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Keywords:

  • curvature-driven growth;
  • geometries;
  • osteoblasts;
  • tissue engineering

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results
  5. 3. Discussion
  6. 4. Conclusion
  7. 5. Experimental Section
  8. Supporting Information
  9. Acknowledgements
  10. Supporting Information

Scaffolds for tissue engineering are usually designed to support cell viability with large adhesion surfaces and high permeability to nutrients and oxygen. Recent experiments support the idea that, in addition to surface roughness, elasticity and chemistry, the macroscopic geometry of the substrate also contributes to control the kinetics of tissue deposition. In this study, a previously proposed model for the behavior of osteoblasts on curved surfaces is used to predict the growth of bone matrix tissue in pores of different shapes. These predictions are compared to in vitro experiments with MC3T3-E1 pre-osteoblast cells cultivated in two-millimeter thick hydroxyapatite plates containing prismatic pores with square- or cross-shaped sections. The amount and shape of the tissue formed in the pores measured by phase contrast microscopy confirms the predictions of the model. In cross-shaped pores, the initial overall tissue deposition is twice as fast as in square-shaped pores. These results suggest that the optimization of pore shapes may improve the speed of ingrowth of bone tissue into porous scaffolds.


1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results
  5. 3. Discussion
  6. 4. Conclusion
  7. 5. Experimental Section
  8. Supporting Information
  9. Acknowledgements
  10. Supporting Information

Three-dimensional scaffolds are needed for tissue engineering applications and may also help to study the effect of the environment on tissue growth in vitro. The material used,1 the fabrication process,2 and the architecture of the scaffold3, 4 are known to influence the biological interactions with the host organism. Although all these parameters are difficult to decouple, quantifying their effects in vitro is necessary to understand the nature of cell and tissue responses and to design optimal scaffolds for in vivo experiments and applications.

Cells are known to adapt to the physical properties of their surroundings by integrating the mechanical equilibrium established at their adhesion sites.5 The resulting mechanical cue is translated into a biochemical signal that triggers biological decisions of the cells.6 As cells are mechanically attached to each other, either directly or via their extracellular matrix, they are also able to synchronize their response on a larger scale. For example, patterning in cell differentiation arises as a response to stiffness7 or strain8 patterns, and the distribution of proliferation activity also correlates with the stress distribution in a layer of cells.9

Cell fate has also been investigated in three-dimensional artificial scaffolds. Adhesion, proliferation, differentiation and mineralization of cells and tissues have been compared in several scaffolds with varying structures.10, 11 Recently, Kumar et al.12 showed that gene expression, and thus cell differentiation, is more affected by the structural properties of the substrate than by its composition. Furthermore, pore size and porosity need to satisfy the compromise between a high permeability that enables cell migration and nutrient diffusion within the scaffold, and a large surface area for cell adhesion and extracellular matrix production.3 Many fabrication processes produce structures with random pores in a large range of sizes and interconnectivities difficult to control. Rapid prototyping techniques are much more accurate in that respect.13 The direct printing of the scaffold enables to control the architecture and thus many mechanical properties of the structure.

Rumpler et al.14 used rapid prototyping to build artificial macro-pores of different controlled geometries and showed that cells locally respond to high curvature by producing tissue. Their hypothesis of local tissue growth proportional to curvature has been confirmed experimentally, not only in pores but also on open surfaces,15 however with the additional observation that tissue does not grow on convexities. The interfacial evolution derived from a curvature-driven tissue growth model matched the experimental observations as well as the in vivo expectations when comparing with the typical geometries involved in bone remodeling (osteon and hemi-osteon).16 An interesting consequence of curvature-driven growth was also observed by Rumpler et al.14 Despite seeing local differences in growth rates in prismatic pores with different convex sections (circle, square and triangle) but identical surface areas, the total tissue growth was found to be independent of the shape. This could be understood using Fenchel's law,17 which states that the average curvature in a convex shape, is inversely proportional to the perimeter. This would imply that the average growth rate, if curvature-controlled, would also be the same.

