ARTICLE IN PRESS

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doi:10.1016/j.bi

Abbreviations

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IL, number of i

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N2, number of

NS, number of

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Biomaterials 28 (2007) 175–186

www.elsevier.com/locate/biomaterials

Review

Stereological methods to assess tissue response fortissue-engineered scaffolds

Yolanda Garciaa,b, Ailish Breena,b, Krishna Burugapallia,b, Peter Dockeryc, Abhay Pandita,b,�

aNational Centre for Biomedical Engineering Science, National University of Ireland, Galway, IrelandbDepartment of Mechanical and Biomedical Engineering, National University of Ireland, Galway, Ireland

cDepartment of Anatomy, National University of Ireland, Galway, Ireland

Received 31 May 2006; accepted 15 August 2006

Available online 8 September 2006

Abstract

The stereological approach can provide an objective unbiased assessment of structural change in biological systems. In this review, we

elucidate the basic principles of stereology and their implementation in the analysis of tissue response to tissue-engineering scaffolds.

A brief outline of tissue response parameters that can be estimated using stereological approach is included. The focus is on frequently

quantified parameters in tissue response, such as host tissue infiltration, inflammatory cell numbers, angiogenesis, fibrous tissue

thickness, areas of calcification, and/or necrosis, among others. Special consideration is given to sampling techniques and how these

techniques can influence the reliability of the obtained results as well as minimizing potential sources of bias.

These basic principles are illustrated with practical examples, where measurements are performed and estimations calculated using

conventional stereological techniques. As the next generation of biomaterials continue to be developed, it is essential that researchers

develop a rigorous and unbiased method of performance quantification.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Stereology; Histomorphometry; Volume fraction; Surface area; Length density; Thickness

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

2. Parameters for evaluation of tissue response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

3. Sampling and experimental design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

3.1. Designing the experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

3.2. Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

3.2.1. Randomizing sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

e front matter r 2006 Elsevier Ltd. All rights reserved.

omaterials.2006.08.037

: AA, partial area; Ai, area of interest; A0, area of counting frame; AT, total area; b, border length; CAT, computerized axial tomography;

ed tomography; d, distance between test lines, grid size; ds, mean tubular diameter; D, distance between sections; HT, host tissue;

ntersections between object of interest and test line; IM, implant matrix; LL, partial length; LM, light microscopy; LT, total length of test

density; M, axis length; MRI, magnetic resonance imaging; N, number of particles; N1, number of particles in the left dissector grid;

particles in the right dissector grid; NV, number density; NMR, nuclear magnetic resonance; Nv, number of particles per unit volume;

slices; PET, positron-emission tomography; PP, partial points; number of points falling on object of interest; PT, test points; number of

n reference space; QA, number of objects per slice plane area; Rdiff, radial diffusion between blood vessels; SA, area of curved or plane

face in space; SV, surface area per unit volume (surface density); TS, thickness of section; Ta, arithmetic thickness; TEM, transmission

copy; TH, harmonic thickness; TT, thickness of sample; V0, volume of the dissector; VT, V, total volume; VV, volume fraction, volume

ing author. Department of Mechanical and Biomedical Engineering, National University of Ireland, Galway, Ireland.

92758; fax: +353 91 563991.

ess: abhay.pandit@nuigalway.ie (A. Pandit).

ARTICLE IN PRESSY. Garcia et al. / Biomaterials 28 (2007) 175–186176

3.2.2. Orientation factor in sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

3.2.3. Stereological probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

3.2.4. Choice of field of vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

4. Stereological parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

4.1. Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

4.1.1. Volume density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

4.1.2. Absolute volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

4.2. Surface density and surface area of blood vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

4.2.1. Surface density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

4.2.2. Surface area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

4.3. Length density and total length of blood vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

4.3.1. Length density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

4.3.2. Total length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

4.4. Cell numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

4.4.1. The dissector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

4.4.2. The fractionator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

4.5. Thickness of fibrous tissue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

1. Introduction

The word stereology was developed to describe themethodologies that allowed access to three-dimensional(3D) information about geometrical structures based uponobservations made on two-dimensional (2D) sections [1]. Amore modern interpretation of the term stereology is as aspatial version of sampling theory [2], whose mainobjective is to make estimations of parameters ofgeometrical structures using sampled information. Thenature of the structure under study in itself does notmatter. It could include any macro or microstructure inbiology or tissue engineering, even holes can also beconsidered as structures. In stereology, quantitation isconsidered the objective method to describe a structure,compare two structures, study structural change and relatestructure to function.

