Anisotropic Diffusive Transport in Annulus Fibrosus: Experimental Determination of the Diffusion Tensor by FRAP Technique

Abstract

The annulus fibrosus (AF) of the intervertebral disc (IVD) exhibits a fiber-organized structure which is responsible for anisotropic and inhomogeneous mechanical and transport properties. Due to its particular morphology, nutrient transport within AF is regulated by complex transport kinetics. This work investigates the diffusive transport of a small solute in the posterior and anterior regions of AF since diffusion is the major transport mechanism for low molecular weight nutrients (e.g., oxygen and glucose) in IVD.

Diffusion coefficient (D) of fluorescein (332 Da) in bovine coccygeal AF was measured in the three major (axial, circumferential, and radial) directions of the IVD by means of fluorescence recovery after photobleaching (FRAP) technique. It was found that the diffusion coefficient was anisotropic and inhomogeneous. In both anterior and posterior regions, the diffusion coefficient in the radial direction was found to be the lowest. Circumferential and axial diffusion coefficients were not significantly different in both posterior and anterior regions and their values were about 130% and 150% the value of the radial diffusion coefficient, respectively. The values of diffusion coefficients in the anterior region were in general higher than those of corresponding diffusion coefficients in the posterior region.

This study represents the first quantitative analysis of anisotropic diffusion transport in AF by means of FRAP technique and provides additional knowledge on understanding the pathways of nutritional supply into IVD.

Appendix

When a tissue sample is bleached over its whole thickness in a FRAP test, the diffusive transport of fluorescent probes occurs within the focal plane of the microscope objective and diffusion is 2D phenomenon. This condition is practically achievable when the thickness of the sample is comparable to the optical slice of the microscope objective (e.g., in membranes or polymeric films). In a FRAP test with CLSM on bulk samples, the bleached region does not extend over the entire thickness of the specimen. Therefore, the presence of a gradient of concentration of fluorescent solute in the direction orthogonal to the focal plane (z-direction) causes fluorescence recovery to be a 3D diffusion phenomenon.9,10,37 If 2D SFA is adopted in the analysis of FRAP test data, the contribution of the diffusive flux in the z-direction is neglected. Consequently, the calculated diffusion coefficient (D) is overestimated. The error in the estimation of D depends on two factors: (1) the ratio of the bleached size (d), in the focal plane, to the thickness (L) of the bleached volume (Fig. 7); and (2) the ratio of the diffusion coefficient in the z-direction to the averaged diffusion coefficient in the focal plane (D⊥/D).

Figure 7

Schematic of the computational domain: (a) the three-dimensional sample is confined between two glass slides (top and bottom) with a cylinder representing bleached volume, obtained by multi-layer bleaching; (b) Cross-sectional view of the sample and the applied boundary conditions

In order to quantify the error in the estimation of the diffusion coefficient by using the 2D SFA approach presented in this work, a numerical analysis was performed to simulate 3D diffusive transport of fluorescent molecules within a bulk sample using finite element method (COMSOL® 3.2, COMSOL Inc., Burlington, MA).

Figure 7a represents a schematic of the computational domain. A cubic sample of 460 μm side is placed between two glass slides. The initial fluorescent solute concentration within the cubic domain was assumed to be uniform with exception for a cylindrical volume, representing the bleached region, in which fluorescent probe concentration is zero. The diameter d of the cylinder was set equal to 28.75 μm in order to simulate the experimental conditions, see Materials and Methods. The height of the cylinder L varied according to the number of the planes being bleached in the simulation.

Since the domain is confined between to glass slides, on the bottom and top surfaces of the cube, an impermeable boundary condition (diffusive flux J = 0) was imposed. Besides, since the cube is large enough with respect to the diameter of the cylinder, the solute concentrations on the lateral surfaces of the cube were assumed to be constant (c = c*), see Fig. 7b.

Numerical simulations were performed using ∼35,000 quadratic Lagrange tetrahedral elements. The degree of anisotropy in diffusion was simulated with the ratio of diffusion coefficient in the z-direction to that in the x–y plane (D⊥/D) varying from one (isotropy) to two. From each simulation a time series of 200 frames, representing the images on the focal plane of the microscope objective (7 μm from the bottom of the sample, see Materials and Methods), were extracted and analyzed by custom-made SFA software (see Materials and Methods).

Figure 8 shows the relative error in the estimation of D _av by Eq. (7) as a function of L/d for different degrees of anisotropy. The relative error increases with the ratio D⊥/D but decreases with L/d. By increasing the height of the bleached volume (cylinder), the relative error decreases (less than 7% for D⊥/D = 2, when L/d = 2) and, theoretically, it could be reduced to zero for sufficiently large values of L/d (or small values of D⊥/D). This indicates that the diffusive flux in the z-direction could be negligible under certain conditions.

Figure 8

Relative error in the estimation of D _av as a function of L/d for different degrees of anisotropy in diffusion (D⊥/D)

In practice only a few layers can be sequentially bleached within the bulk sample before fluorescence recovery occurs in the earlier bleached planes. According to the testing protocol used in this work, four layers were sequentially bleached, generating a cylindrical bleach region of 28 μm diameter and 47 μm height (see Materials and Methods). By measuring the intensity of the fluorescence emission within the bleached volume (cylinder), three regions were identified (measured from the bottom glass slide): (1) from 0 to 17 μm, the fluorescence was completely depleted; (2) from 17 to 27 μm, the fluorescence linearly increased (i.e., recovered) to 50% of the intensity of the surrounding unbleached tissue (I _o ); (3) from 27 to 47 μm, the fluorescence intensity was approximately 50% of the value of I _o (data not shown). Since the light intensity is proportional to the concentration of fluorescent molecules,5 the above information was used as an initial condition for probe concentration in the numerical simulation on mass transport of fluorescent solute, in order to evaluate the relative error committed in the determination of D _av using 2D SFA. Our simulation showed that the highest relative error (in the case D⊥/D = 2) is approximately 18%.