# Patient-Specific Bone Multiscale Modelling, Fracture Simulation and Risk Analysis—A Survey

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## Abstract

**:**

## 1. Interested Parties Concerning this Survey

## 2. Reading this Paper—Textual Organization and Notation

Characteristic Length | Numerical values, surveyed from the literature, that characterize major geometric features of a specific bone lengthscale, see Definition 8. |

Definition | Non-mathematical definitions that may be differently understood by specialists from different fields. Mathematical definitions are not presented here due to their complexity. Rigorous mathematical definitions are found in the references present in Appendix C. |

Highlight | A statement that plays a major role in the argumentation. |

Illustration | A non-mathematical explanation of a physical phenomenon. |

Open Issue | Issues and problems not clearly defined or completely solved within the surveyed literature. |

Remark | Relevant notes. |

**CT**”.

## 3. Motivating Patient-Specific Bone Fracture Simulation

#### 3.1. Ageing Population

#### 3.2. Osteoporosis—A Major Health Concern

**Highlight**

**1.**

#### 3.3. Osteoporosis—Consequences and Costs

## 4. Motivating This Literature Survey

## 5. Collecting Bibliographical References

**Highlight**

**3**(Usage of the Word Model).

**Highlight**

**4**(Usage of the Word Simulation).

**Highlight**

**5**(The Focus of This Survey).

## 6. The Physico-Mathematical Approach to Bone Fracture

**Remark**

**1**(Continuum Mechanics Concepts).

## 7. Modelling Bone as a Continuum

**Remark**

**2**(Continuous Boundary).

**Illustration**

**1**(Domain and Boundary).

- Row II of Figure 1 illustrates the derivation of boundary conditions from accident models which are further described in Section 11.

**Illustration**

**2**(Boundary Condition).

- Row III of Figure 1 illustrates the two types of BCs for solid continuum mechanics problems which are briefly described in Section 11.
- Row IV of Figure 1 illustrates the inputs (body forces in the domain, if present, and surface forces and displacements on the boundary, i.e., as BCs) and outputs (surface forces and displacements everywhere) of solid continuum mechanics problems which are further discussed in Section 11.
- Row V of Figure 1 illustrates the motion and strain-displacement equations which are further described in Section 12.
- Row VI of Figure 1 illustrates the relationships between stresses and strains given by constitutive equations which are further described in Section 10.
- Rows VII to VIII of Figure 1 illustrate different types and categorizations of constitutive equations which are further described in Section 10.

**Remark**

**3.**

- Rows IX to X of Figure 1 illustrate different types and categorizations of material symmetry regarding mechanical properties which are further described in Section 10.
- Rows XI to XII of Figure 1 illustrate homogeneity regarding mechanical properties, i.e., material symmetry and constitutive equations, which are further described in Section 10.
- Row XIII of Figure 1 illustrates the experiment-based categorization of mechanical properties, i.e., homogeneity, material symmetry and constitutive equations, which are further described in Section 10, Section 13 and Section 15.

**Remark**

**4**(Model Accuracy and Validation).

## 8. Categorizing the Surveyed Literature into a Continuum Mechanics Framework

- Column 2 of Table 2 categorizes the surveyed literature based on if the inertial term of the motion equation was neglected or not, see Section 12.
- Column 3 of Table 2 categorizes the surveyed literature based on if the non-linear term of the strain-displacement equation was neglected or not, see Section 12.
- Column 4 of Table 2 categorizes the surveyed literature regarding the constitutive equation of their models, see Section 10.
- Column 5 of Table 2 categorizes the surveyed literature regarding the material symmetry of their models, see Section 10.
- Column 6 of Table 2 categorizes the surveyed literature based on the homogeneity regarding the mechanical properties of their models, see Section 10
- Column 7 of Table 2 categorizes the surveyed literature based on the source of the mechanical properties of their models, see Section 10.
- Column 8 of Table 2 categorizes the surveyed literature based on the BCs imposed on their models, see Section 11.

**Remark**

**5.**

**1.**) There is more than one model in the reference, each set up differently; (

**2.**) The reference considers a multiscale model and each lengthscale is modelled differently (

**3.**) The model is inhomogeneous and each locally homogeneous subdomain is modelled differently.

**Remark**

**6**(Lack of Clarity).

**Remark**

**7**(Unifying Framework for Bone Continuum Modeling).

**Remark**

**8**(Challenges & Limits).

## 9. Patient-Specific Geometry of Bone

**Illustration**

**3.**

**Definition**

**1**(Patient-Specific: in vivo vs. in vitro).

**Remark**

**9**

**Definition**

**2**(Computational Bone Model).

**Open Issue**

**1.**

**Definition**

**3**(Subject-Specific and Specimen-Specific).

**Computed Tomography**(CT), or X-ray computed tomography, is the most used medical imaging technique among the surveyed literature, as demonstrated in Table 2. CT is argued to be the most accurate 3D medical imaging technique for the creation of computational bone models [151]. However, CT is not recommended for routine clinical examinations due to associated high radiation dosages [152].

**Remark**

**10**(CT Resolution).

**HR-pCT**is a high-radiation CT restricted to the peripheral sites of the body, e.g., distal skeleton. HR-pCT provides in vivo imaging with spatial resolution smaller than 100 µm [75,81,153,154]. Similarities between the micro-geometry of peripheral bones and the micro-geometry of non-peripheral bones were discussed and considered by [154,155]. Though HR-pCT is being increasingly used for in vivo bone research, its use has been limited to the distal radius and tibia [156].

**µCT**features a spatial resolution of about 1 µm, higher than that of HR-pCT, enabling delineation of the trabecular microstructure [157,158]. Due to high associated radiation dosages, its usage is restricted to biopsy specimens [153,159]. In comparison to HR-pCT, µCT captures the trabecular porosity more accurately [160].

**Remark**

**11**(QCT and HR-pQCT).

**Dual-Energy X-ray Absorptiometry**(DEXA, DXA) is the clinical standard to diagnose osteoporosis and fracture risk by measuring areal Bone Mineral Density (aBMD) [97,164,165]. DXA can also contain non-BMD parameters that are correlated to bone fracture [128,166]. The main advantage of DXA over CT is that DXA requires minimal radiation exposure. However, ref. [97,163,167] present a “3D-DXA” method capable of assessing the bone femoral shape and density distribution from 2D DXA images. They are based on statistical shape and appearance models and show good correlations between 3D-DXA and CT. However, DXA is still not as accurate as CT in, e.g., predicting femoral strength [97].

**Magnetic Resonance Imaging**(MRI) is the most suitable technique method for in vivo 3D geometry modelling since it emits no harmful ionizing radiation. However, although comparable to CT-based geometry models, MRI-based geometry models are not as accurate as CT-based geometry models [151]. The bone microstructure can be effectively imaged by µMRI [53,83]. µMRI-based models are also effective in assessing mechanical properties, but µCT-based models are still more accurate [48,130].

**Microscopy**provides very fine and detailed images featuring the nano and even sub-nano lengthscale bone geometry. However, this technique is invasive and only able to prove 2D geometry [83,132,133,134]. Microscopy-based 3D geometry models can be created when the third dimension is idealized [134], e.g., when a circle is turned into a cylinder. However, creating the third dimension from scratch is not considered subject-specific.

**Remark**

**12.**

**Remark**

**13.**

**Remark**

**14**(Mesh Generation).

**1.**) Voxel-based meshing defines the mesh contour as the voxel contour [74,83,168,169]. Each voxel turns into a hexahedron-shaped volume, i.e., cube or rectangular cuboid. This type of mesh generation requires no material mapping strategy, see Section 10. However, it may exhibit locations (corners) where stress is concentrated and can only accurately represent the surface of the bone geometry when the mesh is sufficiently fine. Very fine meshes increase the number of nodes and sub-domains and are thus computationally more expensive; (

**2.**) Geometry-based meshing defines the mesh contour based on the surface of the geometry model [119,168]. It requires a material mapping strategy, see Section 10. Geometry-based meshing is difficult to implement computationally, but several commercial software packages (Ansys, Abaqus, Hypermesh, Gmesh, et cetera) already provide it.

## 10. Mechanical Properties Categorization for Computational Bone Models

**Open Issue**

**2.**

**Open Issue 2**aims to hint on what the lack of such justification consists of, by presenting some literature on how $\mathbf{10}{\mathcal{A}}_{\mathbf{2}}$ influences $\mathbf{10}{\mathcal{A}}_{\mathbf{3}}$. Some of these influences are mathematically described by appropriate choices of $\mathbf{10}{\mathcal{B}}_{\mathbf{1}}$, $\mathbf{10}{\mathcal{B}}_{\mathbf{2}}$ and $\mathbf{10}{\mathcal{B}}_{\mathbf{3}}$.

**1.**) bone sample geometry, (

**2.**) bone intensive properties, (

**3.**) phenomenological aspects of bone observation/experimentation and (

**4**) patient-specific characteristics that influence issues (

**1**)–(

**3**).

**1**) Some issues regarding bone sample geometry:

OI 10.11 | Geometric irregularities at the transverse cross-section: contrary to longitudinal geometric irregularities, they contribute significantly to the linearly elastic torsional behaviour of long bones [173,174]. |

OI 10.12 | Bone aspect ratio: long bone failure may be more dependent on deformation rather than on stress [175]. |

OI 10.13 | Microstructure: influences the fatigue life of bone [176,177]. The vascular pattern of bone affects its Young’s Modulus [178]. |

**2**) Some issues regarding bone intensive properties:

OI 10.21 | Temperature: influences the fatigue life of bone [176,177]. |

OI 10.22 | Water content: influences the stiffness, strength and toughness of bone [179,180]; the Young’s Modulus of dead (dry) and living (wet) bones tend to be different [178]. Furthermore, viscoelastic [181] properties of bones are also influenced by the water content. The water content of bone is related to the molecular scale, see Section 13. |

OI 10.23 | Mineral content: porosity and mineral content influence bone Young’s Modulus [182,183,184]. The mineral content of bone is related to the molecular scale, see Section 13 |

OI 10.24 | Bone density: exhibits a p-value based highly significant positive correlation with bone fatigue life [176,177]. Furthermore, density influences bone stiffness and strength [185]. |

OI 10.25 | Porosity: alongside bone mineral content, influences bone Young’s Modulus [182]. |

**3**) Some issues regarding bone phenomenological aspects when under observation/experimentation:

OI 10.31 | Strain-rate: is directly proportional to bone Young’s Modulus under tension and under compression [186] and influences bone compressive strength [187]. |

OI 10.32 | Loading condition: [178] presents a comparison of bone Young’s Modulus for femur and tibia under tension, compression and bending. Experiments performed by [188] exhibited the same mechanical properties for tension and compression in bone. |

OI 10.33 | Stress duration: influences bone Young’s Modulus in a phenomenon labelled elastic after-effect [178]. |

OI 10.34 | Cyclic loading frequency: influences bone Young’s Modulus [189]. |

OI 10.35 | Stress amplitude: influences the fatigue life of bone [176,177]. |

**4**) Some issues regarding patient-specific bone characteristics:

OI 10.41 | Patient age: affects ultimate tensile strength, elastic modulus, maximum deformation, and Brinell hardness [190] and the bone structure in such a way that increases its fracture risk [191]. |

OI 10.42 | Diseases: affect the rate of bone remodelling, see Remark 19, and consequently the percentage of bone mineral content (OI 10.23) and BMD distribution, i.e., the mechanical properties of bone [192,193,194]. Fracture risk analysis in unhealthy, e.g., metastatic, bones is currently even less accurate than fracture risk analysis in healthy bones [195]. |

OI 10.43 | Nutrition: a well-balanced diet (including plant-based diets [196,197,198,199,200]) alongside an adequate intake of Calcium and Vitamin D (sunlight exposure time) may reduce osteoporosis-induced fracture risk and hospital costs [196,201]. |

OI 10.44 | Physical activity: increases not only quality of life [202,203], but also BMD and bone mechanical properties values [204]. Furthermore, regular exercise enhances bone mass and strength, and reduces bone fracture risk [205]. |

**Remark**

**15.**

#### 10.1. Constitutive Equation

**Open Issue**

**3.**

**Elastic materials**, see Appendix C, can be either

**C**auchy-

**L**inear-

**E**lastic (CLE) or

**C**auchy-

**N**on

**L**inear-

**E**lastic (CNLE), see Figure 1 row VIII.

**CLE-materials**display spring-like behaviour according to Hooke’s law: $\mathit{T}\left(\mathit{x}\right(t),t)=\mathbb{C}\mathit{E}\left(\mathit{X}\right(t),t)$. A CLE-material may not comply to Hooke’s Law when there are unknown contributions to the stifness tensor $\mathbb{C}$ that are implicitly, but not explicitly, dependent on the strain tensor $\mathit{E}$. Most of the surveyed literature, as seen in Table 2 column 4, assume that bone is an elastic material; all literature in Table 2(4A) assumes bone complies to Hooke’s Law. Though many materials can be accurately modelled as CLE, the literature on bone mechanical properties rarely reports experimental verifications of CLE-behaviour in bones.

**CNLE-materials**are usually modelled by constitutive equations that correlate stress and strain-energy: $\mathit{T}\equiv \mathit{T}\left(\mathit{x}\right(t),t,W(\mathit{E}\left)\right)$. Non-linear stress-strain correlations may be linearized into affine approximations [207], which are still not linear correlations. Though none of the surveyed literature reports bone to be CNLE, human soft tissue, also present at muscle-bone connection sites, displays Green-elastic (hyperelastic) behaviour [125,208]. For exhibiting quasi-brittle fracture in experiments, bone is sometimes assumed to be a CLE-material [112], e.g., strain measurements performed by [113] have shown this to be a reasonable assumption for femurs. Furthermore, ref. [129,209] assume that the proximal femur behaves as a CLE-material up to fracture, i.e., that the post-yield behaviour, i.e., the plastic behaviour, can be neglected.

**Remark**

**16.**

**Linear Material**Terminology). A material is labelled linear, e.g., Hookean, if it can be accurately modelled by a constitutive equation that exhibits a linear relationship between stress $\mathit{T}$ and strain $\mathit{E}$. Though not all elastic materials are linear, e.g., Green-elastic materials, only elastic materials may be labelled linear. Viscoelastic materials are, misleadingly, labelled linear materials [210,211] even though they are modelled by a constitutive equation that exhibits a linear relationship between stress $\mathit{T}$ and strain-rate $\dot{\mathit{E}}$ instead of between stress $\mathit{T}$ and strain $\mathit{E}$. Figure 1 row X classifies materials into

**L**inear-

**E**lastic (LE) and

**N**on-(

**L**inear-

**E**lastic) (NLE). Emphasis on the subtle distinction between NLE- and CNLE-materials: the first refers to the set of all materials excluding the LE-materials, the latter refers to the set of all elastic materials excluding CLE-materials.

**Plastic materials**, see Appendix C, feature one or more particles that do not return to their unstressed spatial position after unloaded, thus exhibiting long-term memory of previous stresses and strains. Among the NLE constitutive equations, plastic constitutive equations (or elastic–plastic, elastoplastic) are the most frequently used for modelling bone [184,213,214,215,216]. Plastic constitutive equations may accurately predict the failure of vertebrae [217]. Some of the surveyed literature does not explicitly justify the choice of assuming bone as a plastic material [33]. Nevertheless, the entanglement of different molecules that compose bone may justify its plastic behaviour [218].

**Remark**

**17.**

**Elastic-Plastic**, or elastoplastic, materials.

**Elasticity**and

**Plasticity**are modelled by stress-strain constitutive equations. As stated in Remark 3, constitutive equations are not limited to stresses and strains. Constitutive equations can be systems of equations accounting for several phenomena affecting the stress-strain relationship. This papaer presents three such phenomena:

**Viscosity**,

**Porosity**and

**Damageability**.

**Elastic-Viscous materials**, or viscoelastic materials, see Appendix C, exhibit stresses dependent on strain-rate: $\mathit{T}\equiv \mathit{T}(\mathit{x}\left(t\right),t,\mathit{E},\dot{\mathit{E}})$. Other physical phenomena of viscous materials include stress-strain hysteresis, creep and stress relaxation [219] (p. 436), [220]. Phenomena identified by [178], who studies only aspects $\mathbf{10}{\mathcal{A}}_{\mathbf{1}}$, $\mathbf{10}{\mathcal{A}}_{\mathbf{2}}$ were interpreted by [221] as implying that a viscoelastic constitutive equation was an accurate mathematical model for the execution of step $\mathbf{10}{\mathcal{A}}_{\mathbf{3}}$. Usage of viscoelastic constitutive equations may also be justified by the fact that bone mass is $\approx \phantom{\rule{-2.125pt}{0ex}}30\%$ collagen, see Section 13.1, which has been experimentally characterized as viscoelastic [206,222,223]. It has been experimentally verified that biological soft-tissue, which is mostly composed of collagen, can be accurately modelled by the Voigt, Maxwell and Kelvin viscoelastic constitutive equations, see [206,224,225] and references therein.

**Plastic-Viscous materials**, or viscoplastic materials, see Appendix C, are plastic materials that exhibit post-yield strain-rate dependency, which has been experimentally verified at the macroscale [228,229,230,231]. Still, few works ventured to model bone as a viscoplastic material.

**Highlight**

**6.**

**Definition**

**4**(Porous Material).

**Elastic-Porous materials**, or poroelastic materials, see Appendix C, in which fluid flows through porous elastic solids, are modelled by equations from the theories: of elasticity, of viscous fluid flow and of fluid flow through porous media, see [222,234] and references therein. When devising a multiscale poroelastic cortical bone model, ref. [144] found that the fluid flow influences the stiffness of bone. A constitutive equation accounting for the pressure both in the material pores $\mathbf{10}{\mathcal{C}}_{\mathbf{1}}$ and over interconnected fluid compartments $\mathbf{10}{\mathcal{C}}_{\mathbf{2}}$ within a porous solid is studied in [235]; in bone, $\mathbf{10}{\mathcal{C}}_{\mathbf{1}}$ may refer to the collagen-water-hydroxyapatite-lattice lengthscale, see Section 13, and $\mathbf{10}{\mathcal{C}}_{\mathbf{2}}$ may refer to the bone marrow-filled intertrabecular pores, see Section 13.

**Plastic-Porous materials**, or poroplastic materials, see Appendix C, in which fluid flows through porous plastic solids, are modelled by equations from the theories: of plasticity, of viscous fluid flow and of fluid flow through porous media. From a generic poroplastic model for binary mixtures, where the mixture may be assumed as consisting of solid bone and biomaterial, ref. [236] estimated the yield stress associated with the outset of remodelling, see Remark 19

**Definition**

**5**(Damageable Materials).

**Remark**

**18.**

**Remark**

**19**(Bone Remodelling).

**1.**) bone resorption, i.e., bone tissue erosion by osteoclasts; (

**2.**) bone formation, i.e., bone synthesis by osteoblasts. Osteoporosis and several other bone diseases are a consequence of bone remodelling malfunction [244,245], i.e., higher ration of bone resorption in comparison with bone formation.

**Definition**

**6**(Stress Shielding).

#### 10.2. Material Symmetry

**Isotropic materials**, see Appendix C, are the most implemented material symmetry among the surveyed literature, see Table 2(5A). Isotropic materials are easier to implement than anisotropic materials (they possess only two independent constants out of twenty-one possible, triclinic). Patient-specific, e.g., QCT-based, estimation of anisotropic material symmetry is still a non-mature field of research [31,36]. This might be another reason why isotropic materials are more often implemented, especially among patient-specific computational bone models.

**Open Issue**

**4**(Isotropy Assumption).

**Transversal Isotropic materials**, see Appendix C, are the most implemented anisotropic material symmetry among the surveyed literature, see Table 2(5C). Bone exhibited experimental transversely isotropic material symmetry in some works [233,253,254]. Recent works modelled bone as a transversally isotropic material [220,255].

**Orthotropic materials**, see Appendix C, are considered to best describe bone material symmetry. Bone exhibits orthotropic material behaviour in many works, e.g., [44,212,256]. However, small differences between stresses and displacements calculated assuming isotropic and orthotropic patient-specific mechanical properties have been found, e.g., by [44].

**Triclinic materials**, or general anisotropic materials, and other types of material symmetry, see Appendix C, and their application to model bone is still a non-mature field of research. In vitro experiments have not shown such behaviours. That is mainly because triclinic material symmetry could not be experimentally measured and identified [212]; triclinic symmetry could only be assumed. Later, however, experimental methodologies for determination of all triclinic symmetry parameters was presented [257,258].

#### 10.3. Homogeneity Regarding Constitutive Equation and Anisotropy

**Homogeneous materials**, see Appendix C, feature, at any arbitrary pair of points within their spatial domain, mechanical properties of the same numerical value. Though some materials can be accurately modelled as homogeneous, no real-world material fits such description. Computer implementation of a homogeneous material is a mature field of research.

**Inhomogeneous materials**, see Appendix C, feature, at any arbitrary pair of points within their spatial domain, mechanical properties that are not necessarily of the same numerical value. Estimation of inhomogeneous mechanical properties from medical imaging-based geometry models is straightforward and has been performed by many of the references in Table 2(6C).

**Open Issue**

**5.**

**Remark**

**20**(A General Remark on the Physics of Bone Modelling).

#### 10.4. Patient-Specific Mechanical Properties

**Remark**

**21**(vBMD).

**Highlight**

**7**(Patient-Specific Phantomless Estimation of BMD).

**Remark**

**22**(Bone Mechanical Properties from Continuum Micromechanics).

**Open Issue**

**6.**

## 11. Mathematical Model of Bone Trauma-Inducing Accident—The Boundary Conditions

FS 1 | Normal human gait, i.e., walk or run. The individual is in motion through the movements of the legs, e.g., at stance position. |

FS 2 | Tip over, or equilibrium loss. This stage characterizes the fall. The equilibrium loss occurs when the challenge to balance is greater than the ability or strength to stay upright. |

FS 3.1 | 1st environment collision. The first collision between the body and a solid surface from the environment, e.g., the floor. It is usually the most intense and fracture-susceptible collision. The first collision is usually followed by a series of other collisions caused by inertial movements. Picture a bouncing ball; the idea is the same. As long as the inertial forces are greater than the ability to stop them, collisions will follow. |

FS 3.2 | >2nd environment collision. The second collision may have one or more contact points, or zones, between the body and environment, e.g., the individual may hit the floor with both hands or with hip and a hand at the same time. |

FS 3.i | i-th environment collision. The i-th collision may have one or more contact points, or zones, between the body and the environment. |

FS 3.n | n-th environment collision. Similar to the second collision, the n-th, and last, collision may have one or more contact points, or zones, between the body and environment. It is often the least fracture-susceptible collision. The first collisions have already absorbed most of the kinetic energy of the fall. |

FS 4 | Final position. Characterizes the accommodation of the body. Here there is only minor motion. There is no more collision with the environment. The individual has already fallen and looks for a rest position. The accommodating motion is not relevant for fracture. |

**FS 3.1**. Furthermore, all instantaneous motion described by FSs can be considered a quasi-static equilibrium. Thus, it is reasonable to evaluate the motion equation in a quasi-static sense, i.e., equilibrium equation [147,294], see Governing Equation (1).

**Open Issue**

**7**(Body BC vs. Bone BC).

**1.**) if you fall on your hand(s), the wrist is the most fracture-susceptible bone; (

**2.**) if you fall on your back, the spine is the most fracture-susceptible bone; (

**3.**) if you fall on your backside, the hip bone is the most fracture-susceptible bone, (

**4.**) if you fall on your knees, the femur is the most fracture-susceptible bone.

**FS3.1**and

**FS3.n**there are multiple collisions with the environment. It may be argued that only the most intense and fracture-susceptible collision should be modelled for being the most relevant one, however, a sequence of many less intense collisions may also lead the bone to fracture.

