Introduction

The temporomandibular joint (TMJ) is a synovial, bilateral joint with unique morphology and function. The TMJ disc, a fibrocartilaginous tissue, is a major component of jaw function in view of its ability to distribute stress and lubricate the joint.23 , 24 Disc derangement (e.g., dislocation of the disc) is a common clinical finding in patients with TMJ disorders (affecting more than 10 million Americans).31 It has been suggested that degenerative processes predispose the disc to displacement and result in significant changes in disc morphology, biochemistry, function, and material properties.31 , 32

The TMJ disc has a distinctive extracellular matrix (ECM) composition when compared to hyaline cartilage and other fibrocartilaginous tissues [e.g., the intervertebral disc (IVD)]. The TMJ disc is composed primarily of water with a significant amount of collagen (mainly type I) and a very small amount of proteoglycan.4 , 11 , 21 , 22 , 29 The TMJ disc is a large avascular structure with human discs being wider mediolaterally than anteroposteriorly (approximately 19 × 13 mm).28 The nutrients required by disc cells for maintaining healthy matrix are supplied by synovial fluid at the margins of the disc, as well as from nearby blood vessels at the posterior bilaminar zone connection.7 Diffusive transport of nutrient solutes through the ECM plays a key role in maintaining the normal function of TMJ discs while the dysfunctional transport of those solutes is believed to contribute to the degeneration of the disc over time. Solute diffusivities in articular cartilage and IVD have been measured using a variety of techniques, including magnetic resonance imaging (MRI),10 , 12 tracking the net movement of radiolabeled solutes,33 , 34 and fluorescence27 or radiotracer desorption.17 , 18 However, to our knowledge, the solute diffusivities in the TMJ disc have not so far been determined. Moreover, differences in biochemical composition and structure distinguish the disc into three regions: the anterior band, intermediate zone, and posterior band.28 Collagen fibers in the anterior and posterior bands run primarily in the mediolateral direction, while in the intermediate zone, these fibers run predominately in an anteroposterior direction.20 Studies have shown that the mechanical properties of the TMJ disc are region-dependent and anisotropic.7 , 15 Therefore, it is reasonable to hypothesize that the solute diffusion in the TMJ disc is inhomogeneous and anisotropic. The objective of this study was to determine the region-dependent anisotropic diffusion tensor in TMJ discs using a new fluorescence recovery after photobleaching (FRAP) technique coupled with spatial Fourier analysis (SFA).

The FRAP technique is a versatile and widely used tool for the determination of local diffusion properties within solutions, tissues, cells, and membranes.19 Owing to the high spatial resolution, FRAP offers the possibility to microscopically examine a specific region of a sample. For instance, the site-specific (i.e., inhomogeneous) solute diffusion in articular cartilage has been previously studied using FRAP.9 , 16 The basic principle of FRAP is based on photobleaching the fluorescence of molecular probes and analyzing the recovery of the bleached area.1 , 13 , 25 To date, most FRAP analyses are based on the method proposed by Axelrod et al. for isotropic diffusion.1 Another approach for FRAP analysis is based on the SFA of the recovery images and has the advantage of potential detection of anisotropic diffusion.3 , 14 , 26 , 30 , 35 , 38 , 39 Tsay and Jacobson, using SFA of FRAP images, developed a method for the determination of a two-dimensional (2D) diffusion tensor along the fixed coordinate system.38 Most recently, Travascio et al. reported a method for calculating an anisotropic diffusion tensor based on two independent analyses of FRAP images: the Fourier transform (FT) and the Karhunen–Loeve transform (KLT).37 The KLT analysis of FRAP images could determine the principal direction of the diffusion tensor. By combining the FT and KLT analyses, all the components of the anisotropic diffusion tensor can be solved for in a single FRAP experiment. This method has been used to characterize anisotropic diffusion in knee meniscus and intervertebral disc tissues.36 , 37 In this article, we propose a new method solely based on the SFA to completely determine the anisotropic diffusion tensor, avoiding all of the limitations imposed by KLT analysis.

