Imaging the Nonlinear Susceptibility Tensor of Collagen by Nonlinear Optical Stokes Ellipsometry

Abstract

Nonlinear optical Stokes ellipsometric (NOSE) microscopy was demonstrated for the analysis of collagen-rich biological tissues. NOSE is based on polarization-dependent second harmonic generation imaging. NOSE was used to access the molecular-level distribution of collagen fibril orientation relative to the local fiber axis at every position within the field of view. Fibril tilt-angle distribution was investigated by combining the NOSE measurements with ab initio calculations of the predicted molecular nonlinear optical response of a single collagen triple helix. The results were compared with results obtained previously by scanning electron microscopy, nuclear magnetic resonance imaging, and electron tomography. These results were enabled by first measuring the laboratory-frame Jones nonlinear susceptibility tensor, then extending to the local-frame tensor through pixel-by-pixel corrections based on local orientation.

Introduction

Second harmonic generation (SHG) has a rich history in structural analysis of biological tissues (1, 2, 3, 4, 5), having been used to visualize structures including microtubule assemblies in brain tissue (6) and Caenorhabditis elegans embryos (7), collagen organization in tumors (8), pericardial tissue (9), human atrial myocardium (10), rat tails (11, 12), corneas (13), and human skin (14), to cite just a select handful of representative studies. SHG has several properties that make it advantageous in biological imaging. Certain biological structures, including collagen, naturally generate SHG, thereby avoiding the need for exogenous labeling with dyes. In addition, SHG is typically performed using infrared (IR) sources, thereby reducing sample damage and increasing penetration depth through biological tissues. SHG is also an instantaneous and coherent process and does not involve absorption and emission as in other nonlinear optical phenomena such as two-photon excited fluorescence. Consequently, SHG does not suffer from photobleaching effects that can limit two-photon excited fluorescence microscopy.

The unique symmetry properties of SHG make it particularly sensitive to polarization-dependent measurements, which can provide rich quantitative information on local structure and organization. The local-frame second-order nonlinear susceptibility tensor, ${\chi }_l^{\left(2\right)}$, which describes the polarization-dependent process of SHG, is a 3 × 3 × 3 tensor containing 27 elements, 18 of which can be unique for SHG. Comparable linear optical methods such as bright field and birefringence imaging contain only three unique parameters. As a result, polarization-dependent SHG imaging can provide a greater amount of structural information compared to conventional optical imaging methods.

Incorporation of polarization-dependent analysis into SHG imaging has been used for the discrimination of different biological tissues (15), differentiation of tumor and normal tissues (16), and many other biological applications as well as the study of surfaces and crystals (17, 18, 19, 20, 21, 22, 23). Despite the success of previous polarization-dependent SHG imaging approaches, two major limitations exist. The first is the time required for polarization analysis. The most common methods of polarization modulation involve manual rotation of optical elements such as polarizers and waveplates, resulting in long measurement times and significant 1/f noise. The relatively long measurement time also limits the scope of the measurement and makes it particularly challenging for in vivo applications, where artifacts and image blurring can arise from sample movement. A second limitation is related to sample orientation. Recovering structural information that is accessible in polarization-dependent SHG measurements often relies on aligning the sample along the horizontal (H) or vertical (V) axis with respect to the laboratory frame, or rotating the incident polarization to align along one of the axes of the sample (21).

Several attempts have been made to increase the speed of polarization-dependent SHG measurements. In early studies by Stoller et al. (15), polarization modulation was performed at 4 kHz using an electrooptic modulator (EOM) and in more recent studies by Tanaka et al. (24), fast polarization modulation was performed by toggling between two input polarizations also using an EOM. Liquid crystal modulators have also been used for modulation of input polarization, but have relatively slow response times (on the order of milliseconds) (25). Passive polarization modulation has also been performed in studies by Muir et al. (26) through orthogonal pulse-pair generation, where two polarization beam splitters were used to generate a pulse train where every other laser pulse was one of two linear orthogonal polarizations. While these techniques offer relatively fast data acquisition times compared to rotation of optical elements, they are typically able to access only a small subset of the total possible set of polarization states required for full nonlinear optical ellipsometry, ultimately resulting in the inability to uniquely recover the local-frame tensor elements of the sample.

In this work, nonlinear optical Stokes ellipsometric (NOSE) imaging is used to extract three unique local-frame nonlinear susceptibility tensor elements for collagen in biological tissues. NOSE is based on polarization-dependent SHG imaging, and utilizes rapid polarization modulation with EOM and analytical modeling to recover the unique polarization-dependent parameters of the sample. This work builds upon previous studies where NOSE imaging was carried out at video rate on a variety of crystalline samples, enabling recovery of parameters directly related to the laboratory-frame Jones tensor (27). Here, we developed an iterative approach to investigate the orientation angles of the collagen structure and extend the analysis to recovery of the local-frame tensor elements. Two angles (the in-plane azimuthal rotation angle ϕ and out-of-plane polar tilt-angle θ) are recovered in the analysis. The recovery of the orientation information of collagen has been one of the overall goals of polarization-dependent SHG measurements (28). The influence of θ on the polarization-dependent nonlinear optical measurements has been explored and discussed qualitatively (29). However, quantitative recovery of θ at each individual pixel of the image has remained an elusive goal. We also extend NOSE to the analysis of biological tissues and demonstrate analytical models for direct recovery of the local-frame tensor on a pixel-by-pixel basis. The experimental results are compared with bottom-up ab initio atomic modeling of the nonlinear optical (NLO) response of collagen triple helices, providing information about the internal arrangement of the collagen fibrils.

Materials and Methods

Assumptions in the model and parameter reduction

The SHG signal generated by collagen originated from the noncentrosymmetric structure within the collagen fibers. Assuming uniaxial local symmetry (C_∞) for collagen fibers, the only unique nonzero tensor elements are ${\chi }_{zzz}^{\left(2\right)},{\chi }_{zxx}^{\left(2\right)},{\chi }_{xxz}^{\left(2\right)},\text{and}{\chi }_{xyz}^{\left(2\right)}$. However, with the incident laser beam propagating perpendicular to the probed sample plane, ${\chi }_{zzz}^{\left(2\right)},{\chi }_{zxx}^{\left(2\right)},\text{and}{\chi }_{xxz}^{\left(2\right)}$ are strongly preferred to the chiral element, ${\chi }_{xyz}^{\left(2\right)}$. Ab initio calculation of the β-tensors of collagen triple helices indicates the ratio ${\beta }_{xxz}^{\left(2\right)}/{\beta }_{xyz}^{\left(2\right)}$ is ∼−50, reflecting the limited contribution from the chiral terms. Consequently, ${\chi }_{xyz}^{\left(2\right)}$ is excluded from the analysis for simplification and clarity. To determine local-frame tensor elements, knowledge of how the sample is oriented with respect to the laboratory frame is required. As a result of the cylindrical (C_∞) symmetry of a collagen fiber, the twist angle, ψ, has no effect on its polarization-dependent response, leaving only consideration of the in-plane fiber rotation, ϕ, and the out-of-plane tilt angle, θ. Most commonly, the individual collagen fibers are considered to be lying within the image plane, resulting in polar tilt angles, θ, of 90°, defined relative to the optical axis. This assumption holds reasonably well for highly aligned collagen tissues such as tendon samples sectioned along the major axis, and is used in numerous previous studies (15, 29, 30). However, for the meshlike collagen structures such as those found in skin tissues and basement membrane, the collagen fibers can no longer be considered to be lying within the imaging plane at each pixel. Therefore, in this study, three different types of collagen tissues were investigated: mouse tail, porcine ear, and porcine skin, and both ϕ and θ are included in the analysis.