**Remark**

**23**(Multiple Collisions).

**Highlight**

**8.**

## 12. Simulating Bone Fracture

**Remark**

**24**(Governing Equation).

**Governing Equation**

**1**(Motion Equation).

**Governing Equation**

**2**(Strain-Displacement Equation).

**Governing Equation**

**3**(Compatibility Equation).

**Remark**

**25**(Misleading Term—Nonlinear Analysis).

**1.**) material nonlinearity, i.e., when an NLE constitutive equation is used, see Remark 16; (

**2.**) geometric nonlinearity, i.e., the strain-displacement equation does not include the second-order term, see Governing Equation (2); (

**3.**) kinematic non-linearity, i.e., when the displacement BCs depend on the deformations of the structure; and (

**4.**) force nonlinearity, i.e., when the applied forces depend on the deformation of the structure.

**Finite Element Method**(FEM) subdivides the spatial domain into subdomains (or elements) and approximates the governing equations by traditional variational methods over each subdomain [303]. The FEM is by far the most used numerical method in the bone fracture literature [148,160,304]. Most probably because there are many commercial software with friendly user interfaces that facilitate its operation, and because the FEM is a mature field of research which has been optimized for several applications. For instance, FAIM, a finite element solver optimized for solid mechanics simulations of bone, was developed by [305,306].

**Boundary Element Method**(BEM) requires discretization of the boundary only and, for this reason, usually requires a smaller number of DOF than the FEM to achieve accurate results [307,308]. A discretization of the spatial domain into subdomains, commonly labelled subregions by the literature on BEM, is required when the analyzed material is inhomogeneous [307,309], see Section 10. The BEM has been scarcely used in the field of bone fracture. However, some works have used the BEM for bone remodelling simulation [310,311,312,313,314,315].

**Finite Difference Method**(FDM) is simple in formulation, but exhibits some difficulties in modelling complex geometries and, for this reason, has been scarcely used for solid mechanics problems in recent years [316,317]. FDM was used by [318] to simulate bone remodelling, see Definition 19.

**Open Issue**

**8**(Exploring BEM and FDM).

**Remark**

**26**(Inputs for Numerical Methods).

**Linear Elastic Fracture Mechanics**(LEFM), the classical and mature cracking process mathematical model, is restricted to elastic materials. Though largely applied, LEFM is not the most appropriate approach to describe crack propagation in bones [326].

**Elastic-Plastic Fracture Mechanics**(EPFM), though more recommended for materials exhibiting large plastic zones (of the same order of magnitude as the crack size) at the crack tip, has been less successful than LEFM in predicting fracture when large yielding prevails [324]. The

**Cohesive Zone Model**(CZM) is based on considering fracture separation occurring at an extended zone ahead of the crack tip (also labelled “cohesive zone”). Two reasons make the CZM superior to LEFM for bone fracture analysis: (

**1.**) Bone fracture experimental data analysis performed by [327] demonstrated the need for a nonlinear model considering a spatial stress distribution at the fracture zone. (

**2.**) Unlike the LEFM, the CZM can remove stress singularities ahead of the crack tip; i.e., ahead of the furthest extent of damage [143]. Furthermore, both LEFM and EPFM require a pre-existing initial crack, whereas the CZM can be modelled at the interface between continuum elements (spatial sub-domains) [328].

**Remark**

**27**(Animal Bone Modelling and Simulation).

**Definition**

**7**(Specimen).

## 13. The Multiscale Structure of Bone

**Remark**

**28**(Scales Classification).

#### 13.1. Molecular Scale—${H}_{2}O$-CLG-HA Lengthscale

**Characteristic Length 1**(Molecular Scale).

HA mineral crystal and CLG molecule: | HA length | 20–200 nm | [61,139,191,378,379,381,382,383] |

HA width | 15–70 nm | [61,139,191,378,379,382,383] | |

HA thickness | 1.5–5 nm | [61,139,191,378,379,382,383] | |

CLG diameter | 1.5–3.5 nm | [61,139,378] | |

CLG length | 300 nm | [61,139,222,378,384] |

**Remark**

**29**(Proportion of HA, CLG and ${H}_{2}O$ Contents).

#### 13.2. Sub-Nanoscale—Mineralized Collagen Fibrils Lengthscale

**Remark**

**30**(Mineral Within Bone).

**Characteristic Length 2**(Sub-nanoscale).

Mineralized Collagen Fibril: | mCLGf diameter | 20–150 nm | [61,139,378,379,400,401,402] |

mCLGf length | 10,000–30,000 nm | [401,403,404] | |

CLGs gaps | 35–44 nm | [61,139,222,378,379] |

**Highlight**

**9.**

#### 13.3. Nanoscale—Collagen Fiber Lengthscale

**Characteristic Length 3**(Nanoscale).

Collagen Fibers: | CLG fiber diameter | 0.15–0.25/2–3 µm | [378]/[402,406] |

CLG fiber length | ≈10–30 µm | several mCLGf lengths |

**Remark**

**31**(Fibers and Fibrils).

#### 13.4. Sub-Microscale—Lamella Lengthscale

**Characteristic Length 4**(Sub-microscale).

>Lamella: | lamella length | ≈10–30 µm | several CLG fiber lengths |

lamella width | ≈0.15–0.25 µm | several CLG fiber diameters | |

lamella thickness | 3–7 µm | [61,139,191,379] |

#### 13.5. Microscale—Osteon and Trabecula Lengthscale

**Characteristic Length 5**(Microscale).

Osteon and Trabecula: | osteon length | 10,000–20,000/1000–3000 µm | [222]/[379] |

osteon diameter | 200–300 µm | [191,222,379] | |

trabecula length | 1000 µm | [191] | |

trabecula thickness | 50–300 µm | [61,139,191,379] |

**Remark**

**32**(Bone Porosities).

#### 13.6. Mesoscale—Cortical and Trabecular Bone Lengthscale

**Remark**

**33**(Mesoscale—Multiscale Literature).

**Characteristic Length 6**(Mesoscale—Representative Volume Element (RVE)).

#### 13.7. Macroscale—Whole Bone Lengthscale

**Characteristic Length 7**(Macroscale—Whole Bone).

**1.**) long bones; (

**2.**) short bones; (

**3.**) flat bones; (

**4.**) irregular bones; (

**5.**) sesamoid bones. The huge majority literature on bone fracture simulation focuses on long bones, probably due to its beam-like geometry that enables simplified analytical calculations and experimental reproducibility.

## 14. Multiscale Modelling of Bone

**Definition**

**8**(Characteristic Length).

**Definition**

**9**(Higher- and Lower-Scales).

#### 14.1. Continuum Downscaling

**Periodic Boundary Conditions**(PBCs) are the most used BCs for spatial downscaling. As an advantage, they provide the fastest convergence of physical and mechanical properties of $L{S}_{x}$. As a disadvantage, the fact that they restrict the deformation to obey the structural frame periodicity of $L{S}_{x+1}$ imposes unphysical deformation constraints over localization zones (i.e., regions of relative extremely high deformation gradient where micro-cracks occur) [425,426].

**Minimal Kinematic Boundary Conditions**(MKBCs) ensure effective deformation shear strain but overestimate the number of localization zones near the domain boundary [425].

**Tesselation Boundary Conditions**(TBCs) maintain the point-to-point conditions imposed by PBCs while shifting the periodicity frame to correspond to the developing localization zone. In biomaterials, e.g., bones, when transitioning from $L{S}_{x}$ to $L{S}_{x+1}$, TBCs may give the least-error estimation of stresses and strains at $L{S}_{x}$ [426]. Four references feature bone multiscale analyses with downscaling: [53,69,83,143], see Table 3. To transition from the macroscale to the mesoscale, ref. [69] used a displacement interpolation procedure. To transition from the mesoscale to the microscale, ref. [69] transferred mesoscale displacements as BCs to the microscale. Likewise, ref. [143] transitioned from the macroscale to the mesoscale by applying displacements computed from the macroscale strain tensors as BCs on the boundary of the mesoscale RVE.

#### 14.2. Continuum Upscaling

**c**and

**d**. For example, ref. [140,141] present a cascade homogenization procedure for transitioning between several lengthscales. When transitioning from any $L{S}_{x}$ to any corresponding $L{S}_{x-1}$, Equation (4) interprets any spatial discontinuity within the RVE of $L{S}_{x}$ as a uniform volumetric redistribution of ${\mathbf{T}}^{L{S}_{x}}$ and ${\mathbf{E}}^{L{S}_{x}}$ over all space enclosed by the RVE.

#### Molecular Scale as a Non–Continuum Material

**Remark**

**34**(Linking Continuum and Non-Continuum Scales).

**inter**- and

**intra**-molecular interaction there are specific potential energy functions (or simply potentials), some of which are found in the literature referred to in Table 4.

**1.**) an initial-value problem, which requires the atoms’ initial positions $\mathbf{r}\left(0\right)$ and velocities $\dot{\mathbf{r}}\left(0\right)$; or (

**2.**) a boundary-value problem, which requires the atoms’ positions at an initial time instant $\mathbf{r}\left({t}_{i}\right)$ and at a final time instant $\mathbf{r}\left({t}_{f}\right)$. Both (

**1.**) and (

**2.**) require potential energy functions.

**Highlight**

**10**(Multiscale Modelling and Bone Remodelling).

**Open Issue**

**9**(Simulations Coupling 6 or More Lengthscales).

**Remark**

**35**(Scales Jumps).

## 15. Validating Bone Fracture Simulation

**Remark**

**36**(Validation and Experiments—An Iterative Process).

**Remark**

**37**(Validated?).

**Remark**

**38**(Towards Realistic Fracture Predictions).

**1.**) the multiscale structure of bone, see Section 13 and Section 14; (

**2.**) the most robust physico-mathematical approach, see Section 6, Section 7 and Section 8 and Section 12; (

**3.**) the most realistic mechanical properties, see Section 10; (

**4.**) the most realistic BCs, see Section 11; (

**5.**) proper validation, see Section 15.

## 16. Assessing Fracture Risk

#### 16.1. Single-Variable Risk Analysis

**BMD**measurements are the current clinical standard to diagnosis osteoporosis. According to the WHO, women with a BMD that lies 2.5 SD or more below the average value for young healthy women are classified as osteoporotic (T-score $\le 2.5$ SD) [11]. However, BMD alone is unable to accurately assesses fracture risk [454]. Patients classified as osteoporotic will not invariably suffer a fragility fracture; non-osteoporotic patients may also suffer a fragility fracture [11,14,455]. BMD can be used in conjunction with CRFs and available fracture assessment tools to improve the accuracy of fracture predictions. CRFs provide information on fracture risk that are unrelated to BMD [165]. As mentioned in Section 9, DXA is the clinical standard technique to measure BMD. QCT and ultrasound measurements are alternative techniques for the quantification of BMD [14,153].

**TBS**, an acronym for Trabecular Bone Score, is a grayscale-based (or HU-based) texture measurement influenced by the geometry of bone at the meso- and microscales [456]. A low TBS value may indicate thin trabeculae and a highly porous mesostructure, see Figure 4 (LS2.2, LS3.2) [457]. TBS contains structural information that are not captured by BMD measurements [457,458]. TBS can be used, though not very accurately, to assess fracture risk independently of BMD and CRFs [453,459]. TBS has been used in conjunction with BMD alone and with BMD and CRFs in available fracture risk assessment tools such as FRAX, to improve the accuracy of bone fracture predictions [458]. In the study performed by [459], however, TBS did not improve BMD and FRAX fracture predictions. TBS depends on HU variations obtained in vivo, which can have many causes [457] and is most commonly estimated using DXA. For more information on TBS see [14,459,460,461,462].

**BTMs**, an acronym for Bone Turnover Markers, are measurable indicators of bone turnover, e.g., blood and urine tests. Bone turnover, i.e., bone replacement, is the effect, the cause (mechanism) of which is bone remodelling, see Definition 19. Bone turnover refers to the volume of replaced bone per unit time [463]. Deterioration of bone microstructure, i.e., bone structure at the microscale, translates into a high value of bone turnover. BTMs indicate the degree of deterioration of the bone microstructure and, may thus, independently of BMD, predict a person’s fracture risk. Furthermore, BTMs can be used in conjunction with BMD to improve the accuracy of fracture risk assessment tools [464]. The use of BTMs in the osteoporotic risk analysis and in monitoring the efficacy of osteoporosis treatment is rapidly increasing [465,466,467,468].

#### 16.2. Multi-Variable Risk Analysis

**FRAX**(https://www.sheffield.ac.uk/FRAX/), the Fracture Risk Assessment tool, estimates individualized ten-year probability of hip, spine, forearm and proximal humerus osteoporotic fracture [469,470]. FRAX integrates eight main CRFs (prior fragility fracture, parental hip fracture, smoking, systemic glucocorticoid use, excess alcohol intake, body mass index, rheumatoid arthritis, and other causes of secondary osteoporosis), which, in addition to age and sex, contribute to fracture risk analysis independently of BMD. FRAX does not consider risk factors such as BTM and those associated with falls, lower dietary calcium intake and Vitamin D deficiency [464], but has BMD as an optional input variable [165,469,470]. FRAX predictions can become more accurate when used in conjunction with, e.g., BMD and TBS [458]. For more information on FRAX, see [41,91,453,471,472,473,474,475].

**QFracture**(https://qfracture.org/) predicts individual risk of osteoporotic and hip fracture based on several distinct CRFs (age, body mass index, ethnic origin, alcohol intake, smoking status, chronic obstructive pulmonary disease or asthma, any cancer, cardiovascular disease, dementia, diagnosis or treatment for epilepsy, history of falls, chronic liver disease, Parkinson’s disease, rheumatoid arthritis or systemic lupus erythematosus, chronic renal disease, type 1 and 2 diabetes, previous fracture, endocrine disorders, gastrointestinal malabsorption, any antidepressants, corticosteroids, unopposed hormone replacement therapy and parental history of osteoporosis), needing no quantitative measurements [476,477]. When compared with FRAX, QFracture shows some evidence of improved discrimination and calibration for hip fracture [476]. BMD cannot be used in conjunction with QFracture.

**Garvan**, short for Garvan Fracture Risk Calculator [478,479], estimates individualized five- to ten-years risk of total fracture and hip fracture by combining BMD and several CRFs (age, body weight, height, daily physical activity level, daily calcium intake, smoking, history of falls in the preceding 12 months, history of fractures in the past five years, et cetera [480]).

**Open Issue**

**10**(Quantitative Risk Factors).

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Glossary of Symbols

Symbol: | Meaning: |

$i,j,k,l=1,2,3$ | spatial components of Einstein’s summation or notation, or Index/Subscript/Tensor notion |

${T}_{ij}$ | an element of the Cauchy stress tensor $\mathit{T}$ $[N/{m}^{2}]\in {\mathbb{R}}^{3x3}$ |

${b}_{i}$ | an element of the body force vector $\mathit{b}$ $[N/{m}^{3}]\in {\mathbb{R}}^{3x1}$ |

${f}_{i}$ | an element of the surface force vector $\mathit{f}$ $\left[N\right]\in {\mathbb{R}}^{3x1}$ |

${E}_{ij}$ | an element of the strain tensor $\mathit{E}$ $\in {\mathbb{R}}^{3x3}$ |

${n}_{j}$ | an element of the unit normal vector $\mathit{n}$ $\in {\mathbb{R}}^{3x1}$ |

$\rho $ | a density $[g/{m}^{3}]\in {\mathbb{R}}^{1}$ |

${X}_{i}$ | an element of the material point $\mathit{X}$ $\left[m\right]\in {\mathbb{R}}^{3x1}$ [494] (p. 61) |

${x}_{i}$ | an element of the spatial point $\mathit{x}$ $\left[m\right]\in {\mathbb{R}}^{3x1}$ [494] (p. 61) |

${F}_{ij}$ | an element of the deformation gradient $\mathit{F}$ $\in {\mathbb{R}}^{3x3}$ |

${u}_{i}$ | an element of the displacement vector $\mathit{u}$ $\left[m\right]\in {\mathbb{R}}^{3x1}$ |

${C}_{ijkl}$ | an element of the stiffness tensor $\mathbb{C}$ $[N/{m}^{2}]\in {\mathbb{R}}^{3x3x3x3}$ |

W | the strain-energy density function $\left[J\right]\in {\mathbb{R}}^{1}$ |

t | the time $\left[s\right]\in {\mathbb{R}}^{1}$ |

$\theta $ | the temperature $\left[K\right]\in {\mathbb{R}}^{1}$ |

$\mathsf{\Delta}{\theta}_{i}$ | an element of the temperature gradient $\mathsf{\Delta}\mathbf{\theta}$ $[K/s]\in {\mathbb{R}}^{3x1}$ |

$\xi $ | an unknown internal variable $[?]\in {\mathbb{R}}^{?}$ |

Y | a unidirectional Young’s Modulus $[N/{m}^{2}]\in {\mathbb{R}}^{1}$ |

## Appendix B. Glossary of Acronyms

Acronym: | Full form: |

BEM | Boundary Element Method |

BC | Boundary Condition |

MKBC | Minimal Kinematic Boundary Condition |

PBC | Periodic Boundary Condition |

TBC | Tesselation Boundary Condition |

BMD | Bone Mineral Density |

BTM | Bone Turnover Marker |

CLG | CoLlaGen |

mCLGf | mineralized CoLlaGen fibril |

CT | Computed Tomography |

µCT | Micro Computed Tomography |

QCT | Quantitative Computed Tomography |

HR-pQCT | High-Resolution Peripheral Quantitative Computed Tomography |

CZM | Cohesive Zone Model |

DICOM | Digital Imaging and Communications in Medicine |

DOF | Degrees of Freedom |

DXA or DEXA | Dual Energy X-ray Absorptiometry |

FDM | Finite Difference Method |

FEM | Finite Element Method |

EPFM | Elastic-Plastic Fracture Mechanics |

LEFM | Linear Elastic Fracture Mechanics |

FS | Fall Stage |

GDP | Gross Domestic Product |

HA | HydroxyApatite |

HU | Hounsfield Units |

LE | Linear-Elastic |

CLE | Cauchy-Linear-Elastic |

CNLE | Cauchy-NonLinear-Elastic |

NLE | Non-Linear-Elastic |

LS | Lengthscale |

MD | Molecular Dynamics |

MRI | Magnetic Resonance Imaging |

µMRI | Micro Magnetic Resonance Imaging |

PDE | Partial Differential Equation |

RVE | Representative Volume Element |

CRF | Clinical Risk Factor |

QRF | Quantitative Risk Factor |

SD | Standard Deviations |

TBS | Trabecular Bone Score |

TSE | Traction Separation Equation |

WHO | World Health Organization |

## Appendix C. Glossary of PDE, Continuum Mechanics and Theory of Elasticity Concepts

Term: | Mathematical definition: | Physical definition: |

Domain | ([495] p. 222, same as Gebiet), [496] (p. 1, same as Region) | see Illustration 1 |

Boundary | [497] (pp. 25, 28, same as Frontier) | see Illustration 1 |

Boundary Condition | [498] (p. 23) | see Illustration 2 |

Constitutive equation | [499] (p. 170), [219] (p. 69), [500] (p. 1644), [494] (p. 276) | [222] (p. 170), [499] (p. 169), [206] (p. 35), [219] (p. 69), [501] (p. 273), [502] (p. 2), [494] (p. 223), [500] (p. 1642) |

Elastic material (or Cauchy-elastic material) | [503] (p. 170), [504] (p. 207) [499] (p. 175), [502] (p. 117), [494] (p. 297) | [505] (p. 201), [506] (pp. 1, 444), [28] (p. 147) |

Hyperelastic material (or Green-elastic) | [219] (p. 520), [503] (p. 171) [499] (p. 206), [502] (p. 294), [506] (p. 444) | [219] (p. 519), [502] (p. 13), [499] (p. 206), [28] (p. 148), [501] (p. 282) |

Plastic (or elasto-plastic) material | [219] (p. 148), [506] (p. 131), [210] (p. 57) | [219] (p. 1480), [506] (p. 131), [210] (p. 52), [507] (p. 75) |

Viscoelastic material | [503] (p. 174), [211] (p. 5) | [211] (p. 5), [222] (pp. 59, 217), [210] (p. 65) |

Viscoplastic (or elasto-viscoplastic) material | [219] (p. 450) | [506] (p. 133), [210] (p. 65), [219] (p. 435) |

Poroelasticity | [222] (p. 249) | [222] (p. 247) |

Poroplasticity | [508] (p. 226) | [508] (p. 225) |

Poroviscoelasticity | [508] (p. 261) | [508] (p. 261) |

Poroviscoplasticity | [508] (p. 273) | [508] (p. 272) |

Damage mechanics | [509] (p. 8) [510] (pp. 16, 142) | [511] (p. 1) [510] (p. 3) [512] (p. 1) |

Isotropic material | [503] (p. 234), [502] (p. 78), [513] (p. 60), [504] (p. 243) | [514] (p. 25), [250] (p. 41), [505] (p. 203), [499] (p. 170) |

Anisotropic material | [503] (p. 234), [502] (p. 78), [513] (p. 60), [504] (p. 243) | [514] (p. 25), [250] (p. 41), [505] (p. 203) |

Linear elastic anisotropy: triclinic, monoclinic, orthotropic (or rhomibc), trigonal, tetragonal, transversally isotropic (or hexagonal), cubic, isotropic | [250] (p. 44), [222] (p. 150), [257] (p. 10) | [250] (p. 44), [222] (p. 150), [257] (p. 10) |

Homogeneous material | [504] (p. 237), [502] (p. 58,59) | [502] (pp. 58, 59), [514] (p. 25), [505] (p. 203) |

Inhomogeneous (or non-homogeneous, heterogeneous) material | [504] (p. 237) | [514] (p. 25), [505] (p. 203) |

Properties (not only mechanical properties): global and local | [515] (p. 83) | [515] (p. 83), [516] (p. 532) |

Cauchy’s equation of motion (equilibrium equation) | [504] (p. 223), [503] (pp. 153, 204), [499] (p. 148), [494] (pp. 139, 273, 307) | [222] (p. 129), [222] (p. 196) |

Strain-displacement equation (or Lagrange strain tensor, strain) | [503] (p. 272), [505] (p. 84) | [206] (p. 29), [505] (p. 84) |

Stress (or Cauchy stress tensor) | [504] (p. 174), [503] (p. 150) [494] (p. 137) | [206] (p. 25), [222] (p. 122), [505] (p. 157) |

Displacement (or displacement vector) | [503] (p. 272), [494] (p. 297) | [206] (p. 30), [505] (p. 81) |

Body force | [504] (p. 151), [503] (p. 97), [494] (p. 132) | [499] (p. 144), [494] (p. 132) |

Surface force (or surface traction, stress vector, Cauchy traction field) | [503] (p. 97), [494] (p. 133) | [494] (p. 133), [206] (p. 26), [505] (p. 155) |

Materials with memory | [504] (p. 201) | [502, XVIII preface to third edition] |

Hookean Material (or generalized Hooke’s law, Hooke’s law) | [506] (pp. 2, 127), [505] (p. 204), [206] (p. 38) | [222] (p. 58), [211] (p. 4) |

