Wavy nature of collagen fibrils deduced from the dispersion of their second-order nonlinear optical anisotropy parameters

: From P-SHG experiments, second-order nonlinear optical anisotropy parameters ρ = χ ZZZ / χ ZXX of collagen tissues are calculated assuming the same model of supercoiled collagen ﬁbril characterized by a variable angle θ . Dispersion of experimental ρ values is converted into distribution of θ values based on the wavy nature of collagen ﬁbrils deduced from EM studies. For tendon, the results show that the dispersion of experimental ρ values is mainly due to Poisson photonic shot noise assuming a slight ﬁbrillar undulation with θ = 2.2 ° ± 1.8 ° . However for skin and vessels, the dispersion of experimental ρ values is mainly due to a stronger ﬁbrillar undulation with θ = 16.2 ° ± 1.3 ° . The results highlight that this undulation is reduced during the development of liver ﬁbrosis therefore, contributing to the rigidity of the tissue.


Introduction
Second harmonic generation (SHG) microscopy is a label-free technique that relies on a nonlinear optical interaction with hyperpolarizable non-centrosymmetric endogenous fibrillar proteins like collagen and myosin causing scattered coherent radiation at twice the fundamental frequency [1][2][3][4]. Polarization dependence of SHG (P-SHG) microscopy is gaining increase popularity for investigating fibrillar collagen-rich tissues with the desire to extract structural information in both physiological and disease states. Organization of collagen fibrils is usually described by the SHG anisotropy parameter ρ, which is related to the SHG intensity ratios measured for the directions of the excitation fields parallel and perpendicular to the axis of the collagen fibrils [5][6][7][8][9][10][11][12][13].
We previously proposed a simple reliable and fast linear least square (LLS) fitting method to process P-SHG images at pixel-resolution [14]. More recently, we extended this method to retrieve the pixel-resolved sub-microscopic organization of fibrillar collagen molecules taking into account the background Poisson photonic shot noise of the detectors and the poly helical architecture (helix, triple helix, supercoil) of collagen fibrils. Correlation between experimental and theoretical statistical distributions of ρ values through a Monte Carlo simulation was used to estimate the fibrillar disorder in the SHG image. In this previous model, we assumed that it was due to a random tilt of each fibril contained in the pixel [15]. However, with this model we could not accurately report dispersion of experimental ρ values in liver vessels assuming supercoiled tropocollagen molecules twisted at a constant polar angle θ.
The aim of the present work is a significant upgrade of our model to rigorously explain the dispersion of experimental ρ values by considering a pixel by pixel change of polar angle θ. We find that the dispersion of ρ values is the result of a slight fibrillar undulation with θ = 2.2 • ±1.8 • in tendon. However for skin and vessels, the fibrillar undulation is stronger with θ = 16.2 • ±1.3 • . Moreover, in liver vessels, development of fibrosis is characterized by a significant reduction of θ supporting the increase rigidity of the vessels.

Preparation of biological samples
Samples from physiological and pathological tissues were prepared as previously reported and summarized as follows [15]. Physiological samples were prepared from rat extensor digitorum longus (EDL) tendon, rat skin and mouse liver vessels. Adult Wistar rats (200-300 g) were euthanized by CO 2 inhalation. Tissues were dissected, fixed over night with 4% paraformaldehyde in phosphate buffer saline at 4°C, and rinsed at least three times with PBS. For SHG imaging, dissected pieces (100-200 µm thickness) of tissues were mounted in 50% glycerol-PBS solution and stabilized between two coverslips sealed with nail polish. For pathological samples, wild-type C57BL/6J mice received intraperitoneal injection of either 0,5 µg CCl 4 (Sigma, St Louis, MO) dissolved in oil in order to induce liver fibrosis or the olive oil solvent for control mice. Injections were performed 3 times (D1, D3 and D7) the first week followed by single injection every week lasting 8 weeks (W1-W8). Mice were euthanized at D1 or at W10, livers were harvested and fixed in 4% paraformaldehyde. For SHG-imaging, 10 µm thick sections of liver slices mounted between microscope glass slides and coverslips were obtained from the Biosit-H2P2 core (histopathologie.univ-rennes1.fr/) according to established standard procedure. All rats and mice were cared for in accordance to the "Guide for the Care and Use of Laboratory Animals" (Directive 2010/63/UE).

