331

The counting of cells and nuclei in microtome sections

By A. W. MARRABLE(From the Department of Veterinary Anatomy, University of Bristol)

SummaryA quantitative relation between the number of nuclei in a tissue and the number ofnuclear fragments in its resultant microtome sections has been established by a new-method. The general nature of the argument suggested that the correction procedure,although previously restricted to nuclei bounded by membranes, could be applied tomitotic figures and their fragments. With suitable interpretation this was found to beso, and numerical observations are presented which support this extension of themethod.

Depending upon their definitions, some of the various mitotic indices in use aretheoretically subject to fragmentation error. This effect should be considered andonly rejected, as in the example analysed, when it has been proved insignificant.

IntroductionW H E N a tissue is sectioned by a microtome some or all of its constituent cellsare cut into fragments; where serial sections are preserved related fragments arefound in adjacent sections. The number of cell fragments present in a seriesof sections is then greater than the number of whole cells present in the intacttissue. If this is true for whole cells it is in general true for their components,such as the nuclei, which are usually counted with greater ease than the cells.In order, then, to know the actual cell or nuclear number in a given piece oftissue, a correction for fragmentation must be made.

Agduhr (1941) recognized that the degree of fragmentation was a functionof section thickness and nuclear diameter. Using this principle he derived acorrection factor applicable to a model population of spherical nuclei whichhad been sectioned „, „

where JV = the original number of whole nuclei, n = the number of apparentnuclei counted, T = the section thickness (microns), D = the mean nucleardimension at right angles to the plane of section (microns), and a = thenumber of sections within which the representative nucleus wholly or partlyfalls.

Two investigations stemmed independently from Agduhr's work, those ofFloderus (1944) and Abercrombie (1946).

Floderus, examining the accuracy of Agduhr's correction formula by meansof scaled graphical constructions, found it to be valid only when the nucleardiameter was an integral multiple of the section thickness. Using a geometrical

[Quart. J. micr. Sci., Vol. 103, pt. 3, pp. 331-47, 1962.]

332 Marrable—Counting cells and nuclei in sections

argument, Floderus then arrived at a correction formula which, with slightrearrangement, can be written

The model system discussed by Floderus employed a population of spheres,equal in diameter and uniformly distributed. T, D, N, and n are defined asfor equation (i). Nuclear fragments would be zones and major and minorsegments (fig. i), with the hemisphere as a special case. The parameter k is

FIG. I. Diagram showing how a nucleus may be fragmented during serialsectioning, i—6, consecutive microtome sections seen in lateral view. Thespherical model nucleus is cut into 3 fragments; a major zone with anorthogonally 'projected diameter' dx — to the diameter of the sphere; asmall minor segment with diameter d2; and a large minor segment,diameter d3. The thickness k of the smallest visible segment is measured

along the radius perpendicular to the plane of section.

a secondary correction inserted by Floderus because he realized that some ofthe smallest nuclear fragments would be invisible in practice, and thereforenot counted. He defined it as the thickness of the smallest segment that canbe seen, taking his measurement along the radius perpendicular to the surfaceof the section (fig. 1). The evaluation of k for a particular investigation isimportant and will be discussed later. In other respects Floderus's formulais simple; the variable a of Agduhr is omitted as not necessary.

Abercrombie emphasized the significance of fragmentation and cited somecases of its neglect in histological studies. He made clear that in looking ata stained section we see a population of two components, whole nuclei andnuclear fragments. The relative strength of either component will vary with

Marrable—Counting cells and nuclei in sections 333

technical circumstances and in special cases one or other will be absent ornegligible; however, such insignificance may not be presumed without con-sideration. Abercrombie derived a factor which corrected the apparentnuclear number to the true value of whole nuclei:

N ( 3 )

The relationship established by Gray and Scholes (1951) is fundamentallythe same as equation (3). In using it, distinction must be made between thosenuclei which remain whole after microtomy (i.e. uncut nuclei) and equivalentnuclei, the latter being obtained by the theoretical reconstitution of those thatare cut.

In this study it has been convenient to treat first, fragmentation of nucleibounded by nuclear membranes, as in interphase, prophase, and late telo-phase. The method employed, together with some ancillary considerations,suggested that a similar correction factor could be applied to irregular mitoticfigures, as in metaphase, anaphase, and early telophase; this matter is takenup second.

