How Linear Tension Converts to Curvature: Geometric Control of Bone Tissue Growth

This study investigated how substrate geometry influences in-vitro tissue formation at length scales much larger than a single cell. Two-millimetre thick hydroxyapatite plates containing circular pores and semi-circular channels of 0.5 mm radius, mimicking osteons and hemi-osteons respectively, were incubated with MC3T3-E1 cells for 4 weeks. The amount and shape of the tissue formed in the pores, as measured using phase contrast microscopy, depended on the substrate geometry. It was further demonstrated, using a simple geometric model, that the observed curvature-controlled growth can be derived from the assembly of tensile elements on a curved substrate. These tensile elements are cells anchored on distant points of the curved surface, thus creating an actin “chord” by generating tension between the adhesion sites. Such a chord model was used to link the shape of the substrate to cell organisation and tissue patterning. In a pore with a circular cross-section, tissue growth increases the average curvature of the surface, whereas a semi-circular channel tends to be flattened out. Thereby, a single mechanism could describe new tissue growth in both cortical and trabecular bone after resorption due to remodelling. These similarities between in-vitro and in-vivo patterns suggest geometry as an important signal for bone remodelling

Quantitative results: curvature profile and growth rate.

<p>Quantitative analysis of tissue growth in circular pores (<i>CIR</i>) and on semi-circular channels (<i>SC</i>) of 1 mm diameter. A - The average curvature along the perimeter of the circular pore and on a given portion of the semi-circular surfaces is measured on experimental images at different culture times. As theoretically circular pores should be filled in about 432 steps or 25.4 days. B - The projected tissue area (PTA) is normalised by the area of the pore (PA) at D4 (reference) and reported as a function of culture time. In A and B, the full lines correspond to the prediction given by CCTG (; ). A lag time is used to overlap simulated and experimental data ( and ). C - Growth rates are calculated between D7 and D14 with the experimental and the simulated data as well as data simulated on ideal geometries with a radius derived from the experimental images. ANOVA analysis shows no significant differences between the methods used but a statistical difference in the tissue growth rates achieved in CIR and SC (). Dots and error bars represent mean values and standard errors, respectively ().</p

Qualitative results: evolution of the geometry.

<p>A - Evolution of the tissue interface in a circular pore and on a semi-circular surface. Images taken at different culture times (D4, D7, D14 and D21) during in vitro experiments show behaviours comparable to those observed in osteons and hemi-osteons during bone remodelling. B - The superposition of the interfaces obtained experimentally (top) compares to the one derived from CCTG applied to the actual geometry of the experimental pores at D4 (bottom). 7, 14 and 21 days of culture are simulated by 51; 170 and 289 steps for the circle and 34; 153 and 272 steps for the semi-circle respectively ().</p

Experimental protocol.

<p>A - Moulds are produced by rapid prototyping. A build wax (blue) is used to print the mould in 3 D. A support wax (red) is added to reinforce the object while printing, and then removed by dissolution. B - Hydroxyapatite slurry is cast into the moulds, slowly dried and sintered. C - Pre-osteoblast cells are seeded (105 cells/cm²) on the scaffolds and cultured for 28 days. D - Tissue growth is quantified by phase contrast microscopy twice a week by measuring the projected tissue area (PTA) in each pore.</p

Importance of boundary conditions.

<p>Tissue growth (orange) is simulated on different artificial images using the CCTG description (A to F). The predicted PTA is reported as a function of iteration steps. Each initial interface (black) contains a semi-circle with a radius of 0.5 mm. The different boundary conditions show the influence expected on tissue growth rate and organisation. On A, B and C, the model predicts that the sharper the convex corners, the slower the growth. Tissue is eventually deposited on convex surfaces after the surroundings have been filled and the interface has locally become concave (red arrows). Comparing A, D and E reveals that shifting the convex corners upward prolongs the duration of a constant growth rate which is half of the one obtained in a full circle (F). Tissue deposition can expand on the walls until it reaches the convex corners. From this time point (inset), the surface joining the pinning points is minimised, which decreases the curvature and slows the growth.</p

Tissue organisation.

<p>A - Tissue produced in a pore made of 4 adjacent circles and stained for actin stress fibres and myosin IIb. Actin fibres colocalised with myosin IIb are present on the whole surface but their higher density on concave interfaces suggests a local higher stress state of the cells. B - Tissue is made of cells and collagen. Nuclei (<i>red</i>), actin stress fibres (<i>green</i>) and collagen fibres (<i>visualized by polarized microscopy</i>) are oriented parallel to the interface. The white arrows show polarisation direction. C - The homogeneous distribution of nuclei shows that cell density is independent of geometry and suggests a local dependence of cell proliferation on the local curvature. D - An example of a convex HA surface (D35) on which only a mono-layer of tissue was formed.</p