2.4.1.

SHG emission directional analysis

We determine the SHG directional emission ratio ${\mathrm{F}}_{\mathrm{SHG}}/{\mathrm{B}}_{\mathrm{SHG}}$ (or SHG creation ratio) which is reflective of the fibril diameter, the packing density, and regularity relative to the size scale of the SHG wavelength.³¹ This is determined by first measuring the $F/\mathrm{B}$ intensity ratio versus depth, where this axial response arises from a convolution of $F_{\mathrm{SHG}}/B_{\mathrm{SHG}}$ and subsequent propagation through the tissue which is based on the scattering coefficient (${\mu }_s$) and scattering anisotropy ($g$) at the SHG wavelength.³⁰ The measured SHG directional (F/B ratios) values were determined per optical section every $10\mu \mathrm{m}$ of depth using MATLAB software (Mathworks, Natick, Massachusetts). Monte Carlo simulations based on our modified customized MCML code³⁴ for SHG microscopy^30,35 are then used to uniquely extract $F_{\mathrm{SHG}}/B_{\mathrm{SHG}}$ using the scattering coefficients of prostate.³⁶ In the treatment here we used values from the literature, where the ${\mu }_s$ is assumed to be $445{\mathrm{cm}}^{-1}$ and $g$ to be 0.93.

2.4.2.

SHG conversion efficiency analysis

We determine the relative SHG conversion efficiencies from each tissue. This metric is a convolved effect of the collagen concentration (square thereof) and the fibrillar organization relative to the wavelength of SHG emitted light³⁷ and results from the magnitude of the second-order nonlinear susceptibility tensor ${\chi }^{(2)}$ and phase-matching conditions.³⁸ Specifically, better phase matching results in brighter SHG. The relative conversion efficiency is determined by measuring the forward attenuation, i.e., relative SHG intensity as a function of depth through the tissue. Normalization between tissues is necessary to account for local variability within the same tissue (different fields of view) and to make relative comparisons between tissues.³⁰ This metric is dominated by the conversion efficiency and primary filter (squared) based on ${\mu }_s$ at the excitation wavelength. Our SHG MCML framework is then used to decouple the conversion efficiency.³⁰ Here we assumed a ${\mu }_s$ of $335{\mathrm{cm}}^{-1}$ at the laser excitation wavelength and ran a series of forward simulation based on initial guesses until the best fit is found. We have shown previously that the secondary filter effects (i.e., loss of the SHG signal) is negligible compared to the primary filter effect.³⁹

All the simulations were performed at the Wisconsin Center for High Throughput Computing at the University of Wisconsin–Madison and the values were stored in tables, permitting the image processing to be performed in custom MATLAB programs.

2.4.3.

SHG image-based metrics

Multifractal analysis determines if a power-law scaling exists for various statistical moments at different scales, where monofractal and multifractal scaling behaviors are characterized by a single scaling exponential or a nonlinear function of the moments, respectively.⁴⁰ In our specific case, a broader distribution of grayscale values (and feature sizes) would have a broader range of exponential values. The multifractal analysis was performed with the MATLAB tool in the signal-processing toolbox.

Entropy is a statistical measure of randomness that can be used to characterize the texture of the input image. In this context, entropy is defined as $-\mathrm{sum}[p.*\log 2(p)]$, where $p$ contains the normalized histogram count and was calculated with the MATLAB tool in the signal-processing toolbox.

For the maximum overlap discrete wavelet transform, the wavelet and scaling filter coefficients at level $j$ are computed by taking the inverse discrete Fourier transform (DFT) of a product of DFTs. This is performed using the MATLAB tool in the signal-processing toolbox.

3.2.1.

Conversion efficiency measurements by forward attenuation analysis

Figure 2 shows the forward attenuation curves (SHG intensity versus depth) for the AP and VP of the wt and ${\mathrm{Dcn}}^{-/-}$ prostates. First, we note that there are minimal differences between the VP glands of the tissues. This is generally consistent with the observations of the similar morphology of the wt and ${\mathrm{Dcn}}^{-/-}$ in this gland, as shown in Fig. 1, where similar morphologies will have similar phase-matching conditions and similar SHG intensity. The response of the wt AP likely appears similar to both the VP glands due to the normalization process that is necessary to plot these data between different tissues on the same scale. As a result, tissues that have different SHG conversion efficiencies and scattering properties can have similar-looking attenuation curves, and as a consequence, this analysis is best for comparing the same type tissue (i.e., same gland) from normal and pathologic tissues.³⁰ Here, the attenuation curve for the AP ${\mathrm{Dcn}}^{-/-}$ decays much faster than the AP wt. Based on the underlying physics, this means that ${\mathrm{Dcn}}^{-/-}$ has lower conversion efficiency which would correspond to either less collagen or less ordered collagen or thinner collagen bundles, all of which are suggested by the image data in Fig. 1.