Cells in interphase, prophase, and late telophaseThe correction factor established by a more general method

Using the same symbols we consider a model system of spheres evenlydistributed in a space which is cut into equal slices of known thickness. It isassumed initially that for any one case the spheres are the same size.

Case 1. T > D. When the diameter is less than the section thickness thesphere may be within the slice and not be cut (fig. 2, Ai). If cut, it can only becut once giving 2 fragments (fig. 2, A2). There is, therefore, a mixture of wholeand 2-fragment nuclei in the system. The relative strengths of the componentsof the mixture may be assessed, when there are a large number of spheres, byusing probability axioms. If the diameters of the spheres are very small inproportion to the section thickness, then for any one sphere there is a verysmall chance of being cut. This means that, of a large number of spheres, onlya small proportion will be cut. As the diameter approaches the section thick-ness the proportion of cut spheres increases.

Let p2 = proportion of spheres cut into 2-fragments,px = proportion not cut,

D Dthen p2 = — so that pt = 1 — - .

The total number of nuclear objects (fragments and whole nuclei) n

+ f.^. (4)

334 Marrable—Counting cells and nuclei in sections

By rearrangement the number of spheres

TN = T+D

0 o

FIG. 2. Diagram illustrating the fragmentation of nuclei of differentshapes and sizes. i-8, consecutive microtome sections of thickness T, seenin lateral view. Ai, model nucleus, with diameter D < T, uncut bysectioning. A2, nucleus, D < T, cut into 2 fragments. Bi, withT < D < -2.T, cut into 2 fragments. B2, with T < D < zT cut into3 fragments. Ci, ellipsoidal nucleus with major axis perpendicular tothe cutting plane. C2, ellipsoidal nucleus with minor axis perpendicularto cutting plane. C3, ellipsoidal nucleus with axes oblique to the cutting

plane.

Case 2. T < D < 2T. By similar reasoning to case 1, the possible frag-ment numbers are 2 and 3 (fig. 2, Bi and B2).

Let/>3 = proportion of spheres cut into 3-fragments.

Then

and

D-T

D-T

D-T

T I ' \ T

By simplification and rearrangement, again

T+D'

Marrable—Counting cells and nuclei in sections 335

The general case. fT < D < (f-\-i)T. Possible fragment numbers ( / + i )and (/+2), where/ = the integer < DjT.

D-f.TPf+2 — f •

A _ T D-f.T

fz).JV. (s)

TAgain AT = _ _ . „ ,

which is the same as Abercrombie's formula, and like that requires modificationto allow for the invisible fragments. The proposed manner of modification istaken up subsequently in connexion with a related problem.

Consequences of modifying the initial assumptionsThe preceding argument, like those of previous workers, starts with a

number of idealizing assumptions. It is necessary to examine the extent towhich they are fulfilled in the biological situation and to study the effect oftheir modification on the validity of the correction factor.

Size, shape, and orientation. Many nuclei may be treated as spheres sincealthough not perfect their deviations from that norm are not measurable byeyepiece micrometry. Plane-sections through these nuclei are circles which-ever way the nuclei are orientated. It follows that orientation has no effectupon their degree of fragmentation since they always present a similardiameter at right angles to the plane of section. Other nuclei visibly andmensurably deviate from the sphere, and so possess major and minor axes.In this case orientation has an effect, for if the major axis is perpendicular tothe cutting plane (fig. 2, Ci) the probability of being cut is proportionatelygreater than in the reverse case (fig. 2, C2). When a large number of ellipsoidsis randomly orientated in relation to the cutting plane the dimensions per-pendicular to this plane will vary in size between the lengths of the majorand minor axes (fig. 2, C3). The degree of fragmentation will depend on theaverage dimension perpendicular to the cutting plane.

When the nuclei are the same shape but of different sizes the dimensionsperpendicular to the cutting plane will vary and can be arranged in the formof a frequency distribution; the arithmetic mean of this histogram, when sub-stituted in the correction factor, will predict the extent of fragmentationprovided a large number of fragments is counted.

