 Research article
 Open access
 Published:
A computational twoscale approach to cancellous bone remodelling
Advanced Modeling and Simulation in Engineering Sciences volume 11, Article number: 13 (2024)
Abstract
We propose a novel twoscale (mesomacroscale) approach to computationally capture cancellous bone remodelling allowing for efficient and effective numerical implementation. Therein, the macroscale is governed by the wellestablished kinematics and kinetics of onescale continuum bone remodelling. However, the constitutive behaviour is not postulated phenomenologically at the macroscale, but rather follows from the mesoscale. There, for the sake of computational efficiency, the trabecular architecture is idealised as a truss network with the crosssectional area of the trabeculae adapting to mechanical loading. Then, the meso and the macroscale are coupled through up and downscaling. Computational results on benchmark problems from biomechanics demonstrate that the proposed twoscale approach is effective from a modelling perspective and efficient from a computational perspective. In particular, it automatically captures anisotropy resulting from the irregular trabecular architecture at the mesoscale, and, most importantly, enables the direct investigation of different trabecular structures at the mesoscale, thereby serving as a virtual “magnifiying glass”. As an outlook, the proposed twoscale approach to cancellous bone remodelling provides an excellent launch pad for further extension, e.g., by considering more complex trabecular architectures and/or through inclusion of microscale bone cellular activities.
Motivation
It is widely accepted that bone is a living hierarchical material that adapts its internal structure, among others, to mechanical stimuli as evidenced by invivo measurements, see, e.g., [5, 38, 41]. For this remodelling process, osteoclasts and osteoblasts work in a coupled process of bone resorption and subsequent bone formation to repair minor fatigue damages in the tissue and adapt the bone, e.g., to habitual mechanical loading [1, 23, 33,34,35]. Understanding and predicting the remodelling process in human bone is critical, for example to determine fracture risk and/or optimising implant and scaffold integration, see, e.g., [50], and treatment methods. This is especially important in case of degenerative diseases such as osteoporosis, where an imbalance between bone formation and resorption leads to bone loss that weakens the bone.
Meslier and Shefelbine recently reviewed the past 50 years of developments of finite element models for bone adaptation [27]. The majority of early bone adaptation models have been formulated as onescale macroscopic models based on the phenomenological theory of adaptive elasticity [8]. In these models mechanoadaptation responses are based on either strain [8] or energy storage density [15,16,17, 47]. In order to overcome the phenomenological nature of the mechanoadaptive evolution law proposed in these early models, efforts have been made to formulate bone adaptation based on the theory of open system thermodynamics [9, 21, 22]. The latter framework allows to derive the evolution equation for bone mass density (and material orientation) based on thermodynamic system constraints. Several authors applied this theory to simulations of bone remodelling [19, 24, 30, 31, 39, 45].
However, none of these onescale approaches explicitly accounts for the highly irregular trabecular architecture of cancellous bone at the mesoscale. In strong contrast, pixel or voxelbased finite element models in 2D and 3D, respectively, enable for example analysing the purely mechanical behaviour of small cancellous bone specimens [18, 29, 44]. Especially when generated using \(\mu \)CT or HRpQCT measurements, these models prove to be particularly true to the real geometry [4, 10, 49]. Moreover, the \(\mu \)CTgenerated voxelbased finite element models even allow the detailed simulation of the remodelling of trabecular bone samples, see, e.g., [46]. Nevertheless, approaches explicitly resolving the mesoscale are computationally prohibitive for wholebone models, and thus only suitable for studying small bone samples. To aim for computational efficiency, some contributions suggest the use of simplified geometries, but either do not consider the remodelling process [3, 20], or only account for a single specific trabecular structure [13]. In other studies, the trabecular architectures used are not variable and strongly depend on the macroscale finite element discretisation or even a regular pattern [26, 32].
Attempts to incorporate the multiscale nature of bone are made, e.g., by [7, 11, 48] aiming at determining the trabecular structure by solving a material distribution problem based on densitytype design variables at the macro and mesoscale. For given trabecular structure as characterised by \(\mu \)CT, [14] propose an approach to bone remodelling coupling finite elements at the macroscale with a neural network that is trained from voxelbased finite element analysis at the mesoscale.
Given this state of affairs, the objective of this contribution is therefore to develop an efficient yet effective, twoscale approach to computationally capturing cancellous bone remodelling. To this end, we combine continuum concepts of bone remodelling at the macroscale with a model of trabecular bone at the mesoscale. Thereby, to achieve computational efficiency, the sophisticated cancellous architecture is here, however without sacrificing generality, idealised and indeed simplified as a truss network with the crosssectional area of the trabeculae adapting to the local mechanical stimuli. To concurrently couple the trabecular meso and the wholebone macroscale, we adopt suited up and downscaling strategies. To the author’s best knowledge, a simlar twoscale approach has not been attempted before.
Twoscale approach
We propose a twoscale approach (see Fig. 1), where mechanoadaption problems are concurrently considered at the macro and mesoscale. The kinematics and kinetics at the macroscale are based on wellestablished phenomenological continuum bone remodelling, see, e.g., Kuhl and Steinmann [22], where bone is considered a continuum. However, as a novelty, in the current approach the constitutive behaviour is not postulated phenomenologically at the macroscale, but rather follows directly through upscaling from the mesoscale. To this end, each (quadrature) point at the macroscale is assigned a zoomin volume element (ZVE) representing the trabecular mesoscale structure. The boundary conditions imposed to each ZVE are determined via downscaling the deformation at the macroscale. At the mesoscale, for the sake of computational efficiency and simplicity, the cancellous structure is idealised as a trabecular truss network. Through upscaling, the effective stress, the stiffness and bone density are then transferred back to the macroscale. The proposed twoscale approach is schematically depicted in Fig. 1 and is discussed in detail in the following sections.
