Article

Imaging the Nonlinear Susceptibility Tensor of Collagen by Nonlinear Optical Stokes Ellipsometry Ximeng Y. Dow,1 Emma L. DeWalt,1 Shane Z. Sullivan,1 Paul D. Schmitt,1 James R. W. Ulcickas,1 and Garth J. Simpson1,* 1

Department of Chemistry, Purdue University, West Lafayette, Indiana

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 (1–5), 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 twophoton 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

Submitted November 4, 2015, and accepted for publication May 16, 2016. *Correspondence: [email protected] Ximeng Y. Dow and Emma L. DeWalt contributed equally to the work. Editor: Nathan Baker. http://dx.doi.org/10.1016/j.bpj.2016.05.055

ð2Þ

nonlinear susceptibility tensor, cl , 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 (17–23). 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 polarizationdependent 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).

Ó 2016

Biophysical Journal 111, 1361–1374, October 4, 2016 1361

Dow et al.

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 polarizationdependent 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 f and out-of-plane polar tilt-angle q) 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 q on the polarization-dependent nonlinear optical measurements has been explored and discussed qualitatively (29). However, quantitative recovery of q 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 bottomup 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 symme-

1362 Biophysical Journal 111, 1361–1374, October 4, 2016

try (CN) for collagen fibers, the only unique nonzero tensor elements are ð2Þ ð2Þ ð2Þ cð2Þ zzz ; czxx ; cxxz ; and cxyz . However, with the incident laser beam propað2Þ ð2Þ gating perpendicular to the probed sample plane, cð2Þ zzz ; czxx ; and cxxz ð2Þ are strongly preferred to the chiral element, cxyz . Ab initio calculation of ð2Þ the b-tensors of collagen triple helices indicates the ratio bð2Þ xxz =bxyz is ~50, reflecting the limited contribution from the chiral terms. Conseð2Þ quently, cxyz 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 (CN) symmetry of a collagen fiber, the twist angle, j, has no effect on its polarization-dependent response, leaving only consideration of the in-plane fiber rotation, f, and the out-of-plane tilt angle, q. Most commonly, the individual collagen fibers are considered to be lying within the image plane, resulting in polar tilt angles, q, 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 f and q are included in the analysis. .ð2Þ

Solving the local-frame tensor c l fibers

for collagen

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, D, 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:





 D D D D ¼ A , cos4 þ B , cos3 , sin 2 2 2 2




 D D D D D þ C , cos2 , sin2 þ D , cos , sin3 þ E , sin4 ; 2 2 2 2 2 2   X 2  ð2Þ  An ¼  Mnm , cmHH  ;   m¼1 " ! !# 2 2 X X ð2Þ ð2Þ Bn ¼ 4 , Im Mnm , cmHH , Mnm , cmVH ; 2u IL;n

m¼1

m¼1

2  " !  X 2 2 X  ð2Þ  ð2Þ M ,c Mnm , cmHH , Cn ¼ 4 ,    2 , Re  m ¼ 1 nm mVH  m¼1 2 X

! # ð2Þ Mnm , cmVV

;

m¼1

" Dn ¼ 4 , Im

2 X

! ð2Þ

Mnm , cmVV

m¼1

,

2 X

!# ð2Þ

Mnm , cmVH

;

m¼1

2   X 2  ð2Þ  En ¼  Mnm , cmVV  :   m¼1 (1) .

The above equation can be rewritten in the matrix form with A n represent.ð2Þ ing the vector of polynomial coefficients, c j representing the vectorized

Imaging c(2) Tensors of Collagen by NOSE Jones tensor, and the matrix Pn describing the connection between the two, as shown in . An


.ð2Þ .ð2Þ ¼ Pn : c j 5c j :

(2)

The Jones tensors can be further related to the vectorized local-frame ten.ð2Þ sors c l 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: .ð2Þ cj

0 .ð2Þ

¼ Jðq;4;fÞ , c l 
   1 0 0 1 ¼ 5 0 1 0 0

0

0

1

0



 5

1 0

0 0 



1 0

,

.ð2Þ

analysis, and the fitting was performed to recover one universal c l , as shown in Eq. 5, where SH and SV are detector sensitivities for the H and V detector, respectively:

