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Computers & Graphics 32 (2008) 438– 444

Contents lists available at ScienceDirect

Computers & Graphics

0097-84

doi:10.1

� Corr

E-m

(D. Ayal

journal homepage: www.elsevier.com/locate/cag

3D reconstruction and quantification of porous structures

Eduard Verges �, Dolors Ayala, Sergi Grau, Dani Tost

Universitat Politecnica de Catalunya, Spain

a r t i c l e i n f o

Article history:

Received 20 December 2007

Received in revised form

10 March 2008

Accepted 7 April 2008

Keywords:

Bio-CAD modeling

3D digital topology

Pore quantification

93/$ - see front matter & 2008 Published by

016/j.cag.2008.04.001

esponding author.

ail addresses: everges@lsi.upc.edu (E. Verges)

a), sgrau@lsi.upc.edu (S. Grau), dani@lsi.upc.e

a b s t r a c t

In this paper, we describe the methodology that we have designed to quantify the pores distribution in

bone implants and the empirical results that we have obtained with BioCAD designed scaffolds, microCT

and confocal microscopy data. Our method is based on 3D digital topology properties of the porous

structure. We segment the 3D images into three regions: exterior, bone and pore space. Next, we divide

the pore space into pores and connection paths. We compute a graph of the pore space such that each

node of the graph represents a pore, and an arc between two nodes indicates that the two pores are

path-connected. On the basis of the graph and the segmented model, we are able to compute several

properties of the material such as global porosity, effective porosity and radial pore distribution.

& 2008 Published by Elsevier Ltd.

1. Introduction

The reliable analysis of the porosity of biomaterial implants forbone reconstruction is of paramount importance to materialengineers and scientists. Porosity and, specifically, the connectiv-ity between pores determines the blood flow inside the structure,which is one of the main factors of tissue regeneration. Theporosity of a material can be computed physically using aporosimeter by injecting mercury at different pressure levelsinside the material and registering the differential fluid volumethat has filled the material at each pressure level. It can also becomputed ‘‘virtually’’, using BioCAD measurements i.e. computer-aided-design methods on 3D models reconstructed from a stack ofcross-section images of a material sample. Several BioCADprevious papers have addressed the global porosity of thematerials, measured as the ratio between the empty volume andthe total volume of the material. However, less work has beendone on the characterization of the pores distribution inside thematerial. Some authors consider a pore as a connected componentof the empty space [1]. Other authors develop heuristics to defineseveral pores inside one empty connected component [2].However, the criteria used to identify a pore and the methodsproposed to compute the connections between pores are unclearin the bibliography.

In this paper, we describe a new methodology that, startingfrom a voxel model created from a set of cross-section images of abioimplant sample, obtains a meaningful model of the pore space,

Elsevier Ltd.

, dolorsa@lsi.upc.edu

du (D. Tost).

according to a set of user-defined parameters. In Section 2 we firstdefine the properties associated to the porosity of a material. Next,in Section 3, we survey the related work. In Section 4 we overviewthe global methodology and in Section 5 we explain the poreidentification method. Section 6 shows the experimental results,before the conclusions.

2. Definitions

Let V ¼ fvi; i ¼ 1; . . . ;nvg be the voxel model created from a setof 2D images of a bioimplant sample. Two disjoint sets of voxelscan be distinguished inside V: the exterior set, Ve, and thematerial set Vm: V ¼ Ve [ Vm. Moreover, Vm is also composed oftwo disjoint sets of voxels: the set of bioimplant voxels, Vb, andthe set of empty voxels, Vp, that we will call pore space:Vm ¼ Vb [ Vp.

The pore space can be decomposed into n connectedcomponents CPi so, Vp ¼

Sni¼1 CPi. We call V extp the subset of

connected components of the pore space that are connected to theexterior and V intp the subset of those that are totally internal:Vp ¼ V extp [ V intp.