## References

- World Health Organization. World Health Statistics 2016: Monitoring Health for the SDGs, Sustainable Development Goals. 2016. Available online: https://www.who.int/gho/publications/world_health_statistics/2016/en/ (accessed on 2 December 2019).
- United Nations, Department of Economics and Social Affairs, Population Division. World Population Prospects: The 2015 Revision, Key Findings and Advance Tables. Working Paper No. ESA/P/WP. 241. 2015. Available online: http://wedocs.unep.org/handle/20.500.11822/18246?show=full (accessed on 2 December 2019).
- Kaneda, T.; Greenbaum, C.; Patierno, K. 2018 World Population Data Sheet With Focus on Changing Age Structures. 2018. Available online: https://www.prb.org/2018-world-population-data-sheet-with-focus-on-changing-age-structures/ (accessed on 2 December 2019).
- United Nations. World Population Ageing 2017 Highlights. Statistical Papers—United Nations (Ser. A); Population and Vital Statistics Report; UN: New York, NY, USA, 2018; p. 42. [Google Scholar] [CrossRef]
- He, W.; Goodkind, D.; Kowal, P. An Aging World: 2015; U.S. Census Bureau-International Population Reports; Available online: https://www.fiapinternacional.org/en/an-aging-world-2015-international-population-reports-united-states-census-bureau-march-2016/ (accessed on 2 December 2019).
- International Osteoporosis Foundation—IOF. The Latin America Regional Audit—Epidemiology, Costs & Burden of Osteoporosis in 2012. 2012. Available online: https://www.iofbonehealth.org/sites/default/files/media/PDFs/Regional%20Audits/2012-Latin_America_Audit_0_0.pdf (accessed on 2 December 2019).
- Office for National Statistics—UK. Overview of the UK population: November 2018. 2018. Available online: https://www.ons.gov.uk/peoplepopulationandcommunity/populationandmigration/populationestimates/articles/overviewoftheukpopulation/november2018 (accessed on 2 December 2019).
- European Commission. The 2018 Ageing Report: Underlying Assumptions and Projection Methodologies; European Commission: Brussels, Belgium, 2017; Volume 65. [Google Scholar] [CrossRef]
- Ortman, J.M.; Velkoff, V.A.; Hogan, H. An Aging Nation: The Older Population in the United States. Technical Report; 2014. Available online: https://www.census.gov/library/publications/2014/demo/p25-1140.html (accessed on 2 December 2019).
- He, W.; Goodkind, D.; Kowal, P. An Aging World: 2015. Technical Report; 2016. Available online: https://www.census.gov/library/publications/2016/demo/P95-16-1.html (accessed on 2 December 2019).
- World Health Organization. Assessment of Fracture Risk and Its Application to Screening for Postmenopausal Osteoporosis: Report of a WHO Study Group Meeting Held in Rome from 22 to 25 June 1992. Technical Report. 1994. Available online: https://apps.who.int/iris/handle/10665/39142 (accessed on 2 December 2019).
- Osterhoff, G.; Morgan, E.F.; Shefelbine, S.J.; Karim, L.; McNamara, L.M.; Augat, P. Bone mechanical properties and changes with osteoporosis. Injury
**2016**, 47, S11–S20. [Google Scholar] [CrossRef][Green Version] - Woolf, A.D.; Akesson, K. OSTEOPOROSIS, An Atlas of Investigation and Management; Clinical Publishing: Oxford, UK, 2008. [Google Scholar]
- Kanis, J.A. Diagnosis of osteoporosis and assessment of fracture risk. Lancet
**2002**, 359, 1929–2936. [Google Scholar] [CrossRef] - Kanis, J.A.; on behalf of the World Health Organization Scientific Group. Assessment of Osteoporosis at the Primary Health-Care Level; Technical Report; World Health Organization Collaborating Centre for Metabolic Bone Diseases, University of Sheffield: Sheffield, UK, 2007; Available online: https://www.sheffield.ac.uk/FRAX/pdfs/WHO_Technical_Report.pdf (accessed on 4 December 2019).
- Laurence, V.; van Rietbergen, B.; Nicolas, V.; Marie-Thérèse, L.; Hervé, L.; Myriam, N.; Mohamed, Z.; Maude, G.; Nicolas, B.; Valery, N.; et al. Cortical and Trabecular Bone Microstructure Did Not Recover at Weight-Bearing Skeletal Sites and Progressively Deteriorated at Non-Weight-Bearing Sites During the Year Following International Space Station Missions. J. Bone Miner. Res.
**2017**, 32, 2010–2021. [Google Scholar] [CrossRef] - Grimm, D.; Grosse, J.; Wehland, M.; Mann, V.; Reseland, J.E.; Sundaresan, A.; Corydon, T.J. The impact of microgravity on bone in humans. Bone
**2016**, 87, 44–56. [Google Scholar] [CrossRef] - Bhandari, M.; Swiontkowski, M. Management of Acute Hip Fracture. N. Eng. J. Med.
**2017**, 377, 2053–2062. [Google Scholar] [CrossRef] - Hannan, E.L.; Magaziner, J.; Wang, J.J.; Eastwood, E.A.; Silberzweig, S.B.; Gilbert, M.; Morrison, S.; McLaughlin, M.A.; Orosz, G.M.; Siu, A.L. Mortality and locomotion 6 months after hospitalization for hip fracture. Risk factors and risk-adjusted hospital outcomes. JAMA
**2001**, 285, 2736–2742. [Google Scholar] [CrossRef][Green Version] - O’Connor, K.M. Evaluation and Treatment of Osteoporosis. Med. Clin. N. Am.
**2016**, 100, 807–826. [Google Scholar] [CrossRef] - Kling, J.M.; Clarke, B.L.; Sandhu, N.P. Osteoporosis Prevention, Screening, and Treatment: A Review. J. Womens Health
**2014**, 23, 563–572. [Google Scholar] [CrossRef] - Morales-Torres, J.; Gutiérrez-Ureña, S. The burden of osteoporosis in Latin America. Osteoporos. Int.
**2004**, 15, 625–632. [Google Scholar] [CrossRef] - Odén, A.; McCloskey, E.V.; Kanis, J.A.; Harvey, N.C.; Johansson, H. Burden of high fracture probability worldwide: Secular increases 2010–2040. Osteoporos. Int.
**2015**, 26, 2243–2248. [Google Scholar] [CrossRef] - Holzer, L.A.; Leithner, A.; Holzer, G. The Most Cited Papers in Osteoporosis and Related Research. J. Osteoporos.
**2015**, 2015, 12. [Google Scholar] [CrossRef] [PubMed] - Housner, G.W.; Bergman, L.A.; Caughey, T.K.; Chassiakos, A.G.; Claus, R.O.; Masri, S.F.; Skelton, R.E.; Soong, T.T.; Spencer, B.F.; Yao, J.T.P. Structural Control: Past, Present, and Future. J. Eng. Mech.
**1997**, 123, 897–971. [Google Scholar] [CrossRef] - Castiglione, F.; Pappalardo, F.; Bianca, C.; Russo, G.; Motta, S. Modeling Biology Spanning Different Scales: An Open Challenge. BioMed Res. Int.
**2014**, 2014, 1–9. [Google Scholar] [CrossRef] [PubMed][Green Version] - Katz, K.U.; Katz, M.G. Cauchy’s Continuum. Perspect. Sci.
**2011**, 19, 426–452. [Google Scholar] [CrossRef][Green Version] - Chen, W.; Saleeb, A. Constitutive Equations for Engineering Materials: Elasticity and Modeling; Studies in Applied Mechanics; Elsevier Science: Amsterdam, The Netherlands, 2013. [Google Scholar]
- Marsden, J.; Hughes, T. Mathematical Foundations of Elasticity; Dover Civil and Mechanical Engineering; Dover Publications: Mineola, NY, USA, 2012. [Google Scholar]
- Knowles, N.K.; Langohr, G.; Faieghi, M.; Nelson, A.J.; Ferreira, L.M. A comparison of density–modulus relationships used in finite element modeling of the shoulder. Med. Eng. Phys.
**2019**, 66, 40–46. [Google Scholar] [CrossRef] - Yosibash, Z.; Padan, R.; Joskowicz, L.; Milgrom, C. A CT-Based High-Order Finite Element Analysis of the Human Proximal Femur Compared to In-vitro Experiments. J. Biomech. Eng.
**2006**, 129, 297–309. [Google Scholar] [CrossRef][Green Version] - Helgason, B.; Taddei, F.; Pálsson, H.; Schileo, E.; Cristofolini, L.; Viceconti, M.; Brynjólfsson, S. A modified method for assigning material properties to FE models of bones. Med. Eng. Phys.
**2008**, 30, 444–453. [Google Scholar] [CrossRef] - Dall’Ara, E.; Luisier, B.; Schmidt, R.; Kainberger, F.; Zysset, P.; Pahr, D. A nonlinear QCT-based finite element model validation study for the human femur tested in two configurations in vitro. Bone
**2013**, 52, 27–38. [Google Scholar] [CrossRef] - Hambli, R. A quasi-brittle continuum damage finite element model of the human proximal femur based on element deletion. Med. Biol. Eng. Comput.
**2013**, 51, 219–231. [Google Scholar] [CrossRef][Green Version] - Sarvi, M.N.; Luo, Y. A two-level subject-specific biomechanical model for improving prediction of hip fracture risk. Clin. Biomech.
**2015**, 30, 881–887. [Google Scholar] [CrossRef] - Eberle, S.; Göttlinger, M.; Augat, P. An investigation to determine if a single validated density–elasticity relationship can be used for subject specific finite element analyses of human long bones. Med. Eng. Phys.
**2013**, 35, 875–883. [Google Scholar] [CrossRef] [PubMed] - Keyak, J.; Meagher, J.; Skinner, H.; Mote, C. Automated three-dimensional finite element modelling of bone: A new method. J. Biomed. Eng.
**1990**, 12, 389–397. [Google Scholar] [CrossRef] - Viceconti, M.; Davinelli, M.; Taddei, F.; Cappello, A. Automatic generation of accurate subject-specific bone finite element models to be used in clinical studies. J. Biomech.
**2004**, 37, 1597–1605. [Google Scholar] [CrossRef] [PubMed] - Enns-Bray, W.; Bahaloo, H.; Fleps, I.; Pauchard, Y.; Taghizadeh, E.; Sigurdsson, S.; Aspelund, T.; Büchler, P.; Harris, T.; Gudnason, V.; et al. Biofidelic finite element models for accurately classifying hip fracture in a retrospective clinical study of elderly women from the AGES Reykjavik cohort. Bone
**2019**, 120, 25–37. [Google Scholar] [CrossRef] [PubMed] - Falcinelli, C.; Schileo, E.; Pakdel, A.; Whyne, C.; Cristofolini, L.; Taddei, F. Can CT image deblurring improve finite element predictions at the proximal femur? J. Mech. Behav. Biomed.
**2016**, 63, 337–351. [Google Scholar] [CrossRef] - Nishiyama, K.K.; Ito, M.; Harada, A.; Boyd, S.K. Classification of women with and without hip fracture based on quantitative computed tomography and finite element analysis. Osteoporos. Int.
**2014**, 25, 619–626. [Google Scholar] [CrossRef] - Pahr, D.H.; Schwiedrzik, J.; Dall’Ara, E.; Zysset, P.K. Clinical versus pre-clinical FE models for vertebral body strength predictions. J. Mech. Behav. Biomed.
**2014**, 33, 76–83. [Google Scholar] [CrossRef] - Keyak, J.H.; Falkinstein, Y. Comparison of in situ and in vitro CT scan-based finite element model predictions of proximal femoral fracture load. Med. Eng. Phys.
**2003**, 25, 781–787. [Google Scholar] [CrossRef] - Peng, L.; Bai, J.; Zeng, X.; Zhou, Y. Comparison of isotropic and orthotropic material property assignments on femoral finite element models under two loading conditions. Med. Eng. Phys.
**2006**, 28, 227–233. [Google Scholar] [CrossRef] - Zysset, P.; Pahr, D.; Engelke, K.; Genant, H.K.; McClung, M.R.; Kendler, D.L.; Recknor, C.; Kinzl, M.; Schwiedrzik, J.; Museyko, O.; et al. Comparison of proximal femur and vertebral body strength improvements in the FREEDOM trial using an alternative finite element methodology. Bone
**2015**, 81, 122–130. [Google Scholar] [CrossRef] - Gustafson, H.M.; Cripton, P.A.; Ferguson, S.J.; Helgason, B. Comparison of specimen-specific vertebral body finite element models with experimental digital image correlation measurements. J. Mech. Behav. Biomed.
**2017**, 65, 801–807. [Google Scholar] [CrossRef] [PubMed][Green Version] - Chen, G.; Wu, F.; Liu, Z.; Yang, K.; Cui, F. Comparisons of node-based and element-based approaches of assigning bone material properties onto subject-specific finite element models. Med. Eng. Phys.
**2015**, 37, 808–812. [Google Scholar] [CrossRef] [PubMed] - Rajapakse, C.S.; Magland, J.F.; Wald, M.J.; Liu, X.S.; Zhang, X.H.; Guo, X.E.; Wehrli, F.W. Computational biomechanics of the distal tibia from high-resolution MR and micro-CT images. Bone
**2010**, 47, 556–563. [Google Scholar] [CrossRef] [PubMed][Green Version] - Koivumäki, J.E.; Thevenot, J.; Pulkkinen, P.; Kuhn, V.; Link, T.M.; Eckstein, F.; Jämsä, T. Ct-based finite element models can be used to estimate experimentally measured failure loads in the proximal femur. Bone
**2012**, 50, 824–829. [Google Scholar] [CrossRef] - Hussein, A.I.; Louzeiro, D.T.; Unnikrishnan, G.U.; Morgan, E.F. Differences in Trabecular Microarchitecture and Simplified Boundary Conditions Limit the Accuracy of Quantitative Computed Tomography-Based Finite Element Models of Vertebral Failure. J. Biomech. Eng.
**2018**, 140, 021004. [Google Scholar] [CrossRef][Green Version] - Zhang, Y.; Zhong, W.; Zhu, H.; Chen, Y.; Xu, L.; Zhu, J. Establishing the 3-D finite element solid model of femurs in partial by volume rendering. Int. J. Surg.
**2013**, 11, 930–934. [Google Scholar] [CrossRef][Green Version] - Gamez, B.; Divo, E.; Kassab, A.; Cerrolaza, M. Evaluation of fatigue crack growing in cortical bone using the BEM. Int. J. Health Tech. Manag.
**2010**, 11, 202–221. [Google Scholar] [CrossRef] - Podshivalov, L.; Fischer, A.; Bar-Yoseph, P. 3D hierarchical geometric modeling and multiscale FE analysis as a base for individualized medical diagnosis of bone structure. Bone
**2011**, 48, 693–703. [Google Scholar] [CrossRef] - Fish, J.; Hu, N. Multiscale modeling of femur fracture. Int. J. Numer. Methods Eng.
**2017**, 111, 3–25. [Google Scholar] [CrossRef] - Gray, H.A.; Taddei, F.; Zavatsky, A.B.; Cristofolini, L.; Gill, H.S. Experimental Validation of a Finite Element Model of a Human Cadaveric Tibia. J. Biomech. Eng.
**2008**, 130. [Google Scholar] [CrossRef] - Dall’Ara, E.; Eastell, R.; Viceconti, M.; Pahr, D.; Yang, L. Experimental validation of DXA-based finite element models for prediction of femoral strength. J. Mech. Behav. Biomed.
**2016**, 63, 17–25. [Google Scholar] [CrossRef] [PubMed][Green Version] - Grassi, L.; Väänänen, S.P.; Yavari, S.A.; Weinans, H.; Jurvelin, J.S.; Zadpoor, A.A.; Isaksson, H. Experimental validation of finite element model for proximal composite femur using optical measurements. J. Mech. Behav. Biomed.
**2013**, 21, 86–94. [Google Scholar] [CrossRef] [PubMed] - Larrainzar-Garijo, R.; Caeiro, J.; Marco, M.; Giner, E.; Miguélez, M. Experimental validation of finite elements model in hip fracture and its clinical applicability. Revista Española de Cirugía Ortopédica y Traumatología (Engl. Ed.)
**2019**, 63, 146–154. [Google Scholar] [CrossRef] - Wolfram, U.; Gross, T.; Pahr, D.H.; Schwiedrzik, J.; Wilke, H.J.; Zysset, P.K. Fabric-based Tsai–Wu yield criteria for vertebral trabecular bone in stress and strain space. J. Mech. Behav. Biomed.
**2012**, 15, 218–228. [Google Scholar] [CrossRef] [PubMed] - Haider, I.T.; Goldak, J.; Frei, H. Femoral fracture load and fracture pattern is accurately predicted using a gradient-enhanced quasi-brittle finite element model. Med. Eng. Phys.
**2018**, 55, 1–8. [Google Scholar] [CrossRef] - Elham, H.; Iwona, J.; Andrew, Y.; YikHan, L.; Tadeusz, L. Multi-scale modelling of elastic moduli of trabecular bone. J. R. Soc. Interface
**2012**, 9, 1654–1673. [Google Scholar] [CrossRef][Green Version] - Cody, D.D.; Gross, G.J.; Hou, F.J.; Spencer, H.J.; Goldstein, S.A.; Fyhrie, D.P. Femoral strength is better predicted by finite element models than QCT and DXA. J. Biomech.
**1999**, 32, 1013–1020. [Google Scholar] [CrossRef] - Anez-Bustillos, L.; Derikx, L.C.; Verdonschot, N.; Calderon, N.; Zurakowski, D.; Snyder, B.D.; Nazarian, A.; Tanck, E. Finite element analysis and CT-based structural rigidity analysis to assess failure load in bones with simulated lytic defects. Bone
**2014**, 58, 160–167. [Google Scholar] [CrossRef][Green Version] - Zysset, P.K.; Pahr, E.D.P.V.D.H. Finite element analysis for prediction of bone strength. Bonekey Rep.
**2013**, 2. [Google Scholar] [CrossRef][Green Version] - Crawford, R.; Cann, C.E.; Keaveny, T.M. Finite element models predict in vitro vertebral body compressive strength better than quantitative computed tomography. Bone
**2003**, 33, 744–750. [Google Scholar] [CrossRef] - Ota, T.; Yamamoto, I.; Morita, R. Fracture simulation of the femoral bone using the finite-element method: How a fracture initiates and proceeds. J. Bone Miner. Metab.
**1999**, 17, 108–112. [Google Scholar] [CrossRef] [PubMed] - Keyak, J. Improved prediction of proximal femoral fracture load using nonlinear finite element models. Med. Eng. Phys.
**2001**, 23, 165–173. [Google Scholar] [CrossRef] - Eberle, S.; Göttlinger, M.; Augat, P. Individual density-elasticity relationships improve accuracy of subject-specific finite element models of human femurs. J. Biomech.
**2013**, 46, 2152–2157. [Google Scholar] [CrossRef] [PubMed] - Ascenzi, M.G.; Kawas, N.P.; Lutz, A.; Kardas, D.; Nackenhorst, U.; Keyak, J.H. Individual-specific multi-scale finite element simulation of cortical bone of human proximal femur. J. Comput. Phys.
**2013**, 244, 298–311. [Google Scholar] [CrossRef] - Keyak, J.; Sigurdsson, S.; Karlsdottir, G.; Oskarsdottir, D.; Sigmarsdottir, A.; Zhao, S.; Kornak, J.; Harris, T.; Sigurdsson, G.; Jonsson, B.; et al. Male–female differences in the association between incident hip fracture and proximal femoral strength: A finite element analysis study. Bone
**2011**, 48, 1239–1245. [Google Scholar] [CrossRef] [PubMed][Green Version] - Keyak, J.H.; Rossi, S.A.; Jones, K.A.; Skinner, H.B. Prediction of femoral fracture load using automated finite element modeling. J. Biomech.
**1997**, 31, 125–133. [Google Scholar] [CrossRef] - Keyak, J.; Koyama, A.; LeBlanc, A.; Lu, Y.; Lang, T. Reduction in proximal femoral strength due to long-duration spaceflight. Bone
**2009**, 44, 449–453. [Google Scholar] [CrossRef] - Panyasantisuk, J.; Dall’Ara, E.; Pretterklieber, M.; Pahr, D.; Zysset, P. Mapping anisotropy improves QCT-based finite element estimation of hip strength in pooled stance and side-fall load configurations. Med. Eng. Phys.
**2018**, 59, 36–42. [Google Scholar] [CrossRef][Green Version] - Enns-Bray, W.S.; Owoc, J.S.; Nishiyama, K.K.; Boyd, S.K. Mapping anisotropy of the proximal femur for enhanced image based finite element analysis. J. Biomech.
**2014**, 47, 3272–3278. [Google Scholar] [CrossRef] - Enns-Bray, W.; Bahaloo, H.; Fleps, I.; Ariza, O.; Gilchrist, S.; Widmer, R.; Guy, P.; Pálsson, H.; Ferguson, S.; Cripton, P.; et al. Material mapping strategy to improve the predicted response of the proximal femur to a sideways fall impact. J. Mech. Behav. Biomed.
**2018**, 78, 196–205. [Google Scholar] [CrossRef] - Falcinelli, C.; Martino, A.D.; Gizzi, A.; Vairo, G.; Denaro, V. Mechanical behavior of metastatic femurs through patient-specific computational models accounting for bone-metastasis interaction. J. Mech. Behav. Biomed.
**2019**, 93, 9–22. [Google Scholar] [CrossRef] [PubMed] - Arjmand, H.; Nazemi, M.; Kontulainen, S.A.; McLennan, C.E.; Hunter, D.J.; Wilson, D.R.; Johnston, J.D. Mechanical Metrics of the Proximal Tibia are Precise and Differentiate Osteoarthritic and Normal Knees: A Finite Element Study. Sci. Rep.
**2018**, 8. [Google Scholar] [CrossRef] [PubMed] - Sandino, C.; McErlain, D.D.; Schipilow, J.; Boyd, S.K. Mechanical stimuli of trabecular bone in osteoporosis: A numerical simulation by finite element analysis of microarchitecture. J. Mech. Behav. Biomed.
**2017**, 66, 19–27. [Google Scholar] [CrossRef] [PubMed] - Chen, Y.; Dall’Ara, E.; Sales, E.; Manda, K.; Wallace, R.; Pankaj, P.; Viceconti, M. Micro-CT based finite element models of cancellous bone predict accurately displacement once the boundary condition is well replicated: A validation study. J. Mech. Behav. Biomed.
**2017**, 65, 644–651. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hambli, R. Micro-CT finite element model and experimental validation of trabecular bone damage and fracture. Bone
**2013**, 56, 363–374. [Google Scholar] [CrossRef] [PubMed] - Enns-Bray, W.; Ariza, O.; Gilchrist, S.; Soyka, R.W.; Vogt, P.; Palsson, H.; Boyd, S.; Guy, P.; Cripton, P.; Ferguson, S.; et al. Morphology based anisotropic finite element models of the proximal femur validated with experimental data. Med. Eng. Phys.
**2016**, 38, 1339–1347. [Google Scholar] [CrossRef] [PubMed][Green Version] - Falcinelli, C.; Schileo, E.; Balistreri, L.; Baruffaldi, F.; Bordini, B.; Viceconti, M.; Albisinni, U.; Ceccarelli, F.; Milandri, L.; Toni, A.; et al. Multiple loading conditions analysis can improve the association between finite element bone strength estimates and proximal femur fractures: A preliminary study in elderly women. Bone
**2014**, 67, 71–80. [Google Scholar] [CrossRef] - Podshivalov, L.; Fischer, A.; Bar-Yoseph, P. Multiscale FE method for analysis of bone micro-structures. J. Mech. Behav. Biomed.
**2011**, 4, 888–899, Bone Remodeling. [Google Scholar] [CrossRef] - Zeinali, A.; Hashemi, B.; Akhlaghpoor, S. Noninvasive prediction of vertebral body compressive strength using nonlinear finite element method and an image based technique. Phys. Med.
**2010**, 26, 88–97. [Google Scholar] [CrossRef] - Yosibash, Z. p-FEMs in biomechanics: Bones and arteries. Comput. Meth. Appl. Mech. Eng.
**2012**, 249–252, 169–184, Higher Order Finite Element and Isogeometric Methods. [Google Scholar] [CrossRef] - Tanck, E.; van Aken, J.B.; van der Linden, Y.M.; Schreuder, H.B.; Binkowski, M.; Huizenga, H.; Verdonschot, N. Pathological fracture prediction in patients with metastatic lesions can be improved with quantitative computed tomography based computer models. Bone
**2009**, 45, 777–783. [Google Scholar] [CrossRef] [PubMed] - Sternheim, A.; Giladi, O.; Gortzak, Y.; Drexler, M.; Salai, M.; Trabelsi, N.; Milgrom, C.; Yosibash, Z. Pathological fracture risk assessment in patients with femoral metastases using CT-based finite element methods. A retrospective clinical study. Bone
**2018**, 110, 215–220. [Google Scholar] [CrossRef] [PubMed] - Peleg, E.; Beek, M.; Joskowicz, L.; Liebergall, M.; Mosheiff, R.; Whyne, C. Patient specific quantitative analysis of fracture fixation in the proximal femur implementing principal strain ratios. Method and experimental validation. J. Biomech.
**2010**, 43, 2684–2688. [Google Scholar] [CrossRef] [PubMed] - Katz, Y.; Lubovsky, O.; Yosibash, Z. Patient-specific finite element analysis of femurs with cemented hip implants. Clin. Biomech.
**2018**, 58, 74–89. [Google Scholar] [CrossRef] [PubMed] - Trabelsi, N.; Yosibash, Z.; Wutte, C.; Augat, P.; Eberle, S. Patient-specific finite element analysis of the human femur—A double-blinded biomechanical validation. J. Biomech.
**2011**, 44, 1666–1672. [Google Scholar] [CrossRef] [PubMed] - Qasim, M.; Farinella, G.; Zhang, J.; Li, X.; Yang, L.; Eastell, R.; Viceconti, M. Patient-specific finite element estimated femur strength as a predictor of the risk of hip fracture: The effect of methodological determinants. Osteoporos. Int.
**2016**, 27, 2815–2822. [Google Scholar] [CrossRef][Green Version] - Basafa, E.; Armiger, R.S.; Kutzer, M.D.; Belkoff, S.M.; Mears, S.C.; Armand, M. Patient-specific finite element modeling for femoral bone augmentation. Med. Eng. Phys.
**2013**, 35, 860–865. [Google Scholar] [CrossRef][Green Version] - Yosibash, Z.; Trabelsi, N. Patient-Specific Simulation of the Proximal Femur’s Mechanical Response Validated by Experimental Observations. In 13th International Conference on Biomedical Engineering; Lim, C.T., Goh, J.C.H., Eds.; Springer: Berlin/Heidelberg, Germany, 2009; pp. 2019–2022. [Google Scholar]
- Keyak, J.; Kaneko, T.; Tehranzadeh, J.; Skinner, H. Predicting Proximal Femoral Strength Using Structural Engineering Models. Clin. Orthop. Relat. Res.
