quantitative methods study pulmonarypathology

Thorax (I 962), 17, 320.

QUANTITATIVE METHODS IN THE STUDY OFPULMONARY PATHOLOGY

BY

M. S. DUNNILL*From the Department of Medicine, Columbia University, and the Cardiopulmonary Laboratory,

Bellevue Hospital, New York

Clinico-pathological correlation has always beenone of the main aims of the morbid anatomist.However, although clinical physiologists have formany years expressed their results in a quantitativeform, pathologists have progressed mainly, thoughnot entirely, along a descriptive path. Descriptivepathology has reached a high degree of refinementwith the study of ultra-structure by the electronmicroscope and the probing of cellular chemistryby histochemical methods. Little interest hasbeen shown in the field of quantitative morpho-logy, although a quantitative study of gross andmicroscopic pathology, correlated with the findingsof physiopathologists, might well yield muchuseful information. Thompson (1917) pointed outmany applications of this approach to generalbiology, and recently Grant (1961) has drawnattention to the importance of this type of studyin cardiac pathology and has referred to the workof Linzbach (1960). This paper is concerned withthe methods of quantitation as applied to the lung,but the methods mentioned are capable ofadaptation to almost any organ.Pulmonary physiology is a subject which has

undergone enormous advances over the past 30years, mainly since the introduction of cardiaccatheterization as a safe clinical procedure byCournand and Ranges (1941). The relatively slowprogress of pulmonary pathology is probably dueto inadequate descriptions of the various aspectsof emphysema and chronic lung disease. This hasbeen remedied largely by the work on chronicbronchitis of Reid (1954) and the description ofcentrilobular emphysema by Leopold and Gough(1957). What is needed now is precise informationconcerning volumes of normal and abnormalregions of the lung, the surface area of the air-tissue interface, and the number of the variousunits, such as alveoli and alveolar ducts, in normaland emphysematous lungs. The purpose of thispaper is to show how these measurements can be

*Present address: Department of Pathology, The RadcliffeInfirmary, Oxford.

made on the gross specimens available at necropsyand on histological sections with the lightmicroscope.The problems involved bear a striking

resemblance to those of the geologists. Chayes(1956) in a remarkable book sums up the methodsavailable to those interested in the quantitativecomposition of rocks. He remarks in his prefacethat his object is to put " numerical flesh on thebones of definition," a goal that should be theambition of every morbid anatomist. This workrequires the abandonment of such well-worn termsas " extensive fibrosis" and "a mild degree ofemphysema" and their replacement by numericalvalues.

PREPARATION OF ORGANS FOR QUANTITATIVEANALYSIS

The traditional methods of performing anecropsy are often unsuited to the quantitativeapproach which in general requires that organsshould be fixed before they are cut. It is necessaryto measure the volumes of organs in the freshunfixed state. This means, in the case of the lung,very careful removal from the cadaver, to avoidpuncturing the visceral pleura, and careful infla-tion to a volume comparable to that presentduring life. This volume can be estimatedapproximately from measurements of the chestwall. If lung volume studies were performedduring life so much the better, but it is importantin this case to allow approximately 500 ml. forthe volume of tissue, as opposed to air, in eachlung. The physiological measurement of totallung capacity refers only to air volume. Thevolume of the inflated lung can then be measuredby water displacement in a tank, care being taken,by means of a suitable "corking" device, toprevent escape of air from the bronchus duringthe measurement. The procedures of fixation,dehydration, clearing, and embedding all alter thedimensions of an organ and thus careful measure-

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QUANTITATIVE METHODS IN PULMONARY PATHOLOGY

ments are needed both of the volume of the fixedorgan and of the size of the final histologicalsection to determine fixation and processingconstants.

