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The statistical approach in histology
.7. Theoret.Biol. (1962) 3, 111-122
The Statistical Approach in Histology STEPHEN M. SHEA
Department of Pathology, Harvard Medical School, Boston, Mass., U.S.A. (Received 22 November i96x ) A statistical approach to histology is outlined starting from the notion that the field of histology is morphological, and that its data are provided by the exercise of a skill. These data give rise by a variety of operations to certain scales of measurement. Particular attention is drawn to the use of the ordinal scale in histology. The concept of randomness has been utilized in several ways. Volume can be estimated by microscopic random sampling, and relative numbers of cells or nuclei can be determined by enumeration in representative tissue sections, provided systematic bias is allowed for. The observation that particular cells or events appear to occur at particular sites in a tissue may have to be tested against the alternative hypothesis that the spatial distribution is random. There are certain fallacies which may enter into the use of the appropriate tests for randomness and non-randomness. The statistical concept of a random variable underlies probabilistic models of cellular proliferation in tissues that are growing exponentially. Stochastic models of the steady state have not been constructed for cell populations. Models of tissues in a steady state can be constructed in terms of specific hypotheses concerning the component cells, i.e. as regards the distribution of age or intermitotic time. These models give little information about individual cells unless the equilibrium is perturbed and the restoration of equilibrium of the models compared with empirical data.
x. Introduction T h e general principle underlying histology is morphological. Refinem e n t s of technique, such as histochemistry, the use of fluorescent antibodies or radioactive tracers are considered in t e r m s of the distribution in space of the chemical, antigenic, or otherwise defined c o m p o n e n t s they reveal. Tissues are studied as patterns, and the anatomical principle enunciated b y Geoffroy Saint-Hilaire (I818) still holds: " . . . the only general principle which can be applied is given b y the position, the relations and the dependencies of the parts, b y what I n a m e and include u n d e r the t e r m connexions" (de Beer, i954). Morphological criteria in histology, as in comparative anatomy, are best 111
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defined in relation to concrete exemplars or standards. Semantically defined criteria would otherwise involve an endless regress. This is to say, in effect, that morphological criteria are operational ones concerning the exercise of a skill. In Euclidean geometry the two ways in which patterns can be said to be equivalent are by congruence---as between equilateral triangles of equal size--and by similarity--as between equilateral triangles of unequal size. In histology the analogy is with congruence, at least in as much as similar cells or organelles must be of a comparable order of magnitude. However, some variation in outline and in size are acceptable, the outline of one Purkinje cell being mentally accommodated to that of another, for example. This sort of equivalence transcends those of congruence and of similarity, and resembles topological equivalence or homeomorphism in these respects (Patterson, i956 ). 2. Histology and M e a s u r e m e n t
T o assert the autonomy of histological skill is not to rule out the possibility of measurement; on the contrary, histological comparisons provide the empirical basis of histological measurement. What measurement requires is a rule for attaching numbers to quantities, which is provided by the maxim: "No quantity of any kind without a comparison of different quantities of that kind" (Whitehead & Russell, x927). Two caveats must be entered here: First, it is recognized that the term "measurement" is commonly accorded to quantitative assays in which the quantities compared can in some sense be added. Other comparisons are usually referred to as giving "quantal" data, for which, of course, statistical methods of analysis exist. Secondly, addition of empirical quantities is easily provided for in the case of fundamental physical magnitudes, and these magnitudes and functions of them (derived magnitudes) are measured on scales regarded as being truly quantitative. In psychometric measurements, on the other hand, while the stimulus-strength may be precisely known, the empirical datum is the strength of a response, e.g. the tightening of hand-grip in response to a light stimulus of known intensity. There is less agreement as to the meaning of addition of such quantities as sensory responses, as contrasted with the addition of physical quantities. Histological data may involve direct comparisons of sensory responses, but not necessarily their "addition". Nevertheless, it seems appropriate to adopt, for the sake of generality, Stevens's (i946 , i958 ) theory of measurement for the description of measurement in histology. Stevens (i946 , i958 ) places special emphasis on the mathematical properties of the scales of measurement, and extends the concept of measurement to subsume the assay of "quantal" and of "quantitative" data in a single theory of measure-
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ment. For each scale a rule is provided for attaching numbers to data. Stevens (1946 , 1958 ) places less emphasis upon the semantic legitimacy of the concept of addition, and approaches measurement phenomenologically: he examines the scales generated by purported examples of "measurement", and evaluates the empirical operations in terms of their success in generating a scale, and of the mathematical character of the scales they generate. The formal mathematical properties of a scale determine the statistical operations which are permissible, i.e. the use one can make of one's measurements depends on the nature of the scale upon which they are made. This approach has obvious advantages for an accolint of histological measurement from a statistical point of view. Measurement, then, is the assignment of numbers to objects or events in accordance with some rule. As outlined by Stevens (1946 , 1958 ) four kinds of property can be used in the operation of assigning a number. Numbers may be chosen to stand for identity, order, differences or ratios in the items examined. According to how the numbers are chosen they accordingly constitute nominal, ordinal, interval or ratio scales. The empirical operation underlying the nominal scale is comparison with the determination of equality and inequality, i.e. classification. Suppose, for example, that the cells of the pituitary gland are being classified, all members of the same class (e.g. acidophils) are equated with one another. If, as in this example, there are more cells than classes, the numbers of cells in each class can be determined, as well as the class with the largest number of cells (the mode). The permissible statistics with classification, or measurement on a nominal scale, include the mode and the coefficient of association. The empirical operation underlying the ordinal scale of measurement is comparison with the determination of equality and inequality, and of greater and lesser. If these features of histological patterns can be determined, the patterns can be ranked in numerical order. Such a scale of numbers can be transformed in any way that preserves the order of the ranking. In mathematical terms, the numbers can be replaced by a monotonic increasing function. In graphic terms, the scale can be stretched (the numbers could, for example, be doubled or squared). The statistics calculated from such a ranking which remain invariant upon transformation of the scale include the median and the coefficient of rank correlation. Ordinal scales are further discussed in Section 3. They are particularly useful in representing gradual differences in pattern, ranging from minim u m to maximum values. The empirical operation underlying the interval scale is comparison with the determination of equality and inequality, of greater and lesser, and of T.B.
