What structural length scales can be detected by the spectral variance of a microscope image?

Abstract

A spectroscopic microscope, configured to detect interference spectra of backscattered light in the far zone, quantifies the statistics of refractive-index (RI) distribution via the spectral variance (Σ̃²) of the acquired bright-field image. Its sensitivity to subtle structural changes within weakly scattering, label-free media at subdiffraction scales shows great promise in fields from material science to medical diagnostics. We further investigate the length-scale sensitivity of Σ̃ and reveal that, in theory, it can detect RI fluctuations at any spatial frequency whatsoever. Based on a 5% noise floor, Σ̃ detects scales from ~22 to 200–700 nm (exact values depend on sample structure and thickness). In an example involving mass-density distribution characteristic of biological cell nuclei, we suggest the level of chromatin organization, which can be quantified via Σ̃.

Interferometric spectroscopy of scattered light quantifies the statistics of refractive-index (RI) distribution inside weakly scattering media using the spectral variance of their epi-illumination, bright-field microscope image (Σ̃²). Its sensitivity to changes in deeply subdiffractional RI correlation lengths has been previously demonstrated [1]. However, as the RI correlation length is a cumulative statistic of all length scales (LS) present within a sample, a change in its value implies a global structural reorganization at all LS, including those larger than the diffraction limit of light. Thus, when quantifying the LS sensitivity of Σ̃ through a statistic of RI distribution, it remains unclear precisely which structural LS within the sample are detected. Here we investigate two aspects of the LS sensitivity of Σ̃. First, we determine whether there is a fundamental limit to the size of structures it can detect. Second, we identify which LS within complex samples with continuous random RI distribution have the largest contribution to its measured value. Furthermore, in an example of mass-density distribution characteristic of biological cell nuclei, we assess which LS of chromatin organization can be measured via Σ̃.

We model a far-zone, epi-illumination, bright-field microscope with a small numerical aperture of illumination and wavelength-resolved image acquisition. Our sample geometry includes an RI-matched substrate, an RI-mismatched superstrate (e.g., air), and a uniform thickness L, which is smaller than the microscope’s depth of field (for most setups, 0.5–15 μm). We define the continuous, spatially varying RI of a weakly scattering sample as n(r), and its fluctuating part normalized by the mean value (n₁) as n_Δ(r) = (n(r) − n₁)/n₁. Using Born approximation to compute the scattered field inside the sample and ray optics to describe field propagation across its boundaries (theory developed in [1]), the expected value of spectral variance Σ̃² is found through the integral of the power spectral density (PSD, Φ_nΔ) of n_Δ within a frequency-space volume T_3D:

$${\mathrm{\sum }^{\sim }}^2=\frac{{\mathrm{\Gamma }}^2{k}_c^{2}L}{\mathrm{\Delta }k}{\int }_{T_{3D}}{\mathrm{\Phi }}_{{\mathbf{n}}_{\mathrm{\Delta }}}(\mathbf{k}){\mathrm{d}}^3\mathbf{k},$$ (1)

where k_c is the central wavenumber of the instrument’s bandwidth Δk (evaluated inside the sample) and Γ is composed of reflectance and transmission Fresnel coefficients, Γ = 4n₀n₁ (n₀ − n₁)/(n₀ + n₁)³, with n₀ being the RI of a sample’s superstrate. T_3D, as described in detail in [1], includes spatial frequencies k with longitudinal coordinates k_z between 2k₁ and 2k₂ (k₁ and k₂ denoting the lower and upper wavenumbers of the instrument bandwidth), contained within a radial distance k_z NA from the k_z axis (NA denoting the numerical aperture of light collection). For consistency, in this Letter we consider 500–700 nm bandwidth, NA = 0.6, and n₁ corresponding to fixed biological media, n₁ = 1.53 [2,3].

