Vasculature – thin sections

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Review of Publication: C. Mühlfeld (2014) Quantitative morphology of the vascularisation of organs: A stereological approach illustrated using the cardiac circulation. Annals of Anatomy, doi: 10.1016/j.aanat.2012.10.010.

This manuscript (Mühlfeld, 2014) is an important contribution to the literature regarding design-based stereology. That literature has been growing in a sometimes confusing and haphazard way in the last several decades; as a result it can be hard to find explanations of the techniques needed to fit particular applications. It is refreshingly instructive to read a clear explanation of the use of unbiased stereology, in this case to quantify morphometric characteristics of the cardiac vasculature, using thin, one to two micron, sections. Isotropy in three dimensions when making estimates of surface and length is stressed, a requirement that causes bias when overlooked or skipped. For instance, “The most common morphometric approach for analyzing remodelling of the cardiac microcirculation is to count the number of vessel profiles per unit sectional area.” If isotropy is not ensured, overestimates will result if orientations that favor a perpendicular interaction between the area and the vessels are more likely, or underestimates will occur if orientations that favor a parallel interaction between the probe and vessels are favored. The use of the orientator or the isector is recommended to make the tissue isotropic in three dimensions (Mühlfeld, “Sampling of location and orientation, and Fig. 1″). The author points out the proposed techniques can be used to study blood vessels in other tissue and organs. Thick sections, however, are useful when studying tissue geometry that is not isotropic in three dimensions. Isotropic probes can fit in thick sections, eliminating the need for time-consuming and confusing tissue manipulation, i.e. using the isector to randomize three planes to obtain isotropic sections or randomizing only two planes to obtain a vertical section. A warning about the ‘volume-reference’ trap is also tendered; numerical densities should not be directly compared, instead the numerator and denominator of the density should be reported separately. Clear explanations about choosing the appropriate probe, advice about pitfalls, and worked examples make this a valuable paper for conducting unbiased stereology on thin sections to study cardiac vasculature.

There are two fundamental methods for carrying out an unbiased stereological study, the fractionator method or the NvVref method. This review talks about using the latter (Mühlfeld, “Reference Volume”), despite touting the advantages of the former such as the fact that it is not influenced by tissue deformation and does not require the estimation of the reference volume (Mühlfeld, “Sampling of location and orientation”). The procedures from the paper for estimating numerical density (Nv) are described below. The author mentions two ways to obtain the reference volume; weighing the heart and dividing by the density, or using unbiased stereology in the form of the Cavalieri/point-counting probe. The latter method is recommended by the author and by us.

Systematic random sampling, random sampling and smooth fractionator sampling are all mentioned as possibilities for selecting XY-locations in the tissue to study. We recommend systematic random sampling. “The general aim of the sampling procedure is to give every part of the heart an equal chance at being selected for the investigation” (Mühlfeld, “Sampling of location and orientation”).

“The gold standard for obtaining quantitative morphological data is design-based stereology which allows the estimation of volume, surface area, length and number of blood vessels as well as their thickness, diameter or wall composition” (Mühlfeld, “abstract”).


(Mühlfeld, “Morphometric parameters related to vasculature, Volume”)

Volumes of interest in the cardiac vasculature include vascular luminal volume that is proportional to blood volume in the myocardial circulation and vascular wall volume that may reflect changes such as the amount of smooth muscle cells in the tunica media. Point grids (see Cavalieri/point-counting) are used for volume estimation on the thin physical sections talked about in this review and there is not a requirement to make the tissue isotropic.


(Mühlfeld, “Morphometric parameters related to vasculature, Surface”)

The luminal surface of capillaries and its ratio to the surface of cardiomyocytes is important in understanding gas and nutrient diffusion in the heart. Lines are used to estimate surface. Since the tissue is thin in this review, it is suggested to use a line grid, which is similar to the Merz probe except that for the Merz probe half-circle lines are used. That means the tissue must be made isotropic. If it were possible to use thick sections, the isotropic fakir probe would be preferable; since this probe is isotropic the tissue does not have to be randomized in three planes to obtain isotropic sections.


(Mühlfeld, “Morphometric parameters related to vasculature, Volume”)

Thickness of the vascular wall is studied to understand changes in the make up of the wall and in capillaries to look at diffusion distances. It can be calculated in a model-based manner by dividing wall-volume by wall-surface (see ‘estimating width of slabs‘), or estimated directly with orthogonal intercepts.