Solving the local-frame tensor ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$ for collagen fibers

The matrix-based, mathematical framework is described in detail by X. Y. Dow, E. L. DeWalt, J. A. Newman, C. M. Dettmar, and G. J. Simpson (31) from which the specific configuration of phase modulation with postsample quarter-waveplate is used here. In brief, the polarization-dependent analysis yields five trigonometric functions dependent on the phase of modulation of the EOM, Δ, with five corresponding independent observables, termed “polynomial coefficients”. As indicated in Eq. 1, the five polynomial coefficients are directly related to the laboratory frame Jones tensor elements of the sample:

$$\begin{array}{c}{I}_{L,n}^{2\omega }\left(\frac{\Delta }{2}\right)=A\cdot {\text{cos}}^4\left(\frac{\Delta }{2}\right)+B\cdot {\text{cos}}^3\left(\frac{\Delta }{2}\right)\cdot \text{sin}\left(\frac{\Delta }{2}\right)+C\cdot {\text{cos}}^2\left(\frac{\Delta }{2}\right)\cdot {\text{sin}}^2\left(\frac{\Delta }{2}\right)+D\cdot \text{cos}\left(\frac{\Delta }{2}\right)\cdot {\text{sin}}^3\left(\frac{\Delta }{2}\right)+E\cdot {\text{sin}}^4\left(\frac{\Delta }{2}\right),\\ A_n={\left|\sum _{m=1}^2{M}_{\text{nm}}\cdot {\chi }_{\text{mHH}}^{\left(2\right)}\right|}^2,\\ B_n=-4\cdot \mathrm{Im}\left[\left(\sum _{m=1}^2{M}_{\text{nm}}\cdot {\chi }_{\text{mHH}}^{\left(2\right)}\right)\cdot \left(\sum _{m=1}^2{M}_{\text{nm}}\cdot {\chi }_{\text{mVH}}^{\left(2\right)}\right)\right],\\ C_n=4\cdot {\left|\sum _{m=1}^2{M}_{\text{nm}}\cdot {\chi }_{\text{mVH}}^{\left(2\right)}\right|}^2-2\cdot \mathrm{Re}\left[\left(\sum _{m=1}^2{M}_{\text{nm}}\cdot {\chi }_{\text{mHH}}^{\left(2\right)}\right)\cdot {\left(\sum _{m=1}^2{M}_{\text{nm}}\cdot {\chi }_{\text{mVV}}^{\left(2\right)}\right)}^{\ast }\right],\\ D_n=4\cdot \mathrm{Im}\left[\left(\sum _{m=1}^2{M}_{\text{nm}}\cdot {\chi }_{\text{mVV}}^{\left(2\right)}\right)\cdot \left(\sum _{m=1}^2{M}_{\text{nm}}\cdot {\chi }_{\text{mVH}}^{\left(2\right)}\right)\right],\\ E_n={\left|\sum _{m=1}^2{M}_{\text{nm}}\cdot {\chi }_{\text{mVV}}^{\left(2\right)}\right|}^2.\end{array}$$ (1)

The above equation can be rewritten in the matrix form with ${\stackrel{\rightharpoonup }{A}}_n$ representing the vector of polynomial coefficients, ${\stackrel{\rightharpoonup }{\chi }}_{\mathit{j}}^{\left(2\right)}$ representing the vectorized Jones tensor, and the matrix P_n describing the connection between the two, as shown in

$${\stackrel{\rightharpoonup }{A}}_n=P_n.\left({\stackrel{\rightharpoonup }{\chi }}_{\mathit{j}}^{\left(2\right)}\otimes {\stackrel{\rightharpoonup }{\chi }}_{\mathit{j}}^{\left(2\right)}\right).$$ (2)

The Jones tensors can be further related to the vectorized local-frame tensors ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$ through coordinate transformation (assuming that the local field factors yield only an overall scaling factor, consistent with a locally isotropic dielectric environment). This relationship is reflected by the following equation, with the bold character J indicating the expanded form given by the Kronecker product of three matrices:

$${\stackrel{\rightharpoonup }{\chi }}_j^{\left(2\right)}=J_{\left(\theta ,\phi ,\varphi \right)}\cdot {\stackrel{\rightharpoonup }{\chi }}_l^{{\left(2\right)}^{\prime }}=\left\{\left(\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right]\otimes \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right]\otimes \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right]\right)\cdot \left(R\left(\theta ,\psi ,\varphi \right)\otimes R\left(\theta ,\psi ,\varphi \right)\otimes R\left(\theta ,\psi ,\varphi \right)\right)\right\}\cdot {\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}.$$ (3)

Finally, the full set of 27 local-frame tensor ${\stackrel{\rightharpoonup }{\chi }}_{\mathit{l}}^{{\left(2\right)}^{\prime }}$ can be recast using just the unique nonzero tensor elements given by the unprimed vector ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$ through a 27 × 3 symmetry matrix Q:

$${\stackrel{\rightharpoonup }{\chi }}_{\mathit{l}}^{{\left(2\right)}^{\prime }}=Q\cdot {\stackrel{\rightharpoonup }{\chi }}_{\mathit{l}}^{\left(2\right)}.$$ (4)

Most commonly, polarization-dependent SHG measurements are performed to evaluate two classes of sample properties: (1) What is the local sample orientation given the known nonlinear susceptibility tensors in the local frame? (2) What are the local-frame tensors given the orientation angles? Using the theoretical framework described above and by X. Y. Dow, E. L. DeWalt, J. A. Newman, C. M. Dettmar, and G. J. Simpson (31), NOSE analysis can iteratively solve for both the sample orientation and local-frame tensors from one set of polarization-dependent SHG measurements. Based on different sample systems and target applications, two complementary computational approaches were developed, as outlined in Fig. 1.

Figure 1.

General process of solving for a global ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$ and sample orientation using the iterative analysis or for solving for ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$ at every pixel using the pixel-by-pixel analysis.

For fast evaluation of the overall nonlinear optical properties, an iterative algorithm can be used to solve for both a global set of local tensor and sample orientation. A pooled analysis approach was developed to identify a single global set of local-frame ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$ in an image corrected for the differences in orientation on a pixel-by-pixel basis. Vectors and matrices containing q independently oriented data points were constructed in this pooled analysis, and the fitting was performed to recover one universal ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$, as shown in Eq. 5, where S_H and S_V are detector sensitivities for the H and V detector, respectively:

$$\left[\begin{array}{c}{\stackrel{\rightharpoonup }{A}}_{H1}\\ {\stackrel{\rightharpoonup }{A}}_{V1}\\ {\stackrel{\rightharpoonup }{A}}_{H2}\\ {\stackrel{\rightharpoonup }{A}}_{V2}\\ ⋮\\ {\stackrel{\rightharpoonup }{A}}_{Hq}\\ {\stackrel{\rightharpoonup }{A}}_{Vq}\end{array}\right]=\left[\begin{array}{c}\left[\begin{array}{cc}P\cdot S_H& 0\\ 0& P\cdot S_V\end{array}\right]\cdot \left[\begin{array}{c}\left(J_{H,{\varphi }_1}\right)\otimes \left(J_{H,{\varphi }_1}\right)\\ \left(J_{V,{\varphi }_1}\right)\otimes \left(J_{V,{\varphi }_1}\right)\end{array}\right]\\ \left[\begin{array}{cc}P\cdot S_H& 0\\ 0& P\cdot S_V\end{array}\right]\cdot \left[\begin{array}{c}\left(J_{H,{\varphi }_2}\right)\otimes \left(J_{H,{\varphi }_2}\right)\\ \left(J_{V,{\varphi }_2}\right)\otimes \left(J_{V,{\varphi }_2}\right)\end{array}\right]\\ ⋮\\ \left[\begin{array}{cc}P\cdot S_H& 0\\ 0& P\cdot S_V\end{array}\right]\cdot \left[\begin{array}{c}\left(J_{H,{\varphi }_q}\right)\otimes \left(J_{H,{\varphi }_q}\right)\\ \left(J_{V,{\varphi }_q}\right)\otimes \left(J_{V,{\varphi }_q}\right)\end{array}\right]\end{array}\right]\cdot \left(Q\otimes Q\right)\cdot \left({\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}\otimes {\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}\right).$$ (5)

An iterative algorithm to recover the local-frame tensor and refine orientations for the pooled data set can be employed in the pooled analysis. The algorithm iterates between solving for the sample orientation at each pixel and solving for the global set of tensor elements most consistent with the collective polarization-dependent SHG response. In the first step, the unique collagen tensor elements (${\chi }_{xxz}^{\left(2\right)}$, ${\chi }_{zxx}^{\left(2\right)}$, and ${\chi }_{zzz}^{\left(2\right)}$) can be assumed explicitly, based on reported literature values (20, 21, 32, 33), and a nonlinear least-squares fit to sample orientation is performed at each individual pixel. Initial guess values for ϕ were determined by image analysis using the OrientationJ plug-in for ImageJ (34), and initial guess values for θ were chosen arbitrarily to be 40°. After minimization, the recovered values of ϕ and θ were then used to determine the J matrices in Eq. 3. Another nonlinear least-squares fit of the observed polynomial coefficients to the unique tensor elements was performed to recover a global set of local-frame tensor describing the pooled data set. These two steps were then iterated until the system converged on values of ϕ and θ for every pixel, and a global set of elements in ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$.

