Reply - Stereology, magnetic resonance imaging and Eulernumber

The relevance of Euler number to stereologists

The conneulor is one of the most recent methods of moderndesign stereology (Gundersen et al., 1993). It is used toestimate the Euler number x3 [i.e. Euler–Poincare char-acteristic (EPC)] of a structure. The relevant test probescomprise randomly positioned and arbitrarily orientatedpairs of co-registered parallel sections a known distanceapart, bearing sampling windows equipped with a guardframe (known as a shell) which corrects for edge effects andensures that the estimate of x3 is unbiased. The conneulorhas many similarities with the physical disector probe usedfor estimating number (Sterio, 1984). In particular, bothmethods involve deduction as opposed to simple counting.

It is not difficult to appreciate the practical value ofobtaining estimates of, for example, the number of neuronesin the cerebral cortex (Pakkenberg & Gundersen, 1997)using the disector principle. In comparison, Euler number isan abstract quantity. It describes the topology of thesurfaces of a structure of interest and at first sight itsestimation would appear to be of little practical value.However, in certain circumstances the Euler number can beused to count specific features of a structure of interest. Thefeatures in question may be referred to as handles (seeFig. 1), and it is this application that is of special relevanceto stereologists. In particular, Gundersen et al. (1993)realized that by counting the number of handles in aspecimen of bone one is able to derive a value for thenumber of trabeculae, and the conneulor has also beenused by Nyengaard & Marcussen (1993) to count thenumber of glomerular capillaries in the kidney.

In this issue Jernot & Lantuejoul (1999) bring to ourattention the fact that difficulties can arise if one triesadditionally to use the Euler number to count the numberof enclosed cavities in a structure. In particular, the formulathat has been proposed for obtaining a combined count ofthe number of handles and the number of enclosed cavitiesin a structure is not generally applicable [see Eq (1) ofGundersen et al., 1993]. Figure 1 in Jernot & Lantuejoul(1999) illustrates how calculation of the Euler number byassessment of the topology of surfaces does not always givea value identical to that obtained by counting handles andenclosed cavities.

So, what exactly is the relationship between themathematical definition of Euler number and thedefinition that has been adopted for practical purposes bystereologists? Jernot & Lantuejoul (1999) tell us that

mathematically, x3 is the total number of distinct surfacesof a structure minus the sum of the genuses of the surfaces,where genus is defined as the maximum number of closedcurves that can be drawn on a surface without dividing itinto separate parts. On certain occasions, however, a moretangible definition of Euler number can be given. Inparticular, the Euler number of objects with no internalcavities whose surfaces delimit one internal domain1, canbe expressed in terms of the number of discrete objects andthe number of fundamental cuts that can be made throughthese objects, i.e. the cuts that can be made withoutsplitting the objects into separate parts (Fig. 1). For theseobjects the genus is equivalent to the number of handles sothat:

x3 ¼ ðdiscrete objectsÞ ¹ ðhandlesÞ: ð1Þ

Trabecular bone is a structure for which Eq. (1) is valid, andfor the particular case where the specimen comprises onlyone connected component the relationship can be simplifiedto:

x3 ¼ 1 ¹ ðhandlesÞ: ð2Þ

Furthermore, Gundersen et al. (1993) recognized that thenumber of trabeculae in the specimen could be computed as2 – x3. In other words, the first handle represents twotrabeculae and every new handle added to the firstrepresents one additional trabecula.

In addition to discrete objects and handles, enclosedcavities are an inherent aspect of the morphology of a three-dimensional structure and it has been proposed byGundersen et al. (1993), without proof, that Eq. (1) canbe extended so that:

x3 ¼ ðseparate objectsÞ þ ðenclosed cavitiesÞ ¹ ðhandlesÞ ð3Þ

However, Jernot & Lantuejoul (1999) have providedcounter-examples to show that Eq. (3) is not always true.If a structure contains any number of completely enclosedcavities (i.e. like the holes in a Swiss cheese), each of whichhas a genus of zero, these add nothing to the value of x3

and so the relationship holds (Gundersen et al., 1993) butadding toroidal holes, which each have a genus of ¹1,produces inconsistencies. In the case of the porous medium

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1The Euler number of a Klein bottle (popularly described as a bottle with no

inside and no outside) cannot be determined using Eq. (2). The surface of this

object passes through itself and therefore cannot be described as delimiting one

internal domain.

we have studied with magnetic resonance imaging (MRI),enclosed cavities are especially not a problem since the oiland water that provide the signal in the MR images couldnot have flooded any completely enclosed cavity.