Many tissues contain a mixture of nuclei differing in shape, size, andorientation; that is, the situations discussed in the two previous paragraphsare compounded. Here again the fragmentation of any one nucleus dependsupon its size perpendicular to the cutting plane. The overall fragmentationof the mixture will depend upon the frequency distribution of a large number

336 Marrable—Counting cells and nuclei in sections

of such measurements; this distribution is epitomized in the correctionformula by its arithmetic mean.

By pursuing this argument an important generalization can be made whichapplies to the majority of cell nuclei. The degree of fragmentation of anindividual nucleus (or similarly discrete body) is independent of its shape butdependent only upon its linear dimension at right angles to the cutting plane. Inthe case of large numbers of such bodies fragmentation is independent oftheir shapes, but dependent upon the frequency distribution of the stateddimensions. The use of this generalization has made possible a practicalextension of the present counting procedure which will be described later inthis paper.

In the minority case of polylobular nuclei (e.g. granulocytes) fragmentationcould exceed that predicted by this principle.

Spatial arrangement. The disposition of the nuclear models in space isdescribed by Floderus as uniform and by Abercrombie as random. Realnuclei are presumably restricted in position by the confines of their cells,whose limiting membranes they seldom, if ever, touch. We may perhapsimagine them as varying in position about the central point of each cell socompromising these descriptions.

Estimation of the nuclear number using a correction factor

The value finally allotted to the nuclear number depends upon a knowledgeof four quantities, the nuclear diameter, section thickness, the apparentnuclear count, and a correction for invisible fragments.

Nuclear diameter (D). The value substituted in equation (2) or (5) clearlycannot be an individual measurement but must be a statistic which representsa population of differing diameters (Herdan, i960). To decide whichstatistic to use we can consider a model: a large number of equal spheresdistributed in space in a partially random manner. After microtomy we shallsee in the sections a population of nuclear fragments and (in the case whereD < T) residual whole nuclei. Geometrically a single fragment may be aminor or major segment, or where D > T a zone of a sphere (fig. 1). A sec-tion will contain a mixture of such fragments, all of which will have a circularprofile when examined by transmitted light. It is this orthogonally projectedcircle that is measured to give the 'projected diameter'. We see that when themaximum diameters of fragments are so measured, the diameters of majorsegments or zones will all be the same, and equal to the true diameter of awhole nucleus (fig. 1, a\). In contrast, the 'projected diameters' of minorsegments or zones will form a series diminishing in size from somethingdetectably smaller than a hemisphere down to a solid, but infinitely thin,tangential cap sliced from the nuclear membrane (e.g. fig. 1, d2 and d3). Sincethe planes in which the nuclei are cut are assumed to be random, with largenumbers we can expect a rectangular frequency distribution for minor seg-ments when classified by their thicknesses. However, the thickness is not theusual dimension measured microscopically, and when these same segments

Marrable—Counting cells and nuclei in sections 337

are classified according to their 'projected diameters' we get an unequalfrequency distribution. The shape of the new distribution will vary with thenuclear diameter, but if this is fixed the percentage of minor segments in eachsize class can be calculated by simple geometry. Table i shows the frequencydistribution to be expected in a model system of io ju,-diameter spheressectioned in the manner previously discussed.

TABLE I

Theoretical percentage distribution of random minor segments of JO-JU, spheresclassified by their 'projected diameters'

Class limits{microns)

9-108-97-86-7S-64-S3-42 - 31 - 3

0 - 1

% in class

43-6i6-411-4

8-66-65*0

3-72-6

i - 5o-6

Cumulative% smaller thanupper class limit

IOO-O

56-440-038-62O-O

13-48-44-72'1

o-6

It is now possible to construct a theoretical frequency distribution fornuclear measurements derived from such a model system. As an exampleassume that there are 100 nuclei in the tissue, let D = 10 /x and T = 15 ju..This is an example of case 1 (D < T), so we can use equation (4) and write

I X 100A .2X 100

15n= 1 .

I 15/= 1/3 .1 X 100+2/3 • 2 X 100.