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equality of differences. While many psychometric scales are interval scales, it is probable that histological scales that do not meet the requirements of the ratio scale are more often found to be nominal or ordinal, rather than interval scales. The permissible (invariantive) statistics of the interval scale include the median, the mean and the standard deviation and two types of order correlation. The empirical operation underlying the ratio scale is comparison with determination of equality and inequality, of greater and lesser, of equality of differences, and of equality of ratios. Permissible statistics include the median and the mean, two types of order correlation and the coefficient of variation. The only transformation of scale permissible is that of multiplying it by a constant (e.g. expressing distance in/z, m/~, or A). Ratio scales of measurement are involved in enumeration, and are used extensively in cytology, as measures of linear size, volume, refractive index, cell mass, etc., and permit the use of a wide range of statistical operations. 3. N o n - P a r a m e t r i c M e t h o d s
Methods and scales of measurement giving rise t o ranked data are often spoken of as non-parametric, because they do not allow of the estimation of the statistical population parameters, the mean and the standard deviation. They have the compensating advantage that they do not involve the assumption that the variate measured is normally distributed. An example of data which have already been ranked will show how such information can be interpreted. In a study of cases of hemolytic disease of the newborn, Burn & Langley (i956) assigned I6 cases a rank order corresponding to the degree of sudanophilia of the adrenal cortex, as assessed histologically. The infants were also assigned a rank order for anemia, according to the hemoglobin content of the cord blood. These two rankings are remarkably similar. When Kendall's (i955) coefficient of rank correlation, ~,, was calculated, it was found to be equal to +o.73, and it was determined that such a degree of concordance of the rankings was unlikely to have been due to chance (P < o.ooi). Thus a correlation was established between the severity of the anemia and the degree of sudanophilia. Similar results were obtained in a study of histological changes in the diabetic kidney in man (Shea, Robbins & Mallory, x959). Twenty cases were ranked according to the severity of a pathognomonic glomerular lesion, and according to a non-specific arteriolar lesion. The two rankings were correlated with one another to the extent indicated by a value of T of +0"64 (P < o.oo2). In the latter example the ranking was obtained by the method of paired comparisons (Kendall & Babington Smith, i94o; Kendall, i955; and Shea, i958), in which the "determination of greater and lesser" is made as
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between the members of each possible pair of cases. This method has the particular advantage that the consistencyof an observer can be expressed in terms of a coefficient,called~. The method alsoprovides for an estimate of agreement among observers,in terms of a coefficient,calledu, which can take positiveor negative values. Thus the method of paired comparisons enables one to assess whether the procedure has been successful in generating a reproducible ranking. This is often as important as the existence of means of interpretingthe rankings by rank correlationmethods. Together, the method of paired comparisons and rank correlationmethods, which were introduced largely in the context of psychometrics, make it possible to quantify histological data which are refractoryto measurement on the more preciseand powerful ratio scale,but which would be inadequately exploited by the use of the nominal scale of measurement (i.e.classification). 4. Cellular Kinetics in Fixed Tissues
The distribution of cells in tissues has a temporal as well as a spatial significance, if one or more specific phases in the life of a cell can be identified. The first striking demonstration of the temporal significance of histological patterns (apart from embryological observations) was Payling Wright's (I925) calculation of the duration of the phases of the mitotic cycle from the duration of the telophase, as observed in in vivo preparations. In addition to the duration of the telophase, this calculation required the relative frequency with which the different types of mitotic figure are encountered in fixed tissue. Payling Wright (1925) assumed the duration of a mitotic phase to be proportional to the numbers of mitotic figures observed in that phase. A similar assumption makes it possible to estimate the renewal or turnover time of the cell population of a fixed tissue. The mitotic time is taken as constant, and the ratio of cells in mitosis to all cells (the mitotic index) in determined (Hoffman, I947, 1949; Sainte-Marie & Leblond, I958 ). The amount of information that can be obtained is well illustrated by the work of Sainte-Marie & Leblond (i958), who constructed a scheme for the renewal of lymphocytes in the thymus of the rat by determining the relative counts of reticulum-cells and of lymphocytes of three sizes and the mitotic index of each of these four types of cell. Several additional assumptions were necessary--the "life-span" of a cell was taken to be its individual generation time from "birth" to cell division (mitotic time plus intermitotic time), and to be the same for all cells of a type. This is equivalent to making the generation time equal to the renewal time of the cell type. Strictly speaking, renewal time refers to a population of cells or a "corn-
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partment" with a certain number of members. The renewal or turnover time is the time required for the turnover or renewal of a number of cells, of the type in a compartment, equal to the number occupying it. The results obtained by Sainte-Marie and Leblond (1958) were construed to be compatible with a sequence of four generations of large lymphocytes, followed by two of medium-sized lymphocytes and two of small lymphocytes. A more elaborate study of this kind is possible if more than one phase in the cell's history can be marked. In a study of the renewal of intestinal epithelium in the mouse in which mitoses and DNA synthesis determined by H a thymidine autoradiography were both identified, Quastler & Sherman (1959) were able to obtain a fairly detailed estimate of the number of cells in the different compartments (crypts and villi), and subcompartments (stages of the mitotic cycle), of the time spent by the cells in these compartments, and of the rates at which cells were being added (in the crypts), transferred (from crypts to villi), and lost (from villi). Schemes of this kind do not make explicit allowance for variation between cells in the duration of mitosis and of the intermitotic period. Both models cited are based upon generation or turnover times which are characteristic of the compartments rather than of individual cells. Sainte-Marie & Leblond (1958) took the generation time of all cells of a type to be the same, while Quastler & Sherman (1959) expressly confined their analysis to the mean value of the generation time. It is difficult, from material of this kind, to obtain information concerning the frequency distributions of intermitotic and mitotic times. An alternative approach is to work out the implications of specified hypothetical frequency distributions. This has been done (Hoffman, Metropolis & Gardiner, 1955, 1956; Hoffman, 1958 ) for the special case of exponentially growing tissues. The relation between the mitotic and intermitotic times of individual cells and mean values is expressed in terms of the statistical concept of a random variable. The intermitotic time, for example, is defined by a process of random sampling in which every possible value has a calculable, but not necessarily equal chance of selection. The chance of selection of a particular value of a random variable is determined by a frequency function (Kendall et al., 1958 ). If successive mitotic and intermitotic times are made repeatedly to assume values determined by random sampling and by specified frequency functions, theoretical models of the growth of a population of cells can be constructed. This sort of mathematical experiment can be made by use of a digital computer with statistical features, and is called the Monte Carlo method. The frequency functions can be specified in terms of hypothetical frequency distributions.