Estimation of Euler number

There are five key references relevant to the development ofa stereological method for estimating x3. These areHadwiger (1957), DeHoff (1987), Bhanu Prasad et al.(1989), Bhanu Prasad et al. (1990) and Gundersen et al.(1993). Hadwiger (1957) provided a direct way of estimat-ing x3 without having to consider the genus of each of theseparate surfaces of a structure. Hadwiger (1957) showedthat x3 can be determined by monitoring the events thatoccur as a plane sweeps through the structure. Inparticular, one keeps a record of the number of tangentswith a convex surface (i.e. island events), tangents with asaddle surface (i.e. bridge events) and tangents with aconcave surface (i.e. hole events). The terminology ofislands, bridges and holes was provided by Gundersen etal. (1993), based on the ideas of DeHoff (1987), andexamples of island, bridge and hole events are illustrated inFig. 4 of Roberts et al. (1997). The Euler number is given bythe identity (not estimator)

x3 ¼12

ððIÞ ¹ ðBÞ þ ðHÞÞ ð4Þ

which appears as Eq. (3) in Gundersen et al. (1993).Whatever the structure, the approach of Hadwiger (1957)enables x3 to be determined correctly. For example, Eq. (4)can be applied to the torus with one toroidal hole in Fig. 1 ofJernot & Lantuejoul (1999). As the plane sweeps from left to

right from the outside to the centre of the object, first oneencounters a tangent with a convex surface (island), next atangent with a concave surface (hole), next a tangent with asaddle surface (bridge), and finally a tangent with a saddlesurface (bridge). The same events occur in reverse orderas one continues to sweep the plane to the right until itlies outside the object. This gives a total count of twoislands, four bridges and two holes, and a Euler number of(2 – 4 þ 2)/2 ¼ 0, as required.

The approach of Hadwiger (1957) requires that ascanning plane is passed through the whole of the structureof interest and every Euler event recorded. As stereologists,however, we are keen to do the minimum amount of workconsistent with obtaining an unbiased estimate of x3 with aprecision suited to the needs of a particular study. This isachieved by sampling. Euler number density estimates areobtained for a uniform random series of test probes and totalEuler number is obtained by multiplying the average of theestimates by the reference volume. There are two possibi-lities (Bhanu Prasad et al., 1990; Gundersen et al., 1993)and it is useful to clarify the inherent differences betweenthem. Bhanu Prasad et al. (1990) use a 3D probe and countEuler events in a 3D block, whereas Gundersen et al. (1993)use a 2·5D probe. In particular, they deduce the Eulerevents that occur within a 3D block from observations madeon the upper and lower faces of the block. Both methodsincorporate a correction for edge effects that is based on anidea of Bhanu Prasad et al. (1989).

The approach used by Bhanu Prasad and colleagues tocorrect for edge effects when estimating Euler number in 2Dor 3D is known as the shell correction. In 2D, x2 isdetermined within two-dimensional windows, and in 3D x3

is determined for three-dimensional blocks, which samplethe structure of interest. To correct for edge effects a shell is

Fig. 1. (a) The two-dimensional object shown in (i) has two handles and therefore, as shown in (ii), two fundamental cuts are possible. Thethree-dimensional object in (b) has one handle.

