# 3D-printable portable open-source platform for low-cost lens-less holographic cellular imaging

## Image Reconstruction

Following the hologram acquisition, information retrieval of the cellular objects deposited on the object glass plate is performed numerically. The propagation of light fields is completely described by diffraction theory. Hence it is possible to reconstruct amplitude and phase information of the objects from their interference patterns generated on the camera. At the object location the pattern is focused and reveals the shape and morphology of micro-objects. Numerically, arbitrary planes can be re-focused in retrospect yielding access to volumes with a large number of objects in different heights in z-direction which can be studied with acquiring a single image. The propagation of a wave front towards the detector is described by the Fresnel-Kirchhoff diffraction integral

$${U}_{{\rm{\det }}}(X,Y)=-\,\frac{i}{\lambda }\,\int \,\int \,{U}_{{\rm{in}}}(x,y)t(x,y)\frac{\exp (ik|\overrightarrow{r}-\overrightarrow{R}|)}{|\overrightarrow{r}-\overrightarrow{R}|}dxdy,$$
(1)

where $${U}_{{\rm{\det }}}(X,Y)$$ denotes the wave field on the detector, $${U}_{{\rm{in}}}(x,y)$$ is the incident wave, $$t(x,y)$$ the transmission function of the object, and $$\overrightarrow{r}=(x,y,z)$$ and $$\overrightarrow{R}=(X,Y,0)$$ are two points in the object plane respectively detector plane54. The amplitude and phase distribution of the object can be obtained via an inverse integral

$${U}_{{\rm{obj}}}(x,y)=\frac{i}{\lambda }\,\int \,\int \,{U}_{{\rm{ref}}}(X,Y){H}_{{\rm{\det }}}(X,Y)\frac{\exp (\,-\,ik|\overrightarrow{r}-\overrightarrow{R}|)}{|\overrightarrow{r}-\overrightarrow{R}|}dXdY,$$
(2)

with the reference wave $${U}_{{\rm{ref}}}(X,Y)$$ and the intensity distribution of the hologram in the detector plane $${H}_{{\rm{\det }}}(X,Y)$$. For the reconstruction, an open-source plugin for the open-source microscope software Fiji, see “Methods” section, that implements the angular spectrum estimation for small distances in the order of micrometers up to several centimeters, is employed. The amplitude distribution in the reconstructed plane is calculated using two Fourier transforms

$${U}_{{\rm{obj}}}(z)={ {\mathcal F} }^{-1}\{ {\mathcal F} \{{U}_{{\rm{\det }}}\}\,\exp \,(iz\sqrt{{k}^{2}-\frac{4{\pi }^{2}{n}^{2}}{Np}})\},$$
(3)

where z is the height of the reconstructed plane, k the wave vector, n the index of refraction, N the number of pixels and p the pixel size of the CMOS detector. $${\mathcal F}$$ and $${ {\mathcal F} }^{-1}$$ denote the Fourier Transform and the inverse Fourier Transform. The formalism described above is implemented in two open-source reconstruction software packages, see “Methods” section. In the following, we elaborate and identify two easy to implement reconstruction software packages on a standard desktop computer or potentially also on a mobile phone. HoloPy, a software package for python, allows for hologram reconstruction, but also hologram simulation and scattering calculations. The algorithm to reconstruct point source holograms is based on Fresnel-Kirchhoff diffraction1. It considers a background subtracted hologram, experimental parameters including distances and light source wavelengths and then reconstructs the hologram using two Fourier transforms. Alternatively, hologram fitting is provided by HoloPy where the position of a scatterer is simulated in order to produce the same interference pattern instead of image back-propagation55. For the case of a known number of scatterers, this method can be recommended as it allows to reconstruct spherical or cylindrical object shapes. It is less practicable, however, when arbitrarily shaped micro-objects are of interest, as for example folded RBCs. For the latter case, an open-source plugin56 for Fiji, see “Methods” section, is a possible solution with a user-friendly graphical-user-interface, implemented in Java. Phase, amplitude and intensity distribution of micro-objects can be reconstructed at arbitrary heights in z-direction.