Various methods of lung fixation have beentried, usually involving infusion of fluid formalinalong the trachea or main bronchi, and these havebeen reviewed by Heard (1960). Such methods,particularly when performed under considerablepressure, are unsuitable for quantitative studies, ashas been pointed out by Hartung (1962). Theprocedure described by Weibel and Vidone (1961)using formalin steam is preferable to any othermethod. The lung can be expanded to thedesired dimensions in a controlled manner bymeans of "negative" pressures similar to theintrapleural pressures present during life. Thelung retains its shape when removed from thefixing chamber, and there is practically no lateralflattening which is so common with fluid fixation.The final histological slides have a beautifullyclean appearance, and the delicate bronchialepithelium is preserved in a manner not seen withany other method. The volume of the fixed lungcan be measured by water displacement withremarkably little deformity, particularly if ithas been floated in Zenker's solution overnight.The volume of the fixed lung is less than thevolume of the fresh organ, and the expressionVolume of fresh lung 3

Volueo fied ung= f3 will give the volumeVolume of fixed lungfixation constant for conversion of fixed to freshlung. The constant for area measurements will ofcourse be f2 and for linear measurements f.Similar processing constants (p3, p2, and p) canbe obtained by measuring the size of the blockstaken from the fixed specimen and comparingthese measurements with the size of the histo-logical section on the slide. The reciprocals ofthese values will give the constants for convertingdimensions of fresh to fixed material and fixed toprocessed material. It is necessary to carry outthis procedure in every instance as these constantsvary from case to case with the age of theindividual and the disease process.

ASSESSMENT OF VOLUMES ON THE GROSS(FIXED) SPECIMEN

In order to estimate the volumes of normal andof diseased tissue in an organ the methodemployed must be sufficiently accurate, applicableto tissues of considerable complexity, and simpleto use. All the methods available depend uponthe principle first announced by the geologist

Delesse (1848) that in a complex mineral the arealproportions of a section of the mineral areequivalent to the volume proportions. The mathe-matical proof of this principle was not given untilover 100 years later by Chayes (1954), althoughit had been in use as a tool of geologists since1916 when Shand invented his recording micro-meter. The problem thus resolves itself into thedetermination of areas or ratio of areas of variouscomponents on a cut surface.

Rosiwal (1898) extended the principle byproposing that " if the fraction of a line traversinga given component on a section can be determined,this linear fraction provides a good estimate ofthe fraction of the volume occupied by a givencomponent." The length occupied by each com-ponent on the lines was measured, and it can beproved mathematically that the total lengths ofline occupied by the various components areproportional to the areas of the components andhence to the volumes. It was to ease the tedious-ness of this linear measurement in geology thatShand invented his micrometer. On grossanatomical specimens the procedure would,however, be formidable, although integratingeyepieces are available which use the sameprinciple for the analysis of histological sections.Most linear measurements use a system of straight,equidistant, parallel lines, and although this maygive consistent results they are not entirely freefrom bias. In gross pathology camera lucidadrawing and planimetry might be employed inarea estimation, but in an organ as complex asthe lung this would be excessively tedious.

It is, however, possible to perform the task ofarea ratio estimation in a manner which is bothspeedy and accurate, and also free from bias, bythe point-counting method. It has been employedon histological sections by Chalkley (1943), andin petrographic analysis by Chayes (1956), but itdoes not appear to have been used in gross patho-logy. Chalkley approached the problem fromthe concept of a mathematical point movingrandomly through a tissue. The proportion of itspath resting on the various constituents will, if thepath is infinitely prolonged, approach as a limitthe proportions of the fractions of volumeoccupied by these constituents. Chayes (1956), on

the other hand, uses the Delesse principle of theequivalence of areal and volume proportions andthen gives a mathematical proof of the efficacy ofpoint sums as estimators of relative areas. Thus,in Fig. 1, if N random points were placed on thewhole area the ratio of the shaded area to the

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whole area would be the same as the ratio of thenumber of points, n, lying in the shaded area

to N. That is Area of shaded part _ n9 Total area N

Obviously the more points that are assessed thegreater the accuracy of the method. That it ispossible to obtain a very high degree of precisionhas been shown by Chayes and Fairbairn (1951).

