# Strength increase during ceramic biomaterial-induced bone regeneration: a micromechanical study

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## Abstract

Bone tissue engineering materials must blend in the targeted physiological environment, in terms of both the materials’ biocompatibility and mechanical properties. As for the latter, a well-adjusted stiffness ensures that the biomaterial’s deformation behavior fits well to the deformation behavior of the surrounding biological tissue, whereas an appropriate strength provides sufficient load-carrying capacity of the biomaterial. Here, a mathematical modeling approach for estimating the macroscopic load that initiates failure of a hierarchically organized, granular, hydroxyapatite-based biomaterial is presented. For this purpose, a micromechanics model is developed for downscaling macroscopically prescribed stress (or strain) states to the level of the needle-shaped hydroxyapatite crystals. Presuming that the biomaterial fails due to the quasi-brittle failure of the most unfavorably stressed hydroxyapatite needle, the downscaled stress tensors are fed into a suitable, Mohr-Coulomb-type failure criterion, based on which the macroscopic failure load is deduced. The change of the biomaterial’s composition in response to placing it in physiological solution, caused by growth of new bone tissue on the granules’s surfaces, on the one hand, and by resorption of the hydroxyapatite crystals, on the other hand, is taken into account by means of suitable evolution laws. Numerical studies show how the macroscopic load-carrying capacity of the biomaterial is influenced by its design parameters. The presented modeling approach could prove beneficial for the design process of the studied biomaterials (as well as similarly composed biomaterials), particularly in terms of optimizing its mechanical performance.

## Keywords

Continuum micromechanics Elastic limit Multiscale modeling Bone ingrowth Tissue engineering## Abbreviations

- gran
Granule material (RVE II)

- \(\mu \)CT
Micro-computed tomography

- polyHA
Hydroxyapatite polycrystal (RVE I)

- RVE
Representative volume element

- congl
Conglomerate of granules coated with bone tissue (RVE III)

## Latin symbols

- \(\mathbb {A}_\text {HA}^\text {polyHA}\)
Fourth-order strain concentration tensor of hydroxyapatite crystal needles in hydroxyapatite polycrystal

- \(\mathbb {A}_\text {polyHA}^\text {gran}\)
Fourth-order strain concentration tensor of hydroxyapatite polycrystal in granule material

- \(\mathbb {B}_\text {HA}^\text {polyHA}\)
Fourth-order stress concentration tensor of hydroxyapatite crystal needles in hydroxyapatite polycrystal

- \(\mathbb {B}_\text {polyHA}^\text {gran}\)
Fourth-order stress concentration tensor of hydroxyapatite polycrystal in granule material

- \(\mathbb {C}_\text {bone}\)
Fourth-order stiffness tensor of bone tissue

- \(\mathbb {C}_\text {gran}\)
Fourth-order stiffness tensor of granular material

- \(\mathbb {C}_{\text {H}_2\text {O}}\)
Fourth-order stiffness tensor of water

- \(\mathbb {C}_\text {HA}\)
Fourth-order stiffness tensor of hydroxyapatite crystals

- \(\mathbb {C}_{\text {macro}\phi }\)
Fourth-order stiffness tensor of macropores

- \(\mathbb {C}_{\text {meso}\phi }\)
Fourth-order stiffness tensor of mesopores

- \(\mathbb {C}_{\text {micro}\phi }\)
Fourth-order stiffness tensor of micropores

- \(\mathbb {C}_\text {polyHA}\)
Fourth-order stiffness tensor of hydroxyapatite polycrystal

- \(\mathbb {C}_\text {congl}\)
Fourth-order stiffness tensor of bone-scaffold conglomerate

- \({\mathcal {D}}_k\)
Material parameter

*d*Characteristic length of an inhomogeneity within a representative volume element

*e*Crack density parameter

- \({\mathbf {e}}_1,{\mathbf {e}}_2,{\mathbf {e}}_3\)
Unit base vectors of a Cartesian base system

- \({\mathbf {e}}_r,{\mathbf {e}}_\vartheta ,{\mathbf {e}}_\varphi \)
Unit base vectors of a spherical coordinate system

- \(E_\text {HA}\)
Young’s modulus of hydroxyapatite crystals

- \({\mathbf {E}}_\text {congl}\)
Second-order strain tensor of the bone-scaffold conglomerate

- \(E_{\text {congl},ij}\)
Components of \({\mathbf {E}}_\text {congl}\) (\(i,j=1,2,3\))

- \({f}_\text {bone}^\text {congl}\)
Volume fraction of bone tissue within bone-scaffold conglomerate