Notation: In the sequel we use symbolic tensor notation and avoid entirely any use of matrix notation. Therein, any contraction is indicated by a dot, for example the scalar product of two firstorder tensors (vectors) \({\varvec{a}}\) and \({\varvec{b}}\) reads \({\varvec{a}}\cdot {\varvec{b}}\), the scalar product of two secondorder tensors \({\varvec{A}}\) and \({\varvec{B}}\) reads \({\varvec{A}}:{\varvec{B}}\), and the map of a firstorder tensor \({\varvec{b}}\) into a firstorder tensor \({\varvec{a}}\) by a secondorder tensor \({\varvec{A}}\) reads \({\varvec{a}}={\varvec{A}}\cdot {\varvec{b}}\).
Macroscale continuum model
At the macroscale, we consider bone as continuous matter [22, 30, 31, 39, 40], i.e. in terms of effective continuum quantities that do not explicitly resolve any mesoscale features. We here adopt a wellestablished geometrically nonlinear continuum formulation for the sake of modelling rigour, nevertheless, for the range of deformations expected for hard bone tissue exposed to habitual mechanical loading, the response will automatically approach the geometrically linear limit. We here opt for the wellestablished geometrically nonlinear remodelling framework from [22] since it does not pose any additional challenges. Thus, without loss of continuum modelling accuracy, we refrain from first unnecessarily linearising it to a geometrically linear formulation. All continuum quantities at the macroscale are indicated by an overbar.
Kinematics: The kinematics at the macroscale are characterised by the nonlinear deformation map \(\bar{{\varvec{y}}}\) relating the placement \(\bar{{\varvec{X}}}\) of a continuum point in the material configuration \(\bar{{\mathcal {B}}}_{0}\) to its position \(\bar{{\varvec{x}}}\) in the spatial configuration \(\bar{{\mathcal {B}}}_{t} \subset {\mathbb {E}}^d\) (see Fig. 2)
Here \({\bar{t}}\) denotes the macroscale time that shall coincide with the mesoscale time, thus \({\bar{t}}\equiv t\).
The material gradient of the deformation map is denoted the deformation gradient \(\bar{{\varvec{F}}}\)
The differential operator \(\overline{{\text {Grad}}}\) expresses the gradient in terms of derivatives with respect to the material coordinates \(\bar{{\varvec{X}}}\). The deformation gradient \(\bar{{\varvec{F}}}\), a twopoint tensor, linearly maps from the material tangent space \(T\bar{{\mathcal {B}}}_{0}\) to the spatial tangent space \(T\bar{{\mathcal {B}}}_{t}\) at the macroscale.
Kinetics: The kinetics at the macroscale are collectively dictated by the balances of mass and linear momentum [21]
with the (effective) nominal mass density \({\bar{\rho }}_0\) per unit volume in \(\bar{{\mathcal {B}}}_0\), corresponding mass source \({\bar{R}}_0\), and Piola stress \(\bar{{\varvec{P}}}\) (a twopoint tensor mapping from \(T^*\bar{{\mathcal {B}}}_0\) to \(T^*\bar{{\mathcal {B}}}_t\), the material and spatial cotangent spaces), respectively. The differential operator \(\overline{{\text {Div}}}\) expresses the divergence in terms of derivatives with respect to the material coordinates \(\bar{{\varvec{X}}}\) at the macroscale.
Body forces and inertia are here neglected due to the different levels of gravitational and habitual mechanical loading as well as due to the different time scales of the macroscale problem and the bone remodelling process.
Constitutive Expression: A traditional onescale phenomenological bone remodelling approach [22, 30, 31, 39, 40] requires a constitutive model that expresses the mass source (for example as in Harrigan and Hamilton [15]) and the Piola stress (for example as a NeoHookean response) in terms of the nominal mass density and the deformation gradient, i.e.
In our twoscale approach, however, the density evolution in Eq. 3.1 (and thus the constitutive model in Eq. 4.1) as well as the constitutive model in Eq. 4.2 (together with its consistent linearisation \(\bar{{\varvec{A}}}\) so that \({\text {d}}\bar{{\varvec{P}}}=\bar{{\varvec{A}}}:{\text {d}}\bar{{\varvec{F}}}\)) are entirely bypassed by resorting to upscaling from the mesoscale (see below)
Here the macroscale quantities \({\bar{\rho }}_0\), \(\bar{{\varvec{P}}}\) and \(\bar{{\varvec{A}}}\) are equated with corresponding averaged values \(\langle \rho _0\rangle \), \(\langle {\varvec{P}}\rangle \) and \(\langle {\varvec{A}}\rangle \) at the mesoscale. The determination of \(\langle \rho _0\rangle \), \(\langle {\varvec{P}}\rangle \) and \(\langle {\varvec{A}}\rangle \) from the mesoscale truss network model is outlined next.
Mesoscale truss network model
Each continuum point at the macroscale is assigned a zoomin volume element (ZVE) detailing the trabecular structure at the mesoscale. Typical ZVEs are schematically depicted for example in Figs. 1 and 3. We use the terminology ZVE rather than the common terminology “representative volume element (RVE)” to appreciate the lacking scale separation between the solution domain of a ZVE at the mesoscale and a continuum point at the macroscale. Indeed we accept the lack of scale separation to avoid the need to macroscopically resolve the entire cancellous part of a wholebone specimen in all detail, which unfortunately is prohibitive effortwise and moreover also deemed unnecessary. Accepting furthermore severe geometric simplifications^{Footnote 1}, we here consider the trabecular structure at the mesoscale simply as an idealised geometrically linear trabecular truss network^{Footnote 2}. Despite its simplicity this assumption allows already for complex scenarios wherein trusslike trabeculae with (uniformly) evolving crosssection are of different length and orientation, however, possible bending and/or buckling deformations are neglected (which under physiological mechanical loading appears not too much of a restriction). All, admittedly crude, approximations adopted are here assumed for the sake of reducing complexity and increasing computational efficiency (and can/will be relaxed in future developments). In the following the employed finite element setting for the trussnetwork formulation is outlined in all detail for the sake of selfconsistency.
