Medical Biophysics 302E/335G/ st1-07 page 1

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1 Medical Biophysics 302E/335G/ st1-07 page 1 STEREOLOGICAL METHODS - CONCEPTS Upon copletion of this lesson, the student should be able to: -define the ter stereology -distinguish between quantitative and reconstructive stereology -state the Delesse principle -use standard terinology to express stereological relationships -avoid bias when easuring saples of varying cross sectional areas -show how section thickness can affect accuracy (Holes effect) -explain how low contrast (e.g. under staining) can affect accuracy INTRODUCTION Stereology is a body of atheatical ethods used for deriving 3-diensional inforation fro consideration of 2- diensional iages. It is an extrapolation fro 2-D to 3-D space. The 2-diensional iages we coonly deal with are: 1. Photographs of 3-diensional objects (e.g. of particles oving in a odel of an artery, ountains fro an airplane or tissue in a scanning electron icroscope) or 2. Thin sections (2-D) cut fro blocks (3-D) of aterial (e.g. sections for transission electron icroscopy or light icroscope slides or even the cut and polished surface of a piece of rock or bone). There are two aspects of stereology to be considered: 1. Quantitative Stereology is the easureent of the coponents of a structure fro analysis of thin sections. This is a useful technique as the results can be copared using standard statistical tests. For exaple, quantitative easureent of the volue and surface areas of cells or organelles can be used in the estiation of diffusion or etabolic rates. 2. Reconstruction to for a 3-diensional odel of the object can be done by assebling serial sections obtained by MRI, CT, US or histology, or using photographs taken fro different angles (stereo pairs). This is useful for deterining the relative positions of coponents within a structure (eg. the path of a nerve axon in the brain or the arrangeent of capillaries around a uscle fibre). This set of lectures will be concerned ainly with the first of these two related fields of study. QUANTITATIVE STEREOLOGY The basic concepts of quantitative stereology originate fro The French geologist, Delesse, and the English icroscopist, Henry Sorby, independently (as so often happens in science) described what is now called the Delesse Principle: When a section is ade through a solid, the fractional area of a coponent (on the average) is equal to its fractional volue. In this way, the 3-diensional easureent (volue) of an object is represented by a 2-diensional easureent (area) when the section is viewed. Note also that the surface of an object in a 3-D solid (e.g. surface ebrane of a cell) is seen as a line at the surface of a section through it, and that lineal eleents in the solid (e.g. long fine fibres) would appear only as points when sectioned. Such features in a solid are represented in a section through it by iages that have one diension less.

2 Medical Biophysics 302E/335G/ st1-07 page 2 SYMBOLS USED IN QUANTITATIVE STEREOLOGY Standard terinology has (for the ost part) been developed to express stereological relationships. described in the following table can be found in any texts. Definition Sybol Diensions 1. Paraeters of structure Volue (of coponent or structure) Surface area (of coponent or structure) Length of linear feature in space Nuber of objects Diaeter (of spheres) 2. Paraeters of profiles on test areas Area of profiles Boundary length of profiles (perieter) Transections of linear features Nuber of profiles 3. Paraeters on a test line Length (total) of paraeters on a test line Intersections with surface (profile boundary 4. Paraeters on test points in space Point nuber on profiles or structure P 5. Paraeters of test sets Test volue Test area Test line length Test point nuber Section (slice) thickness 6. Densities in space Volue density Surface density Length density in space Nuerical density 7. Densities on test sets Areal density Boundary length density Transection density on area Nuerical density of profiles on area Intercept length density on line Intersection density on line Intercept nuber density on line L L I L N L Point density P P V S J N D A B Q N L I V T A T L T P T t V V S V J V N V A A B A Q A N A

3 Medical Biophysics 302E/335G/ st1-07 page 3 It will be iportant for you to understand the eaning of these sybols when they are used in descriptions or equations. In soe references, the sybols ay have been given ore than one eaning - e.g., "P" could represent the nuber of test points superiposed on the surface being exained, or the nuber of points on a surface (arising fro lineal features or fibres being sectioned). For this reason, the table (taken fro Weibel, 1979a) uses P, Q or I for points, S or A for surface areas and L, J or B for line lengths in various situations. The exact eaning should be obvious when taken in context. THE BASIC RELATIONSHIP BETWEEN OBJECTS AND THEIR IMAGES To exaine the relationship between features in a containing volue and their appearance at a cut surface, think of a fruit cake with cherries in it. A fraction of the surface area at the cut will be cherry. The Delesse principle states that the volue fraction (V V ) equals the ean area fraction (A A ). This would be true for the theoretical ean of an infinite nuber of infinitesially thin sections - but in practical ters, the ean is estiated fro rando or regularly spaced saple sections of finite thickness. SAMPLING BIAS Since a volue is sapled by being sliced, it is iportant not to introduce bias in this sapling. For exaple, suppose one wishes to find the proportion of a volue of tissue that is ade up of nuclei. The volue fraction (V V ) then is the feature or coponent volue (V) divided by the total containing volue (V T ). If the nucleus is represented by a sphere with a diaeter of 5 units positioned centrally in a cube of length 10, V V can be deterined exactly, 4/3 B (5/2) 3 / 10 3 = or fro areas obtained fro sections: ean A A = (85/4)B/1000 = The results agree well considering only 10 slices were taken. Because the test areas are all the sae, it doesn t atter if we take the ean of the ratios for each slice, or the ratio for the su of coponents.

