Multi scale mathematical modeling of coupled phenomena in porous media: application to the bone remodeling; case of healthy and pathological bone - RP7  

Expected results

This work represents an important advance in the modelization of bone remodeling (cortical part only). Contrary of all the articles which are actually published on this subject, our project does not propose an empirical law giving a macroscopic evolution of the geometry. With its multi scales concept, it is based on many processes implied in this phenomenon. This work thus brings a better modelization of bone remodeling.

The objective being to have a sufficiently precise predictive model of bone remodeling which takes into account the main implied biological mechanisms, this model would be optimized by identifying the role of the various mechanical-biological parameters responsible of remodeling.

The direct applications of such a study relate to the cure of fractures and bony repair for example.

Other researchers and/or students can also use the new model developed from this project to predict and design their own models using different settings.

By performing this project, we hope to develop research on human bone globally and on bone remodeling in particular. We hope having at the end of this project a model being a very competitive one, which will be considered as a good investigation tool in the simulation of bony physical properties and in the simulation of bone remodeling.

Steps in accomplishing the project, aiming the fulfillment of the proposed objectives:

Single phase 2009

Activities:

     1.1 Numerical simulation of the increasing of local mineralisation level

     1.2 Improving the Excel version (Fast version) of the SiNuPrOs program by introducing the approximation of the physical  properties at the lamellar and osteonal level

1.3 Building a convivial interface for the Matlab version of the existing informatic program

    1.4 Study of the initial pressure problem (mathematical aspect, convergence of the process, influence over the numerical results)

     1.5 Numerical simulation of the macro-nano retour for diverse femur configurations

    1.6 Determining by numerical simulations of the possible architectures corresponding to the given experimental result    

     1.7 Numerical determination of the corresponding architecture to the micro-deformations obtained experimentally; creating a “map” of such an architecture in a femur  

2.1 Theoretical study of the fluid-structure coupling (existence and unicity of the solution; algorithm of numerical resolve)

2.2 “Multi scale” homogenization of permeability: obtaining and resolving cellular problems; the calculus of homogenized coefficients of permeability at each architectural level of the cortical bone; determining the porosity at each level; numerical study of the influence of the architecture on the permeability at each level

2.3 Numerical simulation of the flow of bony fluid in the osteonal cavity – the aim is to understand why it is necessary to have a minimal number of cells for the mineralization phenomenon to be initiated

2.4 Numerical simulation of the electrical potential – we’ll thus be able to determine the level of mineralization starting from which the piezoelectric effect of collagen disappears

3.1. Elaborating and publishing articles in international papers of specialty

3.2. Participating at national and international conferences

3.3. Elaborating the partial report of the research’s results

3.4. Maintaining up to date the webpage dedicated to the project and of that dedicated to the developed informatic application.

Single phase 2010

Activities:

1.1. Introducing the damage operator. The mathematical objective is to study the effect of this operator over the homogenized behavior ; the mathematical analysis of the problem that results (existence and unicity of the solution, discreetness in space and time, determining the law of homogenized behavior)

1.2. Elaborating a mathematical model of the mineral apposition; underlining a mathematical formalism that, by the three-phase modelized environment (elastic, viscoelastic and fluid) expects to give , by homogenization, two laws of behavior differently homogenized (one viscoelastical and another of flowing type in porous media)

2.1. The study of absence of gravity

2.2. Osteoporosis – we shall use the previous developed modules, at what concerns the permeability calculus  but also the law of behavior that model the mineralized apposition, aiming to determine the  parameters important to diminishing bony mass

2.3. The loss of bony substance and pseudartrosis – we shall use an iterative process over the geometry

2.4. Numerical simulations of these pathologies

3.1. Elaborating and publishing articles

3.2. Participating at national and international conferences

3.3. Elaborating a final research report of the project that will consist besides the resulted conclusions of the research, the proposals and directions towards which future studies will be orientated

3.4. Editing a wide study (monograph)