- \({f}_\text {gran}^\text {congl}\)
Volume fraction of the granules within bone-scaffold conglomerate

- \(f_\text {HA}^\text {polyHA}\)
Volume fraction of the hydroxyapatite needles within hydroxyapatite polycrystal

- \(f_\text {polyHA}^\text {gran}\)
Volume fraction of the microporous hydroxyapatite matrix within granule material

- \({\mathfrak {f}}(\varvec{\sigma })\)
Failure function formulated in terms of stress tensor \(\varvec{\sigma }\)

- \({\mathbf {I}}\)
Second-order unit tensor

- \(\mathbb {I}\)
Fourth-order unit tensor

- \(\mathbb {J}\)
Deviatoric part of the fourth-order unit tensor

- \(k_\text {gran}\)
Bulk modulus of the granule material

- \(k_\text {growth}\)
Formation rate of bone tissue

- \(k_{\text {H}_2\text {O}}\)
Bulk modulus of water

- \(k_\text {HA}\)
Bulk modulus of the hydroxyapatite crystals

- \(k_\text {res}\)
Resorption rate of hydroxyapatite crystals

- \(k_\text {congl}\)
Bulk modulus of the bone-scaffold conglomerate

- \(\mathbb {K}\)
Volumetric part of the fourth-order unit tensor

- \(\ell \)
Characteristic length of a representative volume element

- \({\mathcal {L}}\)
Characteristic length of a structure made up the material defined on the level of a representative volume element

- \({\mathbf {n}}\)
Vector oriented perpendicular to unit base vector \({\mathbf {e}}_r\)

- \(\mathbb {P}_\text {cyl}^\text {polyHA}\)
Fourth-order Hill tensor of cylindrical inclusions in a matrix with stiffness \(\mathbb {C}_\text {polyHA}\)

- \(\mathbb {P}_\text {sph}^\text {polyHA}\)
Fourth-order Hill tensor of spherical inclusions in a matrix with stiffness \(\mathbb {C}_\text {polyHA}\)

- \({\mathbf {Q}}\)
Second-order transformation tensor

- \(\mathbb {Q}\)
Fourth-order tensor defined through Poisson’s ratio \(\nu _\text {polyHA}\)

- \(r_\text {gran}\)
Granule radius

*t*Time variable

## Greek symbols

- \(\overline{\Gamma _\text {gran,1}^k}\)
Material parameter

- \(\overline{\Gamma _\text {gran,1}^\mu }\)
Material parameter

- \(\overline{\Gamma _\text {gran,2}^\mu }\)
Material parameter

- \(\varvec{\varepsilon }_\text {HA}^\text {polyHA}\)
Second-order strain tensor of hydroxyapatite crystals in hydroxyapatite polycrystal

- \(\varvec{\varepsilon }_\text {gran}^\text {congl}\)
Second-order strain tensor of granule material in bone-scaffold conglomerate

- \(\varvec{\varepsilon }_\text {gran}^\text {congl,dev}\)
Second-order deviatoric strain tensor of granule material in bone-scaffold conglomerate

- \(\varvec{\varepsilon }_\text {gran}^\text {congl,vol}\)
Second-order volumetric strain tensor of granule material in bone-scaffold conglomerate

- \(\varvec{\varepsilon }_\text {polyHA}^\text {gran}\)
Second-order strain tensor of hydroxyapatite polycrystal in granule material

- \(\vartheta \)
Angle defining the orientation of the spherical coordinate system (\({\mathbf {e}}_r,{\mathbf {e}}_\vartheta ,{\mathbf {e}}_\varphi \))

- \(\mu _\text {gran}\)
Shear modulus of granule material

- \(\mu _\text {HA}\)
Shear modulus of hydroxyapatite crystals

- \(\mu _\text {congl}\)
Shear modulus of bone-scaffold conglomerate

- \(\nu _\text {gran}\)
Poisson’s ratio of granule material

- \(\nu _\text {HA}\)
Poisson’s ratio of hydroxyapatite crystals

- \(\nu _\text {polyHA}\)
Poisson’s ratio of hydroxyapatite polycrystal

- \(\varvec{\sigma }_\text {HA}^\text {polyHA}\)
Second-order stress tensor of hydroxyapatite crystals in hydroxyapatite polycrystal

- \(\sigma _{\text {HA},ij}^\text {polyHA}\)
Component of \(\varvec{\sigma }_\text {HA}^\text {polyHA}\) (\(i,j=1,2,3\))

- \(\sigma _{\text {HA}}^\text {ult,s}\)
Shear strength of hydroxyapatite crystals

- \(\sigma _{\text {HA}}^\text {ult,t}\)
Tensile strength of hydroxyapatite crystals