## Micro Particle and Cellular Imaging

In the following, we first capture and image standardized PMSs of diameter (6.5 ± 0.2) μm by both the LD and LED-based lens-less DIHM platform and reconstruct the resulting object properties by the Fiji plugin. Second, we investigate anonymized mature human RBCs as micro-objects in the same manner. Third, we image cell suspension culture TBY2s. The recorded holograms and subsequently reconstructed object planes for multiple PMSs and RBCs are depicted in Fig. 2. Laser-based DIHM hologram (a) and reconstruction (b) is presented next to LED-based DIHM hologram (c) and reconstruction results (d). Both insets depict an isolated PMS or RBCs enlarged to five times its original size. It becomes evident that the LD-based platform provides sharper images where a larger number of interference fringes can be captured per object. These fringes overlap within the hologram resulting in a hologram with more grainy texture as compared to the LED-based results in Fig. 2(a,c). In contrast, the LED-based reconstructed image is considerably more washed out resulting in comparably extended objects. Accordingly, for human RBCs imaged by the LD-based platform, the oval disk shape can clearly be resolved as depicted in Fig. 2(e,f). Several RBCs appear to be tilted in their spatial position, resulting in an elliptical shape. This is in stark contrast to the results obtained by the LED-based platform where information retrieval, for example on the cell morphology, are scarce. However, by the LED-based platform, individual cells can clearly be distinguished, thus exemplifying its potential for individual cell counting or tracking. In order to validate both platform’s imaging capabilities also for extended cellular objects, fast growing plant tobacco TBY2s have been prepared and imaged. TBY2s are employed in various fields of plant biology as a model material and are ideally suited for cellular and molecular analyses57. Corresponding results are depicted in Fig. 3. The recorded holograms and reconstructed object planes for an isolated TBY2s are depicted for LD-based DIHM hologram (a) and reconstruction (b) is presented whereas the hologram, obtained by the LED-based DIHM, is depicted in (c) and the corresponding reconstruction in (d). Both platforms allow to successfully access individual cell segments with a length of 50 μm as well as internal structures including cell nuclei and vacuoles. In Fig. 3(b), a dividing cell undergoing mitosis can be observed. For LED illumination, individual vacuoles are not distinguishable. This is not surprising, as the vacuole membrane thickness is around one order of magnitude smaller than the cell wall thickness of (7–10) nm for plant vacuoles58 as compared to (71–87) nm for tobacco leaf cells walls59. However it is possible to identify the nuclei of several cells. Interestingly, TBY2s infer a more complex interference pattern as compared to both PMSs and RBCs, indicating a stronger absorption and thus increased hologram contrast. We found that Fiji revealed a substantially faster reconstruction as compared to HoloPy. The reconstruction of 10 planes of a digital hologram by the Fiji plugin demands 30 seconds computational time on a regular consumer PC as compared to several minutes by HoloPy. Towards larger volumes, the reconstruction time can theoretically be improved by performing computations on a graphics processing unit11 as demonstrated for live imaging60. In the following section, we aim to quantify the theoretical lateral resolution as well as the spatial resolution experimentally achieved by both the LD-based and LED-based DIHM platforms.

## Resolution

In DIHM, the lateral resolution is bounded by the optical assembly numerical aperture (NA) and the illumination wavelength λ1:

$${\delta }_{{\rm{lat}}}=\frac{\lambda }{2NA},$$
(4)

suggesting a shorter wavelength for a higher lateral resolution. For a digital holographic microscope with a pixel number N, with pixel size p, an illumination wavelength λ and a distance s between object and detector plane, this translates to

$${\delta }_{{\rm{lat}}}=\frac{s\lambda }{Np}.$$
(5)

As described in61, a maximum lateral resolution can be achieved at a distance

$${s}_{{\rm{opt}}}=\frac{f}{1+\frac{f\lambda }{N{p}^{2}}}$$
(6)

between object and detector, with the distance f between light source and detector. This originates from the consideration, that a higher resolution is possible, the closer the object is placed to the detector, however at the same time the interference fringes move closer. Here, sopt denotes the distance at which different interference rings are still resolved by different camera pixels. The axial resolution of the system can be calculated according to

$${\delta }_{{\rm{ax}}}=\frac{\lambda }{2{(NA)}^{2}}=\frac{2{s}^{2}\lambda }{{(Np)}^{2}}.$$
(7)