This means that of 100 nuclei one-third or about 34 will be uncut. Whenmeasured by focusing their maximum diameters these whole nuclei willmeasure 10 ft each, and can be allotted to the 9-10-/A class in the frequencydistribution (fig. 3). The remaining two-thirds or about 66 nuclei will becut into 2 segments each. Ignoring the rare case of hemispheres, this willprovide 66 major segments and 66 minor segments. The 'projected diameters'of the major segments will measure 10 /x and so will join the whole nuclei inthe distribution. The diameters of the minor segments will be shared amongthe different size classes in the percentages given by Table 1. The theoreticalhistogram obtained is a small, but complete, distribution possessing distinctcharacteristics which will be present in varying extents in smaller samplestaken from the parent population. It is markedly skew, with an unchallengedmode at the value of the true nuclear diameter. The value of the mean reflects

338 Marrable—Counting cells and nuclei in sections

the diluting effect of the minor segment values. In reality it would not bepossible (or necessary) to resolve the modal class into the three componentsshown in the histogram.

I 70

i> 60

28 msegn

Class

FIG. 3. Theoretical frequency distribution of 'projected diameters' ofnuclei and nuclear fragments in microtome sections. Diameter of spherical

model nuclei = 10 fi. Thickness of sections = 15 p.

In these rigorous circumstances the mode closely represents the actualnuclear diameter and this value could be confidently substituted in the cor-rection factor. But the model is too simple to do more than illustrate a methodand suggest some reasonable interpretations of fragment distributions.Indeed, as Hoffman (1953) explicitly states, it is the converse problem thatrequires solution; given the frequency distribution of fragment diameters,reconstruct the distribution of nuclear diameters.

In reality the nuclei will be different in shape, size, and relative numbers,and the fragment distribution will be correspondingly complex. Often wehave no prior knowledge of the nuclear shapes, but for a particular tissuecareful inspection of a few individual nuclei is informative. For instance, ina homogeneous mass of actively dividing cells examination may show that alate telophase nucleus is spheroidal, whilst a prophase nucleus is larger andellipsoidal; we assume that these stages are connected by growth and thatother nuclei are intermediate in size and shape. Again, in an established tissuethe presence of different distinct cell types may be manifest in the contour ofthe histogram. The lateral resolution sets an ultimate limit to the precisionof linear measurements through the microscope. Thus the class intervals ofhistograms can hardly be smaller than 0-25 /x, and, so constrained, they mustnecessarily be insensitive indicators of size variability.

Marrable—Counting cells and nuclei in sections 339

An example of a real sample distribution is shown in fig. 4. The originalmeasurements were made on the nuclei and nuclear fragments in 7-/X. sectionsof a teleost blastula stained with Heidenhain's haematoxylin. Only a broadmodal region can be recognized and it is not coincident with the largest classas in the theoretical illustration. No fragments smaller than 7 (x, are recordedbut between 7 ju. and 10 JX the shape of the histogram agrees with that of themodel. It was supposed that a complex situation existed with nuclei of dif-ferent sizes and shapes; the arithmetic mean was used as the value of D.

._ 50

I 10

.5 30

i 10 ILCla

9 10 11 12 13 14

FIG. 4. Actual frequency distribution of nuclear and nuclear fragmentdiameters as measured in microtome sections of a teleost blastula. Thick-

ness of sections = 7 /x.

Since there are no irreproachable methods yet available for calculating thetrue mean value of D, our choice of estimate must be guided by considerationsof the kind presented here; sometimes the mode will be used, sometimes themean.

Section thickness (T). It is generally understood that the thickness attributedto a section is subject to considerable error. The value usually quoted is thelongitudinal feed of the microtome chuck according to the maker's calibration.The interference microscope (Barer i960; Hale i960) is capable of measuringoptical thickness to 0-05 fj, but cannot yet be considered as a routine instru-ment. Eranko (1955) reviews some earlier methods of measuring sectionthickness, most of which require special equipment and are not easily appli-cable to serial sections.

Hallen (1955) examined the accuracy of a microtome feed using a precisioninstrument developed in Swedish industry, the microkator. When set to cutat 5 /x, 24 microtome feedings gave a mean advance of 4-84 p. with a standarddeviation of 0-16 \i. He assessed the influence of knife design on the qualityof paraffin sections, using his own modification of the profile microscope tomeasure their thicknesses. Sections cut with the feed described above variedfrom 5-2 /A to 5-9 p. thick, increases of 8-23% over the actual feed, and 4-18%over the nominal feed. Hallen attributes these increases to compression ofthe sections after they had been cut. If this is so, it cannot directly affectfragmentation.