2

3

2 P , SH 7 7 6 0 7 6 7 6 7 6 P,S 7 6 H 7 6 7 6 0 7 ¼6 7 6 6 « 7 6 6 7 6 6. 7 6 6 A 7 6 P , SH 6 Hq 7 4 4. 5 0 A Vq . 6 A H1 6. 6 A V1 6 6. 6 A H2 6 6. 6 A V2 6



#3  " JH;f1 5 JH;f1



, P , SV JV;f1 5 JV;f1 7 7



#7  " 7 0 JH;f 5 JH;f 7 , 2 2 7 P , SV JV;f2 5 JV;f2 7 7, 7 « 7 7 # "



7  JH;fq 5 JH;fq 7 0



5 , P , SV JV;fq 5 JV;fq 0




ðQ5QÞ ,

.ð2Þ

ðRðq; j; fÞ5Rðq; j; fÞ5Rðq; j; fÞÞ , c l :

.ð2Þ .ð2Þ c l 5c l

 : (5)

(3) 0 .ð2Þ cl

Finally, the full set of 27 local-frame tensor can be recast using just .ð2Þ the unique nonzero tensor elements given by the unprimed vector c l through a 27  3 symmetry matrix Q: 0 .ð2Þ cl

.ð2Þ

¼ Q , cl :

(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. 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 .ð2Þ single global set of local-frame c l 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

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 ð2Þ ð2Þ unique collagen tensor elements (cð2Þ xxz , czxx , and czzz ) 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 f were determined by image analysis using the OrientationJ plug-in for ImageJ (34), and initial guess values for q were chosen arbitrarily to be 40 . After minimization, the recovered values of f and q 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 f and q for every pixel, and a global .ð2Þ set of elements in c l . 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 .ð2Þ pixel to reduce the uncertainty in the recovered c l . 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 .ð2Þ used here. A subsequent unique set of c l was calculated using nonlinear least-squares fit at each individual pixel, generating a tensor map of each field of view (FOV). .ð2Þ

.ð2Þ

Relating c l to the molecular tensor, b probing the local order

and

.ð2Þ

.ð2Þ

FIGURE 1 General process of solving for a global c l orientation using the iterative analysis or for solving for using the pixel-by-pixel analysis.

.ð2Þ cl

and sample

at every pixel

Interpreting differences in c l 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 repreð2Þ sentation of cl and the molecular tensor as bð2Þ 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 (20,33,38–41), it is apparent that a distribution of tilt angles of collagen

Biophysical Journal 111, 1361–1374, October 4, 2016 1363

Dow et al. polarizability tensor, bð2Þ 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 ð2Þ collagen fiber, cl is directly proportional to bð2Þ and consequently straight fibers (those with triple-helices aligned and ordered with respect to the fiber ð2Þ axis) should have measured relative values of cl that are similar to theoretð2Þ ically predicted values of b (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 ð2Þ to yield differences between the experimentally observed cl and bð2Þ for an individual helix. Two key assumptions were made to simplify the relað2Þ tionships connecting cl and bð2Þ for the fibers: (1) the formal C3 symmetry of the triple helix was assumed to reasonably approximated by CN 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 bð2Þ onto cð2Þ zzz (one of the three unique nonzero local-frame tensor elements of collagen, considered explicitly as an example), is given by

cð2Þ zzz

8 9 ð2Þ hcos3 q0 i , bz0 z0 z0 > > >

> < =  ð2Þ ð2Þ ð2Þ 0 2 0 2 0 ¼ Ns , þhsin q , cos q , sin j , by0 y0 z0 þ by0 z0 y0 þ bz0 y0 y0 :