- \(\varvec{\sigma }_\text {gran}^\text {congl}\)
second-order stress tensor of granule material in bone-scaffold conglomerate

- \(\varvec{\sigma }_\text {gran}^\text {congl,dev}\)
Second-order deviatoric stress tensor of granule material in bone-scaffold conglomerate

- \(\varvec{\sigma }_\text {gran}^\text {congl,vol}\)
Second-order volumetric stress tensor of granule material in bone-scaffold conglomerate

- \(\sigma _{\text {gran},ij}^\text {congl}\)
Component of \(\varvec{\sigma }_\text {gran}^\text {congl}\) (\(i,j=1,2,3\))

- \(\varvec{\sigma }_\text {polyHA}^\text {gran}\)
Second-order stress tensor of hydroxyapatite polycrystal in granule material

- \(\sigma _{\text {polyHA},ij}^\text {gran}\)
Component of \(\varvec{\sigma }_\text {polyHA}^\text {gran}\) (\(i,j=1,2,3\))

- \(\varvec{\Sigma }_\text {congl}\)
Second-order stress tensor of bone-scaffold conglomerate

- \(\Sigma _{\text {congl},ij}\)
Component of \(\varvec{\Sigma }_\text {congl}\) (\(i,j=1,2,3\))

- \(\Sigma _{\text {congl},11}^\text {ult}\)
Component in direction \({\mathbf {e}}_1\) of the second-order stress tensor of bone-scaffold conglomerate representing the ultimate loading

- \(\phi _\text {macro}^\text {congl}\)
Volume fraction of macropores in bone-scaffold conglomerate

- \(\phi _\text {meso}^\text {gran}\)
Volume fraction of mesopores in granule material

- \(\phi _\text {micro}^\text {polyHA}\)
Volume fraction of micropores in hydroxyapatite polycrystal

- \(\varphi \)
Angle defining the orientation of the spherical coordinate system (\({\mathbf {e}}_r,{\mathbf {e}}_\vartheta ,{\mathbf {e}}_\varphi \))

- \(\psi \)
Angle defining orientation of vector \({\mathbf {n}}\)

## 1 Introduction

The field of bone tissue engineering aims at the reinforcing or even replacing diseased (or for other reasons malfunctioning) bone tissue by scaffold structures that are specifically engineered, for blending in the targeted physiological environment, i.e. the immediate vicinity of bone tissue, as well as possible (Burg et al. 2000; Reichert and Hutmacher 2011). From a mechanical point of view, careful tuning of such scaffold structures (and of the materials which they are made of) is called for because contradictory requirements must be brought in line—scaffold structures must be stiff enough to sustain all relevant mechanical load cases, but also soft enough to facilitate, through mechanobiological couplings (Klein-Nulend et al. 2005; Porter et al. 2009; Velasco et al. 2015), the integration into their bony environment. In this regard, two mechanical properties are of particular interest, both on material and structural levels: the stiffness, governing the elastic deformation behavior and therefore the forces attracted by the involved macro- and microstructures; as well as the strength, indicating the stress level that induces material failure.

In the present paper, we study one specific scaffold material that has been developed as bone replacement material with the human mandible as targeted application area (Komlev et al. 2002, 2003). This biomaterial is produced in form of porous, pre-cracked granules, composed of hydroxyapatite as main constituent, but also including various kinds of pore spaces of distinctively different characteristic lengths. After exposing this biomaterial to the targeted physiological environment prevailing in the immediate vicinity of mandibular bone tissue, two mechanisms are triggered, causing a progressing change of the material’s composition over time. On the one hand, bone tissue grows on the granule surfaces, while, on the other hand, concurrently the hydroxyapatite crystals are resorbed—in the long run, the scaffold material merges with the surrounding bone tissue.

In a first approach to analyzing their mechanical behavior, these granules underwent micro-computed tomography (\(\mu \)CT), and the resulting scans served as basis for combined Finite Element/micromechanics-based simulations (Dejaco et al. 2012, 2016). Here, as a (computationally more efficient) complement, we present a three-step, fully continuum micromechanics-based macro-to-meso-to-micro (stress and strain) downscaling scheme, linking in the end the quasi-brittle failure of single micrometer- or sub-micrometer-sized hydroxyapatite crystal needles to the overall strength of both millimeter-sized biomaterial scaffolds and composites comprising biomaterial scaffold and bone tissue, respectively. For this purpose, a number of homogenization concepts are adapted, extended, and combined, considering the pioneering contributions of Eshelby (1957), Hill (1963, 1965), Laws (1977, 1985), Hervé and Zaoui (1993); and also considering more recent contributions of Deudé et al. (2002), Dormieux et al. (2004), Fritsch et al. (2006), Bertrand and Hellmich (2009). Following Fritsch et al. (2009a, b), we feed the stress of the most unfavorably loaded hydroxyapatite needle into a suitable, Mohr-Coulomb-type failure criterion, and deduce then therefrom the corresponding ultimate macroscopic load bearable by the aforementioned granular, hydroxyapatite-based biomaterial (optionally containing ingrown bone tissue).

