Abstract

We propose a point spread function for three-dimensional localization of nanoparticles. The axial detection range of the point spread function can be simply changed by adjusting the design parameters. In addition, the spatial extent of the point spread function can also be changed by adjusting the design parameters, which is an advantage other point spread functions do not have. We used our point spread functions and the existing point spread functions for dense multi-particle imaging, which proved the advantage that the point spread function with a smaller spatial extent we designed can effectively reduce the overlap between the point spread functions. The three-dimensional process of the fluorescent microsphere penetrating HT-22 cell membrane was successfully recorded, which verified the effectiveness of this method.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

In the development of nano-drugs, researchers hope to observe the movement of nano-drugs inside and outside the cell and then improve the related nano-drugs [1]. Therefore, researchers need a method for three-dimensional tracking of nanoparticles in a large axial range [2]. The existing methods for three-dimensional trajectory tracking of nanoparticles are Z-Stacks, Multifocal Plane Microscopy (MPM), Point spread function engineering, etc. The Z-Stacks obtains three-dimensional position information of particles by axial scanning imaging of samples. Due to the limitation of imaging speed, z-stack leads to low temporal resolution [3]. The MPM uses multiple channels to simultaneously image samples at different depths, which improves the time resolution, but the energy in each channel is much less than that in the single channel [4]. Point spread function (PSF) engineering is widely used in three-dimensional tracking of nanoparticles because of its high time resolution and positioning precision. The point spread function engineering encodes the axial information of nanoparticles into the information of the point spread function which is easily recognized (such as shape, angle, spacing, etc.), to realize the three-dimensional tracking of nanoparticles [5,6]. The double helix PSF(DH-PSF) calculates the axial position of particles by the angle information of two lobes [712], the saddle-point PSF(SP-PSF) [13,14] and the cropped oblique secondary astigmatism PSF (COSA PSF) [15] calculate the axial position of particles by the relative motion of two lobes. These three points spread functions are widely used, but there are also corresponding deficiencies. The axial detection range of the DH-PSF is small. The phase of the SP-PSF is composed of the first 55 Zernike polynomials. The coefficients of each Zernike polynomial need to be obtained by iteration, and the iteration process is easy to fall into a local minimum, so the generation of the phase is complex. The phase of the COSA PSF can be simply obtained by mathematical expression, but the COSA PSF has more side lobe effects like the SP-PSF.

This paper designed a PSF similar to the SP-PSF and the COSA PSF, and this point spread function is named Splicing exponential PSF(SE-PSF), and the Splicing exponential function phase is obtained by splicing Exponential function phase and Defocus phase. The SE-PSF can adjust the axial detectable range and the spatial extent by changing the corresponding parameters in the design process.

2. Method

The basic components of the Splicing exponential function phase are the Exponential function phase and the Defocus phase.

The Defocus phase can be obtained by the Fresnel approximation imaging. According to the Fresnel approximation imaging [16], the PSF intensity distribution of the modulated phase on the Fourier plane can be obtained,

$$I(u,v;z) \propto {\left|{FFT2\left\{ {\exp \left( {i \cdot P(x,y) + \frac{{2i\pi M}}{{\lambda {f_{4f}}}} \cdot (x \cdot \Delta x + y \cdot \Delta y) - \frac{{i\pi \tau {M^2} \cdot z}}{{\lambda f_{4f}^2}} \cdot ({x^2} + {y^2})} \right)} \right\}} \right|^2}.$$
Where, FFT2 denotes Fourier transform, I(u,v;z) is the image plane in the Cartesian coordinate system when the axial depth of the particle is z, P(x,y) is the phase loaded on the Fourier plane in the Cartesian coordinate system(x, y), M is the magnification of the microscopy imaging system, λ is the wavelength, f4f is the focal length of the 4f system, △x and y are the lateral displacements of the particle, the parameter τ=0.81 which is set according to the refractive index of the oil immersion objective lens 1.518. According to Eq. (1), when the PSF axially shifts △z distance, that is,
$$I(u,v;z-\Delta z)\propto \left| {FFT2\left\{ {\exp \left( {i\cdot P(x,y) + \displaystyle{{2i\pi M} \over {\lambda f_{4f}}}\cdot (x\cdot \Delta x + y\cdot \Delta y)-\displaystyle{{i\pi \tau M^2\cdot z} \over {\lambda f_{4f}^2 }}\cdot (x^2 + y^2) + \displaystyle{{i\pi \tau M^2\cdot \Delta z} \over {\lambda f_{4f}^2 }}\cdot (x^2 + y^2)} \right)} \right\}} \right|^2.$$

Therefore, the axial distribution of the PSF can be changed by loading a specific phase (the Defocus phase) on the original phase, which can be expressed as,

$${\psi _d} = \frac{{\pi \tau {M^2} \cdot \Delta z}}{{\lambda f_{4f}^2}} \cdot ({x^2} + {y^2}).$$
The point spread function generated by the Exponential function phase (which is similar to the point spread function of a Cubic spatial phase [17,18] and the Airy -PSF [19]), and the distance between the main lobe and the center point of the image plane is different on the two-dimensional plane of different depths. The Exponential function phase can be expressed as,
$${\psi _e} = \alpha x \cdot \exp (\beta {x^2}) + \alpha y \cdot \exp (\beta {y^2}).$$
Where ψe(x, y) represents the Exponential function phase in Cartesian coordinates (x, y), α and β are the design parameters.

In addition, the phase combination is involved in the design process of the Splicing exponential function phase. The phase after downsampling still retains most of the information to generate the point spread function. Therefore, multiple down-sampled phases can be combined into a phase of the original resolution, and the point spread function generated by the phase will be the combination of the point spread functions generated by multiple down-sampled phases. Based on this principle, by dividing the Fourier plane, multiple phases can be combined to one phase [20] and multiple point spread functions also can be combined to one point spread function. The phase of a combination of different point spread functions (as shown in Fig. 1) can be obtained by the following formula,

$${\psi _c}(\varphi ) = \left\{ {\begin{array}{ll} {{\psi_1}(\varphi );}&{(\bmod (n,S) - 1) + \left[ {\frac{n}{S}} \right] \cdot \frac{{2\pi S}}{N} \le \varphi \le \frac{{2\pi }}{N} \cdot \bmod (n,S) + \left[ {\frac{n}{S}} \right] \cdot \frac{{2\pi S}}{N},\bmod (n,S) = 1}\\ {{\psi_2}(\varphi );}&{(\bmod (n,S) - 1) + \left[ {\frac{n}{S}} \right] \cdot \frac{{2\pi S}}{N} \le \varphi \le \frac{{2\pi }}{N} \cdot \bmod (n,S) + \left[ {\frac{n}{S}} \right] \cdot \frac{{2\pi S}}{N},\bmod (n,S) = 2}\\ {{\psi_3}(\varphi );}&{(\bmod (n,S) - 1) + \left[ {\frac{n}{S}} \right] \cdot \frac{{2\pi S}}{N} \le \varphi \le \frac{{2\pi }}{N} \cdot \bmod (n,S) + \left[ {\frac{n}{S}} \right] \cdot \frac{{2\pi S}}{N},\bmod (n,S) = 3}\\ {\ldots }&{\kern 118pt}{\ldots }\\ {{\psi_s}(\varphi );}&{(\bmod (n,S) - 1) + \left[ {\frac{n}{S}} \right] \cdot \frac{{2\pi S}}{N} \le \varphi \le \frac{{2\pi }}{N} \cdot \bmod (n,S) + \left[ {\frac{n}{S}} \right] \cdot \frac{{2\pi S}}{N},\bmod (n,S) = s}\\ {\ldots }&{\kern 118pt}{\ldots }\\ {{\psi_S}(\varphi );}&{(\bmod (n,S) - 1) + \left[ {\frac{n}{S}} \right] \cdot \frac{{2\pi S}}{N} \le \varphi \le \frac{{2\pi }}{N} \cdot \bmod (n,S) + \left[ {\frac{n}{S}} \right] \cdot \frac{{2\pi S}}{N},\bmod (n,S) = 0} \end{array}} \right..$$
Where ψc(φ) denotes the phase value on the azimuth of φ, N is the phase design parameter, that is, the Fourier plane is divided into N parts, n represents part n of the Fourier plane, S represents a total of S phases involved in the combined phase, s represents the sth phase, (ψ1, ψ2, ψ3, ψs, ψS) represents the phases involved in the combined phase, the mod is a remainder operator, [] is an integer operator (round down).

