Porosity imaged by a vector projection algorithm correlates with fractal dimension measured on 3D models obtained by microCT
D A N I E L C H A P P A R D∗ & I Z A B E L A - C R I S T I N A S T A N C U†
∗GEROM Groupe Etudes Remodelage Osseux et bioMat´eriaux – LHEA, IRIS-IBS Institut de Biologie en Sant´e, CHU d’Angers, LUNAM Universit´e, 49933 Angers, Cedex, France
†Advanced Polymer Materials Group, University Politehnica of Bucharest, 1–7, Gh. Polizu Street, Sector 1, 011061 Bucharest, Romania
Key words. 3D geometry, fractal, microCT, porosity, porous material.
Summary
Porosity is an important factor to consider in a large variety of materials. Porosity can be visualized in bone or 3D syn- thetic biomaterials by microcomputed tomography (microCT).
Blocks of porous poly(2-hydroxyethyl methacrylate) were pre- pared with polystyrene beads of different diameter (500, 850, 1160 and 1560μm) and analysed by microCT. On each 2D binarized microCT section, pixels of the pores which belong to the same image column received the same pseudo-colour according to a look up table. The same colour was applied on the same column of a frontal plane image which was con- structed line by line from all images of the microCT stack. The fractal dimension Dfof the frontal plane image was measured as well as the descriptors of the 3D models (porosity, 3D fractal dimension D3D, thickness, density and separation of material walls. Porosity, thickness Dfand D3Dincreased with the size of the porogen beads. A linear correlation was observed between Dfand D3D. This method provides quantitative and qualitative analysis of porosity on a single frontal plane image of a porous object.
Introduction
Porous materials are nowadays the subject of high interest in material science, petrography and biomedicine. A number of papers have been published to characterize more deeply porous structures and to quantify porosity (Hernandezet al., 2002; Latief & Fauzi, 2012; Wu et al., 2005; Molina et al., 2011). In medicine, bone is a naturally porous structure and a huge amount of work has been done in the last decades on the microarchitecture of this porous and biomechanically very re- sistant tissue, see review in (Chappardet al., 2008). This aspect
Correspondence to: Daniel Chappard, GEROM – LHEA, IRIS-IBS Institut de Biolo- gie en Sant´e, CHU d’Angers, LUNAM Universit´e Nantes Angers Le Mans, 49933 Angers, Cedex, France. Tel: (33) 244 68 83 49; fax: (33) 244 68 84 51; e-mail:
daniel.chappard@univ-angers.fr
has become so important that the definition of osteoporosis according to WHO and several consensus conferences now takes into account the importance of these microarchitectural factors to explain bone frailty (Anonymous, 1993). Recent findings have also stressed the importance of designing new porous biomaterials to enhance invasion by vascular sprouts and osteogenic cells in the centre of the grafted material. The 3D spatial distribution of the pores within a material appear as important characteristics to consider (Ikedaet al., 1999;
Yuanet al., 2001; Ghoshet al., 2008). Nonlinear relationships between the amount of pores and the 3D distribution of poros- ity have been reported for bone and synthetic biomaterials (Chappard et al., 1996; Thomsen et al., 1998; Chappard et al., 1999; N’Diayeet al., 2013). Morphologic methods for studying porosity and its distribution have evolved since the pioneering works of Delesse on porous stones (Delesse, 1847).
He developed the first technique to measure porosity on 2D slabs of polished stones with grids of points. He found that the ratio of points projected on the pores/total number of the points of the grid was similar to the 3D porosity evaluated by other method such as Archimedes’ principle. The development of stereological principles and the availability of microscopic eyepieces with grids have led to a considerable number of works in the bone field since 1970’ (Meunieret al., 1973).
With the development of microcomputed tomography during the last decade (Sasov & Van Dyck, 1998), it is now possible to measure and visualize porosity inside both trabecular or cor- tical bone as well as inside 3D porous scaffolds usable as bone graft biomaterials (Borahet al., 2001; van Lentheet al., 2007;
Simon et al., 2008; Schladitz, 2011; Massai et al., 2014).
