Biomechanics and Modeling in Mechanobiology

, Volume 15, Issue 1, pp 29–42 | Cite as

A mechanostatistical approach to cortical bone remodelling: an equine model

  • X. Wang
  • C. D. L. Thomas
  • J. G. Clement
  • R. Das
  • H. Davies
  • J. W. Fernandez
Original Paper

Abstract

In this study, the development of a mechanostatistical model of three-dimensional cortical bone remodelling informed with in vivo equine data is presented. The equine model was chosen as it is highly translational to the human condition due to similar Haversian systems, availability of in vivo bone strain and biomarker data, and furthermore, equine models are recommended by the US Federal Drugs Administration for comparative joint research. The model was derived from micro-computed tomography imaged specimens taken from the equine third metacarpal bone, and the Frost-based ‘mechanostat’ was informed from both in vivo strain gauges and biomarkers to estimate bone growth rates. The model also described the well-known ‘cutting cone’ phenomena where Haversian canals tunnel and replace bone. In order to make this model useful in practice, a partial least squares regression (PLSR) surrogate model was derived based on training data from finite element simulations with different loads. The PLSR model was able to predict microstructure and homogenised Young’s modulus with errors less than 2.2 % and \(0.6\,\% \), respectively.

Keywords

Mechanostatistical Cortical bone  Bone remodelling Equine Mechanostat 

1 Introduction

Osteoporotic fracture is a costly and frequently fatal condition caused by injuries suffered by people with long-term bone density loss. Changes in bone microstructure and material properties, required for growth, maintenance and repair of bone, occur through the process of bone remodelling, and chronic bone diseases such as osteoporosis are frequently caused by a maladaptation of this process. Macroscopically, anisotropic materials such as bone usually exhibit some form of directionality in its microstructure to optimally resist loading with a heavy directional bias. It is therefore unsurprising to find that cortical bone, which bears \(\sim \)65 % of compressive load in people above 40 years of age (White and Panjabi 1990), consists of osteons, which are structurally units of cylindrical layers running along the axial direction of bone shafts, stacked alongside each other. Bone microfractures from mechanical stress thus propagate along a combination of the radial and circumferential directions in the bone shaft and are guided and dissipated by the presence of osteons, contributing to the resilience of bone structural integrity and highlighting the importance of microstructure inclusion in studies of bone remodelling (Najafi et al. 2007).

Despite the importance of microstructure, modelling of macroscopic bone behaviour rarely includes details at the microscale as constructing a model which captures features across such a wide spatial scale is computationally prohibitive. A popular approach is to adopt a multiscale design, whereby microscale information is homogenised and passed up in spatial scale to inform a macro-level constitutive law. Current macroscale models are usually based on continuum theory or a simplification of microscale geometries. In particular, some attempts have been made to explain anisotropy through a temporal change in load stimulus, which, through several feedback mechanisms involving parameters such as available remodelling surface area and bone deposition and resorption rates, gradually evolves bone density in anatomically discretised regions of a 2D representation of a single bone shaft based on unique regional strain stimuli (Beaupre et al. 1990a, b). Others have suggested taking principles from continuum damage mechanics (CDM), where a remodelling tensor analogous to CDM’s damage tensor characterises homogenised bone microstructure (Doblare and Garcia 2002). The quantitative measure of ‘damage’ is directly related to bone porosity as a measure of voids in the tissue, which increases when stress decreases, and thus, this method provides a directional component to bone remodelling by aligning the damage tensor with the stress tensor. Hierarchical approaches in 3D have also been investigated for the bone remodelling problem, where the macroscale domain is characterised by homogeneous material properties and passes strain and density information down to the microscale (Coelho et al. 2009). The microscale is characterised by a periodic porous geometry in a small neighbourhood area, with the porosity pattern pre-defined and changing depending on the anatomical location of analysis on a proximal femur. Integration of calculated densities at the local microscale across characterised regions then provides density values across macroscale regions of the same type of porous geometry.

Other studies of bone remodelling which have focused on detailed microstructure typically remain at the microstructural level. Amongst these include the construction of a 2D trabecular bone model, where strain stimuli, calculated from a ‘mechanostat’ formulation, are treated distinctly from damage stimuli, and both are used alternatively or together to drive bone remodelling from a finite element (FE) approach (McNamara and Prendergast 2007). An alternative scheme using a particulate continuum method known as smoothed particle hydrodynamics captured subject-specific geometry on a 2D treatment of synchrotron-based human cortical bone, also incorporating the ‘mechanostat’ to calculate density and Young’s modulus changes based on strain stimulation (Fernandez et al. 2013). Subject-specific parameters such as pores or Haversian canal distribution were shown to have a significant effect on the bone remodelling process.

Although cortical bone is the most significant load-bearing component in the skeleton, there are much fewer studies of its strength and evolution based on realistic microstructures in 3D. However, the behaviour of individual Haversian canal units has been reported. A 1D model has been developed, which includes biochemical regulations at the cellular level, reporting in detail on the rates of cutting and closing cone activities on a single ‘bone multicellular unit’ (BMU) through factors such as the available remodelling surface area (Pivonka et al. 2013). Other studies of BMUs suggest that they are aligned by the prevalent local stress, because osteocytes are attracted to sites of low mechanical stimulus, as the area in front of a ‘cutting cone’ in a principal direction is always a low-stimulus area and attracts more osteoclasts (Burger et al. 2003). Furthermore, it has been shown that BMUs must also be driven by damage removal, as otherwise predicted damage repair (without damage as a stimulus) would not match experimental observations (Martin 2007). The suggestion involves apoptotic osteocytes releasing biochemical signals attracting bone resorbing osteoclast cells. BMU directional alignment and action with both principal strain and damage repair as driving factors for bone remodelling has also been used for explaining the osteon orientation in long bones (Martinez-Reina et al. 2014).