We begin the LS sensitivity analysis of Σ̃ by identifying spatial RI fluctuations inside a sample at which frequencies it can detect. To isolate the sensitivity to a given spatial frequency of RI fluctuations, we numerically evaluate Σ̃ measured from a sample composed of periodic structures of that frequency only.

Using MATLAB (The MathWorks Inc.), we generate a zero-mean, statistically isotropic, random, 3D unbounded medium $n_{\mathrm{\Delta }}^{\infty }(\mathbf{r})$, which is characterized by a single spatial frequency k_LS and variance ${\sigma }_n^2$ [Fig. 1(a)]. That is, the expectation of its PSD in spatial-frequency space is an infinitely thin spherical shell with a radius k_LS centered at the origin. Then we define the sample as a part of $n_{\mathrm{\Delta }}^{\infty }(\mathbf{r})$ with thickness $L:n_{\mathrm{\Delta }}(\mathbf{r})=n_{\mathrm{\Delta }}^{\infty }(\mathbf{r})T_L$, where T_L is a windowing function along the z axis. It is important to note that the PSD of n_Δ(r), due to its finite thickness, is no longer a spherical shell and is expressed as Φ_nΔ = |σ_nδ (k − k_LS) ⊗ {T_L}|², where denotes a unitary Fourier transform operator and ⊗ denotes convolution [Fig. 2(a)]. Finally, we obtain Σ̃² by computing the integral of Φ_nΔ within T_3D. For comparison, to accentuate the effect of the finite sample thickness, we repeat the analysis omitting the convolution with {T_L}, which corresponds to bulk media with L = ∞.

Fig. 1.

(a) Cross section of a single spatial-frequency medium with n₁/k_LS of 100 nm. (b) Cross section of n_Δ(r) with an exponential SCF and the corresponding (c) $n_{\mathrm{\Delta }}^l(\mathbf{r})$ and (d) $n_{\mathrm{\Delta }}^h(\mathbf{r})$ for W = 100 nm.

Fig. 2.

(a) Cross section of PSD of $n_{\mathrm{\Delta }}^{\infty }(\mathbf{r})$ (red) with k_LS > 2k₂, and the PSD of the abridged sample n_Δ(r) (blue). (b) $\mathrm{\sum }^{\sim }/{\sigma }_n{\mathrm{\Gamma }}_{01}^2$ as a function of the spatial frequency of RI fluctuations for samples with different thicknesses (all wavenumbers evaluated inside the sample). ${\mathrm{\Gamma }}_{01}^2$ denotes (n₀ − n₁)²/(n₀ + n₁)².

Referring to Fig. 2(b), we establish that Σ̃ can sense RI fluctuations at any spatial frequency whatsoever and has an enhanced sensitivity to frequencies with 1/k_LS between 20 and 40 nm (1/2k₂ = 26 nm and 1/2k₁ = 36 nm). Due to the finite L, weakly scattering structures of any size and shape, including RI fluctuations with subdiffractional frequencies above 2k₂, can still be detected. This is explained by the fact that the Fourier transform of finite structures, due to the convolution with {T_L}, is nonzero inside T_3D. Accordingly, lower values of L correspond to wider {T_L}, resulting in higher sensitivity of Σ̃ to spatial frequencies outside T_3D.

We emphasize that T_3D includes values of |k| only below 2k₂; hence the physics behind the fundamental limit of diffraction remain unbroken. Meanwhile, since the Fourier transform of RI fluctuations of a finite length with frequencies above 2k₂ (as well as below 2k₁) is nonzero inside T_3D, those fluctuations are detectable by Σ̃.

While Σ̃ is most sensitive to 1/k_LS of 20–40 nm, it is not obvious from the experimental perspective which internal structures this LS range refers to. Moreover, for practical cases of samples containing multiple LS, a universal interval of LS having the largest contribution to the measured Σ̃ cannot exist, as it depends on the sample structure. First, it is simply a matter of which LS and in what proportion are present. Second, as shown above, the sensitivity of Σ̃ depends on L. And finally, owing to the non-linear relation between n_Δ(r) and Φ_nΔ, contributions from different internal structures are not independent. Hence, the range of LS predominantly detected by Σ̃ can be found only for a specific structure. Below, we determine the LS sensitivity range for samples with various properties of internal organization, including analytically defined as well as experimentally obtained forms of RI spatial correlation function (SCF).

