Mathematical modelling of the distribution of newly formed bone in bone tissue engineering

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Biomaterials 26 (2005) 6788–6797 www.elsevier.com/locate/biomaterials

Mathematical modelling of the distribution of newly formed bone in bone tissue engineering Laurent Pothuauda, Jean-Christophe Fricaina,b, Stephane Pallua, Reine Bareillea, Martine Renardc, Marie-Christine Durrieua, Michel Dardd, Michel Vernizeaue, Joelle Ame´de´ea, a

INSERM U577, Universite´ Victor Segalen Bordeaux 2, 33076 Bordeaux Cedex, France b UFR Odontologie, Universite´ Victor Segalen Bordeaux 2, Bordeaux, France c CIT, CHU de Bordeaux, Bordeaux, France d Biomet-Merck Biomaterials, Darmstadt, Germany e Biomet-France, Valence, France Received 24 September 2004; accepted 11 April 2005 Available online 13 June 2005

Abstract New bone formation in bone substitutes is usually investigated by histomorphometric global analysis. This study provides a novel mathematical modelling approach of new bone formation in the use of osteoinductive and functionalized biomaterials for bone tissue engineering. We discuss here the repartition and the probability to get new bone formation inside Biphasic Calcium Phosphate (BCP) loaded with autologous osteogenic cells, functionalized with a cyclo RGD peptide, after implantation in rabbits for 2 and 4 weeks. This local analysis allowed us to complement classical global findings and to demonstrate that after 2 weeks of implantation, the probability of new bone formation was significantly higher in RGD-grafted BCP and that new formed bone was largely distributed from the edge to the centre of the implant. While no significant differences were obtained after 4 weeks of implantation between RGD-grafted and non-grafted materials, distribution of new bone formation inside RGD-grafted materials was significantly more homogeneous as demonstrated by our mathematical modelling approach. In conclusion, local analysis of new bone formation inside macroporous substitutes coupled with mathematical modelling constitutes a potential quantitative approach for the evaluation of the osteoconductive and osteoinductive characteristics of such biomaterials. r 2005 Elsevier Ltd. All rights reserved. Keywords: RGD peptide; Animal model; Image analysis; Modelling

1. Introduction With advances in understanding tissue–material interactions [1,2] and bioengineering, several strategies can be exploited to develop efficient bone substitutes, based on macroporous biomaterials, when they are associated with stem cells [3,4] or osteoinductive factors [5]. The application of relevant exploration methods to analyze the amount of new bone formation in such Corresponding author. Tel.: +33 557571737; fax: +33 556900517.

E-mail address: [email protected] (J. Ame´de´e). 0142-9612/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.biomaterials.2005.04.002

biomaterials is absolutely required to evaluate their osteoconductive and osteoinductive properties. The bone defect treatment usually requires the use of bioactive materials such as calcium carbonate [6,7], hydroxyapatite [8], bioglass [9], tricalcium phosphate [10,11], or biphasic ceramics of hydroxyapatite and b-tricalcium phosphate [12,13]. These materials are biocompatible and have osteoconductive properties because they serve as a scaffold for osteoblastic cells [3]. However, none of these materials have osteoinductive properties like autograft which is still the reference process for defect healing. While autogeneous bone