FIG. 1. The proportion of random points lying in theshaded area will be proportional to its area and stillbe independent of its shape.

In practice the lung is bisected through thehilum and the cut surfaces placed face downwardson a wooden board fitted with Perspex runners1 cm. deep. The lung is then cut into 1 cm. thickslices using a specially made 17 in. ham knife.The slices are then laid down on a table in a goodlight and a piece of transparent cellophane, withthe point-counting grid drawn on it, is placedover them. The points lie 1 cm. apart and aresituated at the angles of equilateral triangles ofside 1 cm. (Fig. 2). This distance of separationhas been found to give good precision andreproducibility in the lung, which is the seat ofpulmonary emphysema, although when assessingother organs the points may have to be placedcloser together. The next step is to define thecomponents which are to be evaluated. Inpulmonary emphysema the following componentshave been found satisfactory: (1) Non-parenchyma (bronchi and blood vessels down to2 mm. diameter): this normally occupies about10% by volume of the normal lung and rather lessin emphysema; (2) normal parenchyma; and(3) abnormal air spaces. These components arethen listed and, with the aid of a hand lens, thetissue lying under each point in the grid is assessed.The process is repeated for each slice. If the threecomponents to be measured occupy fractions P,Q, and R of the total of volume V, then thenumber of points p, q, and r counted in these

fractions will be in the same relation to each otheras are the volumes, i.e., p:q:r: :P:Q:R. If thetotal volume of the fixed lung is known, then thevolume occupied by the various components canbe calculated both in the fixed specimen and, usingthe fixation constants, in the fresh specimen. Itis of course important in defining the componentsto be measured to ensure that together they makeup the total lung volume, i.e., P+Q+R=100%V.In most lungs some 2,000 to 3,000 points are

assessed; this may seem an arduous task, but withpractice, and an assistant to record the findings,both lungs can be assessed within 30 to 40 minutes.The method is reproducible by separate observersto within 5%. It can also be used to evaluatequantitative regional differences in the lung. Itshould be noted that this method assumes that,although the total volume of the lung alters withfixation, the volume proportions remain constant.

Before leaving consideration of the gross speci-men, it should be pointed out that measurementof the linear dimensions of emphysematous spaces,particularly centrilobular spaces, is possible at this

4 4-. 4 .4. ,

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+ + + 4- 4 4 4 4 4 +

. 4 4 4. 4 4 4 +.t

+ 4 4 4 + 4 +e+ +44- 4 | + +

4- 4-s v + +

+ + + + + + 4v 4- +

+ + + + + + 4 s+_ A

FIG. 2. A diagram to show the arrangement ofpoints in apoint-counting grid usedfor the estimation of volumeproportions on the gross specimen. The points arearranged at the angles of equilateral triangles of side1 cm.

stage. A frequency distribution diagram of thesize of these spaces can be constructed and alsothe size of the spaces related to their regionaldistribution. What is needed, however, is measure-ment of the number and surfaces of the variouscomponents (alveoli, ducts, emphysematous spaces)

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in the normal and abnormal areas, and for thisanalysis it is necessary to have histologicalmaterial.

SAMPLING OF THE LUNG

In any histological analysis of the lung samplingis all important. In the normal organ fewproblems arise as, apart from a small region nearthe hilum, the parenchyma is fairly uniform. Ina lung which is the seat of emphysema or otherchronic lung disease uniformity may be present,but more often it is strikingly absent. It mightbe possible to examine the entire lung histo-logically, but the labour, time, and expense of thisprocedure render it expedient to select blocks oftissue of moderate dimensions to be submitted fordetailed examination. It is an important premisethat the selection of these samples must ensurethat they give a picture of the whole organ, notjust of the " interesting " areas. This representsa radical departure from the normal method otselecting tissue for histological examination. Thistype of sampling must of course receive somewhathostile criticism until its efficacy is accepted.Objections were levelled at the small samplemethods when they were introduced into botanicaland agricultural research, though to-day they areaccepted in their entirety.Two methods are available. The first is

