According to existing research, in 20% of patients the required tracheal size (diameter and length) could not be measured automatically (using software solution). In this paper, we use Zederbauer\u27s harmonic circle to define the relationship between a person\u27s body height and the diameter and length of their trachea. We then explore differences by gender. The results show that there is a highly significant correlation between measured data and values obtained by harmonic analysis. The final goal is to obtain the mathematical interpretation of the relationship of the tracheal size to the height of man, in order to further apply such sizes in the creation of 3D models that could finally be printed on appropriate 3D printers and with the suitable material

Dinko Milavec

Hrvoje Puretić

Jelenko Lazić

Vlatko Mićković

Zoran Lončar

English

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107Received for publication October 25, 2019IntroductionChanges in tracheal dimensions occur in a variety of conditions, and the knowledge of normal tracheal dimen-sions is essential to the diagnosis of these conditions.1 Trachea is a cartilaginous-membranous tube that with its branches ensures the supply of air to the lungs. It is made of 16–20 cartilages in the shape of a horseshoe (ellipse) and fibroelastic membrane.2 Trachea, as a direct continuation of the larynx goes from the lower edge of the annular cartilage (cricoid car-tilage), i.e. the sixth neck (cervical) to the fourth thoracic vertebra and where it is divided into two main bronchi. The place where the trachea divides into bronchi is called bifurcatie trachea – midpoint trachea (tracheal bifurca-tion) or carina, and it is projected on the front wall of the chest at the level of the angulus steric (sternal angle). Rear projection of the mindpoint of the trachea fits the pointed tip of the adapter (tip of the spinous process) of the third thoracic vertebrae or the line joining the two trigonum spines of the scapula (interscapular line). Observing at the transversal cut, the trachea is not completely round. The anterior and lateral walls of the trachea are rounded and the back is straight. The back wall is called paries mem-branaceus.3 Coll. Antropol. 44 (2020) 2: 107–112Short reportHarmonic Structure of Tracheal BiometricsDinko Milavec1, Jelenko Lazić2, Hrvoje Puretić3, Vlatko Mićković4, Goran Sirovatka4, Zoran Lončar51Clinical Hospital "Dubrava", Zagreb, Croatia2Clinical Hospital "Sveti Duh", Zagreb, Croatia3Department of Pulmology, University Hospital Center, Zagreb, Croatia4University of Applied Sciences, Zagreb, Croatia5Department of Traumatology, Sestre milosrdnice Hospital Center, Zagreb, CroatiaA B S T R A C TAccording to existing research, in 20% of patients the required tracheal size (diameter and length) could not be mea-sured automatically (using software solution). In this paper, we use Zederbauer's harmonic circle to define the relationship between a person's body height and the diameter and length of their trachea. We then explore differences by gender. The results show that there is a highly significant correlation between measured data and values obtained by harmonic analysis. The final goal is to obtain the mathematical interpretation of the relationship of the tracheal size to the height of man, in order to further apply such sizes in the creation of 3D models that could finally be printed on appropriate 3D printers and with the suitable material.Key words: anthropometry, harmonic circle, proportions of the human bodyThere are two parts to the trachea: the neck and the chest. The neck of the trachea is located in the medial line, and the thorax lies slightly more on the right side. There-fore, a midpoint of the trachea is located a little to the right of the midline. The shape of the trachea is in the form of a roller which is slightly flattened at the back. The tra-cheal lumen is not equal in length. Just below the larynx, the trachea is slightly narrowed (thyroid impression). Then the lumen expands to the middle of the trachea, only to narrow again, because the aortic impression arises on the left side. The healthy trachea lumen is always open and wider than the larynx lumen.3The function of the airway is to conduct air to the al-veoli, where gas from the inhaled air and gasses dissolved in the blood of the alveolar capillaries are exchanged. The rings of the cartilage trachea are associated with fibro-elastic and muscle tissue. Theoretically, such a structure exists to prevent loss of tracheal stability. Tracheal stabil-ity means maintaining the shape of the trachea not to bend or recede, but rather to keep its lumen and shape the same when moving the neck and head.According to literature, the average adult trachea mea-sures about 11 cm in length with a diameter that varies from 2 to 2.5 cm.4 However, previously published figures 108D. Milavec et al.: Harmonic Structure of Tracheal Biometrics, Coll. Antropol. 