On the Growth of Bone through Stress Driven Diffusion and Bone Generation Processes

Abstract: In order to prevent or modify bone degeneration due to rest or due to diseases such as osteopenia and osteoporosis, the modeling and remodeling of bone tissue must be better understood. In this thesis it is assumed that the primary condition leading to bone growth is a change of the chemical environment caused by transport of matter resulting from stress driven diffusion. The change in the chemical environment may consist of changes in the concentration of different substances stimulating, for example, bone building osteoblast recruitment or suppression of bone resorbing osteoclast activity. Inspired by a study found in the literature where an experiment is performed on avian bones, two numerical models are developed which can be solved using a regular structural finite element solver. The first model acknowledges that diffusion of matter is affected by the gradient of external potential energies such as heat, electrical or mechanical, of which the mechanical is assumed to be important. The derivation of the statistical mechanics of the molecular diffusing matter leads, according to Einstein (1905), to a partial differential equation similar to Fick’s law but with an added diffusio-mechanical coupling term. The diffusio-mechanical term is a function of the hydrostatic pressure due to bending of the bone. Since bone growth takes place at the outer bone surface, the hypothesis is that substances promoting bone growth are transported from the medullarycavity to the outer surface, the periosteum, of the long skeletal bone. From comparison with experiments, it is found that bone growth to a higher extent takes place where high concentration of matter arises rather than where the mechanical stress is high. It is also seen that bone growth depends on load frequency. The second model also starts from the basic assumptions made in the first model regarding the preservation of energy. To derive the governing coupled diffusio-mechanical partial differential equations, the matterrequired for the generation of bone is assumed to be transported easily through the bone or surrounding fluids. Therefore, to simplify the analysis, the governing equations may be put on a non-conservative form. The Ginzburg-Landau theory is used to formulate the expression for the phase transformation like process for the generation of bone. The available energies are elastic strain energy due to bending, the concentration gradient energy and a chemical potential. The model uses a phase field variable to describe the state of the bone, and the model shows that high loading initiate bone growth whereas low loading makes the bone contract. The models use normalized input data, but in order to make full use of the results the actual diffusion coefficient of interest must be known,and hence an approach to determine diffusion coefficient in bone tissue is developed. By means of conductivity measurements together with an analytical solution, which is fitted to the experimental data using a Kalman filter, diffusion coefficients can be extracted. With known diffusion coefficients it is possible to evaluate the normalized results from the numerical models. Finally a model to evaluate the physiological status of the bone by looking only at a small portion of the cross-section of a bone is presented. The approach uses the size and shape of the pores of a representative area of the bone crosssectionto determine a value of an effective diffusion coefficient of matter and an effective Young’s modulus of the bone. A database with parameters used in the method must be established once with finite element analysis. The model can then be used by anyone, and no knowledge of finite element analysis is required. The calculated values can be used to evaluate how much the porosity is affecting the bone status.