**2005**, 437, 219–228. [Google Scholar] [CrossRef] - Yosibash, Z.; Mayo, R.P.; Dahan, G.; Trabelsi, N.; Amir, G.; Milgrom, C. Predicting the stiffness and strength of human femurs with real metastatic tumors. Bone
**2014**, 69, 180–190. [Google Scholar] [CrossRef] - Zohar, Y.; David, T.; Nir, T. Predicting the yield of the proximal femur using high-order finite-element analysis with inhomogeneous orthotropic material properties. Philos. Trans. R. Soc. A
**2010**, 368. [Google Scholar] [CrossRef] - Grassi, L.; Väänänen, S.P.; Ristinmaa, M.; Jurvelin, J.S.; Isaksson, H. Prediction of femoral strength using 3D finite element models reconstructed from DXA images: Validation against experiments. Biomech. Model Mechanobiol.
**2017**, 16, 989–1000. [Google Scholar] [CrossRef] [PubMed][Green Version] - Nazemi, S.M.; Amini, M.; Kontulainen, S.A.; Milner, J.S.; Holdsworth, D.W.; Masri, B.A.; Wilson, D.R.; Johnston, J.D. Prediction of local proximal tibial subchondral bone structural stiffness using subject-specific finite element modeling: Effect of selected density–modulus relationship. Clin. Biomech.
**2015**, 30, 703–712. [Google Scholar] [CrossRef] [PubMed] - Wang, X.; Sanyal, A.; Cawthon, P.M.; Palermo, L.; Jekir, M.; Christensen, J.; Ensrud, K.E.; Cummings, S.R.; Orwoll, E.; Black, D.M.; et al. Prediction of new clinical vertebral fractures in elderly men using finite element analysis of CT scans. J. Bone Miner. Res.
**2012**, 27, 808–816. [Google Scholar] [CrossRef] [PubMed][Green Version] - Bessho, M.; Ohnishi, I.; Matsumoto, T.; Ohashi, S.; Matsuyama, J.; Tobita, K.; Kaneko, M.; Nakamura, K. Prediction of proximal femur strength using a CT-based nonlinear finite element method: Differences in predicted fracture load and site with changing load and boundary conditions. Bone
**2009**, 45, 226–231. [Google Scholar] [CrossRef] - Bessho, M.; Ohnishi, I.; Matsuyama, J.; Matsumoto, T.; Imai, K.; Nakamura, K. Prediction of strength and strain of the proximal femur by a CT-based finite element method. J. Biomech.
**2007**, 40, 1745–1753. [Google Scholar] [CrossRef] - Tuncer, M.; Hansen, U.N.; Amis, A.A. Prediction of structural failure of tibial bone models under physiological loads: Effect of CT density–modulus relationships. Med. Eng. Phys.
**2014**, 36, 991–997. [Google Scholar] [CrossRef] - Wille, H.; Rank, E.; Yosibash, Z. Prediction of the mechanical response of the femur with uncertain elastic properties. J. Biomech.
**2012**, 45, 1140–1148. [Google Scholar] [CrossRef] - Nishiyama, K.K.; Gilchrist, S.; Guy, P.; Cripton, P.; Boyd, S.K. Proximal femur bone strength estimated by a computationally fast finite element analysis in a sideways fall configuration. J. Biomech.
**2013**, 46, 1231–1236. [Google Scholar] [CrossRef] - Dragomir-Daescu, D.; Op Den Buijs, J.; McEligot, S.; Dai, Y.; Entwistle, R.C.; Salas, C.; Melton, L.J.; Bennet, K.E.; Khosla, S.; Amin, S. Robust QCT/FEA Models of Proximal Femur Stiffness and Fracture Load During a Sideways Fall on the Hip. Ann. Biomed. Eng.
**2011**, 39, 742–755. [Google Scholar] [CrossRef][Green Version] - Katz, Y.; Dahan, G.; Sosna, J.; Shelef, I.; Cherniavsky, E.; Yosibash, Z. Scanner influence on the mechanical response of QCT-based finite element analysis of long bones. J. Biomech.
**2019**, 86, 149–159. [Google Scholar] [CrossRef] - Weinans, H.; Sumner, D.R.; Igloria, R.; Natarajan, R.N. Sensitivity of periprosthetic stress-shielding to load and the bone density–modulus relationship in subject-specific finite element models. J. Biomech.
**2000**, 33, 809–817. [Google Scholar] [CrossRef] - Cody, D.D.; Hou, F.J.; Divine, G.W.; Fyhrie, D.P. Short Term In Vivo Precision of Proximal Femoral Finite Element Modeling. Ann. Biomed. Eng.
**2000**, 28, 408–414. [Google Scholar] [CrossRef] [PubMed] - Long, Y.; Leslie, W.D.; Luo, Y. Study of DXA-derived lateral–medial cortical bone thickness in assessing hip fracture risk. Bone Rep.
**2015**, 2, 44–51. [Google Scholar] [CrossRef] [PubMed][Green Version] - McErlain, D.D.; Milner, J.S.; Ivanov, T.G.; Jencikova-Celerin, L.; Pollmann, S.I.; Holdsworth, D.W. Subchondral cysts create increased intra-osseous stress in early knee OA: A finite element analysis using simulated lesions. Bone
**2011**, 48, 639–646. [Google Scholar] [CrossRef] [PubMed] - Schileo, E.; Taddei, F.; Malandrino, A.; Cristofolini, L.; Viceconti, M. Subject-specific finite element models can accurately predict strain levels in long bones. J. Biomech.
**2007**, 40, 2982–2989. [Google Scholar] [CrossRef] [PubMed] - Schileo, E.; Taddei, F.; Cristofolini, L.; Viceconti, M. Subject-specific finite element models implementing a maximum principal strain criterion are able to estimate failure risk and fracture location on human femurs tested in vitro. J. Biomech.
**2008**, 41, 356–367. [Google Scholar] [CrossRef] - Taddei, F.; Cristofolini, L.; Martelli, S.; Gill, H.; Viceconti, M. Subject-specific finite element models of long bones: An in vitro evaluation of the overall accuracy. J. Biomech.
**2006**, 39, 2457–2467. [Google Scholar] [CrossRef] - Hölzer, A.; Schröder, C.; Woiczinski, M.; Sadoghi, P.; Scharpf, A.; Heimkes, B.; Jansson, V. Subject-specific finite element simulation of the human femur considering inhomogeneous material properties: A straightforward method and convergence study. Comput. Meth. Prog. Biol.
**2013**, 110, 82–88. [Google Scholar] [CrossRef] - Yosibash, Z.; Trabelsi, N.; Hellmich, C. Subject-Specific p-FE Analysis of the Proximal Femur Utilizing Micromechanics-Based Material Properties. Int. J. Multiscale Comput. Eng.
**2008**, 6, 483–498. [Google Scholar] [CrossRef][Green Version] - Austman, R.L.; Milner, J.S.; Holdsworth, D.W.; Dunning, C.E. The effect of the density–modulus relationship selected to apply material properties in a finite element model of long bone. J. Biomech.
**2008**, 41, 3171–3176. [Google Scholar] [CrossRef] - Michalski, A.S.; Edwards, W.B.; Boyd, S.K. The Influence of Reconstruction Kernel on Bone Mineral and Strength Estimates Using Quantitative Computed Tomography and Finite Element Analysis. J. Clin. Densitom.
**2019**, 22, 219–228. [Google Scholar] [CrossRef] [PubMed] - Helgason, B.; Gilchrist, S.; Ariza, O.; Vogt, P.; Enns-Bray, W.; Widmer, R.; Fitze, T.; Pálsson, H.; Pauchard, Y.; Guy, P.; et al. The influence of the modulus–density relationship and the material mapping method on the simulated mechanical response of the proximal femur in side-ways fall loading configuration. Med. Eng. Phys.
**2016**, 38, 679–689. [Google Scholar] [CrossRef] [PubMed][Green Version] - Taddei, F.; Schileo, E.; Helgason, B.; Cristofolini, L.; Viceconti, M. The material mapping strategy influences the accuracy of CT-based finite element models of bones: An evaluation against experimental measurements. Med. Eng. Phys.
**2007**, 29, 973–979. [Google Scholar] [CrossRef] - Schileo, E.; Balistreri, L.; Grassi, L.; Cristofolini, L.; Taddei, F. To what extent can linear finite element models of human femora predict failure under stance and fall loading configurations? J. Biomech.
**2014**, 47, 3531–3538. [Google Scholar] [CrossRef] [PubMed] - Yosibash, Z.; Katz, A.; Milgrom, C. Toward verified and validated FE simulations of a femur with a cemented hip prosthesis. Med. Eng. Phys.
**2013**, 35, 978–987. [Google Scholar] [CrossRef] [PubMed] - Van Rietbergen, B.; Huiskes, R.; Eckstein, F.; Rüegsegger, P. Trabecular Bone Tissue Strains in the Healthy and Osteoporotic Human Femur. J. Bone Miner. Res.
**2003**, 18, 1781–1788. [Google Scholar] [CrossRef] - Villette, C.C.; Phillips, A.T.M. Microscale poroelastic metamodel for efficient mesoscale bone remodelling simulations. Biomech. Model Mechanobiol.
**2017**, 16, 2077–2091. [Google Scholar] [CrossRef][Green Version] - Garcia, D.; Zysset, P.K.; Charlebois, M.; Curnier, A. A three-dimensional elastic plastic damage constitutive law for bone tissue. Biomech. Model Mechanobiol.
**2009**, 8, 149–165. [Google Scholar] [CrossRef] - Johnson, J.E.; Troy, K.L. Validation of a new multiscale finite element analysis approach at the distal radius. Med. Eng. Phys.
**2017**, 44, 16–24. [Google Scholar] [CrossRef] - Bhattacharya, P.; Altai, Z.; Qasim, M.; Viceconti, M. A multiscale model to predict current absolute risk of femoral fracture in a postmenopausal population. Biomech. Model Mechanobiol.
**2019**, 18, 301–318. [Google Scholar] [CrossRef][Green Version] - Wills, C.R.; Olivares, A.L.; Tassani, S.; Ceresa, M.; Zimmer, V.; Ballester, M.A.G.; del Río, L.M.; Humbert, L.; Noailly, J. 3D patient-specific finite element models of the proximal femur based on DXA towards the classification of fracture and non-fracture cases. Bone
**2019**, 121, 89–99. [Google Scholar] [CrossRef] [PubMed] - Luo, Y.; Ahmed, S.; Leslie, W.D. Automation of a DXA-based finite element tool for clinical assessment of hip fracture risk. Comput. Meth. Prog. Biol.
**2018**, 155, 75–83. [Google Scholar] [CrossRef] [PubMed] - Nasiri, M.; Luo, Y. Study of sex differences in the association between hip fracture risk and body parameters by DXA-based biomechanical modeling. Bone
**2016**, 90, 90–98. [Google Scholar] [CrossRef] [PubMed] - Rajapakse, C.S.; Gupta, N.; Evans, M.; Alizai, H.; Shukurova, M.; Hong, A.L.; Cruickshank, N.J.; Tejwani, N.; Egol, K.; Honig, S.; et al. Influence of bone lesion location on femoral bone strength assessed by MRI-based finite-element modeling. Bone
**2019**, 122, 209–217. [Google Scholar] [CrossRef] - Rajapakse, C.S.; Hotca, A.; Newman, B.T.; Ramme, A.; Vira, S.; Kobe, E.A.; Miller, R.; Honig, S.; Chang, G. Patient-specific Hip Fracture Strength Assessment with Microstructural MR Imaging–based Finite Element Modeling. Radiology
**2017**, 283, 854–861. [Google Scholar] [CrossRef][Green Version] - Wang, M.; Zimmermann, E.A.; Riedel, C.; Busseb, B.; Li, S.; Silberschmidt, V.V. Effect of micro-morphology of cortical bone tissue on fracture toughness and crack propagation. Procedia Struct. Integr.
**2017**, 6, 64–68. [Google Scholar] [CrossRef] - Idkaidek, A.; Koric, S.; Jasiuk, I. Fracture analysis of multi-osteon cortical bone using XFEM. Comput. Mech.
**2018**, 62, 171–184. [Google Scholar] [CrossRef] - Ural, A.; Mischinski, S. Multiscale modeling of bone fracture using cohesive finite elements. Eng. Fract. Mech.
**2013**, 103, 141–152. [Google Scholar] [CrossRef] - Abueidda, D.W.; Sabet, F.A.; Jasiuk, I.M. Modeling of Stiffness and Strength of Bone at Nanoscale. J. Biomech. Eng.
**2017**, 139. [Google Scholar] [CrossRef] - Lin, L.; Samuel, J.; Zeng, X.; Wang, X. Contribution of extrafibrillar matrix to the mechanical behavior of bone using a novel cohesive finite element model. J. Mech. Behav. Biomed.
**2017**, 65, 224–235. [Google Scholar] [CrossRef][Green Version] - Idkaidek, A.; Jasiuk, I. Cortical bone fracture analysis using XFEM—Case study. Int. J. Numer. Methods Biomed. Eng.
**2017**, 33, e2809. [Google Scholar] [CrossRef] [PubMed] - Vaughan, T.; McCarthy, C.; McNamara, L. A three-scale finite element investigation into the effects of tissue mineralisation and lamellar organisation in human cortical and trabecular bone. J. Mech. Behav. Biomed.
**2012**, 12, 50–62. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hamed, E.; Lee, Y.; Jasiuk, I. Multiscale modeling of elastic properties of cortical bone. Acta Mech.
**2010**, 213, 131–154. [Google Scholar] [CrossRef] - Fritsch, A.; Hellmich, C. ‘Universal’ microstructural patterns in cortical and trabecular, extracellular and extravascular bone materials: Micromechanics-based prediction of anisotropic elasticity. J. Theor. Biol.
**2007**, 244, 597–620. [Google Scholar] [CrossRef] - Hellmich, C.; Barthélémy, J.F.; Dormieux, L. Mineral–collagen interactions in elasticity of bone ultrastructure—A continuum micromechanics approach. Eur. J. Mech. A/Solids
**2004**, 23, 783–810. [Google Scholar] [CrossRef] - Crolet, J.; Aoubiza, B.; Meunier, A. Compact bone: Numerical simulation of mechanical characteristics. J. Biomech.
**1993**, 26, 677–687. [Google Scholar] [CrossRef] - You, T.; Kim, Y.R.; Park, T. Two-Way Coupled Multiscale Model for Predicting Mechanical Behavior of Bone Subjected to Viscoelastic Deformation and Fracture Damage. J. Eng. Mater. Technol.
**2017**, 139, 021016. [Google Scholar] [CrossRef] - Perrin, E.; Bou-Saïd, B.; Massi, F. Numerical modeling of bone as a multiscale poroelastic material by the homogenization technique. J. Mech. Behav. Biomed.
**2019**, 91, 373–382. [Google Scholar] [CrossRef] - Polgar, K.; Viceconti, M.; Connor, J.J. A comparison between automatically generated linear and parabolic tetrahedra when used to mesh a human femur. Proc. Inst. Mech. Eng. H
**2001**, 215, 85–94. [Google Scholar] [CrossRef] - Li, S.; Abdel-Wahab, A.; Silberschmidt, V.V. Analysis of fracture processes in cortical bone tissue. Eng. Fract. Mech.
**2013**, 110, 448–458. [Google Scholar] [CrossRef][Green Version] - Kaczmarczyk, L.; Pearce, C. Efficient numerical analysis of bone remodelling. J. Mech. Behav. Biomed. Mater.
**2011**, 4, 858–867. [Google Scholar] [CrossRef] [PubMed] - Lee, Y.; Ogihara, N.; Lee, T. Assessment of finite element models for prediction of osteoporotic fracture. J. Mech. Behav. Biomed.
**2019**, 97, 312–320. [Google Scholar] [CrossRef] [PubMed] - Bartsch, C. Atomistische und Kopplungsmodelle in der Elastizitätstheorie. Master’s Thesis, TU Berlin, Berlin, Germany, 2014. [Google Scholar]
- Wong, A.K. A comparison of peripheral imaging technologies for bone and muscle quantification: A technical review of image acquisition. J. Musculoskelet. Neuronal Interact.
**2016**, 16, 265–282. Available online: https://www.ncbi.nlm.nih.gov/pubmed/27973379 (accessed on 11 December 2019). [PubMed] - Rathnayaka, K.; Momot, K.I.; Noser, H.; Volp, A.; Schuetz, M.A.; Sahama, T.; Schmutz, B. Quantification of the accuracy of MRI generated 3D models of long bones compared to CT generated 3D models. Med. Eng. Phys.
**2012**, 34, 357–363. [Google Scholar] [CrossRef] [PubMed][Green Version] - Zysset, P.; Qin, L.; Lang, T.; Khosla, S.; Leslie, W.D.; Shepherd, J.A.; Schousboe, J.T.; Engelke, K. Clinical Use of Quantitative Computed Tomography–Based Finite Element Analysis of the Hip and Spine in the Management of Osteoporosis in Adults: the 2015 ISCD Official Positions—Part II. J. Clin. Densitom.
**2015**, 18, 359–392. [Google Scholar] [CrossRef] [PubMed] - Engelke, K.; Libanati, C.; Fuerst, T.; Zysset, P.; Genant, H.K. Advanced CT based In Vivo Methods for the Assessment of Bone Density, Structure, and Strength. Curr. Osteoporos. Rep.
**2013**, 11, 246–255. [Google Scholar] [CrossRef] - Iori, G.; Schneider, J.; Reisinger, A.; Heyer, F.; Peralta, L.; Wyers, C.; Gräsel, M.; Barkmann, R.; Glüer, C.C.; van den Bergh, J.P.; et al. Large cortical bone pores in the tibia are associated with proximal femur strength. PLoS ONE
**2019**, 14, 1–18. [Google Scholar] [CrossRef] - Alcantara, A. Osteoporosis Diagnosis through Multiscale Modeling of Bone Fracture using the Boundary Element Method and Molecular Dynamics; Research Project FAPESP; The São Paulo Research Foundation: São Paulo, Brazil, 2018. [Google Scholar]
- Alsayednoor, J.; Metcalf, L.; Rochester, J.; Dall’Ara, E.; McCloskey, E.; Lacroix, D. Comparison of HR-pQCT- and microCT-based finite element models for the estimation of the mechanical properties of the calcaneus trabecular bone. Biomech. Model Mechanobiol.
**2018**, 17, 1715–1730. [Google Scholar] [CrossRef][Green Version] - Jiang, Y.; Zhao, J.; Liao, E.Y.; Dai, R.C.; Wu, X.P.; Genant, H.K. Application of micro-ct assessment of 3-d bone microstructure in preclinical and clinical studies. J. Bone Miner. Metab.
**2005**, 23, 122–131. [Google Scholar] [CrossRef] - Landis, E.N.; Keane, D.T. X-ray microtomography. Mater Charact.
**2010**, 61, 1305–1316. [Google Scholar] [CrossRef] - Irie, M.S.; Rabelo, G.D.; Spin-Neto, R.; Dechichi, P.; Borges, J.S.; Soares, P.B.F. Use of Micro-Computed Tomography for Bone Evaluation in Dentistry. Braz. Dent. J.
**2018**, 29, 227–238. [Google Scholar] [CrossRef] [PubMed][Green Version] - Pahr, D.H.; Zysset, P.K. Finite Element-Based Mechanical Assessment of Bone Quality on the Basis of In Vivo Images. Curr. Osteoporos. Rep.
**2016**, 14, 374–385. [Google Scholar] [CrossRef] [PubMed][Green Version] - Fuller, H.; Fuller, R.; Pereira, R.M.R. High resolution peripheral quantitative computed tomography for the assessment of morphological and mechanical bone parameters. Rev. Bras. Reumatol. (Eng. Ed.)
**2015**, 55, 352–362. [Google Scholar] [CrossRef] [PubMed] - Kohlbrenner, A.; Haemmerle, S.; Laib, A.; Koller, B.; Ruegsegger, P. Fast 3D multiple fan-beam CT systems. In Developments in X-Ray Tomography II; Bonse, U., Ed.; International Society for Optics and Photonics (SPIE): Bellingham, WA, USA, 1999; Volume 3772, pp. 44–54. [Google Scholar] [CrossRef]
- Humbert, L.; Martelli, Y.; Fonollà, R.; Steghöfer, M.; Di Gregorio, S.; Malouf, J.; Romera, J.; Barquero, L.M.D.R. 3D-DXA: Assessing the Femoral Shape, the Trabecular Macrostructure and the Cortex in 3D from DXA images. IEEE Trans. Med Imaging
**2017**, 36, 27–39. [Google Scholar] [CrossRef] [PubMed] - El Maghraoui, A.; Roux, C. DXA scanning in clinical practice. QJM-Int. J. Med.
**2008**, 101, 605–617. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kanis, J.A.; Borgstrom, F.; De Laet, C.; Johansson, H.; Johnell, O.; Jonsson, B.; Oden, A.; Zethraeus, N.; Pfleger, B.; Khaltaev, N. Assessment of fracture risk. Osteoporos. Int.
**2005**, 16, 581–589. [Google Scholar] [CrossRef] - Broy, S.B.; Cauley, J.A.; Lewiecki, M.E.; Schousboe, J.T.; Shepherd, J.A.; Leslie, W.D. Fracture Risk Prediction by Non-BMD DXA Measures: The 2015 ISCD Official Positions Part 1: Hip Geometry. J. Clin. Densitom.
**2015**, 18, 287–308. [Google Scholar] [CrossRef] - Väänänen, S.P.; Grassi, L.; Flivik, G.; Jurvelin, J.S.; Isaksson, H. Generation of 3D shape, density, cortical thickness and finite element mesh of proximal femur from a DXA image. Med. Image Anal.
**2015**, 24, 125–134. [Google Scholar] [CrossRef][Green Version] - Lenaerts, L.; van Lenthe, G.H. Multi-level patient-specific modelling of the proximal femur. A promising tool to quantify the effect of osteoporosis treatment. Philos. Trans. R. Soc. A
**2009**, 367, 2079–2093. [Google Scholar] [CrossRef] - Cyganik, L.; Binkowski, M.; Kokot, G.; Rusin, T.; Popik, P.; Bolechala, F.; Nowak, R.; Wrobel, Z.; John, A. Prediction of Young’s modulus of trabeculae in microscale using macro-scale’s relationships between bone density and mechanical properties. J. Mech. Behav. Biomed.
**2014**, 36, 120–134. [Google Scholar] [CrossRef][Green Version] - Zhang, J.; Yan, C.H.; Chui, C.K.; Ong, S.H. Fast segmentation of bone in CT images using 3D adaptive thresholding. Comput. Biol. Med.
**2010**, 40, 231–236. [Google Scholar] [CrossRef] [PubMed] - Amorim, P.; Moraes, T.; Silva, J.; Pedrini, H. InVesalius: An Interactive Rendering Framework for Health Care Support. In Advances in Visual Computing; Bebis, G., Boyle, R., Parvin, B., Koracin, D., Pavlidis, I., Feris, R., McGraw, T., Elendt, M., Kopper, R., Ragan, E., et al., Eds.; Springer: Cham, Switzerland, 2015; pp. 45–54. [Google Scholar]
- Liao, S.H.; Tong, R.F.; Dong, J.X. Anisotropic finite element modeling for patient-specific mandible. Comput. Meth. Prog. Biol.
**2007**, 88, 197–209. [Google Scholar] [CrossRef] [PubMed] - Kennedy, J.G.; Carter, D.R. Long Bone Torsion: I. Effects of Heterogeneity, Anisotropy and Geometric Irregularity. J. Biomech. Eng.
**1985**, 107, 183–188. [Google Scholar] [CrossRef] [PubMed] - Kennedy, J.G.; Carter, D.R.; Caler, W.E. Long Bone Torsion: II. A Combined Experimental and Computational Method for Determining an Effective Shear Modulus. J. Biomech. Eng.
**1985**, 107, 189–191. [Google Scholar] [CrossRef] - Sammarco, G.; Burstein, A.H.; Davis, W.L.; Frankel, V.H. The biomechanics of torsional fractures: The effect of loading on ultimate properties. J. Biomech.
**1971**, 4, 113–117. [Google Scholar] [CrossRef] - Carter, D.; Hayes, W. Fatigue life of compact bone—I effects of stress amplitude, temperature and density. J. Biomech.
**1976**, 9, 27–34. [Google Scholar] [CrossRef] - Carter, D.; Hayes, W.; Schurman, D. Fatigue life of compact bone—II. Effects of microstructure and density. J. Biomech.
**1976**, 9, 211–218. [Google Scholar] [CrossRef] - Smith, J.; Walmsley, R. Factors affecting the elasticity of bone. J. Anat.
**1959**, 93, 503–523. Available online: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1244544/ (accessed on 2 December 2019). - Nyman, J.S.; Roy, A.; Shen, X.; Acuna, R.L.; Tyler, J.H.; Wang, X. The influence of water removal on the strength and toughness of cortical bone. J. Biomech.
**2006**, 39, 931–938. [Google Scholar] [CrossRef][Green Version] - Granke, M.; Does, M.D.; Nyman, J.S. The Role of Water Compartments in the Material Properties of Cortical Bone. Calcif. Tissue Int.
**2015**, 97, 292–307. [Google Scholar] [CrossRef][Green Version] - Sasaki, N.; Enyo, A. Viscoelastic properties of bone as a function of water content. J. Biomech.
**1995**, 28, 809–815. [Google Scholar] [CrossRef] - Currey, J. The effect of porosity and mineral content on the Young’s modulus of elasticity of compact bone. J. Biomech.
**1988**, 21, 131–139. [Google Scholar] [CrossRef] - Currey, J. The mechanical consequences of variation in the mineral content of bone. J. Biomech.
**1969**, 2, 1–11. [Google Scholar] [CrossRef] - Burstein, A.; Zika, J.; Heiple, K.; Klein, L. Contribution of collagen and mineral to the elastic-plastic properties of bone. J. Bone Jt. Surg.
**1975**, 57, 956–961. [Google Scholar] [CrossRef] - Burr, D.B. The relationships among physical, geometrical and mechanical properties of bone, with a note on the properties of nonhuman primate bone. Am. J. Phys. Anthropol.
**1980**, 23, 109–146. [Google Scholar] [CrossRef] - Hansen, U.; Zioupo, P.; Simpson, R.; Currey, J.D.; Hynd, D. The Effect of Strain Rate on the Mechanical Properties of Human Cortical Bone. J. Biomech. Eng.
**2008**, 130, 011011. [Google Scholar] [CrossRef] - Galante, J.; Rostoker, W.; Ray, R.D. Physical properties of trabecular bone. Calcif. Tissue Res.
**1970**, 5, 236–246. [Google Scholar] [CrossRef] - Bargren, J.H.; Bassett, C.L.; Gjelsvik, A. Mechanical properties of hydrated cortical bone. J. Biomech.
**1974**, 7, 239–245. [Google Scholar] [CrossRef] - Black, J.; Korostoff, E. Dynamic mechanical properties of viable human cortical bone. J. Biomech.
**1973**, 6, 435–438. [Google Scholar] [CrossRef] - Vinz, H. Change in the mechanical properties of human compact bone tissue upon aging. Polym. Mech.
**1975**, 11, 568–571. [Google Scholar] [CrossRef] - Sabet, F.A.; Raeisi Najafi, A.; Hamed, E.; Jasiuk, I. Modelling of bone fracture and strength at different length scales: A review. Interface Focus
**2016**, 6, 20150055. [Google Scholar] [CrossRef] [PubMed] - Grynpas, M. Age and disease-related changes in the mineral of bone. Calcif. Tissue Int.
**1993**, 53, S57–S64. [Google Scholar] [CrossRef] [PubMed] - Roschger, P.; Paschalis, E.; Fratzl, P.; Klaushofer, K. Bone mineralization density distribution in health and disease. Bone
**2008**, 42, 456–466. [Google Scholar] [CrossRef] [PubMed] - Kuhlencordt, F.; Dietsch, P.; Keck, E.; Kruse, H.P. Generalized Bone Diseases: Osteoporosis Osteomalacia Ostitis Fibrosa; Springer: Berlin/Heidelberg, Germany, 1986. [Google Scholar] [CrossRef]
- Derikx, L.C.; Verdonschot, N.; Tanck, E. Towards clinical application of biomechanical tools for the prediction of fracture risk in metastatic bone disease. J. Biomech.
