Linear least square (LLS) method for pixel-resolution analysis of polarization dependent SHG images of collagen fibrils

Denis Rouède; Jean-Jacques Bellanger; Jérémy Bomo; Georges Baffet; François Tiaho

doi:10.1364/OE.23.013309

1. Introduction

Second harmonic generation (SHG) imaging microscopy is a powerful tool to image intrinsic subcellular signals from endogenous proteins such as microtubule, myosin and collagen in living tissues [1–3]. SHG signal from triple helix collagen molecule has been suggested to originate from peptide bounds [4] inducing a dominant uniaxial molecular nonlinear optical hyperpolarizability β along each alpha-helix. For well aligned fibrils of collagen tissues, the relationship between macroscopic nonlinear susceptibility $χ^{(2)}$ and β is straightforward enabling estimation of the alpha-helix pitch angle based on theoretical modeling of the polarization dependence of SHG intensity [5, 6]. Recently, involvement of an additional molecular contribution from the methylene groups in the pyrrolidine rings has been suggested in order to obtain pitch angle values close to that of type I collagen X-rays diffraction [7, 8]. More recently theoretical models have been developed in order to take into account molecular orientation distribution in the determination of the macroscopic nonlinear susceptibility $χ^{(2)}$ coefficients and of the pitch angle [9–11]. Determination of the macroscopic nonlinear susceptibility coefficients are usually obtained from fitting of the polarization dependent SHG intensity using nonlinear least square (NLLS) method [10, 12–15]. The main drawback of NLLS is that it is time consuming necessitating a few hours to process a 512 × 512 pixel image. However, substantial reduction of the processing time up to few hundred milliseconds has been recently reached based on Fourier transform (FT) analysis of the angular anisotropy signature of SHG signal [16]. In this study, we propose an alternative method based on linear least square (LLS) fitting of the polarization dependent SHG image at pixel-resolution level. Firstly, we validate this method on isolated collagen gels fibrils and on well-aligned fibrils of xenopus gastrocnemius muscle tendon. Secondly, we use the model to determine fibrillar collagen remodeling due to cell and extracellular matrix (ECM) interactions. We show that computing time for LLS method is comparable to that obtained with FT method but LLS method provides an analytic solution of the nonlinear susceptibility $χ^{(2)}$ coefficients. As a consequence, LLS method is very easy to perform for non-expert in numerical signal processing and could be extended to the case of more complex molecular orientation distribution. Moreover, the results suggest that, in addition to the peptide group, a second molecular nonlinear optical hyperpolarizability $β$ contributes to the SHG signal.

2. Experimental methods

2.1. Preparation of biological sample

Isolated cell-free collagen gel fibrils were obtained at room temperature (18-20°C) as follow. Type I collagen (Sigma-Aldrich, St-Quentin-Fallavier, France) solution of 3.0 mg/ml was diluted in William’s E Medium without L-Glutamin to obtain a collagen solution of 1.5 mg/ml, and the pH was adjusted to 7.4 by NaOH 0.01N. This solution was poured into 24-well plates (400 µL), and incubated at 37°C. After 10 minutes, the gels were polymerized and an equal volume of medium was added. Gels were fixed overnight in phosphate buffer saline (PBS) containing 4% paraformaldehyde (PFA) at 4°C, and rinsed at least three times with PBS. This fixation procedure was undertaken to harden the collagen gel and prevents laser beam induced motion artifact during imaging.

Achille tendons were obtained from gastrocnemius muscle of adult Xenopus laevis animals (national breeding facility of xenopus animals in Rennes, France). Muscles and their attached Achille tendons were dissected in marked modified Ringer solution (MMR), fixed overnight in MMR containing 4% PFA at 4^◦C, and rinsed at least three times with MMR. We used 100 µm-thick slices obtained by slicing horizontally agarose-embedded tendon, glued on the stage of a vibroslicer (LEICA, VT 1200S).

