Superresolved labeling nanoscopy based on temporally flickering nanoparticles and the K-factor image deshadowing

Tali Ilovitsh; Yossef Danan; Asaf Ilovitsh; Amihai Meiri; Rinat Meir; Zeev Zalevsky

doi:10.1364/BOE.6.001262

1. Introduction

The spatial resolution of an optical system is limited by the phenomenon of diffraction to approximately half the wavelength of the light [1–3]. In visible light microscope the diffraction limit is about 200–300 nm in the lateral direction. When imaging subcellular structures at smaller sizes, they will appear as a diffraction-limited spots in the shape of a point spread function (PSF) given by an Airy function. Localization microscopy is one category of Super resolution (SR) techniques that enable to achieve sub diffraction limit accuracy [4]. These methods, like photoactivated localization microscopy (PALM) [5] and stochastic optical reconstruction microscopy (STORM) [6], use optical control to activate a sparse subset of fluorophores labeling a structure, measure their fluorescence and determine the location of each molecule to a much higher accuracy than conventional optical methods [7]. The precision of these techniques is limited by the number of photons collected from each emitter and by assuming that there are no multiple single molecules within a diffraction-limited spot. Low photon count and high density of activated emitters increase the localization error. Several methods were presented with the aim of localizing overlapping PSFs. One utilizes a maximum likelihood technique assuming an increased number of point sources within the recorded PSF in the localization algorithm, however it requires the use of a graphics processing unit (GPU) analysis [8–10]. Another paper uses a statistical deconvolution technique that iterates through the observed PSF with a guess-work of overlapping PSFs. This approach is very slow and requires ~10 times more computation time/frame than the other methods of single-emitter fitting [11].

The method proposed in this paper is aimed for noise reduction and enables the detection of overlapping PSFs, which increases the density of activated molecules. Therefore, it will allow faster data acquisition rates and improve localization precision. Our proposed technique is an alternative approach for molecule imaging based on the use of gold nano particles (GNPs) as biomarkers [12–16]. GNPs exhibit the localized surface plasmon resonance (SPR) effect, which is manifested by enhanced absorption and scattering at a specific optical frequency when are under optical illumination that matches this resonant wavelength [17]. However, the GNPs heat during the process and when used as biomarkers, the heat is harmful for most biological samples. This heating restricts the laser intensity and therefore increases the shot noise in the obtained image [18]. In addition, when the particles are to be detected in cells or tissues, they need to be discriminated from high background noise [19]. This dictates working in poor signal-to-noise ratio (SNR) conditions. Various imaging methods measure the scattering from GNPs tagged biological samples. These include dark-field illumination [20], differential interference contrast and video enhancement [21], and total internal reflection [22]. However, none addresses the low SNR restriction and they are all based on the use of a high laser intensity.

The proposed method has two steps, first a temporally sequenced labeling (TSL) routine that was described in a previous work [23] is used to image the sample. The TSL technique increases the SNR, enables the extraction of signal even in poor photon count and improves the localization accuracy of a single particle. The reconstructed TSL image undergoes a second step of signal processing using the K-factor algorithm [24]. This algorithm reduces the overlap between closely spaced PSFs, therefore allows their detection and in addition improves the localization precision of each PSF, making this approach applicable for superresolution imaging.

2. Theoretical background

The TSL is the process in which the GNPs that label a biological sample are excited using a modulated laser beam with a wavelength that matches the GNPs SPR wavelength. The modulated illumination results in a temporal flickering of the scattered light from the GNPs at a known frequency. The mathematical analysis of the technique is presented in [23]. Briefly, the captured intensity images are a temporal sequence of the light scattered from the sample that contains the GNPs. The intensity of each image is proportional to the sum of a time sample of the modulated signal and the additive noise. The obtained signal is convolved with the modulation signal and after the convolution, the elements that are at the frequency of the modulation signal are recovered (i.e. the image data), whereas components at different frequencies (i.e. the noise), are attenuated significantly. This allows the extraction of the signal from the wide spectrum noise. The next step is to apply the K-factor algorithm on the obtained TSL image. The K-factor is a nonlinear image decomposition that divides the image pattern into a nonlinear set of contrast-ordered pseudo-image factors whose joint product reassembles the original image [25,26]. The uniqueness of the decomposition is that factors that contain noise elements are distinct from those containing the image structure of interest. In this decomposition, the first few components, which have the highest contrast depth, contain mainly the desired image information while the higher orders contain mostly noise components. Therefore, it enables a distinct separation between noise and data elements. The algorithm is an iterative technique, which reduces an image The reconstructed image I(x,y) can be described mathematically as:

