1. Introduction

Polarization-sensitive optical coherence tomography (PS-OCT) is a depth-resolved polarimetric imaging technology [1] which, when coupled with appropriate processing algorithms, is capable of quantifying anisotropic tissue properties such as form-birefringence, form-biattenuance [2], and optic axis orientation. Whereas tissue form-birefringence (Δn) and form-biattenuance (Δχ) provide information related to constituent fiber diameters, volume fraction, and refractive index mismatch between fibers and surrounding matrix [2–4], optic axis orientation (θ) provides the direction of constituent fibers relative to a fixed reference direction (i.e., horizontal in the laboratory frame).

Measurement of optic axis orientation using PS-OCT has been previously reported in non-scattering retardation waveplates and highly scattering fibrous tissue. Hitzenberger et al. [5] demonstrated a method for quantifying optic axis orientation in a Berek’s compensator. Methods given by Zhang et al. [6] and Park et al. [7] were used to measure optic axis orientation in other non-biological media with fiber-based PS-OCT instruments. Jiao and Wang [8,9] successfully quantified the optic axis orientation using tendon rotated within the plane normal to incident light, and several papers [5,6,10–12] have shown B-scan images of cumulative optic axis orientation in anisotropic tissues including skin, retina, cornea, muscle, and tendon. However, because anisotropy in superficial regions is not considered during calculation of optic axis orientation in deeper regions, these B-scan images of optic axis orientation do not represent the anatomical fiber direction with respect to a laboratory reference. In addition, these B-scan images can be difficult to interpret due to their speckled appearance. Recently, both Todorovic et al. [13] and Guo et al. [14] modeled specimens as stacked, multiple layered retarders with arbitrary optic axis orientation for each layer. By employing Jones or quaternion algebraic techniques, cumulative effects of overlying layers were removed and differential optic axis orientation was measured.

In this paper, we present a similar method for measuring depth-resolved optic axis orientation [θ(z)] deep within multiple layered tissue using enhanced polarization-sensitive optical coherence tomography (EPS-OCT) [15]. Importantly, using bulk-optic EPS-OCT, the depth-resolved optic axis orientation [θ(z)] unambiguously represents the actual anatomical fiber direction in each layer with respect to a fixed laboratory reference and can be measured with high sensitivity and accuracy. Characterization of the anatomical fiber direction in connective tissues with respect to a fixed reference is important because functional and structural characteristics such as tensile and compressive strength are directly related to the orientation of constituent collagen fibers.

Collagen organization in cartilage has been previously studied using PS-OCT [16] and the utility of intervertebral disc cartilage as a model tissue on which to demonstrate the depth-resolved polarimetric imaging ability of PS-OCT was recognized by Matcher et al. [17]. Their work and others [18,19] effectively describe intervertebral disc structure and its elegant biomechanical role in shock-absorption and flexibility; therefore we provide only a brief summary here. Intervertebral discs (Fig. 1) are located between spinal vertebrae and consist of the annulus fibrosis enclosing an inner gel-like nucleus pulposis. Annulus fibrosis is composed of axially concentric rings (i.e., lamellae) of dense type I collagen fibers (fibrocartilage), the orientation of which is consistent within a single lamella but approximately perpendicular to fibers in neighboring lamellae, forming a lattice-like pattern. Regular orientation of collagen fibers within a single lamella is responsible for form-birefringence [Δn(z)], and alternating fiber directions between successive lamellae correspond to alternation of optic axis orientation [θ(z)] within the annulus fibrosis.

 

Fig. 1. Schematic of intervertebral disc and annulus fibrosis showing alternating fiber directions in the laboratory frame (H and V) [20], the incident beam, and scan location (dashed red line). AF: annulus fibrosis; NP: nucleus pulposis.

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2. Theory

Transformations in the polarization state of light can be described using either linear algebra techniques involving Jones or quaternion vectors/matrices [7,8,14] or three-dimensional geometric techniques involving normalized Stokes vector trajectories on the Poincaré sphere [15,21]. A linearly anisotropic, homogenous element such as a single lamella in the annulus fibrosis can be modeled with the Jones matrix

where phase retardation (δ) and relative-attenuation (ε) [2] between orthogonal eigenpolarizations (eigenvectors) are contained in the diagonal matrices J _ret and J _att respectively, and attenuation common to both eigenpolarizations is neglected. J _rot is a rotation matrix which defines the linear eigenvectors relative to the laboratory frame by the orientation of the optic axis (θ).

