# Mathematical Analysis Techniques of Frontal Sinus Morphology with Emphasis on Homo.

код для вставкиСкачатьTHE ANATOMICAL RECORD 291:1455–1478 (2008) Mathematical Analysis Techniques of Frontal Sinus Morphology, with Emphasis on Homo HERMANN PROSSINGER* Department for Anthropology, University of Vienna, Vienna, Austria ABSTRACT Of all the paranasal sinuses, frontal sinus (FS) morphology, volumes, outlines, and cross-sectional areas vary most and so their statistical noise presents particular challenges. To assess and control this statistical noise requires a suite of mathematical techniques that: model their volume and cross-sectional area ontogeny, determine the uniqueness and fractal dimensions of their outlines (useful in forensics), smooth their outlines via Singular Value Decomposition (SVD), and model their expansion via percolation cluster models (PCMs). Published data sets of FS outlines, cross-sectional areas and volumes of Neanderthal and modern crania (obtained via CT-imaging techniques) are utilized here for application of these novel mathematical methods, which necessitate a modeling approach. Results show that: FS noisiness can be explained as cluster growth, their fractal outlines have properties similar to closed random walks (Brownian bridges) about predeﬁned curves, and the PCMs can simulate the emergence of lamellae. The statistical properties derived from the analysis techniques presented here suggest that an emergence of the lamellae via PCMs (with pinning and quenching-correlated noise) resolves the masticatory stress debate by showing that the lamellae are indeed responses to masticatory stresses, but these are of so low a level that they cannot be measured with strain gauges. PCMs and Brownian bridges, deﬁned by local rules, lead to the emergence of macroscopically observable morphologies. The methodologies presented here contribute to research in emergence phenomena and are not conﬁned to morphological analyses of frontal sinuses. Anat Rec, 291:1455–1478, 2008. Ó 2008 Wiley-Liss, Inc. Key words: frontal sinuses; fractals; singular value decomposition; supraorbital torus; masticatory stress; Eden model; pinning; sigmoid function; paranasal sinuses; discrete Fourier transform; stationary time series; autocorrelation function; Ljung-Box statistic; Yule-Walker algorithm *Correspondence to: Hermann Prossinger, Department for Anthropology, University of Vienna, Althanstrasse 14, A-1091 Vienna, Austria. Fax: 43-1-4277-9547. E-mail: [email protected] univie.ac.at Ó 2008 WILEY-LISS, INC. Received 22 April 2008; Accepted 23 April 2008 DOI 10.1002/ar.20783 Published online in Wiley InterScience (www.interscience.wiley. com). 1456 PROSSINGER INTRODUCTION Historical Précis Almost contemporaneously with the pioneering work by Mihalkovićs (1896), Paulli (1900) investigated the pneumatizations of the crania of all (then) known mammals. He was most emphatic in denying the existence of one single (functional) role of these pneumatizations. Rather, he claimed that they resulted from differential growth rates during the ontogeny of the individual. He also pointed out that the convention of naming the sinuses after the bone in which they (predominantly) reside is inaccurate. The frontal sinuses in primates, although in the os frontalis, are ethmoidally derived, for example. Despite his claim of no function of the pneumatizations, he does concede that the pneumatization reduces the bony mass without compromising strength. (A view that several morphologists do not concur with: more about this later). A generation after Paulli, Weinert (1925) undertook another comprehensive overview of the frontal sinuses in a large number of vertebrate species. In a long two-part article, he made a remarkable morphological analysis of the frontal sinuses, based on a large data set that encompassed crania from reptiles to primates. In the primates, certain patterns in the paranasal sinuses seem to be observed (Anon et al., 1996; Koppe and Nagai, 1999), predominantly in the maxillary sinuses (Rae and Koppe, 2000); they are, in the case of frontal sinuses, tenuous at best (Borovanský, 1936; Bugyi, 1959). The frontal sinuses of three specimens of H. heidelbergensis—Broken Hill/Kabwe (Woodward, 1921), Petralona (Kokkoros and Kanellis, 1960), and Steinheim (Berckheimer, 1933)—are very large (Fig. 1). It has been speculated (Prossinger et al., 2003) that the African and European specimens may be different clades, but with a sample size of two (Europe) and one (Africa), implications derived from such cladistics would be questionable indeed. Below, I present statistical evidence for how extremely (statistically) noisy the morphology of frontal sinuses in Homo is. Conventional morphometric assessment (as used by Buckland-Wright, 1990, e.g.), it will be argued, is therefore of limited value. Alternatively, a suite of mathematical/statistical analysis techniques that offers progress is presented in this review. Reasons for Analyzing Frontal Sinus Morphology 1. Morphology assists in cladistic and/or phylogenetic reconstructions. That the morphology of paranasal sinuses might be different in different species of humans seems to have been ﬁrst expressed by Schwalbe (1899). Many researchers base their work on the paradigm that differences in sinus morphology help deﬁnes clades and can be used to reconstruct phylogenies (Rae and Koppe, 2004). 2. Improved knowledge of ontogeny (Sieglbauer, 1947; Schmid, 1973; Lanz and Wachsmuth, 1985) may support or reject clinical/epidemiological intervention strategies (Scuderi et al., 1993). In Homo, the frontal sinuses communicate with the nasal cavity via very small openings located between the lamellae of the ethnoturbinals (Moore, 1981; p. 267). Whenever the frontal sinuses become infected, treatment is extraordinarily difﬁcult. A simple search with Google Scholar for ‘‘frontal sinuses’’ yields in excess of 15,900 citations. The overwhelming majority of these refer to clinical studies and/or reports about intervention strategies and attendant difﬁculties. Almost all clinicians report that the large variability of frontal sinus morphology necessitates interventions that are casespeciﬁc. If morphologists cannot succeed in characterizing regularities that supply deﬁnitive help to the clinician, the converse question arises: why is the morphology of the frontal sinuses so (statistically) noisy? 3. Currently, the evidence for the highly variable morphology of the frontal sinuses (Donald et al., 1994) suggests that the 2D projection outlines are unique to each individual. They thus, so it is claimed, can be used in forensic sciences as a ﬁngerprint technique (Quatrehomme et al., 1996; Nambiar et al., 1999; Riberiro, 2000; Christensen, 2004a). 4. The assessment of the morphology, especially geometric extent and spatial distribution (Fig. 1), of the frontal sinuses contribute to the debate as to whether the mechanical stiffness coupled with weight decrease is achievable by these structures (Shea, 1986). As all cranial sinuses are cavities within bony structures, they do indeed decrease the weight. Biomechanics researchers are interested in clarifying the question whether these cavities are where they are and have the morphology they do, because putative reasons would explain how they ‘‘lighten’’ the skull without comprising its stiffness/ rigidity. There is a long, drawn-out debate involving the issue of stiffness and frontal sinus extent. Homo erectus and H. heidelbergensis, for example, have prominent browridges (Schwartz and Tattersall, 2002). Consequently, their frontal sinuses are very extended (Fig. 1). One group of researchers interprets these prominent browridges as being a mechanical bar that stiffens the upper facial cranium (Demes, 1982, 1987; Russell, 1985; Spencer and Demes, 1993; Prossinger et al., 2000a). The measurements by Hylander (Hylander et al., 1991a,b; Hylander and Johnson, 1992; Hylander and Ravosa, 1992) and colleagues (Moss and Young, 1960; Ross and Hylander, 1996; Ravosa et al., 2000a,b) have drawn this view into question, basing their conclusion on the absence of measurable strains that they attempted to detect in vivo with strain gauges. They view the frontal sinuses to be the result of a supraorbital browridge formation that is an ontogenetic/phylogenetic trait, varying with species, and not a response to masticatory stresses. In the previous millennium, Bookstein et al. (1999) could show that the internal curvature of the frontal bone maintains a constant curvature over a time span in excess of 2 million years. They interpreted this as a straightforward result of mechanical pressure exerted by the growing brain, and therefore argued that the extended browridges (and, by implication, the large frontal sinuses) function as (hollow) MATHEMATICAL ANALYSES OF FRONTAL SINUS MORPHOLOGY Fig. 1. The frontal sinuses of three Homo heidelnergensis specimens: (a) Kabwe/Broken Hill, (b) Petralona, and (c) Steinheim. Yellow: the frontal sinuses; blue: the sphenoidal sinuses; red: the neuronal cavity. In all three specimens, the frontal sinuses have an enormous size. The lateral extent of the frontal sinus of the African specimen is much smaller than in the European ones. In all specimens, the lamellas can clearly be seen. 1457 mechanical beams absorbing masticatory stresses during chewing highly ﬁbrous foods. Note that prominent browridges are observed in Homo with a large masticatory apparatus (Fig. 1). A response, disagreeing with this interpretation was soon published (Ravosa et al., 2000b). Prossinger et al. (2000a) disagreed, pointing out that the morphological evidence for phylogenetic browridge formation is lacking. The absence of evidence for one argument is not validation for a different