The total porosity (TP) is the volume ratio between the porespace and the total material: TP ¼ VolðVpÞ=VolðVmÞ [2,3]. Theeffective porosity (EP) measures the porous volume through whichfluid can flow and it is the volume ratio between that part of thepore space connected to the outside and the total material:EP ¼ VolðV extpÞ=VolðVmÞ.

Other properties of the pore space are pore size andpermeability. The pore size has a major impact on new tissueformation because it determines the ingrowth of blood vesselsand cells inside the tissue. Permeability measures the ability to

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Fig. 1. Pores typology. (a) The red pore is absorbed by the blue one; (b) the red

pore has less intersection with the blue one and it is kept as a pore; (c) the yellow

pore is a bridge between the red and the blue ones, and it is removed from the

graph.

Fig. 2. Voxel data set example. The gray voxels are the pores voxels and the

numbers represent the distance map of each voxel.

Fig. 3. Voxel data set example. The different maximum local voxels are grouped on

connected components. Each group has a different color.

Fig. 4. Voxel data set example. The green and the blue voxels belong to two

different pores. Note that the orange group has failed to be a pore candidate due to

its size, and its voxels are added to the connection voxels set (in gray).

Fig. 5. Voxel data set example. The computed graph: because of the existence of

connection voxels between the two pores, a connection arc is created between the

two nodes in the graph.

Table 1Some data sets and results obtained using our method

Attribute Scaffold Confocal MicroCT

Mod. size 1323 10242� 1 2562

� 73

TP (%) 16 41.2 74

CC 4 420 116

Mean 7 6.95 3.2

Std. dev. 1.5 7.2 2.2

Min. size 5.5 0.5 0.5

Max. size 9.5 75.5 26.5

N. pores 216 966 2485

N. arcs 1260 1316 4646

SizeMin 1 1 1

CandidateMin (%) 25 25 25

PoreMin (%) 25 25 25

Time (s) 29 1440 7354

E. Verges et al. / Computers & Graphics 32 (2008) 438–444 439

transmit fluids through pores when subjected to a pressuredifference. To compute these properties require to model the porespace.

From a morphological point of view, any connected componentof the pore space can be interpreted as a set of disjoint pores

connected through disjoint connection stretches. The pores arelocal openings inside the empty region. We call pore voxels the setof voxels belonging to a pore and connection voxels thosebelonging to a connection stretch. Therefore, a connected compo-nent of the pore space can be defined as CP ¼ ð

Snpj¼1 pjÞ [ CV , being

pj; j ¼ 1; . . . ;np the set of np pores and CV the set of connectionvoxels of the connected component CP.

The axial radii of a pore are one half the maximum axialdistances between any pair of points inside the pore. The radial

distribution of pores (RDP) represents the histogram of pore axialradii.

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E. Verges et al. / Computers & Graphics 32 (2008) 438–444440

3. Related work

Related work can be classified into two main strategies: thesefocusing on the study of the bioimplant or trabecular bone (TB)region, and those addressing the pore space.

Within the first group, several methods perform a character-ization of the amount of rod (linear) and plate (surface) elementsof a sample, which is a measure of its quality [4,5]. Other worksare interested in the measurement of the anisotropy or orientationof trabeculae [6]. Moreover, the TB pattern parameter [7], relatedwith connectivity, is computed as a ratio between perimeter andarea differences of a region of interest and the same region dilatedone voxel. This method can be extended to 3D in order to computethe structure model index (SMI) using areas and volumes. The SMIis related with connectivity and the amount of rod and plateelements [4]. A different approach consists of computing thedigital skeleton of the structure and analyzing it. One of thesemethods [8] begins with a surface skeleton computation [9] andthen performs a classification of points in it [10] in order toidentify rods and plates and the ratio of rod–plate junctionelements. Another work [5] also computes a surface skeleton [11]and, adding several parameters, classifies elements as rods andplates, thin and thick and inner and outer. Computing 3D-lineskeletons with its associated graph is also used to evaluatemechanical properties through the analysis of the graph in termsof connection and termini nodes and branches [12].