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added to the respective windows and blocks. In 2D the shellcomprises two adjacent orthogonal edges of the window (seeFig. 2a) (Bhanu Prasad et al., 1989). In 3D the shellcomprises three adjacent orthogonal faces of the block (seeFig. 2b) (Bhanu Prasad et al., 1990). To obtain unbiasedestimates of Euler number density from the analyses ofwindows or blocks, a value of 1 is subtracted from the Eulernumber calculated for the union of the shell and the portionof the material that lies within the window or block. Theaverage of the measurements of Euler number per unit area,or per unit volume, for several appropriately sampledwindows, or blocks, is an unbiased estimate of the Eulernumber density of the material. The approach of Bhanu

Prasad et al. (1990) for estimating x3 can be used incombination with imaging techniques such as confocalmicroscopy or MRI. Either one can record the number ofisland, bridge and hole events which occur as the scanningplane moves systematically through the union of the shelland the material within the block or, if one is able to make acomputer-based 3D reconstruction of the union of the shelland the material within the block, simply count the numberof discrete objects and handles by visual inspection.

The conneulor method is again based on the approach ofHadwiger (1957), and like Bhanu Prasad et al. (1989)considers samples of the structure of interest. What isdifferent is that the test probe comprises two co-registeredwindows on parallel sections (i.e. a disector), rather than anentire 3D block. Instead of calculating the Euler number ofthe material within the block and its associated shell, theEuler events that occur within the volume swept outbetween the two windows are deduced from the informationpresent in the two 2D windows, and the 2D shell correctionis applied to correct for edge effects within each window(Fig. 2c). The approach is analogous to the tangent countmethod of DeHoff (1987), which also uses disectors, butnow a shell correction is incorporated to properly accountfor edge effects. Of course, the relevant pairs of sectionscomprising the sampling probe must be sufficiently closetogether to ensure that correct deductions are made aboutthe Euler events that lie between them and that no Eulerevents are missed because they occur, unseen, in the spacebetween the two sections. The 2·5D-test probes should berandomly positioned but it is not necessary that they areorientated isotropically as was suggested by DeHoff (1987).Ohser & Nagel (1996) propose a technique for determiningEuler number from measurements made in 2D onco-registered sections, but this is not the conneulor.Furthermore, x3 cannot be calculated as the absolute valueof the difference in x2 computed for the upper and lowerwindows of the conneulor using the 2D shell correction.

Hadwiger (1957), Bhanu Prasad (1989) and Gundersenet al. (1993) have, in turn, provided stereologists withmethods for measuring x3 using exhaustive sectioning, andestimating x3 without bias by sampling with blocks (i.e. a3D probe) and sampling with co-registered pairs of sections(i.e. a 2·5D probe), respectively.

The conneulor method, ingeneously devised by Gunder-sen et al. (1993) as an amalgamation of the tangent countmethod of DeHoff (1987), the shell correction of BhanuPrasad et al. (1989) and the insight (shared with Zhao &MacDonald, 1993) that the test probes can take anyarbitrary orientation, is likely to be the method that will bemost widely used by stereologists. If there are no enclosedcavities then x3 is an unbiased estimate of the number ofdiscrete objects and the handles they contain [i.e. Eq. (1)].Probably, however, the most practical application of theconneulor is to study structures which are a single

Fig. 2. The solid lines in (a), and the shaded faces in (b), representthe 2D, and 3D, shells employed by Bhanu Prasad et al. (1989),and Bhanu Prasad et al. (1990), for estimating Euler number den-sity in two, and three, dimensions, respectively. The conneulor testprobe is shown in (c). The upper and lower sampling windows areeach equipped with the 2D shell of Bhanu Prasad et al. (1989).

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connected component with no internal cavities and to applyEq. (2).

Interesting parallels exist between the family of methodsfor estimating number, consisting of the unbiased countingframe of Gundersen (1977), the unbiased brick of Howard etal. (1985) (see also Cruz-Orive, 1980, and Gundersen,1981) and the physical disector of Sterio (1984), and thefamily of methods for estimating Euler number using 2D or3D shell corrections and the conneulor. These parallels areshown in Table 1.