one. The impasse may be resolved by a subtle effect in the percolation cluster model (PCM) with quenched noise (see later). The lamella structure of the frontal sinuses strongly suggests that the extended browridges are not independent of masticatory stress (Prossinger et al., 2000a). The electronically prepared images (Recheis et al., 1999; Prossinger et al., 2000b) shown in Fig. 2 of the interior of the Petralona cranium exemplify the apositional hypotheses. Its frontal sinus is enormous in extent. The supraorbital torus formation hypothesis claims that there is no mechanical necessity for generating such a large sinus. The masticatory stress hypothesis claims that the small internal vault alone could not withstand the forces occurring during chewing. In H. heidelbergensis, the large masticatory apparatus, so the argument, necessitates the existence of a second mechanical vault that is considerably distanced from the ﬁrst, otherwise the vault structure would be too weak (Prossinger et al., 2000a). The large distance between inner and outer table of the frontal bone generates such a required double-vault structure, hence, by implication, the existence of large frontal sinuses. The masticatory stress hypothesis thereby ‘‘explains’’ why the frontal sinuses, which are ethmodially derived, pneumatize the frontal bone. The lamellar structure in the frontal sinus demands a close inspection (Fig. 2). The lamellae are predominantly orientated parallel to the midsagittal plane, and thus perpendicular to the occlusal surface of dentition. They are roughly(!) periodically spaced, yet have no further ascertainable regularity. They are interconnected by horizontal lamellae, in a way reminiscent of rib reinforcements in vault architecture. It seems difﬁcult to claim that such a complicated rib structure is ‘‘merely’’ a phylogenetic trait of H. heidelbergensis or Australopithecines with massive dentition. (The latter also have such lamellae distribution morphology; Prossinger, unpublished.) A parsimonious explanation would be the lamellae distribution morphology is a reinforcement structure for a ribbed vault architecture needed to absorb masticatory stresses. In the Kabwe (Fig. 1a) cranium, the ‘‘vault ribbing’’ is also prominent, even though the frontal sinuses do not extend as far laterally as in Petralona (Fig. 1b) and Steinheim (Fig. 1c). 1458 PROSSINGER 5. The processes active at a cellular level help to clarify how the sinuses emerge and expand during ontogeny (Fig. 3) and are then maintained during the remaining lifespan of the individual. These putative processes need not be genetically controlled or indicators of heritability. Rather, it is conceivable that mechanisms depend on the physiology of the sinus surfaces, much like the deposition growth during molecular beam epitaxy of semiconductor materials research, sedimentation processes in geology, and bacterial colony growth (many examples in Barabasi and Stanley, 1995 and in Vicsek, 2001). Any Putative Relations Between Function and Morphology? Moore (1981) is of the opinion that there is little point in speculating about the many features that are observed in frontal sinuses and in looking for an explanation. In part, I concur. One goal of this review is to show to what extent some of these ‘‘many’’ features are statistically reliably quantiﬁable. A suite of methods, many of which derive from those developed by physicists, statisticians and biologists investigating numerous biological growth phenomena as presented in Vicsek (2001), can be used for (and have successfully been applied to) the analysis of frontal sinus morphology. As will become clear in the descriptions of the Methods, none are restricted to analyzing the morphological features of frontal sinuses. Some are most suitable for many morphological analyses of cross sections and outlines. Further research into the morphological assessment of anatomical structures needs to take the many novel statistical techniques presented here into account. These methods, evidently successful (as manifest by the robustness of the estimators), point toward explanations that involve an uncommon perception of sinus ontogeny. This latter claim is at odds with Moore: a suitable suite of analysis techniques may point to physiologically satisfactory explanations. MATERIALS Frontal Sinus Volumes One data set consisted of ﬁve H. sapiens crania from the Department of Anatomy Collection, University of Vienna and an Avar cemetary (Lippert, 1969) ranging from 2 years of age to adulthood (Fig. 4) and the other of four H. neanderthalensis crania from 3 years of age to adulthood (Fig. 5; Gorjanovic-Kramberger, 1899; Klaatsch and Hauser, 1908; Blanc, 1939; Bartucz et al., 1940). Each cranium was CT-scanned, and the method of ﬂood-ﬁlling the cavity in the scan (Prossinger et al., 2003, 2005) was used to estimate frontal sinus volumes. Fig. 2. The frontal sinus of the Petralona fossil in two mutually perpendicular cross sections; (a) longitudinal and (b) coronal. These images are electronically prepared from a CT-scan. In (a), the internal table of the frontal bone has been stippled a dark gray. It has been electronically removed in (b) so that the lamella structure can be assessed. Frontal Sinus Cross-Sectional Areas A data set published by Szilvàssy (1982), namely the cross-sectional areas of the frontal sinuses of young H. sapiens (105 boys and 87 girls, aged 3–11 years), is combined with of 50 adult male and 50 adult female frontal sinus cross sections published in 1973 (Szilvàssy, 1973). MATHEMATICAL ANALYSES OF FRONTAL SINUS MORPHOLOGY Fig. 3. An illustration of the pneumatization process leading to the frontal sinus morphology in Homo sapiens. The smallest cranium is of a 2-year-old girl, the medium-sized of a 15-year-old adolescent, and the largest a 35-year-old woman. All imaged frontal sinuses have the openings to the ethnoturbinals in common. The growth of the cranium thus accompanies the corresponding expansion of the frontal sinus lobes. This ontogenetic process is one of the modeling tasks explored in this review article. This sample of 292 individuals is ethnically homogeneous. Graph 1 shows the distribution of the cross sections by sex. Frontal Sinus Outlines In the course of his thesis work, Kritscher (1980) made roentgenograms of many crania in the Caldwell projection. From the roentgenograms, he traced the outlines, which were orientated so that the midsagittal plane of each cranium is perpendicular to the top border of the page. These tracings have been scanned; of these, 25 of Chinese (Fig. 6) are used herein. METHODS Analyzing Morphology Mathematically This review limits itself to presenting the various methods that assess morphology via quantiﬁable morphological features. As the vast literature (Vlcek, 1967; Tillier, 1977) has shown, classical approaches to frontal sinus morphologies, describing overall/summative features, like dimensions, angles, etc. or solely verbal assessments, invariably demonstrates the absence of any generalizable patterns (Strek et al., 1992). This review, therefore, concentrates on methods of morphological analysis that depart from classical descriptive 1459 Fig. 4. Images from CT-scans of H. sapiens crania at four different ages: (a) 2 years old, (b, c) 8/9 years old, (d) 11 years old, and (e) adult. Details in Table 1. approaches. ‘‘Traditional’’ morphological assessments, such as those inventorizing height, width, surface area, and arithmetically derived variables (such as ratios), allow at best a limited description of frontal sinus morphology. More often than not, noisiness will render such assessments as inconclusive. Mathematical modeling techniques, on the other hand, promise to overcome the limitations of traditional ones. Frontal Sinus Volume Ontogeny as Sigmoids Frontal sinuses do not grow; rather, they expand (Maresh, 1940; Fairbanks, 1990; Weiglein, 1999). It is, of course, unreasonable to monitor the expansion in a single individual: repeated CT-scans would be unethical (because of the X-ray radiation exposure). Instead, one samples the ontogeny by using CT-scans of crania of deceased individuals and assumes that this sample is (in some way) representative for the frontal sinus volumes of contemporaneous individuals whose sinuses we cannot measure. Figure 3 shows a graphical representation of how, as the cranium grows, the frontal sinuses expand. This expansion need not be the same mathematical function as the increase in braincase volume or dimensions of the facial cranium. Nonetheless, many 1460 PROSSINGER Fig. 5. Images from CT-scans of H. neanderthalensis crania at three/four different ages: (a) 3 (?) years old, (b) 15 (?) years old, (c) adult, and (d) adult. Details in Table 1. geometric aspects of the frontal sinuses (volumes, crosssection areas, and outlines) can best be described with a sigmoid function VðtÞ ¼ V1 1 : 1 þ aert West et al. (2001) have shown that many ontogenetic trajectories derived from biological principles are sigmoid functions. If so, one challenge is to see how well the parameters V1 , a, and r can be estimated from the data. Based on the assumption that the ontogeny of frontal sinuses is an expansion that is most rapid during adolescence, and then asymptotically approaches the (constant) adult volume, a sigmoid regression analysis is promising. Modeling Statistical Noisiness of Frontal Sinus Cross-Section Ontogeny Morphological characterization of the frontal sinuses of Homo is fraught with difﬁculties. Attempts at a verbal description are usually unsuccessful, because the features detected are rarely (if ever) found in another specimen. Because the data sets of frontal cross sections are so noisy, a more reﬁned estimation technique is neces- Graph 