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grafts have drawbacks such as morbidity of the second operation and restricted quality of material. For these reasons, osteoblastic cells or growth factors like Bone Morphogenic Protein have been associated with calcified bone substitute to make osteoconductive material [3,4,14]. One difficulty of this approach is to have a good material autologous bone cells adhesion, colonization and differentiation [15]. To promote material colonization and cells biosynthetic activity, different solutions have been proposed such as perfusion culture [16] or grafting RGD peptide onto the material [17]. RGD peptide is a sequence common to many of the adhesive matrix molecules. So, this sequence is the most widely recognized by cells. In previous studies, Pallu et al. [17] and Verrier et al. [18] have shown that cyclo-DfKRG increase in vitro bone marrow stromal cells adhesion, differentiation and mineralization through activation of different Kinases. Furthermore, Porte´-Durrieu et al. [19,20] have shown that cyclo-DfKRG peptide grafting onto Ti–6Al–4 V or hydroxyapatite increased in vitro osteoprogenitor cells adhesion. In vivo, RGD peptide increases bone formation and contacts in alveolar bone, femur and tibia [21–23]. Moreover, whereas literature exhaustively described the use of RGD-containing peptides to promote osseointegration [24], tissue colonization inside macroporous implant and peri-implant new formed bone were almost always evaluated by standard histomorphometric measurements [12,21,25–27] which give mainly access to mean and global evaluation. Nevertheless, according to new bioengineering strategies, local bone formation inside the macropores in straightened contact with the materials should also be extended to increase our knowledge of the benefits of these strategies in terms of osteoconductive and osteoinductive properties, but nowadays no specific evaluation method has been developed. In the present study, specific interest has been devoted to the development and use of a new local evaluation method based on image processing applied to histomophometric data of bone formation inside a Biphasic Calcium Phosphate macroporous material cellularized with autologous osteogenic cells, grafted or not with a cyclic RGD peptide. This histomophometric-based approach gives access to new characteristics of bone formation in macroporous implants, such as the distribution of new formed bone inside the macropores, as well as the repartition of the new formed bone according to the distance at the external edge of the implant and at the internal surface. Furthermore, a mathematical model has been proposed for fitting the distribution of new formed bone inside the macropores. This model permits to quantify the homogenization of the bone formation inside the material, which was compared between grafted and non-grafted materials.

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2. Materials and methods 2.1. Preparation of biomaterials Materials were composed of 40% hydroxyapatite and 60% b-tricalcium phosphate and were shaped as cylinders with a length of 10 mm and a diameter of 6 mm (BIOCETIS, Boulogne/mer, France). The porosity of these materials was 75710% with pore size of 4507100 mm. These materials were first functionalized by covalent grafting with a cyclo-RGD peptide (cyclo-DfKRG) (BiometMerck Biomaterials, Darmstadt, Germany) according to Porte-Durrieu et al. [19]. Surface modification was carried out in a dry and air-free chamber in order to avoid surface contamination by water and carbon compounds from the surrounding atmosphere and hence to ensure reproducibility and stability of the biomolecule covering. The strategy of peptide immobilization involves: (i) grafting of an aminofunctional organosilane (APTES) onto the surface of material; (ii) substitution of the terminal amine for a hetero-bifunctional cross-linker SMP in order to; (iii) react the ‘‘outer’’ maleimide group with a peptide, thanks to the thiol group in the mercapto group [19]. The distribution of the peptide was homogeneous due to covalent grafting. Such grafting gives better spatial distribution, reproducibility, and homogeneity compared to simple adsorption. This grafting homogeneity has been previously checked by X-ray photoelectron spectroscopy for different types of substrate [28]. In addition, grafted and non-grafted materials were loaded with autologous rabbit osteoprogenitor cells arising from bone marrow stromal cells. Osteoprogenitor cells were isolated from rabbit marrow stroma cells according to a previously described methodology [29,30]. Rabbit bone marrow was obtained by aspiration from the iliac crest rabbit. Cells were separated into single suspension, centrifuged, incubated in a humidified atmosphere, isolated, and then cultured for 1 week onto both the grafted and non-grafted materials before implantation. 2.2. Experimental animal model Two groups of six New Zealand rabbits were randomly identified, cared for according to the European Guidelines for the care and the use of laboratory animals (Directive 24/11/86, 86/ 609/CEE) and respectively programmed for histomorphometry based evaluation after 2 and 4 weeks of implantation (W2, W4). Recommendations of surgical procedures concerning animal implantation were respected [31]. The animals were anaesthetized before implantation. Knee joints of both sides were exposed through a lateral incision. A pre-hole was drilled in the rabbit condyle, and increased with growing diameter wicks up to 7 mm. For each animal, left and right condyles were respectively implanted with cellularized RGD-grafted and cellularized nongrafted biomaterials. At W2 or W4, animals were sacrificed and the condyle pieces including biomaterials were removed and fixed in formalin (10%) for histological procedure. 2.3. Histological method The condyle samples were cut longitudinally in two parts with the use of a manual saw. The samples were washed, dehydrated