This study is motivated by an interest in the dynamic evolution of cortical bone in 3D based on realistic geometries. To model bone remodelling phenomena such as the temporal change of bone material properties and evolution of ‘cutting cones’, geometric rules were developed and coupled with linear elasticity analysis in a microscale FE model. To retain the relevance of the results in the microscale when macroscale geometries are provided, we also propose the initial steps to link the microscale model to the macroscale domain, using a mechanostatistical approach to minimise computational cost by implementing partial least squares regression (PLSR) to develop surrogate models of observations, such as Young’s modulus and stress distributions. The objectives of this study are to (1) propose a set of geometric rules to guide the 3D evolution of ‘cutting cones’, (2) introduce a ‘memory effect’ for the strain stimulus on the ‘mechanostat’ for bone density evolution to allow bone to gradually adapt to significantly different loads, and (3) develop an efficient micro- to macro-link via a surrogate model approach. Due to the vast quantity of human cortical bone remodelling information reported in the literature and cadaver studies only providing information at time of death, we have adopted an equine model. Animal models are useful in that they provide in vivo remodelling rates and strain boundary conditions. Despite the popularity of murine models, this study adopted an equine model due to the absence of Haversian canals in mice bone. The equine model is highly translational to the human condition due to similar Haversian systems, availability of in vivo bone strain and biomarker data, and furthermore, equine models are recommended by the US Federal Drugs Administration (FDA) for comparative joint research (United States. Department of Health and Human Services 2005).

This study presents (1) the development of a 3D cortical bone algorithm using an equine model; (2) model predictions of Haversian microstructure, stress and homogenised Young’s modulus for two specimens; and (3) the PLSR mechanostatistical trained model with microstructure, stress and modulus predictions.

2 Methods

2.1 Mesh construction

The bone mesh construction pipeline is summarised in Fig. 1. Micro-computed tomography (CT) images of an equine bone biopsy sample of dimensions \(4 \,\hbox {mm} \times 3.5\, \hbox {mm} \times 2\, \hbox {mm}\) were obtained at \(5\, \upmu \hbox {m}\) resolution using an Xradia Micro-XCT imaging device, and the images were used to create a voxel mesh for finite element analysis (see Table 1 for more imaging details). Images were auto-segmented using the snake evolution algorithm in ITK-SNAP (Yushkevich et al. 2006). The resulting mesh was cleaned to remove redundant and stray non-connected elements and decimated to reduce element density in the software package Rapidform XOR2 (www.rapidform.com). Following this, the resulting STL mesh was imported into HyperMesh (www.altairhyperworks.com) where a voxel mesh with cubic hexahedral elements of dimensions \(5\, \upmu \hbox {m} \times 5\, \upmu \hbox {m} \times 5\, \upmu \hbox {m}\) was constructed and exported as an ABAQUS (Simulia, www.3ds.com) compatible input file. Each element was assigned an isotropic material ‘state’ of either cortical bone or Haversian canal properties. The element size was chosen in order to capture the key Haversian features and retain computational efficiency for mechanics simulations.
Table 1

X-ray micro-CT imaging conditions

X-ray energy

60 kV, 10 W

Exposure time for each projection

60 s

Total number of projections

721

Objective magnification

4\(\times \)

Source to sample distance

120 mm

Detector to sample distance

40 mm

Pixel numbers

1024 \(\times \) 1024 \(\times \) 1024

Effective voxel size

\(5~\upmu \!\hbox {m}\)

Fig. 1

Diagram of the mesh building process. (1) Xradia Micro-XCT images collected from a bone biopsy sample of dimensions were used to build a 3D geometry (2), with Haversian features extracted (3) in ITK-SNAP. A voxel mesh with cubic elements was then constructed (4), with sufficient resolution to capture the necessary detail in microstructure. Representative samples of unique Haversian canal arrangements (5) were then chosen for the FE simulation, coupled with bone remodelling simulations

2.2 Mechanostat model

The isotropic voxel mesh was incorporated into an FE model of linear elasticity in ABAQUS. Mesh elements were categorised into either ‘cortical’ or ‘canal’ state elements based on the voxel mesh geometry and the relevant material properties assigned based on these states. The well-known ‘mechanostat’ model (Frost 2000) was used to guide bone growth and loss. As summarised in Fig. 2, the model describes the effect of different magnitudes of strain stimuli on density changes in bone by sorting strains sensed by cortical elements into ‘zones’ with defined strain magnitude bounds. More specifically, the ‘resorption zone’ for strains is found for magnitudes below L1, where bone density decreases; the ‘quiescent zone’ is found for magnitudes between L1 and L2, for which the bone is adapted to the sensed strain and maintained in homoeostatic equilibrium, and thus, no bone density changes occur; and the ‘growth zone’ is found for magnitudes between L2 and L3, where bone growth occurs. Strains above L3 are defined to be caused by loads above the elastic yield limit for cortical bone and thus cause bone damage, and so behave in the same manner as strains in the resorption zone. Mathematically, strain inputs into the mechanostat model produce a strain stimulus, which is used to drive density change via the forward Euler formulation
$$\begin{aligned} \rho _{\mathrm{new}} =\rho _{\mathrm{old}} +c_1 \epsilon _\mathrm{s}{\Delta }t \end{aligned}$$
(1)
where \(\rho _{\mathrm{old}}\) is the current density, \(\rho _{\mathrm{new}}\) is the newly calculated density; \(c_1 \) is a rate constant for density remodelling set to 3.225 kg \(\hbox {m}^{-3} \,\hbox {s}^{-1}\), scaled to three times the rate adapted in Fernandez et al. (2013) as the equine data were observed to be approximately three times that of human remodelling rates; \(\Delta t\) is the time stepping in the current iteration set to 14,400 s or 1 day; and \(\epsilon _\mathrm{s}\) is the strain stimulus determined by the mechanostat. The mechanostat is defined as
$$\begin{aligned} \epsilon _\mathrm{s} =\left\{ {\begin{array}{ll} \epsilon _\mathrm{VMI} -L1, &{} \qquad \epsilon _\mathrm{VMI} <L1 \\ 0, &{} \qquad L1<\epsilon _{\mathrm{VMI}} <L2 \\ \epsilon _\mathrm{VMI} -L2,&{} \qquad L2<\epsilon _\mathrm{VMI} <L3 \\ L3-\epsilon _\mathrm{VMI} ,&{} \qquad \epsilon _\mathrm{VMI} >L3 \\ \end{array} } \right. \end{aligned}$$
(2)
where the mechanostat input parameter \(\epsilon _{\mathrm{VMI}}\) is the von Mises input strain, and the initial values L1, L2 and L3 are 3000, 5000 and \(7500\, \upmu \!\upvarepsilon \), respectively, adapted from equine bone growth rate experiments in trotting, cantering and galloping specimens (Davies 1995, Chapter 8).
Fig. 2