 figure: Fig. 1.

Fig. 1. Schematic diagram of the phase combination method based on sector segmentation.

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Under the conditions of NA = 1.4, wavelength 514 nm, and magnification M = 100, the SE-PSF (α=70, β=0.5) with an axial detection range of 20µm is designed as an example. By combining the two symmetric Exponential function phases (as shown in Fig. 2(a)) through Eq. (5), the Double exponential function phase (as shown in Fig. 2(b)) can be obtained. By adding the Negative defocus phase (△z = -10µm, the calculation of the Negative defocus phase based on Eq. (3)) and the Positive defocus phase (△z = 10µm, the calculation of the Positive defocus phase based on Eq. (3)) to the Double exponential function phase and the Double exponential function phase after 90° rotation, the Negative defocus double exponential function phase (as shown in Fig. 2(c)) and the Positive defocus double exponential function phase (as shown in Fig. 2(d)) can be obtained. Equation (5) is used again to combine the Negative defocus double exponential function phase with the Positive defocus double exponential function phase, and finally, the Splicing exponential function phase (as shown in Fig. 2(e)) can be obtained. The sidelobe energy of the SE-PSF obtained by the Splicing exponential function phase is too high, which is not suitable for practical application. Therefore, we use the optimization method to generate the Optimized splicing exponential function phase (as shown in Fig. 2(f)). The generation process of the Optimized splicing exponential function phase can refer to Fig. 2(g).

 figure: Fig. 2.

Fig. 2. (a) The exponential function phase and the exponential point spread function (α=70, β=0.5). (b) The Double exponential function phase and the Double exponential point spread function (α=70, β=0.5). (c) The Negative double exponential function phase and the Negative double exponential point spread function (α=70, β=0.5). (d) The Positive double exponential function phase and the Positive double exponential point spread function (α=70, β=0.5). (e) The Splicing exponential function phase and the Splicing exponential point spread function (α=70, β=0.5). (f) The Optimized splicing exponential function phase and the Optimized splicing exponential point spread function (α=70, β=0.5). (g) The generation process of the Optimized splicing exponential function phase.

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3. Optimization

An optimization algorithm (as shown in Fig. 3) based on phase inversion is introduced here, and the algorithm is a modification of the Iterative Fourier Transform Algorithm (IFTA) or Gerchberg Saxton Algorithm (GSA) [7,21]. The optimization algorithm is carried out on the optical axis, so the relationship between the initial phase and the electric field distribution on the image plane can be expressed as,

$${A_z}(u,v) = FFT2\left\{ {\exp \left( {i \cdot {P^0}(x,y) - \frac{{i\pi \tau {M^2} \cdot z}}{{\lambda f_{4f}^2}} \cdot ({x^2} + {y^2})} \right)} \right\}.$$
Where, Az is the vector electric field distribution when the particle at different depths z, P°(x,y) is the initial phase loaded on the Fourier plane in the Cartesian coordinate system(x, y).

 figure: Fig. 3.

Fig. 3. Iterative process

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Therefore, the modulation function corresponding to the vector electric field distribution of the particle at different depths z can be expressed as,

$$\exp (i\cdot P_z(x,y)) = {{IFFT2\left\{ {A_z(u,v)} \right\}} \left/ {\exp \left( {-\displaystyle{{i\pi \tau M^2\cdot z} \over {\lambda f_{4f}^2 }}\cdot (x^2 + y^2)} \right)}\right.}.$$
Where IFFT2 denotes Fourier inverse transform. The vector electric field distribution also can be expressed as,
$${A_z}(u,v) = |{{A_z}(u,v)} |\cdot \exp (i \cdot Arg({A_z}(u,v))).$$
Under different depths z (the axial depth of the PSF is divided into T △z, T is odd), by replacing the original electric field distribution amplitude with the ideal amplitude |Az|, we can get a new electric field vector distribution Az. Substituting the newly obtained vector amplitude Az into Eq. (7) obtains an approximate phase Pz about the ideal amplitude intensity. The optimized phase after one iteration can be expressed as,
$${P^1} = Arg\left( {\frac{1}{{\sum\nolimits_{t = 1}^T {{a_t}} }} \cdot \sum\nolimits_{t = 1}^T {{a_t} \cdot {P_z}(x,y)} } \right).$$
Where at is the weight coefficient at different depths z, and its function is to make the strength of the PSF at different z basically the same. The weight coefficient at is the inverse ratio of the maximum peak intensity of the PSF at depths z to the maximum peak intensity of the PSF at the 0 focal plane, which can be expressed as,
$${a_t} = \frac{{\max \{{{I_{(T + 1)/2}}} \}}}{{\max \{{{I_t}} \}}}.$$
The phase after the first iteration is brought into Eq. (1) to obtain the new vector electric field distribution, and the above steps are repeated until the iterative phase is optimal.

The phase of Fig. 2(e) is optimized to the phase shown in Fig. 2(f). The side lobe of the optimized SE-PSF is greatly reduced, and the main lobe energy is more concentrated.

4. Comparison

By adjusting the parameters α and β, the axial detection range of the SE-PSF can be changed. Parameter α is used to control the spatial extent of the SE-PSF, and parameter β controls the motion velocity of the main lobe of the SE-PSF. The larger the parameter α, the larger the spatial extent of the SE-PSF. And the smaller the parameter β is, the faster the main lobe of the SE-PSF moves. The relationship between the parameters α, β, and the axial detection range is shown in Fig. 4(a) when the wavelength is 514nm in a 100× microscopy imaging system (NA=1.4). Here we set the parameter β to 0.5 and change only the parameter α. It can be seen from Fig. 4(a) that the larger the parameter α, the larger the axial detection range, and the axial detection range is proportional to parameter α. Therefore, when the velocity of the main lobe is constant, the axial detection range can be increased by increasing the spatial extent of the SE-PSF. By changing the parameter β and set the parameter α to 40, we can also find that the axial detection range is proportional to the parameter β (as shown in Fig. 4(a)). This shows that slowing down the velocity of the main lobe can increase the axial detection range when the spatial extent of the SE-PSF is constant. In the 100× microscope imaging system (NA=1.4), when the wavelength is 514nm, the relationship between the axial detection range zr of the SE-PSF and the parameters α and β can be expressed as,

$${z_r} = 0.3115\alpha + 10.38\beta - 5.923.$$
Computing Fisher's information matrix, the square root of the Cramer-Rao lower bound($\sqrt {CRLB}$) [22] provides a fundamental theoretical limit on localization precisions. The vector can be obtained by calculating the square root of the diagonal of the Fisher's information inverse matrix,
$$\sqrt {CRL{B_i}} = \sqrt {\sum\nolimits_{k = 1}^{{N_p}} {{{({\raise0.5ex\hbox{$\scriptstyle {(\frac{{\partial \mu (k)}}{{\partial {\theta _i}}}) \times (\frac{{\partial \mu (k)}}{{\partial {\theta _i}}})}$}\kern-0.1em\bigg/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle {({\mu (k) + B} )}$}})}^{ - 1}}} } .$$
where μ is the PSF represented by the number of photons, k is the pixel on the detector, θ is the 3D position of the particle, and Np is the total number of pixels, B is the average background noise per pixel.

 figure: Fig. 4.

