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Biomaterials 26 (2005) 6788–6797

www.elsevier.com/locate/biomaterials

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

Laurent Pothuauda, Jean-Christophe Fricaina,b, Stephane Pallua, Reine Bareillea,Martine Renardc, Marie-Christine Durrieua, Michel Dardd,

Michel Vernizeaue, Joelle Amedeea,�

aINSERM U577, Universite Victor Segalen Bordeaux 2, 33076 Bordeaux Cedex, FrancebUFR Odontologie, Universite Victor Segalen Bordeaux 2, Bordeaux, France

cCIT, CHU de Bordeaux, Bordeaux, FrancedBiomet-Merck Biomaterials, Darmstadt, Germany

eBiomet-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–materialinteractions [1,2] and bioengineering, several strategiescan be exploited to develop efficient bone substitutes,based on macroporous biomaterials, when they areassociated with stem cells [3,4] or osteoinductive factors[5]. The application of relevant exploration methods toanalyze the amount of new bone formation in such

e front matter r 2005 Elsevier Ltd. All rights reserved.

omaterials.2005.04.002

ing author. Tel.: +33557571737; fax: +33 556900517.

ess: joelle.amedee@bordeaux.inserm.fr (J. Amedee).

biomaterials is absolutely required to evaluate theirosteoconductive 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 andb-tricalcium phosphate [12,13]. These materials arebiocompatible and have osteoconductive propertiesbecause they serve as a scaffold for osteoblastic cells[3]. However, none of these materials have osteoinduc-tive properties like autograft which is still the referenceprocess for defect healing. While autogeneous bone

ARTICLE IN PRESSL. Pothuaud et al. / Biomaterials 26 (2005) 6788–6797 6789

grafts have drawbacks such as morbidity of the secondoperation and restricted quality of material. For thesereasons, osteoblastic cells or growth factors like BoneMorphogenic Protein have been associated with calci-fied bone substitute to make osteoconductive material[3,4,14]. One difficulty of this approach is to have a goodmaterial autologous bone cells adhesion, colonizationand differentiation [15]. To promote material coloniza-tion and cells biosynthetic activity, different solutionshave been proposed such as perfusion culture [16] orgrafting RGD peptide onto the material [17]. RGDpeptide is a sequence common to many of the adhesivematrix molecules. So, this sequence is the most widelyrecognized by cells. In previous studies, Pallu et al. [17]and Verrier et al. [18] have shown that cyclo-DfKRGincrease in vitro bone marrow stromal cells adhesion,differentiation and mineralization through activation ofdifferent Kinases. Furthermore, Porte-Durrieu et al.[19,20] have shown that cyclo-DfKRG peptide graftingonto Ti–6Al–4V or hydroxyapatite increased in vitroosteoprogenitor cells adhesion. In vivo, RGD peptideincreases bone formation and contacts in alveolar bone,femur and tibia [21–23].Moreover, whereas literature exhaustively described

the use of RGD-containing peptides to promoteosseointegration [24], tissue colonization inside macro-porous implant and peri-implant new formed bone werealmost always evaluated by standard histomorpho-metric measurements [12,21,25–27] which give mainlyaccess to mean and global evaluation. Nevertheless,according to new bioengineering strategies, local boneformation inside the macropores in straightened con-tact with the materials should also be extended toincrease our knowledge of the benefits of these strategiesin terms of osteoconductive and osteoinductive proper-ties, but nowadays no specific evaluation method hasbeen developed.In the present study, specific interest has been devoted

to the development and use of a new local evaluationmethod based on image processing applied to histomo-phometric data of bone formation inside a BiphasicCalcium Phosphate macroporous material cellularizedwith autologous osteogenic cells, grafted or not witha cyclic RGD peptide. This histomophometric-basedapproach gives access to new characteristics ofbone formation in macroporous implants, such as thedistribution of new formed bone inside the macropores,as well as the repartition of the new formed boneaccording to the distance at the external edge ofthe implant and at the internal surface. Furthermore,a mathematical model has been proposed for fittingthe distribution of new formed bone inside the macro-pores. This model permits to quantify the homogeniza-tion of the bone formation inside the material,which was compared between grafted and non-graftedmaterials.