Mechanostat strain curve, showing the rate of change of density against strain

Table 2 shows the raw equine data from the work of Davies (Davies 1995, Chapter 8) that was used to inform the thresholds and remodelling rates in the current mechanostat. Two groups of horses (exercised and control) underwent rest, trot, canter, gallop and rest again for 40 day durations. The data show that bone growth rates during trotting and cantering phases in the exercised group did not exhibit any significant statistical difference to the rates in the control group, while rates during the galloping phase were significantly higher than the control group. Also notably, during the 40-day rest period after the galloping phase, the growth rate was also significantly higher than the control group. Bone remodelling was determined from bone markers on histological sections taken from the third metacarpal bone as shown in Fig. 3, with strain gauges for the exercise regimes placed on the surface of the dorsal cortex. Note that significant bone remodelling only occurred at gallop speeds of \(17\, \hbox {m}\,\hbox {s}^{-1}\) and above, corresponding to \(6000\, \upmu \!\upvarepsilon \) and a growth rate of \(12.7 \,\upmu \hbox {m}\) per day. No significant remodelling was present at cantering speeds as low as \(10\, \hbox {m}\, \hbox {s}^{-1}\) with strains of \(4000\, \upmu \!\upvarepsilon \) and a bone growth rate of \(2.2\, \upmu \hbox {m}\) per day. Due to the small window of strain between \(6000\, \upmu \!\upvarepsilon \) and a damage zone of \(7500\,\upmu \!\upvarepsilon \), we lowered L2 to \(5000\, \upmu \!\upvarepsilon \) in our model.
Fig. 3

Equine experimental strain recording method and biomarker data from Davies (Davies 1995, Chapter 8). (Left) Histological section of equine bone tissue, showing bone growth in woven bone; the markers oxytetracycline and fluorescein complexone were administered alternatively after periods of approximately 30–40 days, leaving distinct yellow–orange and yellow–green (respectively, of the chemical marker administered) bands which indicates the pore boundary at the time. (Right) Schematic of the equine third metacarpal bone; dark grey rectangle shows the location where the histological section was taken

Table 2

Cortical bone growth rates in horse specimens obtained from biostaining (Davies 1995, Chapter 8)

Days

Growth rate per day, \(\upmu \!\hbox {m}\pm \hbox {SD}\)

Significance (control vs. exercised)

Control

Exercised

40

\(1.8 \pm 1.1\)

\(2.6 \pm 2.6\) (Trot)

p = 0.510

40

\(3.2 \pm 4.3\)

\(2.2 \pm 3.3\) (Canter)

p = 0.680

40

\(2.1 \pm 2.9\)

\(12.7 \pm 8.6\) (Gallop)

p = 0.017

40

\(1.2 \pm 0.7\)

\(4.3 \pm 2.5\) (Rest)

p = 0.014

Individuals in the control horse group were left to their natural behaviour, while those in the exercised group underwent activities of increasing physical intensity

The resulting density is used in a fitted power law model of the form
$$\begin{aligned} E=A\rho ^{b} \end{aligned}$$
(3)
where \(E\) is the elastic modulus, initially set to a value calculated from an average-estimated equine compressive elastic modulus of 12,802.5 MPa (Skedros et al. 2006), \(b\) is the exponent parameter set to 1.54 (Keller et al. 1990) and \(A\) is determined from Eq. (3) with an initial density value calculated from \(1.938\, \hbox {g}\,\hbox {cm}^{-3}\) (McCarthy et al. 1990). Note that both initial and calculations from the values given depend on the number of canal elements in relation to the number of cortical elements in the model. Dramatic growth and resorption (to and from cortical and canal states) were restricted to occur only on the cortical elements at the interface between cortical bone and Haversian canals. When an interface element reaches a defined growth limit, all surrounding canal elements change state to cortical bone with an initial elastic modulus equal to the resorption limit. Likewise, when an interface element reaches a defined resorption limit, it is converted into a canal element with an elastic modulus of 2 MPa.

With initial values of \(E\) and \(\rho \) for cortical elements calculated, and 2 MPa for canal elements, the model was adapted for an expected load, which results in a \(4000\,\upmu \!\upvarepsilon \) displacement, or halfway between the initial values of L1 and L2, by running a ‘conditioning’ simulation to obtain unique ‘mechanostats’ for each cortical element, with the intended effect that the expected load causes von Mises strains in cortical bone elements to be in the centre of their individual ‘mechanostat’s’ quiescent zone, allowing for the evaluation of higher and lower load perturbations.

