2.1.

Capillary Morphogenesis Model

A model for capillary morphogenesis as first described by Nehls and Drenckhahn was implemented in Ref. 41. Primary human umbilical vein endothelial cells (HUVECs) were grown on microcarrier beads imbedded in a crosslinked fibrin hydrogel and given pro-angiogenic signaling factors naturally released by fibroblast cells cultured on top of the gel. HUVECs were first expanded in cell culture flasks to passage number three and then cultured on the surface of 150 to 210 μm diameter carrier beads (Cytodex). Beads were immersed in a $2.5\mathrm{mg}/\mathrm{ml}$ solution of fibrinogen (bovine, Sigma) in EBM-2 media (Lonza) at approximately $100\text{beads}/\mathrm{ml}$ solution. The solution was then pipetted into a 35 mm diameter Petri dish with type 1 glass bottom (MatTek) containing 20 μl of thrombin (Sigma). A clotting cascade forms a fibrin gel thus imbedding the carrier beads within a 3-D extracellular matrix. Once the gel is formed, normal human lung fibroblast cells (NHLFs) grown out to passage five were seeded onto a Transwell container (Corning) that is placed on top of the gel allowing signaling factors released by the fibroblasts to diffuse down into the gel while keeping the fibroblasts mechanically separated from the gel (Fig. 1). Next, 2 ml of EGM-2 media (Lonza) was added to each sample with media exchanged every other day. Capillary sprouting from the carrier beads begins by day two with large interconnected capillary networks forming by day seven. Imaging was performed on day seven.

Fig. 1

Schematic of our tissue culture system and lens-free computational imaging apparatus. NHLF fibroblasts are cultured on the top surface of a Transwell insert, which is placed within a 35 mm glass-bottom Petri dish. HUVECs were first cultured onto microcarrier beads, which were then embedded within a crosslinked fibrin hydrogel. Soluble signals released from the fibroblasts stimulate the HUVECs to spontaneously form capillaries. The insert was removed prior to imaging. A partially coherent fiber-coupled LED light source illuminates the Petri dish, which is placed above the CMOS chip to record the holographic image of the sample over a large field-of-view and extended depth-of-field; $z_1$ is the distance between the light source and the object; $z_2$ is the distance between the objects (capillaries) and the active area of the detector, which is changed by placing or removing a glass coverslip underneath the sample. A computer reconstructs lens-free images according to described algorithms.

2.2.

Lens-free On-Chip Imaging Setup

The lens-free on-chip imaging setup, shown in Fig. 1, is composed of a partially coherent light source and an opto-electronic sensor array (complementary metal-oxide-semiconductor (CMOS) Model #MT9P031, Micron Technology; pixel size: 2.2 μm, 5 mega pixels). The sample is placed directly on the top of the sensor chip such that the distance, $z_2$, between the objects and the active area of the detector is $\sim 1$ to 2 mm. The sample is illuminated using a near infrared light emitting diode (LED, $\lambda =950\mathrm{nm}$) that is butt-coupled to a multimode optical fiber that has a core-diameter of 100 μm. The distance, $z_1$, between the light source and the sample is typically $\sim 10\mathrm{cm}$, and its placement does not require sensitive alignment.

An infrared LED was selected in our imaging experiments since it was found to reduce the background noise from multiple scattering within the fibrin gel, and this also minimizes the phase-wrapping problems in the reconstructed lens-free phase images as phase delay is inversely proportional to the wavelength. Since the light impinging on the sample is partially coherent both temporally and spatially,^13,14 the unperturbed portion of the illumination interferes with the waves scattered by the objects. The sensor records this interference pattern, i.e., an in-line hologram of the objects placed on the chip. Owing to its unique geometrical configuration where $z_1\gg z_2$, in-line holograms are recorded with unit magnification over a large imaging FOV that equals the active area of the sensor-array, in this case $24{\mathrm{mm}}^2$ (Fig. 2), which can further increase to e.g., $>15{\mathrm{cm}}^2$ with a different choice of sensor-array.¹⁶

Fig. 2

A full FOV hologram of a sample placed on the sensor array. The unit magnification holographic recording scheme permits imaging and monitoring capillaries over a large FOV of $24{\mathrm{mm}}^2$, that equals the active area of the CMOS detector array.

2.3.