After introducing the fundamental modeling concept, together with the chosen model representation of the studied biomaterial, see Sect. 2, a mathematical model for downscaling of the mechanical loading, from the macroscopic to the hydroxyapatite needle scale, is presented, see Sect. 3. Then, a suitable failure criterion is elaborated in Sect. 4.1, and numerical studies show how the macroscopic mechanical loading inducing single hydroxyapatite needle failure changes with varying biomaterial composition. In order to simulate bone regeneration (which occurs after having placed the biomaterial in the targeted physiological environment), involving bone growth and scaffold resorption, suitable evolution laws are introduced, and the effects of different material input parameters on the model-predicted development of the load-carrying capacity over time are studied, see Sect. 5. A brief discussion closes the paper, see Sect. 6.

## 2 Material and methods

### 2.1 Characterization of the multi-porous hydroxyapatite tissue engineering scaffold material

Several morphological features of these granules can be observed, see the column on the left-hand side of Fig. 1. Firstly, the granules contain pores of two different characteristic lengths: small pores, with a characteristic length ranging from less than one to several micrometers (Dejaco et al. 2016)—these pores are termed “micropores” hereafter; and large pores, with a characteristic length of several hundred micrometers—these pores are termed “mesopores” hereafter. A composite of randomly oriented hydroxyapatite crystals and the micropores constitutes the “base material” of the granules. Increasing the observation scale by several orders of magnitude, one can discern, besides the mesopores, cracks pervading the granule body. Finally, the scaffold material is made up of the above described granules, with pore space in-between—due to the characteristic length of these pores, which is approximately equal to the granule diameter, they are termed “macropores” in the remainder of this paper.

### 2.2 Fundamentals of continuum micromechanics: the representative volume element

A method particularly well suited for modeling the mechanical behavior of the material described in Sect. 2.1 is continuum micromechanics (Hill 1963; Zaoui 1997, 2002), where a material is understood as a micro-heterogeneous body filling a macro-homoge-neous representative volume element (RVE) with characteristic length \(\ell \), \(\ell \,\gg \,d\), *d* standing for the characteristic length of inhomogeneities within the RVE, and \(\ell \,\ll \,{{\mathcal {L}}}\), \({{\mathcal {L}}}\) standing for the characteristic lengths of geometry or loading of a structure built up by the material defined on the RVE. It should be noted the aforementioned requirements of “much larger” (\(\gg \)) and “much smaller” (\(\ll \)), respectively, have been shown to be already satisfied if the respective characteristic lengths are separated by a factor of two to three and five to ten, respectively (Drugan and Willis 1996; Kohlhauser and Hellmich 2013).

In general, the microstructure within an RVE is too complicated to be described in complete detail. Therefore, quasi-homogeneous subdomains with known physical properties (such as volume fractions and mechanical properties) are reasonably chosen. They are called material phases. The homogenized (upscaled) behavior of the material on the observation scale of the RVE, i.e. the relation between homogeneous deformations acting on the boundary of the RVE and resulting macroscopic (average) stresses, can then be estimated from the mechanical behavior of the material phases, their volume fractions within the RVE, their characteristic shapes, and their interactions. If a single material phase is micro-heterogeneous itself, its mechanical behavior can be estimated by introduction of RVEs within this phase, with characteristic lengths \(\ell _1\,\le \,d\), comprising again inhomo- geneities with characteristic length \(d_1\,\ll \,\ell _1\), and so on. Such an approach is referred to as multi-step homogenization and provides, eventually, access to “universal” phase properties at sufficiently low observation scales (Fritsch and Hellmich 2007).