1. Distribution of the total frontal sinus cross-sectional areas for 155 males and 137 females from an Eastern Austrian population. Light gray: adults; dark gray: 3–11 years old. sary. The ontogeny of the frontal sinus cross section can also be modeled as a sigmoid function (Prossinger, 2001) AðtÞ ¼ K 1 ; 1 þ aert with K the asymptotic value of the cross-sectional area, r a measure of the rate of expansion at the point of inﬂection, and a a parameter related to the age at which maximum expansion occurs via t0 ¼ ln r : a As evidenced in the data sets published by Szilvassy (1973, 1982) and shown in Graph 2, noisiness needs to be controlled. To do so, note that, at each subadult age MATHEMATICAL ANALYSES OF FRONTAL SINUS MORPHOLOGY 1461 Fig. 6. The outlines of 25 H. sapiens frontal sinuses in Caldwell projection obtained by tracing roentgenograms made by Kritscher (1980); they are roughly ordered by increasing cross-sectional area. The right outline no. 11 (marked by the symbol ) is used in many of the following ﬁgures and graphs. All tracings have been made with the Frankfort Horizontal parallel to the top of the page. group, the data set consists of more than one crosssectional area. There are nk such areas for each age cohort k (k 5 1. . .9 for the ages t1. . .t9 5 3. . .9 years), both for the boys and for the girls. Let Ak denote the mean area of the Ajk areas of the j (j 5 1. . .nk) individuals in each age cohort k. Prossinger and Bookstein (2003) presented the rather involved reasoning that implies that frontal sinus cross sections at each age cohort are well approximated by a lognormal distribution. Because, to a good approximation, varðln AÞ 1 A 2 varðAÞ; one can ﬁt the sigmoid with common lognormal variance by ﬁtting the observed data to a curve with the cost function nk 2 1 X Wk ¼ 2 Ajk Aðtk Þ Ak j¼1 for each (child) cohort k with weightings A2 for the k residuals of each age cohort. The cost criterion for the adult cross-sectional areas is Wadult ¼ 1 nX adult 2 Aadult m¼1 ðAm Aðtm ÞÞ2 (where tm (m 5 1. . .50) are the ages at death of the adults in the data set) because, as will be shown, the variation of age of the individual hardly inﬂuences the ﬁtted sigmoid. The total cost function to be minimized, pooled for each sex, is therefore Wsex 8 9 9 X > > > > > ¼: Wk þ Wadult > > >! min : ; k¼1 Graph 2. The ontogeny of total frontal sinus cross-sectional areas for 155 males and 137 females from the distribution shown in Graph 1. The sigmoids are estimated according to the methods explained in the text. Note that the dots for the 3–11 years old do not express their multiplicity; more than one individual can have the same total frontal sinus cross-sectional area. Refer to Graph 1. Fourier Descriptors of Frontal Sinus Outlines In the following three sections, various outline analysis methods are presented. First, the method of obtaining the outline data set(s) of frontal sinuses is exempliﬁed on Kritscher’s (Kritscher, 1980) tracings of roentgenograms of 25 Chinese skulls (Fig. 6). The details are exempliﬁed with one particular right outline (no. 11, marked with the symbol ). The center of mass of the ﬂood-ﬁlled outline (Fig. 7a) is the origin of the coordinate system in the subsequent calculation steps. The outline has been digitized, so it is effectively a polygon. In the next algorithmic step, points on the polygon that are nearest to integer multiples of some angle (2-degree angle in this example) are selected (Fig. 7b). A linear interpolation determines the arrowhead of the vector at 2-degree angle (the tail being the origin; see Fig. 8). Each outline j is therefore characterized by 180 vectors. The k (k 5 0. . .179) norms rjk of the vectors are a function rj 5 rj(a) of the angle—one function for each individual j (j 5 1. . .25). The differences between the norm at each angle and the circle with a radius equal to 1462 PROSSINGER Fig. 8. An illustration of how the 2-degree angle-polygon derived from a digitized outline is constructed. The points of the digitized outline (300 dpi resolution) are dots and circles. Gray arrows point from the center of mass to points on the outline that straddle integral multiple of 2-degree angle; the corresponding dots are drawn as small circles. The vectors that are integral multiples of 2-degree angle are drawn in black. Their tips do not (in general) point to integral multiples of 2-degree angle; rather, these tips are found through triangulation (the error due to linearization of the digitized outline is exceedingly small because of the smallness of the angles). Fig. 7. Digitized outline of the right frontal sinus of Chinese specimen no. 11 from Fig. 6. (a) The frontal sinus in cross section as a disk, with the center of mass marked as a dot. (b) The radial vectors, in 2-degree angle spacings from the center of mass to points interpolated on the digitized outline, as clariﬁed in Fig. 8. In this, and in all other outlines, the 0-degree angle vector is the horizontal one pointing to the left. the mean norm (Fig. 9) estimate a periodic function (Graph 3). One can ﬁt harmonics to this function by using the discrete Fourier transform (Press et al., 1992). For every individual j, rj ðaÞ ¼ 179 179 1 X 1 X Amj eixm a ¼ Amj ðcos xm a i sin xm aÞ 180 m¼0 180 m¼0 where Amj is amplitude of the mth harmonic pwith ﬃﬃﬃﬃﬃﬃﬃ p (angular) frequency xm ¼ m 2p32 360 ¼ m 90 (and i ¼ 1). There are at most 89 such harmonics (Nyquist theorem) and the 0th harmonic is the mean of the function estimation. For the outline in this example, the Fourier spectrum is shown in Graph 10. Note that in this spectrum, the second harmonic has a very large amplitude because the outline is periodic in 180-degree angle. The second largest harmonic is the third harmonic, corresponding to a periodicity of 90-degree angle. The Fourier spectrum shows that a few harmonics are markedly larger than all the others. A cursory inspection seems to indicate that the other harmonics constitute the noise spectrum. The inverse of the Fourier transform using only the large amplitude harmonics (those with Am 0.1 in this example) shows a smooth outline (Graph 3) and the others the noisy ﬂuctuations around the smoothed outline. Further results of this method of analysis will be discussed later. Using only the harmonics with the largest amplitudes is part of a standard toolkit in function smoothing. A graph of the Fourier spectrum of all 25 outlines (Graph 11) demonstrates one special application to frontal sinus outlines. First, observe that less than 20 harmonics (around 15) capture the salient features of the outlines, and the remaining ones describe each outline’s noisiness (see a more detailed argument later). Second, observe that there is no pattern in the 20 ﬁrst harmonics for the 25 individuals. The Fourier spectrum of each individual is unique (a generalization derived from the 50 outlines of 25 individuals). Forensic scientists use this ‘‘ﬁngerprint’’ as an identiﬁcation methodology (Nambiar et al., 1999; Christensen, 2004b). Not all forensic scientists use the full outline in their investigations; some use an outline arc (details can be found in Christensen, 2004b). Using the full outline as described here has a major advantage; it is a method free of coordinate orientation artifacts. Further research shows that the uniqueness of the ‘‘outline ﬁngerprint’’ converges to a stable value at around 2-degree angle, hence the choice of this angle in this article. Because there are two outlines (a left and a right one) per individual, the probability of all the Fourier harmonics ‘‘ﬁngerprints’’ in two different crania being identical is then halved. 1463 MATHEMATICAL ANALYSES OF FRONTAL SINUS MORPHOLOGY Graph 3. The result of smoothing the 2-degree angle-polygon deviations from mean curves as shown in Fig. 9. The 2-degree anglepolygons are deﬁned as shown in Fig. 7b; (a) the deviations as deﬁned in Fig. 9a. (b) the deviations as deﬁned in Fig. 9b. The small circles are the actual deviations, the black curves are the superposition of the ﬁrst ﬁve harmonics (those with the largest amplitudes; refer to Graph 10). The residuals are graphed as noise. The abscissa for the noise curves has not been drawn through the origin of the graph. The noise graphs have been shifted for clarity only, they ﬂuctuate about an axis actually passing through the origin. Note the difference in scale of the ordinates. The noise curves are not iid, as explained in the text. Fig. 9. Deviations of a 2-degree angle-polygonized outline from two different curves. (a) The curve is a circle with radius equal to the mean distance of the outline from its center of mass. (b) The curve is the ﬁrst SVD smoothed outline (shown in Fig. 10a). SVD as a Method of Smoothing Noisy Outlines The sample of the 25 Chinese outlines (Fig. 6) are mathematically 25 radial vector sets that form a (180 3 25) matrix 8 r11 > > > .. > > > . > > > A¼> > > > > .. > > > > . : r180 1 r1j ... ... rij r180 j ... ... r1 25 .. . .. . r180 25 9 > > > > > > > > > > : > > > > > > > > ; In A, each column vector (r1j . . . . . . rij . . . . . . r180j)T j 5 1. . . 25 consists of the norms of the 2-degree anglevectors, scaled for Centroid Size (Slice, 2005). We calculate the singular value decomposition (SVD) of A (Leon, 1998) namely, 8 A ¼ :... ~ uk 8 r > 1 9 > > > > ...; > > > > : r25 9 > > 8 > > > : :: ~ > vl > > > ; 9T :: ; (There are 25 singular values rj because 25 individuals were used for constructing the decomposition.) The matrix A1 ¼ r1~ u1~ vT1 can be obtained by setting all singular values to zero except r1. This matrix A1 is the ﬁrst approximation of A with respect to the Frobenius norm (Leon, 1998). The column vectors of this matrix A1 are the 25 outlines that are the closest common ﬁt for the 25 individuals. In fact, they are practically identical, as can be seen in Fig. 10a. Therefore, SVD is a powerful smoothing technique that produces a mean outline of the 25 (fractal) frontal sinus outlines. One investigation approach is to compare these mean outlines for the left versus the right lobes to assess any possible asymmetries, either by sex or by population. If one calculates A2 ¼ r1~ u1~ vT1 þ r2~ u2~ vT2 by setting rj 5 0 (for j 5 3. . . 