Consequences of brain refractive index mismatch for two-photon microscopy

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Astronomy uses direct wavefront measurement

Several methods have been developed over the years to determine the aberrations present in an optical wavefront. For individual optical components such as lenses, one can analyze their imperfections by passing light from a collimated laser beam or from a point source through the component, causing its aberrations to be imprinted in the beam wavefront. This wavefront can then be analyzed using a Shack-Hartmann-Sensor (Hartmann, 1900; Shack & Platt, 1971), a pyramid sensor (Ragazzoni, 1996), a curvature sensor (Roddier, 1988) or an interferometer such as a Twyman-Green interferometer (Twyman & Green, 1916) or a shearing interferometer (Bates, 1947).
These methods (except interferometry) are generally also applicable to measuring atmospheric aberrations using the light coming from an individual star of sufficient brightness, or from an artificial guide star created by exciting fluorescence in high atmospheric layers using a strong, focused laser beam (Tyson, 1997). Similarly in ophthalmology, a test beam reflected off of the retina can generate a test beam containing the aberrations (Prieto, et al., 2000). In fluorescence microscopy it might sometimes be possible to introduce a strongly fluorescent test bead near the region(s) to be imaged and use its fluorescence signal with one of the above methods (Azucena, et al., 2011), but the introduction of beads is certainly not a universally feasible option.

Why direct wavefront sensing is hard in microscopy

Unfortunately, none of these methods seem particularly suited for determining the optimal wavefront for focusing inside three-dimensional biological samples where subsequent optimized imaging at different depths can be required. The root of the problem is that different layers of the sample will generally participate in any wavefront measurement, presenting the wavefront sensor not with one clean wavefront, but with a superposition of a multitude of wavefronts, which cannot be separated and which hinder each other’s measurement. While the Shack-Hartmann sensor has been successfully adapted to laterally extended objects for solar adaptive optics (Rimmele & Radick, 1998), dealing with wavefronts from different depths in the sample is not possible with this method. To separate wavefronts coming from different depths in biological samples, coherence gated wavefront sensing (CGWS) has been developed, where an interferometer illuminated by a broad-band spatially coherent source performs depth discrimination of backscattered light, phase shifting is used to reconstruct the complex field and a virtual Shack-Hartmann sensor is used for phase unwrapping (Feierabend, et al., 2004; Rueckel, et al., 2006). While this technique has proven successful for measurement of aberrations when imaging through the skull of transparent zebrafish, it is relatively complex to implement on an existing microscope. What’s more, it requires a scattering sample with a random distribution of scatterers, but scattering must be weak enough for single scattering to still be dominant in the depths where aberrations are to be measured (Binding & Rückel, in preparation).
For deep brain imaging in rodents, the short scattering length seems to place a severe limitation on the usefulness of CGWS. In particular, the speckle size of the coherence-gated electromagnetic field decreases quickly with penetration depth (Markus Rückel and Jinyu Wang, both private communication). Initial tests optimizing two-photon fluorescence signal in the rodent cortex with a deformable mirror by manually optimizing spherical aberration and lowest-order astigmatism did not show significant improvements (Binding, 2008). It was therefore unclear whether optical aberrations do not play a significant role in this system, or whether they are just complicated to measure and to correct.

High-speed in vivo rat brain imaging shows blood flow

The camera in our system was fast enough to take the two images needed for optical sectioning before the respiration and heartbeat caused the cortex to move more than a small fraction of a wavelength. Even the movement of individual red blood cells in thin veins as well as the movement of leukocytes on the surface of larger vessels at the brain surface could be observed in real time with our OCT imaging frequency of 33 Hz (Figure 2-2 as well as Movie 1 & Movie 2).

Importance of defocus correction for high-NA OCT and OCM

Most scanning OCT systems are inherently limited to extremely low NA objectives, since their axial scan range is limited by the depth of field of the objective. In ff-OCT the sample is moved relative to the objective for z-scanning, so the scan range is only limited by the working distance of the objective and not by its NA. Higher NA can be used to increase the lateral resolution, so we chose objectives with an NA of 0.8. Even higher NAs are in principle usable, but a pair of identical objectives with larger NA and sufficiently long working distance was not available to us. As explained in more detail below, a new problem arises with medium and high NA objective ff-OCT systems when the refractive index of the sample does not perfectly match the refractive index of the immersion medium. When imaging deeper layers, the additional optical path difference causes the coherence volume to move out of the depth of field of the objective; this defocus considerably decreases OCT signal (an effect also called “confocal effect of OCT”, see Sheppard, et al., 2004) and limits imaging depth when uncorrected. To remove defocus, we implemented an automatic reference arm length scan which kept the imaging plane fixed inside the sample while moving the coherence volume. The optimal reference arm length was taken to be the one maximizing the total OCT signal.
Without defocus correction (i.e. at fixed reference arm length optimized on the cover glass), only 10% of the signal above background remains at around 120µm imaging depth (Figure 2-3, black dashed line). However, optimizing the reference arm length to compensate defocus at 200 µm imaging depth, we were able to recover signal more than twice as strong (Figure 2-3, gray solid line). Optimizing at 300 and 400 µm imaging depth (Figure 2-3, gray dashed and dotted lines), we found new optimal reference arm lengths which again boost signal strength by more than a factor of two. As expected, these reference arm lengths were only optimal in the vicinity of the depths where they were determined and cause low signal levels when used for imaging near the surface. During the refractive index measurement we found that optimal reference arm length actually varied linearly with depth inside tissue (see chapter 1). This implied that defocus-corrected imaging at arbitrary depths could be achieved without re-optimizing defocus at every single depth. With open-loop defocus correction based on the measured defocus slope, OCT signal falls off a lot more slowly (Figure 2-3, black solid line); the 10% level is reached at about a 300µm depth, indicating a 2.5-fold increase in imaging depth compared to imaging without defocus correction.
In summary, signal level and penetration depth in ff-OCT imaging with high-NA objectives benefit greatly from defocus correction, underlining the importance of the automated correction integrated in deep-OCM.