Neither the disector nor the conneulor assume infinitelythin sections, and the associated point rule (Miles, 1978) maybe applied to projections of the structure of interest (Gundersenet al., 1993). Projection effects, however, need to be carefullyconsidered in the application of the optical disector (Gundersen1986; Gundersen et al., 1988; see also West et al., 1991) forcounting cells by actively scanning a sampling windowthrough a 3D block of a structure of interest (for example,by adjusting the depth of the plane of focus in a light orconfocal microscope). The step-wise optical disector (Gunder-sen et al., 1988) uses the counting rule described by Sterio(1984). The continuous optical disector uses an associatedtransect counting rule (Reed and Howard, personal commu-nication) that is analogous to the associated point rule. Aunique transect is associated with each particle (e.g. the planeof maximal focus or ‘clarity’ of a particle). The particle iscounted if this transect lies between the top and bottom facesof the optical disector and is correctly sampled by the 2Dcounting frame of Gundersen (1977). In the case where anucleus can be identified in the particle, the associated pointrule rather than the associated profile rule applies.

Estimation of Euler number by magnetic resonance imaging

The synthetic porous medium studied by Roberts et al.(1997) was formed by heating silica grains between 200and 500 mm in diameter to 710 8C for 15 min andsubsequently cooling at a rate of 3 8C min¹1. Jernot &Lantuejoul (1999) suggest that this regime would causeonly small contact zones to develop between the originalparticles. However, qualitative inspection of several physicalsections cut from the specimen indicates that the originalgrains were rarely discernible after sintering (T. Fens,personal communication). Furthermore, data obtainedfrom previous stereological investigations of the samespecimen reported in Roberts et al. (1994) indicate thatthe sintering produced large alterations (see, for example,colour plate VII in Roberts et al., 1994). Prior to imaging,the specimen had been flooded with a mixture of oil anddegassed water and chemically selective MRI was used toproduce separate images of the oil and water phases(Roberts et al., 1994). The mean volume of the oil-filledpores was 203 × 10¹3 mm¹3, which is over seven timesgreater than the value of 27 × 10¹3 mm¹3 obtained for thewater-filled pores. We therefore question the validity ofestimating the Euler number density of our synthetic porousmedium as ¹ 2VV/v, where VV is the volume fraction of thesolid phase and v is the mean volume of the solid particlesprior to sintering, as proposed by Jernot & Lantuejoul(1999). The approach assumes that only very mild sinteringwill have occurred and this is clearly untrue. Finally,whereas the calculations made in Jernot & Lantuejoul(1999) refer to the solid phase of the porous medium, theEuler number density measurements quoted in Table 1 ofRoberts et al. (1997) refer to the fluid phase, which occupies36·1% (CE ¼ 3·1%) of the cell. Contrary to the suggestion ofJernot & Lantuejoul (1999), we did not ignore thetopological changes at the edge of sampling windows, butapplied the conneulor exactly as described by Gundersen etal. (1993).

Our motivation in demonstrating the application of theconneulor method for estimating Euler number density in aporous medium using an MR microscope was to bring theconneuler method to the attention of MR scientists,microscopists and petrochemists, and to obtain pilot datafor a specimen that was being studied extensively by ourcolleagues in industry. The combination of MR imagingwith the efficient and unbiased sampling strategies ofmodern design stereology undoubtedly represents a power-ful and versatile approach for non-invasive quantification ofgeometric parameters.

References

Bhanu Prasad, P., Lantuejoul, C. & Jernot, J.P. (1990) Use of theshell correction method for the quantification of three-dimen-sional images. Trans. R. Microsc. Soc. 1, 297–300.

Table 1. Summary of the stereological methods available forestimating number and Euler number.

Dimension Number Euler number

2D 2D counting frame 2D shell correction(Gundersen, 1977) (Bhanu Prasad et al.,

1989)3D 3D unbiased brick* 3D shell correction

(Cruz-Orive, 1980) (Bhanu Prasad et al.,(Gundersen, 1981) 1990)(Howard et al., 1985)

3D from 2D Disector Conneuler(i.e. 2.5D) (Sterio, 1984) (Gundersen et al., 1993)

*Cruz-Orive (1980) suggests that ‘direct enumeration of particleswithin a three-dimensional sample of the structure by anyunbiased counting rule [e.g. the natural extension to three dimen-sions of the ‘tiling method’ of Gundersen (1977), or the ‘associatedpoint’ technique of Miles (1978)] would require, say, a three-dimensional reconstruction of the particles and careful control ofedge effects’.