25), then one obtains a better approximation of A (in the Frobenius norm sense). In this case, the 25 column vectors rTj of A2 represent a very interesting approximation of these 25 frontal sinus outlines, as can be seen in Fig. 10b. One can take advantage of a remarkable feature: Each ﬁrst approximation outline must intersect the second approximation outline in four 1464 PROSSINGER points; however, these four intersection points of all 25 outlines are remarkably close together; they are (almost) common to all outlines (Fig. 10b). The four points obtained this way can be used to deﬁne surrogate landmarks. A discussion of the biological implications can be found in Prossinger (2005). Again, the distribution of such surrogate landmarks (and their number) in comparisons of left/right lobes, by sex, in a sample population is a novel approach to morphological assessment. One can introduce a further methodological advancement. Rather than look at the differences in radial vectors with the unit circle, investigate the differences against the ﬁrst and second SVD-smoothed outlines (shown for the 1st SVD-smoothed outline in Fig. 9b). Not the uniqueness (in the ﬁngerprint sense), but rather the patterns in the harmonics relative to the common form of a population (as morphologists—rather than forensic scientists—are wont to do) is the essence of the summary morphological description. In particular, the amplitudes of the higher harmonics reveal common features, because the overall periodicity of the common form is already parceled out. Note that, in the example drawn in Fig. 9b, deviations are most pronounced on one side of the lobe, whereas in the deviation from a circle (Fig. 9a), the deviations are large and quite periodically distributed. Analyzing Frontal Sinus Outlines as Fractals and Random Walks The residuals remaining (after approximating the outline with only the largest amplitude harmonics) ‘‘look’’ like noise (Graph 3). Alas, they are not. The autocorrelation function for many lags (Brockwell and Davis, 2002) reveals that there are a large number of periodicities remaining (not shown). In fact, the time series of the residuals (in both cases) after parceling out the ﬁrst four harmonics is not even stationary. Despite the seemingly reassuring apparent ‘‘randomness’’ of the residuals’ time series (Graph 3), it must be stressed that the ﬁrst four harmonics reveal considerable morphology, yet there is apparently more information to be extracted. Why is this so? The answer lies in the fact that the outlines have fractal properties (Prossinger, 2004, 2005). Because of the high resolution of the digitizing process, the sum of the distances between the outline pixels is a good measure of the circumference U of the outline. The number of pixels enclosed by the outline is its crosssectional area A (in pixel units; the dpi scale factor of the scan can be used to convert to physical cross-sectional area in cm2). For smooth outlines, A U2, so the graph A(U) should be parabolic. For objects with fractional dimension (fractals), the exponent is not an integer; the area increases less rapidly with circumference (because of the self-similar jaggedness of the outline). Each frontal sinus outline 2-degree angle-polygon is a sequence r(j) (j 5 1 . . . n) of radii. We are interested how the radial component r(j) covaries with its neighbor F (0 < F 2p) steps away. We look at the differences DrU ðjÞ ¼ rðj þ UÞ rðjÞ: The periodicity r(j 1 n) 5 r(j) ensures that DrF(j) is deﬁned Vj. We estimate the variance of these differences for a given F, viz, Fig. 10. Two outcomes of SVD smoothing method for the 2-degree angle-polygonized right outlines of the 25 Chinese specimens from Fig. 6. (a) The 25 outlines using only the ﬁrst singular value. The 25 outlines are essentially identical. The outline is the best (in the Frobenius sense) representation of a common form. (b) The 25 outlines using the ﬁrst and second singular values only (details in the text). The variation among the outlines is in distinct regions; all 25 outlines have four points in common. VarðDrU Þ ¼ hDr2 iU ¼ n 1 X ðDrU ðjÞ hDriÞ2 n 1 j¼1 where hDri denotes the (arithmetic) mean over all DrF(j) (observing that this average is independent of F). The 1=2 standard deviation C(F) 5 hDr2iU is a function of F. If C(F) varies as C(F) FF, then the graph of ln C(F) 5 ln 1=2 (hDr2iU ) versus ln (F) should be a straight line with slope F. If F is integer, then the curve is smooth (a onedimensional geometric object); if not, then F is a measure of the fractal dimension of the outline (Baumann et al., 1997). Geometric outlines that derive from biologi- MATHEMATICAL ANALYSES OF FRONTAL SINUS MORPHOLOGY 1465 Fig. 12. An image of the changes in the perimeter list generated by the Percolation Cluster Model (PCM) algorithm with P0 5 0.72 after 80 iterations. Sites are drawn as squares or circles. The starting perimeter list is the straight, horizontal set of sites at the base of the drawing. After 80 iterations, the cluster consists of 80 sites (white) and the perimeter list consists of 25 sites (gray). (a) After an additional 25 iterations, 18 more sites (drawn as circles) have been generated; they are adjacent to sites from the perimeter list one iteration step earlier. Not all iterations generate an addition to the perimeter list, because P0 5 0.72. In one case, a perimeter site has been added at the recently added perimeter site (identiﬁable by two stacked circles). (b) The cluster (96 sites drawn as white squares) and the perimeter list (25 sites drawn in gray) after the iterations shown earlier (in a) just before the next iteration. Note that the perimeter site (square) at k 5 5 is isolated after only 80 iterations. cal morphologies are statistical fractals; they do not have a ﬁxed F for all F, but only over some range. For outlines to be considered statistical fractals, then the linear region should be over some reasonably large interval [FA,FB]. Generating Frontal Sinus Outlines as Fractals via PCMs Fig. 11. Three samples of Brownian bridges around (a) a circle, (b) an ellipse, and (c) two crossed ellipses. (Brownian bridges are random walks that return to the starting point.) Each drawn sample is one of the 609 Brownian bridges generated from 10,000 random walks. The circles are the 180 positions of steps around the curves. This method derives from predictions of two spreading models: the Eden model (Eden, 1961) and its generalization, the PCM (Gaylord and Wellin, 1995). In both models, a perimeter list is deﬁned on a lattice, and a seed site is chosen randomly (from the perimeter list). A site that is adjacent to (but not an element of) the perimeter list is added to the perimeter list. This process is iterated. In the PCM, a predeﬁned probability P0 determines whether the site adjacent to the randomly chosen perimeter site is added to the perimeter list. In both cases, the resulting union of all perimeter lists (after many iterations) is called a cluster, and its ﬁnal perimeter list is sometimes called the outline. These two remarkably simple algorithms can generate a wealth of different clusters. Depending on the geometry of the original perimeter list, the clusters show unexpected geometries. If the PCM is used (note that the Eden model is the PCM with P0 5 1), then changing the predeﬁned value of P0 not only changes the geometry (primarily the fractal dimension) of the cluster but also introduces other biologically relevant features, especially in the geometrical properties of the outline, as will be shown. 1466 PROSSINGER TABLE 1. The specimens used for estimating the sigmoids V(t) that model ontogenetic expansion of the frontal sinuses, as shown in Graph 4 Label Homo neanderthalensis Subalyuk Le Moustier I Krapina C Guattari I (Monte Cerceo) Homo sapiens SAP-A ZW-715 ZW-759 SAP-254 SAP-2 Age (years) Sex Reference Frontal sinus volume (mL) 3 15 Adult Adult ? # $ ? Bartucz et al. (1940) Klaatsch and Hauser (1909) Gorjanovic-Kramberger (1899) Blanc (1939) 2 8/9 8/9 11 Adult $ ? ? # # Dept. Anat. Collection, University of Vienna Avar Cemetery (Lippert, 1969) Avar Cemetery (Lippert, 1969) Dept. Anat. Coll., University of Vienna Inst. Anthrop. Collection, University of Vienna 0.064 5.3 9.34 9.09 0.0 0.84 2.09 5.11 13.9 The sex and the ages at death of the specimens are estimated. Details can be found in the references. The ages of the adults are simulated to estimate uncertainties (see Graphs 5 and 6). TABLE 2. The parameter estimates in the sigmoids V(t) 5 V‘ (1 1 ae2rt)21 for Homo neanderthalensis and H. sapiens Sigmoid parameter estimates Homo neanderthalensis Homo sapiens V1 (mL) a r (year21) Range in simulations 9.39 13.9 9332 2316 0.6267 0.6559 9.223 < V1 < 9.375 11.50 < t0 < 12.03 The age t0 when maximum frontal sinus expansion occurs is related to the parameter r via t0 5 a21 ln r . The uncertainties in the parameters are exempliﬁed by simulating a range of ages and volumes for the input data (see Graphs 5 and 6) and their distributions are shown in Graph 7. In Fig. 12 a simple model—very much simpler than the one used in the frontal sinus outline investigations presented in this article—depicts the outcome of the PCM algorithm. The original perimeter list is a set of sites along a straight line, and the top squares (shaded gray) form