Kinematics: At the mesoscale, for the sake of computational efficiency, the trabecular structure is described as a truss network where the trabeculae are considered as truss elements with uniform (but adaptable) crosssectional area that is sufficiently smaller than their length. For the notation and details of truss network kinematics we refer to the appendix.
Statics: For a linear elastic truss network the total energy storage in one element reads
Note, due to the (engineering) strain \(\epsilon ^e\) (defined in the appendix) being constant in an element there is no volume integral appearing over the energy storage density per volume
but \(w^e\) is rather simply multiplied with the total volume \(V^e=A^eL^e\) of an element. Here, \(E^{\textrm{s}}\), \(A^e\) and \(L^e\) denote the elastic modulus of the solid material that is assumed constant and given for the trabeculae, the uniform element crosssectional area and the element length, respectively.
Then, the derivative of the total energy storage \(W^e\) with respect to the element vector of displacements \({\varvec{u}}^e\) (defined in the appendix) renders the element vector of internal forces
with \({\varvec{b}}^e\) the element straindisplacement operator (defined in the appendix) and the uniform normal force in an element
The element stiffness tensor follows furthermore from linearisation as
with the uniform (longitudinal) tangent stiffness of a truss element
Note that the element crosssectional area evolves due to the energy storage density \(A^e=A^e(w^e)\), see below. Consequently, with \(w^e=w^e(\epsilon ^e)\) the crosssectional area \(A^e\) implicitly depends on the strain \(\epsilon ^e\).
Finally, in the absence of external forces the assembly of all element contributions results in the global residual and its linearisation
so that the Newton update for the global vector of displacements \({\varvec{u}}\) reads
Crosssectional area evolution: To consider the remodelling process at the mesoscale, some considerations must be made in advance. Previous findings have shown that bone remodelling changes mainly the size of individual trabeculae, whereas the overall architecture of the trabecular bone at the mesoscale remains mostly the same. Thus, increased mechanical stimulation causes trabeculae to become thicker without changing the actual number of trabeculae and their connectivity [28]. In contrast, bone resorption may completely remove individual trabeculae, resulting in a modified network with reduced connectivity. Accordingly, our mesoscale approach considers the adaption of the crosssections of individual truss elements to mechanical stimulus. More precisely, truss elements subjected to high mechanical loading become thicker, while truss elements subjected to insufficient mechanical loading become thinner until they may be completely removed^{Footnote 3}.
Based on these considerations, we here propose a simple phenomenological evolution equation for the crosssectional areas in which the energy storage density in a truss element is compared to a socalled attractor state. Similar to the theory of Frost [12] and Skerry [42] regarding a habitual setpoint strain, which initiates either bone formation or resorption, the mesoscale attractor state \(w^\textrm{a}\) describes the biological homeostasis to which the system is driving. With this, we model the temporal change in crosssectional area \({\dot{A}}^e\) as
with the parameter c (dimension volume per energy and time) controlling the velocity of the remodelling process. Note that evolution equation Eq. (14) is the simplest possible approach and can be easily extended, for example by including the activity of osteocytes and and osteoblasts at the cellular scale [33,34,35].
The temporal discretisation of the crosssectional area evolution equation over a time step \(\Delta t\) is performed with the implicit Euler time stepping scheme. Suppressing explicit indication of the new time point \(t_{n+1}\), the time discrete evolution of Eq. (14) reads as
The linearisation of the updated \(A^e\) with respect to \(\epsilon ^e\) as needed in the expression for the (longitudinal) tangent stiffness \(k^e\) of a truss element results eventually as
Here, concretely, \(\partial R^e/\partial A^e=c\,[w^ew^\textrm{a}]\) and \(\partial R^e/\partial \epsilon ^e=c\, E^{\textrm{s}}A^e \epsilon ^e=c\, n^e\).
Next, we detail the down and upscaling between macro and mesoscale.
DownScaling: To relate the macro to the microscale, we impose affine displacement boundary conditions, i.e. prescribed boundary displacements^{Footnote 4} to the ZVE see Fig. 3. Thereby, the displacements \({\varvec{u}}_a\) of the truss network node points on the discrete ZVE boundary \(\partial V_\mathrm {{ZVE}}\) (constituting the set of boundary node points \({{{\mathcal {N}}}}^{\textrm{b}}\)) are prescribed in terms of the deformation gradient \(\bar{{\varvec{F}}}\) from the macroscale
With these boundary conditions given, the equilibrium problem for \({\varvec{r}}\doteq {\varvec{0}}\) is solved for the displacements \({\varvec{u}}_a\) of the truss network node points within the ZVE, thus completing the downscaling.
Remark: As an aside, the deformation gradient \(\bar{{\varvec{F}}}\) at the macroscale relates to the volume average of the deformation gradient \(\langle {\varvec{F}}\rangle \) at the mesoscale as \(\bar{{\varvec{F}}}=\langle {\varvec{F}}\rangle \), thus
Here, \(V_\mathrm {{ZVE}}\) represents the volume of the ZVE, see Fig. 3, and \({\varvec{A}}_a\) denotes the discrete vectorial area element (outwards pointing unit normal multiplied by the discrete area element \(A_a\)) at boundary node point \(a\in {{{\mathcal {N}}}}^{\textrm{b}}\). \(\square \)
UpScaling: To relate the meso to the macroscale, first averaged values for the nominal density \(\langle \rho _0\rangle \) and the Piola stress \(\langle {\varvec{P}}\rangle \) are computed for the ZVE
Here \(V_{\textrm{T}}=\sum _{e=1}^{n_{el}}V_e\), \(\rho _{0}^\textrm{s}\) and \({\varvec{f}}_a\) denote the volume occupied by the trabeculae within the ZVE, the density of the solid material (assumed constant and given for the trabeculae) and the reaction forces (resulting from the assembly of internal forces) at the boundary node points \(a\in {{{\mathcal {N}}}}^{\textrm{b}}\).