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in acetone and then infiltrated and embedded in a methyl metacrylate–butyl metacrylate solution according to the Wolf’s technique [32]. Thin sections were cut using a Young microtome K equipped with a diamond saw, and then coloured with the light green colouring of the Masson’s trichrome. The mean numbers of exploited thin sections were 4.071.3 for the 2-week samples, and 3.270.9 for the 4-week samples. 2.4. Digitization procedure The green coloured thin sections were digitized by using a dedicated high-resolution scanner (NIKON, Super Coolscan 4000). A non-interpolated resolution of 0.16 mm/pixel was used, and all the other options of digitization were inhibited (Fig. 1a). 2.5. Algorithm development All the algorithms of image processing were developed in C-writing code (Microsoft Developer Studio, Microsoft

Corporation), and specific scripts were used in order to automatically analyze the overall set of images. 2.6. Global analysis The Region of Interest (ROI) was manually selected such as it fitted the circumference of the implant. Then, a binary mask was created representing the surface of the ROI (Fig. 1b). This surface, SROI, was measured by counting of pixels. A model of colorimetry (LAB-model) was employed in order to decompose the initial colour (RGB) image into three grey level imagecomponents (VISIOLAB, BIOCOM, France): (L)—luminosity; (A)—component encoding the initial colour from green to red (Fig. 1c); (B)—component encoding the initial colour from blue to green. While the calcified tissue was initially coloured in green, surface representing the bone tissue, SBONE, was obtained by thresholding applied to the component (A) of the previous model (Fig. 1d). The initial colour image was converted into an 8-bit grey level image (Fig. 1e), in which the contrast between the material (dark grey levels) and the other parts (pore space, bone) was sufficient to extract the surface of the material, SMAT, by global thresholding (Fig. 1f). Finally, the surface of the pore space inside the implant, SPORE, was evaluated as follows: S PORE ¼ SROI  S MAT . The mean and standard deviation of the global bone formation ratio, F ¼ 100 SBONE =SPORE (expressed in %), were calculated on all the samples (several images per sample) at a particular time (W2, W4). 2.7. Local analysis In addition, the pore space of the implant was segmented into individual pores by using a semi-automatic approach based on the technique of the distance map [33]. This local analysis permitted the evaluation of the probability distribution, P( f ), of local bone formation ratio ( f—expressed in %) inside each pore. Then, this probability distribution was characterized by its first (m) and second (s2 ) moments, or theoretical mean (m) and standard deviation (s ): Z m¼ f Pð f Þ df ð%Þ, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Z s¼

ð f  mÞ2 Pð f Þ df ð%Þ.

ð1Þ

The mean probability distribution was evaluated on all the samples (several images per sample) at a particular time (W2, W4). 2.8. Mathematical modelling At the start of the bone formation process, all the pores are empty, meaning that the probability to find an empty pore is maximal: Fig. 1. Green coloured thin section (a) of bone ingrowth inside macroporous hydroxyapatite/b-tricalcium phosphate biomaterial, 4 weeks after implantation in the femoral condyle of rabbit. This material was functionalized by cyclo-DfKRG peptide grafting and cellularized with autologous osteoprogenitor cells. The initial image was segmented into three components (binary mask images): the total ROI component (b), that was manually selected; the new formed bone component (d) obtained after use of a specific colorimetry model (c); and, the material component (f) obtained after grey level conversion of the initial image (e).

Pð0Þ ¼ 1, Pð f Þ ¼ 0 8f 40.

ð2Þ

As soon as new formed bone appears inside the pore space, the probability distribution evolutes like a decreased exponential function :   f Eð f Þ ¼ k1 exp  (3) k2

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characterizing that:

2.10. Statistical analysis

 

The statistical evaluation was performed using S-PLUS 2000 (Mathsoft Engineering & Education Inc., Cambridge, MA, USA). The statistical differences between RGD-grafted and non-grafted samples were evaluated by using paired Wilcoxon’s signed rank test, in pairing both grafted and nongrafted samples of the same animal (left/right sides). Statistical differences between W2 and W4 were evaluated by using the (unpaired) Mann–Whitney U test. The statistical differences between probability distributions, P( f ), at the same investigated time, were evaluated by applying paired Wilcoxon’s signed rank test to the functions f Pð f Þ.

the probability to find an empty pore decreases fast; the probability to find a local bone formation ratio f is a decreased function of f.