I (x, y) = \prod_{n = 1}^{M} f_{n} (x, y)

where M is the number of factors that reconstruct the image, and f_n are the pseudo-image factors that are given by:

f_{n} (x, y) = \frac{1 + k^{n} g_{n} (x, y)}{1 + k^{n}}

where the parameter k controls the contrast depth at each level with a value is between 0 and 1, and g_n is a binary image computed as:

g_{n} (x, y) = {\begin{matrix} 1 & \frac{I (x, y)}{\prod_{j = 1}^{n - 1} f_{j} (x, y)} \geq \frac{1}{1 + k^{n}} \\ 0 & O . W \end{matrix}

The full reconstruction of the image involves the multiplication of all f_n, n = 1…M factors. However, in a previous paper [27] we had shown that the first few factors of the decomposition, marked by f_h, where h<M, contain most of the image data together with noise, while high order factors contains mostly noise together with some fine spatial information associated with low contrast levels. By multiplying the original image with the first few harmonies, the image data will be de-emphasized in the reconstruction and noise will be reduced. The received image R_h(x,y):

R_{h} (x, y) = I (x, y) \prod_{n = 1}^{h < M} f_{n} (x, y)

The optimal choice of the algorithms parameters k and M, as well as the number of factors h that the original image will be multiplied by, depends entirely on the nature of the image and therefore is determined per application.

The K-factor algorithm narrows the width of each PSF and reduces additive noise. Therefore, applying the algorithm on noisy images, that contain overlapping PSFs, reduces the saddle generated between two adjacent PSFs and enables their detection.

3. Simulation results

Monte-Carlo simulations were used in order to generate mock data sets of samples that contain randomly positioned GNPs with peak emission at wavelength of λ = 532nm. The TSL method was simulated creating a temporal sequence of intensity images, with parameters that matches the experimental setup; the frame rate was 12 frames/s and ν₀ = 3Hz. Shot noise is a Poisson process with an expected value which corresponds to the noiseless pixel values and a standard deviation (STD) that equals the square root of the value of each pixel. Background noise was introduced by adding a sample from a Poisson distribution random variable with variance N_b (assumed constant across the field of view) [28].

The position of the each GNP, the distance between GNPs, the number of detected photons N and the background noise parameter N_b were allowed to vary. All analysis was performed in MATLB. The images were convolved with the modulation signal according to Eq. (2) followed by time averaging (TA) in order to reconstruct the image. As a second step, the reconstructed TSL image was processed using the K-factor algorithm. The result was compared to a simple TA of the images for the same time period. The K-factor algorithm parameters were chosen according to previous work [24] to be k = 0.9, n = 48, h = 8.

Figure 1(a) is the simulated sample with random diffraction limited spots originating from scattering of light from the GNPs with added background noise and shot noise so that the SNR was -20dB. The SNR was calculated by

Fig. 1 Simulation results. (a) is the simulated sample with random diffraction limited spots originating from scattering from the GNPs with added background noise and shot noise so that the SNR was -20dB. (b) is the image after applying the proposed TSL technique, the SNR of the reconstructed image is 30dB. (c) is the K-factor algorithm applied to the TSL image, the SNR is 100dB. (d), (e) and (f) show the cross section of the dashed red line that passes through the center of an emitter in the left part of (a), (b) and (c) respectively.

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S N R_{o r} = 10 \log_{10} \sqrt{\frac{\frac{1}{N} \sum_{i = 1}^{N} I_{s i g} {_{i}}^{2}}{\frac{1}{N} \sum_{i = 1}^{N} I_{n o i s e} {_{i}}^{2}}}

Figure 1(b) is the image after applying the proposed TSL technique, the SNR of the reconstructed image is 30dB. Figure 1(c) is the K-factor algorithm applied to the TSL image, the SNR is 100dB. It is clearly seen that the data is indistinguishable in the original image, and becomes visibly seen after applying the proposed TSL technique. The K-factor algorithm narrows the PSF of each particle which yields a further improvement. Figures 1(d)-1(f) show the cross section of the dashed red line that passes through the center of an emitter in the left part of Figs. 1(a)-1(c) respectively.