> > > > ð2Þ ð2Þ : þhsin2 q0 , cos q0 , cos2 j0  , bð2Þ ; 0 0 0 þ b 0 0 0 þ b 0 0 0 xxz

xzx

zxx

(6) 0

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. 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. In theory, the internal distribution of the triple helix with respect to the fið2Þ ber axis can be probed through the connection of the measured cl of the fiber to the molecular structural information of a single collagen triple helix. In ð2Þ the experiments described in this work, the cl of the fiber can be measured, while the molecular tensor bð2Þ 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, bð2Þ and experimental results (36,42). Consequently, theoretical predictions of bð2Þ for a single collagen triple helix were combined with experimental ð2Þ methods to determine cl for a fiber, for us to gain insight into helical organization within a fiber. For a collagen triple helix specifically, the nonlinear

2 .ð2Þ cl

.ð2Þ

¼ Ns , K , b

hcos3 q0 i

 6 2 0 6 sin q , cos q0 6 ¼ Ns , 6 2 6  4 2 0 sin q , cos q0 2

0

0

The primed coordinates on b (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, Ns is the 0 number density of the helices, q is the tilt angle away from the fiber 0 axis, and j is the rotation about the fiber axis. Collagen fibers have CN 0 symmetry, and as a result the dependence on j disappears. In addition, the equalities dictated by the effective CN of a collagen triple helix ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ (by0 y0 z0 ¼ bx0 x0 z0 , by0 z0 y0 ¼ bx0 z0 x0 , and bz0 y0 y0 ¼ bz0 x0 x0 ) result in the simplified expression shown in



1  3 0  ð2Þ , cos q , bz0 z0 z0 2   ð2Þ þ sin2 q0 , cos q0 , bx0 x0 z0 bx0 x0 z0   ð2Þ 1  þ , sin2 q0 , cos q0 , bz0 x0 x0 bz0 x0 x0 : 2

cð2Þ zzz ¼ Ns ,

(7)

ð2Þ Similar expressions can also be derived for cxxz and cð2Þ zxx , and a conversion matrix K can be constructed to bridge the calculated molecular tensor of a ð2Þ model collagen triple helix, bð2Þ , and the experimentally determined cl , as shown in

   3 2 , sin2 q0 , cos q0 sin2 q0 , cos q0 2 ð2Þ 3   7 bz0 ;z0 ;z0    7  sin2 q0 , cos q0 7 6 ð2Þ 7 þ cos q0 i  sin2 q0 , cos q0 7 , 4 bz0 ;x0 ;x0 5: 2 7 ð2Þ   2 0 5 0 bx0 ;x0 ;z0   2 0  sin q , cos q 0 0  sin q , cos q þ hcos q i 2 (8)

1364 Biophysical Journal 111, 1361–1374, October 4, 2016



Imaging c(2) Tensors of Collagen by NOSE

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 mm 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 mL 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.

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

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.

Biophysical Journal 111, 1361–1374, October 4, 2016 1365

Dow et al. 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, D, 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 D, as described in detail in Simpson et al. (44).

TABLE 1 The Ratios of Molecular Tensors Recovered from Ab Initio Calculation ð2Þ

ð2Þ

bz0 z0 z0 =bz0 x0 x0 TDDFT-B3LYP TDHF LC-BLYP LC-BOP ONIOMa

4.5 25 14 13 2.97

a

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 mm 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 mm sections of porcine skin (two using 40 magnification, and one using 10 magnification) and four FOVs of 10 mm 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 polarizationdependent 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 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 d-function distribution is not sufficient to compare the average experimental local-frame tensor elements to the molecular level ab initio calculations. 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 ð2Þ ð2Þ cl;zzz =cl;zxx value was plotted as a function of the spread of the normal distribution, sq, given different input values of ð2Þ ð2Þ bz0 z0 z0 =bz0 x0 x0 . The range of sq-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 ð2Þ ð2Þ from the singularity of the calculated cl;zzz =cl;zxx . Different from the d-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 ð2Þ ð2Þ ð2Þ ð2Þ by the weak order limit, where cl;zzz ycl;zxx þ cl;xxz þ cl;xzx (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