This approach allows inference of geometrical para-meters such as volume, surface area, number, thicknessand spacing. The wide applicability of this approach isdue to its reliance on basic geometric and statisticalfacts. In biology, stereology provides a spatial frame-work upon which to lay physiological and molecularinformation. Fields such as neurobiology, reproductivebiology and cancer cell biology have applied stereologicalapproaches to the interpretation of a wide variety ofproblems [3–8].

Stereology is based on sampling and the mode ofsampling/slicing varies according the questions asked(Table 1). This may simply involve using a knife or bymore sophisticated medical imaging devices such ascomputed tomography (CT) or magnetic resonance ima-ging (MRI). This approach produces thicknesses ofsamples in the order of centimetres to millimetres.Slice dimensions thinner than this generally requiresome microscopic approach. Traditional frozen or wax

histological sectioning methods produce sample thick-nesses in the order of 5–20 mm. Optical, plastic embeddingmethods or indeed frozen sectioning can produce muchthinner slices in the 0.5–50 nm range, appropriate forelectron microscopy [9].The choice of slicing mode is dependant on the

dimensions of the compartment of interest, for example,if brain nuclei are the object of interest, then macro slicingmethods may be appropriate. On the other hand, if sub-cellular resolution is required, then the sectioning/samplingmust have a greater degree of spatial resolution. Thephysical size of the object of interest will help to determinethe initial sampling mode e.g. in some compartments brainnuclei may be in the order of mm-cm, cell nuclei 5–10 mmand mitochondria 0.25–1 mm.A major problem of sampling/slicing is the loss of

dimension; a 3D object becomes a 2D area, a surface (2D)becomes a line (1D), a line (1D) becomes a point (0D) anda number (points 0D) is lost. A stereological approachprovides solutions to this.Most stereological procedures require some kind of

isotropy. Sections must be isotropic uniform random(IUR) planes, or the structure itself must be isotropic.Biological structures are often anisotropic. The biologistoften prefers orientation. Modern stereology providessolutions.To date, the use of stereology on wound healing and

tissue engineering has been limited [10–13] and mainlyconfined to the area of bone tissue engineering [14–18]. Inthis paper, we outline some basic stereological principles ina simplified manner to encourage the greater utilization ofthese techniques in tissue engineering. The scientificadvantages of using modern stereological methods aredisplayed in Table 2 [19,20].The stereological methodology in this review is mainly

focused on conventional histopathological processing

ARTICLE IN PRESS

Table 1

Slicing methods extracted from Mayhew, T.M., with permission from

Blackwell Publishing [9]

Mode of slicing Magnitude of typical slice thickness

Mechanical

Macroscopic Millimetres

Microscopic

Paraffin embed, LM Micrometres

Plastic embed, LM Micrometres

Plastic embed, TEM Nanometres

Optical

Microscopic

Confocal Micrometres

Medical imaging

MRI (NMR) Millimetres

CT (CAT) scan Millimetres

PET Millimetres

Table 2

Scientific advantages of stereological methods in evaluating tissue

response in tissue engineering

Scientific advantages of stereology

Effective interpretation tool of functional morphology [34]

Unbiased [3,34]

Repeatibility of results [3]

Experimental groups comparisons [3]

Efficient [34]

Well-established theoretical background [3]

Easy implementation when using strict sampling strategy [3]

Inexpensive [3,34]

Table 3

Tissue response parameters that can be quantified using a stereological

approach

Tissue response parameters

Porosity of scaffold

Degree and quality of host tissue infiltration into scaffold

Scaffold degradation

Nature and distribution of vasculature

Number and distribution of inflammatory cells (including neutrophils,

monocytes, macrophages, fibroblasts, lymphocytes, plasma cells,

eosinophiles, mast cells and giant cells)

Thickness and quality of fibrous tissue surrounding scaffold

Other parameters, including calcification, necrosis and apoptosis

Y. Garcia et al. / Biomaterials 28 (2007) 175–186 177

methods, mechanical slicing and the estimation of standardtissue response parameters.

2. Parameters for evaluation of tissue response

In order to maximize the information gathered throughthe stereological approach, it is necessary to define thequestions to be addressed in terms of host tissue (HT)interaction with an implanted scaffold.

There are four prerequisites for a tissue response to takeplace: The first is the scaffold; the second, is an initiation ofthe wound-healing mechanisms in response to injury due tosurgery (acute inflammatory response). The third is thepersistence of the response over time due to scaffold(chronic inflammatory response); and finally, the elimina-tion of the scaffold from the host after its intended function(scaffold degradation) [21]. Stereological tools allow thequantitative estimation of these complex events.

Table 3 summarizes the tissue response parameters thatcan be estimated using a stereological approach.