The phenomenon of 'thick and thin' sectioning, familiar to those who workwith serial sections, cannot be explained entirely on this basis. One section isapparently thin, while its neighbour is remarkably thick, and this alternation

34° Marrable—Counting cells and nuclei in sections

is repeated. Again, when ribbon sectioning is resumed after interruption, thefirst section is often visibly thicker than either its successor or predecessor. Itseems that the block is subject to compression by the longitudinal componentof the force delivered by the knife. Under some circumstances, easy to recog-nize but difficult to define, the compressed block will recover, so providingexcess material evident in the thick section. Fragmentation will be affected,and if a single section is used for a nuclear count its real thickness may differsignificantly from the feed value substituted in equation (3). When the frag-ments in a large sequence of sections are counted and the correction formulaapplied to their sum, it is reasonable to use the feed value if it is thought of asthe mean thickness of the sections.

The value of Tj( T-\-D) is more sensitive to a given relative change in sectionthickness with low correction factors than with high. For counting purposessections are of the same order of thickness as the nuclear diameter. When thesensitivity of the correction factor is examined at an arbitrary value of, say,0-5, it is seen that an unknown change in section thickness of ±10% may pro-duce a + 5 % or a —6% error in the number of estimated nuclei (but seeEranko).

Apparent nuclear count (n). Despite the aid of hand tallies and eyepiecegraticules, personal observational error usually intrudes in these counts. Itsextent may be assessed by replication but the labour is great. When alterna-tive lens combinations are used for comparative counts on the same section,notably different results may be obtained. The larger count is not alwaysobtained with the higher resolution objective, for this suffers from a shallowerdepth of field. It follows that, for a specified optical arrangement, somenuclear fragments are invisible. If the factor T/(T-\-D) is applied to this un-modified count an over-correction will result in an underestimation of the realnumber of nuclei.

Corrections for invisible fragments. The manner in which Floderus intro-duced this secondary correction is consonant with the rest of his analysis. Itwas as if the nuclei were reduced in diameter and therefore were fragmentedless; they were reduced by twice that amount (k) of the radius resident withinthe real invisible fragment (fig. 1). This scheme may be misleading in otherrespects as it substitutes imaginary small nuclei in a situation where there arein fact larger ones (equation (2)).

The value of k can be estimated directly by focusing the upper and lowerlimits of one of the smallest visible fragments and recording the accompanyingmovement of the micrometer adjustment. A well-known optical correction isnecessary and may be significant if the specimen is impregnated with a mediumof different refractive index from air (Brattgard, 1954; Galbraith, 1955).Floderus says that when using an objective X 40 and an ocular X 15 he foundk = 0-5 i*., although it is not clear how he obtained this value.

Another approach is convenient if a sample of nuclear fragment diametershas already been made for the purpose of finding D. With the assumption ofspherical nuclei the value of k can be calculated from the diameter of the

Man able—Counting cells and nuclei in sections 341

smallest fragment recorded. This method has the advantage that the fragmentdiameter being in the plane of section needs no correction for the refractivitiesof different media.

A third method also utilizes the fragment frequency histogram like thatshown in fig. 3. Such a distribution may be plotted cumulatively showing thepercentage of fragments falling below a chosen diameter (fig. 5). On geo-metrical grounds we can expect to see fragments of all diameters down to 1 /x

- iuu

> 90

jj 80" 70

5 60

nts

E 40

I 30

E 20

°. 1O

/-

/-•-

/

of fragn9 10 11 12 13 14

FIG. 5. Theoretical cumulative percentage distribution of 'projecteddiameters' of nuclei and nuclear fragments in microtome sections.Diameter of spherical model nuclei = IO/J.. Thickness of sections = 15/i.

and below in the microscopic fields. When they are not found in a reasonablylarge sample it must be due to their invisibility. If the size of the smallestfragment seen is known, then from the cumulative curve we can read off thepercentage of invisible fragments falling below that value; for example fromthe figure, about 11 % of all fragments are below 7 /x in diameter. The missingpercentage of fragments can now be added to the apparent nuclear countbefore equation (3) is applied.