In order to identify whether a particular LS has a significant contribution to Σ̃, we “remove” it from the sample and calculate the consequent change in Σ̃ (for convenience, particle sizes rather than RI frequencies are measured, [4]). Thus, for lower LS analysis, we introduce a perturbed medium $n_{\mathrm{\Delta }}^l(\mathbf{r})$, which represents the original n(r) convolved with a 3D Gaussian filter G(r) with FWHM $W:n_{\mathrm{\Delta }}^l(\mathbf{r})=(n_{\mathrm{\Delta }}^{\infty }(\mathbf{r})\otimes G(\mathbf{r}))T_L$, as a result of which LS lower than W are removed from n(r). Similarly, $n_{\mathrm{\Delta }}^h(\mathbf{r})$ represents n(r) with high LS removed: $n_{\mathrm{\Delta }}^h(\mathbf{r})=n_{\mathrm{\Delta }}(\mathbf{r})-n_{\mathrm{\Delta }}^l(\mathbf{r})$ [Figs. 1(b)–1(d)]. Using Eq. (1), we obtain the spectral variance of a microscope image that would be measured from a sample if all LS lower (Σ̃^l2) or higher (Σ̃^h2) than W were removed:

$$\begin{array}{l}{\mathrm{\sum }^{\sim }}^{l2}=\frac{{\mathrm{\Gamma }}^2{k}_c^{2}L}{\mathrm{\Delta }k}{\int }_{T_{3D}}{\mid \mathcal{F}\{n_{\mathrm{\Delta }}^{\infty }\}\mathcal{F}\{G\}\otimes \mathcal{F}\{T_L\}\mid }^2{\mathrm{d}}^3\mathbf{k},\\ {\mathrm{\sum }^{\sim }}^{h2}=\frac{{\mathrm{\Gamma }}^2{k}_c^{2}L}{\mathrm{\Delta }k}{\int }_{T_{3D}}{\mid \mathcal{F}\{n_{\mathrm{\Delta }}^{\infty }\}[1-\mathcal{F}\{G\}]\otimes \mathcal{F}\{T_L\}\mid }^2{\mathrm{d}}^3\mathbf{k}.\end{array}$$ (2)

Since an analytical solution for Σ̃^l and Σ̃^h cannot be obtained, we evaluate Eq. (2) numerically. As before, we generate a zero-mean random 3D array with the desired form of SCF using MATLAB. Then we multiply the Fourier transform of the medium either by {G(r)} or by [1 − {G(r)}] (to remove short and long LS, respectively), after which we convolve it with {T_L} to account for the finite thickness. We obtain Σ̃^l2 and Σ̃^h2 by computing the integral of the squared absolute value of the resultant array within T_3D.

Importantly, when a Gaussian particle described by W is contained inside $n_{\mathrm{\Delta }}^{\infty }(\mathbf{r})$, only a fraction of that feature may be included in the truncated medium $n_{\mathrm{\Delta }}^{\infty }(\mathbf{r})T_L$. Hence we define its effective size W_eff as W multiplied by the expected value of the particle fraction inside the longitudinal interval $(-L/2,L/2):(-1+e^{-C^2})/(C\sqrt{\pi })+\text{Erf}(C)$, with Erf denoting the error function and $C=2L\sqrt{ln2}/W$.

We first consider the sample to have an exponential SCF. We calculate the spectral variance measured after removing low (Σ̃^l) or high (Σ̃^h) LS relative to that measured from the original sample (Σ̃). Results of this LS perturbation analysis, shown in Fig. 3, lead to two important conclusions. First, from the region of the steepest decline in Σ̃^h/Σ̃ and Σ̃^l/Σ̃, we find that Σ̃ is most sensitive to LS below 200 nm. Second, we determine that the limited L most significantly affects the sensitivity of Σ̃ to large structural LS. This results in a qualitative difference in the behavior of Σ̃^l and Σ̃^h calculated for different values of L: whereas for L = ∞ they saturate at W_eff > 200 nm, for thin samples they remain slowly varying functions of W_eff. This is in agreement with the single-k_LS analysis, which established that smaller L expands the range of detected LS.

Fig. 3.

Decrease in Σ̃ measured from samples with different thicknesses when (a) low and (b) high LS are perturbed. Data shown for RI correlation length of 100 nm. Dotted horizontal line indicates the threshold of a 5% change in Σ̃.

Whereas we have shown that theoretically Σ̃ can sense all LS, in experiments its sensitivity is limited by the signal-to-noise ratio, which varies largely depending on the instrumentation and scattering power of the studied sample. Thus, to give an estimate of the LS sensitivity of Σ̃, we define it as the range of W_eff for which perturbing the corresponding higher or lower LS within the sample causes at least a 5% change in Σ̃. That is, the lower (r_min) and upper (r_max) limits of LS sensitivity are the values of W_eff such that Σ̃^l (W_eff = r_min) = 0.95Σ̃ and Σ̃^h (W_eff = r_max) = 0.95Σ̃. Applying this threshold, we find the range of LS detected by Σ̃ to be 22–240 nm for L = 0.5 μm, 23–281 nm for L = 3 μm, and 22–171 nm for L = ∞. For comparison, thresholds of a 1% and a 25% change in Σ̃ would define corresponding LS sensitivity ranges of 9–360 nm and 56–124 nm for L = 0.5 μm, 10–360 nm and 56–124 nm for L = 3 μm, and 10–212 and 53–116 nm for L = ∞.

Naturally, since the range of LS detected by Σ̃ depends on the LS composition of the sample, r_min and r_max change as a function of RI correlation length l_c (Fig. 4). While there is no analytical relationship describing r_min and r_max in terms of l_c and L, their qualitative behavior is as follows. As l_c increases, the amount of large LS increases, and hence the range of detected LS shifts upward, increasing r_min and r_max (note that r_max cannot reach L and therefore saturates below it at L/l_c ≤ 3). Expectedly, the effect of a finite L on r_max is much greater in magnitude than that on r_min. In contrast, at small l_c the sensitivity of Σ̃ to shorter LS is emphasized, which is quantified by values of r_min lower than that for L = ∞ and by r_max approaching its value corresponding to L = ∞ when L/lc ≫ 3.

Fig. 4.

Dependence of (a) r_min and (b) r_max corresponding to the 5% threshold on L and l_c. Black dashed lines indicate r_min = 22 nm and r_max = 171 nm corresponding to L = ∞.

We next employ the present LS perturbation analysis to determine which structures inside biological cells are quantified by the spectral content of their microscope image. We model RI distribution typical to biological cell nuclei using mass-density SCF measured from transmission electron microscopy (TEM) images reported in [5]. Following standard TEM protocol, human colonic cell nuclei were stained with specific to DNA osmium tetroxide, sectioned, and imaged. The gray-scale image intensity, assumed to be proportional to the local density of chromatin, was used to compute the mass-density SCFs of 36 micrographs, after which an average SCF of nuclear material was obtained [5]. Since the RI of biological media is a linear function of mass density [2], the SCF of RI distribution equals that of mass-density distribution with a constant prefactor. The prefactors of mass-density SCF are unknown (due to the variability in the depth of staining), and therefore we normalize the RI SCF to 1 at the origin (r = 0).

