Theoretical model to determine the effects of geometrical factors on the resorption of calcium phosphate bone substitutes

Abstract

A theoretical approach was used to determine the effect of geometrical factors on the resorption rate of calcium phosphate bone substitutes that are either dense, microporous, and/or contain spherical macropores. Two cases were considered: (a) macroporous blocks that can be invaded by resorbing cells either directly because the structure is fully open-porous, or indirectly after some resorption of the macropores walls and/or interconnections. (b) Microporous or dense blocks/granules that cannot be invaded by resorbing cells, i.e. can only be resorbed from the outside to the inside, layer by layer. The theoretical approach was based on five assumptions: (i) the pores are spherical; (ii) the pores are ordered according to a face-centered cubic packing; (iii) the resorption is surface-controlled; (iv) the resorption is only possible if the surface can be accessed by blood vessels of 50 μm in diameter; and (v) the resorption time of a given amount of calcium phosphate is proportional to the net amount of material. Based on these assumptions, the calculations showed that the resorption time of a macroporous block could be minimized at a specific pore radius. This pore radius depended (i) on the size of the bone substitute and (ii) on the interpore distance. Typical radii were in the range of 100–400 μm. These values are similar to the numerous pore size optima mentioned in the scientific literature. For microporous or dense blocks/granules, the model suggested that a relatively small radius should be preferred. Such a radius leads to an optimum combination of a high surface area favorizing resorption and the presence of large intergranular gaps favorizing blood vessel ingrowth. In that case, the optimum of granule radius is around 100–200 μm. Finally, a very good agreement was found between the predictions of the model and experimental data, i.e. the model explained in all but two cases the results with an accuracy superior to 80%. In conclusion, the model appears to be a useful tool to better understand in vivo results, and possibly better define the geometry and distribution of the pores as well as the size of a bone substitute.