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 10mm and a diameter of 6mm (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) (Biomet-

Merck 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 aminofunc-

tional 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 7mm.

For each animal, left and right condyles were respectively

implanted with cellularized RGD-grafted and cellularized non-

grafted 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 0.16mm/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

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).

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 image-

components (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:

SPORE ¼ SROI � SMAT. The mean and standard deviation of the

global bone formation ratio, F ¼ 100SBONE=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 distribu-

tion, 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, ortheoretical mean (m) and standard deviation (s ):

m ¼

Zf Pð f Þdf ð%Þ,

s ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZð f � mÞ2 Pð f Þdf

sð%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:

Pð0Þ ¼ 1,

Pð f Þ ¼ 0 8f40. ð2Þ

As soon as new formed bone appears inside the pore space,

the probability distribution evolutes like a decreased exponen-

tial function :

Eð f Þ ¼ k1 exp �f

k2

� �(3)

ARTICLE IN PRESSL. Pothuaud et al. / Biomaterials 26 (2005) 6788–6797 6791

characterizing that:

�

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:

Gð f Þ ¼ k3 exp �0:5f � f 0

k4

� �2 !. (4)

In this study, we have hypothesized that the probability

distribution P( f ) , characterizing the bone formation process,

could be fitted to the following model:

P0ðf Þ ¼ k1 exp �f

k2

� �þ k3 exp �0:5

f � f 0k4

� �2 !(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 least-

square fit approach.

The percentage weight (G%,) of the Gaussian component

was then evaluated as follows:

G% ¼ 100

RGð f ÞdfRP0ð f Þdf

ð%Þ. (6)

2.9. Distance analysis

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:

Rðd iÞ ¼ 100SBONEðd iÞ

SROIðd iÞð%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 dpfrom the interface of the material:

RðdpÞ ¼ 100SBONEðdpÞ

SROIðdpÞð%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).

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 non-

grafted 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 investi-

gated time, were evaluated by applying paired Wilcoxon’s

signed rank test to the functions f Pð f Þ.

3. Results

3.1. Global analysis

Classical global findings (Table 1) revealed that newbone formation was higher in RGD-grafted BCP after 2weeks (F ¼ 21:3 5:7%) when compared to non-grafted materials (F ¼ 12:3 5:0%) (p ¼ 0:02), whereasno significant differences occurred between these twomaterials after 4 weeks of implantation. Between W2

and W4, bone formation increased significantly withnon-grafted samples (F ¼ 12:3 5:0% versusF ¼ 21:1 5:2%; p ¼ 0:03), while no significant changeappeared between W2 and W4 in the case of RGD-grafted samples.

3.2. Local analysis and mathematical modelling

At W2, 50% of the pores were empty (without anynew bone element) with non-grafted samples, while only20% of the pores were empty with RGD-graftedsamples (Fig. 2a). The probability to find a local boneformation (inside a pore) higher than 10% was moreimportant with RGD samples than with non-graftedmaterials. The mean and standard deviation values(m s) calculated from the probability distribution P( f )were 18.972.6% with RGD-grafted samples versus10.471.1% 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 aparticular local bone formation ratio ( f ) was similarwith both materials (Fig. 2b), with no significantdifference (m s): 15.972.3% with RGD-grafted sam-ples 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) withRGD-grafted materials than with non-grafted samples.The most probable ratio of local bone formation insidea 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

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Table 1

Global analysis

F ¼ 100 � SBONE=SPORE(%)

W2 W4 p(W2/W4)

�RGD 12.375.0 21.175.2 p ¼ 0:03(n ¼ 5) (n ¼ 6)

+RGD 21.375.7 18.172.7 ns (0.3)

(n ¼ 5) (n ¼ 6)

p(7RGD) p ¼ 0:02 ns (0.2)

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.

