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We acquired polarization-sensitive reflectance images in freshly excised skeletal muscle samples. The obtained raw images varied depending on the incident and detection polarization states. The Stokes vectors were measured for incident light of four different polarization states, and the whole Mueller matrix images were also calculated. We found that the images obtained in skeletal muscles exhibited different features from those obtained in a typical polystyrene sphere solution. The back-reflected light in muscle maintained a higher degree of polarization along the axis perpendicular to muscle fiber orientation. Our analysis indicates that the unique muscle sarcomere structure plays an important role in modulating the propagation of polarized light in whole muscle.
©2008 Optical Society of America
1. Introduction
Skeletal muscle tissues are responsible for many critical physiological functions such as locomotion and body temperature maintenance; they also served as a source of nutrient reserve in times of suboptimal dietary intake. Whole muscle is made of a collection of muscle fibers structured by connective tissues. Sarcomere is the fundamental structural and contractile unit that is responsible for muscle contractions [1, 2]. Many skeletal muscle diseases result from mutations in sarcomeric proteins [3]. Under a microscope, a sarcomere appears as repetitive band units due to a periodic distribution of optical refractive indices. We previously found [4] that unpolarized light propagation in bulk muscle is strongly affected by the periodic sarcomere structures. As a result, optical reflectance patterns obtained in muscle samples have distinct features compared to isotropic or other fibrous tissues [5]. Such patterns can be simulated using a Monte Carlo model when sarcomere diffraction is included [5].
In this study, we studied polarization-dependent reflectance images in fresh skeletal muscles. Optical polarization is actively involved in light-tissue interactions. In many cases, the polarization states of incident light are altered by tissue optical properties [6, 7]. For example, the size and shape of the scatterers in biological tissues affect how polarized light is scattered. Therefore, polarized light scattering has been used to extract quantitative morphological information from living tissues [8–10]. Optical polarization can significantly enhance optical imaging capability by providing target-specific “polarization contrast” [11–15]. Such polarization-specific responses are usually not discernable with non-polarized detection. Though polarized light has been previously used to study sarcomere dynamics in single muscle fiber or bundles [16], no study has investigated polarization-sensitive responses in whole muscle.
We implemented a polarization-sensitive imaging system to acquire reflectance images in skeletal muscles illuminated with light in different polarization states. The acquired polarization images as well as the corresponding Stokes vector and whole Mueller matrix images exhibit unique features that provide further evidence of the importance of sarcomere structures in modulating light propagation in whole muscles.
2. Material and method
Bovine Sternomandibularis muscles were used in this study. The samples were excised from the animal immediately after slaughtering at the Meat Science Laboratory at the University of Missouri-Columbia. After removing the surrounding fat tissue, we mounted the muscle on a sample holder by fixing both ends so that the muscle length was unchanged during the experiment. The muscle samples were mounted against a thin cover glass to ensure a flat imaging surface.
Our experimental apparatus is illustrated in Fig. 1. A 633nm linearly polarized He-Ne laser was used as the light source. A half waveplate (HW) was used to rotate the polarization direction of the incident light so that it was aligned with the linear polarizer P1. A variable waveplate (VW) after P1 was adjusted to obtain either linearly or circularly polarized light. The polarized light was incident upon the sample through a small 1.0 mm hole at the center of a 45° mirror M2. The incident beam diameter at the sample (S) surface was about 1.0 mm. The mirror M2 redirected backscattered light from the sample toward the camera through a quarter waveplate (QW) and a linear polarizer (P2). The CCD used was an 8-bit video camera (Allen Bradley 2801 YF, 640×480) equipped with a 50mm, f/2.8 imaging lens. A fixed aperture of f/8 was used throughout the experiments. The camera aperture accepted photons within 1.2° of normal over a 26.5×19.9mm2 imaging area. A reference coordinate system was defined so that the muscle fiber orientation (y-axis) was aligned with the V-polarization direction.
Fig. 1. A schematic of the experimental setup. LS: a 10mW He-Ne laser; HW: half wave plate; M1: mirror; P1: polarizer; VW: variable wave plate; M2: mirror; S: sample; QW: quarter wave plate; P2: polarizer; CCD: imaging camera.