Fig. 4. (a) The relationship between the parameter α and the axial detection range. (b) Comparison of the localization precisions.

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Under the same axial detection range (10µm in 100× microscopy imaging system, NA=1.4, λ=514nm), we compared the three-dimensional localization precision of the SE-PSF, the SP-PSF, and the COSA PSF. In the calculation, we set the detected number of photons as 3000, set the average background noise B as 15, and add Gaussian noise and Poisson noise as interference. As shown in Fig. 4(b), the three-dimensional localization precision of the SE-PSF (α=40, β=0.35) is inferior to that of the SP-PSF and the COSA PSF at both ends. In the average three-dimensional localization precision, the SP-PSF is the best, because the SP-PSF is designed based on the optimal localization precision. However, in the middle part, the axial localization precision of the SE-PSF (α=40, β=0.35) is more stable and accurate than the SP-PSF and the COSA PSF.

Compared with other SE-PSFs, the SE-PSF (α=40, β=0.35) with a larger spatial extent (as shown in Fig. 5(a)) have better localization precisions. Therefore, when designing the SE-PSF, after determining the axial detection range, higher localization precisions can be ensured by selecting the SE-PSF with a larger spatial extent (i.e. increasing parameter α and decreasing parameter β). However, in dense multi-particle imaging, the spatial extent may be more important than the localization precisions. The PSF with a smaller spatial extent can reduce the overlap between the PSFs and is more convenient for subsequent processing. In the same axial detection range, the spatial extent of the SE-PSF can be adjusted by the parameters, which is not available for the SP-PSF and the COSA PSF. Therefore, compared with the COSA PSF and the SP-PSF, the SE-PSF with a smaller spatial extent can effectively reduce the overlap between PSFs in dense multi-particle imaging. We also compared the SE-PSF with various Double-helix point spread functions(DH-PSFs), as shown in Supplement 1.

 figure: Fig. 5.

Fig. 5. (a) Five phases involved in comparison and their point spread functions. (b) Comparison of the main lobes strength.

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In addition, the main lobes strength of the SE-PSFs are significantly stronger than that of the COSA-PSF and the SP-PSF (as shown in Fig. 5(b)), indicating that the SE-PSF is more suitable for photon-limited environments than the other two. Especially, it has brighter main lobes than the COSA-PSF and the SP-PSF under low exposure.

5. Localization

In this paper, the axial position of the particle is determined by calculating the distance and direction between the two main lobes. In general, a single SE-PSF has two Gaussian spots, and there are four Gaussian spots at the focal plane position. Therefore, we use the Gaussian mixture model to fit the Gaussian center of the obtained spot (using the fitgmdist function in MATLAB). The specific process is:

1.Get the image of the SE-PSF.

2.The binary image of SE-PSF is obtained by processing the image (removing noise) and threshold segmentation (determine the distribution area of SE-PSF on the image plane).

3.The gray value of each pixel in the SE-PSF distribution area is recorded as the number of occurrences of the pixel coordinates,

$$P = \{{{p_1},{p_2},{p_3},\ldots ,{p_N}} \},\left\{ \begin{array}{l} {p_1} = \{{({{x_1},{y_1}} ),({{x_1},{y_1}} ),({{x_1},{y_1}} ),\ldots ,({{x_1},{y_1}} )} \},card({{p_1}} )= g{v_1}\\ {p_2} = \{{({{x_2},{y_2}} ),({{x_2},{y_2}} ),({{x_2},{y_2}} ),\ldots ,({{x_2},{y_2}} )} \},card({{p_2}} )= g{v_2}\\ {p_3} = \{{({{x_3},{y_3}} ),({{x_3},{y_3}} ),({{x_3},{y_3}} ),\ldots ,({{x_3},{y_3}} )} \},card({{p_3}} )= g{v_3}\\ \begin{array}{llll} {\begin{array}{llll} {\begin{array}{ll} {}&{} \end{array}}&{}&{}&{} \end{array}}&{}&{}&{} \end{array}\ldots \\ {p_N} = \{{({{x_N},{y_N}} ),({{x_N},{y_N}} ),({{x_N},{y_N}} ),\ldots ,({{x_N},{y_N}} )} \},card({{p_N}} )= g{v_N} \end{array} \right..$$
where p1, p2, p3 and pN represents the coordinate set at pixel 1,2,3 and N; card is an operator used to calculate the number of elements in set p1, p2, p3 and pN; gv1, gv2 and gv3, gvN represent the gray value at pixel 1,2,3 and N; and P is the total coordinate set in the SE-PSF distribution area.

4.Before fitting with Gaussian mixture model, it is difficult to determine whether the shape of SE-PSF near the focal plane is four Gaussian spots or two Gaussian spots. So, we need to open the AIC option (an option in the fitgmdist function) in the Gaussian mixture model. AIC can determine whether the model is a four-Gaussian model or a double-Gaussian model. The coordinate set P is brought into the Gaussian mixture model for fitting. If it is a four-Gaussian model, we judge that the particle is in the focal plane. If it is a double Gaussian model, we obtain the center coordinates of two Gaussian spots. Through the direction of the connection between the two central coordinates, we can determine whether the particle is above or below the focal plane. Through the distance between the two center coordinates, we can get the exact distance between the particle and the focal plane. The transverse position coordinates of particle can be directly obtained by averaging the center coordinates of each Gaussian spot.

When the particle is on any side of the focal plane, the distance between the two main lobes of SE-PSF (α=19, β=1.5) is proportional to the distance between the particle and the focal plane (as shown in Fig. 6).

 figure: Fig. 6.

Fig. 6. Relationship between the distance between the two main lobes and the axial position of the particle

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6. Experiment

We verified the effectiveness of the SE-PSF through experiments. The experimental setup is shown in Fig. 7(a). The 2D charge-coupled device sCMOS (Hamamatsu Orca-Flash4.0V2) was used to record the light profile of the sampling region. The phase mask was loaded onto the liquid crystal spatial light modulator (LC-SLM, Meadowlark Optics, 1920×1152) placed in the Fourier plane and schematically shown in Fig. 7(a). An oil immersion objective (NA=1.4, oil immersion, OLYMPUS, 100×) was used in the experiments. Fluorescence was excited using a 488nm wavelength laser, and the excited fluorescence was filtered by a dichroic mirror (DM, MD498 (Reflection Band:452∼490nm; Transmission Band: 505∼ 800nm)) and modulated by the LC-SLM located on the Fourier plane. And we used fluorescent microspheres with a diameter of 100nm as the sample. The sample was moved 0.01μm/step along the Z-axis from -5μm to 5μm by using a Nano displacement platform (OPLAN Nano Z100), and the images of the SE-PSF were recorded (as shown in Fig. 7(b)). According to the obtained images, we can establish the measured stack of the SE-PSF for the maximum likelihood estimation localization of the particle [13]. However, this paper determines the axial position of the particle by calculating the distance and direction between the two main lobes, which is a simple and fast method.

 figure: Fig. 7.