Note that for a twodimensional case we consider the volume \(V_\textrm{ZVE}\) and the crosssectional area \(A^e\) as being extruded to the third dimension by assuming a unit thickness.
Then the linearisation of the average Piola stress \(\langle {\varvec{P}}\rangle \) with respect to the deformation gradient at the macroscale \(\bar{{\varvec{F}}}\), i.e.
follows in terms of the fourthorder tangent stiffness (or elasticity) tensor
which is here denoted (by misuse of the average operator notation) as \(\langle {\varvec{A}}\rangle \). Its detailed derivation is outlined in the appendix.
Finally, the nominal density \(\langle \rho _0\rangle \), the Piola stress \(\langle {\varvec{P}}\rangle \) and its linearisation \(\langle {\varvec{A}}\rangle \) as averaged over the ZVE are equated with their counterparts \(\bar{\rho }_0\), \(\bar{{\varvec{P}}}\) and \(\bar{{\varvec{A}}}\) at the macro scale, thus completing the upscaling.
Algorithmic implementation
The algorithmic implementation is based on the previous set of equations. To this end, the finite element method is used for the spatial discretisation at the macro and the mesoscale. In a nutshell, the stepbystep algorithmic flow follows as

1.
The external mechanical load is incremented and applied to the spatially discretised specimen at the macroscale.

2.
The macroscale deformation gradient \(\bar{{\varvec{F}}}\) at each macroscale quadrature point is determined from the macroscale nodal displacements.

3.
The displacements at the boundary of the discrete trussnetwork ZVE at the mesoscale are prescribed from the macroscale deformation gradient \(\bar{{\varvec{F}}}\).

4.
The truss network mechanoadaption problem is solved at the mesoscale.

5.
The effective stress, nominal bone density, and tangent stiffness tensor are determined from averaging over the ZVE and transferred back to the macroscale.

6.
Using the global (assembled) tangent stiffness at the macroscale, an iteratively updated estimation of the macroscale deformation is computed and steps 2–6 are repeated until equilibrium at the macroscale is obtained.

7.
The outer loop starting with step 1 stops at the end of the external load history.
Benchmark problems
The performance of the proposed twoscale approach is qualitatively demonstrated using two benchmark problems, see, e.g.,Schmidt et al. [40]. These are restricted to two space dimensions for the sake of simplicity and demonstration. The first example considers a uniform macroscale (unit square) specimen under compression to study the influence of the underlying trabecular truss network structure at the mesoscale. The second example analyses a (twodimensional) macroscale proximal femur head under habitual mechanical loading, similar to, e.g., Carter and Beaupré [2], however taking into account the evolving trabecular truss network structure at the mesoscale. Since in both examples the focus of the study is merely on the qualitative behaviour of the twoscale model, all units are omitted. Nevertheless, to qualitatively analyse the evolution of the effective nominal bone density at the macroscale and the underlying trabecular truss network at the mesoscale, the dimensionless simulation times are also provided.
Uniform compression problem
Due to its simplicity, first a uniform (unit square) compression problem is used to study continuum bone remodelling at the elementary level of a single macroscale point (the quadrature point). Since this homogeneous example has already been considered frequently for the onescale approach under uniaxial loading in the literature, we consider it a benchmark and chose to compare the results obtained for the twoscale approach for exactly the same loading and boundary conditions.
To this end, a single^{Footnote 5} finite element with bilinear shape functions and \(2 \times 2\) Gauss quadrature rule is considered at the macroscale, see Fig. 4a. Each macroscale quadrature point is assigned a ZVE at the mesoscale, wherein we analyse two different types of trabecular truss network structures. The structured ZVE consists of 81 equidistantly distributed node points and 208 trabeculae (truss elements) displaying fourfold symmetry horizontally, vertically, and diagonally. The unstructured ZVE consists of 80 node points and 205 trabeculae, wherein node points are randomly shifted. Note that the structured ZVE could be divided into 16 equal unit cells consisting of only 9 node points and 16 trabeculae, however, to better compare it to the unstructured ZVE, we consider the entire ZVE with 81 node points and 208 trabeculae. Moreover, while the number of node points and trabeculae of both ZVEs is similar, the unstructured ZVE results in slightly more node points in vertical than in horizontal direction.
A compressive load in vertical, Fig. 4b, and horizontal direction, Fig. 4c, is applied at the macroscale node points that increases stepwise over time, Fig. 4d. As a reference, we also considered the entirely phenomenological onescale continuum bone remodelling approach at the macroscale as detailed for example in Schmidt et al. [40]^{Footnote 6}. The dimensionless set of parameters of the here advocated twoscale approach and of the reference onescale continuum bone remodelling approach is given in Table 1. Note that we specify an initial (effective) nominal density \(\bar{\rho }_{0}^\star =\bar{\rho }_0(t=0)\) at time \(t=0\). Consequently, the initial trabecular crosssection area \(A^\star =A^e(t=0)\) is consistently calculated by assuming it initially the same for all trabeculae.