44 (2020) 2: 107–112for normal tracheal dimensions show considerable varia-tion in attempts to establish normal ranges for these mea-surements.1 It has been shown that tracheal dimensions vary by age, gender and population.5,6 Measurements on chest radiographs have shown that the coronal diameter of the normal trachea ranges from 13 to 25 mm in men and from 10 to 21 mm in women, and the sagittal diam-eter of the normal trachea ranges from 13 to 27 mm in men and from 10 to 23 mm in women.7 Some studies have reported the association between tracheal width and body height which makes it possible to predict the tracheal dimensions from simple external measurements.6,8 However, according to the study by Ma et al.9, in 20% of patients the required tracheal size (di-ameter and length) could not be measured automatically (using software solution) from other measurements.In a previous paper, we have shown that the method of calculation in the canon of the harmonic circle is ad-vantageous in defining sizes required by anthropometric analyses or those that we have no ability of measuring, as it leads to a high degree of correlation with actual measurements on a representa tive sample.10 In this paper, we use the harmonic circle to define the relationship between a person's body height and the di-ameter and length of their trachea. We then explore dif-ferences by gender. The final goal is to obtain the math-ematical interpretation of the relationship of the tracheal size to the height of man, in order to further apply such sizes in the creation of 3D models that could finally be printed on appropriate 3D printers and with the suitable material. Material and MethodIn this research we observed 79 people, whose head, neck and lungs were recorded using computer tomogra-phy (CT).11 In our sample, the organs were normally de-veloped, without any deformation of the trachea and bron-chi. The type of CT device on which we recorded the subjects was MSCT Simens Somatom definition AS. We divided patients by gender into 42 males and 37 females for measurement of height and weight. We selected can-didates between 40 and 50 years of age The anthropomet-ric measurements were taken by a protocol described in detail elsewhere.12 From CT recordings tracheal length (TL), anteropos-terior (APD) – sagittal cut and laterolateral (LLD) – fron-tal cut of the trachea were measured in males and fe-males. We further calculated the ratio of tracheal length to body height for each patient separately. From this ratio, we calculated the mean that we used as a benchmark. Furthermore, we derived a value from Zederbauer har-monic circle, that is a linear combination of the sizes shown and calculated on the circle.13,14 We compared the calculated values with the measured methods using sta-tistical methods. Finally, we determined whether such a coefficient resulted in the true trachea size for a given height of a person. Harmonic circleA significant contribution to the definition of the length of body parts is given by the Zederbauer harmonic cir-cle,13,14 which is shown in Figure 1. The construction of the harmonic circle is based according to Zederbauer13 on a regular triangle with hypotenuse a, and sides b, as it is shown in Figure 1. After we draw over the sides of the triangle the respective squares, we can finally draw the circle of radius R that passes through the peaks of a such geometrical construction. Mathematical relations between these geometrical values give harmonic numbers. In the Zederbauer’s circle the standing height of man is used as diameter, this height being equal to the sum of eight head lengths, and after that, in a geometrical combination, other characteristic body dimensions are inserted. As pre-viously described, the derived lengths of different body segments already showed strong correlation with human anthropometric measures.10, 15The mathematical relations in the harmonic circle are described in Table 1.Fig. 1. Harmonic circle.ResultsLength measurements and diameters of trachea Table 2 presents the obtained measures (mean, SD and range) in mm for males. The mean values for BH, TL, APD and LLD were 1736.15 mm, 132.69 mm, 17.56 mm and 18.04 mm, respectively. The ranges of the examined vari-ables were 1650 –1890 (BH), 120 – 150 (TL), 16 – 19 (APD) and 17 – 21 (LLD).If we return to the harmonic circle of the mean value of the standing height, we will associate it with the variable H that in the harmonic circle represents the diameter H = 2 R. For the so defined variable R we get the derived values of the harmonic variables in the circle shown in Table 3. The calculated harmonic sizes are BH – 1736.15 mm, TL – 137.24 mm, APD – 17.36 mm and LLD – 19.41 mm.109D. Milavec et al.: Harmonic Structure of Tracheal Biometrics, Coll. Antropol. 