Fig. 7. (a) Experimental setup employing an oil immersion objective. (b) The images of the SE-PSF (α=19, β=1.5) on the sCMOS when the particle is in different axial positions. (c) The experimental diagrams of the dense multi-particle. (d) Three-dimensional movement of fluorescent microspheres reaching the surface of the HT-22 cell membrane.

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In addition, we performed dense multi-particle imaging experiments to compare the imaging effects of the three PSFs (COSA PSF, SP-PSF, SE-PSF). The fluorescent microspheres (488/515) with a diameter of 100nm were fixed by agar. We use the three phases to image the region where fluorescent microspheres gather (as shown in Fig. 7(c)). As shown in Fig. 7(c), compared with the phases of the COSA PSF and the SP-PSF, the individual fluorescent microspheres in the fluorescent microsphere aggregation region can be well distinguished using the phase of the SE-PSF imaging, which facilitates the subsequent 3D coordinate calculation. Therefore, the SE-PSF with a smaller spatial extent has an obvious advantage in experiments with high-density multiparticle localization, which has a higher resolution. Under the same conditions, the images formed by the SE-PSF are also clearer. Although the SE-PSF with a small spatial extent is adopted, when the particle density is too large, the SE-PSFs will also overlap with each other.

To assess the feasibility of this approach on particle tracking in live cells, we studied preliminarily the endocytic process of the neural cells using the SE-PSF (Visualization 1). The HT-22 cells were cultured in phenol-red DMEM containing 10% FCS and labeled with membrane dye Nile Red (3μg/mL). The surface morphology of the labeled cell membrane can be constructed by slice scanning (because of the overlap between cells, here we only construct a part of the cell membrane surface). Then cells were incubated with fluorescent microspheres for 10mins before imaging. During the imaging process, the cell sample dish was maintained at 37°C by using an objective warmer. Total imaging time was less than 30min. To prevent the fluorescence bleaching caused by long imaging time, we reduced the intensity of the excitation light when the fluorescent microspheres reached the cell membrane surface. We have tracked the 3D dynamics of fluorescent microspheres reaching the cell membrane surface and entering the cell. Figure 7(d) shows a typical image of the fluorescent microspheres at a time point. Based on this experiment, it seems reasonable that the SE-PSF could be used effectively for live-cell tracking experiments. Some nanomedicines need to have good penetrating effects on specific cells, so this method can be used to determine the penetrating effects of different types of nanocarriers [2325]. We will present more detailed cell application results in future studies.

7. Conclusion

In summary, we propose a new point spread function that can be used for the three-dimensional localization of nanoparticles. The axial detection range of the SE-PSF can be adjusted by the design parameters. In addition, the spatial extent of the SE-PSF can also be changed by adjusting the design parameters, especially the SE-PSF with a smaller spatial extent can effectively reduce the overlap of nanoparticle images and realize the 3D localization of dense multi-particles. The superiority of the designed SE-PSF is demonstrated by comparison experiments with the COSA PSF and the SP-PSF. The three-dimensional process of fluorescent microspheres penetrating HT-22 cell membrane was successfully recorded by SE-PSF, which verified the effectiveness of this method.

Funding

Projects of International Cooperation of Jiangsu Province (No. BZ2020004); Jiangsu Planned Projects for Postdoctoral Research Funds (No.2018K020A, No.2018K040B, No.2018K114C); Natural Science Foundation of Shandong Province (No. ZR2019BF012, No. ZR2020QF103); the Strategic Priority Research Program (C) of the CAS (Grant No. XDC07040200); National Natural Science Foundation of China (No. 11804371).

Disclosures

The authors declare no conflicts of interest.

Data availability

All data underlying the results presented in this paper are available herein.

Supplemental document

See Supplement 1 for supporting content.

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12. H. Li, F. Wang, T. Wei, X. Miu, W. Huang, and Y. Zhang, “Particles 3D tracking with large axial depth by using the 2π-DH-PSF,” Opt. Lett. 46(20), 5088–5091 (2021). [CrossRef]  

13. Y. Shechtman, S. J. Sahl, A. S. Backer, and W. E. Moerner, “Optimal Point Spread Function Design for 3D Imaging,” Phys. Rev. Lett. 113(13), 133902 (2014). [CrossRef]  

14. Y. Shechtman, L. E. Weiss, A. S. Backer, S. J. Sahl, and W. E. Moerner, “Precise Three-Dimensional Scan-Free Multiple-Particle Tracking over Large Axial Ranges with Tetrapod Point Spread Functions,” Nano Lett. 15(6), 4194–4199 (2015). [CrossRef]  

15. Y. Zhou and G. Carles, “Precise 3D particle localization over large axial ranges using secondary astigmatism,” Opt. Lett. 45(8), 2466–2469 (2020). [CrossRef]  

16. S. Quirin, S. R. Pavani, and R. Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proceedings of the National Academy of Sciences 109(3), 675–679 (2012). [CrossRef]  

17. S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8(4), 302–306 (2014). [CrossRef]  

18. Y. Zhou, M. Handley, G. Carles, and A. R. Harvey, “Advances in 3D single particle localization microscopy,” APL Photonics 4(6), 060901 (2019). [CrossRef]  

19. Y. Zhou, P. Zammit, V. Zickus, J. M. Taylor, and A. R. Harvey, “Twin-Airy Point-Spread Function for Extended-Volume Particle Localization,” Phys. Rev. Lett. 124(19), 198104 (2020). [CrossRef]  

20. L. Zhu, M. Sun, D. Zhang, J. Yu, J. Wen, and J. Chen, “Multifocal array with controllable polarization in each focal spot,” Opt. Express 23(19), 24688–24698 (2015). [CrossRef]  

21. H. Li, X. Yun, Y. Zhang, F. Wang, and W. Huang, “Optimization of Fresnel-zones-based Double Helix Point Spread Function and measurement of particle diffusion coefficient,” Opt. Commun. 502, 127411 (2022). [CrossRef]  

22. J. Chao, E. Sally Ward, and R. J. Ober, “Fisher information theory for parameter estimation in single molecule microscopy: tutorial,” J. Opt. Soc. Am. A 33(7), B36–57 (2016). [CrossRef]  

23. B. D. Chithrani, A. A. Ghazani, and W. C. W. Chan, “Determining the size and shape dependence of gold nanoparticle uptake into mammalian cells,” Nano Lett. 6(4), 662–668 (2006). [CrossRef]  

24. R. Toy, P. M. Peiris, K. B. Ghaghada, and E. Karathanasis, “Shaping cancer nanomedicine: the effect of particle shape on the in vivo journey of nanoparticles,” Nanomedicine 9(1), 121–134 (2014). [CrossRef]  

25. S. E. A. Gratton, P. A. Ropp, P. D. Pohlhaus, J. C. Luft, V. J. Madden, M. E. Napier, and J. M. DeSimone, “The effect of particle design on cellular internalization pathways,” Proceedings of the National Academy of Sciences of the United States of America 105, 11613–11618 (2008).