The resulting temporal evolution of the (effective) nominal density \(\bar{\rho }_0\) is depicted in Fig. 5a, c. As a trend, initially a lower (effective) nominal density is observed during the first loading phase when comparing the twoscale to the onescale approach. This behaviour reverts as the load level increases, such that the (effective) nominal density for the twoscale approach is larger at the end of the load history. While the twoscale approach using the structured ZVE responds the same to compression in vertical and horizontal direction, thus indicating isotropic behaviour, compression in vertical direction using the unstructured ZVE here leads to a significantly larger (effective) nominal density than in horizontal direction, thus indicating anisotropic behaviour. Obviously, the anisotropy results from the uneven distribution of node points and truss elements in the vertical and horizontal directions. Indeed, 134 of 205 truss elements experience an increase in crosssectional area when loaded in vertical direction, whereas only 100 do when loaded in horizontal direction. It is also interesting to observe that the truss elements at the boundary of the unstructured ZVE that are perfectly aligned with either the vertical or horizontal loading direction carry the major part of the load by adapting most, while the internal trusses carry less load and thus also rather reduce their crosssection. As a contrast, for the structured ZVE all trusses aligned with the loading direction, regardless whether at the boundary or inside the ZVE, carry the same load and consequently adapt the same. This may be considered a sanity check.
Considering finally also the displacements at the macroscale node points in Fig. 5b, d, it is observed that for the presented examples and the chosen material parameters the terminating nodal displacements using the twoscale approach are consistently smaller compared to the onescale approach; due to the predicted higher (effective) nominal density regardless of the underlying ZVE at the mesoscale.
Note that although the evolution equations for the nominal bone mass density in the onescale approach and the crosssectional areas in the twoscale approach share formal similarities, and although the (upscaled) bone mass density on the macroscale is similar on average for both approaches, the stresses in the onescale approach scale nonlinearly (indeed quadratically) with the solid skeleton volume fraction, whereas the truss resultants (forces) scale linearly with the crosssectional area in the twoscale approach. As a result, we can either tune the two approaches to predict on average similar nominal bone mass density at the macroscale or to predict on average similar macroscale displacements (or equivalently effective macroscale stiffness). Since we are primarily interested in the nominal bone mass density, we here opted for the former.
Remark: For the sake of interest and as a sanity check we demonstrate in Fig. 6 the evolution of the crosssectional areas when employing the structured ZVE in simple shear loading resulting in the diagonal loadcarrying truss elements becoming thicker while the horizontal and vertical truss elements vanishing immediately, obviously since they do not carry load in this case.
Remark: As a proof of concept for the applicability of our twoscale approach to 3D problems we consider the 3Dextension of the above uniform compression problem. For the sake of demonstration we employ a threedimensional (initially isotropic) bcctype mesoarchitecture of the ZVE. Here, the nine nodes of the truss network for the ZVE are positioned at the corners of a unit cube and in its center and are connected by 20 truss elements as showcased in Fig. 7. The macroscale is discretised by a single trilinear finite element with the uniform surface load applied in horizontal direction by the same protocol as in Fig. 4d. Fig. 7 displays the evolution of the (effective) nominal density \(\bar{\rho }_0\) (panel a) and the evolution of the underlying mesoarchitecture of the ZVE into an anisotropic structure (panel b). Obviously, the truss elements along the space diagonals and orthogonal to the load direction vanish over time while those in load direction considerably increase their crosssectional area, resulting in the corresponding increase of \(\bar{\rho }_0\) (panel a).
Proximal femur head
Here, the advocated twoscale approach is applied to a simplified biomechanical problem by qualitatively analysing a twodimensional section of a proximal femur head. The macroscale finite element mesh consists of 1988 bilinear continuum elements with \(2 \times 2\) Gauss quadrature rule and 2124 node points, see Fig. 8 (left). Consistent linearisation of the averaged Piola stress resulting in Eq. (20) assures a quadratic rate of convergence for the incrementaliterative Newton scheme with only a few iterations in each load increment. Taken together, as a benefit of the severe geometric assumptions regarding the geometry of the underlying trabecular structure, the computational burden of the advocated twoscale approach remains slim.
Habitual daily loading at the femur head can be typified by three characteristic load cases representing the midstance, extreme abduction, and extreme adduction phase of the gait cycle. Here, since the accurate determination of the magnitude and spatiotemporal distribution of habitual daily loading as in Christen et al. [6] is not in the scope of this contribution, the loads as detailed in Carter and Beaupré [2] are for simplicity simultaneously applied as single forces at the corresponding node points of the macroscale discretisation. Obviously, a different mechanical load in terms of magnitude, direction, distribution and location of application critically influences the resulting bone density distribution. The following results are thus merely qualitative at this point and are thus intended to demonstrate the overall applicability, performance and behaviour of the advocated twoscale approach.
Again, the entirely phenomenological onescale approach detailed in Schmidt et al. [40] serves as reference. To this end, for the onescale approach, the initial nominal density of macroscale elements at the external boundary is set to \(\bar{\rho }_{0}^\star = 1.8\) representing cortical bone, and to \(\bar{\rho }_{0}^\star = 0.55\) for the internal macroscale elements representing cancellous bone. A smooth transition zone between cortical bone, including the regions colored from red to yellow in Fig. 8 (right), and cancellous bone, including the regions colored from green to blue in Fig. 8 (right), is represented by a sigmoidal function. For the twoscale approach only macroscale quadrature points with an initial nominal density less than \(\bar{\rho }_{0}^\star = 1\), i.e. only the regions colored from green to blue in Fig. 8 (right), are assigned a ZVE. For these, we again consider either the structured or the unstructured mesoscale ZVE. Thus, macroscale elements with relatively high initial nominal density representing cortical bone are captured via the onescale approach, whereas macroscale elements representing cancellous bone are described either via the onescale or the novel twoscale approach, thereby considering different ZVEs. The utilised set of dimensionless parameters is given in Table 2.