F1 biliary epithelial cells were obtained from liver of 10-day-old Fischer rats [17]. F1 cells were grown in culture medium composed of William’s E Medium without L-Glutamin containing penicillin (100 IU/ml), streptomycin (100 µg/ml) and supplemented with 7.5 foetal calf serum (FCS). Type I collagen (Sigma-Aldrich, St-Quentin-Fallavier, France) at 3 mg/ml was diluted in culture medium to obtain an extracellular matrix (ECM) collagen solution of 1.5mg/ml. F1 cells were immediately added at a concentration of 1.25 10⁵ cells/ml to this solution, and the pH was adjusted to 7.4 by NaOH 0.01N. This mix of cells and collagen was poured into 24-well plates (400µL) and incubated at 37°C. After 10 minutes, the gels were polymerized and an equal volume of medium was added. After 4 days of cells culture, F1 cells containing gels were fixed overnight in PFA 4% at 4°C, and rinsed at least three times with phosphate buffer saline (PBS). This fixation procedure prevents motion artifact due to live cells stress on collagen fibrils.

Collagen gels (with or without cells) and tendon were mounted in the appropriate buffer solution and stabilized between two coverslips in a POC-R2 tissue culture chamber system (PeCon, Erbach, Germany) before imaging.

2.2. Imaging system and theoretical simulation

Images were acquired on PIXEL (http://pixel.univ-rennes1.fr/) facility of, University of Rennes1 (France) based on a Leica TCS SP2 confocal scanning head coupled to a DMIRE2 inverted microscope and equipped with an IR 80 MHz femtosecond laser (MAITAI, Spectra Physics). A laser beam tuned at 940 nm was focalized through a 60X water immersion objective (LUMFl 60XW, NA = 1.1, Olympus) to apply 5-10 mW excitation mean power at the sample. The dichroic filter wheel of the microscope was removed and replaced by a computer control PR50CC Newport rotation stage (precision 0.1°) equipped with an achromatic zero-order Quartz-MgF2 half-wave plate in order to adjust the input linear polarization angle $α$ of the incident IR electric field. SHG signal was collected in the forward direction using 0.90 NA multi-immersion condenser (model 501000, Leica). BG39 band pass and 470 ± 5 nm interference filters were placed in front of the PMT. For F1 cells imaging, flavin autofluorescence were epidetected using BG39 band pass and 500-650 nm band pass filters.

Image analysis and simulations were performed using MALAB (MathWorks, Natick, MA, USA). All SHG experimental polarization stacks were obtained for input polarization angles $α$ uniformly distributed between 0° and 180° with 10° increments. This incremental step is a good sampling compromise for accurate acquisition of SHG parameters and quality of the recording, avoiding motion artifact and cell damage by the laser. Data were obtained from at least 4 samples in each experimental condition (cell-free fibrils, F1 cells containing fibrils and Achille tendon) and at least 4 ROIs per sample. Automatic thresholding was performed on the polarization stacks before fitting with LLS method. The total image processing time for a 512 × 512 pixels image is about 3 s, that includes stack reading, image thresholding, LLS computing time and image display. The LLS computing time is about 20% of the total.

3. Theoretical model

For well aligned collagen and myosin molecules with helical arrangement along the main fibril axis z₀, the macroscopic susceptibility tensor is considered to have a cylindrically symmetry along z₀ resulting into a 6mm point group tensor symmetry with three independent macroscopic nonlinear susceptibility coefficients $χ_{33}$ , $χ_{31}$ and $χ_{15}$ in the fibril coordinate system x₀, y₀, z₀ [18]. Assuming that well aligned fibrils lie in the microscope stage x, z and that light propagates in the y direction of the x, y, z, laboratory coordinate system (see Fig. 1), theoretical polarization dependence of SHG intensity $I^{t h}$ is usually given by [2, 6, 10, 12–16, 19]

I^{t h} = c (\sin^{2} 2 (α - φ) + {(a \sin^{2} (α - φ) + b \cos^{2} (α - φ))}^{2})

where

φ

and

α

are respectively angles of fibril main axis and of input linear polarization angle relative to z direction.

a = χ_{31} / χ_{15}

,

b = χ_{33} / χ_{15}

and

c = K χ_{15}^{2}

are coefficients independent of

φ

assuming that in the excitation volume all fibrils have the same orientation. K is a positive scaling factor proportional to the intensity of the excitation fundamental field. In the case of a dominant axial molecular hyperpolarizability β,

χ_{33} = N β 〈 \cos^{3} θ 〉

,

χ_{31} = χ_{15} = N β 〈 \cos θ \sin^{2} θ 〉 / 2

where N is the density of nonlinear dipoles and brackets < > indicates an orientational average over the distribution of nonlinear dipoles orientation. It follows that

\cos^{2} θ = b / (2 + b)

where θ is the helix pitch angle (see Fig. 1) [6].

