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Real-time brain-wide multi-planar microscopy for simultaneous cortex and hippocampus imaging at the cellular resolution in mice

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Abstract

Interactions between the cerebral cortex and the deep cerebellar nuclei play important roles in cognitive processes. However, conventional microscopes fail to dynamically record cellular structures in distinct brain regions and at different depths, which requires high resolution, large field of view (FOV), and depth of field (DOF). Here we propose a single-photon excited fluorescence microscopy technique that performs simultaneous cortex and hippocampus imaging, enabled by a customized microscope and a chronic optical window. After we implant a glass microwindow above the hippocampus, the surface of the hippocampus is shifted to the superficial plane. We demonstrate that the proposed technique is able to image cellular structures and blood vessel dynamics in the cortex and the hippocampus in in vivo experiments, and is compatible with various mesoscopic systems.

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

1. Introduction

Recent studies have revealed various cognitive processes that involve the interaction of multiple brain regions. The formation of episodic memory involves the interplay of hippocampus and prefrontal cortex [13], while spatial navigation is related to hippocampal-parietal cortical interactions [47], and posttraumatic stress disorder (PTSD) is also associated with amygdala, medial prefrontal cortex, and hippocampus [8]. In adult mice, these nuclei are situated more than 1 mm beneath the dura, and spread across the whole brain. In order to record the dynamic behavior of different brain modules at the fine scale, one requires an optical system with cellular resolution, brain-wide FOV, video-rate acquisition, as well as the capability of multi-depth imaging.

To achieve deep mouse brain imaging, a variety of techniques have been proposed. Optical fiber photometry [9,10] and gradient index (GRIN) lens [11] show their superiority in deep brain imaging. However, optical fiber photometry collects the signal of a bulk of neurons and fails to provide the spatial resolution. GRIN-lens-based endomicroscope reaches depths of several millimeters, while the FOV is just several hundred microns. Multi-photon microscopy (MPM) also has the potential to image at 1 mm depth with diffraction-limited resolution [12]. However, limited by the laser repetition rate, a tradeoff exists between the FOV and the imaging speed. Although several methods have been proposed to enhance the volumetric imaging speed while retaining the FOV, including dual-plane imaging by remote focusing [13], random-access scanning [14], reverberation two-photon microscopy [15], and spatial light modulator (SLM) combined with MPM [16], these techniques are limited by a restricted FOV of 1 mm2, thereby precluding the simultaneous observation of areas large enough to encompass multiple cortical areas. The 2-photon random access mesoscope (2p-RAM) [17] provides diffraction-limited resolution in a cylindrical volume ($\mathrm{\Phi }$5 mm × 1 mm), while the frame rate is just 1.9 Hz for 4.4 mm × 4.2 mm FOV, thus limiting its application of dynamic imaging.

Many custom-built imaging systems have been designed for imaging a whole mouse brain with large FOV. The Mesolens has been reported with an FOV greater than 6 mm and with submicron resolution, while the Mesolens microscope needs approximate 200 seconds for a full FOV (6 mm), full resolution image [18], due to the point-scanning strategy. The real-time, ultra-large-scale, high-resolution (RUSH) macroscope has a 1.2 cm × 1 cm FOV, cellular resolution and video-rate imaging [19], which is applicable to in vivo whole mouse brain imaging. Recently, the COSMOS macroscope has demonstrated the ability of recording in-focus projections of a 1 cm × 1 cm × 1.3 mm volume at video rate for revealing widespread population encoding of actions [20]. However, scattering in tissue precludes the high-quality hippocampus imaging [21]. Thus, the existing imaging systems and techniques cannot achieve simultaneous cortex and hippocampus imaging with cellular resolution, brain-wide FOV and video-rate acquisition.

To address this problem, we propose a microscopic system that enables simultaneous recording of the cortical and hippocampal dynamics at the fine resolution. First, we implant a customized optical window above the mouse brain. To eliminate scattering, we ablate the skin, skull, and some white matter above the region of interest [2225]. We also insert an additional glass column above the hippocampus, shifting its surface to the superficial plane. Second, we build a wide-field fluorescence mesoscope with off-the-shelf optical components, and show that our system is compatible with various mesoscopic systems. Finally, we demonstrate that our technique is able to image the cellular structures of microglia, neurons, and vasculature dynamics in mouse cortex and hippocampus simultaneously.