Four polarization states were achieved for the incident light: horizontal linear (H), vertical linear (V), 45° linear (P), and right circular (R). The polarization accuracy of the entire system was calibrated by using a mirror as the sample [17]. The measured polarization extinction ratios were >40dB for linearly polarized light and >32dB for circularly polarized light. To obtain the Stokes vector images for each incident polarization state, we measured four polarization components of the optical reflectance: horizontal linear (H), vertical linear (V), 45° linear (P), and right circular (R). The Stokes vector was calculated as:
where the symbol I i indicates the reflectance intensity with incident polarization state of i. For example, I H is the reflectance intensity with H-polarized incident light. The degree of polarization (DOP) can be calculated from the Stokes vector as:
And the Mueller matrix can be calculated from the four Stokes vectors:
where S i indicates the Stokes vector for incident light with a polarization state of i. A total of 16 images were captured in the experiment with four different polarization incident and detection states. For convenience, each raw image was labeled with two capital letters: the first term represented the input polarization state, and the second term represented the output polarization state. For example, ‘HV’ represented a vertically-polarized (V) reflectance image with horizontally-polarized (H) incident light.
3. Results and discussion
Figure 2 shows polarization images of a muscle sample acquired with 16 different combinations of incident and detection polarization states. The incident point is at the center, and the muscle was mounted such that the muscle fiber was oriented along the vertical direction (y-axis) of the image. To better illustrate the relationship between the muscle orientation and intensity decay, the images are shown in a pseudo-color depiction of the equi-intensity distribution. In other words, the same color in the image represents all pixels of the same intensity.
Fig. 2. Polarization-sensitive reflectance images in skeletal muscle. The incident light was located at the center of the image. The image size was 26.5mm by 19.9mm. The muscle fibers were along the vertical direction (y-axis). The H-polarization direction was along the horizontal direction (x-axis).
Figure 2 shows that the equi-intensity profiles of all the acquired images appear to have a rhombus shape, similar to what was observed in our previous non-polarized imaging study [5]. Differences exist, however, among images acquired with different polarizations. The most significant difference is found between the HH and VV images. The VV image has the strongest signal among all the images, and its inner equi-intensity profile is elongated along the vertical direction. The HH image shows very strong rhombus profiles. The equi-intensity profiles of other images have patterns that lie in between those of the HH and VV images. A diagonal symmetric relationship between incident polarization and detection polarization can be identified from the images. For example, HV and VH are very similar to each other.
Similar to the reflectance image obtained with unpolarized light, the equi-intensity profiles of polarization-sensitive images shown in Fig. 2 can be well fitted using the following equation [5]:
The patterns described by the above equation transition from a rhombus to an ellipse as the parameter q increases from 1 to 2. The parameters a and b indicate the axis length along the xand y-axis, respectively. The three parameters a, b, and q can be estimated by using the Levenberg-Marquet (LM) nonlinear fitting algorithm [18].
Fig. 3. Fitting results of the parameter q and axis ratio (a/b) for the HH, HV, and VV images shown in Fig. 2 obtained at different distances along the y-axis from the incident point.
From the numerical fitting results shown in Fig. 3, the equi-intensity patterns obtained in HH, HV, and VV images are indeed found to be different. At small distances from the incident point, the HH image is almost an exact rhombus with a q value approaching 1.0. In contrast, the VV image has an exact elliptical pattern with a q value of 2.0. The HV images have a pattern between a rhombus and an ellipse. As the distances from the incident point increase, the q-value in the HH image increases; while the q-value in the VV image decreases. The q-value in the HV image remains quite stable at the entire 2–8 mm distance from the incident point. At larger distances, the fitted q-values in all HH, HV, and VV images converge to a value of ~1.4, which is similar to the value obtained in an unpolarized reflectance image [5]. The ratio of the two axes in all three images decreases with the distance. The decrease in VV, from >2.5 at 2 mm to ~1.5 at 8 mm, is the most significant. The trend implies that the anisotropic effect was reduced at larger distances because of multiple scattering.