An important parameter for evaluation of a nativescaffold is its porosity, since porosity aids the infiltration ofHT into the scaffold matrix. Upon the introduction of thescaffold into a host, the rate of infiltration of HT into the

porous structure is of interest. Changes in volume of theimplanted scaffold as a function of time may provideinformation about in vivo degradation of the scaffold.Simple point counting methods can permit access to this

information.A quantitative assessment of the extent of angiogenesis is

important in the evaluation of tissue/wound healingresponse. It is vital for the host to establish neo-vasculaturewith the scaffold matrix for sustenance and metabolicfunction of the infiltrating tissue [22,23]. There are anumber of stereological approaches, which provide usefulnumerical adjectives to help describe these complexvascular beds. In particular length density can provide anindication of physiologically relevant information such adthe radial diffusion distance.Quantification of inflammatory cells in terms of pre-

sence/absence and time frame of occurrence [21] indicatesthe nature of inflammatory reaction and scaffold integra-tion [24–31]. For instance, the initial acute inflammatoryphase of tissue response is predominated by recruitment ofneutrophils and monocytes, while in the chronic inflam-matory phase macrophages take over the function ofphagocytosis and antigen presentation. More advancedstereological probes such as the dissector can provideunbiased estimates of numerical density and if volume isknown estimates of number.Initial wound healing response also triggers the deposi-

tion of fibrous tissue rich in fibroblasts (evident between 48and 72 h), collagen, ground substances and blood capil-laries known as granulation tissue surrounding theimplanted scaffold. This layer, upon maturation is typicallyreplaced by normal HT (original histological configurationof the implantation site) [21,22]. Evaluation of thethickness and quality of this layer gives an indication ofthe degree of acceptance of the implanted scaffold.

3. Sampling and experimental design

Stereology is a science based on the principles ofgeometry and statistics.Adequate sampling is the key to success in the

stereological approach. Attention to detail is vital from

ARTICLE IN PRESSY. Garcia et al. / Biomaterials 28 (2007) 175–186178

experimental design to the application of the appropriateprobe. The modern stereological approached providesunbiased estimations of structural parameters. In order toreduce potential bias, proper selection of experimentaldesign and sampling is essential. In this section, we outlineguidelines for experimental design and sampling methods.

3.1. Designing the experiment

An important aim in designing a research animalexperiment is to obtain statistically relevant results withthe use of a minimal number of animals. Cruz-Orive et al.suggest a conservative starting point of five animals perexperimental group (n ¼ 5). The reason being that ‘‘ifsomething is found to increase (or decrease) in all fivecases, then the probability that this is due to chance isP ¼ ð1=2Þn¼5o0:05, and the experiment could be conclu-sive’’ [13,32]. The same magic number, five, can be appliedto the number of blocks per animal, sections per block[13,32] and/or fields per slide to obtain statistically relevantdata. This cascading system implies a time consuming andlaborious quantitation process that is commonly avoidedby evaluating the data of a pilot study. In the pilot studyonly account one parameter is estimated per experimentalgroup this will provide an estimate of the number of slidesneeded to attain statistically relevant data on a givenparameter. This concept is illustrated in Section 4.2.1 withan example of surface density. It is reported that the totalcount for any object of interest need not exceed 200in order to attain accuracy, beyond this number; thestatistical error is negligible [13,32].

3.2. Sampling

Most often, bias and error in stereological estimationsare direct consequences of inadequate sampling [33]. Astereology-compatible sample must be representative of theentire structure [34]. In other words, all the parts of thespecimen under study should have the same chance ofbeing selected [4]. Therefore, the selection of samplingmethods must be random and unbiased in order to obtainestimates close to the true nature of the structure. Unbiasedsampling procedures must also be applied uniformly atevery stage of tissue processing, sectioning and selection offields of view.

3.2.1. Randomizing sampling

Randomization of sampling can either be independent orsystematic. In the independent approach, successive loca-tions in the specimen are arbitrarily selected and measured[48]. Randomization is achieved by the random applicationof a test grid on images or using random number tables[19]. However, this method is not considered efficient as thenumber of samples needed to get estimations close to thetrue values are quite high [34]. In a systematic approach,samples are drawn at regular intervals with a randomstarting point [33,34,42]. Tinsley et al. [6] use this method

by exhaustively sectioning the whole organ obtainingribbons of 5 sections and collecting samples every 60sections. This method is efficient and widely used instereology [35–37].

