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

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<sub>j+1</sub>/L<sub>j</sub>=2<sup>λ</sup>, where L<sub>j+1</sub>, and L<sub>j</sub> denote the mean lengths of branches at Horsfield’ order of <i>j</i> + 1 and <i>j</i>) and power laws between diameters and lengths of branches. Fluid dynamic parameters of fractal trees were assumed to be interrelated among powers and <i>λ</i>. 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<sub>l</sub>L<sub>l</sub> = constant(c), where r<sub>l</sub> and L<sub>l</sub> 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<sub>l</sub>) = -(1+<i>λ</i>)/(1+5<i>λ</i>)log(h<sub>l</sub>/c), where r<sub>l</sub> and h<sub>l</sub> 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 <i>λ</i> = 0.5, the diameter <i>R</i> of a cartilaginous bronchial branch was determined by log(<i>R</i>) = F<m> - 3/7log(h/c), where F(m) denotes a function of m, and <i>h</i> 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

A new geometrical and mechanical relation in the respiratory system with airflow limitation—From the perspective of analytical respiratory mechanics

“…From the viewpoint of the secondary pulmonary lobule (SPL), the terminal branch of the bronchial tree should be the lobular bronchiole because each SPL is supplied by a single lobular bronchiole of about 1 mm in diameter [2,3]. The lung is composed of a large number (N) of SPLs by integration of the single fractal bronchial tree [4]. Thus, the volume variable (V [L]) of a respiratory system is a function of lobular volume variables ;…”

“…A pulmonary lobe is composed of a large number of SPLs, which a single bronchial tree integrates to the whole lung as a complicated system. Recent rapid progress in new technologies concerning pulmonary functional images has displayed regional and temporal functional images of the lung in vivo, which should be recognized as the complicated system of a large number of SPLs [2][3][4]. Thus, it is now demanded that physiologists reconstruct analytical respiratory mechanics based on SPLs in place of the classical vectorial, which will help us to accurately understand new pulmonary functional images produced by new technologies.…”

Self-Similarities of Pulmonary Arterial Tree and a New Integrated Model of Pulmonary Circulation with the Name of Fractal Phasic Perfusion (FPP) Model

“…By applying the rules of integration to PAT with BT, we assumed a power relationship between the diameters at a bifurcation ( 1 r , 2 r , and 3 r ) in Figure 2(a), as follows: 1 2 3 n n n r r r = + . Biologists with an interest in the self-similarities of biological branching structures including PAT have found Horton's branching law to be applicable to PAT [5] is related to those of daughter branches (r 2 and r 3 ) by a certain power (n) at every branching point; (b) In Horsfield's ordering system, the first edge starts from the terminal, and when the edges of order j and order k come together at a vertex, the third edge is assigned to one order greater than the greater of j and k, or to j + 1 if j = k; (c) There is the selfsimilarity of branching known as a statistical law of Horton among branches with Horsfield ordering. Horsfield's version of Horton's law implies a geometrical self-similarity (modified from ref.…”

A Stochastic Optimal Control Theory to Model Spontaneous Breathing

“…The alveolar pressure is transformed into airway pressure by the pulmonary lobule, and goes into the environment by producing airflows through the fractal bronchial tree ( Figure 1). It is important to note in Figure 1 that there are two origins of fluctuations in this process: in the respiratory rhythm generator (the neural center of respiration) and in the fractal airway modulator (the phasic asynchronous contractions of airway smooth muscles in the lobular bronchioles) [6]. Then, based on that bronchial flow F(t) is composed of N-number of phasic lobular flow (q), a tidal volume V T is defined as following, …”

“…By using the lobule and fractal bronchial tree model (LBT model) [6], the state variable of spontaneous noisy breathing was expressed by the polar coordinate x, which is composed of two dimensional variables τ and θ. The biological variability of θ may be determined through the variability in the amplitude of tidal volume V T based on (3.1.3).…”