If porosity can be visualized in 3D, it is difficult to evaluate whether the pores are regularly arranged or not and also to ap- preciate the regularity of the distribution of the pores in a given volume.
The aim of the present study was to develop a method that allows visualization of porosity across a porous material on a frontal plane image and to quantify the complexity of this parameter by fractal analysis.
Preparation of porous blocks
We used 2-hydroxyethyl-methacrylate (HEMA) and polystyr- ene beads to prepare porous biomaterials. Polystyrene (PS) beads were prepared by a solvent-evaporation technique described by Kenteprozidou and Kiparissides and modified as previously described (Stancuet al., 2007). All reagents were purchased from Sigma-Aldrich (St Quentin-Fallavier, France).
Four diameters of PS beads (500, 850, 1160 and 1560μm) were sieved and used as porogens. The mean size of the beads was verified by scanning electron microscopy. The PS beads were placed into 10-mL polyethylene test tubes which were filled with HEMA. Polymerization of the monomer was performed using benzoyl peroxide (0.125% w/v) and N,N- dimethyl-p-toluidine (0.03% w/v). Blocks were totally poly- merized after 24 h at room temperature; they were removed from the molds and transferred in a bath of dichloromethane (solvent of PS beads and nonsolvent for pHEMA matrix). The PS beads were dissolved within 3 days and the porous pHEMA blocks were then dried at 40°C in an oven for 48 h. For each type of porogen bead, a series of five blocks was prepared and analysed; quantitative measurements were averaged.
Microcomputed analysis (microCT)
A Skyscan 1172 microCT (Bruker MicroCT, Kontich, Belgium) was used. The porous cylinders were fixed on the sample holder with plasticine and analysed at a magnification of 26× (one pixel corresponding to 11.41μm) at 69 kV and 100μA.
Reconstruction of the sections from the projection images was done with the NRecon software (Bruker) and morphometric analysis with CtAn (release 1.13.5.1, Bruker) after global thresholding. On the 2D section images, a square region of interest was drawn. It was reported on all the 2D sections and reconstruction was done after global thresholding and binarizing the images. The surface of the square was the same for all blocks but the height depended on the number Hof sections in the stack. Such reconstruction provided 3D rect- angular parallelepiped volumes with a constant cross-section surface (side of the square=6160μm) and a slightly variable height (always>7400μm) depending onH. The following parameters were determined on the 3D models reconstructed and imaged with a surface rendering algorithm.
(1) Porosity defined as the fractional amount of the pores in the volume of interest (Po, in %),
(2) mean thickness of the pHEMA material between the pores (Mat.Th, inμm),
(3) mean separation between two pHEMA walls which is an indirect measure of the pore size (Mat.Sp, inμm), (4) mean density of material walls (Mat.N in mm−1),
by ‘cube counting’ method, an extrapolation of classical
‘box-counting’ method described by Kolmogorov in 2D (Russ, 1994).
The nomenclature used to describe the morphometric pa- rameters follows the recommendations of the American Soci- ety for Bone and Mineral Research (Dempsteret al., 2013).
Frontal imaging with porosity vectors
The binarized stack of 2D sections was then transferred to a lab-made software written in Matlab (Math Works, Natick, MA, USA) release 7.10 and illustrated on Figure 1.
On the binarized images, pores were visible in white and the surrounding material in black. For each binarized imageh of the stack, the following algorithm was applied. Considering I(i,j) the matrix of the binarized image h, for each column j, the total number n of white pixel was determined and the maximum and minimumnwere obtained on the whole image. This numbernwas used to produce a new colorized image with a LUT (lookup table) ranging from blue (the smallestnvalue) to red (the highestnvalue). In each column j, all the white pixels were painted with the same value calculated according to the LUT and the subsequent colorized mages were stored for 3D reconstruction with VG Studiomax (Volume Graphics GmbH, Heidelberg, Germany). A ‘frontal plane’ image F(h,j) was produced by reporting the values of all pixels of imageh. The other following images of the stack (for h=1 toH), were analysed using the same procedure and the frontal plane image was finally composed ofjcolumns andH lines (equal to the number of binarized images of the stack).
