A mathematical model of protease-antiprotease homeostasis failure in chronic obstructive pulmonary disease (COPD)

Abstract

Chronic obstructive pulmonary disease (COPD), the fourth leading cause of death worldwide, has a puzzling etiology. Although it is a smoking-associated disease, only a minority of smokers develop it. Moreover, the disease continues to progress in COPD patients, even after smoking ceases. This article proposes a mathematical model of COPD that offers one possible explanation for both observations. Building on a conceptual model of COPD causation as resulting from protease-antiprotease imbalance in the lung, leading to ongoing proteolysis (digestion) of lung tissue by excess proteases, we formulate a system of seven ordinary differential equations (ODEs) with 18 parameters to describe the network of interacting homeostatic processes regulating the levels of key proteases (macrophage elastase (MMP-12) and neutrophil elastase (NE)) and antiproteases (alpha-1-antitrypsin and tissue inhibitor of metalloproteinase-1). We show that this system can be simplified to a single quadratic equation with only two parameters to predict the equilibrium behavior of the entire network. The model predicts two possible equilibrium behaviors: a unique stable "normal" (healthy) equilibrium or a "COPD" equilibrium with elevated levels of MMP-12 and NE (and of lung macrophages and neutrophils) and reduced levels of antiproteases. The COPD equilibrium is induced in the model only if cigarette smoking increases the average production of MMP-12 per alveolar macrophage above a certain threshold. Following smoking cessation, the predicted COPD equilibrium levels of MMP-12 and other disease markers decline, but do not return to their original (presmoking) levels. These and other predictions of the model are consistent with limited available human data.