**2015**, 48, 761–766. [Google Scholar] [CrossRef][Green Version] - Grant, J.D. Food for thought … and health: Making a case for plant-based nutrition. Can. Fam. Phy.
**2012**, 58, 917–919. Available online: https://www.ncbi.nlm.nih.gov/pubmed/22972718 (accessed on 2 December 2019). - Weaver, C.M.; Proulx, W.R.; Heaney, R. Choices for achieving adequate dietary calcium with a vegetarian diet. Am. J. Clin. Nutr.
**1999**, 70, 543s–548s. [Google Scholar] [CrossRef] - Anderson, J.J. Plant-based diets and bone health: nutritional implications. Am. J. Clin. Nutr.
**1999**, 70, 539s–542s. [Google Scholar] [CrossRef][Green Version] - Lanou, A.J. Soy foods: Are they useful for optimal bone health? Ther. Adv. Musculoskel. Dis.
**2011**, 3, 293–300. [Google Scholar] [CrossRef][Green Version] - Weikert, C.; Walter, D.; Hoffmann, K.; Kroke, A.; Bergmann, M.; Boeing, H. The Relation between Dietary Protein, Calcium and Bone Health in Women: Results from the EPIC-Potsdam Cohort. Ann. Nutr. Metab.
**2005**, 49, 312–318. [Google Scholar] [CrossRef] - Weaver, C.M.; Bischoff Ferrari, H.A.; Shanahan, C.J. Cost-benefit analysis of calcium and vitamin D supplements. Arch. Osteoporos.
**2019**, 14, 50. [Google Scholar] [CrossRef][Green Version] - Caputo, E.L.; Costa, M.Z. Influence of physical activity on quality of life in postmenopausal women with osteoporosis. Rev. Bras. Reumatol. (Eng. Ed.)
**2014**, 54, 467–473. [Google Scholar] [CrossRef] [PubMed][Green Version] - Xu, X.; Ji, W.; Lv, X.Q.; Zhu, Y.C.; Zhao, J.X.; Miao, L.Z. Impact of physical activity on health-related quality of life in osteoporotic and osteopenic postmenopausal women: A systematic review. Int. J. Nurs. Sci.
**2015**, 2, 204–217. [Google Scholar] [CrossRef][Green Version] - Yuan, Y.; Chen, X.; Zhang, L.; Wu, J.; Guo, J.; Zou, D.; Chen, B.; Sun, Z.; Shen, C.; Zou, J. The roles of exercise in bone remodeling and in prevention and treatment of osteoporosis. Prog. Biophys. Mol. Biol.
**2016**, 122, 122–130. [Google Scholar] [CrossRef] [PubMed] - Daly, R.M.; Via, J.D.; Duckham, R.L.; Fraser, S.F.; Helge, E.W. Exercise for the prevention of osteoporosis in postmenopausal women: an evidence-based guide to the optimal prescription. Braz. J. Phys. Ther.
**2019**, 23, 170–180. [Google Scholar] [CrossRef] - Fung, Y.C. Biomechanics—Mechanical Properties of Living Tissues; Springer: New York, NY, USA, 1993. [Google Scholar]
- Masson, R.; Bornert, M.; Suquet, P.; Zaoui, A. An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals. J. Mech. Phys. Solids
**2000**, 48, 1203–1227. [Google Scholar] [CrossRef][Green Version] - Weiss, J.A.; Maker, B.N.; Govindjee, S. Finite element implementation of incompressible, transversely isotropic hyperelasticity. Comput. Meth. Appl. Mech. Eng.
**1996**, 135, 107–128. [Google Scholar] [CrossRef] - Juszczyk, M.M.; Cristofolini, L.; Viceconti, M. The human proximal femur behaves linearly elastic up to failure under physiological loading conditions. J. Biomech.
**2011**, 44, 2259–2266. [Google Scholar] [CrossRef] - Wriggers, P. Nonlinear Finite Element Methods; Springer: Berlin, Germany, 2008. [Google Scholar]
- Flügge, W. Viscoelasticity; Springer: Berlin/Heidelberg, Germany, 1975. [Google Scholar] [CrossRef]
- Rietbergen, B.V.; Odgaard, A.; Kabel, J.; Huiskes, R. Direct mechanics assessment of elastic symmetries and properties of trabecular bone architecture. J. Biomech.
**1996**, 29, 1653–1657. [Google Scholar] [CrossRef] - Burstein, A.H.; Currey, J.D.; Frankel, V.H.; Reilly, D.T. The ultimate properties of bone tissue: The effects of yielding. J. Biomech.
**1972**, 5, 35–44. [Google Scholar] [CrossRef] - David, C.; James, W.D.; Ralph, M. Multiscale modelling and nonlinear finite element analysis as clinical tools for the assessment of fracture risk. Philos. Trans. R. Soc. A
**2010**, 368, 2653–2668. [Google Scholar] [CrossRef][Green Version] - An, B. Constitutive modeling the plastic deformation of bone-like materials. Int. J. Solids Struct.
**2016**, 92–93, 1–8. [Google Scholar] [CrossRef] - Natali, A.N.; Carniel, E.L.; Pavan, P.G. Constitutive modelling of inelastic behaviour of cortical bone. Med. Eng. Phys.
**2008**, 30, 905–912. [Google Scholar] [CrossRef] [PubMed] - Nguyen, L.H.; Schillinger, D. A multiscale predictor/corrector scheme for efficient elastoplastic voxel finite element analysis, with application to CT-based bone strength prediction. Comput. Meth. Appl. Mech. Eng.
**2018**, 330, 598–628. [Google Scholar] [CrossRef] - Mabrey, J.D.; Fitch, R.D. Plastic deformation in pediatric fractures: Mechanism and treatment. J. Pediat. Orthop.
**1989**, 9, 310–314. Available online: https://journals.lww.com/pedorthopaedics/Abstract/1989/05000/Plastic_Deformation_in_Pediatric_Fractures_.10.aspx (accessed on 11 December 2019). [CrossRef] - De Souza Neto, E.; Peric, D.; Owen, D. Computational Methods for Plasticity: Theory and Applications; Wiley: Hoboken, NJ, USA, 2011. [Google Scholar]
- Pawlikowski, M.; Barcz, K. Non-linear viscoelastic constitutive model for bovine cortical bone tissue. Biocybern. Biomed. Eng.
**2016**, 36, 491–498. [Google Scholar] [CrossRef] - Wirtz, D.C.; Schiffers, N.; Pandorf, T.; Radermacher, K.; Weichert, D.; Forst, R. Critical evaluation of known bone material properties to realize anisotropic FE-simulation of the proximal femur. J. Biomech.
**2000**, 33, 1325–1330. [Google Scholar] [CrossRef] - Cowin, S.C.; Doty, S.B. Tissue Mechanics; Springer: New York, NY, USA, 2007. [Google Scholar]
- Ji, B.; Gao, H. Mechanical properties of nanostructure of biological materials. J. Mech. Phys. Solids
**2004**, 52, 1963–1990. [Google Scholar] [CrossRef] - Fung, Y. Elasticity of soft tissues in simple elongation. Am. J. Physiol. Legacy Content
**1967**, 213, 1532–1544. [Google Scholar] [CrossRef][Green Version] - Fung, Y.C. Structure and Stress-Strain Relationship of Soft Tissues. Am. Zool.
**1984**, 24, 13–22. [Google Scholar] [CrossRef][Green Version] - Ojanen, X.; Tanska, P.; Malo, M.; Isaksson, H.; Väänänen, S.; Koistinen, A.; Grassi, L.; Magnusson, S.; Ribel-Madsen, S.; Korhonen, R.; et al. Tissue viscoelasticity is related to tissue composition but may not fully predict the apparent-level viscoelasticity in human trabecular bone—An experimental and finite element study. J. Biomech.
**2017**, 65, 96–105. [Google Scholar] [CrossRef][Green Version] - Pawlikowski, M.; Jankowski, K.; Skalski, K. New microscale constitutive model of human trabecular bone based on depth sensing indentation technique. J. Mech. Behav. Biomed.
**2018**, 85, 162–169. [Google Scholar] [CrossRef] [PubMed] - Fondrk, M.; Bahniuk, E.; Davy, D.; Michaels, C. Some viscoplastic characteristics of bovine and human cortical bone. J. Biomech.
**1988**, 21, 623–630. [Google Scholar] [CrossRef] - Schwiedrzik, J.J.; Zysset, P.K. An anisotropic elastic-viscoplastic damage model for bone tissue. Biomech. Model Mechanobiol.
**2013**, 12, 201–213. [Google Scholar] [CrossRef] [PubMed] - Gupta, H.S.; Fratzl, P.; Kerschnitzki, M.; Benecke, G.; Wagermaier, W.; Kirchner, H.O. Evidence for an elementary process in bone plasticity with an activation enthalpy of 1 eV. J. R. Soc. Interface
**2007**, 4, 277–282. [Google Scholar] [CrossRef] - Johnson, T.; Socrate, S.; Boyce, M. A viscoelastic, viscoplastic model of cortical bone valid at low and high strain rates. Acta Biomater.
**2010**, 6, 4073–4080. [Google Scholar] [CrossRef] - Lee, C.S.; Lee, J.M.; Youn, B.; Kim, H.S.; Shin, J.K.; Goh, T.S.; Lee, J.S. A new constitutive model for simulation of softening, plateau, and densification phenomena for trabecular bone under compression. J. Mech. Behav. Biomed.
**2017**, 65, 213–223. [Google Scholar] [CrossRef] - Turner, C.; Burr, D. Basic biomechanical measurements of bone: A tutorial. Bone
**1993**, 14, 595–608. [Google Scholar] [CrossRef] - Cowin, S.C. Bone poroelasticity. J. Biomech.
**1999**, 32, 217–238. [Google Scholar] [CrossRef] - Penta, R.; Merodio, J. Homogenized modeling for vascularized poroelastic materials. Meccanica
**2017**, 52, 3321–3343. [Google Scholar] [CrossRef][Green Version] - Grillo, A.; Prohl, R.; Wittum, G. A poroplastic model of structural reorganisation in porous media of biomechanical interest. Contin. Mech. Thermodyn.
**2016**, 28, 579–601. [Google Scholar] [CrossRef] - Krajcinovic, D. Damage mechanics. Mech. Mater.
**1989**, 8, 117–197. [Google Scholar] [CrossRef] - Sandino, C.; McErlain, D.D.; Schipilow, J.; Boyd, S.K. The poro-viscoelastic properties of trabecular bone: A micro computed tomography-based finite element study. J. Mech. Behav. Biomed.
**2015**, 44, 1–9. [Google Scholar] [CrossRef] [PubMed] - Wolff, J. Das Gesetz der Transformation der Knochen—1892; Reprint; Pro Business: Berlin, Germany, 2010. [Google Scholar]
- Maquet, P.; Wolff, J.; Furlong, R. The Law of Bone Remodelling; Springer: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Fazzalari, N.L. Bone remodeling: A review of the bone microenvironment perspective for fragility fracture (osteoporosis) of the hip. Semin. Cell Dev. Biol.
**2008**, 19, 467–472. [Google Scholar] [CrossRef] [PubMed] - Cowin, S.C.; Hegedus, D.H. Bone remodeling I: theory of adaptive elasticity. J. Elast.
**1976**, 6, 313–326. [Google Scholar] [CrossRef] - Huiskes, R.; Ruimerman, R.; van Lenthe, G.H.; Janssen, J.D. Effects of mechanical forces on maintenance and adaptation of form in trabecular bone. Nature
**2000**, 405, 704–706. [Google Scholar] [CrossRef] - Ralston, S.H. Bone structure and metabolism. Medicine
**2013**, 41, 581–585. [Google Scholar] [CrossRef] - Lerebours, C.; Buenzli, P.R.; Scheiner, S.; Pivonka, P. A multiscale mechanobiological model of bone remodelling predicts site-specific bone loss in the femur during osteoporosis and mechanical disuse. Biomech. Model Mechanobiol.
**2016**, 15, 43–67. [Google Scholar] [CrossRef][Green Version] - Engh, C.; Bobyn, J.; Glassman, A. Porous-coated hip replacement. The factors governing bone ingrowth, stress shielding, and clinical results. J. Bone Jt. Surg. Brit. Vol.
**1987**, 69-B, 45–55. [Google Scholar] [CrossRef] - Frost, H. Wolff’s Law and bone’s structural adaptations to mechanical usage: An overview for clinicians. Angle Orthod.
**1994**, 64, 175–188. Available online: http://wedocs.unep.org/handle/20.500.11822/18246?show=full (accessed on 2 December 2019). [CrossRef] - Andreaus, U.; Giorgio, I.; Madeo, A. Modeling of the interaction between bone tissue and resorbable biomaterial as linear elastic materials with voids. Z. Angew. Math. Phys.
**2015**, 66, 209–237. [Google Scholar] [CrossRef][Green Version] - Van Houtte, P. Anisotropic Plasticity. In Numerical Modelling of Material Deformation Processes: Research, Development and Applications; Hartley, P., Pillinger, I., Sturgess, C., Eds.; Springer: London, UK, 1992; pp. 84–111. [Google Scholar] [CrossRef]
- Ting, T.C. Anisotropic Elasticity: Theory and Applications; Oxford University Press: Oxford, UK, 1996. [Google Scholar]
- Baca, V.; Horak, Z.; Mikulenka, P.; Dzupa, V. Comparison of an inhomogeneous orthotropic and isotropic material models used for FE analyses. Med. Eng. Phys.
**2008**, 30, 924–930. [Google Scholar] [CrossRef] [PubMed] - Taghizadeh, E.; Reyes, M.; Zysset, P.; Latypova, A.; Terrier, A.; Büchler, P. Biomechanical Role of Bone Anisotropy Estimated on Clinical CT Scans by Image Registration. Ann. Biomed. Eng.
**2016**, 44, 2505–2517. [Google Scholar] [CrossRef] [PubMed][Green Version] - Uten’kin, A.A.; Ashkenazi, E.K. The anisotropy of compact bone material. Polym. Mech.
**1972**, 8, 614–618. [Google Scholar] [CrossRef] - Yoon, H.S.; Katz, J.L. Ultrasonic wave propagation in human cortical bone—I. Theoretical considerations for hexagonal symmetry. J. Biomech.
**1976**, 9, 407-IN3. [Google Scholar] [CrossRef] - Atsumi, N.; Tanaka, E.; Iwamoto, M.; Hirabayashi, S. Constitutive modeling of cortical bone considering anisotropic inelasticity and damage evolution. Mech. Eng. J.
**2017**, 17-00095, advpub. [Google Scholar] [CrossRef][Green Version] - Carter, D.; Hayes, W. The Compressive Behavior of Bone as a Two-Phase Porous Structure. J. Bone Jt. Surg.
**1977**, 59, 954–962. [Google Scholar] [CrossRef] - François, M.; Geymonat, G.; Berthaud, Y. Determination of the symmetries of an experimentally determined stiffness tensor: Application to acoustic measurements. Int. J. Solids Struct.
**1998**, 35, 4091–4106. [Google Scholar] [CrossRef][Green Version] - Sedlák, P.; Seiner, H.; Zídek, J.; Janovská, M.; Landa, M. Determination of All 21 Independent Elastic Coefficients of Generally Anisotropic Solids by Resonant Ultrasound Spectroscopy: Benchmark Examples. Exp. Mech.
**2014**, 54, 1073–1085. [Google Scholar] [CrossRef] - Goulet, R.; Goldstein, S.; Ciarelli, M.; Kuhn, J.; Brown, M.; Feldkamp, L. The relationship between the structural and orthogonal compressive properties of trabecular bone. J. Biomech.
**1994**, 27, 375–389. [Google Scholar] [CrossRef][Green Version] - Netz, P.; Eriksson, K.; Stromberg, L. Non-Linear Properties of Diaphyseal Bone: An Experimental Study on Dogs. Acta Orthop. Scand.
**1979**, 50, 139–143. [Google Scholar] [CrossRef] - Lakes, R.S. Dynamical Study of Couple Stress Effects in Human Compact Bone. J. Biomech. Eng.
**1982**, 104, 6–11. [Google Scholar] [CrossRef] [PubMed] - Melnis, A.É.; Knets, I.V.; Moorlat, P.A. Deformation behavior of human compact bone tissue upon creep under tensile testing. Mech. Compos. Mater.
**1980**, 15, 574–579. [Google Scholar] [CrossRef] - Yang, J.; Lakes, R.S. Experimental study of micropolar and couple stress elasticity in compact bone in bending. J. Biomech.
**1982**, 15, 91–98. [Google Scholar] [CrossRef] - Currey, J.D. The effects of drying and re-wetting on some mechanical properties of cortical bone. J. Biomech.
**1988**, 21, 439–441. [Google Scholar] [CrossRef] - Skedros, J.G.; Mason, M.W.; Bloebaum, R.D. Differences in osteonal micromorphology between tensile and compressive cortices of a bending skeletal system: Indications of potential strain-specific differences in bone microstructure. Anat Rec.
**1994**, 239, 405–413. [Google Scholar] [CrossRef] - Barak, M.M.; Weiner, S.; Shahar, R. Importance of the integrity of trabecular bone to the relationship between load and deformation of rat femora: an optical metrology study. J. Mater. Chem.
**2008**, 18, 3855–3864. [Google Scholar] [CrossRef] - Yan, J.; Daga, A.; Kumar, R.; Mecholsky, J.J. Fracture toughness and work of fracture of hydrated, dehydrated, and ashed bovine bone. J. Biomech.
**2008**, 41, 1929–1936. [Google Scholar] [CrossRef] - Brynk, T.; Hellmich, C.; Fritsch, A.; Zysset, P.; Eberhardsteiner, J. Experimental poromechanics of trabecular bone strength: Role of Terzaghi’s effective stress and of tissue level stress fluctuations. J. Biomech.
**2011**, 44, 501–508. [Google Scholar] [CrossRef] - Zannoni, C.; Mantovani, R.; Viceconti, M. Material properties assignment to finite element models of bone structures: A new method. Med. Eng. Phys.
**1999**, 20, 735–740. [Google Scholar] [CrossRef] - Taddei, F.; Pancanti, A.; Viceconti, M. An improved method for the automatic mapping of computed tomography numbers onto finite element models. Med. Eng. Phys.
**2004**, 26, 61–69. [Google Scholar] [CrossRef] - Pegg, E.C.; Gill, H.S. An open source software tool to assign the material properties of bone for ABAQUS finite element simulations. J. Biomech.
**2016**, 49, 3116–3121. [Google Scholar] [CrossRef] [PubMed][Green Version] - Alcantara, A.C.S. Implementierung verschiedener Algorithmen zur automatisierten Berechnung und Zuweisung von Materialgesetzen von CT-Daten auf FE-Netze (eng. Implementation of Various Algorithms using Matlab for an Automated Calculation and Assignment of Material Mapping of Computed Tomography Data onto Finite Element Meshes). Bachelor’s Thesis, Hochschule Merseburg, Merseburg, Germany, 2017. [Google Scholar]
- Les, C.M.; Keyak, J.H.; Stover, S.M.; Taylor, K.T.; Kaneps, A.J. Estimation of material properties in the equine metacarpus with use of quantitative computed tomography. J. Orthop. Res.
**1994**, 12, 822–833. [Google Scholar] [CrossRef] [PubMed] - Hounsfield, G. Computed medical imaging. Science
**1980**, 210, 22–28. [Google Scholar] [CrossRef] [PubMed][Green Version] - Knowles, N.K.; Reeves, J.M.; Ferreira, L.M. Quantitative Computed Tomography (QCT) derived Bone Mineral Density (BMD) in finite element studies: A review of the literature. J. Exp. Orthop.
**2016**, 3, 36. [Google Scholar] [CrossRef][Green Version] - Schileo, E.; Dall’Ara, E.; Taddei, F.; Malandrino, A.; Schotkamp, T.; Baleani, M.; Viceconti, M. An accurate estimation of bone density improves the accuracy of subject-specific finite element models. J. Biomech.
**2008**, 41, 2483–2491. [Google Scholar] [CrossRef] - Rajapakse, C.S.; Kobe, E.A.; Batzdorf, A.S.; Hast, M.W.; Wehrli, F.W. Accuracy of MRI-based finite element assessment of distal tibia compared to mechanical testing. Bone
**2018**, 108, 71–78. [Google Scholar] [CrossRef] - Ho, K.Y.; Hu, H.H.; Keyak, J.H.; Colletti, P.M.; Powers, C.M. Measuring bone mineral density with fat–water MRI: Comparison with computed tomography. J. Magn. Reson. Imaging
**2013**, 37, 237–242. [Google Scholar] [CrossRef] - Lee, Y.H.; Kim, J.J.; Jang, I.G. Patient-Specific Phantomless Estimation of Bone Mineral Density and Its Effects on Finite Element Analysis Results: A Feasibility Study. Comput. Math. Methods Med.
**2019**, 2019, 10. [Google Scholar] [CrossRef][Green Version] - Helgason, B.; Perilli, E.; Schileo, E.; Taddei, F.; Brynjólfsson, S.; Viceconti, M. Mathematical relationships between bone density and mechanical properties: A literature review. Clin. Biomech.
**2008**, 23, 135–146. [Google Scholar] [CrossRef] - Morgan, E.F.; Bayraktar, H.H.; Keaveny, T.M. Trabecular bone modulus–density relationships depend on anatomic site. J. Biomech.
**2003**, 36, 897–904. [Google Scholar] [CrossRef] - Kopperdahl, D.L.; Morgan, E.F.; Keaveny, T.M. Quantitative computed tomography estimates of the mechanical properties of human vertebral trabecular bone. J. Orthop. Res.
**2002**, 20, 801–805. [Google Scholar] [CrossRef] - Huang, H.L.; Tsai, M.T.; Lin, D.J.; Chien, C.S.; Hsu, J.T. A new method to evaluate the elastic modulus of cortical bone by using a combined computed tomography and finite element approach. Comput. Biol. Med.
**2010**, 40, 464–468. [Google Scholar] [CrossRef] [PubMed] - Cong, A.; Buijs, J.O.D.; Dragomir-Daescu, D. In situ parameter identification of optimal density–elastic modulus relationships in subject-specific finite element models of the proximal femur. Med. Eng. Phys.
**2011**, 33, 164–173. [Google Scholar] [CrossRef] [PubMed][Green Version] - Keller, T.S. Predicting the compressive mechanical behavior of bone. J. Biomech.
**1994**, 27, 1159–1168. [Google Scholar] [CrossRef] - Hellmich, C.; Kober, C.; Erdmann, B. Micromechanics-Based Conversion of CT Data into Anisotropic Elasticity Tensors, Applied to FE Simulations of a Mandible. Ann. Biomed. Eng.
**2007**, 36, 108–122. [Google Scholar] [CrossRef] - Blanchard, R.; Dejaco, A.; Bongaers, E.; Hellmich, C. Intravoxel bone micromechanics for microCT-based finite element simulations. J. Biomech.
**2013**, 46, 2710–2721. [Google Scholar] [CrossRef][Green Version] - Hasslinger, P.; Vass, V.; Dejaco, A.; Blanchard, R.; Örlygsson, G.; Gargiulo, P.; Hellmich, C. Coupling multiscale X-ray physics and micromechanics for bone tissue composition and elasticity determination from micro-CT data, by example of femora from OVX and sham rats. Int. J. Comput. Methods Eng. Sci. Mech.
**2016**, 17, 222–244. [Google Scholar] [CrossRef] - Blanchard, R.; Morin, C.; Malandrino, A.; Vella, A.; Sant, Z.; Hellmich, C. Patient-specific fracture risk assessment of vertebrae: A multiscale approach coupling X-ray physics and continuum micromechanics. Int. J. Numer. Methods Biomed. Eng.
**2016**, 32, e02760. [Google Scholar] [CrossRef][Green Version] - Fritsch, A.; Hellmich, C.; Dormieux, L. Ductile sliding between mineral Crystals. followed by rupture of collagen crosslinks: Experimentally supported micromechanical explanation of bone strength. J. Theor. Biol.
**2009**, 260, 230–252. [Google Scholar] [CrossRef][Green Version] - Eberhardsteiner, L.; Hellmich, C.; Scheiner, S. Layered water in crystal interfaces as source for bone viscoelasticity: Arguments from a multiscale approach. Comput. Methods Biomech. Biomed. Engin.
**2014**, 17, 48–63. [Google Scholar] [CrossRef][Green Version] - Morin, C.; Hellmich, C. A multiscale poromicromechanical approach to wave propagation and attenuation in bone. Ultrasonics
**2014**, 54, 1251–1269. [Google Scholar] [CrossRef] [PubMed][Green Version] - Pahr, D.H.; Dall’Ara, E.; Varga, P.; Zysset, P.K. HR-pQCT-based homogenised finite element models provide quantitative predictions of experimental vertebral body stiffness and strength with the same accuracy as μ FE models. Comput. Methods Biomech. Biomed. Eng.
**2012**, 15, 711–720. [Google Scholar] [CrossRef] [PubMed] - Kuhl, E.; Menzel, A.; Steinmann, P. Computational modeling of growth. Comput. Mech.
**2003**, 32, 71–88. [Google Scholar] [CrossRef] - Sarvi, M.N.; Luo, Y.; Sun, P.; Ouyang, J. Experimental Validation of Subject-Specific Dynamics Model for Predicting Impact Force in Sideways Fall. J. Biomed. Sci. Eng.
**2014**, 7, 405–418. [Google Scholar] [CrossRef][Green Version] - Sarvi, M.N.; Luo, Y. Improving the prediction of sideways fall-induced impact force for women by developing a female-specific equation. J. Biomech.
**2019**, 88, 64–71. [Google Scholar] [CrossRef] - Panyasantisuk, J.; Pahr, D.H.; Zysset, P.K. Effect of boundary conditions on yield properties of human femoral trabecular bone. Biomech. Model Mechanobiol.
**2016**, 15, 1043–1053. [Google Scholar] [CrossRef] - Van den Kroonenberg, A.J.; Hayes, W.C.; McMahon, T.A. Dynamic Models for Sideways Falls From Standing Height. J. Biomech. Eng.
**1995**, 117, 309–318. [Google Scholar] [CrossRef] - Bergmann, G.; Deuretzbacher, G.; Heller, M.; Graichen, F.; Rohlmann, A.; Strauss, J.; Duda, G. Hip contact forces and gait patterns from routine activities. J. Biomech.
**2001**, 34, 859–871. [Google Scholar] [CrossRef] - Altai, Z.; Qasim, M.; Li, X.; Viceconti, M. The effect of boundary and loading conditions on patient classification using finite element predicted risk of fracture. Clin. Biomech.
**2019**, 68, 137–143. [Google Scholar] [CrossRef][Green Version] - Varga, P.; Schwiedrzik, J.; Zysset, P.K.; Fliri-Hofmann, L.; Widmer, D.; Gueorguiev, B.; Blauth, M.; Windolf, M. Nonlinear quasi-static finite element simulations predict in vitro strength of human proximal femora assessed in a dynamic sideways fall setup. J. Mech. Behav. Biomed.
**2016**, 57, 116–127. [Google Scholar] [CrossRef][Green Version] - Kim, N. Introduction to Nonlinear Finite Element Analysis; Springer: New York, NY, USA, 2014. [Google Scholar]
- Reddy, J. An Introduction to Nonlinear Finite Element Analysis: With Applications to Heat Transfer, Fluid Mechanics, and Solid Mechanics; Oxford University Press: Oxford, UK, 2015. [Google Scholar]
- Parashar, S.K.; Sharma, J.K. A review on application of finite element modelling in bone biomechanics. Perspect. Sci.