2.3 Fading memory extension

Table 2 shows that bone continues to grow in the rest period immediately after the galloping phase, demonstrating that bone has a memory of previously sensed strains and that the ‘mechanostat’ quiescent zone adapts to these past strains. Motivated by this, a ‘fading memory’ model was proposed, which postulates that bone retains a limited memory of recent strain stimuli, and so that the ‘mechanostat’ input strain, given as the von Mises input strain \(\epsilon _{\mathrm{VMI}}\) in Eq. (2), is derived from weighted values of a past history of ‘sensed’ von Mises strains given by \(\epsilon _{\mathrm{VM}}\). A weighting scheme \(W\) is defined with a memory of \(N\) historical iteration steps, equal to 30 days, based on normalised sampled values from \(N\) equally spaced intervals on an exponential growth function, such that
$$\begin{aligned} \mathop \sum \limits _{i=1}^N W_i =1 \end{aligned}$$
(4)
A strain history \(H\) of \(N\) history values was then initialised with von Mises strains resulting from the conditioned load. After each simulation iteration, equal to precisely 1 day, a new strain history \(H_t \) is updated from the old strain history \(H_{t-1} \) by
$$\begin{aligned} H_{i,t} =\left\{ {\begin{array}{ll} H_{i+1,t-1} , &{} \qquad i<N \\ \epsilon _{\mathrm{VM}} ,&{}\qquad i=N \\ \end{array} } \right. \end{aligned}$$
(5)
with \(\epsilon _{\mathrm{VM}}\) as the currently sensed von Mises strain and \(H_{i,t-1}\) as the past von Mises strains. Thus, \(\epsilon _{\mathrm{VMI}}\) is equal to the sum of the weighted current strain history:
$$\begin{aligned} \epsilon _{\mathrm{VMI},t} =\mathop \sum \limits _{i=1}^{N} W_{i} \times \left( {H_{i,t} } \right) \end{aligned}$$
(6)
The new mechanostat zone limits L1, L2 and L3 are updated as
$$\begin{aligned} \hbox {L}_t =\hbox {L}_{t-1} +\epsilon _{\mathrm{VMI},t} -\epsilon _{\mathrm{VMI},t-1} \end{aligned}$$
(7)

2.4 Cutting cone evolution

The evolution of Haversian canals characterised by osteoclast activity at the head and osteoblast activity at the tail, also known as the ‘cutting cone’, is based on a geometric analysis of canal elements. Binary TIFF image stacks of the model were generated with binary colour values assigned for each element, represented as a pixel, for cortical and canal elements, respectively. The image stack was then passed into Fiji (fiji.sc/Fiji), generating an image skeleton with the Skeletonize3D plugin (fiji.sc/Skeletonize3D) and analysing the shape of the skeleton (fiji.sc/AnalyzeSkeleton). The skeleton shape analysis allowed the determination of Haversian canal ends, their current evolution directions, and whether they are determined to be ‘cutting regions’ or ‘closing regions’. As shown in Fig. 4, ‘cutting regions’ are defined as the areas where the angle between the evolution direction and the direction from the canal end to the closest point of load application in the model is less than \(90^{\circ }\), and ‘closing regions’ are assigned to all other canal ends. During each simulation iteration, closing regions close with a rate of \(1.9 \times 10^{-6} \hbox {mm}\, \hbox {s}^{-1}\) (Lee 1964) and cutting regions extend at a rate of \(2.083 \times 10^{-6} \hbox {mm}\, \hbox {s}^{-1}\) (Pivonka et al. 2013), with the extension width influenced by the average width of the Haversian canal, which contains this canal end. The canal extending direction is calculated through a summation of the eigenvectors, weighted by their eigenvalues, of the Cauchy’s strain tensors of all surrounding cortical bone elements of a cutting region. Specifically, given \(S\) cortical interface bone elements surrounding a cutting region, the growth direction \({\mathbf {G}}\) of the cutting region is given by the vector
$$\begin{aligned} {\mathbf {G}}=\mathop \sum \limits _{s=1}^{S} \mathop \sum \limits _{j=1}^{3} e_{s,j} {\mathbf {v}}_{s,j} \end{aligned}$$
(8)
where \(e_s\) and \(v_s\) are found by eigenvalue decomposition of the \(s\hbox {th}\) Cauchy’s strain tensor
$$\begin{aligned} {\mathbf {C}}_s =\left( {{\begin{array}{lll} {\epsilon _{11} }&{} {\epsilon _{12} }&{} {\epsilon _{13} } \\ {\epsilon _{21} }&{} {\epsilon _{22} }&{} {\epsilon _{23} } \\ {\epsilon _{31} }&{} {\epsilon _{32} }&{} {\epsilon _{33} } \\ \end{array} }} \right) _{s} \end{aligned}$$
(9)
In this manner, the ‘cutting cone’ is directed towards zones of highest strain where bone is most likely being replaced, which typically aligns in the longitudinal direction in bone.
Fig. 4

Assignment of canal evolution behaviour to Haversian canals (red) based on their geometry. The loads (yellow arrows) are applied to nodes on the surface of the model (green line). (Left and Right) Cutting behaviour is assigned when the angle (yellow sectors) between the blue and black arrows is less than or equal to \(90^{\circ }\). The blue arrow is defined as the centre of the canal end region (circular red region) to the tip of the canal, which is found through geometric analysis of the canal end. The black arrow is defined as the closest point of load application from the centre of the canal end region. (Centre) Closing behaviour is assigned when the angle (yellow sectors) is greater than \(90^{\circ }\). Mathematically, representing the direction of the blue arrow as a vector \(x\) defines the closing direction as \(-x\), represented by the white arrow