Figure 4 shows the calculated Stokes vectors of the reflectance image at the four different incident polarization states H, V, P, and R. As can be seen, the second Stokes component S1 shows the same “cross-like” patterns for all four incident polarization states, although their intensity distributions are quite different. Because the S1 Stokes component represents the difference in the H and V polarization components of the back-reflected light, the results indicate that the reflected light has a larger H-polarized component along the y-axis or the muscle fiber orientation and a larger V-polarized component perpendicular to the muscle fibers. Their relative weights, however, are significantly different. In the S1 component of the Stokes vector obtained with H incidence, the V component along x-axis is barely larger than the H component; while, in the case of V incidence, the V component is clearly dominant. The Stokes vectors obtained with P and R incident light are almost the same. There are no significant patterns in the S2 and S3 Stokes components obtained for all four different incident polarization states. It is interesting to note that, with circularly polarized incident light, the reflectance Stokes vector has a strong pattern in the S1 component and a plain S3 component. This implies that the incident circularly polarized light is converted into linearly polarized light, likely resulting from muscle birefringence [19].
Fig. 4. Stokes vectors of the reflectance images in a muscle for 4 different incident polarization states: H, V, P, and R. The images were calculated from the raw images in Fig. 2 using Eq. (1). The S1, S2, and S3 images were normalized with the S0 image. The color map shown was used for S1, S2, and S3 images only. The muscle fibers were along the vertical direction. The Hpolarization direction was along the horizontal direction.
Fig. 5. Mueller matrices of the reflectance images in a muscle sample. The images were calculated from the raw images in Fig. 2 by using Eq. (3). The muscle fibers were along the vertical direction. The H-polarization direction was along the horizontal direction. All images were normalized with the M11 image. As a comparison, the Mueller matrices obtained in a polystyrene (1.093 µm in diameter) solution were also shown. Please note that the M11 images used their own color maps.
The Mueller matrix images (Fig. 5) of the muscle sample were calculated using Eq. (3). As a comparison, the Mueller matrices obtained in a polystyrene (Polysciences, Inc., Warrington, PA) solution are also shown. The diameter of the polystyrene sphere was 1.093 µm. A volume concentration of 0.12% was used in the study with calculated scattering coefficient of 41cm-1 and anisotropy (the g-parameter) of 0.93. The Mueller matrix images obtained in the polystyrene solution are similar as those reported before [20–23]. The M11 component of the Mueller matrix represents the unpolarized measurements. As expected, the equi-intensity profiles of the M11 element in polystyrene solution are a set of circles centered at the incident point with the reflectance intensity decreasing with the radial distance. In contrast, the M11 component in muscle has the typical rhombus profile as discussed before.
The equi-intensity profiles of the M12 and M21 elements in both muscle and polystyrene solution have similar shapes of quatrefoils. The intensity values along the x-axis are negative, and those along the y-axis are positive. A close examination indicates that the intensity distributions along the x- and y-axis are similar in polystyrene solution, while the signal along the x-axis is ~3 times larger than that along the y-axis in muscle. Similarly, in polystyrene solution, the M22 has a symmetric cross-like pattern with almost identical intensity distributions along the x- and y-axis. In skeletal muscle, however, the M22 pattern along the muscle fiber orientation (the y-axis) is very weak, while the signal along the x-axis is dominant. The other typical patterns shown in M23, M32, M33, and M44 from polystyrene solution do not present in the images obtained in muscle. Instead there appear to be some residual patterns along the x-axis in the muscle images.
In an isotropic medium (such as polystyrene solution), the patterns of the polarization-sensitive reflectance image can be described by the single-scattering approximation as discussed in [24]. For example, the typical quatrefoils patterns shown in the M12 and M21 images can be predicted by using the standard Mie scattering matrix. The appearance of such patterns in the muscle sample suggests that scattering by spheroidal particles still exists in this complex tissue. All our imaging results, however, identify a clear preference along the x-axis, the direction perpendicular to the muscle fibers. To obtain further insight into such anisotropic effect, we calculated degree-of-polarization (DOP) images for the 4 different incident polarization states H, V, P, and R.
Fig. 6. Images of the degree of polarization (DOP) calculated from the Stokes vector images in Fig. 4 by using Eq. (2). The muscle fibers were along the vertical direction. The H-polarization direction was along the horizontal direction. As a comparison, the DOP images obtained in a polystyrene solution were also shown.
