Diffusion anisotropy in collagen gels and tumors: the effect of fiber network orientation

Abstract

The interstitial matrix is comprised of cross-linked collagen fibers, generally arranged in nonisotropic orientations. Spatial alignment of matrix components within the tissue can affect diffusion patterns of drugs. In this study, we developed a methodology for the calculation of diffusion coefficients of macromolecules and nanoparticles in collagenous tissues. The tissues are modeled as three-dimensional, stochastic, fiber networks with varying degrees of alignment. We employed a random walk approach to simulate diffusion and a Stokesian dynamics method to account for hydrodynamic hindrance. We performed our analysis for four different structures ranging from nearly isotropic to perfectly aligned. We showed that the overall diffusion coefficient is not affected by the orientation of the network. However, structural anisotropy results in diffusion anisotropy, which becomes more significant with increase in the degree of alignment, the size of the diffusing particle, and the fiber volume fraction. To test our model predictions we performed diffusion measurements in reconstituted collagen gels and tumor xenografts. We measured fiber alignment and diffusion with second harmonic generation and multiphoton fluorescent recovery after photobleaching techniques, respectively. The results showed for the first time in tumors that the structure and orientation of collagen fibers in the extracellular space leads to diffusion anisotropy.

Figure 1

(a) Formulation of the mathematical model. Particles diffuse inside the fibrous medium (Random Walk Domain). A second computational domain is constructed for the calculation of hydrodynamic interactions (Stokesian Dynamics Domain). At each time step of the random walk, the position of the particle is mapped to the Stokesian dynamics domain and its diffusion coefficient is calculated by the solution of a Stokesian dynamics problem. The diffusion coefficient is returned to the random walk domain and the particle moves to a new, randomly-chosen position. Periodic boundary conditions are applied, and in the case of collision with a fiber, the displacement is rejected. (b) Typical fiber structures employed in this study.

Figure 2

Overall diffusion coefficients, D, as a function of the ratio of the particle radius over the fiber radius for the fiber structures employed in the study and for three fiber volume fractions 0.01, 0.03, and 0.05. The diffusion values of the fiber networks are the average of four realizations. Standard deviations were too small to be distinguished in the plot and were omitted. D₀ is the diffusion coefficient in solution.

Figure 3

Diffusion anisotropy as a function of the ratio of the particle radius over the fiber radius for the fiber structures employed in the study and for two fiber volume fractions: (a) 0.01, and (b) 0.05. D_parallel is the diffusion coefficient parallel to the preferred fiber direction (D_zz) and D_transverse is the diffusion coefficient transverse to the preferred fiber direction (average value of D_xx and D_zz). The value of the fiber networks is the average of four realizations. Standard deviations were too small to be distinguished in the plot and were omitted.

Figure 4

Diffusion anisotropy as a function of the fiber volume fraction for highly aligned networks and for particles of three different sizes. The ratio λ of the particle radius over the fiber radius is 0.05, 0.2, and 0.5.

Figure 5

Effect of steric and hydrodynamic interactions on the overall diffusion coefficient, D, as a function of the fiber volume fraction. We consider 1), only steric interactions; 2), steric interactions plus long-range hydrodynamic interactions; and 3), steric and hydrodynamic interaction plus the correction for short-range (lubrication) effects. The ratio λ of the particle radius over the fiber radius is 0.2 and 3.0.

Figure 6

Effect of steric and hydrodynamic interactions on the diffusion anisotropy of moderately, highly, and perfectly aligned networks.

Figure 7

Effects of area fraction and fiber alignment on overall diffusion measured in collagen gels. The top panel shows an SHG image of the collagen gel overlaid with the points at which overall diffusion was measured. It also shows the regions that were sampled for fiber alignment studies. The histograms on the middle panel show the relative alignment of fibers in Regions A and B. The bottom-right panel shows the strong correlation between fiber area fraction and overall diffusion. The bottom-left panel shows that there is no significant difference between overall diffusion measurements obtained from the highly aligned Region A compared to the unaligned Region B.

Figure 8

In vivo evidence of diffusion anisotropy in tumors. The top-left panel shows an SHG image of the tumor interstitium. Collagen fibers are visible in gray. The red dashed lines show the planes along which line FRAP measurements were taken. The bottom-left panel shows a histogram of the relative alignment of collagen fibers in the network. The histogram shows that most fibers are aligned at 100 ° to the horizontal. The top-right panel shows a recovery curve obtained from line FRAP measurements. The bottom-right panel shows the relative diffusivities obtained from line FRAP. Diffusion coefficients at 100 ° (parallel to the collagen fibers) are 1.5 times larger than those performed at 190 ° (orthogonal to the collagen fiber alignment).