Incident light (represented by Jones vector E _in) is transformed by the element J _s into exiting light, E _out=J _s(δ,ε,θ)E _in. In the equivalent Poincaré sphere geometry, normalized Stokes vectors representing the incident (S _in) and exiting (S _out) light polarizations are displaced along a trajectory which can be decomposed into movement in two orthogonal planes: 1) a rotation around eigen-axis $\widehat{\beta }$ with angle equal to the phase retardation (δ), and 2) a collapse toward $\widehat{\beta }$ which is related to the relative-attenuation (ε) by Eq. (13) in Ref [2]. Eigen-axis ($\widehat{\beta }$) is a vector in the equatorial (Q-U) plane of the Poincaré sphere and is related to the optic axis orientation (θ) by

where q̂ is the unit-vector defining the Q axis of the three-dimensional Cartesian coordinate system containing the Poincaré sphere and βu is the U component of $\widehat{\beta }$ .

When incident light (E _in) propagates to a given depth and is reflected back in double-pass, the exiting light (E _{dp_out}) is given by the equation E _{dp_out}=${\mathbf{J}}_{\mathrm{s}}^{\mathrm{T}}$(δ,ε,θ)J _s(δ,ε,θ)E _in=J _s(2δ,2ε,θ)E _in. Likewise, the double-pass trajectory between S _in and S _{dp_out} on the Poincaré sphere is simply twice the rotation (2δ) and collapse (2ε) as the single-pass trajectory about the same eigen-axis ($\widehat{\beta }$).

Multiple layered fibrous tissue such as the annulus fibrosis is modeled as a stack of K linearly anisotropic, homogeneous elements, each with arbitrary phase retardation (δ_k ), relative-attenuation (ε_k ), optic axis orientation (θ_k ), and corresponding kth Jones matrix [J _s(k)(δ_k,ε_k,θ_k ), Eq. (1)]. Incident polarized light (E _in) propagating to the rear of the kth intermediate element and back out in double-pass (E _{dp_out (k)}) is represented by

For the most superficial layer (k=1), Eq. (3) becomes E _{dp_out(1)}=${\mathbf{J}}_{\mathrm{s}\left(1\right)}^{\mathrm{T}}$ J _s(1) E _in, and J _s(1)(δ ₁,ε ₁,θ ₁) can be recovered using matrix algebra [7,8]. Likewise, δ ₁, ε ₁, and θ ₁ can be found using a nonlinear fit [2] to the trajectory between S _in and S _{dp_out(1)} on the Poincaré sphere and Eq. (2).

For the next layer (k=2), Eq. (3) becomes E _{dp_out(2)}=${\mathbf{J}}_{\mathrm{s}\left(1\right)}^{\mathrm{T}}$ ${\mathbf{J}}_{\mathrm{s}\left(2\right)}^{\mathrm{T}}$ J _s(2) J _s(1) E _in. Again, matrix algebra and knowledge of J _s(1) allows recovery of J _s(2)(δ₂,ε₂,θ₂ ) [13]. Likewise, δ₂, ε₂ , and θ ₂ can be found using a nonlinear fit to the trajectory between S _{dp_out(1)} and S _{dp_out(2)} after compensation of anisotropy in the superficial layer (reverse rotation by -δ ₁ and reverse collapse by -ε ₁ with respect to $\widehat{\beta }$ ₁). This process is repeated for successively deeper layers in the stack to determine δ_k, ε_k , and θ_k for all K layers as summarized by Todorovic et al [13]. To avoid confusion with the multiple layer analysis presented here, we note Ref [13] contains a typographical error in the unlabeled equation directly above Fig. 1 in which the term to the right of the final ellipsis should read “[J _st(i-1)]-1”(no transpose).

In the context of PS-OCT imaging, the polarization state detected after double-pass to the rear of the kth intermediate element [E _{dp_out(k)}, Eq. (3)] is also transformed by optics in the instrument (e.g., beamsplitter, optical fiber, polarization modulator, retroreflector), adding complexity to the optic axis orientation analysis. With inclusion of a Jones matrix (J _c) which encompasses instrumental transformations, Eq. (3) becomes