In the pooled analysis described in Eq. 5, it was assumed that one global set of local-frame tensor elements could be used to model collagen at every pixel to reduce the uncertainty in the recovered ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$. However, this method may not be applicable for structurally diverse samples, such as the biological tissues considered here. A pixel-by-pixel analysis approach was also employed to assess intrasample variation. In this approach, the orientation angles at each individual pixel recovered from the pooled analysis were used here. A subsequent unique set of ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$ was calculated using nonlinear least-squares fit at each individual pixel, generating a tensor map of each field of view (FOV).

Relating ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$ to the molecular tensor, ${\stackrel{\rightharpoonup }{\beta }}^{\left(2\right)}$ and probing the local order

Interpreting differences in ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$ can provide information regarding the structure of the collagen fibers within tissues. The structural hierarchy of collagen is illustrated in Fig. 2, where a model amide chromophore, N-methylacetamide (NMA) (a), is considered to be the dominant NLO chromophore within the collagen triple helix (b), consistent with previous experimental and computational determinations (35, 36). Triple helices tightly pack together to form collagen fibrils (c), and individual collagen fibrils assemble to form a larger collagen fiber (d). A hyperellipsoid representation of ${\chi }_l^{\left(2\right)}$ and the molecular tensor as ${\beta }^{\left(2\right)}$ is also shown in Fig. 2, which describes the relative local-frame tensor elements of the structures (37). From the breadth of work on analysis of collagen fiber substructure (20, 33, 38, 39, 40, 41), it is apparent that a distribution of tilt angles of collagen triple helices away from the principal fiber axis can be expected. This is visualized conceptually in Fig. 2 where the individual fibrils are shown exhibiting a distribution in tilt angles within the fiber.

Figure 2.

Hyperellipsoid representation of hyperpolarizability tensor of (a) NMA; (b) a collagen triple helix; (c) a collagen fibril; and (d) the second-order susceptibility tensor of collagen, which exhibits a distribution in fibril tilt angles. To see this figure in color, go online.

In theory, the internal distribution of the triple helix with respect to the fiber axis can be probed through the connection of the measured ${\chi }_l^{\left(2\right)}$ of the fiber to the molecular structural information of a single collagen triple helix. In the experiments described in this work, the ${\chi }_l^{\left(2\right)}$ of the fiber can be measured, while the molecular tensor ${\beta }^{\left(2\right)}$ of the triple helix is challenging to unambiguously access experimentally. Previous work has shown good agreement between ab initio predictions of molecular nonlinear susceptibility tensors, ${\beta }^{\left(2\right)}$ and experimental results (36, 42). Consequently, theoretical predictions of ${\beta }^{\left(2\right)}$ for a single collagen triple helix were combined with experimental methods to determine ${\chi }_l^{\left(2\right)}$ for a fiber, for us to gain insight into helical organization within a fiber. For a collagen triple helix specifically, the nonlinear polarizability tensor, ${\beta }^{\left(2\right)}$ was calculated using the symmetry additive model (36, 37), in which ab initio calculations are used to first predict the nonlinear polarizability of NMA, which serves as a model system for the amide bonds of a protein. The amide contributions were then coherently summed for a single collagen triple helix. The validity of the symmetry-additive model has been confirmed with experimental hyper-Rayleigh scattering experiments of polypeptides (43), and has been used previously for SHG measurements of collagen (35). If the triple helices are all perfectly aligned within a collagen fiber, ${\chi }_l^{\left(2\right)}$ is directly proportional to ${\beta }^{\left(2\right)}$ and consequently straight fibers (those with triple-helices aligned and ordered with respect to the fiber axis) should have measured relative values of ${\chi }_l^{\left(2\right)}$ that are similar to theoretically predicted values of ${\beta }^{\left(2\right)}$ (20, 33).

Transformation of the triple helix and fiber reference frames indicate that variation in the tilt angles of the helices away from the fiber axis is expected to yield differences between the experimentally observed ${\chi }_l^{\left(2\right)}$ and ${\beta }^{\left(2\right)}$ for an individual helix. Two key assumptions were made to simplify the relationships connecting ${\chi }_l^{\left(2\right)}$ and ${\beta }^{\left(2\right)}$ for the fibers: (1) the formal C₃ symmetry of the triple helix was assumed to reasonably approximated by C_∞ symmetry in an ensembles of triple helices; and (2) the chiral-specific contributions to the observed polarization-dependence were assumed to be negligible, consistent with the results of the modeling calculations. Within the validity of these assumptions, the projection of individual elements of ${\beta }^{\left(2\right)}$ onto ${\chi }_{zzz}^{\left(2\right)}$ (one of the three unique nonzero local-frame tensor elements of collagen, considered explicitly as an example), is given by

$${\chi }_{zzz}^{\left(2\right)}=N_s\cdot \left\{\begin{array}{l}\langle {\text{cos}}^3{\theta }^{\prime }\rangle \cdot {\beta }_{z^{\prime }{z}^{\prime }{z}^{\prime }}^{\left(2\right)}\\ +\langle {\text{sin}}^2{\theta }^{\prime }\cdot \text{cos}{\theta }^{\prime }\cdot {\text{sin}}^2{\psi }^{\prime }\rangle \cdot \left({\beta }_{y^{\prime }{y}^{\prime }{z}^{\prime }}^{\left(2\right)}+{\beta }_{y^{\prime }{z}^{\prime }{y}^{\prime }}^{\left(2\right)}+{\beta }_{z^{\prime }{y}^{\prime }{y}^{\prime }}^{\left(2\right)}\right)\\ +\langle {\text{sin}}^2{\theta }^{\prime }\cdot \text{cos}{\theta }^{\prime }\cdot {\text{cos}}^2{\psi }^{\prime }\rangle \cdot \left({\beta }_{x^{\prime }{x}^{\prime }{z}^{\prime }}^{\left(2\right)}+{\beta }_{x^{\prime }{z}^{\prime }{x}^{\prime }}^{\left(2\right)}+{\beta }_{z^{\prime }{x}^{\prime }{x}^{\prime }}^{\left(2\right)}\right)\end{array}\right\}.$$ (6)

The primed coordinates on β (x^′, y^′, and z^′) indicate the local reference frame of the triple helix, where the unprimed coordinates (x, y, and z) indicate the local reference frame of the collagen fiber. In Eq. 6, N_s is the number density of the helices, θ^′ is the tilt angle away from the fiber axis, and ψ^′ is the rotation about the fiber axis. Collagen fibers have C_∞ symmetry, and as a result the dependence on ψ^′ disappears. In addition, the equalities dictated by the effective C_∞ of a collagen triple helix (${\beta }_{y^{\prime }{y}^{\prime }{z}^{\prime }}^{\left(2\right)}={\beta }_{x^{\prime }{x}^{\prime }{z}^{\prime }}^{\left(2\right)}$, ${\beta }_{y^{\prime }{z}^{\prime }{y}^{\prime }}^{\left(2\right)}={\beta }_{x^{\prime }{z}^{\prime }{x}^{\prime }}^{\left(2\right)}$, and ${\beta }_{z^{\prime }{y}^{\prime }{y}^{\prime }}^{\left(2\right)}={\beta }_{z^{\prime }{x}^{\prime }{x}^{\prime }}^{\left(2\right)}$) result in the simplified expression shown in

$${\chi }_{zzz}^{\left(2\right)}=N_s\cdot \left\{\frac{1}{2}\cdot \langle {\text{cos}}^3{\theta }^{\prime }\rangle \cdot {\beta }_{z^{\prime }{z}^{\prime }{z}^{\prime }}^{\left(2\right)}+\langle {\text{sin}}^2{\theta }^{\prime }\cdot \text{cos}{\theta }^{\prime }\rangle \cdot {\beta }_{x^{\prime }{x}^{\prime }{z}^{\prime }}^{\left(2\right)}{\beta }_{x^{\prime }{x}^{\prime }{z}^{\prime }}+\frac{1}{2}\cdot \langle {\text{sin}}^2{\theta }^{\prime }\cdot \text{cos}{\theta }^{\prime }\rangle \cdot {\beta }_{z^{\prime }{x}^{\prime }{x}^{\prime }}^{\left(2\right)}{\beta }_{z^{\prime }{x}^{\prime }{x}^{\prime }}\right\}.$$ (7)