In this article, we first discuss the theoretical basis of this new analysis. Next, we apply this analysis to computer-simulated anisotropic FRAP experiments to validate the method, as well as make a comparison to the method proposed by Travascio et al. in terms of accuracy and robustness. Finally, we also apply this analysis to characterize anisotropic solute diffusion in different regions of the TMJ disc using the new FRAP technique.

Theory

Diffusional transport is governed by Fick’s second law and in two-dimensions, the partial differential equation becomes6

$$ {\frac{\partial }{\partial t}}C(x,y,t) = \nabla \cdot \left\{ {{\mathbf{D}}\left[ {\nabla C(x,y,t)} \right]} \right\} $$
(1)

where (x, y) are the spatial coordinates of the imaging system, and t is the time. The function C(x, y, t) represents the spatial concentration distribution of solutes in the x–y plane, and D is the diffusion tensor which is assumed symmetric. Within the small observation field of FRAP experiments, it is assumed that the diffusion tensor (D) is independent of position and time. Therefore, Eq. (1) could be rewritten as30

$$ {\frac{\partial }{\partial t}}C(x,y,t) = D_{xx} {\frac{{\partial^{2} C(x,y,t)}}{{\partial x^{2} }}} + 2D_{xy} {\frac{{\partial^{2} C(x,y,t)}}{\partial x\partial y}} + D_{yy} {\frac{{\partial^{2} C(x,y,t)}}{{\partial y^{2} }}} $$
(2)

where D xx , D xy , and D yy are the components of diffusion coefficient tensor (D) in the (x, y) coordinate system.

Given that \( \tilde{C}(u,v,t) = \int_{ - \infty }^{\infty } {\int_{ - \infty }^{\infty } {C(x,y,t)e^{ - i2\pi (ux + vy)} dxdy} } \) is the Fourier transform of C(x, y, t), with an arbitrary initial condition and a boundary condition that C(x, y, t) is a constant as (x, y) → ±∞, Eq. (2) can be solved by using 2D SFA in the frequency domain3 , 30 , 35 , 37 , 38

$$ \tilde{C}(u,v,t) = \tilde{C}(u,v,0)e^{{ - 4\pi^{2} \left( {u^{2} + v^{2} } \right)D(u,v)t}} $$
(3)

where \( \tilde{C}(u,v,0) \) is the initial solute concentration, and D(u, v) represents the diffusion coefficient in Fourier space (with frequencies u and v) and is defined as

$$ D(u,v) = {\frac{{u^{2} D_{xx} + 2uvD_{xy} + v^{2} D_{yy} }}{{u^{2} + v^{2} }}} $$
(4)

The function D(u, v) can be determined by curve fitting the light intensity of a time series of fluorescence recovery images (in the Fourier space) to Eq. (3).

The average of D(u, v) over the arc of a circumference (Integral curve L: u 2 + v 2 = a 2, a is an arbitrary constant, see Fig. 1) over an angle range from α to β can be given as:

Figure 1
figure 1

Diffusion coefficient D(u, v) in frequency domain (u, v). The averages of D(u, v) over the arcs from 0 to π, 0 to π/2 and 0 to π/4 can be used to determine the components (D xx , D yy and D xy ) of the diffusion tensor. On the ‘Ring 4’: \( D_{{^{{0,{\frac{\pi }{4}}}} }} = [D ( 4 , 0 )+ D ( 4 , 1 )+ D ( 4 , 2 )+ D ( 3 , 2 )+ D ( 3 , 3 )]/5 \); \( D_{{^{{0,{\frac{\pi }{2}}}} }} = [D ( 4 , 0 )+ D ( 4 , 1 )+ D ( 4 , 2 )+ D ( 3 , 2 )+ D ( 3 , 3 )+ D ( 2 , 3 )+ D ( 2 , 4 )+ D ( 1 , 4 )+ D ( 0 , 4 )]/9 \); and \( \begin{aligned} D_{{^{0,\pi } }} = & \left[ {D ( 4 , 0 )+ D ( 4 , 1 )+ D ( 4 , 2 )+ D ( 3 , 2 )+ D ( 3 , 3 )+ D ( 2 , 3 )+ D ( 2 , 4 )+ D ( 1 , 4 )+ D ( 0 , 4 )} \right. \\ & { + }{{\left. {D( - 1,4) + D( - 2,4) + D( - 2,3) + D( - 3,3) + D( - 3,2) + D( - 4,2) + D( - 4,1) + D( - 4,0)} \right]} \mathord{\left/ {\vphantom {{\left. {D( - 1,4) + D( - 2,4) + D( - 2,3) + D( - 3,3) + D( - 3,2) + D( - 4,2) + D( - 4,1) + D( - 4,0)} \right]} { 1 7}}} \right. \kern-\nulldelimiterspace} { 1 7}} \\ \end{aligned} \)

