Phase-Retrieved Tomography enables Mesoscopic imaging of Opaque Tumor Spheroids

We present a new Phase-Retrieved Tomography (PRT) method to radically improve mesoscopic imaging at regimes beyond one transport mean-free-path and achieve high resolution, uniformly throughout the volume of opaque samples. The method exploits multi-view acquisition in a hybrid Selective Plane Illumination Microscope (SPIM) and Optical Projection Tomography (OPT) setup and a three-dimensional Gerchberg-Saxton phase-retrieval algorithm applied in 3D through the autocorrelation sinogram. We have successfully applied this innovative protocol to image optically dense 3D cell cultures in the form of tumor spheroids, highly versatile models to study cancer behavior and response to chemotherapy. We have thus achieved a significant improvement of resolution in depths not yet accessible with the currently used methods in SPIM/OPT, while overcoming all registration and alignment problems inherent to these techniques.


Experimental SPIM-OPT Setup:
The images presented in this work were acquired with a combined SPIM/OPT setup, shown in For SPIM various continuous wave diode lasers are being used. In this work the output of a 635nm diode laser is used. The laser beam (colored with blue in Sup. Fig. 1) is initially expanded (BE) and then is directed to a cylindrical achromat doublet (CL) through which it is focused in a horizontal line on the corner mirror (CM). After the mirror the formed light sheet is imaged through a 2x telescope (T) to the back focal plane of the illumination objective (IO) (Mitutoyo, Plan Apo, 10x/0.28, WD=34.0mm). The telescope is placed in such a way that two conjugate planes are formed on the mirror and the back focal plane of the objective, for a better and easier adjustment of the light sheet. The formed light sheet is established orthogonally to the detection axis, intersecting with the focal plane of the detection objective.
The emitted light (colored with green in Sup. Fig. 1) is collected by a second 10x/0.28 detector objective (DO) (Plan Apo, Mitutoyo, Japan) and is projected through an apochromatic doublet tube lens (TL) (ITL200, Thorlabs) on a thermoelectrically cooled, electron multiplying CCD camera (1004x1002 pixels sensor, pixelsize: 8μm) (Ixon DV885, ANDOR Technology). Right after the objective an iris (ID) is placed in order to control the NA of the detection and thus define the depth of field and a filter wheel with appropriate fluorescence filters. For the DRAQ7 emission a 650 nm long-pass filter is used to acquire the signal.
The sample is stabilized inside a FEP tube with a solidifying agent (CyGel), and then is mounted on the sample holder which has 4 degrees of freedom. Four motorized software controlled stages allow the micrometric translation along x, y, and z-axes and rotation around the vertical y-axis.
For refractive index matching, the sample is inserted inside a chamber filled with water.
In case wide field tomographic imaging is required instead of selective plane excitation and detection, the white LED Lamp is used to perform Optical Projection Tomography.

Phase-Retrieved Tomography -Flow Chart protocol
The protocol proposed for correctly reinterpreting misaligned datasets is schematically described in the flowchart of Supplementary Figure 2. The sample is placed into the setup described in the previous paragraph, and although the SPIM measurements are not strictly required, this reduces the out of focus contribution of the Average Intensity Projection (AIP). In the diagram, the Direct Projection block could be fed with different kind of projections obtained at a certain angle : parallel bright field projections, AIP of fluorescence excited by SPIM or simple fluorescence signal exciting the whole sample. Speckle patterns produced by a sample hidden behind a scattering curtain are also a feasible choice, since it has been proved experimentally 1, 2 and numerically 3, 4, that the camera image of both the speckle pattern and the object itself share the same autocorrelation features. Further considerations about the possibility of using speckle patterns are discussed in the last section of this document. For each acquisition, we calculate the twodimensional autocorrelation of the unfiltered camera image, stacking them to obtain the autocorrelation sinogram. After the whole rotation is accomplished, it is possible to backproject the autocorrelation sinogram, in order to obtain the three-dimensional autocorrelation of the whole object. It is worth noticing that several techniques can be used to backproject the sinogram 5, allowing for comparative studies of different approaches. The reconstructed three-dimensional autocorrelation then is used as starting point for a phase retrieval problem, which results to the reconstruction of the object.