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.

L. Pothuaud et al. / Biomaterials 26 (2005) 6788–67976792

significantly higher with RGD-grafted materials, andinterestingly, f0 was almost stabilized on the whole ofRGD-grafted samples ( f 0 ¼ 13:8 0:4%, coefficient ofvariation ¼ 2.9%), while without RGD, this stabilityseemed not yet reached ( f 0 ¼ 8:6 5:4%, coefficientof variation ¼ 63%). Then, while no significant differ-ences were detected after 4 weeks of implantation withboth RGD-grafted BCP and non-grafted materials,

distribution of new bone formation was more homo-geneous in RGD-grafted materials than in non-graftedsamples 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% (ormore) for small pores (500 pixel surface) to lower than20% for smallest pores (2500 pixel surface).

3.3. Distance analysis

At W2, the bone formation was higher with RGD-grafted samples than with non-grafted samples, what-ever the distance at the edge of the implant (Fig. 5a)even in the centre of the implant. The gradient of boneformation was decreased from the edge of the implant toits centre. This profile was similar with both grafted ornon-grafted samples. At W4, the bone formation seemedcomparable with both groups of samples, except closeto the edge of the implant where the gradient ofbone formation seemed higher with non-grafted samples(Fig. 5b). Moreover, an involution of the peripheral newformed bone at W2 and W4 with RGD-grafted materialswas observed. Anyway, bone formation was alwayshigher 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 newformed bone was generated starting from the surface ofthe material, with a centripetal gradient of boneformation inside the pore. At W2 (Fig. 6a) the boneformation was higher with RGD-grafted materials thanwith non-grafted samples, whatever the distance at theedge of the pore. At W4 (Fig. 6b) the bone formationwas similar with both groups of materials, whatever thedistance at the edge of the pore.

4. Discussion

In this study we have evaluated the in vivo effects ofthe grafting of cyclo-DfKRG in the early phase ofintegration of biomaterials made with macroporoushydroxyapatite/b-tricalcium phosphate colonized withautologous bone marrow stromal cells, implanted in thefemoral condyle of rabbits. These short times werechosen to explore the potential of our new developedlocal evaluation method to detect early significantdifferences between these two groups of materialswhereas most of the papers described effectiveness ofRGD coatings only after 2 or 3 months [21,34]. Twoweeks after implantation, the bone formation inside theimplant was higher with the grafted samples (+RGD)than with the non-grafted samples (�RGD), with asimilar 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).

L. Pothuaud et al. / Biomaterials 26 (2005) 6788–6797 6793

peptides in calcium phosphate ceramics is efficient onnew bone formation soon after implantation and thatRGD peptides act in vitro in stimulating cell adhesionup to 3 h [19]. Four weeks after implantation, thereremained no significant difference between the twogroups studied (7RGD), while an involution of theperipheral new formed bone was observed between 2weeks (W2) and 4 weeks (W4) after implantation in thecase of +RGD samples. Most of the new formed bonewas generated starting from the surface of the material,with a centripetal gradient of cicatrization inside thepore, decreasing starting from the edge of the pore untilits centre. These data were completed by a local analysisof bone formation ratio inside the macropores of theimplant. The probability distribution P( f ) was fitted asa summation of an exponential component E( f )—representing the invasion of the macropores by the newformed bone—and a Gaussian component G( f )—representing the homogenization of the bone formationinside the macroporous implant (Fig. 2). Mathematicalmodelling was used in order to exploit the local analysisperformed inside each pore of the macroporous implant,and has shown an accelerated homogenization of theratio of local bone formation inside the pores in the caseof +RGD samples. This mathematical modelling has