systematic sampling, and this involves takingblocks of tissue at given intervals throughout thelung. Its main disadvantage is that, if the diseasedareas have a similar type of arrangement to thesampling pattern, the sample will be unrepresenta-tive of the lung as a whole, and any resultsdeduced from such samples will be hopelesslywide of the true state of affairs. The alternativeis random sampling, the selection of blocks bymeans of a random number table. In theory thismethod has the property of securing a small groupof blocks of tissue possessing the same charac-teristics as the entire lung, i.e., the same propor-tion in which each special feature is present orabsent. In practice in a totally random sample,where each of the sample units has an equalchance of selection, a highly unrepresentativeselection is possible, e.g., two adjacent blocks oftissue from the upper lobe and no blocks fromthe lower lobe in a given slice of lung. To preventthe occurrence of this type of selection a com-promise between random and systematic samplingis employed. This is the principle of stratifiedrandom sampling which retains the advantages ofrandom sampling but imposes some restrictions on

the fluctuations encountered in a simple randomsample. The theory and justification of stratifiedrandom sampling have been rigorously treated anddiscussed by Hendricks (1956).The method used in the lung employs grids,

consisting of squares of side 1 cm. (Fig. 3), drawnon pieces of transparent cellophane. Each square

FIG. 3. A diagram to show the design of the grid used instratified random sampling of the lung. The squaresare of side 1 cm. The perforations at the corners ofeach square are large enough to admit a pin head.

is numbered, and the corners of the squares areperforated. The grids are made of such a size thateach will cover entirely a slice of lung. A gridis placed over each slice. If it is proposed to take40 blocks for histological study from a pair oflungs and there are five slices for each lung, fourblocks will be taken from each slice. In factall the slices will not be of the same size, and thenumber of blocks taken from each slice will bedetermined by the size of the slice. Furthermore,employing a system of stratification meansselecting the first block in each slice on a randomnumber basis, and subsequent blocks are chosenby addition of a constant number. This avoidshaving two adjacent samples. The selected squareson the grid are marked with a pin which passesthrough the perforation at one corner into theunderlying specimen. When all the blocks havebeen marked the grids are removed, leaving thepins in situ, and the blocks are cut out. Blocksare taken, using an arbitrary convention, with thepin situated at the top right-hand corner of theblock. As near as possible blocks should be of astandard size; 2.8 x 2 x 1 cm. is convenient, butwhatever their size each block must be measuredcarefully so that it can be compared with the sizeof the final section on the slide to determine theprocessing constant referred to above. Samplesare not taken from the hilar regions where largebronchi and blood vessels are present.

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ANALYSIS OF HISTOLOGICAL SECTIONSVOLUME PROPoRTIoNs.-The methods available

for volumetric analysis all depend on the Delesseprinciple which has already been discussed. Thereare two types of integrating eyepiece available forthis work. The first employs the linear measure-ment principle of Rosiwal (1898) and is somewhattedious if used for any length of time. The seconddepends on the point-counting principle alreadymentioned when dealing with the gross specimen.This was the method used by Chalkley (1943). Theinstrument devised by Chalkley employed onlyfour points. This seems a very small number, buttoo many points in one microscopic field would bedifficult to assess. Twenty-five would seem to be areasonable number. Such eyepieces are availableand are easy and rapid to operate. The points arearranged at the angles of equilateral triangles(Fig. 4).

FIG. 4. A diagram of the integrating eyepiece of the pointcounting variety. Points are arranged at the anglesof equilateral triangles. A peripheral rim of themicroscopicalfield is left out of the grid as this is theregion ofgreatest spherical aberration where the exactlocation ofapoint on a section is difficult to determine.This is the design of the Zeiss integrating eyepiece.

