A microfluidic chamber-based approach to map the shear moduli of vascular cells and other soft materials.

There is growing interest in quantifying vascular cell and tissue stiffness. Most measurement approaches, however, are incapable of assessing stiffness in the presence of physiological flows. We developed a microfluidic approach which allows measurement of shear modulus (G) during flow. The design included a chamber with glass windows allowing imaging with upright or inverted microscopes. Flow was controlled gravitationally to push culture media through the chamber. Fluorescent beads were conjugated to the sample surface and imaged before and during flow. Bead displacements were calculated from images and G was computed as the ratio of imposed shear stress to measured shear strain. Fluid-structure simulations showed that shear stress on the surface did not depend on sample stiffness. Our approach was verified by measuring the moduli of polyacrylamide gels of known stiffness. In human pulmonary microvascular endothelial cells, G was 20.4 ± 12 Pa and decreased by 20% and 22% with increasing shear stress and inhibition of non-muscle myosin II motors, respectively. The G showed a larger intra- than inter-cellular variability and it was mostly determined by the cytosol. Our shear modulus microscopy can thus map the spatial distribution of G of soft materials including gels, cells and tissues while allowing the visualization of microscopic structures such as the cytoskeleleton.

Homeostatic maintenance via degradation and repair of elastic fibers under tension.

Cellular maintenance of the extracellular matrix requires an effective regulation that balances enzymatic degradation with the repair of collagen fibrils and fibers. Here, we investigate the long-term maintenance of elastic fibers under tension combined with diffusion of general degradative and regenerative particles associated with digestion and repair processes. Computational results show that homeostatic fiber stiffness can be achieved by assuming that cells periodically probe fiber stiffness to adjust the production and release of degradative and regenerative particles. However, this mechanism is unable to maintain a homogeneous fiber. To account for axial homogeneity, we introduce a robust control mechanism that is locally governed by how the binding affinity of particles is modulated by mechanical forces applied to the ends of the fiber. This model predicts diameter variations along the fiber that are in agreement with the axial distribution of collagen fibril diameters obtained from scanning electron microscopic images of normal rat thoracic aorta. The model predictions match the experiments only when the applied force on the fiber is in the range where the variance of local stiffness along the fiber takes a minimum value. Our model thus predicts that the biophysical properties of the fibers play an important role in the long-term regulatory maintenance of these fibers.

Entropy Production and the Pressure-Volume Curve of the Lung.

We investigate analytically the production of entropy during a breathing cycle in healthy and diseased lungs. First, we calculate entropy production in healthy lungs by applying the laws of thermodynamics to the well-known transpulmonary pressure-volume (P-V) curves of the lung under the assumption that lung tissue behaves as an entropic spring similar to rubber. The bulk modulus, B, of the lung is also derived from these calculations. Second, we extend this approach to elastic recoil disorders of the lung such as occur in pulmonary fibrosis and emphysema. These diseases are characterized by particular alterations in the P-V relationship. For example, in fibrotic lungs B increases monotonically with disease progression, while in emphysema the opposite occurs. These diseases can thus be mimicked simply by making appropriate adjustments to the parameters of the P-V curve. Using Clausius's formalism, we show that entropy production, ΔS, is related to the hysteresis area, ΔA, enclosed by the P-V curve during a breathing cycle, namely, ΔS=ΔA∕T, where T is the body temperature. Although ΔA is highly dependent on the disease, such formula applies to healthy as well as diseased lungs, regardless of the disease stage. Finally, we use an ansatz to predict analytically the entropy produced by the fibrotic and emphysematous lungs.

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 + cß ln(NaI)], where ß = 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.

Explosive electric breakdown due to conducting-particle deposition on an insulating substrate.

We introduce a theoretical model to investigate the electric breakdown of a substrate on which highly conducting particles are adsorbed and desorbed with a probability that depends on the local electric field. We find that, by tuning the relative strength q of this dependence, the breakdown can change from continuous to explosive. Precisely, in the limit in which the adsorption probability is the same for any finite voltage drop, we can map our model exactly onto the q-state Potts model and thus the transition to a jump occurs at q = 4. In another limit, where the adsorption probability becomes independent of the local field strength, the traditional bond percolation model is recovered. Our model is thus an example of a possible experimental realization exhibiting a truly discontinuous percolation transition.

Enhanced flow in small-world networks.

The proper addition of shortcuts to a regular substrate can lead to the formation of a complex network with a highly efficient structure for navigation [J. M. Kleinberg, Nature 406, 845 (2000)]. Here we show that enhanced flow properties can also be observed in these small-world topologies. Precisely, our model is a network built from an underlying regular lattice over which long-range connections are randomly added according to the probability, Pij ∼ r−α ij , where rij is the Manhattan distance between nodes i and j, and the exponent α is a controlling parameter. The mean two-point global conductance of the system is computed by considering that each link has a local conductance given by gij ∝ r−C ij , where C determines the extent of the geographical limitations (costs) on the long-range connections. Our results show that the best flow conditions are obtained for C = 0 with α = 0, while for C ≫ 1 the overall conductance always increases with α. For C ≈ 1, α = d becomes the optimal exponent, where d is the topological dimension of the substrate. Interestingly, this exponent is identical to the one obtained for optimal navigation in small-world networks using decentralized algorithms.

Oil displacement through a porous medium with a temperature gradient.

We investigate the effect of a temperature gradient on oil recovery in a two-dimensional pore-network model. The oil viscosity depends on temperature as µ(o) [Please see text] e(B/T), where B is a physicochemical parameter, depending on the type of oil, and T is the temperature. A temperature gradient is applied across the medium in the flow direction. Initially, the porous medium is saturated with oil, and then another fluid is injected. We have considered two cases representing different injection strategies. In the first case, the invading fluid viscosity is constant (finite viscosity ratio), while in the second one, the invading fluid is inviscid (infinite viscosity ratio). Our results show that for the case of finite viscosity ratio, recovery increases with ΔT independent of strength or sign of the gradient. For an infinite viscosity ratio, a positive temperature gradient is necessary to enhance recovery. Moreover, we show that for ΔT>0, the percentage of oil recovery generally decreases (increases) with B for a finite (infinite) viscosity ratio. Finally, we also extend our results for infinite viscosity ratio to a three-dimensional porous media geometry.

Transport on exploding percolation clusters.

We propose a simple generalization of the explosive percolation process [Achlioptas et al., Science 323, 1453 (2009)], and investigate its structural and transport properties. In this model, at each step, a set of q unoccupied bonds is randomly chosen. Each of these bonds is then associated with a weight given by the product of the cluster sizes that they would potentially connect, and only that bond among the q set which has the smallest weight becomes occupied. Our results indicate that, at criticality, all finite-size scaling exponents for the spanning cluster, the conducting backbone, the cutting bonds, and the global conductance of the system, change continuously and significantly with q. Surprisingly, we also observe that systems with intermediate values of q display the worst conductive performance. This is explained by the strong inhibition of loops in the spanning cluster, resulting in a substantially smaller associated conducting backbone.