Image acquisition and analysis
Images were acquired on a custom made SHG microscope part of the BIOSIT facility https://biosit.univ-rennes1.fr. It is based on an inverted microscope (IX71, Olympus, Japan) and the laser source is a tunable IR 80 MHz femtosecond Ti:Sa laser (MAITAI, Spectra Physics). High NA water immersion objective (Olympus UPLSAPO 60XW NA = 1.20, WD = 0.28 mm) was used for applying 10-20 mW of 740 nm excitation at the sample. PSF was obtained from 0.17 µm diameter fluorescent micro beads (Molecular Probes PS-Speck Microscope Point Source Kit (P7220)) and estimated to be 0.4×1.2 µm. SHG signal was collected in forward direction using high NA objective (Olympus, LUMFl 60XW, NA = 1.1). SHG signal is detected using high sensitivity single photon GasAsP photomultipliers (H7421-40, Hamamatsu). Photons are counted by a general-purpose acquisition USB module (NI-USB 6363, National Instruments). It is important to note that in addition to tissue clearing obtained with 50% glycerol treatment, all polarization dependent SHG images were acquired at a depth < 20 µm from the tissue surface to minimize depolarization [16] and birefringence effects [17,18]. All experimental P-SHG stacks were obtained for input polarization angles uniformly distributed between 0°and 180°with 20°i ncrements. This incremental step is a good sampling compromise for accurate acquisition of P-SHG parameters and quality of the recording, avoiding motion artifact. Image analysis and simulations were performed using MALAB (MathWorks, Natick, MA, USA).

Theory
Nonlinear molecular response of collagen tissues can be described by the hyperpolarizability response of tropocollagen triple helix molecules [19] that assemble in a quarter-staggered array to form microfibrils [20]. Moreover, it is known that collagen fibrils can be straight as in tendon or supercoiled as in skin and vessels [21][22][23][24][25][26][27]. In this latter case, the triple helices are wrapped with an angle θ around the main axis Z of the fibril in the fibrillar coordinate system (X, Y, Z), ( Fig. 1(a)). (X 0 , Y 0 , Z 0 ) and (X, Y, Z) are the reference frames of respectively the triple helix and of the supercoil collagen fibril and Z 0 and Z are their respective main directions. θ and ϕ refer to the polar and azimuthal angles and P is the pitch corresponding to a full turn of the coiling. Schematic diagram illustrating the waving of collagen fibrils in tendon (b) and in skin or vessels (c) with increasing pixel by pixel dispersion std θ of supercoil angles θ from left to right. (d) Truncated normal (Gaussian) distribution of supercoil angles θ with parametersθ and std θ of the normal distribution. (e) Schematic view of intra pixel disorder. In this illustration, the pixel contains three fibrils different polar angles (θ f 1 , θ f 2 , θ f 3 ). Note that each fibril is represented by one of its triple helix molecule. (f) Schematic view of the microscope stage. Z is the main fibrillar direction supposed to be in the plane of the microscope stage, and forming an angle γ with the fixed direction z of the laboratory coordinate system (x, y, z). Polarization of the incident IR beam (POL) makes an angle α with z. Light propagates along Y = y.
In the following, we choose to represent straight and supercoiled fibrils by the same model of fibril with a variable supercoil angle θ according to the collagen tissue : θ 0 • for tendon and θ 17 • for skin and vessels. This is consistent with the fact that collagen fibrils are more or less coiled (for review see [21]). Moreover, waving or undulating collagen fibrils were also observed in liver [28] and in tendon from EM studies [24,[29][30][31]. From EM results, we hypothesize that the wavy nature of collagen fibrils could be modeled by a pixel by pixel dispersion of the supercoil angle θ. A schematic diagram illustrating the waving of collagen fibrils is shown for tendon ( Fig. 1(b)) and for skin or vessels ( Fig. 1(c)). The increasing dispersion of supercoil angles θ from left to right corresponds to an increasing fibrillar undulation. This fibrillar waving is simply modeled here by a truncated normal (Gaussian) distribution of supercoil angles θ centered around mean value θ with dispersion std θ ( Fig. 1(d)). These parameters correspond to the parameters of the normal distribution without truncation which should not be confused with the mean and standard deviation of the truncated normal distribution.
In the hypothesis of a possible intra pixel fibrillar angular disorder, we next assume that each pixel may contain N f fibrils having triple helices with different polar angles θ f ( Fig. 1(e)), always measured with the same common direction Z supposed to be in the plane of the microscope stage ( Fig. 1(f)). These angles are randomly chosen in the same truncated normal distribution of parameters θ and stdθ for simplicity. In this context, we calculated the second-order nonlinear optical susceptibility χ (2) at pixel level by summing the contribution of all triple helices in the focusing volume. We found (see Appendix) that for an incident IR beam propagating in the Y direction ( Fig. 1(f)), only two independent second-order nonlinear optical susceptibility coefficients χ ZZZ and χ ZXX = χ XXZ = χ XZX mainly contribute to the SHG signal when expressed in coordinate system (X, Y, Z). The resulting SHG anisotropy parameter ρ th = χ ZZZ / χ ZXX is given by where h(θ f ) and cos 2 ϕ f are also defined in the Appendix. These coefficients correspond respectively to the summation made on the N f fibrils f contained in the pixel and to a mean value made on the azimuthal angle ϕ on the part of the fibril f that is contained in the transverse PSF. ρ 0 is defined as the anisotropy parameter of the triple helix [9,12,32]. Under these conditions, the SHG intensity is given by the usual formula [5][6][7][8][9][10][11][12][13][14][32][33][34][35][36][37][38][39][40]] where α and γ are the angles of respectively the directions of the input polarization (POL) and of the main fibril axis Z with a common direction z of the laboratory coordinate system (x, y, z), ( Fig. 1(f)).
On the other hand, we have also previously shown that it is important to take into account the contribution of Poisson photonic shot noise of the detection system to minimize correlation errors between experimental and theoretical results [15]. For this reason, a Poisson noise anisotropy parameter, named ρ poiss in the following, is deduced from ρ th for correlation with experimental anisotropy parameters ρ exp calculated by LLS fitting method of the P-SHG stack [14]. The procedure to calculate ρ poiss starts with the calculation of I 2ω (α) at given ρ th considering the proportionality factor as unity in Eq. (2). A Poisson noise P-SHG intensity curve I poiss (α) is then simulated by generating for each α (0°, 20°,40°. . . 180°, see section Materials and methods) a Poisson distributed random variable with a Poisson parameter (mean of the Poisson distribution) equals to N ph × I 2ω (α)/I 2ω . N ph and I 2ω respectively correspond to the averages on α of the number of photons per pixel and of I 2ω (α). Finally, ρ poiss is deduced from I poiss (α) using LLS fitting method [14].  The result shows that ρ th increases with θ similarly to the case of increase fibrillar tilting [11]. We next model the impact of stdθ on ρ th for tendon (θ = 0 • , Fig. 2(b)) and skin (θ = 17 • , Fig. 2(c)). The impact of intra pixel fibrillar angular disorder is also taken into account considering a variable number N f (1,3,9) of fibrils in the pixel with different polar angles θ f . The main finding is that increasing std θ, reflecting a growing angular waving of the fibrils, results in an increase of mean value of ρ th and its dispersion std ρ th . However, increasing N f progressively reduced std ρ th towards zero. Taking into account Poisson photonic shot noise, we next model the evolution of ρ poiss as a function of the P-SHG stack mean photon number per pixel N ph for tendon (θ = 0 • , Fig. 2(d)) and skin (θ = 17 • , Fig. 2(e)) assuming no angular dispersion std θ = 0. For both cases, we observe that, whereas std ρ poiss decreases approximately as the inverse of the square root of N ph (as classically expected), the corresponding estimation of mean ρ poiss decreases asymptotically toward ρ 0 . One should notice that a bias due to Poisson noise is clearly visible for N ph < 40.

Model simulation results
This bias is further highlighted in Fig. 2(f) for well aligned straight fibrilsθ = 0, stdθ = 0. In accordance with Figs. 2(d) and 2(e), this bias is significantly reduced for N ph > 50. This shows that N ph must be greater than 100 to reduce the bias below 1% in order to accurately determine ρ 0 from experimental results.

Experimental results
Our objective in this section is to determine the organization of collagen fibrils in different tissues (rat tendon, rat skin, liver vessels of fibrotic mice) by adjusting the pair of valuesθ, stdθ of the theoretical distribution of supercoil angles θ ( Fig. 1(d)) on the basis of the best match between experimental ρ exp and theoretical ρ poiss distributions, obtained by minimizing the Kolmogorov-Smirnov distance between the two empirical cumulative distribution functions associated to ρ exp and ρ poiss [41]. P-SHG images and maps of experimental anisotropy parameters ρ exp are shown respectively in the first (a, g, m, s) and second (b, h, n, t) column of Fig. 3 for all studied tissues. Superimposed distributions of experimental ρ exp (red color) and theoretical ρ poiss (blue color) are shown as a function of N ph in the third column (c, i, o, u) of Fig. 3. ρ poiss distribution is obtained by minimizing for each interval of ten photons the Kolmogorov-Smirnov distance d k with ρ exp distribution. By doing this, ρ poiss is calculated assuming the same mean number of photons N ph (mean value of the interval) and thus the same Poisson parameter (see section Theory) for all pixels of each interval. A Kolmogorov-Smirnov distance D k is then deduced from the global distributions of ρ exp and ρ poiss and a coefficient R k = 1 − D k (R k ∈ [0 1]), herein named correlation coefficient is deduced. Superimposed histograms of ρ th (green), ρ poiss (blue) and ρ exp (red) are given in the fourth column (d, j, p, v) of Fig. 3. By assigning the same best matching values of θ and stdθ to the corresponding pixels of each ten photons interval, maps of θ and stdθ are shown in the fifth (e, k, q, w) and sixth (f, l, r, x) columns of Fig. 3. So, the distributions of mean number of photons N ph are re-encoded into distributions of supercoil angles θ. For all simulations, the best match between ρ exp and ρ poiss is obtained assuming no intra pixel angular disorder (N f = 1, see Discussion).
For rat EDL tendon illustrated in Figs. 3(a)-3(f), the average distribution of θ is characterized byθ = 3.8, stdθ = 3.0 (R k = 0.995). This suggests a weak angular waving of collagen fibrils. The statistical parameters of the 9 samples studied are shown in Table 1. Moreover, histograms of Fig. 3(d) show that dispersion of ρ exp is mainly due to Poisson photonic shot noise since dispersion of ρ th is negligible. In addition, maps ofθ and stdθ shown in Figs. 3(e) and 3(f) indicate that in plane fibrils (blue color) correspond to high N ph values ( Fig. 3(a), red color). Finally, from the plateau of Fig. 3(c) obtained at high photon number, the anisotropy parameter of the triple helix is deduced ρ 0 = 1.27 (see Eq. (1) and also the legend of Fig. 2(d)). One should notice that mean value of ρ exp obtained over all pixels is ρ exp = 1.34 ± 0.03 (see Table 1). This value is slightly smaller than 1.4 ± 0.03 found in rat-tails tendon taking into account birefringence and diattenuation [18] suggesting that SHG images were acquired at about 5µm depth (see Materials and Methods). Since it has been shown that in 50% glycerol cleared tissue (like our rat EDL tendon and skin samples) polarization curves remain unchanged up to 200 µm [16], we assume that birefringence and diattenuation should have a marginal effect onθ and stdθ in our experimental conditions, and therefore have not been taken into account in this study.
For rat skin illustrated in Figs. 3(g)-3(l), we found that dispersion of ρ exp is characterized by a larger average supercoil angleθ = 16.2 and a stronger fibrillar waving stdθ = 6.0 (R k =0.98), in agreement with the theoretical simulation of Fig. 2(c). The statistical parameters of the 9 samples studied are shown in Table 1.
For fibrotic mouse liver vessels, as illustrated in Figs. 3(s)-3(x), we found thatθ = 13.4 is significantly reduced (p = 0.006) compared to control vessels, while stdθ = 8.5 is almost  All values correspond to mean values for n samples.θ and stdθ correspond to mean value and dispersion of the normal distribution of supercoil angles θ that best describes the dispersion ofρ exp . R k is the correlation coefficient between ρ exp and ρ poiss distributions. All angles are in degrees. ** has the following t-test statistical meaning: 0.001 < p < 0.01, between fibrotic and control liver vessels. unchanged (R k = 0.97). The statistical parameters from 23 samples studied are also shown in Table 1. For tendon, skin and liver vessels, a good inverse correlation between stdθ and N ph is observed (Fig. 3, third column; compare also first and last columns) indicating a lower fibrillar undulation at high photon level. In other words, fibrillar undulation increases dispersion of ρ exp . The results show that the fibrillar undulation stdθ increases from tendon to vessels (see Fig. 3, last column and also Table 1), which is consistent with a reduction of the spatial variation of the SHG intensity along each fibril as can be seen in first column of Fig. 3. The results also suggest for the first time to our knowledge that the distribution of experimental anisotropy parameters ρ exp in vessels can be explained by a variation of the supercoil angles θ. Since the mechanical properties of collagen fibrils in the toe and heel regions of the stress-strain curves is characterized by straightening of molecular kinks [42][43][44][45][46], the significant reduction of average angles θ in fibrotic vessels compared to control ones should corresponds to collagen fibrils elongation. This should result in an enhancement of the elastic Young's modulus and therefore contributing to the stiffness of the fibrotic vessels.

