) normal strains (i.e., compressive and tensile strain) alone to be osteogenic and (ii) only tensile normal strain to be osteogenic. When normal strain is considered, new bone is formed away from neutral axis, namely at periosteal points 1–10 and 17–27, out of which points 1–10 only have osteogenesis in accordance with the in vivo study (Fig. 3b). When only tensile normal strain is considered, the model predicts new bone formation away from the neutral axis along periosteal points 1–10 and endocortical points \(3^{\prime }\hbox {--}9^{\prime }\) and \(16^{\prime } \hbox {--} 23^{\prime }\) similar to experimental observation (Fig. 3c), whereas it fails around the neutral axis along endocortical points \(26^{\prime } \hbox {--} 35^{\prime }\) and \(10^{\prime } \hbox {--} 14^{\prime }\) (Fig. 3c). If compressive normal strain were considered alone, new bone would be formed along posterior-medial sites in contrast with the experimental observation. The above findings confirm that normal strain may not be the sole stimulus for osteogenesis.

3.2 Shear strain

Shear strain as a sole stimulus in the model predicts new bone formation near the neutral axis along periosteal points 11–18 and 30–35, and endocortical points \(10^{\prime } \hbox {--} 23^{\prime }\) and \(27^{\prime } \hbox {--} 36^{\prime }\), among which points \(27^{\prime } \hbox {--}36^{\prime }\) and \(19^{\prime } \hbox {--} 23^{\prime }\) only have new bone formed in the experimental study (Fig. 3d). Thus, shear strain as a stimulus fails to completely model osteogenesis.

3.3 Normal plus shear strain

When the effects of normal strains (both tensile and compressive) and solid shear strains are added, the site-specific new bone formation at endocortical surface was found close to experimental observation (Fig. 4a); however, the model falsely predicts osteogenesis at periosteal points 12–17, 19–26 and 29–35 (Fig. 4a). Thus, this combination also fails to accurately model osteogenesis.

3.4 Tensile normal strain plus shear strain

The results improve when tensile strain is considered in combination with shear strain with an assumption that endocortical surface is more mechano-responsive as compared to periosteal surface (Carter et al. 1998; Carpenter 2006; Feldman et al. 2007; Birkhold et al. 2016) (Fig. 4b). The sites of new bone formation predicted by the model are close to that noticed during the in vivo study (Fig. 4b) except for only 5 endocortical sites \(13^{\prime } \hbox {--} 17^{\prime }\) (out of total 36 points compared) where the model overestimates new bone formation.

3.5 Interstitial fluid shear stress

Canalicular fluid flows from compression to tension side when bones are subjected to mechanical loading (Turner and Pavalko 1998; Tate et al. 2000). In vitro studies on the bone cells (such as osteoblasts and osteocytes) have found that the fluid flow is a stimulus to osteogenic activities (Huo et al. 2008; Lu et al. 2012). Accordingly, when the interstitial fluid flow is considered osteogenic, the model predicts new bone formation near neutral axis along periosteal points 12-17 and 29-35 and endocortical points \(10^{\prime } \hbox {-} 17^{\prime }\), \(20^{\prime } \hbox {-} 23^{\prime }\) and \(27^{\prime } \hbox {-} 35^{\prime }\), which is similar to that obtained due to solid shear strain (Fig. 4c). The model, however, fails to fit the experimental new bone distribution along the periosteal points 1–10 (Fig. 4c).

The model predicts the peak fluid velocity and shear stress as \(18 \, \upmu \hbox {m}/\hbox {s}\) and \(10 \hbox { dyne}/\hbox {cm}^{2}\) respectively, which are close to \(20.37 \, \upmu \hbox {m}/\hbox {s}\) and \(18 \hbox { dyne}/\hbox {cm}^{2}\), respectively, calculated from empirical relations reported in several in vivo studies (Price et al. 2011; Wang et al. 2013; Jing et al. 2014). The difference in fluid shear magnitude may be attributed to the difference in loading condition; nevertheless, it is in the range of shear stress required to induce calcium signaling and new bone formation (Weinbaum et al. 1994). Moreover, the model’s prediction of maximal flow near minimal strain sites aligns with the findings of several in silico studies (Steck et al. 2000; Tate et al. 2000; Hamilton et al. 2004).

