We read with interest and appreciate the paper by Sabo *et al*. (1)
. Despite that this study is an impressive contribution introducing the principles of the fractal geometry in the human pathology, the following observations are made.

Although irregularity in shape and complexity in behavior are the major properties of any biological system, including all natural and pathological subcellular components, cells, and tissues, an important and surprising feature of the human vascular system, from the largest blood vessels to the finest branchings, is the self-similarity; in other words, it is a hierarchical structure made up of never identical parts that are in some way self-similar to the whole. Self-similarity is a fundamental property of objects in space and processes in time that have been called fractals by the French physicist Benoit Mandelbrot (2)
. Fractals are the concern of a new geometry, whose primary aim is to describe the great variety of natural systems. They are mainly characterized by four properties: (*a*) irregular, rough, or fragmented shape; (*b*) self-similar structure; (*c)* scaling, which means that measured properties depend on the scale at which they are measured; and (*d*) not-integer or fractal dimension (2
, 3
, 4
, 5)
.

On the basis of these general observations, we make the following suggestions.

(*a*) Although intratumoral MVD1
has proven to be an important prognostic indicator for many malignant neoplasms, it has been obtained in a subjective manner, so that any mathematical or statistical analysis made on its data cannot be representative of a general behavior.

(*b*) The human vascular system is characterized by a complex three-dimensional ramified structure that can be detected on histological sections as two-dimensional irregular and fragmented forms. Fractal dimension that represents the space-filling properties of an irregular object cannot be correctly considered a measure of the vascular tortuosity (which is an architectural characteristic of an ramified structure), but can be used as a valid index of both the amount and the irregular distribution of the blood vessels in the parenchymal tissue (Fig. 1)⇓
. This demonstrates that different spatial distributions of an equal number of blood vessels in the parenchyma correspond to different fractal dimensions (Fig. 1)⇓
.

(*c*) It is not clear what is the correlation between MVD and its counterpart fractal dimension. This observation derives from the fact that the higher the MVD calculated in the “hot spot” microscopic field, the higher is the space filled by the vascular system *versus* the parenchymal tissue. Because fractal dimension represents an index of the space-filling property of an irregular structure, and its value is a rational number comprised between 0, which is the Euclidean dimension of a point (*i.e*., a single vessel in the hot spot microscopic field) and 2, which corresponds to the Euclidean dimension of a plane (*i.e*., the blood vessels tended to fill entirely the hot spot microscopic field), it is reasonable that a positive direct correlation between these two parameters can be found.

(*d*) The most important property of a ramified fractal structure is the self-similarity. Self-similarity can be geometrical or statistical. We can construct geometrically self-similar objects whose pieces are smaller, exact duplications of the whole objects (2)
. The geometric form of these objects is specified by means of an algorithm that instructs us how to construct the object. Typical examples of geometrically self-similar structures are the Kock curves, the Cantor set, or the Sierpinsky triangle.

The pieces of biological objects are rarely exact reduced copies of the whole object. Usually, the pieces are “kind of like” the whole (3
, 5)
. Rather than being geometrically self-similar, they are statistically self-similar. That is, the statistical properties of the pieces are proportional to the statistical properties of the whole. A number of studies have shown that, in biological tissues, fractal patterns or self-similar structures could be observed only within a “scaling window” of the measure length which must be experimentally established. In “Material and Methods” of this paper, it is not clear how this scaling window, in which the value of the fractal dimension is invariant with respect to changes of magnification, has been evaluated. This is a fundamental index of the scale-independence characteristic of every fractal property, in contrast with Euclidean parameter (*i.e*., areas or perimeter) that vary significantly, changing the scale of observation.

(*e*) The human vascular system is a dynamical structure dependent by a high number of variables that locally influence the development and growth of blood vessels and render its three-dimensional configuration irregularly distributed in the surrounding parenchyma. This complex geometry renders necessary not only the quantification of the area surface filled by the blood vessels (MVD) and the area of the single vessels, which are both morphometrical indices, but also the measure of the complexity of its final architecture. The latter can be obtained in a two-dimensional analysis by calculating the spatial distribution of the blood vessels or in a three-dimensional analysis measuring the degree of tortuosity or vascular branching by means of its fractal dimension. This observation derives from the necessity to distinguish the term complexity, which regards the dynamical set of variables determining a particular structural configuration, which results complex, from irregularity, roughness, or tortuosity, which are qualitative properties of any natural shape.

Finally, it is our opinion that fractal geometry can be viewed as a new way to observe the great variety of natural forms and correctly measure their important qualitative properties. To quote the founder of this field, Benoit Mandelbrot, “Fractals provide a workable new middle ground between the excessive geometric order of Euclid and the geometric chaos of roughness and fragmentation” (2) .

This paper discuss some important points and, although the results are impressive, we as well as other morphologists need these additional details if we are to replicate the results.

## Footnotes

- Received May 21, 2001.
- Accepted June 15, 2001.