1 Introduction

Bone is a biomaterial with a complex hierarchical structure characterised by at least three distinct length scales: (i) the cellular scale (10–20 \({\upmu }\)m); (ii) the tissue scale (2–5 mm); and (iii) the whole organ scale (4–45 cm) (Rho et al. 1998; Weiner and Wagner 1998). Several interactions exist between these scales, which affect bone remodelling, bone material properties and bone structural integrity. The activity of bone-resorbing and bone-forming cells during bone remodelling leads to changes in material properties at the tissue scale which subsequently affect the distribution of loads at the structural, whole organ scale (Fig. 1). Besides, changes in bone shape and microarchitecture modify the stress/strain distribution and bone surface availability, which provide mechanical and geometrical feedbacks onto the bone cells and, eventually, affect bone remodelling (Martin 1972; Lanyon et al. 1982; Frost 1987). Due to the complexity of these interactions, the interpretation of experimental data at a single scale is difficult. Predicting the evolution of multifactorial bone disorders, such as osteoporosis, necessitates a comprehensive modelling approach in which these multiscale interactions are consistently integrated.

Fig. 1
figure 1

Multiscale representation of bone. a Scheme of the couplings in the bone remodelling process; b femur bone geometry (organ scale); c midshaft cross section depicting coordinate axes and the sectional forces used for beam theory (tissue to organ scales); d representative volume element (RVE) of cortical bone used to define bone cell densities, bone volume fraction and specific surface (cellular to tissue scales)

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Various mathematical models of bone remodelling have been proposed in the literature. Biomechanics models estimating tissue-scale stress and strain distributions from musculoskeletal models and average material properties, such as bone density, are often used in conjunction with remodelling algorithms based on Wolff’s law. These remodelling algorithms locally increase or decrease bone density depending on the tissue’s mechanical state (Carter and Hayes 1977; Carter and Beaupré 2001; Fyhrie and Carter 1986; Weinans et al. 1992; Van der Meulen et al. 1993; Pettermann et al. 1997). Such models may also include damage accumulation due to fatigue loading and damage repair (Prendergast and Taylor 1994; McNamara and Prendergast 2007; García-Aznar et al. 2005). Other models focus at the microstructural scale (\(\mathrm{{\upmu }}\)m to mm) and describe the evolution of trabecular bone microarchitecture through resorption and formation events at the bone surface induced by the local mechanical state (Huiskes et al. 2000; Ruimerman et al. 2005; Van Oers et al. 2008; Christen et al. 2012, 2013). Most of these mathematical models focus on the biomechanical aspects of bone remodelling and do not consider hormonal regulation or biochemical coupling between bone cells.

Fig. 2
figure 2

Flow chart of bone remodelling simulations taking into account (a) the global mechanical loading, (b) the bone cell population model and (c) the bone material and geometry adaptation

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In this paper, we propose a novel multiscale modelling approach of bone remodelling combining and extending several mathematical models into a consistent framework. This framework enables (i) the consideration of biochemical and cellular interactions in bone remodelling at the cellular scale (Lemaire et al. 2004; Pivonka et al. 2008; Buenzli et al. 2012; Pivonka et al. 2013), (ii) the evolution of material properties at the tissue scale based on bone cell remodelling activities regulated by mechanical feedback (Scheiner et al. 2013) and bone surface availability (Pivonka et al. 2013; Buenzli et al. 2013), and (iii) the determination of the stress/ strain distributions from the tissue scale to the microstructural scale by a combination of generalised beam theory and micromechanical homogenisation (Hellmich et al. 2008; Scheiner et al. 2013; Buenzli et al. 2013).

This modelling approach is applied to simulate the temporal evolution of a human femoral bone at the midshaft (Fig. 1), subjected to various deregulating circumstances such as osteoporosis and changes in mechanical loading. An initial state of normal bone remodelling is first assumed, in which the tissue across the midshaft cross section remodels at site-specific turnover rates without changing its average material properties. Osteoporosis is then simulated by hormonal changes deregulating the biochemical coupling between osteoclasts and osteoblasts. These hormonal changes are calibrated so as to reproduce realistic rates of osteoporotic bone loss. The strength of the resorptive and formative responses of bone cells to mechanical feedback is calibrated so as to reproduce rates of bone loss and recovery in cosmonauts undertaking long-duration space flight missions. A scan of a femur cross section is used as initial condition for our simulations. This illustrates the potential of our modelling approach to be used as a predictive, patient-specific diagnostic tool for estimating the deterioration of bone tissues. Here, we use the model to investigate the interplay between geometrical and mechanical feedbacks in inducing site-specific bone loss in osteoporosis, which is characterised by endocortical bone loss, cortical wall thinning and the expansion of the marrow cavity (Feik et al. 1997; Bousson et al. 2001; Zebaze et al. 2010).

2 Description of the model

Figures 1 and 2 summarise the general approach of our model. We consider a portion of human femur near the midshaft. This portion of bone is assumed to carry loads corresponding to a total normal force \({\varvec{N}}\) and total bending moment \( {\varvec{M}}\) (Fig. 1c). These loads are distributed unevenly across the midshaft cross section depending on the site-specific bone microstructure, particularly on the cortical porosity (Zebaze et al. 2010; Buenzli et al. 2013). This load distribution determines a site-specific mechanical stimulus which is sensed and transduced by bone cells (Fig. 2a). This mechanical feedback is incorporated in a cell population model as biochemical signals leading to changes in the balance between osteoclasts and osteoblasts (Fig. 2b). In addition, microstructural parameters such as bone volume fraction (\({f_\text {bm}}\)) and bone specific surface influence the propensity of bone cells to differentiate and become active (Martin 1984; Lerebours et al. 2015). This geometrical feedback is included in the cell population model via a dependence of the bone turnover rate on the bone volume fraction. The activities of osteoclasts and osteoblasts modify the tissue microstructural parameters (bone volume fraction, bone specific surface), which in turn induce changes in the load distribution (Fig. 2c). In the following, we introduce in more detail the multiple scales and related variables involved in this model workflow. Table 1, in the section “Model parameter values” of the Appendix, lists all the parameters of the model.

2.1 Load distribution from the organ scale to the cellular scale

Loading is composed of body weight and muscle forces exerted onto bone via tendons and direct action of muscles. These forces can be calculated from bone shape, muscle and tendon attachment, and gait analysis data using musculoskeletal models (Lloyd and Besier 2003; Viceconti et al. 2006; Martelli et al. 2014). Continuum mechanics provides the link between external forces exerted onto a structure, and the strain and stress distribution in the structure (Salencon 2001).

Tissue-scale properties within the framework of continuum mechanics are average mechanical properties over microstructural material phases and pores (presented in detail in the next sections). The corresponding tissue-scale stresses and strains may significantly deviate from the microscopic, cellular scale, stresses and strains acting in the different material phases composing the tissue due to so-called strain and stress concentration effects (Zaoui 2002; Hill 1963). Microscopic stress and strain distributions in the bone matrix are likely to be sensed directly by bone cells, particularly by osteocytes (Scheiner et al. 2013). However, as osteocytes form an extensive interconnected network (Marotti 2000; Buenzli and Sims 2015), they may also sense larger scale stress and strain distributions. We will let either the tissue-scale or the microscopic mechanical state of bone act onto the bone cells to investigate how this influences the site-specific evolution of bone tissue microstructures.

In the following, we present first how stress and strain distributions can be calculated at the tissue scale using beam theory. We then present how these tissue-scale stress and strain distributions are employed as site-specific loading boundary conditions to the continuum micromechanical model of Hellmich et al. (2008) for the calculation of microscopic stress and strain distributions effective at the cellular level.

2.1.1 Determination of tissue-scale stress and strain distributions based on beam theory

The continuum mechanical field equations allow the calculation of tissue-scale strain and stress distributions in bone. Given that the length of the femur L is significantly larger (45–50 cm) than its diameter D (3–5 cm) at the midshaft (Fig. 1b), the continuum mechanical field equations can be approximated using beam theory formulated for small strains and generalised to materials of non-uniform properties, an approach we have used previously in Buenzli et al. (2013).

This approach requires the knowledge of the total external forces, i.e. the normal force \({\varvec{N}}\) and the bending moment \({\varvec{M}}\) carried by the femur cross section. \({\varvec{N}}\) and \({\varvec{M}}\) can be estimated for different physical activities by using musculoskeletal models (Vaughan et al. 1992; Forner-Cordero et al. 2006). In our simulations, we take constant values for \({\varvec{N}}\) and \({\varvec{M}}\) comparable with the maximum ground reaction force and knee and hip moments that occur during a gait analysis, estimated as: \({\varvec{N}} = (N_{x}, 0, 0), ~N_{x} = - 700 \text {N}\), and \({\varvec{M}} = M ~\hat{{\varvec{m}}}\), M =  50 Nm, where \(\hat{{\varvec{m}}}\) is a unit vector along the antero-posterior axis of the cross section determined from the microradiograph (Vaughan et al. 1992; Forner-Cordero et al. 2006; Ruff 2000; Cordey and Gautier 1999) (Fig. 6c). The x-axis is the femur’s longitudinal axis, and (y,z) is the plane transverse to x at the midshaft (Fig. 1b, c).