### 2.3 Micromechanical modeling

Having in mind the concept of “separation of scales”, as introduced in Sect. 2.2, the following three-level micromechanical representation emerges for the biomaterial under investigation:

On *hierarchical level I*, a microporous, overall isotropic, hydroxyapatite polycrystal is composed of spherical micropores (with volume fraction \(\phi _\text {micro}^\text {polyHA}\)), which interact mutually with randomly oriented cylindrical hydroxyapatite crystals (with volume fraction \(f_\text {HA}^\text {polyHA}=1-\phi _\text {micro}^\text {polyHA}\)). Typically, the microporosity amounts to \(\phi _\text {micro}^\text {polyHA}=0.445\) (Dejaco et al. 2012). The characteristic length of the polycrystalline RVE I is in the order of \(10\,\upmu \)m, see the bottom row of Fig. 1, with a scanning electron micrograph of the granule nano-structure on the left-hand side and the corresponding RVE I on the right-hand side. In terms of stiffness upscaling, the mutual mechanical interaction of all phases within RVE I calls for a self-consistent homogenization scheme, as introduced by (Fritsch et al. 2006), giving access to the stiffness tensor of the microporous hydroxyapatite polycrystal, \(\mathbb {C}_\text {polyHA}\), based on the composition and morphology of RVE I, as well as on the stiffness tensors of the hydroxyapatite crystals, \(\mathbb {C}_\text {HA}\), and of the micropores, \(\mathbb {C}_{\text {micro}\phi }\).

On *hierarchical level II*, penny-shaped cracks (with vanishing volume fraction) and spherical mesopores (with volume fraction \(\phi _\text {meso}^\text {gran}\)) are embedded in the polycrystal matrix with properties arising from the structure of RVE I, this matrix filling within RVE II the volume fraction \(f_\text {polyHA}^\text {gran}=1-\phi _\text {meso}^\text {gran}\). Typically, the mesoporosity comes to \(\phi _\text {meso}^\text {gran}=0.189\) (Dejaco et al. 2012). The characteristic length of RVE II is in the order of 1 mm, see the middle row in Fig. 1, with a micro-computed tomography (\(\mu \)CT) image of the microstructure within a granule on the left-hand side and the corresponding RVE II on the the right-hand side. The distinctive matrix-inclusion morphology of RVE II—i.e. cracks and mesopores can be considered as inclusions embedded in the hydroxyapatite polycrystal matrix—suggests the use of a Mori-Tanaka-type homogenization scheme (Mori and Tanaka 1973; Benveniste 1987) for stiffness homogenization; mathematical treatment of the penny-shaped cracks has been dealt with by Deudé et al. (2002), Dormieux et al. (2004). The stiffness tensor of the pre-cracked, mesoporous granule material, \(\mathbb {C}_\text {gran}\) is then governed by the composition and morphology of RVE II, as well as by the stiffness tensors of the hydroxyapatite polycrystal matrix, \(\mathbb {C}_\text {polyHA}\), accessible from stiffness homogenization across RVE I, and of the mesopores, \(\mathbb {C}_{\text {meso}\phi }\), and by the density of cracks, quantified by the so-called crack density parameter *e* (Budianksy and O’Connell 1976).

On *hierarchical level III*, a macroporous conglomerate material consisting of mesoporous, cracked hydroxapatite granules and newly grown bone tissue emerges, see the top of Fig. 1: granules with the stiffness of RVE II described above and filling volume fraction \(f_\text {gran}^\text {congl}\), are surrounded by layers of newly grown bone tissue, with volume fraction \(f_\text {bone}^\text {congl}\) and stiffness derived from the ultrasonic tests of Ashman and van Buskirk (1987). These coated spherical elements are assembled, in mutual contact, to a granular conglomerate with macropores, with volume fraction \(\phi _\text {macro}^\text {congl}\), in-between. At the time of granule implantation, no bone tissue has been formed yet, and this initial configuration is characterized by \(f_\text {bone}^\text {congl}\,=\,0\). For estimating the macroscopic stiffness tensor of the bone-scaffold conglomerate, \(\mathbb {C}_\text {congl}\), the homogenization approach for an *n*-layered spherical inclusion proposed by Hervé and Zaoui (1993) is specialized for \(n=1\) (relating to bone tissue), adapted for the case that the stiffness of this layer is transversally isotropic, see (Bertrand and Hellmich 2009), and further combined with a self-consistent homogenization scheme, in order to account for mutually interacting coated spheres with porous space in-between—in absence of any explicit “matrix phase”. This homogenization step is thus based on the composition and morphology of RVE III, as well as on the stiffness tensors of the granule material, \(\mathbb {C}_\text {gran}\), accessible from stiffness homogenization across RVE II, of the bone tissue, \(\mathbb {C}_\text {bone}\), and of the macropores, \(\mathbb {C}_{\text {macro}\phi }\); the underlying mathematical framework is described at length in (Scheiner et al. 2016).

## 3 Downscaling of stresses from macro- to microscale

- 1.
From the macroporous bone-scaffold conglomerate to the pre-cracked and mesoporous granules (see Sect. 3.2);

- 2.
From the pre-cracked and mesoporous granules to the microporous, polycrystalline hydroxyapatite matrix (see Sect. 3.3); and

- 3.
From the microporous, polycrystalline hydroxyapatite matrix to single, arbitrarily oriented hydroxyapatite crystal needles (see Sect. 3.4).