The final theoretical state of the bone formation process would be characterized by a specific local bone formation ratio f 0 in each pore, or more realistically P( f ) would be characterized by a Gaussian distribution:   ! f f0 2 Gð f Þ ¼ k3 exp 0:5 . (4) k4 In this study, we have hypothesized that the probability distribution P( f ) , characterizing the bone formation process, could be fitted to the following model:     ! f f f0 2 P ðf Þ ¼ k1 exp  þ k3 exp 0:5 k2 k4 0

3. Results 3.1. Global analysis

(5)

where k1, k2, k3, and k4 are the fit-coefficients. This mathematical modelling was performed by using Sigmaplot software (SPSS Inc., Chicago, IL, USA), and the coefficients (k1, k2, k3, k4) were evaluated following a leastsquare fit approach. The percentage weight (G%,) of the Gaussian component was then evaluated as follows: R Gð f Þ df R G% ¼ 100 ð%Þ. (6) P0 ð f Þ df

Classical global findings (Table 1) revealed that new bone formation was higher in RGD-grafted BCP after 2 weeks (F ¼ 21:3 5:7%) when compared to nongrafted materials (F ¼ 12:3 5:0%) (p ¼ 0:02), whereas no significant differences occurred between these two materials after 4 weeks of implantation. Between W2 and W4, bone formation increased significantly with non-grafted samples (F ¼ 12:3 5:0% versus F ¼ 21:1 5:2%; p ¼ 0:03), while no significant change appeared between W2 and W4 in the case of RGDgrafted samples.

2.9. Distance analysis

3.2. Local analysis and mathematical modelling

The repartition of new formed bone, R(di) expressed in %, in function of the distance at the edge of the implant (di), was evaluated as the ratio of the elementary surfaces SBONE(di) and SROI(di) located at the particular distance di from the edge of the implant:

At W2, 50% of the pores were empty (without any new bone element) with non-grafted samples, while only 20% of the pores were empty with RGD-grafted samples (Fig. 2a). The probability to find a local bone formation (inside a pore) higher than 10% was more important with RGD samples than with non-grafted materials. The mean and standard deviation values (m s) calculated from the probability distribution P( f ) were 18.972.6% with RGD-grafted samples versus 10.471.1% with non-grafted samples (p ¼ 0:02). At W4, roughly 30% of the pores were empty with both groups of samples and the probability to find a particular local bone formation ratio ( f ) was similar with both materials (Fig. 2b), with no significant difference (m s): 15.972.3% with RGD-grafted samples versus 18.471.9% with non-grafted samples (p ¼ 0:2). The percentage weight of the Gaussian function component (G%) was higher at W2 (Fig. 3a) with RGD-grafted materials than with non-grafted samples. The most probable ratio of local bone formation inside a pore ( f0) (in relation to the Gaussian component) was not well stabilized with both groups of samples (Fig. 3b). At W4 (Figs. 3c and d), G% remained

Rðd i Þ ¼ 100

SBONE ðd i Þ ð%Þ. SROI ðd i Þ

(7)

The normalization by the elementary surface of ROI, SROI(di), was made in order to take into account the geometry of the sample. The mean repartition function was evaluated on all the samples (several images per sample) at a particular time (W2, W4). Similarly, the repartition of new formed bone, R(dp) expressed in %, in function of the distance at the edge of the pore (dp), was evaluated as the ratio of the elementary surfaces SBONE(dp) and SROI(dp) located at a particular distance dp from the interface of the material: Rðd p Þ ¼ 100

SBONE ðd p Þ SROI ðd p Þ

ð%Þ.

(8)

The normalization by the elementary surface of ROI, SROI(dp), was made in order to take into account the geometry of the pore space. The mean repartition function was evaluated on the whole of samples (several images per sample) at a particular time (W2, W4).

