Macro- and mesomorphology of frontal sinuses in humans: Noisiness models relating to their ontogeny

ELSEVIER

ANNALS OF ANATOMY

Ann Anat 186 (2004):443-449 http:llwww.elsevier.de

Macro- and mesomorphology of frontal sinuses in humans: Noisiness models relating to their ontogeny Hermann Prossinger Institute for Anthropology, University of Vienna, A-1190 Vienna, Austria

Summary. The debate about the role of the frontal sinuses remains unresolved. In this paper, I explore several statistical and fractal features of the frontal sinus outlines in order to better assess the spatial models of supraorbital torus formation. Although the analyses are restricted to a suite of techniques dealing with two-dimensional projections of the frontal sinuses, here I report many features that explain why it is so difficult to describe the spatial models. I show how the outline circumference scales with enclosed area, how the fractal dimensions of the outlines are distributed, how the method of singular value decomposition is used to define surrogate landmarks, and how a principal component analysis of vectors between these landmarks indicates directions of high variability in some of the inferior-distal directions. All these analysis techniques reveal regularities underlying the statistical noise. I believe that uncovering them necessitates a fresh look at the biology of noisy ontogeny phenomena.

Key words: Frontal sinus - Fractal - Ontogeny - Singular value decomposition - Landmarks - Morphology - Brownian Bridge

Introduction The role of the frontal sinuses in H o m o and in Australopithecus is currently debated (Prossinger et al. 2000; Ravosa et al. 2000). The two major positions concerning their functional role are: (1) the masticatory stress hy-

Correspondence to: H. Prossinger E-mail: [email protected]

pothesis (Demes 1982, 1987; Spencer and Demes 1993; Bookstein et al. 1999; Prossinger et al. 2000) and (2) spatial models of supraorbital torus formation (Moss and Young 1960; Shea 1986; Ravosa 1991). Doubtless, this controversy is also debated among primatologists, sensu strictu (Hylander et al. 1991). Furthermore, one asks whether all paranasal sinuses have one or the other (or partially one and partially the other) of these roles in primates (see also Preuschoft et al. 2002). The problem of advocating either of these two positions is confounded by describing the increase in size of any sinus as "growth" (Enlow and Hans 1996; Liebermann 2000). Actually, sinuses do not grow; rather, the surrounding tissue does - in such a way that the ontogeny of the sinus appears to be a growth process (Prossinger and Bookstein 2003). Tissue growth is driven either by the stresses incurred by the living skull or by processes needed to model the morphology of neighbouring functional entities (Prossinger et al. 2003). Nonetheless, describing sinuses necessitates, as shown in this article, a suite of advanced analysis methods and I point to some features that must warrant consideration when trying to understand the biology of sinus emergence and expansion. More specifically: I first investigate how the circumference of the outlines varies with cross-sectional area, looking for any left/right asymmetries. I then compare the fractal dimensions of the outlines with random walk simulations on closed paths. Third, I use a Singular Value Decomposition (SVD) technique to identify four landmarks on the left and four on the right outline of each individual. A Principal Component Analysis (PCA) shows some statistical orientation properties of the outlines' geometry. 0940-9602/04/186/05-06-443 $30.00•0

Materials and Methods Materials. The Vienna Natural History Museum houses a large collection of crania with known sex, provenience and ethnic affiliation. Kritscher (1980) radiographed 711 of these in an occipito-frontal orientation and made tracings of the frontal sinuses outlines. Careful imaging control avoided distortion, which therefore hardly influenced the statistically "noisy" variation. Here I

use the outline tracings of 25 Chinese frontal sinuses (Fig. 1), which were scanned and digitized with the software package tpsDig (Rohlf 2001). Methods. Because of the high resolution of the digitizing process, the sum of the distances between the pixels/dots along the outline is a good measure of the circumference U. The number of pixels enclosed by the outline is its cross-sectional area A. I approximate each outline with a polygon having 180 corners by:

Fig. 1. The 2 ° polynomials approximating the 50 Chinese frontal sinus outlines (each vertex represented by a dot), the four surrogate landmarks on each outline (small open circles), the centres of mass (larger open circles) and the vectors (defined in Fig. 9) used in the Principal Component Analysis.

Fig. 2. The interpolation method. Points on the outline closest to integral multiples of 2 ° are shown as open circles, other points as black dots. The grey vectors point to the corners used for the interpolation, the black arrows to the interpolated polygonal vertices.