3.6 Normal strain plus interstitial fluid shear stress

Fluid flow-induced shear stress combined with tensile normal strain predicts site-specific new bone formation similar to the in vivo experiment (Fig. 4d). It may support the idea that (i) fluid shear acts as a secondary stimulus of osteogenic activity along with normal strain and (ii) different components of loading-induced mechanical environment may act collectively to adapt bone.

The experimental new bone thickness has been compared site-specifically with model’s predictions for (i) tensile strain plus solid shear strain and (ii) tensile strain plus fluid shear stress in Fig. 5. The three cases have approximately similar new bone distribution, except for a few endocortical points. The computational model predicts an overall increase in cortical area approximately by 6.7 % (from 0.5042 to \(0.538 \hbox { mm}^{2}\)) for the above two cases, which is close to that of 4.75 % found during the in vivo study. The error between computational and experimental new bone thickness shows that the model only underestimates or overestimates the new bone formation at periosteal points 10–11 and endocortical points \(12^{\prime } \hbox {-} 16^{\prime }\), respectively (Fig. 5c, d). The osteogenic potentials of shear strain and fluid flow-induced shear strain are also site-specifically compared at periosteal and endocortical surfaces (Fig. 6a, b). Both showed similar site-specific osteogenic response in terms of new bone thickness.

Fig. 5
figure 5

Experimental versus computational new bone thickness when the tensile normal strain is considered along with the shear strain or the fluid shear at: a periosteal and b endocortical surfaces. The error (experimental–computational) in new bone thickness is shown at: c periosteal and d endocortical surfaces

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Fig. 6
figure 6

Computationally obtained new bone thickness due to the shear strain as compared to that due to the fluid shear at: a periosteal and b endocortical surfaces

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Fig. 7
figure 7

Schematic representation of the new bone formation noticed during in vivo studies in response to different loading conditions such as a four-point bending (adapted from Roberts et al. 2009); b three-point bending (adapted from Sakai et al. 2011); c three-point bending (adapted from Matsumoto et al. 2008); d cantilever bending (adapted from Srinivasan et al. 2002); e cantilever bending (adapted from LaMothe 2004); f axial compression (adapted from Sugiyama et al. 2010)

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4 Discussion and conclusions

This study suggests that shear strain or fluid shear stress may be responsible for osteogenesis near the sites of minimal normal strain; however, two or more stimuli (e.g., normal strain with shear strain or fluid shear) may be required to accurately model osteogenesis. This is in contrast to most of the previous in silico studies (Martínez and Cerrolaza 2006; Chennimalai Kumar et al. 2010), where a single stimulus such as normal strain energy density was typically used. These models had limitations to fit those experiments where new bone formation was present near minimal strain sites. The current study fills this gap. The comparison of shear strain and fluid shear stress in terms of new bone thickness suggests that they are equivalent in terms of their osteogenic potentials. Studies done by Skedros et al. (2003); Srinivasan and Gross (2000) also confirm the same. It provides a new perspective that macro-level shear strain can be used to estimate osteogenic potential of micro-level fluid shear, thus reducing computational and analytical complexities of the bone remodelling models. For example, Fig. 7 shows new bone distribution on tibial cross section (adapted from various in vivo studies) corresponding to different loading conditions. In all cases, osteogenesis can be seen away from as well as near the neutral axis, which may be modelled incorporating normal as well as shear components of mechanical environment. The present model would be helpful to model such experiments; however, it may fail in case of four-point bending, as there is no solid shear strain and new bone formation is still present around the neutral axis at periosteal surface. Incorporation of fluid flow may be required in that case to closely model osteogenesis. Moreover, as the loading frequency, number of cycles, time period and rest-inserted loading also affect the fluid flow and hence osteogenesis (Pereira et al. 2015; Malachanne et al. 2011), consideration of fluid flow in addition to mechanical components may improve the computational model (Pereira and Shefelbine 2014). Past research efforts have shown that different mechanical stimuli may opt different mechanisms of mechano-transduction to initiate remodelling activity at cellular level. For example, normal strain can be directly sensed by mechano-receptors located at osteoblast cell membrane which may increase the calcium signaling and in turn the new bone formation. Strain also restructures the cytoskeleton of bone cell to regulate bone remodelling activities. In addition, tissue deformation/strain allows the interstitial fluid to flow through the osteocyte network. This fluid flow imparts shear stress on cell membrane of osteocytes which releases different signaling molecules such as prostaglandin (\(\hbox {PGE}_{2}\)) and NO in order to communicate with osteoblasts to initiate new bone formation (Turner and Pavalko 1998). Thus, incorporation of normal strain and fluid flow in the model may be useful in capturing the characteristics of cellular modelling activities.

