Stereologic methods are used to obtain quantitative information about three-dimensional structures based on observations from section planes or—to a limited degree—projections. Stereologic methods, which are used in biologic research and especially in the research of normal and pathologic kidneys, will be discussed in this review. Special emphasis will be placed on modern stereologic methods, free of assumptions of the structure, size, and shape, *etc.*, so-called UFAPP (unbiased for all practical purposes) stereologic methods. The basic foundation of all stereology, sampling, will be reviewed in relation to most of the methods discussed. Estimation of error variances and some of the basic problems in stereology will be reviewed briefly. Finally, a few comments will be made about the future directions for stereology in kidney research.

# Stereologic Methods and Their Application in Kidney Research

- Free

## Abstract

Nephrons and cells are functional units in a kidney. The number, size, and distribution of nephrons, cells, and other components contain important information about the function and organization of the kidney being studied. Therefore, it is important that the various structural components are measured correctly. Quantification of these components is also important when examining how kidneys react to trauma, chemicals, and disease.?

Using different kinds of imaging modalities (light or electron microscopes, confocal laser scanning microscopes, ultrasound, computed tomography, and magnetic resonance imaging), the researcher can study the three-dimensional kidney on two-dimensional images. The problem that confronts the researcher is that a single kidney structure can generate widely differing sections or projections, and several different kidney structures may generate similar sections or projections.

A two-dimensional section through a three-dimensional kidney results in an irreversible loss of qualitative information and a reversible quantitative change of information. A section through a single podocyte, which results in many unconnected profiles of the podocyte, is an example of loss of qualitative information (that all podocyte profiles represent just one podocyte). The apparent thickness of the glomerular basement membrane in two dimensions may be larger than the true thickness in three dimensions, which is an example of a quantitative change of information.

Two-dimensional projections of a three-dimensional kidney result in a qualitative change of information and a quantitative increase of information, both of which are irreversible. In total projection of thick kidney sections, false connections may appear between tubules and capillaries, resulting in an irreversible change of qualitative information. In similar thick sections using projection, too many kidney cells are seen, illustrating an irreversible increase in quantitative information.

The problems in estimating structural parameters such as number, size, and distribution in a three-dimensional kidney from two-dimensional sections or projections of the kidney may be solved by using stereologic methods. Stereologic methods are practical tools based on sound mathematical and statistical principles. Stereology may also be regarded as a sampling theory for populations having a geometric structure (^{1}).

During the past decade, stereology has developed to the extent that it is now the technique of choice whenever three-dimensional structural kidney quantities need to be extracted from planar measurements performed on sections. The purpose of this article is threefold: (*1*) to present the newer stereologic methods and review some of the older, design-based methods for use in bioscience with special emphasis on kidney research; (*2*) to describe the biologic significance of stereologic methods in kidney research; and (*3*) to facilitate the understanding of how stereology has changed from being primarily a descriptive tool to becoming a confirmatory science in kidney research.

## Sampling Strategy

One of the greatest advantages of modern stereology is that it can be used to minimize the workload in sampling while still providing reliable, quantitative information about the whole structure of interest. The task of sampling comprises several problems because of the loss of information that three-dimensional tissue undergoes when viewed in two-dimensional sections. In this sense, it may be of interest to briefly mention the different sampling modalities in stereology, which ensure that the slices or structures of interest are sampled with the same probability. Assume that the kidney is cut into a number, *n*, of arbitrary slices. (*1*) Independent, uniformly random sampling of the slices means that a random number between 1 and *n* is chosen and the corresponding slice is sampled. This continues until a fixed sample size (number) of slices is chosen. If the same random number is chosen twice, it is ignored and a new random number is chosen (Figure 1). (*2*) Systematic, uniformly random sampling of

of the slices means that one of the first *m* slices is taken at random and from then on every *m*^{th} slice is chosen from the ordered set of slices, arranged in either their natural order or arranged in a smooth order (Figure 2). (*3*) Cluster sampling of structures means that method (*1*) or (*2*) is used to sample clusters among the arbitrarily arranged clusters, with a probability *p*. In the clusters sampled, the structures of interest are sampled by method (*1*) or (*2*) with probability *q*, making a total sampling fraction of *f* × *q* (Figure 3A). (*4*) Stratified random sampling means that the slices are divided into strata and every stratum is studied using method (*1*) or (*2*) (Figure 3B).

A prerequisite for all stereologic methods is that all structures under study can unambiguously be identified (^{2}^{,}^{3}). In many stereologic methods, a density of number, length, surface area, and volume is estimated in the reference volume. The reference volume is the total volume of the tissue under study. To say anything about changes in the totals of the three-dimensional structures, it is essential to multiply the densities by the total volume of the reference volume. The densities are just ratios that do not allow any conclusions about changes in total number, total length, total surface area, *etc*. Examples of dubious conclusions based solely on the estimation of densities have been reviewed previously (^{4}^{,}^{5}^{,}^{6}). The term “reference trap” is applied to the cases in which wrong conclusions have been drawn from densities alone. The simplest way to determine the total volume of a kidney is to weigh it and then transform the weight to a proper volume. Kidney volume can also be estimated using the Cavalieri principle, and if smaller kidney structures are the reference volume then other stereologic principles are available (see Estimation of Volume).

A different way to avoid the reference trap is to use the fractionator principle for tissue sampling (^{3}) (Figure 4). This principle has been known outside the stereologic field for several years and may be regarded as uniformly random cluster sampling. A predetermined fraction of the entirely sliced kidney object is sampled, typically in a multistage sampling system

.. The desired structural quantity (*X*) is estimated in the predetermined fraction (*X*_{i}) of the object. The total amount of the structure can then be estimated from:

where *F*_{1} to *F*_{n} are the inverse sampling fractions at each sampling level. The sign:= indicates that the equations represent estimates. This quite simple sampling scheme of the fractionator makes it a very powerful sampling method. If the fractionator is used for estimation of number, it becomes especially useful because the result will be independent of any kind of tissue deformation. This has been of particular interest in kidney research because older and pathologic kidneys are usually more fibrotic than normal kidneys, in which case different tissue deformation in the normal and “sick” kidneys can be expected. It has, for instance, previously been proposed by Brenner *et al.* (^{7}) that the reason only 30 to 40% of the human diabetic population developed diabetic kidney disease was that they had a genetically determined smaller number of glomeruli than the rest of the diabetic patients. To test such a hypothesis, the problem about differential shrinkage in the two kidney populations needs to be addressed. Use of the fractionator will completely eliminate this problem (^{8}).

## Orientation of Section Planes

When beginning a stereologic study, it is important to determine how the sections should be cut. Are sections with an arbitrary orientation sufficient, will vertical sections be needed, or can the stereologic problem only be solved by isotropic, uniform random (IUR) sections (Figure 5)? The advantage of arbitrary and vertical sections is that it is usually possible to keep track of the architecture of the tissue being studied.

All estimations of zero-dimensional numbers such as number of structures, as well as the estimation of three-dimensional volume by point counting, including the Cavalieri principle, may be performed on sections with arbitrary orientation. When the stereologic methods require *isotropic test lines*, the ingenious invention of vertical slices by Baddeley *et al.* (^{9}) will often be the best choice to fulfill this requirement. Stereologic methods requiring isotropic test lines include surface estimation using the cycloid test system described by Baddeley, and local volume estimation with the nucleator (^{10}), the planar rotator (^{11}), the optical rotator (^{12}), or the selector (^{13}). The estimation of the volume-weighted mean volume of structure (^{14}) can also be performed using isotropic test lines.