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Hoffman et al. (i956) experimented with combinations of one hypothetical frequency distribution for the duration of mitosis, with one or other of three different hypothetical frequency distributions for the intermitotic time. They studied the "growth" of cell populations from a single cell, paying particular attention to the growth of the population in size, and to the mitotic index. It was found that there were wide fluctuations in both, particularly in the early stages of growth. The characteristics of a population were found to depend upon the frequency distributions of the mitotic and intermitotic times. These determined, for example, the age distribution of the cells of the population, which tended to a limiting form as the population grew older. The numbers of mitoses in "clones" of ioo cells approximated to a Poisson distribution. Stochastic or probabilistic models of this kind have not apparently been used in the study of populations of cells in the later stages of growth (Harris, i959), such as those cell populations which are in a steady state. Deterministic methods have been applied by yon Foerster (I959) to populations in a steady state. The counting of elements of a population in a steady state does not give information about any assumptions concerning the individual elements, unless perturbations of the steady state are introduced. Specific assumptions concerning the age distribution of the elements of a population in a steady state have been shown (yon Foerster, I959) to influence the return to equilibrium to be predicted after the steady state has been perturbed. The comparison of the theoretical effect of perturbation with that of empirical perturbation will then enable the assumptions concerning age-distribution to be tested. This would appear to apply equally well to stochastic as to classical theoretical models. When the specific activity of tissue DNA can be determined it is possible to estimate the parameters of the distributions of generation times (for cells destined to divide) and of life-span (for cells destined to die) (Rigas, i958 , x959). This method is applicable to tissues in the steady state, or growing exponentially. The equations for other states are excessively complex and impractical. The method requires also quantitative autoradiography and was described in connection with the study of populations of leukemic cells. The duration of mitosis, which plays such a large part in the calculations of Sainte-Marie and Leblond (x958) and of Hoffman (I947, I949) can be calculated either from direct observation of living cells in vitro (Lambert & Hanes, i9i 3 ; Strangeways, I92Z) or by the use of irradiation to prevent cells entering prophase (Knowlton & Widner, x95o) or colchicine to arrest mitosis at metaphase. The fallacies in the use of colchicine have been outlined by Leblond & Walker (I956), and Leblond (x959), but if properly used it appears to give valuable results. In correct dosage colchicine causes
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a linear increase in the accumulation of metaphases over a period of a few hours, the rate of which is a measure of the mean mitotic time (Stevens Hooper, i959). The renewal of cell populations in a variety of tissues is reviewed by Leblond & Walker (i956), and more recently the subject has been dealt with in considerable technical and mathematical detail, in a hematological context, in a symposium edited by Stohlman (1959). 5- Enumeration and Volume Determination in Tissue Sections
Differential counts of cells in tissue sections, such as those of hemopoietic tissues, and of pituitary (Kindred, 1942; Kirkman, 1937) were originally treated in the same way as differential cell counts upon stained smear preparations on glass. There is a fallacy implied in this procedure because cellular structures of the same order of magnitude as the thickness of the tissue section will be encountered the more frequently the larger they are, even if their number per unit volume is the same. A correction has been proposed by Agduhr (1941) and a simpler correction by Abercrombie (1946). Abercrombie (1946) reviews the subject, and proposes the following correction in the case of a count of cell nuclei : M P = A - L+M where P is the corrected number of nuclei, A is the crude count of nuclei, M is the thickness of the section (in/~), and L is the average length of the nuclei normal to the plane of section. Abercrombie (1946) discusses the correction to be made in estimates of nuclear length from micrometry of the apparent nuclear population. Geometrical considerations show that very thin sections underestimate the mean nuclear length by 2 1 ~ ; sections comparable in thickness to the nuclei underestimate the mean nuclear length by about I 1%. The percentage errors in nuclear count when these values are inserted in Abercrombie's (1946) correction formula are about halved for ordinary paraffin sections. The 21% underestimate of nuclear length, when inserted in the correction formula, in cases in which the sections are "ultra-thin" would result in a comparable overestimate of the corrected nuclear count, P. Spheroidal organelles comparable in length to the thickness of the ultra-thin section would not be so greatly underestimated in length as larger structures. Sainte-Marie & Leblond (i958), in their work on the renewal of the lymphocytes of the rat thymus, corrected the differential cell counts in accordance with Abercrombie's (1946) formula. Counts of the larger
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cellular structures will require this correction even more with the use of ultra-thin sections. While correct enumeration is sensitive to the thickness of a section, by an apparent paradox nseful estimates of the relative volume occupied by a class of tissue components can be made from the examination of sections of undetermined thickness, without correction. This is done by the method of Chalkley (i943). In Chalkley's (i943) method a fine pointer, or several pointers, are inserted in the focal plane of the eye-piece of a microscope. The section of tissue is examined by recording, for several hundred observations, the identity of the tissue component intercepted by the tip of the pointer. Movements of the slide and of the focal point (within the limits of the section) are made at random. The coefficient of variation of these volume ratios is less than 3 ~/o. 6. Randomness of Spatial Distribution Randomness in the spatial distribution of cells capable of division; or random variation of the generation time, or, with cells having a fixed generation time, random distribution of the phases or "ages" of the cells (Hoffman, Metropolis & Gardiner, I956 ) should lead to a random spatial distribution of mitoses. In the case of regenerating rat liver the spatial distribution of mitoses 48 hours after partial hepatectomy is a random one (Brues & Marble, i937; Harkness, I952 ). This can be demonstrated by counting the number of mitoses per unit area, and comparing the observed distribution with the expected figures calculated for a Poisson distribution (Brues & Marble, I937; Harkness, x952). However, Harkness (i95z) observed at 24 hours a relative excess of mitoses in the periportal areas, in agreement with the observations of yon Meister (I894) and Milne (I9O9) in rabbits. When the numbers of mitoses observed in 64 × 64/zwere compared by Harkness (i952) with the expected figures calculated for a Poisson distribution, no significant difference was observed. When the unit area was x9z × i92/z the observed distribution departed significantly from a Poisson distribution. If a distribution is really random, no significant departure from a Poisson distribution should be observed no matter what the unit of area chosen for counting. To establish the randomness of a spatial distribution it is not sufficient to compare counts per unit area with a Poisson distribution, using only one unit of area for the purpose. We owe the discovery of this fallacy to Harkness (i952), who confirmed the non-randomness of mitoses in regenerating liver at 24 hours, and whose analysis confirms the observation of Brues ~g Marble (i937) that mitose8 occur at random at 48 hours.