2. Methods and experimental setup

2.1 Strategy of multi-planar imaging

The principle of our simultaneous multi-planar imaging is shown in Fig. 1(a) and Fig. 1(b). First, we removed the skull of the mice and ablated a part of the cortical matter to expose the hippocampus. The surface of the hippocampus is approximate 1 mm below the dura, while the layer 2/3 neurons in the cortex are approximate 100 to 300 μm below the dura. In order to shift the neurons in the hippocampus to the superficial cortex, we inserted some media with higher refractive index above the hippocampus. According to Snell’s law of refraction, the apparent depth ${H_a}$ in refractive media is related with the real depth ${H_r}$ as: ${H_a} = {H_r}{n_i}/{n_o},$ where ${n_i}$ is the refractive index of the image space and ${n_o}$ is the refractive index of the refractive media. To compensate for the depth difference $\mathrm{\Delta }H = {H_r} - {H_a},$ the height of the media that we need to insert is calculated as: ${H_r} = \mathrm{\Delta }H{n_o}/({n_o} - {n_i}).$ Assuming that a dry objective is used for detection at the wavelength around 515 nm, we estimate the refractive indices as ${n_i} = 1$ and ${n_o} = 1.5.$ The depth between the hippocampus and superficial cortex is approximately 0.9 mm [26], and the thickness of the coverslip is 0.17 mm. Therefore, we design a customized optical window for multi-planar imaging. A glass column of 0.9 mm thickness and 2 mm diameter is stuck to the bottom of the cover glass at the position of the mouse hippocampus, and a second glass column with an optimal thickness of 1.8 mm is placed on the top to further compensate for the depth difference, as shown in Fig. 1(a) and Fig. 1(b).

 figure: Fig. 1.

Fig. 1. The principle of the proposed multi-plane imaging. (a) Schematic of the experimental setup showing the chronic window implanted above the dentate gyrus. (b) Schematic diagram of the proposed multi-planar imaging. (c) The scheme of the custom-built wide-field mesoscope, with brain-wide FOV and cellular resolution.

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2.2 Optical mesoscope setup

Our method is compatible with various mesoscopic systems with moderate numerical apertures. One example is the Real-time, Ultra-large-scale, High-resolution (RUSH) microscope we proposed earlier. The customized objective is designed with a 1-centimeter FOV and submicron resolution. An alternative method we show here is a wide-field mesoscope constructed from off-the-shelf components (shown in Fig. 1(c)). The excitation source is a CW laser (MBL-III-473-100 mW, CNI), whose central wavelength is at 473 nm. A beam expander and a paired 4f-system with Lens1 (f = 50 mm) and Lens2 (f = 150 mm) are utilized to expand the beam to 9 mm. After focused by Lens3 (f = 150 mm) and reflected by a microprism (2 mm × 2 mm × 2 mm), the beam passes through the excitation objective lens with 2x/0.5 NA (MVPLAPO 2 XC, Olympus) and excites the fluorophores in the samples [27]. The fluorescence is collected by the same objective lens and refocused on the scientific Complementary Metal Oxide Semiconductor (sCMOS, pixel size: 6.5 μm, ORCA-flash4.0, Hamamatsu) by the tube lens (MVPLAPO 1X, Olympus). A dual-band filter (wavelength: 520 ± 12.5 nm/630 ± 46.5 nm, #87-241, Edmund) is placed in the emission light path to eliminate the reflected laser light. The FOV of the system is approximate 6 mm, and each pixel in the sCMOS corresponds to 3.25 μm on the image plane.

2.3 Animal surgery

Chronic craniotomy was performed according to the procedures described in a previous report in accordance with the guidance of the Animal Care and Use Committees of Tsinghua University. To be brief, we created a window of 6 mm in diameter, and then aspirated the cortex matter via a 0.9-mm-diameter (19 gauge) blunt needle connected to a vacuum pump. The central position we ablated was approximately 1.5 mm distant from the sagittal suture and 2 mm distant from the lambdoid suture. When seeing the fibers from the ventral [22], we stopped ablating the cortex. A chronic window was assembled by adhering a ϕ9 mm glass coverslip to a glass column using tissue adhesive (3M Vetbond), after which the window was implanted above the mouse cortex. Lastly, we mounted an aluminum head post to the skull, affixing it with the dental cement. For the surgical procedure, the mouse was anesthetized using isoflurane (3% in oxygen for induction and 1.5% to 2% for surgery and imaging) and a respiratory frequency of 1 Hz was maintained. The body temperature was kept at 37.5 °C with a feedback-controlled blanket and eye ointment was applied. In chronic imaging experiments, we also allowed two weeks for the mouse to recover. The mouse implanted by the chronic window was alive for more than two months.