Similar expressions can also be derived for ${\chi }_{xxz}^{\left(2\right)}$ and ${\chi }_{zxx}^{\left(2\right)}$, and a conversion matrix K can be constructed to bridge the calculated molecular tensor of a model collagen triple helix, ${\beta }^{\left(2\right)}$, and the experimentally determined ${\chi }_l^{\left(2\right)}$, as shown in

$${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}=N_s\cdot K\cdot {\stackrel{\rightharpoonup }{\beta }}^{\left(2\right)}=N_s\cdot \left[\begin{array}{ccc}\langle {\text{cos}}^3{\theta }^{\prime }\rangle & \langle {\text{sin}}^2{\theta }^{\prime }\cdot \text{cos}{\theta }^{\prime }\rangle & 2\cdot \langle {\text{sin}}^2{\theta }^{\prime }\cdot \text{cos}{\theta }^{\prime }\rangle \\ \frac{\langle {\text{sin}}^2{\theta }^{\prime }\cdot \text{cos}{\theta }^{\prime }\rangle }{2}& \frac{-\langle {\text{sin}}^2{\theta }^{\prime }\cdot \text{cos}{\theta }^{\prime }\rangle }{2}+\langle \text{cos}{\theta }^{\prime }\rangle & -\langle {\text{sin}}^2{\theta }^{\prime }\cdot \text{cos}{\theta }^{\prime }\rangle \\ \frac{\langle {\text{sin}}^2{\theta }^{\prime }\cdot \text{cos}{\theta }^{\prime }\rangle }{2}& \frac{-\langle {\text{sin}}^2{\theta }^{\prime }\cdot \text{cos}{\theta }^{\prime }\rangle }{2}& -\langle {\text{sin}}^2{\theta }^{\prime }\cdot \text{cos}{\theta }^{\prime }\rangle +\langle \text{cos}{\theta }^{\prime }\rangle \end{array}\right]\cdot \left[\begin{array}{c}{\beta }_{z^{\prime },z^{\prime },z^{\prime }}^{\left(2\right)}\\ {\beta }_{z^{\prime },x^{\prime },x^{\prime }}^{\left(2\right)}\\ {\beta }_{x^{\prime },x^{\prime },z^{\prime }}^{\left(2\right)}\end{array}\right].$$ (8)

Experimental methods

Sample preparation

Mouse tails were obtained from the Purdue University Center for Cancer Research’s Transgenic Mouse Core Facility (under Purdue Animal Care and Use Committee protocol No. 1111000314). Porcine skin and ear samples were gifted from Professor Jonathan Wilker (Purdue University, West Lafayette, IN). Porcine ear samples were obtained from the surface of the ear, and were likely composed largely of skin tissue. The samples were frozen and thinly cryosectioned at 5 and 10 μm to minimize effects from birefringence, and thaw-mounted to glass microscope slides. Sections were stored at −80°C before analysis. Immediately before imaging, frozen sections were allowed to come to room temperature and 10 μL of phosphate buffered saline was added to each section and sealed with a coverslip to prevent sample dehydration during NOSE imaging.

Instrumentation

SHG and laser transmittance signals were collected using a custom microscope as described in detail in DeWalt et al. (27) and shown schematically in Fig. 3. Briefly, the excitation was accomplished by a mode-locked Ti:Sapphire laser (Spectra-Physics, Mountain View, CA) operating at 80 MHz with a pulse duration of 100 fs. A wavelength of 800 nm and average powers of 60–140 mW were used during data acquisition. Beam scanning was performed to sample through the field of view with a resonant mirror operating at 8 kHz (EOPC, Fresh Meadows, NY) in the fast axis and a galvanometer mirror (Cambridge Technology, Bedford, MA) in the slow axis. The beam was passed through an EOM (Conoptics, Danbury, CT) at 45° from its fast-axis. A Soleil-Babinet compensator (Thorlabs, Newton, NJ) was placed after the EOM to correct for polarization changes induced by the beam path and optical components. The beam was then focused onto the sample by a 40×, 0.75 numerical aperture (NA) objective (Nikon, Melville, NY) or a 10× 0.3 NA objective (Nikon) and collected by a 10× 0.35 NA long working distance objective (Qioptiq, Waltham, MA). Fundamental light was separated from SHG using a dichroic mirror and directed through a horizontal polarizer, and collected in the transmittance direction by a photodiode (DET-10A; Thorlabs). The reflected SHG signal was passed through a quarter waveplate rotated 22.5° from its fast-axis and then separated into its horizontal and vertical components with a Glan-Taylor polarizer and vertically and horizontally polarized SHG intensities were detected by two separate photomultiplier tubes (PMTs) (H12310-40; Hamamatsu, Hamamatsu City, Japan) with bandpass filters (HQ 400/20M-2P; Chroma Technology, Bellows Falls, VT) and a colored glass KG3 filter to further reject the fundamental light. This configuration allowed simultaneous detection of multiple channels of SHG and laser transmittance.

Figure 3.

Instrument and timing schematic for NOSE instrumentation capable of rapid polarization modulation and synchronous digitization. To see this figure in color, go online.

Synchronous data acquisition and polarization modulation

Data acquisition and polarization modulation were performed synchronously with the laser. The principle of synchronous data acquisition and polarization modulation has been described in detail in Muir et al. (26), and is summarized briefly here: Three channels were digitized synchronously with the laser using PCIe digitizer cards (ATS-9350; AlazarTech, Pointe-Claire, Quebec, Canada) with the 80 MHz signal from the laser as the master clock. The laser clock was sent through a custom timing control module to produce a 10 MHz signal to be used to communicate with all timing-dependent components including the resonant mirror, the digitizer cards, and the function generator used to drive EOM. The signal transients from each individual detector response from every laser pulse were digitized synchronously with the laser. A custom 3–13-ns electronic digital delay circuit was added to allow adjustment of the relative delay between the laser pulse and signal arrival at the digitizer cards. The EOM was driven by a function generator synchronized with the 10 MHz phase lock loop at 8 MHz, resulting in a total of 10 unique elliptical polarizations, with every laser pulse cycling among one of the 10 polarizations. Custom software (MATLAB, Natick, MA) was developed to reconstruct 10 polarization images. To ensure all 10 polarizations were sampled in each pixel within one resonant mirror trajectory, laser pulses were binned to generate 316 × 316 pixel images. The first step in the extraction of tensor elements from the sample is characterization of the input polarization. The time-varying phase angle of the EOM, Δ, was measured for every imaging session simultaneously with SHG through measurement of the transmitted polarized fundamental light. A nonlinear fit of the experimental transmitted laser intensity to the theoretical polarized laser transmittance intensity was performed to extract Δ, as described in detail in Simpson et al. (44).

NOSE imaging

NOSE imaging was performed on several FOVs for three tissue types: porcine skin, porcine ear, and mouse tail. Average powers between 60 and 140 mW and acquisition times between 30 and 100 s were used, depending on the sample. NOSE imaging was performed on 5 and 10 μm sections of mouse tail, with one FOV imaged using 40× magnification and two FOVs using 10× magnification. NOSE imaging was performed for three FOVs of 10 μm sections of porcine skin (two using 40× magnification, and one using 10× magnification) and four FOVs of 10 μm sections of porcine ear (two using 40× magnification, and two using 10× magnification). For each imaging session, 10 polarization-dependent SHG images were acquired for each detector (20 total) along with 10 polarization-dependent laser transmittance images. The total time required for acquisition of the full polarization-dependent data set for all three detectors was 30–100 s for a single FOV, corresponding to an average pixel dwell time of <1 ms.