This is a much more precise procedure than thatemployed on the gross specimen, and furthermoredifferent parameters are evaluated. In most casesof emphysema the following might well beassessed: alveolar air, alveolar duct air, abnormalair space, tissue, and vessels. The position of eachpoint in the eyepiece is then recorded accordingto which constituent it lies in on the histologicalsection. A hand-operated counting device, similarto those used for differential blood counts, has theadvantage that results can be written down every100 points instead of the laborious procedure of

recording on paper the position of each point asit is observed. This method is simple, rapid, andindependent of the shapes of the various com-ponents. The entire section can be covered withabout 500 points if a low-power objective is used.The objective chosen will depend on what is finallyrequired. Chalkley used this method to determinenuclear cytoplasmic ratios, and obviously for thiswork a very high power is needed. The assess-ment of the volume proportions in the lungrequires the recording of between 500 and 1,000points per section for consistent results. Theselection of the microscopic fields is important aslarge bronchi and blood vessels (>2 mm.diameter) must be avoided, since these are classedas non-parenchyma. When the situation of thepoints on each slide has been observed andrecorded the totals are added up. The propor-tions of the sums of the points in each componentare then taken to equal the volume proportions ofeach component.

SURFACE AREA ESTIMATION.-One of the mostimportant quantitative measurements to be madein the lungs is that of the internal surface area orair-tissue interface. The method of measuringthis by the mean linear intercept is both simpleand accurate. Its development is interesting.Tomkeieff (1945) appears to have been the firstto have drawn attention to the method of findingthe surface area of small objects from their area ofprojection. As with so many of these methods hewas at first interested in the grain size of rocks.This was developed by Campbell and Tomkeieff(1952) to include a number of small bodies suchas alveoli in the lung. They state that "if anumber of similar bodies, each of volume V andsurface area A, are enclosed at random in a unitvolume V and this volume is traversed by anumber of lines of total length T, if also L is thetotal length of these lines which is interior to theconvex bodies and N is the number of intercepts,then

L Volume of convex bodies LXAT Unit volume 4NV

and in the limit IA 4NV T

Campbell and Tomkeieff (1952) also demon-strated that the bodies need not be similar to oneanother in shape or entirely enclosed in a unitvolume. It is necessary, however, that the bodiesshould be randomly disposed towards the lines of

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traverse. This method was employed by Duguid,Hulse, Richardson, and Young (1953) to calculatethe respiratory surface area of the lung inexperimental animals.

Short (1950), in studying alveolar epithelium inrelation to growth of the lung, gave a similarformulation for the calculation of the internalsurface area of the lung. The proof that heoffered was somewhat different and he appearedunaware of Tomkeieff's work. Hennig (1956aand b; 1957) also arrived at this formulationindependently, and developed it further to includethe estimation of surface area of orientatedstructures such as intestinal villi.

In the application of this method to the lungcertain practical points must be noted. Thetraverses best employed for this_work are crossedhair lines, of equal and known length (Fig. 5),

FIG. 5. A diagram of a lung field with crossed hair linespresent in the eyepiece for use in the mean linearintercept methodfor the determination of the area ofthe air-tissue interface. A cut into the large vesseldrawn in the lower part of thefield would count as halfan intercept as would a cut out of the vessel. Thetraverse passing through an alveolar partition isregistered as a full intercept.

fitted to an eyepiece. The use of the two hairlines, at right angles to one another, compensatesfor any deformation that may occur to the sectionduring cutting and mounting. The selection ofmicroscopical fields on the section presents a

problem. It is possible to count every field on any

given section, but this is extremely time-consumingand some form of sampling is needed. A simplestratified random sampling system can be used,similar to that described for the selection ofblocks. The micrometer gauges on the mechanicalstage of the microscope are employed togetherwith a table of six digit random numbers. The

first three digits are used to give the referenceson the vertical scale and the second three thereference on the horizontal scale. Using thismethod 10 fields are easily selected. In order toavoid adjacent fields two readings can be madeon the horizontal scale to every one on the verticalscale. The next point that arises is how to dealwith traverses which cut structures other thanalveolar walls. The situation is shown in Fig. 5.Here a cut through an alveolar wall counts as asingle intercept. A cut into a blood vessel wall,however, counts as a half, as does a cut out ofa blood vessel wall. The number of intercepts iscounted on both the horizontal and the verticalhair line for each field and the sum for all thesections obtained. The mean linear intercept L.is then calculated from m, the sum of all theintercepts, L, the length of the traverses, and N,the number of times the traverses are placed onthe lung, i.e., in this case twicq for each field

