Systematic uniform random sampling (SURS) selects sections, tissue blocks or measurement positions at fixed intervals after a uniformly random start. The spacing distributes observations across the specimen; the random start prevents the investigator from choosing which locations enter the sample.
In stereology, SURS answers the question “where should measurements be made?” It does not, by itself, determine how to estimate cell number, volume, surface area or length. Those measurements require suitable probes and estimators. Within a broader stereological sampling design, SURS provides a repeatable way to select material without choosing fields because they look representative.
What Systematic Uniform Random Sampling Means
Each word describes part of the design. Systematic means that sampling positions follow a regular interval or grid. Uniform means that eligible sampling units have equal inclusion probabilities at that stage. Random means that the starting position is selected uniformly from the possible positions within one sampling interval.
For an ordered series of sections, choose an integer interval, k, then draw a starting section, r, uniformly from 1 through k. Sample sections r, r + k, r + 2k, and so on through the complete series. Each section then has an inclusion probability of 1/k. The mathematical basis for this selection and inverse probability weighting appears in Cruz-Orive’s treatment of systematic block sampling.
Uniform does not mean that every selected section contains the same amount of tissue. Nor does it mean that all possible combinations of sections can be selected. Once the start is drawn, the remaining positions are determined. This dependence distinguishes SURS from simple random sampling.
How to Select Sections Using SURS
Define the complete sampling series
Begin by stating what the estimate should describe: an entire organ, a named anatomical region, a material specimen or a predefined compartment. Identify the complete sequence through which that target can occur. Keep section numbering tied to cutting order, rather than renumbering only the sections that reach a slide.
A practical section log should distinguish between sections deliberately not collected and sections lost during preparation. Otherwise, “every tenth section” can become every tenth surviving section, which is a different design.
Choose the interval before drawing the start
Consider an illustrative specimen represented by 240 consecutive sections. Suppose the pilot design calls for approximately 12 sampled sections. An interval of 20 gives that number:
Sampling interval: k = 240 ÷ 12 = 20.
Draw one integer uniformly from 1 to 20. If the result is 7, select:
7, 27, 47, 67, 87, 107, 127, 147, 167, 187, 207 and 227.
The section sampling fraction is 1/20, or 5%. Every section belongs to exactly one of the 20 possible starting sequences. Starting at section 1 for every specimen is not an equivalent procedure, even though the resulting sections remain evenly spaced.
Separate section count from physical spacing
If the microtome advances 40 µm for each section, sampling every twentieth section gives a nominal distance of 800 µm between sampled section planes. That distance is not the same quantity as the thickness of a mounted, processed section.
| Setting | Value | Meaning |
|---|---|---|
| Section interval | 20 sections | Select every twentieth section |
| Random start | 7 | First selected section in this example |
| Section sampling fraction | 1/20 | Inclusion probability at the section stage |
| Microtome advance | 40 µm | Nominal cutting advance per section |
| Distance between sampled planes | 800 µm | 20 × 40 µm |
The numbers illustrate the procedure, not a recommended sampling intensity. Choose the interval through sample size assessment and pilot studies, rather than copying a section count from an unrelated specimen.
Why Regular Spacing Can Improve Precision
Independent random sampling can place several observations close together while leaving other stretches sparsely sampled. Systematic spacing distributes the same sampling effort more evenly. Where neighboring sections have similar measurements, this can reduce redundant observations and improve precision. The benefit depends on the structure and sampling design; it is not a fixed efficiency multiplier.
Periodicity is an important exception. A sampling interval that matches a repeating structural pattern can repeatedly encounter similar parts of that pattern and miss others. Uniform randomization still protects the expectation of an appropriate estimator, but individual estimates may vary greatly between starting positions. Both efficiency and this periodicity problem are treated in Gundersen and Jensen’s research on systematic sampling.
For a layered material, a useful pilot check is to compare the planned spacing with the repeat distance of the layers. If they coincide, reconsider the design before collecting the main sample. Moving a few inconvenient observations afterward is not a statistical remedy.
Applying SURS to Microscopic Fields
Sampling sections does not remove the need to sample positions within them. Establish the region boundary using predefined criteria, then position a regular grid with a random offset in both the horizontal and vertical directions. Visit the required grid positions without moving them toward clearer staining or denser populations.
For a rectangular grid, the offset is drawn within one grid cell. With 400 µm spacing in each direction, draw the horizontal and vertical offsets independently within a 400 µm interval. After that placement, the grid fixes the remaining positions.
When the grid places counting frames, the area sampling fraction is the frame area divided by the area associated with one grid step. This relationship is implemented in the optical fractionator protocol for developing human forebrain.
For an illustrative 80 × 80 µm counting frame on a 400 × 400 µm grid:
Area sampling fraction = (80 × 80) ÷ (400 × 400) = 0.04 = 1/25.