This paper aims at understanding how tissue production can be enhanced simply by controlling the geometry of the surface by exploring non-convex pore geometries. The model of curvature-driven growth as implemented in a previous work15 was first used to predict growth in pores with cross-, star- (non-convex) and square- (convex) shaped sections. The simulations predict higher initial growth rates in non-convex shapes and even a two-fold increase by growing cells in a cross-shaped pore compared with a square-shaped pore. To verify these predictions, straight sided pores with cross- and square-shaped sections are designed in hydroxyapatite scaffolds and incubated with MC3T3-E1 pre-osteoblast cells for in vitro tissue culture. Not only the motion of the tissue-medium interfaces and the evolution of their curvature profiles compare well to the model, but the quantitative analysis of tissue production also matches the outcomes from the curvature-controlled growth model. Understanding the mechanisms involved in such a phenomenon is of high interest for developing tools to design scaffolds with the optimal geometry and meeting the numerous criteria for tissue engineering and clinical applications.

2. Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results
  5. 3. Discussion
  6. 4. Conclusion
  7. 5. Experimental Section
  8. Supporting Information
  9. Acknowledgements
  10. Supporting Information

2.1. Predictions of the Theoretical Model

The model of curvature-controlled growth proposed in previous studies was applied to non-convex geometries to predict potential changes of growth behavior in comparison to the simple shapes used up to now.14, 15 Figure 1 shows the growth behavior expected in square-, star- and cross-shaped pores normalized with respect to the perimeter of their section (Pmedium = 4.71 mm). Although the interface between tissue and medium tends to adopt a circular shape in all cases (Figure 1(a)), the initial kinetics of growth is expected to be significantly affected by the geometry of the straight sided pore (Figure 1(b)). For example, tissue is predicted to grow twice as fast in a cross-shaped pore as in a square-shaped pore or any other convex shape (Figure 1(c)).

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Figure 1. The computational simulation of curvature-driven growth was run on artificial images representing square-, star-, and cross- shaped pores of medium size (Pmedium = 4.71 mm). a) The tissue-medium interface evolves toward a circular shape. b) Initial kinetics of growth are significantly affected by the geometry of the pore section before reaching a circular interface. c) Initial growth rates calculated on the 40 first steps of the simulation suggest that a two-fold increase in tissue deposition can be expected in cross-shaped pores compared to square-shaped pores. Adapted from Bidan et al.23

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2.2. Evolution of the Shape in the in vitro System

In order to verify the predictions obtained with the model of curvature-driven growth, tissue was cultured in 2 mm thick hydroxyapatite (HA) plates containing straight sided pores of square- (convex) and cross- (non-convex) shaped sections normalized with respect to their perimeter. Once seeded on the scaffolds, the MC3T3-E1 pre-osteoblasts proliferate and start to produce collagenous extracellular matrix (ECM). Tissue deposition was followed in each pore by phase contrast microscopy over 28 days and quantified in terms of projected tissue area (PTA).

Three sets of 5 pores of each size (Pmedium = 4.71 mm, Plarge = 6.28 mm) and each shape (Sq stands for square and Cr for cross) have been independently seeded and showed similar results. The paper presents the data from one representative set.

Figure 2(a) shows phase contrast images taken at 4 different time points during culture (D2, D7, D14 and D21). As already observed by Rumpler et al. in convex shapes,14 tissue deposition starts in the corners whereas no growth occurs on flat surfaces until the surrounding tissue deposition modifies the local geometry. In cross-shaped pores, the concave regions of the branches are also quickly filled with tissue. However, the 4 convex points (indicated by red circles on the figure) seem to act as ‘flat surfaces’ since no growth occurs until local curvature becomes positive through the global interfacial evolution. This effect of the sign of curvature has already been pointed out in previous studies.15 In both cases, the tissue-medium interface evolves toward a circle, as predicted by the model of curvature-driven growth applied to the actual geometry of the pore, derived from the experimental images (Figure 2(b)).

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Figure 2. Tissue growth in square- and cross-shaped pores. a) Phase contrast images of the pore taken 2, 7, 14, and 21 days after seeding the MC3T3-E1 on the scaffolds. b) The superposition of the interfaces obtained experimentally is compared with the predictions of the curvature-driven growth simulation applied to the actual geometry of the experimental pore at D2. 7, 14 and 21 days of experiments are obtained with 36, 120, and 204 steps of simulation with r = 8.5pxl, α = 12steps, t0 = 4 d. c) Curvature profiles of the tissue–medium interface measured at D2, D7, D14, and D21 in a square- (i) and a cross-shaped pore (ii). d) Curvature profiles are measured on the interfaces predicted by the curvature-driven growth model after 7, 14, and 21 days of culture in ideal square- (iii) and cross-shaped pores (iv). The curvature measurements were smoothed using a mask size of r = 14.5pxl.

