1. Introduction

Photon migration in tissue can be described by radiative transport theory or its diffuse approximation [1]. In many studies, biological tissues can be treated as an isotropic turbid medium where light scattering is independent of incident direction. However, this treatment is invalid in anisotropic tissues such as tendon [2], dentin [3], skin [4] and skeletal muscle [5]. Since these anisotropic tissues are abundant in human and animals, a thorough understanding on light propagation in such tissues is critical for correctly interpreting optical measurements obtained in them.

Anisotropic tissues such as tendon usually have a predominant axis that is determined by the fibrous collagen structures. Nickel et al. [4] found that the equi-intensity reflectance distribution at skin surface resembled an ellipse whose long axis was perpendicular to the collagen fiber orientation at small detection distances, but parallel to the fibers at larger distances. Kienle et al. [6] found that the reduced optical scattering coefficient was higher when measured perpendicularly to the fibers. Both anisotropic diffuse equation (ADE) [7] and continues time random walk (CTRW) theory [8] predicted the elliptical shape of the equi-intensity reflectance in fibrous samples and its alignment with fiber orientations, which have been also confirmed in several additional experimental studies [9–11].

Skeletal muscles are the other major anisotropic tissues in human and are essential for many important physiological functions. In addition to their fibrous appearance, skeletal muscle fibers are composed of repetitive units called “sarcomere” which are the fundamental structural and contractile units in striated muscle [12]. In fact, many muscle diseases are associated with mutations in sarcomeric proteins [13]. Light propagation in striated muscle was found to be affected by the sarcomere structures [14, 15] and a unique 2D reflectance pattern was documented in skeletal muscles [5]. Subsequent studies have revealed that different striated muscles have somewhat different 2D reflectance patterns due to variations in muscle structural properties [16].

Time or path-length resolved optical measurements have been widely used to study light propagation in tissue [17]. In addition to the use of short-pulse lasers, an alternative implementation is to use “coherence gating” based low coherent interferometry (LCI) to achieve path-length-resolved measurements. Popescu & Dogariu [18] measured optical reflectance in scattering samples using a Michelson-based interferometric implementation of LCI and found that experimental measurements can be described by a time-resolved diffuse model. Mach-Zehnder-based LCI [19] was also applied to study path-length resolved Doppler shift of multiply scattered photons in turbid media.

In this study, we used a single-mode fiber optic Mach-Zehnder LCI to investigate the angular-dependent light propagation in anisotropic tissues including tendon and skeletal muscle. Measurements were conducted at different distances from the incidence and different angles from the fiber orientation in tissue. The experimental and measurement details are described in Section 2. The Section 3 shows the measured reflectance data which revealed both similarities and significant differences in tendon and skeletal muscle. In the discussion section (Section 4), we reanalyzed the results shown in Section 3 and showed that anisotropic diffuse theory can explain our observation in tendon, but not in skeletal muscle.

2. Materials and methods

Fresh bovine skeletal muscle and tendon samples were obtained from the meat science laboratory at the University of Missouri-Columbia. All surface fat on the tissue samples were carefully removed before measurements.

Tissue samples were scanned by using a single mode fiber optic Mach-Zehnder interferometer (Fig. 1 ) that is similar to those reported previously [19]. The light source was a 4 mW superluminescent diode (SLD-561, Superlum Inc., Russia) with a central wavelength of λ_o= 1287.0 nm and spectral bandwidth Δλ_FWHM=32.7 nm (with calculated coherence length of 22.4 μm in air). The 4 mW light was split by the fiber coupler A with 90% of the source power delivered to the sample, and the remaining 10% used as the reference light. In the sample arm, the incident light was focused onto the sample by using an f=25 cm lens which produced a 41 μm diameter incident spot. The detection head was angled at 60° from the sample surface and consisted of two identical lenses (f= 8 mm, N.A. = 0.28), resulting a detection diameter of 9.4 μm on the sample surface. The collected photons were coupled into a 2×2 single-mode fiber coupler B. The reference light from the coupler A was reflected back from a retro-reflector and also coupled into the coupler B. A translational stage was used to move the retro-reflector to vary the path-length of the reference arm. The interference fringes were divided into two equal parts with a π phase difference [20] and detected by using a balanced photo-detector (2117-FC, New focus Inc., USA). The signal was amplified, band-pass filtered and acquired by a data acquisition card into a computer for processing. The system had a detection sensitivity of −123 dB.