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Bhanu Prasad, P., Lantuejoul, C., Jernot, J.P. & Chermant, J.L.(1989) Unbiased stimation of the Euler–Poincare characteristic.Acta Stereol. 8, 101–106.

Cruz-Orive, L.M. (1980) On the estimation of particle number. J.Microsc. 120, 15–27.

DeHoff, R.T. (1987) Use of the disector to estimate the Eulercharacteristics of three-dimensional micro structures. ActaStereol. 6, 133–140.

Gundersen, H.J.G. (1977) Notes on the estimation of the numericaldensity of arbitrary profiles: the edge effect. J. Microsc. 111, 219–223.

Gundersen, H.J.G. (1981) Stereologi: eller hvordan tal for runlig formog indhold opnas ved iagttagelse af struktureer pa snitplaner.Laegeforeningens Forlag, Copenhagen.

Gundersen, H.J.G., (1986) Stereology of arbitary particles. A reviewof unbiased numbers and size estimators and the presentation ofsome new ones, in memory of William R. Thompson. J. Microsc.,143, 3–45.

Gundersen, H.J.G., Bagger, P., Bendtsen, T.F., Evans, S.M., Korbo, L.,Marcussen, N., Moller, A., Nielsen, K., Nyengaard, J.R., Pakken-berg, B., Sorensen, F.B., Vesterby, A. & West, M.J. (1988) Thenew stereological tools: Disector, fractionator, nucleator andpoint sampled intercepts and their use in pathological researchand diagnosis. APMIS, 96, 857–881.

Gundersen, H.J.G., Boyce, R.W., Nyengaard, J.R. & Odgaard, A.(1993) The conneulor: unbiased estimation of connectivityusing physical disectors under projection. Bone, 14, 217–222.

Hadwiger, H. (1957) Vorleungen uber Inhalt, Oberflache undIsoperimetrie. Springer-Verlag Berlin, pp. 237–243.

Howard, C.V., Reid, S., Baddeley, A.J. & Boyde, A. (1985) Unbiasedestimation of particle density in the tandem scanning reflectedlight microscope. J. Microsc. 138, 203–212.

Jernot, J.P. & Lantuejoul, C. (1999) Comments on Estimation ofthe connectivity of a synthetic porous medium. J. Microsc. 193, 97–99.

Miles, R.E. (1978) The sampling, by quadrats, of planar aggregates.J. Microsc. 113, 257–267.

Nyengaard, J.R. & Marcussen, N. (1993) The number of glomerularcapillaries estimated by an unbiased and efficient stereologicalmethod. J. Microsc. 177, 27–37.

Ohser, J. & Nagel, W. (1996) The estimation of the Euler–Poincarecharacteristic from observations on serial sections. J. Microsc.184, 117–126.

Pakkenberg, B. & Gundersen, H.J.G. (1997) Neocortical neuronnumber in humans. Effect of sex and age. J. Comparative Neurol.384, 312–320.

Roberts, N., Nesbitt, G. & Fens, T. (1994) Visualization andquantification of oil and water phases in a synthetic porousmedium using NMRM and stereology. Nondestruct. Test. Eval. 11,273–291.

Roberts, N., Reed, M. & Nesbitt, G. (1997) Estimation of theconnectivity of a synthetic porous medium. J. Microsc. 187,110–118.

Sterio, D.C. (1984) The unbiased estimation of number and sizes ofarbitrary particles using the disector. J. Microsc. 134, 127–136.

West, M.J., Slomanika, L. & Gundersen, H.J.G. (1991) Unbiasedestimation of the total number of neurons in the subdivisions ofthe rat hippocampus using the optical disector. Anat. Rec. 231,482–497.

Zhao, H.Q. & MacDonald, I.F. (1993) An unbiased and efficientprocedure for 3D connectivity measurement as applied to porousmedia. J. Microsc. 172, 157–162.

NEIL ROBERTS

Magnetic Resonance and Image Analysis Research Centre (MARIARC),University of Liverpool, PO Box 147, Liverpool, L69 3BX, U.K.

MATT REED

Department of Fetal and Infant Toxico-Pathology,University of Liverpool,

PO Box 147, Liverpool, L69 3BX, U.K.

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