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The evolution of the geometry was quantified in terms of curvature. Figure 2(c) presents the curvature profiles measured on experimental images taken at different time points. The behavior compares well with the interfacial curvature profiles measured on images obtained with the curvature-driven growth model applied to ideal square and cross shapes (Figure 2(d)). The 4 peaks of curvature characterizing the corners of a square-shaped pore vanish as the tissue grows; the curvature profile of the interface flattens and becomes characteristic of a circle. In a cross-shaped pore, 8 peaks correspond to the 8 concave corners and 4 regions of highly negative curvature are related to the 4 convex corners. As in the square, all the peaks–whatever their sign–tend to vanish and curvature profiles become smooth as tissue is deposited. Note that differences in curvature values are due to the imperfections of the HA scaffolds, and that the final profiles are not totally smooth due to the discrete character of the binarized images.

2.3. Structure

The organization of the cells and collagen fibers within the newly formed tissue was investigated qualitatively using immunofluorescence methods. On Figure 3(a), nuclei staining (red) reveals the homogeneous distribution of cells within the tissue whereas actin fibers (green) are mostly concentrated and highly oriented along the tissue-medium interface.

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Figure 3. a) Whatever the geometry of the pore, cells are homogeneously distributed in the tissue (nuclei in red) but actin concentration is much higher at the interface which tends to be circular. b) Polarized microscopy reveals collagen fibers having the same orientation as the cells, i.e., parallel to the tissue–medium interface. c) The geometrical construction that considers tissue as an assembly of tensile elements representing contractile cells15 applies to convex and non-convex geometries. d) Tissue stained for actin fibers reveals stretched cells organized along the interface as predicted by the chord model.

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In larger pores, polarized microscopy enables to image the preferential orientation of the fibrous extracellular matrix deposited by the cells. As shown on Figure 3(b), collagen fibers deep in the tissue are oriented parallel to the substrate, whereas those at the interface have a direction similar to the cells.

2.4. Kinetics of Bone-Like Tissue Growth

Kinetics of new tissue formation within the pores of the scaffolds was followed in the in vitro system by measuring the projected tissue area (PTA) on phase contrast images taken twice a week. As predicted by the model of curvature-driven growth in ideal shapes (Figure 1), Figure 4(a) and 4(b) reveal that after any time of culture, more tissue has been produced in the cross- than in the square-shaped pores, and this for two different pore sizes (Plarge = 6.28 mm and Pmedium = 4.71 mm respectively). Reporting tissue growth rates calculated between D4 and D14 also confirms that initial growth rates are almost two times faster in cross- than in square-shaped pores, independent of the size (Figure 4(c) and 4(d)).

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Figure 4. Growth kinetics measured in square- and cross-shaped pores of large (a) and medium (b) size. Experimental and simulated growths are reported in terms of projected tissue area (PTA) (α = 12steps, t0 = 4d). Growth rates are calculated between D4 and D14 with experimental and simulated data in large (c) and medium (d) pores. ANOVA analysis shows no significant differences between the methods used (Exp, Sim) but a significant difference in the tissue growth rates achieved in square and cross (p < 0.05). Dots and error bars represent mean values and standard errors respectively (n = 5).

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Additionally, Figure 4 shows that the model of curvature-driven growth applied to the actual geometry of the pores, also predicts quantitatively the growth behavior for the first two weeks of cell culture. The model is fitted with a single parameter which sets the time scale of the simulation and is calculated with the experimental tissue growth rate in a square-shaped pore. For this set of experiments, 12 steps of simulation represent 1 day of culture (α = 12steps). A lag time t0 = 4d accounting for the time that cells need to settle was incorporated to overlap the curves.

3. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results
  5. 3. Discussion
  6. 4. Conclusion
  7. 5. Experimental Section
  8. Supporting Information
  9. Acknowledgements
  10. Supporting Information

In this study, a curvature-driven growth model14, 15 was applied to different (non-convex) geometries. The growth behaviors obtained by computational simulations were verified experimentally using an in vitro tissue culture system that offers the possibility to vary the geometry of a substrate in a controlled way, independently of the chemistry. Not only the qualitative and quantitative geometrical evolution of the tissue-medium interface, but also the faster tissue generation by MC3T3-E1 cells in non-convex-shaped pores (cross) could be derived from the simple hypothesis that the local growth rate is locally proportional to the curvature (if it is positive).

Despite a well-defined experimental protocol, some limitations remain. The hydroxyapatite scaffolds produced by casting and sintering present the expected geometry on the millimeter scale, but the roughness of the surfaces is difficult to control, especially in non-convex shapes. This drawback also justifies the necessity of a computational tool able to quantify the geometry in terms of curvature profile and apply the curvature-controlled growth model directly on experimental images, and therefore take into account the interfacial defects.

3.1. Convex Versus Partially Non-Convex Pores

To get a simple analytical estimate of the growth rate based on local curvature, we use the following considerations. In non-convex-shaped pores, the growth law can be written as:

  • equation image((1))

Or in terms of projected area:

  • equation image((2))

with equation image being the positive curvature averaged over the perimeter P of the section.

The calculation of equation image is based on the demonstration of Fenchel's law (see Supporting Information (SI)). As equation image is proportional to P+ave(P), which turns to be constant and characteristic of the shape but not the size neither the proportions, equation image is defined as a constant dimensionless value characteristic of the “non-convexity” of the shape. equation image for convex shapes and equation image for non-convex shapes.

In cross-shaped pores, the curvature is negative in four points, positive in the eight right angle corners and null elsewhere. In squares, the curvature is positive in the 4 right angle corners and null elsewhere. If the negative curvature plays no role in the growth rate, then the positive curvature averaged on the perimeter in the cross is twice the one of a square and growth should be twice faster. Both simulation and experiments meet this prediction.

3.2. Gradual Slowdown of Growth Rates

The patterns obtained with the curvature-controlled growth model and the ones observed in the experiments can also be derived from the simple geometrical construction using tensile elements.15 In essence, this model represents a cell by its internal actin filaments (stress fibers) connecting adhesion sites of the cell. This “chord model” explains intrinsically not only the absence of growth on convexities but also the faster tissue growth and the higher tissue organization in non-convex shapes (Figure 3(a)). This approach is further supported by actin stained tissues showing cells locally oriented parallel to the tissue-medium interface (Figure 3(a) and 3(d)). Considering that this preferential organization is also followed by the collagen fibers synthesized by the cells during tissue growth (Figure 3(b)), one could transfer the geometrical construction to the tissue level in a similar way to the cable model of Bischof et al.24

The curvature-driven growth model predicts a decrease in tissue growth rate in cross-shaped pores as soon as the tissue-medium interface becomes convex. However, as already observed in similar experiments, the tissue growth slows down after 18 days of culture, independently of the shape of the pore. Three main hypotheses have been proposed to explain this phenomenon.15 i) Ageing and differentiation affect the proliferative activity of the cells and thus their ability to produce tissue. ii) Not only cells mature, but also the ECM they produce. With maturing collagen cross-links the matrix could become locally denser, which may be implemented into the model as a gradual reduction of the growth rate. iii) Considering the projected tissue area to quantify growth supposes that tissue grows homogeneously all along the height of the three-dimensional pore, which is unlikely. Therefore, one needs two principal curvatures to describe the geometry of the interface along the vertical axis, and they are likely to be of opposite sign.

To discuss this last point further, one can analytically estimate the impact of the convexity appearing in the third dimension in a cylinder of radius R0. As depicted on Figure 5(a), if the inward curvature is approximated by a circle (red), the two principal curvatures at the point M are:

  • equation image((3))

where R is the radius of the pore at time t and L is the depth of tissue deposition. Mean curvatures equation image in each point of the interface in the middle plane are lower than the one measured on the projection (κ1) and, therefore,

  • equation image((4))
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Figure 5. The effect of the curvature in the third dimension may be partly responsible for the slowdown of growth observed experimentally on the projected plane. a) Schematic representation of tissue repartition in the pore with the associated geometrical descriptors in the middle plane. b) A numerical derivation was done with λ = 0.01 mm.timeunit − 1, L = 2 mm in pores of medium and large sizes. The dashed line indicates a linear growth.