44 (2020) 2: 107–112Table 4 presents slightly lower obtained measures (mean, SD and range) in mm for females. The mean values for BH, TL, APD and LLD were 1649.28 mm, 127.57 mm, 17.32 mm and 17.35mm, respectively. The ranges of the examined variables were 1550 –1780 (BH), 105 – 148 (TL), 15 – 20 (APD) and 15 – 19.8 (LLD). By using the same calculation as for males, we ob-tained the derived harmonic values for females shown in Table 5. The calculated harmonic sizes are BH –1649.28 mm, TL – 130.38 mm, APD – 16.49 mm and LLD – 18.43 mm.A correlation analysis with two sets of measured and calculated values was performed. The results are present-ed in Table 6. It can be seen that the Pearson correlation coefficient is 0.999, which indicates an extremely high degree of correlation. Further rank correlation analyses (Kendall's tau_b and Spearman's rho), given in Table 7, confirmed a highly significant correlation between the measured and the calculated harmonic values.Volume of tracheaThe cross-section of the trachea is the ellipse of the known tracheal size (diameter AP and diameter LL), and therefore we were able to calculate the surface section. The cross-section of the trachea is approximately an el-lipse and in the present work we abstracted its imperfec-tions caused by different structures and thicknesses of the tissue at the front and rear part of the trachea.16 The area of the section is given by the formula presented below (al-though it is generally known) to show how one can calcu-late the surface section and from that to calculate the volume showing that mathematized access to medical problems exist as a realistic option.The surface of ellipse is as follows:From this formula, the surface of the trachea can be calculated as:P = abπ Formula 1Then, let a = Diameter AP (PAP) and let b = Diameter LL (PLL) define the ellipse as the base of the cylinder of a given surface according to the Formula 1, the height of which is in reality the length of the trachea so that the cross-sectional area of the trachea will be:P = PAP * PLL * π Formula 2And the cylinder with the elliptic base V = abπ * h Formula 3will for trachea be:Vm = PAP * PLL * DD * π Formula 4TABLE 1HARMONIC SIZES AS HARMONIC NUMBERSBasic harmonic sizes Derived harmonic sizesLabel Formula Calculated values Label Formula Calculated valuesa 1a 1 r= - = -2 12 2b a0.207bb = 220.707 d= - = -5 12 2R a0.618RR = 521.118 b + r= - = -2 12 2b a0.914a = Hypotenuse of the harmonic triangle; b = Cathetus of the harmonic triangle side; R = Radius of the harmonic circle. Source: Muftić et al.15 and Mičković et al.10TABLE 2MALES - MEASURED SIZES IN MMAn An (mean) SD Var Min MaxBody height BH 1736.15 68.01207 4625.641 1650 1890Trachea length TL 132.69 10.53079 110.8974 120 150Trachea diameter APD 17.56 0.970725 0.942308 16 19Trachea diameter LLD 18.04 0.964033 0.929359 17 21110D. Milavec et al.: Harmonic Structure of Tracheal Biometrics, Coll. Antropol. 44 (2020) 2: 107–112TABLE 5FEMALES - CALCULATED SIZES IN MMAn Measured values An Harmonic sizes Harmonic sizes (H) Calculated harmonic values AnBH 1649.28 2R H 1649.28TL 127.57 R*b 0.07905378H 130.38APD 17.32 2R/100 0.01 H 16.49LLD 17.35 b 0.01118 H 18.43TABLE 6PEARSON CORRELATIONMean values of measured sizes Calculated harmonic sizesMean values of measured sizesPearson Correlation 1 .999**Sig. (2-tailed) .000N 25 25Calculatedharmonic sizesPearson Correlation .999** 1Sig. (2-tailed) .000N 25 25TABLE 3MALES - CALCULATED SIZES IN MMAn Measured values An Harmonic sizes Harmonic sizes (H) Calculated harmonic values AnBH 1736.15 2R H 1736.15TL 132.69 R*b 0.07905378H 137.24APD 17.56 2R/100 0.01 H 17.36LLD 18.04 b 0.01118 H 19.41TABLE 4FEMALES - MEASURED SIZES IN MMAn An (mean) SD Var Min MaxBody height BH 1649.28 66.38632 4407.143 1550 1780Trachea length TL 127.57 11.17493 124.8791 105 148Trachea diameter APD 17.32 1.301732 1.694505 15 20Trachea diameter LLD 17.35 1.242671 1.544231 15 19.8On the other hand, we have calculated the values of AP Diameter and LL Diameter in the canon of the Zeder-bauer circle, which are functionally related to the height of the BH, so it follows thatPAP = α1 * H PLL = α2 * H DD = α3 * H Formula 5And if we include these calculated values in Formula 4, we obtain the following:Vr = α1 α2 α3 * H3 * π Formula 6The results of the tracheal volume calculation are ex-pressed in cm3 and presented in Table 8.We can see that the difference between the calculated and the measured results amounts to 3% of volume for the female data, and 10% for the male data where there is a slightly larger deviance in the calculated length of the trachea. For both males and females the calculated values give larger volume with acceptable variation.ConclusionIn this paper, we presented a new approach of the analysis of anthropometric dimensions of the parts of the human body that are not available for direct mea-111D. Milavec et al.: Harmonic Structure of Tracheal Biometrics, Coll. Antropol. 