the ‘‘surface’’ (i.e., the perimeter; note that it need not be connected) that is generated after a (ﬁnite) number of iterations. The circles above the perimeter are the sites that have been chosen for the ‘‘growth’’ of the cluster. However, the ﬁgure does not show the temporal sequence of when the sites had been chosen. It only shows that repeated calls to the perimeter sites have not resulted in every site being chosen to deﬁne the subsequent ‘‘surface’’ or outline. It often happens that a site can be selected before all other nonvisited sites have been visited (exempliﬁed in Fig. 12 by some circles being stacked). Furthermore, in the PCM, randomly chosen perimeter sites need not be added to the perimeter list to form the new ‘‘surface’’ of the cluster; they will only be added with a probability P0. RESULTS Frontal Sinus Volume Ontogeny Sigmoid regression can quantify the volumetric expansion of H. sapiens and H. neanderthalensis frontal sinuses. Data and regression parameters are listed in Tables 1 and 2, and Graph 4 shows the estimated sigmoids. Frontal sinus volumes are very difﬁcult to obtain (Prossinger et al., 2000b). Using four (H. neanderthalensis) and ﬁve (H. sapiens) data points gives a seemingly reliable parameter estimate (Table 2). However, it is well known that the frontal sinuses are statistically extremely noisy (see later), so one should be suspicious of the apparently successful modeling. The analysis presented here does not, therefore, imply that the asymptotic frontal sinus volume V1 in H. neanderthalensis is smaller than that of H. sapiens. Quite the contrary, the observation of the large Neanderthaler browridge (Tattersall, 1995; Schwartz and Tattersall, 2002) leads one to expect a different morphological outcome. Rather, this example shows the sensitivity of the parameter estimation to the data by addressing the assumptions made in the published literature. There is a long, involved debate about the age-at-death of the LeMoustier I adolescent (Thompson and Nelson, 2005). Graphs 5 and 6 show how simulating a variation in the ‘‘raw’’ data inﬂuences the parameter estimates. All numerical simulations show that the parameter estimates are markedly sensitive to age estimates of the time of maximal growth rate (mathematically: the point of inﬂection)—around the age of death of the Le Moustier I individual. Because of technical sophistication, the uncertainties in volumes determined by the ﬂood-ﬁlling algorithms applied to CT-scans (Prossinger et al., 2000b) are less than 1%, so the ageat-death estimates are what drives the uncertainty in the parameter estimates V1, a, and r. However, summary histograms (Graph 7) show that these parameter estimates are quite robust. Graph 7a shows that the asymptotic volume derived from the Neanderthal varies by less than 61 mL in a total of 24 simulated uncertainties; Graph 7b shows that the putative age-at-death of the adult does not inﬂuence the estimation of sigmoid parameters, and the uncertainty of the age-at-death of one adolescent individual results in a ﬂuctuation of the age at maximal expansion by 61/4 year, less than the published uncertainty in the estimated age-at-death of Le MATHEMATICAL ANALYSES OF FRONTAL SINUS MORPHOLOGY 1467 Graph 4. Sigmoid regressions of Homo neanderthalensis (large circles) and H. sapiens (small circles) modeling volume ontogeny of the frontal sinuses illustrated in Figs. 4 and 5. Moustier I (Thompson and Nelson, 2005). Sigmoid functions are not only model biological processes very well but also they supply robust estimators. Frontal Sinus Cross-Sectional Area Ontogeny Applying the methods of controlling for noise to the Szilvassy data sets, one obtains the following sigmoid functions: For the males, the regression is 1 A# ðtÞ ¼ 12:299 ; t0# ¼ 12:53 years; 1 þ 37:15e0:2885t and for the females it is A$ ðtÞ ¼ 10:451 1 ; t0$ ¼ 9:51 years 1 þ 129:18e0:5110t Both ﬁtted sigmoids are shown in Graph 2. The asymptotic areas for the males (K# 5 12.30 cm2) and the females (K$ 5 10.45 cm2) are close to the averages (12.32 and 10.31 cm2) as published by Szilvassy (1981). The logarithm of the interpolating sigmoid and the logarithms of the cross-sectional areas are graphed in Graph 8. The ﬁts are very good, despite the noisiness of the data. The signiﬁcance of the difference in models is tested with a log-likelihood test: test the hypothesis H1 (interpolating the two sexes separately with a total of eight parameters K#, K$, a#, a$, r#, r$, r#2 , and r$2) against the hypothesis H0 (the complete data set is to be interpolated with the four parameters Kall, aall, rall, and r2all ) by calculating the statistic 8 9 > > k ¼ 2 ln> :LH0 LH1 > ;; where L is the likelihood of H, the product of the probabilities of the observations over each datum of the dataset. k has approximately a v2-distribution with 4 (5 8–4) degrees of freedom. In the complete data set, ln LH0 5 Graph 5. Fluctuations in regression sigmoids due to variation of age-at-death as input: Homo neanderthalensis. (a) Varying the estimated age-at-death of the adult specimens. (b) Varying the estimated age-at-death of the adolescent specimen. (c) Varying the estimated ages at death of the adults and the adolescent. 1468 PROSSINGER Graph 6. Fluctuations in regression sigmoids due to variation of age-at-death as input: Homo sapiens. (a) Varying the estimated age-at-death of the adult. (b) Varying the estimated age-at-death of the adolescent and the adult. (c) Varying the estimated volume of the 8/9 years old and the estimated age-atdeath of the adult. (d) Varying the estimated age-at-death of the 8/9 years old. 2358.571 and ln LH1 5 2348.571, k 5 19.30 (P < 0.001, highly signiﬁcant; P 0.001 at v24 5 18.5). Is there a signiﬁcant laterally asymmetry of the crosssections? This tests the hypothesis: is it to be expected that the left lobes to have a larger/smaller cross-sectional area? A paired two-tailed t test shows that there is a difference only at the 6.1% signiﬁcance level (females) and 7.4% signiﬁcance level (males). Although the total cross-sectional areas are lognormally distributed, the left and right lobes are not (Graph 9). Fractality of Frontal Sinus Outlines Graph 12 shows the relation A(U) obtained from the 25 left and right Chinese frontal sinus outlines in the functional form A 5 z1 Uz (left: z 5 1.897 6 0.071; right: z 5 1.864 6 0.059). The exponent is signiﬁcantly different for 2 (ﬁrst successful outcome of a test for fractality) and that the fractal ‘‘growth’’ is statistically the same for both the left and right lobes. The graphs in Graph 13 superimpose the fractal dimensions of the 25 Chinese left and 25 Chinese right frontal sinus outlines. The fractal dimensions so obtained are consistent and no trend separating left from right outlines can be ascertained. The 2-degree angle-polygon outlining the right sinus lobe of Crapina C (Gorjanovic-Kronberger, 1899; Wolpoff, 1999; Schwartz and Tattersall, 2002), together with the arrows that deﬁne this polygon, is shown in Fig. 13 (compare with Fig. 5c). The log–log function shows that the outline scales self-similarly over a scale factor of e2 7.38 (both for right and left lobe outlines; corresponding to an angle range 28–288)—a scale factor that is very MATHEMATICAL ANALYSES OF FRONTAL SINUS MORPHOLOGY Graph 7. Distribution of the estimators of the sigmoid regression parameters. (a) The distribution of the estimates of the asymptotic frontal sinus volume obtained with the variations shown in Graph 5. Light gray: variation in Graph 5a; medium gray: variation shown in Graph 5b; dark gray: variation shown in Graph 5c. (b) The distribution of the age of maximal frontal sinus expansion obtained with the variations shown in Graph 6. Light gray: variation in Graph 6a; medium gray: variation shown in Graph 6b; dark gray: variation shown in Graph 6c; black: variation shown in Graph 6d. Graph 8. The demonstration of the regression quality of sigmoids when using weighted residuals for lognormally distributed data points. The logarithms of the total frontal sinus cross-sectional areas are graphed, together with the logarithms of the regressions sigmoids that estimate the frontal sinus ontogeny. The ﬁts are remarkably good, especially for such statistically very noisy data. large for a biological object! The fractal dimension estimate of the right frontal sinus lobe outline is 0.660 6 0.011 (r2adjusted 5 0.9997); for the left one (not shown) the fractal dimension estimate is even better: 0.6197 6 0.0063 (r2adjusted 5 0.9997). Frontal Sinus Outlines as Random Walks Another way of assessing the distribution of the fractal dimension of outlines is to compare those with the properties 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 61 step while he moves forward by 2-degree angle (in this simulation). A further condition is that the drunkard return 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 10,000 random sequences rjk (k 5 1. .181) of 11s and 21s, collect those j for which P.181 k51 rjk 5 0 — a total of 609 Brownian bridges were found in this simulation. Using these bridges and adding steps of width sf generated 609 random walks ranj and random walk outline points ranjk 5 rk 1 sf rjk (k 5 1. . .180; j 5 1. . .609) were obtained. If rk 5 const, one simulates a lurching drunk trying to follow a circle. With 1469 rk ¼ P ; 1 þ e cosðk 2 Þ one simulates a random walk along an ellipse with eccentricity e and parameter P. Another case is the sum of two ‘‘crossed’’ ellipses, viz., rk ¼ P1 P2 þ 1 þ e1 cosðk 2 Þ 1 þ e2 cosðk 2 þ XÞ (two eccentricities e1 and e2, two parameters P1 and P2, and an angle X between the two major axes). Figure 11 shows the three closed paths chosen for this analysis, together with one such random walk for each. Graph 15 shows the distribution of the fractal dimensions of these 3 3 609 Brownian bridges. Notice, how the fractal dimension distribution of the Brownian bridges about the ‘‘crossed’’ ellipses straddles the fractal dimensions of the 25 left and 25 right Chinese frontal sinus outlines. Also, notice that the distribution of fractal dimensions of the Chinese outlines exhibits no left/ right asymmetry. 