**2016**, 8, 696–698. [Google Scholar] [CrossRef][Green Version] - Macdonald, H.M.; Nishiyama, K.K.; Kang, J.; Hanley, D.A.; Boyd, S.K. Age-related patterns of trabecular and cortical bone loss differ between sexes and skeletal sites: A population-based HR-pQCT study. J. Bone Miner. Res.
**2011**, 26, 50–62. [Google Scholar] [CrossRef] [PubMed] - Boyd, S.K.; Müller, R.; Zernicke, R.F. Mechanical and Architectural Bone Adaptation in Early Stage Experimental Osteoarthritis. J. Bone Miner. Res.
**2002**, 17, 687–694. [Google Scholar] [CrossRef] [PubMed] - Brebbia, C.A.; Dominguez, J. Boundary Elements: An Introductory Course; WIT Press: Southampton, UK, 1994. [Google Scholar]
- Wrobel, L.C.; Aliabadi, M. The Boundary Element Method, Applications in Thermo-Fluids and Acoustics; The Boundary Element Method; Wiley: Hoboken, NJ, USA, 2002. [Google Scholar]
- Brebbia, C.; Telles, J.; Wrobel, L. Boundary Element Techniques: Theory and Applications in Engineering; Springer: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Sollero, P.; Noritomi, P.; da Silva, J.V.L. Transversely Isotropic Bone Remodeling Using Boundary Element Method. In Proceedings of the II. International Conference on Computational Bioengineering, Lisbon, Portugal, 14–16 September 2005. [Google Scholar]
- Prada, D.; Galvis, A.; Sollero, P. Superficial 3D mesh generation process using multimedia software for multiscale bone analysis. In Proceedings of the 18th International COnference on Boundary Element and Meshless Techniques-BETEQ; EC Ltd.: Bucharest, Romania, 2017; pp. 126–131. [Google Scholar]
- Martinez, M.; Power, M.A.H. A Boundary Element Method For Analysis Of Bone Remodelling. WIT Trans. Model Simul.
**1996**, 14, 8. [Google Scholar] [CrossRef] - Martínez, G.; Aznar, J.M.G.; Doblaré, M.; Cerrolaza, M. External bone remodeling through boundary elements and damage mechanics. Math Comput. Simul.
**2006**, 73, 183–199. [Google Scholar] [CrossRef] - Sadegh, A.; Luo, G.; Cowin, S. Bone ingrowth: An application of the boundary element method to bone remodeling at the implant interface. J. Biomech.
**1993**, 26, 167–182. [Google Scholar] [CrossRef] - Prada, D.M.; Galvis, A.F.; Alcântara, A.C.; Sollero, P. 3D Boundary element meshing for multiscale bone anisotropic analysis. Eur. J. Comput. Mech.
**2018**, 27, 425–442. [Google Scholar] [CrossRef] - Dow, J.O.; Jones, M.S.; Harwood, S.A. A generalized finite difference method for solid mechanics. Numer. Methods Partial Differ. Equ.
**1990**, 6, 137–152. [Google Scholar] [CrossRef] - Hong, J.H.; Mun, M.S.; Lim, T.H. Strain rate dependent poroelastic behavior of bovine vertebral trabecular bone. KSME Int. J.
**2001**, 15, 1032–1040. [Google Scholar] [CrossRef] - Hosokawa, A. Numerical simulation of cancellous bone remodeling using finite difference time-domain method. AIP Conf. Proc.
**2012**, 1433, 233–236. [Google Scholar] [CrossRef] - Bahrieh, M.; Fakharzadeh, A. A survey on bone metastasis by finite difference method. In Proceedings of the 2015 2nd International Conference on Knowledge-Based Engineering and Innovation (KBEI), Tehran, Iran, 5–6 November 2015; pp. 468–472. [Google Scholar] [CrossRef]
- Hosokawa, A. Finite-Difference Time-Domain Simulations of Ultrasound Backscattered Waves in Cancellous Bone. In 5th International Conference on Biomedical Engineering in Vietnam; Toi, V.V., Lien Phuong, T.H., Eds.; Springer: Cham, Switzerland, 2015; pp. 289–292. [Google Scholar]
- Keyak, J.H.; Rossi, S.A. Prediction of femoral fracture load using finite element models: An examination of stress- and strain-based failure theories. J. Biomech.
**2000**, 33, 209–214. [Google Scholar] [CrossRef] - Anderson, T. Fracture Mechanics: Fundamentals and Applications, 4th ed.; CRC Press: Boca Raton, FL, USA, 2017. [Google Scholar]
- Knott, J. Fundamentals of Fracture Mechanics; Butterworths: London, UK, 1973. [Google Scholar]
- Jin, Z.H.; Sun, C. A comparison of cohesive zone modeling and classical fracture mechanics based on near tip stress field. Int. J. Solids Struct.
**2006**, 43, 1047–1060. [Google Scholar] [CrossRef][Green Version] - Park, K.; Paulino, G.H. Cohesive Zone Models: A Critical Review of Traction-Separation Relationships Across Fracture Surfaces. Appl. Mech. Rev.
**2013**, 64, 060802. [Google Scholar] [CrossRef] - Libonati, F.; Vergani, L. Understanding the structure–property relationship in cortical bone to design a biomimetic composite. Compos. Struct.
**2016**, 139, 188–198. [Google Scholar] [CrossRef][Green Version] - Cox, B.N.; Yang, Q. Cohesive zone models of localization and fracture in bone. Eng. Fract. Mech.
**2007**, 74, 1079–1092. [Google Scholar] [CrossRef] - Elices, M.; Guinea, G.; Gómez, J.; Planas, J. The cohesive zone model: Advantages, limitations and challenges. Eng. Fract. Mech.
**2002**, 69, 137–163. [Google Scholar] [CrossRef] - Begonia, M.; Dallas, M.; Johnson, M.; Thiagarajan, G. Comparison of strain measurement in the mouse forearm using subject-specific finite element models, strain gaging, and digital image correlation. Biomech. Model Mechanobiol.
**2017**, 16, 1243–1253. [Google Scholar] [CrossRef] - Pereira, A.; Javaheri, B.; Pitsillides, A.; Shefelbine, S. Predicting cortical bone adaptation to axial loading in the mouse tibia. J. R. Soc. Interface
**2015**, 12, 1–14. [Google Scholar] [CrossRef][Green Version] - Ramezanzadehkoldeh, M.; Skallerud, B. MicroCT-based finite element models as a tool for virtual testing of cortical bone. Med. Eng. Phys.
**2017**, 46, 12–20. [Google Scholar] [CrossRef] - Sreenivasan, D.; Tu, P.; Dickinson, M.; Watson, M.; Blais, A.; Das, R.; Cornish, J.; Fernandez, J. Computer modelling integrated with micro-CT and material testing provides additional insight to evaluate bone treatments: Application to a beta-glycan derived whey protein mice model. Comput. Biol. Med.
**2016**, 68, 9–20. [Google Scholar] [CrossRef] - Thiagarajan, G.; Lu, Y.; Dallas, M.; Johnson, M. Experimental and finite element analysis of dynamic loading of the mouse forearm. J. Orthop. Res.
**2014**, 32, 1580–1588. [Google Scholar] [CrossRef] [PubMed][Green Version] - Yang, H.; Xu, X.; Bullock, W.; Main, R. Adaptive changes in micromechanical environments of cancellous and cortical bone in response to in vivo loading and disuse. J. Biomech.
**2019**, 89, 85–94. [Google Scholar] [CrossRef] [PubMed] - Liu, C.; Carrera, R.; Flamini, V.; Kenny, L.; Cabahug-Zuckerman, P.; George, B.M.; Hunter, D.; Liu, B.; Singh, G.; Leucht, P.; et al. Effects of mechanical loading on cortical defect repair using a novel mechanobiological model of bone healing. Bone
**2018**, 108, 145–155. [Google Scholar] [CrossRef] [PubMed] - Oliviero, S.; Lu, Y.; Viceconti, M.; Dall’Ara, E. Effect of integration time on the morphometric, densitometric and mechanical properties of the mouse tibia. J. Biomech.
**2017**, 65, 203–211. [Google Scholar] [CrossRef] [PubMed] - Chennimalai Kumar, N.; Dantzig, J.; Jasiuk, I.; Robling, A.; Turner, C. Numerical modeling of long bone adaptation due to mechanical loading: Correlation with experiments. Ann. Biomed. Eng.
**2010**, 38, 594–604. [Google Scholar] [CrossRef][Green Version] - Hojjat, S.P.; Beek, M.; Akens, M.; Whyne, C. Can micro-imaging based analysis methods quantify structural integrity of rat vertebrae with and without metastatic involvement? J. Biomech.
**2012**, 45, 2342–2348. [Google Scholar] [CrossRef] - Stadelmann, V.; Potapova, I.; Camenisch, K.; Nehrbass, D.; Richards, R.; Moriarty, T.; Chang, Y. In vivo MicroCT monitoring of osteomyelitis in a rat model. BioMed Res. Int.
**2015**, 2015. [Google Scholar] [CrossRef][Green Version] - Torcasio, A.; Zhang, X.; Duyck, J.; Van Lenthe, G. 3D characterization of bone strains in the rat tibia loading model. Biomech. Model Mechanobiol.
**2012**, 11, 403–410. [Google Scholar] [CrossRef][Green Version] - Vickerton, P.; Jarvis, J.; Gallagher, J.; Akhtar, R.; Sutherland, H.; Jeffery, N. Morphological and histological adaptation of muscle and bone to loading induced by repetitive activation of muscle. Proc. R. Soc. B
**2014**, 281, 20140786. [Google Scholar] [CrossRef][Green Version] - Wehner, T.; Steiner, M.; Ignatius, A.; Claes, L.; Aegerter, C. Prediction of the time course of callus stiffness as a function of mechanical parameters in experimental rat fracture healing studies—A numerical study. PLoS ONE
**2014**, 9, e115695. [Google Scholar] [CrossRef] - Tsafnat, N.; Wroe, S. An experimentally validated micromechanical model of a rat vertebra under compressive loading. J. Anat.
**2011**, 218, 40–46. [Google Scholar] [CrossRef] [PubMed] - Newham, E.; Kague, E.; Aggleton, J.A.; Fernee, C.; Brown, K.R.; Hammond, C.L. Finite element and deformation analyses predict pattern of bone failure in loaded zebrafish spines. J. R. Soc. Interface
**2019**, 16, 20190430. [Google Scholar] [CrossRef] [PubMed][Green Version] - Rothstock, S.; Kowaleski, M.; Boudrieau, R.; Beale, B.; Piras, A.; Ryan, M.; Bouré, L.; Brianza, S. Biomechanical and computational evaluation of two loading transfer concepts for pancarpal arthrodesis in dogs. Am. J. Vet. Res.
**2012**, 73, 1687–1695. [Google Scholar] [CrossRef] [PubMed] - Arias-Moreno, A.; Ito, K.; van Rietbergen, B. Micro-Finite Element analysis will overestimate the compressive stiffness of fractured cancellous bone. J. Biomech.
**2016**, 49, 2613–2618. [Google Scholar] [CrossRef] - Bright, J.; Gröning, F. Strain accommodation in the zygomatic arch of the pig: A validation study using digital speckle pattern interferometry and finite element analysis. J. Morphol.
**2011**, 272, 1388–1398. [Google Scholar] [CrossRef] - Lei, T.; Xie, L.; Tu, W.; Chen, Y.; Tan, Y. Development of a finite element model for blast injuries to the pig mandible and a preliminary biomechanical analysis. J. Trauma Acute Care Surg.
**2012**, 73, 902–907. [Google Scholar] [CrossRef] - Li, S.; Abdel-Wahab, A.; Demirci, E.; Silberschmidt, V. Penetration of cutting tool into cortical bone: Experimental and numerical investigation of anisotropic mechanical behaviour. J. Biomech.
**2014**, 47, 1117–1126. [Google Scholar] [CrossRef][Green Version] - Qi, L.; Wang, X.; Meng, M. 3D finite element modeling and analysis of dynamic force in bone drilling for orthopedic surgery. Int. J. Numer. Methods Biomed. Eng.
**2014**, 30, 845–856. [Google Scholar] [CrossRef] - Sabet, F.; Jin, O.; Koric, S.; Jasiuk, I. Nonlinear micro-CT based FE modeling of trabecular bone Sensitivity of apparent response to tissue constitutive law and bone volume fraction. Int. J. Numer. Methods Biomed. Eng.
**2018**, 34, e2941. [Google Scholar] [CrossRef] - Stricker, A.; Widmer, D.; Gueorguiev, B.; Wahl, D.; Varga, P.; Duttenhoefer, F. Finite element analysis and biomechanical testing to analyze fracture displacement of alveolar ridge splitting. BioMed Res. Int.
**2018**, 2018. [Google Scholar] [CrossRef] - Avery, C.; Bujtár, P.; Simonovics, J.; Dézsi, T.; Váradi, K.; Sándor, G.; Pan, J. A finite element analysis of bone plates available for prophylactic internal fixation of the radial osteocutaneous donor site using the sheep tibia model. Med. Eng. Phys.
**2013**, 35, 1421–1430. [Google Scholar] [CrossRef] [PubMed] - Müller, R.; Henss, A.; Kampschulte, M.; Rohnke, M.; Langheinrich, A.; Heiss, C.; Janek, J.; Voigt, A.; Wilke, H.; Ignatius, A.; et al. Analysis of microscopic bone properties in an osteoporotic sheep model: A combined biomechanics, FE and ToF-SIMS study. J. R. Soc. Interface
**2019**, 16, 20180793. [Google Scholar] [CrossRef] [PubMed][Green Version] - Rehman, S.; Garner, P.; Aaron, J.; Wilcox, R. The use of preserved tissue in finite element modelling of fresh ovine vertebral behaviour. Comput. Methods Biomech. Biomed. Eng.
**2013**, 16, 1163–1169. [Google Scholar] [CrossRef] [PubMed] - Guillean, T.; Zhang, Q.H.; Tozzi, G.; Ohrndorf, A.; Christ, H.J.; Tong, J. Compressive behaviour of bovine cancellous bone and bone analogous materials, microCT characterisation and FE analysis. J. Mech. Behav. Biomed.
**2011**, 4, 1452–1461. [Google Scholar] [CrossRef] [PubMed] - Jungmann, R.; Szabo, M.; Schitter, G.; Yue-Sing Tang, R.; Vashishth, D.; Hansma, P.; Thurner, P. Local strain and damage mapping in single trabeculae during three-point bending tests. J. Mech. Behav. Biomed.
**2011**, 4, 523–534. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ng, T.; R. Koloor, S.; Djuansjah, J.; Abdul Kadir, M. Assessment of compressive failure process of cortical bone materials using damage-based model. J. Mech. Behav. Biomed.
**2017**, 66, 1–11. [Google Scholar] [CrossRef] - Ridha, H.; Thurner, P. Finite element prediction with experimental validation of damage distribution in single trabeculae during three-point bending tests. J. Mech. Behav. Biomed.
**2013**, 27, 94–106. [Google Scholar] [CrossRef] - Tozzi, G.; Zhang, Q.H.; Tong, J. 3D real-time micromechanical compressive behaviour of bone-cement interface: Experimental and finite element studies. J. Biomech.
**2012**, 45, 356–363. [Google Scholar] [CrossRef] - Zhang, G.; Xu, S.; Yang, J.; Guan, F.; Cao, L.; Mao, H. Combining specimen-specific finite-element models and optimization in cortical-bone material characterization improves prediction accuracy in three-point bending tests. J. Biomech.
**2018**, 76, 103–111. [Google Scholar] [CrossRef] - Zhang, G.J.; Yang, J.; Guan, F.J.; Chen, D.; Li, N.; Cao, L.; Mao, H. Quantifying the Effects of Formalin Fixation on the Mechanical Properties of Cortical Bone Using Beam Theory and Optimization Methodology with Specimen-Specific Finite Element Models. J. Biomech. Eng.
**2016**, 138, 094502. [Google Scholar] [CrossRef] - Zhang, M.; Gao, J.; Huang, X.; Gong, H.; Zhang, M.; Liu, B. Effects of scan resolutions and element sizes on bovine vertebral mechanical parameters from quantitative computed tomography-based finite element analysis. J. Healthc. Eng.
**2017**, 2017. [Google Scholar] [CrossRef] [PubMed][Green Version] - Harrison, S.; Chris Whitton, R.; Kawcak, C.; Stover, S.; Pandy, M. Evaluation of a subject-specific finite-element model of the equine metacarpophalangeal joint under physiological load. J. Biomech.
**2014**, 47, 65–73. [Google Scholar] [CrossRef] [PubMed] - Huynh Nguyen, N.; Pahr, D.; Gross, T.; Skinner, M.; Kivell, T. Micro-finite element (μFE) modeling of the siamang (Symphalangus syndactylus) third proximal phalanx: The functional role of curvature and the flexor sheath ridge. J. Hum. Evol.
**2014**, 67, 60–75. [Google Scholar] [CrossRef] [PubMed] - Cox, P.; Fagan, M.; Rayfield, E.; Jeffery, N. Finite element modelling of squirrel, guinea pig and rat skulls: Using geometric morphometrics to assess sensitivity. J. Anat.
**2011**, 219, 696–709. [Google Scholar] [CrossRef] [PubMed] - Liu, J.; Shi, J.; Fitton, L.; Phillips, R.; O’Higgins, P.; Fagan, M. The application of muscle wrapping to voxel-based finite element models of skeletal structures. Biomech. Model Mechanobiol.
**2012**, 11, 35–47. [Google Scholar] [CrossRef] [PubMed] - Strazic Geljic, I.; Melis, N.; Boukhechba, F.; Schaub, S.; Mellier, C.; Janvier, P.; Laugier, J.P.; Bouler, J.M.; Verron, E.; Scimeca, J.C. Gallium enhances reconstructive properties of a calcium phosphate bone biomaterial. J. Tissue Eng. Regen. Med.
**2018**, 12, e854–e866. [Google Scholar] [CrossRef] [PubMed] - Mengoni, M.; Voide, R.; de Bien, C.; Freichels, H.; Jerôme, C.; Leonard, A.; Toye, D.; Müller, R.; van Lenthe, G.; Ponthot, J. A non-linear homogeneous model for bone-like materials under compressive load. Int. J. Numer. Methods Biomed. Eng.
**2012**, 28, 334–348. [Google Scholar] [CrossRef][Green Version] - Pramudita, J.; Kamiya, S.; Ujihashi, S.; Choi, H.Y.; Ito, M.; Watanabe, R.; Crandall, J.; Kent, R. Estimation of conditions evoking fracture in finger bones under pinch loading based on finite element analysis. Comput. Methods Biomech. Biomed. Eng.
**2017**, 20, 35–44. [Google Scholar] [CrossRef] - Chen, J.; He, Y.; Keilig, L.; Reimann, S.; Hasan, I.; Weinhold, J.; Radlanski, R.; Bourauel, C. Numerical investigations of bone remodelling around the mouse mandibular molar primordia. Ann. Anat.
**2019**, 222, 146–152. [Google Scholar] [CrossRef] - Doostmohammadi, A.; Karimzadeh Esfahani, Z.; Ardeshirylajimi, A.; Rahmati Dehkordi, Z. Zirconium modified calcium-silicate-based nanoceramics: An in vivo evaluation in a rabbit tibial defect model. Int. J. Appl. Ceram. Technol.
**2019**, 16, 431–437. [Google Scholar] [CrossRef] - Ren, L.M.; Arahira, T.; Todo, M.; Yoshikawa, H.; Myoui, A. Biomechanical evaluation of porous bioactive ceramics after implantation: Micro CT-based three-dimensional finite element analysis. J. Mater. Sci. Mater. Med.
**2012**, 23, 463–472. [Google Scholar] [CrossRef] [PubMed] - Karunratanakul, K.; Kerckhofs, G.; Lammens, J.; Vanlauwe, J.; Schrooten, J.; Van Oosterwyck, H. Validation of a finite element model of a unilateral external fixator in a rabbit tibia defect model. Med. Eng. Phys.
**2013**, 35, 1037–1043. [Google Scholar] [CrossRef] [PubMed] - Ni, Y.; Wang, L.; Liu, X.; Zhang, H.; Lin, C.Y.; Fan, Y. Micro-mechanical properties of different sites on woodpecker’s skull. Comput. Methods Biomech. Biomed. Eng.
**2017**, 20, 1483–1493. [Google Scholar] [CrossRef] [PubMed] - McCartney, W.; MacDonald, B.; Ober, C.; Lostado-Lorza, R.; Gómez, F. Pelvic modelling and the comparison between plate position for double pelvic osteotomy using artificial cancellous bone and finite element analysis. BMC Vet. Res.
**2018**, 14, 100. [Google Scholar] [CrossRef] [PubMed] - Weiner, S.; Traub, W. Bone structure: From angstroms to microns. FASEB J.
**1992**, 6, 879–885. [Google Scholar] [CrossRef] [PubMed] - Rho, J.Y.; Kuhn-Spearing, L.; Zioupos, P. Mechanical properties and the hierarchical structure of bone. Med. Eng. Phys.
**1998**, 20, 92–102. [Google Scholar] [CrossRef] - Keaveny, T.M.; Morgan, E.F.; Yeh, O.C. Standard Handbook Of Biomedical Engineering And Design; Chapter Bone Mechanics; McGraw-Hill: New York, NY, USA, 2003. [Google Scholar]
- Schwarcz, H.P. The ultrastructure of bone as revealed in electron microscopy of ion-milled sections. Semin. Cell Dev. Biol.
**2015**, 46, 44–50. [Google Scholar] [CrossRef] - Fratzl, P.; Weinkamer, R. Hierarchical Structure and Repair of Bone: Deformation, Remodelling, Healing. In Self Healing Materials: An Alternative Approach to 20 Centuries of Materials Science; van der Zwaag, S., Ed.; Springer: Dordrecht, The Netherlands, 2007; pp. 323–335. [Google Scholar] [CrossRef]
- McNally, E.A.; Schwarcz, H.P.; Botton, G.A.; Arsenault, A.L. A Model for the Ultrastructure of Bone Based on Electron Microscopy of Ion-Milled Sections. PLoS ONE
**2012**, 7, e29258. [Google Scholar] [CrossRef][Green Version] - Schwarcz, H.P.; McNally, E.A.; Botton, G.A. Dark-field transmission electron microscopy of cortical bone reveals details of extrafibrillar crystals. J. Struct. Biol.
**2014**, 188, 240–248. [Google Scholar] [CrossRef] - Kuhn, L. Bone Mineralization. In Encyclopedia of Materials: Science and Technology; Buschow, K.J., Cahn, R.W., Flemings, M.C., Ilschner, B., Kramer, E.J., Mahajan, S., Veyssière, P., Eds.; Elsevier: Oxford, UK, 2001; pp. 787–794. [Google Scholar] [CrossRef]
- Hellmich, C.; Ulm, F.J.; Dormieux, L. Can the diverse elastic properties of trabecular and cortical bone be attributed to only a few tissue-independent phase properties and their interactions? Biomech. Model Mechanobiol.
**2004**, 2, 219–238. [Google Scholar] [CrossRef] - Buehler, M.J. Nature designs tough collagen: Explaining the nanostructure of collagen fibrils. Proc. Natl. Acad. Sci. USA
**2006**, 103, 12285–12290. [Google Scholar] [CrossRef] [PubMed][Green Version] - Parenteau-Bareil, R.; Gauvin, R.; Berthod, F. Collagen-Based Biomaterials for Tissue Engineering Applications. Materials
**2010**, 3, 1863–1887. [Google Scholar] [CrossRef][Green Version] - Shoulders, M.D.; Raines, R.T. Collagen Structure and Stability. Annu. Rev. Biochem.
**2009**, 78, 929–958. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hodge, A.J.; Petruska, J.A. Recent studies with the electron microscope on ordered aggregates of the tropocollagen macromolecule. In Aspects of Protein Structure; Ramachandran, G.N., Ed.; Academic Press: New York, NY, USA, 1963; pp. 289–300. [Google Scholar]
- Orgel, J.P.R.O.; Irving, T.C.; Miller, A.; Wess, T.J. Microfibrillar structure of type I collagen in situ. Proc. Natl. Acad. Sci. USA
**2006**, 103, 9001–9005. [Google Scholar] [CrossRef] [PubMed][Green Version] - Lees, S.; Prostak, K.S.; Ingle, K.; Kjoller, K. The loci of mineral in turkey leg tendon as seen by atomic force microscope and electron microscopy. Calcif. Tissue Int.
**1994**, 55, 180–189. [Google Scholar] [CrossRef] [PubMed] - Lees, S.; Bonar, L.C.; Mook, H.A. A study of dense mineralized tissue by neutron diffraction. Int. J. Biol. Macromol.
**1984**, 6, 321–326. [Google Scholar] [CrossRef] - Bonar, L.C.; Lees, S.; Mook, H. Neutron diffraction studies of collagen in fully mineralized bone. J. Mol. Biol.
**1985**, 181, 265–270. [Google Scholar] [CrossRef] - Lees, S.; Prostak, K.S. The locus of mineral crystallites in bone. Connect. Tissue Res.
**1988**, 18, 41–54. [Google Scholar] [CrossRef] - Lees, S.; Prostak, K.S. Visualization of crystal-matrix structure. In situ demineralization of mineralized turkey leg tendon and bone. Calcif. Tissue Int.
**1996**, 59, 474–479. [Google Scholar] [CrossRef] - Sasaki, N.; Tagami, A.; Goto, T.; Taniguchi, M.; Nakata, M.; Hikichi, K. Atomic force microscopic studies on the structure of bovine femoral cortical bone at the collagen fibril-mineral level. J. Mater. Sci. Mater. Med.
**2002**, 13, 333–337. [Google Scholar] [CrossRef] - Pidaparti, R.; Chandran, A.; Takano, Y.; Turner, C. Bone mineral lies mainly outside collagen fibrils: Predictions of a composite model for osternal bone. J. Biomech.
**1996**, 29, 909–916. [Google Scholar] [CrossRef] - Hellmich, C.; Ulm, F.J. Micromechanical Model for Ultrastructural Stiffness of Mineralized Tissues. J. Eng. Mech.
**2002**, 128, 898–908. [Google Scholar] [CrossRef] - Kurfürst, A.; Henits, P.; Morin, C.; Abdalrahman, T.; Hellmich, C. Bone Ultrastructure as Composite of Aligned Mineralized Collagen Fibrils Embedded Into a Porous Polycrystalline Matrix: Confirmation by Computational Electrodynamics. Front. Phys.
**2018**, 6, 125. [Google Scholar] [CrossRef][Green Version] - Weiner, S.; Wagner, H.D. The Material Bone: Structure-Mechanical Function Relations. Annu. Rev. Mater. Sci.
**1998**, 28, 271–298. [Google Scholar] [CrossRef] - Craig, A.S.; Birtles, M.J.; Conway, J.F.; Parry, D.A. An Estimate of the Mean Length of Collagen Fibrils in Rat Tail-Tendon as a Function of age. Connect. Tissue Res.
**1989**, 19, 51–62. [Google Scholar] [CrossRef] - Reznikov, N.; Shahar, R.; Weiner, S. Bone hierarchical structure in three dimensions. Acta Biomater.
**2014**, 10, 3815–3826. [Google Scholar] [CrossRef] - Birk, D.E.; Nurminskaya, M.V.; Zycband, E.I. Collagen fibrillogenesis in situ: Fibril segments undergo post-depositional modifications resulting in linear and lateral growth during matrix development. Dev. Dyn.
**1995**, 202, 229–243. [Google Scholar] [CrossRef] - Lai, Z.B.; Yan, C. Mechanical behaviour of staggered array of mineralised collagen fibrils in protein matrix: Effects of fibril dimensions and failure energy in protein matrix. J. Mech. Behav. Biomed.
**2017**, 65, 236–247. [Google Scholar] [CrossRef] - Wang, Y.; Ural, A. Effect of modifications in mineralized collagen fibril and extra-fibrillar matrix material properties on submicroscale mechanical behavior of cortical bone. J. Mech. Behav. Biomed.
**2018**, 82, 18–26. [Google Scholar] [CrossRef] - Reznikov, N.; Chase, H.; Brumfeld, V.; Shahar, R.; Weiner, S. The 3D structure of the collagen fibril network in human trabecular bone: Relation to trabecular organization. Bone
**2015**, 71, 189–195. [Google Scholar] [CrossRef] - Fish, J. Practical Multiscaling, 1st ed.; John Wiley& Sons Inc.: Hoboken, NJ, USA, 2014. [Google Scholar]
- Tomasz Sadowski, P.T.E. Multiscale Modeling of Complex Materials: Phenomenological, Theoretical and Computational Aspects, 1st ed.; CISM International Centre for Mechanical Sciences 556; Springer: Wien, Austria, 2014; Volume 556. [Google Scholar] [CrossRef]
