Massimiliano M.
Villone
^{a},
Pasquale
Memmolo
^{bc},
Francesco
Merola
^{b},
Martina
Mugnano
^{b},
Lisa
Miccio
^{b},
Pier Luca
Maffettone
^{ab} and
Pietro
Ferraro
*^{b}
^{a}Dipartimento di Ingegneria Chimica, dei Materiali e della Produzione Industriale, University of Naples “Federico II”, Piazzale Tecchio 80, 80125 Napoli, Italy
^{b}National Research Council of Italy, Institute of Applied Sciences and Intelligent Systems “E. Caianiello”, Via Campi Flegrei 34, Pozzuoli, Naples, Italy. E-mail: pietro.ferraro@cnr.it
^{c}National Research Council of Italy, Institute for Microelectronics and Microsystems, Via Pietro Castellino 111, Naples, Italy

Received
1st September 2017
, Accepted 31st October 2017

First published on 31st October 2017

We report a reliable full-angle tomographic phase microscopy (FA-TPM) method for flowing quasi-spherical cells along microfluidic channels. This method lies in a completely passive optical system, i.e. mechanical scanning or multi-direction probing of the sample is avoided. It exploits the engineered rolling of cells while they are flowing along a microfluidic channel. Here we demonstrate significant progress with respect to the state of the art of in-flow TPM by showing a general extension to cells having almost spherical shapes while they are flowing in suspension. In fact, the adopted strategy allows the accurate retrieval of rotation angles through a theoretical model of the cells' rotation in a dynamic microfluidic flow by matching it with phase-contrast images resulting from holographic reconstructions. So far, the proposed method is the first and the only one that permits to get in-flow TPM by probing the cells with full-angle, achieving accurate 3D refractive index mapping and the simplest optical setup, simultaneously. Proof of concept experiments were performed successfully on human breast adenocarcinoma MCF-7 cells, opening the way for the full characterization of circulating tumor cells (CTCs) in the new paradigm of liquid biopsy.

Finally, we performed quantitative phase-contrast reconstructions of the entire sequence of recorded holograms and implemented a new computational algorithm, described in the next section, to retrieve the 3D orientation of all imaged rolling cells to be used as input for the filtered back-projection tomographic reconstruction algorithm.

It is worth remarking that as the in-flow cell clusters are seen to have their longest dimension almost oriented along the vorticity (where the experimental device is 5 times longer than its height), the hypothesis of negligible confinement still holds for aggregates. From the values of the experimental parameters, we make a preliminary estimation of the Reynolds number Re = ρvD/η, with ρ as the suspending liquid density, v as the flow characteristic velocity, D as the particle characteristic length, and η as the suspending liquid viscosity. Since such estimation yields a value in the order of 10^{–3}, in the system, inertial forces can be considered negligible with respect to viscous ones; thus we can model the fluid motion through the mass and momentum balance equations in the Stokes formulation, which reads as follows:

∇·u = 0 | (1) |

−∇p + η∇^{2}u = 0 | (2) |

For what concerns the boundary conditions, far from the particle we apply the unperturbed shear flow velocity u_{∞} (see above), whereas on the particle boundary ∂P(t), the rigid-body motion condition is imposed in the form

u = ω × r | (3) |

The hydrodynamic torque acting on the particle also needs to be specified. Since inertia is neglected, the particle is torque free, i.e., the total torque T on the spherical surface is zero. In mathematical language

(4) |

The fact that inertia is neglected implies that no initial condition for the velocity field u needs to be specified. The initial orientation of the particle can be chosen by assigning p(t = 0) = p_{0}.

The solution of eqn (1) and (2) with the boundary conditions of eqn (3) and (4) enables to compute the time evolution of the u- and p-fields and of the angular velocity ω, which, in turn, allows to compute the particle orientation dynamics p(t). Such data can then be compared with the experimental cell/cluster orientation. For the solution of the equations presented above, the finite element method is used. The fluid domain is discretized through a mesh of quadratic tetrahedra. On the particle boundary, the mesh aligns with the tetrahedral element faces (quadratic triangles). During the simulations, the elements of the volume mesh progressively deform due to the particle rotation.

Any time the “quality” of the mesh elements in the domain becomes unacceptable, a re-meshing is performed, as technically detailed in ref. 38. At every time step of the simulation, once the angular velocity ω is computed, the orientation of the particle is updated by using the quaternion formalism.^{39} The open-source software Gmsh is used for the design of the computational domain and discretization through a mesh made of quadratic tetrahedra. Moreover, the numerical solution of the mathematical model of the system, given by eqn (1)–(4), is calculated through a non-commercial finite element numerical code written in Fortran.