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$$ D_{\alpha ,\beta } = {\frac{{\int\limits_{{L:\,u^{2} + v^{2} = a^{2} }} {D(u,v)ds} }}{{\int\limits_{{L:\,u^{2} + v^{2} = a^{2} }} {ds} }}} = {\frac{1}{\beta - \alpha }}\int\limits_{\alpha }^{\beta } {(D_{xx} \cos^{2} \xi + 2D_{xy} \sin \xi \cos \xi + D_{yy} \sin^{2} \xi )d\xi } $$
(5)

The auxiliary variable ξ is a function of the frequencies u and v 35 , 37 , 38

$$ \xi = \tan^{ - 1} (v/u),\quad (\alpha \le \xi \le \beta ). $$
(6)

Based on Eq. (5), the average of D(u, v) over the arc from 0 to π can be related to the components of the diffusion tensor D as35 , 37

$$ D_{0,\pi } = {\frac{{D_{xx} + D_{yy} }}{2}} = {\frac{{tr({\mathbf{D}})}}{2}} $$
(7)

Similarly, the average of D(u, v) over the arc from 0 to \( {\frac{\pi }{2}} \) can be calculated as37

$$ D_{{0,{\frac{\pi }{2}}}} = {\frac{{D_{xx} + D_{yy} }}{2}} + {\frac{{2D_{xy} }}{\pi }} = {\frac{{tr({\mathbf{D}})}}{2}} + {\frac{{2D_{xy} }}{\pi }} $$
(8)

The average of D(u, v) over the arc from 0 to \( {\frac{\pi }{4}} \) can be calculated as

$$ D_{{0,{\frac{\pi }{4}}}} = {\frac{{D_{xx} + D_{yy} }}{2}} + \left( {{\frac{{D_{xx} - D_{yy} }}{\pi }}} \right) + {\frac{{2D_{xy} }}{\pi }} = {\frac{{tr({\mathbf{D}})}}{2}} + \left( {{\frac{{D_{xx} - D_{yy} }}{\pi }}} \right) + {\frac{{2D_{xy} }}{\pi }} $$
(9)

According to Eqs. (7)–(9), all the components of the diffusion tensor (i.e., D xx , D yy , and D xy ) can be obtained and expressed as the functions of \( D_{{0,{\frac{\pi }{4}}}} \), \( D_{{0,{\frac{\pi }{2}}}} \), and D 0,π :

$$ \left\{ \begin{gathered} D_{xx} = D_{0,\pi } + {\frac{{\pi \left( {D_{{0,{\frac{\pi }{4}}}} - D_{{0,{\frac{\pi }{2}}}} } \right)}}{2}} \hfill \\ D_{yy} = D_{0,\pi } - {\frac{{\pi \left( {D_{{0,{\frac{\pi }{4}}}} - D_{{0,{\frac{\pi }{2}}}} } \right)}}{2}} \hfill \\ D_{xy} = {\frac{\pi }{2}}\left( {D_{{0,{\frac{\pi }{2}}}} - D_{0,\pi } } \right) \hfill \\ \end{gathered} \right. $$
(10)

Using all the components of the diffusion tensor two principal diffusivities (eigenvalues) of the 2D diffusion tensor can be calculated by

$$ \left\{ \begin{gathered} D_{{{\text{Eig\_Min}}}} = {\frac{{D_{xx} + D_{yy} - \sqrt {(D_{xx} - D_{yy} )^{2} + 4D_{xy}^{2} } }}{2}} \hfill \\ D_{{{\text{Eig\_Max}}}} = {\frac{{D_{xx} + D_{yy} + \sqrt {(D_{xx} - D_{yy} )^{2} + 4D_{xy}^{2} } }}{2}} \hfill \\ \end{gathered} \right. $$
(11)

where D Eig_Min is the relatively smaller of the two eigenvalues, and D Eig_Max is the relatively larger one. The ratio of D Eig_Min to D Eig_Max can be used to represent the anisotropy of the diffusion.