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Discussion
The concept of randomness is a fertile one in histology. Chalkley's method (1943) is essentially a method of selecting points in space at random, and of using the results to estimate the relative volumes of tissue components. The demonstration that a spatial distribution is not random (Harkness, I95z ) may be necessary to confirm observations relating to particular sites in tissues. Failure of one test to demonstrate non-randomness does not necessarily mean a spatial distribution is random; it may be that the test selected is at fault. For the purpose of enumerating cells, tissue sections are samples biased in favour of the larger structures, and this bias must be corrected or fallacious conclusions will be drawn from counts of relative numbers of cells or nuclei (Abercrombie, i946 ). The concept of a random variable underlies stochastic models of cell proliferation (Hoffman et al. I955, I956; Hoffman, I958 ). These models have not yet been applied to cell populations in a steady state. Models of tissues in a steady state can be constructed in terms of the turnover of cells in various "compartments" (histological sites or cytological states) if a clearly defined phase in the life of a cell, such as mitosis, can be identified. The spatial distribution of mitoses has a temporal significance. If another phase, such as the period of D N A synthesis, can in addition be marked with radioactive tracers, even more information can be obtained. Von Foerster (i959) provides a very general discussion of populations in terms of the ages of the elements (cells). He points out that the problem is the reverse of that encountered in statistical estimation. In statistical estimation one seeks to obtain information about a population from information about the individual elements of a sample. In studying changing populations, one seeks to obtain as much information as possible about individual elements from information about the population as a whole. This may be done by describing the age of the elements in terms of substates representing age-groups. There is an intimate relation between the duration of a particular state and the number of elements in that state. When substates are age-groups, direct determination of the age of elements is equivalent to the classification and enumeration of the population by age-groups; both procedures present the same difficulties. Direct determination of a maximum value for the age of some elements is made possible by labelling cells at a known time before sacrifice, as in the work of Quastler & Sherman (i959). Another approach is to formulate specific hypotheses about the age distribution. Von Foerster (x959) has defined the relations that must exist between the functions of parameters controlling intrinsically or extrinsically determined losses of cells, and cell productiort
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in the steady state. Information about these parameters cannot be obtained by counting elements in the steady state. However, by introducing perturbations of the steady state and counting elements as a function of time, the return to an equilibrium is followed. It is also predicted by functions of parameters determining cell loss and cell production, and the age distribution of the population. Von Foerster (i959) made certain restrictive assumptions about intrinsic losses and cell production, and studied primarily the relation between perturbation and extrinsic removal mechanisms; but the method is quite general. This theoretical approach has in a sense been anticipated in the experimental work of Brues, Drury & Brues (I936) and Brues & Marble (i937) on the regenerating liver of the partially hepatectomized rat (Higgins & Anderson, i93I ). ]]rues, Drury & Brues (I936) studied the restoration of numbers of nuclei, while Brues & Marble (I937) studied the incidence of mitosis with time after partial hepatectomy. They did not, however, speculate on the nature of the distribution of the intermitotic time. The growth of regenerating liver is exponential during the first 72 hours (Brues & Marble, I937) but the growth rate decays exponentially from an initially established maximal value, unlike the exponential growth with constant characteristics of the models of Hoffman et aL (I955, i956 ) and Hoffman (x958). To account for the phenomena, a model would have to specify the changing distribution of intermitotic time, or equivalently of the ages of cells at the period of maximal growth, and for a certain period after the onset of regenerative growth. Assuming for the moment intrinsic and extrinsic losses to be negligible during this period, the partially hepatectomized liver represents in yon Foerster's (i959) terms a population perturbed in two ways : (a) Randomly--in that the numbers of cells in the various age groups have been altered, presumably to an equal extent; and (b) parametrically, to the extent that the "generation coefficient" has been altered. This latter parameter is non-linear, that is, it is dependent on the state of the population. Von Foerster (i959) believes that knowledge of the mathematical functional nature of such parameters obtained from perturbation of the steady state may help in the construction of models for the steady state itself. REFERENCES ABERCROMBIE,M. (I946). Anat. Rec. 94, 239. AGDUHR, E. (x94x). Anat. Rec. 80, I9I. BRtrr~, A. M., DRURY, D. R. & BRuEs, M. C. (I936). Arch. Path. 2z, 658. Baul~s, A. M. & MARBLE, B. B. (x937). a7. Exp. Med. 65, ~5. BURN, J. C. & LANOLEY,F. A. (I956). ft. Path. Bact. 72~ 47. CHALKLEY, H. W. (x943). ft. Nat. Cancer Inst. 4, 47. DE B~sm, SIR G. (1954). "Embryos and Ancestors", znd ed. Oxford University Press, Oxford, p. x26.
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FOERSTER, H. YON (I959). In "The Kinetics of Cellular Proliferation", Ed. F. Stohlman, Jr. Grune & Stratton, New York, pp. 382-407. GEOFVaOY SAINT-HILAIaE,E. (I818). "Philosophie Anatomique". Paris. pp. xxv-xxvi. Cited in translation by de Beer (1954). HARKNESS, R. D. (1952). ft. Physiol. xx6, 373. HARRIS, T. E. (I959). In "The Kinetics of Cellular Proliferation". Ed, F. Stohlman, Jr. Grune & Stratton, New York, pp. 368-38i. HIGGINS, G. M. & ANDERSON,R. M. (1931). Arch. Path. x2, 186. HOFFMAN,J. G. (x947). Science xo6, 343HOFFMAN,J. G. (1949). Bull. Math. Biophysics xx, 139. HOFFMAN, J. G. (1958). Biometrics x4, 139. HOFFMAN, J. G., METROI'OLIS,N. & GARDINER,V. (1955). Science 122, 465. HOFFMAN, J. G., METROPOLIS,N. & GAaDINER,V. (1956). ft. Nat. Cancer Inst. x7, 175. KEI~DALL,D. G. (1956). Proc. Third Berkeley Symposium, University of California Press, 4, I49. K~DALL, M. G. (1955)- "Rank Correlation Methods". Charles Griffin, London. KENDALL,M. G. & STUART,A. (1958). "The Advanced Theory of Statistics", Vol. I, Ed. 3. Charles Griffin, London. KENDALL,M. G. & BABINGTONSMITH,B. (I94O). Biometrika 3 x, 324. KINDRED, J. E. (1942). Am. ft. Anat. 71, 2o7. KIRKMAN, H. (1937). Am. ft. Anat. 61, 233. KNOWLTON, N. P. & WlDNER, W. R. (I95o). Cancer Research to, 59. LAMBERT, R. A. & HANES, F. M. (1913). girchows. Arch. path. Anat. 2Ix, 89. LEBLONI), C. P. (1959). In "Kinetics of Cellular Proliferation", Ed. F. Stohlman, Jr. Grune & Stratton, New York, pp. 31--49LEBLOND, C. P. & WALKER,B. E. (1956). Physiol. Rev. 36, 225. MEISTER, V. YON (I894). Beitr. path. Anat. x5, 1. MmNE, L. S. (19o9)..7. Path. Bact. I3, 127. PATTERSON,E. M. (1956). "Topology". Oliver & Boyd, Edinburgh and London. QUASTLErt, H. & SHERMAN,F. G. (1959). Experimental Cell Research 17, 420. RIGAS, D. A. (1958). Bull. Math. Biophysics 2o, 33. RmAS, D. A. (1959). In "The Kinetics of Cellular Proliferation", Ed. F. Stohlman, Jr. Grune & Stratton, New York, pp. 408-43 o. SAINTE~MARIE,G. & LEBLOND, C. P. (I958)- Proc. Soc. Exper. Biol. & Med. 97, 263. SHEA, S. M. (1958). Arch. Path. 65, 77SHEA, S. M., ROBEINS, S. L. & MALLORY,G. K. (1959). Arch. Path. 68, 447STEVENS,S. S. (1946). Science xo3, 677. STEVENS, S. S. (1958). Science x27, 383. STEVENSHOOPER, C. (1959). Unpublished observations cited by Leblond (1959). STOHLMA~, F., JR. (1959). Editor. "The Kinetics of Cellular Proliferation", Grune & Stratton, New York. STaANGEWAYS,T. S. P. (1922). Proc. Roy. Soc. B, 94, I37. WmTEHEAD, A. N. & RUSSELL, B. (1927). "Principia Mathematica", 2nd Ed., VoI. 3. Cambridge University Press, p. 261. WRIGHT, G. PAYLING,(1925). ft. Roy. Microscopical Soc. p. 414 ,
The Statistical Approach in Histology STEPHEN M. SHEA
Department of Pathology, Harvard Medical School, Boston, Mass., U.S.A. (Received 22 November i96x ) A statistical approach to histology is outlined starting from the notion that the field of histology is morphological, and that its data are provided by the exercise of a skill. These data give rise by a variety of operations to certain scales of measurement. Particular attention is drawn to the use of the ordinal scale in histology. The concept of randomness has been utilized in several ways. Volume can be estimated by microscopic random sampling, and relative numbers of cells or nuclei can be determined by enumeration in representative tissue sections, provided systematic bias is allowed for. The observation that particular cells or events appear to occur at particular sites in a tissue may have to be tested against the alternative hypothesis that the spatial distribution is random. There are certain fallacies which may enter into the use of the appropriate tests for randomness and non-randomness. The statistical concept of a random variable underlies probabilistic models of cellular proliferation in tissues that are growing exponentially. Stochastic models of the steady state have not been constructed for cell populations. Models of tissues in a steady state can be constructed in terms of specific hypotheses concerning the component cells, i.e. as regards the distribution of age or intermitotic time. These models give little information about individual cells unless the equilibrium is perturbed and the restoration of equilibrium of the models compared with empirical data.