For a single-mode-fiber-based PS-OCT instrument, J _c represents an unstable phase retardation between arbitrary elliptical eigenvectors. In this case, eigenvectors of Jones matrices in Eq. (4) vary in an unknown fashion and measurement of the anatomical fiber direction with respect to the laboratory frame is distorted by the optical fiber birefringence [22]. Advanced calibration techniques may allow fiber-based PS-OCT instruments to overcome distortion in the optical fiber, but these have yet to be demonstrated. A generic bulk-optic PS-OCT instrument has stable J _c with linear eigenvectors in the laboratory frame; therefore J _c reduces to a simple phase retardation (δ_c , due to the beamsplitter and retroreflector) between horizontal and vertical interference fringes. Additionally, our bulk-optic EPS-OCT instrument incorporates a liquid crystal variable retarder (LCVR) [15] to modulate the launched polarization state incident on the specimen by applying a voltage-controlled phase retardation (δ _LCVR). The optic axis of the LCVR is horizontal, thus the total systematic phase retardation (δ _LCVR+δ_c ) can be compensated by subtraction of δ _LCVR+δ_c from the relative phase of the detected horizontal and vertical interference fringe signals, allowing unambiguous and undistorted measurement of the anatomical fiber direction (θ_k ) absolutely referenced to the laboratory frame.

3. Methods

Instrumentation used for this study was a previously reported bulk-optic EPS-OCT system [15]. Mode-locked Ti:Al₂O₃ laser source parameters were: λ₀=830 nm, Δλ_FWHM=55 nm, and power incident on specimen=3.0 mW. The instrument was shot-noise limited and A-scan frequency was 30 Hz. An achromatic lens (f=8 mm) was used to focus incident light on the specimen surface. A constant instrumental phase retardation (δ_c =58°) between horizontal and vertical interference fringes was determined empirically from the image of a mirror.

3.1 Crossed tendon specimen

To validate the procedure (Sec. 2) for measuring fiber direction (θ_k ) with respect to the fixed laboratory frame in multiple layered tissue with EPS-OCT, we prepared a specimen consisting of overlapped mouse tail tendon fascicles placed on a microscope slide with a random orientation. The slide was placed in the EPS-OCT sample path and an en face photograph of the crossed tendon specimen was recorded using a dissection microscope and digital camera oriented carefully in the laboratory frame. The specimen was imaged with EPS-OCT in three locations designated by numbered dots in Fig. 2. Dots #1 and #2 correspond to locations where fascicles were not overlapping (K=1), and dot #3 corresponds to the location were the two fascicles overlapped and therefore represents multiple layered tissue (K=2).

 

Fig. 2. Photograph of the crossed tendon specimen showing orientation in the laboratory frame (H and V) and three locations imaged using EPS-OCT.

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At each location, a cluster of 36 A-scans was acquired on a 6×6 grid within a small square region (25 µm×25 µm). Acquisition was repeated for M=3 launched polarization states distributed uniformly around the U-V plane of the Poincaré sphere as set by the voltage-controlled LCVR in the sample path. After subtraction of the constant instrumental phase retardation (δ_c+δ_LCVR,m ), normalized Stokes parameters representing depth-resolved polarization data [S _m(z)] were calculated and averaged to reduce polarimetric speckle noise (σ _speckle) for each cluster as described previously [2]. A multistate nonlinear algorithm [2] was applied to determine phase retardation (δ ₁), relative-attenuation (ε ₁), and eigen-axis ($\widehat{\beta }$ ₁) for fascicles at locations #1 and #2 and for the top fascicle (k=1) at location #3. S _m(z) for the bottom fascicle (k=2) at location #3 were compensated by δ ₁, ε ₁, and $\widehat{\beta }$ ₁ as described in Sec. 2 and the multistate nonlinear algorithm was applied to determine δ ₂, ε ₂, and $\widehat{\beta }$ ₂. Eq. (2) was used to calculate fiber direction (θ_k ) from the estimated eigen-axis ($\widehat{\beta }$ _k). EPS-OCT-measured θ_k was compared to the orientation measured in the digital photograph using a software-based angle dimensioning tool (Canvas, ACD Systems Intl, British Columbia, Canada).

3.2 Ex vivo porcine annulus fibrosis specimen

A section of spine with four intact thoracic vertebrae was harvested from a freshly-euthanized Yucatan mini-pig (female, 6-month-old, 35 kg) obtained from an unrelated study. Intervertebral disc was exposed on the dorsal side and placed in the EPS-OCT instrument. Care was taken to insure that the exposed surface was normal to the incident beam. Figure 1 shows the incident beam (normal to specimen surface) and B-scan pattern on the intervertebral disc specimen.