The requirement of *isotropic test planes* can be met with the use of IUR sections. The first practical method for making IUR sections was invented by Mattfeldt and Mall (^{15}), but the use of the method was limited because it required a mechanical randomizing device. The breakthrough came in 1990, when Mattfeldt *et al.* introduced the orientator (^{16}). The orientator has become the gold standard in producing IUR sections because it is easy to use and does not require any expensive equipment. If smaller specimens are used, the isector (^{17}) becomes a very efficient alternative because it is not necessary to keep track of any IUR plane of the specimen during the embedding process. Isotropic test planes are needed in the estimation of axial length density of tubes, *e.g.*, capillary length (^{15}^{,}^{18}^{,}^{19}). If it is possible to visualize the tubes in projections, thick vertical sections may be used (^{20}^{,}^{21}^{,}^{22}). The most efficient estimator of membrane thickness and its distribution, derived from orthogonal intercepts, requires IUR sections (^{23}), whereas another less efficient method (^{24}) may be performed on vertical sections. The bias due to membrane loss when estimating surface of anisotropic mitochondrial cristae (^{16}) in, for example, kidney tubule cells may be significantly reduced using IUR sections. The volume-weighted second moment of structures (^{14}), which is used for the estimation of variability of the volume estimate, can be exclusively estimated on IUR sections, enabling estimation via a fast approximation (^{25}). If each section of a surface contains information about the angle at which the surface was sectioned, *e.g.*, luminal microvilli in the kidney tubules, local vertical windows in combination with a cycloid test system and IUR sections may be used to the estimate surface area (^{9}). The spatial arrangement of kidney structures can be estimated using so-called “second-order” stereologic methods (^{26}^{,}^{27}^{,}^{28}) based on the nucleator principle (^{10}) or the rotator principle (^{11}). The estimation of spatial distances requires isotropy of the section except when the distribution of number related to number (^{29}), or the distribution of volume, is investigated. Finally, the use of digitizers or image analyzers for estimating surface area using the equation *S*_{v} = (4/π) × *B*_{A} requires IUR sections.

## Some Basic Stereologic Problems

When a kidney is cut into small blocks before histologic embedding, artificial edges (xy-direction of the kidney blocks) and artificial surfaces (top and bottom of the kidney blocks) will be produced. If the structure of interest, *e.g.*, a tubule cell, hits the knife, the structure will be cut through, or fall out of, the tissue block. After the histologic embedding, structures may still fall out of the embedded kidney during cutting. In this case, some of the structures may *a priori* have a sampling probability of zero. During the sampling of the structures on the histologic sections, the probability that any of the studied structures is close to the artificial edges should be zero. The two-dimensional problem with the edges can be avoided by making complete sections through the kidney (^{30}^{,}^{31}) or can be dealt with as described in Figure 6. Every test system used on sections with artificial edges should be surrounded by a “guard area” (^{3}^{,}^{32}). This will prevent sampling of structures close (less than about two times their radius) to the artificial edges, as described previously (^{8}^{,}^{33}^{,}^{34}^{,}^{35}). There are several examples of the use of biased counting rule in kidney research before the invention of the two-dimensional counting rule (^{36}^{,}^{37}). Kidney sections cut less than a “safety distance” from kidney slices with artificial surfaces should be evaluated differently from the rest of the kidney sections (^{3}^{,}^{35}).

The probability that a section will hit certain structures depends on the size of the structure (^{38}^{,}^{39}). A three-dimensional probe is required for number-weighted sampling of structures. A two-dimensional section through a kidney hits the structures with a probability proportional to the height of the structure perpendicular to the section. One-dimensional test lines hit the structures with a probability proportional to the surfaces of the structure, and zero-dimensional test points hit the structures with a probability proportional to the volume of the structures. All sections, whether they are physical or optical, have a certain thickness, which influences the probability of hitting the structure: The thicker the section the greater the probability of hitting the structure (^{40}). Computations of overestimation of volume, surface area, and length of a specific model in thick sections have been performed in a previous study estimating length and volume of microvilli in the brush-border zone of proximal tubules from a rat kidney (^{17}).

Loss of fragments of a structure, or “lost caps” (^{40}), means that fragments of the structure actually or apparently disappear. Fragments may be lost due to mechanical action during the cutting of the tissue, or due to a chemical reaction during the histologic process. Fragments may also be invisible due to weak tissue contrast, or it may be impossible to identify small fragments using strict morphologic criteria. The number and size of lost caps are generally unknown. Lost caps will frequently be small and peripheral and will therefore not affect the estimates of different dimensions of structures equally. Lost caps in two dimensions are in general smaller than the average size of profiles, in which case the relative importance of lost caps is greatest for total number of profiles, less for total circumference, and the least for total area. The effect and significance of bias caused by lost caps are unpredictable.

Overprojection, or “Holmes effect,” is caused by the positive thickness, *t*, of transparent slices, because all size measurements of nontransparent structures are greater than or equal to the real size (^{40}^{,}^{41}). The number of profiles per area, *Q*_{A}, is proportional to the numerical density of structures, *N*_{V}, and the average height, *h*, of the structures perpendicular to the thin section. In transparent thick slices containing nontransparent profiles of the structures being studied, overprojection will increase *Q*_{A} with *t*. This bias may in theory be corrected (^{40}).

In practice, this requires considerably more work than simply estimating *N*_{V} directly as discussed in the next section. Overprojection affects various stereologic methods to a different degree. The estimators of number presented later will not be influenced by overprojection. All estimators of size will be affected by overprojection depending on the thickness of the slices. If the volume of a convex, nontransparent object is to be estimated using the principle of Cavalieri (see Estimation of Volume), the bias resulting from overprojection may be minimized by disregarding the slice with the largest area of the object from the calculation.

Tissue deformations such as shrinkage and swelling will influence all stereologic size estimators whether it is distance, surface area, or volume. There is no “smart” unbiased way to obtain information about tissue deformation during tissue fixation and processing. The area of a piece of kidney tissue before and after fixation/processing may be estimated, and the tissue deformation on a volume basis can be calculated as:

This formula requires that IUR sections be used, or assumes that the tissue deformation in the z-axis is equal to the tissue deformation in the x- and y-axis. Another labor-intensive technique to estimate tissue deformation is to weigh a kidney piece, process it, and cut it up exhaustively. The weight of the kidney piece before processing is transformed to a volume and compared with the volume of the cut-up piece estimated by the principle of Cavalieri. Considering the theoretical and practical difficulties in estimating tissue deformation, it is important to recognize that deformation of kidney tissue is markedly reduced in plastic embedding materials when compared with paraffin (^{42}^{,}^{43}). This probably also holds true for carefully prepared Vibratome sections in the x-y axis (^{44}^{,}^{45}).

## Estimation of Number of Objects, Associated Features of Objects, and Connected Sets of Objects

### Number of Objects

Number estimation of kidney structures may have great impact when it is used to study genesis, development, growth, and transformation of the kidney, to get a better understanding of how the objects/particles were formed. In this way, hyperplastic, hypertrophic, and interstitial growth can be differentiated, and it is possible to examine the mechanisms involved in the differentiation of kidney structures. Furthermore, number estimation is highly useful in generating information about degeneration, intercellular communication, and connectivity (^{46}^{,}^{47}).