Tissue-scale mechanical properties correspond to spatial averages over a so-called representative volume element (RVE) of the tissue. In cortical bone, an appropriate tissue RVE is of the order of \(10\,\times \,2\,\times \,2\,\text {mm}^{3}\), a size large enough to contain a large number of pores, but small enough to retain site-specific information and to not be influenced by macroscopic features such as overall bone shape (Hill 1963; Zaoui 2002). We denote by \(\mathbb {C}^\text {tissue}({\varvec{r}}, t)\) the local bone tissue stiffness tensor defined at the RVE scale, where \({\varvec{r}}\) denotes the location in bone of the RVE (Fig. 1c), and the dependence on time t reflects the fact that bone remodelling may modify the local mechanical properties of the tissue. This tissue-scale stiffness tensor is assumed to relate the tissue-scale stress tensor \({\varvec{\sigma }}^\text {tissue}\) and strain tensor \({\varvec{\varepsilon }}^\text {tissue}\) pointwise according to Hooke’s law:

$$\begin{aligned} {\varvec{\sigma }}^\text {tissue}({\varvec{r}}, t) = \mathbb {C}^\text {tissue}({\varvec{r}}, t):{\varvec{\varepsilon }}^\text {tissue}({\varvec{r}}, t). \end{aligned}$$
(1)

Beam theory is based on the so-called Euler–Bernoulli kinematic hypothesis, which asserts that material cross sections initially normal to the beam’s neutral axis remain planar, undeformed in their own plane, and normal to the neutral axis in the beam’s deformed state (Timoshenko and Goodier 1951; Bauchau and Craig 2009; Hjelmstad 2005). These assumptions are expected to be well satisfied near the femoral midshaft under small deformations generated by bending and compression or tension. Furthermore, no shear force, torsional loads or twisting along the beam axis are assumed. These assumptions, Eq. (1), and the fact that bone is an orthotropic material (Hellmich et al. 2004) imply that the only nonzero components of the stress tensor are the normal stresses \(\sigma ^\text {tissue}_{xx}=\mathbb {C}^\text {tissue}_{1111}\varepsilon ^\text {tissue}_{xx}\), \(\sigma ^\text {tissue}_{yy}=\mathbb {C}^\text {tissue}_{1122}\varepsilon ^\text {tissue}_{xx}\) and \(\sigma ^\text {tissue}_{zz}=\mathbb {C}^\text {tissue}_{1133}\varepsilon ^\text {tissue}_{xx}\), where the normal stresses \(\sigma ^\text {tissue}_{yy}\) and \(\sigma ^\text {tissue}_{zz}\) are induced by compression or tension along the beam axis x by the Poisson effectFootnote 1 (see  Buenzli et al. 2013 for more details). The Euler–Bernoulli hypothesis implies that the tissue strain tensor reduces to the single nonzero component \(\varepsilon ^\text {tissue}_{xx}\) and that:

$$\begin{aligned} \varepsilon ^\text {tissue}_{xx}(y,z,t) = \varepsilon _1(t) - \kappa _3(t) y + \kappa _2(t) z, \end{aligned}$$
(2)

where \(\varepsilon _1\) is the sectional axial strain, and \(\kappa _2\) and \(\kappa _3\) are the sectional beam curvatures about the z- and y-axes, respectively (Bauchau and Craig 2009). The three unknowns \(\varepsilon _1\), \(\kappa _2\) and \(\kappa _3\) are determined by the constraints that (i) the integral of \(\sigma ^\text {tissue}_{xx}\) over the midshaft cross section must give the total normal force \(N_{x}\), and (ii) the integral of the stress moment \((0,y,z)\,\times \, \sigma ^\text {tissue}_{xx}\hat{{\varvec{x}}}\) must give the total bending moment \({\varvec{M}} = M ~\hat{{\varvec{m}}}\) [the axes origin in the (y, z) plane is set at the modulus-weighted centroid of the section, also called normal force centre (Bauchau and Craig 2009)]. Explicit formulas for \(\varepsilon _1, \kappa _2\) and \(\kappa _3\) as functions of \(\mathbb {C}^\text {tissue}\), \({\varvec{N}}\) and \({\varvec{M}}\) are presented in the section “Generalised Beam theory for inhomogeneous materials” of the Appendix. We refer the reader to Bauchau and Craig (2009), Sect. 6.3 and Buenzli et al. (2013) for their derivation.

2.1.2 Determination of microscopic stress and strain distributions based on micromechanical homogenisation theory

Bone tissue stiffness \(\mathbb {C}^\text {tissue}\) is strongly influenced by the tissue’s microstructure, in particular its porosity, or equivalently, its bone volume fraction \({f_\text {bm}}\). Bone volume fraction is a microstructural parameter defined at the tissue scale as the volume fraction of bone matrix in the RVE (Fig. 1d): \({f_\text {bm}}= \text {BV}/\text {TV}= 1 - \text {porosity}\), where \(\text {BV}\) is the volume of bone matrix in the RVE and \(\text {TV}\) is the tissue volume, i.e. the total volume of the RVE (Dempster et al. 2013). In Buenzli et al. (2013), we used an explicit power-law relationship \(\mathbb {C}^\text {tissue}_{1111}({f_\text {bm}}) \propto {f_\text {bm}}^3\) based on experimental relationships between bone stiffness and bone mineral content (Carter and Hayes 1977; Hernandez et al. 2001). While regression approaches based on power-law relations are able to account for material properties in one principal direction, they are less accurate in estimating material properties in other principal directions.

Here, we follow a different approach taken by Hellmich and colleagues using the framework of continuum micromechanics (Hill 1963, 1965; Zaoui 1997, 2002; Hellmich et al. 2004, 2008; Fritsch and Hellmich 2007; Fritsch et al. 2009). Mechanically, bone tissue can be considered as a two-phase material: a bone matrix phase (‘\(\text {bm}\)’) consisting of mineralised bone matrix, and a vascular phase (‘\(\text {vas}\)’) consisting of vascular components, cells, extracellular matrix and other soft tissues present in Haversian canals and in the marrow.

Continuum micromechanics provides a framework to estimate the tissue-scale stiffness tensor \(\mathbb {C}^\text {tissue}({f_\text {bm}})\) from the microscopic stiffness properties of bone matrix and vascular pores, and considerations regarding pore microarchitecture and phase interactions (Hellmich et al. 2008). The advantage of this approach is to provide (i) accurate three-dimensional estimates of \(\mathbb {C}^\text {tissue}\) and (ii) estimates of the microscopic stress and strain distributions of the bone matrix without recourse to costly microfinite element analyses of the tissue microstructure (Fritsch et al. 2009). Using the concept of continuum micromechanics is justified in bone due to the separation of length scales between the RVE size and the characteristic sizes of the two-phase microstructures (Hellmich et al. 2008; Scheiner et al. 2013). We summarise below the premises upon which this approach is based.

The tissue-scale stress and strain tensors \({\varvec{\sigma }}^\text {tissue}\) and \({\varvec{\varepsilon }}^\text {tissue}\) correspond to spatial averages over the RVE of the microscopic (cellular-scale) stress and strain tensors. Assuming that each phase within the RVE is homogeneous, these spatial averages can be expressed as sums over the phases:

$$\begin{aligned} {\varvec{\sigma }}^\text {tissue}({\varvec{r}}, t)&\equiv \frac{1}{\text {TV}} \int _{\text {TV}} {\varvec{\sigma }}^\text {micro}\mathrm {d}V =\sum \limits _{k}f_k\, {\varvec{\sigma }}^\text {micro}_k , \end{aligned}$$
(3)
$$\begin{aligned} {\varvec{\varepsilon }}^\text {tissue}({\varvec{r}}, t)&\equiv \frac{1}{\text {TV}} \int _{\text {TV}} {\varvec{\varepsilon }}^\text {micro}\mathrm {d}V =\sum \limits _{k}f_k\, {\varvec{\varepsilon }}^\text {micro}_k, \end{aligned}$$
(4)

where \(f_k({\varvec{r}}, t)\) is the volume fraction of phase k (‘bm’, ‘vas’), \({\varvec{\sigma }}^\text {micro}_k({\varvec{r}}, t)\) and \({\varvec{\varepsilon }}^\text {micro}_k({\varvec{r}}, t)\) are the microscopic stress and strain tensors in phase k. We emphasise that all these quantities still depend on the tissue-scale location \({\varvec{r}}\) of the RVE in bone, while microscopic inhomogeneities are encoded in the phase index k. It can be shown that due to the linearity of the constitutive equations, the phase strain tensor \({\varvec{\varepsilon }}^\text {micro}_k\) is related linearly with the tissue-scale strain tensor:

$$\begin{aligned} {\varvec{\varepsilon }}^\text {micro}_k = \mathbb {A}_k : {\varvec{\varepsilon }}^\text {tissue}, \end{aligned}$$
(5)

where \(\mathbb {A}_k\) is a fourth-order tensor called the strain concentration tensor (Zaoui 2002; Hellmich et al. 2008; Fritsch et al. 2009). Assuming that Hooke’s law also holds for each phase at the microscopic scale, \({\varvec{\sigma }}^\text {micro}_k = \mathbbm {c}^{\text {micro}}_k : {\varvec{\varepsilon }}^\text {micro}_k\) (with \(\mathbbm {c}^{\text {micro}}_k\) the stiffness tensor of phase k), one obtains from Eqs. (3) and (5):

$$\begin{aligned} {\varvec{\sigma }}^\text {tissue}&= \sum _k f_k ~\mathbbm {c}^{\text {micro}}_k : {\varvec{\varepsilon }}^\text {micro}_k \\&= \Big (\sum _k f_k ~\mathbbm {c}^{\text {micro}}_k: \mathbb {A}_k\Big ) : {\varvec{\varepsilon }}^\text {tissue}\equiv \mathbb {C}^\text {tissue}: {\varvec{\varepsilon }}^\text {tissue}, \nonumber \end{aligned}$$
(6)

where

$$\begin{aligned} \mathbb {C}^\text {tissue}= {f_\text {bm}}~\mathbbm {c}^{\text {micro}}_\text {bm}:\mathbb {A}_\text {bm}+ {f_\text {vas}}~\mathbbm {c}^{\text {micro}}_\text {vas}: \mathbb {A}_\text {vas}. \end{aligned}$$
(7)

