Cavalieri Estimator

The Cavalieri estimator is used to estimate volume. The estimator is often used in conjunction with other estimators that estimate density such as number density, length density, or surface density.

Begin by preparing a set of sections. The Cavalieri estimator is prone to overprojection if the set of slices are thick. This is explained in the section on volume formulas. The sections are sampled using the fractionator principle. A rule of thumb is to use 10 to 15 sections. Suppose that the tissue is sectioned into 80 sections. If every 6th section is used, then around 13 sections are used. This is usually an adequate number of sections. The fraction of sections used is the section sampling fraction or ssf. More information on doing this sampling step is in the section on the fractionator in 1-dimension.

The Cavalieri estimator is usually performed using a point grid. The points need to be appropriately spaced. The points are usually spaced equally both across and down. A rule of thumb is to count 200 to 250 points. In our example 13 sections are being used. If an average of 18 counts is made per section, then this rule is met. Some experimentation is usually required to determine an adequate spacing of the point grid. Regions that have complicated profiles or profiles that are thin require closer points than simple blob shapes. The distance between points can be used to compute the intensity of the probe.

Figure 1. Point grid overlaid onto tissue.

Toss the point probe randomly onto each section. Count how many points hit the region of interest. Keep a record of how many points hit each section. The results are used to compute the volume and the CE.

Figure 2. CE estimation for point counting across sections.

The CE calculation requires the determination of the shape factor. This value can be determined either by estimation or by inspection. It has been erroneously claimed that the shape factor must be known exactly. That is not true as has been demonstrated in the section on point counting CE. The shape factor is defined as the boundary length divided by the square root of the profile area. The Cavalieri estimator has already estimated the profile area. The boundary length must now be estimated. The shape factor varies little between similar organs so once the shape factor has been determined the same value can be used in other Cavalieri estimators.

To make the formula easier to understand here is an example.

Sampling Parameter Value
Slice Interval 6
Grid spacing 400 microns
Block advance 50 microns

Figure 3. Sampling fractions used in the example.

Here are the counts:

Section P A B C
1 6 36 66 96
2 11 121 176 242
3 16 256 352 336
4 22 484 462 594
5 21 441 567 525
6 27 729 675 837
7 25 625 775 800
8 31 961 992 775
9 32 1024 800 480
10 25 625 375 175
11 15 225 105  
12 7 49    
Totals 238 5576 5345 4860

Figure 4. Volume estimate and CE estimate for the Cavalieri estimator.

The shape factor was assigned a value of 6 in this example. It is possible to assign a value from estimation or by inspection using a nomogram. The shape factor value is not as critical as one might suppose at first. To see the effect on the CE by using a shape factor determine by a rough guess see the section on the efficiency of point counting CE.

In summary the steps of the estimator are:

  1. Prepare a set of Cavalieri sections.
  2. Sample the sections with a fractionator.
  3. Select a spacing for the point probe.
  4. Randomly toss the point probe onto each section.
  5. Count how many points hit the region of interest.
  6. Process all sections keeping a tally of counts per section.
  7. Possibly estimate the shape factor.
  8. Calculate the volume and the CE.