N.Lexamined, Lm =

mThe surface area of the air-tissue interface S

is then given by S 4X vp where VLm

volume of the processed lung calculated from thefresh lung by applying the volume constants forprocessing and fixation; X= fraction of lungcontaining parenchyma as previously defined;p2=constant for converting areas of processed tofixed tissue; and f2 =constant for converting areasof fixed to fresh tissue.

This method has the great advantage that it ispossible to calculate the air-tissue interface inconditions such as emphysema where the bodiesare by the very nature of the disease not uniformin size. Duguid et al. (1953) used an electronicrecording device for the counting of the intercepts.While this is an ingenious method it is a littledifficult to see how such a method can allow forsmall blood vessels, and also how recording of ahit on a piece of alveolar debris can be avoided.In fact, counting by eye is both rapid andconsistent.ESTIMATION OF NUMBERS.-One of the serious

deterrents to a quantitative approach to morbidanatomy has been the lack of a suitable methodof estimating accurately the number of units in agiven volume from counting the number of unitsin a given area on a histological slide. Thisproblem has been very beautifully solved byWeibel and Gomez (1962). They have shown that

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the number, N, of randomly distributed bodies in

a unit volume is given by N= where n=fyp

the number of transections of the bodies on aunit area in a random section; p=the fraction ofthe entire volume occupied by the bodies; andf8=a constant depending on the shape but not thesize of the bodies.The principle is only applicable to sections

which are infinitely thin in comparison with thesize of the bodies under investigation. In the caseof alveoli, of average diameter 250 /A, a histologicalsection of thickness 5 I is satisfactory. If, how-ever, it is desired to count the number of cells inan organ such as the pituitary, a 5 /A thick sectionwould obviously be inadequate as the sectionwould be of the same order of thickness as thediameter of the cell being counted (approximately15 u). Thus for cytological counts electron micro-scope sections would be required. An alternativemight be to use optically plane objectives andcount only those cells in the plane in focus. Theshape constant ,B has been determined mathe-matically by Weibel and Gomez (1962) for variousbodies. The shape of a normal alveolus describedby Hartroft and Macklin (1944) as a truncatedcone surmounted by a cone was taken, and thisgave a value of 1.55.An eyepiece grid is used to count the number

of alveoli seen in a known area. The dimensionsof the grid are measured and then the numberof transections seen in the area outlined by thegrid is counted. A convention is used similar tothat used in counting white blood cells, i.e., alveolicutting the top line and the left-hand line of thegrid are counted as being within the area andthose cutting the right-hand and bottom line areconsidered outside the area. On any given slidethe number of alveoli are counted in 10 fields,selected by the same method of sampling as thatused in the mean linear intercept method. Themean number of alveoli per field is then calculatedfrom the counts on all the sections, the number perunit area calculated, and thus the number per unitvolume deduced. As the volume of the lungparenchyma is known the total number of alveolican be calculated. Weibel and Gomez (1962) havegiven a figure of 296.106 as the number of alveoliin the normal adult lung.

This method has been used by Dunnill (1962)for the determination of the number of abnormalair spaces in the lung. A different shape constant,,B, is used which, if these spaces are considered asspheres, is approximately 1.37.