Fig. 1 Schematic diagram of collagen fibril orientation parameters. Input polarization angle $α$ , fibril orientation $φ$ relative to the laboratory x, y, z coordinate system. Helix pitch angle θ corresponds to the orientation of the molecular hyperpolarizability β relative to the fibril main axis. Incident IR beam is in the y direction.

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The problem is then to obtain estimated values $\tilde{a}, \tilde{b}, \tilde{c}, \tilde{φ}$ of respectively $a, b, c, φ$ from experimental SHG intensity $I^{\exp}$ obtained at different input polarization angles $α$ . It is worth to note that Eq. (1) has a mathematical intrinsic ambiguity since the same result is obtained by exchanging a and b and adding π/2 phase to $φ$ . To avoid this difficulty, we can rewrite $I^{t h}$ as a sum of cosine and sine frequency components over angle $α$ . Using trigonometric power formulas, Eq. (1) can be rewritten

I = A + B \cos 2 (φ - α) + C \cos 4 (φ - α)

where

A = c (4 + 3 a^{2} + 3 b^{2} + 2 a b) / 8

,

B = c (4 b^{2} - 4 a^{2}) / 8

and

C = c (a^{2} + b^{2} - 2 a b - 4) / 8

[16]. Using trigonometric addition formulas, Eq. (2) results in

I^{t h} = C_{1} + C_{2} \cos 2 α + C_{3} \sin 2 α + C_{4} \cos 4 α + C_{5} \sin 4 α

with

| \begin{matrix} C_{1} = A = c (4 + 3 a^{2} + 3 b^{2} + 2 a b) / 8 \\ C_{2} = B \cos 2 φ = c (4 b^{2} - 4 a^{2}) / 8 \cos 2 φ \\ C_{3} = B \sin 2 φ = c (4 b^{2} - 4 a^{2}) / 8 \sin 2 φ \\ C_{4} = C \cos 4 φ = c (a^{2} + b^{2} - 2 a b - 4) / 8 \cos 4 φ \\ C_{5} = C \sin 4 φ = c (a^{2} + b^{2} - 2 a b - 4) / 8 \sin 4 φ . \end{matrix}

Equation (3) represents a linear development of Eq. (1) on its parameters C_i (i = 1…5) that can be deduced using general linear least square (LLS) method. To this aim, Eq. (3) can be written in matrix form

I^{t h} = A C

with

I^{t h} = [\begin{matrix} I_{1}^{t h} \\ ⋮ \\ I_{k}^{t h} \\ ⋮ \\ I_{K}^{t h} \end{matrix}] A = [\begin{matrix} 1 & \cos 2 α_{1} & \sin 2 α_{1} & \cos 4 α_{1} & \sin 4 α_{1} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 1 & \cos 2 α_{k} & \sin 2 α_{k} & \cos 4 α_{k} & \sin 4 α_{k} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 1 & \cos 2 α_{K} & \sin 2 α_{K} & \cos 4 α_{K} & \sin 4 α_{K} \end{matrix}] C = [\begin{matrix} C_{1} \\ ⋮ \\ C_{5} \end{matrix}],

where

α_{k}

(k = 1…K) is the incident polarization angle and K the number of measurements. Coefficients

C

can be estimated by

\tilde{C}

solving the LLS problem

\tilde{C} = \arg \min_{C} \sum_{k = 1}^{k = K} {| I^{\exp} (α_{k}) - I^{t h} (α_{k}) |}^{2} = \arg \min_{C} {‖ I^{\exp} - A C ‖}^{2}

For K≥5, Eq. (7) admits the unique solution [20]

\tilde{C} = {(A^{T} A)}^{- 1} A^{T} I^{\exp}

where A^t is the transpose of A.

{(A^{T} A)}^{- 1}

is defined when columns of A are linearly independent and that is always verified for values of

α_{k}

uniformly distributed between 0° and 180° i.e. for

α_{k} = k π / K, k = 0, .., K - 1

.