Digital Holographic Reconstruction

Lens-free images are reconstructed using digital beam propagation techniques based on the angular spectrum approach.⁴² In this approach, lens-free images are convolved with the impulse response of free space propagation. This operation is done in the Fourier domain, by multiplying the 2D Fourier transform of lens-free images with the transfer function of free space propagation, which can be expressed as:⁴²

When the above-mentioned digital beam-propagation is performed on the measured lens-free holograms, the reconstructed images exhibit artifacts due to the “twin-image” noise. Stated differently, since imaging sensors are only sensitive to the intensity of an optical field, the initial phase of the complex field at the hologram plane, which is unknown, is assumed to be zero before digital propagation, giving rise to artifacts in the reconstructed image. In order to obtain a refined reconstructed image that does not suffer from this twin-image noise, phase recovery algorithms should be used to retrieve the unknown phase at the hologram plane. One way of achieving this is to use a size-constrained iterative phase recovery algorithm,¹⁴ where the object size is used as additional information (i.e., size-constraint) in the reconstruction process. The object size can generally be estimated from the initial reconstruction despite the contamination due to the twin-image noise, as the boundaries of the objects can still be determined. For lens-free imaging of capillaries, however, the complex morphology of the sample makes it challenging to estimate the capillary shape and size, hampering the use of size-constrained phase recovery algorithms. As a result, here we utilized an alternative phase-recovery algorithm, multi-height phase recovery (MHPR),^24,25,43 which does “not” require the knowledge of the object size or shape as additional information, but instead uses multiple intensity measurements.

2.4.

Multi-Height Phase Recovery

In this iterative phase-retrieval technique, multiple in-line holograms (i.e., intensity measurements) are recorded for the same object, where each measurement is performed at a different $z_2$ distance. For lens-free imaging of capillaries, two intensity measurements were sufficient to effectively retrieve the phase of the optical field at the hologram plane. The change in $z_2$ distance is achieved by placing or replacing a coverslip with a thickness of, for example, $\sim 150\mu \text{m}$ as a spacer between the sample and the sensor. This operation may also cause a slight translation and rotation of the sample with respect to the sensor array between two successive lens-free image acquisitions. Therefore, the two measured lens-free holograms are first registered to each other in order to compensate for any translational or rotational motion. The exact $z_2$ distances do not need to be known a priori, since an auto-focus algorithm¹⁷ as described below is utilized to estimate this parameter. After image registration, the MHPR algorithm is invoked to retrieve the missing phase of the sample holograms. The algorithm works by propagating the measured fields back and forth between the two planes of measurement. At each iteration, the algorithm enforces the recorded (i.e., measured) amplitude at the corresponding height (i.e., $z_2$), while keeping the updated phase for the next iteration. This way, the missing phase is retrieved in $\sim 10$ iterations without modifying the measured amplitude values, resulting in a refined lens-free image where the twin-image noise is significantly suppressed.

2.5.

Digital Autofocusing Algorithm

An important attribute of lens-free holography is that it enables imaging of samples over a long depth-of-field, as long as the holograms of objects at large distances above the sensor (e.g., 1 to 5 mm) have sufficient signal-to-noise ratio.^21,22 This ability is particularly useful for imaging capillary morphogenesis due to the inherent 3-D nature of the sample. Imaging an extended depth-of-field is achieved by reconstructing the holograms at different depths-of-interest, which is equivalent to focusing a conventional microscope objective lens at different depths. However, digitally selecting the object-to-detector distance ($z_2$), which maximizes the contrast and the signal-to-noise ratio of the object, can be a tedious task when imaging objects over a large volume. Towards this end, we implemented an autofocus algorithm to automatically determine the best plane of focus for imaging different capillaries across the sample volume.¹⁷

This autofocus algorithm is based on the fact that the edges of the object should be the sharpest at the plane of best focus. To estimate the sharpness of the edges, first a Sobel operator is used to detect edges in the vertical and horizontal directions. Second, the edge images are combined by using the two-norm (i.e., the square root of the sum of the squares). Third, the variance of the resultant image is calculated, where a high variance indicates existence of sharp edges. Therefore our autofocus algorithm scans several $z_2$ distances spaced by e.g., 1 μm distances in order to find the $z_2$ distance with the maximum edge variance, i.e., the best focus. It should be noted that this auto-focus algorithm is invoked twice in order to estimate the $z_2$ distances for both measurement planes. For a typical sample, the difference in the $z_2$ distances of the two lens-free holograms does not exceed 50 μm. Consequently, once the $z_2$ distance for one of the lens-free holograms is determined, the search space for the $z_2$ distance of the second hologram is rather small, achieving fast convergence.