3. Experimental results

3.1 Resolution measurements

To evaluate the performance of our system, we measured the lateral intensity profiles of fluorescent beads at different depths, as shown in Fig. 2. A step substrate is made of two microscope slides, with a 1.0 mm depth difference. We spray 1-μm-diameter green fluorescent microspheres (T14792, Thermo Fisher Scientific) onto the substrate, and then place a glass block on the lower step (Layer 2, as shown in Fig. 2(a)). In our RUSH system, we show that the beads on the two layers can be imaged simultaneously and their lateral intensity profiles are shown in Fig. 2(b). Considering the optical aberrations, we plot the line profiles along two perpendicular axes. After the Gaussian fitting, the full widths at half maximum of the beads on two layers are 2.5 μm and 3.0 μm, respectively, indicating that cellular resolution can be obtained in both layers.

 figure: Fig. 2.

Fig. 2. (a) The scheme of imaging fluorescent microspheres. (b) The experimental PSFs in the x-y plane of Layer 2 in the left figure and Layer 1 in the right figure. Scale bar: 10 μm. (c, d) Lateral intensity profiles of the fluorescent microspheres on Layer 1 and Layer 2, respectively.

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3.2 In vivo imaging of microglia cells in mice brains

Our proposed technique can perform in vivo imaging of microglia cells in Cx3Cr1-GFP mice (JAX no. 005582), as is shown in Fig. 3. The cellular structures of microglia cells across the cortex and in the hippocampus are captured by the RUSH microscope in a single snapshot (shown in Fig. 3(a)). Two subregions of the superficial cortex (indicated by a yellow box) and the hippocampus (indicated by a red box) are zoomed in, showing the fine details. Though these hippocampal cells are approximate 0.9 mm deeper than the cortical cells, they have similar structures in images. The diameters of the microglia cells in the cortex and hippocampus are 7.2 μm and 9.6 μm, respectively. However, the maximum fluorescence intensity of the hippocampal microglia in Fig. 3(e) is nearly approximate 37% less than that of the cortical microglia in Fig. 3(d).

 figure: Fig. 3.

Fig. 3. In vivo imaging of the Cx3Cr1-GFP mouse brain by the RUSH macroscope. (a) Large-FOV, high-resolution image of the Cx3Cr1-GFP mouse brain. Scale bar: 1 mm. (b, c) Magnified views of the subregions marked (a). Scale bar: 50 μm. (d, e) Signal intensity along the red dash lines in (b, c). Background subtraction is applied for all brain images in this article.

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To illustrate the vertical location of our implantation, we sacrifice the Cx3Cr1-GFP mouse, harvest its brain, and slice it into 150-μm-thick coronal slices after fixation overnight. In Fig. 4, we annotate the locations of the glass column, the cortex and the hippocampus. The result verifies that the deep region we observe is the hippocampal tissue.

 figure: Fig. 4.

Fig. 4. A coronal slice of the Cx3Cr1-GFP mouse brain. Scale bar: 1 mm.

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3.3 In vivo imaging of neurons in mice brains

We show that our system enables in vivo multi-planar neuro-imaging in Thy1-YFP mice (JAX no. 003782). Figure 5(a) shows the whole mouse brain imaging by the RUSH microscope. Subregions of the hippocampus and cortex labeled in Fig. 5(a) are shown in Fig. 5(b) and Fig. 5(c), where several spine-like structures are clearly distinguished and labeled by yellow circles. Besides, the axons are also distinguishable in both subregions (indicated by red arrows), showing the superior performance of this system. For our customized microscope shown in Fig. 6(a), the lateral resolution is restricted by the pixel size of the sCMOS detector, whereby only the soma can be visualized in each subregion (Figs. 6(b) and Figs. 6(c)), with a lower signal-to-background ratio. Both results show that our system has cellular resolution and has the potential to be applied in in vivo calcium imaging.

 figure: Fig. 5.