\tilde{C}

substitution for

C

in Eq. (4) allows determination of coefficients

a = χ_{31} / χ_{15}

,

b = χ_{33} / χ_{15}

and

φ

. In order to obtain from Eq. (4) analytical expression of a, b and

φ

as a function of parameters

C

, we can introduced two coefficients R and R’ defined as

\begin{array}{l} R \equiv \sqrt{\frac{C_{2}^{2} + C_{3}^{2}}{C_{1}^{2}}} = \sqrt{\frac{{(4 b^{2} - 4 a^{2})}^{2}}{{(4 + 3 a^{2} + 3 b^{2} + 2 a b)}^{2}}} \\ R' \equiv \sqrt{\frac{C_{4}^{2} + C_{5}^{2}}{C_{1}^{2}}} = \sqrt{\frac{{(a^{2} + b^{2} - 2 a b - 4)}^{2}}{{(4 + 3 a^{2} + 3 b^{2} + 2 a b)}^{2}}} . \end{array}

For collagen and myosin where

a, b \in [0, 2]

[6, 12, 21], Eq. (9) results in

R = \frac{4 | b^{2} - a^{2} |}{4 + 3 a^{2} + 3 b^{2} + 2 a b}, R' = \frac{4 - {(a - b)}^{2}}{4 + 3 a^{2} + 3 b^{2} + 2 a b} .

Finally, a and b can be obtained from Eq. (10) with an uncertainty concerning the sign of the absolute value that is positive for collagen (b>a) and negative for myosin (a>b). That corresponds to the mathematical intrinsic ambiguity of Eq. (1) for the determination of

φ

. For collagen (b>a), straightforward calculation gives

a = \sqrt{2} \times {(\frac{1 + 3 R'}{1 - R' - R} - \sqrt{\frac{1 - R' + R}{1 - R' - R}})}^{1 / 2}, b = \sqrt{2} \times {(\frac{1 + 3 R'}{1 - R' + R} - \sqrt{\frac{1 - R' - R}{1 - R' + R}})}^{1 / 2} .

For myosin (a<b), then a and b have to be permuted in Eq. (11). Estimation

\tilde{φ}

of

φ

is directly obtained from Eq. (4)

\tilde{φ} = \frac{1}{2} \arg ({\tilde{C}}_{2} + i {\tilde{C}}_{3})

restricted to

φ \in (0, π)

and where arg denotes the argument of the complex number.

4. Results and discussions

Validation of LLS theoretical model has been performed on gels of type I collagen fibrils. It is well established that spontaneous entropy-driven process is responsible for the assembly of procollagen molecules into fibrils of 50 to 100 nm diameter and of 1 to 20 μm length (for review see [22]). Typical SHG image from collagen gels is shown in Fig. 2(a).

Fig. 2 Polarization dependent SHG image analysis of type I collagen fibrils. (a) Average intensity of SHG polarization stack. (b) SHG intensity variation with polarization angle $α$ from the 3 ROIs indicated in (a). Theoretical determination of (c) angular orientation $φ$ , (d, e) nonlinear susceptibility coefficient ratios $χ_{33} / χ_{15}$ , $χ_{31} / χ_{15}$ and (f) collagen helix pitch angle $θ$ . Note that the vertical scales for $φ$ and $θ$ are in degree. LLS computing time to obtain SHG parameters was found to be 40 ms (image size is 512 × 512 pixels, pixel computing time is 10 μs). Scale bars are 10 μm.

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Straight fibrils with different orientation can be observed ensuring that in the focusing volume, all molecules are oriented in the same direction. Polarization dependence of SHG intensity for three different fibrils (ROIs 1-3) is shown in Fig. 2(b). Pixel by pixel orientation $φ$ determined by LLS fitting is shown in Fig. 2(c). The root-mean-square error of the angular orientation of each straight fibril is about 15° compared to its physical orientation which is in agreement with theoretical analysis of Fig. 5(d) and with previous report [16]. Pixel by pixel determination of nonlinear susceptibility coefficient ratios $χ_{33} / χ_{15}$ and $χ_{31} / χ_{15}$ are shown respectively in Fig. 2(d) and 2(e). Their mean values obtained from 4 different cultures (28 ROIs) are respectively 1.6 ± 0.2 and 1.1 ± 0.2. Helix pitch angle $θ$ is shown in Fig. 2(f) and mean value is found to be 48.3° ± 1.5° (see also Table 1).

Table 1. Quantification of SHG parameters using LLS method for different experimental conditions. Fibril corresponds to collagen gel fibrils; ECM corresponds to extracellular matrix collagen fibrils. P values are obtained from student t-test (the number n of ROIs is above 16 for each experimental condition).