For many years, the most worked-on issue in stereology was the problem concerning number estimation in three-dimensional space (^{48}). Before 1984, researchers had to rely on indirect measurements using model-based methods and single sections (^{49}^{,}^{50}^{,}^{51}^{,}^{52}^{,}^{53}^{,}^{54}^{,}^{55}^{,}^{56}^{,}^{57}). Cruz-Orive was the first to overcome the barrier for the estimation of numerical density (^{2}) using a volume probe consisting of serial sections cut with a random starting position, but at an arbitrary orientation. The final breakthrough came with the description of the disector principle (^{39}), even though the basic principles behind the disector had been known for decades by isolated or unrecognized researchers (^{58}). The disector comprised, for the first time, a complete rule for uniform sampling “by the book” of three-dimensional objects and other discrete events according to cardinality. The disector is a three-dimensional probe that samples structures proportional to their number without regard to size or shape of the structures. The disector comprises an integral test system (^{59}) and a counting rule: The number of structures sampled by the counting frame disappearing from one section plane to another, *Q*^{-}, is counted (Figure 7). The integral test system comprises the two-dimensional counting frame (^{32}) with a certain area, *a*(frame), the number Σ*P* of points hitting the reference volume being studied, and two parallel planes a known distance, *h*, apart. If the number of sampled structures is related to the volume between the two parallel section planes, it is possible to estimate the numerical density of the structures:

One of the great advantages of the disector is that it not only allows the use of physical sections, but it also allows optical sections (Figure 8, optical disector) in certain aspects of object sampling, so that the need for aligning the two physical sections is avoided (^{3}). By multiplying the numerical density with the reference volume determined using the Cavalieri principle, the total number of structures is derived. The fractionator/disector principle estimates the total number of structures in the last sampling fraction (no numerical density!) and then multiplies this value by the inverse total sampling fraction to get the total number independent of tissue deformation. It may be worth mentioning here that if the estimation of volume by the Cavalieri principle and the estimation of numerical density by the disector are performed on the same histologic sections, then section thickness *t* cancels out. The estimate of the number of structures will then be independent of tissue deformation just like the fractionator/disector combination (^{60}). Also, the fractionator/disector method can now be performed in an optical manner with some precautions (^{61}). The pitfall in the disector counting principle is illustrated in Figure 9.

A question of great interest for kidney scientists concerns the number of glomerular cells, which can be changed by aging and disease (^{62}). The numerical density of glomeruli in the cortex of the kidney may be estimated using a physical disector, and the numerical density of all glomerular cells may be estimated using an optical disector on the same approximately 25-μm-thick sections. Consequently, the total number of glomerular cells may be estimated using this “double disector” (^{63}). The problems with lost caps and overprojection (^{40}) in these thick sections are probably without importance (^{63}). If the aim is to estimate the total number of each glomerular cell type, *i.e.*, glomerular and parietal epithelial cells, endothelial cells, and mesangial cells, an electron microscopic approach is needed (^{64}). The total number of glomerular cells is still counted in an optical disector by use of light microscopy (LM), but the number of individual glomerular cells is counted in physical disectors of unknown thickness using an electron microscope (EM) providing unbiased estimates of their correct proportions. This very elegant LM and EM approach is possible because the relative frequency of each cell type per glomerulus obtained by EM can be multiplied by the total number of cells per glomerulus obtained by LM to estimate the total number of each cell type. Trivially, it is a prerequisite, as it is for all uses of the disector method, that each cell type unambiguously can be identified on at least one micrograph (section plane). Other applications of the disector principle on kidney objects include number estimation of glomeruli (^{8}^{,}^{35}^{,}^{65}^{,}^{66}^{,}^{67}^{,}^{68}^{,}^{69}^{,}^{70}^{,}^{71}^{,}^{72}), and of tubule cells (^{73}).

### Number of Associated Features of Objects

The second entity that may be estimated with the disector method is the number of associated features of objects, or “external units,” meaning specialized, functional, and anatomical units on the surface of any structure. This could include the slender processes or microvilli on the proximal tubule cells that have the important biologic function of enlarging the luminal cell surface for better reabsorption of the glomerular ultrafiltrate and are the place for the degradation of low molecular peptides (^{74}). The number of microvilli can be estimated with the disector because all microvilli have one and only one connection to the luminal surface of a proximal tubule cell. In such a case, a number-weighted sample of microvilli may be obtained using physical disectors at the EM level. If the mean volume of a proximal tubule cell is known (^{73}), the number of microvilli per cell may be estimated under different experimental situations.

### Number of Connected Sets of Objects (ConnEulor)

The third entity estimated using the disector is the number of connected objects. Using the zero-dimensional Euler-Poincare characteristic (Euler number), estimation of number with the disector is not restricted to isolated objects but may also be useful in estimating number of elements in a histologically complex network (Figure 10, A through D). For the estimation of Euler number, χ, in a network, the number of isolated parts, the number of extra connections (the so-called connectivity), and the number of enclosed cavities in the network must be taken into account (^{75}^{,}^{76}^{,}^{77}^{,}^{78}):

There is one isolated part and no enclosed cavities in a capillary network; thus, the equation for a capillary network boils down to χ = 1 - connectivity. The Euler number is a zero-dimensional entity and may then be sampled by the disector (^{39}^{,}^{79}) for estimating the numerical density of capillaries, *W*_{V}, as described by Nyengaard and Marcussen (^{80}). Figure 10E illustrates the counting of bridges, islands, and holes in practice using a physical disector:

An interesting and very well investigated network in the kidney is the glomerular capillary network. In a glomerular capillary network, there are no problems at the edges because a whole glomerular profile can be sampled. However, the numerical density can be estimated regardless of edge problems on all other capillary systems or networks using the ConnEulor (^{81}), which again is based on the additivity of Euler number under set-intersection (^{82}).

This estimator of capillary number has shown new aspects of glomerular capillary growth following normal development (Figure 11), nephrectomy, diabetes, and lithium feeding of rats. A lengthening of the rat glomerular capillary network has been observed in normal growth and after pathologic growth following nephrectomy (^{83}^{,}^{84}) and pathologic growth after the induction of diabetes (^{85}). The main capillary variable following glomerular growth in these experimental settings was capillary number (^{86}^{,}^{87}^{,}^{88}^{,}^{89}). However, glomerular hypertrophy after puromycin aminonucleoside poisoning is not followed by an increase in glomerular capillary number (^{90}). The Euler number has been used in the number estimation of renin granules in the afferent arterioles of rats and mice, taking into consideration that some renin granule profiles in one section may connect with other profiles in another section. The finding was that the number of renin granules increased approximately 20 times in angiotensin-converting enzyme inhibitor-treated rats (^{91}), indicating a positive feedback mechanism. In normal mice, the number of renin granules per afferent arteriole was several thousand, whereas Beige mice had a few very large granules per juxtaglomerular granular cell, illustrating the inability of Beige mice to form granules (^{92}).

## Estimation of Volume

The capillary luminal volume is related to the volume of capillary blood available for gas exchange within the tissue. The volume of a cell may reflect the viability of the cell, because a changed cell volume often indicates a diseased cell.

Probably the first straightforward stereologic method to be published was in the material sciences, where Delesse in 1848 estimated the area fraction of a particular mineral in a sample of rock. Delesse indicated that the volume fraction of the mineral in the whole rock was equal to the area fraction of the mineral on a cut surface of a representative sample of the rock. This corresponds to the fact that a three-dimensional structure is represented as an area on a set of two-dimensional plane sections. All point counting methods are based on the principle that test points “feel” volume, and point counting is still one of the cornerstones in many stereologic applications. The old principle of point counting provides an estimate of volume density without any assumptions about the structure, and it does so more efficiently than delineating the structure by digitizers or image analyzers (^{93}^{,}^{94}).