Reasons for the invisibility of small fragments

Some fragments fall below the limit of lateral resolution of the opticalsystem. In order to provide a reasonably large field for counting, combinedwith a medium depth of field, I have used objectives with numerical aperturesof 0-65 and 0-95. A monochromator is not usually available but most workersuse a colour filter to enhance contrast in stained specimens; this will confinethe wavelength to some extent. Taking the wavelength of light passed by agreen filter as 5,000 A, such lenses will resolve detail down to a separation ofabout 0-5 p. and 0-35 /JL respectively. It is evident from fig. 4 that fragmentsmuch larger than 0-35 /x were missing from the distribution; it is thereforenot only the resolution limit that causes invisibility of fragments.

342 Marrable—Counting cells and nuclei in sections

When examining a stained preparation with transmitted light, provided thenumerical aperture is relatively large the refraction effects are small comparedwith those of absorption, and may be ignored. The image seen by the micro-scopist is a consequence of differential absorption by parts of the object, thecomparative effect being known as contrast. The amount of light absorbedby a structure depends upon its thickness, chemical constitution, and mole-cular orientation, as well as upon the direction and degree of polarization ofthe beam.

The validity of laws of light absorption, when applied to such heterogeneoussystems as stained biological tissues impregnated with resin, has been ex-tensively studied and criticized by those using the techniques of micro-densitometry and microspectrophotometry (e.g. Caspersson, 1950; Keohaneand Metcalf, 1959; King and Roe, 1953; Walker and Richards, 1959).Applications of the Beer-Lambert relationships must be made with careespecially when relative absorptions are taken as indices of relative con-centrations.

Caspersson showed that when optically anisotropic molecules were illu-minated with plane-polarized light the extinction varied with the orientationof the plane of polarization. Commoner and Lipkin (1949) concluded thatwhere a molecule has a low degree of symmetry the extinction would varywith the molecular orientation when in non-polarized light.

Some system of micellar or molecular orientation has long been expectedin the nuclear membrane (Frey-Wyssling, 1948). Electron-microscope studieshave shown it to be complex and probably composed of two layers, each layeritself containing orientated molecules (Callan, 1955; Robertson, 1959;Wischnitzer, i960). After its transformation by fixing and staining, remnantsof this organization will probably persist. If this is so, the absorption of lightduring its transit of the nuclear membrane is not likely to conform to theunmodified Beer-Lambert equations and no simple interpretation can bemore than an approximation.

With this cautionary background we can say that when a beam of parallellight passes through a hollow spherical nuclear membrane there is differentialabsorption because of the varying geometrical paths traversed between thelimiting cases of axial and tangential rays. The resulting luminous contrastbetween the centre and perimeter of the sphere is the 'circular' nuclear mem-brane seen by the observer. In the case of a major segment the contrast willbe similar to a whole nucleus since it includes the tangential zone. The con-trast between the perimeter and centre of a minor segment will be less as thelight path difference is smaller. For spheres of a given size we can expect thatat a certain fragment diameter the contrast will be negligible, so that minorsegments smaller than this will not be seen. King (1959) illustrates a micro-photometer trace (A = 257 m/x) across part of a squashed ascites cell. Thenuclear membrane gives an absorption maximum, the shape of which is ingood agreement with the present interpretation. The visibility of a stainednucleus or fragment will also depend upon the staining and distribution of its

Marrable—Counting cells and nuclei in sections 343

contents, and in some cases the absorption of the contents will be the dominanteffect.

Cells in metaphase, anaphase, and early telophaseThe extended use of the correction factor

In order to find the total number of cells present in a sample, or in a wholeorganism, those cells engaged in mitotic division must be included; in activelyhyperplastic tissue they will form an important component of the population.During prophase, the nuclear membrane is still intact and the fragmentationof such nuclei may be treated by the method already described. With therupture of the nuclear membrane the analogy with the model system isdestroyed, as the chromosomal aggregates present a sequence of less familiarshapes, changing from the equatorial plate of metaphase to the disjoint telo-phase groups.