View Table

The overall performance of the model has been tested next on collagen tissue from xenopus gastrocnemius tendon. This tissue has been chosen because it is predominantly of type I collagen [23] and fibrils are straight and well aligned within each voxel. Typical SHG image is shown in Fig. 3(a). Pixel by pixel quantification of polarization dependent SHG parameters is shown in Fig. 3(b) and 3(e). We found the following averaged mean values $χ_{33} / χ_{15} = 1.5 \pm 0.1$ , $χ_{31} / χ_{15} = 0.9 \pm 0.2$ and helix pitch angle $θ = 49.8 ° \pm 0.7 °$ (4 samples and 16 ROIs, see also Table 1). These values are also in agreement with previous reports [6, 10, 11, 13, 21, 24, 25]. Surprisingly, statistical analysis indicates that polarization dependent SHG parameters determined by LLS method are significantly different between in vitro collagen gel fibrils and in situ collagen tendon.

Fig. 3 Polarization dependent SHG image analysis of collagen tendon. (a) Average intensity of SHG polarization stack. (b) Angular orientation $φ$ of collagen fibrils. Nonlinear susceptibility coefficient ratios of (c) $χ_{33} / χ_{15}$ and (d) $χ_{31} / χ_{15}$ . (e) Collagen helix pitch angle $θ$ . Note that the horizontal scales for $φ$ and $θ$ are in degree. LLS pixel computing time to obtain SHG parameters was found to be 250 ms (image size is 396 × 139 pixels, pixel computing time is 10 μs). Scale bars are 10 μm.

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We next applied our model to extracellular matrix (ECM) collagen fibrils remodeled by F1 biliary epithelial cells. Typical images of respectively SHG signal from ECM collagen and auto-fluorescence from spheroid forming cells are illustrated in Fig. 4(a) and 4(b). The merge image is shown in Fig. 4(c).

Fig. 4 ECM collagen fibrils remodeling by F1 cells studied by TPEF and SHG polarization analysis. (a, b, c) are respectively average intensity of SHG polarization stack, TPEF and merged images. (d) Angular orientation $φ$ of collagen fibrils. Nonlinear susceptibility coefficient ratios of (e) $χ_{33} / χ_{15}$ and (f) $χ_{31} / χ_{15}$ and (g) collagen helix pitch angle $θ$ . Note the difference of color between the delimited ROI and elsewhere only for (d). Note also that the horizontal scales for $φ$ and $θ$ are in degree. LLS computing time to obtain SHG parameters was found to be 400 ms (image size is 512 × 512 pixels, pixel computing time is 9 μs). Scale bars are 10 μm.

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One can observed that collagen fibrils in the ECM are aligned between spheroid cells (delimited ROI) and misaligned elsewhere as confirmed with pixel-by-pixel angular orientation $φ$ analysis of collagen fibrils (Fig. 4(d)). Hence, homogeneous angle of about 40° (blue pixel color) is observed in the main direction linking bottom and top spheroids (delimited ROI). Despite heterogeneity of the overall pixel-based orientation, more uniform nonlinear coefficient ratios are obtained within delimited ROI and elsewhere (Fig. 4(e),4(f)) and as a consequence, helix pitch angle is also homogeneous (Fig. 4(g)). This result clearly indicates that F1 cells affect fibrillar orientation without modifying nonlinear susceptibility coefficients. The following averaged values were obtained $χ_{33} / χ_{15} = 1.3 \pm 0.1$ , $χ_{31} / χ_{15} = 0.7 \pm 0.1$ and helix pitch angle $θ = 51.6 ° \pm 0.6 °$ (4 samples and 18 ROIs, see also Table 1). These values are significantly different from those of cell free gel fibrils and tendon. Furthermore, one can observed localized higher values of nonlinear susceptibility coefficients ratios in the vicinity of spheroids suggesting angular tilt of collagen fibrils [10]. The discrepancy of polarization dependent SHG parameters values ( $χ_{33} / χ_{15}$ , $χ_{31} / χ_{15}$ and pitch angle $θ$ ) between ECM of F1 cells and cell-free collagen gel fibrils (see Table 1) could arise from a different 1/ molecular hyperpolarizability β 2/ supramolecular organization within the excitation volume 3/ signal to noise ratio (S/N). Considering that collagen fibrils originate from the same chemical compound of type I collagen, one can assume that the molecular hyperpolarizability is the same for the two experimental conditions. Despite a different organization of extra cellular collagen fibrils between spheroids (delimited ROI) and elsewhere, uniform nonlinear coefficient ratio is obtained (see Fig. 4(e) and 4(f)) suggesting that the supramolecular organization within the excitation volume is the same. However, we have noticed that the experimental conditions of Fig. 2 and Fig. 4 needed different PMT gains leading to different S/N values as shown in Fig. 5.

