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

SiNuPrOs: Simulations Numériques des Propriétés de l’Os

SiNuPrOs is a model of the human cortical bone that takes into accounts the architectural, multiphysics and multi scale aspects of this medium. We have built this model in which bony architecture is organized in a hierarchical way, each level contributing to the overall properties. The present modelization of the cortical bone leans on previous studies (J.M. Crolet, B. Aoubiza, S. Dasser, N. Pernin, R. Mahraoui, ...) but this one is more complete because it takes into account firstly many parameters characterizing the complex architecture of cortical bone which were ignored before and secondly the interaction fluid-structure and evolving behaviors law. This model contains five architectural levels, 28 parameters and provides detailed information at each level of the bony architecture, including micro and nano levels.

Mathematical homogenization theory is used as a modelization tool and finite element analysis is used for computing the mechanical properties of the cortical bone at each architectural level.

Many configurations can be tested, covering a large number of possible situations and allowing comparative analyses between macroscopic and microscopic properties.

Bony architecture

Human cortical bone can be considered as a composite medium, the fibbers being the osteons embedded in the interstitial system (IS) i.e. a medium made of surmineralized osteons which have been destroyed during the remodelling phase. The most important entities are the mineralization and the osteon, which, by its tubular geometry, is characterized by a diameter and a porosity. The channels called Haversian channels are oriented according to the longitudinal direction and are linked by Volkmann channels located in transverse plans. The following parameters have to be considered: distance between two osteons, distance between an osteon and the IS (i.e. the thickness of the cement line), distance between two Volkmann channels and the Volkmann porosity.   

The osteon is not homogeneous. Being composed by concentric lamellae encased one in the others, it is parameterized by the lamellar and interlamellar thicknesses. These lamellae are crossed by small channels (canaliculae) and this canalicular volume is taken into account simultaneously for the current osteon and the IS.  

The nonhomogeneous lamella is considered as a composite, the fibers (the collagen fibers which one considers as cylinders) being embedded in a medium made of hydroxyapatite (Hap) crystals and fluid. Then it is necessary to introduce the cylinder diameter, the distance between two such cylinders and their orientation according the longitudinal axis.

Concerning the mineralization, an abstract entity, (the Elementary Volume of Mineral Content or EVMC) has been introduced; it is made of a particular set of Hap crystals. This EVMC has no physiological reality but it represents a typical elementary volume of the crystalline structure. The associated parameters are the percentages of EVMC and linked water respectively in the current osteon and in the IS. A coefficient of nanoscopic anisotropy is added.

According to this description, the SiNuPrOs model contains 18 structural parameters, distributed as following: 6 for the Haversian system, 4 for the osteon, 3 for the lamellar architecture and 5 for the mineral structure. All the physical properties (10 parameters) have to be added: density of collagen and Hap, Young's moduli of collagen and Hap, Poisson's ratios of collagen and Hap, piezoelectric and dielectric tensors of collagen, dielectric tensors for the EVMC and the fluid. This geometrical and architectural description makes the SiNuPrOs model particularly complete.    

The present model has been implemented in Matlab and the basic cells discretization has been made with Modulef. The data used by the SiNuPrOs model determine the architectural composition at each level and in a second step, several techniques of homogenization are applied in order to determine the physical properties at each level. The inputs of SiNuPrOs are only the mechanical properties of basic components (collagen and Hap crystals) and a given architecture (described by the parameters mentioned above). The outputs are the physical properties at each level of the architecture.

So, its main interest is the evaluation of physical properties at each level, for a given configuration. It can also be used to seek, by successive tests, configurations corresponding to already measured properties. The calculation time being relatively long, from 10 minutes for a mono-architecture to 45 minutes for architectures with 3 or 4 types of osteons, the research of such a configuration could take hours. This point is not neglectable knowing that many experimental values can be found in the literature but none give any information on the architectural characteristics of the studied sample. For this reason we have built a fast version of this software (SiNuPrOs-Fast) based on a database of results and on the approximation techniques. The results are instantaneously obtained and that's why the Fast version is extremely convenient. The researched configuration being thus obtained, it has then to be validated by the original version. The main interest of such a Fast version is the research of a possible architectural configuration for a sample whose the elastic diagonal properties have been experimentally measured.

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