Imaging myelin fibers in vivo in cortex

Myelin imaging using deep-OCM was not only possible in slices, but also in vivo in the rat cortex. By visual inspection of several image stacks from different animals, we found that fibers had their highest concentration in depths of up to 150 µm, consistent with previous studies found on (Mikula, et al., 2007; Trotts, et al., 2007).
Since fiber density was depth dependent, we took the maximum image intensity per slice as a measure of signal decrease with depth. To reduce the influence of individual pixels, spatial filtering with a Gaussian kernel the size of the diffraction limited PSF was performed before taking the maximum. The signal level reached the constant noise level at a depth of 400 µm (Figure 3-5b), consistent with the visual finding of individual fibers down to 340 µm (Figure 3-5a).

Table of contents :

1 Introduction
1.1 We want to image the brain
1.2 Optical aberrations cause imperfect imaging
1.3 Aberrations can be thought of as a phase term on the wavefront
1.4 Many imaging systems can be limited by aberrations
1.4.1 Wide-field microscopy
1.4.2 Confocal microscopy
1.4.3 Two-photon microscopy
1.4.4 Structured illumination microscopy
1.4.6 Stimulated emission depletion
1.4.7 Optical coherence tomography
1.5 Astronomy uses direct wavefront measurement
1.5.1 Why direct wavefront sensing is hard in microscopy
1.6 The sample refractive index sets the scale for aberrations
1.7 Our refractive index measurements led us to develop deep-OCM
1.8 Deep rat brain imaging is limited by aberrations
1.9 Image analysis allows indirect wavefront measurement
1.9.1 Metric-based, imaging-model-agnostic methods
1.9.2 Metric-based modal wavefront sensing
1.9.3 Pupil segmentation
1.9.4 Phase diversity
2 Deep-OCM
2.1 Details of the setup
2.2 Animal preparation and treatment
2.3 High-speed in vivo rat brain imaging shows blood flow
2.4 Importance of defocus correction for high-NA OCT and OCM
3 Myelin imaging with deep-OCM
3.1 Deep-OCM shows myelin
3.1.1 Sensitivity to fiber orientation
3.2 Imaging myelin fibers in vivo in cortex
3.3 Myelin imaging in the peripheral nervous system
3.4 Discussion
4 Rat brain refractive index
4.1 Introduction
4.2 Measuring refractive index using high-NA OCT
4.2.1 OCT signal strength is sensitive to refractive-index-induced defocus
4.2.2 Dispersion and high NA complicate the refractive index measurement
4.2.3 Modeling the image formation process in high-NA OCT
4.2.4 Assuming a dispersion function allows calculation of the refractive index .
4.2.5 Choosing a suitable metric increases penetration depth
4.3 Results
4.3.1 Rat brain refractive index as a function of rat age
4.3.2 Importance of dispersion and high NA
4.4 Discussion
4.4.1 A model of defocus in OCT taking high NA and dispersion into account
4.4.2 Value and (non-)dependence of brain refractive index
4.4.3 Limits to the measuring precision
4.4.4 Potential systematic errors
4.4.5 Comparison with recent measurements
5 Consequences of brain refractive index mismatch for two-photon microscopy
5.1 Discussion
6 Maximum-A-Posteriori Focus and Stigmation (MAPFoSt)
6.1 Introduction
6.2 Materials and methods
6.2.1 The MAPFoSt algorithm
6.2.2 Data analysis
6.2.3 Experiments
6.2.4 Simulating image pairs
6.3 Results
6.3.1 Simulations show bias-free aberration estimation
6.3.2 SEM imaging experiments
6.4 Discussion
7 General Discussion
7.1 Correcting aberrations adds complexity
7.2 Defocus correction in high-NA OCM is worth it
7.3 Two-photon rat-brain imaging suffers from spherical aberration
7.4 The race is still on
8 Acknowledgements
9 Appendices
9.1 Deep-OCM motor placement
9.2 Derivations for MAPFoSt
9.2.1 Calculating the MTF and its derivatives
9.2.2 Calculating the MAPFoSt posterior and profile posterior
9.3 The heuristic SEM autofocus and auto-stigmation algorithm
9.4 Modal wavefront sensing for SEM
10 Literature
11 List of Acronyms
12 French summary / Résumé substantiel de cette thèse


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