Equation (7) provides a relationship between the tissue-scale stiffness, \(\mathbb {C}^\text {tissue}\), and the microscopic properties of the phases composing the tissue, \(f_k, \mathbbm {c}^{\text {micro}}_k\) and \(\mathbb {A}_k\). Because mineral content across bone tissues only varies little on average (Scheiner et al. 2013; Fritsch and Hellmich 2007), \(\mathbbm {c}^{\text {micro}}_\text {bm}\) can be assumed constant and homogeneous, i.e. independent of \({\varvec{r}}, t\). The elastic modulus \(\mathbbm {c}^{\text {micro}}_\text {vas}\) is likewise assumed independent of \({\varvec{r}}, t\) and taken as that of water (Scheiner et al. 2013). Both \(\mathbbm {c}^{\text {micro}}_\text {bm}\) and \(\mathbbm {c}^{\text {micro}}_\text {vas}\) have been measured experimentally, and their values are listed in Table 1. The strain concentration tensors \(\mathbb {A}_k\) can be estimated by solving so-called matrix-inclusion problems of elasticity homogenisation theory, which use assumptions on phase shape within the RVE and phase interactions (Eshelby 1957; Laws 1977). For bone, accurate multi-scale homogenisation schemes were developed and validated experimentally (Hellmich et al. 2008; Fritsch et al. 2009; Morin and Hellmich 2014). These schemes provide explicit expressions for \(\mathbb {A}_k\) depending on the phase volume fractions \(f_\text {bm}\) and \(f_\text {vas}\). Because \(f_\text {vas}= 1-{f_\text {bm}}\), this defines both the \({f_\text {bm}}\) dependence of \(\mathbb {C}^\text {tissue}\) via Eq. (7), and a method to estimate the strains and stresses in the bone matrix at the microscopic level from those known at the tissue level:

$$\begin{aligned} {\varvec{\varepsilon }}^\text {micro}_\text {bm}({\varvec{r}}, t)&= \mathbb {A}_\text {bm}({f_\text {bm}}) : {\varvec{\varepsilon }}^\text {tissue}\end{aligned}$$
(8)
$$\begin{aligned} {\varvec{\sigma }}^\text {micro}_\text {bm}({\varvec{r}}, t)&= \mathbbm {c}^{\text {micro}}_\text {bm}: \big (\mathbb {A}_\text {bm}({f_\text {bm}}) : {\varvec{\varepsilon }}^\text {tissue}\big ) \nonumber \\&\equiv \mathbb {B}_\text {bm}({f_\text {bm}}): {\varvec{\sigma }}^\text {tissue}, \end{aligned}$$
(9)

where Hooke’s law, Eq. (1), was used in the last equality in Eq. (9). The stiffness tensor \(\mathbb {C}^\text {tissue}({f_\text {bm}})\), the strain concentration tensor \(\mathbb {A}_\text {bm}({f_\text {bm}})\) and the stress concentration tensor \(\mathbb {B}_\text {bm}({f_\text {bm}})\) can be evaluated numerically at each location \({\varvec{r}}\) in the femur midshaft cross section and each time t based on the value of \({f_\text {bm}}({\varvec{r}}, t)\) and the expressions given in Fritsch and Hellmich (2007) and Scheiner et al. (2013).

Combined with beam theory, this procedure enables us to completely determine, at each time t, the spatial distribution across the femur midshaft of (i) the tissue-scale stress and strain tensors, \({\varvec{\sigma }}^\text {tissue}\), \({\varvec{\varepsilon }}^\text {tissue}\); and (ii) the microscopic stress and strain tensors of bone matrix, \({\varvec{\sigma }}_\text {bm}^\text {micro}\), \({\varvec{\varepsilon }}_\text {bm}^\text {micro}\).

In this paper, we will consider both the tissue-scale strain energy density (SED), \(\varPsi ^\text {tissue}\),  and microscopic SED of the bone matrix, \(\varPsi _\text {bm}^\text {micro}\), as local mechanical quantities sensed and transduced by bone cells. These SEDs are defined by:

$$\begin{aligned}&\varPsi ^\text {tissue}({\varvec{r}}, t) = \tfrac{1}{2} {\varvec{\varepsilon }}^\text {tissue}: \mathbb {C}^\text {tissue}: {\varvec{\varepsilon }}^\text {tissue}, \end{aligned}$$
(10)
$$\begin{aligned}&\varPsi _\text {bm}^\text {micro}({\varvec{r}}, t) = \tfrac{1}{2} {\varvec{\varepsilon }}_\text {bm}^\text {micro}: \mathbbm {c}^{\text {micro}}_\text {bm}: {\varvec{\varepsilon }}_\text {bm}^\text {micro}. \end{aligned}$$
(11)

The SEDs defined in Eqs. (10) and (11) will be used to formulate biomechanical regulation in the bone remodelling equations. In the literature, biomechanical regulation is commonly based on the SED since this quantity is scalar and it integrates both microstructural state and loading environment (Fyhrie and Carter 1986; Mullender et al. 1994; Ruimerman et al. 2005).

2.2 Bone tissue remodelling

The tissue is assumed to be remodelled by a population of active osteoclasts (\(\text {OC}_{\mathrm{a}}\)) and active osteoblasts (\(\text {OB}_{\mathrm{a}}\)). Active osteoclasts are assumed to resorb bone at rate \(k_\text {res}\) (volume of bone resorbed per cell per unit time). Active osteoblasts are assumed to secrete new bone matrix at rate \(\text {k}_{\mathrm{form}}\) (volume of bone formed per cell per unit time). These cellular resorption and formation rates are taken to be constant and uniform. However, the bone volume fraction \({f_\text {bm}}({\varvec{r}}, t)\) of the tissue may evolve with site-specific rates depending on the balance between the populations of active osteoclasts and active osteoblasts (Martin 1972; Buenzli et al. 2013):

$$\begin{aligned} \tfrac{\partial }{\partial t}{f_\text {bm}}({\varvec{r}}, t) = \text {k}_{\mathrm{form}}\text {OB}_{\mathrm{a}}- k_\text {res}\text {OC}_{\mathrm{a}}. \end{aligned}$$
(12)

In Eq. (12), \(\text {OC}_{\mathrm{a}}({\varvec{r}}, t)\) and \(\text {OB}_{\mathrm{a}}({\varvec{r}}, t)\) denote the average densities of active osteoclasts and active osteoblasts in the tissue located at \({\varvec{r}}\) (number of cells in the RVE/TV, Fig. 1d). The site-specific remodelling rate \(\chi _\text {BV}({\varvec{r}}, t)\) of the tissue at \({\varvec{r}}\) (also called turnover rate) can be defined as the volume fraction of bone in the RVE that is resorbed and refilled in matched amount per unit time. This corresponds to the minimum of the volume fraction of bone resorbed per unit time, \(k_\text {res}\text {OC}_{\mathrm{a}}\), and volume fraction of bone formed per unit time, \(\text {k}_{\mathrm{form}}\text {OB}_{\mathrm{a}}\) (Parfitt 1983, Sect. II.C.2.c.ii):

$$\begin{aligned} \chi _\text {BV}({\varvec{r}}, t) = \min \{k_\text {res}\text {OC}_{\mathrm{a}}, \text {k}_{\mathrm{form}}\text {OB}_{\mathrm{a}}\}. \end{aligned}$$
(13)

Any imbalance between resorption and formation in Eq. (12) corresponds to surplus resorption or surplus formation with respect to the baseline of bone properly turned over in Eq. (13), as shown in Fig. 3.

Fig. 3
figure 3

Schematic representation of the definition of bone turnover provided by Eq. (13) in cases of balanced remodelling, remodelling with bone loss and remodelling with bone gain

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Fig. 4
figure 4

Proposed cell population model of bone remodelling taking into account several developmental stages of osteoblasts and osteoclasts together with biochemical regulation, biomechanical regulation (via strain energy density, \(\varPsi \)) and geometrical regulation (via the turnover function, \(\overline{\chi _\text {BV}}({f_\text {bm}})\))

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Equation (12) enables us to track site-specific modifications of the midshaft tissue microstructure through \({f_\text {bm}}({\varvec{r}}, t)\), from which stress and strain distributions across the midshaft can be estimated at both the tissue scale and the microscopic, cellular scale, by means of Eqs. (34)–(35) and (8)–(11).

2.3 Bone cell population model

It remains to specify how the populations of active osteoclasts \(\text {OC}_{\mathrm{a}}({\varvec{r}}, t)\) and active osteoblasts \(\text {OB}_{\mathrm{a}}({\varvec{r}}, t)\) evolve in the RVE located at \({\varvec{r}}\) under mechanobiological, geometrical and biochemical regulations. For this, we use a continuum cell population model based on rate equations, originally developed by Lemaire et al. (2004), and later refined and extended by Pivonka and co-workers (Pivonka et al. 2008, 2010; Buenzli et al. 2012; Pivonka et al. 2013; Scheiner et al. 2013; Pivonka et al. 2012).