1470 PROSSINGER Graph 11. The discrete Fourier transform spectra of the deviations of the 2-degree angle-polygons from a mean circle (as deﬁned in Fig. 9a) of all the 25 right Chinese frontal sinus outlines shown in Fig. 6. The pattern of large amplitudes is unique for every individual; the harmonics beyond 20 contribute to noise (for details, refer to the text). Graph 9. The distribution of left and right frontal sinus cross-sectional areas of 155 males and 137 females from the population shown in Graph 1. Light gray: right lobes; dark gray: left lobes. A paired t test shows that there is no signiﬁcant asymmetry in the cross-sectional area distribution. Graph 12. The relation between circumference and enclosed area of the 25 left and 25 right Chinese frontal sinus outlines shown in Fig. 6. Stars: right frontal sinus outlines; squares: left frontal sinus outlines. Note that there are two regression functions graphed: Aleft U1.897 and Aright U1.864, but they are so close that they cannot be graphically resolved. The two exponents are the fractal dimensions of the left/right outlines. Fractal Features of Frontal Sinus Outlines Generated by PCMs Graph 10. The spectra of the discrete Fourier transforms of the deviations from the mean curves as shown in Fig. 9. Main graph: for the deviations as deﬁned in Fig. 9a. Inset: the deviations as deﬁned in Fig. 9b. Note the difference in amplitudes in the two deviation conditions. There are 2 3 89 harmonics; only half are graphed, as the other half is symmetric. The 0th harmonic is the average of the ﬂuctuations, which has been parceled out before calculating the discrete Fourier transforms. The resulting outlines generated by the PCM are fractals (Czirok, 2001; Prossinger, 2005). Graph 16 shows the outlines of a cluster after 25,000 and then 75,000 iterations with two different perimeter list geometries, both with P0 5 0.75. As is to be expected, the cluster becomes larger as the number of iterations increases and the number of sites that deﬁne the perimeter length also becomes larger. Note that the ‘‘length’’ of the perimeter is deﬁned as the number of lattice sites in the perimeter list after the last iteration, not the length (in a geometric sense). In Graph 16b, there are perimeter sites that are ‘‘inside’’ the cluster. One way of ﬁnding the geometric perimeter, i.e., the biologically relevant MATHEMATICAL ANALYSES OF FRONTAL SINUS MORPHOLOGY 1471 Graph 13. The distribution of the fractal dimensions of the 25 left and 25 right Chinese frontal sinus outlines shown in Fig. 6. The individual numbers have been reordered, left and right separately to demonstrate the distribution of these fractal dimensions. one, using PCM, is demonstrated later in the case of the Crapina C outlines. Because of the limitation of (randomly) choosing a site in the perimeter list, and not any arbitrary lattice site, the elements of perimeter list are correlated. One feature of the growth is the pinning effect (Barabasi and Stanley, 1995): some elements of the perimeter list are not chosen frequently enough (because of chance) and this part of the perimeter lags behind as the perimeter moves ‘‘outward.’’ (An example is the k 5 5 site in Fig. 12) Whenever pinning sites are close, the entire ‘‘surface’’ lags behind, as shown at P1 and at P2 in Graph 16b. Computer simulations have revealed that such pinning effects become more pronounced when the modeling of physical and biological systems become more realistic. In bacteria growth modeling, for example, the pinning occurs predominantly at any inhomogeneity of the agar gel; in some physical systems, pinning is enhanced because of small irregularities in the density of the substrate or because of disorders in a medium or dislocations in a crystal. Consider the example of the propagation of a liquid through a paper towel (Barabasi and Stanley, 1995, p. 119–122). The lag in ﬂow because of inhomogeneities of the paper results in a quenched noise phenomenon. The pinning due to quenched noise is not to be confused with limiting the expansion of the cluster due to a barrier (Graph 17). A barrier is straightforward to incorporate in PCMs: it is a topology of absent lattice sites. The quenched noise effect enhances pinning, whereas the existence of a barrier does not. In Graph 17b, the deep groove in the perimeter is due to the lag as the perimeter sites ‘‘try’’ to ‘‘catch up’’ after having successfully surrounded the barrier. At P1 and P2 in Graph 16b, the lags are due to the pinning effect— there is no barrier at these lags. The ‘‘growth’’ of the cluster can be used to model the expansion of sinuses. In other words, pneumatization as a PCM is modeled with some P0. Cranial bones grow by periosteal intramembranous ossiﬁcation (Martin et al., 1998). Using PCM as a method of morphological assessment is justiﬁed by noting that the shaping of the pneumatization occurs via osteonal remodeling: osteoblasts deposit bony material and, therefore, build up the bone, whereas osteoclasts sculpt it by removing bony material. In the cranium, bones have respond to strains/stresses differently than do long bones (Rawlinson et al., 2000): location-dependent patterning has traditionally been ascribed to physiology, but Skerry (2000) claims this issue is unresolved. This unresolved issue is addressed by looking at the outcomes of algorithmically simulations of osteonal remodeling as a PCM process. The bony Fig. 13. A simulation of the ontogeny of the two Crapina C frontal sinus outlines using a PCM with P0 5 0.55 and a total of 175,000 iterations. The fractals generated by the PCM are shown in Graph 18. The outermost fractal is splined once onto the left frontal sinus outline, once onto the right one. The previous (ontogenetically: ‘‘earlier’’) fractals are also splined, with a scale factor proportional to the ratio of their length to the length of the outermost fractal. The black dots are the corners of the 2-degree-angle-polygon (shown as arrow tips in Graph 14), the contours of the digitized outlines derived from Fig. 5c are drawn in mauve. Color-coding of the fractals is the same as in Graph 18. material is the lattice, the growth of the perimeter list is the expansion of the sinus(es) because of the osteoclasts removing bony material with a higher probability than the osteblasts depositing it. The difference between deposition and removal probability must satisfy the condition 0 < P0 < 1. Therefore, inhomogeneities in the bone (though not necessarily the composition of the bony material!) result in quenched noise sculpting. In the context of frontal sinus morphology, the most important sources of inhomogeneity are the slight, repeated compressions of bony material during mastication, as the resulting compression waves nonuniformly compress the frontal bone. As has been noted earlier, the inhomogeneities that produce pinning via the quenched noise phenomenon can be (indeed: are) very slight. As the osteoclasts remove bone, the pinning results in (locally) delaying/ halting the outcomes of their activity: wherever the lattice sites are denser, more new sites must be added to the perimeter list (at constant P0). In other words, the apparent slowing of osteoclastic activity outcomes and the attendant removal of less bony material is modeled by inhomogeneities (primarily a higher density of lattice sites) in the lattice in the PCM, and the slowing of sinus expansion is due to masticatory stresses. The pinning sites are therefore the observed bone lamellae in the sinuses (Fig. 2). The emergence of lamellae in the pneumatizations is modeled as pinning and quenched noise phenomena in PCMs. One can show how this effect comes about in the generation of the lamellar morphology of the frontal sinuses in Crapina C (Fig. 5c). First, generate a cluster using a PCM with P0 5 0.55. ‘‘Snapshots’’ of the perimeter list after 75,000, 145,000, and 175,000 iterations (with a suitably complicated initial perimeter list geometry, details can be requested from the author) are shown in Graph 18. We note how the pinning sites become more 1472 PROSSINGER Graph 15. The distribution of the fractal dimensions of three sets of 609 Brownian bridges around curves shown in Fig. 11, and the fractal dimensions of the 25 left and 25 right Chinese frontal sinus outlines shown in Fig. 6. White bars: Brownian bridges about a circle; light gray bars: Brownian bridges about an ellipse; black: Brownian bridges about crossed ellipses (see Fig. 11c). Inset: the distribution of the left (labeled L) and the right (labeled R) fractal dimensions of the 25 left and 25 right Chinese outlines, and the fractal dimensions of the Brownian bridges about crossed ellipses (Fig. 11c). The left/right fractal dimensions straddle the Brownian bridges about the crossed ellipses very well. Note the scale of the ordinates. scaling factor for the spline is the ratio of the length of each perimeter list to the longest one. This procedure not only models lamellae emergence but also reconstructs the ontogeny of Crapina C (Fig. 13). DISCUSSION Graph 14. The 2-degree angle-polygon approximating the right frontal sinus outline of Crapina C (Fig. 5c) and the log–log graph of the standard deviation C(F) as deﬁned in the text. The circles are the points used for estimated the linear regression. The self-similarity of the outline is a statistical