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(1) determining the centre of mass of the noncircular disk enclosed by the outlines (Fig. 1) and (2) using those pixels in the digitised outline that straddle integral multiples of 2° to interpolate along the straight line segment, resulting in vectors from the centre of mass at n . 2° (n = 0 . . . 179) angles (Fig. 2). Figure 3 shows the resulting vector plot for the right outline of individual no. 11. The fractal dimension of each outline can be found by the method of correlations. Note that the outline polygon can be

considered a list of 180 distances rn = r(n. ha) = r(n. 2°) from the centre of mass. At each lag (angle) O, one looks at the difference between the length r(O) and the length of the neighbour j steps away, A r c ( j ) = r ( j + O ) - r ( O ) , and then calculates the square root of the variance at O, viz. ~

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where (Ar) denotes the (arithmetic) mean over all ] (observing that this average is independent of 4). If the expression (Ar2}~ varies as ~F, then ln((Ar2)~) versus ln(qS) should be linear, ff F is non-integer, then the outline is a fractal and F is a measure of its ffactal dimension (Baumann et al. 1997). Geometric outlines

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that derive from biological morphologies are statistical fractals; they do not have a constant F for all q~, but only over some reasonable range [~A, ~s], as shown in Prossinger (in press). To better assess the distribution of the fractal dimensions, I compared them with those of a closed random walk. Consider the following analogy: a drunkard, while attempting to follow a closed path, lurches randomly from side to side. Each lurch to the side is + 1 step while he moves forward by 2 ° (in this simulation). A further condition is that the drunkard returns exactly to the position he started from: the "lurching drunk" has performed a closed random walk (a Brownian bridge). One simulates closed random walks by using a (suitably chosen) step size. In detail: from 10000 random sequences rjk (k = 1 . . . 181) of + ls and -ls, I collected those j with x--q81 2-,k=1r jk = 0 - a total of 609 such sequences. I generated 609 random walks ranj by adding the steps of width sf and obtained random walk outline points ranjk = rk + s f . rjk (k = 1 ... 180; j = 1 -.. 609). If rk = constant, one simulates a lurching drunk trying to follow a circle. With rk

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Fig. 5. The power-law relation between circumference U (in pixel-pixel distance units) and enclosed area A (in pixel 2 units) for the left (squares) and right (stars) outline polygons. The two interpolation functions are practically indistinguishable (left: dashed, right: continuous). Parameter estimates are given in the text.

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Fig. 6. The 25 left (squares) and 25 right (stars) fractal dimensions of the outline polygons. The values have been sorted. Overall, left fractal dimensions are slightly larger than right ones, but the overlap is considerable (see also the distribution histogram in Figure 8, inset).

one simulates a random walk along an ellipse with eccentricity e and parameter p. Another case is the sum of two "crossed" ellipses, viz. Pl P2 rk -- 1 + el cos(k • 2 °) -t 1 + e2 cos(k- 2 ° + £2' (two eccentricities, two parameters, and an angle I2 between the two major axes). Figure 7 shows the 3 closed paths chosen for this analysis, together with one such random walk for each. Because of the wildly varying geometry of frontal sinus outlines, one can also use these "2 ° polygons" to find "surrogate" landmarks (Prossinger, in press): first, normalize each polygon (divide by its Centroid Size) and construct a column vector with the 180 radial distances; second, construct two 180 × 25 dimensional matrices Alen and Aright (there are 25 individuals), and, third, calculate the SVD for each side: A = U . ~ . V v (Leon 1998). The (sorted) nonzero entries of 1~ are the singular values trj Q"= 1 . . . 25). The matrices A1 = al/71~l (Ul is the first column of U, vl the first column of V) and A2 = alffl~l q- O'2/~2~2 are the first and second approximations of A (in the Frobenius norm sense). The columns in the approximation matrices are (smoothed) approximations of the outlines. Each second approximation outline intersects the first approximation outline in four points (Fig. 4). One observes the remarkable property that the

Fig. 7. Fractal outlines simulated by random walks around closed paths (Brownian bridges). Left: a circle; middle: an ellipse (p = 1, = 0.60, sf= 0.05); right: two crossed ellipses (Pl = 11.5, P2 = 11.0, el = 0.75, e2 = 0.55, sf= 0.75, g2 = 150°). The steps of the "lurching drunk" are eatenated small open circles. All random walks (illustrated is Brownian bridge no. 111 of a total sample size of 609) resemble frontal sinus outlines. 446

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Fractal dimension Fig. 8. Histograms of fractal dimension frequencies of random walks, as defined in the text, and the fractal dimensions of the outline polygons (inset). Abscissa: fractal dimensiofi; ordinate: relative frequencies. The simulated distributions are based on 609 Brownian bridge's for each of the 3 outlines. The distributions of the different Brownian bridges have different shades of grey (white: random walk around a circle; grey: random walk around an ellipse; black: random walk around a crossed ellipse). Inset: The distributions of the fractal dimensions of the Brownian bridges that straddle the distributions of the left (white; labelled L) and the right (light grey; labelled R) outline polygons. Note the differences in scale.

four intersections of the first and second approximations are the same (within the resolution of digitising - better than 0.05%) for all outlines on one side. These intersections define 8 directions, which identify 4 points on each left and each right outline. These 8 points I call surrogate landmarks. Together with the two centres of mass, there are now 10 points per individual for a 2-D geometric analysis of the frontal sinus region (Fig. 1).