44 (2020) 2: 107–112TABLE 7RANK CORRELATION COEFFICIENTSTests Calculated harmonic sizesKendall's tau_b Mean values of measured sizesCorrelation Coefficient .990**Sig. (2-tailed) .000N 25Correlation Coefficient .973**Sig. (1-tailed) .000N 18Spearman's rho Mean values of measured sizesCorrelation Coefficient .999**Sig. (2-tailed) .000N 25Correlation Coefficient .995**Sig. (1-tailed) .000N 18** Correlation is significant at the 0.01 level.TABLE 8TRACHEAL VOLUME IN CM3Volume from measured quantities Vm Volume from  calculated quantities Vr Relative differenceMale 132.05 145.27 10.02%Female 120.43 124.48 3.36%surement in all situations. It represents an advancement and better feasibility in determining sizes of body parts in relation to similar studies9 in the percentage of matched cases between the measured and calculated values. The problems of deformation of body parts and their replacement by 3D printed models require a math-ematical approach to the estimation of their sizes. The trachea has proven to be one such component that can (in the future) be replaced partly or as a whole, for which measurement problems may occur in various diseases or injuries. This paper demonstrates an approach to the estimation and calculation of anthropometric trachea sizes showing a high degree of correlation with the ac-tual measured values on a representative sample. It can be helpful in forensics, biomedical engineering, ergo-nomics and surgery.R E F E R E N C E S1. BREATNACH E, GC ABBOTT GC, FRASER RG, Am J Roent-genol, 142 (1984) 903. — 2. KRMPOTIĆ-NEMANIĆ J, MARUŠIĆ A, Anatomija čovjeka (Medicinska naklada, Zagreb, 2004). — 3. KEROS P, Temelji anatomije čovjeka (Medicinska naklada, Zagreb, 1977). — 4. FERNANDEZ LG, SCOTT H. NORWOOD SH, BERNE JD, Tracheal, laryngeal, and oropharyngeal injuries. In: JUAN A. ASENSIO JA, TRUNKEY DD (Eds) Current Therapy of Trauma and Surgical Critical Care (Elsevier, 2008). — 5. GRISCOM TN, WOHL MEB, ARJ, 146 (1986) 233. — 6. DATTA D, KUNDU D, PAL S, DAS A, Int J Med Res Rev 7 (2019) 36. — 7. BOISELLE P, Clin Chest Med, 29 (2008) 181. — 8. GRISCOM NT, WOHL ME, The American Review of Respiratory Dis-ease, 131 (1985) 840. — 9. MA S, ADJAVON S, BOUCHIHA N, CAS-TELLI C, FISCHLER M, MELLOT F, LE GUEN M, Minerva Aneste-siologica, 85 (2019) 1281. — 10. MIĆKOVIĆ V, KARLAK I, LAZIĆ J, LONČAR Z, DOMIJAN M, SIROVATKA G, Coll Antropol, 42 (2018) 281. — 11. GAMSU G, WEBB R, Am J Roentgenol, 139 (1982), 321. — 12. RUDAN P, MUFTIĆ O, MAVER H, TABORŠAK D, KOVAČEVIĆ M, GRGIĆ Z, Practicum of Biological Anthropology (HAD, Zagreb, 1983). — 13. ZEDERBAUER E, Die Harmonie im Weltall, in der Natur und Kunst (Orion, Wien & Leipzig, 1917). — 14. MUFTIĆ O, Harmonijska antropometrija kao osnova za primjenjenu dinamičku antropometriju (Zbornik radova Skupa o konstruiranju FSB, Zagreb, 1984). — 15. MUFTIĆ O, JURČEVIĆ LULIĆ T, GODAN B, Sigurnost 53 (2011) 1. – OSMANLIEV D, BOWLEY N, HUNTER DM, PRIDE NB, Am Rev Respir Dis, 26 (1982) 179.V. MićkovićUniversity of Applied Sciences Zagreb, Vrbik 8, 10000 Zagreb, Croatiae-mail: vmickovic@tvz.hr112D. Milavec et al.: Harmonic Structure of Tracheal Biometrics, Coll. Antropol. 44 (2020) 2: 107–112HARMONIJSKA STRUKTURA BIOMETRIJE DUŠNIKAS A Ž E T A KIstraživanja pokazuju da kod 20 % pacijenata nije moguće automatski softverskim programom izmjeriti potrebne veličine dušnika (promjer i duljina). U ovom radu pomoću Zederbauerove harmonijske kružnice definiramo odnos tjele-sne visine čovjeka i promjera i duljine njegovog dušnika. Potom razlučujemo taj odnos prema spolu. Odredili smo i volu-men dušnika kao važnu veličinu u matematičkom modelu. Konačni cilj i put kojim treba krenuti je matematičko defini-ranje odnosa dušnika prema visini čovjeka u cilju daljnje primjene takvih veličina u kreiranju 3D modela koji bi se konačno mogli isprintati na odgovarajućim printerima i u odgovarajućem materijalu, za potrebe biomehaničkog inžinjerstva, kirurgije i forenzike.

Harmonic Structure of Tracheal Biometrics

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