fractal (a self-similar object) from 28 to 288. The variables C(F) and F are explained in the text. pronounced as the perimeter becomes longer: these are the predicted lags. We then generate 2-degree angle-polygons of the observed outlines (one left and one right) of Crapina C, with the method demonstrated in ‘‘Frontal Sinus Outlines as Fractals and Random Walks’’ (earlier). The left polygon is then splined onto the digitized left outline of Crapina C, likewise the right outline, along with the fractals (with lengths 1,706, 2,268, 2,549). Each Attempting to estimate the ontogeny of frontal sinus volumes via a sigmoid function is fraught with difﬁculty: the sample size is often too small but never too large. Because only a few well-behaved points ensure a good sigmoid ﬁt, readers should be warned of the false reassurance in Graph 1. The case of the very noisy cross-sectional areas of the outlines show most dramatically how challenging it is to properly estimate a biologically meaningful sigmoid. However, the sigmoids modeling the volume ontogeny do reveal several things. First, they show how sigmoids behave with variation in measurement error—they are quite robust, in fact, for biologically relevant estimators. Second, there is a debate of the age-at-death of Le Moustier I and the sex of the individual because of the observed gracility of the cranium (Thompson and Nelson, 2005); one can justify why the debate is crucial in this context. Only the volumes near the point of inﬂection drive the parameter estimates of the sigmoid function. Accurate age-at-death determination of adults is not very important for estimating the parameters. Intriguing is the consistency of the frontal sinus data despite their paucity. Although Neanderthals have large browridges, their ontogeny seems to indicate that the frontal sinuses are much smaller than those of H. sapiens. In the debate about ‘‘accelerated ontogeny’’ of Neanderthals, the sigmoids indicate that their frontal sinus ontogeny may be chronically comparable (in some statistical measure) with H. sapiens’s. MATHEMATICAL ANALYSES OF FRONTAL SINUS MORPHOLOGY Graph 16. Two clusters generated by the Percolations Cluster Model (PCM) algorithm after (a) 25,000 and (b) 75,000 iterations with a predeﬁned probability P0 5 0.75. The initial perimeter list consisted of two straight strings, perpendicular to each other. SVD smoothing has revealed the common underlying form of the 25 Chinese outlines. The smoothed shape (Fig. 10b) is closer to that generated by two crossed ellipses than by a single ellipse (Fig. 11). The observed distribution of fractal dimensions of Chinese outlines is closer to the fractal dimensions of random walks around crossed ellipses than around a single ellipse; the fractal dimensions of these random walks comfortingly conﬁrm the underlying common form detected by SVD smoothing. Random walks Si are not iid-noise (Brockwell and Davis, 2002) while their ﬁrst differences Xi 5 Si11 2 Si are. Brownian bridges are a subset of random walks, but their noise properties are more involved (Bookstein, personal communication). Details of the noise properties of 1473 Graph 17. Two clusters generated by the PCM algorithm after (a) 45,000 (b) 175,000 iterations with a predeﬁned probability P0 5 0.55 in the presence of a barrier. In both cases, the initial perimeter list consisted of a straight string of sites, roughly along the ﬁrst median of the coordinate system. For illustration purposes, the two barriers have different geometries; in both cases, after sufﬁciently many iterations, the barrier will be engulfed by the cluster (one not shown). Brownian properties on SVD-smoothed frontal sinus outlines have not yet been resolved and are a topic of ongoing investigations. The noisiness of the frontal sinuses is one reason why this article reviews so many different analysis techniques. The noisy signals do contain biologically relevant information, but it cannot be extracted all with one assessment methodology. The sigmoids found by Prossinger and Bookstein (2003) do reveal biologically consistent, even interesting, relevant insights. The histo- 1474 PROSSINGER Graph 18. Three clusters generated by the PCM algorithm after 75,000, 145,000, and 175,000 iterations with a predeﬁned probability P0 5 0.55. The initial perimeter list consisted of a straight string of sites, roughly along the ﬁrst median of the coordinate system. grams of left and right cross-sectional areas indicate that they are not well approximated by a lognormal distribution, whereas their sum (the total cross-sectional area) is. One implication of this is: the observed lognormality of the total cross-sectional area of both lobes together, yet its absence for each lobe population points out that septum morphology also exhibits noisiness. Rarely is the septum close to being planar, and it is also rarely orientated parallel to the midsagittal plane. In projection, therefore, the parts of the outlines of the individual lobes near the midsagittal plane are even noisier than the other parts. In Fig. 4e, the septum is not detectable in the projection. The frontal sinus lobe projections in Figs. 5b,c mask some of the septum; its morphology can only be assessed in 3D CTscans. The clusters generated by PCMs that model the putatively earlier outlines of the Crapina C lobes when the individual was younger must have been separated by a smaller distance than is rendered in Fig. 13. Further studies of how the lobes move apart—in all species of Homo—are needed. Such studies will also contribute to our knowledge of the ontogeny of the septum, its morphology, and further details of the pneumatization process in the frontal bone. In general, morphologists try to bridge the gap between two diverging challenges: to ﬁnd the unique morphology of an individual specimen and to extract general features inherent in the specimens of a population. Frontal sinuses present, in this regard, probably (although this probability has yet to be calculated!) the most extreme challenge. The noisiness ensures that each individual’s frontal sinus outlines are unique enough to warrant using them for identiﬁcation purposes. Although human digit ﬁngerprints and iris patterns are similarly unique, this uniqueness of the frontal sinus outlines is frustrating for the researcher attempting to extract biological generalities in the face of extreme noisiness (pun intended!). The methods presented here (many of which had been developed and ﬁrst applied to the identiﬁcation of cell outline morphologies and extraction of statistical descriptors of bacteria growth colony outlines) show how this challenge can be met. The ontogeny of the frontal bone seems to have wider biological implications than the patterns of the iris. How many Fourier components need to be retained for the identiﬁcation of an individual? Answer: retain all those frequencies that do not constitute iid noise. Those Fourier components that contribute to iid noise can be detected with a suite of tests (Brockwell and Davis, 2002): (1) the maximum and minimum values of the sample autocorrelation function, (2) the sample value of the Ljung-Box statistic, (3) the sample value of the turning-point statistic, (4) the sample value of the difference sign statistic, (5) the rank test (Kendall and Stuart, 1976) for the existence of a trend, and (6) testing for the possibility to ﬁt an autoregressive model with the YuleWalker algorithm. These tests have been carried out in the case of the 25 Chinese outlines, but the details are too involved to be presented here. As a rule of thumb, some 20 harmonics need to be retained. The successful characterization of general biological principles that lead to highly individualized pneumatization morphologies is, as is to be expected, unorthodox. This review shows that a suite of statistical techniques does exist and that they can successfully assess morphology, but only by modeling the pneumatization (i.e., its cellular process). It is interesting that a reductionist modeling approach can extract biologically relevant macromorphological features that conventional morphometrics methods fail to ﬁnd. The PCMs presented here offer an added bonus: they may resolve the masticatory stress debate. Chewing results in compression waves (albeit of very small amplitude) propagating through the facial skull, including the outer frontal bone. Bone modeling (osteonal remodeling) is a competition between osteoblast and osteoclast activity, which is best modeled by PCM. The resulting pinning sites are stochastically distributed along the perimeter. Wherever they form, however, they lag more and more behind as the perimeter list lengthens. The lamellae (the set of pinning sites) therefore have an observable morphology (Fig. 2b). They apparently form rib-reinforced architecture. The ‘‘masticatory stress debate’’ (Prossinger et al., 2000a; Ravosa MATHEMATICAL ANALYSES OF FRONTAL SINUS MORPHOLOGY et al. 2000b) has heretofore been inconclusive, because the observed large frontal sinuses imply a mechanical response, yet the stresses could not be detected in vivo (Hylander and Ravosa, 1992). The PCMs show that lamellae emerge because of mechanical stresses and strains, but their amplitudes are so small that they can hardly be detected with strain gauges applied to the browridges and faces of small primates. And, to boot, the strain gauges were not attached to the lamellae. A careful morphological analysis, which necessitates modeling of fractals, resolves this debate. The lamellae, then, are the responses to masticatory stress, not the large frontal sinuses per se. On the other hand, the mechanical strains/stresses, although responsible for the emergence of the lamellae, do not explain why the pneumatization