We use the information contained in the experimental SCF by following two approaches. First, we fit an exponential SCF to the experimental (r² value of the fit between 39 and 1000 nm is 0.99) and find the exponential correlation length l_c to be 156 nm. We then follow the previously described analysis for two values of L, 1.5, and 6.0 μm (mimicking the thickness of squamous and columnar epithelial cell nuclei). Second, to evaluate how well the model of an exponentially correlated medium applies to biological cells, we use the experimental SCF directly. Since the experimental SCF is only defined at r ≥ 39 nm (resolution of TEM micrographs), we extend it to r = 0 using the fitted exponential SCF at r < 39 nm. We then generate random media with SCF equal to the experimental and perform LS sensitivity analysis.

An excellent match between Σ̃^l/Σ̃ and Σ̃^h/Σ̃ calculated based on analytical and experimental SCFs is obtained for all values of W_eff (Fig. 5). Applying the 5% threshold, for L = 1.5 μm we find that Σ̃ detects intranuclear structures from 25 to 427 ± 11 nm based on the analytical SCF, and from 25 to 441 ± 11 nm based on the experimental (uncertainty intervals correspond to the standard error between 20 samples; error is not shown when it is <0.5 nm). For L = 6 μm, we find that Σ̃ detects structures from 23 to 324 ± 5 nm (for analytical SCF) and from 23 to 334 ± 5 nm (for experimental). Additionally, from Σ̃^l and Σ̃^h calculated for both thicknesses and types of SCFs, we note that the largest contribution to Σ̃ measured from biological cell nuclei comes from structures smaller than 200 nm in size (Fig. 5).

Fig. 5.

Relative change in Σ̃ when (a) lower (Σ̃^l/Σ̃) and (b) higher (Σ̃^h/Σ̃) LS are perturbed. Calculation performed for samples with SCF, which is experimentally measured from TEM images (blue markers for samples with thickness 1.5 μm and green for 6.0 μm) and analytically defined as exponential (blue solid line for L = 1.5 and green-dashed for L = 6.0 μm).

To summarize, we establish that the interferometric spectroscopy of scattered light has a unique ability to detect spatial RI fluctuations of any frequency inside a thin, weakly scattering sample. Most importantly, its sensitivity to subdiffractional spatial frequencies (above 2k₂) is limited not by the diffraction of light but by the signal-to-noise ratio of the detector.

Based on a 5% noise floor, we report the range of LS Σ̃ detects inside complex media with an exponential SCF. The smallest detectable LS is determined to be ~λ/26 (22 nm) for all tested values af L and l_c. The sensitivity to large LS, in turn, is found to be highly dependent on L. As a result, the largest detectable LS is ~λ/3 (200 nm) for samples with short RI correlation lengths (l_c < 100 nm) and reaches values as high as ~λ for l_c > 100 nm. Precise sensitivity limits for a wide range of sample parameters relevant to the case of isolated biological cells are reported.

We emphasize that the present analysis does not address the sensitivity of Σ̃ to material rearrangement processes such as macromolecular aggregation or decondensation. Mass-preserving structural rearrangement is characterized by alterations in the shape of PSD and a constant total power of RI fluctuations. Thus the sensitivity of Σ̃ to such processes is best evaluated through its functional dependence on parameters of the shape of PSD, which is analogous to that reported in [1].

Finally, we substantiate the relevance of the presented analysis and its conclusions to biological media, which are not necessarily quantified by an exponential SCF. We establish that spectral variance of an epi-illumination, bright-field microscope image, utilized for the quantification of subdiffractional intracellular structures in techniques such as partial wave spectroscopic microscopy [6], measures LS between 25 and 400 nm, which, in terms of nuclear organization, correspond to structures from a chromatin fiber to nucleoli and chromatin aggregates barely resolvable by a conventional microscope. Accordingly, we conclude that the largest contribution to Σ̃ measured from biological cell nuclei must come from structures smaller than 200 nm in size. Large LS such as nuclear size or shape would have virtually no contribution to the monitored signal.