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W2

W4

p(W2/W4)

RGD

12.375.0 (n ¼ 5) 21.375.7 (n ¼ 5) p ¼ 0:02

21.175.2 (n ¼ 6) 18.172.7 (n ¼ 6) ns (0.2)

p ¼ 0:03

+RGD p(7RGD)

ns (0.3)

Mean and standard deviation values of the global bone formation ratio (F), 2 weeks (W2) and 4 weeks (W4) after implantation with (+RGD) and without (RGD) cyclo-DfKRG peptide grafting.

distribution of new bone formation was more homogeneous in RGD-grafted materials than in non-grafted samples as demonstrated by the Gaussian component. A decreased relationship was observed between local bone formation inside a pore and the surface of this pore (available space, s; Fig. 4) ranging from about 40% (or more) for small pores ( 500 pixel surface) to lower than 20% for smallest pores ( 2500 pixel surface). 3.3. Distance analysis At W2, the bone formation was higher with RGDgrafted samples than with non-grafted samples, whatever the distance at the edge of the implant (Fig. 5a) even in the centre of the implant. The gradient of bone formation was decreased from the edge of the implant to its centre. This profile was similar with both grafted or non-grafted samples. At W4, the bone formation seemed comparable with both groups of samples, except close to the edge of the implant where the gradient of bone formation seemed higher with non-grafted samples (Fig. 5b). Moreover, an involution of the peripheral new formed bone at W2 and W4 with RGD-grafted materials was observed. Anyway, bone formation was always higher at the periphery of the implant. Finally, new bone formation was also evaluated from the edge of the pore to its centre. Most of the new formed bone was generated starting from the surface of the material, with a centripetal gradient of bone formation inside the pore. At W2 (Fig. 6a) the bone formation was higher with RGD-grafted materials than with non-grafted samples, whatever the distance at the edge of the pore. At W4 (Fig. 6b) the bone formation was similar with both groups of materials, whatever the distance at the edge of the pore.

4. Discussion Fig. 2. Mean probability distribution P( f ), expressing the probability to find a local bone formation ratio ( f ) inside a pore, evaluated on the whole of samples for non-grafted (RGD) and grafted (+RGD) materials after 2 weeks (W2) and 4 weeks (W4) of implantation. The mean (m) and standard deviation (s) were evaluated from the first and second moments of the corresponding probability distribution, respectively.

significantly higher with RGD-grafted materials, and interestingly, f0 was almost stabilized on the whole of RGD-grafted samples ( f 0 ¼ 13:8 0:4%, coefficient of variation ¼ 2.9%), while without RGD, this stability seemed not yet reached ( f 0 ¼ 8:6 5:4%, coefficient of variation ¼ 63%). Then, while no significant differences were detected after 4 weeks of implantation with both RGD-grafted BCP and non-grafted materials,

In this study we have evaluated the in vivo effects of the grafting of cyclo-DfKRG in the early phase of integration of biomaterials made with macroporous hydroxyapatite/b-tricalcium phosphate colonized with autologous bone marrow stromal cells, implanted in the femoral condyle of rabbits. These short times were chosen to explore the potential of our new developed local evaluation method to detect early significant differences between these two groups of materials whereas most of the papers described effectiveness of RGD coatings only after 2 or 3 months [21,34]. Two weeks after implantation, the bone formation inside the implant was higher with the grafted samples (+RGD) than with the non-grafted samples (RGD), with a similar centripetal gradient of cicatrization decreasing, starting from the edge of the implant until its centre. These data suggested that incorporation of RGD

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Fig. 3. Mathematical modelling of the probability distribution P( f ) in relation to non-grafted (RGD) and grafted (+RGD) samples, after 2 weeks (W2) (a,b) and 4 weeks (W4) (c,d) of implantation. G% expressed the percentage weight of the Gaussian component G( f ), and f0 its central value (most probable local bone formation ratio).