Materials and Methods

Computer Simulation of Anisotropic FRAP Experiment

The 2D anisotropic diffusion process of the fluorescent molecule recovery after photobleaching was simulated using finite element analysis software (COMSOL Multiphysics 3.4, COMSOL Inc., Burlington, MA). At the boundaries of the sample (i.e., simulation domain, 4 × 4 mm2), the concentration of the fluorescent molecules satisfied the constancy assumption (C = C*). Initially, the concentration distribution of fluorescent molecules was assumed to be uniform (C = C*) outside the photobleached spot and zero within the photobleached area. The diameter of the bleached area was set to 26.71 μm according to the optimum value of 8 for the ratio of the imaging frame size (213.68 μm) to the initial diameter of the bleached spot as suggested by Travascio et al. 37 The mesh used in the simulation experiments included 8,784 quadratic Lagrange triangular elements. The convergence criterion for the solution was the relative error tolerance of less than 0.001. The simulation was stopped when the concentration in the bleached spot reached 95% of C*. For data analysis purposes, an image sequence containing 101 frames (0.6 s/frame) representing the FRAP experiment was extracted from the central region of the simulation domain and digitalized to 128 × 128 pixels (Fig. 2a). The frame size was set to 213.68 × 213.68 μm2, which was used in the following TMJ disc FRAP experiment.

Figure 2
figure 2

(a) The computational domain with the initial and boundary conditions for FRAP simulation experiments. (b) The orientation of pre-defined anisotropic diffusion tensor generated using various rotation angles. θ is the rotation angle from the principal diffusion tensor coordinate (X 0, Y 0) to the observed diffusion tensor coordinate (X, Y)

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Three groups of anisotropic diffusion were investigated in the simulation experiments and there were seven pre-defined diffusion tensors in each group. In the first group, a base diffusion tensor of \( {\mathbf{D}}_{0} = \left[ {\begin{array}{*{20}c} {10} & 0 \\ 0 & {15} \\ \end{array} } \right] \) μm2/s was selected and rotated at seven different angles θ i , i = 0–6 (θ 0–6 = 0, π/8, π/6, π/4, π/3, 3π/8, and π/2) using a rotation matrix R i defined as

$$ {\mathbf{R}}{}_{i} = \left[ {\begin{array}{*{20}c} {\cos \theta_{i} } & { - \sin \theta_{i} } \\ {\sin \theta_{i} } & {\cos \theta_{i} } \\ \end{array} } \right] $$
(12)

The seven pre-defined diffusion tensors, see Fig. 2b, were calculated by

$$ {\mathbf{D}}_{i} = {\mathbf{R}}_{i}^{\text{T}} {\mathbf{D}}_{0} {\mathbf{R}}_{i} \quad (i = 0{-}6) $$
(13)

In the second and third group, the base diffusion tensors were \( {\mathbf{D}}_{0} = \left[ {\begin{array}{*{20}c} {10} & 0 \\ 0 & {20} \\ \end{array} } \right] \) and \( {\mathbf{D}}_{0} = \left[ {\begin{array}{*{20}c} {10} & 0 \\ 0 & {30} \\ \end{array} } \right] \) μm2/s, respectively. The computer simulations of these three groups allowed investigation into the effects of anisotropy and orientation of the diffusion tensors on the accuracy of the new SFA based FRAP data analysis.

In order to study the effect of experimental noise on this anisotropic FRAP data analysis, photon noise was added to the ideal simulated FRAP images. Although photon noise has a Poisson distribution, for large photon fluxes (e.g., during FRAP experiments) Gaussian noise represents a close approximation of Poisson noise.5 Similar to previous studies,37 – 39 different magnitudes of Gaussian noise were added to the purely simulated recovery images to estimate how image noise affected the analysis. The magnitude of the Gaussian noise was characterized by its standard deviation, σ. Two magnitudes of spatial Gaussian noise (i.e., σ = 5 and 10) were generated by ImageJ software (Version 1.39f, by Wayne Rasband, National Institutes of Health, USA) and were then superimposed onto the simulated recovery images. For each level of σ investigated, ten FRAP experiments were simulated.