x. Introduction T h e general principle underlying histology is morphological. Refinem e n t s of technique, such as histochemistry, the use of fluorescent antibodies or radioactive tracers are considered in t e r m s of the distribution in space of the chemical, antigenic, or otherwise defined c o m p o n e n t s they reveal. Tissues are studied as patterns, and the anatomical principle enunciated b y Geoffroy Saint-Hilaire (I818) still holds: " . . . the only general principle which can be applied is given b y the position, the relations and the dependencies of the parts, b y what I n a m e and include u n d e r the t e r m connexions" (de Beer, i954). Morphological criteria in histology, as in comparative anatomy, are best 111
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defined in relation to concrete exemplars or standards. Semantically defined criteria would otherwise involve an endless regress. This is to say, in effect, that morphological criteria are operational ones concerning the exercise of a skill. In Euclidean geometry the two ways in which patterns can be said to be equivalent are by congruence---as between equilateral triangles of equal size--and by similarity--as between equilateral triangles of unequal size. In histology the analogy is with congruence, at least in as much as similar cells or organelles must be of a comparable order of magnitude. However, some variation in outline and in size are acceptable, the outline of one Purkinje cell being mentally accommodated to that of another, for example. This sort of equivalence transcends those of congruence and of similarity, and resembles topological equivalence or homeomorphism in these respects (Patterson, i956 ). 2. Histology and M e a s u r e m e n t
T o assert the autonomy of histological skill is not to rule out the possibility of measurement; on the contrary, histological comparisons provide the empirical basis of histological measurement. What measurement requires is a rule for attaching numbers to quantities, which is provided by the maxim: "No quantity of any kind without a comparison of different quantities of that kind" (Whitehead & Russell, x927). Two caveats must be entered here: First, it is recognized that the term "measurement" is commonly accorded to quantitative assays in which the quantities compared can in some sense be added. Other comparisons are usually referred to as giving "quantal" data, for which, of course, statistical methods of analysis exist. Secondly, addition of empirical quantities is easily provided for in the case of fundamental physical magnitudes, and these magnitudes and functions of them (derived magnitudes) are measured on scales regarded as being truly quantitative. In psychometric measurements, on the other hand, while the stimulus-strength may be precisely known, the empirical datum is the strength of a response, e.g. the tightening of hand-grip in response to a light stimulus of known intensity. There is less agreement as to the meaning of addition of such quantities as sensory responses, as contrasted with the addition of physical quantities. Histological data may involve direct comparisons of sensory responses, but not necessarily their "addition". Nevertheless, it seems appropriate to adopt, for the sake of generality, Stevens's (i946 , i958 ) theory of measurement for the description of measurement in histology. Stevens (i946 , i958 ) places special emphasis on the mathematical properties of the scales of measurement, and extends the concept of measurement to subsume the assay of "quantal" and of "quantitative" data in a single theory of measure-
THE STATISTICAL APPROACH IN H I S T O L O G Y
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ment. For each scale a rule is provided for attaching numbers to data. Stevens (1946 , 1958 ) places less emphasis upon the semantic legitimacy of the concept of addition, and approaches measurement phenomenologically: he examines the scales generated by purported examples of "measurement", and evaluates the empirical operations in terms of their success in generating a scale, and of the mathematical character of the scales they generate. The formal mathematical properties of a scale determine the statistical operations which are permissible, i.e. the use one can make of one's measurements depends on the nature of the scale upon which they are made. This approach has obvious advantages for an accolint of histological measurement from a statistical point of view. Measurement, then, is the assignment of numbers to objects or events in accordance with some rule. As outlined by Stevens (1946 , 1958 ) four kinds of property can be used in the operation of assigning a number. Numbers may be chosen to stand for identity, order, differences or ratios in the items examined. According to how the numbers are chosen they accordingly constitute nominal, ordinal, interval or ratio scales. The empirical operation underlying the nominal scale is comparison with the determination of equality and inequality, i.e. classification. Suppose, for example, that the cells of the pituitary gland are being classified, all members of the same class (e.g. acidophils) are equated with one another. If, as in this example, there are more cells than classes, the numbers of cells in each class can be determined, as well as the class with the largest number of cells (the mode). The permissible statistics with classification, or measurement on a nominal scale, include the mode and the coefficient of association. The empirical operation underlying the ordinal scale of measurement is comparison with the determination of equality and inequality, and of greater and lesser. If these features of histological patterns can be determined, the patterns can be ranked in numerical order. Such a scale of numbers can be transformed in any way that preserves the order of the ranking. In mathematical terms, the numbers can be replaced by a monotonic increasing function. In graphic terms, the scale can be stretched (the numbers could, for example, be doubled or squared). The statistics calculated from such a ranking which remain invariant upon transformation of the scale include the median and the coefficient of rank correlation. Ordinal scales are further discussed in Section 3. They are particularly useful in representing gradual differences in pattern, ranging from minim u m to maximum values. The empirical operation underlying the interval scale is comparison with the determination of equality and inequality, of greater and lesser, and of T.B.