peptides in calcium phosphate ceramics is efficient on new bone formation soon after implantation and that RGD peptides act in vitro in stimulating cell adhesion up to 3 h [19]. Four weeks after implantation, there remained no significant difference between the two groups studied (7RGD), while an involution of the peripheral new formed bone was observed between 2 weeks (W2) and 4 weeks (W4) after implantation in the case of +RGD samples. Most of the new formed bone was generated starting from the surface of the material, with a centripetal gradient of cicatrization inside the pore, decreasing starting from the edge of the pore until its centre. These data were completed by a local analysis of bone formation ratio inside the macropores of the implant. The probability distribution P( f ) was fitted as a summation of an exponential component E( f )— representing the invasion of the macropores by the new formed bone—and a Gaussian component G( f )— representing the homogenization of the bone formation inside the macroporous implant (Fig. 2). Mathematical modelling was used in order to exploit the local analysis performed inside each pore of the macroporous implant, and has shown an accelerated homogenization of the ratio of local bone formation inside the pores in the case of +RGD samples. This mathematical modelling has

argued the assumption that peptide grafting should therefore increase the early osteoconduction by showing an homogenization of the ratio of local bone formation in the case of grafted samples (Fig. 3). A similar evolution, but slowed down, could concern the case of non-grafted materials. The surface characteristics of biomaterials are important factors conditioning their biocompatibility and bioactivity in the bone repair process. There are many possibilities and variations to improve these characteristics of surface, by using surface coating techniques, and/or by loading the surface with osteoblastic cells or growth factors. RGD peptide grafting has been widely studied in vitro, while only little is known on its in vivo effects. Ferris et al. [22] have evaluated the quality and quantity of the new formed bone in response to titanium rods coated with the peptide sequence Arg–Gly– Asp–Cys (RGDc) and implanted in rat femurs. They have shown a thicker shell of new formed bone around RGD-modified implants after 2 weeks of implantation compared to control implants without RGD coating. Schliephake et al. [21] have tested the interest of RGD peptide coating of titanium materials implanted in dog alveolar crests. They observed only a weak evidence that RGD coating on titanium increases bone formation.

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Fig. 4. Mean local bone formation f ðsÞ in a pore in function of the available space of this pore, evaluated on the whole of samples for non-grafted (RGD) and grafted (+RGD) materials after 2 weeks (W2) and 4 weeks (W4) of implantation.

These two studies [21,22] have investigated periimplant bone formation and bone/implant contact around compact titanium materials. In our study, we have investigated macroporous implants colonized with autologous bone marrow stromal cells because such biomaterials have proved their ability to improve both osteoconductive and osteoinductive characteristics [35,36]. Several interesting data could be underlined from our study. We have shown earlier the difference between grafted (+RGD) and non-grafted (RGD) materials when compared to the other in vivo studies carried out with RGD-coated materials. At W2, bone formation was higher in the +RGD samples whatever the distance to the edge of the implant or to the edge of the pore. At W4, only the mathematical modelling showed a homogenization of the ratio of local bone formation in case of grafted samples (+RGD). Both groups being cellularized, this could be due to the higher activities of osteoprogenitor cells in contact with the grafted peptide, as it was previously shown in vitro [20]. In fact, the efficiency of RGD peptide grafting has been demonstrated by measuring the adhesion between 1 and 24 h of osteoprogenitor cells isolated from human bone marrow stroma cells. The osteoprogenitor cells were seeded at a density of 2 104 cell/m2 on each substrate (grafted and

Fig. 5. Mean bone repartition R( f ), expressing the percentage of new formed bone (% of occupied ROI surface) in function of the distance at the edge of the implant (di), evaluated on the whole of samples for non-grafted (RGD) and grafted (+RGD) materials after 2 weeks (W2) and 4 weeks (W4) of implantation.

non-grafted). There were about two times more cells in grafted substrates after 24 h compared to non-grafted substrates. This in vitro study has demonstrated that there were more viable cells in presence of RGD peptide, RGD peptide favouring cell attachment and promoting cell differentiation also [20]. The osteogenic cell distribution in our material should promote bone formation in almost all the pores, particularly with RGDgrafted samples for which cells are supposed to be highly adherent. The local analysis applied to the macroporous material has shown that with peptide grafting, new formed bone was mainly in contact with the surface of the implant, which revealed the osteoconductive properties of the biomaterial used. However, bone formation was always higher at the periphery of the implant. In fact, this effect could be explained like the superposition of two dissociated processes: due to osteogenic cells loaded onto the surface of the biomaterial at the side of the bone defect; and due to osteoblastic cells and extracellular matrix at the side of adjacent bone. A similar evolution of new bone formation occurred with the RGD-non grafted samples but 2 weeks later (W4). The data reported in Table 1 concern the global bone formation ratio, which is the ratio of the new formed