Imaging c(2) Tensors of Collagen by NOSE

calculations, previous calculations, and previous HRS meað2Þ surements suggesting the presence of a dominating bz0 z0 z0 tensor element, there is still no reasonable solution for the ð2Þ ð2Þ recovery of cl;zzz =cl;zxx in the range of 1.4–1.7 when ð2Þ assuming a dominating bz0 z0 z0 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 ð2Þ ð2Þ (corresponding to an effective cl;zzz =cl;zxx ¼ 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 ð2Þ ð2Þ and the experimentally recovered cl;zzz =cl;zxx 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 ð2Þ ð2Þ bz0 z0 z0 =bz0 x0 x0 y3 (49) could not be made to recover the experimentally observed local frame tensor elements in either the monomodal or bimodal orientation distributions. It is worth revisiting the assumptions employed in the .ð2Þ analysis. Although variation in c l is solely attributed to differences in sq0 in the analysis presented here, differences .ð2Þ in the experimentally measured c l 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 framerate 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.

FIGURE 11 Bimodal distribution of the triple-helix tilt angles with respect to the fiber axis.

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Dow et al.

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 q and f 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 q that can substantially complicate and/or bias polarization analyses that do not include recovery of q. The azimuthal angle f is relatively straightforward to recover from either texture analysis of the images or simple models for the polarization-dependence. However, determination of q 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 polarð2Þorientation combined with recovery of the local. frame c l 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 .ð2Þ

nonlinear susceptibility tensor, c l . Computational algorithms for iteratively solving for the vectorized tensor of .ð2Þ

unique local-frame elements, c l , the in-plane rotation of collagen fibers, f, and the out-of-plane tilt angle, q, were

1372 Biophysical Journal 111, 1361–1374, October 4, 2016

implemented. In addition, methods for pooling data and .ð2Þ

calculating a global c l

for an entire image were demon.ð2Þ

strated, as well as an approach for generating c l for a single pixel to generate images with contrast based on dif.ð2Þ

ferences in c l . 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 polarizationdependent 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 .ð2Þ

with the recovered per-pixel c l 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 bz0 z0 z0 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 .ð2Þ

experimentally recovered ensemble-averaged c l 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. SUPPORTING MATERIAL Supporting Materials and Methods and three figures are available at http:// www.biophysj.org/biophysj/supplemental/S0006-3495(16)30700-7.

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.

Imaging c(2) Tensors of Collagen by NOSE developed computational algorithms and wrote the article; and G.J.S. designed research, developed theoretical and mathematical models, and wrote the article.

16. Han, X., R. M. Burke, ., E. B. Brown. 2008. Second harmonic properties of tumor collagen: determining the structural relationship between reactive stroma and healthy stroma. Opt. Express. 16:1846– 1859.

ACKNOWLEDGMENTS

17. DeWalt, E. L., V. J. Begue, ., G. J. Simpson. 2013. Polarizationresolved second-harmonic generation microscopy as a method to visualize protein-crystal domains. Acta Crystallogr. D Biol. Crystallogr. 69:74–81.

The authors acknowledge the Jonathon Amy Facility for Chemical Instrumentation at Purdue University for their support in developing the data acquisition electronics. The authors also thank Kevin Kerian for providing mouse tail samples and for sectioning the biological tissues used in this work. The authors acknowledge support from National Institutes of Health grant No. R01GM-103401, and grant No. R01GM-103910 from the National Institute of General Medical Sciences.

REFERENCES 1. Fine, S., and W. P. Hansen. 1971. Optical second harmonic generation in biological systems. Appl. Opt. 10:2350–2353. 2. Freund, I., and M. Deutsch. 1986. Second-harmonic microscopy of biological tissue. Opt. Lett. 11:94–96.

18. Plocinik, R. M., and G. J. Simpson. 2003. Polarization characterization in surface second harmonic generation by nonlinear optical null ellipsometry. Anal. Chim. Acta. 496:133–142. 19. Filippidis, G., K. Troulinaki, ., N. Tavernarakis. 2009. In vivo polarization dependant second and third harmonic generation imaging of Caenorhabditis elegans pharyngeal muscles. Laser Phys. 19:1475– 1479. 20. Tuer, A. E., S. Krouglov, ., V. Barzda. 2011. Nonlinear optical properties of type I collagen fibers studied by polarization dependent second harmonic generation microscopy. J. Phys. Chem. B. 115:12759–12769. 21. Su, P.-J., W.-L. Chen, ., C.-Y. Dong. 2011. Determination of collagen nanostructure from second-order susceptibility tensor analysis. Biophys. J. 100:2053–2062. 22. Latour, G., I. Gusachenko, ., M.-C. Schanne-Klein. 2012. In vivo structural imaging of the cornea by polarization-resolved second harmonic microscopy. Biomed. Opt. Express. 3:1–15.