3.2.2. Orientation factor in sampling

Isotropy is essential to avoid bias and error in moststereological estimations. Most biological materials areanisotropic [9,13]. A few tissues however, do exhibitisotropy (e.g. liver parenchyma or a cell suspension) arandomized section of such sample would result in an IUR

sample. In principle, all stereological calculations can becarried out only on IUR samples. However, mostbiological structures tend to vary with spatial positionand orientation, thus being heterogeneous and anisotropic.Examples of such samples include skeletal muscle, myo-cardium, placenta, bone, cartilage, skin, intestine [9,13,38].In order to obtain unbiased information from an aniso-tropic sample, special processing techniques are adapted tomake the sections/slice uniform/isotropic [39]. Thesetechniques are based on the randomization of the orienta-tion of the organ or the randomization of the orientationprobe. The former is achieved by cutting the anisotropicspecimen at random followed by two more subsequentsections perpendicular to the first plane (the Isector orOrientator methods) [39].The tissue engineer or pathologist often prefers a chosen

orientation this is especially important in anisotropiclayered structures like cartilage and skin. To overcomethis samples obtained by this method are called vertical

uniform random (VUR) sampling [13,19,40]. In this methodsamples are plane sections longitudinal to a fixed (butarbitrary) axial direction. To compensate for this loss,sampling must be made isotropic by the adaptation of astereological probe (see Section 3.2.3) in an isotropicorientation. Randomization principles presented in Section3.2.1 can be applied in both IUR and VUR samplingmethods.

3.2.3. Stereological probes

A test system (e.g. counting frame, grid, etc.) (Figs. 3, 5and 6) is a stereological tool/probe used to gatherinformation from slices, which in practice is a grid withdifferent designs of known points, lines and areas super-imposed on an image with random positioning [19,39].Cycloid arcs (curved surfaces in three dimensions) arespecifically designed to compensate for the pre-selected,non-random vertical axis in VUR samples when estimatingarea and length densities. The additional introduction ofconfocal microscopy and image analysis software allowsthe manipulation of 3D structures and virtual orientation[37] allowing a greater degree of flexibility in therandomization of the probe.

3.2.4. Choice of field of vision

The magnification of any field of view should be limitedto the minimum magnification that allows resolution of the

ARTICLE IN PRESSY. Garcia et al. / Biomaterials 28 (2007) 175–186 179

objects of interest [9,13]. The fields of view at the chosenmagnification must be randomized based on the principlespresented in Section 3.2.1 systematic random samplingoffers the simplest and most practical approach (the use ofstage co-ordinates can allow a simple mapping template tobe constructed to record position of samples and tofacilitate positioning of section).

4. Stereological parameters

Considering that preliminary steps and appropriatesampling have been enforced, a set of sections are readyto be measured. The challenge faced is determining whichstereological parameter is relevant to the particularquestions being addressed. The stereological parametersfall into two categories. These are absolute (e.g. volume,surface area, length and number) or relative (e.g. volumedensity, surface density and number density). Table 4provides a summary of the stereological methods that canbe applied to quantify parameters of tissue response totissue-engineered scaffolds.

To begin, a test-system needs to be superimposed ontoan image before counting, so that stereological formulaecan be subsequently applied [5,6]. The following sectionsexplain the appropriate formulae and test systems neededin determining volumes, surface areas, lengths, numbersand thicknesses. Suitable practical examples are alsopresented as guidelines for application of the stereologicalformulae.

4.1. Volume

4.1.1. Volume density

Volume density is one of the most useful stereologicaltools in the interpretation of tissue response. Parameterssuch as porosity of scaffold, HT infiltration into theporous scaffold, collagen deposition/granulation tissue, cell

Table 4

A guide to the stereological methods used in evaluating specific tissue respons

Stereological parameters Tissue response parameters

Volume fraction Collagen deposition/granulat

Necrotic/calcified areas

Remaining implant

Host tissue integration and in

ECM synthesis

Surface area Angiogenesis

Fibrous reaction

Length fraction Angiogenesis

Number Cell counts: inflammatory or

Angiogenesis: counting profil

Thickness Fibrous capsule

Epithelial coverage

Dermis, cartilage, etc.

density, necrotic and/or calcified areas, remaining implantand ECM synthesis can be estimated amongst others.Volume density, also known as volume fraction (VV)

(Eq. (1)), is a relative parameter best estimated by pointcounting. In this case, the ratio of the number of test points(PP) (in a test-system) falling on the measured object to allthe points (PT) falling in the containing object [43,44] iscalculated (Fig. 1).

VV ¼PP

PT. (1)

According to the Delesse Principle [39], the expectedvolumetric value of an object varies proportionally to theobserved object’s area when cut randomly. Therefore, theratio between points, lengths, areas and volumes isequivalent, and the volume density (VV), area density(AV) or length density (LV), can be calculated by pointcounting samples [39,45].

PP

PT¼ VV ¼ AV ¼ LV. (2)

Due to the recent development on image analysissoftware for stereology, an alternative method of calculat-ing area density can be achieved by digitally tracinganatomical landmarks of interest. These structures will beexpressed as relative parameters per total area [46].