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Fig. 6. Mean bone repartition R( f ), expressing the percentage of new formed bone (% of occupied ROI surface) in function of the distance at the edge of the pore (dp), evaluated on the whole of samples for nongrafted (RGD) and grafted (+RGD) materials after 2 weeks (W2) and 4 weeks (W4) of implantation.

bone area and the pore area, as measured in twodimensional (2D) thin slices. This ratio cannot be higher than the biological ratio, generally known as ‘‘bone porosity’’. For example, a ratio ranging from 12% to 22% is coherent with what is generally observed in human bone. Our developments are based on the evaluation of local bone formation inside each pore from a 2D thin slice. Based on stereological considerations, it is well known that such an evaluation is correlated to the true bone formation in the 3D pore space, although not exactly the same in absolute value. The decreased relationship observed between local bone formation and available pore space (Fig. 4) could only be an effect of the 2D evaluation. Only 3D imaging techniques, such as synchrotron radiation-based imaging, for example [37], would permit access to and characterization of the true pore space without any cutting effect. The global results (Table 1) as well as the repartition of new formed bone (Figs. 5 and 6) relate an involution process at W4 compared to W2. This involution process is mainly observed at the periphery of the implant (Fig. 5) and should be due to an accelerated bone formation process at W2 induced by edge effects with

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pre-existing outside bone. Hence, the potential additional interest of the mathematical modelling approach would be to characterize the homogenization of bone formation process without any quantification artefact due to the existence of a peripheral involution effect. The main limitation of this work is that the short time of investigation does not allow one to clearly verify these statements. Hence, it is now necessary to complement these preliminary results with additional experiments taking into account longer time of investigations. In the present work, we have focussed on short investigation times in order to explore the effect of osteoprogenitor cells loaded on grafted or non-grafted materials. This effect had more chance to be observed in a short time, such as it has been observed in the present study. As a consequence, the main limitation of this work is the lack of longer investigation times that would permit a better conclusion concerning the evolution of the bone formation process in our experimental model. Nevertheless, the preliminary results obtained in the present work are encouraging and have demonstrated the interest of our new local quantification approach for bone repair study in macroporous implants. Furthermore, the exploitation of mathematical modelling has led to higher level of bone formation process characteristics, concerning in particular the homogeneity of new formed bone distribution inside the macroporous material. This quantification is based on implicit hypothesis linked to the validity of the mathematical model used (Eq (5)). This hypothesis needs to be verified now from larger sets of samples and for a longer time of observation. Numerical simulation would also constitute an additional tool to evaluate the field of validity of this mathematical model. However, the preliminary finding that bone formation would be distributed homogenously around a specific ratio value (13.870.4%, for grafted samples) constitutes a very interesting point. Further investigations should consist in studying the relationship between this ratio value and the structure characteristics of the material used (porosity, pore size distribution, pore connection, etc.).

5. Conclusions With bioengineering technologies [1,2,38,39], the contribution of sophisticated imaging and image analysis techniques is essential to evaluate the various components intervening in the manufacturing of hybrid artificial bone substitutes. In this study, we have demonstrated the potential of global and local quantification tools based on the morphological histomorphometry approach. Global analysis gives access to mean bone formation ratio on the overall surface studied and only general trends can be concluded from such global evaluation approach. On the contrary, local analysis

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gives access to more precise characteristics such as already been demonstrated for the evaluation of bone structure [40]. In the case of bone formation inside biomaterials, local analysis allows one to evaluate the distribution of local ratio of bone formation as well as the repartition of the new formed bone in contact with the material surface. These local characteristics can be related to the osteoconductive and osteoinductive properties of biomaterials, and would constitute potent quantitative information for tissue engineering. Furthermore, the modelling approach based on statistical and mathematical techniques applied to local characteristics could permit the development of more efficient prognostic tools for tissue engineering, in both ex vivo histomorphometry-based investigations or non-invasive magnetic resonance relaxometry applications [41].

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