To highlight important biochemical couplings and regulations in osteoclastogenesis and osteoblastogenesis, several differentiation stages of osteoclasts and osteoblasts are considered. These biochemical interactions are mediated by several signalling molecules whose binding kinetics are explicitly considered in the model, such as transforming growth factor \(\upbeta \) (\(\text {TGF} \upbeta \)), receptor activator of nuclear factor \(\kappa \)B (RANK) and associated ligand RANKL, osteoprotegerin (OPG) and parathyroid hormone (PTH). The biochemical network of these couplings and regulations is summarised in Fig. 4.

Active osteoclasts (\(\text {OC}_{\mathrm{a}}\)s) denote cells attached to the bone surface that actively resorb bone matrix. These cells are assumed to differentiate from a pool of osteoclast precursor cells (\(\text {OC}_{\mathrm{p}}\)s) by the binding of RANKL to the RANK receptor, expressed on \(\text {OC}_{\mathrm{p}}\)s, which induces intracellular \(\text {NF} \kappa \hbox {B}\) signalling. Osteoclast precursors are assumed to differentiate from a pool of uncommitted osteoclasts progenitors (\(\text {OC}_{\mathrm{u}}\)), such as haematopoietic stem cells, under the action of macrophage colony stimulating factor (MCSF) and RANKL signalling (Roodman 1999; Martin 2004).

Active osteoblasts (\(\text {OB}_{\mathrm{a}}\)s) denote cells at the bone surface that actively deposit new bone matrix. These cells are assumed to differentiate from a pool of osteoblast precursor cells (\(\text {OB}_{\mathrm{p}}\)s). This activation is inhibited in the presence of \(\text {TGF} \upbeta \). Osteoblast precursors are assumed to differentiate from a pool of uncommitted osteoblasts progenitors (\(\text {OB}_{\mathrm{u}}\)), such as mesenchymal stem cells or bone marrow stromal cells, upon \(\text {TGF} \upbeta \) signalling (Roodman 1999).

The rate equations governing the evolution of the tissue-average cell densities are given by:

$$\begin{aligned} \tfrac{\partial }{\partial t}{{\text {OC}_{\mathrm{p}}}}({\varvec{r}}, t) =&\,{\mathcal {D}_{\text {OC}_{\mathrm{u}}}}\big ({\text {MCSF}}, {\text {RANKL}}(\varPsi ,{\text {PTH}})\big )\, \text {OC}_{\mathrm{u}}({f_\text {bm}})\nonumber \\&- {\mathcal {D}_{\text {OC}_{\mathrm{p}}}}\big (\text {RANKL}(\varPsi ,\text {PTH})\big )\, \text {OC}_{\mathrm{p}}, \end{aligned}$$
(14)
$$\begin{aligned} \tfrac{\partial }{\partial t}{\text {OC}_{\mathrm{a}}}({\varvec{r}}, t) =&\,\mathcal {D}_{\text {OC}_{\mathrm{p}}}\big (\text {RANKL}(\varPsi ,\text {PTH})\big )\, \text {OC}_{\mathrm{p}}\nonumber \\&- \mathcal {A}_{\text {OC}_{\mathrm{a}}}(\text {TGF} \upbeta )\, \text {OC}_{\mathrm{a}}, \end{aligned}$$
(15)
$$\begin{aligned} \tfrac{\partial }{\partial t}\text {OB}_{\mathrm{p}}({\varvec{r}}, t) =&\,\mathcal {D}_{\text {OB}_{\mathrm{u}}}(\text {TGF} \upbeta )\, \text {OB}_{\mathrm{u}}({f_\text {bm}}) + \mathcal {P}_{\text {OB}_{\mathrm{p}}}(\varPsi ) \, \text {OB}_{\mathrm{p}}\nonumber \\&- \mathcal {D}_{\text {OB}_{\mathrm{p}}}(\text {TGF} \upbeta )\, \text {OB}_{\mathrm{p}}, \end{aligned}$$
(16)
$$\begin{aligned} \tfrac{\partial }{\partial t}\text {OB}_{\mathrm{a}}({\varvec{r}}, t) =&\,\mathcal {D}_{\text {OB}_{\mathrm{p}}}(\text {TGF} \upbeta )\, \text {OB}_{\mathrm{p}}\nonumber \\&- A_{{\text {OB}_{\mathrm{a}}}} {\text {OB}_{\mathrm{a}}}, \end{aligned}$$
(17)

where \({\mathcal {D}}_i\) is the differentiation rate of cell type i (\(i=\text {OC}_{\mathrm{u}}, \text {OC}_{\mathrm{p}}, \text {OB}_{\mathrm{u}}, \text {OB}_{\mathrm{p}}\)) modulated by signalling molecules, \(\mathcal {A}_{\text {OC}_{\mathrm{a}}}\) is the apoptosis rate of active osteoclasts modulated by \(\text {TGF} \upbeta \), \(A_{\text {OB}_{\mathrm{a}}}\) is the (constant) apoptosis rate of active osteoblasts, \(\mathcal {P}_{\text {OB}_{\mathrm{p}}}\) is the proliferation rate of osteoblast precursor cells and \(\varPsi \) is the strain energy density, taken to be either \(\varPsi ^\text {tissue}\) or \(\varPsi ^\text {micro}_\text {bm}\).

The concentrations of the signalling molecules are governed by rate equations expressing mass action kinetics of receptor–ligand binding reactions. Since time scales involved in cell differentiation and apoptosis are much longer than characteristic times of receptor–ligand binding reactions, the signalling molecule concentrations can be solved for in a quasi-steady state (adiabatic approximation) (Buenzli et al. 2012; Pivonka et al. 2012).

Explicit expressions for the signalling molecules concentrations and their modulation of the cell differentiation and apoptosis rates depending on receptor–ligand binding are presented in the sections “Differentiation rates and signalling molecules in the cell populations model” and “Recalibration of the model” of the Appendix. Below, we comment in more detail on new features of Eqs. (14)–(17) that are included to model the geometrical and biomechanical feedbacks on bone cell populations.

2.3.1 Geometrical feedback and turnover rate

The turnover rate of different bone microstructures is believed to be significantly influenced by geometric variables in particular by the availability of bone surfaces (Parfitt 1983; Martin 1972). Active osteoclasts and active osteoblasts require a surface to resorb or form bone matrix. In a fixed RVE, the more surface there is, the more cells process bone matrix, and the higher the turnover rate. A remarkable relationship between the density of bone surface \(\text {S}_{\text {V}}\) and bone volume fraction \({f_\text {bm}}\) has been exhibited in bone tissues across wide ranges of porosities (Martin 1984; Fyhrie and Kimura 1999; Lerebours et al. 2015). This relationship \(\text {S}_{\text {V}}({f_\text {bm}})\) is particularly interesting from a computational modelling perspective as it enables to model the influence of bone surface density through an implicit dependence on bone volume fraction. Turnover rate may also be influenced by bone volume fraction in another way. Indeed, turnover rate is influenced by the biochemical microenvironment, such as the availability of precursor cells. In the femur midshaft, the availability of precursor cells depends on pore size and the radial position in the midshaft (i.e. periosteal, intracortical and endosteal), which are both well characterised by bone volume fraction. These two geometrical influences (surface density and pore size/location) can thus be represented by a single dependence of turnover rate on bone volume fraction.

Fig. 5
figure 5

a Plot of the phenomenological relationship \(\overline{\chi _\text {BV}}({f_\text {bm}})\) between turnover rate and bone volume fraction assumed in the model, the grey data points are the ones given by Parfitt (1983). b Dependence of \(\text {OC}_{\mathrm{u}}\) and \(\text {OB}_{\mathrm{u}}\) upon \({f_\text {bm}}\) based on the prescribed bone turnover function. \(\text {OB}_{\mathrm{u}}\)(\({f_\text {bm}}\)) and \(\text {OC}_{\mathrm{u}}\)(\({f_\text {bm}}\)) serve as input values in the bone cell population model

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In our model, different turnover rates, Eq. (13), can be achieved in microstructures of different bone volume fractions by assuming that \(\text {OB}_{\mathrm{u}}\) and \(\text {OC}_{\mathrm{u}}\) are functions of \({f_\text {bm}}\). Indeed, the functions \(\text {OB}_{\mathrm{u}}({f_\text {bm}})\) and \(\text {OC}_{\mathrm{u}}({f_\text {bm}})\) introduce a dependence of the active bone cell populations \(\text {OC}_{\mathrm{a}}\) and \(\text {OB}_{\mathrm{a}}\) upon \({f_\text {bm}}\) via Eqs. (14)–(17). We emphasise that an explicit geometrical influence of microstructure is only introduced here on the population of uncommitted cells. This is consistent with the conclusions of Pivonka et al. (2013) who investigated different possibilities on how geometrical feedback can affect various stages of bone cell differentiation. This is also consistent with the interpretation that the microstructure’s surface area plays an important role for the initiation of new BMUs (basic multicellular units), occurring through \(\text {OB}_{\mathrm{u}}\) and \(\text {OC}_{\mathrm{u}}\) differentiation, rather than for the activation of osteoclasts and osteoblasts. The activation of osteoclasts and osteoblasts depends instead on temporary BMU cavity internal surfaces that only comprise a small fraction of the microstructure’s bone surface.