Quantum chemical calculations

The predicted nonlinear polarizability of collagen was generated by calculating the NLO properties of NMA, which was used as a model system for the amide bonds within proteins. The symmetry additive model was then applied, in which the amide contributions from the amino-acid linkages in collagen were coherently summed (35, 45). The input geometry of the NMA was optimized using density functional theory (DFT) calculations with the BL3YP functional (46). The SHG hyperpolarizability of NMA was calculated with a driving frequency of 800 nm using the time-dependent Hartree-Fock (TDHF) method (47) as well as the time-dependent DFT (TDDFT) method. All calculations were performed using GAMESS (Ver. May.01.2014.R1) with the 6-311++G^∗∗ basis set. Calculations were also performed using the long-range corrected Becke 88 exchange functional with the Lee Yang Parr correlation functional (LC-BLYP) and the Becke 88 exchange with a one-parameter progressive correlation (LC-BOP) (48). Long-range corrected functionals were studied to reduce the overestimation of hyperpolarizabilities associated with conventional DFT under the local density approximation (49). A collagenlike peptide, (Pro-Pro-Gly)₁₀ (Protein Data Bank (PDB): 1K6F), was chosen as a model for a single collagen triple helix due to its similarity to the collagen triple helix structure (35). The amide contributions predicted from the quantum chemical calculations of NMA were coherently summed for the collagenlike peptide structure using the NLOPredict plugin in Chimera (37) to generate the triple helix ${\beta }^{\left(2\right)}$ tensor. The results of the ratios of ${\beta }_{z^{\prime }{z}^{\prime }{z}^{\prime }}^{\left(2\right)}/{\beta }_{z^{\prime }{x}^{\prime }{x}^{\prime }}^{\left(2\right)}$ for a collagen triple helix calculated using this approach are listed in Table 1. These results indicate that ${\beta }_{z^{\prime }{z}^{\prime }{z}^{\prime }}^{\left(2\right)}$ is the major contributor to the nonlinear optical response of the triple helix, while ${\beta }_{z^{\prime }{x}^{\prime }{x}^{\prime }}^{\left(2\right)}$ and ${\beta }_{x^{\prime }{x}^{\prime }{z}^{\prime }}^{\left(2\right)}$ are significantly smaller in magnitude in all calculations. These results are in good qualitative agreement with previously reported calculations for collagen fibers, which suggest that the ${\beta }_{z^{\prime }{z}^{\prime }{z}^{\prime }}^{\left(2\right)}$ tensor element is significantly larger in magnitude than other contributing elements (20, 49). Previous studies have found good agreement between the experimentally measured hyper-Rayleigh scattering (HRS) intensity and the calculated tensors for collagen triple helix. Long-range correction functionals were also employed to evaluate the effect of electron correlation by comparing the results from TDHF and long-range corrected methods. The trend of dominating ${\chi }_{l,zzz}^{\left(2\right)}$ was observed for both TDHF and long-range corrected sets of calculations even though the recovered amplitude is different between the long-range corrected calculations and TDHF. This result indicates that the electron correlation effect does has some weak influence on the recovered molecular tensors; however, the trend of the molecular tensor elements dictating the polarization-dependence were still reasonably recovered using TDHF.

Table 1.

The Ratios of Molecular Tensors Recovered from Ab Initio Calculation

	${\beta }_{z^{\prime }{z}^{\prime }{z}^{\prime }}^{\left(2\right)}/{\beta }_{z^{\prime }{x}^{\prime }{x}^{\prime }}^{\left(2\right)}$
TDDFT-B3LYP	4.5
TDHF	−25
LC-BLYP	−14
LC-BOP	−13
ONIOMa	2.97

^aResults using the ONIOM scheme were adopted from previous work by de Wergifosse et al. (49).

While it may initially appear overly simplistic to represent the hyperpolarizability of collagen using calculations for just the NMA building block, previous computational and experimental studies support the validity of this approach (37, 49, 50, 51). For all but the aromatic amino acids, the hyperpolarizabilities of polypeptides formed from the common amino acids are dominated by the amide unit connecting amino acid groups, which exhibits a relatively large hyperpolarizability. In the case of collagen, incorporation of aromatic amino acids is low, with the major composition dominated by proline, hydroxyproline, and glycine. Close to electronic resonance, significant perturbations to the hyperpolarizability of the complex can arise from interchromophore coupling. However, far from resonance, these perturbations become insignificant, converging to the results obtained in the absence of coupling, consistent with the symmetry additive model used herein for collagen. Because full high-level quantum chemical calculations are prohibitively expensive on an intact triple helix but achievable on the amide chromophore, this stepwise approach represents a reasonable compromise between computational cost and accuracy in the recovered fibril hyperpolarizability. Although previous studies have reported errors in the absolute hyperpolarizabilities in Hartree-Fock (HF) calculations of charge-transfer chromophores from neglect of electron correlation effects, the overall polarization dependence does not appear to be as sensitively dependent on the level of theory (52). Similarly, solvent effects have also been shown to impact the magnitude of the hyperpolarizability in charge-transfer chromophores, but less information is available on the impact of such factors on the polarization-dependence (53). However, it is reasonable to assume that an isotropic solvent would scale all tensor elements proportionally. Consequently, errors in the ab initio calculations are not likely to be the dominant source of uncertainty in the analysis.

Results and Discussion

A set of representative results illustrating the measurement and analysis pipeline is shown in Fig. 4. The initial step in NOSE analysis involves characterization of the input polarization using the polarization-modulated transmitted IR beam. A nonlinear fit of the transmitted IR intensity as a function of the phase delay value is used to recover the relative phase delay of each of the 10 unique polarization states. The recovered phase values are then used in processing the SHG images. The perturbation in polarization from the dichroic mirror was determined experimentally to be sufficiently subtle to have no quantitative impact on the measured polarization-dependence (within experimental error). Details of the characterization can be found in the Supporting Material. Using the phase values recovered previously, the polarization-dependent SHG intensity information was converted to Fourier coefficients through linear fitting. This process is shown in Fig. 4 for a single FOV of porcine ear for the vertically polarized SHG detector. The set of 10 SHG images are shown in Fig. 4 a, and a representative fit to a single pixel to extract Fourier coefficients is shown in Fig. 4 b. Fourier coefficient images are shown in Fig. 4 c, with a unique color assigned to each coefficient. The Fourier images were overlaid to generate a single color map depicting the differences in Fourier coefficients as a function of location within the sample, shown in Fig. 4 d. Fourier coefficient color maps for three representative FOVs of each collagen sample are shown in Fig. 5 for both the horizontal and vertical detectors. To more easily visualize the relative differences in coefficients, Fourier coefficient a is displayed on an intensity scale that is two times less sensitive than the other four coefficients in Figs. 4 and 5. Subsequent conversion to polynomial coefficients was accomplished through a simple matrix multiplication before solving for local tensor elements, as described previously in DeWalt et al. (27). Contrast in the Fourier coefficient color maps of the collagenous tissues is largely dependent on the orientation of the fibers within the FOV. Accordingly, a coordinate transformation between the laboratory frame and the local frame is necessary to determine the local-frame tensor elements in ${\chi }_l^{\left(2\right)}$ and compare the orientation-independent properties of the samples. Two complementary approaches for the recovery of the local-frame tensors are demonstrated in this work.

Figure 4.

The process of linear fitting to recover Fourier coefficients for collagen fibers in a porcine ear sample. A set of 10 unique polarization-dependent images for the vertical PMT are shown in (a) and a representative linear fit to recover Fourier coefficients for a single pixel is shown in (b). The five Fourier coefficient images were assigned a unique color (red, green, blue, cyan, and magenta) (c) before being merged into a single five-color image, shown in (d). To see this figure in color, go online.

Figure 5.

Fourier coefficient color maps for mouse tail, porcine ear, and porcine skin for the horizontal and vertical detectors (H-PMT and V-PMT, respectively). To see this figure in color, go online.

