Mueller matrix polarimetry and polar decomposition of articular cartilage imaged in reflectance

Abstract

Articular cartilage birefringence relates to zonal architecture primarily of type II collagen, which has been assessed extensively in transmission, through thin tissue sections, to evaluate cartilage repair and degeneration. Mueller matrix imaging of articular cartilage in reflection is of potential utility for non-destructive imaging in clinical and research applications. Therefore, such an imaging system was constructed to measure laser reflectance signals, calibrated, and tested with optical standards. Polar decomposition was chosen as a method to extract fundamental optical parameters from the experimental Mueller matrices, with performance confirmed by simulations. Adult bovine articular cartilage from the patellofemoral groove was found to have ∼0.93 radians retardance, low diattenuation of ∼0.2, and moderately high depolarization of 0.66. Simulations showed that variation in depolarization drives inaccuracy of depolarization and retardance maps derived by polar decomposition. These results create a basis for further investigation of the clinical utility of polarized signals from knee tissue and suggest potential approaches for improving the accuracy of polar decomposition maps.

Fig. 1.

Schematic of the Mueller matrix experimental setup for reflectance imaging. Abbreviations are indicated in the legend.

Fig. 2.

(A) Illumination intensity before and after generation of polarization states. (B) Linear polarization intensity at 15° polarizer rotation intervals. Error bars represent standard deviation of 10 consecutive measurements.

Fig. 3.

Representative images (A,B) and histograms (C,D) of a Spectralon surface in the absence (A,C) and presence (B,D) of the speckle reducer. The scale bar for both images is indicated.

Fig. 4.

Mueller matrix decomposition parameters of (A,B,C) spectralon diffused reflectance standard, (D,E,F) quarter-wave plate and (G,H,I) spectralon coupled with quarter-wave plate. (A,D,G) Depolarization, (B,E,H) retardance and (C,F,I) diattenuation maps were compared between the objects. Scale and colormap bars are indicated.

Fig. 5.

Normalized Mueller matrix images of a cartilage explant. The scale bar is indicated. The dashed area in M11 indicates the extent of cartilage tissue, cut with a razor blade to make a defined corner.

Fig. 6.

MMD parameters of bovine articular cartilage as (A) depolarization, (B) retardance, (C) diattenuation and (D) azimuth maps. Scale bar is indicated.

Fig. 7.

Retardance, diattenuation, and depolarization values are insensitive to small added noise and filtering. The raw images (A) unaltered, (B) with standard deviation = 0.05 noise added, and (C) median filtered at a radius of 6 pixels were processed to produce optical maps. The mean ± standard deviation of pixel values are presented for each map. Scale bar and grayscale colorbars (one per column) are indicated.

Fig. 8.

Results of MMD on a simulated specimen with uniform retardance R=1.57 and variable depolarization Δ centered at values between 0 to 1 with Gaussian-distributed random pixel variation creating pixel value standard deviations of 0.05 or 0.2, on a mean of Δ=0.3. Error bars represent the reconstructed R map standard deviation across all 40 × 40 pixels.

Fig. 9.

Simulation results for polar decomposition. Input and polar decomposition (PD) output (A) retardance maps, R, (B) diattenuation maps, D, and (C) depolarization maps, Δ, with a range of pixel standard deviation values (expressed as coefficient of variation, %) for input Δ and D. The mean ± standard deviation of each map is displayed in or below each image.

Fig. 10.

Simulation results for PD of a field of view of 50 × 50 pixels, with constant retardance R but varying depolarization, Δ, and diattenuation, D. More pixels had singular matrix calculations with higher variation in Δ, leading to inaccurate R. Meanwhile the depolarization values were highly skewed from the input mean of 0.3. Colorbars with pixel values are presented in the histograms. The depolarization index histogram contains an inset to visualize pixel values from 0 to 1.