occurs in the ﬁrst place. The pneumatization itself must be due to phylogenetic/ontogenetic processes (Leicher, 1928). The pneumatization of the diploë between inner and outer table of the frontal bone is described as a difference in growth rates of the two growth ﬁelds (Enlow, 1975; Liebermann, 2000). What happens within the diploë needs to be ‘‘described’’ (more precisely: mathematically modeled) as well. It is therefore reasonable—even necessary—to analyze frontal sinus morphology using PCMs. Such analyses should give insight into the emergence of phylogenetic trees, or at least clades. Morphological assessment using PCMs also explains another observed feature: why the lamellae distribution in a population is (statistically) noisy yet has an observed quasi-regularity (Figs. 1 and 2) in each single individual. The PCM algorithm shows that the pinning sites occur stochastically. It is their lagging that is enhanced by mechanical (in the case of frontal sinuses!) feedback mechanisms, the quenched noise effect. No two lamellae patterns can be expected to be the same, the probability (the product of two very small probabilities) for that is miniscule. Using PCM algorithms, therefore, shows why the projection of frontal sinus outlines can be used as ‘‘ﬁngerprints’’ unique to an individual. In projection, the lamellae generate the (stochastically emerging) outlines. PCMs supply a justiﬁcation for the statistics of fractal forms. PCMs do not predict that the volume distributions are as noisy as the outlines. If pneumatization is primarily phylogenetically driven, then sigmoids are good models for volume ontogeny. The outlines, as projections of the 3D-fractals, become noisy and the highly variable outline morphology generates the uniqueness of the individual! In this review, Moore’s skepticism can therefore be addressed: although we cannot derive a function from the traditional assessment of frontal sinus morphology, attempts at deriving statistical estimators that are applicable to noisy data sets point to physiological and phenomenological insights (including insights as to the ‘‘roles’’ and ‘‘functions’’ of the frontal sinuses) that had heretofore remained inaccessible. CONCLUSIONS This article reviews a suite of analysis techniques, not an inventory of conventional descriptive morphological features of specimens or populations. The presented techniques show that frontal sinuses are not only in- 1475 triguing objects of study but also remain elusive when conventional techniques, which have been successful for other features of the skeleton, are applied. Novacek (1993) cautiously claims that sinuses may merely be spaces between struts and pillars in the cranium or that there are many possible functions. The analysis techniques presented here have more to offer than the highly verbose morphological assessments so ubiquitous in the literature. Three sequence analysis methods (Fourier descriptors, random walks, and fractal dimension determination) and, fourth, the PCMs provide morphological analysis novelty. Each of these four can be employed to focus on possible answers (plural!) to the elusive question of the frontal sinuses’ role(s). Indeed, the techniques presented here allow one to ascertain whether there could be one, several, or no role of the observed morphology of these pneumatizations. Here, it has been shown, for example, how the pinning effects in PCMs can explain the emergence of lamellae and their (statistical) noisiness—both in one single individual and also within groups. Why are models of the morphology of frontal sinuses needed, rather than conventional, verbose descriptors? More speciﬁcally: need one model in order to statistically describe? The rationale for theory is the necessity of logical consistency, not simpliﬁcation or ad hoc narratives. Simpliﬁcation is most parsimoniously achieved with models; they should be designed to be consistent with theory (albeit not an ad hoc one!) and demonstrate the salient features observed. It is properly constituted theory that deﬁnes what salient is. Ad hoc theories, alas, are at odds with the deﬁnition of salient, as they cannot identify—let alone specify—saliency. Whenever, in any scientiﬁc undertaking, models are successful, their parameters constitute the sine-qua-non information extractable from the observed natural phenomena. Describing roughness with simple summary statistics, for example, is not very appropriate (because roughness is statistically too noisy). Describing how roughness varies (better: changes) by ﬁnding the numerical values of the statistical estimators is an endeavor more appropriate and more rigorous for fractal morphologies, including anatomical ones (Graph 15). The intent of modeling fractal dimensions of frontal sinus outlines did not a priori include supplying a resolution of the masticatory stress debate. Pinning and quenched noise effects are salient features that are not added to the models of pneumatization in an ad hoc manner; they are emergent properties. The noisiness of frontal sinus morphology has necessitated a shift in how anatomists approach difﬁcult-toresolve morphology questions. The methodology needed to assess morphology now demands model building; otherwise, the statistical variances do not converge. Fractals do not have converging variances; so repeated measurements of ever-larger data sets are not worthwhile. Only parameters of an underlying model can be estimated by applying the model to the data—to boot, it must be a model not derived from an ad hoc theory. Sinuses are cavities; the outlines are projections of their lateral extent. A scale change that is the same in all directions will change the morphology of self-afﬁne objects. Scale changes that differ in different directions do not change the morphology of interfaces (Barabasi and Stanley, 1995). The latter changes behave like fractals. The projection of the fractal interface bone/pneuma- 1476 PROSSINGER tization will produce a fractal outline. But an inherent limitation of outline analysis remains: it misses out on processes that take place in the direction perpendicular to the projection plane. Although morphological features detected in the analysis of outlines are not misleading, one would like to know what happens in the (third) dimension perpendicular to the projection. Even though, in the case of frontal sinuses, where cavity dimensions perpendicular to the Caldwell projection are not large, analyses incorporating this dimension promise to reveal a considerable wealth of further information—information (not only morphological) that doubtless is highly desirable in its own right. The frontal sinuses are cavities whose enveloping surfaces follow the curvature of the inner and outer tables of the frontal bone. Envelope morphology needs to be investigated. Statistical analysis of envelope morphology necessitates, however, 3D-morphometrics. The septum not only has fractal surfaces (these are the fractals of the pneumatizations) but also a highly irregular orientation distribution within the frontal bones of a population. Assessment of septum morphology and orientation distribution is only possible in 3D. The perimeter generated by a PCM is rougher than the surfaces observed in biological specimens (Figs. 4 and 5). This is because perimeter and cluster growth in PCMs generate uncorrelated surface structures. The introduction of correlated noise (Barabasi and Stanley, 1995) smoothes the surface to some degree and—most importantly—removes the isolated perimeter sites within the cluster. In the case of pneumatizations, PCMs with correlated noise are only meaningful in 3D, because the ﬁnite-distance correlations are not restricted to correlations in the outlines’ plane. For all these reasons, the motto for the next round of morphological assessment(s) must henceforth be: 3D analysis techniques are needed! ACKNOWLEDGEMENTS Many researchers and colleagues have given (invaluable) assistance during the development and tailoring of the mathematical methods presented here: their (often critical) advice was not restricted to mathematics and statistics; quite the contrary, many were morphological practitioners whose criticisms helped hone the statistical/mathematical approaches by enhancing the understanding of the biological aspects of frontal sinuses. The author thanks (in alphabetical order): Leslie Aiello (Wenner-Gren Foundation for Anthropological Research, New York, USA), Fred Bookstein (Department of Statistics, University of Washington, Seattle, USA), Phillip Gunz (Department of Human Evolution, Max Planck Institute for Evolutionary Anthropology, Leipzig, Germany), Thomas Koppe (Institute of Anatomy, Ernst Moritz Arndt University, Greifswald, Germany), Les Marcus (y), Roberto Macchiarelli (Université Poitiers, France), Philipp Mitteröcker (Konrad Lorenz Institute for Evolution and Cognition Research, Altenburg, Austria), Gerd Müller (Department of Theoretical Biology, University of Vienna, Austria), Todd Rae (Department of Anthropology, University of Durham, UK), Wolfgang Recheis (Universitätsklinik für Radiodiagnostik, University of Innsbruck, Austria), Silvia Scherbaum (Library for the Biological Sciences, University of Vienna, Austria), Dennis Slice (Department of Anthropology, University of Vienna, Austria), Chris Stringer (Department of Palaeontology, Natural History Museum, London, UK), Ian Tattersall (Division of Anthropology, American Museum of Natural History, New York, USA), Maria Teschler-Nicola (Department of Anthropology, Natural History Museum, Vienna, Austria), Philipp Tobias (Department of Anatomy, University of Witwatersrand, Republic of South Africa), and Lothar Wicke (Vienna, Austria). 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