Practical example: Volume fraction estimations can beused to determine the amount of HT infiltrating into theimplant matrix (IM). Fig. 1 represents an image of scaffoldmatrix infiltrated with HT. A point counting test grid(50� 50 cm) containing 35 grid intersections is overlaid onthe image. To estimate the volume density of HT within theimplant, the intersections falling on the HT (PP(HT)) arecounted and also intersections falling on the IM (PP(IM)).The volume fractions of HT and IM are calculatedusing Eq. (1).

e parameters

Stereological methods

ion tissue Grid or pixel count

-growth

VUR sections and cycloid test system

Measuring borders

VUR sections and cycloid test system

host tissue Dissector/fractionator

es

Orthogonal lines

ARTICLE IN PRESS

Fig. 1. Volume fraction estimation of host inflammatory tissue infiltrate into a subcutaneously implanted porous scaffold. Stain is Masson’s Trichrome.

Magnification: 1000� . Test grid size: 50 mm� 50 mm (green), PT ¼ 35, PP(IM) ¼ 10 (green arrows), PP(HT) ¼ 35�10 ¼ 25, VV(HT) ¼ 25/35 ¼ 71.4% of

host inflammatory tissue infiltrate within the implant matrix.

Y. Garcia et al. / Biomaterials 28 (2007) 175–186180

4.1.2. Absolute volume

If, in the interpretation of the results, absolute numbersare essential, then this step must be carried out prior to anyother analysis and taken into consideration when sampling.The absolute volume of an explant can be calculated byweight and specific gravity or by liquid displacement usingthe Archimedes’ Principle [9,39]. However, these methodscannot be applied, firstly, if isolation of the organ inquestion is not feasible or practical (e.g. brain cortex) [9],secondly, if the organ does not displace water after beingimmersed (e.g. lungs); thirdly, if the organ absorbs water(e.g. some tissue-engineered scaffolds) [34,39] and finally, ifthe structures are microscopic [9,39]. In such cases, volumecan be estimated using Cavalieri’s Principle (Fig. 2), whichstates: ‘‘the volume of an arbitrary shaped object can beestimated in an unbiased manner from the product ofthe distance between planes and the sum of the areason systematic random parallel sections through theobject’’ [39,47].

In the application of this method, the object (organ orblock) has to be serially sectioned at known distances, D

that are parallel to an originally randomly chosen sectionplane (Fig. 2). Only one face of each section can bemeasured and the areas (Ai) are determined by pointcounting [9]. In point counting, a suitable test-system isplaced on the image and the area occupied by an object ismeasured by counting the points that fall onto the object(P). Each point has an area associated to it given by thegrid size (d2). Finally, the application of Eq. (3) will give anunbiased estimation of the object’s absolute volume,irrespective of the shape of the object and of the cutting

direction chosen [9]. Only 4–8 systematic sections willguarantee a coefficient error of the estimate of about 5%when using the Cavalieri’s estimator [32]

V ¼ TT �Xi

i¼1

Ai, (3)

TT ¼ NS�D, (4)

Ai ¼X

P� d2. (5)

In tissue engineering, scaffold degradation may bequantified by estimating the volume changes at theimplantation site of the scaffold as a function of time.This provides valuable information about the in vivodegradation profile.

4.2. Surface density and surface area of blood vessels

The main application of these methods in tissue responseis the evaluation of angiogenesis, which provides informa-tion about the surface area of the blood vessel network.However, the same principle applies to any relevanttubular/fibrillar structure (collagen fibres, actin filaments,axons).

4.2.1. Surface density

Surface density (SV), surface fraction or area fractionrepresents the amount of surface area (SA) contained in areference volume (V). Its calculation is based on theprobability that a randomly placed area or profile/border

ARTICLE IN PRESS

Fig. 2. Cavalieri’s Principle adopted from Mandarim de Lacerda. The

absolute volume can be calculated by the ‘‘product of the distance between

planes (TS) and the sum of the areas (Ai) on systematic random sections

through the object’’ [39].

Fig. 3. Measuring surface area density of capillaries in rabbit dermis.

Immunohistochemistry (eNOS antibody) was performed on paraffin-

embedded sections. Section thickness: 5 mm. Magnification: 400� .

Distance between cycloids d: 40mm. Number of intersections: 16 (I),

b ¼ 1005mm. Cycloid height: 20 mm, LT ¼ 2000mm (40� 10� 5),

SV ¼ 2� ð16=2000Þ ¼ 0:016ð1=mmÞ.