The acquired B-scan image was 0.52 mm deep and 0.25 mm wide, consisting of 360 A-scans divided laterally into 10 uniformly-spaced clusters of 36 A-scans each. Calculation of depth-resolved polarization data [S _m(z)] for each cluster proceeded as described for the crossed tendon specimen, except M=6 launched polarization states were used. The OCT intensity B-scan image of the annulus fibrosis was divided into K layers, each representing one lamella with thickness Δz_k . Index of refraction n=1.40 was used to convert from optical thickness to physical thickness. For the most superficial layer (k=1), the multistate nonlinear algorithm [2] was applied to determine phase retardation (δ ₁), relative-attenuation (ε ₁), and eigen-axis (β̂₁) from the depth-resolved polarization data [S _m(0…Δz ₁)]. S _m(z) for deeper layers (k=2…K) were compensated by δ ₁, ε ₁, and $\widehat{\beta }$ ₁ as described in Sec. 2. Estimation of δ_k, ε_k , and $\widehat{\beta }$_k and compensation of successively deeper layers for each lateral cluster allowed measurement of local form-birefringence (Δn_k =λ₀ δ_k /2πΔz_k ), form-biattenuance (Δχ_k =λ₀ ε_k /2πΔz_k ), and collagen fiber direction [θ_k , Eq. (2)] in the annulus fibrosis.

4. Results

4.1 Crossed tendons

EPS-OCT-measured values of fiber direction (θ_k ) and the orientation angles measured in the digital photograph are summarized in Table 1. Figure 3 shows the angle dimensioning measurements superimposed on a zoomed-in view of Fig. 2. Standard deviation of polarimetric speckle noise [15] was approximately σ _speckle ≈7° for all locations and did not increase significantly for the bottom fascicle in location #3.

Table 1. Fiber directions (θ_k ) in degrees (°) measured counterclockwise with respect to the horizontal laboratory frame.

View Table

 

Fig. 3. Orientations of the top (blue) and bottom (red) fascicle are measured with respect to the horizontal (green) in this digital photograph using a software-based angle dimensioning tool. Numbered dots (cyan) indicate the locations of EPS-OCT imaging.

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4.2 Annulus fibrosis

The OCT intensity B-scan image of porcine annulus fibrosis (K=11) is shown in Fig. 4(a). Depth-resolved anatomical fiber direction (θ_k ) and birefringence (Δn_k ) are represented by false-colors in the corresponding B-scan images shown in Figs. 4(b) and 4(c). Figure 5 shows a graph of mean θ_k for each lamella (k) averaged across the 10 lateral clusters. Standard deviation in θ_k is denoted by error bars. Mean and standard deviation in depth-resolved birefringence (Δn_k , gray bars) and biattenuance (Δχ_k , white bars) for each lamella (k) are shown in Fig. 6. Mean σ _speckle increased from approximately 7° for k=1 to approximately 10° for k=11.

 

Fig. 4. (a) OCT intensity B-scan image and corresponding (b) depth-resolved anatomical fiber direction and (c) birefringence images of ex vivo porcine annulus fibrosis (K=11). Image dimensions are 0.52 mm deep by 0.25 mm wide.

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Fig. 5. Mean and standard deviation in depth-resolved anatomical fiber direction (θ_k ) for each lamella (k) in ex vivo porcine annulus fibrosis. As expected, an approximate 90° change in fiber direction is apparent between successive layers.

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Fig. 6. Mean and standard deviation in birefringence (Δn_k , gray) and biattenuance (Δχ_k , white) for each lamella (k) in ex vivo porcine annulus fibrosis.

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5. Discussion

To our knowledge, this work represents the first reported implementation of a depth-resolved polarimetric method (e.g., PS-OCT) to unambiguously characterize anatomical fiber directions (θ_k ) deep within multiple layered anisotropic tissues with absolute referencing to the laboratory frame. The EPS-OCT approach builds on previous algorithms [13,14] and a bulk-optic (or free-space) implementation avoids distortion and unstable polarization transformations introduced by fiber-based PS-OCT instruments [22]. Although the authors acknowledge that a non-fiber-based instrument is expected to have less clinical impact on noninvasive diagnosis of diseases, nevertheless, the application of a bulk-optic instrument in a laboratory setting could have significant impact on the basic understanding of tissue ultrastructure and the diseases or traumas that affect it.