If the number of alveoli and their volumeproportion in the lung are known the surface areaof the alveoli can be calculated. The surface S ofany body will be related to its volume V by theequation S=KVi where K is a constant relatedto the shape of the body. Weibel (1962) has founda value of 4.8 for this constant in the case of thealveolus. The mean volume of single alveolus Vwill be obtainable from the volume V occupiedby the lung parenchyma, y the proportion of thisvolume occupied by the alveoli, and N the totalnumber of alveoli by the following equation

Vy.V

From this it follows that the total surface S ofthe alveolar air-tissue interface will be given by

S = NK (^Y)The implications of this are interesting as it

provides a good check, in the normal lung, onthe surface area calculated from the mean linearintercept method. In the emphysematous lung it iseven more useful as it provides an estimate of thearea of the air-tissue interface of the alveoli asopposed to the total air-tissue interface due toalveoli and emphysematous spaces, which isestimated by the mean linear intercept method.

ASSESSMENT OF RESULTSIn a normal lung the standard deviation of the

mean for each of the histological measurements(mean linear intercept, alveolar transections, etc.)will be relatively small, as the lung parenchymawill be of uniform composition. This applies notonly to the microscopic samples obtained on anygiven slide but also to the combination of meanstaken from each slide. The standard error of themean, which depends on the square root of thenumber of the observations, can thus be reducedto a very small figure with relatively few observa-tions. In an emphysematous lung the standarddeviation between samples on any given slide, dueto the nature of the disease, is bound to be largebecause in some fields there may be few alveoli oreven none at all. Frequency distribution histo-grams of the intercepts on a normal and on anemphysematous lung (Fig. 6) illustrate this point.The assessment of the standard error of the

mean in these two cases will be different. In thenormal lung, with its uniform distribution, thestandard error, a,, may be calculated simply from

the equation ao = where or = the standard

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NO.N80

70

60

50

40

30

20

1 0

F FIELDS

(N ORMAL LUNG 200 FIELD"

2 3 4 5 6 7 8 9 loll 1213 1415 16 1718 19INTERCEPTS PER FIELD

FIG. 6a

EMPHYSEMATOUS LUNG (200 FrELDS)

N OF FIELDS60

50

40

30

20

I o

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16INTERCEPTS PER FIELD

FIG. 6b

FIG. 6. Freqiency distribution diagrams of intercepts in(a) a normal lung showing a relatively narrow dis-tribution of intercepts; and (b) an emphysematouslung; there is a much wider spread of the distributionof the intercepts.

deviation and n=the number of observations (i.e.,number of fields counted for alveolar transectionsor linear intercepts).

In the pathological lung, with its wide variationbetween samples on the histological section andbetween the various sections, the followingformula for finding the standard error crx of themean from a stratified random sample, recom-mended by Arkin and Colton (1956), is used:

a ( Ns a ) where ax =standardN2

error of the mean for each slide; N.= numberof observations in each slide; and N=numberof observations in the entire lung.

Obviously in a pathological lung many moreobservations will have to be made than in anormal lung, but the actual number will vary fromcase to case and will depend on how evenly thedisease process is distributed throughout the lung.The number of blocks taken for study by stratifiedrandom sampling to obtain any given standarderror cannot be predetermined, but in a lung whichis the seat of 50% emphysema, as determined bythe point-counting method on the gross specimen,20 blocks of tissue will give a standard error foralveolar transections of approximately 5%. Thiswvill not be the error of the estimate of the surfacearea of the air-tissue interface or number ofalveoli, because these will depend also on the errorinvolved in measuring the lung volume and thatinvolved in ascertaining the volume proportions.The estimation of the error in the determination

of the volume proportions by the point-countingmethod, either in the gross specimens or on thehistological slide, will be estimated somewhatdifferently. The standard error, r, of a percentage

area will be given by r= VA (l00-A) where A=N

the area estimated expressed as a percentage ofthe whole area and N=the total number of pointscounted.