Pooled analysis for determination of global ${\chi }_l^{\left(2\right)}$ and sample orientation

In the pooled analysis, the algorithm is designed to recover the one most probable global set of tensor elements that describes the overall local-frame nonlinear optical susceptibility of the SHG-active pixels within the field of view. The polarization-dependent response from an entire FOV was used in each iterative analysis. The in-plane rotation angles, ϕ, and out-of-plane tilt angle, θ, were identified at each pixel by iteratively solving for the global tensor elements and the local sample orientation, as described in detail in Solving the Local-Frame Tensor ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$ for Collagen Fibers. Initial guess values for the angle ϕ of individual pixels within an image were obtained using the OrientationJ plugin in ImageJ (34), in which local orientation is determined based on image texture analysis of the integrated SHG intensity images. Guess values for θ were set to be 40°, chosen to ensure solutions in the quadrant between 0° and 90°. After convergence of the iterative analysis, a set of three local-frame tensor elements for collagen was obtained from each FOV as well as the orientation angle ϕ and θ at each pixel.

The quantitative results from the polarization analysis agree well with the established differences in morphology of the collagen structures for the different tissues. Representative results for ϕ from one FOV of a mouse tail section are shown in Fig. 6. The out-of-plane tilt angle was recovered at each pixel for all FOVs, the normalized histogram for the out-of-plane tilt angle θ for the three different types of tissue are plotted in Fig. 7, and the y axis of the plot was truncated to better visualize the difference among the three different types of tissues. Even though all three types of samples share a major peak at ∼90°, skin and ear tissues have a significant higher variance of out-of-plane tilt angles compared to the tail tissues. The result suggests that 79% of the pixels from the tail tissue have a θ-angle between 85° and 95°, while the ear and skin tissues have 28 and 43% of pixels that fall into this range, respectively. In all cases, good agreement was observed between angles obtained using OrientationJ and those obtained independently from the per-pixel polarization measurements using the iterative tensor analysis.

Figure 6.

Orientation images of the azimuthal angle ϕ for a single FOV of mouse tail section (left) from OrientationJ and (right) from the iterative analysis. To see this figure in color, go online.

Figure 7.

The out-of-plane tilt angle of the collagen fibers from three different types of samples. To see this figure in color, go online.

A set of the three nonzero elements in ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$ for collagen was also obtained for each FOV for each sample, and are reported in Fig. 8 along with the 95% confidence intervals from the sample set of three/four FOV for each tissue type. Relative standard deviations (SDs) in the nonlinear fit to retrieve individual tensor elements from the products were ∼17%. The average ratios of the tensor elements ${\chi }_{zzz}/{\chi }_{zxx}$ and ${\chi }_{xxz}/{\chi }_{zxx}$ for tail, ear, and skin samples are estimated to be (1.45, 1.04), (1.55, 1.18), and (1.65, 1.24), respectively. These pooled results are in good agreement with previously reported values (20, 54, 55).

Figure 8.

Normalized local-frame tensor elements for the three tissue types for (a) pooled analysis and (b) pixel-by-pixel analysis. Error bars indicate 95% CI within each sample set. To see this figure in color, go online.

Pixel-by-pixel recovery of ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$ to assess intrasample variation

From simple inspection of the SHG images, it is evident that the collagen fibers exhibit substantial structural variations within the different fields of view for the different tissues. Therefore, it is arguably more meaningful to recover the local-frame tensor elements at each pixel rather than just the pooled response. Accordingly, a pixel-by-pixel analysis was used to generate a local-frame tensor map for each set of images in addition to the pooled analysis described in the preceding section for determining orientation maps of ϕ and θ. In the pooled analysis, OrientationJ was used to provide an initial estimation of the azimuthal orientation angles, with the final polar and azimuthal angles recovered subsequently using the iterative algorithm at each pixel through a nonlinear fit. For the pixel-by-pixel analysis, only the recovered orientation angles from polarization analysis were further used in subsequent analysis. Values for ϕ and θ recovered from the pooled analysis were used directly to build the J_θϕ coordinate transformation matrix, and the intensity at each pixel was analyzed by a nonlinear fit to minimize the differences in theoretical and experimental polynomial coefficients, as described in Solving the Local-Frame Tensor ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$ for Collagen Fibers. In this analysis, three unique tensor elements were calculated at every pixel in the FOV. Compared to pooled analysis, higher mean relative SD in the recovered tensor elements was observed in the per-pixel analysis (∼61%), estimated by the mode of the histograms of the errors at each pixel. This is largely attributed to the substantial reduction in measurement integration time when considering a single pixel versus the combined pixels in the pooled analysis. Typical data acquisition times for complete polarization analysis were ∼1.5 μs per pixel, with minimum acquisition times of 150 ns possible for video-rate measurements. Fortunately, the use of synchronous digitization coupled with fast polarization modulation greatly reduces the 1/f noise and enables recovery of sufficient signal/noise (SNR) at each pixel to acquire statistically significant results. In addition, the uncertainties associated with the recovered ϕ- and θ-angles are also propagated through the analysis and reflected in the final tensor results.

Images of calculated tensor elements for a representative FOV of each collagen sample are shown in Fig. 9. Each image in Fig. 9 was generated by assigning the three tensor images, ${\chi }_{xxz}^{\left(2\right)}$, ${\chi }_{zxx}^{\left(2\right)}$, and ${\chi }_{zzz}^{\left(2\right)}$, a unique color (green, blue, and red) before merging them into a single RGB image. Overall intensity of the sample is represented by the brightness at each pixel, where the hue represents the relative ratios among ${\chi }_{xxz}^{\left(2\right)}$, ${\chi }_{zxx}^{\left(2\right)}$, and ${\chi }_{zzz}^{\left(2\right)}$. The mean values of ${\chi }_{xxz}^{\left(2\right)}$, ${\chi }_{zxx}^{\left(2\right)}$, and ${\chi }_{zxx}^{\left(2\right)}$ were calculated for each FOV of each sample, and are shown and compared to the results from the pooled analysis in Fig. 8 b. The different hue across the image indicates significant structural variation. For example, in the mouse tail tissue, an area with relatively high ${\chi }_{zxx}^{\left(2\right)}$ values was observed. The mean values of the tensor elements ${\chi }_{zxx}^{\left(2\right)}$ and ${\chi }_{zzz}^{\left(2\right)}$ are similar between the pooled and pixel-by-pixel analysis, while the SD of the result is significantly larger in the single pixel analysis, reflecting variance among different pixels within the FOV. The average ratios ${\chi }_{zzz}^{\left(2\right)}/{\chi }_{zxx}^{\left(2\right)}$ and ${\chi }_{xxz}^{\left(2\right)}/{\chi }_{zxx}^{\left(2\right)}$ recovered in the pixel-by-pixel analysis are also similar to the results from pooled analysis. The values of ${\chi }_{zzz}^{\left(2\right)}/{\chi }_{zxx}^{\left(2\right)}$ and ${\chi }_{xxz}^{\left(2\right)}/{\chi }_{zxx}^{\left(2\right)}$ for mouse tail, porcine ear, and porcine skin samples were (1.38, 1.19), (1.82, 1.40), and (1.80, 1.37), respectively. Images of the tensor elements ratio are included in the Supporting Material. It is worth mentioning that for the fibers that align coparallel with the x axis in the laboratory frame (corresponding to ϕ = 0°), higher uncertainties are observed for the recovered ϕ-angles and subsequently the recovered tensor elements. This is attributed to the fact that when the percent error in the nonlinear fit is estimated relative to the mean value, division of a small ϕ-angle yielded a relatively high uncertainty.

Figure 9.

Images representing the relationships between the three unique local frame tensor elements, ${\chi }_{xxz}^{\left(2\right)}$, ${\chi }_{zxx}^{\left(2\right)}$, and ${\chi }_{zzz}^{\left(2\right)}$ for the three tissue types. Images were created by assigning each tensor image acquired from the pixel-by-pixel analysis a unique color (green, blue, and red for ${\chi }_{xxz}^{\left(2\right)}$, ${\chi }_{zxx}^{\left(2\right)}$, and ${\chi }_{zzz}^{\left(2\right)}$, respectively) and overlaying them on the same intensity scale. Image brightness represents overall SHG intensity and the hue of the image represents the relative magnitudes of ${\chi }_{xxz}^{\left(2\right)}$, ${\chi }_{zxx}^{\left(2\right)}$, and ${\chi }_{zzz}^{\left(2\right)}$. To see this figure in color, go online.