FRAP Experiment on Porcine TMJ Disc

Nine fresh porcine TMJ discs from five young adult pigs (~8–10 months) were used for testing. Razor blades were used to prepare tissue slices from the intermediate, posterior, anterior, lateral, and medial regions of the discs (Fig. 3). One tissue slice (about 4 × 4 mm2) was obtained from each region. The final specimens with ~650 μm thickness were obtained from each tissue slice using a microtome (SM24000, Leica Microsystems GmbH, Wetzlar, Germany) with a freezing stage (Model BFS-30, Physitemp, Clifton, NJ). The tissue slice was microtomed on both inferior and superior surfaces to remove the natural concave shape of the tissue to maintain a flat cartilage surface for the FRAP experiment. Diffusion tensors for FITC-dextran molecules with a molecular weight of 4 kDa (FD4, λex 490 nm; λem 520 nm, Sigma–Aldrich®, St. Louis, MO, USA) were measured with FRAP in the horizontal X–Y plane in Fig. 3 (parallel to the surface of the TMJ disc). Correspondingly, the three components (i.e., D xx , D yy , and D xy ) and two principal diffusivities (D Eig_Min and D Eig_Max) of the anisotropic diffusion tensor were obtained from the FRAP experiment.

Figure 3
figure 3

The schematic of FRAP experiments on porcine TMJ discs. Tissue specimens were prepared from the anterior, posterior, intermediate, lateral, and medial regions of the discs. The diffusion tensor of 4 kDa FITC-Dextran was measured with FRAP in the horizontal X–Y plane (parallel to the surface of the TMJ disc)

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Specimens were immersed in FITC-Dextran solution (0.1 mM) for 48 h (tissue swelling was not significant) to allow the concentration distribution of fluorescent solutes to reach equilibrium throughout the tissue. The FRAP experiment was performed on a Leica TCS-SP2 Confocal Microscope System (Leica Microsystems, Inc., Exton, PA) at room temperature (22 °C). The specimen was photobleached using an Ar-488 nm laser to create a circular bleached spot with a diameter about 27 μm. A Multi-Layer Bleaching (MLB) protocol was used to ensure a 2D fluorescent recovery in the X–Y plane by minimizing the diffusion in the Z-direction along the thickness of the specimen.35 The depth of the bleaching spot was about 70 μm from the surface into the specimen, and fluorescence recovery was observed on the focal plan at 7 μm beneath the surface of the specimen. A HC PlanAPO 20×/0.7NA dry objective (Leica Microsystems, Inc., Exton, PA) was used to collect fluorescent recovery images of 128 × 128 pixels (213.68 × 213.68 μm2). For each experiment, 100 frames of recovery images, plus five images prior to bleaching, were acquired at a rate of 0.6 s per frame. In order to minimize the contribution of the fluorescence emission of the background, pre-bleaching images were averaged and then subtracted from the post-bleaching image series.

Images and Data Processing

For both computer-simulated and experimental FRAP tests, the image processing was completed using custom written codes in MATLAB (MATLAB 7.0, The MathWorks Inc., Natick, MA). Based on Eq. (3), the code performed a 2D Fourier Transform and nonlinear curve-fit to yield solutions for D(u, v), which was the diffusion coefficient in the frequency domain. Based on Eq. (5), averaging D(u, v) on ‘Ring 4’ over different angle ranges generated D 0,π , \( D_{{0,{\frac{\pi }{2}}}} \), and \( D_{{0,{\frac{\pi }{4}}}} . \) The selection of Ring 4 was suggested by previous studies,37 and the frequency couples (u, v) for Ring 4 are shown in Fig. 1. Using Eq. (10), the code calculated all the components of the diffusion tensor: D xx , D yy and D xy . Finally, the calculation of the principal diffusivities, D Eig_Min and D Eig_Max, were completed according to Eq. (11). The average values, as well as the ratio of the two principal diffusivities for each specimen, were determined.