- Ostoja-Starzewski, M. Microstructural Randomness and Scaling in Mechanics of Materials; Modern Mechanics and Mathematics; CRC Press: Boca Raton, FL, USA, 2007. [Google Scholar]
- Feldesman, M.R.; Kleckner, J.G.; Lundy, J.K. Femur/stature ratio and estimates of stature in mid- and late-pleistocene fossil hominids. Am. J. Phys. Anthropol.
**1990**, 83, 359–372. [Google Scholar] [CrossRef] - Àwengen, D.F.; Kurokawa, H.; Nishihara, S.; Goode, R.L. Measurements of the Stapes Superstructure. Ann. Otol. Rhinol. Laryngol.
**1995**, 104, 311–316. [Google Scholar] [CrossRef] [PubMed] - Wei, S.; Siegal, G.P. Atlas of Bone Pathology, 1st ed.; Atlas of Anatomic Pathology; Springer: New York, NY, USA, 2013. [Google Scholar]
- Almora-Barrios, N.; De Leeuw, N.H. Molecular Dynamics Simulation of the Early Stages of Nucleation of Hydroxyapatite at a Collagen Template. Cryst. Growth Des.
**2012**, 12, 756–763. [Google Scholar] [CrossRef] - Zhou, Z.; Qian, D.; Minary-Jolandan, M. Clustering of hydroxyapatite on a super-twisted collagen microfibril under mechanical tension. J. Mater. Chem. B
**2017**, 5, 2235–2244. [Google Scholar] [CrossRef] - Barrios, N.A. A Computational Investigation of the Interaction of the Collagen Molecule with Hydroxyapatite. Ph.D. Thesis, University College London, London, UK, 2010. [Google Scholar]
- Peter Fratzl, P.F. Collagen: Structure and Mechanics, 1st ed.; Springer: Berlin, Germany, 2008. [Google Scholar]
- Liu, Y.; Luo, D.; Wang, T. Hierarchical Structures of Bone and Bioinspired Bone Tissue Engineering. Small
**2016**, 12, 4611–4632. [Google Scholar] [CrossRef] [PubMed] - Budarapu, P.R.; Rabczuk, T. Multiscale Methods for Fracture: A Review. J. Ind. Sci.
**2017**, 97, 339–376. [Google Scholar] [CrossRef] - Nair, A.K.; Gautieri, A.; Chang, S.W.; Buehler, M.J. Molecular mechanics of mineralized collagen fibrils in bone. Nat. Commun.
**2013**, 4, 1724. [Google Scholar] [CrossRef][Green Version] - Nair, A.K.; Gautieri, A.; Buehler, M.J. Role of Intrafibrillar Collagen Mineralization in Defining the Compressive Properties of Nascent Bone. Biomacromolecules
**2014**, 15, 2494–2500. [Google Scholar] [CrossRef] - Kanouté, P.; Boso, D.P.; Chaboche, J.L.; Schrefler, B.A. Multiscale Methods for Composites: A Review. Arch. Comput. Methods Eng.
**2009**, 16, 31–75. [Google Scholar] [CrossRef] - Benedetti, I.; Aliabadi, M. Multiscale modeling of polycrystalline materials: A boundary element approach to material degradation and fracture. Comput. Meth. Appl. Mech. Eng.
**2015**, 289, 429–453. [Google Scholar] [CrossRef][Green Version] - Ostoja-Starzewski, M. Material spatial randomness: From statistical to representative volume element. Probab. Eng. Mech.
**2006**, 21, 112–132. [Google Scholar] [CrossRef] - Nguyen, V.P.; Stroeven, M.; Sluys, L.J. Multiscale Continuous and Discontinuous Modeling of Heterogeneous Materials: A Review on Recent Developments. J. Multiscale Modell. (JMM)
**2011**, 3, 229–270. [Google Scholar] [CrossRef] - Mesarovic, S.D.; Padbidri, J. Minimal kinematic boundary conditions for simulations of disordered microstructures. Philos. Mag.
**2005**, 85, 65–78. [Google Scholar] [CrossRef] - Joseph, G.; Jörg, B.; Volker, U. On boundary conditions for homogenization of volume elements undergoing localization. Int. J. Numer. Methods Eng.
**2017**, 113, 1–21. [Google Scholar] [CrossRef] - Ricardez-Sandoval, L.A. Current challenges in the design and control of multiscale systems. Can. J. Chem. Eng.
**2011**, 89, 1324–1341. [Google Scholar] [CrossRef] - Qu, T.; Verma, D.; Shahidi, M.; Pichler, B.; Hellmich, C.; Tomar, V. Mechanics of organic-inorganic biointerfaces—Implications for strength and creep properties. MRS Bull.
**2015**, 40, 349–358. [Google Scholar] [CrossRef] - Bhowmik, R.; Katti, K.S.; Katti, D.R. Mechanics of molecular collagen is influenced by hydroxyapatite in natural bone. J. Mater. Sci.
**2007**, 42, 8795–8803. [Google Scholar] [CrossRef] - Chandra, N.; Namilae, S.; Srinivasan, A. Linking Atomistic and Continuum Mechanics Using Multiscale Models. AIP Conf. Proc.
**2004**, 712, 1571–1576. [Google Scholar] [CrossRef] - Rafii-Tabar, H.; Hua, L.; Cross, M. A multi-scale atomistic-continuum modelling of crack propagation in a two-dimensional macroscopic plate. J. Condens. Matter Phys.
**1998**, 10, 2375–2387. [Google Scholar] [CrossRef] - Guin, L.; Raphanel, J.L.; Kysar, J.W. Atomistically derived cohesive zone model of intergranular fracture in polycrystalline graphene. J. Appl. Phys.
**2016**, 119, 245107. [Google Scholar] [CrossRef] - Enayatpour, S.; van Oort, E.; Patzek, T. Thermal shale fracturing simulation using the Cohesive Zone Method (CZM). J. Nat. Gas Sci. Eng.
**2018**, 55, 476–494. [Google Scholar] [CrossRef] - Lawrimore, W.; Paliwal, B.; Chandler, M.; Johnson, K.; Horstemeyer, M. Hierarchical multiscale modeling of Polyvinyl Alcohol/Montmorillonite nanocomposites. Polymer
**2016**, 99, 386–398. [Google Scholar] [CrossRef] - Paggi, M.; Wriggers, P. A nonlocal cohesive zone model for finite thickness interfaces—Part I: Mathematical formulation and validation with molecular dynamics. Comput. Mater. Sci.
**2011**, 50, 1625–1633. [Google Scholar] [CrossRef] - Plimpton, S. Fast Parallel Algorithms for Short-Range Molecular Dynamics. J. Comput. Phys.
**1995**, 117, 1–19. [Google Scholar] [CrossRef][Green Version] - De Leeuw, N.H. A computer modelling study of the uptake and segregation of fluoride ions at the hydrated hydroxyapatite (0001) surface: Introducing a Ca
_{10}(PO_{4})_{6}(OH)_{2}potential model. Phys. Chem. Chem. Phys.**2004**, 6, 1860–1866. [Google Scholar] [CrossRef] - Yeo, J.; Jung, G.; Tarakanova, A.; Martín-Martínez, F.J.; Qin, Z.; Cheng, Y.; Zhang, Y.W.; Buehler, M.J. Multiscale modeling of keratin, collagen, elastin and related human diseases: Perspectives from atomistic to coarse-grained molecular dynamics simulations. Extreme Mech Lett.
**2018**, 20, 112–124. [Google Scholar] [CrossRef] - Cornell, W.D.; Cieplak, P.; Bayly, C.I.; Gould, I.R.; Merz, K.M.; Ferguson, D.M.; Spellmeyer, D.C.; Fox, T.; Caldwell, J.W. A Second Generation Force Field for the Simulation of Proteins, Nucleic Acids, and Organic Molecules. J. Am. Chem. Soc.
**1995**, 117, 5179–5197. [Google Scholar] [CrossRef][Green Version] - Griebel, M.; Knapek, S.; Zumbusch, G. Numerical Simulation in Molecular Dynamics. In Numerics, Algorithms, Parallelization, Applications; Volume 5 of Texts in Computational Science and Engineering; Springer: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
- Hirschfelder, J.O.; Curtiss, C.F.; Bird, R.B. Molecular Theory of Gases and Liquids; Wiley: New York, NY, USA, 1964. [Google Scholar]
- Rabone, J.A.L.; De Leeuw, N.H. Interatomic potential models for natural apatite Crystals: Incorporating strontium and the lanthanides. J. Comput. Chem.
**2006**, 27, 253–266. [Google Scholar] [CrossRef] - Hauptmann, S.; Dufner, H.; Brickmann, J.; Kast, S.M.; Berry, R.S. Potential energy function for apatites. Phys. Chem. Chem. Phys.
**2003**, 5, 635–639. [Google Scholar] [CrossRef][Green Version] - Walser, R.; Hünenberger, P.H.; van Gunsteren, W.F. Comparison of different schemes to treat long-range electrostatic interactions in molecular dynamics simulations of a protein crystal. Proteins
**2001**, 43, 509–519. [Google Scholar] [CrossRef] - Scheiner, S.; Pivonka, P.; Hellmich, C. Coupling systems biology with multiscale mechanics, for computer simulations of bone remodeling. Comput. Meth. Appl. Mech. Eng.
**2013**, 254, 181–196. [Google Scholar] [CrossRef] - Scheiner, S.; Pivonka, P.; Hellmich, C. Poromicromechanics reveals that physiological bone strains induce osteocyte-stimulating lacunar pressure. Biomech. Model Mechanobiol.
**2016**, 15, 9–28. [Google Scholar] [CrossRef] [PubMed][Green Version] - Pastrama, M.I.; Scheiner, S.; Pivonka, P.; Hellmich, C. A mathematical multiscale model of bone remodeling, accounting for pore space-specific mechanosensation. Bone
**2018**, 107, 208–221. [Google Scholar] [CrossRef] [PubMed] - Klein-Nulend, J.; van der Plas, A.; Semeins, C.M.; Ajubi, N.E.; Frangos, J.A.; Nijweide, P.J.; Burger, E.H. Sensitivity of osteocytes to biomechanical stress in vitro. FASEB J.
**1995**, 9, 441–445. [Google Scholar] [CrossRef] [PubMed] - Colloca, M.; Blanchard, R.; Hellmich, C.; Ito, K.; van Rietbergen, B. A multiscale analytical approach for bone remodeling simulations: Linking scales from collagen to trabeculae. Bone
**2014**, 64, 303–313. [Google Scholar] [CrossRef][Green Version] - Mirzaali, M.J.; Schwiedrzik, J.J.; Thaiwichai, S.; Best, J.P.; Michler, J.; Zysset, P.K.; Wolfram, U. Mechanical properties of cortical bone and their relationships with age, gender, composition and microindentation properties in the elderly. Bone
**2016**, 93, 196–211. [Google Scholar] [CrossRef] - Zimmermann, E.A.; Launey, M.E.; Ritchie, R.O. The significance of crack-resistance curves to the mixed-mode fracture toughness of human cortical bone. Biomaterials
**2010**, 31, 5297–5305. [Google Scholar] [CrossRef][Green Version] - Zimmermann, E.A.; Gludovatz, B.; Schaible, E.; Busse, B.; Ritchie, R.O. Fracture resistance of human cortical bone across multiple length-scales at physiological strain rates. Biomaterials
**2014**, 35, 5472–5481. [Google Scholar] [CrossRef] - Chandran, M. Fracture Risk Assessment in Clinical Practice: Why Do It? What to Do It with? J. Clin. Densitom.
**2017**, 20, 274–279. [Google Scholar] [CrossRef] - Kanis, J.A.; Johnell, O.; De Laet, C.; Jonsson, B.; Oden, A.; Ogelsby, A.K. International Variations in Hip Fracture Probabilities: Implications for Risk Assessment. J. Bone Miner. Res.
**2002**, 17, 1237–1244. [Google Scholar] [CrossRef] - Curtis, E.M.; Moon, R.J.; Harvey, N.C.; Cooper, C. The impact of fragility fracture and approaches to osteoporosis risk assessment worldwide. Bone
**2017**, 104, 29–38. [Google Scholar] [CrossRef] [PubMed][Green Version] - Pothuaud, L.; Carceller, P.; Hans, D. Correlations between grey-level variations in 2D projection images (TBS) and 3D microarchitecture: Applications in the study of human trabecular bone microarchitecture. Bone
**2008**, 42, 775–787. [Google Scholar] [CrossRef] [PubMed] - Bousson, V.; Bergot, C.; Sutter, B.; Levitz, P.; Cortet, B.; The Scientific Committee of the GRIO (Groupe de Recherche et d’Information sur les Ostéoporoses). Trabecular bone score (TBS): available knowledge, clinical relevance, and future prospects. Osteoporos. Int.
**2012**, 23, 1489–1501. [Google Scholar] [CrossRef] [PubMed] - Shevroja, E.; Lamy, O.; Kohlmeier, L.; Koromani, F.; Rivadeneira, F.; Hans, D. Use of Trabecular Bone Score (TBS) as a Complementary Approach to Dual-energy X-ray Absorptiometry (DXA) for Fracture Risk Assessment in Clinical Practice. J. Clin. Densitom.
**2017**, 20, 334–345. [Google Scholar] [CrossRef] - Mirzaei, A.; Jahed, S.A.; Nojomi, M.; Rajaei, A.; Zabihiyeganeh, M. A study of the value of trabecular bone score in fracture risk assessment of postmenopausal women. Taiwan J. Obstet. Gynecol.
**2018**, 57, 389–393. [Google Scholar] [CrossRef] - Martineau, P.; Leslie, W. Trabecular bone score (TBS): Method and applications. Bone
**2017**, 104, 66–72. [Google Scholar] [CrossRef] - Bousson, V.; Bergot, C.; Sutter, B.; Thomas, T.; Bendavid, S.; Benhamou, C.L.; Blain, H.; Brazier, M.; Breuil, V.; Briot, K.; et al. Trabecular Bone Score: Where are we now? Jt. Bone Spine
**2015**, 82, 320–325. [Google Scholar] [CrossRef] - Harvey, N.; Glüer, C.; Binkley, N.; McCloskey, E.; Brandi, M.L.; Cooper, C.; Kendler, D.; Lamy, O.; Laslop, A.; Camargos, B.; et al. Trabecular bone score (TBS) as a new complementary approach for osteoporosis evaluation in clinical practice. Bone
**2015**, 78, 216–224. [Google Scholar] [CrossRef][Green Version] - Parfitt, A. Misconceptions (2): Turnover is always higher in cancellous than in cortical bone. Bone
**2002**, 30, 807–809. [Google Scholar] [CrossRef] - Sahana, S.; Nitin, K.; Joseph, B.; Nihal, T.; Thomas, P. Bone turnover markers: Emerging tool in the management of osteoporosis. Ind. J. Endocrinol. Metab.
**2016**, 20, 846–852. [Google Scholar] [CrossRef] - Vasikaran, S.; Eastell, R.; Bruyère, O.; Foldes, A.J.; Garnero, P.; Griesmacher, A.; McClung, M.; Morris, H.A.; Silverman, S.; Trenti, T.; et al. Markers of bone turnover for the prediction of fracture risk and monitoring of osteoporosis treatment: a need for international reference standards. Osteoporos. Int.
**2011**, 22, 391–420. [Google Scholar] [CrossRef] [PubMed] - Swaminathan, R. Biochemical markers of bone turnover. Clin. Chim. Acta
**2001**, 313, 95–105. [Google Scholar] [CrossRef] - Afsarimanesh, N.; Mukhopadhyay, S.; Kruger, M. Sensing technologies for monitoring of bone-health: A review. Sens. Actuators A Phys.
**2018**, 274, 165–178. [Google Scholar] [CrossRef] - Afsarimanesh, N.; Mukhopadhyay, S.; Kruger, M. State-of-the-art of sensing technologies for monitoring of bone-health. Smart Sens. Measur. Instrum.
**2019**, 30, 7–31. [Google Scholar] [CrossRef] - Kanis, J.A.; Johnell, O.; Oden, A.; Johansson, H.; McCloskey, E. FRAX™ and the assessment of fracture probability in men and women from the UK. Osteoporos. Int.
**2008**, 19, 385–397. [Google Scholar] [CrossRef][Green Version] - Kanis, J.A.; Harvey, N.C.; Johansson, H.; Odén, A.; Leslie, W.D.; McCloskey, E.V. FRAX Update. J. Clin. Densitom.
**2017**, 20, 360–367. [Google Scholar] [CrossRef] - Edwards, B.J. Osteoporosis Risk Calculators. J. Clin. Densitom.
**2017**, 20, 379–388. [Google Scholar] [CrossRef] - Fuleihan, G.E.H.; Chakhtoura, M.; Cauley, J.A.; Chamoun, N. Worldwide Fracture Prediction. J. Clin. Densitom.
**2017**, 20, 397–424. [Google Scholar] [CrossRef] - Dimai, H.P. Use of dual-energy X-ray absorptiometry (DXA) for diagnosis and fracture risk assessment; WHO-criteria, T- and Z-score, and reference databases. Bone
**2017**, 104, 39–43. [Google Scholar] [CrossRef] - Donaldson, M.G.; Palermo, L.; Schousboe, J.T.; Ensrud, K.E.; Hochberg, M.C.; Cummings, S.R. FRAX and Risk of Vertebral Fractures: The Fracture Intervention Trial. J. Bone Miner. Res.
**2009**, 24, 1793–1799. [Google Scholar] [CrossRef] - Okazaki, N.; Burghardt, A.J.; Chiba, K.; Schafer, A.L.; Majumdar, S. Bone microstructure in men assessed by HR-pQCT: Associations with risk factors and differences between men with normal, low, and osteoporosis-range areal BMD. Bone Rep.
**2016**, 5, 312–319. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hippisley-Cox, J.; Coupland, C. Predicting risk of osteoporotic fracture in men and women in England and Wales: Prospective derivation and validation of QFractureScores. BMJ
**2009**, 339, b4229. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hippisley-Cox, J.; Coupland, C. Derivation and validation of updated QFracture algorithm to predict risk of osteoporotic fracture in primary care in the United Kingdom: prospective open cohort study. BMJ
**2012**, 344, e3427. [Google Scholar] [CrossRef] [PubMed][Green Version] - Nguyen, N.D.; Frost, S.A.; Center, J.R.; Eisman, J.A.; Nguyen, T.V. Development of a nomogram for individualizing hip fracture risk in men and women. Osteoporos. Int.
**2007**, 18, 1109–1117. [Google Scholar] [CrossRef] - Nguyen, N.D.; Frost, S.A.; Center, J.R.; Eisman, J.A.; Nguyen, T.V. Development of prognostic nomograms for individualizing 5-year and 10-year fracture risks. Osteoporos. Int.
**2008**, 19, 1431–1444. [Google Scholar] [CrossRef] - Nguyen, N.D.; Pongchaiyakul, C.; Center, J.R.; Eisman, J.A.; Nguyen, T.V. Identification of High-Risk Individuals for Hip Fracture: A 14-Year Prospective Study. J. Bone Miner. Res.
**2005**, 20, 1921–1928. [Google Scholar] [CrossRef] - Nguyen, T.V.; Eisman, J.A. Fracture Risk Assessment: From Population to Individual. J. Clin. Densitom.
**2017**, 20, 368–378. [Google Scholar] [CrossRef] - Curtis, E.M.; Moon, R.J.; Harvey, N.C.; Cooper, C. Reprint of: The impact of fragility fracture and approaches to osteoporosis risk assessment worldwide. Int. J. Orthop. Trauma Nurs.
**2017**, 26, 7–17. [Google Scholar] [CrossRef][Green Version] - Kanis, J.A.; Harvey, N.C.; Johansson, H.; Odén, A.; McCloskey, E.V.; Leslie, W.D. Overview of Fracture Prediction Tools. J. Clin. Densitom.
**2017**, 20, 444–450. [Google Scholar] [CrossRef][Green Version] - Leslie, W.D.; Lix, L.M. Comparison between various fracture risk assessment tools. Osteoporos. Int.
**2014**, 25, 1–21. [Google Scholar] [CrossRef] - Nguyen, T.V. Individualized Assessment of Fracture Risk: Contribution of “Osteogenomic Profile”. J. Clin. Densitom.
**2017**, 20, 353–359. [Google Scholar] [CrossRef] [PubMed] - Forgetta, V.; Keller-Baruch, J.; Forest, M.; Durand, A.; Bhatnagar, S.; Kemp, J.; Morris, J.A.; Kanis, J.A.; Kiel, D.P.; McCloskey, E.V.; et al. Machine Learning to Predict Osteoporotic Fracture Risk from Genotypes. bioRxiv
**2018**, 413716. [Google Scholar] [CrossRef] - DVO-Leitlinie. Methodenreport der DVO-Leitlinie 2014 zur Prophylaxe, Diagnostik und Therapie der Osteoporose bei Männern ab dem 60. Lebensjahr und bei postmenopausalen Frauen. 2014. Available online: http://www.dv-osteologie.org/uploads/Leitlinie%202014/DVO-Leitlinie%20Osteoporose%202014%20Kurzfassung%20und%20Langfassung%20Version%201a%2012%2001%202016.pdf (accessed on 12 December 2019).
- Thomasius, F.; Baum, E.; Bernecker, P.; Böcker, W.; Brabant, T.; Clarenz, P.; Demary, W.; Dimai, H.P.; Engelbrecht, M.; Engelke, K.; et al. DVO Leitlinie 2017 zur Prophylaxe, Diagnostik und Therapie der Osteoporose bei postmenopausalen Frauen und Männern/S-3 DVO Guidelines 2017 in prophylaxis, diagnosis and therapy of osteoporosis in postmenopausal women und men. Osteologie
**2018**, 27, 154–160. [Google Scholar] [CrossRef][Green Version] - Reber, K.C.; König, H.H.; Becker, C.; Rapp, K.; Büchele, G.; Mächler, S.; Lindlbauer, I. Development of a risk assessment tool for osteoporotic fracture prevention: A claims data approach. Bone
**2018**, 110, 170–176. [Google Scholar] [CrossRef] - Black, D.; Steinbuch, M.; Palermo, L.; Dargent-Molina, P.; Lindsay, R.; Hoseyni, M.; Johnell, O. An assessment tool for predicting fracture risk in postmenopausal women. Osteoporos. Int.
**2001**, 12, 519–528. [Google Scholar] [CrossRef] - Järvinen, T.L.; Michaëlsson, K.; Jokihaara, J.; Collins, G.S.; Perry, T.L.; Mintzes, B.; Musini, V.; Erviti, J.; Gorricho, J.; Wright, J.M.; et al. Overdiagnosis of bone fragility in the quest to prevent hip fracture. BMJ
**2015**, 350, h2088. [Google Scholar] [CrossRef][Green Version] - Hamdy, R.C. Osteoporosis—Assessing Fracture Risk. J. Clin. Densitom.
**2017**, 20, 271. [Google Scholar] [CrossRef] - Baim, S. The Future of Fracture Risk Assessment in the Management of Osteoporosis. J. Clin. Densitom.
**2017**, 20, 451–457. [Google Scholar] [CrossRef] - Gurtin, M.; Fried, E.; Anand, L. The Mechanics and Thermodynamics of Continua; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Carathéodory, C. Vorlesungen über Reelle Funktionen, 2nd ed.; Springer: Berlin, Germany, 1927. [Google Scholar]
- Motteler, Z.; Miranda, C. Partial Differential Equations of Elliptic Type; Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge; Springer: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Willard, S. General Topology; Addison-Wesley series in mathematics; Dover Publications: Mineola, NY, USA, 2004. [Google Scholar]
- Henry, D.; Hale, J.; Pereira, A. Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations; London Mathematical Society Lecture Note Series; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Ogden, R. Non-Linear Elastic Deformations; Dover Civil and Mechanical Engineering; Dover Publications: Mineola, NY, USA, 1997. [Google Scholar]
- Chaboche, J. A review of some plasticity and viscoplasticity constitutive theories. Int. J. Plast.
**2008**, 24, 1642–1693. [Google Scholar] [CrossRef] - Malvern, L. Introduction to the Mechanics of a Continuous Medium; Prentice-Hall series in engineering of the physical sciences; Prentice-Hall: Upper Saddle River, NJ, USA, 1969. [Google Scholar]
- Truesdell, C.; Noll, W. The Non-Linear Field Theories of Mechanics, 3rd ed.; Springer: Berlin/Heidelberg, Germany, 2004. [Google Scholar] [CrossRef]
- Smith, D.R. An Introduction to Continuum Mechanics—After Truesdell and Noll; Springer: Dordrecht, The Netherlands, 1993. [Google Scholar] [CrossRef]
- Truesdell, C. A First Course in Rational Continuum Mechanics; Academic Press Inc.: Cambridge, MA, USA, 1991; Volume 1. [Google Scholar]
- Lai, W.M.; Rubin, D.; Krempl, E. Introduction to Continuum Mechanics, 4th ed.; Butterworth-Heinemann: Boston, MA, USA, 2010. [Google Scholar] [CrossRef]
- Fung, Y. Foundations of Solid Mechanics; International series in dynamics; Prentice-Hall: Upper Saddle River, NJ, USA, 1965. [Google Scholar]
- Lubliner, J. Plasticity Theory; Dover books on engineering; Dover Publications: Mineola, NY, USA, 2008. [Google Scholar]
- Coussy, O. Poromechanics, 2nd ed.; Wiley: Hoboken, NJ, USA, 2004. [Google Scholar]
- Kachanov, L. Introduction to Continuum Damage Mechanics, 1st ed.; Mechanics of Elastic Stability; Springer: Berlin, Germany, 2010. [Google Scholar]
- Murakami, S. Continuum Damage Mechanics: A Continuum Mechanics Approach to the Analysis of Damage and Fracture, 1st ed.; Solid Mechanics and Its Applications 185; Springer: Dordrecht, The Netherlands, 2012. [Google Scholar]
- Krajcinovic, D.; Lemaitre, J. Continuum Damage Mechanics Theory and Application; CISM International Centre for Mechanical Sciences; Springer: Vienna, Austria, 2014. [Google Scholar]
- Zhang, W.; Cai, Y. Continuum Damage Mechanics and Numerical Applications; Jointly published with zhejiang university press, 2011 ed.; Advanced Topics in Science and Technology in China; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
- Truesdell, C. The Elements of Continuum Mechanics; Springer: Berlin/Heidelberg, Germany, 1984. [Google Scholar] [CrossRef]
- Lekhnitskik, S.G. Theory of Elasticity of an Anisotropic Elastic Body; Holden-Day: Toronto, ON, Canada, 1963. [Google Scholar]
- Gamelin, T.; Greene, R. Introduction to Topology, 2nd ed.; Dover Books on Mathematics; Dover Publications: Mineola, NY, USA, 2013. [Google Scholar]
- Cowin, S.C. Bone Stress Adaptation Models. J. Biomech. Eng.
**1993**, 115, 528–533. [Google Scholar] [CrossRef]