3. Zipfel, W. R., R. M. Williams, ., W. W. Webb. 2003. Live tissue intrinsic emission microscopy using multiphoton-excited native fluorescence and second harmonic generation. Proc. Natl. Acad. Sci. USA. 100:7075–7080.

23. Gusachenko, I., V. Tran, ., M. C. Schanne-Klein. 2012. Polarizationresolved second-harmonic generation in tendon upon mechanical stretching. Biophys. J. 102:2220–2229.

4. Zoumi, A., A. Yeh, and B. J. Tromberg. 2002. Imaging cells and extracellular matrix in vivo by using second-harmonic generation and two-photon excited fluorescence. Proc. Natl. Acad. Sci. USA. 99: 11014–11019.

24. Tanaka, Y., E. Hase, ., T. Yasui. 2014. Motion-artifact-robust, polarization-resolved second-harmonic-generation microscopy based on rapid polarization switching with electro-optic Pockells cell and its application to in vivo visualization of collagen fiber orientation in human facial skin. Biomed. Opt. Express. 5:1099–1113.

5. Chen, X., O. Nadiarynkh, ., P. J. Campagnola. 2012. Second harmonic generation microscopy for quantitative analysis of collagen fibrillar structure. Nat. Protoc. 7:654–669. 6. Dombeck, D. A., K. A. Kasischke, ., W. W. Webb. 2003. Uniform polarity microtubule assemblies imaged in native brain tissue by secondharmonic generation microscopy. Proc. Natl. Acad. Sci. USA. 100: 7081–7086. 7. Mohler, W., A. C. Millard, and P. J. Campagnola. 2003. Second harmonic generation imaging of endogenous structural proteins. Methods. 29:97–109. 8. Brown, E., T. McKee, ., R. K. Jain. 2003. Dynamic imaging of collagen and its modulation in tumors in vivo using second-harmonic generation. Nat. Med. 9:796–800. 9. Belisle, J., T. Zigras, ., R. L. Leask. 2010. Second harmonic generation microscopy to investigate collagen configuration: a pericarditis case study. Cardiovasc. Pathol. 19:e125–e128. 10. Tsai, M. R., Y. W. Chiu, ., C. K. Sun. 2010. Second-harmonic generation imaging of collagen fibers in myocardium for atrial fibrillation diagnosis. J. Biomed. Opt. 15:026002. 11. Theodossiou, T. A., C. Thrasivoulou, ., D. L. Becker. 2006. Second harmonic generation confocal microscopy of collagen type I from rat tendon cryosections. Biophys. J. 91:4665–4677.

25. Lien, C.-H., K. Tilbury, ., P. J. Campagnola. 2013. Precise, motionfree polarization control in second harmonic generation microscopy using a liquid crystal modulator in the infinity space. Biomed. Opt. Express. 4:1991–2002. 26. Muir, R. D., S. Z. Sullivan, ., G. J. Simpson. 2014. Synchronous digitization for high dynamic range lock-in amplification in beam-scanning microscopy. Rev. Sci. Instrum. 85:033703. 27. DeWalt, E. L., S. Z. Sullivan, ., G. J. Simpson. 2014. Polarizationmodulated second harmonic generation ellipsometric microscopy at video rate. Anal. Chem. 86:8448–8456. 28. Duboisset, J., D. Aı¨t-Belkacem, ., S. Brasselet. 2012. Generic model of the molecular orientational distribution probed by polarizationresolved second-harmonic generation. Phys. Rev. A. 85:043829. 29. Han, Y., J. Hsu, ., E. O. Potma. 2015. Polarization-sensitive sum-frequency generation microscopy of collagen fibers. J. Phys. Chem. B. 119:3356–3365. 30. Sivaguru, M., S. Durgam, ., K. C. Toussaint, Jr. 2010. Quantitative analysis of collagen fiber organization in injured tendons using Fourier transform-second harmonic generation imaging. Opt. Express. 18: 24983–24993.