argued the assumption that peptide grafting shouldtherefore increase the early osteoconduction by showingan homogenization of the ratio of local bone formationin the case of grafted samples (Fig. 3). A similarevolution, but slowed down, could concern the case ofnon-grafted materials.The surface characteristics of biomaterials are im-

portant factors conditioning their biocompatibility andbioactivity in the bone repair process. There are manypossibilities and variations to improve these character-istics of surface, by using surface coating techniques,and/or by loading the surface with osteoblastic cells orgrowth factors. RGD peptide grafting has been widelystudied in vitro, while only little is known on its in vivoeffects. Ferris et al. [22] have evaluated the quality andquantity of the new formed bone in response to titaniumrods coated with the peptide sequence Arg–Gly–Asp–Cys (RGDc) and implanted in rat femurs. Theyhave shown a thicker shell of new formed bone aroundRGD-modified implants after 2 weeks of implantationcompared to control implants without RGD coating.Schliephake et al. [21] have tested the interest of RGDpeptide coating of titanium materials implanted in dogalveolar crests. They observed only a weak evidence thatRGD coating on titanium increases bone formation.

ARTICLE IN PRESS

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.

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.

L. Pothuaud et al. / Biomaterials 26 (2005) 6788–67976794

These two studies [21,22] have investigated periimplantbone formation and bone/implant contact around com-pact titanium materials. In our study, we have investi-gated macroporous implants colonized with autologousbone marrow stromal cells because such biomaterialshave proved their ability to improve both osteoconductiveand osteoinductive characteristics [35,36].Several interesting data could be underlined from our

study. We have shown earlier the difference betweengrafted (+RGD) and non-grafted (�RGD) materialswhen compared to the other in vivo studies carried outwith RGD-coated materials. At W2, bone formationwas higher in the +RGD samples whatever the distanceto the edge of the implant or to the edge of the pore. AtW4, only the mathematical modelling showed a homo-genization of the ratio of local bone formation in case ofgrafted samples (+RGD). Both groups being cellular-ized, this could be due to the higher activities ofosteoprogenitor cells in contact with the grafted peptide,as it was previously shown in vitro [20]. In fact, theefficiency of RGD peptide grafting has been demon-strated by measuring the adhesion between 1 and 24 h ofosteoprogenitor cells isolated from human bone marrowstroma cells. The osteoprogenitor cells were seeded at adensity of 2� 104 cell/m2 on each substrate (grafted and

non-grafted). There were about two times more cells ingrafted substrates after 24 h compared to non-graftedsubstrates. This in vitro study has demonstrated thatthere were more viable cells in presence of RGD peptide,RGD peptide favouring cell attachment and promotingcell differentiation also [20]. The osteogenic cell dis-tribution in our material should promote bone forma-tion in almost all the pores, particularly with RGD-grafted samples for which cells are supposed to be highlyadherent. The local analysis applied to the macroporousmaterial has shown that with peptide grafting, newformed bone was mainly in contact with the surface ofthe implant, which revealed the osteoconductive proper-ties of the biomaterial used. However, bone formationwas always higher at the periphery of the implant. Infact, this effect could be explained like the superpositionof two dissociated processes: due to osteogenic cellsloaded onto the surface of the biomaterial at the side ofthe bone defect; and due to osteoblastic cells andextracellular matrix at the side of adjacent bone. Asimilar evolution of new bone formation occurred withthe 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 non-

grafted (�RGD) and grafted (+RGD) materials after 2 weeks (W2)

and 4 weeks (W4) of implantation.