The frontal plane image was saved with the colorized LUT and with a similar monochromatic grey LUT. This grey image was analysed by the FracLac plugin developed for ImageJ, to obtain the box plot fractal dimension (of the frontal plane image Df) (Rasband, 1997–2014; Karperien, 1999–2013). The grey LUT was designed asymmetric, thus favouring the holes which appear in white because the FracLab plugin only works on grey images. This avoids the ImageJ plugin to transform the coloured image into a grey level image in the manner of a colourblind, giving the same level of grey to blue than red.
Statistical analysis
Statistical analysis was done with Systat 13 (Systat Inc., San Jos´e, CA, USA). Results are expressed as mean±standard error of the mean. Differences between groups were analysed by the Kruskall–Wallis nonparametric analysis variance. Linear re- gression analysis was performed and the Pearson’s coefficient of correlation was determined. A difference was considered as significant whenp<0.05.
h
(A) (B)
(C)
i
j
H
Frontal plane
Fig. 1. Principle of the algorithm used here. (A) A virtual parallelepiped of material containing cylindrical holes was prepared by overimposingHidentical 2D cross sections (ofilines andjcolumns) and containing three holes. Each image is figured as a thin slab. Vectors, projected on the first imagehare colorized according to the number of pixels of the pores; the frontal plane is constructed section by section fromh=1 toH(pixels of the first line of the frontal plane are figured). (B) Image of a cross section with coloured vectors, (C) image of the frontal plane composed ofH=250 identical sections.
Results and discussion
Blocks of porous material
The 3-day dissolution period in the solvent completely removed PS and provided cylinders of homogenous networks with spherical pores similar to the diameter of the porogen beads as previously described (Stancu et al., 2007). The 3D models of the different types of blocks are illustrated on Figure 2. MicroCT evidenced the different types of porosity oc- curring with the various types of beads. The image of a frontal plane prepared on a simplified model appears on Figure 1, whereas typical frontal plane images from the different groups of porous blocks are illustrated on Figure 3. Because the size of the square used to compute the volume of interest was kept constant for all the analysed blocks, both the morphological aspect and the quantitative results obtained on the frontal plane images of the different groups are comparable. For the porous scaffold obtained using the smallest porogen beads, the frontal plane image appeared relatively uniform, presenting numerous small domains with densely packed pores (red areas) and thick areas of polymer material (blue areas). Increasing the diameter of the pores was associated with an increase of the microarchitectural heterogeneity, with large pores coexisting with/separated by large areas of dense material (Fig. 3). The porogen spheres in the polymer were randomly stacked and their characteristics vary from
the well-known arrangements such as the centred cubic and the face-centred cubic stackings described for the atom lattices (Pavlidis & Lathouwers, 2013). An explanation could be that the irregular arrangement may depend of the small variation in size of the beads themselves, even if the sieving was done carefully. This is also a plea to use fractal geometry to measure some characteristics of these random beds of spheres.
The heterogeneity of the pores is especially important in bone pathology where an increased altered microarchitecture (with increased porosity) is responsible for decreased biomechanical properties leading to bone frailty (Mulleret al., 2014). Fractal geometry has been little used in the characterization of bioma- terials. Porosity has been calculated by the box-plot method on scanning electron microscopic images (SEM) of highly porous polyε-caprolactone scaffolds (Guarinoet al., 2010). However, the high depth of focus of SEM prevents accurate dimensional measurements excepted if stereopairs images are used (Minnichet al., 1999). When porosity is evenly distributed, the use of microCT slices allows accurate analysis with Euclidean or fractal algorithms (N’Diayeet al., 2013; N’Diaye et al., 2014).
Quantitative analysis
Porosity significantly decreased as the pore diameter increased (p=0.006) (Fig. 4). On the contrary Mat.Th increased signif- icantly when pores were larger (p<0.0001) but the density
Fig. 2. MicroCT reconstruction of the four types of porous pHEMA blocks (A: 500, B: 850, C: 1160 and D: 1560μm).