References

  • View by:

  1. J. Jeevanandam, Y. S. Chan, and M. K. Danquah, “Nano-formulations of drugs: Recent developments, impact and challenges,” Biochimie 128-129, 99–112 (2016).
    [Crossref]
  2. K. Shin, Y. H. Song, Y. Goh, and K. T. Lee, “Two-Dimensional and Three-Dimensional Single Particle Tracking of Upconverting Nanoparticles in Living Cells,” Int J Mol Sci 20(6), 1424 (2019).
    [Crossref]
  3. A. Dupont and D. C. Lamb, “Nanoscale three-dimensional single particle tracking,” Nanoscale 3(11), 4532–4541 (2011).
    [Crossref]
  4. X. Wang, H. Yi, I. Gdor, M. Hereld, and N. F. Scherer, “Nanoscale Resolution 3D Snapshot Particle Tracking by Multifocal Microscopy,” Nano Lett. 19(10), 6781–6787 (2019).
    [Crossref]
  5. A. von Diezmann, Y. Shechtman, and W. E. Moerner, “Three-Dimensional Localization of Single Molecules for Super Resolution Imaging and Single-Particle Tracking,” Chem. Rev. 117(11), 7244–7275 (2017).
    [Crossref]
  6. H. Shen, L. J. Tauzin, R. Baiyasi, W. Wang, N. Moringo, B. Shuang, and C. F. Landes, “Single Particle Tracking: From Theory to Biophysical Applications,” Chem. Rev. 117(11), 7331–7376 (2017).
    [Crossref]
  7. S. R. P. Pavani and R. Piestun, “High-efficiency rotating point spread functions,” Opt. Express 16(5), 3484–3489 (2008).
    [Crossref]
  8. R. Berlich and S. Stallinga, “High-order-helix point spread functions for monocular three-dimensional imaging with superior aberration robustness,” Opt. Express 26(4), 4873–4891 (2018).
    [Crossref]
  9. G. Grover, K. DeLuca, S. Quirin, J. DeLuca, and R. Piestun, “Super-resolution photon-efficient imaging by nanometric double-helix point spread function localization of emitters (SPINDLE),” Opt. Express 20(24), 26681–26695 (2012).
    [Crossref]
  10. H. Li, D. Chen, G. Xu, B. Yu, and H. Niu, “Three dimensional multi-molecule tracking in thick samples with extended depth-of-field,” Opt. Express 23(2), 787–794 (2015).
    [Crossref]
  11. C. Roider, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Axial super-localisation using rotating point spread functions shaped by polarisation-dependent phase modulation,” Opt. Express 22(4), 4029–4037 (2014).
    [Crossref]
  12. H. Li, F. Wang, T. Wei, X. Miu, W. Huang, and Y. Zhang, “Particles 3D tracking with large axial depth by using the 2π-DH-PSF,” Opt. Lett. 46(20), 5088–5091 (2021).
    [Crossref]
  13. Y. Shechtman, S. J. Sahl, A. S. Backer, and W. E. Moerner, “Optimal Point Spread Function Design for 3D Imaging,” Phys. Rev. Lett. 113(13), 133902 (2014).
    [Crossref]
  14. Y. Shechtman, L. E. Weiss, A. S. Backer, S. J. Sahl, and W. E. Moerner, “Precise Three-Dimensional Scan-Free Multiple-Particle Tracking over Large Axial Ranges with Tetrapod Point Spread Functions,” Nano Lett. 15(6), 4194–4199 (2015).
    [Crossref]
  15. Y. Zhou and G. Carles, “Precise 3D particle localization over large axial ranges using secondary astigmatism,” Opt. Lett. 45(8), 2466–2469 (2020).
    [Crossref]
  16. S. Quirin, S. R. Pavani, and R. Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proceedings of the National Academy of Sciences 109(3), 675–679 (2012).
    [Crossref]
  17. S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8(4), 302–306 (2014).
    [Crossref]
  18. Y. Zhou, M. Handley, G. Carles, and A. R. Harvey, “Advances in 3D single particle localization microscopy,” APL Photonics 4(6), 060901 (2019).
    [Crossref]
  19. Y. Zhou, P. Zammit, V. Zickus, J. M. Taylor, and A. R. Harvey, “Twin-Airy Point-Spread Function for Extended-Volume Particle Localization,” Phys. Rev. Lett. 124(19), 198104 (2020).
    [Crossref]
  20. L. Zhu, M. Sun, D. Zhang, J. Yu, J. Wen, and J. Chen, “Multifocal array with controllable polarization in each focal spot,” Opt. Express 23(19), 24688–24698 (2015).
    [Crossref]
  21. H. Li, X. Yun, Y. Zhang, F. Wang, and W. Huang, “Optimization of Fresnel-zones-based Double Helix Point Spread Function and measurement of particle diffusion coefficient,” Opt. Commun. 502, 127411 (2022).
    [Crossref]
  22. J. Chao, E. Sally Ward, and R. J. Ober, “Fisher information theory for parameter estimation in single molecule microscopy: tutorial,” J. Opt. Soc. Am. A 33(7), B36–57 (2016).
    [Crossref]
  23. B. D. Chithrani, A. A. Ghazani, and W. C. W. Chan, “Determining the size and shape dependence of gold nanoparticle uptake into mammalian cells,” Nano Lett. 6(4), 662–668 (2006).
    [Crossref]
  24. R. Toy, P. M. Peiris, K. B. Ghaghada, and E. Karathanasis, “Shaping cancer nanomedicine: the effect of particle shape on the in vivo journey of nanoparticles,” Nanomedicine 9(1), 121–134 (2014).
    [Crossref]
  25. S. E. A. Gratton, P. A. Ropp, P. D. Pohlhaus, J. C. Luft, V. J. Madden, M. E. Napier, and J. M. DeSimone, “The effect of particle design on cellular internalization pathways,” Proceedings of the National Academy of Sciences of the United States of America 105, 11613–11618 (2008).

2022 (1)

H. Li, X. Yun, Y. Zhang, F. Wang, and W. Huang, “Optimization of Fresnel-zones-based Double Helix Point Spread Function and measurement of particle diffusion coefficient,” Opt. Commun. 502, 127411 (2022).
[Crossref]

2021 (1)

2020 (2)

Y. Zhou and G. Carles, “Precise 3D particle localization over large axial ranges using secondary astigmatism,” Opt. Lett. 45(8), 2466–2469 (2020).
[Crossref]

Y. Zhou, P. Zammit, V. Zickus, J. M. Taylor, and A. R. Harvey, “Twin-Airy Point-Spread Function for Extended-Volume Particle Localization,” Phys. Rev. Lett. 124(19), 198104 (2020).
[Crossref]

2019 (3)

Y. Zhou, M. Handley, G. Carles, and A. R. Harvey, “Advances in 3D single particle localization microscopy,” APL Photonics 4(6), 060901 (2019).
[Crossref]

X. Wang, H. Yi, I. Gdor, M. Hereld, and N. F. Scherer, “Nanoscale Resolution 3D Snapshot Particle Tracking by Multifocal Microscopy,” Nano Lett. 19(10), 6781–6787 (2019).
[Crossref]

K. Shin, Y. H. Song, Y. Goh, and K. T. Lee, “Two-Dimensional and Three-Dimensional Single Particle Tracking of Upconverting Nanoparticles in Living Cells,” Int J Mol Sci 20(6), 1424 (2019).
[Crossref]