Y. Garcia et al. / Biomaterials 28 (2007) 175–186 181

(b) will intersect a test-system (cycloid or straight) ofparallel lines separated by a known distance (d) [32,39,47].This infers that the higher the surface area the higher theprobability that the surface will intersect the lines in the testsystem. Therefore, surface density can be estimated bydividing the number of intersections (I) by the total lengthof the test line (LT)

SV ¼SA

V, (6)

b ¼p2� d � I , (7)

SV ¼ 2�I

LT. (8)

There are two situations when IUR sections are notsuitable to estimate SV, which is when the histologicalstructure of the specimen is highly ordered or stratified andneeds to be preserved (e.g. placenta, skin, brain cortex,cartilage and many epithelia) or when the SV is of amacroscopic specimen [32]. In the first case, VUR samples

are used combined with a test system of cycloids orientedwith minor axis parallel to the vertical section (Fig. 3). Theuse of this method is equivalent to a collection of IUR linesin a 3D space [9,19].

Practical example: Fig. 3 shows how to measure surfacedensity of capillaries in a sample of wound tissue. As thewound model used was a dermal rabbit ear ulcer model, wehave to take into account the stratified nature of skin. Forthis reason, a cycloid arc test system is used. The image iscalibrated as per magnification of the field of view, and acycloid arc of known radius is placed on the image.The height of the cycloid is measured and length of

cycloid arc is calculated as twice this value. The total lengthof test lines (LT) can now be calculated by multiplying thenumber of arcs by the length of cycloid arc and by thenumber of test lines. The number of intersections (I)between blood vessels and test lines are counted. Thus, thesurface density of blood vessels in that field of view can becalculated, with the use of Eq. (8). As the distance betweentest lines is known, the total border length of blood vesselscan be calculated using Eq. (7).

Estimation of minimal number of fields of view: Asmentioned in Section 3.1, a pilot set of data from onetreatment group will direct the user as to the minimumnumber of sections which need to be measured for eachtreatment group in the study. In Table 5, we display datacollected for the surface density parameter from foursections in one treatment group. The number of intersec-tions between blood vessels and the test system for fivefields of view on each section are counted (I). Thecumulative surface density of blood vessels for thetreatment group is measured by expressing the sum of allthe intersections (I total) as a function of the sum of totallengths of test line in the sample set (L total). The number

ARTICLE IN PRESS

Table 5

Raw data displaying the method of determining cumulative surface area

density, from intersections counted on five fields of view on four separate

sections in one animal/treatment group

Section Field of view I I total L total SA

1 1 16 16 2000 0.008

2 21 37 4000 0.00925

3 13 50 6000 0.008333

4 22 72 8000 0.009

5 20 92 10 000 0.0092

2 1 9 101 12 000 0.008417

2 15 116 14 000 0.008286

3 13 129 16 000 0.008063

4 17 146 18 000 0.008111

5 21 167 20 000 0.00835

3 1 14 181 22 000 0.008227

2 27 208 24 000 0.008667

3 22 230 26 000 0.008846

4 13 243 28 000 0.008679

5 17 260 30 000 0.008667

4 1 24 284 32 000 0.008875

2 18 302 34 000 0.008882

3 9 311 36 000 0.008639

4 24 335 38 000 0.008816

5 16 351 40 000 0.008775

The final SV value in bold represents the surface density of the total data

set in one treatment group. The italicized value is first to fall within the 5%

confidence interval of the final surface density value. It is understood that

sufficient sections have been recorded to limit statistically significant

variance. This table shows that a minimum of three sections (208 points

counted) are needed in order to obtain statistically relevant data.

Fig. 4. Cumulative surface area density values are plotted for each image

and consecutive field of view in the same treatment group. As the variance

in surface area density values diminishes, the graph is highlighted to show

the point at which the values are deviating less than 75% (lines) of the

final cumulative surface area density.

Y. Garcia et al. / Biomaterials 28 (2007) 175–186182

of field of views are plotted (Fig. 4) against the set ofcumulative surface densities, which displays the variancein the data set. The first value to fall in the 75% range ofthe final cumulative surface density determines the least

number of samples to be measured to ensure a 95%confidence interval. This procedure can be repeated for anygiven parameter, such as volume, length, thickness etc.

4.2.2. Surface area

The number of intersections (I) between the lines and thesurface area of interest multiplied by the length unit (L) ofthe superimposed lines (test system) will give the surfacearea (SA) [32,50]. This method has been frequently used inthe calculation of total capillary surface area within anorgan [50].

SA ¼ I � L. (9)

Additionally, the area enclosed by a contour can betraced yielding to relative errors of less than 0.5% andallowing area and volume mapping, preserving shapes andstructures for posterior analysis [51].If the volume of tissue has been determined according to

the Cavalieri principle, the total surface area of bloodvessels in that volume of tissue can be estimated by usingEq. (6), provided the surface density has been calculated.It is worth bearing in mind that the image may represent

different types of blood vessels, which might interfere withthe particular question the user wishes to address. It maybe more appropriate to distinguish capillaries fromarterioles and venules in elucidating the biologicalresponse.