To determine functions \(\text {OB}_{\mathrm{u}}({f_\text {bm}})\) and \(\text {OC}_{\mathrm{u}}({f_\text {bm}})\) that lead to physiological turnover rates at different bone volume fractions, we assume that in a ‘normal healthy state’, a phenomenological relationship \(\overline{\chi _\text {BV}}({f_\text {bm}})\) can be defined between turnover rate and bone volume fraction, such that \(\chi _\text {BV}({\varvec{r}}, t) = \overline{\chi _\text {BV}}\big ({f_\text {bm}}({\varvec{r}}, t)\big )\). A normal healthy state is defined as a steady state in which resorption and formation are balanced in an individual with normal physical activity and normal hormonal levels. Taking as input such a phenomenological relationship \(\overline{\chi _\text {BV}}({f_\text {bm}})\), one can determine for each \({f_\text {bm}}\) the values \(\text {OC}_{\mathrm{u}}({f_\text {bm}})\) and \(\text {OB}_{\mathrm{u}}({f_\text {bm}})\) that lead to (i) balanced remodelling and (ii) an actual steady-state turnover rate \(\chi _\text {BV}\) such that \(\chi _\text {BV}= \overline{\chi _\text {BV}}({f_\text {bm}})\). From Eqs. (12) and (13), this imposes the following two constraints to be solved for \(\text {OC}_{\mathrm{u}}\)(\({f_\text {bm}}\)) and \(\text {OB}_{\mathrm{u}}\)(\({f_\text {bm}}\)):

$$\begin{aligned} \overline{\chi _\text {BV}}({f_\text {bm}})&= \text {k}_{\mathrm{form}}\overline{\text {OB}_{\mathrm{a}}}\big (\text {OC}_{\mathrm{u}}({f_\text {bm}}), \text {OB}_{\mathrm{u}}({f_\text {bm}})\big ) \end{aligned}$$
(18)
$$\begin{aligned}&= k_\text {res}\overline{\text {OC}_{\mathrm{a}}}\big (\text {OC}_{\mathrm{u}}({f_\text {bm}}), \text {OB}_{\mathrm{u}}({f_\text {bm}})\big ), \end{aligned}$$
(19)

where the bars on the right-hand side indicate steady-state values of the cell density variables.

By analogy with the relationship \(\text {S}_{\text {V}}({f_\text {bm}})\), we assume in this paper that the phenomenological relationship \(\overline{\chi _\text {BV}}({f_\text {bm}})\) is a dome-shaped function whose height is adjusted to Parfitt’s reported measurements of turnover rate in cortical and trabecular bone (Parfitt 1983, Table 1 and Table 7),Footnote 2 see Fig. 5a. The relationship satisfies \(\overline{\chi _\text {BV}}(0)=\overline{\chi _\text {BV}}(1)=0\) since there is no turnover possible without the presence of bone surfaces. The maximum of bone turnover is assumed to occur at \({f_\text {bm}}\) = 0.35, corresponding to typical trabecular or trabecularised bone microstructures in the midshaft femur. With this assumed relationship \(\overline{\chi _\text {BV}}({f_\text {bm}})\), the constraints (18)–(19) were solved numerically for \(\text {OB}_{\mathrm{u}}\)(\({f_\text {bm}}\)) and \(\text {OC}_{\mathrm{u}}\)(\({f_\text {bm}}\)) at each value of \({f_\text {bm}}\in [0,1]\) by using a trust-region dogleg method. The functions \(\text {OC}_{\mathrm{u}}({f_\text {bm}})\) and \(\text {OB}_{\mathrm{u}}({f_\text {bm}})\) obtained by this procedure are shown in Fig. 5b. These functions are used as input in Eqs. (14)–(17) in all our simulations. This ensures that in the normal healthy state, each RVE of the midshaft cross section located at \({\varvec{r}}\) remodels at rate \(\overline{\chi _\text {BV}}({f_\text {bm}}({\varvec{r}}))\) without changing bone volume fraction. The functions \(\text {OC}_{\mathrm{u}}({f_\text {bm}})\) and \(\text {OB}_{\mathrm{u}}({f_\text {bm}})\) are assumed to hold unaffected in the various deregulating circumstances considered later on (i.e. osteoporosis and altered mechanical loading). During these deregulating circumstances, the value of the actual turnover rate, Eq. (13), differs from the value of \(\overline{\chi _\text {BV}}\) and depends implicitly on the hormonal and mechanical states that influence the active cell populations \(\text {OB}_{\mathrm{a}}\) and \(\text {OC}_{\mathrm{a}}\).

The explicit calibration of the cell population model, Eqs. (14)–(17), to site-specific tissue remodelling rates is a significant novelty compared to our previous temporal model (Pivonka et al. 2013; Scheiner et al. 2013). This modification was made necessary to consistently describe the site-specific evolution of bone in the spatio-temporal framework of Buenzli et al. (2013) while retaining a cell population model that includes biochemical regulations.

2.3.2 Mechanical feedback and initial bone microstructure stability

A mechanical feedback is included in the cell population model such that underloaded regions of bone promote osteoclastogenesis and overloaded regions of bone promote osteoblastogenesis (Frost 1987, 2003). These responses are viewed as consequences of biochemical signals transduced from a mechanical stimulus sensed by osteocytes (Bonewald 2011). Osteocytes are known to express RANKL, which regulates osteoclast generation, and sclerostin, which regulates osteoblast generation via Wnt signalling (Bonewald and Johnson 2008). Following Scheiner et al. (2013), the resorptive response of the mechanical feedback is assumed to act by an increase in the microenvironmental concentration of RANKL, whereas its formative response is assumed to act by an increase in the proliferation rate of osteoblast precursors.

The exact nature of the mechanical stimulus sensed by osteocytes is still a matter of debate. It may include lacuno-canalicular extracellular fluid shear stress on the osteocyte cell membrane, extracellular fluid pressure, streaming potentials and direct deformations of the osteocyte body induced by bone matrix strains (Knothe Tate 2003; Bonewald and Johnson 2008; Bonewald 2011). Due to the extensive network of osteocyte connections in bone (Buenzli and Sims 2015), average bone matrix strains at a higher scale may also be sensed by the osteocyte network. Below, we assume that the mechanical stimulus to the bone cell population model is described by a local strain energy density, \(\varPsi ({\varvec{r}}, t)\). This strain energy density will be taken to be either the microscopic, cellular-scale strain energy density of bone matrix, \(\varPsi ^\text {micro}_\text {bm}({\varvec{r}}, t)\), or the average tissue-scale strain energy density, \(\varPsi ^\text {tissue}({\varvec{r}}, t)\), defined, respectively, in Eqs. (11) and (10).

To predict with our model the evolution of a real scan of midshaft femur under various deregulating circumstances, it is important to assume that the bone scan represents a stable state initially in the absence of any deregulation. In particular, this initial bone state is assumed mechanically optimal. This can be ensured by choosing the local mechanical stimulus acting onto the bone cells, \(\mathrm{{\mu }}({\varvec{r}}, t)\), as a normalised difference between the current SED and the SED of the initial bone microstructure \(\varPsi ({\varvec{r}}, 0)\):

$$\begin{aligned} \mu ({\varvec{r}}, t) = \frac{\varPsi ({\varvec{r}}, t) - \varPsi ({\varvec{r}}, 0)}{\varPsi ({\varvec{r}}, 0) + K} \end{aligned}$$
(20)

The normalisation by \(\varPsi ({\varvec{r}}, 0)\) in the denominator in Eq. (20) ensures that the mechanical stimulus is not over-emphasised away from the neutral axis where SED takes high values. The small positive constant \(K=1 \cdot 10^{-6}\,\text {GPa}\) is added to keep mechanical stimulus well defined near the neutral axis where \(\varPsi ({\varvec{r}}, 0) \approx 0\) (see also Discussion Sect. 4.3).

When negative, \(\mathrm{{\mu }}({\varvec{r}}, t)\) in Eq. (20) is assumed to promote \(\beta _\text {RANKL}^\text {mech}\), the production rate of RANKL (Fan et al. 2004; Wang et al. 2004):

$$\begin{aligned} \beta _\text {RANKL}^\text {mech}(\varPsi ) = {\left\{ \begin{array}{ll} - \kappa \cdot \mu ({\varvec{r}}, t), &{}\quad \text {if } \mu ({\varvec{r}}, t) \le 0\\ 0, &{}\quad \text {if } \mu ({\varvec{r}}, t) > 0 \end{array}\right. } \end{aligned}$$
(21)

where \(\kappa \) is a parameter describing the strength of the biomechanical transduction (see Sect. 2.6). This results in increased RANKL signalling in underloaded conditions (see Eq. (28) in Appendix), and so in increased osteoclast generation in Eqs. (14)–(15).

Fig. 6
figure 6

a Microradiograph of a midshaft femur cross section (courtesy of C. David L. Thomas and John G. Clement, Melbourne Femur Collection). b Binarisation of the microradiograph and determination of the local \({f_\text {bm}}\) values: (i) at the periosteal region; (ii) in the intracortical region; and (iii) at the endosteal region; c Bone volume fraction distribution extracted from the radiograph and interpolated. The dashed line represents the location of the neutral axis. The origin of the coordinate system (y, z) is taken at the normal force centre, NC. The grey lines are the 10 mm along which we are studying the evolution of the model in the Results Section

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When positive, \(\mathrm{{\mu }}({\varvec{r}}, t)\) in Eq. (20) is assumed to promote \(\mathcal {P}_{\text {OB}_{\mathrm{p}}}\), the proliferation rate of pre-osteoblasts in Eq. (16) (Kaspar et al. 2002; Weyts et al. 2003; Robling et al. 2006):

(22)

where \(\lambda \) is a parameter describing the strength of the biomechanical transduction. The first term in (22) accounts for a transit-amplifying stage of osteoblast differentiation occurring in the absence of mechanical stimulation (Buenzli et al. 2012). The proliferation rate is assumed to saturate to the value \(\mathcal {P}_{\text {OB}_{\mathrm{p}}}= 2 P_{\text {OB}_{\mathrm{p}}}\) in highly overloaded situations to ensure the stability of the population of \(\text {OB}_{\mathrm{p}}\)s (Buenzli et al. 2012; Scheiner et al. 2013).

