New 3D Ellipsoid Method Visualizes Tissue Polarization

Published in the journal Frontiers of Optoelectronics, the study presents the Global-Polarization Stokes Ellipsoid, or GPSE, as a way to visualize the complete polarization information of biological tissues. The Mueller matrix is a 4×4 mathematical representation that describes how a medium modifies the polarization state of light, but its sixteen numerical elements can be difficult to interpret intuitively. The new method projects the outgoing polarization states of all fully polarized light onto the Poincaré sphere, creating a three-dimensional ellipsoidal shape that reflects the inherent optical properties of the sample.

While Stokes vectors are commonly used to describe polarization, they are highly dependent on the specific incident polarization state chosen for measurement. In contrast, the Mueller matrix is independent of the incident state and provides a more stable and comprehensive description of a sample’s polarization characteristics. However, visualizing the Mueller matrix has been rarely proposed in previous literature. The authors aimed to extend the intuitive projection methods used for Stokes vectors to the Mueller matrix, thereby establishing a more robust framework for analyzing the inherent features of biological samples from the perspective of polarization optics.

The experimental setup utilized a Mueller matrix imaging system based on the dual rotating retarder method. This configuration included two detection pathways: forward transmission and backward scattering. The illumination source was a 3-watt LED emitting at 633 nanometers with a bandwidth of 20 nanometers. The system employed a polarization state generator and a polarization state analyzer, each consisting of a lens, a polarizer, and a quarter-wave plate. The quarter-wave plates were controlled by rotation servo motors to generate various polarization states. A CMOS camera captured 30 intensity images with diverse polarization states, which were then used to calculate the Fourier coefficients and subsequently the sixteen elements of the Mueller matrix. The system was calibrated using standard samples, with systematic error controlled within 1 percent.

The study examined porcine liver, bovine skeletal muscle, and porcine fat as measurement samples. These tissues were selected because they represent different optical and structural properties. Skeletal muscle exhibits significant anisotropy due to its orderly arranged fibers, while fat is considered an isotropic medium. Liver tissue contains both anisotropic connective tissues and isotropic hepatic lobule tissues. The samples were preserved in phosphate-buffered saline at 4 degrees Celsius and sliced into thin sections of 0.06 centimeters for transmission measurements, while samples for backward detection were cut to a thickness of 0.5 centimeters. All samples were cut into square pieces measuring 1.5 centimeters by 1.5 centimeters.

The GPSE is defined by projecting the outgoing Stokes vectors of all fully polarized incident states onto the Poincaré sphere. When a sufficient number of polarization points are selected on the sphere, their corresponding outgoing projection points can be connected to form a topological space. In most cases, this projection exhibits an ellipsoidal shape. The study defines three basic characteristic parameters of the GPSE: V, E, and D-dagger. The V parameter represents the polarization maintenance ability of the sample, calculated as the cube root of the product of the long, middle, and short axes of the ellipsoid. A V value of 1 indicates complete polarization maintenance, while a value of 0 indicates complete depolarization.

The E parameter describes the eccentricity of the GPSE, reflecting the difference in polarization maintenance ability among various fully polarized states. An E value of 0 corresponds to a spherical GPSE, indicating consistent polarization maintenance across all states, while values closer to 1 indicate significant differences due to anisotropy. The D-dagger parameter describes the position of the GPSE center relative to the origin of the coordinate system, which may also be related to tissue anisotropy. These parameters were derived from the geometric properties of the ellipsoid, providing a quantitative measure of the tissue’s optical behavior.

Monte Carlo simulations were employed to further explore the relationship between GPSE characteristics and tissue anisotropy. The simulations used a sphere-cylinder birefringence model to independently regulate two main sources of tissue anisotropy: intrinsic birefringence and form birefringence due to cylindrical scattering. By varying the refractive index difference, the full width at half maximum of the cylindrical scatterer distribution, and the fiber content, the researchers observed how the GPSE responded to changes in tissue structure.

Experimental results revealed notable distinctions in the GPSEs of the different tissues. The GPSE of bovine skeletal muscle showed a higher polarization maintenance ability for linearly polarized light aligned with or perpendicular to the muscle fibers, compared to other polarization states. In contrast, the GPSE of porcine fat was more spherical, with the three symmetry axes coinciding with the S1, S2, and S3 axes, indicating almost no linear or circular phase retardance. The liver tissue displayed a more complex GPSE, with the hepatic lobule region showing a circular projection on the S1-S2 plane, similar to isotropic tissues, while the connective tissue region formed an ellipse, similar to anisotropic tissues.

The study also demonstrated the application of the GPSE system in monitoring the dynamic dehydration process of bovine skeletal muscle. Using a forward imaging setup, the team monitored the tissue every five minutes over a period of 90 minutes. The diagonal elements of the Mueller matrix gradually enhanced during dehydration, indicating an increased polarization maintenance ability. The volume of the GPSE increased, reflecting this change. The E and D-dagger parameters initially increased and then decreased, suggesting that the anisotropy of the muscle showed an overall trend of first increasing and then decreasing during dehydration. This trend was attributed to the tightening of fiber arrangement and increased birefringence in the early stages, followed by a decrease in scattering coefficient and refractive index difference in the later stages.

A significant advantage of the GPSE parameters over traditional Mueller matrix polar decomposition parameters is their immunity to the phase wrapping effect. The linear phase retardance parameter from Mueller matrix polar decomposition can exhibit anomalous oscillations due to the periodic nature of phase retardance. The GPSE parameters, however, are monotonically correlated with birefringence and can directly reflect the actual phase retardance due to tissue anisotropy without the need for additional dual-wavelength measurements. This makes the GPSE system a more stable and accurate tool for extracting anisotropic-related parameters from biological tissues.

The findings of this study provide a new perspective on perceiving polarization characteristics of biological tissues. The GPSE method enables a more intuitive identification of the optimal incident polarization state for improving discrimination among different tissue types or during dynamic tissue processes. By offering a three-dimensional visualization of the Mueller matrix, the method facilitates a deeper understanding of the microstructural and optical properties of tissues. The robustness and accuracy of the GPSE system were confirmed through both experimental observations and Monte Carlo simulations, highlighting its potential for further detailed evaluation of physiological mechanisms in tissue processes.

The research contributes to the field of biomedical optics by providing a comprehensive and intuitive tool for analyzing polarization properties. The study’s demonstration of the GPSE system in monitoring tissue dehydration underscores its utility in tracking dynamic biological processes.

The authors emphasize that the GPSE system provides a reliable detection basis for evaluating physiological mechanisms, such as birefringence effects and fiber orientation distribution. The method’s capacity to handle the full range of polarization states makes it a powerful tool for characterizing the optical properties of diverse biological samples.

Subject of Research: Stereoscopic spatial graphical method of Mueller matrix: Global-Polarization Stokes Ellipsoid

Article Title: Stereoscopic spatial graphical method of Mueller matrix: Global-Polarization Stokes Ellipsoid

Article References: Zhang, X., Song, J., Fan, J., Zeng, N., He, H., Tuchin, V. V., & Ma, H. (2024). Stereoscopic spatial graphical method of Mueller matrix: Global-Polarization Stokes Ellipsoid. Frontiers of Optoelectronics, 17(1), Article 29. https://doi.org/10.1007/s12200-024-00132-4

Image Credits: AI Generated

DOI: 10.1007/s12200-024-00132-4

Keywords: Stereoscopic, spatial, graphical, method, Mueller, matrix, Global-Polarization, Stokes, Ellipsoid, scientific research