Methods concerned with the pore space compute the TP from thevoxelized model by simple voxel counting [1,3]. Devising a completerepresentation of the pore space is addressed in very few papers. In[1], a pore is defined simply as a six-connected component of the

Fig. 6. (a) 3D view of the scaffold original data; (b) cross section of the segmented dat

section of the classified voxel model: voxels belonging to a pore are white and connectio

nodes are gray ellipsoids.

pore space. The pore space is subdivided into connected components(connected component labeling, CCL) one corresponding to each poreand the surface of the pores is extracted using the marching cubes(MC) triangularization. In this work, the authors conclude that, sincethe pore space is generally highly connected, in most cases, theydetect only one pore. A more precise modeling of the pore space isperformed in [2] in order to evaluate the permeability and pore sizeparameters. This method is based on an initial skeletonization of thepore space and the application of a two-pass heuristic. First, the porespace is filled with non-overlapping spheres centered at skeletonpoints, and, then, these spheres are inflated until any voxel of thepore space belongs to one of them. These inflated spheres areconsidered individual pores. However, this method can create poreswith strange shapes that may have local narrowings. Othertechniques to compute porosity are based on stochastic geometry(Monte Carlo methods) [13] and fractal analysis [14].

4. Global methodology

Our method is composed of three main steps: segmentation,computation of the connected components of the pore space andconstruction of the pores graph.

4.1. Segmentation and TP computation

In the first step, we classify the voxel model into three regions:Ve (exterior), Vb (bioimplant) and Vp (pore space). This segmenta-tion process varies depending on the characteristics of the datasets and includes geometrical pattern separation, threshold

a: black voxels belong to the scaffold and white ones to the pore space; (c) cross

n voxels are orange; (d) 3D view of the generated graph: edges are orange lines and

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E. Verges et al. / Computers & Graphics 32 (2008) 438–444 441

segmentation and image filtering. When the image data set iscompletely inside the material such as in the confocal microscopydata (see Section 6), only Vb and Vp are detected. Once the modelhas been segmented, we are able to compute the TP on the basis ofthe number of voxels of the pore space in relation to the numberof voxels of the material: TP ¼ jVpj=ðjVpj þ jVbjÞ.

4.2. CCL and EP computation

In the second step, we separate the pore space into its set ofconnected components CPi; i ¼ 1; . . . ;n using a customized CCLalgorithm [15] that scans the volume slice to slice, updating a listof 26-connected components. Next, we check for each of thecomponents if there exists at least one voxel of the componentthat is connected to the exterior. The set of the connectedcomponents connected to Ve is V extp. Obviously, this process isperformed only for those data sets that fulfill Vea;. Then the EP iscomputed as EP ¼ jV extpj=ðjVpj þ jVbjÞ.

4.3. Pores graph construction and RDP computation

The third step of the process consists of identifying the poresinto each connected component and of computing the adjacencybetween pores. More precisely, we compute a graph such thateach node is a pore with its associated voxels and each arcbetween nodes indicates that the two pores are connected. Weconsider two types of arcs: adjacency arcs, that connect pores thattouch one another, and connection arcs, that connect non-adjacentpores through a path of connection voxels. The computation of thegraph is described in Section 5. The RDP is then computed with asimple topological traversal of the graph.

Fig. 7. (a) 3D view of the confocal original data; (b) cross section of the segmented data

section of the classified voxel model: voxels belonging to a pore are white and connec

5. Pores graph construction

5.1. Pores definition

According to the bibliography, the relevant aspects of a poreare the maximality (pores touch the bioimplant volume at morethan one point), the locality (pores are local growths in the poreregion) and the convexity (pores do not have strong localnarrowings). We thus define a pore p of a connected componentCP of the pore space as a union of maximal balls inscribed into CP

such that:

�

: bl

tion

The voxels of a pore are path-connected.

� The radii of the balls are local maxima of the distance map of

CP.

� For any pair of pores pj, pk of CP, pj \ pk ¼ ;. � The maximal balls that form a pore have a volume of

intersection, in relation to the volume of the smallest ball,higher than a given threshold.