Fig. 5. In vivo imaging of the Thy1-YFP mouse brain by the RUSH macroscope. (a) Large-FOV, high-resolution image of the Thy1-YFP mouse brain. Scale bar: 1 mm. (b, c) Magnified views of the subregions in hippocampus and cortex marked in (a). Scale bar: 50 μm.

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 figure: Fig. 6.

Fig. 6. In vivo imaging of the Thy1-YFP mouse brain by the custom-built wide-field fluorescence mesoscope. (a) Large-FOV, high-resolution image of the Thy1-YFP mouse brain. Scale bar: 1 mm. (b, c) Magnified views of the subregions in hippocampus and cortex marked in (a). Scale bar: 50 μm.

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3.4 In vivo imaging of blood vessels in mice brains

We also demonstrate the blood vessels imaging of C57BL/6J mice brains in vivo. Prior to imaging, Dextran-conjugated fluorescein (D1823, Thermo Fisher Scientific) was injected for brain vasculature labeling by tail vein injection. In Fig. 7(a), we show the image of superficial cortex and hippocampus. In the subregions at the hippocampus and superficial cortex (marked in the red and yellow box), the vasculature in both regions can be visualized clearly (in Fig. 7(b) and Fig. 7(c)). Figure 7(d) and Fig. 7(e) plot the intensity fluctuation of selected vessels labeled by the dash line in Fig. 7(b) and Fig. 7(c) with a frame rate of 10 Hz. The spontaneous intensity fluctuation can be caused by vasodilation and blood cell flow.

 figure: Fig. 7.

Fig. 7. In vivo blood vessel imaging of the mouse brain by the RUSH macroscope. (a) Large-FOV, high-resolution blood vessel image of the mouse brain. Scale bar: 1 mm. (b, c) Magnified views of the subregions marked in (a). Scale bar: 100 μm. (d, e) Temporal traces of vasculature intensities labeled in (b, c). $\mathrm{\Delta }I/I$ is the relative intensity change of the vessel.

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3.5 In vivo chronic imaging in mice brains

To further investigate the effectiveness of our technology in chronic mice imaging, we allowed two weeks for the mice to recover from the optical window implantation. In Fig. 8(a1)-Fig. 8(a3), we show Thy1-YFP mouse brain images on Day 14, 30 and 60 after the optical window implantation. From the hippocampus in Fig. 8(b1)-Fig. 8(b3) and superficial cortex in Fig. 8(c1)–8(c3), the neurons in all regions are visible, showing that our microwindow implantation is applicable to chronic imaging. Figure 8(a1)-Fig. 8(c1) and Fig. 8(a3)-Fig. 8(c3) are acquired by the custom-built wide-field fluorescence mesoscope and Fig. 8(a2)-Fig. 8(c2) are acquired by the RUSH macroscope. We test the post-surgery behaviors of mice (n≥3), and no distinguishable abnormality is observed.

 figure: Fig. 8.

Fig. 8. In vivo chronic imaging of the mouse brain (14, 30 and 60 days post craniotomy). (a1-a3) Large-FOV, high-resolution neuron images of the mouse brain. Scale bar: 1 mm. (b1-b3) Magnified views of the hippocampal regions. (c1-c3) Magnified views of the superficial cortical region. Scale bar 100 μm.

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4. Discussions and Conclusions

In this article, we propose a multi-planar, brain-wide microscopic system that enables single-shot imaging of the cortex and deep cerebral nuclei. By changing the diameter and height of the glass column, this method is applicable to neurons at different depths, and could easily combine with different microscopic modalities. However, the glass columns placed in the path of a converging beam not only displace the focus by a certain amount, but also introduce optical aberrations and increase reflection, which degrade the image quality in the deep layer.

