  During the progression of pulmonary fibrosis, initially isolated regions of
high stiffness form and grow in the lung tissue due to collagen deposition by
fibroblast cells. We have previously shown that ongoing collagen deposition may
not lead to significant increases in the bulk modulus of the lung until these
local remodeled regions have become sufficiently numerous and extensive to
percolate 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 a
two-dimensional elastic network model of lung tissue. Random collagen
deposition at a single site is modeled by increasing the elastic constant of
the spring at that site by a factor of 100.
  By contrast, correlated collagen deposition is represented by stiffening the
springs encountered along a random walk starting from some initial spring, the
rationale being that excess collagen deposition is more likely in the vicinity
of an already stiff region. A combination of random and correlated deposition
is modeled by performing random walks of length $N$ from randomly selected
initial 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 the
fraction of stiff springs, $c$, can be described by a strikingly simple set of
empirical equations.
  Our results suggest that the spatial correlation of collagen deposition in
the fibrotic lung has a strong effect on the rate of lung function decline and
on the mechanical environment in which the cells responsible for remodeling
find themselves.