It is obvious that the smaller the area to beestimated the larger the number of points thatmust be counted for any given error. Further-more, it can be seen that, as the error depends onthe square root of the number of observations, todecrease the error by half requires four times thenumber of observations. In fact the estimation ofthe error of the volume proportions, as determinedby the point-counting integrating eyepiece, isrendered very simple because the manufacturersof these eyepieces provide a normogram whichenables the error for any volume proportion to bedetermined. These normograms can be used todetermine the number of points which must beassessed for differing proportions to be within agiven range of error.

CONCLUSIONThese remarks have been concerned entirely

with the methods available for quantitativeevaluation of pathological changes in the lung.The methods themselves are, however, adaptableto other organs and systems. An attempt hasbeen made to analyse the available methods, most

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of which are in the geological literature and havereceived mathematical proof. The adaptation ofinstrumentation from the geological to the histo-logical field is simple, and adequate types ofintegrating eyepiece are readily available. Itwould seem that the time is ripe for furtherdevelopment in the field of quantitative morpho-logy. In certain areas of clinico-pathologicalcorrelation, notably in the lung, an abundance ofthis type of quantitative information is needed.

SUMMARYProgress in the correlation of pulmonary patho-

logy with pulmonary function studies is dependenton the collection of quantitative morphologicaldata. This is especially important in the variousforms of emphysema which have been studiedextensively by sophisticated physiological tech-niques. Methods are described for preparing andsampling the pathological lung and for deter-mining quantitatively (1) the volume proportionsof normal and abnormal lung, (2) the surface areaof the air-tissue interface of normal and abnormallung, and (3) the numbers of normal alveoli and ofabnormal air spaces in emphysema.

I am most grateful to Drs. E. R. Weibel andD. Gomez for introducing me to their countingprinciple before its publication. It is a pleasure toacknowledge the considerable help and encourage-ment received from them and from Drs. AndreCournand and Dickinson W. Richards. The work

was supported by grants from the Health ResearchCouncil of the City of New York, the United StatesPublic Health Service (USPHS-H-5741 (RI)), and theNew York Heart Association.

REFERENCESArkin, H., and Colton, R. R. (1956). Statistical Methods, 4th ed.

(rev.), p. 117. Barnes and Noble, New York.Campbell, H., and Tomkeieff, S. A. (1952). Nature (Lond.), 170, 117.Chalkley, H. W. (1943). J. nat. Cancer Inst., 4, 47.Chayes, F. (1954). J. Geology, 62, 92.

(1956). Petrographic Modal Analysis. Wiley, New York.and Fairbairn, H. W. (1951). American Mineralogist, 36, 704.

Cournand, A., and Ranges, H. A. (1941). Proc. Soc. exp. Biol.(N.Y.), 46, 462.

Delesse, A. (1848). Annales des Mines, 13, 378.Duguid, J. B., Hulse, E. V., Richardson, M. W., and Young, A. E.

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Sect. V, 38, 51.Hartung, W. (1962). Amer. Rev. resp. Dis., 85, 287.Heard, B. E. (1960). Ibid., 82, 792.Hendricks, W. A. (1956). The Mathematical Theory of Sampling.

p. 120. Scarecrow Press, New Brunswick, N.J.Hennig, A. (1956a). Mikroskopie, 11, 1.-(1956b). Ibid., 11, 206.-(1957). Ibid., 12, 174.Leopold, J. G., and Gough, J. (1957). Thorax, 12, 219.Linzbach, A. J. (1960). Amer. J. Cardiol., 5, 370.Reid, L. McA. (1954). Lancet, 1, 275.Rosiwal, A. (1898). Verh. Kongr. Geol. Reichsanst. Wien, 6, 143.Short, R. H. D. (1950). Phil. Trans. B, 235, 35.Thompson, D'Arcy W. (1917). On Growth and Form, 1st ed. Uni-

versity Press, Cambridge.Tomkeieff, S. A. (1945). Nature (Lond.), 155, 24.Weibel, E. R. (1962). Personal communication.- and Gomez, D. M. (1962). J. appl. Physiol., 17, 343.- and Vidone, R. A. (1961). Amer. Rev. resp. Dis., 84, 856.

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