Spherical aberration exists all across the microwindow, and will be significant with the increase of numerical aperture and glass plate thickness [28]. To illustrate the effect of the spherical aberration, we simulated the point-spread-function (PSF) of a simplified system, in which focused rays pass through a glass plate. In Fig. 9(a), we show that the peak intensity increases when NA < 0.2, and then decreases dramatically when NA > 0.2, but the central the full widths at half maximum (FWHM) continuously decreases from 0.1 NA to 0.5 NA. This means high-angle rays induce significant aberrations and spread energy to the side lobes. In Fig. 9(b), we show the PSFs with glass columns of various thicknesses in the case of 0.3 NA, where the peak intensity decreases with the glass column thickness, but the FWHM does not change significantly. We also investigate the enclosed energy in different radii, and plot the radius with half energy in Fig. 9(c)-Fig. 9(d). In order to choose the optimal system parameters, we need to strike a trade-off between the resolution of both the superficial layer and the deep layer, and the light-collection efficiency. The large NA leads to high light-collection efficiency, high surface resolution, but degrades deep resolution, so we choose NA = 0.3 to 0.5 for the purpose of neuroimaging (with soma diameter ∼10 μm). However, our system is more tolerant to variations in glass thickness. According to our simulation results, a glass column thinner than 10 mm is acceptable for neural soma imaging, which indicates our method has the potential to image other subcortical nuclei less than 3 mm below the superficial cortex. For example, we calculate that around 15.8% intensity decay results from spherical aberration in microglia imaging, showed in Fig. 3, while other factors that lead to intensity decay include interface reflection and vignetting.

 figure: Fig. 9.

Fig. 9. (a) Simulated intensity distributions with various NAs of the objective, in the case of a 3-mm-thick glass column. (b) Simulated intensity distributions with glass column of various thicknesses, in the case of 0.3 NA. (c)-(d) Calculated radius that enclosed half the total energy, with different NAs of the objective and varying glass thicknesses.

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Vignetting occurs at the edges of the microwindows, because the wall of the glass column acts as an additional aperture and intercept light beam. In our microwindows, the ring area close to the boundary has lower brightness and deteriorated image quality. Derived with geometrical optics, $r = {r_0} - {H_r}{\theta _o}$ is the expression of the radius of the unvignetted area, wherein ${\theta _o}$ is the boundary refractive angle of rays and ${r_0}$ is the radius of the glass column. As shown in Fig. 10(a)-(b), we consider the boundary rays and derive boundary refractive angle as: ${\theta _o} = \textrm{NA}{n_i}/{n_o}$, where NA is the numerical aperture of the objective. According to Snell’s law, the radius of the unvignetted area is shown as: $r = {r_0} - \mathrm{\Delta }H{n_i}\textrm{NA}/({{n_o} - {n_i}} ).$ The vignetted area can be further reduced by changing the diameter of the top glass column (Fig. 10(c)) or the shape of the glass columns (Fig. 10(d)) and the radius of the vignetted area is derived as: $r = {r_0} - \mathrm{\Delta }H{n_i}\textrm{NA}/{n_o}.$ Fig. 10(e)-Fig. 10(f) show the relationship among the depth difference $\mathrm{\Delta }H,$ the numerical aperture of the objective NA, and the vignetted radius. This result indicates that deeper tissue imaging requires a large window or a low-NA objective, but a large window also induces more damage to the animals, and a low-NA objective results in reduced resolution. Our further study will combine this method with the GRIN lens to investigate deeper nuclei.

 figure: Fig. 10.

Fig. 10. Different configurations of our system. (a) In the unvignetted area, the glass column does not block rays that enter the entrance pupil of the microscope. (b) The top column has the same diameter as the bottom column. (c) The top column has a larger diameter than the bottom column. (d) Shape of the columns can be designed to reduce vignetting. (e) Vignetted radius with different depth for (b). (e) Vignetted radius with different depths for (c) and (d).

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The reflection of both the excitation light and emission light reduces the detected fluorescence intensity. For small-angle blue and green light, each water-air interface contributes around 4% energy loss, and it adds up to around 16% for one glass column. Reflection can be alleviated through the use of refractive index matching medium between glass gaps and the application of the anti-reflection coating on the columns.

Chromatic aberration changes the focal depths of different wavelengths, which needs to be considered in multi-color imaging. We analyze the two widely-used fluorescent proteins enhanced green fluorescent protein (eGFP) and mCherry, whose central wavelengths are 507 nm and 610 nm, respectively. Assuming the thickness of the glass column ${H_r}$ is 2.7 mm, the focal shift induced by chromatic aberration is approximate 6 μm, which is smaller than the neuron soma and the depth field of our system (∼ 10 μm). This result indicates that our system can be easily adapted to multi-color imaging with an additional excitation laser.