Gaussian fitting was performed on the TA image, the TSL image and on the K-factor filtered image. Each image contained randomly positioned particles, which were fitted using the non-linear least-squares minimization routine lsqnonlin in MATLAB to fit a model of the form of:

I_{psf} (x, y) = \frac{N}{2 {πσ}^{2}} e^{- \frac{{(x - x_{0})}^{2} + {(y - y_{0})}^{2}}{2 σ^{2}}}

Where N is the number of detected photons,σ is the standard deviation of the Gaussian and (x₀,y₀) is the emitters coordinates. The algorithm initially detected the position of each particle as the pixel with the highest intensity in its region, and preformed a fit that resulted in the best estimate of the position of that particular particle [29]. The process is repeated for each particle yielding a set of L localization positions (◯_i,ŷ_i) that can be compared with the known positions (x_i,y_i). The root mean square (RMS) localization error is computed as:

e r r_{r m s} = \sqrt{\frac{1}{L} \sum_{i = 1}^{L} [{(\hat{x_{i}} - x_{i})}^{2} + {(\hat{y_{i}} - y_{i})}^{2}]}

Figure 2(a) presents the RMS error in localization of a single diffraction limited GNP using a simple TA of the images (red line), using only TSL (blue line) and applying the K-factor to the reconstructed TSL image (black line). Background noise was not added.

Fig. 2 (a) RMS error in localization of a single diffraction limited GNP using a simple TA of the images (red line), using only TSL (blue line) and applying the K-factor to the reconstructed TSL image (black line). No background noise was added.

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The localization improvement (%) was calculated by:

I m p r o v e m e n t = 100 \times \frac{R M S_{t i m e - a v e r a g i n g} - R M S_{T S L + K - F a c t o r}}{R M S_{t i m e - a v e r a g i n g}}

Since the shot noise is a Poisson random process with rate that depends on the total number of photons detected and is proportional to √N, an increased number of emitted photons results in a more accurate localization precision [30]. The improvement using the TSL technique alone at low photon count was ~37%, while the improvement when adding the K-factor algorithm was as high as ~67%. When the number of emitted photons increases the proposed method reached a precision of ~5nm, while the TSL performance became similar to that of a simple TA since shot noise is less pronounced in that case.

Another important factor is the effect of the proposed technique on overlapping PSFs. In order to test this, a set of closely spaced emitters was generated. The minimal resolvable distance between two points without added background noise was 2.5σ, and therefore it was chosen as the distance between the emitters. Shot noise and background noise were added to the image and the influence of the proposed method on the SNR is illustrated in Fig. 3(a).

Fig. 3 (a) Image processing influence of 3 different methods on the SNR. Simple TA of the images (Red line). Using only TSL (Blue line). The K-factor applied to the reconstructed TSL image (Black line). (b) RMS error in localization of two GNPs at a distance of 2σ for different SNRs, using only TSL (blue line) and applying the K-factor to the reconstructed TSL image (black line).

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The x-axis is the SNR of the original image and the y-axis is the SNR of the image after preforming three different image processing steps: 1. Using a simple TA of the images (red line). 2. Using only TSL (blue line) and 3. Applying the K-factor to the reconstructed TSL image (black line). As can be seen, the SNR improvement is the highest at poor SNR of -40dB. The TSL method can reconstruct the signal even at such low SNR that the signal is almost indistinguishable and reaches 30dB SNR. The K-factor algorithm further increases the effect by narrowing the PSF and further denoising the result to yield in 100dB SNR.

Shot noise and background noise were added so that the particles were undistinguishable among the noise, and the image data was recovered and localized. The proposed method of TSL + K-factor, enabled the detection of particles at a minimal distance of 1.5σ, compared to 2.5σ using TA - an improvement of 65%. The minimal distance for the TSL on its own was 2σ. The RMS localization error for a distance of 2σ and for a variety of SNRs is demonstrated in Fig. 3(b). Using TA, the particles were indistinguishable. With compare to the TSL on its own, even at poor SNR conditions, our proposed technique yielded an improvement by a factor of ~2 and reached a precision of ~5nm.