The evolution of the (effective) nominal density distribution is highlighted for specific simulation time points in Fig. 9. Note that at \(t=0\) the onescale approach (top row) and the twoscale approach using either the structured or the unstructured ZVE (second and third row) have the same nominal bone density distribution. However, over time the nominal bone density distribution starts deviating for the different one and twoscale approaches. Thereby, only small differences are apparent when comparing the results of the twoscale approach with structured and unstructured ZVE in Fig. 9 (second and third row). These minor differences are mainly evident in the highly loaded zone. Overall, however, the (effective) nominal density distribution is pretty similar when comparing the one and twoscale approaches that consistently predict a slight relative increase in (effective) nominal density in the highly loaded zone on the medial side and reduced bone density in the femoral neck region. Likewise, the cortical bone shell at the boundary of the femur head is predicted as similar.
Nevertheless, using the twoscale approach, we can not only analyse the evolution of the (effective) nominal density, but also, as a virtual “magnifying glass”, the evolution of the trabecular structures in the individual ZVEs. To this end, the evolution of the trabecular structures at the mesoscale is exemplified at selected quadrature points of the macroscale elements A and B, see Fig. 10.
As indicated by the distribution of the (effective) nominal density, macroscale element B is located in a less loaded region. The (effective) nominal density there decreases from \(\bar{\rho }_{0}^\star = 0.5705\) to \(\bar{\rho }_0(t=100) \approx 0.3959\). Overall thus, the crosssectional areas of the trabeculae become smaller. Especially when using the structured ZVE, a uniform thinning of the trabeculae over time is evident.
By contrast, when focusing on the macroscale element A located in a highly loaded region, the dominant load direction is clearly highlighted, as trabeculae lying in this direction become significantly thicker. However, this also entails a thinning of the trabeculae perpendicular to this principal loading direction. This effect is especially apparent in the unstructured ZVE. Using the unstructured ZVE thus leads to an increase in (effective) nominal density from \(\bar{\rho }_{0}^\star = 0.5503\) up to \(\bar{\rho }_0(t=100) \approx 0.8030\), whereas the (effective) nominal density only results in \(\bar{\rho }_0(t=100) \approx 0.7867\) when using the structured ZVE.
Summary and conclusion
A wellestablished onescale continuum bone remodelling framework is here extended to a twoscale approach to account in a simple manner for the highly irregular structure of cancellous bone at the mesoscale. The advocated twoscale approach relies on up and downscaling without postulating scaleseparation when defining the effective response at the macroscale as resulting from the mesoscale. Here, the rationale is to find a compromise between the prohibitive computational cost when globally resolving the detailed trabecular mesostructure and the attempt to take the trabecular mesostructure into account at least to a certain extent. As a result, we merely consider a zoomin rather than the common representative volume element. Moreover, within the ZVE the trabecular structure at the mesoscale is idealised and thus severely simplified as a truss network. The energy storage driven evolution of the trabecular crosssectional area is a consequence of lacking biological homeostasis due to mechanical over or underloading. The novel twoscale approach was applied to an elementary uniform and a proximal femur head benchmark problem. Therein, the twoscale approach proved capable to capture the effective macroscale changes as emerging from bone remodelling at the mesoscale and helps understanding how the structure of the trabecular network affects the effective macroscale constitutive response. Noteworthy, in this approach anisotropy is a natural outcome as depending on the regular or irregular structure of the trabecular network. Also observe that neither the onescale approach nor the novel twoscale approach is affected by the widespread numerical checkerboard problem [43], and therefore no smoothing is required in the postprocessing.
Similar to the established onescale approach to continuum bone remodelling, the novel twoscale approach enables analysing the evolution of the (effective) nominal bone density as response to mechanical over and underloading. However, the proposed twoscale approach also allows analysing the evolution of the trabecular structure at the mesoscale, which is deemed crucial to properly evaluate the quality of bone. As an example, consider the evolution of the trabecular structure in the highly loaded macroscale region of the femur head benchmark. There, even though the effective nominal density is rather high, the risk of a fracture cannot be precluded, since only the trabeculae in the dominant direction of loading become thicker, whereas the trabeculae perpendicular to this direction display reduced loadcarrying capacity. Consequently, despite an overall high effective bone density, the risk of a fracture is increased in case of a sudden change in load direction, which can occur for example in the event of a sideways fall.
Summarising, the proposed twoscale approach to bone remodelling is effective from a modelling perspective, efficient from a computational perspective, and easily extendable, for example to take the bone cell responses at the cellular scale into account (we will pursue a corresponding threescale approach in a followup contribution, see also Scheiner et al. [37], Martin et al. [25]). The present twoscale approach opens the door to directly investigate and assess various trabecular structures at the mesoscale for the same bone sample at the macroscale. We thus believe that the proposed twoscale approach provides a promising basis for further development and investigation. However, as of now it still lacks quantitative assessment and can thus not readily translate into clinical practice. To achieve this, we will exploit in a larger collaborative effort medical datasets for identification and calibration of crucial parameters such as, for example, the here involved mesoscale attractor stimulus.
Taken together, the added value of this contribution is indeed the proposed computationally highly efficient twoscale methodology coupling the discrete trabecular mesostructure of cancellous bone with the evolution of the upscaled nominal bone mass density at the continuum macroscale. In particular, even though we here considered elementary truss networks representing the trabecular mesostructure for the sake of computational efficiency and ease of analysis, our advocated twoscale approach is conceptually entirely general and allows considering different mesostructures, including realistic ones obtained from imaging. As an example, the reduction of \(\mu \)CT images to their skeleton and subsequent discretization as a truss network is on our agenda, again especially for its computational efficiency, and will constitute the topic of a separate contribution. Likewise, extension to open and/or closedcell foamlike mesostructures is a valid future possibility, however the increased geometrical complexity comes at elevated computational costs.
Availability of data and materials
All computations were performed using an inhouse Matlab code. Input/output data can be made available on request.