 

Fig. 1 A schematic diagram of the fiber optic Mach-Zehnder low coherence interferometer. SLD: superluminescent diode, PL: pilot light for aiming. PD: balance photo-detector. AF: amplifier & band-pass filter. DAQ: data acquisition board.

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For convenience, a reference coordinate was defined with the x-y plane on the sample surface as shown in Fig. 1. A focused incident beam was perpendicular to the sample surface with its incident location defined as the origin (0, 0). Tissue samples were positioned so that the y-axis was aligned with the orientation of fibers in the sample. Two parameters (d and α) were used to define the measurement position on the sample surface. The measurement distance (d) was defined as the distance between detection position and the origin; whereas the measurement angular position (α) was defined as the angle formed between the measurement plane and sample fiber orientation (y-axis).

The sample was placed on a rotational stage so that the orientation (α) of the fibers can be adjusted. A cover glass was placed on the sample surface to define the measurement surface and tissues samples were in touch with the cover glass during the measurement. The detection fiber was scanned across the sample to measure reflectance at different distances (d) away from the origin with a step size of 0.05 cm (e.g., 0.00, 0.05, 0.10, 0.20 cm) and at different angular positions (α=0°, 22.5°, 45°, 67.5°, 90°) relative to the fiber orientation. The maximal scanning distance of the reference arm was 1.1 cm (corresponding to a 2.2 cm path-length in air) where detected signals reached noise floor. At every measurement position, eight measurements were repeated and averaged to improve the signal-to-noise ratio.

The detected signal at each detection position (d, α) is $S_{d,\alpha }(l)\propto 2{\left(I_r{I}_s\right)}^{1/2}$, where I _r and I _s are light intensities from the reference arm and sample arm, respectively. The effects of detector (PD in Fig. 1) responsivity and amplifier gain (AF in Fig. 1) were removed from the acquired raw signals. The optical path-length resolved reflectance r_d,α(l) measured at distance d and orientation angle α is calculated as: $r_{d,a}\left(l\right)=S_{d,\alpha }^2(l)/\left(4I_r{I}_{in}\right)$, where I _in is the incident light intensity.

The total reflectance R_d,α and the mean path-length $〈l_{d,\alpha }〉$ at position (d, α) were calculated as:

3. Results

The measurements obtained in a tendon sample are shown in Fig. 2 . There is a reflection from the cover glass and tissue interface which was used to define the global zero point (zero path-length) in all measurements. Figure 2(a) shows examples of raw path-length resolved reflectance measured in tendon. At measurement angles 0°, 22.5°, 45° and 67.5°, signals with sufficient signal-to-noise were obtained at 0.0 to 0.2 cm away from the incident point. In general, the reflectance had smaller maximal amplitudes at smaller angles and larger distances. When measuring perpendicularly to the fibers (α= 90.0°), the reflectance was detectable up to d = 0.40 cm. Figure 2(b) shows the mean optical path-length as a function of measurement distance in tendon. The mean path-length generally increased with measurement distance. When measured at α = 90°, the mean path-length was the shortest among all other angles. The curves at angular position α = 0.0°, 22.5°, and 45.0° were similar. All curves in Fig. 2(b) appeared to follow quadratic or power functions.When plotting the total reflectance (Eq. 1) with the mean path-length, all measurements at different distances and angles merged into one line (Fig. 2(c)) which can be fit using an exponential decay function with the best fit exponential decay constant of 9.5 cm⁻¹ (with coefficient of determination R²=0.98). As shown in Fig. 2(d), the transition distance Δl increased nearly linearly with the mean path-length at α = 0.0°, 22.5°, and 45.0°. At α = 67.5°, the transition distance increased initially until reaching a mean path-length of ~0.45 cm, and maintained a value of ~0.6 cm thereafter. At α = 90.0°, the transition distance initially had similar increasing trend as at other angles, but leveled out soon at a value of ~0.3 cm at measurement distance d ≥ 0.05 cm.