Medical Biophysics 302E/335G/500 20070109 st1-07 page 4 Suppose, however, that the containing volue is another concentric sphere (like a cell around a nucleus) with twice the diaeter.

This error is introduced because the saller containing areas have been given the sae weighting as the larger ones. Since they also have the sallest area ratios, the results are biased.

4 Medical Biophysics 302E/335G/ st1-07 page 4 Suppose, however, that the containing volue is another concentric sphere (like a cell around a nucleus) with twice the diaeter. The exact V V would be 1/2 3 = Fro the ean of the area ratios we would get: which is a rather poor estiate even with the first section not counted ( if it is). This error is introduced because the saller containing areas have been given the sae weighting as the larger ones. Since they also have the sallest area ratios, the results are biased. What value is obtained if the areas for each section are added first and then the ratio is taken? There is nothing fancy about the atheatics used here, but it serves to ephasize the iportance of using techniques properly. Would it ake any difference if the nucleus were not concentric in the cell? THE IMPORTANCE OF SECTION THICKNESS Iage loss due to lack of contrast Frequently, the contrast between the feature and the containing volue is poor, and a certain thickness of tissue is necessary to see it. In this case, if the surface within the section is oblique, the position of the edge ay be difficult to deterine. Thicker sections ay iprove contrast, but this will introduce a different error known as the Holes effect:

$In reality, we usually exaine a projected iage - created fro the full thickness of the section. If the feature is wider within the section than at the surface, the area fraction is overestiated.$

5 Medical Biophysics 302E/335G/ st1-07 page 5 The Holes Effect The atheatics used in quantitative stereology is based on the easureent of a coponent where it is exposed at the cut surface. In reality, we usually exaine a projected iage - created fro the full thickness of the section. If the feature is wider within the section than at the surface, the area fraction is overestiated. As an exaple, consider a sphere of diaeter D which is sliced into sections of thickness t. In the figure below, the solid lines indicate the proper (top surface) easureent, and the broken lines the apparent diaeter which is actually used for calculations. The shaded portion represents the overestiated volue. Notice that the % error increases as D/t gets saller. As an exaple, one could estiate the volue of the yolk in a sliced egg fro the exposed surface at the top of each slice. If the yolk is popped out of each section, however, and projected, it is ipossible to tell if the area is an overestiate. The forula for correcting (for spheres) is, for the opaque phase: V V = A A /(1 + K) where K = 3t/2D t = section thickness D = particle diaeter There is nothing agic about the forula. The extra area observed fro the sections is equal to the projected area of the sphere (Br 2 ) so the extra volue calculated is Br 2 t. The true volue is 4/3 B r 3 so the overestiate, expressed as a ratio, is 3t/4r which is equal to the correction factor K above. It is often assued that iage contrast copensates for the Holes effect, but this will depend on the degree of contrast. In any case, the thickness of the section relative to the diensions of the coponent is iportant.

6 Medical Biophysics 302E/335G/ st1-07 page 6 IMPORTANCE OF THICKNESS OF SECTIONS (OR DEPTH OF FOCUS) IN ORIENTED STRUCTURES The exaination of oriented structures in sections ay present special probles for quantitative analysis. Features such as uscle cells viewed under the light icroscope, or collagen fibres using transission electron icroscopy will vary in size depending on the thickness of the section and the angle at which it is sectioned. If the section is noral to the long axis of the feature, the results will be accurate as long as the section is thick enough to provide sufficient contrast. On the other hand, if the features are cut obliquely, the area fraction ay be overestiated due to the Holes effect, especially in thick sections. In thinner sections the area fraction will be reduced due to lack of contrast at the obliquely cut edges. REFERENCES Bolender, R.D. and Pentcheff, N.D. (1985), Coputer Progras for Biological Stereology. Washington Research Foundation, Seattle, Washington. Underwood, E.E. (1970), Quantitative Stereology, Addison-Wesley Publishing. Willias, M.A. (1977), Quantitative Methods in Biology, in: Practical Methods in Electron Microscopy, Vol 6, Part II, ed. A.M. Glauert, North-Holland Publishing. Weibel, E.R. (1963), Principles and Methods for the Morphoetric Study of the Lung and Other Organs. Lab. Invest. 12:p131. Weibel, E.R. (1979), Stereological Methods I, Practical Methods for Biological Morphoetry. Acadeic Press. Weibel, E.R. (1979), Stereological Methods II, Theoretical Foundations. Acadeic Press also: check the net for refs. e.g.,

Development of a Computer Application to Simulate Porous Structures

Development of a Computer Application to Simulate Porous Structures Vol. Materials 5, No. Research, 3, 00Vol. 5, No. 3, 75-79, 00. Developent of a Coputer Application to Siulate Porous Structures 00 75 Developent of a Coputer Application to Siulate Porous Structures S.C.