The relevance of Euler number to stereologists

The conneulor is one of the most recent methods of modern design stereology (Gundersen et al., 1993). It is used to estimate the Euler number x3 [i.e. Euler–Poincaré characteristic (EPC)] of a structure. The relevant test probes comprise randomly positioned and arbitrarily orientated pairs of co-registered parallel sections a known distance apart, bearing sampling windows equipped with a guard frame (known as a shell) which corrects for edge effects and ensures that the estimate of x3 is unbiased. The conneulor has many similarities with the physical disector probe used for estimating number (Sterio, 1984). In particular, both methods involve deduction as opposed to simple counting. It is not difficult to appreciate the practical value of obtaining estimates of, for example, the number of neurones in the cerebral cortex (Pakkenberg & Gundersen, 1997) using the disector principle. In comparison, Euler number is an abstract quantity. It describes the topology of the surfaces of a structure of interest and at first sight its estimation would appear to be of little practical value. However, in certain circumstances the Euler number can be used to count specific features of a structure of interest. The features in question may be referred to as handles (see Fig. 1), and it is this application that is of special relevance to stereologists. In particular, Gundersen et al. (1993) realized that by counting the number of handles in a specimen of bone one is able to derive a value for the number of trabeculae, and the conneulor has also been used by Nyengaard & Marcussen (1993) to count the number of glomerular capillaries in the kidney. In this issue Jernot & Lantuéjoul (1999) bring to our attention the fact that difficulties can arise if one tries additionally to use the Euler number to count the number of enclosed cavities in a structure. In particular, the formula that has been proposed for obtaining a combined count of the number of handles and the number of enclosed cavities in a structure is not generally applicable [see Eq (1) of Gundersen et al., 1993]. Figure 1 in Jernot & Lantuéjoul (1999) illustrates how calculation of the Euler number by assessment of the topology of surfaces does not always give a value identical to that obtained by counting handles and enclosed cavities. So, what exactly is the relationship between the mathematical definition of Euler number and the definition that has been adopted for practical purposes by stereologists? Jernot & Lantuéjoul (1999) tell us that mathematically, x3 is the total number of distinct surfaces of a structure minus the sum of the genuses of the surfaces, where genus is defined as the maximum number of closed curves that can be drawn on a surface without dividing it into separate parts. On certain occasions, however, a more tangible definition of Euler number can be given. In particular, the Euler number of objects with no internal cavities whose surfaces delimit one internal domain, can be expressed in terms of the number of discrete objects and the number of fundamental cuts that can be made through these objects, i.e. the cuts that can be made without splitting the objects into separate parts (Fig. 1). For these objects the genus is equivalent to the number of handles so that: