Generalized 3D quasi-phase-matching model of image contrast in second harmonic generation microscopy of fibrillar collagen architectures

Second harmonic generation (SHG) microscopy is a powerful tool in assessing collagen structure, especially with respect to differentiating the respective architectures of normal and diseased tissues. An under-explored area is exploiting SHG to determine the sub-resolution aspects of the collagen fibril size, polarity, and packing (∼50-100 nm diameters). Due to the phase-matching and associated coherence of SHG, these structural aspects are encoded in the wavelength dependence of the spatial emission and relative conversion efficiency, denoted the creation attributes. As a means to extract this information, we present a generalized 3D computational/theoretical treatment based on quasi-phase-matching (QPM), which can predict the SHG emission pattern and relative conversion efficiency using collagen models based on 3D biomimetic fibril architectures. Specifically, we incorporate random rather than purely periodic structures and non-ideal phase-matching <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:mrow><mml:mo>(</mml:mo> <mml:mrow><mml:mi>Δ</mml:mi> <mml:mi>k</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn></mml:mrow> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> </mml:math> conditions. By exploration of parameter space, and comparison with imaging data, we can place bounds on the fibril architecture without the use of structural biology tools. The resulting predicted fibril sizes of real tissues are in good agreement with known values from electron microscopy. Moreover, by examining the role of heterogeneity, we have identified the contribution of small and large fibrils and clustering therein to the creation attributes, and the regimes where these dominate the spatial emission pattern. These simulations also resulted in good agreement with prior work on the wavelength dependence of SHG conversion efficiency, where the fibril size and packing are sufficient to reproduce experimental data without invoking a two-state model. This level of agreement provides validation of the model and also points to the need for this approach to treat the SHG responses due to the intrinsic complexity of many tissues.