Fig. 3. Images of the frontal planes made from the different types of porous pHEMA blocks prepared with PS beads (A: 500, B: 850, C: 1160 and D: 1560μm). The regions with maximal porosity are red, minimal porosity in blue according to the LUT.
of the material separation walls between the pores decreased simultaneously (p=0.004 for Mat.N). Mat.Sp increased in parallel of the pore size but no difference among groups could be evidenced. Very similar findings have been reported for bone microarchitecture and high correlations are reported be- tween trabecular thickness and texture analysis parameters determined on plain X-ray images of the bones by fractal or Euclidian algorithms (Lespessailleset al., 1998; Guggenbuhl et al., 2006; Pothuaudet al., 2008; Le Corrolleret al., 2013;
Ollivieret al., 2013). D3Dwas clearly increased as a function
of the bead size (p<0.0001) and so was Df(p=0.04). The coloured image of the frontal plane appears as an easy mor- phological method to characterize the complexity of the porous structures. Dfwas significantly correlated with D3D(r=0.67;
p=0.001) (Fig. 4C).
When the pseudo-coloured sections were reconstructed with a volume rendering program, the aspect is ill-defined with this series of porous blocks because the material is visi- ble in black and the depth of field alters the visual perception.
However, this possibility is interesting if the porous material
○ D3D Fractal dimension ● Df Fractal frontal plane
Size of PS beads
500 850 1160 1560 1.2
1.3 1.4 1.5
2.2 2.3 2.4 2.5 2.6
(B)
● Po (Porosity, in %)
Size of PS beads
○ Mat.Th (Thickess, in μm)
500 850 1160 1560
60 70 80 90
0 50 100
(A)
2.2 2.3 2.4 2.5 2.6 2.7
1.2 1.3 1.4 1.5 1.6
D3D Fractal dimension Df Fractal frontal plane
(C)
Fig. 4. (A) Evolution of porosity (l Po) and thickness (◦Mat.Th) of the pHEMA separation walls between the PS beads in the different types of porous blocks. (B) Fractal dimensions (l Dfand◦D3D) in the different blocks. (C) Linear correlation between Dfand D3Din the whole series of blocks (r=0.67, y=0.39×x+0.48,p=0.001).
Fig. 5. Application of the algorithm to bones samples from a control rat (A) and a rat with a disuse-induced bone loss after a botulinum toxin injection (C). The frontal plane is replaced by the anterior periosteal surface of the bone. The regions with maximal porosity are red, minimal porosity in blue according to the LUT. The corresponding 3D models of the inner structure of the trabecular bone appear for the control (B) and BTX rat (D). Porosity is respectively 58.9% and 78.8%; D3Dis respectively 2.391 and 2.691 and Dfis respectively 1.437 and 1.581.
is outlined by a continuous layer. In this case, the pixels of the boundary have the value of the hidden porosity within the object. An interesting application of the method, to vali- date its efficiency, is to study the inner porosity of bone from animal models of bone loss. Therefore, Figure 5 illustrates the projected porosity of the secondary spongiosa on the periosteal surface of the femur cortex. Rats having received a single injec- tion of botulinum toxin in the quadriceps develop paralysis and a massive bone loss due to disuse within one month (Chappard et al., 2001; Marchand-Liboubanet al., 2013). The developed method clearly evidences the differences between control and
treated animals. This technique will be used in the future to evaluate the increase in bone porosity in other types of bones having a more complex shape.
Mapping cortical thickness at the surface of the cortical bone has also been presented as a visualization of cortical thin- ning in different areas of the upper femoral extremity (Treece et al., 2010; Pooleet al., 2012). In the present study we also found that the complexity of the projection image evaluated by Df was well correlated with D3D measured on the whole volume of interest and these findings prove the importance and potential of this method. To conclude, the complexity of
precursors) in a material reveals marked differences in produc- ing highly heterogeneous regions (large pores coexisting with dense separation walls) when large beads of porogen are used.
Acknowledgements
This work was made possible by grants from Contrat Region Pays de la Loire: Bioregos2 program (France) and PNII-PCCA- 183/2012 (Romania). Many thanks to Mrs. Lechat for secre- tarial assistance.
Conflict of Interest
Authors have no conflict of interest.
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