**Figure 2.**Hounsfield-Scale for different kinds of tissues (adapted from © Institut für Anatomie, Universität Bern (https://elearning.medizin.unibe.ch/morphomed/radioanatomie/ct-mrt-des-rumpfs/ct-mrt-einf%C3%BChrung/hounsfield-skala).

**Table 1.**Main interested parties involved in bone fracture and which sections of this paper are most interesting for each party.

Specialization, Interested | Related Sections |
---|---|

Biology | Section 3. Motivating Patient-Specific Bone Fracture Simulation |

Section 4. Motivating this Literature Survey | |

Section 6. The Physico-Mathematical Approach to Bone Fracture | |

Section 7. Modelling Bone as a Continuum | |

Section 8. Categorizing the Surveyed Literature into a Continuum Mechanics Framework | |

Section 9. Patient-Specific Geometry of Bone | |

Section 10. Mechanical Properties Categorization for Computational Bone Models | |

Section 13. The Multiscale Structure of Bone | |

Medicine | Section 3. Motivating Patient-Specific Bone Fracture Simulation |

Section 4. Motivating this Literature Survey | |

Section 6. The Physico-Mathematical Approach to Bone Fracture | |

Section 7. Modelling Bone as a Continuum | |

Section 8. Categorizing the Surveyed Literature into a Continuum Mechanics Framework | |

Section 9. Patient-Specific Geometry of Bone | |

Section 10. Mechanical Properties Categorization for Computational Bone Models | |

Section 11. Mathematical Model of Bone Trauma-inducing Accident—The Boundary Conditions | |

Section 15. Validating Bone Fracture Simulation | |

Section 16. Assessing Fracture Risk | |

Physics | Section 3. Motivating Patient-Specific Bone Fracture Simulation |

Section 4. Motivating this Literature Survey | |

Section 6. The Physico-Mathematical Approach to Bone Fracture | |

Section 7. Modelling Bone as a Continuum | |

Section 8. Categorizing the Surveyed Literature into a Continuum Mechanics Framework | |

Section 9. Patient-Specific Geometry of Bone | |

Section 10. Mechanical Properties Categorization for Computational Bone Models | |

Section 11. Mathematical Model of Bone Trauma-inducing Accident—The Boundary Conditions | |

Section 12. Simulating Bone Fracture | |

Section 13. The Multiscale Structure of Bone | |

Section 14. Multiscale Modelling of Bone | |

Section 15. Validating Bone Fracture Simulation | |

Section 16. Assessing Fracture Risk | |

Engineering | Section 3. Motivating Patient-Specific Bone Fracture Simulation |

Section 4. Motivating this Literature Survey | |

Section 6. The Physico-Mathematical Approach to Bone Fracture | |

Section 7. Modelling Bone as a Continuum | |

Section 8. Categorizing the Surveyed Literature into a Continuum Mechanics Framework | |

Section 9. Patient-Specific Geometry of Bone | |

Section 10. Mechanical Properties Categorization for Computational Bone Models | |

Section 11. Mathematical Model of Bone Trauma-inducing Accident—The Boundary Conditions | |

Section 12. Simulating Bone Fracture | |

Section 13. The Multiscale Structure of Bone | |

Section 14. Multiscale Modelling of Bone | |

Section 15. Validating Bone Fracture Simulation | |

Section 16. Assessing Fracture Risk | |

Government | Section 3. Motivating Patient-Specific Bone Fracture Simulation |

Section 4. Motivating this Literature Survey | |

Section 16. Assessing Fracture Risk | |

Philanthropy | Section 3. Motivating Patient-Specific Bone Fracture Simulation |

Section 4. Motivating this Literature Survey | |

Section 16. Assessing Fracture Risk |

**Table 2.**Categorization of surveyed literature that modelled bone as a solid continuum [30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147].

**Table 3.**Bone multiscale modelling [30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,290,419,420].

Interaction | $\mathit{CLG}$–$\mathit{CLG}$ | $\mathit{CLG}$–$\mathit{HA}$ | $\mathit{CLG}$–${\mathit{H}}_{2}\mathit{O}$ | $\mathit{HA}$–$\mathit{HA}$ | $\mathit{HA}$–${\mathit{H}}_{2}\mathit{O}$ | ${\mathit{H}}_{2}\mathit{O}$–${\mathit{H}}_{2}\mathit{O}$ |
---|---|---|---|---|---|---|

Reference | [413,415] apud [439] | [440,441] | [440,441] | [442] apud [437], [414] apud [443] | [440,441] | [413] apud [444] |

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## Share and Cite

**MDPI and ACS Style**

Alcântara, A.C.S.; Assis, I.; Prada, D.; Mehle, K.; Schwan, S.; Costa-Paiva, L.; Skaf, M.S.; Wrobel, L.C.; Sollero, P.
Patient-Specific Bone Multiscale Modelling, Fracture Simulation and Risk Analysis—A Survey. *Materials* **2020**, *13*, 106.
https://doi.org/10.3390/ma13010106

**AMA Style**

Alcântara ACS, Assis I, Prada D, Mehle K, Schwan S, Costa-Paiva L, Skaf MS, Wrobel LC, Sollero P.
Patient-Specific Bone Multiscale Modelling, Fracture Simulation and Risk Analysis—A Survey. *Materials*. 2020; 13(1):106.
https://doi.org/10.3390/ma13010106

**Chicago/Turabian Style**

Alcântara, Amadeus C. S., Israel Assis, Daniel Prada, Konrad Mehle, Stefan Schwan, Lúcia Costa-Paiva, Munir S. Skaf, Luiz C. Wrobel, and Paulo Sollero.
2020. "Patient-Specific Bone Multiscale Modelling, Fracture Simulation and Risk Analysis—A Survey" *Materials* 13, no. 1: 106.
https://doi.org/10.3390/ma13010106