The two first conditions account for the maximality and locality ofthe pores, the third is motivated by the fact that we need tounequivocally assign each voxel to one pore at most, and thefourth accounts for the convexity.

5.2. Pores graph computation

In order to identify the pores according to the definition givenabove, we apply a four-step process to each connected compo-nent, CP, of the pore space. The first step consists of computing the

ack voxels belong to the bioimplant and gray ones to the pore space; (c) cross

voxels are orange; (d) 3D view of the generated graph.

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E. Verges et al. / Computers & Graphics 32 (2008) 438–444442

distance map. This gives as a result a new voxel model, storing foreach voxel of each CP its smallest distance to a voxel that does notbelong to the CP (see Fig. 2). We use the scanning method of Silveret al. with different distance metrics [16].

Next, we identify the local maxima of the distance map asthose voxels of the CP having a distance value greater or equal tothat of all their neighboring voxels (see Fig. 3). We consider arasterized sphere associated to each local maxima with thedistance value as its radius.

The third step computes the pores (nodes of the graph) of eachCP. Initially, every rasterized sphere is considered as a potentialpore and, therefore, added to the graph as a node. In this stepseveral pore candidates can be discarted if they fail somerequirement such as minimum radius (related with the SizeMin

parameter explained in the following). Thus, at this stage of theprocess, nodes can share voxels. We subdivide the set of voxelsbelonging to each node of the graph into subsets depending on thenumber of pores that share them. Then, a node of the graph hasone set of non-shared voxels and as many other sets of voxels asother combination of nodes interfering with it. During this thirdstep, we grow some pores, shrink others and remove other ones sothat, at the end of the step, all the remaining pores are disjoint andfulfill the definition. The voxels not belonging to any maximal ballremain unclassified at this stage of the process. In the fourth step,we either insert them into one pore or label them as connectionvoxels. In this fourth step, we also compute the arcs of the graph(see Figs. 4 and 5).

The algorithm is based on the pore size and, for any pore p, ituses the function InitialSizeðpÞ, related to the initial radius of thesphere and CurrentSizeðpÞ related to the number of voxelsassociated to the current grown or shrunk pore p.

Fig. 8. (a) 2D view of the MicroCT original data; (b) cross section of the segmented data:

pore space; (c) cross section of the classified voxel model: voxels belonging to a pore a

The spheres obtained in the second step of this process areinserted into an initial list, LE, which is sorted in descending orderwith respect its radius (the corresponding local maximum value).

The algorithm takes the biggest sphere, emax, of LE andinitializes a list of pore candidates, LP , with this sphere, emax.Then it considers the set of spheres that share voxels with emax,E ¼ fe 2 LEjemax \ ea;g and for each sphere e 2 E, processed indescending radius order, it performs the two following tests:

�

blac

re w

Bridge: The sphere is not a candidate to be a pore but to be partof the connection voxels set. The condition that a sphere mustfulfill to be a Bridge is that 9e0 2 LE; e

0 \ ea;; e0 \ emax ¼ ; andInitialSizeðe0Þ4InitialSizeðeÞ.

� PoreCandidate: The sphere is a candidate to be a pore. The

condition that a sphere must fulfill is that InitalSizeðeÞ4SizeMin

and 8p 2 LP ;CurrentSizeðenpÞ=CurrentSizeðeÞ4CandidateMin.

The algorithm proceeds as follows. If e is a Bridge, all its voxels notshared with other spheres are labeled as connection voxels and e

is removed from LE. If e is not a Bridge, then if it is a PoreCandidate,it is inserted into LP and if it is not a PoreCandidate it is absorbedby the pore p that causes it to fail the poreCandidate test, and e isremoved from LE.

At this point, several pores in LE have grown or remainedunaltered while others have disappeared and the set of connectionvoxels has grown. Fig. 1 illustrates the algorithm’s pores typology.