Fig. 2

(a) Forward SHG emission attenuation curves (intensity versus depth) for the four tissue types, normalized by peak power per series every $10\mu \mathrm{m}$. Each line represents the $\text{mean}\pm \mathrm{SEM}$ of $n=3$. No significant change is detected in the VP lobes. (b) Best-fit Monte Carlo simulations to the experimental data of the ${\mathrm{AP}}^{+/+}$ and ${\mathrm{AP}}^{-/-}$ based on the primary and secondary filers and guesses to the conversion efficiency. The $R^2$ values between the data and best simulation are 0.86 and 0.69 for the wt and ${\mathrm{Dcn}}^{-/-}$, respectively. (c) Bar graph of the extracted SHG conversion efficiencies for the AP gland, where the ${\mathrm{Dcn}}^{-/-}$ has significantly lower SHG conversion efficiencies compared to wt ($p=0.0046$).

We use the SHG MCML framework to extract the relative SHG conversion efficiencies of the normal and ${\mathrm{Dcn}}^{-/-}$ AP glands, where the simulation includes both the primary ($335{\mathrm{cm}}^{-1}$) and secondary filters ($445{\mathrm{cm}}^{-1}$; although the former dominates the response).³⁹ The data and best fit simulations are shown in Fig. 2(b). Based on the published values of the optical properties,³⁶ we have extracted relative conversion efficiencies of $10.82\pm 1.35$ and $4.78\pm 1.29$ for the normal and ${\mathrm{Dcn}}^{-/-}$ tissues, respectively. These are statistically different ($p=0.0046$). Moreover, these values are consistent with the wt having a more ordered collagen structure and/or thicker collagen fibers, resulting in better phase matching and brighter SHG.

3.2.2.

Fibril size and distribution via SHG emission directionality

The measured $F$ versus $B$ as function of depth response is used to extract the $F_{\mathrm{SHG}}/B_{\mathrm{SHG}}$ emission direction, which is a subresolution parameter that arises from the fibril size and packing relative to ${\lambda }_{\mathrm{SHG}}$. Figure 3(a) shows the averaged response from three tissues for the wt and Dcn AP lobes. We note that the ${\mathrm{Dcn}}^{-/-}$ AP lobe has a significantly lower measured F/B versus depth ratio compared to wt, which we have shown is reflective of lower fibril organization. To quantify this difference, we use the SHG MCML framework to decouple the initial emission directionality $F_{\mathrm{SHG}}/B_{\mathrm{SHG}}$ from the scattered component at ${\lambda }_{\mathrm{SHG}}$. The best fit simulations for the wt and ${\mathrm{Dcn}}^{-/-}$ using published values for ${\mu }_s$ and $g$^30,36 are given in Fig. 3(a) ($R^2$ values of 0.88 and 0.79, respectively). We then extracted $F_{\mathrm{SHG}}/B_{\mathrm{SHG}}$ values of 3.35 versus 2.85 for the wt and ${\mathrm{Dcn}}^{-/-}$, respectively ($p=0.046$). The lower value for the null is consistent with the structure having either smaller or more disordered fibrils relative to the wt. Thus, our SHG analysis is consistent with structural analyses of the null in other tissues (skin etc.) by SEM, which directly showed a more disordered fibril structure.²³

Fig. 3

(a) The measured forward/backward SHG as a function of depth (solid symbols and solid lines) for the ${\mathrm{AP}}^{+/+}$ and ${\mathrm{AP}}^{-/-}$ where each line represents the $\text{mean}\pm \mathrm{SEM}$ of $n=3$ along with the best-fit simulation based on the reduced scattering coefficient at 445 nm (open symbols and lines). The $R^2$ values between the data and best simulation are 0.88 and 0.79 for the wt and ${\mathrm{Dcn}}^{-/-}$, respectively. The best-fit SHG emission directionality and significance ($p=0.046$) are shown in (b). The results are consistent with the ${\mathrm{Dcn}}^{-/-}$ having decreased fibril organization.

3.3.1.

Entropy and multifractal dimension

To characterize the distribution of contrast levels from the collagen architecture between the wt and ${\mathrm{Dcn}}^{-/-}$ AP gland tissues, we determined the image entropy as a statistical measure. Here the entropy ranges from 0 to 1 where higher levels correspond to a larger distribution of gray scale levels, where in this formalism the absolute values are calculated. Figure 4(a) shows the extracted image entropy levels for the wt ($0.87\pm 0.03$) and ${\mathrm{Dcn}}^{-/-}$ ($0.77\pm 0.02$), where the latter had significantly decreased entropy ($p=0.036$). The larger entropy for the wt indicates that the SHG arises from a wider distribution of larger collagen structures. This is consistent with the higher $F_{\mathrm{SHG}}/B_{\mathrm{SHG}}$ values extracted for the wt [Fig. 3(b)], which arise from larger and more orderly distributed fibrils than those of the ${\mathrm{Dcn}}^{-/-}$.

Fig. 4

(a) Entropy of the forward-directed SHG for the ${\mathrm{AP}}^{+/+}$ and ${\mathrm{AP}}^{-/-}$, where the higher value for the ${\mathrm{AP}}^{+/+}$ is consistent with larger structures ($p=0.036$). (b) The multifractal dimension for the ${\mathrm{AP}}^{+/+}$ and ${\mathrm{AP}}^{-/-}$ where the larger values for the ${\mathrm{AP}}^{+/+}$ are further consistent with larger, more organized structures.

We next used multifractal analysis to further probe the distribution of fibers in each tissue. Figure 4(b) shows the resulting multifractal analysis of the forward collected SHG for the wt and ${\mathrm{Dcn}}^{-/-}$ tissues. Here, $d(h)$ is the normalized probability of the Holder singularity exponent $h$. We point out that the wt has a broader distribution and extends to higher values in the distribution. This is consistent with larger ordered structures in the fibrillar architecture in the wt and is further consistent with the entropy analysis. The respective multifractal distributions for the backward collected SHG are similar to the forward SHG (not shown).

3.3.2.

Quantification of coherence and effect on segmented image features

Figure 5 shows representative $F$ and $B$ collected images for the wt and ${\mathrm{Dcn}}^{-/-}$ from the AP glands. For both tissues, the forward collected images showed clear fibrillar behavior, whereas the backward collected images have many punctate or segmented spots mixed in with clear, contiguous fibers. However, these segmented fibers are much more evident in ${\mathrm{Dcn}}^{-/-}$. We have previously reported this phenomenon in other tissues and shown that it arises from destructive interference within the focal volume in the backward emitted SHG.^32,43 This arises because the coherence length of backward emitted SHG is shorter than the forward direction, as the phase mismatch $\mathrm{\Delta }k$ is larger, where the coherence length is given as $2\pi /\mathrm{\Delta }k$. In this framework, more disordered fibrils, with larger phase mismatch, as those seen in the ${\mathrm{Dcn}}^{-/-}$, will lead to significant numbers of “segmented” fibers.

Fig. 5

Representative forward and backward collected SHG images of the ${\mathrm{Dcn}}^{-/-}$ AP (a) and (b) wt AP. The segmented features in the backward channel arise from destructive interference from shorter coherence length structures, which are more pronounced in the ${\mathrm{Dcn}}^{-/-}$. $\text{Scale bar}=50\mu \mathrm{m}$.

We can use the extent of these segmented features to further compare the collagen architecture in wt and ${\mathrm{Dcn}}^{-/-}$ AP glands. Our approach uses the maximal overlap discrete wavelet transform to probe the distribution of sizes of the features in the images. We have calculated the resulting energy coefficients from the transform in each level, where each scale corresponds to $2^{^n}$ pixels. The results for the forward and backward images comparing the wt and ${\mathrm{Dcn}}^{-/-}$ are shown in Figs. 6(a) and 6(b), respectively, along with the statistical differences between the tissues. In the forward collected SHG, there are no significant differences between the tissues, although the lower scales have values $p\sim 0.1$ and higher $p$ values at larger scales. This corresponds to somewhat slightly dissimilar sizes of features in the tissues at small size and increasing similarity at larger-sized structures, comprising larger collections of fibers. This is consistent with the phase matching as small features (less than ${\lambda }_{\mathrm{SHG}}$) result in both $F_{\mathrm{SHG}}$ and $B_{\mathrm{SHG}}$, whereas larger features result only in forward emitted SHG.

Fig. 6

Calculated energy coefficients and associated $p$ values from the maximum overlap discrete wavelet transform from the forward (a) and backward (b) SHG from the ${\mathrm{AP}}^{+/+}$ and ${\mathrm{AP}}^{-/-}$ glands. There are no statistical differences in the forward direction, where in contrast there are differences in the backward SHG between the ${\mathrm{AP}}^{+/+}$ and ${\mathrm{AP}}^{-/-}$ at low levels, corresponding to small differences in the distribution of small structures in the tissues, consistent with the increased segmented features in the ${\mathrm{Dcn}}^{-/-}$ in Fig. 5(a).

Statistical differences are seen in the analysis of the backward collected data. The differences at low and medium scales are highly significant, where these diminish at larger scales. We stress that smaller sizes in the backward channel result from smaller features (and smaller $F_{\mathrm{SHG}}/B_{\mathrm{SHG}}$ values), whereas larger structures arise from multiple scattering of the forward SHG, rather than backward emitted SHG (i.e., $B_{\mathrm{SHG}}$). We further note that the segmented features arise from subresolution fibril sizes (and packing therein), but they can be discerned by SHG through the differences in coherence length.³¹ In this case, we next note that this analysis completely agrees with the emission direction analysis in the prior section, where the ${\mathrm{Dcn}}^{-/-}$ had lower $F_{\mathrm{SHG}}/B_{\mathrm{SHG}}$ values, corresponding to less ordered/smaller fibrils. We also note that while we have seen these segmented features in other tissues^32,43 the ${\mathrm{Dcn}}^{-/-}$ prostate is the most striking example to date.