### 3.1 Definition of mechanical input parameters

As for the underlying main elementary constituent, i.e. hydroxyapatite, the respective stiffness tensor, \(\mathbb {C}_\text {HA}\), is defined via the bulk modulus, \(k_\text {HA}\), and the shear modulus, \(\mu _\text {HA}\), \(\mathbb {C}_\text {HA}\,=\,3k_\text {HA}\mathbb {K}+2\mu _\text {HA}\mathbb {J}\), with \(\mathbb {K}\) being the volumetric part of the fourth-order unit tensor \(\mathbb {I}\), and \(\mathbb {J}\) the corresponding deviatoric part, \(\mathbb {K}+\mathbb {J}\,=\,\mathbb {I}\). Numerical values for \(k_\text {HA}\) and \(\mu _\text {HA}\) are found based on the experiments performed by Katz and co-workers (Katz and Ukraincik 1971; Gilmore and Katz 1982), who revealed the Young’s modulus and Poisson’s ratio of hydroxyapatite, \(E_\text {HA}\,=\,114\) GPa and \(\nu _\text {HA}\,=\,0.27\), see also (Hellmich and Ulm 2002; Hellmich et al. 2004). Through standard relations of continuum mechanics, \(k\,=\,E/[3(1-2\nu )]\) and \(\mu \,=\,E/[2(1+\nu )]\) (Mang and Hofstetter 2000), one finally obtains \(k_\text {HA}\,=\,82.61\) GPa and \(\mu _\text {HA}\,=\,44.88\) GPa. Furthermore, all pore spaces are assumed to be drained at all times, thus \(\mathbb {C}_{\text {micro}\phi }\,=\,\mathbb {C}_{\text {meso}\phi }\,=\,\mathbb {C}_{\text {macro}\phi }\,=\,0\).

### 3.2 From the macroporous bone-scaffold conglomerate to the pre- cracked, mesoporous granules (Fig. 1, hierarchical level III)

*not*homogeneous throughout the inclusion.

### 3.3 From the pre-cracked, mesoporous granules to the microporous hydroxyapatite polycrystal (Fig. 1, hierarchical level II)

*e*is the so-called crack density parameter (Budianksy and O’Connell 1976), \(e\,=\,{\mathcal {N}}(r_\text {crack})^3\), with \({\mathcal {N}}\) as the number of cracks per volume, and \(r_\text {crack}\) as the (average) crack radius, and \(\mathbb {Q}\) is a tensor defined via the Poisson’s ratio of the microporous hydroxyapatite polycrystal, \(\nu _\text {polyHA}\), through (Dormieux et al. 2004)

### 3.4 From the microporous hydroxyapatite polycrystal to the single hydroxyapatite crystal (Fig. 1, hierarchical level III)

\(\phi _\text {macro}^\text {congl}\) | \(\min {\sigma _{\text {HA},ij}^\text {polyHA}}\) (MPa) | \(\max {\sigma _{\text {HA},ij}^\text {polyHA}}\) (MPa) |
---|---|---|

0.3 | \(-47.24\) | 19.41 |

0.4 | \(-54.31\) | 23.45 |

0.5 | \(-63.99\) | 29.38 |

## 4 Estimates for the macroscopic strength of hydroxyapatite-based granular biomaterials

### 4.1 Failure criterion suitable for hydroxyapatite needles

Based on the three-step scheme presented in Sect. 3, a macroscopically applied mechanical loading, prescribed in terms of macroscopic strains \({\mathbf {E}}_\text {congl}\) or macroscopic stresses \(\varvec{\Sigma }_\text {congl}\), were downscaled to the corresponding stress state experienced by a single, arbitrarily oriented hydroxyapatite needle, \(\varvec{\sigma }_\text {HA}^\text {polyHA}(\vartheta ,\varphi )\).