 

Fig. 2 (a) Examples of raw data measured at 0.1 cm away from incidence; (b) the mean path-length as a function of measurement distance; (c) the total reflectance and (d) the transition distance as a function of the mean path-length in tendon.

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Figure 3(a) shows examples of raw optical path-length resolved reflectance obtained in skeletal muscle. The corresponding mean path-lengths at different measurement positions were shown in Fig. 3(b). Similar to the results obtained in tendon, the mean path-length increased with the measurement distance. At α = 90°, the reflectance signal was detectable at much longer measurement distances than at other angles. The curves measured at α = 0.0°, 22.5°, 45°, and 67.5° had similar mean path-length; whereas, the curve at α = 90° had shorter mean path-lengths than all the others.

 

Fig. 3 (a) Examples of raw data measured at 0.1 cm from incidence; (b) the mean path-length as function of measurement distance; (c) the total reflectance and (d) the transition distance as function of the mean path-length obtained in skeletal muscle.

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Figure 3(c) shows the total reflectance obtained in the skeletal muscle as a function of the mean optical path-length. Excluding the first a few points at the smallest measurement distance, all curves measured at different angular positions (α) seemed to have similar exponential decay slopes. However, at the same mean path-length, the total reflectance measured at smaller angles was higher than those measured at large angles. The transition distances as a function of the mean path-length were shown in Fig. 3(d). The curves shown were similar to those obtained in tendon, except that they reached a plateau at slightly shorter mean path-length. Again, the transition distance at α = 90° was the smallest (0.2~0.3 cm) among all other angles and it stopped increasing at the shortest mean path-length of merely d = 0.05 cm. The three curves measured at 0°, 22.5° and 45° were similar; whereas the result at α = 67.5° was in between of this group and the result at α = 90°.

4. Discussion

The raw path-length resolved reflectance clearly showed angular dependence in both tendon and muscle tissues. But it is interesting to note that the total reflectance versus mean path-length showed no anisotropic pattern in tendon (Fig. 2(c)). All data points measured at different angles were well fitted with one exponential curve. In other words, the reflectance measured at different angles had the same value at the same mean path-length.

To explain this observation, we examined the anisotropic diffusion theory [7] where a diffuse tensor was applied to describe the incident angle dependent diffusion process. For fibrous samples with a predominant fiber orientation (y-axis in Fig. 1), two different diffuse coefficients can be applied: D_y represents the diffuse coefficient along the fibers and D represents that perpendicular to the fibers (along x- and z-axis). The anisotropic diffuse equation can be written as:

To confirm these analyses, we reanalyzed our experimental results to find whether there was a single scale factor to transform the y-axis so that the mean path-length and reflectance became isotropic. We started with the mean path-length obtained at 0° and 90° in Fig. 2(b), By applying a scaling factor D/D_y=2.43, the 0° curve can be best fitted using the same power function obtained from the 90° curve. The same transformation factor 2.43 was then applied to all data in Fig. 2(b) & (c). The results were shown in Fig. 4 as a function of the corrected distance (x²+2.43²y²)^1/2. It is clear that the angular dependency disappeared in both graphs as predicted from the anisotropic diffuse theory. Figure 4 confirmed that the total reflectance (Eq. 1) vs. mean path-length curve was independent of measurement angle α in tendon tissue as shown in Fig. 2(c).

 

Fig. 4 The mean path-length (a) and total reflectance (b) after transforming the y-axis by y’=2.43y for the same tendon data shown in Fig. 2(b) & (c), respectively.

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The muscle samples showed quite different reflectance vs. mean path-length curves (Fig. 3(c)). Despite the fact that all curves in Fig. 3(c) followed a similar decaying trend, they had different y-axis offset values. The transformation procedure mentioned above failed to eliminate the angular dependency in both mean path-length and total reflectance as shown in Fig. 5 . Although the mean path-lengths at 0° and 90° were matched by using the obtained transformation factor D/D_y=2.03, the corresponding reflectance curves were not matched using the same scaling factor (Fig. 5(b)).