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PTA(t) can be derived numerically from Equation (4) and some results presented in Figure 5(b) show, indeed, a slowing down of the growth that is not predicted by the two dimensional model but appears in experiments.

Although the observations and calculations proposed in this paper are simple, they have important consequences for the design of tissue engineering scaffolds. For example, the results suggest that for purely convex channels, the rate of new tissue growth in a single pore is independent of size and geometry. This implies that pore shape can be modified to satisfy other criteria (e.g. strength, fatigue resistance, permeability, etc)25 without changing the rate of tissue ingress. Moreover, introducing non-convexities into the pore shapes can greatly increase the growth rate (by a factor of 2 in the case of cross-like pores) giving a new opportunity to optimize the architecture of scaffolds for tissue repair.

3.3. Towards Optimizing Pore Geometry in a Scaffold

Integrating a scaffold in a host organism often implies to produce as much tissue as possible in a short time. In that respect, a lot of highly non-convex pores would be useful. However, having small pores filling fast and completely with agglomerates of cells is also not desired. Indeed, diffusion of nutrients would be impaired and cell viability affected. Moreover, cells also need space to migrate and lay down extracellular matrix. Pores should then be large enough to guarantee a good permeability and leave room for the formation of new tissue and angiogenesis.26

As shown in this study, the geometry of individual pores not only influences density, permeability and the amount of tissue produced in the scaffold, but also the speed and repartition of tissue deposition. In the cross-shaped pore, for example, tissue is generated with a high rate in the branches in a first stage, which could help anchoring the scaffold faster in the host organism. As the interface smoothens and becomes circular (convex), the growth rate slows down, leaving time and space for exchanges through the pore. Experimentally, the slowdown occurs a bit earlier for the reasons discussed above.

These principles apply to single pores and, when up-scaling them to scaffolds with multiple pores, one has to consider that geometry also determines the number of pores npores which geometrically fit in a scaffold with defined size Ascaff and porosity ϕ:

  • equation image((5))

Therefore, the global tissue growth rate in a scaffold can be estimated as (based on Equation (2)):

  • equation image((6))

This can be rewritten in terms of the circularity C, which is a dimensionless shape factor that can be used to describe the pores. This value depends on the geometry but not on the size:

  • equation image((7))

where P is the pore perimeter. Figure 6(a) classifies pore shapes with respect to their “non-convexity” and their “circularity”, two geometrical parameters that influence respectively tissue growth rate in an individual pore and the global porosity of a scaffold made of those pores. The global tissue growth rate can then be expressed as a function of the scaffold and the pores characteristics:

  • equation image((8))
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Figure 6. a) Pore shapes can be classified using the “non-convexity” which determines tissue growth rate in the pore and the “circularity” which influences the number of pores fitting in a scaffold. b) Contour plot representing the influence of pore geometry and pore size on the total tissue growth rate in a 20 mm2 scaffold with a porosity of 0.9. Grey level decreases with the growth rate. For each shape, the dot shows the perimeter corresponding to an inner radius of 150 μm (considered as the limit for good permeability properties). In that respect, smaller perimeters are not relevant for tissue engineering purposes.

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Equation (8) shows that the global tissue growth rate in the scaffold is a product of independent terms characterizing i) cell activity, ii) scaffold properties, iii) pore size and iv) pore geometry.

Figure 6(b) shows how the total tissue growth rate depends on the size and the geometry of the pores. The initial tissue growth rate is considered, i.e. the growth rate achieved until the interface becomes convex. In a plate-like scaffold of a given area (20 mm2) and given porosity (0.9), small and non-convex pores give rise to higher growth rates (white areas on the bottom right).

However, a common concern in tissue engineering is the permeability of the scaffold to guarantee cell migration as well as nutrient and waste diffusion necessary for cells to survive. Tissue engineering literature suggests that pores should be at least 300 μm large to ensure a good permeability of the scaffold.27 For each shape, the size of the inner circle is taken as a limitation for pore size. An inner radius of 150 μm leads to the minimum relevant perimeter. Maximum realistic initial growth rates are estimated for each shape (small shapes on Figure 6(b)). Considering this aspect, the fastest initial growth rate can be obtained using regular crosses with thick branches. The circular interface being quickly reached, the amount of tissue produced at that high rate is however low. The remaining space and the slower growth from this time point could be profitable for angiogenesis and facilitate diffusion as the pore is closing.