Iteration scheme for deriving the macroscopic loading of the biomaterial inducing quasi-brittle failure in the most unfavorably stressed hydroxyapatite needle

Iteration steps | |
---|---|

1. | Choice of initial value for \(\Sigma _{\text {congl},11}\). |

2. | Computation of macroscopic stress tensor according to Eq. (5). |

3. | Downscaling of macroscopic stress tensor to the level of hydroxyapatite needles as function of the needle orientation, \(\varvec{\sigma }_\text {HA}^\text {polyHA}(\vartheta ,\varphi )\), \(\vartheta \,=\,0\ldots \pi \), \(\varphi \,=\,0\ldots 2\pi \), by means of Eqs. (2–13). |

4. | Calculation of the corresponding normal and shear stress component experienced by single hydroxyapatite needles, for any needle orientation, \(\vartheta \,=\,0\ldots \pi \), \(\varphi \,=\,0\ldots 2\pi \), and for any tangential plane, \(\psi \,=\,0\ldots 2\pi \), by means of Eqs. (14–18). |

5. | Evaluation of the failure criterion given by Eqs. (19) and (20), respectively: |

–If \({\mathfrak {f}}_\text {HA}(\varvec{\Sigma }_\text {congl})\,<\,0\), then \(|\Sigma _{\text {congl},11}|\) is increased; return to step 2. | |

–If \({\mathfrak {f}}_\text {HA}(\varvec{\Sigma }_\text {congl})\,=\,0\), then the load iteration is completed, and the current magnitude for \(\Sigma _{\text {congl},11}\) induces failure of the material. | |

–If \({\mathfrak {f}}_\text {HA}(\varvec{\Sigma }_\text {congl})\,>\,0\), then \(|\Sigma _{\text {congl},11}|\) is decreased; return to step 2. |

### 4.2 Computation of composition-dependent macroscopic loading inducing hydroxyapatite needle-failure

Furthermore, the magnitude of the macroscopic, material failure-inducing loading increases with increasing bone volume fraction (\(f_\text {bone}^\text {congl}\)), as well as with decreasing volume fraction of the macropores (\(\phi _\text {macro}^\text {congl}\)), see Fig. 7d.

## 5 Development of failure-inducing macroscopic stresses during bone regeneration

*t*is the time after placing the scaffold in a bony environment, \(k_\text {growth}\) is the bone growth rate, and \(r_\text {gran}\) is the radius of the granules. For the presently studied material, the typical bone growth rate is \(k_\text {growth}=4\pm 3\,\upmu \)m/week (Scheiner et al. 2016). The time-dependent bone tissue volume fraction defined by Eq. (21) enters, as given in great detail in (Scheiner et al. 2016), the material functions \(\overline{\Gamma _\text {gran,1}^k}\), \(\overline{\Gamma _\text {gran,1}^\mu }\), \(\overline{\Gamma _\text {gran,2}^\mu }\), and \({\mathcal {D}}_k\) appearing in Eqs. (2–4). In this way, Eq. (21) induces time-dependency in the strength of the investigated conglomerate material. In the same way, Eq. (21) induces time-dependencies in the homogenized stiffness expressions relating to the RVEs depicted in Fig. 1, as given in (Scheiner et al. 2016).

Values chosen for model input parameters, in order to study corresponding variations in the macroscopic failure load of bone tissue-coated hydroxyapatite-based scaffolds

Model input parameter | Unit | Numerical value (s) |
---|---|---|

Initial microporosity \(\phi _{\text {micro},0}^\text {polyHA}\) | (–) | 0.445 |

Mesoporosity \(\phi _\text {meso}^\text {gran}\) | (–) | 0.189 |

Macroporosity \(\phi _\text {macro}^\text {congl}\) | (–) | 0.3, 0.4, 0.5 |

Crack density parameter | (–) | 0, 10, 25 |

Granule radius \(r_\text {gran}\) | (\(\upmu \)m) | 300, 500, 1000 |

Bone formation rate \(k_\text {form}\) | (\(\upmu \)m/week) | 4, 7, 10 |

Scaffold resorption rate \(k_\text {res}\) | (week\(^{-1}\)) | 0, 0.008, 0.016 |

The scaffold resorption rate, in turn, governs both the load-carrying capacity of the bone-coated granules when the macroporosity is completely filled with new bone tissue, as well as the long-term development afterwards. Zero resorption implies that the value of \(\Sigma _{\text {congl},11}^\text {ult}\) is maintained at a constant (maximum) level (related to complete filling of the pore space by bone matrix), whereas a non-zero resorption rate causes a long-term decrease of \(\Sigma _{\text {congl},11}^\text {ult}\) after reaching the aforementioned maximum value, see Fig. 8b. It should be noted that this long-term decrease is caused by assuming that bone growth merely occurs on the outer surface of the (bone-covered) granules. However, in reality, it can be assumed that after substantial resorption of the hydroxyapatite crystals the physiological solution enters the micro- and mesopore spaces, leading eventually also to bone formation within the granule body. The omission of this potential additional bone formation effect can be deemed as limitation of our model.

Similar to the granule radius, the bone formation rate only influences the time span until the whole macropore space is filled with bone tissue; the long-term development of the load-carrying capacity is unaffected, see Fig. 8c.