8
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S T E P H E N M . SHEA
equality of differences. While many psychometric scales are interval scales, it is probable that histological scales that do not meet the requirements of the ratio scale are more often found to be nominal or ordinal, rather than interval scales. The permissible (invariantive) statistics of the interval scale include the median, the mean and the standard deviation and two types of order correlation. The empirical operation underlying the ratio scale is comparison with determination of equality and inequality, of greater and lesser, of equality of differences, and of equality of ratios. Permissible statistics include the median and the mean, two types of order correlation and the coefficient of variation. The only transformation of scale permissible is that of multiplying it by a constant (e.g. expressing distance in/z, m/~, or A). Ratio scales of measurement are involved in enumeration, and are used extensively in cytology, as measures of linear size, volume, refractive index, cell mass, etc., and permit the use of a wide range of statistical operations. 3. N o n - P a r a m e t r i c M e t h o d s
Methods and scales of measurement giving rise t o ranked data are often spoken of as non-parametric, because they do not allow of the estimation of the statistical population parameters, the mean and the standard deviation. They have the compensating advantage that they do not involve the assumption that the variate measured is normally distributed. An example of data which have already been ranked will show how such information can be interpreted. In a study of cases of hemolytic disease of the newborn, Burn & Langley (i956) assigned I6 cases a rank order corresponding to the degree of sudanophilia of the adrenal cortex, as assessed histologically. The infants were also assigned a rank order for anemia, according to the hemoglobin content of the cord blood. These two rankings are remarkably similar. When Kendall's (i955) coefficient of rank correlation, ~,, was calculated, it was found to be equal to +o.73, and it was determined that such a degree of concordance of the rankings was unlikely to have been due to chance (P < o.ooi). Thus a correlation was established between the severity of the anemia and the degree of sudanophilia. Similar results were obtained in a study of histological changes in the diabetic kidney in man (Shea, Robbins & Mallory, x959). Twenty cases were ranked according to the severity of a pathognomonic glomerular lesion, and according to a non-specific arteriolar lesion. The two rankings were correlated with one another to the extent indicated by a value of T of +0"64 (P < o.oo2). In the latter example the ranking was obtained by the method of paired comparisons (Kendall & Babington Smith, i94o; Kendall, i955; and Shea, i958), in which the "determination of greater and lesser" is made as
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between the members of each possible pair of cases. This method has the particular advantage that the consistencyof an observer can be expressed in terms of a coefficient,called~. The method alsoprovides for an estimate of agreement among observers,in terms of a coefficient,calledu, which can take positiveor negative values. Thus the method of paired comparisons enables one to assess whether the procedure has been successful in generating a reproducible ranking. This is often as important as the existence of means of interpretingthe rankings by rank correlationmethods. Together, the method of paired comparisons and rank correlationmethods, which were introduced largely in the context of psychometrics, make it possible to quantify histological data which are refractoryto measurement on the more preciseand powerful ratio scale,but which would be inadequately exploited by the use of the nominal scale of measurement (i.e.classification). 4. Cellular Kinetics in Fixed Tissues
The distribution of cells in tissues has a temporal as well as a spatial significance, if one or more specific phases in the life of a cell can be identified. The first striking demonstration of the temporal significance of histological patterns (apart from embryological observations) was Payling Wright's (I925) calculation of the duration of the phases of the mitotic cycle from the duration of the telophase, as observed in in vivo preparations. In addition to the duration of the telophase, this calculation required the relative frequency with which the different types of mitotic figure are encountered in fixed tissue. Payling Wright (1925) assumed the duration of a mitotic phase to be proportional to the numbers of mitotic figures observed in that phase. A similar assumption makes it possible to estimate the renewal or turnover time of the cell population of a fixed tissue. The mitotic time is taken as constant, and the ratio of cells in mitosis to all cells (the mitotic index) in determined (Hoffman, I947, 1949; Sainte-Marie & Leblond, I958 ). The amount of information that can be obtained is well illustrated by the work of Sainte-Marie & Leblond (i958), who constructed a scheme for the renewal of lymphocytes in the thymus of the rat by determining the relative counts of reticulum-cells and of lymphocytes of three sizes and the mitotic index of each of these four types of cell. Several additional assumptions were necessary--the "life-span" of a cell was taken to be its individual generation time from "birth" to cell division (mitotic time plus intermitotic time), and to be the same for all cells of a type. This is equivalent to making the generation time equal to the renewal time of the cell type. Strictly speaking, renewal time refers to a population of cells or a "corn-
II6
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M. SHEA
partment" with a certain number of members. The renewal or turnover time is the time required for the turnover or renewal of a number of cells, of the type in a compartment, equal to the number occupying it. The results obtained by Sainte-Marie and Leblond (1958) were construed to be compatible with a sequence of four generations of large lymphocytes, followed by two of medium-sized lymphocytes and two of small lymphocytes. A more elaborate study of this kind is possible if more than one phase in the cell's history can be marked. In a study of the renewal of intestinal epithelium in the mouse in which mitoses and DNA synthesis determined by H a thymidine autoradiography were both identified, Quastler & Sherman (1959) were able to obtain a fairly detailed estimate of the number of cells in the different compartments (crypts and villi), and subcompartments (stages of the mitotic cycle), of the time spent by the cells in these compartments, and of the rates at which cells were being added (in the crypts), transferred (from crypts to villi), and lost (from villi). Schemes of this kind do not make explicit allowance for variation between cells in the duration of mitosis and of the intermitotic period. Both models cited are based upon generation or turnover times which are characteristic of the compartments rather than of individual cells. Sainte-Marie & Leblond (1958) took the generation time of all cells of a type to be the same, while Quastler & Sherman (1959) expressly confined their analysis to the mean value of the generation time. It is difficult, from material of this kind, to obtain information concerning the frequency distributions of intermitotic and mitotic times. An alternative approach is to work out the implications of specified hypothetical frequency distributions. This has been done (Hoffman, Metropolis & Gardiner, 1955, 1956; Hoffman, 1958 ) for the special case of exponentially growing tissues. The relation between the mitotic and intermitotic times of individual cells and mean values is expressed in terms of the statistical concept of a random variable. The intermitotic time, for example, is defined by a process of random sampling in which every possible value has a calculable, but not necessarily equal chance of selection. The chance of selection of a particular value of a random variable is determined by a frequency function (Kendall et al., 1958 ). If successive mitotic and intermitotic times are made repeatedly to assume values determined by random sampling and by specified frequency functions, theoretical models of the growth of a population of cells can be constructed. This sort of mathematical experiment can be made by use of a digital computer with statistical features, and is called the Monte Carlo method. The frequency functions can be specified in terms of hypothetical frequency distributions.