A similar type of mechanical feedback was implemented in purely temporal settings in Scheiner et al. (2013). The initial SED distribution, \(\varPsi ({\varvec{r}}, 0)\), is calculated from Eqs. (10)–(11) and from the initial bone volume fraction distribution, \({f_\text {bm}}({\varvec{r}}, 0)\), determined on the bone scan (described in the next section).

2.4 Initial distribution of bone volume fraction from microradiographs

The initial microstructural state of the midshaft bone cross section can be derived from high-resolution bone scans such as microcomputed tomography (micro-CT) scans or microradiographs. Since Haversian canals have an average diameter of about 40 \(\mathrm{{\upmu }}\)m, at least 10-\(\mathrm{{\upmu }}\)m resolution is required to evaluate intracortical bone volume fractions with sufficient accuracy.

In the simulations presented in Sect. 3, we used the microradiograph represented in Fig. 6a where the pixel size is 7 \(\mathrm{{\upmu }}\)m. The femur sample was collected from a 21-year-old subject. The microradiograph was digitised and binarised by a thresholding operation based on pixel grey level. Bone matrix is assigned the value 1 irrespective of the degree of mineralisation, and intracortical pores are assigned the value 0. The distribution of the bone volume fraction, \({f_\text {bm}}({\varvec{r}}, 0)\), across the midshaft was determined by calculating the volume of bone matrix in a disc of 2 mm diameter, centred at each pixel of the binarised image and divided by the disc’s area. For the points near the periosteal surface, only the portion of the disc contained into the subperiosteal area was used for this calculation (see Fig. 6b). The discrete values of \({f_\text {bm}}\) defined at each pixel contained in the subperiosteal region were then interpolated into a continuous function, \({f_\text {bm}}({\varvec{r}}, 0)\), using MATLAB’s 2D cubic interpolation procedure. The result is shown in Fig. 6c. A similar exclusion was not performed at the endosteal surface since this surface is less well defined, in opposition to the periosteal surface, due to the presence of trabecular-like structures. Bone matrix volume fractions near the endosteal surface are averages of intracortical bone regions and regions in the bone marrow cavity.

Fig. 7
figure 7

a Evolution of the total bone mass in the cross section with time while simulating osteoporosis: calibration of the PTH infusion. Osteoporosis is characterised by a bone loss of 1 %/year (Parfitt and Chir 1987; Nordin et al. 1988; Szulc et al. 2006). b Evolution of the total bone mass in the cross section with time while simulating a spaceflight mission: calibration of the mechanical parameters, \(\lambda \) and \(\kappa \)

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2.5 Numerical simulations

The multiscale mechanobiological model of bone remodelling presented in this paper is governed by a coupled system of (i) distributed ODEs describing the evolution of bone cell populations at each location \({\varvec{r}}\) in the midshaft femur (Eqs. (14)–(17)) and (ii) non-local and tensorial algebraic equations determining the mechanical state of the tissue RVE at \({\varvec{r}}\), both at the tissue scale and at the microscopic scale (Eqs. (2)–(11)). The model is initialised with a bone volume fraction distribution across the midshaft femur deduced from high-resolution bone scans (Fig. 6a) and with steady-state populations of cells fulfilling the site-specific turnover rate conditions Eqs. (18)–(19). This initial state is thereby constructed to be a steady state of the model, in which the biochemical, geometrical and mechanobiological regulations of resorption and formation are balanced.

To solve this non-local spatio-temporal problem numerically, we use a staggered iteration scheme in which we first solve the mechanical problem (i.e. tissue-scale SED and microscopic SED), assuming constant material properties, and then solve the bone cell population model and evolve the bone volume fraction at each location \({\varvec{r}}\) of the femur midshaft assuming constant mechanical feedback for a duration \(\varDelta \)t. After \(\varDelta \)t, the mechanical problem is recalculated based on the updated bone volume fraction distribution, \({f_\text {bm}}\)(\({\varvec{r}}\), t+\(\varDelta \)t), and this procedure is iterated. The ODEs are solved using a standard stiff ODE solver (MATLAB, ode15s). The spatial discretisation is a regular grid with steps \(\varDelta y = \varDelta z = 0.8\,\text {mm}\). Due to the separation of time scales between changes in the local mechanical environment (years) and changes in bone cell populations (days), the mechanical stimulus requires updating after durations \(\varDelta \)t = 2 years. The accuracy of the numerical results depending on \(\varDelta \)t is studied in the section “Update frequency of mechanical state in the numerical algorithm” of the Appendix.

2.6 Model calibration

The model presented in this paper contains: (i) biomechanical parameters associated with the estimation of \(\varPsi ({\varvec{r}},t)\), and (ii) parameters associated with the bone cell population model. Biomechanical parameters as well as biochemical parameters were determined and validated in other studies (Scheiner et al. 2013; Pivonka et al. 2013, 2008; Buenzli et al. 2012) (Table 1). The origin and robustness of parameter estimates are discussed in the section “Model parameter values” of the Appendix. Here we calibrate the newly introduced parameters: (a) mechanical coupling parameters \(\lambda \) and \(\kappa \) (Eqs. (21) and (22)), and (b) biochemical parameters related to the simulation of osteoporosis.

2.6.1 Calibration of the hormonal deregulation for osteoporosis simulation

In our previous temporal model (Scheiner et al. 2013), osteoporosis was modelled by an increase in systemic PTH together with a reduction in the biomechanical transduction parameters: \(\lambda \) and \(\kappa \). In this paper, we simulate age-related bone loss using a single parameter perturbation, i.e. an increase in systemic PTH concentration. This increase is calibrated so as to obtain a loss of total bone cross-sectional area in the femur midshaft of 1 % per year Footnote 3 (Parfitt and Chir 1987; Nordin et al. 1988; Szulc et al. 2006). The total bone cross-sectional area is defined by the integral of \({f_\text {bm}}({\varvec{r}}, t)\) over the midshaft cross section. In the model, a rate of bone loss of 1 %/year was obtained by an increase in systemic concentration of PTH from 2.907 to 2.954 pM (1.62 % increase) (see Fig. 7a).

2.6.2 Calibration of mechanobiological feedback

The rate of change in bone mass due to mechanical feedback is determined in the model by the biomechanical transduction parameters \(\lambda \) (in Eq. (22)) and \(\kappa \) (in Eq. (21)). To calibrate these parameters, we used data gathered from mechanical disuse and re-use experiments. It has been shown that cosmonauts undertaking long mission space flights lose bone mass at a rate of approximately 0.3 % per month (Vico et al. 2000). This microgravity-induced bone loss is only slowly recovered after return to Earth. No significant bone gain is observed after 6-month exposure to normal gravity on Earth (Vico et al. 2000; Collet et al. 1997).

In our multiscale model, microgravity is simulated as a 80 % reduction in the normal mechanical loads experienced by the femur, i.e. \({\varvec{N}}_{\text {microgravity}} = 0.2 {\varvec{N}}\) and \({\varvec{M}}_{\text {microgravity}} = 0.2 {\varvec{M}}\). Based on these reduced loads, the parameter \(\kappa \) has been calibrated such that \(1.8\,\%\) of total bone cross-sectional area is lost after 6 months. We found such rate of loss with \(\kappa = 18\) pM/day when the mechanical stimulus is based on the microscopic SED, \(\varPsi ^\text {micro}_\text {bm}({\varvec{r}}, t)\) (see Fig. 7b), and \(\kappa = 19\) pM/day when the mechanical stimulus is based on the tissue-scale SED, \(\varPsi ^\text {tissue}({\varvec{r}}, t)\). After return to Earth, rates of bone recovery are too low to be detected after 6 months (Collet et al. 1997). We performed a parametric study investigating various strengths of \(\lambda \). Using parameter values of \(\lambda > 1\) in our model results in significant bone gain after 6 months, while \(\lambda \le 1\) results in small bone gain. Based on these results, we use \(\lambda = 0.5\) for both the microscopic and tissue-scale mechanical stimuli.

3 Results

In this section, we present numerical simulations of the evolution of the midshaft femur cross section subjected to either (i) changes in mechanical environment (Sect. 3.1) or (ii) hormonal deregulation simulating osteoporosis (Sect. 3.2). We also investigate how site-specific bone loss may depend on whether mechanical stimulus is sensed at the microscopic, cellular scale or at the tissue scale.

Fig. 8
figure 8

a Difference between the bone volume fraction distribution in the cross section after a 6-month spaceflight mission with the initial distribution. b Difference between the bone volume fraction distribution in the cross section after a 40 years of simulated osteoporosis with the initial distribution. c Bone volume fraction distribution in the cross section after 40 years of simulated osteoporosis. d Microradiograph of a human femur cross section from an 89-year-old individual. The dashed lines highlight regions with sharp transition between porous and compact tissue. The arrows point out regions with high porosity along the antero-posterior axis

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Fig. 9
figure 9

Evolution of a bone volume fraction; b mechanical stimulus, \(\mathrm{{\mu }}({\varvec{r}}, t)\), at the microscopic scale; c mechanical stimulus, \(\mathrm{{\mu }}({\varvec{r}}, t)\), at the tissue scale; and d turnover rate along the y and z-axis during the simulation of osteoporosis

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3.1 Bone loss due to mechanical disuse

Figure 8a represents site-specific changes of the femur midshaft cross section simulated by the model assuming a 80 % reduction in the normal mechanical loading. This reduction in mechanical loading may represent microgravity in long spaceflight missions (see Sect. 2.6) or prolonged bed rest. The Figure depicts the difference between the bone volume fraction distribution after 6 months of mechanical disuse and the initial bone volume fraction. It can be seen that bone loss is site-specific with more bone loss occurring near the endosteal surface. Close to the neutral axis, only limited loss of bone is observed.