4.3. Length density and total length of blood vessels

Length is another stereological parameter frequentlyused in the assessment of angiogenesis [50], and relatesinformation on the length and convolution of capillaries.Again, the same principle applies to any relevant tubular/fibrillar structure.

4.3.1. Length density

Length density is a measurement of the length of vesselper unit volume of tissue (LV) [50]. The underlyingprinciple of measuring length in a volume of tissue isbased on the number of occasions that the IUR sectionstransect a filament-like structure embedded in a volume toproduce profiles Q [9].

LV ¼L

V¼ 2�

Q

A. (10)

Q/A denotes the number of profiles divided by the sliceplane area (A) (number of test points multiplied by theassociated area) [9,32]. When using image analysis soft-ware, the profiles (Q) may be traced to calculate the area ina previously calibrated image, and the Eq. (10) applied [51].In the estimation of length density, the IUR samples and

the cycloid test arc are the standard methods used. Inpractice, the slices through the specimen must be vertical,uniform random and with the same thickness, TS [9].

LV ¼ð2� ILÞ

TS. (11)

ARTICLE IN PRESSY. Garcia et al. / Biomaterials 28 (2007) 175–186 183

Even if LV cannot be estimated from vertical sections, inpractice it is possible when using a cycloid test arc undercertain conditions. In the first instance the sections must beVUR with the same thickness and the second conditionstates that the minor axis of the cycloid arcs must beparallel to the horizontal axis of the vertical sections [9,49].

Practical example: In Fig. 5 capillary length density isestimated in a stratified skin section. The intersections arecounted as in Fig. 3, and applied to Eq. (11). In addition,the radial diffusion distance (Rdiff) between blood vesselscan be calculated using the Eq. (12). This distance gives ameasure of the diffusion zone around the capillaries [52]and thus may be important in determining efficiency of thecapillary network

Rdiff ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p� LV

p . (12)

4.3.2. Total length

The total length of capillaries in a volume of tissue canbe estimated by multiplying length density value by thevolume of tissue (Eq. (10)).

4.4. Cell numbers

The relative and absolute number and distribution ofinflammatory cell types (neutrophils, lymphocytes, plasmacells, eosinophils, macrophages and giant cells), and oftissue cell types (fibroblasts, keratinocytes) or to countprofiles of blood vessels could be estimated by stereologicalmethods.

Number is a dimensionless entity. Therefore, whencounting objects, one should use the smallest uniquefeature in the object that is readily identifiable (e.g. nucleusif we want to count cells) [53]. However, that object could

Fig. 5. Measuring length density of capillaries in rabbit dermis.

Immunohistochemistry (eNOS antibody) was performed on paraffin-

embedded sections. Section thickness: 5 mm (T). Magnification: 400� .

Number of intersections: 22 (I). Cycloid height: 20 mm, LT ¼ 1680mm(6� 7� 40), IL ¼ 22=1680 ¼ 0:013 (1/mm). LV ¼ ð2� 0:013Þ=5 ¼0:005ð1=mm2Þ, Rdiff ¼ 7:97mm.

be present in more than one section if the sections are thinenough [34]. Conventional object counting of each slidewill result in an overestimation. Stereology provides twosolutions to this problem:

4.4.1. The dissector

The dissector has been created to prevent overestimationof particle counting. ‘‘The dissector consists of two parallelplanes aimed at sampling top points of particles inbetween. Since a particle, no matter its size and shape,has a single top point, the number of top points betweenthe two planes represents the number of particles in there,with no further qualification. In practice, one simplycounts particles that are hit by one of the planes of thedissector (reference plane—Fig. 6a), but not by the other(look-up plane—Fig. 6b). The volume of the dissectorequals the area of an unbiased counting frame placed in thereference plane, times the distance between the two planes’’

Fig. 6. The dissector method measures the number density, NV, of convex

features in 3D space: (a) reference plane and (b) look-up plane. Stain is

Movat’s Pentachrome. Magnification: 1000� . Calibration: A0 ¼ 40�

40mm2; h ¼ 5mm, V 0 ¼ 8000mm3, N1 ¼ 7, N2 ¼ 15. Relationship:

NV ¼ N=V 0. Geometric property: NV ¼ 7=8000mm3 ¼ 0:000875mm�3.Geometric property: NV ¼ 15=8000mm3 ¼ 0:001875mm�3.

