Use of fractal geometry to propose a new mechanism of airway-parenchymal interdependence

Author(s)
Kyongyob Min^{*},
Keita Hosoi,
Yoshinori Kinoshita,
Satoshi Hara,
Hiroyuki Degami,
Tetsuo Takada,
Takahiko Nakamura

Affiliation(s)

Respiratory Division, Department of Internal Medicine, Itami City Hospital, Hyogo, Japan.

Respiratory Division, Department of Internal Medicine, Itami City Hospital, Hyogo, Japan.

ABSTRACT

The topic of airway-parenchymal interdependence (API) is of great importance to those interested in identifying factors that influence airway patency. A carefully designed experiment has raised questions about the classical concept of API. This paper proposes a new mechanism of API. The pulmonary lobe is an aggregated body consisting of many Miller’s lobular polyhedrons and a fractal bronchial tree. The fractal cartilaginous bronchial tree was assumed to be characterized by both Horton’s ratio (L_{j+1}/L_{j}=2^{λ}, where L_{j+1}, and L_{j} denote the mean lengths of branches at Horsfield’ order of *j* + 1 and *j*) and power laws between diameters and lengths of branches. Fluid dynamic parameters of fractal trees were assumed to be interrelated among powers and *λ*. A non-cartilaginous lobular bronchiole is adjoined to the edge of a lobular polyhedron, and is encircled by an inextensible basement membrane to reflect a reversible relationship of r_{l}L_{l} = constant(c), where r_{l} and L_{l} denote the diameter and the length of a lobular bronchiole, respectively. API at the level of the lobu-lar bronchiole was described by log(r_{l}) = -(1+*λ*)/(1+5*λ*)log(h_{l}/c), where r_{l} and h_{l} denote the diameter of the lobular bronchiole and the parenchymal parameter relating the size of the lobular polyhedron, respectively. If the distribution in sizes of the lobular polyhedrons was described by a Weibull’s probability density function characterized by the shape parameter m as well as the fractal parameter *λ* = 0.5, the diameter *R* of a cartilaginous bronchial branch was determined by log(*R*) = F - 3/7log(h/c), where F(m) denotes a
function of m, and *h* denotes the mean size of the polyhedrons in the lobe. As a conclusion, API can be described by a combination of both lobular API and corresponding adaptive changes in the degree of contraction of airway smooth muscles.

The topic of airway-parenchymal interdependence (API) is of great importance to those interested in identifying factors that influence airway patency. A carefully designed experiment has raised questions about the classical concept of API. This paper proposes a new mechanism of API. The pulmonary lobe is an aggregated body consisting of many Miller’s lobular polyhedrons and a fractal bronchial tree. The fractal cartilaginous bronchial tree was assumed to be characterized by both Horton’s ratio (L

Cite this paper

Min, K. , Hosoi, K. , Kinoshita, Y. , Hara, S. , Degami, H. , Takada, T. and Nakamura, T. (2012) Use of fractal geometry to propose a new mechanism of airway-parenchymal interdependence.*Open Journal of Molecular and Integrative Physiology*, **2**, 14-20. doi: 10.4236/ojmip.2012.21003.

Min, K. , Hosoi, K. , Kinoshita, Y. , Hara, S. , Degami, H. , Takada, T. and Nakamura, T. (2012) Use of fractal geometry to propose a new mechanism of airway-parenchymal interdependence.

References

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[1] Macklem, P.T. (1998) The mechanism of breathing. American Journal of Respiratory and Critical Care Medicine, 157, S88-S94.

[2] Sumikawa, H., Johkoh, T., Colby, T.V., Ichikado, K., Suga, M., Taniguchi, H., Kondoh, Y., Ogura, T., Arakawa, H., Fujimoto, K., Inoue, A., Mihara, N., Honda, O., Tomiyama, N., Nakamura, H. and Müller, N.L. (2008) Computed tomography findings in pathological usual interstitial pneumonia: Relationship to survival. American Journal of Respiratory and Critical Care Medicine, 177, 433-439. doi:10.1164/rccm.200611-1696OC

[3] Noble, P.B., Sharma, P.K., McFawn, P.K. and Mitchell, H.W. (2005) Airway narrowing in porcine bronchi with and without lung parenchyma. The European Respiratory Journal, 26, 804-811.