3.2 Simulation of osteoporosis due to hormonal deregulation

Figure 8b and c represents the site-specific changes of the midshaft cross section that occur after 40 years of simulated osteoporosis when the mechanical feedback acting onto the bone cell population model is based on the microscopic SED. Figure 8b depicts the difference between the \({f_\text {bm}}\) distribution at the end of the simulation and the initial distribution. Figure 8c depicts the \({f_\text {bm}}\) distribution at the end of the simulation. Bone loss occurs everywhere in the cross section except at the medial and lateral sides. The loss is site specific with higher rates of loss in the endocortical region and around the neutral axis, close to the antero-posterior axis. This pattern of bone loss is consistent with the high porosity commonly observed in these regions in osteoporotic subjects (Fig. 8d, arrows). The simulation exhibits a sharp transition between a very porous endocortical region and a dense intracortical region towards the periosteum. Although perhaps less pronounced, such a transition is also observed in the microradiograph of Fig. 8d (dashed lines). In contrast to the osteoporosis simulation, the simulation of mechanical disuse (Fig. 8a) shows that bone was lost all over the cross section, with little change around the neutral axis. In both simulations, bone was lost predominantly in the endocortical region.

Fig. 10
figure 10

Bone volume fraction along the y- and z-axes, the initial distribution and the distributions after 40 years of simulated osteoporosis, without mechanical regulation, with mechanical regulation based on microscopic SED and with mechanical regulation based on tissue-scale SED

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In Fig. 9, we show how the distributions of the following quantities evolved along the y- and the z-axes during the simulation of osteoporosis: (a) the bone volume fraction, (b) the microscopic mechanical stimulus, \(\mathrm{{\mu }}_{\text {bm}}^{\text {micro}}\), used as mechanical stimulus in this simulation, (c) the tissue-scale mechanical stimulus, \(\mathrm{{\mu }}^{\text {tissue}}\), not used as mechanical stimulus in this simulation and (d) the turnover rate. Along both axes, the regions in which bone volume fraction transitions from low to high (3 \(< y < \) 6 mm and \(-7 < z < -5\) mm) are resorbed at higher rate, due to the higher values of \(\chi _\text {BV}\) in these regions (Fig. 9d). As time progresses, bone volume fraction is strongly reduced in the endocortical region, leading to an expansion of the marrow space and a reduction in cortical wall width. This is accompanied by a shift of the maximum of \(\chi _\text {BV}\) towards the periosteum. Along the y-axis (near the neutral axis), bone is lost at a high rate not only in the endocortical region but also near the periosteum, as can be seen by the gradual increase in turnover rate in the whole cortical width (Fig. 9d). In contrast, along the z-axis, bone is lost at a high rate only at the endosteum where turnover rate maintains a well-defined peak. The intracortical region (\(z < -7\) mm) is preserved even after 40 years of simulated osteoporosis.

3.2.1 Microscopic vs tissue-scale mechanical stimulus

Comparing Fig. 9b and c, we can observe that the values of the mechanical adaptation stimuli are strongly dependent on the length scale at which they are calculated, i.e. tissue scale or microscopic scale. Along the z-axis, \(\mathrm{{\mu }}^{\text {micro}}_{\text {bm}}\) is always positive (Fig. 9b), whereas, \(\mathrm{{\mu }}^{\text {tissue}}\) takes negative values in the endocortical region (Fig. 9c). Regions with high values of \(\mathrm{{\mu }}^{\text {micro}}_{\text {bm}}\) and positive values of \(\mathrm{{\mu }}^{\text {tissue}}\) correlate with regions where the bone matrix is preserved. Regions with low values of \(\mathrm{{\mu }}^{\text {micro}}_{\text {bm}}\) and negative values of \(\mathrm{{\mu }}^{\text {tissue}}\) correlate with regions where the bone matrix is resorbed. The Figures also show a qualitative and quantitative difference in mechanical stimuli \(\mathrm{{\mu }}^{\text {micro}}_{\text {bm}}\) and \(\mathrm{{\mu }}^{\text {tissue}}\) between the y- and z-axes. The mechanical stimulus is asymmetric between the antero-posterior axis and lateral–medial axis due to the assumed bending loading state. Along the y-axis, no important variation can be observed between endocortical and periosteal regions. Along the z-axis, both stimuli exhibit much lower values in the endocortical region than at the periosteum or in the marrow cavity. We note that the mechanical stimuli are not zero in the marrow even when \({f_\text {bm}}= 0\) due to the assumed vascular stiffness.

Figure 10 compares the evolution of bone volume fraction during simulated osteoporosis when the mechanical stimulus acting onto the bone cells is either (i) absent, (ii) based on the microscopic mechanical stimulus, \(\mathrm{{\mu }}^{\text {micro}}_{\text {bm}}\), or (iii) based on the tissue-scale mechanical stimulus, \(\mathrm{{\mu }}^{\text {tissue}}\).Footnote 4 All cases exhibit strong endocortical bone loss with little difference in the expansion rate of the medullary cavity. A slightly steeper endosteal wall is created along the z-axis during the simulation using tissue-scale mechanical stimulus, and a region with very low bone volume fraction (\({f_\text {bm}}\simeq 0.1\)) is preserved in the medullary cavity during the simulation with microscopic mechanical stimulus. Intracortical bone towards the periosteum is preserved along the z-axis by both mechanical stimuli, but it is resorbed more strongly along the y-axis in the simulation with tissue-scale mechanical stimulus.

4 Discussion

4.1 Endocortical bone loss

The loss of endocortical bone, with its associated expansion of the marrow cavity and cortical wall thinning, is a trait common to several bone disorders and deregulations of remodelling. It is observed in biochemical disorders, such as in osteoporosis, vitamin D deficiency and hyperparathyroidism (Feik et al. 1997; Parfitt 1998; Bousson et al. 2001; Thomas et al. 2005; Szulc et al. 2006; Zebaze et al. 2010; Busse et al. 2013; Hirano et al. 2000; Burr et al. 2001; Turner et al. 2002). It is also observed during disruptions of normal mechanical loading, such as in prolonged bed rest, long-term space missions and trauma-induced paralysis such as spinal cord injury (Leblanc et al. 2007; Rittweger et al. 2009; Vico et al. 2000; Lang et al. 2004; Kiratli et al. 2000; Eser et al. 2004), and in animal experiments involving muscle paralysis or hindlimb disuse induced by tail suspension (Warner et al. 2006; Ausk et al. 2012, 2013; Bloomfield et al. 2002; Judex et al. 2004).

Our numerical simulations of osteoporosis and mechanical disuse are consistent with these experimental findings. All Figs. 8, 9 and 10 highlight the strong site-specificity of bone loss under deregulations of bone remodelling. The endocortical region systematically undergoes the most significant loss. This similarity arises despite the fact that in the simulation of mechanical disuse, the deregulation is non-uniform in the cross section (due to the uneven distribution of mechanical loads), whereas in the simulation of osteoporosis, the hormonal deregulation is uniform in the cross section.

The precise mechanisms that underlie the predominant loss of bone in the endocortical region are still poorly understood (Raisz and Seeman 2001; Thomas et al. 2005; Squire et al. 2008; Ausk et al. 2012). Mechanical adaptation has been suggested as a potential mechanism (Frost 1997; Burr 1997; Thomas et al. 2005; Jepsen and Andarawis-Puri 2012). Bone loss induced by mechanical disuse redistributes mechanical loads towards the periosteum, where bone volume fraction is higher. This could unload endocortical regions and thereby accelerate their resorption. Reduced physical activity and muscle strength in ageing subjects support this hypothesis (Frost 1997). However, the ubiquity of endocortical bone loss in situations in which mechanical loading is not significantly modified suggests that other mechanisms are at play. The morphological influence of the tissue microstructure on the rate of bone loss has been suggested to be another important factor (Martin 1972; Squire et al. 2008; Zebaze et al. 2010; Buenzli et al. 2013). Cortical bone has little bone surface available to bone cells, but this surface expands during bone loss, which could increase the activation frequency of remodelling events. If remodelling is imbalanced, this may lead to an acceleration of bone loss and to an increase of the available surface until the tissue microstructure becomes so porous that its surface area reduces with further loss (Martin 1972; Raisz and Seeman 2001).

We have shown previously the possibility of this morphological mechanism to explain cortical bone trabecularisation in both temporal (Pivonka et al. 2013) and spatio-temporal settings (Buenzli et al. 2013). The spatio-temporal model proposed in the present work incorporates both mechanical adaptation and a morphological feedback of the microstructure on turnover rate. In Fig. 10, our simulations of osteoporosis conducted with and without mechanical feedback suggest that the rate at which the medullary cavity expands and the cortical wall thins is only marginally dependent on mechanical adaptation. This rate is primarily due to the high turnover rates present in the endocortical region (Fig. 9d), i.e. due to the morphological influence of microstructure on the rate of loss. This proposed mechanism is consistent with the observation that distinct conditions exhibit endocortical bone loss, whether mechanical loading is disrupted or not.