Statistical Analysis

An independent t-test was performed on the principal diffusivities, D Eig_Min and D Eig_Max, to examine the anisotropy of solute diffusion in each disc region. One-way ANOVA and Tukey’s post hoc tests were performed on the average diffusivities to examine the regional effect on the diffusion rate in porcine TMJ discs. This effect was also examined through the ratio of the principal diffusivities. SPSS 16.0 software (SPSS Inc., Chicago, IL) was used for all statistical analysis and significant differences were reported at p-values <0.05.

Results

Validation Using Computer-Simulated FRAP Experiments

2D anisotropic FRAP experiments were simulated by using pre-defined components of the anisotropic diffusion tensor. Numerically simulated FRAP digital images were transformed into the frequency domain using SFA. The principal diffusivities of the anisotropic diffusion process were calculated based on our proposed method. The accuracy of the new method was assessed by the relative error (ε) which is defined as37

$$ \varepsilon = {\frac{{\left| {{\text{Pre-defined}}\;{\text{Value}}-{\text{Calculated}}\;{\text{Value}}} \right|}}{{{\text{Pre-defined}}\;{\text{Value}}}}} \times 100(\% ) $$
(14)

The effects of anisotropic ratio (D 0xx /D 0yy ) and diffusion tensor orientation (θ) on the accuracy of the new method were evaluated. The relative errors (ε) of the two principal diffusivities for three different anisotropic ratios (D 0xx :D 0yy  = 1:1.5, 1:2.0, and 1:3.0) with seven diffusion tensor orientations (θ = 0–π/2) are shown in Fig. 4. Overall, the relative errors of D Eig_Min were less than 1.5% for all simulated cases (Fig. 4a), while the relative errors of D Eig_Max were less than 1% (Fig. 4b). This result indicated that the accuracy of the new method was not significantly sensitive to D 0xx :D 0yy , and θ. The highest accuracy was achieved when the diffusion tensors were oriented in alignment with the axes of the coordinate system (i.e., θ = 0 or π/2). The accuracy slightly decreased when the orientation of the diffusion tensor deviated from the axes of the coordinate system. The accuracy of this method increased when the anisotropic ratio (D 0xx :D 0yy ) decreased.

Figure 4
figure 4

Effects of the orientation (θ) and anisotropic ratio (D 0xx /D 0yy ) of the diffusion tensor on the relative error (ε) for the determination of (a) D Eig_Min and (b) D Eig_Max

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The effects of imaging noise (σ) on the accuracy of the new method were also investigated for the cases of D 0xx :D 0yy  = 1:1.5, 1:2.0 and 1:3.0, with θ = π/4. The typical FRAP images with and without spatial Gaussian noise (σ = 0, 5, 10) as well as the corresponding curve-fits to determine D(u, v) are shown in Fig. 5. The relative errors of the two principal diffusivities increased with increasing σ. The average relative errors were less than 8% at σ = 5, while the average values were less than 12.5% at σ = 10 (Fig. 6). In this study, the accuracy of the new method was not sensitive to the anisotropic ratio.

Figure 5
figure 5

Effect of the imaging noise on the determination of the D(u, v). (a) Typical computer-simulated FRAP images without and with spatial Gaussian noise (σ = 0, 5, 10). (b) Typical curve-fits to determine D(4, 4) for the simulated FRAP image sequences without and with spatial Gaussian noise (σ = 0, 5, 10)

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Figure 6
figure 6

Effect of the imaging noise on the precision of determined diffusion tensors with different anisotropy: (a) The relative error (ε) for D Eig_Min and (b) the relative error (ε) for D Eig_Max

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FRAP Experiments on Porcine TMJ Discs

The two principal diffusivities of the 2D anisotropic diffusion tensor of 4 kDa FTIC-Dextran in the five regions of porcine TMJ disc were determined using our new method and listed in Table 1. The characteristic diffusivity defined as (D Eig_Min + D Eig_Max)/2 was calculated to represent the average solute diffusion rate in each disc region. The ratio of D Eig_Min to D Eig_Max was also calculated to evaluate the anisotropy of solute diffusion in the disc.