Notes
Indeed, geometrically, cancellous bone is rather characterised by shelllike structures, which, when captured exactly, today still result, however, in a prohibitive computational burden. Resorting to a twoscale approach allows us generically incorporating the architecture of the trabecular structure at the mesoscale and thus requires formulating corresponding evolution equations for its remodelling. Here, for the sake of simplicity and to highlight the methodology, we opted to consider a truss network to characterise the trabecular architecture at the mesoscale. In this case, a phenomenological evolution equation for the truss crosssections, i.e. geometrical quantities, is a natural option with the crosssectional evolution driven by the energy storage density in the solid skeleton at the mesoscale, see below. We like to stress though that the methodology is neither restricted to more involved trabecular architectures at the mesoscale such as, e.g., composed by shelllike structures, nor to more mechanobiologically inspired driving forces following, e.g., from upscaling a bone cell population model operating at the underlying microscale. Likewise, within individual trabeculae remodelling need not render a uniform change in their geometry. In this regard the advocated trussnetwork approach is indeed a compromise between accuracy and computational efficiency and is thus considered only a first step towards more realistically capturing the “true” cancellous bone and its adaption to mechanical stimuli. Clearly, any of these extensions that are certainly on our agenda for the future come at the expense of way higher computational costs.
The restriction to a geometrically linear setting is justified by the small level of external loading (as normalised by the bone stiffness) and in addition entirely avoids rather technical and in the current context unnecessary difficulties when formulating geometrically nonlinear truss networks.
We emphasize that the constitutive equations employed at the mesoscale are inspired by  but not the same as those usually used at the macroscale. In a onescale approach the evolution of the bone mass density, a state quantity at the macroscale, is considered. In the twoscale approach the evolution of truss crosssections, geometrical quantities at the mesoscale, is considered whereby the nominal bone mass density at the macroscale follows as a postprocessed quantity from upscaling. These are two rather different concepts even if the evolution equations at macro and mesoscale share formal similarities. The orientation of individual trabeculae (before deformation) is admittedly unaffected by the present idealised approach. Potential reorientations of trabeculae due to remodelling are here, however, deemed a secondary effect, given the geometric simplifications made anyway.
It is noted that other boundary conditions for the ZVE are indeed possible, among them the limiting cases of Voigt and Reuss bounds, traction boundary conditions, as well as the perhaps most accurate, periodic boundary conditions. For a more indepth discussion we refer to [36] and references therein. Displacement boundary conditions are here merely chosen for the sake of simplicity and demonstration, however without loss of generality. Both traction and periodic boundary conditions require more technicalities when it comes to their implementation, see e.g. our review paper on computational homogenisation [36] among many other accounts on the matter, but do not pose conceptual difficulties per se. It is acknowledged that the selection of either displacement, periodic, or traction boundary conditions results in an effective upscaled response that is either too stiff, believed to be about right, or too soft.
Since this benchmark problem renders homogeneous response in all macroscale fields, discretisation by only a single finite element is sufficient. Nevertheless, as a sanity check of our implementation, we confirmed that arbitrary discretisation densities (1 \(\times \) 1, 2 \(\times \) 2, 3 \(\times \) 3, \(\ldots \) bilinear elements) result, as expected, in exactly the same evolution for the nominal bone mass density and nominal stress components at the macroscale quadrature points as well as in the same overall displacements.
For the onescale approach, the relation between the solid and the initial nominal material properties is based on the initial volume fraction \(\bar{\upsilon }^\star _0 :=\bar{\rho }^\star _0/\rho ^\textrm{s}_0\) of the initial nominal density \(\bar{\rho }^\star _0\) and the solid bone density \(\rho ^\textrm{s}_0\), and a homogenisation exponent \(\bar{N}\). Then, the current nominal stored energy density follows as \(\bar{\psi }_0 = [\bar{\upsilon }_0]^{\bar{n}} [\bar{\upsilon }^\star _0]^{\bar{N}\bar{n}} \psi _0^\textrm{s}\) with \(\psi _0^\textrm{s}\) the stored energy in the solid bone and \(\bar{\upsilon }_0 :=\bar{\rho }_0/\rho ^\textrm{s}_0\) the current volume fraction. Moreover, the nominal mass source reads as \(\bar{R}_0 = \bar{c}\, [\bar{\upsilon }_0^{\bar{n}\bar{m}} [\bar{\upsilon }^\star _0]^{\bar{N}\bar{n}+\bar{m}} \psi ^\textrm{s}_0\bar{\psi }_0^\textrm{a}]\). For the here considered homogenisation exponent \(\bar{N}=\bar{n}\) we obtain \(\bar{\psi }_0 = [\bar{\upsilon }_0]^{\bar{n}} \psi _0^\textrm{s}\) and \(\bar{R}_0 = \bar{c} \,[\bar{\upsilon }_0^{\bar{n}\bar{m}} [\bar{\upsilon }^\star _0]^{\bar{m}} \psi ^\textrm{s}_0\bar{\psi }_0^\textrm{a}]\), for a detailed derivation and justification of the material parameters selected see Schmidt et al. [40]. The Piola stress entering the macroscale equilibrium equation then follows as \(\bar{{\varvec{P}}}=[\bar{\upsilon }_0]^{\bar{n}}\partial \psi ^\textrm{s}_0/\partial \bar{{\varvec{F}}}\), i.e. as nonlinearily (with \(\bar{n}\) set to 2 indeed quadratically) scaled by the evolving current volume fraction \(\bar{\upsilon }_0\). Observe that this is different to the truss normal forces \(n^e=E^{\textrm{s}}\epsilon ^eA^e\) that only scale linearly with the evolving crosssectional area \(A^e\) in the twoscale approach. Obviously, we can not expect a onetoone correspondence of the one and twoscale models, also as the latter incorporates more mesoscale detail.
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Acknowledgements
The support of Paul Scheuerlein for the initial 3D implementation is gratefully acknowledged. IS and PS acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under 377472739/GRK 2423/12019. AP and IS acknowledge support by the Bavarian Academic Forum (BayWISS)  Doctoral Consortium “Health Research”. PP acknowledges support from the Australian Research Council (IC190100020, DP230101404).
