Use of a mathematical model of autopoietic maintenance of pulmonary alveolar integrity to show a possible relationship between epithelial-mesenchymal transition and the genesis of emphysema and fibrosis

Rationale: There is accumulating evidence that a group of stem/progenitor cells (SPCs) maintain alveolar epithelial integrity. Pulmonary emphysema is characterized by the histological finding of the loss of alveolar epithelial integrity along with corresponding bronchiolar fibrosis. Objectives: Based on the concept of autopoiesis (the capacity to produce oneself), we proposed a mathematical model in the maintenance of alveolar epithelial integrity as related to the genesis of pulmonary emphysema and fibrosis. Methods: A tessellation automaton model was used to describe the autopoietic dynamics of the bronchiolo-alveolar epithelial surface. The alveolar septal volume en-closed by the epithelial surface is a distributed system of discrete elements, which move by random walk in the manner of Brownian motion. Assuming that the numbers of components and events in the automaton are large, an approximate theoretical treatment in terms of differential equations is possible, allowing a set of partial differential equations to be produced. Results: 1) Assuming the loss of progenitor cells through the epithelial-mesenchymal transition (EMT), a sharp bifurcation between two qualitatively distinct regions of the phase space (one that is repaired completely, and another that has disappeared entirely) clearly appeared. 2) Thus, from the system of discrete and spatial partial differential equations, we obtained a system of ordinary differential equations in equilibrium conditions that defined a close relationship between the degree of emphysema, the density of alveolar septal fibroblasts, and the mean concentration of SPCs. Conclusions: A mathematical model of the autopoietic maintenance of the alveolar epithelial surface suggested a close relationship between alveolar emphysema and fibrosis and EMT in lungs affected by chronic obstructive pulmonary disease.

Determinants of airway hyperresponsiveness—Balance of tonic and phasic contractility of airway smooth muscles of lobular bronchioles

Airway hyperresponsiveness (AHR) is a characteristic feature of asthma, and generally correlates with severity of asthma. Understanding the protection mechanism against excessive airway narrowing and how it breaks down is fundamental to solving the problem of asthma. In this paper we have proposed a stochastic modeling the airway smooth muscle bundle for reproducing AHR such as an increased sensitivity of the airways to an inhaled constrictor agonist, a steeper slope of the dose-response curve, and a greater maximal response to agonist. A large number N of contractile muscle cells was assumed to repeat themselves in between contraction and relaxation asynchronously. Dynamic equilibrium of statistic physics was applied to the system of ASM bundle. Thus, the relation of dose to response of a piece of ASM bundle was described by Φ=tanh(βH) , where β was Boltzman factor and H represented energy of contraction induced by constrictor agents. Each of adjacent pair contractile cells was assumed to have Ising-type of antimagnetic interactions of preference energy J (for the condition of contraction-relaxation) between them. A motion equation for a piece of ASM bundle was described by Φ=N(H-zJΦ , which explained existence of combined tonic and phasic contractions. Based on observations of Venegas et al. [4], airway responsiveness was assumed to be assessable by total volume of the ventilation defects (TVD) of 13NN PET-CT images. Interactions via propagation of Ca ion waves between ASM bundles would cause percolation probability by PΦ=(1+tanh(βH))2/4 along the tree, then the relation of dose βH to TVD was described by TVD=PΦ[1-(1-PΦ)3/PΦ3]-TVD0. TVD0 represented the protection mechanism against excessive airway narrowing, which was determined by the ratio of amplitudes between tonic and phasic contractions, thus the balance of amplitudes between tonic and phasic contractions of peripheral lobular smooth muscles would be the determinant of AHR.

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

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 (Lj+1/Lj=2λ, where Lj+1, and Lj 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 rlLl = constant(c), where rl and Ll denote the diameter and the length of a lobular bronchiole, respectively. API at the level of the lobu-lar bronchiole was described by log(rl) = -(1+λ)/(1+5λ)log(hl/c), where rl and hl 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.