In fact, any static mitotic figure can be stylized as a cylinder. At metaphaseits height is small compared with its diameter. During anaphase the heightincreases, the diameter decreases, and by telophase a tall slim cylinder hasbeen produced by continuous transformation. The appropriate new model isa population of cylinders uniformly distributed in space, their sizes varyingcontinuously between certain limits. For spheres and ellipsoids the dimen-sions at right angles to the plane of section control the degree of fragmenta-tion, and this condition will operate for cylinders. For a large collection ofmitoses, random in orientation, the arithmetic mean of these dimensions isthe statistic to use in the correction formula, which for practical conveniencemay be partially resymbolized as

l (6)where T and D are as before, m = the number of mitotic fragments counted,and M = the actual number of mitotic nuclei.

A notable feature of mitotic cells is the high density of the nuclear materialin the chromosomes and the corresponding concentration of dye in stainedsections. The visibility of small chromosomal fragments is thus greater thandimensionally similar fragments of vesicular nuclei, and in counting mitoseswe may expect a smaller error from this effect.

Experimental evidence

To test the above conclusions a study was made of the mitotic cells in a lateblastula of a teleost fish. The actual number of dividing cells was establishedby means of graphic reconstruction and compared with the estimate obtainedby using the corrected fragment number.

A complete series of sections was studied, and for each section the positionsof mitotic fragments were mapped by projection on to drawing paper. Everyfragment was allotted a number on the drawing and, with the aid of thisreference system and further subjective study, mitotic figures could be

344 Marrable—Counting cells and nuclei in sections

recognized as a whole or their reciprocating fragments could be identified inthe neighbouring sections. Three things were noted about each mitosis—thephase, its orientation in relation to the plane of section, and the number offragments into which it had been cut. This information is presented inTable 2. There was a total of 275 mitotic cells present in the blastula andthese were cut into a total of 586 fragments, i.e. 2-1 fragments per average cell.From this information the necessary correction factor was 0-469, and thiscould now be compared with the following estimates obtained from measure-ments upon mitoses.

TABLE 2

Three-way classification of total number of dividing cells in a teleost blastulaaccording to mitotic phase, orientation with respect to the cutting plane, and

degree of fragmentation

Uncut.2-fragments .3-fragments .4-fragments .Unclassified .

TOTALS .

Metaphase

perp. obi. para.

15 4 99 104 290 7 40 0 00 0 0

24 115 42

Anaphase

perp. obi. para.

0 0 10 8 104 10 00 0 00 0 1

4 18 12

Telophase

perp. obi. para.

0 0 00 19 104 14 06 5 0

0 0 0

10 38 10

Unclassified

perp. obi. para.

0 0 00 2 10 0 00 0 00 0 0

02 1

Totals

291924311

276

perp. = perpendicular, obi. = oblique, para. = parallel.

Three separate samples of 10 measurements were made upon dividing cellsin the same sections. They were chosen from those with their long axesparallel to the cutting plane but without regard to mitotic phase. Height andwidth were measured to the nearest micron. The arithmetic mean of the 20measurements in each sample was taken as the statistic of size, perpendicularto the cutting plane, of the mitotic figures. The means were substituted for Din the correction factor Tj( T-\- D) with T = 7 fj. (the nominal value). The threeindependent correction factors so obtained are listed in Table 3 with theirpooled value. The results of using such corrections on the fragment number,m = 586 in equation (6) are shown as the percentage error of the true numberof mitotic nuclei (M = 275).

TABLE 3

Estimated correction factors and the resultant percentage errors in M

Sample

1

2

3Pooled

Mean maximumdimension(microns)

9-6ic-49-79-9

Correctionfactor

0'422O'4O20-4190-414

Percentageerror

— IO-2— 14-2- 1 0 - 5- 1 1 - 6

Marrable—Counting cells and nuclei in sections 345

From this first evidence the method seems valuable as the size of the erroris biologically acceptable and the necessary sample of measurements relativelysmall. The systematic underestimate of M may have important origins innon-random orientation, a significant number of invisible fragments, and atendency to overestimate in linear micrometry. In undisputed cases of non-random orientation of nuclei (e.g. muscles, nerves) measurements taken in theplane of the counting sections will not sample the population dimensionsperpendicular to that plane. In these circumstances the latter measurementsmust be obtained from a second series of sections of similar material. Aber-crombie discusses this last matter in its relation to vesicular nuclei.

DiscussionThe method used to establish the correction factor has proved valuable for

its generality and, as a consequence, in its extension to the counting ofmitoses. The practical observations made on mitotic cells support the theo-retical expectation. A partial answer may now be made to Abercrombie'sinteresting speculation—'I wonder whether the size of mitotic figures is notsometimes relevant in the very numerous investigations of mitotic rate'(p. 242).