The sampling of tissue to be used in a stereologic study has within the last decade benefited greatly from the implementation, in a stereologic context, of the ancient principle of Cavalieri (^{95}). The Cavalieri principle is quite general and defines how to estimate the total volume of any physical object, *V*(obj). By cutting the entire object into slices with a known thickness, *t*, and counting the number of points, *P*, associated with a known area, *a*/*p*, and hitting the structure on the cut surfaces of the slices, the total volume may be estimated:

In case the object is cut into more than approximately 10 slices, depending on the structure being studied, a systematic, uniformly random sample of the slices should be taken. When calculating the total volume, the equation above should be multiplied by the inverse-sampling fraction. The advantage of the Cavalieri principle is that it does not rely on any assumptions with regard to the object. The use of the Cavalieri principle requires simply: (*1*) that the position of the first slice hitting the object must be random; (*2*) that the slices are parallel; and (*3*) that the thickness of the slices is constant. The Cavalieri principle has been used to estimate the volume of a kidney, kidney cortex, glomerulus, or any kidney structure in rats (^{70}^{,}^{73}^{,}^{90}) and humans (^{66}^{,}^{69}^{,}^{96}^{,}^{97}) and in human kidney biopsies (^{98}). If the volume of a kidney is to be estimated using “noninvasive” techniques, stereology can help again, because the use of computed tomography (^{99}) or magnetic resonance imaging (^{100}), in combination with the Cavalieri principle, may estimate kidney volume or renal cortical volume (^{101}^{,}^{102}). If ultrasound scanning is used (^{103}^{,}^{104}), the Pappus estimator with coaxial sections (^{105}) may greatly facilitate the kidney volume estimation.

There is probably not any structure in the kidney that has not been point counted, as illustrated in Figure 12. If the task is to estimate, *e.g.*, the total volume of mitochondria in proximal tubule cells, *V*(mito), the Cavalieri principle and point counting or a simple weight of the kidney can be used to estimate the volume of the kidney, *V*(kid). Light microscopy and point counting are used to estimate the volume density of proximal tubule cells in the kidney, *V*_{V}(prox/kid). Finally, the volume density of mitochondria in proximal tubule cells, *V*_{V}(mito/prox), is obtained by EM and point counting:

This is the classical hierarchical (^{106}) design for the estimation of total volumes on systematic, uniformly random sampled sections. Numerous examples of the use of this principle in the kidneys exist in the literature.

For the estimation of *mean* volumes of any kidney structures, three different definitions of volumes that can be estimated with stereologic methods should be considered: (*1*) number-weighted volume; (*2*) volume-weighted volume; and (*3*) star volume.

The number-weighted mean volume is the usual way of sizing structures on an average. It is obtained when the structures are sampled with equal probability using the disector, *i.e.*, the objects are selected according to their number regardless of their size and shape, and their volume is estimated. Once the structures are sampled number-weighted, their mean volume may be estimated with three fundamentally different methods:

- The local volume estimators comprise the nucleator (
^{10}); the planar rotator (^{11}); the optical rotator (^{12}); and the selector (^{13}). - A global volume estimator uses the principle of Cavalieri (see above).
- A ratio of global estimators can generate an indirect estimate of mean volume.

### Local Volume Estimators of Number-Weighted Mean Volume

The main purpose for the use of the local volume estimators is to estimate cell volume. The advantage of the local volume estimators is that they may provide the distribution of volume of the individual structures under study. In general, the rotator(s) is the most efficient way to estimate individual volume; the nucleator is the second most efficient individual volume estimator, and the selector comes in third. Both the nucleator and rotator(s) rely on a unique point, a nucleolus, as a reference from where the volume estimation is performed. This feature makes them ideal for optical sectioning using LM or confocal laser scanning microscopy (CLSM). If the aim of the investigation is to estimate the individual kidney cell volume using the nucleator or rotator(s), it is important to realize that individual glomerular cell or tubule cell borders are not usually distinguishable on ordinary LM sections. It is necessary either to use antibodies against individual cell membranes and thick (30 to 80 μm) LM/CLSM sections or to use EM sections. The selector does not need a reference point for volume estimation. None of the local number-weighted volume estimators has thus far been used in kidney research (planar rotator is illustrated in Figure 8), and interested readers are referred to authors working in other organs: planar rotator (^{107}); nucleator (^{29}); and selector (^{108}).

### Indirect Estimate of Number-Weighted Mean Volume

If it is sufficient with the volume of average kidney structures, the indirect estimate of mean volume obtained by the volume density and the number density of the structures will be the most efficient method:

where *V*_{V} and *N*_{V} are obtained by point counting and disector-sampling, respectively. The estimation of number-weighted mean glomerular volume has attracted considerable attention in kidney research (^{109}^{,}^{110}^{,}^{111}^{,}^{112}^{,}^{113}^{,}^{114}^{,}^{115}^{,}^{116}^{,}^{117}). Some researchers have used model-based techniques to estimate glomerular volume. However, design-based stereologic methods are available for the estimation of number-weighted mean glomerular volume. They include point counting in combination with the disector method. If the distribution of number-weighted mean volume of glomeruli is wanted, the Cavalieri principle is the right choice. This may also be the method of choice in the special case when only a kidney biopsy is available (^{98}). In all other cases, the disector principle and point counting will be the most efficient method to obtain the mean glomerular volume of average glomeruli. The indirect estimate of mean glomerular volume has been used in rats (^{70}^{,}^{71}^{,}^{72}) and in humans (^{35}^{,}^{118}^{,}^{119}). The indirect method has also been used in estimating volumes of glomerular epithelial cells, endothelial cells, mesangial cells, and parietal epithelial cells in rats (^{120}^{,}^{121}) using EM. The mean volume of rat tubule cells has been estimated using the optical disector on thick, and point counting on thin, LM plastic sections (^{73}).

Number-weighted mean cell volume may be estimated in the special cases in which cells are cultured and grown on plastic dishes or when cells are in suspension. In the case of a flat support, the number of cells per area of dish is estimated using the two-dimensional counting frame (^{32}). The cells are embedded in the plastic dish and vertical sections (^{9}) are generated perpendicular to the monolayer. Using LM or EM, the mean height of the monolayer is estimated using point counting on the vertical sections as:

where *a*/*p* is the area associated with each test point in the grid, Σ*P* is the number of points hitting the cells, and *L* is the length of the monolayer base. The mean cell volume is then the number of cells per area of dish multiplied by the mean height of the monolayer. This method has been used in characterizing the transport capabilities of Golgi cisternae in baby hamster kidney cells (^{122}^{,}^{123}). In the case of cells in suspension, a known number of cells are Millipore-filtered or centrifuged to make a pellet of defined shape so its volume can be measured (^{124}^{,}^{125}). Again, vertical sections are generated and the cell volume density is estimated by point counting. The mean cell volume is obtained from the total pellet volume, *V*(pellet), multiplied by the cell volume density, *V*_{V}(cell/pellet), and divided by the number of cells counted on the filter *N*(cell):

### Local Estimator of Volume-Weighted Mean Volume

The term volume-weighted means that the structures are sampled in proportion to their volume: The greater the volume the more often they are sampled. The volume-weighted mean volume

may be estimated with point-sampled linear intercepts (^{14}^{,}^{126}) on a single section:

where

is the average of the cubed intercept length across the structure of interest through the sampling point. This estimator of

was the first in a family of stereologic methods for local volume estimation, also comprising the nucleator, the rotator(s), the selector, and star volume, the development of which was based on an integral geometric formula (^{127}^{,}^{128}^{,}^{129}). The volume-weighted sampling of

can help disclose new information about the intraindividual variation of the structure of interest because:

where *CV*_{N}^{2}(ν) represents the coefficient of variation in the ordinary number-weighted distribution.