Remarkable effects are revealed when varying the crack density parameter *e*, see Fig. 8d. In particular, decreasing the crack density parameter leads to a significantly increased stiffness of the granule material. Considering that in a composite material (such as the material studied in this paper) stiffer constituents attract larger fractions of a macroscopically applied loading than softer constituents, which attract lower fractions of the macroscopic loading, a granule material containing less cracks hence transfers higher stresses to the hydroxyapatite needles than those with more cracks. This eventually implies that increasing the crack density in the granule material leads to an increased load-carrying capacity. This possibly counterintuitive conclusion straightforwardly suggests that future extensions of the here presented model should comprise formulation of a failure criterion related to the bone tissue as well, in order to improve the significance of the model-predicted load-carrying capacity.

## 6 Discussion and concluding remarks

In this paper, a continuum micromechanics-based model was presented for estimating the macroscopic loading acting onto a hydroxyapatite-based granular biomaterial, developed for application as bone replacement material (in mandibular bone), that leads to quasi-brittle failure of the material’s main constituent, i.e. hydroxyapatite crystals. The parametric studies presented in Sect. 5 show how the load-carrying capacity of the studied biomaterial develops over time once placed in the targeted physiological environment, i.e. the immediate vicinity of mandibular bone, considering the growth of new bone tissue on the surface of the scaffold material, and resorption of the hydroxyapatite needles. Thereby, main emphasis was on highlighting the influence of specific design parameters of the production process, e.g. the exact chemical composition of the biomaterial might influence the rates of bone ingrowth and scaffold resorption, the crack density may be related to the production process, and the macroporosity can be tuned based on the packing density of the granules. From a practical point of view, the presented modeling approach allows to determine from when onwards a particular area of the mandible including an implant composed of the studied biomaterial can be used for mastication if the mechanical loading acting onto this mandibular region is approximately known, e.g. from Finite Element simulations (Korioth et al. 1992; Meijer et al. 1993; Choi et al. 2005; Hellmich et al. 2008; Bevilacqua et al. 2011).

However, the simulation results also point out two model restrictions. On the one hand, the kind of counterintuitive observation was made that a severely cracked granule material implies that the bone tissue growing on the granule surface attracts most of themacroscopically applied stress. Thus, our model suggests that increasing the crack density leads to an increasing load-carrying capacity, owing to the fact that the employed failure criterion considers only the most unfavorably stressed hydroxyapatite needle contained in the granules, neglecting however the stress experienced by the newly formed bone tissue. On the other hand, our model does not consider that the dissolution of hydroxyapatite needles would eventually lead to morphological changes in the microporous hydroxyapatite matrix (hierarchical level I in compare Fig. 1), implying that physiological solution could enter the micropore space, facilitating there bone tissue formation. Given that the described model inadequacies become relevant only after a certain (not yet quantifiable) time span, but not directly after scaffold implantation, leads to the conclusion that our model is particularly accurate for early-age bone-scaffold conglomerates (with respect to the time instant when the granules are placed into the targeted physiological environment), while the prediction accuracy presumably diminishes over time. This restriction constitutes the basis for reasonable future research directions.

Finally, it is also important to discuss the relevance of traditional fracture mechanics approaches, typically focusing on the prediction of crack propagation, thus assuming the existence of an initial crack (Müller et al. 2002; Näser et al. 2007; Kolednik et al. 2010; Ott et al. 2010), in the context of the material studied in this paper. Actually, it seems to be a worthwhile subject of future research actitivities to extend the micromechanics-based assessment of specific, microscopically sized material constituents (as demonstrated in this paper) towards traditional fracture mechanics, see e.g. (Pichler et al. 2007; Pichler and Dormieux 2009b, a). Such extension would be particularly relevant for mature bone-scaffold conglomerates containing already a substantial amount of bone tissue, allowing for studying the effects of crack emergence and propagation in the bone tissue—given that a respective failure criterion has been formulated, see e.g. (Fritsch et al. 2009b)—as well as of biologically driven crack healing.

## Notes

### Acknowledgments

Open access funding provided by Vienna University of Technology. Partial financial support by the European Research Council (ERC), in the framework of the project *Multiscale poromicromechanics of bone materials, with links to biology and medicine* (project number FP7-257023), as well as the partial financial support by the Russian Science Foundation (Grant Number 15-13-00108), are gratefully acknowledged. Furthermore, COST-action MP1005, *NAMABIO—From nano to macro biomaterials (design, processing, characterization, modeling) and applications to stem cells regenerative orthopedic and dental medicine* has provided means for a sustainable cooperation over several years.

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