Fig 1.
Schematic illustration of the physical disector method for quantitative stereological estimation of numerical volume densities.
In this example, the numerical volume density of particles (green spheres) within a reference compartment (grey cube) is to be estimated (A). For a simplified presentation of the physical disector principle, the particles to be counted are equally sized spheres, evenly distributed within their reference compartment. A stack of parallel, equally thick sections (s) is cut from the reference compartment containing the particles of interest (B). The thickness of each of these sections is “d”. Two sections (a “look-up” section S1, and a “reference” section S2) are sampled with a known distance (h) between S1 and S2 (to avoid to miss particles completely located between the S1 and the S2 section, a disector height h of approximately 1/3rd of the mean particle height perpendicular to the S1-S2 section planes is chosen). The distance h between the “look-up” section S1, and the reference section S2 is equal to the product of the number of sections between S1 and S2 +1 and the mean thickness (d) of these sections (here: h = 6d). C: A physical disector is a 3-dimensional test-system (probe) of known volume used for direct and unbiased counting of particles. Within the aligned, congruent (2D) focus planes of the “reference” and the “look-up” section, each one area is defined, in which particles being hit by either the reference section and/or the look-up section are sampled for counting. Here, an unbiased counting frame (cf) [2, 3] with “allowed” (green) and “forbidden” (red) lines is used. The disector volume in which the particles are counted is defined by the area of the counting frame (Acf) and the distance between the focus planes of the “reference” and the “look-up” section (i.e., disector height, h). D: Using the unbiased counting frame, particles are sampled for counting, if their section profiles in the reference section are either entirely within the counting frame or if they touch an “allowed”line but none of the “forbidden”lines of the counting frame. Particles whose section profiles hit one of the “forbidden”lines of the counting frame in the reference section are excluded from the analysis. Only sampled particles that hit the reference section but are not present in the “look-up”section are counted (sampled particles sectioned by the “reference“- and the “look-up”section are not counted). The process of counting might then be repeated with interchanged roles of the “reference“- and the “look-up”section, thereby doubling the effectiveness of the counting procedure. Thus, in the present example, four particles (green section profiles in the “reference“- or the “look-up”section) are counted in a corresponding reference compartment volume of two disector volumes (2 x h x Acf).
Table 1.
Legend to Eq 1.
Fig 2.
Morphology of paraffin- and plastic-sections.
A-I: Perspective view on the relief of sections of paraffin sections and GMA/MMA- or Epon- plastic sections containing murine kidney tissue. Section areas with profiles of embedded tissue are indicated by arrows, section areas containing embedding medium without tissue are indicated by hashes (#), and the surface of the glass slides the sections are mounted on are indicated by asterisks (*). A-C. Freshly cut paraffin- (A), GMA/MMA- (B) and Epon- (C) sections mounted on glass slides after stretching of the sections on warm water-baths. Insets show paraffin, GMA/MMA- and Epon-blocks. D-F: HE-stained paraffin- (D) and GMA/MMA-sections (E) and Toloidine-blue-stained Epon- (F) section prior to mounting of cover-slips. D: Note that in paraffin-sections, the paraffin is removed during the processing of the section. Left image: Deparaffinized section prior to staining. Right image: HE-stained paraffin section. E, F: In plastic sections, the embedding medium (#) remains present in the section. Bars in A, B, inset to C, D and E = 5 mm. Bars in C and F = 0.5 mm. G-I: Scanning electron microscopic images of paraffin- (G), GMA/MMA- (H) and Epon- (I) sections. Note the uneven section surface and the absence of embedding medium in the paraffin-section, as compared to plastic sections. In both GMA/MMA- and in Epon-sections, the section surfaces of areas with and without embedded tissue are evenly smooth and at the same level. Bars = 20 μm. J-L: Orthogonal sections of paraffin- (J), GMA/MMA- (K) and Epon- (L) sections. J, K, L (left image): Light microscopic images. L (right image): Transmission electron microscopic image. Bars = 5 μm. Note the uneven surface of the paraffin-section, and the even level of the surface of plastic-sections in areas with and without (#) embedded tissue.
Fig 3.
Schematic illustration of section thickness determination of orthogonally re-embedded sections.