[4] Mandelbrot, B.B. (1983) The fractal geometry of nature. Freeman, New York. doi:10.1183/09031936.05.00065405

[5] Suwa, N. and Takahashi, T. (1971) Morphological and morphometrical analysis of circulation in hypertension and ischemic kidney. München.

[6] Kamiya, A. and Takahashi, T. (2007) Quantitative assessments of morphological and functional properties of biological trees based on their fractal nature. Journal of Applied Physiology, 102, 2315-2323. doi:10.1152/japplphysiol.00856.2006

[7] Weibel, E.R. (1991) Fractal geometry: A design principle for living oraganisms. American Journal of Physiology, 261, L361-L369.

[8] MacDonald, N. (1983) Trees and networks in biological models. Part III. Branching structures: Description, biophysics, and simulations. John Wiley & Sons Ltd., Chichester.

[9] Min, K., Kawai, M., Tamoto, A., Mozai, T. and Uchida, E. (1987) Geometrical analysis on the pulmonary lobular polyhedron (PLP) and consideration of the way of arrangement of PLPs in the lung. Japanese Journal of Thoracicic Diseases , 25, 722-730.

[10] Weibull, W. (1951) A statistical distribution function of wide applicability. Journal of Applied Mechanics, 18, 293-297.

[11] NIST/SEMATECH (2010) Weibull’s distribution, e-Handbook of statistical methods. http://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm

[12] Horsfield, K., Dart, G., Olson, D.E., Filley, G.F. and Cumming, G. (1971) Models of the human bronchial tree. Journal of Applied Physiology, 31, 207-217.

[13] Shreve, R.L. (1966) Statistical law of stream numbers. Journal of Geology, 74, 17-37. doi:10.1086/627137

[14] Horton, R.E. (1945) Erosional development of streams and their drainage basins: Hydrophysical approach to quantitative morphology. Bulletin Geological Society of America, 56, 275-370. doi:10.1130/0016-7606(1945)56[275:EDOSAT]2.0.CO;2

[15] Horsfield, K. and Cumming, G. (1976) Morphology of the bronchial tree in the dog. Respiration Physiology, 26, 173-181. doi:10.1016/0034-5687(76)90095-5

[16] Raabe, O.G., Yeh, H.C., Schum, G.M. and Phalen, R.F. (1976) Tracheobronchial geometry: Human, dog, rat, hamster. Lovelace Foundation, Albuquerque.

[17] Horsfield, K. and Thurlbeck, A. (1981) Relation between diameter and flow in branches of bronchial the tree. Bulletin of Mathematical Biology, 43, 681-691.

[18] Seow, C.Y. (2005) Are you pulling my airway? European Respiratory Journal, 26, 759-761. doi:10.1183/09031936.05.00099905

[19] Saam, B.T. (1996) Magnetic resonance imaging with laser-polarized noble gases. Nature Medicine, 2, 358-359.

[20] De Lange, E.E., Mugler III, J.P., Brookeman, J.R., Knight-Scott, J., Truwit, J.D., Teates, C.D., Daniel, T.M., Bogorad, P.L. and Cates, G.D. (1999) Lung air spaces: MR imaging evaluation with hyperpolarized 3He gas. Radiology, 210, 851-857.

[21] Coxson, H.O., Mayo, J., Lam, S., Santyr, G., Parraga, G. and Sin, D.D. (2009) New and current clinical imaging techniques to study chronic obstructive pulmonary disease. American Journal of Respiratory Critical Care Medicine, 180, 588-597. doi:10.1164/rccm.200901-0159PP

[22] Brouns, I., Van Genechten, J., Hayashi, H., Gajda, M., Gomi, T., Burnstock, G., Timmermans, J.P. and Adriaensen, D. (2002) Dual sensory innervation of pulmonary neuroepithelial bodies. American Journal of Respiratory Cell Molecular, 28, 275-28. doi:10.1165/rcmb.2002-0117OC