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During the progression of pulmonary fibrosis, initially isolated regions ofhigh stiffness form and grow in the lung tissue due to collagen deposition byfibroblast cells. We have previously shown that ongoing collagen deposition maynot lead to significant increases in the bulk modulus of the lung until theselocal remodeled regions have become sufficiently numerous and extensive topercolate in a continuous path across the entire tissue [Bates {\it et al.}2007 {\it Am. J. Respir. Crit. Care Med.} {\bf 176} 617]. In the present study,we investigate whether spatial correlations influence the bulk modulus in atwo-dimensional elastic network model of lung tissue. Random collagendeposition at a single site is modeled by increasing the elastic constant ofthe spring at that site by a factor of 100.  By contrast, correlated collagen deposition is represented by stiffening thesprings encountered along a random walk starting from some initial spring, therationale being that excess collagen deposition is more likely in the vicinityof an already stiff region. A combination of random and correlated depositionis modeled by performing random walks of length $N$ from randomly selectedinitial sites, the balance between the two processes being determined by $N$.We found that the dependence of bulk modulus, $B(N,c)$, on both $N$ and thefraction of stiff springs, $c$, can be described by a strikingly simple set ofempirical equations.  Our results suggest that the spatial correlation of collagen deposition inthe fibrotic lung has a strong effect on the rate of lung function decline andon the mechanical environment in which the cells responsible for remodelingfind themselves.

A network model of correlated growth of tissue stiffening in pulmonary fibrosis