(A) A block of plastic-embedded tissue (e.g., Epon or GMA/MMA) is serially sectioned on a microtome. (B) From the series of consecutive sections (here N°1–10), section pairs are sampled for physical disector analysis (here sections N° 2 and N°4). C: From the remaining sections of the series, one section not used for disector analysis is sampled (here N°3) and re-embedded in plastic-embedding medium, vertically to its original section plane. The block with the re-embedded section is then sectioned with a microtome (for light microscopy, respectively with an ultra-microtome for electron microscopic examination). (D) The thickness of the orthogonally re-embedded section (d) is measured at randomly sampled locations, as the direct (orthogonal) distance between the upper and the lower cut-border of the section profile of the orthogonally re-embedded section, using light- (LM) or transmission electron-microscopy (TEM).
Fig 4.
Effect of over-projection in determination of section thicknesses using vertically re-embedded sections.
Two sections with the equal factual thickness d are re-embedded. (A) Ideal orthogonal re-embedding. The re-embedded section is re-sectioned exactly at 90° to its original section plane. The thickness h of the section of the re-embedded section does not affect the measured thickness dmeas of the re-embedded section, and dmeas is equal to the factual thickness of the section d. (B) Non-orthogonal (oblique) re-embedding. The re-embedded section is obliquely (≠ 90°) re-sectioned to its original section plane. Here, the measured thickness (dmeas) of the (obliquely) re-embedded section exceeds the true thickness (d) of the re-embedded section S. This effect results from to the oblique sectioning angle (indicated in blue) and increases with the degree of deviation from the 90° angle and also from overprojection (indicated in red), increasing with the thickness of the sections cut from the obliquely re-embedded section. However, if GMA/MMA- or Epon sections are approximately vertically re-embedded in Epon and re-sectioned to 70–90 nm thin ultra-thin sections, the effect of overprojection in these ultra-thin sections will only marginally affect the measured thickness of the orthogonally re-embedded sections.
Fig 5.
Schematic illustration of section thickness determination of orthogonally re-embedded sections and correction for non-vertical embedding.
Compare to Fig 3. (A) A block of plastic-embedded tissue (e.g., Epon or GMA/MMA) is serially sectioned on a microtome. A section (S) sampled for determination of the section thickness is mounted flat on a calibration foil (F) of known thickness (ttF). (B) The section-calibration foil stack is re-embedded in plastic-embedding medium, vertically to the original section plane. The block with the re-embedded section and calibration foil is then sectioned with a microtome (for light microscopy, respectively with an ultra-microtome for electron microscopic examination). In the section of the section-calibration foil stack, the thickness of the orthogonally re-embedded section (mtS) is measured as the direct (orthogonal) distance between the upper and the lower cut-border of the section profile of the orthogonally re-embedded section. The thickness of the orthogonally re-embedded calibration foil (mtF) is measured accordingly. Depending on the angle (α) of the section plane relative to the level of the re-embedded section/calibration foil stack, the measured thicknesses of the section and the calibration foil exceed the true thicknesses of the section (ttS, unknown) and the foil (ttF, known). The blue section plane (SPvert) indicates an orthogonal section plane (α = 90°), the red section plane (SPobl) is cut at an oblique angle (α≠90°). C: The true factual thickness of the section (ttS) can be calculated geometrically, using the measured thicknesses of the calibration foil (mtF), the re-embedded section (mtS), and the known true thickness of the calibration foil (ttF). The mathematical equations (1–5) used for calculation of the angles and distances used for calculation of ttS are displayed.
Fig 6.
Thicknesses of GMA/MMA- and of Epon sections were determined by spectral reflectance measurement and by light- and/or electron-microscopic measurement of the thicknesses of orthogonally re-embedded sections. (A) Sections were cut from five GMA/MMA-blocks and five Epon-blocks containing perfusion-fixed murine kidney tissue, (archive material) from 2005–2016 (different, independent casts). (B, C) From each GMA/MMA block, three series of each 10 consecutive sections were cut at nominal section thicknesses of 1, 2, and 3 μm. From each Epon block, a series of 10 sections with a nominal section thicknesses of 0.5 μm was cut (in C, only one section series of a GMA/MMA-block is shown). From each section series, 2–3 sections were randomly sampled (here: N°3 and N°7) for section thickness determination by orthogonal re-embedding (ORE) in Epon. GMA/MMA sections sampled for ORE were halved, whereas sampled Epon sections were completely re-embedded (not halved). The remaining sections were mounted on glass slides. (D) Per case, six spectral reflectance measurements (SR) of the section thickness were performed at measuring points/locations (mp) of the section (s), where no embedded tissue (t) was present. (E) From the halved GMA/MMA sections, each one half was mounted on a glass slide for spectral reflectance section thickness measurements, while the second section halves (s) were mounted flat on calibration foils (cf) and orthogonally re-embedded (ORE) in Epon and re-sectioned for verification of the section thickness by light microscopy (LM) of HE- or toluidin-blue stained sections. The thicknesses of orthogonally re-embedded Epon sections were determined by light- and by transmission electron-microscopic (TEM) measurements.