Various mitotic counts are used to assess the proliferative activities oforganisms; some have a meaning limited to the confines of the investigation.Bullough (1955) counts the 'Average number of mitoses arrested by colchicinein 4 hr. in unit lengths (1 cm.) of sections 7 \x, thick'. Gelfant (1959) counts thenumber of metaphase figures in five i-cm unit lengths of epidermis sectionedat 7 /x thick. Workers such as these have been wary of a general mitotic indexand have frequently used their counts only as comparisons with controls.Doljanski (i960) records the number of mitoses per 1,000 cells. Hoffmandefines the mitotic index as 'the ratio of the number of cells seen in mitosis tothe total number of cells present'. It is difficult to tell whether Messier andLeblond (i960) express the number of mitotic cells as a percentage of the non-mitotic cells or of the total number of cells, and a similar difficulty is asso-ciated with their radioactive index.

Because of fragmentation the results of mitotic counts upon smears andupon sections may not be comparable. For in sections the correction factorfor dividing cells and for interphase cells may be different and so affect thevalue of the mitotic index. Even when this fact is realized in the original workit can be lost sight of in quotation or when incorporated in calculations (e.g.Fry, Lesher, and Kohn, 1961). One field where such calculations have be-come urgent is that of haemopoietic analysis. Leblond and Sainte-Marie(i960) do not allow for fragmentation when computing mitotic indices in thethymus, although the cells are similar in size to the section thickness (5 JU.).

We can examine one such problem using the experimental teleost blastula.Table 4 sets out the data and the calculation of a mitotic index (M.I. = cellsin metaphase, anaphase, and early telophase as a percentage of the totalnumber of cells) both with and without a correction for fragmentation. In

346 Marrable—Counting cells and nuclei in sections

this instance the results are different but possibly not significantly so. Inother more highly differentiated tissues a greater error might be caused.

TABLE 4

(a) Calculation of mitotic index without correction for fragmentation

Number of mitotic fragments = 586Number of non-mitotic fragments = 2,194Total number = 2,780

M.I. = 586/2780 X 100 = 21-1%

(b) Calculation of mitotic index with correction for fragmentation

Number of mitotic fragments =586Mean dimension mitotic cell = 9-6 /x (Table 3, sample 1)Section thickness = 7-0 p

Correction factor = = 0-4227 + 9'6

Number of mitotic cells = 0-422 X 586 = 247Number of non-mitotic fragments = 2,194Mean dimension non-mitotic nucleus = 9-9 JX (fig. 4)Correction for invisible fragments, k = 1-45 fj,

Correction factor = —— = 0-5007 + (o -92Xi -45 )

Number of non-mitotic cells = 0-500X2194 = 1,097Total number of cells = 1,344

M.I. = — X 100 = 18-4%1344

The influence of fragmentation upon the measurement of nuclear sizes hasbeen less appreciated than its effect upon counting. Helweg-Larsen (1952)did not reckon explicitly with its manifestations in his extensive study ofnuclear sizes, although the consequences were probably minimized since byexploratory focusing he chose the largest diameters to measure. The evidencefor Jacobj's (1925) concept of nuclear size classes might be re-examinedfor the effects of fragmentation.

Certain comparisons may be made between automatic and human countingin sections. The discriminatory ability of man is at present subtler; Taylor(i960) states that the flying-spot film-scanner may count nucleoli but notpoorly stained or out-of-focus cytoplasm. Presumably the histograms ob-tained from such systems also contain a fragment component; Taylor showsa large number of particles (over 250) in the o-i-/x class. It is clear from hisaccount that the machine is more consistent than man and, a matter of greatimportance, not subject to fatigue. It would be interesting to see if the scannercould be empirically calibrated to give the same count as a visual one on acontrol section, so combining in this setting the best attributes of visual andmachine counting.

Marrable—Counting cells and nuclei in sections 347

I thank the following gentlemen for their generous advice: Professor C. W.Ottaway, Head of the Department of Veterinary Anatomy, and Drs. R. R.Ashdown, E. H. Batten, and G. Herdan, all of this University.

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