The anatomical fiber direction B-scan created using EPS-OCT [Fig. 4(b)] does not exhibit high-contrast speckle noise common to previous optic axis orientation B-scan images [5,6,14]. This is accomplished by 1) lateral ensemble averaging and 2) launching multiple incident polarization states to reduce σ _speckle and maximize polarimetric signal-to-noise ratio (PSNR) [15], and 3) fitting of noisy depth-resolved polarization data [S _m(z)] to a noise-free model. The effects of speckle noise are reduced to small errors in the fiber direction estimates (θ_k ) provided by the multistate nonlinear algorithm and thus do not appear as high-contrast speckle noise in our fiber direction B-scan image.

Expected uncertainty in estimates of δ and ε was discussed previously [2,15] and is based on σ _speckle. We expect uncertainty in θ ₁ (the top layer) to be no more than 3.5° for σ _speckle=7° and Table 1 indicates uncertainty may be even lower. Accurate measurement of θ_k in deeper layers is directly dependent on accuracy in preceding layers; therefore one would expect errors to accumulate rapidly as deeper layers are analyzed. However, the standard deviation in estimated θ_k across each lamella (error bars in Fig. 5) shows only marginal increase versus k. The robustness of the multistate nonlinear algorithm and insensitivity to noise with which EPS-OCT can measure depth-resolved fiber direction (θ_k ) is thus demonstrated by the consistency in calculated θ_k (across each lamella) after iterating through more than K=10 layers. Because of noise, estimated eigen-axis ($\widehat{\beta }$_k) showed a slight deviation from the equatorial plane after compensation of previous layers. Deviation was removed by projecting the estimated $\widehat{\beta }$_k into the equatorial (Q-U) plane before compensation of the subsequent layers. The number of incident polarization states (M) was anticipated prior to recording data. For annulus fibrosis lamellae (lower phase retardation than tendon specimens), we insured sufficient PSNR by including additional incident polarization states (M=6). Detailed analysis of the optimum number of incident polarization states (M) requires further study.

Although layer-by-layer compensation of relative-attenuation (ε_k ) was accomplished for crossed tendon and annulus fibrosis specimens, it was not necessary for accurate determination of θ_k or δ in the annulus fibrosis specimen studied here. ε_k in each layer of annulus fibrosis was small and the possibility of relative-attenuation accumulating versus depth is cancelled by the approximately orthogonal orientation of alternating fiber directions in successive lamellae. For layers with larger relative-attenuation (ε_k , e.g, tendon [2]), compensation of both δ_k and ε_k in superficial layers is necessary for accurate determination of δ_k, ε_k , and θ_k in deeper layers.

A possible constraint of PS-OCT-based approaches for characterizing fiber direction in anisotropic tissue is that current techniques are sensitive only to the component of fiber orientation within the plane perpendicular to the probing beam. Although not a limitation of this study due to the approximately planar structure of annulus fibrosis over the small region imaged, future characterization of fiber direction in more dimensionally-complex tissues (e.g., skin, articular cartilage) should be analyzed with prior knowledge of this constraint.

In this study, lamella thickness (Δz_k ) was determined by inspection of the OCT intensity image [Fig. 4(a)] because contrast was sufficient to identify lamellae as alternating bright and dark bands. However, potentially the most effective contrast mechanism for identifying lamellae in annulus fibrosis is the fiber direction (θ_k ) due to its approximately 90° change between successive layers [Figs. 4(b) and 5]. Matcher et al [17] demonstrated using PS-OCT that the abrupt change in cumulative phase retardation vs. depth due to discontinuity in fiber direction could be used to identify the boundary between the two most superficial lamellae (k=1 and 2) in bovine annulus fibrosis.

Alternate methods for characterization of collagen fiber direction have been reported. These include x-ray diffraction [23], electron microscopy [24,25], a microwave-based method [26], and second-harmonic-generation (SHG) techniques [27,28]. For in vivo or in situ measurements, SHG polarimetry [29] is the most promising of these techniques. Yasui et al [30] assessed the sensitivity of SHG polarimetry and other polarimetric methods to collagen fiber orientation in tendon and concluded that SHG polarimetry can effectively detect the absolute collagen fiber orientation; although a quantitative value for the sensitivity of SHG polarimetry was not given. We have shown that EPS-OCT provides high accuracy (~3°) tomographic images of collagen fiber direction deep (>0.5 mm) within multiple layered (K > 10) anisotropic tissue such as annulus fibrosis cartilage. EPS-OCT shows promise as a tool for high resolution, nondestructive, three-dimensional characterization of fibrous tissue ultrastructure.