By equation (6), the theoretical resolution for both developed platforms can be estimated. For p = 1.12 μm, N = 2464, f = 30 mm, λLED = 430 nm and λLD = 405 nm, respectively, lateral resolutions of δLED = 0.92 μm and δLD = 0.87 μm are theoretically possible by the selected, optimum platform design. To evaluate the experimentally achieved resolution, a 1951 United States Air Force (USAF) microscopic imaging test target on a glass microscopic slide serves as a reference object consisting of groups of horizontal and vertical lines with decreasing spatial frequency. The resulting reconstructed amplitude images acquired with the LED (a) and LD (b) setup are depicted in Fig. 4. With LED illumination, element 1 of group 7 is the last resolvable element corresponding to a resolution of 128 line pairs/mm and a line width of 3.91 μm. For LD illumination, element 3 of group 8 is still resolvable, leading to a resolution of 322.5 line pairs/mm and a line width of 1.55 μm. This is mostly a result of the higher temporal and spatial resolution of the laser in comparison to the used LED, as well as the smaller wavelength. The resolving power with LD illumination is thus significantly higher.

Finally, to quantify the achieved image contrast and thus evaluate the capability of the developed DIHM setups, in the following the intensity of a single USAF element is averaged along its axis and plotted (see red and blue rectangle in Fig. 4(a,b). Then, the contrast of consecutive extreme points is calculated by $$K=\frac{{I}_{{\rm{\max }}}-{I}_{{\rm{\min }}}}{{I}_{{\rm{\max }}}+{I}_{{\rm{\min }}}}$$. Each element consists of 5 dark lines which leads to 4 contrast values. The average contrast is then displayed in dependence on the resolution in line pairs/mm, with the error bars ranging from the minimum to the maximum value. The resulting image contrast of different elements of the respective USAF image is depicted in Fig. 4(c,d). Results obtained by both light sources indicate decreasing contrast values with increasing line pairs. The maximum resolvable elements are as shown by the the intensity profile plots for horizontal and vertical elements 3 of group 8 in the inset of Fig. 4(c). For LED, the maximum resolvable element is element 1 of group 7 as depicted in the inset of Fig. 4(d). We compared our contrast values to reported values62, where an unspecified laser or a LED with a central wavelength of 470 nm (25 μm pinhole) was used for image acquisition. The reported value for laser illumination coincides with our results. With LED illumination, the reported value reaches a higher resolution, group 7 element 3 is still resolvable. This could be due to a higher signal-to-noise ratio of the employed camera, the smaller bandwidth of only 10 nm as well as the use of advanced reconstruction algorithms. The achieved spatial resolution of both setups is suited to image and detect individual micro-particles and cellular objects including ensembles of microscopic biological samples. For the LD-based DIHM, sharper interference pattern and a higher number of interference fringes could be captured. Thus, more object information is retrieved yielding a crisper reconstructed image including more details. We attribute this to the LD’s higher second-order temporal coherence as well as spatial coherence. The LED’s spatial coherence could be increased by reducing the pinhole diameter which would require increased LED biasing currents which then require efficient cooling of the LED and thereby increasing the physical dimensions of the setup. Our DIHMs allow to image cellular objects with dimensions smaller than 10 μm as well as microscopy of larger objects with a spatial resolution of 3.91 μm. To ensure maximum possible resolution, a LD-based DIHMs is recommended, whereas for applications where high temporal stability, compact set-up, ruggedness and reduced costs a required, LED-based DIHMs is recommended.

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## Acknowledgements

The authors thank Markus Langhans for assistance with cell cultivation and preparation, Christoph Weber for 3D printing the Raspberry Pi housing, Alexander Rohrbach for helpful discussions as well as W. Elsäßer for support. S.B. acknowledges support by the German Research Foundation (DFG) (389193326). Article processing charge was funded by the DFG and the Open Access Publishing Fund of Technische Universität Darmstadt.

## Author information

S.A., M.v.W. and S.B. developed the experimental idea. S.A. and M.v.W. constructed the first prototype platform, S.A. and M.v.W. performed the experiments, acquired first images of PMSs and cells and analyzed the acquired data. S.A. developed the second prototype platform, performed all experiments and the image reconstruction. S.A. and S.B. prepared the figures and wrote the manuscript. All authors interpreted the data. S.B. initiated and supervised the project.

Correspondence to Stefan Breuer.

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