4.2 Model formulation of morphological feedback

In the cell population model of Pivonka et al. (2013), the morphological influence of the tissue microstructure was included through the specific surface of the tissue (Martin 1984; Lerebours et al. 2015) normalised by its initial value. This normalisation allowed to maintain the same cell behaviour in both cortical and trabecular bones. However, it leads to a turnover rate that is initially independent of bone volume fraction, and so the same in cortical and trabecular bone. The morphological feedback proposed in the present model differs by (i) avoiding a dependence on an initial reference state (i.e. absence of normalisation to allow microstructure-dependent turnover rates) and (ii) by referring to turnover rate (a dynamic biological quantity) instead of specific surface (a morphological characterisation of the microstructure).

While specific surface can be estimated directly from high-resolution scans of bone tissues (Chappard et al. 2005; Squire et al. 2008; Lerebours et al. 2015), quantitative links between \(\text {S}_{\text {V}}\) and cell numbers remain unclear (Martin 1972; Parfitt 1983; Pivonka et al. 2013). The direct reference to turnover rate, in the present model, makes the model more accurate, due to the straightforward link between turnover rate and cell populations (see Eq. (13)). Unfortunately, turnover rate is rarely measured experimentally by cell counts or volumes of bone resorbed and re-formed. It is most commonly characterised by measurements of serum concentrations of bone resorption and/or formation markers (Szulc et al. 2006; Burghardt et al. 2010; Malluche et al. 2012), which are difficult to relate quantitatively to cell numbers or bone volume at a particular bone site. While the phenomenological relationship \(\overline{\chi _\text {BV}}({f_\text {bm}})\) that we assumed between turnover rate and microstructure remains to be studied quantitatively, such a relationship has been suggested in several studies (Felsenberg and Boonen 2005; Burghardt et al. 2010; Malluche et al. 2012).

4.3 Nature of the mechanical stimulus

The nature of the mechanical stimulus sensed by bone cells and transduced into signals prompting resorption or formation has been a matter of discussion for many years. A number of computational studies simulating mechanical adaptation of bone microstructure suggested that the strain energy density could be a good candidate. Ruimerman et al. (2005) tested several mechanical stimuli and concluded the SED gave best results when comparing simulations outcomes with biological parameters such as porosity, trabecular number or adaptability to external loading. However, Levenston and Carter (1998) argued that the drawback of using the SED is that it does not lead to a different response when bone is stimulated in tension or in compression. In the literature, most computational models use the SED because it is a scalar representing both microstructure and mechanical loading (Fyhrie and Carter 1986; Mullender et al. 1994; Van Rietbergen et al. 1999; Van Oers et al. 2008; Scheiner et al. 2013). Quantitative criteria based on experimental observations are still lacking, especially ones testing the tensorial aspects of mechanical loading conditions. For our purpose of studying tissue-scale average changes in bone volume fraction, these tensorial aspects are likely to be secondary. Hence, we have based our mechanical stimulus on the strain energy density (see below for a discussion of the scale). We note here that other mechanical quantities have also been proposed and studied for their magnitude and possible influence onto osteocytes, such as fluid shear stress and fluid pressure in the lacuno-canalicular system (Knothe Tate et al. 1998; Burger et al. 2003; Tan et al. 2007; Bonewald and Johnson 2008; Adachi et al. 2009b).

Mechanical adaptation also relies on the comparison of the current mechanical state with a reference state. The definition of this reference state remains unclear (Frost 1987; Carter and Beaupré 2001). Our choice is to take as mechanical reference state the initial distribution of the strain energy density in the midshaft femur. This choice introduces a memory of the stimulus “normally” experienced in a certain region of the tissue. This memory effect leads to a position-dependent reference state which can be interpreted as taking into account different sensitivities of the mechano-sensing cells depending on where they are located (Skerry et al. 1988; Turner et al. 2002; Robling et al. 2006).

4.4 Neutral axis and site-specific bone adaptation

A common issue in models of mechanical adaptation is the risk to resorb too much bone in regions that are naturally unloaded. Such regions may exist when bending moment is large enough with respect to compressive or tensile forces. In the human midshaft femur, a neutral axis runs approximately along the antero-posterior axis (Lanyon and Rubin 1984; Cordey and Gautier 1999; Thomas et al. 2005; Martelli et al. 2014). To prevent excessive resorption in such regions, some models have considered torsional loads (Van der Meulen et al. 1993; Carpenter and Carter 2008), average values of periodic dynamic loads under which the neutral axis moves (Van der Meulen et al. 1993; Carter and Beaupré 2001), or a residual background of mechanical stimulus modelling muscle twitching and other background mechanical forces (Mittlmeier et al. 1994; Carpenter and Carter 2008).

Such additional features were not introduced explicitly in our model. The strength of the mechanical stimulus around the neutral axis remained weak in our simulation of mechanical disuse (Fig. 8a). This is due to the fact that stimulus sensitivity is prescribed according to the initial state. The neutral axis did not move substantially during the simulation, and so the difference in strain energy density remained small. Resorption around the neutral axis was pronounced in our simulation of osteoporosis (Figs. 8b, c and 9) due to hormone-induced remodelling imbalance. Resorption was limited by the duration of the simulated osteoporosis (40 years) and the calibration of the overall bone loss according to experimental data.

The bending moment exerted onto the femur at the midshaft creates a strong asymmetry in the local mechanical state. Over time, this asymmetry leads to a cortical wall thickness which differs between the y-axis (antero-posterior axis) and the z-axis (lateral–medial axis), as seen in Fig. 9a. Asymmetries in cortical wall thickness and bone volume fraction, in osteoporotic patients, are commonly observed (Fig. 8d) (Feik et al. 2000; Thomas et al. 2005; Zebaze et al. 2010).

4.5 Microscopic versus tissue-scale mechanical regulation

Mechanical deformations of bone matrix can be sensed by osteocytes at the microscopic, cellular scale by deformation of the cell body, transmitted either through direct contact with the matrix or through changes in fluid flow or hydrostatic pressure (Weinbaum et al. 1994; Knothe Tate 2003; Adachi et al. 2009b, a; Bonewald 2011). However, osteocytes are highly connected to one another and to other cells in the vascular phase by an extensive network (Marotti 2000; Kerschnitzki et al. 2013; Buenzli and Sims 2015). While signal transmission mechanisms in this network remain to be determined, it is possible that the network integrates deformations of both the matrix and vascular phases before transducing them into a biochemical response, enabling a mechanical sensitivity of the network to tissue-average stresses and strains.

The uncertainty of the scale at which mechanical stimulus is sensed in bone has motivated many computational studies to estimate stress concentration effects in bone microstructures (Hipp et al. 1990; Kasiri and Taylor 2008; Gitman et al. 2010). However, few studies have explored the changes that occur during simulated bone loss when using microscopic or tissue-scale mechanical stimulus.

Our simulations of osteoporosis show that most of the difference between the mechanical stimulus at the microscopic and tissue scales occurs near the endosteum and neutral axis (Fig. 9b, c). Changes in bone volume fraction were similar in both simulations. Stress concentration effects captured in the microscopic mechanical stimulus (but not in the tissue-scale mechanical stimulus) resulted in maintaining a region of low porosity (\({f_\text {bm}}\simeq 0.1\)) near the medullary cavity and in widening the transition between endocortical and intracortical bone volume fractions (Fig. 10).

Osteoporotic human femur midshafts exhibit a wide range of variability, reflecting the multiple factors influencing bone loss (Feik et al. 1997, 2000; Thomas et al. 2005; Zebaze et al. 2010). The expansion of the medullary cavity and thinning of the cortical wall are commonly reported, but other changes in midshaft tissue microstructures have been studied less systematically. Depending on the subject and their specific condition, the transition between porous endocortical bone and dense intracortical bone may be sharp or wide, and highly porous microstructures near the endosteum may be found or not (Feik et al. 1997, Figure 6).

Our model possesses several limitations which prevent at this stage to draw definite conclusions about the mechanical regulation of the tissue. The mechanical state is calculated only based on bone volume fraction. Other microstructural parameters such as the connectivity of the microstructure are not accounted for. Loss of connectivity is observed in osteoporotic trabecular bone (Parfitt et al. 1987; Mosekilde 1990; Raisz and Seeman 2001), which could lead to mechanical disuse and so increase in resorption. Periosteal apposition is often reported and believed to result from a compensation of endocortical bone loss in osteoporotic patients (Szulc et al. 2006; Russo et al. 2006; Jepsen and Andarawis-Puri 2012). Our simulations assumed the periosteal surface to be fixed, which could limit the expansion rate of the medullary cavity. Finally, our simulation of osteoporosis assumed a constant level of physical activity. A reduction in physical activity with age could further limit the preservation of bone matrix by mechanical feedback.

5 Summary and conclusions

In this paper, a novel spatio-temporal multiscale model of bone remodelling is proposed. This model bridges organ, tissue and cellular scales. It takes into account biochemical, geometrical and biomechanical feedbacks. The model is applied to simulate the evolution of a human femur midshaft scan under mechanical disuse and osteoporosis. It enables us to investigate how these scales and feedbacks interact during bone loss. Our numerical simulations revealed the following findings:

  • Endocortical bone loss during both osteoporosis and mechanical disuse is driven to a large extent by site-specific turnover rates.

  • Mechanical regulation does not influence significantly the expansion rate of the medullary cavity.

  • Mechanical regulation helps preserve cortical bone near the periosteum. It explains site-specific differences in the bone volume fraction distribution in the midshaft cross section during osteoporosis such as increased porosity near the neutral axis, and thicker cortical wall along the medial–lateral axis of the femur midshaft, due to the anisotropy of the mechanical stimulus in the presence of bending moments.

  • The inclusion of stress concentration effects in the mechanical stimulus sensed by the bone cells has a pronounced effect on porosity in the endocortical region.

Our methodology provides a framework for the future development of patient-specific models to predict loss of bone with age or deregulating circumstances.