2018 (1)

2017 (2)

A. von Diezmann, Y. Shechtman, and W. E. Moerner, “Three-Dimensional Localization of Single Molecules for Super Resolution Imaging and Single-Particle Tracking,” Chem. Rev. 117(11), 7244–7275 (2017).
[Crossref]

H. Shen, L. J. Tauzin, R. Baiyasi, W. Wang, N. Moringo, B. Shuang, and C. F. Landes, “Single Particle Tracking: From Theory to Biophysical Applications,” Chem. Rev. 117(11), 7331–7376 (2017).
[Crossref]

2016 (2)

J. Jeevanandam, Y. S. Chan, and M. K. Danquah, “Nano-formulations of drugs: Recent developments, impact and challenges,” Biochimie 128-129, 99–112 (2016).
[Crossref]

J. Chao, E. Sally Ward, and R. J. Ober, “Fisher information theory for parameter estimation in single molecule microscopy: tutorial,” J. Opt. Soc. Am. A 33(7), B36–57 (2016).
[Crossref]

2015 (3)

2014 (4)

C. Roider, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Axial super-localisation using rotating point spread functions shaped by polarisation-dependent phase modulation,” Opt. Express 22(4), 4029–4037 (2014).
[Crossref]

Y. Shechtman, S. J. Sahl, A. S. Backer, and W. E. Moerner, “Optimal Point Spread Function Design for 3D Imaging,” Phys. Rev. Lett. 113(13), 133902 (2014).
[Crossref]

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8(4), 302–306 (2014).
[Crossref]

R. Toy, P. M. Peiris, K. B. Ghaghada, and E. Karathanasis, “Shaping cancer nanomedicine: the effect of particle shape on the in vivo journey of nanoparticles,” Nanomedicine 9(1), 121–134 (2014).
[Crossref]

2012 (2)

S. Quirin, S. R. Pavani, and R. Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proceedings of the National Academy of Sciences 109(3), 675–679 (2012).
[Crossref]

G. Grover, K. DeLuca, S. Quirin, J. DeLuca, and R. Piestun, “Super-resolution photon-efficient imaging by nanometric double-helix point spread function localization of emitters (SPINDLE),” Opt. Express 20(24), 26681–26695 (2012).
[Crossref]

2011 (1)

A. Dupont and D. C. Lamb, “Nanoscale three-dimensional single particle tracking,” Nanoscale 3(11), 4532–4541 (2011).
[Crossref]

2008 (1)

2006 (1)

B. D. Chithrani, A. A. Ghazani, and W. C. W. Chan, “Determining the size and shape dependence of gold nanoparticle uptake into mammalian cells,” Nano Lett. 6(4), 662–668 (2006).
[Crossref]

Backer, A. S.

Y. Shechtman, L. E. Weiss, A. S. Backer, S. J. Sahl, and W. E. Moerner, “Precise Three-Dimensional Scan-Free Multiple-Particle Tracking over Large Axial Ranges with Tetrapod Point Spread Functions,” Nano Lett. 15(6), 4194–4199 (2015).
[Crossref]

Y. Shechtman, S. J. Sahl, A. S. Backer, and W. E. Moerner, “Optimal Point Spread Function Design for 3D Imaging,” Phys. Rev. Lett. 113(13), 133902 (2014).
[Crossref]

Baiyasi, R.

H. Shen, L. J. Tauzin, R. Baiyasi, W. Wang, N. Moringo, B. Shuang, and C. F. Landes, “Single Particle Tracking: From Theory to Biophysical Applications,” Chem. Rev. 117(11), 7331–7376 (2017).
[Crossref]

Berlich, R.

Bernet, S.

Carles, G.

Y. Zhou and G. Carles, “Precise 3D particle localization over large axial ranges using secondary astigmatism,” Opt. Lett. 45(8), 2466–2469 (2020).
[Crossref]

Y. Zhou, M. Handley, G. Carles, and A. R. Harvey, “Advances in 3D single particle localization microscopy,” APL Photonics 4(6), 060901 (2019).
[Crossref]

Chan, W. C. W.

B. D. Chithrani, A. A. Ghazani, and W. C. W. Chan, “Determining the size and shape dependence of gold nanoparticle uptake into mammalian cells,” Nano Lett. 6(4), 662–668 (2006).
[Crossref]

Chan, Y. S.

J. Jeevanandam, Y. S. Chan, and M. K. Danquah, “Nano-formulations of drugs: Recent developments, impact and challenges,” Biochimie 128-129, 99–112 (2016).
[Crossref]

Chao, J.

Chen, D.

Chen, J.

Chithrani, B. D.

B. D. Chithrani, A. A. Ghazani, and W. C. W. Chan, “Determining the size and shape dependence of gold nanoparticle uptake into mammalian cells,” Nano Lett. 6(4), 662–668 (2006).
[Crossref]

Danquah, M. K.

J. Jeevanandam, Y. S. Chan, and M. K. Danquah, “Nano-formulations of drugs: Recent developments, impact and challenges,” Biochimie 128-129, 99–112 (2016).
[Crossref]

DeLuca, J.

DeLuca, K.

DeSimone, J. M.

S. E. A. Gratton, P. A. Ropp, P. D. Pohlhaus, J. C. Luft, V. J. Madden, M. E. Napier, and J. M. DeSimone, “The effect of particle design on cellular internalization pathways,” Proceedings of the National Academy of Sciences of the United States of America 105, 11613–11618 (2008).

Dupont, A.

A. Dupont and D. C. Lamb, “Nanoscale three-dimensional single particle tracking,” Nanoscale 3(11), 4532–4541 (2011).
[Crossref]

Gdor, I.

X. Wang, H. Yi, I. Gdor, M. Hereld, and N. F. Scherer, “Nanoscale Resolution 3D Snapshot Particle Tracking by Multifocal Microscopy,” Nano Lett. 19(10), 6781–6787 (2019).
[Crossref]

Ghaghada, K. B.

R. Toy, P. M. Peiris, K. B. Ghaghada, and E. Karathanasis, “Shaping cancer nanomedicine: the effect of particle shape on the in vivo journey of nanoparticles,” Nanomedicine 9(1), 121–134 (2014).
[Crossref]

Ghazani, A. A.

B. D. Chithrani, A. A. Ghazani, and W. C. W. Chan, “Determining the size and shape dependence of gold nanoparticle uptake into mammalian cells,” Nano Lett. 6(4), 662–668 (2006).
[Crossref]

Goh, Y.

K. Shin, Y. H. Song, Y. Goh, and K. T. Lee, “Two-Dimensional and Three-Dimensional Single Particle Tracking of Upconverting Nanoparticles in Living Cells,” Int J Mol Sci 20(6), 1424 (2019).
[Crossref]

Gratton, S. E. A.

S. E. A. Gratton, P. A. Ropp, P. D. Pohlhaus, J. C. Luft, V. J. Madden, M. E. Napier, and J. M. DeSimone, “The effect of particle design on cellular internalization pathways,” Proceedings of the National Academy of Sciences of the United States of America 105, 11613–11618 (2008).

Grover, G.

Handley, M.

Y. Zhou, M. Handley, G. Carles, and A. R. Harvey, “Advances in 3D single particle localization microscopy,” APL Photonics 4(6), 060901 (2019).
[Crossref]

Harvey, A. R.

