Funkcjonalna adaptacja kości jako zagadnienie optymalnego sterowania

The functional adaptation of bone is a process of bone tissue remodeling induced by variable in time mechanical demands that the skeleton has to satisfy. It is a very complex but highly organized process composed of events at micro-level (molecular and cellular) but having effects in macro-scale (variation of bone internal structure and external shape). Mathematical models of this phenomenon proposed in the literature represent formulas postulated on the basis of the results of medical observations or laboratory investigations and describe locally the evolution of a material in space and time. In the present paper a use is made of the hypothesis of optimal response of bone, proposed earlier by the author, what enables derivation (instead of postulation) the remodeling rules from a very general and global assumption. It turns out that such a formulation has many similarities to engineering optimal control problems. The link between the postulated local adaptation rules and those derived from the global assumption is also discussed.Funkcjonalna adaptacja kości jest procesem polegającym na przebudowie tkanki kostnej wywołanej zmieniającymi się w czasie wymaganiami mechanicznymi, jakie musi spełniać szkielet kostny. Proces ten jest niezwykle złożony, ale doskonale zorganizowany i składa się z szeregu zjawisk zachodzących na poziomie mikro (molekularnym i komórkowym) lecz mających efekt na poziomie makro (zmiana zewnętrznego kształtu kości oraz jej struktury wewnętrznej). Matematyczne modele tego zjawiska, postulowane w oparciu o obserwacje medyczne i badania laboratoryjne, opisują lokalną ewolucję materiału w czasie i przestrzeni. W tej pracy zastosowano hipotezę optymalnej odpowiedzi kości zaproponowaną wcześniej przez autora w celu wyprowadzenia (zamiast postulowania) związków rządzących przebudową kości w oparciu o bardzo ogólne założenia. Okazuje się, że takie sformułowanie ma wiele wspólnego z zagadnieniami optymalnego sterowania. W pracy zaprezentowano przykład zastosowania omawianego podejścia oraz przeprowadzono krótką dyskusję na temat związków między postulowanymi modelami i wyprowadzonymi w oparciu o przyjętą hipotezę

Optimality conditions in modelling of bone adaptation phenomenon

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.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

INVESTIGATION AND MODELING OF BONE FRACTURE HEALING

Abstract An initial stage of bone fracture healing, a clot formation is discussed in this paper. This is an important step in a healing process and it determines future formation of a callus and resulting tissue formation and remodeling. Preliminary results of an animal experiment and theoretical considerations are presented. A biomechanical model of considered phenomenon is proposed and a simple numerical example is discussed. In this example square domain of a porous tissue surrounding broken bone is considered. In this domain micro-cracks are generated. Their distribution changes linearly with a distance from a surface contacting with a bone. The open pores are filled with a physiological liquid. One of the edges of the considered domain is in contact with the fractured bone. It is assumed that blood leeks from the fracture in bone, mixes with the fluids in the pores and propagates into the porous tissue. After a certain time period blood solidification starts what results in a clot formation. It is assumed that the shape of the clot is determined by an assumed level of tissue saturation with blood component at the moment when solidification starts. It follows from this investigations that after minor improvements of the theoretical description and experimental determination of the parameters necessary to perform calculations this model can be used in future to predict conditions necessary for correct and fast bone fracture healing

A mixture model with evolving mass densities for describing synthesis and resorption phenomena in bones reconstructed with bio-resorbable materials

The multiform bio-mechanical phenomena occurring in bones grafted with the addition of artificial materials urge for the formulation of models which are sophisticated enough to describe their complexity. In the present paper we present a continuum poro-elastic mixture model in which two apparent mass densities are introduced to describe, at a macroscopic length scale, situations in which bone tissues and artificial materials coexist and interact. We focus on the final healing stage process when the bone remodelling becomes the dominant phenomenon. Artificial materials used are obviously to be bio- compatible and must resist to externally applied mechanical loads. More recently in order to favour bone tissue re-growth in grafts, which improves the long term performances of grafted bones, it has been conceived to use substitute materials which may be, similarly to bone tissue, bio-resorbed by osteoclasts and eventually replaced by newly synthesised living tissue. To account for resorption and synthesis phenomena suitable evolution equations are introduced for Lagrangian mass densities of the mixture constituents in which an integrodifferential operator defined on deformation fields appears. This operator is chosen to model some features of the coupling between mechanical compliance and biological bone tissue activity. The obtained system of integrodifferential equations is not trivial also when one considers one dimensional cases. Treating this simplified situations will allow us to individuate more easily some important remodelling scenarios. The numerical simulations which we present here show that the introduced model is promising and deserves to be developed to give previsions in more realistic applications

A continuum model for the bio-mechanical interactions between living tissue and bio-resorbable graft after bone reconstructive surgery

We introduce a two-constituent porous continuum as a model describing the long-term growth/resorption phenomena in bone tissues grafted with bio-resorbable materials as driven by mechanical loads. The proposed model is able to account for the interplay between mechanical and biological phenomena which are known to be important for the bone tissue synthesis and the resorption of both bone tissue and bio-material. In particular, in the presented model the Lagrangian apparent mass densities of the natural bone and of the artificial material evolve in time according to precise ordinary differential equations. These latter are obtained by postulating a growth/resorption law and suitable constitutive equations conceived to account for the influence on bone resorption and synthesis of the action of different applied external loads as mediated by biological stimulus. The considered constitutive equations are chosen on the basis of the known biological phenomena occurring in bone resorption and synthesis. We present some numerical simulations for rod-bones subjected to axial external load. These numerical simulations allow for the description of the most desirable situation in which a gradual resorption of the artificial material takes place together with the contemporary formation of new bone, finally giving rise to an almost complete replacement of the artificial material with natural living tissue. (C) 2011 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved

A 2-D continuum model of a mixture of bone tissue and bio-resorbable material for simulating mass density redistribution under load slowly variable in time

The bio-mechanical phenomena occurring in bones grafted with the inclusion of artificial materials demand the formulation of mathematical models which are refined enough to describe their not trivial behavior. A 3D theoretical model, previously developed and used in 1D space, is employed to investigate and explain possible effects resulting from 2D interactions, which may not be present in 1D case so more realistic situations are approached and discussed. The enhanced model was used to numerically analyze the physiological balance between the processes of bone apposition and resorption and material resorption in a bone sample under plain stress state. The specimen was constituted by a portion of bone living tissue and one of bio-resorbable material and was acted by an in-plane loading condition. The signal intensity between sensor cells and actor cells was assumed to decrease exponentially with their distance; the effects of adopting two different laws, namely an absolute and a quadratic functions, were compared. Ranges of load magnitudes were identified within which physiological states are established. A parametric analysis was carried out to evaluate the sensitivity of the model to changes of some critical quantities within physiological ranges, namely resorption rate of bio-material, load level and homeostatic strain. In particular the spatial distribution of mass densities of bone tissue and of resorbable bio-material and their time evolution were considered in order to analyze the biological effects due to the parameter's changes. Synthetically, these biological effects can be associated to different ratios between bone and bio-material densities at the end of the process and to different delays in the bone growth and material resorption. These numerical analyses allowed for finding the most desirable situations in which a gradual resorption of the artificial graft occurs together with the simultaneous formation of new bone, finally leading to an almost complete substitution of the bio-resorbable material with living tissue. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim