Evolution of bone tissue based on angiogenesis as a crucial factor: New mathematical attempt

Bednarczyk, Ewa; Lekszycki, Tomasz

doi:10.1177/10812865211048925

Cited by 8 publications

(3 citation statements)

References 52 publications

(64 reference statements)

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“…The use of a variational principle is a highly effective and safe conceptual tool for developing complex theories regarding generalized continua [21][22][23][24][25][26][27][28], or approximate theories of structural components, alongside natural boundary conditions [29]. Additionally, it may serve as a preparatory step towards the establishment of a comprehensive thermo-mechanical formulation, a synthetic viewpoint to facilitate the proof of many mathematical theorems, and the theoretical basis for the application of modern numerical techniques like, for instance, the finite element method [30][31][32][33][34][35][36][37][38]. Variational principles are a widely accepted method of organizing information about dynamic systems and developing their evolution equations [7,[39][40][41][42][43][44][45][46].…”

Section: Introductionmentioning

confidence: 99%

A variational formulation for three-dimensional linear thermoelasticity with ‘thermal inertia’

Giorgio,

Placidi

2024

Meccanica

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A variational model has been developed to investigate the coupled thermo-mechanical response of a three-dimensional continuum. The linear Partial Differential Equations (PDEs) of this problem are already well-known in the literature. However, in this paper, we avoid the use of the second principle of thermodynamics, basing the formulation only on a proper definition (i) of kinematic descriptors (the displacement and the entropic displacement), (ii) of the action functional (with kinetic, internal and external energy functions) and (iii) of the Rayleigh dissipation function. Thus, a Hamilton–Rayleigh variational principle is formulated, and the cited PDEs have been derived with a set of proper Boundary Conditions (BCs). Besides, the Lagrangian variational perspective has been expanded to analyze linear irreversible processes by generalizing Biot’s formulation, namely, including thermal inertia in the kinetic energy definition. Specifically, this implies Cattaneo’s law for heat conduction, and the well-known Lord–Shulman model for thermo-elastic anisotropic bodies is then deduced. The developed variational framework is ideal for the perspective of analyzing the thermo-mechanical problems with micromorphic and/or higher-order gradient continuum models, where the deduction of a coherent system of PDEs and BCs is, on the one hand, not straightforward and, on the other hand, natural within the presented variational deduction.

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Section: Introductionmentioning

confidence: 99%

A variational formulation for three-dimensional linear thermoelasticity with ‘thermal inertia’

Giorgio,

Placidi

2024

Meccanica

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“…The bottom line is that the structure of bone tissue maintains a delicate balance between providing strength and support to the body while minimizing the cost of nutrients and required energy for the living cells operating in it (Bednarczyk and Lekszycki 2016 , 2022 ). By adjusting the distribution of bone mass and arranging its microstructure in an efficient composite structure, bone tissue is able to achieve this balance and maintain its function and integrity over time.…”

Section: Introductionmentioning

confidence: 99%

An orthotropic continuum model with substructure evolution for describing bone remodeling: an interpretation of the primary mechanism behind Wolff’s law

Giorgio,

dell’Isola,

Andreaus

et al. 2023

Biomech Model Mechanobiol

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We propose a variational approach that employs a generalized principle of virtual work to estimate both the mechanical response and the changes in living bone tissue during the remodeling process. This approach provides an explanation for the adaptive regulation of the bone substructure in the context of orthotropic material symmetry. We specifically focus upon the crucial gradual adjustment of bone tissue as a structural material that adapts its mechanical features, such as materials stiffnesses and microstructure, in response to the evolving loading conditions. We postulate that the evolution process relies on a feedback mechanism involving multiple stimulus signals. The mechanical and remodeling behavior of bone tissue is clearly a complex process that is difficult to describe within the framework of classical continuum theories. For this reason, a generalized continuum elastic theory is employed as a proper mathematical context for an adequate description of the examined phenomenon. To simplify the investigation, we considered a two-dimensional problem. Numerical simulations have been performed to illustrate bone evolution in a few significant cases: the bending of a rectangular cantilever plate and a three-point flexure test. The results are encouraging because they can replicate the optimization process observed in bone remodeling. The proposed model provides a likely distribution of stiffnesses and accurately represents the arrangement of trabeculae macroscopically described by the orthotropic symmetry directions, as supported by experimental evidence from the trajectorial theory.

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“…The problem of bone adaptation and remodeling has been considered since Wolff’s observations in 1892 that mathematical laws can describe changes in the architecture of bones [ 21 , 22 ]. Many mathematical models have been developed [ 23 , 24 , 25 , 26 , 27 , 28 , 29 ] dependent on various mechanical stimuli inducing bone adaptation. Among them can be mentioned diverse instances based on strain energy density [ 30 , 31 ], tissue damage [ 32 , 33 , 34 ], daily stress stimulus [ 35 ], effective stress [ 36 ], and strain [ 37 ].…”

Section: Introductionmentioning

confidence: 99%

Bone Remodeling Process Based on Hydrostatic and Deviatoric Strain Mechano-Sensing

et al. 2022

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A macroscopic continuum model intended to provide predictions for the remodeling process occurring in bone tissue is proposed. Specifically, we consider a formulation in which two characteristic stiffnesses, namely the bulk and shear moduli, evolve independently to adapt the hydrostatic and deviatoric response of the bone tissue to environmental changes. The formulation is deliberately simplified, aiming at constituting a preliminary step toward a more comprehensive modeling approach. The evolutive process for describing the functional adaptation of the two stiffnesses is proposed based on an energetic argument. Numerical experiments reveal that it is possible to model the bone remodeling process with a different evolution for more than one material parameter, as usually done. Moreover, the results motivate further investigations into the subject.

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Evolution of bone tissue based on angiogenesis as a crucial factor: New mathematical attempt

Cited by 8 publications

References 52 publications

A variational formulation for three-dimensional linear thermoelasticity with ‘thermal inertia’

A variational formulation for three-dimensional linear thermoelasticity with ‘thermal inertia’

An orthotropic continuum model with substructure evolution for describing bone remodeling: an interpretation of the primary mechanism behind Wolff’s law

Bone Remodeling Process Based on Hydrostatic and Deviatoric Strain Mechano-Sensing

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