 

Fig. 5 The mean path-length (a) and total reflectance (b) after transforming the y-axis by y’=2.03y for the same muscle data shown in Fig. 3(b) & (c), respectively.

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A closer examination over Fig. 2 and 3 revealed some interesting differences between the results obtained in tendon and muscle. In tendon, the total reflectance measured at a position along 0° was equal to that measured at a larger distance along 90° (Fig. 2(c) and Fig. 2(b)). In other words, the equi-intensity profiles were elongated toward 90°, exactly as that predicted from fibrous scatterings [4, 22] at small detection distances. However, Fig. 3(b) and (c) suggested that similar total reflectance was obtained at roughly the same detection distance along 0° and 90° in muscle. In other words, the equi-intensity profiles in muscle were more symmetric at 0° and 90° than that in tendon within the small measurement distances of ~4 mm. It is expected that photons were still scattered toward 90° by the cylindrical shaped muscle fibers. Thus the more symmetric equi-intensity patterns implied the existence of other mechanisms that scattered photons toward 0° in muscle. As revealed in previous studies [5, 15], sarcomere diffraction clearly fits this profile. It is worth mentioning that skeletal muscle and fibrous tissues also had different equi-intensity reflectance profiles at measurement distances much larger than those used in this study [4, 5]. When measured far away from the incidence, the equi-intensity profiles in fibrous samples form an ellipse that is stretched along the fibers (0°) because of the smaller scattering coefficient along the fibers [22]. Such elliptical patterns (elongated along fibers) were not observed in muscle. On the contrary, the patterns in some types of muscles [16] tend to elongate perpendicularly to the fibers at larger distance.

The transition distance also had strong angular dependency in both tendon and muscle as shown in Fig. 2(d) and 3(d). It was interesting that the transition distance was much smaller and did not change much with mean path-length in both tendon and muscle when measured at 90°, i.e. the reflectance reached maximal values at smaller path-lengths at 90°. Because of the smaller mean path-length, the reflectance decayed slower with detection distance and can be detected at much larger distance at 90° as shown in Fig. 2 & 3. However, the same aforementioned y-axis transformation didn’t eliminate the angular dependency in transition distance in both tendon and muscle. Since the transition distance was primarily determined by photons experiencing few number of scattering events, the diffuse equation may not be applicable to them.

The aforementioned results were replicated in three tendon samples and six muscle samples, all of which were from different animals. Specifically, angular dependency in reflectance was eliminated in all tendon samples when representing reflectance as a function of mean path-length (as illustrated in Fig. 2(c)). However, the same phenomenon was not observed in any of the six muscle samples measured in this study. We would like to also point out that reflectance was measured by using a probe positioned at 60° from the sample surface (or 30° from the normal). It has been shown that surface reflectance has similar angular distributions in both isotropic and anisotropic tissues [23]. Therefore we suspect similar results will be obtained by using other detection angles.

5. Conclusion

In summary, a Mach-Zehnder low coherent interferometer was used to study the light propagation in tendon and skeletal muscle tissues. The raw path-length resolved reflectance showed strong angular dependency in both tissues. Quantitative analyses revealed both similarities and significant differences in tendon tissue and skeletal muscles. When reflectance was represented as a function of mean path-length, the angular dependency was eliminated in tendon but not in muscle. Experimental observations in tendon sample corroborated the existing knowledge on light scattering in fibrous materials. These results suggested that anisotropic diffuse model may be applied to measure optical properties in fibrous tissues such as tendon. However, it cannot satisfactorily explain optical reflectance measured in skeletal muscles. Additional scattering mechanisms such as sarcomere effect need to be considered in order to fully describe photon migration in muscle. On the other hand, if the periodic sarcomere structures influence light propagation in muscle, optical techniques may be developed to detect some sarcomere related muscle diseases [13].