Y. Zhou, P. Zammit, V. Zickus, J. M. Taylor, and A. R. Harvey, “Twin-Airy Point-Spread Function for Extended-Volume Particle Localization,” Phys. Rev. Lett. 124(19), 198104 (2020).
[Crossref]

Y. Zhou, M. Handley, G. Carles, and A. R. Harvey, “Advances in 3D single particle localization microscopy,” APL Photonics 4(6), 060901 (2019).
[Crossref]

Hereld, M.

X. Wang, H. Yi, I. Gdor, M. Hereld, and N. F. Scherer, “Nanoscale Resolution 3D Snapshot Particle Tracking by Multifocal Microscopy,” Nano Lett. 19(10), 6781–6787 (2019).
[Crossref]

Huang, W.

H. Li, X. Yun, Y. Zhang, F. Wang, and W. Huang, “Optimization of Fresnel-zones-based Double Helix Point Spread Function and measurement of particle diffusion coefficient,” Opt. Commun. 502, 127411 (2022).
[Crossref]

H. Li, F. Wang, T. Wei, X. Miu, W. Huang, and Y. Zhang, “Particles 3D tracking with large axial depth by using the 2π-DH-PSF,” Opt. Lett. 46(20), 5088–5091 (2021).
[Crossref]

Jeevanandam, J.

J. Jeevanandam, Y. S. Chan, and M. K. Danquah, “Nano-formulations of drugs: Recent developments, impact and challenges,” Biochimie 128-129, 99–112 (2016).
[Crossref]

Jesacher, A.

Jia, S.

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8(4), 302–306 (2014).
[Crossref]

Karathanasis, E.

R. Toy, P. M. Peiris, K. B. Ghaghada, and E. Karathanasis, “Shaping cancer nanomedicine: the effect of particle shape on the in vivo journey of nanoparticles,” Nanomedicine 9(1), 121–134 (2014).
[Crossref]

Lamb, D. C.

A. Dupont and D. C. Lamb, “Nanoscale three-dimensional single particle tracking,” Nanoscale 3(11), 4532–4541 (2011).
[Crossref]

Landes, C. F.

H. Shen, L. J. Tauzin, R. Baiyasi, W. Wang, N. Moringo, B. Shuang, and C. F. Landes, “Single Particle Tracking: From Theory to Biophysical Applications,” Chem. Rev. 117(11), 7331–7376 (2017).
[Crossref]

Lee, K. T.

K. Shin, Y. H. Song, Y. Goh, and K. T. Lee, “Two-Dimensional and Three-Dimensional Single Particle Tracking of Upconverting Nanoparticles in Living Cells,” Int J Mol Sci 20(6), 1424 (2019).
[Crossref]

Li, H.

Luft, J. C.

S. E. A. Gratton, P. A. Ropp, P. D. Pohlhaus, J. C. Luft, V. J. Madden, M. E. Napier, and J. M. DeSimone, “The effect of particle design on cellular internalization pathways,” Proceedings of the National Academy of Sciences of the United States of America 105, 11613–11618 (2008).

Madden, V. J.

S. E. A. Gratton, P. A. Ropp, P. D. Pohlhaus, J. C. Luft, V. J. Madden, M. E. Napier, and J. M. DeSimone, “The effect of particle design on cellular internalization pathways,” Proceedings of the National Academy of Sciences of the United States of America 105, 11613–11618 (2008).

Miu, X.

Moerner, W. E.

A. von Diezmann, Y. Shechtman, and W. E. Moerner, “Three-Dimensional Localization of Single Molecules for Super Resolution Imaging and Single-Particle Tracking,” Chem. Rev. 117(11), 7244–7275 (2017).
[Crossref]

Y. Shechtman, L. E. Weiss, A. S. Backer, S. J. Sahl, and W. E. Moerner, “Precise Three-Dimensional Scan-Free Multiple-Particle Tracking over Large Axial Ranges with Tetrapod Point Spread Functions,” Nano Lett. 15(6), 4194–4199 (2015).
[Crossref]

Y. Shechtman, S. J. Sahl, A. S. Backer, and W. E. Moerner, “Optimal Point Spread Function Design for 3D Imaging,” Phys. Rev. Lett. 113(13), 133902 (2014).
[Crossref]

Moringo, N.

H. Shen, L. J. Tauzin, R. Baiyasi, W. Wang, N. Moringo, B. Shuang, and C. F. Landes, “Single Particle Tracking: From Theory to Biophysical Applications,” Chem. Rev. 117(11), 7331–7376 (2017).
[Crossref]

Napier, M. E.

S. E. A. Gratton, P. A. Ropp, P. D. Pohlhaus, J. C. Luft, V. J. Madden, M. E. Napier, and J. M. DeSimone, “The effect of particle design on cellular internalization pathways,” Proceedings of the National Academy of Sciences of the United States of America 105, 11613–11618 (2008).

Niu, H.

Ober, R. J.

Pavani, S. R.

S. Quirin, S. R. Pavani, and R. Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proceedings of the National Academy of Sciences 109(3), 675–679 (2012).
[Crossref]

Pavani, S. R. P.

Peiris, P. M.

R. Toy, P. M. Peiris, K. B. Ghaghada, and E. Karathanasis, “Shaping cancer nanomedicine: the effect of particle shape on the in vivo journey of nanoparticles,” Nanomedicine 9(1), 121–134 (2014).
[Crossref]

Piestun, R.

Pohlhaus, P. D.

S. E. A. Gratton, P. A. Ropp, P. D. Pohlhaus, J. C. Luft, V. J. Madden, M. E. Napier, and J. M. DeSimone, “The effect of particle design on cellular internalization pathways,” Proceedings of the National Academy of Sciences of the United States of America 105, 11613–11618 (2008).

Quirin, S.

G. Grover, K. DeLuca, S. Quirin, J. DeLuca, and R. Piestun, “Super-resolution photon-efficient imaging by nanometric double-helix point spread function localization of emitters (SPINDLE),” Opt. Express 20(24), 26681–26695 (2012).
[Crossref]

S. Quirin, S. R. Pavani, and R. Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proceedings of the National Academy of Sciences 109(3), 675–679 (2012).
[Crossref]

Ritsch-Marte, M.

Roider, C.

Ropp, P. A.

S. E. A. Gratton, P. A. Ropp, P. D. Pohlhaus, J. C. Luft, V. J. Madden, M. E. Napier, and J. M. DeSimone, “The effect of particle design on cellular internalization pathways,” Proceedings of the National Academy of Sciences of the United States of America 105, 11613–11618 (2008).

Sahl, S. J.

Y. Shechtman, L. E. Weiss, A. S. Backer, S. J. Sahl, and W. E. Moerner, “Precise Three-Dimensional Scan-Free Multiple-Particle Tracking over Large Axial Ranges with Tetrapod Point Spread Functions,” Nano Lett. 15(6), 4194–4199 (2015).
[Crossref]

Y. Shechtman, S. J. Sahl, A. S. Backer, and W. E. Moerner, “Optimal Point Spread Function Design for 3D Imaging,” Phys. Rev. Lett. 113(13), 133902 (2014).
[Crossref]

Sally Ward, E.

Scherer, N. F.

X. Wang, H. Yi, I. Gdor, M. Hereld, and N. F. Scherer, “Nanoscale Resolution 3D Snapshot Particle Tracking by Multifocal Microscopy,” Nano Lett. 19(10), 6781–6787 (2019).
[Crossref]

Shechtman, Y.