In particular, Fig. 3b highlights how to use the recovered transversal orientation of the imaged cell in Fig. 3a, named φ, to calculate the corresponding rolling angle ϑ. Actually, we investigated two experimental situations in which tomographic reconstructions of single rolling cells as well as a rolling cluster of cells are obtained. In fact, the proposed FA-TPM method can be easily adapted to recover the rolling angles of a cluster of cells in flow. Considering a number of cells N in a cluster, the 3D pose of the cluster can be calculated by identifying the centroids of each cell and calculating the orientation of the N-sided polygon obtained connecting these centroids. In Fig. 4a, we report the 3D RI reconstruction of the central slice of the single MCF-7 cell reported in Fig. 3, while a different representation of the tomogram, highlighting three ranges of the RI, i.e. 1.334–1.355 (blue), 1.355–1.375 (green) and 1.375–1.400 (red), is shown in Fig. 4b. This last representation aims to identify the membrane region as well as the cell nucleus which is typically characterized by the highest RI values. Moreover, Fig. 4(c) and (d) report the central slice and the three-level tomogram of an MCF-10A cell, used as the healthy cell control in terms of tomographic signatures. Finally, in Fig. 5 we demonstrate the FA-TPM method for clusters of two and three MCF-7 cells. Notice that, also in this case, the experimental transverse orientations of the entire clusters perfectly fit the theoretical cell's transversal orientation dynamics calculated by the simulations (see Fig. 5a and d).

Fig. 4 FA-TPM images for single quasi-spherical cells. (a and c) Tomographic reconstruction of the central slices of MCF-7 and MCF-10A, respectively; (b and d) alternative representations of the tomograms by using three RI ranges as thresholds. The gray cube, with a volume equal to 10 μm^{3}, is used as the 3D scale bar for the tomograms. In movie S1,† the entire FA-TPM process is reported for the case of the single MCF-7 cell. |

Fig. 5 FA-TPM images for clusters of MCF-7 cells. (a and d) The theoretical curves of the normalized cluster orientations vs. the rolling angles are fitted with the estimated cluster orientations and the corresponding central slices of the tomographic reconstructions are obtained and shown in (b and e) for the cases of two and three MCF-7 cells per cluster, respectively. QPMs of the two- and three-cell clusters are reported as subfigures in (a) and (d), respectively. The black dots identify the centroid of each cell and their connection describes a 3D polygon that is used to calculate the orientation of the cluster. (c and f) report the representations of the tomograms by using the same RI ranges used in Fig. 4b. Movie S2 and S3† report the entire FA-TPM process for the two- and three-cell clusters, respectively. The white scale bars in the inset figures of (a) and (d) are 10 μm. The gray cube in (b), with a volume equal to 10 μm^{3}, is used as the 3D scale bar for the tomograms. |

The central slices in Fig. 5b and e show different RI density distributions; thus the representations in Fig. 5c and f report on the fragmentation of the higher RI regions. In particular, the cell at the right of Fig. 5f shows a very small region of RI > 1.375. Definitely, all the information contained in the tomographic reconstructions can be used for cancer diagnosis. Typically, pathologists examine the tissue structure and cellular morphometry, including the cell's size and the nucleus distribution, to ascertain the diagnosis. In the case of tomographic reconstructions, one can calculate all morphometric features of cells, thus helping in the objectivity of the diagnosis.

For this motivation, we evaluated the main biophysical characteristics of the analyzed cells, i.e. the cellular volume and the average RI. Notice that the nucleus size could also be retrieved by tomographic reconstruction by detecting the higher density regions of the cell, i.e. the higher RI distributions. However, a perfect knowledge of the RI range of the nucleus is necessary for its identification. Cellular volumes and average RIs for all cells in Fig. 4 and 5 are reported in Table 1. All results reported in Fig. 3–5 are obtained by using the functions and scripts of the image processing and 3D visualization toolboxes in the MATLAB language. In particular, the 3D renders are obtained by extracting the isosurface data from the tomographic reconstructions (MATLAB function “isosurface”).

Cell. vol. (μm^{3}) |
Avg. RI | |
---|---|---|

Single MCF-10A | 871.1 | 1.361 |

Single MCF-7 | 4849.5 | 1.365 |

Two MCF-7 cluster (L) | 6071.7 | 1.369 |

Two MCF-7 cluster (R) | 6203.1 | 1.368 |

Three MCF-7 cluster (L) | 4989.5 | 1.364 |

Three MCF-7 cluster (C) | 2787.9 | 1.366 |

Three MCF-7 cluster (R) | 3790.5 | 1.357 |

They are in agreement with the values reported in the literature.^{41,42} As expected, the cell at the right of Fig. 5f has the lower RI.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7lc00943g |

This journal is © The Royal Society of Chemistry 2018 |