Discussion
The novelty of this study is that it addresses the origin of the dispersion of experimental anisotropy parameters ρ exp values measured from P-SHG experiments in collagen tissues. To this aim, a simple model is used to convert dispersion of ρ into distributions of supercoil angles θ. We hypothesize that the angular dispersion of supercoil angles θ could be correlated with fibrillar undulations observed at EM level [24,[28][29][30][31]. The results of our model given in Table 1 agree with those of EM studies [22,23,27]. Hence, for tendon we foundθ = 2.2 • ±1.8 • versusθ ≤ 5 • for rat tendon [21]. For skin, we foundθ = 16.2 • ±1.3 • versusθ = 18.4 • ±2.8 • for rat skin [23,26]. In addition, other EM studies in dermis have shownθ 18 • [24,27]. For liver vessels, we found θ = 16.3 • ±2.1 • versusθ = 17 • ±3.6 • for bovine aorta [23]. In our model, a good correlation (R k >0.95) between ρ exp and ρ poiss is obtained assuming no intra pixel fibrillar disorder, otherwise a drastic reduction of R k occurs as previously shown [15]. For example, for liver vessels and for skin, R k is reduced respectively from 0.96 to less than 0.9 and from 0.98 to 0.92. The absence of intra pixel fibrillar disorder suggests that all fibrils are aligned within each pixel, and this could be explained by interfibrillar cross-linking properties underlying the assembly of fibrils into fibers (bundle of fibrils).
It is worth to note that in the analysis of all SHG images, pixels have been thresholded above 5 photons. As a consequence the greatest value of supercoil angular dispersion calculated from the model is found in liver vessels and is about std θ = 7.5 • . It is also useful to recall that supercoil angle θ is determined from fibril main axis that is supposed to be in the plane of the microscope stage. This assumption hold in our experimental conditions for collagen tendon and skin since experimental anisotropy parameters ρ exp values found (respectively ∼1.3 in tendon and ∼1.5 in skin) at high photon level correspond to values found for in plane fibrils [11,12,47]. Moreover, the dispersion of ρ exp is fully explained by Poisson photonic shot noise (blue and red histograms of Fig. 3 fully superimpose). For liver vessels, ρ exp values found at high photon level (5% of the pixels) are similar to those found in skin suggesting that corresponding fibrils are in plane. For the remaining pixels (95%), Poisson photonic shot noise alone cannot explain dispersion of ρ exp assuming that fibrils are in plane (compare blue and red histograms of Figs. 3(p) and 3(v)). Indeed, dispersion of ρ exp that is correlated to dispersion of θ is greater in vessels compared to skin (see Table 1) suggesting an additional contribution due to tilted fibrils that is not considered by the model. Assuming that std θ is similar in skin and control liver vessels, the differential value of ρ exp could be explained by a tilt of less than 5°(assuming that tilt angular dependence of ρ is approximately given by Fig. 2(a)). Another way to estimate the contribution of tilted fibrils in std θ is to compare the values found in skin and liver vessels. The observed difference of about 2°(see Table 1) can be attributed to tilt. Alltogether, neglecting the tilt contribution in our model migth lead to an error below 5°in the estimation of θ.
