Accordion-like collagen fibrils suggested by P-SHG image modeling : implication in liver fibrosis

Second-order non-linear optical anisotropy parameter ρ = χ33 / χ31 is calculated for collagen-richt issues considering both a single dominant molecular hyperpolarizability tensor element β333 = β at single helix level and a priori known submicrometric triple helical organization of collagen molecules. Modeling is further improved by taking account of Poisson photonic shot noise of the detection system and simple supra-molecular fibrillar arrangements in order to accurately simulate the dispersion of ρ values in collagen-rich tissues such as tendon, skin and liver vessels. From combined P-SHG experiments and modeling, we next correlate experimental and theoretical statistical distributions of ρ. Our results highlight that the dispersion of experimental ρ values is mainly due to (i) Poisson photonic shot noise in tendon and skin, which proves to have a preponderant effect in P-SHG experiments (ii) variance of supercoil angles of accordion-like fibrils in vessels that is further reduced during the development of liver fibrosis therefore contributing to the rigidity of the tissue. These results open new avenue for future modeling correlating 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.


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 as much structural information in physiological as well as in disease state. Organization of collagen tissue using P-SHG is usually described by anisotropy parameter coefficient that is defined by the ratio of the two independent second-order nonlinear optical susceptibility tensor coefficients and that are involved in the nonlinear optical interaction (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 hierarchical organization (helical, triple-helical) of nonlinear molecules by correlating the experimental and theoretical statistical distributions of values through a Monte Carlo simulation taking into account the background Poisson photonic shot noise of the detectors (15). However, we failed to explain rigorously the dispersion of values in some tissues such as mouse liver vessels.
The aim of the present article is to solve this problem. For this purpose, we first model the distribution of values considering Poisson photonic shot noise of the detectors and different fibrillar architectures (intra and inter pixel arrangements of straight or supercoil fibrils). Then by correlating experimental and theoretical results, we find that the dispersion of values is dominated by Poisson photonic shot noise in collagen tendon and skin characterized by quasi-

THEORY
Second harmonic electric fields emitted at 2ω originate from nonlinear polarization induced by mixing of intense electric fields at ω in a medium characterized by a macroscopic second-order nonlinear optical susceptibility . Our strategy to calculate for each pixel is first to calculate the second order nonlinear optical susceptibility of an individual fibril and then to add the contributions of all the fibrils inside the pixel. can be obtained from the associated molecular hyperpolarizability tensor by averaging the contributions of all nonlinear dipoles of the fibril. Throughout the rest of the document, we assume that has a single coefficient that has been verified experimentally on 12 different tissues (see Supporting Material S1) and that is dominated by the peptide bonds along the collagen single helix scaffold.
Mechanism of fibril formation is well known (19), it is a hierarchical helical organization process of peptide bonds related as schematized in Fig. 1. Starting from ( Fig. 1 a), the hierarchical organization is a multi-step process. The first process is the formation of a single helix (H) polypeptide chain around single helix axis z1 characterized by a polar angle ( Fig. 1 b). The second process is the formation of a triple helix (3H) around triple helix axis z2 characterized by a polar angle (Fig. 1 c) and resulting from the association of three polypeptide chains called tropocollagen (collagen molecule). Staggered and cross-linked assembly of these collagen molecules explain the formation of straight fibrils as in tendon (19). A supercoil (SC) process ( Fig. 1 d) is taken into account in dermis where triple helical tropocollagen is twisted at constant polar angle around supercoiled fibril axis z3, forming supercoiled micro fibrils or fibrils (19). Another process may be considered if the fibril is tilted (T) with a polar angle relative to the microscope stage ( Fig. 1 e and f). Taking account of the hierarchical four processes, we found after calculation (see Supporting Material S2) that only three tensor elements and of mainly contribute to the SHG signal, and they are given by when written in microscope stage coordinate system (x4, y4, z4), (see Fig. 1 f). In this equation, are the polar angles of respectively helix (H), triple helix (3H), supercoil (SC) and tilt (T) processes and and are the azimuthal angles of respectively SC and T processes. In this equation, we assume that fibrillar number density of nonlinear dipoles is equal to unity. Since the pitch PSC = 1 µm (20,21) of the supercoil helix is greater than the transverse PSF (0.4 µm, see Materials and Methods), < > stands for an average over at constant obtained on the part of the supercoil fibril that is contained in the PSF.
Based on staggering and cross-linking properties of fibrillar collagen molecules and the non-centrosymmetric requirement for the development of the SHG process, we assume that each pixel consists of Nf identical fibrils, such that theoretical anisotropy parameter of each pixel is and SHG intensity of each pixel is given by the usual formula (1,(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(22)(23)(24)(25)(26)(27)(28) ( In this equation, and are angles of respectively z4 and incident laser input polarization (POL) directions with fixed z direction of the microscope stage (see Fig. 1 f) and the scaling factor k is proportional to the setup geometry and to the square of the incident IR laser intensity. In order to take account of a possible fibrillar disorder that may be present in collagen tissues, we next assume that each pixel of the SHG image may contain fibrils with a different fibrillar architecture i.e. different values. Three simple cases that accurately simulate the dispersion of values observed in collagen-rich tissues such as tendon, skin and liver vessels are considered. A diagram regrouping the three cases is shown in Fig. 2 where four pixels and three fibrils per pixel (Nf = 3) are displayed for both straight and supercoiled fibrils. So we assume that the fibrillar architecture may be due to a constant change (case 1) or to a dispersion (case 2) of tilt angle from in plane position In these two cases, supercoiled fibrils are regular and characterized by a constant supercoil angle . For supercoiled fibrils, we hypothesize that in addition to the two previous cases, fibrillar architecture may be due to a variation of the supercoil angle along main fibril axis (case 3), producing irregular supercoiled fibrils with an accordion-like shape. It is worth noting that in cases 2 and 3, values of and vary with and leading to a greater dispersion of these parameters in case of increased fibrillar disorder.
Since we have recently shown that the experimental anisotropy parameter of each pixel , obtained by fitting P-SHG data with Eq. 2 is highly dependent on Poisson photonic shot noise of the detection system (14), it is of paramount importance to take Poisson noise into account in the calculation of the theoretical anisotropy parameter called in the following. is obtained from a random P-SHG intensity curve generated from Eq. 2 with Poisson parameter proportional to (see Supporting Material S3 for more details). In summary, three anisotropy parameters and are determined for each pixel of the SHG image: is directly calculated from Eq. 1 while and are obtained by LLS fitting of Eq. 2 on respectively experimental and Poisson noise P-SHG intensity curves (details of the LLS fitting procedure is also described in Supporting Material S3).

MODEL SIMULATION RESULTS
In order to rigorously explain dispersion of experimental values in collagen-rich tissues, we model in the following the impact of the inter-pixel fibrillar disorder on the dispersion of theoretical values for the three cases of fibrillar arrangements described above. We first consider the case of straight fibrils in Fig. 3, then that of the supercoiled fibrils in Fig. 4. The three cases of fibrillar architecture are recalled in inset for each figure. Since we assume that there is no intra-pixel fibrillar disorder, only one fibril is represented for each pixel and therefore, each fibril represented in each inset belongs to a different pixel. Theoretical results can be summarized as follows.
For straight fibrils, the case of pixels with in plane ordered fibrils is considered first. Tendon collagen could be the closest tissue corresponding to this case. The evolution of anisotropy parameter as a function of the P-SHG stack mean photon number per pixel Nph is shown in Fig. 3 a. Simulation is obtained with , corresponding to the experimental mean values found in rat tendon for Nph = 100 (see Table 1). The curve decreases quasi exponentially with increase Nph toward and a bias due to Poisson noise is clearly visible below Nph = 40. Normalized histograms highlight the decrease of both mean (white crosses) and dispersion of with increasing Nph (Fig. 3 b). Note that the effect of and on is not shown in Fig. 3 a and b. Increasing their values within the range of Table 1 results in respectively decrease and increase of . More importantly, we noticed that their dispersion within the same range had a minor impact on the dispersion of for both straight and supercoiled fibrils and for this reason and are considered constant in the following. The case of pixels with out of plane straight fibrils and constant tilt angle (case 1) is shown in Fig. 3 c. Histograms of are characterized by an increase of both mean (white crosses) and dispersion of with increasing from 0° to 30°.
The case of pixels with disordered straight fibrils with random tilt angles (case 2) around is shown in Fig. 3 d. Histograms of are characterized by a larger increase of both mean (white crosses) and dispersion of with increasing from 0° to 30°.
Simulation is obtained for angles that are distributed within a normal distribution around as indicated in the inset. Correlation between and is more precisely shown in Fig. 3 e for different values of Nph (20, 50, 100) and for the two cases of fibrillar architecture considered: constant tilt angle (continuous line curves, case 1) and random tilt angle, (dashed line curves, case 2) and for values of either or varying from 0° to 15°. The major finding of the simulation is that dispersion of increases more rapidly for fibrils with random tilt angles (case 2) compared to fibrils with constant ones (case 1) as clearly shown by the difference of slope between the continuous and dashed line curves.