The great advantage of the volume-weighted mean volume is that the measurements can be performed on thin single sections. The drawback is that conclusions from

are ambiguous with respect to

.. An enlarged volume-weighted mean volume can be the result of a greater number-weighted volume, a greater intraindividual variation, or both. This is known, for example, in lithium-treated rats. Two studies demonstrated that lithium-treated rats had a larger volume-weighted mean volume of glomeruli than control rats, which was caused by a 10-fold increase in intraindividual variation of glomerular volume (^{70}^{,}^{130}). The number-weighted mean volume of glomeruli was similar in the control and lithium-treated rats. A size distribution of glomeruli revealed many smaller glomeruli and some enlarged glomeruli in lithium-treated rats. When lithium-treated glomeruli were disector-sampled and serially sectioned, it was found that some were atubular, thus explaining the impairment of kidney function. Atubular glomeruli have also been found in cisplatin-treated rats (^{65}), and in patients with renal artery stenosis (^{66}) and chronic pyelonephritis (^{69}). In normal and diabetic rats,

of juxtamedullary glomeruli is greater than that of midcortical and superficial glomeruli (^{131}). Another important role for point-sampled linear intercepts would be in cancer-grading of kidney tumors, because point-sampled linear intercepts provide information of both mean size and variability of size. So far, only one study has used this method to classify renal tumors (^{132}). In bladder tumors, the volume-weighted mean nuclear volume of nuclei correlates well with the prognosis of the patients (^{133}^{,}^{134}).

### Local Estimator of Star Volume

Star volume

is related to

by being a volume-weighted size estimator. Star volume is defined as the mean volume of all parts of an object, which can be seen in all directions from a particular point without any interference from other structures (^{14}^{,}^{135}). The measurements are performed in the same manner as for point-sampled intercepts, and it is a very efficient detector of changes in volume and shape of complex structures. The estimate is taken over a uniform sample of points. If the object is convex, the star volume is equal to the number-weighted mean volume. A straight tubular structure will get a smaller star volume if it bends, and a marrow separated by trabeculae will get a greater star volume when holes appear in the trabeculae. Studies using the estimator of star volume detected mesangial matrix expansion in glomeruli by demonstrating an increase in the size of the matrix star volume in kidney biopsies from type 1 diabetic patients. (^{136}^{,}^{137}^{,}^{138})

## Estimation of Length of Tubes

The linear or tubular structures in the kidney include dynamic components of the cytoskeleton (microtubules, microfilaments) and tubular structures that provide directional flow and possess highly specialized functions aimed at adjusting the final composition of the glomerular ultrafiltrate (proximal and distal tubules, collecting duct) or carry oxygen and nutrients to the cells (blood vessels).

The first attempt to estimate length of tubes using stereologic methods was performed by Hennig in 1963, when he estimated the length of a nephron (proximal/distal tubules) in a rat kidney. Since then, the estimation of length of tubes has been described extensively (^{15}^{,}^{17}^{,}^{18}^{,}^{19}^{,}^{20}^{,}^{21}^{,}^{24}^{,}^{139}^{,}^{140}) in the stereologic literature. The only requirement that needs to be met when estimating length of tubes is that isotropic, uniform, and random sections must be used, or, alternatively, that the tube axes must be isotropic. If the tubes have a certain thickness, *i.e.*, diameter, one more requirement must be met: There must be, on average, a one-to-one relationship between the occurrence of an isolated tube profile and a planar intersection of the tube axes. This requires that the length of a tube must be much longer than its diameter. On two-dimensional sections, the length of a structure is represented as profiles of the structure and is sampled uniformly by isotropic area probes. That is why the basic formula for estimation of tube length density, *L*_{V}, relies on the simple fact that test planes “feel” curve length:

where *Q*_{A} is the number of tube profiles per two-dimensional test frame area. The two-dimensional counting rule is illustrated in Figure 7.

An elegant example of length estimation of capillaries outside the kidney (in rat soleus) using projection of vertical slices (^{19}^{,}^{20}) onto a cycloid test system and a specific staining method (^{141}) has been performed by Batra *et al.* (^{142}). A recent publication reported that the length of tubes may be estimated on uniformly sampled thick sections with an arbitrary direction using computer-generated, virtual isotropic test planes inside the sections (^{143}).

Before the development of the orientator and the isector, most researchers assumed isotropy of the tubular structure under study. On the basis of this assumption, two previous studies demonstrated an increase in length of rat glomerular capillaries after diabetes (^{85}) and uninephrectomy (^{144}), and three studies found an increase in rat proximal tubule length in short-term diabetes (^{145}^{,}^{146}^{,}^{147}). Although the assumption of isotropy of glomerular capillaries seems reasonable, this assumption may not necessarily be fulfilled with the proximal and distal tubules. Estimation of rat glomerular capillary length on IUR sections has been performed recently (^{86}^{,}^{87}^{,}^{88}). Estimation of the length of microvilli in the proximal tubule brush border is an exceptional example in which length may be measured directly on EM micrographs when the plane of the section contains the long axis of straight microvilli (^{17}).

## Estimation of Diameter and Cross-Sectional Area of Tubes

The mean diameter and average cross-sectional area of tubes carrying fluid (vessels, proximal and distal tubules, collecting duct) have significant implications for the flow of the fluid, given that the potential rate of flow in such a tube is directly proportional to the fourth power of its radius.

The sampling of tubes should ideally be performed on isotropic, uniform and random sections, because arbitrary sections require the unrealistic assumption that all tubes have the same diameter. The further sub-sampling of tubes should be performed using the two-dimensional unbiased counting frame (^{32}). The possibility remains that the estimation of diameter may be carried out on vertical sections using cycloids (see Estimation of Surface Area). Each time a cycloid his a tube, the diameter is measured (*i.e.*, if a cycloid hits a tube twice, the tube diameter is measured twice). This provides a surface-weighted diameter,

,, and the ordinary length-weighted diameter,

,, is calculated by computing the harmonic mean of the surface-weighted diameter measurements (Helle Clausen, personal communication):

where *m* is the number of times tubes were hit by cycloids and *d*_{i} is the observed diameter. The diameter of a tube in an IUR or vertical section is measured as the longest diameter of the tube profile perpendicular to the longest axis of the tube profile. For tube profiles that are figure-8-shaped, two diameters are measured. Fuzzy tube profiles that represent grazing cuts should be discarded from diameter measurements because they will underestimate the diameter. A different, and for many purposes better, approach to estimate tube mean diameter is to calculate it from tube surface area density and tube length density:

Tubes should be sampled in the same way as mentioned above to estimate their cross-sectional area. Point counting may estimate the cross-sectional area of tubes directly, without making shape assumptions. The following relationship exists between average tubular cross-sectional area,

, tubular length, tubular volume, and mean diameter:

Note that

unless the diameter is constant and that the SD of the diameter distribution can be calculated as SD

(^{148}). All of the estimates of diameter and its variation require the cross section of the tubes to be circular. Another important aspect here is that if the diameter/cross-sectional area of the tubes is to relate to its size in natural life, the tubes should ideally be perfusion-fixed or snap-frozen.