Table 1 The results (mean ± SD) of the diffusion tensor of 4 kDa FTIC-Dextran in the five regions of porcine TMJ discs
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The average characteristic diffusivity across the disc was 26.05 ± 4.32 μm2/s. Regional differences in characteristic diffusivity were observed in the anteroposterior and mediolateral directions (Fig. 7a). In the anteroposterior direction, the anterior region (30.99 ± 5.93 μm2/s) had significantly higher characteristic diffusivity than the intermediate region (20.49 ± 5.38 μm2/s, p < 0.0003) and posterior region (20.97 ± 2.46 μm2/s, p < 0.00005). In the mediolateral direction, the medial region (26.51 ± 6.30 μm2/s) and lateral region (26.71 ± 5.50 μm2/s) had higher characteristic diffusivities than the intermediate region, although this trend was not statistically significant.

Figure 7
figure 7

The results of statistical analysis of determined diffusion tensors in porcine TMJ disc. (a) Characteristic diffusivities [(D Eig_Min + D Eig_Max)/2] were significantly region dependent (* p < 0.05 compared to Anterior). (b) Significant differences between D Eig_Min and D Eig_Max were detected in all disc regions (* p < 0.05). (c) The ratio of D Eig_Min to D Eig_Max was not region dependent

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Figure 7b showed that there was a significant difference between D Eig_Min and D Eig_Max in each disc region (p < 0.001). The ratio of D Eig_Min to D Eig_Max was approximately 0.45–0.51 (with 1.0 indicating isotropic) across the disc without significant regional differences (Fig. 7c). These results indicated that the diffusion of 4 kDa FTIC-Dextran was highly anisotropic in the porcine TMJ disc.

Discussion

In this study, a new method solely based on the SFA of FRAP images was developed to completely determine the anisotropic diffusion tensor in hydrated soft tissues. This method was used to determine the region-dependent anisotropic diffusion tensor in porcine TMJ discs.

The accuracy and robustness of our new method was validated by the computer-simulated FRAP experiments. Our method demonstrated several key advantages over the previous studies on the SFA based FRAP methods for anisotropic diffusion. Most of all, it makes it possible to deal with any orientation of the diffusion tensor (θ = 0–π/2). The complete diffusion tensor, including both diagonal and off-diagonal components, can be determined using this new method.

Compared to the method recently reported by Travascio et al.,37 our method is more accurate and robust. In their method, an anisotropic diffusion tensor was determined based on two independent analyses of FRAP images: the Fourier transform and the Karhunen–Loeve transform. The KLT analysis of FRAP images can determine the orientation of the diffusion tensor and by combining the FT and KLT analyses, Travascio et al.’s method can determine all the components of the anisotropic diffusion tensor in a single FRAP experiment. In this study, we proposed a method solely based on the SFA to completely determine the anisotropic diffusion tensor which avoids all of the limitations imposed by KLT analysis. For example, owing to the limitations of the KLT, Travascio’s method is not applicable when the orientation of the diffusion tensor coincides with the fixed coordinate system (i.e., θ = 0 or π/2).37 More importantly, the major limitation of their method, as recognized by the authors, is the lack of robustness to deal with the imaging noise which is unavoidable in real experiments. In their simulation results, the average relative errors reached 40% at σ = 10 noise level, while the average values reached 20% at σ = 5 level.37 Under the same noise conditions, the average relative errors of our method were less than 12.5% at σ = 10 level and less than 8% at σ = 5 level. It is apparent that our new method is more robust against the noise and our method only needs one image analysis (i.e., the SFA) resulting in fewer numerical errors for the determination of the diffusion tensor. In our simulation without imaging noise, the relative errors of D Eig_Min were less than 1.5% (Fig. 4a) while under the similar conditions, the relative errors reached 14% in simulations performed by Travascio et al. 37 In the above accuracy comparison, it should be noted that the sizes of the bleaching spot and imaging frame in this study were only about half the values of those parameters used in the study by Travascio et al. 37 This implies that the imaging sampling might also affect the precision of the results.