2.5 PLSR surrogate model construction

Partial least squares regression (PLSR) is a method of relating \(n\) sets of information from one matrix \({\mathbf {X}}\) (the predictor matrix) to another matrix \({\mathbf {Y}}\) (the response matrix), where \({\mathbf {X}}\) contains \(m\) orthogonal predictor parameters and is of size \(n\times m\) and \({\mathbf {Y}}\) contains \(p\) responses and is of size \(n\times p\) (Wold et al. 2001). Mathematically, both \({\mathbf {X}}\) and \({\mathbf {Y}}\) are decomposed into a row eigenvector of independent variables pre-multiplied by a column vector of scores or projections on the eigenvector, in the form
$$\begin{aligned} {\mathbf {X}}= & {} {\mathbf {w}}_{X} {\mathbf {p}}^{\mathrm{T}}+{\mathbf {E}}_X , \nonumber \\ {\mathbf {Y}}= & {} {\mathbf {w}}_{Y} {\mathbf {q}}^{\mathrm{T}}+{\mathbf {E}}_Y , \end{aligned}$$
(10)
where \({\mathbf {w}}\) is the scores vector, \({\mathbf {p}}\) and \({\mathbf {q}}\) are the eigenvectors, and \({\mathbf {E}}\) is the residual term from the decomposition. PLSR aims to maximise covariance between \({\mathbf {w}}_X\) and \({\mathbf {w}}_Y\) such that
$$\begin{aligned} {\mathbf {w}}_{Y} ={\mathbf {r}}^{\circ } {\mathbf {w}}_X , \end{aligned}$$
(11)
where \(^\circ \) is the binary entry-wise product operator and \(r^{\mathrm{T}}=[r_1 ,\ldots ,r_n ]\) is a vector of factors calculated to maximise the covariance. Because \({\mathbf {w}}_Y \) can be used to predict \({\mathbf {Y}}\) and \({\mathbf {w}}_X \) can be used to predict \({\mathbf {X}}\), it follows that \({\mathbf {X}}\) can be used to predict \({\mathbf {Y}}\).

PLSR models are developed as the \({\mathbf {Y}}\) responses are usually too expensive to determine experimentally for every single combination of predictors that could exist in \({\mathbf {X}}\). In this study, the predictors are different combinations of normal and shear loads applied on the model while the responses are elastic modulus or von Mises stress predictions on each element, with the expectation that load information passed from the macroscale (whole bone model) to the microscale (Haversian level model) could be used as predictors, which allows rapid estimation of Haversian shape and homogenised Young’s modulus.

2.6 Bone remodelling pipeline

Figure 5 shows the structure of the ABAQUS Python script used to run the bone remodelling simulation. In a loop where the user specifies the number of simulated days for the model to run, the model is submitted to ABAQUS Finite Element Analysis (FEA), and the results obtained used to drive the evolution of the ‘mechanostat’, analysed for geometry and producing any changes in cutting and closing cones. The loop ends when the maximum number of simulated days is reached.
Fig. 5

Structure of bone remodelling Python scripts complementary to ABAQUS mechanical FEA (the ‘Load bone’ step)

3 Results

In all cases unless otherwise specified, the conditioned load magnitude applied was the same, calculated by applying a \(0.4\,\%\) compressive strain to the top surface as shown in Fig. 6 and fixing the bottom surface in both translation and rotation, obtaining the resulting reaction force for each node on the fixed surface of the model, averaging these reaction forces and applying the averaged force on the opposite face to the fixed surface. The high and low load magnitudes are \(156.25\,\%\) and \(37.5\,\%\) of this averaged force, respectively, and unless otherwise stated are applied perpendicularly and in the negative \(z\) direction. These high and low loads were chosen to produce strains which fall into the growth and resorption zones of the mechanostat, respectively.
Fig. 6

Loading schematic for microscale bone sample. Yellow arrows (top) represent applied load locations, while blue (bottom) represents applied boundary conditions, restricting nodal translation and rotation. Loads and boundary conditions were applied on a per-node basis across the entire surfaces. Adapted from ABAQUS (Simulia, www.3ds.com)

3.1 Canal evolution behaviour

Prediction of Haversian bone evolution is shown in Fig. 7, representing up to 10 days of remodelling. A cutting cone of a Haversian canal (solid blue circle) is directed towards different paths based on the applied surface traction vector. All loads caused an initial growth towards the positive \(z\) direction; however, a pure shear load (left column) was shown to halt the canal evolution progress by day 4. Amongst the other loads, the tensile load (middle column) caused the ‘cutting cone’ to grow towards the positive \(z\) direction, while the purely normal compressive load (right column) influenced the growing branch of the canal to evolve towards a neighbouring branch by day 10.
Fig. 7

Examples of canal evolution directions as a result of different traction force vectors (purple) applied along the surface (green) in the first row; red pie sectors indicate the angle from the \(x\)-axis, while yellowpie sectors indicate the angle from the \(xy\) plane. The same bone model was loaded with three different traction vectors. The region in the solid blue circle shows a cutting cone in action. (Left column) Application of a pure shear force, \(45^{\circ }\) from the \(x\)-axis; (middle column) application of a tensile load \(45^{\circ }\) from the \(x\)-axis and \(67.5^{\circ }\) from the \(xy\) plane; (right column) application of a pure normal compressive force to the \(xy\) plane

3.2 Effect of load magnitude on remodelling rate

Haversian remodelling in response to an increased load of 56.25 % and decreased load of 62.5 %, with respect to the quiescent load, is shown in Fig. 8. Specifically, increasing and decreasing the stimulus loading gave rise to increased and decreased rates of Young’s modulus growth, respectively. The Young’s modulus values were calculated by homogenising (averaging) the Young’s Modulus of individual elements across the entire model. The modulus is constantly affected by an increasing Young’s modulus as a background effect from the ‘cutting cones’. The three load cases showed different initial rates of Young’s Modulus change, causing the initial curves to diverge. After five simulated days, however, any divergence becomes insignificant.
Fig. 8