The calculation uses the counting frame, not the entire camera image. If the displayed image extends beyond the frame, those extra pixels do not enlarge the sampled counting area.
At the region boundary, apply the estimator’s prescribed boundary and counting rules. Do not require every frame to sit entirely inside the tissue unless the design explicitly accommodates that restriction. A frame near an edge is not defective simply because it contains less tissue.
How Sampling Fractions Enter an Estimate
In a multistage design, each sampling stage contributes a fraction. The optical fractionator combines section sampling, area sampling and sampling through section thickness with disector counting. The original optical fractionator study of rat hippocampal neurons demonstrates the combination of a three dimensional counting probe and systematic fractionator sampling.
For a simplified example with constant fractions, suppose the section fraction is 1/20, the area fraction is 1/25 and the thickness fraction is 1/2. Their product is:
Total sampling fraction = 1/20 × 1/25 × 1/2 = 1/1,000.
If 180 eligible objects are counted using the appropriate disector rules, the corresponding estimate is:
Estimated total = 180 × 1,000 = 180,000 objects.
This arithmetic illustrates weighting, not a complete laboratory protocol. Variable section thickness, additional block sampling or unequal sampling fractions require the estimator appropriate to those conditions.
Most importantly, regularly sampled two dimensional cell profiles do not become an unbiased cell count simply by multiplying them by a sampling factor. The disector principle for counting particles addresses object selection; SURS addresses the locations at which that counting occurs.
Choosing Sampling Effort and Assessing Precision
Use the pilot to identify where uncertainty enters the estimate. Record results by section as well as the pooled total. Ask whether measurements change strongly along the sectioning axis, whether most fields contain few events, and whether delineating the region is repeatable.
If the main concern is variation between sections, evaluate a shorter section interval. If sections are adequately distributed but observations within them are sparse, evaluate closer field spacing. These are different changes, with different costs. Do not assume that counting more objects in one section substitutes for sampling additional positions along the specimen.
The coefficient of error, or CE, describes the relative sampling uncertainty of an estimate. Its calculation must match the estimator and sampling design. Treating systematically selected observations as independent random observations can give an inappropriate precision estimate. Sampling density also affects which variance approximation is suitable, a central issue in research on variance estimation for systematic stereological samples.
Set a precision goal in relation to the scientific comparison, rather than treating one CE threshold as universal. Keep specimen sampling separate from within specimen measurement effort: additional microscope fields are not additional independently sampled animals, patients or material specimens.
Common Implementation Errors
Choosing a convenient starting section
A protocol should specify both the randomization range and how the draw is made. “Start near the beginning” leaves room for judgment. “Draw an integer uniformly from 1 to 20” does not. Record the outcome so that another analyst can reconstruct the selected sequence.
Replacing empty or damaged fields informally
Distinguish a valid zero from an observation that cannot be measured. An intact sampled field with no target objects contributes a zero. A torn section or unreadable image represents missing information. Do not turn one into the other, or replace either with a nearby field that looks more useful.
Write a missing material procedure before the main study. Record the location, reason and extent of each loss. Where replacement or adjusted weighting is contemplated, check that it preserves a valid design instead of assuming that proximity makes the substitute equivalent.
Changing sampling density in response to the observed result
Use pilot findings to set the main sampling settings. Avoid making a grid denser only where an unexpected result appears, unless an appropriate adaptive design was specified beforehand. Otherwise, measurement intensity becomes entangled with the outcome being measured.
Different predefined compartments can use different sampling intensities, but retain their sampling fractions and combine estimates with the correct weights. A pooled average does not automatically account for unequal sampling.
SURS Does Not Randomize Section Orientation
Position and orientation are separate design choices. A series of parallel sections can have a uniformly random starting position without having an isotropic orientation. Regular spacing tells you nothing about the distribution of section directions.
Whether orientation randomization is needed depends on the quantity and probe. Surface and length estimation require particular care with directional structure. Where equal representation of spatial directions is required, use an appropriate orientation design, such as isotropic uniform random sections. Adding a random start to fixed anatomical sections does not satisfy that requirement.
What to Record in the Study Protocol
Make the selection procedure reconstructable. Record the reference region, section order, sampling interval, randomization method and realized start. For field sampling, retain the grid spacing, offsets, frame or probe dimensions, boundary rules and any compartment dependent settings.
Keep the selected section numbers and position records alongside raw measurements. Document missing material, departures from the protocol, all sampling fractions and the method used to estimate precision. A software setting labeled “random” is not enough documentation on its own.
The working rule is straightforward: define the eligible material, randomize the start, follow the interval and preserve the resulting selection. SURS is effective because the investigator chooses the design before seeing which observations it delivers, not because every delivered observation looks typical.