### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Results
- 3. Discussion
- 4. Conclusion
- 5. Experimental Section
- Supporting Information
- Acknowledgements
- 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

- Top of page
- Abstract
- 1. Introduction
- 2. Results
- 3. Discussion
- 4. Conclusion
- 5. Experimental Section
- Supporting Information
- Acknowledgements
- 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.

### 3. Discussion

- Top of page
- Abstract
- 1. Introduction
- 2. Results
- 3. Discussion
- 4. Conclusion
- 5. Experimental Section
- Supporting Information
- Acknowledgements
- 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:

- ((1))

Or in terms of projected area:

- ((2))

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

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 *R*_{0}. 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:

- ((3))

where *R* is the radius of the pore at time *t* and *L* is the depth of tissue deposition. Mean curvatures in each point of the interface in the middle plane are lower than the one measured on the projection (κ_{1}) and, therefore,

- ((4))

*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 *n*_{pores} which geometrically fit in a scaffold with defined size *A*_{scaff} and porosity ϕ:

- ((5))

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

- ((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:

- ((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:

- ((8))

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 mm^{2}) 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):

- ((9))

with ϕ_{i} the contribution of the shape *i* to the global density.

### 5. Experimental Section

- Top of page
- Abstract
- 1. Introduction
- 2. Results
- 3. Discussion
- 4. Conclusion
- 5. Experimental Section
- Supporting Information
- Acknowledgements
- 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* = 8.5*pxl*, i.e. times of the size of a cell (about 50 μ*m* 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 (*P*_{medium} = 4.71 *mm, P*_{large} = 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 10^{5} 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 CO_{2} (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:

- ((10))

where *A* is the number of pixels in the mask and on the outer side of the interface, *A*_{tot} 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* = 14.5*pxl* 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).