Abstract
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 et al 2007 Am. J. Respir. Crit. Care
Med. 176 617). This model, however, did not include the possibility of
spatially correlated deposition of collagen. 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. For c <
0.3, B(N, c) exhibits exponential growth from its initial value
according to B(N, c) approximate to B-0 exp (2c) 1 + c(ss) ln (N-I(a)), where ss = 0.994 +/- 0.024 and a(I) = 0.54 +/- 0.026. For intermediate concentrations of stiffening, 0.3 <= c <= 0.8, another exponential rule describes the bulk modulus as B(N, c) = 4B(0) exp
a(II)(c - c(c)), where a(II) and c(c) are parameters that depend on
N. For c > 0.8, B(N, c) is linear in c and independent of N, such that B(N, c) = 100 B-0 - 100a(III) (1 - c) B-0, where a(III) = 2.857. For
small concentrations, the physiologically most relevant regime, the
forces in the network springs are distributed according to a power law. When c = 0.3, the exponent of this power law increases from -4.5, when N = 1, and saturates to about -2, as N increases above 40. These 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.
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