Discussion of the triple-helix distribution with respect to the fiber axis

While the primary focus of this study is centered on recovery of fiber orientation (azimuthal and polar) and recovery of the local-frame tensor on a per-pixel basis, it is interesting to consider different models for connecting those results to molecular-level structural information given the opportunity afforded by access to the local tensor at each position. As discussed in Relating ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$ to the Molecular Tensor, ${\stackrel{\rightharpoonup }{\beta }}^{\left(2\right)}$ and Probing the Local Order, this distribution can also be investigated by comparing the ${\beta }^{\left(2\right)}$ tensors recovered from ab initio calculation with the experimentally determined ${\chi }_l^{\left(2\right)}$. In the limit of perfect alignment between the triple helices and the fiber axis, similar values of ${\chi }_{l,zzz}^{\left(2\right)}/{\chi }_{l,zxx}^{\left(2\right)}$ and ${\beta }_{z^{\prime }{z}^{\prime }{z}^{\prime }}^{\left(2\right)}/{\beta }_{z^{\prime }{x}^{\prime }{x}^{\prime }}^{\left(2\right)}$ are expected. Within the validity of the measurements and calculations, differences between ${\chi }_l^{\left(2\right)}$ and ${\beta }^{\left(2\right)}$ tensors can be attributed to the tilt-angle distribution of the collagen triple helices.

The most straightforward case of a δ-function distribution for the tilt angle was considered first. Fig. 10 a plots the ${\chi }_{l,zzz}^{\left(2\right)}/{\chi }_{l,zxx}^{\left(2\right)}$ as a function of the tilt angle θ, given different input values of ${\beta }_{z^{\prime }{z}^{\prime }{z}^{\prime }}^{\left(2\right)}/{\beta }_{z^{\prime }{x}^{\prime }{x}^{\prime }}^{\left(2\right)}$. The inputs were calculated either from the symmetry additive approach described by Perry et al. (50) for a suite of different computational methods, or from previously reported values by de Wergifosse et al. (49) based on an ONIOM calculation. The rectangular vertical bar at ∼15–20° represents the expected tilt-angle range reported based on scanning electron microscopy (SEM), NMR, and electron tomography, and the horizontal dotted line represents the experimental recovered value for ${\chi }_{l,zzz}^{\left(2\right)}/{\chi }_{l,zxx}^{\left(2\right)}$. The additional vertical bars in the figure are the artifacts from the singularity of the calculated ${\chi }_{l,zzz}^{\left(2\right)}/{\chi }_{l,zxx}^{\left(2\right)}$. The ab initio calculation indicates that ${\beta }_{z^{\prime }{z}^{\prime }{z}^{\prime }}^{\left(2\right)}$ is the dominating component for the molecular polarizability while ${\beta }_{z^{\prime }{x}^{\prime }{x}^{\prime }}^{\left(2\right)}\cong {\beta }_{x^{\prime }{x}^{\prime }{z}^{\prime }}^{\left(2\right)}$ values are more than four or five times smaller—in which case the tilt angle required to yield an experimentally observed ${\chi }_{l,zzz}^{\left(2\right)}/{\chi }_{l,zxx}^{\left(2\right)}$ of 1.4–1.7 is calculated to be >50°. The unrealistically large value for the tilt angle is difficult to reconcile with the fibril tilt angles observed previously using SEM, NMR imaging, and electron tomography. NMR studies of fibril distribution within collagen fibers in a sheep tendon have found that the fibrils lie along the fiber axis, but can have tilt-angle distributions of 19° (38). Previous SEM measurements of rat tail tendons have found that individual collagen fibrils lie primarily along the fiber axis, but also exhibit regions of interweaving fibrils and regions exhibiting random fibril order (39, 40). A study of fibril structures in cornea using electron tomography found that microfibril substructures adopted tilt angles of 15° with respect to the fibril axis (41). Constant tilt angles between 0 and 19° for collagen helices and/or fibrils within collagen fibers have been measured for several different collagen structures using a variety of techniques (20, 33, 38, 41). This discrepancy indicates that the simple δ-function distribution is not sufficient to compare the average experimental local-frame tensor elements to the molecular level ab initio calculations.

Figure 10.

(a) The ratio of ${\chi }_{l,zzz}^{\left(2\right)}/{\chi }_{l,zxx}^{\left(2\right)}$ as a function of the spread of the tilt-angle distribution of triple helices given different input values of ${\beta }_{z^{\prime }{z}^{\prime }{z}^{\prime }}^{\left(2\right)}/{\beta }_{z^{\prime }{x}^{\prime }{x}^{\prime }}^{\left(2\right)}$. (b) The ratio of ${\chi }_{l,zzz}^{\left(2\right)}/{\chi }_{l,zxx}^{\left(2\right)}$ as a function of tilt angle of triple helices given different input values of ${\beta }_{z^{\prime }{z}^{\prime }{z}^{\prime }}^{\left(2\right)}/{\beta }_{z^{\prime }{x}^{\prime }{x}^{\prime }}^{\left(2\right)}$.

The next most obvious selection for an orientation distribution was a normal distribution centered about the primary fiber axis. The degree of order within the collagen fiber can be probed by recovering the spread of the tilt-angle distribution, as shown in the inset of Fig. 10 b. Similarly, the ${\chi }_{l,zzz}^{\left(2\right)}/{\chi }_{l,zxx}^{\left(2\right)}$ value was plotted as a function of the spread of the normal distribution, σ_θ, given different input values of ${\beta }_{z^{\prime }{z}^{\prime }{z}^{\prime }}^{\left(2\right)}/{\beta }_{z^{\prime }{x}^{\prime }{x}^{\prime }}^{\left(2\right)}$. The range of σ_θ-values that correspond to the most probable tilt angle of 15–20° were calculated and illustrated in the figure by the rectangular vertical bar at ∼5–8°. The additional vertical lines in the figure are the artifacts from the singularity of the calculated ${\chi }_{l,zzz}^{\left(2\right)}/{\chi }_{l,zxx}^{\left(2\right)}$. Different from the δ-function model, the recovered ratio of local frame tensor elements approach an asymptotic value of 3 as the tilt-angle distribution approaches a broad disordered limit. Conceptually, the asymptotic value of 3 is explained by the weak order limit, where ${\chi }_{l,zzz}^{\left(2\right)}\cong {\chi }_{l,zxx}^{\left(2\right)}+{\chi }_{l,xxz}^{\left(2\right)}+{\chi }_{l,xzx}^{\left(2\right)}$ (56). The independence of the previously reported values from the ONIOM calculations on the distribution width can be understood based on this same asymptotic behavior. Both the HRS measurements of a fibril suspension and the ab initio calculations fortuitously recovered ratios for collagen within experimental error of the weak-order limit (49, 57). Consequently, any broadening of the distribution in triple helix tilt angles will not result in significant changes to the results already at the asymptotic limit. Despite these calculations, previous calculations, and previous HRS measurements suggesting the presence of a dominating ${\beta }_{z^{\prime }{z}^{\prime }{z}^{\prime }}^{\left(2\right)}$ tensor element, there is still no reasonable solution for the recovery of ${\chi }_{l,zzz}^{\left(2\right)}/{\chi }_{l,zxx}^{\left(2\right)}$ in the range of 1.4–1.7 when assuming a dominating ${\beta }_{z^{\prime }{z}^{\prime }{z}^{\prime }}^{\left(2\right)}$ value for a monotonic normal distribution centered about the principal fiber axis.