In summary, we have demonstrated the simultaneous in vivo imaging of the superficial cortex and the 2-mm-diameter FOV hippocampus by applying microwindow implantation and extended depth of field technology. This method facilitates high-quality, in vivo imaging of the cortex and the hippocampus with depth difference of as much as 900 μm. Through our experiments, we investigated the performance of the method and validated the performance of simultaneous cortical and hippocampal imaging in living mice brains.

Funding

National Natural Science Foundation of China (61831014, 61971256, 62088102); National Key Research and Development Program of China (2020AA0105500).

Acknowledgments

We thank Yingjun Tang for her assistance with the preparation of the brainslices.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (10)

Fig. 1.
Fig. 1. The principle of the proposed multi-plane imaging. (a) Schematic of the experimental setup showing the chronic window implanted above the dentate gyrus. (b) Schematic diagram of the proposed multi-planar imaging. (c) The scheme of the custom-built wide-field mesoscope, with brain-wide FOV and cellular resolution.
Fig. 2.
Fig. 2. (a) The scheme of imaging fluorescent microspheres. (b) The experimental PSFs in the x-y plane of Layer 2 in the left figure and Layer 1 in the right figure. Scale bar: 10 μm. (c, d) Lateral intensity profiles of the fluorescent microspheres on Layer 1 and Layer 2, respectively.
Fig. 3.
Fig. 3. In vivo imaging of the Cx3Cr1-GFP mouse brain by the RUSH macroscope. (a) Large-FOV, high-resolution image of the Cx3Cr1-GFP mouse brain. Scale bar: 1 mm. (b, c) Magnified views of the subregions marked (a). Scale bar: 50 μm. (d, e) Signal intensity along the red dash lines in (b, c). Background subtraction is applied for all brain images in this article.
Fig. 4.
Fig. 4. A coronal slice of the Cx3Cr1-GFP mouse brain. Scale bar: 1 mm.
Fig. 5.
Fig. 5. In vivo imaging of the Thy1-YFP mouse brain by the RUSH macroscope. (a) Large-FOV, high-resolution image of the Thy1-YFP mouse brain. Scale bar: 1 mm. (b, c) Magnified views of the subregions in hippocampus and cortex marked in (a). Scale bar: 50 μm.
Fig. 6.
Fig. 6. In vivo imaging of the Thy1-YFP mouse brain by the custom-built wide-field fluorescence mesoscope. (a) Large-FOV, high-resolution image of the Thy1-YFP mouse brain. Scale bar: 1 mm. (b, c) Magnified views of the subregions in hippocampus and cortex marked in (a). Scale bar: 50 μm.
Fig. 7.
Fig. 7. In vivo blood vessel imaging of the mouse brain by the RUSH macroscope. (a) Large-FOV, high-resolution blood vessel image of the mouse brain. Scale bar: 1 mm. (b, c) Magnified views of the subregions marked in (a). Scale bar: 100 μm. (d, e) Temporal traces of vasculature intensities labeled in (b, c). $\mathrm{\Delta }I/I$ is the relative intensity change of the vessel.
Fig. 8.
Fig. 8. In vivo chronic imaging of the mouse brain (14, 30 and 60 days post craniotomy). (a1-a3) Large-FOV, high-resolution neuron images of the mouse brain. Scale bar: 1 mm. (b1-b3) Magnified views of the hippocampal regions. (c1-c3) Magnified views of the superficial cortical region. Scale bar 100 μm.
Fig. 9.
Fig. 9. (a) Simulated intensity distributions with various NAs of the objective, in the case of a 3-mm-thick glass column. (b) Simulated intensity distributions with glass column of various thicknesses, in the case of 0.3 NA. (c)-(d) Calculated radius that enclosed half the total energy, with different NAs of the objective and varying glass thicknesses.
Fig. 10.
Fig. 10. Different configurations of our system. (a) In the unvignetted area, the glass column does not block rays that enter the entrance pupil of the microscope. (b) The top column has the same diameter as the bottom column. (c) The top column has a larger diameter than the bottom column. (d) Shape of the columns can be designed to reduce vignetting. (e) Vignetted radius with different depth for (b). (e) Vignetted radius with different depths for (c) and (d).
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