We have addressed the sensitivity of our adjustment method between experimental ρ exp and theoretical ρ poiss distributions by mapping correlation coefficient r k as a function ofθ and stdθ for tendon, skin and liver vessels obtained for each interval of ten photons. Maps of r k corresponding to the data of Fig. 3 are shown in Fig. 4. At low photon level, several couples ofθ, stdθ values (brown color) are close to the same maximum r k value but at high photon level, the results show that the intra fibrillar structure of fibrils of tendon are straight while those of skin and vessels are supercoiled. Moreover, at low photon level, fibrotic liver vessels have lower supercoilθ values (14.7°) than control vessels (16.3°) (see also Table 1), suggesting that supercoil angles might be regulated by physio pathological conditions of the tissue that need to be determined. The reduction ofθ, that results in increasing the stiffness of the extracellular matrix, is probably induced by mechanical traction along fibril axis [48][49][50][51]. As a consequence, an increase rigidity of vascular wall and a subsequent rise in intra-portal tracts and central veins tone are expected [52]. Fig. 4. Maps of correlation coefficient r k between ρ exp and ρ poiss distributions as a function of θ and stdθ for intervals of ten photons for (a) Rat EDL tendon (b) Rat skin (c) Control liver vessel (d) Fibrotic liver vessel. Data correspond to samples of Fig. 3. Note that to avoid a great reduction of the number of pixels for the last map, its range of photons has been increased. Note that number of photons N ph and of pixels N pix are indicated for the eight maps. All angles are in degrees. Note also that maps in panel (c) are characterized by rough lines. This is due to the fact that control mouse liver vessels are characterized by fewer collagen-based pixels compared to fibrotic ones (see also Fig. 3). Consequently, the standard deviations of the r k statistics which are estimated from less data are greater, which explains the roughness of the lines.
We have noticed that for most collagen tissues studied, the average number of photons N ph for a SHG image is between 20 and 50 (see Table 1). In this range of photon numbers, the Poisson photonic shot noise impacts the values of the experimental anisotropy parameter ρ exp (see Figs. 2(d)-2(f)). However, thresholding above 50 photons to avoid Poisson photonic shot noise will result in a great loss of information concerning the wavy nature of fibrils. Alternatively, increasing the acquisition time will improve the signal to noise ratio, but the disadvantage is probably the presence of more artifacts related to mechanical drift. For example, with our experimental setup, the acquisition of a 512 × 512 pixels SHG image with a duration of one second gives ∼ 20 photons per pixel due to the 5 MHZ bandwidth limit of the high sensitivity single photon GasAsP photomultiplier (H7421-40) provided by Hamamatsu. In a usual P-SHG acquisition, a compromise between acquisition duration and signal to noise ratio is mandatory. We found that a maximum photons number around 100 is a good compromise. Overall, our study indicates that an accurate estimate of fibrillar disorder from a P-SHG experiment is possible regardless of the number of photons per pixel as long as Poisson photonic shot noise is considered.
Our model enables an efficient conversion of the dispersion of ρ values into a dispersion of supercoil angles θ. However, it does not take into account either chiral components of the nonlinear susceptibility tensor that appear for tilted fibrils [53] or axial components of the electric fields induced by high numerical aperture (NA) [54,55] while both factors could impact an accurate determination of ρ. Concerning chirality, we assume its contribution to be negligible in the determination of ρ because we estimated that collagen fibrils of tendon and skin are in plane and those of liver vessels have tilt angles less than 5°. Since the simulation is based on the plane wave approximation, it does not consider axial components of the electric fields induced by high numerical aperture (NA) objective like those used in our experiment (60XW, NA = 1.2, see section Materials and methods). For vessels, we found a reduction of ρ exp values of about 0.15 when using low NA objective (20X, NA = 0.75). This is in agreement with the theoretical prediction of about 0.05 obtained for the same NA variation [54,55]. Therefore, ρ exp and ρ 0 are overestimated by using high NA objective, and in consequence supercoil angles are also overestimated. According to Fig. 2(a), fluctuation of ρ of 0.15 corresponds to an angle variation of about 5°that still remains a good estimate of θ for all collagen tissues, and in good agreement with values found in EM studies.

Conclusion
In this study, we have calculated the second-order nonlinear optical anisotropy parameter ρ = χ ZZZ / χ ZXX for collagen-rich tissues assuming the same model of wavy collagen fibril with a variable supercoil angle θ. The results highlight that the wavy nature of collagen fibrils can be deduced from distributions of experimental ρ values taking into account the Poisson photonic shot noise of the detection system. Furthermore, the results suggest that the stiffness of fibrotic liver is due to a remodeling of the extracellular matrix which reduces the average value of θ. These results open new avenue for future modeling that correlates the dispersion of ρ values in P-SHG experiments and the fibrillar architecture as well as the mechanical stiffness of patho-physiological extracellular matrices in collagen tissues.
