Reviewed manuscript: Nyengaard, J.R., (1999) Stereologic Methods and Their Application in Kidney Research. Journal American Society Nephrology 10, 1100 – 1123.

Note that our comments/suggestions are italicized.

This review (Nyengaard, 1999) is a useful guide for using design based stereology to “study the three dimensional kidney on two dimensional images” and addresses the problem that when viewed at the microscopic level, “a single kidney structure can generate widely differing sections or projections, and several different kidney structures may generate similar sections or projections (Nyengaard, second paragraph). Examples of bias that can arise when not using design based stereology are given. In thin cross-section, it is not known which profiles of pedicels belong to what podocyte and a glomerular basement membrane will look thicker if it is sectioned orthogonally. In low-magnification projections through a thick section false connections among tubules and among capillaries may appear and too many kidney cells may be seen (Nyengaard, third and fourth paragraph). Although this review is approaching two decades in age, it for the most part holds up due to the efficiency and unbiased nature of modern stereological probes that Nyengaard, in 1999, was saying had become the “technique of choice” (Nyengaard, sixth paragraph, first sentence).

SAMPLING (Nyengaard, Sampling Strategy)

It is pointed out that random sampling should be used at the section-sampling level. We recommend systematic random sampling (Nyengaard, Fig. 2). Systematic random sampling should also be used at the intra-section level (Nyengaard, Fig. 3).

The seemingly obvious but often overlooked or unattained need to be able to unambiguously identify the structures in question is stated.

The NvVref method and the fractionator method are discussed. The problem with the NvVref method is noted; densities contain only relative information and are subject to the ‘reference trap’ so reference volumes must be obtained. For this reason the fractionator method (Nyengaard, equation 1), that requires no knowledge of a reference volume and is not subject to differential volumes due to different size artifact, is recommended.

ISOTROPY (Nyengaard, Orientation of Section Planes)

The necessity to not favor any orientation of the probe relative to the object being probed is discussed. “All estimation 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” (Nyengaard, Sampling Strategy, second paragraph, first sentence). Arbitrary orientation means you can section the tissue in the direction you prefer. In the case of number estimates, the leading edge of the particle is counted, and it does not matter from which direction you approach a leading edge; any direction will be equally unbiased. Regarding volume estimates with point counting, the estimate is just as unbiased no matter which direction you section the volume. For surface, length, thickness and particle size estimates, isotropy of interaction with the probe is achieved by arranging it so that no orientation is favored. This review talks about assuring that the tissue is random in three dimensions by using isotropic sections (Nyengaard, Fig. 5) or randomizing the tissue in two planes to get vertical sections and then using a special cycloid line segment to probe for surface or cycloid plane to probe for length (Nyengaard, Orientation of Section Planes, paragraph two and three). This advice still holds today if only thin sections can be obtained, but the advent around the same time as the publication of this review of probes that are themselves isotropic has eliminated the need for isotropic or vertical sections when probing length and surface. These isotropic probes, Spaceballs to estimate length and Isotropic Fakir to estimate surface, are three dimensional probes and therefore must be accommodated by using thick sections. When estimating particle volume or particle surface, for example with the Nucleator or Surfactor, the only way to ensure an isotropic interaction is to use isotropic or vertical sections. When estimating the thickness of a structure using the orthogonal intercepts probe, isotropic sections should be used.


The need to use the whole region of interest, if possible, is noted (Nyengaard, Fig. 6). The existence of artifact created by sectioning is mentioned (Nyengaard, Some Basic Stereological Problems, third paragraph). We recommend using guard zones that are big enough to cover the artifact.


“A three-dimensional probe is required for number-weighted sampling of structures” (Nyengaard, Some Basic Stereological Problems, second paragraph, second sentence). This is because the leading edge of the particle whose number is being estimated is counted in order to normalize particles of different sizes and orientations to each other. To find the leading edge, for example the top of the cell or the top of the nucleus when estimating cell number, at least two physical sections are required (physical fractionator) but many optical sections is often more efficient (optical fractionator).

OVERPROJECTION EFFECT (Nyengaard, Some Basic Stereological Problems, fourth and fifth paragraphs)

When estimating volume using the Cavalieri/point-counting method, use thin sections relative to the extent of the region to minimize the overprojection or Holmes effect.

TISSUE DEFORMATION (Nyengaard, Some Basic Stereological Problems, last paragraph)

Estimates of length, surface and volume are affected by tissue deformation. It is pointed out that any correction factors should be used with isotropic sections unless the shrinkage is the same in the X,Y and Z dimensions. It is suggested to tackle this problem during histological processing: “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” (Nyengaard, Some Basic Stereological Problems, last paragraph, second to last sentence).