The model predicts osteogenesis also at those sites where mechanical integrity is least challenged, e.g. around the neutral axis. Previous studies have explained that asymmetric bone morphology allows the bone to be bent in a predictable direction, i.e. about the neutral axis of bending during locomotion, and thus bone sacrifices its strength for predictable strain environment (Bertram and Biewener 1988). It permits the bone to adjust its morphology against any long-lasting stimulus (Main 2007). The new bone addition near neutral axis will not affect overall strain distribution; rather, it will fortify the environment to ensure that bones are loaded in a predictable fashion (Judex et al. 1997). This can be considered as a compromise between maintaining optimum strain level while ensuring that bone upholds a sufficient safety factor to withstand atypical loading (Wallace et al. 2014). Osteogenesis may, thus, have potential to align with the bone ’s neutral axis. This aspect was ignored in previous models.

A few in vivo studies noticed that peak circumferential strain gradient (an indirect measure of maximal fluid flow) closely correlates with new bone formation near the neutral axis (Gross et al. 1997; Judex et al. 1997). In the present study, maximum osteogenic response due to fluid flow has indeed been observed near the neutral axis and thus the model’s findings are in agreement with aforementioned studies. These findings may be useful in designing weight-bearing exercises where induced neutral axis will align with the sites requiring enhanced structural strength.

It has been noticed that fluid shear stress on osteocytic cell processes and fluid drag force on pericellular matrix fibres (i.e. tethering elements between cell transmembrane proteins and proteins in the extracellular matrix which accommodate the space between the cell body and the lacunae wall) both can potentially induce new bone formation (Pereira et al. 2015). Based on some previous work (You et al. 2001), however, canalicular fluid shear stress/force may be assumed to be proportional to pericellular drag force and flow velocity. The canalicular fluid stress has therefore been solely considered in the present model.

One limitation of this study is that only one histological cross section from a single animal has been used as a baseline to be compared with model output. As individual animals may have slightly different bone geometry and osteogenic, biological environment, there may be difference in site-specific new bone formation. The robustness of the model may, therefore, be improved by incorporating data from a group of animals, rather than a single animal currently used. Nevertheless, the baseline histological cross section used for the current model represents a typical new bone distribution near minimal strain sites/neutral axis observed in other in vivo studies where tibia of C57Bl/6J mice of different age groups was given cantilever bending (LaMothe et al. 2004; Srinivasan et al. 2003; Srinivasan et al. 2010; and Srinivasan et al. 2002). In addition, the presence of new bone formation near the minimal strain sites/neutral axis has been observed in several other in vivo studies (Wallace et al. 2014; Judex et al. 1997; Gross et al. 1997; Mosley et al. 1997). The model’s accuracy may also be tested in future to predict osteogenesis at different sites along tibial length as these sites will experience different strain patterns.

The model does not incorporate possible bone resorption at several locations which might have been occurred as a result of, for example, damage imparted by high-magnitude strain. The model assumes that osteocytes positioned in the vicinity of tibial surfaces receive and integrate the signals from the other osteocytes in the network and communicate with lining cells/surface osteoblasts to initiate remodelling activity. Thus, the local effect of the fluid flow has been considered in the model (Pereira et al. 2015) although the flow should ideally be estimated across the osteocyte network. The model overlooks this effect not only to make the simulation computationally easy but also to remain consistent with other stimuli such as normal and shear strains, which have been considered at the endocortical and periosteal surfaces only. Incorporation of network’s effect has been taken as a future work. Moreover, signaling molecules such as prostaglandins (\(\hbox {PGE}_{2}\)), nitric oxide (NO), sclerostin and ATP diffuse through extracellular fluid and communicate with neighbouring osteoblasts/osteocytes and, thus, are considered as potential regulators of new bone formation (Robling et al. 2006; Lu et al. 2012; Klein-Nulend et al. 2012). This paracrine mechanism has not been considered in the current model to reduce complexity and has, however, been taken as a future work.