For supercoiled fibrils, the simple case of in plane ordered fibrils with regular supercoil angle is considered first. Skin collagen could be the closest tissue corresponding to this case. The evolution of as a function of Nph is shown in Fig. 4 a. The and corresponding to mean values found in rat skin for Nph =100 (see Table 1). As in tendon, the curve decreases with increasing Nph toward higher . Similarly, a bias due to Poisson noise is also observed below Nph = 40. The case of out of plane regular supercoiled fibrils with constant tilt angle (case 1) is shown in Fig. 4 b. Tilting the fibrils has a similar increasing effect on and std as in tendon (compare with Fig. 3 c). The case of supercoiled fibrils with fibrillar disorder due to dispersion of tilt angles (case 2) is represented later in Fig. 4 d. The histograms are similar to that obtained for straight fibrils in Fig. 3 d except that they are shifted to higher values. Finally, the case of fibrillar disorder due to variation of supercoil angles around mean position (case 3), producing irregular supercoiled fibrils with an accordion-like shape is considered next in Fig. 4  , both mean and dispersion of increase with increasing dispersion of and but the effect of the latter is more drastic.

EXPERIMENTAL RESULTS
In order to determine the origin of the dispersion of experimental anisotropy parameters , we choose to correlate our architecture-based model of theoretical anisotropy parameters to in situ experimental ones obtained from typical examples of respectively straight ( tendon) and supercoiled skin, liver vessels) collagen fibrils. All following theoretical simulations were obtained using hierarchical angles and corresponding to the best match between and found in rat tendon and skin for Nph = 100 (see Table 1).
For rat extensor digitorum longus tendon, P-SHG image and map of are shown respectively in Fig. 5 a and b. Distribution of decreases with Nph ( Fig. 5 c) as expected from theoretical analysis (see Fig. 3). For this tissue, we obtain over the entire SHG image (see also   5 f, red and blue histograms) indicates that some fibrils are probably tilted and therefore out of plane , corresponding to case 1 of inter pixel fibrillar architecture (see Fig. 2). As the dispersion of the experimental values decreases with Nph (see Fig. 5 c and e), the fibrillar architecture is estimated for each value of Nph, i.e. between pixels having an identical value of Nph. The mismatch between distributions of and is calculated for each interval of ten photons by minimizing the Kolmogorov-Smirnov distance (29) between the two empirical cumulative distribution functions associated to and as a function of the tilt angle ( Fig. 5 g, green   curve). From dk, we have defined a correlation coefficient that is used to determine the best match between distributions of (red curves) and (magenta curves) as shown in fig. 5 g-i. As a result of this correction, we obtain over the entire SHG image and Rk increases from 0.89 to 0.99 without and with correction of tilt angle . Altogether, these results show that rat tendon is characterized by preponderant ordered straight fibrils and that dispersion of values is mainly described by effect of Poisson noise. Small contribution of fibrillar arrangement associated with constant tilt angles (case 1) is often observed for pixels with low Nph values.
We next extend our study to tissues characterized by supercoiled collagen fibrils such as skin and liver vessels. In a previous study, we showed that the distribution of mean values of experimental anisotropy parameters was well described by our theoretical model, but we did not explain their dispersions rigorously (15). Despite taking into account both Poisson noise and fibrillar disorder due to the tilt of fibrils corresponding to case 2 of inter pixel fibrillar architecture (see Fig. 2), dispersion of experimental values were always greater than theoretical ones (15). Based on our new theoretical modeling indicating that dispersion of can be further enhanced by introducing the dispersion on supercoil angles corresponding to case 3 of inter pixel fibrillar architecture (see Fig. 2), we next test this hypothesis.
For rat skin, P-SHG image and normalized histogram of (red color) are shown respectively in Fig. 6 a and   and (magenta color) histograms (Fig. 6 b) is obtained for a mean dispersion of tilt angles calculated over the entire SHG image and corresponding to (yellow color) Fig. S3 of Supporting Material S4 for more details). These results suggest that the dispersion of values in rat skin is mainly due to both Poisson noise with a contribution due to fibrillar disorder associated with a dispersion of tilt angles . For control mouse liver vessels P-SHG image analysis is shown in Fig. 6 c and d. Normalized histogram of (red color) is characterized by with mean Nph ~ 23 over the entire SHG image (see also Table 2). Compared to skin, while mean value is similar (p = 0.1), dispersion is significantly greater (p<0.001) suggesting greater fibrillar disorder. Extrapolating from the experimental values (red asterisks), the theoretical curves of Fid. 4 d suggest that dispersion of should originate from dispersion of supercoil angles rather than tilt angles . Thus, the only way to achieve the best match between dispersion of and is to take into account additional disorder of supercoil angles assuming accordion-like supercoiled fibrils. Theoretical simulation reveals that the best match (Rk = 0.95) between (red color) and (magenta color) histograms ( Fig. 6 d) is obtained for calculated over the entire SHG image and corresponding to (yellow color) histogram (see also Fig. S4 of Supporting Material S4 for more details). Similar disorder of supercoil angles is also found in rat aorta media (see Table 2). We have previously shown that fibrosis in mouse liver vessels is characterized by a significant reduction in the mean value of when taking into account pixels with low Nph (5<Nph<50), compared to that obtained in mouse liver control and this has been interpreted as a reduction of fibrillar disorder (15). However, this interpretation did not consider the difference between the dispersions of and , that we address in the following. Fibrotic CCl4-treated mouse liver vessels P-SHG image analysis is shown in Fig. 6 e and f. Normalized histogram of (red color) is characterized by with mean Nph ~ 25 over the entire SHG image (see also Table 2). While mean values between control and fibrotic vessels are not significantly different (p = 0.45), dispersion of is significantly reduced (p<0.05) in fibrotic vessel compared to control one. Theoretical simulations reveal that the reduction of the dispersion of in fibrotic vessel is due to a significant (p<0.001) reduction of the inter pixel disorder of the supercoil angles from 7° to 5° (see also Table 2), that corresponds to the best match (Rk = 0.95) between (red color) and (magenta color) histograms of Fig. 6 f (see also Fig. S4 of Supporting Material S4 for more details).
Altogether, these results suggest for the first time to our knowledge that distribution of experimental anisotropy parameter in vessels is mainly driven by a variation of supercoil angles within a single accordion-like supercoiled fibril. This disorder is reduced in fibrotic livers, suggesting more efficient cross-linking and an increase stiffening compared to control vessels.

DISCUSSION
Our analytical calculation of the second-order nonlinear optical susceptibility tensor in collagen-rich tissues is based on the approximation of a single element in the associated molecular hyperpolarizability tensor . Within this approximation, we found that the best (2 ) estimated value of single helix angle (H) determined from collagen tendon is (see Table 1). This value is similar to that previously found in collagen tendon (5,30,31) but is about 7° higher than value deduced from X-rays diffraction studies (20,22). Although peptide bonds have a major contribution to SHG signal (5,6,11,26), this discrepancy suggests other contributions to SHG signal as previously reported (9,13,32). Interestingly, our assumption of an equivalent unique single molecular hyperpolarizability tensor element is further validated by experimental results showing that in several collagen tissues (see Supporting Material S1). Moreover, our estimation of triple helix (3H) angle in collagen tendon is in agreement with X-rays diffraction studies (20). We also assumed that supercoiled fibrils characteristics of skin and vessels, correspond to an additional helical supercoil (SC) arrangement obtained from straight fibrils with a mean supercoil angle . This was necessary to accurately simulates distribution of experimental anisotropy parameter in these tissues as previously reported (12). In this context, the two independent nonlinear optical susceptibility coefficients , contributing to the SHG signal as well as the associated anisotropy parameter were calculated as a function of the known hierarchical poly-helical assembly of the collagen straight or supercoiled fibrils. In order to correlate dispersions of experimental and theoretical distributions of anisotropy parameters and , Poisson photonic noise and fibrillar architecture have been considered. Thus, by minimizing the Kolmogorov-Smirnov distance between the two empirical cumulative distribution functions associated to and fine fibrillar arrangement has been deduced in different physio-pathological collagen tissues (tendon, skin, liver vessels). We considered only fibrillar disorder originating from either tilt (case 2) or supercoil angular (case 3) dispersions resulting from pixel to pixel random variation. Since we observed that the discrepancy between distributions of and decreases with number of photons Nph in all tissues (see Figs 5, S3, S4, S5 d and e), we assumed that the fibrillar disorder also decreases with Nph. We therefore fitted the experimental data for each Nph value with a Kolmogorov-Smirnov test procedure to determine fibrillar disorder for each pixel of a 512×512 SHG image. We found that, for ordered straight and regular supercoil fibrils as tendon and skin, dispersion of is mainly due to Poisson shot noise. However, we found that the dispersion of in skin is slightly larger and associated to a slight dispersion of the tilt angles (see Table 2). The novelty of this study is that dispersion of values in liver vessels and aorta media might originate from variation of supercoil angles along fibrillar main axis. Regular supercoiled fibrils with mean values of have been observed by ultrastructural SEM studies in skin cornea and vessels, however dispersion of has not been reported to our knowledge (19). While P-SHG studies have also reported the presence of regular supercoiled fibrils with constant supercoil angle in the same tissues (12,15), our present study suggests that accordion-like supercoiled fibrils characterized by dispersion of supercoil angles are responsible of the dispersion of values in liver vessels. This assumption is supported by the fact that experimental values for control mouse live vessels (red asterisks) are on the dotted line curves of Fig. 4 d for all values of Nph. For this tissue, we obtain calculated over the entire SHG image. Interestingly, similar results are found in rat aorta media (see td θ SC = 4.5° Table 2). For CCl4-treated mouse liver vessels, results show a significant (p<0.05) reduction of the dispersion of the supercoil angles, , compared to control ones (see Table 2). This result is in agreement with our previous results suggesting less disorder in CCl4-treated fibrotic vessels (15). While in that study we focused our analysis on the variations of the mean values of , in the present work, we accurately consider their dispersions. It turns out that dispersion of supercoil angles well described both mean and standard deviation of values in liver vessels. Altogether, these results suggest that liver fibrosis is characterized by a remodeling of collagen fibrils favoring increase fibrillar alignment. Since it has been shown that collagen fibrillar alignment is involved in cross-linking and stiffening of the extracellular matrix (33)(34)(35)(36), this remodeling will result in an increase in rigidity of vascular wall and a subsequent rise in intraportal tracts and central veins tone as already suggested (37). We also noticed that for most collagen tissues studied, the average number of photons Nph for a SHG image is between 20 and 50 (see Table 2) and that the lowest photon numbers are associated with the more disordered tissues. In this range of photon numbers, the Poisson noise impacts the values of the experimental anisotropy parameter and this concerns both the dispersion and the bias of its values (see Figs 3 a and 4 a). However, thresholding above 50 photons to avoid Poisson noise will result in a great loss of information regarding the estimate of the fibrillar disorder. On the other hand, increasing the acquisition time to improve the signal-to-noise ratio makes it possible to reduce the dispersion of values, but the disadvantage is probably the presence of more artifacts related to the mechanical drift. For example, with our experimental setup, the acquisition of a SHG image with a duration of one second gives 20 photons per pixel due to the 5MHZ bandwidth limit of the high sensitivity single photon GasAsP photomultiplier (H7421-40) provided by Hamamatsu. In a usual P-SHG acquisition, a compromise needs to be done between acquisition duration and signal to noise ratio. We found that a maximum photons number around 100 is a good compromise. In any case, even at low photon numbers, there is no difficulty in distinguishing between the dispersion of values between straight and supercoiled fibrils (see Figs 3 a and 4 a). 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 noise is considered.

CONCLUSION
In this study, we have calculated theoretical second-order non-linear optical anisotropy parameter for collagen-rich tissues considering the fibrillar disorder. The major contribution of this work, combining theoretical modeling and P-SHG experiment, concerns the structural origin of the dispersion of the values of anisotropy parameter in collagen tendon, skin and pathophysiological liver vessels. Our study reveals that fibrillar disorder significantly increases the dispersion of values and that its rigorous determination from a P-SHG experiment should consider the Poisson noise of the detection system. The novelty of this study is that dispersion of values in liver vessels and aorta media might originate from variation of supercoil angles along fibrillar main axis suggesting the presence of accordion-like fibrils. In addition, this study paves the way for future modeling correlating dispersion of values, fibrillar disorder and mechanical stiffness of diseases tissues.

SUPPORTING MATERIAL
Supplemental Material S1-S4 are available at http...     but theoretical curves in yellow and magenta color are obtained for out of plane fibrils . was calculated by minimizing the Kolmogorov-Smirnov distance dk between the two empirical cumulative distribution functions associated to and for each interval of ten photons (green curve of panel g with values on right y-axis). The overall correction corresponds to a Kolmogorov-Smirnov mean correlation coefficient between and distributions associated with a mean tilt angle over the entire SHG image (see also Table 2). Note the good superposition of the red and magenta curves as well as red and magenta histograms. (yellow color) gives the best correspondence with the greatest Kolmogorov-Smirnov correlation coefficient Rk between (red color) and (magenta