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Hoffman et al. (i956) experimented with combinations of one hypothetical frequency distribution for the duration of mitosis, with one or other of three different hypothetical frequency distributions for the intermitotic time. They studied the "growth" of cell populations from a single cell, paying particular attention to the growth of the population in size, and to the mitotic index. It was found that there were wide fluctuations in both, particularly in the early stages of growth. The characteristics of a population were found to depend upon the frequency distributions of the mitotic and intermitotic times. These determined, for example, the age distribution of the cells of the population, which tended to a limiting form as the population grew older. The numbers of mitoses in "clones" of ioo cells approximated to a Poisson distribution. Stochastic or probabilistic models of this kind have not apparently been used in the study of populations of cells in the later stages of growth (Harris, i959), such as those cell populations which are in a steady state. Deterministic methods have been applied by yon Foerster (I959) to populations in a steady state. The counting of elements of a population in a steady state does not give information about any assumptions concerning the individual elements, unless perturbations of the steady state are introduced. Specific assumptions concerning the age distribution of the elements of a population in a steady state have been shown (yon Foerster, I959) to influence the return to equilibrium to be predicted after the steady state has been perturbed. The comparison of the theoretical effect of perturbation with that of empirical perturbation will then enable the assumptions concerning age-distribution to be tested. This would appear to apply equally well to stochastic as to classical theoretical models. When the specific activity of tissue DNA can be determined it is possible to estimate the parameters of the distributions of generation times (for cells destined to divide) and of life-span (for cells destined to die) (Rigas, i958 , x959). This method is applicable to tissues in the steady state, or growing exponentially. The equations for other states are excessively complex and impractical. The method requires also quantitative autoradiography and was described in connection with the study of populations of leukemic cells. The duration of mitosis, which plays such a large part in the calculations of Sainte-Marie and Leblond (x958) and of Hoffman (I947, I949) can be calculated either from direct observation of living cells in vitro (Lambert & Hanes, i9i 3 ; Strangeways, I92Z) or by the use of irradiation to prevent cells entering prophase (Knowlton & Widner, x95o) or colchicine to arrest mitosis at metaphase. The fallacies in the use of colchicine have been outlined by Leblond & Walker (I956), and Leblond (x959), but if properly used it appears to give valuable results. In correct dosage colchicine causes
x18
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M. SHEA
a linear increase in the accumulation of metaphases over a period of a few hours, the rate of which is a measure of the mean mitotic time (Stevens Hooper, i959). The renewal of cell populations in a variety of tissues is reviewed by Leblond & Walker (i956), and more recently the subject has been dealt with in considerable technical and mathematical detail, in a hematological context, in a symposium edited by Stohlman (1959). 5- Enumeration and Volume Determination in Tissue Sections
Differential counts of cells in tissue sections, such as those of hemopoietic tissues, and of pituitary (Kindred, 1942; Kirkman, 1937) were originally treated in the same way as differential cell counts upon stained smear preparations on glass. There is a fallacy implied in this procedure because cellular structures of the same order of magnitude as the thickness of the tissue section will be encountered the more frequently the larger they are, even if their number per unit volume is the same. A correction has been proposed by Agduhr (1941) and a simpler correction by Abercrombie (1946). Abercrombie (1946) reviews the subject, and proposes the following correction in the case of a count of cell nuclei : M P = A - L+M where P is the corrected number of nuclei, A is the crude count of nuclei, M is the thickness of the section (in/~), and L is the average length of the nuclei normal to the plane of section. Abercrombie (1946) discusses the correction to be made in estimates of nuclear length from micrometry of the apparent nuclear population. Geometrical considerations show that very thin sections underestimate the mean nuclear length by 2 1 ~ ; sections comparable in thickness to the nuclei underestimate the mean nuclear length by about I 1%. The percentage errors in nuclear count when these values are inserted in Abercrombie's (1946) correction formula are about halved for ordinary paraffin sections. The 21% underestimate of nuclear length, when inserted in the correction formula, in cases in which the sections are "ultra-thin" would result in a comparable overestimate of the corrected nuclear count, P. Spheroidal organelles comparable in length to the thickness of the ultra-thin section would not be so greatly underestimated in length as larger structures. Sainte-Marie & Leblond (i958), in their work on the renewal of the lymphocytes of the rat thymus, corrected the differential cell counts in accordance with Abercrombie's (1946) formula. Counts of the larger
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tI 9
cellular structures will require this correction even more with the use of ultra-thin sections. While correct enumeration is sensitive to the thickness of a section, by an apparent paradox nseful estimates of the relative volume occupied by a class of tissue components can be made from the examination of sections of undetermined thickness, without correction. This is done by the method of Chalkley (i943). In Chalkley's (i943) method a fine pointer, or several pointers, are inserted in the focal plane of the eye-piece of a microscope. The section of tissue is examined by recording, for several hundred observations, the identity of the tissue component intercepted by the tip of the pointer. Movements of the slide and of the focal point (within the limits of the section) are made at random. The coefficient of variation of these volume ratios is less than 3 ~/o. 6. Randomness of Spatial Distribution Randomness in the spatial distribution of cells capable of division; or random variation of the generation time, or, with cells having a fixed generation time, random distribution of the phases or "ages" of the cells (Hoffman, Metropolis & Gardiner, I956 ) should lead to a random spatial distribution of mitoses. In the case of regenerating rat liver the spatial distribution of mitoses 48 hours after partial hepatectomy is a random one (Brues & Marble, i937; Harkness, I952 ). This can be demonstrated by counting the number of mitoses per unit area, and comparing the observed distribution with the expected figures calculated for a Poisson distribution (Brues & Marble, I937; Harkness, x952). However, Harkness (i95z) observed at 24 hours a relative excess of mitoses in the periportal areas, in agreement with the observations of yon Meister (I894) and Milne (I9O9) in rabbits. When the numbers of mitoses observed in 64 × 64/zwere compared by Harkness (i952) with the expected figures calculated for a Poisson distribution, no significant difference was observed. When the unit area was x9z × i92/z the observed distribution departed significantly from a Poisson distribution. If a distribution is really random, no significant departure from a Poisson distribution should be observed no matter what the unit of area chosen for counting. To establish the randomness of a spatial distribution it is not sufficient to compare counts per unit area with a Poisson distribution, using only one unit of area for the purpose. We owe the discovery of this fallacy to Harkness (i952), who confirmed the non-randomness of mitoses in regenerating liver at 24 hours, and whose analysis confirms the observation of Brues ~g Marble (i937) that mitose8 occur at random at 48 hours.
