BazTech - Yadda

1

Stress–strain characteristic of human trabecular bone based on depth sensing indentation measurements

Pawlikowski M., Skalski K., Bańczerowski J., Makuch A., Jankowski K.

Biocybernetics and Biomedical Engineering

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2017

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Vol. 37, no. 2

271--280

EN

In the paper a relation between stress and strain for trabecular bone is presented. The relation is based on the results of depth sensing indentation (DSI) tests which were performed with a spherical indenter. The DSI technique allowed also to determine three measures of hardness, i.e. Martens hardness (HM), nanohardness (HIT), Vickers hardness (HV) and Young modulus EIT of the trabecular bone tissue. The bone samples were harvested from human femoral heads during orthopaedical procedures of hip joint implantation. In the research the Hertzian approach is undertaken. The constitutive relation is then formulated in the elastic domain. The values of hardness and the Young modulus obtained from the DSI tests are in good agreement with those found in literature. The stress–strain relation is formulated to implement it in the future in finite element analyses of trabecular bone. Such simulations allow to take into account the microstructural mechanical proper- ties of the trabecular tissue as well as remodelling phenomenon. This will make it possible to analyse the stress and strain states in bone for engineering and medical purposes.

2

On a class of bone cell-based remodelling laws with spatial fading influence of stimuli

Lekszycki T.

Engineering Transactions

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2001

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Vol.49, iss.2/3

155-164

EN

In the present paper the elms of cell-based bone remodeling laws is considered. The fundamental assumption is that of fading in space influence on actor cells osteocytes functioning as sensors. The actor cells - osteoblasts and osteoclasts are responsible for the changes of bone micro-structure. The model proposed in the previous publications of other authors, is based on the resumption of exponential influence function and density of strain energy as the stimulus to which the osteocytes are sensitive, see [11, 12, 14]. As the result of the adaptation according to such remodeling law the porous material is created. The topology of the micro-structure of this material is dependent on the mechanical loading conditions and the characteristics of the interactions of cells. The aim of the present work was to examine if this phenomenon is characteristic only for this specific law or represents a rather general property associated with the hypothesis of fading influence of the cells. Different influence functions were examined for different functionals selected to represent the stimulus. It follows from the these considerations that the fading influence of the cells plays fundamental role for the remodeling process and the creation of trabecular structure. Such structures were obtained for several adaptation laws based on different influence functions and functionals representing the stimulus. They were compared with the results obtained for the adaptation law proposed and discussed in [11, 12, 14]. The numerical calculations suggest that the idea of spatial fading influence of the cells can be possibly combined in future with the results of the research on the biological mechanisms of the bone remodeling to propose more sophisticated models

3

Optimality conditions in modelling of bone adaptation phenomenon

Lekszycki T.

Journal of Theoretical and Applied Mechanics

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1999

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nr 3

607-623

EN

Continuous bone remodeling consists in simultaneous resorption of tissues and synthesis of a new matrix. If, due to variable external or internal conditions, the equilibrium is disrupted, significant rearrangments of the micro-structure and bone shape are possible. Many mathematical and computational models of this adaptation phenomenon can be assigned one of two categories; namely, theoretical models originating from the theory of adaptive elasticity and computational models making use of the optimization theory. In the present paper the approach based on the hypothesis of optimal response of a bone is proposed. It enables derivation of various adaptation laws associated with extremum of the objective functional under a set of appropriate constraints and makes a bridge between the aforementioned categories. In order to illustrate possible application of the proposed general approach the specific formulation is presented and mathematical relations governing the adaptation process are derived. Four numerical examples illustrating some of possible applications of the presented relations are included.

PL

Na przebudowę kości mają zasadniczy wpływ dwa procesy: resorpcja tkanek oraz synteza nowej matrycy. W stanie ustalonym są one w równowadze, lecz gdy na skutek zmiennych warunków zewnętrznych któryś z nich zaczyna przeważać może nastąpić nawet znaczna zmiana struktury wewnętrznej i zewnętrznego kształu kości. W literaturze poświęconej problemowi modelowania zjawiska adaptacji kości można wyróżnić dwa charakterystyczne podejścia, jedno oparte na teorii adaptacyjnej sprężystości i drugie wykorzystujące matematyczne metody optymalizacji. W niniejszej pracy zaproponowano nowe sformułowanie wykorzystujące hipotezę optymalnej reakcji układu. Łączy ono w sobie wiele zalet obu wspomnianych metod. W celu zilustrowania ogólnej idei wyprowadzono konkretne, proste prawo adaptacji. Przedstawiono też kilka przykładów numerycznych ilustrujących niektóre z możliwych zastosowań omawianych związków teoretycznych.