12. Roth, S., and I. Freund. 1980. Coherent optical harmonic generation in rat-tail tendon. Opt. Commun. 33:292–296.

31. Dow, X. Y., E. L. DeWalt, ., G. J. Simpson. 2016. Unified Theory for Polarization Analysis in Second Harmonic and Sum Frequency Microscopy. Biophys. J. 111:1553–1568.

13. Han, M., G. Giese, and J. Bille. 2005. Second harmonic generation imaging of collagen fibrils in cornea and sclera. Opt. Express. 13:5791– 5797.

32. Yew, E. Y. S., and C. J. R. Sheppard. 2007. Second harmonic generation polarization microscopy with tightly focused linearly and radially polarized beams. Opt. Commun. 275:453–457.

14. Lutz, V., M. Sattler, ., F. Fischer. 2012. Impact of collagen crosslinking on the second harmonic generation signal and the fluorescence lifetime of collagen autofluorescence. Skin Res. Technol. 18:168–179.

33. Tuer, A. E., M. K. Akens, ., V. Barzda. 2012. Hierarchical model of fibrillar collagen organization for interpreting the second-order susceptibility tensors in biological tissue. Biophys. J. 103:2093–2105.

15. Stoller, P., K. M. Reiser, ., A. M. Rubenchik. 2002. Polarizationmodulated second harmonic generation in collagen. Biophys. J. 82:3330–3342.

34. Sage, D. 2012. OrientationJ: ImageJ’s plugin for directional analysis in images. Biomedical Imaging Group, E´cole Polytechnique Fe´de´rale de Lausanne. http://bigwww.epfl.ch/demo/orientation/.

Biophysical Journal 111, 1361–1374, October 4, 2016 1373

Dow et al. 35. Loison, C., and D. Simon. 2010. Additive model for the second harmonic generation hyperpolarizability applied to a collagen-mimicking peptide (Pro-Pro-Gly)10. J. Phys. Chem. A. 114:7769–7779.

50. Perry, J. M., A. J. Moad, ., G. J. Simpson. 2005. Electronic and vibrational second-order nonlinear optical properties of protein secondary structural motifs. J. Phys. Chem. B. 109:20009–20026.

36. Haupert, L. M., E. L. DeWalt, and G. J. Simpson. 2012. Modeling the SHG activities of diverse protein crystals. Acta Crystallogr. D Biol. Crystallogr. 68:1513–1521. 37. Moad, A. J., C. W. Moad, ., G. J. Simpson. 2007. NLOPredict: visualization and data analysis software for nonlinear optics. J. Comput. Chem. 28:1996–2002.

51. Wampler, R. D., A. J. Moad, ., G. J. Simpson. 2007. Visual methods for interpreting optical nonlinearity at the molecular level. Acc. Chem. Res. 40:953–960. 52. Sim, F., S. Chin, ., J. E. Rice. 1993. Electron correlation effects in hyperpolarizabilities of P-nitroaniline. J. Phys. Chem. 97:1158–1163.

38. Fechete, R., D. E. Demco, ., G. Navon. 2003. Anisotropy of collagen fiber orientation in sheep tendon by 1H double-quantum-filtered NMR signals. J. Magn. Resonance. 162:166–175.

53. Ferrighi, L., L. Frediani, ., K. Ruud. 2006. Density-functional-theory study of the electric-field-induced second harmonic generation (EFISHG) of push-pull phenylpolyenes in solution. Chem. Phys. Lett. 425:267–272.

39. Provenzano, P. P., and R. Vanderby, Jr. 2006. Collagen fibril morphology and organization: implications for force transmission in ligament and tendon. Matrix Biol. 25:71–84.

54. Hovhannisyan, V. A., P.-J. Su, ., C.-Y. Dong. 2009. Quantifying thermodynamics of collagen thermal denaturation by second harmonic generation imaging. Appl. Phys. Lett. 94:233902.