Fig. 5 Effect of S/N on experimental and theoretical determination of SHG parameters from polarization analysis using LLS and NLLS methods. (a, b, c) Theoretical (continuous lines) and experimental results (discrete values) for respectively nonlinear susceptibility coefficient ratios of $χ_{31} / χ_{15}$ , $χ_{33} / χ_{15}$ and collagen helix pitch angle $θ$ . (d) Theoretical estimation of angular fibril orientation $φ$ . Note that continuous lines are mean values of estimated SHG parameters obtained for 10⁴ independent trials and that full error bars are two standard deviations estimations units of SHG parameters. Cyan and yellow colors are for LLS and NLLS respectively. Dotted lines with white color correspond to theoretical values $b = χ_{33} / χ_{15} = 2$ , $a = χ_{31} / χ_{15} = 1.5$ , $θ = arc \cos {(b / (2 + b))}^{1 / 2} = 45 °$ and $φ = 90 °$ . Note also that experimental data have been obtained from gel of collagen fibrils (cross), collagen tendon (filled circle) and ECM collagen fibrils remodeled by F1 cells (asteriks). LLS and NLLS computing time to obtain SHG parameters were found to be respectively 70 ms and 40 mins.

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Therefore, we next determine the impact of S/N on the accuracy of the estimated SHG parameters. To this aim, we first generate a theoretical polarization dependent SHG intensity curve using Eq. (1) obtained with polarization angles $α$ uniformly distributed between 0° and 180° incremented by 10°. Theoretical values $χ_{33} / χ_{15} = 2.0$ , $χ_{31} / χ_{15} = 1.5$ and $φ^{t h} = 90 °$ , corresponding to the best fit of the experimental data, were used for the theoretical curve. These values are also in agreement with those reported by Su et al. [7]. This theoretical curve was first scaled such that the square root of its maximum value was equal to a given S/N value. Assuming photonic shot noise, a Poisson random intensity value was simulated independently for each points of this theoretical curve. Individual Poisson distributions were centered on the corresponding scaled theoretical intensity values (it is worth to note that a Poisson distribution is completely determined when its mean value is imposed). SHG parameters were estimated from this noisy polarization curve using either LLS or NLLS implemented with lsqcurvefit (optimization toolbox) using $χ_{33} / χ_{15} = 2.0$ , $χ_{31} / χ_{15} = 1.5$ and $φ^{t h} = 90 °$ as initial guesses. Mean values of SHG parameters, obtained after 10⁴ independent trials, were finally plotted in Fig. 5 as a function of S/N. Each experimental S/N value of Fig. 5 was estimated from a polarization stack within a particular ROI with fibrils of the same orientation. For each polarization angle $α$ , S/N_α was calculated from the ratio of the mean pixel intensity over the standard deviation from all pixels of the ROI. Experimental S/N value of this ROI was then defined by the maximum of these values (SN = max_α(S/N_α). For collagen gel fibrils, muscle tendon and extracellular matrix of F1 cells, experimental results of Fig. 5 are from 16 to 28 different ROIs and from four different samples. Both LLS and NLLS methods give almost identical results showing that both estimated nonlinear coefficient ratios $χ_{33} / χ_{15}$ and $χ_{31} / χ_{15}$ increase with S/N and reach a plateau for S/N ≈6 (Fig. 5(a) and 5(b)). This demonstrates that accuracy of SHG nonlinear coefficients ratio is strongly dependent of S/N. It follows that the estimated helix pitch angle $θ$ decreases with S/N from 55° to 45° for S/N varying from 1 to 6 (Fig. 5(c)). In contrast to nonlinear coefficient ratios curves, fibril orientation $φ$ curve (Fig. 5(d)) exhibits no bias (mean value of the difference between estimated and theoretical values). We have also verified that nonlinear coefficient ratios curves are independent of fibril orientation $φ$ . STD of estimated SHG parameters $χ_{33} / χ_{15}$ , $χ_{31} / χ_{15}$ , and $θ$ are almost identical for LLS and for NLLS (Fig. 5(a)-5(c)). However, STD of estimated angular fibril orientation $φ$ is greater for LLS than for NLLS at low S/N value (Fig. 5(d)). Altogether these results show that the accurate measurement of the nonlinear SHG plateau parameters necessitates S/N of the order of 6. Indeed, a good correlation (R = 0.9, p<0.001) was found between nonlinear susceptibility coefficient ratios ( $χ_{33} / χ_{15}$ , $χ_{31} / χ_{15}$ ) and S/N. This explains the difference obtained between mean values of nonlinear susceptibility coefficient ratios between samples from the same type I collagen for different S/N. It follows from this result that $χ_{31} \neq χ_{15}$ , suggesting a possible additional molecular contribution to the peptide bound as suggested by Rocha-Mendoza et al. [7, 26]. Consequently, taking into account a second molecular contribution will modify the formula of the helix pitch angle [7]. However, regardless of the number of molecular contribution to the nonlinear optical susceptibility tensor, discrimination of different types or composition of collagen using modification of pitch angle needs to take into account S/N variation. Also, increasing the focusing depth under the gel surface should reduce S/N (because of diffusion and absorption of the incident IR beam) and therefore accuracy of SHG parameters should be affected. Finally, polarization dependent SHG parameters determined by LLS method in straight fibrils of predominantly type I collagen (gel and tendon) are in good agreement with previous results obtained with NLLS method. The main advantage of LLS compared to both NLLS [6, 10,11, 13, 21, 24, 25] and FT [16] methods is that it provides an analytic solution. As a consequence, estimated SHG parameters are obtained in a few hundred milliseconds with no need for initial conditions and for correction of fibril angle $φ$ because the mathematical intrinsic ambiguity of Eq. (1) has been a priori solved. As a result, non-expert in numerical signal processing can easily realize automatic fitting of SHG polarization stacks with LLS method (the MATLAB code is available on request).