In the (rat) kidney, there is a great pressure gradient across the afferent and efferent arteriole (^{149}), which suggests that the diameter/cross-sectional area of the afferent/efferent arterioles reflects the resistance of the renal vasculature (^{150}). Ichikawa *et al.* (^{151}) postulated that the higher resistance of the renal vasculature seen in immature rats is caused by a smaller diameter of the afferent/efferent arterioles. John *et al.* (^{152}) could not find any evidence for this theory in growing dogs after direct measurement of the diameter of the afferent/efferent arterioles. The spontaneously hypertensive rat, which is an inbred race of Wistar-Kyoto rats, has been reported to have higher resistance of the renal vasculature caused by approximately 15% reduction in diameter of preglomerular arterioles as measured directly on sections following a perfusion technique and a vascular casting technique (^{153}^{,}^{154}^{,}^{155}^{,}^{156}).

The diameter of tubes can be calculated from length and surface area as shown in previous studies for rat glomerular capillaries (^{86}^{,}^{87}^{,}^{88}), and the cross-sectional area of tubes can be obtained by estimating the volume and length as shown for rat proximal tubules (^{145}^{,}^{157}) and rat glomerular capillaries (^{85}). The diameter of microvilli in the rat proximal tubule brush border has also been estimated previously (^{17}).

## Estimation of Surface Area

The surface area of cells is modulated closely to their function. For example, an increase in the transport capacity of a distal tubule cell is related to an increase in the surface area of the basolateral plasma membrane (^{158}^{,}^{159}). The capillary surface area represents the area available for transport of oxygen (peritubular capillary, vasa recta) to the tissues surrounding it or the surface area available for filtration (glomerular capillaries).

On two-dimensional sections or projections, the area of a structure is reduced to a structure boundary and is sampled by three-dimensional isotropic line probes. As test lines “feel” surface area, estimation of surface area density, *S*_{V}, is performed by counting intersections (*I*_{L}) between isotropic test lines and the surface under study:

The requirement for isotropic test lines, as mentioned previously, can be greatly facilitated using the vertical section technique (^{9}) as illustrated in Figure 13. Before the invention of vertical sections, the surface area of the plasma membrane of kidney tubule cells was mainly estimated on electron micrographs of tubule cross sections as exemplified by the estimation of luminal surface area density of rat proximal tubule cells (^{160}) and rat distal tubule cells (^{161}^{,}^{162}^{,}^{163}^{,}^{164}). Mathiasen *et al.* (^{165}) showed that if the surface area of the basolateral membrane was estimated on tubule cross sections instead of vertical sections, the surface area was underestimated by approximately 20%. Finally, it has recently been shown that the surface area of different objects can be estimated using thick transparent slices with an arbitrary direction and randomized virtual linear test lines generated by a computer (^{166}).

Assuming isotropy of glomerular epithelial cells, it was reported that the surface area of the glomerular epithelial cell plasma membrane was decreased by approximately 25% 10 days after administration of puromycin aminonucleoside (^{120}^{,}^{121}). The filtration surface area of the glomerular capillary network was found to be increased in short-term (^{85}^{,}^{88}) and in long-term experimental diabetes in the rat (^{167}). Using kidney biopsies, the glomerular capillary surface area has been shown to be increased in short-term type 1 (insulin-dependent) diabetic patients (^{168}). In addition, a strong correlation has been demonstrated between whole kidney GFR and glomerular filtration surface area in type 1 diabetic patients (^{169}^{,}^{170}^{,}^{171}^{,}^{172}^{,}^{173}). In immature rats, the GFR is low compared to that in mature rats measured both in absolute and relative terms (^{174}^{,}^{175}^{,}^{176}). There is a considerable increase in the glomerular capillary surface area or glomerular filtration surface area from immature to mature rats, but this is probably not the single most important parameter in explaining the increased GFR (^{87}^{,}^{177}^{,}^{178}).

## Estimation of Membrane or Barrier Thickness

The membrane or barrier width, *T*, between two biologic compartments may represent important information about the ease and potential of transport between the two compartments. The reconstruction of the distribution of reciprocal membrane thickness *T*^{-1} (by so-called unfolding procedures) is of great interest for membranes involved in diffusion processes, because the mean value of *T*^{-1} is related to the diffusion capacity for these membranes (^{179}).

In contrast to number estimation on sections, where the probability of hitting a structure is proportional to the height of the structure, membrane width estimation is essentially independent of the membrane thickness. Several methods for the estimation of membrane thickness using model-based sampling have been employed with alternating success (^{179}^{,}^{180}^{,}^{181}^{,}^{182}). In 1978, the first estimator of membrane width (^{24}) based on design-based sampling and uniformly oriented intercept lengths was introduced. However, the use of orthogonal intercept lengths has further improved the efficiency of membrane width estimation (^{23}).

In kidney research, much attention has focused on the glomerular basement membrane width in diabetic humans (^{138}^{,}^{183}^{,}^{184}^{,}^{185}^{,}^{186}) and in diabetic animals (^{187}^{,}^{188}^{,}^{189}^{,}^{190}) in studies of the initial and long-term diabetic microangiopathy leading to impaired renal function. Also, the width of epithelial foot processes in glomeruli (^{191}^{,}^{192}^{,}^{193}) has been studied in nephrotic conditions, which result in foot process fusion. Experimental short-term hypertension in the rat also causes a decrease in the width of epithelial foot processes in glomeruli (^{194}). An example of increased glomerular epithelial slit length has been reported in long-term subtotal nephrectomy of the rat (^{195}).

## Estimation of Error Variance

Before discussing sampling or error variance, it is crucial to understand the difference between bias and sampling variance as shown in Figure 14. One of the best ways to illustrate the importance of estimation of sampling variance is to use the previously described example estimating glomerular number in autopsy kidneys from control subjects and diabetic patients (^{8}^{,}^{35}). The observed total variation

of the number of glomeruli among control subjects and diabetic patients ranged from 23 to 34%. The mean number of glomeruli in nine type 1 diabetic patients with proteinuria (685 × 10^{3}; 0.27) (*mean; CV*) was not significantly different from the nine type 1 diabetic patients without proteinuria (561 × 10^{3}; 0.23). One might argue that the reason these two mean values did not differ was that the variation of the method was too great. However, when the coefficient of error of the stereologic procedure (*CE*_{ste}:= *SEM/mean*) was estimated, it was found to be 4.8% due to the counting of only 107 glomeruli per kidney. In this case, *CV*_{tot} stems primarily from the biologic variation *CV*_{bio}, because:

The consequence is that in this case it does not pay off to work harder and do more counting. Even if tens of thousands of glomeruli were counted per kidney and *CE*_{ste} was reduced to below 1%, the observed total variation would only be reduced from 23 to 22%.

An equally important reason for estimating sampling variance is to find out which sampling step in the individual kidney—the tissue blocks, the sections, or the stereologic estimation—is responsible for the greatest error variance to “Do more less well” (^{197}). The formula above is based on the basic principles for nested sampling (see, for example, Chapter 10 in reference ^{198}), which was introduced to stereologists by Shay (^{199}) and by Nicholson (^{200}). This principle has been used in practical stereologic methods estimating total glomerular filtration surface area and total glomerular mesangial volume in rats (^{201}) and glomerular basement membrane thickness in rats (^{197}).