NUMBER (Nyengaard, Estimation of Number of Objects…)

“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 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)” (Nyengaard, Estimation of Number of Objects…, second paragraph, sixth and seventh sentences). Using the disector along with systematic random sampling, numerical densities of objects can be estimated in an unbiased way (NvVref method), or if the fractionator method is used, estimates can be obtained without the need to know the total volume. The advantage of the optical fractionator (Nyengaard, Fig. 8) over using the optical disector with the NvVref method is pointed out enthusiastically with a rare appearance by the exclamation point; “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” (Nyengaard, Estimation of Number of Objects…, third paragraph, third sentence).

It is recommended in this review to use the physical disector to estimate the number of glomeruli and the optical disector to estimate the number of cells that make up the glomeruli, in the same, about twenty-five micron thick, sections. This is the so-called double-disector. The thin pairs of contiguous sections needed for the physical fractionator will actually be optical; inside of the thick physical sections needed for the optical fractionator. If electron microscopy is needed to identify a cell type, glomerular and parietal epithelial cells, endothelial cells and mesangial cells are given as examples, the physical disector method can be used to estimate the percentage of a given cell type. This percentage can be multiplied by the estimate of total cell number obtained with the optical fractionator at the light level (Nyengaard, Estimation of Number of Objects…, fourth paragraph).

Pointed out also is the fact that the number of ‘associated features’ or ‘external objects’ on the surface of a structure can be estimated in an unbiased way. The example given is the use of the physical disector at the E.M. level to estimate the number of microvilli. Each microvillous has one and only one connection to the luminal surface of a proximal tubule cell; making this point perfect to be used when counting the associated features (Nyengaard, Number of Associated Features of Objects).

NUMBER OF CAPILLARIES (Nyengaard, Number of Connected Sets of Objects (ConnEulor))

The number of units in a complex network can be estimated in an unbiased way using the Euler number and the connectivity probe. “For the estimation of Euler number, χ, in a network, the number of isolated parts (islands), the number of extra connections (bridges), and the number of enclosed cavities (holes) in the network must be taken into account” (Nyengaard, Number of Connected Sets of Objects (ConnEulor), first paragraph, last sentence and equation (5)). Connectivity is defined as (1- χ ) and χ is (islands – bridges + holes). Connectivity is the number of cuts you could make in the capillary network without forming two separate networks (Nyengaard, Fig. 10c), that is for example the number of capillaries in a capillary network. A physical disector is used and if one capillary cross-section is seen in the reference but not the look-up section, an island is counted. If one cross-section is seen in the reference section and it turns into two cross sections in the look-up section, a bridge is counted. And if one cross-section on the reference section turns into a cross section with an isolated part in its middle the rare hole is counted (Nyengaard, Fig. 10c).

REGION VOLUME (Nyengaard, Estimation of Volume)

Capillary luminal volume is related to the volume of blood available for gas exchange. Cavalieri/point-counting is recommended to estimate volumes of regions such as this, and has been used to estimate the volume of the kidney, kidney cortex, and glomerulus. We are reminded to use systematic random sampling, as we should for all unbiased stereology. The area-fraction-fractionator probe is recommended to obtain estimates of percent by volume. For instance, the Cavalieri point-counting probe or even water displacement can be used to estimate the volume of the whole kidney, then the area-fraction-fractionator probe can be used first at the light level to estimate the percent by volume of proximal tubule cells and then at the E.M. level to estimate the volume fraction of mitochondria in proximal tubule cells. The mitochondria volume estimate is equal to the estimate of kidney volume times the estimate of the percent of proximal tubule cells in the kidney times the estimate of the percent of mitochondria in proximal tubule cells (Nyengaard, equation 8).

PARTICLE VOLUME (Nyengaard, Local Volume Estimators of Number-weighted Mean Volume)

The difference between number-weighted, volume-weighted, and (surface weighted) star volume is noted; in most cases number-weighted sampling is desired. The planar rotator, optical rotator, nucleator and selector are given as candidates to estimate the volume of cells in the kidney. To estimate the volume of glomeruli, a combination of the disector to pick the glomeruli for study without regard to size or orientation and point-counting to estimate the volume can be used (Nyengaard, equation 9). This indirect method that is a variation of the selector is given as a choice for estimating the volume of cells also, but we think that point-counting can be problematic on cells as it is often hard to discern their membrane. The most common choice today for number weighted sampling and estimation of particle volume is to use a disector to pick the cell for study (this is number weighted selection) and use the nucleator probe to estimate cell size. If isotropic or vertical sections are not used, note this and do not claim the estimate is unbiased, instead graph the areas in a histogram and explain that arbitrary (preferential) sections were used, possibly introducing bias.