40. Kannus, P. 2000. Structure of the tendon connective tissue. Scand. J. Med. Sci. Sports. 10:312–320. 41. Holmes, D. F., C. J. Gilpin, ., K. E. Kadler. 2001. Corneal collagen fibril structure in three dimensions: structural insights into fibril assembly, mechanical properties, and tissue organization. Proc. Natl. Acad. Sci. USA. 98:7307–7312. 42. Toth, S. J., P. D. Schmitt, ., G. J. Simpson. 2015. Ab initio prediction of the diversity of second harmonic generation from pharmaceutically relevant materials. Cryst. Growth Des. 15:581–586. 43. Wanapun, D., R. D. Wampler, ., G. J. Simpson. 2008. Polarizationdependent two-photon absorption for the determination of protein secondary structure: a theoretical study. Chem. Phys. Lett. 455:6–12. 44. Simpson, G. J., C. A. Dailey, ., R. M. Everly. 2005. Direct determination of effective interfacial optical constants by nonlinear optical null ellipsometry of chiral films. Anal. Chem. 77:215–224. 45. Haupert, L. M., E. L. DeWalt, and G. J. Simpson. 2012. Modeling the SHG activities of diverse protein crystals. Acta Crystallogr. Biol. Crystallogr. 68:1513–1521.

55. Tiaho, F., G. Recher, and D. Roue`de. 2007. Estimation of helical angles of myosin and collagen by second harmonic generation imaging microscopy. Opt. Express. 15:12286–12295. 56. Simpson, G. J., and K. L. Rowlen. 1999. An SHG magic angle: dependence of second harmonic generation orientation measurements on the width of the orientation distribution. J. Am. Chem. Soc. 121:2635– 2636. 57. Deniset-Besseau, A., J. Duboisset, ., M.-C. Schanne-Klein. 2009. Measurement of the second-order hyperpolarizability of the collagen triple helix and determination of its physical origin. J. Phys. Chem. B. 113:13437–13445. 58. Gutsmann, T., G. E. Fantner, ., P. K. Hansma. 2003. Evidence that collagen fibrils in tendons are inhomogeneously structured in a tubelike manner. Biophys. J. 84:2593–2598. 59. Hulmes, D. J., T. J. Wess, ., P. Fratzl. 1995. Radial packing, order, and disorder in collagen fibrils. Biophys. J. 68:1661–1670.

46. Becke, A. D. 1993. A new mixing of Hartree-Fock and local-densityfunctional theories. J. Chem. Phys. 98:1372–1377.

60. Han, Y., V. Raghunathan, ., N.-H. Ge. 2013. Mapping molecular orientation with phase sensitive vibrationally resonant sum-frequency generation microscopy. J. Phys. Chem. B. 117:6149–6156.

47. Karna, S. P., and M. Dupuis. 1991. Frequency dependent nonlinear optical properties of molecules: formulation and implementation in the HONDO program. J. Comput. Chem. 12:487–504.

61. Franchi, M., V. Ottani, ., A. Ruggeri. 2010. Tendon and ligament fibrillar crimps give rise to left-handed helices of collagen fibrils in both planar and helical crimps. J. Anat. 216:301–309.

48. Kamiya, M., H. Sekino, ., K. Hirao. 2005. Nonlinear optical property calculations by the long-range-corrected coupled-perturbed KohnSham method. J. Chem. Phys. 122:234111. 49. de Wergifosse, M., J. de Ruyck, and B. Champagne. 2014. How the second-order nonlinear optical response of the collagen triple helix appears: a theoretical investigation. J. Phys. Chem. C. 118:8595–8602.

62. Brasselet, S., D. Aı¨t-Belkacem, ., S. Brasselet. 2010. Influence of birefringence on polarization resolved nonlinear microscopy and collagen SHG structural imaging. Opt. Express. 18:14859–14870.

1374 Biophysical Journal 111, 1361–1374, October 4, 2016

63. Bancelin, S., C. Aime´, ., M.-C. Schanne-Klein. 2014. Determination of collagen fibril size via absolute measurements of second-harmonic generation signals. Nat. Commun. 5:4920.