It is worth mentioning that all of the below-mentioned estimations of *CE*_{ste} are only approximations because systematic sampling is used, but such approximations will in practice fulfill the purposes mentioned above.

### Error Variance of the Cavalieri Principle

The sampling variance of the Cavalieri method is related to the point counting variance expressed in the “noise,” *Var*_{Noise}(Σ*P*), and the variance of the area estimation using systematic uniformly random sampling, *Var*_{SURS}(Σ*a*) (^{202}). The contribution to the error variance caused by the noise is:

where

is an average shape factor that can be found for different shapes in Figure 18 in Gundersen and Jensen (^{95}), *n* is the number of sections, and Σ*P* is the total number of points hitting the structure being studied. The contribution to the error variance caused by the systematic uniformly random sampling is:

where *i* is the section number. The error variance of the Cavalieri estimate is then:

In practice, the contribution from *Var*_{Noise}(Σ*P*) is dominating and *Var*_{SURS}(Σ*a*) may often be ignored since it decreases by *n*^{4}. As an example, 200 points counted on eight rather complicated kidney sections may provide a

.

### Error Variance of Number Estimation

The estimation of error variance of the disector and fractionator method used for number-weighted sampling of objects can be performed in the same manner as the estimation of error variance of the Cavalieri method based on the quadratic approximation formula (^{202}). The contribution to error variance caused by the noise equals:

where Σ*Q*^{-} is the total number of objects counted. Using systematic uniformly random sampling, the contribution to the error variance of the estimate from the set of systematic sections may be predicted as for the Cavalieri principle except that Σ*P* is replaced by Σ*Q*^{-}. Considering that the noise accounts for by far the largest contribution to the error variance of the fractionator and disector, *CE* may be calculated very quickly as:

Number estimation is more demanding than volume estimation. The above equation means that if 100 to 200 objects are counted, then it is possible to achieve a *CE* around 0.1.

When the Euler number is to be estimated, a similar approach can be used for estimation of error variance. The reason is again that the main contribution to the error variance comes from the noise, which may be estimated as (^{81}):

In a study by Gundersen *et al.* (^{81}), Figure 5 depicts a nomogram for reading the expected error variance as a function of the number of bridges and the fraction that the bridges constitute of all observations.

### Error Variance of Ratios

The estimators of length density, surface area density, volume density, and number density are all ratios, in which case their observed sampling variances also may be approximated by the variances of the ratio-of-sums (see references ^{203} or ^{204} for the use in a stereologic context).

where

is the ratio of the wanted parameters and *n* is the number of sampling items (often sections). The formula is based on the assumptions that there is independence between the sampling items and requires that *CE*(Σ*x*) is rather small (≤0.1).

### Error Variance of Local Volume Estimators

The variation of the local volume estimators, nucleator, rotator(s), selector, star volume, and volume-weighted mean volume, can be judged by estimation of the error variance from the actual measurements. The individual estimate of each object, ν_{i}, is registered, and the coefficient of variation is computed as

.. This is the intraindividual variation in the size of the objects. The coefficient of error of the estimated mean size of the objects based on *n* individual estimates is:

## Future Tasks for Kidney Stereology

Today, many kidney scientists still find it difficult to use stereology due to their fear of numbers and lack of understanding of statistics. One of the big leaps forward in making stereology easier to use was the invention of computer-assisted stereology: The EM, CLSM, or LM microscope is equipped with a video camera connected to a computer with a video frame grabber. A large range of test systems (frames, lines, curves, points) and specific stereologic probes at user-defined intensities and in any combination may then be superimposed on the video images. The practical procedure of sampling and counting is in most cases much easier and quicker to perform. This is especially the case if the counting procedure requires systematic uniformly random sampling. The area of interest, *e.g.*, kidney cortex or tubule cell cytosol, is delineated at a relatively low magnification, and the sampling in the x- and y-axis using a computer-controlled, motorized microscope stage is performed at a higher magnification. The LM or CLSM microscopes can in addition be equipped with an electronic microcator, which measures the level of the focal plane along the z-axis with a precision of 0.5 μm and works in a feedback loop with the software. It is then possible for the researcher to have full control of the sampling in the x-, y-, and z-axis in thick (> 25 μm) sections.

Some of the new stereologic methods such as the optical rotator (^{12}) can only work in practice by the use of computer-assisted stereology. In the computerized environment of today and tomorrow, several new developments will take place. Neither point counting nor delineating is the most efficient estimator of the area of an arbitrary kidney profile. Using the two-dimensional nucleator (^{10}), the researcher defines a “central” point and clicks intersections at the boundary with a set of radiating test lines. The number of test lines (directions) determines the precision. Using the area information, a subroutine may proceed to a dedicated estimator using a two-dimensional isotropic test net for unbiased boundary estimation.

A novel concept of spatial IUR sampling with virtual probes in thick (>25 μm) sections with an arbitrary orientation may be applied to all stereologic estimators (^{143}). The computer supplies the necessary isotropy of the probes, and the intersection between the focal plane and the virtual probe is visible and moving when focusing up and down in the thick section. The great advantage in practice is that the sections may be made in an orientation that makes the kidney objects well defined. There are problems that must be considered when using spatial IUR sampling with virtual probes and observation of spatial distributions. The three-dimensional test volume needs to be well defined and of simple shape, and it should allow tight control of the sampling fraction as a function of distance from the primary object.

Another much more complicated problem is overprojection associated with the use of thick sections and LM. The images of the focal plane from thick sections contain out-of-focus information, which reduces the quality of the image. The use of thick sections and CLSM reduces overprojection considerably for objects far away from the focal plane but CLSM works only in fluorescence mode. There are no mathematical tricks to overcome the problem of overprojection in LM of thick sections. It may be possible to carry out optical sectioning of thick sections with focal restoration of the images, taking advantage of: (*1*) digital optical sectioning; (*2*) the information in the focal planes, which is impaired by known, deterministic, and nonstochastic mechanisms; and (*3*) overprojection in transmission microscopy, which in general only adds further information to that belonging to the focal plane.

The estimation of “shape,” *e.g.*, of kidney cells, may have significant implications for the prognosis of cancers in the kidney. So far, no one has come up with stereologically valid estimators of shape. It will be quite impossible to invent design-based estimators of shape, but valid model-based estimators of shape may be useful if the model is tested in an appropriate way.

The physical disector consists of two adjacent thin sections in which each field is compared to the corresponding one in the other section, which until now is possible only by using two identical projecting microscopes. Because the process of finding the corresponding field takes quite a long time, the optical disector is used in many cases with LM and CLSM. In some cases, however, the physical disector has to be used because of difficulties with penetration of antibodies and other markers. Only the physical disector can be used with conventional EM. The core of the “split-screen disector” computer procedure is the statistical solution to the alignment of two or many sets of nonmatching fix points or landmarks. In practice, the computer screen shows the sampling section and look-up section at the same time and a proper counting principle may be applied.