LENGTH OF TUBULES (Nyengaard, Estimation of Length of Tubes)

Examples of tubes of interest in the kidney are given; microtubules, microfilaments, proximal and distal tubules, collecting ducts and blood vessels. The need for vertical or isotropic sections is stressed and the basic formula for length per volume estimation is given (Nyengaard, equation 14). This is one of the places where this review is dated. The idea of making the probe isotropic in thick sections instead of using vertical or isotropic sections is mentioned; it is reported that isotropic virtual planes had been recently used to estimate glomerular capillary length (Nyengaard, Estimation of Length of Tubes, last paragraph, second to last sentence). But the Spaceballs probe in thick sections to estimate length has become much more popular because the rules are easier to follow than for isotropic virtual planes and just like isotropic virtual planes, arbitrarily oriented sections, sections oriented anyway you prefer may be used.

DIAMETER AND CROSS SECTIONAL AREA OF TUBES (Nyengaard, Estimation of Diameter and Cross-Sectional Area of Tubes)

The mean diameter and cross sectional area of vessels, proximal and distal tubules, and collecting ducts influence the flow of fluid. Isotropic sections must be used and only profiles where the tube is in ‘perfect’ cross-section should be used, and these should be picked with a two-dimensional disector, such as the fractionator probe. Point-counting can be used to estimate cross sectional area and the longest diameter of the tube profile should be measured. Or model-based methods can be used; the surface per volume and length per volume can be used to calculate the tube cross sectional diameter (Nyengaard, Estimation of Diameter and Cross-Sectional Area of Tubes, equation 16), and that diameter may in turn be used to calculate the cross sectional area (Nyengaard, Estimation of Diameter and Cross-Sectional Area of Tubes, equation 17).


Surface area of cells influence their function, the example of the proximal convoluted tubule cell and the effect the surface of its basolateral plasma membrane has on transport capacity is given. Surface of structures is important also, the surface of glomerular capillaries also affects filtration. The basic formula for estimating surface per volume is given (Nyengaard, Estimation of Surface Area, equation 18). When estimating surface, as when estimating length, isotropy must be assured by some combination of the arrangement of the probe and the tissue. It is noted that when the surface area of the basolateral membrane on tubule cross sections was estimated in thin sections that were not vertical or isotropic, there was an underestimation of about 20% compared to estimates of surface on thin vertical sections using cycloids (Nyengaard, Estimation of Surface Area, third paragraph, Mathiasen reference). The use of vertical sections is discussed, but this review also anticipates the use of thick sections when estimating surface (Nyengaard, Estimation of Surface Area, third paragraph, last sentence). To estimate surface, it is easier to use thick sections and an isotropic probe, called the isotropic fakir.

THICKNESS (Nyengaard, Estimation of Membrane or Barrier Thickness)

Membrane thickness, for instance of the glomerular basement membrane, has implications for transport and is related to diffusion capacity. This review suggests using the orthogonal intercepts probe to estimate thickness instead of model based stereology.


In order to tell if the determination that two means are ‘not significantly different’ is due to too much variation in the sampling process, the equation that says the total coefficient of variance (CV = standard deviation/mean) has two components, biological and sampling, is useful (Nyengaard, Estimation of Error Variance, equation 19). The CV can be calculated from the actual data. The sampling variance term, called the coefficient of error or CE, can be estimated . In this case the Gundersen CE is used to estimate the sampling error (Nyengaard, Estimation of Error Variance of the Cavalieri Principle and of Number Estimation). Then biological variance can be calculated and you can compare the biological variance to the sampling variance to see if it is worth sampling more. If the biological variance dominates, no amount of further sampling will tease out any differences. If the sampling variance dominates, more sampling may help. The Gundersen CE has two terms, a within-sections term (Nyengaard, Estimation of Error Variance, equation 20 for Cavalieri estimation of volume and equation 23 for disector estimation of number) and an among-sections term (Nyengaard, Estimation of Error Variance, equation 21). Although some believe the two terms are useful for both volume and number estimates (West, 2012, Chapter 10) Nyengaard says that the within-sections term “accounts for by far the largest contribution to the error variance of the fractionator and disector” (Nyengaard, Estimation of Error Variance, sentence just before equation 24, see also Schmitz, 1998, last equation). For local volume estimators, it is recommended to use the actual measurements to calculate the Coefficient of Error (Nyengaard, Estimation of Error Variance, equation 27), although no explanation is given as to why this is equivalent to repeated estimates.


Nyengaard, J.R. (1999) Stereologic Methods and Their Application in Kidney Research. Journal American Society Nephrology 10, 1100 – 1123.

Schmitz, C. (1998) Variation of Fractionator Estimates and its Prediction. Anat Embryol 198: 371-397.

West, M.J. (2012) Basic Stereology for Biologists and Neuroscientists, Cold Spring Harbor Laboratory Press, Cold Spring Harbor, NY