Results I determined the relationship A = A(U) by nonlinear interpolation, assuming a power function of the form A = z l U z. If the outlines were some regular, smooth curve, one expects z = 2. I found, for the left outlines, z = 1.897 + 0.071, and for the right ones, z = 1.864 + 0.059. The area increased more slowly with increasing circumference than did a smooth outline - a hallmark of a fractal outline. Furthermore, this deviation (z < 2) reflects the peculiar expansion geometry of frontal sinus outlines (the "cauliflower effect"): the sinus, as it expands, develops bulges and lobes, which contribute disproportionately to the circumference of its projection. The left and right power tractions were essentially the same, as shown in Figure 5. Figure 6 shows the fractal dimensions of all 25 left and right frontal sinus outlines. Although the left outlines had somewhat larger values, there is considerable overlap; the difference is obviously insignificant. A conventional t-test could not be used, because, as shown below, the distributions of the left and right exponents are not Student t-distributions.

Using the results of simulating the drunkard, I calculated the fractal dimension of each of the 3 x 609 Brownian bridges and looked at their distribution for each closed path (Fig. 8). Observe that the distributions of fractal dimensions of the outline polygons lie between the random walk about an ellipse and the one for a crossed ellipse (Fig. 8, inset). (Of course, I had experimented with various parameter values so as to straddle the observed distributions of Fl~t and F~ight; the parameters are listed in the caption to Fig. 8). The eight surrogate landmarks, together with the centres of mass, allow a PCA of the components of the 5 vectors (Fig. 9) between surrogate landmarks. The first two eigenvalues explained 49.6% and 40.6% of the variance. The 1 st PC correlated strongly with the y-components of gl, g3, ~4, and ~5 and the 2na PC with the x-components of ~i and ~3 (Table 1).

Fig. 9. The definition of vectors between surrogate landmarks, as used in the PCA and in Table 1.

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Table 1. The result of a Principal Component Analysis (PCA) of the 5 vectors (as defined in Figure 9): correlations with the first two eigenvectors. Four y-components of these vectors correlate highly with the 1~t PC, two x-components with the 2 nd PC. The highly correlating components are highlighted (in white).

and which lateral (outer-outer and inferior-inferior) variability is greatest.

Vector

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The frontal sinuses are so very noisy that forensic investigators consider them distinct enough to be a substitute for fingerprints (Christensen, personal communication). Nonetheless, here I present a canvas of unusual geometric properties underlying this high variability. This underlying geometric canvas constrains putative models of biological erosion and deposition processes taking place within the sinus (Prossinger and Bookstein 2003). So, a renewed look at the masticatory stress versus supraorbital torus formation debate is in order: some processes constrain the lateral variability and the long-range correlations of the ffactal outline; all the same, the vertical variability is unusually high. The geometric methodology presented in this article cannot, by necessity, ascertain the putative causes, but they do indicate what geometric features must be considered when deciding between the two debated modes of explanation.

Discussion Despite the fact that frontal sinuses are very noisy (in a statistical sense), the suite of approaches presented here can find some geometrically intriguing aspects of their morphology, which have implications for the study of their ontogeny and need to be considered in the debate about the role (or: formation) of the frontal sinuses. First, observe that, as the frontal sinus expands, the circumference of the outline increases more rapidly than it would if the outline were an object with a smooth geometry. This is evidence that the expansion processes must be local (a hallmark of fractals), even if there is an overall, long-range correlation with other regions of the same frontal sinus lobe. Elsewhere, I have described (Prossinger, in press) how the expansion mechanism can be modelled as a local, cellular-level process (a Percolation Cluster Model). Second, note that the fractal dimensions of the outlines lie within a narrow range of values between two distributions generated by random walks along closed paths: an elliptical or a crossed-ellipses path with suitably chosen parameters. Third, the design matrix was used to find points, which, if indeed surrogate landmarks, cannot be discerned by the eye of an anatomist. The precision of the coordinates of the intersection points depends on the smallness of the angle used to construct the interpolating polygon; no test is presented here, as the finite resolution of the scanning does not allow an opening angle less than 2 ° for all scanned outlines. These surrogate points may relate to biological processes which define anatomical features; further investigations are necessary to clarify this. They are points where the variation among the outlines is least when all (normalized) outlines are compared. Lastly, a PCA of these surrogate landmarks shows which vertical (outer-outer, inferior-inferior, distal-distal)

Acknowledgements. I thank Herbert Kritscher, (Naturhistorisches Museum Wien, Vienna) for permission to use the roentgenogram tracings and E Bookstein (Michigan Center for Biological Information, University of Michigan, Ann Arbor) for advice.

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