Now, before the process continues with the next sphere of LE,all the shared voxels in the pore candidates of LP have to beassigned unequivocally to one pore. To do so, the algorithmtraverses all the sets of shared voxels and assigns them to the

k voxels are exterior, gray voxels belong to the bioimplant and white ones to the

hite and connection voxels are orange; (d) 3D view of the generated graph.

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Fig. 9. Radial distribution of pores. (a) MicroCT, (b) confocal, (c) scaffold.

E. Verges et al. / Computers & Graphics 32 (2008) 438–444 443

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E. Verges et al. / Computers & Graphics 32 (2008) 438–444444

biggest pore candidate that owns them. As a result, there are porecandidates that have lost voxels and, for this reason, another testis applied to each p in LP in order to classify it definitively as apore. The test for p to be a pore is that CurrentSizeðpÞ=

InitialSizeðpÞ4PoreMin. If p is not classified as a pore anymorethen its shared voxels are assigned to its largest parent, its ownvoxels are added to the connection voxels set, and p is removedfrom the graph.

The parameters SizeMin, CandidateMin and PoreMin need to bevalidated by an expert and their value affects substantially to theresult.

The process continues with the next sphere of LE until this listis empty.

The fourth step deals with the voxels of the connection voxelsset (remaining voxels set), which is the union between the set ofvoxels not yet labeled and the set of voxels coming from porecandidates that finally have not been classified as pores. This stepcomputes the arcs of the graph.

First, a CCL process is applied to this set of remaining voxels,which performs a partition of it into its non-overlappingconnected components of remaining voxels, that we refer asCCRV. Then, the algorithm checks the adjacency of each of theseCCRV to all the nodes (pores) of the graph and creates a connection

arc between any two nodes adjacent to the same CCRV. Therefore,any connection arc has a set of voxels associated with it that arelabeled as connection voxels. When a CCRV is adjacent to only onenode, its voxels are added to this node. Finally, the methodidentifies the adjacency arcs by searching, for each couple ofnodes, a voxel of one of them that is adjacent to a voxel of theother.

6. Experimental results

This method has been implemented in our software platformHipoVis [17], using the FeatureGraph model designed by Camposet al. [18]. In order to check the validity of our method, we havefirst tested it against several phantom models featuring differentpore configurations. We have also evaluated the methodologywith geometrical scaffold structures (shown in Fig. 6) usedfrequently in the design of implants [19]. We have generated thevoxel models corresponding to these scaffolds and have appliedour method. After that, we have compared the computed poresdistribution with the real scaffold’s known pores distribution. Theresults show that the computed pore model matches perfectly thegiven scaffold’s pore distribution: we have obtained the samepores with the same radii and the voxels have been unequivocallylabeled (see Figs. 2–5). The phantom model consists of 54 pores ofsize 9.5, 54 pores of size 5.5 and 108 pores of size 6.5. In addition,we have applied our method to the real acquired data: themicroCT-scanned bioimplant (shown in Fig. 8) and a slice of abioimplant scanned via confocal microscopy (shown in Fig. 7). Fig.9 shows the radial distribution histogram (RDP) of the three datasets. Table 1 shows some of the results obtained. It shows thevoxel model size, TP, number of connected components (CC),mean pore radius, standard deviation for the pore radius,minimum and maximum pore radius, number of pores andnumber of arcs. It also shows the value of the parameters SizeMin,CandidateMin and PoreMin, for each model (see Section 5.2), aswell as its computation time (see Figs. 6–9).

7. Conclusions and future work

We have proposed a new method for computing the porousinternal structure of biomaterials and bioimplants. This methodproduces both the pore size and its spatial location, and incontrast with other methods it provides connectivity informationamong pores as well. We have checked our method with severalsamples obtaining encouraging results, but we plan to keepevaluating it with more data sets and test configurations. Inaddition, we are currently analyzing how to reduce the computa-tional cost of our method and to improve its efficiency.Furthermore, our next research will focus on the computation ofnew material properties such as the permeability.

Acknowledgments

This work has been partially funded by the Project MAT2005-07244-C03-03 and the Project 02-1/07 of the Institut deBioenginyeria de Catalunya (IBEC).

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