Evolution of the average Young’s Modulus over a simulated period of 11 days from different load magnitudes applied on the same bone model in the \(-z\) direction. The anomaly seen in the conditioned load at \(t=10\) was caused by the mechanostat triggering a small remodelling event which caused a cutting cone to behave slightly differently to the others, accumulating canal elements earlier than the other load conditions and thereby causing the homogenised Young’s modulus to fall

3.3 Sensitivity analysis

Model sensitivity was assessed by perturbing applied load angle, load magnitude, density rate and the ‘mechanostat’ boundaries by \(\pm 10\,\% \) and obtaining results at the 11th simulated day to examine the sensitivity of these parameters, as summarised in Table 3. Angle was perturbed from the reference pure normal load in the \(-z\) direction by \(\pm 9^{\circ }\) with a conditioned load magnitude and measured according to the reference, while sensitivity tests of the other parameters were done in both the higher (growth) and lower (resorption) load zones and used as reference standards. Two variables were tested for sensitivity: the homogenised Young’s modulus \(E\), and a parameter termed ‘shape’, a binary variable which records the material state of each element in the model (canal or cortical bone). While the model was stable to most variable changes, the loading angle was the most sensitive input variable, where a \(10\,\% \) perturbation caused \(0.4\,\% \) variation in \(E\) values and \(2.44\,\% \) variation in shape values. The model was less sensitive to the other parameters; in order of decreasing sensitivity, \(E\) was subsequently most sensitive to changes in magnitude \((0.07\,\%)\), density rate \((0.05\,\%)\); then, the mechanostat boundary limits \((0.04\,\%)\), while shape was most sensitive to changes in density rate \((0.13\,\%)\), ‘mechanostat’ boundary limits \((0.10\,\%)\), and magnitude \((0.06\,\%)\).
Table 3

Table of percentage deviations presented by a model sensitivity analysis, through the perturbation of four key parameters: magnitude, density rate, the mechanostat limits and the angle of applied load

Resorption zone and growth zone refer to the zones immediately adjacent to the quiescent region (see Fig. 2) where resorption and growth occur, respectively. The Young’s modulus \(E\) given is a homogenised value calculated as an average from the whole model. All loads were applied in the normal direction \((-z)\) except in the angle sensitivity analysis

3.4 Fading memory

The effect of the fading memory function in the model is illustrated in Fig. 9. The model was initially conditioned to a 0.4 % strain and stimulated with a high load for 12 simulated days and then perturbed with a low load for the remainder of the simulation. Young’s modulus evolution diverged from the memory-informed model, which showed a comparatively lesser increase. After 12 days, the low load perturbation took effect, and the model without memory showed an immediate and more unrealistic decrease in the rise of Young’s modulus, while the memory-informed model showed a continued increase.
Fig. 9

Effect of fading memory on model perturbations. Perturbation from high to low load occurred at \(t=12\) days

3.5 PLSR predictions

The PLSR surrogate model was trained on two separate specimens with 21 remodelling mechanics simulations for each specimen, with normal and shear load ranges of approximately 0–1.9 and 0–0.35 mN, respectively. Separate PLSR models were built for Young’s modulus and stress, where element-based Young’s modulus predictions were used as additional information to inform the homogenised Young’s modulus of the entire micro-specimen. Additionally, Young’s modulus thresholds were used to identify mismatched elements in the shape and stress data for error statistics. During model training, 1 dataset of the 21 was left out of the training population for prediction purposes, and this was repeated for each of the 21 datasets. A shape mismatch example is shown in Fig. 10, with the circled regions on the PLSR-predicted image (red) highlighting the shape prediction error when contrasted with the ideal calculated shape (blue).
Fig. 10

Ideal calculated versus PLSR-predicted shape. (Left in blue) Example calculated shape from bone remodelling algorithm for a loading scenario. (Right in red) Predicted shape using PLSR under the same loading scenario. Circled areas emphasise where the shape was predicted differently

Table 4 summarises the loading spectrum, homogenised Young’s modulus and shape errors for the PLSR models on two bone specimens, where ‘shape mismatches’ refers to the percentage of non-matching material states. The average error measured across both specimens is \(0.16\,\%\) and \(0.30\,\% \) for the homogenised Young’s modulus and shape mismatches, respectively. The average error for stress was considerably higher, with a significant proportion of elements having stress error predictions of greater than \(20\,\% \). Across the two specimens, \(76.1\,\% \) of elements exhibited errors in stress less than \(20\,\% \).
Table 4

Table of percentage errors between the calculated model and the PLSR-predicted model for two bone specimens

Magnitude

Angle from \(xy\) plane

\(0^{\circ }\)

\(15^{\circ }\)

\(30^{\circ }\)

\(45^{\circ }\)

\(60^{\circ }\)

\(75^{\circ }\)

\(90^{\circ }\)

Bone specimen 1

\(0.375F_\mathrm{C} \)

   \(E_{\mathrm{Homogenised}} \)

\(0.02\)

\(0.06 \)

\({<}0.01 \)

\({<}0.01\)

\(0.03 \)

\(0.06\)

\(0.25 \)

   Shape mismatches

\(0.38 \)

\(0.04 \)

0.04

0.18

\(0.15 \)

\(0.10 \)

\(0.36 \)

\(F_\mathrm{C} \)

   \(E_{\mathrm{Homogenised}} \)

\(0.50 \)

\(0.12 \)

\(0.13 \)

\(0.21\)

\(0.15 \)

\(0.12 \)

\(0.30 \)

   Shape mismatches

\(0.66 \)

\(0.29 \)

\(0.17 \)

\(0.07 \)

