title: A network model of correlated growth of tissue stiffening in pulmonary fibrosis.
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 [Bateset al. 2007Am. J. Respir. Crit. Care Med.176617]. 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 lengthNfrom randomly selected initial sites, the balance between the two processes being determined byN. We found that the dependence of bulk modulus,B(N, c), on bothNand the fraction of stiff springs,c, can be described by a strikingly simple set of empirical equations. Forc< 0.3,B(N, c) exhibits exponential growth from its initial value according toB(N, c) ≈B0exp(2c)[1 +cβln(NaI)], whereβ= 0.994 ± 0.024 andaI= 0.54 ± 0.026. For intermediate concentrations of stiffening, 0.3 ≤c≤ 0.8, another exponential rule describes the bulk modulus asB(N, c) = 4B0exp[aII(c-cc)], whereaIIandccare parameters that depend onN. Forc> 0.8,B(N, c) is linear incand independent ofN, such thatB(N, c) = 100B0- 100aIII(1 -c)B0, whereaIII= 2.857. For small concentrations, the physiologically most relevant regime, the forces in the network springs are distributed according to a power law. Whenc= 0.3, the exponent of this power law increases from -4.5, whenN= 1, and saturates to about -2, asNincreases 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.
authors.affiliation: Department of Biomedical Engineering, Boston University, Boston, MA 02215.
authors.forename: Cláudio L N
authors.initials: CL
authors.lastname: Oliveira
authors.affiliation: Department of Medicine, University of Vermont, Burlington, VT 05405.
authors.forename: Jason H T
authors.initials: JH
authors.lastname: Bates
authors.affiliation: Department of Biomedical Engineering, Boston University, Boston, MA 02215.
authors.forename: Béla
authors.initials: B
authors.lastname: Suki