According to the helical symmetry of the collagen triple helix molecule, its nonlinear optical hyperpolarizability β (2) is described by only two independent tensor elements β Z 0 Z 0 Z 0 and [9,12,32], when written in coordinate system (X 0 , Y 0 , Z 0 ) where Z 0 is the main axis of the triple helix ( Fig. 1(a)). Considering that the triple helix can be wrapped with an angle θ around the collagen main fibril axis Z in coordinate system (X, Y, Z) associated to the fibril ( Fig. 1(a)), components of β (2) expressed in this reference frame are given by the following tensor transformation [56] β IJK = R II 0 is any component of the Euler matrix R(Ω) that transforms between (X 0 , Y 0 , Z 0 ) and (X, Y, Z) coordinate systems [57] R(Ω) = cos ϕ cos θ − sin ϕ cos ϕ sin θ sin ϕ cos θ cos ϕ sin ϕ sin θ Euler angles θ, ϕ refer to polar and azimuthal angles ( Fig. 1(a)). As a rule, macroscopic second-order nonlinear optical susceptibility χ (2) at pixel level is obtained from β (2) by summing the contribution of all the nonlinear molecular sources within the focusing volume [58][59][60]. In the case where the incident IR beam light propagates in the Y direction ( Fig. 1(f)), the only components of χ (2) that participate to the SHG intensity are χ ZZZ , χ ZXX , χ XXZ = χ XZX , and χ XXX , χ XZZ , χ ZXZ = χ ZZX when expressed in coordinate system X, Y, Z, and if restricted to the plane wave approximation E ω Y = 0 [61]. In consequence, only the corresponding components of β (2) need to be calculated. Assuming that collagen main fibril axis Z lies in the plane of the microscope stage ( Fig. 1(f)), we found that only four coefficients β ZZZ , β ZXX = β XXZ = β XZX , β XXX , β XZZ = β ZXZ = β ZZX are independent, and they are given by 3 cos θsin 2 θ cos θ sin 2 θcos 2 ϕ cos θ − 3 cos θsin 2 θcos 2 ϕ sin 3 θcos 3 ϕ 3 sin θ cos ϕ − 3sin 3 θcos 3 ϕ sin θcos 2 θ cos ϕ sin θ(1 − 3cos 2 θ) cos ϕ Considering that each pixel may contain N f fibrils f (supposed identical for simplicity) but with different polar angles θ f (see Fig. 1(e)) measured from the same direction Z, the corresponding second-order nonlinear optical susceptibility coefficients are deduced where N is the triple helix concentration and h(θ f ) dϕ denotes mean value on the part of each fibril f of pitch P that is contained in the transverse PSF. Indeed, for skin and vessels, the pitch P is about 1 µm [22,62], which is greater than the size of the transverse PSF (w T = 0.4µm, see section Materials and methods) such that ∆ϕ = 2π × w T /P. For tendon, P∼20 µm is estimated from the SHG images by the average distance between two maximums along each fibril (see Fig. 3(a)).
Otherwise, ϕ f 0 is randomly chosen for each fibril f. χ XXX and χ XZZ are considered here weak because mean values g(ϕ) f are distributed around zero. In the particular case of an average of ϕ over a full turn of the coiling, these coefficients are exactly zeros. Moreover, we have verified numerically that they contribute generally to less than 10% to the SHG signal in our experimental conditions (θ f ≤ 25 • for all experimental results), and for this reason they have been neglected in the expression of the SHG intensity. Theoretical SHG anisotropy parameter ρ th = χ ZZZ / χ ZXX of each pixel is deduced from Eq. (6) ρ th = ρ 0 cos θ f + (3 − ρ 0 ) cos θ f sin 2 θ f cos θ f − (3 − ρ 0 ) cos 2 ϕ f cos θ f sin 2 θ f , where ρ 0 = β Z 0 Z 0 Z 0 /β Z 0 X 0 X 0 is defined as the anisotropy parameter of the triple helix. Equation (7) corresponds to Eq. (1) of the main text. When all the fibrils within the pixel have the same polar angle θ f = θ (no intra pixel disorder), then N f = 1 and h(θ f ) = h(θ) in the above equation. If in addition, the average of ϕ is carried out on a full turn of the supercoil helix then <cos 2 ϕ> = 1 2 , and the resulting expression of ρ th is identical to Eq. (1) of Ref. [12].