It has for many years been a problem for any biologic researcher to combine image analysis and (computer-assisted) stereology. The strict criteria for making measurements in stereology have not been satisfactorily solved in the field of image analysis as yet. The dominating problem is the initial automatic “segmentation” of the image into precisely defined and meaningful compartments (structure profiles). Improved software and hardware may alter this schism somewhat. However, it is my opinion that in the foreseeable future most, if not all, stereologic counting procedures will require an expert researcher as well as a “super”-computer. The use of noninvasive scanning techniques (ultrasound, magnetic resonance imaging, computed tomography, *etc.*), in combination with powerful three-dimensional image analysis and manipulation, amplifies the need for stereology to design proper sampling procedures with spatial test systems. Absolute numbers and sizes can be obtained directly from such imaging techniques. The whole kidney, large kidney vessel systems, and kidney tumors can be directly imaged in three dimensions and entirely immersed in a spatial grid by virtual imaging techniques. The relevant structural parameters may be estimated properly if either the object or the test system is shifted, and eventually rotated according to stereologic rules.

As such, computer-assisted stereology offers many potential advantages and possibilities, but it also carries a great risk. Some researchers do not pay attention to the criteria for making the different stereologic estimates, *e.g.*, sampling of tissue blocks, orientation of sections or test systems, *etc.*, because they rely on the “wisdom” of the computer taking care of everything.

Previous methods for obtaining quantitative structural information of kidneys have in many cases been troubled by being model- or assumption-based. Because the reality and ideal rarely agree and the underlying assumptions are untested, the validity of the conclusions drawn is unknown. It is hoped that modern stereology as described here and in other reviews (^{205}^{,}^{206}^{,}^{207}^{,}^{208}) will advance kidney science beyond that stage.

## Glossary of Terms

**Bias:** A systematic difference between the average estimate and the true value (Figure 14).

**Cavalieri principle:** An unbiased principle for estimating the volume of any object. With a random start, the object is cut into slices of a known and fixed thickness. The volume is estimated by multiplying the distance between the slices by the total cut area of the slices (see Equation 6).

**Connectivity:** The maximal number of cuts through an object, which does not produce two disconnected objects. These extra or redundant connections are equal to the topologically defined number of units in a network (see Equations 5 and 6 and Figure 10).

**Cycloid:** A curve describing the exact position of a given point on the periphery of a disc rolling along a straight line (see Figure 13). A cycloid arc has a length distribution proportional to the sine of the angle to the vertical axis. Cycloids are primarily used for surface estimation on vertical section planes.

**Density:** Many stereologic estimators report the ratio of the amount of an object phase to the total reference volume. These ratio estimates are known as densities. Examples are volume density, *V*_{V}, surface density, *S*_{V}, length density, *L*_{V}, and numerical density, *N*_{V}.

**Disector:** A three-dimensional stereologic probe for sampling of objects according to their number. It consists of a pair of physical or optical section planes separated by a known distance and an unbiased two-dimensional counting frame. The objects are sampled if they are present in the counting frame in one section plane and not in the other plane (see Equations 4 and 6 and Figures 7, 8, 9, 10).

**Euler number:** A zero-dimensional entity that can be used to estimate the number of units in a network. A topologic definition of a unit, *i.e.*, a loop in the network, is used for the estimation (see Figure 10 and Equations 5 and 6).

**Error variance:** The random fluctuation between individual unbiased estimates and the true value. This sampling variance is a random error and therefore not a bias (see Equations 19 through 27 and Figure 14).

**Fractionator:** A sampling methodology in which a known and predetermined fraction of an object is sampled, often in several steps of different sampling fractions. In the final sample, the structural quantity (usually number) is estimated. The total structural quantity (number) is estimated by multiplying the inverse sampling fraction by the structural quantity in the final sample (see Figure 4 and Equation 1).

**Isector:** A simple method for generating isotropic section planes by embedding the small object into a spherical mold. Section planes through the spherically embedded and rolled object will be isotropic (see Figure 5 and the section, Orientation of Section Planes).

**Isotropic design:** Isotropy means that all directions have equal probability of being chosen. A design with completely random orientation is therefore an isotropic design, which is necessary if the stereologic estimators require isotropic test planes (see Figure 5 and the section, Orientation of Section Planes). An isotropic design is usually combined with uniform random sampling generating isotropic, uniform random (IUR) section planes.

**Lost caps:** Objects are physically lost or optically not visible in a thin section or close to the edges of a thick section (see the section, Some Basic Stereologic Problems).

**Nucleator:** A two-step, number-weighted, local size estimator. First, the objects are sampled in proportion to their number using the disector. Second, the volume of each object is estimated by measuring the distance from a unique point, *e.g.*, the nucleolus in a cell, to the boundary. The average estimate is the ordinary, number-weighted, mean volume (see the section, Estimation of Volume).

**Number-weighted:** Each object has equal statistical weight equal to one regardless of its shape and size.

**Noise:** The independent variance of a stereologic counting procedure, *e.g.*, point counting and object counting, is called the noise. The sum of the error variance caused by the noise and by the systematic sampling procedure is the error variance of the systematic stereologic estimators (Equations 20 through 25).

**Orientator:** A two-step procedure for generating isotropic section planes in large objects. First, the object is placed flat on a circle with equidistant divisions along the perimeter. A random number decides the direction of the first cut. Second, the isotropic section is obtained using a random number and a circle with sine-weighted divisions along the perimeter (see Figure 5 and the section, Orientation of Section Planes).

**Overprojection:** An effect of positive section thickness of transparent slices, which makes all measurements of nontransparent structures larger. The best way to avoid overprojection bias is to make section planes as thin as possible (see the section, Some Basic Stereologic Problems).

**Point-sampled intercepts:** A two-step procedure for estimation of the volume-weighted volume. First, the objects are sampled in proportion to their volume using test points. Second, the volume of each object is estimated by measuring the length of an isotropic intercept in the object passing through the test point. The average estimate is an unusual, volume-weighted, mean volume (see Equation 12).

**Rotator:** A number-weighted, local size (volume) estimator performed in two steps like the nucleator except that details of the measurements and computations are performed differently (see Figure 8).

**Star volume:** Defined as the average volume of the object that can be seen unobscured from a random point within the object. Star volume is estimated by the use of point-sampled intercepts.

**Systematic, uniformly random:** Sampling performed with a systematic and a random component. If the goal is to sample approximately six slices from a population of 100 numbered slices, the systematic component is the decision of choosing a sampling periodicity of 16. The random sampling component is to look up a random number between 1 and 16, *e.g.*, 5. The following slices are then sampled systematic, uniformly random: 5, 21, 37, 53, 69, and 85 (see Figure 2).

**Uniformly random:** Every object in the whole population has an equal probability of being sampled or every position in the object has an equal probability of being hit (see Figure 1).

**Vertical design:** A vertical direction is identified or generated in the object, which is rotated at random around this vertical axis (see Figure 5). On the properly sampled and uniformly positioned section planes, the vertical direction must be identifiable. A range of stereologic estimators requiring isotropic test lines may then be applied (see Figure 5 and the section, Orientation of Section Planes).

**Volume-weighted:** Each object has a statistical “weight” according to its volume. The larger the volume the higher the probability of being chosen (see the section, Estimation of Volume).

Acknowledgments

This study was supported by the Beckett Foundation, the Danish Diabetes Association, the Danish Heart Foundation, the Danish Medical Association Research Fund, Fonden til Lægevidenskabens Fremme, the Novo Nordic Foundation, and the Aarhus University Research Foundation. I thank Hans Jørgen G. Gundersen for his invaluable comments. The superb technical assistance of A. M. Funder, A. Larsen, and M. Lundorf is highly appreciated.

American Society of Nephrology

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