The failure of the simple monotonic distribution models describing the tilt-angle of the triple helices in collagen motivated the consideration of alternatives. The next simplest distribution was a constrained bimodal distribution, in which prior knowledge of fibril arrangement in collagen was explicitly incorporated. The inhomogeneity of the fibril distribution has been suggested in previous studies using an atomic force microscope, where collagen fibrils with different packing densities were observed in the fiber bundle (58, 59). Fibrous collagen domains of opposite orientation were observed in the tail tendon by Han et al. (60). In addition, a study using SEM has observed collagen fibrils with a twisting angle of nearly 180° at the fibrillar crimp (61). From previous measurements performed using SEM, NMR, and electron tomography, the most probable helix tilt angle was reported to be ∼17°. From the fiber diffraction studies, collagenous material was observed to be comprised of both a well-ordered fraction and a relatively disordered fraction (59). If it is reasonably assumed that the less-ordered fraction exhibits a broad distribution in orientations (corresponding to an effective ${\chi }_{l,zzz}^{\left(2\right)}/{\chi }_{l,zxx}^{\left(2\right)}=3$), the net distribution can be modeled as a sum of a normal distribution centered about zero with a mode at 17° and a broadly distributed antiparallel component. For such a distribution the only unknown parameter is the relative volume fraction of the ordered versus disordered contributions to the net fiber polarization-dependence. Combining both contributions and the experimentally recovered ${\chi }_{l,zzz}^{\left(2\right)}/{\chi }_{l,zxx}^{\left(2\right)}$ for the tail, ear, and skin samples, the relative ratios between the coparallel and antiparallel fibrils were calculated to be 0.43, 0.43, and 0.42, respectively, corresponding to ∼43% of the collagen fibrils oriented coparallel with the fiber axis while ∼57% of the fibrils are antiparallel with the fiber axis, as indicated in Fig. 11. Previously reported values of ${\beta }_{z^{\prime }{z}^{\prime }{z}^{\prime }}^{\left(2\right)}/{\beta }_{z^{\prime }{x}^{\prime }{x}^{\prime }}^{\left(2\right)}\cong 3$ (49) could not be made to recover the experimentally observed local frame tensor elements in either the monomodal or bimodal orientation distributions.

Figure 11.

Bimodal distribution of the triple-helix tilt angles with respect to the fiber axis.

It is worth revisiting the assumptions employed in the analysis. Although variation in ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$ is solely attributed to differences in σ_θ′ in the analysis presented here, differences in the experimentally measured ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$ may also be due to the influence of birefringence. The presence of strong sample birefringence can affect the relative magnitude of the unique tensor elements, which can complicate quantitative polarization-dependent SHG analyses (62). As the collagen thickness approaches and exceeds the forward coherence length, interference effects have the potential to impact the overall intensity of the observed SHG (57, 63). Therefore, all analysis was based exclusively on the polarization-dependence derived from the relative intensities. Provided the birefringence of the sample is relatively small (which was confirmed experimentally by measurement of the polarized laser transmittance), the interference effects will be identical for all polarizations, resulting in an overall scaling of each term in the recovered set of 10 observables in Eq. 2. Consistent with this prediction, the polarization-dependence of Z-cut quartz is nicely recovered in X. Y. Dow, E. L. DeWalt, J. A. Newman, C. M. Dettmar, and G. J. Simpson (31), despite the presence of substantial interference effects from the bulk crystal.

The ability to perform polarized laser transmittance imaging simultaneously with SHG imaging illustrated in this and De Walt et al. (27) served a dual purpose, both to quantify the polarization-state of the electrooptic modulator from global analysis across the image and assess on a per-pixel basis the potential significance of birefringence. In all the measurements described in this work, the measured effects from local birefringence were negligible relative to other sources of measurement uncertainty.

The ability to quickly recover local-frame tensor and local orientation angles at every location in the field of view has potential applications to aid in tissue analysis and diagnosis by nonlinear optical microscopy. While the images presented in the preceding figures were signal-averaged with 0.3–1 ms per pixel measurement times, polarization images of collagen samples can be acquired at video rate (15 frames per s) (27) with as little as ∼150 ns per pixel total measurement time. The high achievable frame-rate coupled with image stabilization algorithms can allow for similar signal averaging advantages even in samples with positions changing in time (e.g., for in vivo analysis of skin). A representative video acquired at 7.5 frames per s of Fourier coefficients color maps for a mouse tail is shown in the Supporting Material. These timeframes open up the possibilities of quantitative polarization imaging in real-time for in vivo analysis and high-throughput measurements.

Significantly, the high SNR afforded by rapid polarization modulation and corresponding reduction in 1/f noise enabled independent determination of both polar and azimuthal rotation angles θ and ϕ and the full set of significant independent local-frame tensor elements at each pixel. Previous efforts to map collagen networks in complex matrices such as skin are often complicated by local variations in the polar angle θ that can substantially complicate and/or bias polarization analyses that do not include recovery of θ. The azimuthal angle ϕ is relatively straightforward to recover from either texture analysis of the images or simple models for the polarization-dependence. However, determination of θ generally requires a SNR that is challenging to routinely achieve with alternative modulation schemes.

While the primary focus of this study is squarely on the experimental determination of the fiber azimuthal and polar orientation combined with recovery of the local-frame ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$ tensor on a per-pixel basis, it is worthwhile to assess the significance of the proposed distributions linking the molecular and macroscopic structures. Notably, the final orientation distribution that is consistent with the experimental observables is remarkably sensitive to the finer details of the bottom-up calculations. As such, the preceding analysis and the corresponding proposed orientation distribution should be viewed cautiously; improvements in the calculations or independent measurements of individual triple helix segments representative of collagen could result in significant changes to the proposed orientation distributions. However, the sensitivity of the final polarization dependence on the reliability of the quantum chemical calculations is clearly advantageous, suggesting that polarization-dependent SHG measurements of collagenous tissues inherently possess high sensitivity to molecular-scale structure and interactions. These results contribute further to the growing body of literature, suggesting the clear opportunities engendered by polarization-dependent SHG microscopy measurements.

Conclusions

NOSE microscopy was used to generate local-frame tensor maps of collagen-rich tissues together with determination of both polar and azimuthal fiber orientations at each pixel. Rapid polarization modulation microscopy measurements were fit and decomposed to produce a set of five images of the Fourier components of the modulation frequency, displayed as RGBCM colormaps. From the set of 10 unique observables (five for each detector), analytical models enabled the direct recovery of the unique local-frame nonlinear susceptibility tensor, ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$. Computational algorithms for iteratively solving for the vectorized tensor of unique local-frame elements, ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$, the in-plane rotation of collagen fibers, ϕ, and the out-of-plane tilt angle, θ, were implemented. In addition, methods for pooling data and calculating a global ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$ for an entire image were demonstrated, as well as an approach for generating ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$ for a single pixel to generate images with contrast based on differences in ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$. The high SNR afforded by rapid polarization modulation allowed for the per-pixel recovery of the local-frame tensor as well as the fiber polar and azimuthal orientation angles. This work represents the first demonstration to our knowledge of the recovery of the polar tilt angle of the collagen fibers using an all-optical approach based on polarization analysis (e.g., without z-sectioning). The ability to solve for the polar tilt angle from a set of polarization-dependent SHG microscopy measurements acquired at up to video rate accesses three-dimensional structural information at the speeds consistent with the timeframe required for clinical decision-making.

In addition to the model-independent local frame measurements, molecular level ab initio analysis was combined with the recovered per-pixel ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$ maps to probe the internal orientational information of helices within the collagen fibers. The ab initio results suggested a triple-helix tensor dominated by the β_z′z′z′ tensor element describing the nonlinear polarizability along the helix axis, consistent with previously reported calculations. Three different models of the fibril tilt angles were explored to bridge the experimentally recovered ensemble-averaged ${\stackrel{\rightharpoonup }{\chi }}_l^{\left(2\right)}$ tensor with the molecular tensor, which was also in good agreement with previous measurements of collagen. The assumption of a narrow orientation distribution resulted in values of the triple helix tilt angles that differed substantially from independently determined values as determined by SEM and x-ray fiber diffraction. Assumption of a monomodal Gaussian distribution could not be made to recover the experimental observables. Consequently, a bimodal distribution that consists of coparallel and broadly distributed antiparallel components was proposed, in which roughly half of the measured contributions to the local frame response were attributed to a more poorly ordered antiparallel component. Independent evidence based on fiber diffraction also suggests the presence of a significant fraction of weakly ordered collagen, consistent with the recovered distribution.

Author Contributions

X.Y.D. and E.L.D. wrote the article, designed and performed research, developed computational algorithms, and analyzed data; S.Z.S. and J.R.W.U. performed ab initio calculations and wrote the article; P.D.S. developed computational algorithms and wrote the article; and G.J.S. designed research, developed theoretical and mathematical models, and wrote the article.

Editor: Nathan Baker.