4. Materials and methods

Materials

GNPs Synthesis: GNPs were prepared using sodium citrate according to the known methodology described by Enustun and Turkevic [31]. 0.414 mL of 1.4M HAuCl₄ solution in 200mL water was added to a 250mL single-neck round bottom flask and stirred in an oil bath on a hot plate until boiled. 4.04 mL of a 10% sodium citrate solution (0.39M sodium citrate tribasic dihydrate 98%, Sigma cas 6132-04-3) was then quickly added. The solution was stirred for 5min, and then the flask was removed from the hot oil and placed aside until cooled.

GNPs Conjugation: In order to prevent aggregation and to stabilize the particles in physiological solutions, O-(2-Carboxyethyl)-O′-(2-mercaptoethyl)heptaethylene glycol (PEG7) (95%, Sigma-Aldrich, Israel Ltd.) was absorbed onto the GNPs. This layer also provides the chemical groups required for antibody conjugation (-COOH). First, the solution was centrifuged to dispose of excess citrate. PEG7 solution was then added to the GNP solution, stirred overnight and put in a centrifuge in order to dispose of excess PEG. In order to increase cell-uptake rate, stabilized GNPs were further coated with glucose. Excess EDC (N-ethyl-N -(3-dimethylaminopropyl) carbodiimide) and NHS (N-hydroxysuccinimide) (Thermo Fisher Scientific, Inc, Rockford, IL) were added to the solution, followed by addition of Glucose-2 (2GF)(D-( + )-Glucosamine hydrochloride, Sigma-Aldrich, Israel Ltd.). NHS and EDC form an active ester intermediate with the -COOH functional groups, which can then undergo an amidation reaction with the glucose –NH₂ group. Glucosamine molecule C-2 (2GF-GNP): D-( + )-Glucosamine hydrochloride (3 mg; Sigma Aldrich) was added to the activated linker-coated GNPs.

Cell uploading with GNP: HEK 293 cells were cultured in 5 ml glucose-free DMEM medium containing 5% FCS, 0.5% Penicillin and 0.5% glutamine. Cells were centrifuged and a saline solution containing GNPs was added in excess. The cells were then incubated at 37°C for 1 hour. After incubation, the cells were centrifuged twice (7 minutes in 1000 rpm) to wash out unbound nanoparticles.

5. Experimental results

The experiment setup is described in Fig. 4.

Fig. 4 The experiment setup is composed of a function generator that modulates a green laser at wavelength of 532nm. The modulated beam illuminated the sample and the scattered light as a function of time was imaged using a microscope and recorded with a CMOS camera.

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A function generator (AFG3022B by Tektronix) was used to create a square wave with a known frequency of 3Hz (that fulfils the Nyquist sampling criteria as the frame rate of the camera was 12 frames/s) and a duty cycle of 50%. This signal was connected to a modulation port of a green laser at 532nm (Photop DPGL-2100F). The modulated beam illuminated the sample and the scattered light as a function of time was imaged using the Olympus BX51 microscope with a 40x objective lens and recorded with the CMOS camera (PixeLink PL-A741-E). The effective pixel size was 167.5nm. Two samples were used in order to test the proposed method; the first is a sample that contains individual GNPs immobilized on a coverslip (the sample preparation is described in the materials and methods section). Particles characteristics were measured using transmission electron microscopy (TEM) where their diameter was verified to be 20nm [Fig. 5(a)]. The scattering spectrum of the GNPs was measured using a visible spectrophotometer (Cary 5000 by Agilent) and showed a peak at ~532nm [Fig. 5(b)]. The GNPs scattering and absorption peaks have the same wavelength of ~532nm, and it was verified by measuring their absorption spectrum (using the NanoDrop2000c by Thermo-Scientific), which showed a same peak as in Fig. 5(b).

Fig. 5 Characterization of GNPs. (a) TEM image of 20nm GNPs. (b) Scattering spectrum of the GNPs.

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The images of the sample were taken with parameters of the highest gain of the camera (17.7dB), low exposure time (10ms) and low laser power (10mW) to mimic high background and shot noise conditions. The SNR of the original image was calculated to be -15dB. An example of such a recorded sequence is presented in Fig. 6. An example of a GNP is marked in a red frame and it is clearly seen that is almost indistinguishable.