(Mühlfeld, “Morphometric parameters related to vasculature, Length”)

Total length of capillaries is a direct way to look at the vascularization of the myocardium and is related to the vascular resistance. Vascular resistance also depends on the degree of branching (see ‘Number’ below). The reciprocal of length per volume is the area surrounding one capillary profile and can help to think about the amount of myocardium being supplied by one capillary. Length per unit volume is estimated on these thin sections by counting the number of intersections per unit volume and multiplying by two (see ‘length probed with the image plane’). This requires isotropic sections however. To avoid having to spin the tissue blocks randomly in three planes, thick sections should be used if possible so that the spaceballs probe can fit into the sections. Spaceballs is an isotropic probe so preferentially oriented sections can be used. The author points out however that for length, it is important to have thin, two to three micron sections, to ensure unambiguous identification of vessel profiles. Of course, if it can be shown that the vessels themselves are isotropic in space, using the image plane as the probe can be done without isotropic sections.

An excellent example of the ‘volume-reference’ trap is given (Mühlfeld, “Morphometric parameters related to vasculature, Length”). For illustration, the author asks us to consider a hypertrophied heart that has no changes in length of blood vessels. If we report that the numerical density of length per volume went down, who knows if the length went down, the volume went up, or some combination happened. To avoid the volume reference trap when using the fractionator method, report length and volume separately; and another probe will need to be used to estimate the volume, such as Cavalieri/point-counting. To avoid the volume reference trap when using the NvVref method, as was considered in this paper, always multiply the numerical density by the reference volume. These precautions are ‘frequently ignored in the scientific literature’ (Mühlfeld, “Morphometric parameters related to vasculature, Length”).

Also pointed out is the fact that any estimates of length can be significantly affected by artifact shrinkage during histological processing (Mühlfeld, Morphometric parameters related to vasculature, Length).


(Mühlfeld, “Morphometric parameters related to vasculature, Length”)

“Using the model assumption of blood vessels as being right circular cylinders, simple geometric formulae can be applied to calculate the mean diameter”. For instance you can ‘work backwards’ from the estimated volume and estimated length, using the formula for the volume of a cylinder, to calculate the diameter (see ‘model-based stereology’).

d = 2 * square root [V/π L]

d is the diameter of the vessels

V is the estimated volume of the vessels

L is the estimated length of the vessels (height of the cylinder)

Alternatively, the author suggests that direct measurements can be made on cross-sections. We recommend considering use of the nucleator on cross sections of vessels that would give you an estimate of the area.


(Mühlfeld, “Morphometric parameters related to vasculature, Number”)

The phrase ‘number of blood vessels’ needs definition since the vasculature could be thought of as one continuous vessel. In this review, the author suggests using the connectivity assay. The physical disector must be used; pairs of thin sections are compared and ‘bridges’ and ‘islands’ are marked. An island is a piece of vessel that appears in one section but not the next, and a bridge appears in one section and then splits into two pieces in the next section. The Euler number is the sum of the number of islands minus the number of bridges. Connectivity of the capillary network can be thought of as the maximum number of cuts that could be made without separating the network into two parts (Nyengaard, 1999, “Fig. 10”), and is mathematically defined as ‘one minus the Euler number’. The numerical density of ‘capillaries per volume’ is:

Wv = (1 – ∑Euler number) / (2 * t * ∑a(frame))                 (Nyengaard, “equation 6”)

Wv is connectivity per volume or ‘capillaries per volume’

Euler number = I – B

I = islands

B = bridges

divide by ‘2’ if you are counting in both directions

t = thickness of section

a = area of counting frame

Volume Fraction of Vessel Wall Components

(Mühlfeld, “Morphometric parameters related to vasculature, Vessel wall composition”)

The percentage by volume of vessel wall components such as lipids and smooth muscle cells is of interest . Point counting is used to estimate the percent by volume; and that is multiplied by the reference volume. For the fractionator version of this probe please see Area Fraction Fractionator.


Løkkegaard, A, Nyengaard, J. R., and M.J. West (2001) Stereological Estimates of Number and Length of Capillaries in Subdivisions of the Human Hippocampal Region Hippocampus 11, 726 – 740.

Mühlfeld, C. (2014) Quantitative morphology of the vascularisation of organs: A Stereological Approach Illustrated Using the Cardiac Circulation. Annals of Anatomy, doi: 10.1016/j.aanat.2012.10.010.

Nyengaard, J.R. (1999) Stereologic Methods and Their Application in Kidney Research. J. Am. Soc. Nephrology, 10, 110 – 1123.