Phase-Retrieved Tomography -Theoretical Support
Our proposed method is based on the calculation of the three-dimensional autocorrelation of the investigated specimen and the use of a three-dimensional phase retrieval to form the final reconstruction, as shown in previous works for two dimensions 6. Calculating a 3D autocorrelation by the Radon transform of 2D camera autocorrelations requires that the autocorrelation of the projection of the specimen is equal to the projection of its three-dimensional autocorrelation at each angle. In the following section, we prove that the two quantities are identical. For simplicity, we treat the 2D case considering a two-dimensional object to be reconstructed starting from its 1D projections.
Considering , as the spatial coordinates in which the specimen exists and , as their respective translational coordinates, let us define the following quantities: Object autocorrelation: Projection of the Object (at angle = 0): Projection of the Autocorrelation ( = 0): The object is finite and limited in space, defined in a closed region , ∈ R, and every integral considered is defined up to the region boundaries. In this representation 2 is the 1D camera detection at angle = 0.
To be able to reconstruct the object autocorrelation by calculating the autocorrelations of the object projections literally means that the following relation must be satisfied: Explicitly we can write: Both terms are integrated along y, so we can compare the arguments of the integrals: Now we focus on the integration along the translation It is possible however, to make the following consideration regarding the argument in brackets; for finite objects in space, a linear translation does not influence their definite integral, i.e. the projection is preserved for translations along the axis perpendicular to the detection axis. This means that integrating in the translation η, projects the object into its perpendicular axis, eliminating the dependence in (one coordinate could be seen as the translation of the other) and hence: This implies that: Which exactly proves our initial claim. Theoretically then, it is possible to calculate the autocorrelation of the object by the Inverse Radon transformation of the autocorrelation of the projections.

Phase-Retrieved Tomography -Numerical Validation
To test the validity of the proposed method, we performed a numerical validation using a three dimensional Shepp-Logan phantom, a commonly used model for testing the performance of reconstruction algorithms. We used a freely available tool for the generation of such phantom in the Matlab environment 7, creating a cubic volume of 128 pixels per side. The whole process is shown in Supplementary Figure 3 Since we have a better estimation for the three-dimensional object's autocorrelation rather than the object itself, we can exploit this information with a three-dimensional implementation of a phase retrieval algorithm, in order to accomplish the reconstruction and retrieve the real object.
As can be seen in panel I, the reconstruction obtained with Phase-Retrieved Tomography (PRT) matches the original object (panel A) even if the dataset was perturbed with strong vibrational noise.
Finally, to monitor the convergence of the reconstruction it is possible to calculate the recovery error, defined by the Euclidean distance between the autocorrelation and the autocorrelation of the retrieved object at each step (Supplementary Figure 4). In an ideal case, when the phase is fully retrieved, this quantity is minimized and the two autocorrelations converge. It is possible to notice that already after 5000 steps of HIO the solution is already stable, while the eventual addition of 1000 steps of Error Reduction (ER) 6 does not alter significantly the results. In any case, we prefer to do not use ER steps due to the fact that the algorithm might force sharpness in the reconstructed volume.

Phase-Retrieved Tomography -Resolution and Cross sections
In this section we present the results obtained with PRT in comparison to that of normal OPT-SPIM reconstructions. The results of classical OPT-SPIM reconstruction, corrected for the not centered axis of rotation, lead to a misaligned recontruction affected by rotational noise (Supp. Gerchberg-Saxton algorithm 14 used to retrieve its Fourier phase, allowing the reconstruction of the hidden object. Although mathematically the phase retrieval process works particularly well at every dimensionality 15, except for the lack of uniqueness in 1D problems, for optical imaging purposes (to the best of our knowledge) it has been used only in 2D implementations.
In this scenario, it is worth taking into account another key aspect of this work. Because of its design, the PRT protocol can potentially be implemented for imaging hidden three-dimensional specimens behind scattering curtains or around corners. In principle, in fact, PRT can tackle the current lack of techniques for high-resolution 3D imaging of hidden objects by exploiting speckle pattern resulting from the object positioned at different angles rather than single projections. The speckle pattern sequence generated by a specimen fluorescing behind a curtain that encloses it still contain the autocorrelation information needed to correctly perform the reconstruction.
Preliminary numerical studies 3 4, in fact, already shown the possibility to correctly calculate the autocorrelation sinogram of a three-dimensional object hidden behind a random phase scrambling media. Such an A-sinogram, is identical to the A-sinogram of the object itself and can be processed via PRT methods to correctly retrieve the hidden three-dimensional object. We are already facing the challenges of a 3D hidden imaging reconstruction and currently we are on the process of testing it in experimental measurements.