A. von Diezmann, Y. Shechtman, and W. E. Moerner, “Three-Dimensional Localization of Single Molecules for Super Resolution Imaging and Single-Particle Tracking,” Chem. Rev. 117(11), 7244–7275 (2017).
[Crossref]

Y. Shechtman, L. E. Weiss, A. S. Backer, S. J. Sahl, and W. E. Moerner, “Precise Three-Dimensional Scan-Free Multiple-Particle Tracking over Large Axial Ranges with Tetrapod Point Spread Functions,” Nano Lett. 15(6), 4194–4199 (2015).
[Crossref]

Y. Shechtman, S. J. Sahl, A. S. Backer, and W. E. Moerner, “Optimal Point Spread Function Design for 3D Imaging,” Phys. Rev. Lett. 113(13), 133902 (2014).
[Crossref]

Shen, H.

H. Shen, L. J. Tauzin, R. Baiyasi, W. Wang, N. Moringo, B. Shuang, and C. F. Landes, “Single Particle Tracking: From Theory to Biophysical Applications,” Chem. Rev. 117(11), 7331–7376 (2017).
[Crossref]

Shin, K.

K. Shin, Y. H. Song, Y. Goh, and K. T. Lee, “Two-Dimensional and Three-Dimensional Single Particle Tracking of Upconverting Nanoparticles in Living Cells,” Int J Mol Sci 20(6), 1424 (2019).
[Crossref]

Shuang, B.

H. Shen, L. J. Tauzin, R. Baiyasi, W. Wang, N. Moringo, B. Shuang, and C. F. Landes, “Single Particle Tracking: From Theory to Biophysical Applications,” Chem. Rev. 117(11), 7331–7376 (2017).
[Crossref]

Song, Y. H.

K. Shin, Y. H. Song, Y. Goh, and K. T. Lee, “Two-Dimensional and Three-Dimensional Single Particle Tracking of Upconverting Nanoparticles in Living Cells,” Int J Mol Sci 20(6), 1424 (2019).
[Crossref]

Stallinga, S.

Sun, M.

Tauzin, L. J.

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Supplementary Material (2)

NameDescription
Supplement 1       Supplement 1
Visualization 1       The three-dimensional process of fluorescent microspheres penetrating HT-22 cell membrane.

Data availability

All data underlying the results presented in this paper are available herein.

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Figures (7)

Fig. 1.
Fig. 1. Schematic diagram of the phase combination method based on sector segmentation.
Fig. 2.
Fig. 2. (a) The exponential function phase and the exponential point spread function (α=70, β=0.5). (b) The Double exponential function phase and the Double exponential point spread function (α=70, β=0.5). (c) The Negative double exponential function phase and the Negative double exponential point spread function (α=70, β=0.5). (d) The Positive double exponential function phase and the Positive double exponential point spread function (α=70, β=0.5). (e) The Splicing exponential function phase and the Splicing exponential point spread function (α=70, β=0.5). (f) The Optimized splicing exponential function phase and the Optimized splicing exponential point spread function (α=70, β=0.5). (g) The generation process of the Optimized splicing exponential function phase.
Fig. 3.
Fig. 3. Iterative process
Fig. 4.
Fig. 4. (a) The relationship between the parameter α and the axial detection range. (b) Comparison of the localization precisions.
Fig. 5.
Fig. 5. (a) Five phases involved in comparison and their point spread functions. (b) Comparison of the main lobes strength.
Fig. 6.
Fig. 6. Relationship between the distance between the two main lobes and the axial position of the particle
Fig. 7.
Fig. 7. (a) Experimental setup employing an oil immersion objective. (b) The images of the SE-PSF (α=19, β=1.5) on the sCMOS when the particle is in different axial positions. (c) The experimental diagrams of the dense multi-particle. (d) Three-dimensional movement of fluorescent microspheres reaching the surface of the HT-22 cell membrane.

Equations (13)

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I ( u , v ; z ) | F F T 2 { exp ( i P ( x , y ) + 2 i π M λ f 4 f ( x Δ x + y Δ y ) i π τ M 2 z λ f 4 f 2 ( x 2 + y 2 ) ) } | 2 .
I ( u , v ; z Δ z ) | F F T 2 { exp ( i P ( x , y ) + 2 i π M λ f 4 f ( x Δ x + y Δ y ) i π τ M 2 z λ f 4 f 2 ( x 2 + y 2 ) + i π τ M 2 Δ z λ f 4 f 2 ( x 2 + y 2 ) ) } | 2 .
ψ d = π τ M 2 Δ z λ f 4 f 2 ( x 2 + y 2 ) .
ψ e = α x exp ( β x 2 ) + α y exp ( β y 2 ) .
ψ c ( φ ) = { ψ 1 ( φ ) ; ( mod ( n , S ) 1 ) + [ n S ] 2 π S N φ 2 π N mod ( n , S ) + [ n S ] 2 π S N , mod ( n , S ) = 1 ψ 2 ( φ ) ; ( mod ( n , S ) 1 ) + [ n S ] 2 π S N φ 2 π N mod ( n , S ) + [ n S ] 2 π S N , mod ( n , S ) = 2 ψ 3 ( φ ) ; ( mod ( n , S ) 1 ) + [ n S ] 2 π S N φ 2 π N mod ( n , S ) + [ n S ] 2 π S N , mod ( n , S ) = 3 ψ s ( φ ) ; ( mod ( n , S ) 1 ) + [ n S ] 2 π S N φ 2 π N mod ( n , S ) + [ n S ] 2 π S N , mod ( n , S ) = s ψ S ( φ ) ; ( mod ( n , S ) 1 ) + [ n S ] 2 π S N φ 2 π N mod ( n , S ) + [ n S ] 2 π S N , mod ( n , S ) = 0 .
A z ( u , v ) = F F T 2 { exp ( i P 0 ( x , y ) i π τ M 2 z λ f 4 f 2 ( x 2 + y 2 ) ) } .
exp ( i P z ( x , y ) ) = I F F T 2 { A z ( u , v ) } / exp ( i π τ M 2 z λ f 4 f 2 ( x 2 + y 2 ) ) .
A z ( u , v ) = | A z ( u , v ) | exp ( i A r g ( A z ( u , v ) ) ) .
P 1 = A r g ( 1 t = 1 T a t t = 1 T a t P z ( x , y ) ) .
a t = max { I ( T + 1 ) / 2 } max { I t } .
z r = 0.3115 α + 10.38 β 5.923.
C R L B i = k = 1 N p ( ( μ ( k ) θ i ) × ( μ ( k ) θ i ) / ( μ ( k ) + B ) ) 1 .
P = { p 1 , p 2 , p 3 , , p N } , { p 1 = { ( x 1 , y 1 ) , ( x 1 , y 1 ) , ( x 1 , y 1 ) , , ( x 1 , y 1 ) } , c a r d ( p 1 ) = g v 1 p 2 = { ( x 2 , y 2 ) , ( x 2 , y 2 ) , ( x 2 , y 2 ) , , ( x 2 , y 2 ) } , c a r d ( p 2 ) = g v 2 p 3 = { ( x 3 , y 3 ) , ( x 3 , y 3 ) , ( x 3 , y 3 ) , , ( x 3 , y 3 ) } , c a r d ( p 3 ) = g v 3 p N = { ( x N , y N ) , ( x N , y N ) , ( x N , y N ) , , ( x N , y N ) } , c a r d ( p N ) = g v N .