All these calculations assume that the totality of the scaffold area can be covered by assembling pores of the same shape. This statement is true for squares, triangles and regular crosses with k = 0.33. However, it is known that the maximum density achieved by packing circles in a hexagonal arrangement on a surface is 0.90 and star shapes are also not likely to be packed in an optimal arrangement. It is therefore relevant to envisage scaffolds containing pores of different geometries and sizes. Few large highly non-convex shapes can promote the anchoring of the scaffold at the early time points and facilitate diffusion as growth progresses, whereas smaller (mainly convex) pores fitted in between can provide additional surfaces for cells to deposit tissue with a slower rate, but that will also support integration of the implant in the host body. The total initial tissue growth rate obtained in such a scaffold can be estimated by adapting Equation (8):

  • equation image((9))

with ϕi the contribution of the shape i to the global density.

4. Conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results
  5. 3. Discussion
  6. 4. Conclusion
  7. 5. Experimental Section
  8. Supporting Information
  9. Acknowledgements
  10. Supporting Information

This work lays stress on the determining role of the geometry of a substrate on the kinetics of tissue deposition. We show that tissue growth can be promoted simply by tuning the curvature of the surfaces where cells deposit their extracellular matrix. A simple geometrical model based on the tensile behavior of the cells, which leads to curvature-controlled growth, can predict both the kinetics achieved and the distribution of tissue deposition. As such simple principles could be of high interest for tissue engineering, we propose some methods to optimize pore design when considering a porous scaffold intended for tissue repair.

5. Experimental Section

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results
  5. 3. Discussion
  6. 4. Conclusion
  7. 5. Experimental Section
  8. Supporting Information
  9. Acknowledgements
  10. Supporting Information

Curvature-Driven Growth Simulation: A model for curvature-driven tissue growth was proposed by Rumpler et al.14 and implemented by Bidan et al.15 in a Matlab (Matlab 7.8.0 R2009a, MathWorks, Natick, MA) code based on a method for measuring curvature on digital images.18

This computational simulation is run on binarized images of the pores in which the scaffold is black and the medium is white. Each iteration consists in a) attributing a value of effective curvature to each pixel representing what cells sense from the geometry of the surface, b) transforming the white pixels having a positive effective curvature to black and represent tissue deposition in concave regions. The process is then repeated to simulate curvature-controlled growth. To make a quantitative comparison with the experimental results, this simple model only requires the input of a single parameter accounting for the number of iterations needed to simulate one day of culture. This value is calculated using the experimental growth rate measured in a convex shape, the square in this study.

In order to be equivalent to the geometrical interpretation described previously,15 when the computational tool is used for modeling purposes, the mask radius is set to r = 8.5pxl, i.e. equation image times of the size of a cell (about 50 μm for an elongated osteoblast).

Production of the Hydroxyapatite (HA) Scaffolds: 2 mm thick HA scaffolds containing straight sided pores are produced by slurry casting as mentioned in previous studies.14, 15 Pore sections represent squares or crosses and are normalized with respect to their perimeter (Pmedium = 4.71 mm, Plarge = 6.28 mm). Molds are designed using the computer-aided design (CAD) software Alibre Design (Alibre Inc., Richardson, TX) and produced with a three-dimensional wax printer, Model Maker II (Solidscape Inc., Merrimack, NH) as described by Manjubala et al.19 The molds are then filled with a HA slurry made of methacrylamide (MAM) monomers (15 g), N-N′-Methylenebisacrylamide (BMAM) (5 g), water (75 g), Dextran (12.5 g) and HA powder (300 g), and cross-linked with ammonium persulfate and N,N,N′,N′-Tetramethylethylenediamine (TEMED). The structures are slowly air dried by heating the samples to 50 °C at a rate of 5 °C per day and then are held at this temperature for one day. The dried samples are then pre-sintered at 600 °C for 48 h to remove the wax molds and are finally sintered at 1100 °C for 24 h.20

Cell Culture: Murine pre-osteoblastic cells MC3T3-E1 (provided by the Ludwig Boltzmann Institute of Osteology, Vienna, Austria) are seeded with a density of 105 cells.cm−2 on the surface of the HA scaffolds and cultured for 28 days in α-MEM (Sigma-Aldrich, St. Louis, MO) supplemented with fetal calf serum (PAA laboratories, Linz, Austria) (10%), ascorbic acid (Sigma-Aldrich, St. Louis, MO) (0.1%) and gentamicin (Sigma-Aldrich, Steinheim, Germany) (0.1%) in a humidified atmosphere with CO2 (5%) at 37 °C.