\(0.07 \)

\(0.10 \)

\(2.16 \)

\(1.5625F_\mathrm{C} \)

   \(E_{\mathrm{Homogenised}} \)

\(0.15 \)

\(0.37 \)

\(0.24 \)

\(0.09\)

\(0.05 \)

\(0.42 \)

\(0.16 \)

   Shape mismatches

\(0.38 \)

\(0.35 \)

\(0.22 \)

\(0.09 \)

\(0.07 \)

\(0.44 \)

\(0.40 \)

Bone specimen 2

\(0.375F_\mathrm{C}\)

   \(E_{\mathrm{Homogenised}} \)

\(0.08 \)

\(0.57 \)

\(0.19 \)

\(0.05 \)

\(0.04 \)

\(0.38 \)

\(0.21 \)

   Shape mismatches

\(0.23 \)

\(0.95 \)

\(0.37 \)

\(0.25 \)

\(0.22 \)

\(0.39 \)

\(0.14 \)

\(F_\mathrm{C} \)

   \(E_{\mathrm{Homogenised}} \)

\(0.13 \)

\(0.22 \)

\(0.08 \)

\(0.16 \)

\(0.25 \)

\(0.11 \)

\(0.06 \)

   Shape mismatches

\(0.12 \)

\(0.41 \)

\(0.28 \)

\(0.24 \)

\(0.17 \)

\(0.10 \)

\(0.14 \)

\(1.5625F_\mathrm{C} \)

   \(E_{\mathrm{Homogenised}} \)

\(0.16 \)

\(0.04 \)

\(0.36 \)

\(0.02 \)

\(0.16 \)

\(0.03 \)

\(0.07 \)

   Shape mismatches

\(0.24 \)

\(0.40 \)

\(0.33 \)

\(0.21 \)

\(0.20 \)

\(0.28 \)

\(0.25 \)

Reference for error calculation is the conditioned load applied compressively normal to the surface opposite to the fixed surface. \(F_{C}\) conditioned load, \(E_{{Homogenised}}\) homogenised Young’s modulus percentage difference

4 Discussion

The presented cortical bone model captures evolving Haversian structures in 3D at the microscale level. Equine data were used due to the similarity of its Haversian canal structures as compared to human bone microstructure, and the availability of in vivo strains and remodelling rates (from biomarkers) adapted from equine gait analysis. Furthermore, the use of equine models is recommended by the US FDA and is useful for translational studies. The model was informed by Frost’s ‘mechanostat’ with a fading memory component to allow modification of the set point, where bone adapts to a newly applied load. Young’s modulus was shown to increase consistently according to the applied loads. A partial least squares regression (PLSR) surrogate model was trained on two specimens using 21 mechanics remodelling simulations and predicted shape and homogenised Young’s modulus with errors less than \(0.4\,\%\) and \(0.2\,\% \), respectively. The error in predicted stress was much higher, with \(23.9\,\% \) of elements showing errors greater than \(20\,\%\).

There are a number of limitations that should be considered when interpreting the results of this work. Firstly, the model did not include a specific osteocyte distribution to act as a mechanosensitive sensor. However, assuming a healthy distribution of osteocytes (no apoptosis), all elements in the model could sense strain, which would not likely to influence model predictions but does presently limit our ability to evaluate osteocyte-related diseases. Secondly, we have assumed that a linear statistical model in the form of a linear PLSR model is sufficient for capturing nonlinear processes. While the predicted errors in shape and Young’s modulus in this study were less than \(2.2\,\% \), a nonlinear PLSR model may be more suitable for stress predictions and needs to be evaluated in future versions of this work. Thirdly, we have presented only two representative pieces of cortical bone and to make this useful in a full multiscale model, additional specimens of varying porosity would need to be added and used for model training. Lastly, although we have attempted to evaluate each section of this study as thoroughly as possible through comparison with the literature, expected behaviours and provided in vivo data, it is difficult to validate bone growth quantities directly as the history of loading is difficult to quantify. Validation may be possible through surrogate measures such as through quantitative density measurements from CT scans.

As observed in Fig. 7, the predicted Haversian canal pathways are consistent with images showing fine filamentary structures in 3D micro-CT data (see Fig. 11) from both equine and human cortical bone. In the model, application of a pure shear load (Fig. 7, left column) caused the cone to stop cutting; an angled tensile load (Fig. 7, middle column) caused growth resembling the anatomical structures found in sample Haversian canals. Interestingly, application of a pure compressive normal load caused one branch of a bifurcated canal to merge into a neighbouring canal, resembling the formation of ‘super-osteons’ (Bell et al. 2001) as described by Bell et al. (2001) or exhibiting Volkmann’s canal-type behaviour. Haversian ‘cutting cones’ were guided by weighted strain eigenvectors sensed at the tip of the ‘cutting cone’ directing the canal path towards the most loaded or damaged zones. This is consistent with the idea that Haversian canals are always remodelling to remove damaged bone and tend to align with the longitudinal direction of loading. The slight deviations in Haversian geometry predicted by our model are consistent with the filamentary paths observed in micro-CT images of bone (see Fig. 11) and may reflect the slightly altered loading patterns over time that horses and humans experience as part of daily loading stimulus.
Fig. 11

a 3D micro-CT data of equine cortical bone. b Complex bifurcation and cross-linking patterns observed in the filamentary Haversian canal structures, reconstructed from the pores in (a)

Homogenised Young’s modulus responded in a manner consistent with the ‘mechanostat’ as observed in Fig. 8. Load magnitudes were chosen such that the resulting strains fell into the resorption and growth zones of the ‘mechanostat’, respectively, and the loads were applied in the \(-z\) direction (see Fig. 7 for axes alignment with respect to the model). In all three load cases, Young’s modulus increases due to the independent closing cone phenomenon which dominating the Young’s modulus changes; at each simulated day, this behaviour causes a certain number of canal elements to turn into cortical elements, increasing the average Young’s modulus. Initially, increased load resulted in a higher rate of Young’s modulus increase, while decreased load reduced the Young’s modulus rate of increase, as expected from Equation (2) and shown on the figure as the initial curve divergence between the different loadings. However, the fading memory model increasingly diminishes the effect of the high and low loads, causing the mechanostat to adapt to these loads by six days and preventing further significant divergence of the models.