This study represents the first measurement of solute diffusion in TMJ disc tissues. The average diffusivity of 4 kDa FITC-Dextran in porcine TMJ discs (26 μm2/s) was approximately 16% of the diffusivity in water (162 μm2/s, based on the Stokes–Einstein equation with the hydraulic radius of 1.4 nm) and was comparable to the diffusivity of the similar sized solute (3 kDa FITC-Dextran) in porcine articular cartilage (75 μm2/s) measured by FRAP.16 Most recently, Benavides et al. reported that there is a region-dependent water diffusion in porcine TMJ disc while using diffusion tensor MRI.2 In the anteroposterior direction, the mean water diffusivity was higher in the anterior and posterior bands compared with the intermediate zone. In the mediolateral direction, the mean water diffusivity was higher in the medial and lateral aspects than in the center. Our study of 4 kDa FITC-Dextran diffusion showed exactly the same trends.

The results of this study indicated that the diffusion rates of uncharged dextran molecules vary by region in the TMJ disc. The anterior region had higher characteristic diffusivity than the intermediate region and posterior region in the anteroposterior direction, while the medial and lateral regions had higher characteristic diffusivities than the intermediate region in the mediolateral direction. Previous studies have shown that solute diffusivity in cartilaginous tissues depended on tissue water content in such a manner that a decrease in the solute diffusivity occurred as tissue water content decreased,9 , 35 , 37 and these findings were also observed in our study. For the porcine TMJ disc, it has been reported that along the anteroposterior axis, the anterior band (74.5 ± 2.9%) had higher water content than the intermediate zone (73.7 ± 3.1%) and posterior band (70.1 ± 4.0%), while along the mediolateral axis, the medial region (75.3 ± 2.1%) had a higher water content than the central (71.3 ± 3.7%) and lateral (71.3 ± 4.1%) regions.8 The correlation between the diffusivity and tissue water content suggests that the diffusive transport in porcine TMJ discs is dependent on tissue composition. The inhomogeneous distribution of tissue components results in region-dependent diffusivity in TMJ disc tissues.

The results of this study also demonstrated that solute diffusive transport in porcine TMJ discs is highly anisotropic with the ratio of D Eig_Min to D Eig_Max around 0.45–0.51 across the disc. Studies on knee meniscus and IVD annulus fibrosus showed that the anisotropic solute diffusion in fibrocartilage is due to the orientation of collagen fibers.36 , 37 The orientation of the largest principal component (D Eig_Max) of the diffusion tensor is parallel to the direction of the bundles of collagen fibers, while the smallest principal component (D Eig_Min) is perpendicular to the fibers. Inside the porcine TMJ disc, Detamore et al. reported that collagen fibers were well organized and had a ring-like structure around the periphery and were oriented anteroposteriorly through the intermediate zone.8 Our results suggested that the diffusive transport in TMJ disc is not only dependent on tissue composition but also on tissue structure, especially collagen structure.

In this study, we measured diffusivities of 4 kDa FTIC-Dextran (hydraulic radius equivalent to insulin)16 to examine the region-dependent anisotropic diffusion in porcine TMJ discs. However, it is important to note that the dextran molecules have a linear structure that may not duplicate the molecular structure of other physiologically relevant molecules in the TMJ disc, which may be more globular in structure. Pluen et al. have shown that flexible macromolecules have greater mobility than similarly sized globular molecules in a random fiber matrix.26 It is also important to note that the diffusion tensor was determined in the horizontal plane in this study. The diffusion in the vertical direction (i.e., superior–inferior direction) could be the main pathway for nutrient transport through the TMJ disc. Therefore, in order to fully understand solute transport in the TMJ disc, it is necessary to determine the 3D diffusion tensor in each disc region and correlate it to tissue composition and structure.

In summary, a new method solely based on the SFA has been developed to determine the 2D anisotropic solute diffusion tensor in the fibrous tissues using FRAP. The accuracy and robustness of this method was validated using computer-simulated FRAP experiments. The new method was applied to determine the 2D diffusion tensor of 4 kDa FITC-Dextran in the porcine TMJ discs. It was determined that the diffusion of this solute in TMJ discs was inhomogeneous and anisotropic. These findings suggested that diffusive transport in TMJ disc is dependent on tissue composition (e.g., water content) and structure (e.g., collagen orientation). This study provides a new method to quantitatively investigate the relationship between transport properties and tissue composition and structure. The transport properties obtained are crucial for future development of numerical models studying nutritional supply within the TMJ disc.