Fig 7.
Mounting of sections (here: Epon sections) on calibration foils for subsequent orthogonal re-embedding.
(A) Freshly sectioned Epon sections (encircled by a white dotted line) floating in the water bath (blue collecting basin) of the ultra-microtome. (B) Detail enlargement of Epon sections demonstrating a pink interference color. The grey-black material in the center of the section is the embedded tissue. (A, C) A section is carefully transferred to a stripe of calibration foil (arrow), using a horse hair (arrowhead in C).
Fig 8.
Light- and electron-microscopic images of sections of re-embedded GMA/MMA sections mounted on calibration foils.
Compare to Fig 3. The thicknesses (mtF) of the calibration foils (F) and the thicknesses (mtS) of the re-embedded sections (*) are indicated. (A) Light microscopic image of a re-embedded, 3 μm thick GMA/MMA section of mouse kidney tissue mounted on a calibration foil (ACLAR®, Plano, Germany) of 198 μm (true) thickness. #: Tension fold at the interface of the calibration foil and the surrounding Epon resin. GMA/MMA section, HE-staining, 200 x magnification. For calibration, the image of an object micrometer (distance between scale lines: 10 μm) photographed under identical conditions is displayed (bar = 20 μm). The insets to A show detail enlargements of the profile of the orthogonally re-embedded tissue section (top inset, bar = 5 μm) and the lower surface of the calibration foil (bottom inset, bar = 5 μm). (B) Transmission electron microscopic image of a re-embedded, 1 μm thick GMA/MMA section of mouse kidney tissue mounted on a calibration foil (LIST-MAGETIK®, Germany) of 49 μm (true) thickness. 1000 x magnification. For calibration, the image of a standard cross-grating calibration grid (width of squares: 0.463 μm) photographed under identical conditions is displayed (bar = 10 squares). The inset to B shows a detail enlargement of the profile of the orthogonally re-embedded tissue section (10,000 x magnification, bar = 0.463 μm).
Fig 9.
Spectral reflectance section thickness measurement.
Thicknesses of unstained GMA/MMA or Epon sections mounted on borosilicate glass slides were measured with a F20 optical reflectometer (Filmetrics®, USA) using the “contact stage” mode. The glass slide is placed on the stage with the mounted section facing the opening of the fiber optic cable (A, B). The opening of the fiber optic cable (inset to B) has a diameter of ~250 μm. (C) Detail enlargement of one half of a GMA/MMA section (s, arrow) mounted on a glass slide (gs) for spectral reflectance measurement of the section thickness (the second half is orthogonally re-embedded in Epon for verification of the section thickness by microscopic measurement). The tissue (t) present in the section is indicated (arrow). Per case, section thickness measurements were performed at six different locations (indicated by red circles) of the section, where no embedded tissue was present. Measurement details (D, E) and results are directly displayed at the monitor of the connected computer (A, arrow). (D) Reflectance spectrum. (E) Fast Fourier Transform (FFT)-intensity plot. The blue line on the graph represents the measured reflectance data, whereas the red line on the graph shows the calculated reflectance (based on the indicated refractive indices, extinction coefficients, and approximate thicknesses of the Epon-, or GMA/MMA sections and the borosilicate-glass slides). A successful measurement is indicated by an overlap of the wavelengths of the maxima and the minima of the calculated and the measured reflectance curve.
Table 2.
Measurement settings and sample specifications for thickness measurement of GMA/MMA and Epon sections with the F20 optical reflectometer (Filmetrics®, USA).
Table 3.
Mean deviation of spectral reflectance section thickness measurement values in series of (consecutive) plastic sections (inter-section variability of section thickness).
Table 4.
Deviation of section thickness measurement values determined by spectral reflectance measurement and orthogonal re-embedding of (identical) sections.