L. Pothuaud et al. / Biomaterials 26 (2005) 6788–6797 6795

bone area and the pore area, as measured in two-dimensional (2D) thin slices. This ratio cannot be higherthan the biological ratio, generally known as ‘‘boneporosity’’. For example, a ratio ranging from 12% to22% is coherent with what is generally observed inhuman bone.Our developments are based on the evaluation of

local bone formation inside each pore from a 2D thinslice. Based on stereological considerations, it is wellknown that such an evaluation is correlated to the truebone formation in the 3D pore space, although notexactly the same in absolute value. The decreasedrelationship observed between local bone formationand available pore space (Fig. 4) could only be an effectof the 2D evaluation. Only 3D imaging techniques, suchas synchrotron radiation-based imaging, for example[37], would permit access to and characterization of thetrue 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 involutionprocess at W4 compared to W2. This involution processis mainly observed at the periphery of the implant(Fig. 5) and should be due to an accelerated boneformation process at W2 induced by edge effects with

pre-existing outside bone. Hence, the potential addi-tional interest of the mathematical modelling approachwould be to characterize the homogenization of boneformation process without any quantification artefactdue to the existence of a peripheral involution effect.The main limitation of this work is that the short time ofinvestigation does not allow one to clearly verify thesestatements. Hence, it is now necessary to complementthese preliminary results with additional experimentstaking into account longer time of investigations.In the present work, we have focussed on short

investigation times in order to explore the effect ofosteoprogenitor cells loaded on grafted or non-graftedmaterials. This effect had more chance to be observed ina short time, such as it has been observed in the presentstudy. As a consequence, the main limitation of thiswork is the lack of longer investigation times that wouldpermit a better conclusion concerning the evolution ofthe bone formation process in our experimental model.Nevertheless, the preliminary results obtained in thepresent work are encouraging and have demonstratedthe interest of our new local quantification approach forbone repair study in macroporous implants. Further-more, the exploitation of mathematical modelling hasled to higher level of bone formation process character-istics, concerning in particular the homogeneity of newformed bone distribution inside the macroporousmaterial. This quantification is based on implicithypothesis linked to the validity of the mathematicalmodel used (Eq (5)). This hypothesis needs to be verifiednow from larger sets of samples and for a longer timeof observation. Numerical simulation would also con-stitute an additional tool to evaluate the field ofvalidity of this mathematical model. However, thepreliminary finding that bone formation would bedistributed homogenously around a specific ratio value(13.870.4%, for grafted samples) constitutes a veryinteresting point. Further investigations should consistin studying the relationship between this ratio value andthe structure characteristics of the material used(porosity, pore size distribution, pore connection, etc.).

5. Conclusions

With bioengineering technologies [1,2,38,39], thecontribution of sophisticated imaging and image analy-sis techniques is essential to evaluate the variouscomponents intervening in the manufacturing of hybridartificial bone substitutes. In this study, we havedemonstrated the potential of global and local quanti-fication tools based on the morphological histomorpho-metry approach. Global analysis gives access to meanbone formation ratio on the overall surface studied andonly general trends can be concluded from such globalevaluation approach. On the contrary, local analysis

ARTICLE IN PRESSL. Pothuaud et al. / Biomaterials 26 (2005) 6788–67976796

gives access to more precise characteristics such asalready been demonstrated for the evaluation of bonestructure [40]. In the case of bone formation insidebiomaterials, local analysis allows one to evaluate thedistribution of local ratio of bone formation as well asthe repartition of the new formed bone in contact withthe material surface. These local characteristics can berelated to the osteoconductive and osteoinductiveproperties of biomaterials, and would constitute potentquantitative information for tissue engineering. Further-more, the modelling approach based on statistical andmathematical techniques applied to local characteristicscould permit the development of more efficient prog-nostic tools for tissue engineering, in both ex vivohistomorphometry-based investigations or non-invasivemagnetic resonance relaxometry applications [41].

Acknowledgements

The authors would like to thank Jean-Patrick Chenu(DETERCA, Bordeaux, France) for his help during thechirurgical procedures, Franck Villars, Jean-LouisPariente (CIT—Centre d’Innovations Cliniques, CHUBordeaux, France) for their assistance in the digitizationprocedure and histological experiments. This work wassupported by INSERM, CNRS and the RNTS.

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