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

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) ≈ B0exp(2c)[1 + cβ ln(NaI)], where β = 0.994 ± 0.024 and aI = 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) = 4B0exp[aII (c - cc )], where aII and cc 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) = 100B0 - 100aIII (1 - c)B0, where aIII = 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.