Imaging: Each pore is imaged every 3 to 4 days using a phase contrast microscope (Nikon Eclipse TS100, Japan) equipped with a digital camera (Nikon Digital sight DS 2Mv). All pictures are taken with a 4× objective, yielding a final image resolution of 205 pixels per mm.

Image Analysis: The digital phase contrast images are semi-automatically binarized using ImageJ (National Institutes of Health, Bethesda).21 The contrast in the images is sufficient to enable scaffold and tissue (represented in black in the binarized images) to be distinguished from the medium (represented in white).

Measurement of Tissue Growth: Tissue growth in the pores is quantified by determining the projected tissue area (PTA) formed in the pores. As this measurement is two-dimensional, it is only a proxy for quantifying the volume of growth into the depth of the pore. The free section of a pore, corresponding to the white regions in the binarized images, decreases with time. The PTA is then calculated by subtracting the binarized image at an initial time point from the image at the time of interest, and then calculating the remaining area. As cells need time to settle on the scaffold and start tissue deposition, the initial pore section is taken on the second day after seeding (D2).

Curvature Measurement: The curvature profile of the interface between the tissue and the medium on each binarized image is calculated using Frette's algorithm18, 22 implemented in a custom made Matlab code (Matlab 7.8.0 R2009a, MathWorks, Natick, MA) as described in.15 Briefly, the algorithm first locates the pixels on the tissue-medium interface in the binarized image and the local curvature κ associated with an interface pixel is then estimated from the ratio of the number of black to white pixels lying within a given radius from the interface with the formula:

  • equation image((10))

where A is the number of pixels in the mask and on the outer side of the interface, Atot is the number of pixels in the mask and r is the mask radius. The calculation is made for all pixels on the interface on each side of the border. The local curvature in one position of the interface is taken as the mean value of the curvatures measured on the outer pixel and the inner pixel. In the limit of a perfectly smooth interface and an infinitely small radius, this ratio corresponds to the local curvature. In the context of this paper, concave surfaces have a positive curvature.

To quantify interfacial geometry at different time points, the local curvature is given as a function of the position along the interface normalized with respect to its perimeter. In order to reduce the noise induced on the curvature profiles by both the roughness of the experimental interfaces and the digitalization, the mask radius r of the computational tool is set to r = 14.5pxl and the resulting profile is then smoothed using a running average algorithm with a sampling proportion of 5% of the total length of the perimeter.

Immunofluorescence Staining: Scaffolds are washed with phosphate buffered saline (PBS), fixed with 4% paraformaldehyde for 5 min and permeabilized overnight with 1% Triton-X100 (Sigma-Aldrich, Steinheim, Germany) at room temperature. Once washed in PBS, the tissue is stained for actin stress fibers by incubating with Alexa-Fluor 488–phalloidin (Invitrogen, Molecular Probes) (3 × 10−7 M) for 90 min. Nuclei are then stained with TO-PRO 3 692–661 (Invitrogen, Molecular Probes) (3 × 10−6 M) for 5 min. Fluorescent images of the stress fibers are obtained using a confocal laser scanning microscope (Leica TCS SP5).

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results
  5. 3. Discussion
  6. 4. Conclusion
  7. 5. Experimental Section
  8. Supporting Information
  9. Acknowledgements
  10. Supporting Information

Supporting Information is available from the Wiley Online Library or from the author.

Acknowledgements

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results
  5. 3. Discussion
  6. 4. Conclusion
  7. 5. Experimental Section
  8. Supporting Information
  9. Acknowledgements
  10. Supporting Information

We acknowledge funding from the Leibniz prize of PF running under DFG contract number FR2190/4-1. C.B. is a member of the Berlin-Brandenburg School for Regenerative Therapies (GSC 203).

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results
  5. 3. Discussion
  6. 4. Conclusion
  7. 5. Experimental Section
  8. Supporting Information
  9. Acknowledgements
  10. Supporting Information

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