Qualitative bone remodelling behaviour is consistent with other studies, which have reported bone material adaptation (Beaupre et al. 1990a; Fernandez et al. 2013), whereby more bone atrophy or a lower bone density occur when lower loads are applied compared to higher loads. However, the model does not yet show an absolute decrease in bone strength when low loads are applied due to the dominant effect of the closing cones and the relative lack of cutting cones in any particular sampled section of cortical bone, which is a current limitation of the present work. To offset the heavy influence of closing cones on the homogenised Young’s modulus on the model, new canals must be seeded occasionally for a more realistic result for both anatomical structure and mechanical strength.

The memory effect was shown to predict smoother changes in Young’s modulus and continued bone growth even after loading stimulus was removed, consistent with the equine data as presented in Table 2. As seen in Fig. 9, during the first 12 days of simulation, with the high load representing the ‘galloping phase’, while initially both models exhibited a similar Young’s modulus growth, the model without the memory effect shows a greater rise in Young’s modulus than the memory-informed model after four days. This is due to the diminishing effect of the higher load on the memory-informed model, which eventually adapts to the load change. When the lower load, representing the ‘resting phase’, takes effect after 12 days, the Young’s modulus immediately stops increasing in the model without memory. In contrast, in the memory-informed model, this resting phase is now a combination of the current low stimulus and the recent high stimuli so Young’s modulus continues to increase. This result is consistent with the data observed in Table 2, which shows significantly that Young’s modulus continues to increase long after the galloping-phase regime is complete and well into the resting phase.

Table 3 shows the results of sensitivity analysis, identifying the applied load angle as the most sensitive parameter with the other parameters (load magnitude, density change rate and mechanostat limits) less sensitive by an order of magnitude. Despite being the most sensitive parameter, perturbing the applied load angle by \(10\,\% \) still only resulted in only a \(2.44\,\% \) change in shape and \(0.4\,\% \) change in homogenised Young’s modulus, indicating that angle is still a relatively insensitive parameter and that the model is very resilient to errors in parameter estimation in general. It also shows that the anatomical microstructure is the major influence on remodelling predictions. Apart from the loading angle, remodelling parameters are therefore not required to be extremely accurate for simulations of bone remodelling in the current model. However, as seen by the very divergent microstructural changes due to applied load orientation in Fig. 7, the loading angle can be identified as a parameter of key importance for microstructural changes.

The PLSR model was shown to be a good predictor of shape and homogenised Young’s modulus, with maximum errors of 2.2 % and 0.6 %, respectively (see Table 4), suggesting its usefulness as a prediction tool for Haversian geometries and homogenised material properties for macro-level whole bone models. Figure 9 shows the areas where the PLSR-predicted shape is different to the ideal calculated shape, and it is apparent that the highest errors in shape prediction arise in regions where cutting cone activity occurs, as this remodelling behaviour is significantly more dynamic than the bone density changes from the ‘mechanostat’ aspect of the model. However, PLSR is known to be less accurate at predicting spatial fields, which is confirmed by present predictions of von Mises stress in this study. Specifically, 23.9 % of elements showed above 20 % error in stress prediction, which indicates that nonlinear regression methods may be more suitable in predicting these values. These large errors were primarily a result of the PLSR model predicting the wrong material state (cortical bone elements for canal elements and vice-versa) for elements around the cortical–canal interface.

The presented equine model was informed by bone growth rates and bone strain data from in vivo measurements. The observed threshold nature that linked remodelling rates to distinct speeds and bone strains supports the use of the classical bone remodelling ‘mechanostat’. This is in contrast to other popular models, such as murine models, which lack Haversian canals and present a challenge when controlling their loading environment, and human models, which only present limited end of life data where the history of loading is unknown. The prediction ability of the presented surrogate model can inform whole bone stiffness (Young’s modulus) changes rapidly and accurately without expensive iterative computation. Moreover, the influence of disease and drug effects at the micro-level can also be incorporated and additional synthetic data generated for scenarios not experimented on. Translation to the human may be achieved by scaling equine remodelling rates and strain thresholds to human physiological ranges, as these parameters are generally higher for horses. The results of this study are currently being developed for inclusion in the open-source IUPS Physiome project repository (Hunter et al. 2005) for evaluation and sharing by the scientific community.

Notes

Acknowledgments

This work was funded by the Ministry of Business, Innovation and Employment (MBIE) of New Zealand. The authors of this work acknowledge Benedicta Arhatari of the Physics Department of LaTrobe University for the micro-CT work.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • X. Wang
    • 1
  • C. D. L. Thomas
    • 2
  • J. G. Clement
    • 2
  • R. Das
    • 3
  • H. Davies
    • 4
  • J. W. Fernandez
    • 1
    • 5
  1. 1.Auckland Bioengineering InstituteThe University of AucklandAucklandNew Zealand
  2. 2.Melbourne Dental SchoolUniversity of MelbourneParkvilleAustralia
  3. 3.Department of Mechanical EngineeringThe University of AucklandAucklandNew Zealand
  4. 4.Faculty of Veterinary and Agricultural ScienceUniversity of MelbourneParkvilleAustralia
  5. 5.Department of Engineering ScienceThe University of AucklandAucklandNew Zealand

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