As shown in Fig. 6, the DOP images from the muscle sample and the polystyrene solution are significantly different. With linearly polarized incident light, the reflected light in the polystyrene solution maintains a certain polarization along the direction orthogonal to the original polarization direction. For example, the incident H polarization is aligned with the x-axis in Fig. 6, while the corresponding reflected light has a high degree of polarization along the y-axis within a small area. In polystyrene solution, a circularly polarized incident light can maintain its polarization state better than a linearly polarized light during the propagation, which is often referred to as the polarization memory effect [14]. Therefore a circularly polarized incident light produces polarization-preserving backscattered light over a large area as shown in Fig. 6. In muscle, the x-axis preference appears again in the DOP images. With V-polarized incident light, the back-reflected light preserves polarization over a much longer distance along the x-axis: the DOP is close to 0.5 even at 5 mm distance from the incident location. The polarization memory effect disappears in muscle with circularly polarized incident light. Even with P- and R-polarized incident light, the backscattered photons maintain higher polarization at locations along the x-axis.
It is known that multiple scattering depolarizes incident polarized light, especially linearly polarized light. Therefore the DOP of the backscattered light can be used as an indicator of the number of scatterings. At a larger distance from the incidence, the DOP is generally close to zero in both samples due to multiple scatterings. Those photons exiting at a smaller distance from the incidence experience fewer scatterings and thus maintain certain polarization. At small distance from the incident, the orientation of DOP patterns in polystyrene sample can be explained by using single Mie scattering theory [24]. For example, a spherical particle tends to scatter more polarization maintained V-polarized incident light to the orthogonal direction (x-axis in Fig. 6). In muscle, such tendency is greatly enhanced for V-polarized incident light, which indicates a much smaller scattering probability along the x-axis for this particular polarization. In other words, V-polarized incident photons experiences longer pathlength along the y-axis and are subject to more attenuations, which is supported by the greater than 1.0 axis ratio of a/b as shown in Fig. 3. In addition, the backscattered light along the x-axis is primarily V-polarized as shown in Fig. 4.
Fig. 7. Diffraction efficiencies for the first 3 diffraction orders calculated by using coupled wave theory. The curves shown in solid lines are for TM polarization component, while those shown in dash lines are for TE polarization components. The geometry of the calculation is also illustrated in the figure. Please note that the TE direction is aligned with the muscle fiber orientation (y-axis) and the V-polarization in our coordinate system.
The above phenomenon is most likely caused by the diffraction effect of the unique muscle sarcomere structures. To provide some quantitatively evidence, we calculated the diffraction efficiencies of the muscle fiber when light is scattered back to the surface within a plane perpendicular to muscle fibers, i.e., along the x-axis (Fig. 7), by using a three dimensional coupled wave theory [5, 25]. A physical sarcomere structure model and properties proposed by Thornhill et al. [5, 26] was used in the calculation. The sarcomere length used was 2.8 µm. Other sarcomere lengths led to the same conclusion. As shown in Fig. 7, the periodic sarcomere structure clearly diffracts much less V-polarized (TE polarization) light away than H-polarized light (TM polarization). Therefore, most of the light reaching back to the surface along the x-axis is V-polarized as observed in this study.
4. Conclusion
We acquired polarization-sensitive reflectance images in fresh bovine sternomandibularis muscle. The whole Mueller matrix images were computed and compared with those obtained in a well-studied polystyrene solution. Our experimental results indicate that the propagation of polarized light in muscle is significantly different from that in media of spherical particles. Although more quantitative studies are needed, experimental evidence indicates that the scattering by spherical particles, muscle-fiber-induced birefringence, and sarcomere-induced diffraction may all contribute to the results observed in this study. The most important difference between striated muscle and other biological tissues is the periodic sarcomere structures. Their strong effects must be taken into consideration when applying optical measurements to muscle tissues. In addition, because sarcomere is a critical component for normal muscle functions as well as meat quality, its distinct effect on light transport should be explored to develop optical techniques that can assess its structural and functional properties noninvasively.
Acknowledgment
This project was supported in part by a National Science Foundation grant CBET-0643190 and the National Research Initiative (NRI) of the USDA Cooperative State Research, Education and Extension Service under grant number 2006-35503-17619.
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