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Open Access funding enabled and organized by Projekt DEAL. IS and PS acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under 377472739/GRK 2423/12019. AP and IS acknowledge support by the Bavarian Academic Forum (BayWISS)  Doctoral Consortium “Health Research”. PP acknowledges support from the Australian Research Council (IC190100020, DP230101404).
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PS: Conceptualisation, methodology, editing, supervision, funding acquisition. IS: Methodology, implementation, simulations, visualisation, editing. PP: Conceptualisation, methodology, editing, supervision. AP: Conceptualisation, methodology, simulations, visualisation, editing, supervision, funding acquisition.
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Appendix
Appendix
Here, for the sake of completeness and in order to fix notation, we detail the underlying kinematics of the truss network model employed as well as the derivation of the fourthorder tangent stiffness tensor.
Kinematics of truss networks: A truss network consists of a set of node points \({{{\mathcal {N}}}}\) (globally) numbered by \(a=1,\cdots n_{np}\) connected by elements numbered by \(e=1,\cdots n_{el}\). The coordinates and displacements of the node points are
Node points are assigned to elements via a connectivity list that reads
Then the element vectors of coordinates and displacements follow as
Here, for two firstorder tensors (vectors) \({\varvec{a}}\in {\mathbb {R}}^d\) and \({\varvec{b}}\in {\mathbb {R}}^d\) the notation \([{\varvec{a}}\backslash {\varvec{b}}]\in {\mathbb {R}}^{2d}\) refers to a firstorder tensor (vector) composed from \({\varvec{a}}\) and \({\varvec{b}}\) and mapping from \({\mathbb {R}}^{2d}\) to \({\mathbb {R}}\).
Next, with \({\varvec{I}}\) the secondorder unit tensor, we introduce the projection operator
Here, for two secondorder tensors \({\varvec{A}}\in {\mathbb {R}}^d\times {\mathbb {R}}^d\) and \({\varvec{B}}\in {\mathbb {R}}^d\times {\mathbb {R}}^d\) the notation \([{\varvec{A}}{\varvec{B}}]\in {\mathbb {R}}^d\times {\mathbb {R}}^{2d}\) refers to a tensorial object composed from \({\varvec{A}}\) and \({\varvec{B}}\) and mapping from \({\mathbb {R}}^{2d}\) to \({\mathbb {R}}^d\).
Then the element length and director (a unit vector) follow as
The (engineering) strain in an element, i.e. the elongation of an element divided by its original length, is constant and reads
Here, the straindisplacement operator is defined as
Note finally that \([{\varvec{d}}^e\;\backslash +{\varvec{d}}^e]\cdot {\varvec{u}}^e={\varvec{d}}^e\cdot [{\varvec{u}}_b{\varvec{u}}_a]\).
Derivation of fourthorder tangent stiffness tensor: The ZVE tangent stiffness \({\varvec{k}}\) together with the linearised increments of nodal displacements \({\text {d}}\!{\varvec{u}}\) and nodal forces \({\text {d}}\!{\varvec{f}}\) are partitioned according to the prescribed (boundary) nodal displacements at the ZVE boundary and the retained (internal) nodal displacements within the ZVE, denoted as \({\varvec{u}}^{\textrm{p}}\) and \({\varvec{u}}^{\textrm{r}}\), respectively, (see Fig. 3)
Here, \({\text {d}}\!{\varvec{f}}^{\textrm{p}}\) denotes the linearised increment of the nodal (reaction) forces at the prescribed (boundary) nodes of the ZVE, the corresponding quantities at the retained (internal) nodes are zero to maintain equilibrium. Then, the reduced tangent stiffness \(\widehat{{\varvec{k}}}\) relating the linearised increments of the prescribed nodal displacements \({\text {d}}\!{\varvec{u}}^{\textrm{p}}\) and the corresponding linearised increments of nodal forces \({\text {d}}\!{\varvec{f}}^{\textrm{p}}\) at the ZVE boundary follows as
With nodes \(a, b\in {{{\mathcal {N}}}}^{\textrm{b}}\) on the ZVE boundary, the reduced tangent stiffness \(\widehat{{\varvec{k}}}\) as well as \({\text {d}}\!{\varvec{u}}^{\textrm{p}}\) and \({\text {d}}\!{\varvec{f}}^{\textrm{p}}\) can be further partitioned into nodal contributions as
Note that the nodal stiffness \(\widehat{{\varvec{k}}}_{ab}\in {\mathbb {R}}^d\times {\mathbb {R}}^d\) is a secondorder tensor relating the firstorder tensors (vectors) \({\text {d}}\!{\varvec{u}}_b\in {\mathbb {R}}^d\) and \({\text {d}}\!{\varvec{f}}_a\in {\mathbb {R}}^d\), thus
With this preliminary at hand, the linearised increment of the averaged Piola stress \({\text {d}}\langle {\varvec{P}}\rangle \) follows as
Substituting eventually \({\text {d}}\!{\varvec{u}}_b ={\text {d}}\!\bar{{\varvec{F}}}\cdot {\varvec{X}}_b\) renders the final expression
Here, the expression in square brackets denotes the required fourthorder tangent stiffness tensor \(\langle {\varvec{A}}\rangle \).
Note the usage of the special dyadic product \({\overline{\otimes }}\) of two secondorder tensors defined as \(\left[ {\varvec{a}} \, {\overline{\otimes }} \,{\varvec{B}} \right] _{iJkL} := a_{ik} \, B_{JL}\).
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Steinmann, P., Schmidt, I., Pivonka, P. et al. A computational twoscale approach to cancellous bone remodelling. Adv. Model. and Simul. in Eng. Sci. 11, 13 (2024). https://doi.org/10.1186/s40323024002671
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DOI: https://doi.org/10.1186/s40323024002671