Fig. 6 Sequence of recorded images of the scattered light from the sample having −15dB SNR. A 532 nm laser illuminated the sample and was modulated using modulation frequency of 3Hz. The presented images were captured with an exposure time of 10msec. The red frame marks a single GNP that is almost indistinguishable.

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Figure 7(a) is the TA of the images, where the GNPs are barely seen. The TSL technique was applied to the sequence of recorded images [Fig. 7(b)]. The GNPs become visible, however, areas with high GNPs density appear as large spots, where conventional localization analysis is not applicable. After applying the K-factor algorithm with parameters of k = 0.9, n = 48, h = 8 on Fig. 7(b), the PSFs narrows, thus, individual GNPs can be detected in areas with high concentration [Fig. 7(c)].

Fig. 7 Expeirmental results. (a) A simple TA of the images of the sample. (b) The reconstructed image of the sample using TSL. (c) The K-factor algorithm applied on the reconstructed image of the sample using TSL.

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The second sample used was of Human Embryonic Kidney (HEK) 293 cells [32] that were injected with the same 20nm GNPs immobilized on a coverslip using a well-established procedure [33–35]. The images of this sample were in poor SNR conditions of -25dB, where the GNPs were indistinguishable among the noise as [Fig. 8(a)]. A bright field image of the sample was taken using the Olympus BX51 microscope using X40 objective lens [Fig. 8(b)]. The TSL technique was applied to the sequence of recorded images and the result was an image of the GNPs in the observed sample [Fig. 8(c)]. The reconstructed image after applying the K-factor algorithm on is presented in Fig. 8(d), where the GNPs are clearly observed as brighter and sharper. Figure 8(e) is the superimposing of Fig. 8(b) and Fig. 8(d), where the GNPs are marked in red. The overlap between the locations of the GNPs to that of the cells indicate that the GNPs are concentrated within and on top of the cells.

Fig. 8 Expeirmental results. (a) A single image from the experimental set, where the GNPs are indistinguishable from the noise. (b) A bright field image of the sample. (c) The reconstructed image of the sample using TSL. (d) The reconstructed image of the sample using both TSL and K-factor. (e) The superimposing of (b) and (d), where the GNPs are marked in red.

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A localization routine was used to determine the location of the GNPs in the images. The localization was done in ImageJ [36] using the molecule-localization plugin.

In order to calculate the localization precision, a set of 100 TSL images of the same sample were taken under the same conditions exactly. The localization routine was applied to the images and the standard deviation of particles coordinates was calculated. Due to the low SNR of the TA image, we couldn't preform a localization of the TA image. The standard deviation σ of the TSL images was 11.2nm, which corresponds to resolution of (FWHM = 2.35σ) 26.32nm. The K-factor algorithm applied on the TSL image yielded a standard deviation of the TSL images was 4.1nm, which corresponds to resolution of 9.6nm, an improvement by a factor of ~2.5.

The use of the TSL technique improves the resolution of the original image and the proposed method adds a further improvement, making it attractive for single molecule localization applications.

6. Conclusions

The TSL technique is an alternative approach for imaging a sample that is labeled with GNPs that provides a tool intra-cellular processes study. It allows a high noise immunity which makes it ideal for biological applications. When combined with the K-factor image decomposition algorithm on images acquired using TSL, the obtained image has a significantly higher SNR and improved single particle localization precision. The localization routine was done using a standard method of least-squares fitting. The combined approach had maximum impact at poor SNRs of -40dB and reached SNR of 100dB. In addition, it improves the localization precision of individual particles by up to 67% with compared to a simple TA. Another important aspect is its ability to detect closely spaced particles at distances of 1.5σ, an improvement of 65%. The obtained localization precision of the experimental data was 9.6nm at such an SNR that the image data was indistinguishable, and there wasn't an ability to perform localization of the image.

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Superresolved labeling nanoscopy based on temporally flickering nanoparticles and the K-factor image deshadowing

Abstract

1. Introduction

2. Theoretical background

3. Simulation results

4. Materials and methods

Materials

5. Experimental results

6. Conclusions

References and links

Cited By

Figures (8)

Equations (8)

Biomedical Optics Express