5. Conclusion

This study proposes a new method for pixel by pixel analysis of nonlinear susceptibility $χ^{(2)}$ coefficients and fibril orientation from polarization dependent SHG images. This method is based on a linear least square fitting that provides an analytic solution. This method is fast (few hundred milliseconds for a 512 × 512 pixel image) and very easy to perform for non-expert in numerical signal processing. This report highlights the importance of signal to noise ratio for accurate determination of nonlinear susceptibility $χ^{(2)}$ coefficients. The results suggest that, in addition to the peptide group, a second molecular nonlinear optical hyperpolarizability $β$ contributes to the SHG signal. This study also highlights a particular remodeling of the microarchitecture of the extracellular matrix collagen fibrils by F1 cells and this could be important for fulfilling important physiological functions such as cell migration and communication.

Acknowledgement

This work was supported by Région Bretagne, Rennes Métropole, Conseil Général d’Ille-et-Villaine, CRITT Santé Bretagne, CBB Developpement Bretagne, Ministère de l’Enseignement Supérieur et de la Recherche and the European Union Federal Funds FEDER.

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	fibril	tendon	ECM	P value fibril/tendon	P value fibril/ECM	P value tendon/ECM
$χ_{33} / χ_{15}$ ± std	1.6 ± 0.2	1.5 ± 0.1	1.3 ± 0.1	0.01(**)	<0.001(***)	<0.001(***)
$χ_{31} / χ_{15}$ ± std	1.1 ± 0.2	0.9 ± 0.2	0.7 ± 0.1	0.002(**)	<0.001(***)	<0.001(***)
$θ$ ± std (in deg)	48.3 ± 1.5	49.8 ± 0.7	51.6 ± 0.6	<0.001(***)	<0.001(***)	<0.001(***)

Linear least square (LLS) method for pixel-resolution analysis of polarization dependent SHG images of collagen fibrils

Abstract

1. Introduction

2. Experimental methods

2.1. Preparation of biological sample

2.2. Imaging system and theoretical simulation

3. Theoretical model

4. Results and discussions

5. Conclusion

Acknowledgement

References and links

Cited By

Figures (5)

Tables (1)

Equations (12)

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