ARTICLE IN PRESS

Fig. 7. Capsule thickness estimation. Randomized vertical section of

subcutaneous implantation site showing the fibrous tissue. Stain is

Masson’s Trichrome. Magnification: 400� . Parallel lines, with an inter-

line distance of 50mm, were overlaid on the image providing intersection

points marked with the base of the yellow arrows. Orthogonal lines were

drawn from the intercepts across the capsule. Values: (1) 108.55mm, (2)

112.77mm, (3) 103.71mm, (4) 82.71mm. Mean value is: 101.93mm for this

field of view. Arithmetic thickness (Ta) ¼ 80.06. Harmonic mean thickness

(TH) ¼ 86.52. The measurements were taken using image analysis software

(Image Pros Media Cybernetics, MD, USA).

Y. Garcia et al. / Biomaterials 28 (2007) 175–186184

[32]. The section orientation is not critical and theparameters obtained are more numerical densities in avolume than a number by itself [9]. In order to practicallyperform the counting, two microscopes have to be usedsimultaneously, one for each plane (look-up plane andreference plane), or a double-screen computer (as far as thefields can be accurately superimposed), or printouts of bothimages [32]. The optimal dissector height is less (30% or1/4–1/3) than the minimum height of the objects of interest[9,19]. Sections between 3 and 5 mm are consideredadequate in general [5,50]. In this way, undercounting ofobjects will be prevented [43,54].

Confocal microscopy can accurately implement opticaldissector avoiding the previously mentioned labor-inten-sive process [5,9,32,50,51,55,56] by creating optical planeswithin thick tissue sections [53].

4.4.2. The fractionator

The fractionator applies systematic sampling [32,57].Therefore a specimen will be sectioned in a few fragmentsat a fixed distance between them, D. If for example, wemeasure the number of objects (Q) in an area (Ai) every 5thfragment, then the total number of objects (n) in thestructure is determined by [34]

n ¼ 5� Ai �D�X

Q. (13)

This is a direct counting procedure, which unlike thedissector does not require the measurement of the referencespace [32,57].

Similar to the optical dissector, several commercialsystems developed for confocal microscopy are availableto assist in the application of optical fractionator for theestimation of population size [51].

4.5. Thickness of fibrous tissue

When characterizing fibrous reaction after implantation,it is useful to measure the thickness of fibrous capsule.Similarly, the thickness of the epithelial coverage, dermallayer, cartilaginous layer or any membrane may be usefulin quantifying a body response.

Thickness on random slices does not represent truethickness, as the angle of sectioning in different sampleswill vary, giving an overestimation of the real thickness [9].The stereological approach to counteract this problemconsists of using orthogonal intercepts in IUR samples (seepractical example) [9]. The arithmetic mean of theseorthogonal intercept measurements multiplied by the p/4or 8/3p provides an estimate of the arithmetic thickness(Ta) or harmonic thickness (TH), respectively.

The inherent associated error when using VUR sampleshas to be taken into consideration when interpreting results

Ta ¼ L�p4, (14)

TH ¼ L�8

3p. (15)

Practical example: Fig. 7 presents a test system withparallel lines overlaid intersecting with fibrous tissue.Orthogonal lines to the chosen aspect of the membraneare drawn across from the intersections giving a mean ofthe intercept lengths (L). The arithmetic thickness andharmonic mean thickness are calculated using Eqs. (14)and (15), respectively.

5. Conclusions

The main aim of this review is to emphasize the need toapply objective quantitative methods to describe tissueresponse to tissue-engineering scaffolds. The dynamicnature and chronology of events associated with tissue-scaffold interaction necessitates the application of robustmethods, to obtain 3D information from 2D histologicalsections. A stereological approach provides an accuratemeans to obtain such data. The use of statistical principleslike sample size, randomization and isotropy, makesstereological estimates more accurate and objective.Biomedical researchers are often daunted by the tedious

nature of the counting regimen involved in quantitativeestimations. However, proper planning and design ofmodern stereological methodologies can drastically reducethe time, effort and cost required to obtain quantitative 3Dinformation. The stereological approach is efficient inachieving accurate information from minimum amount ofslices and counts. Furthermore, the increasing availabilityand affordability of powerful computing have made datacollection, storage and analysis effortless [41,50,58].

ARTICLE IN PRESSY. Garcia et al. / Biomaterials 28 (2007) 175–186 185

Stereology has been applied successfully in otherdisciplines of biology. However, its potential has not beenrealized in the evaluation of tissue response to tissue-engineering scaffolds. In order to understand the dynamicsand complexity of tissue-scaffold interactions in anobjective quantitative manner, we recommend the stereo-logical guidelines provided in this review.

Acknowledgements

The authors would like to thank Mr. Jerome Henry, forhis help in image analysis and Dr. Dale Feldman(University of Alabama at Birmingham) for introducingthe concepts of stereology. The following agencies fundedthe work: Higher Education Authority; Faculty ofEngineering, National University of Ireland, Galway;Health Research Board; and Enterprise Ireland—Technol-ogy Development grant.

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