I20
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Discussion
The concept of randomness is a fertile one in histology. Chalkley's method (1943) is essentially a method of selecting points in space at random, and of using the results to estimate the relative volumes of tissue components. The demonstration that a spatial distribution is not random (Harkness, I95z ) may be necessary to confirm observations relating to particular sites in tissues. Failure of one test to demonstrate non-randomness does not necessarily mean a spatial distribution is random; it may be that the test selected is at fault. For the purpose of enumerating cells, tissue sections are samples biased in favour of the larger structures, and this bias must be corrected or fallacious conclusions will be drawn from counts of relative numbers of cells or nuclei (Abercrombie, i946 ). The concept of a random variable underlies stochastic models of cell proliferation (Hoffman et al. I955, I956; Hoffman, I958 ). These models have not yet been applied to cell populations in a steady state. Models of tissues in a steady state can be constructed in terms of the turnover of cells in various "compartments" (histological sites or cytological states) if a clearly defined phase in the life of a cell, such as mitosis, can be identified. The spatial distribution of mitoses has a temporal significance. If another phase, such as the period of D N A synthesis, can in addition be marked with radioactive tracers, even more information can be obtained. Von Foerster (i959) provides a very general discussion of populations in terms of the ages of the elements (cells). He points out that the problem is the reverse of that encountered in statistical estimation. In statistical estimation one seeks to obtain information about a population from information about the individual elements of a sample. In studying changing populations, one seeks to obtain as much information as possible about individual elements from information about the population as a whole. This may be done by describing the age of the elements in terms of substates representing age-groups. There is an intimate relation between the duration of a particular state and the number of elements in that state. When substates are age-groups, direct determination of the age of elements is equivalent to the classification and enumeration of the population by age-groups; both procedures present the same difficulties. Direct determination of a maximum value for the age of some elements is made possible by labelling cells at a known time before sacrifice, as in the work of Quastler & Sherman (i959). Another approach is to formulate specific hypotheses about the age distribution. Von Foerster (x959) has defined the relations that must exist between the functions of parameters controlling intrinsically or extrinsically determined losses of cells, and cell productiort
THE S T A T I S T I C A L APPROACH I N HISTOLOGY
I2I
in the steady state. Information about these parameters cannot be obtained by counting elements in the steady state. However, by introducing perturbations of the steady state and counting elements as a function of time, the return to an equilibrium is followed. It is also predicted by functions of parameters determining cell loss and cell production, and the age distribution of the population. Von Foerster (i959) made certain restrictive assumptions about intrinsic losses and cell production, and studied primarily the relation between perturbation and extrinsic removal mechanisms; but the method is quite general. This theoretical approach has in a sense been anticipated in the experimental work of Brues, Drury & Brues (I936) and Brues & Marble (i937) on the regenerating liver of the partially hepatectomized rat (Higgins & Anderson, i93I ). ]]rues, Drury & Brues (I936) studied the restoration of numbers of nuclei, while Brues & Marble (I937) studied the incidence of mitosis with time after partial hepatectomy. They did not, however, speculate on the nature of the distribution of the intermitotic time. The growth of regenerating liver is exponential during the first 72 hours (Brues & Marble, I937) but the growth rate decays exponentially from an initially established maximal value, unlike the exponential growth with constant characteristics of the models of Hoffman et aL (I955, i956 ) and Hoffman (x958). To account for the phenomena, a model would have to specify the changing distribution of intermitotic time, or equivalently of the ages of cells at the period of maximal growth, and for a certain period after the onset of regenerative growth. Assuming for the moment intrinsic and extrinsic losses to be negligible during this period, the partially hepatectomized liver represents in yon Foerster's (i959) terms a population perturbed in two ways : (a) Randomly--in that the numbers of cells in the various age groups have been altered, presumably to an equal extent; and (b) parametrically, to the extent that the "generation coefficient" has been altered. This latter parameter is non-linear, that is, it is dependent on the state of the population. Von Foerster (i959) believes that knowledge of the mathematical functional nature of such parameters obtained from perturbation of the steady state may help in the construction of models for the steady state itself. REFERENCES ABERCROMBIE,M. (I946). Anat. Rec. 94, 239. AGDUHR, E. (x94x). Anat. Rec. 80, I9I. BRtrr~, A. M., DRURY, D. R. & BRuEs, M. C. (I936). Arch. Path. 2z, 658. Baul~s, A. M. & MARBLE, B. B. (x937). a7. Exp. Med. 65, ~5. BURN, J. C. & LANOLEY,F. A. (I956). ft. Path. Bact. 72~ 47. CHALKLEY, H. W. (x943). ft. Nat. Cancer Inst. 4, 47. DE B~sm, SIR G. (1954). "Embryos and Ancestors", znd ed. Oxford University Press, Oxford, p. x26.
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FOERSTER, H. YON (I959). In "The Kinetics of Cellular Proliferation", Ed. F. Stohlman, Jr. Grune & Stratton, New York, pp. 382-407. GEOFVaOY SAINT-HILAIaE,E. (I818). "Philosophie Anatomique". Paris. pp. xxv-xxvi. Cited in translation by de Beer (1954). HARKNESS, R. D. (1952). ft. Physiol. xx6, 373. HARRIS, T. E. (I959). In "The Kinetics of Cellular Proliferation". Ed, F. Stohlman, Jr. Grune & Stratton, New York, pp. 368-38i. HIGGINS, G. M. & ANDERSON,R. M. (1931). Arch. Path. x2, 186. HOFFMAN,J. G. (x947). Science xo6, 343HOFFMAN,J. G. (1949). Bull. Math. Biophysics xx, 139. HOFFMAN, J. G. (1958). Biometrics x4, 139. HOFFMAN, J. G., METROI'OLIS,N. & GARDINER,V. (1955). Science 122, 465. HOFFMAN, J. G., METROPOLIS,N. & GAaDINER,V. (1956). ft. Nat. Cancer Inst. x7, 175. KEI~DALL,D. G. (1956). Proc. Third Berkeley Symposium, University of California Press, 4, I49. K~DALL, M. G. (1955)- "Rank Correlation Methods". Charles Griffin, London. KENDALL,M. G. & STUART,A. (1958). "The Advanced Theory of Statistics", Vol. I, Ed. 3. Charles Griffin, London. KENDALL,M. G. & BABINGTONSMITH,B. (I94O). Biometrika 3 x, 324. KINDRED, J. E. (1942). Am. ft. Anat. 71, 2o7. KIRKMAN, H. (1937). Am. ft. Anat. 61, 233. KNOWLTON, N. P. & WlDNER, W. R. (I95o). Cancer Research to, 59. LAMBERT, R. A. & HANES, F. M. (1913). girchows. Arch. path. Anat. 2Ix, 89. LEBLONI), C. P. (1959). In "Kinetics of Cellular Proliferation", Ed. F. Stohlman, Jr. Grune & Stratton, New York, pp. 31--49LEBLOND, C. P. & WALKER,B. E. (1956). Physiol. Rev. 36, 225. MEISTER, V. YON (I894). Beitr. path. Anat. x5, 1. MmNE, L. S. (19o9)..7. Path. Bact. I3, 127. PATTERSON,E. M. (1956). "Topology". Oliver & Boyd, Edinburgh and London. QUASTLErt, H. & SHERMAN,F. G. (1959). Experimental Cell Research 17, 420. RIGAS, D. A. (1958). Bull. Math. Biophysics 2o, 33. RmAS, D. A. (1959). In "The Kinetics of Cellular Proliferation", Ed. F. Stohlman, Jr. Grune & Stratton, New York, pp. 408-43 o. SAINTE~MARIE,G. & LEBLOND, C. P. (I958)- Proc. Soc. Exper. Biol. & Med. 97, 263. SHEA, S. M. (1958). Arch. Path. 65, 77SHEA, S. M., ROBEINS, S. L. & MALLORY,G. K. (1959). Arch. Path. 68, 447STEVENS,S. S. (1946). Science xo3, 677. STEVENS, S. S. (1958). Science x27, 383. STEVENSHOOPER, C. (1959). Unpublished observations cited by Leblond (1959). STOHLMA~, F., JR. (1959). Editor. "The Kinetics of Cellular Proliferation", Grune & Stratton, New York. STaANGEWAYS,T. S. P. (1922). Proc. Roy. Soc. B, 94, I37. WmTEHEAD, A. N. & RUSSELL, B. (1927). "Principia Mathematica", 2nd Ed., VoI. 3. Cambridge University Press, p. 261. WRIGHT, G. PAYLING,(1925). ft. Roy. Microscopical Soc. p. 414 ,