2.1.

Diffraction Phase Microscope

Our DPM (Fig. 1) is directly inspired from the system published in Ref. 24. This interference microscope combines both off-axis and common-path interferometry techniques to produce fringe patterns [Fig. 1(a)] from which the phase images [Fig. 1(d)] can be reconstructed. The optical system is very compact and stable and does not require an expensive laser source [for instance, a laser-diode (Thorlabs, GmbH, Germany) with a wavelength $\lambda =532\mathrm{nm}$]. It can be coupled to a fast image recording system (e.g., a CMOS camera, Hamamatsu, Japan, ORCA-Flash 4.0) with a $2048\times 2048\text{pixel}$ grid. A Köhler illumination³³ is required to obtain an extremely even illumination and to avoid any perturbation of the sample image by the image of the light source at the image plane (IP) [Fig. 1(f)]. The interferences are obtained by combining a transmission grating (G) ($110\text{grooves}/\mathrm{mm}$), localized at the IP of the microscope, with a spatial filter (Thorlabs, custom-made) placed at the Fourier plane of lens $L_1$ that selects the first-order beam (imaging field) created by the grating and low-pass filters the zeroth-order beam (reference). The two beams are recombined through a second Fourier lens $L_2$. This $4f$ lens system adds a $5.9\times$ magnification ($f_1=25.4\mathrm{mm}$, $f_2=150\mathrm{mm}$). The spatial filter consists of two circular apertures with diameters of 1 and $15\mu \mathrm{m}$ [Fig. 1(f)]. The objective lens (O) $40\times$ (Olympus, SPlan40, $\mathrm{NA}=0.7$) allows a field of view of $75\mu \mathrm{m}$. Prior to image capture, $65\mu \mathrm{L}$ of the solution containing the cells are poured between two glass coverslips glued by a Gene Frame Seal (Thermo Scientific AB-0577). Images are captured on the CMOS camera within the next 15 min of preparation at room temperature $T\sim 22°\mathrm{C}$. A reference image of the background next to the area containing the cell is also recorded for each image.

Fig. 1

Diffraction phase microscopy (DPM) principle: (a) untreated DPM image of a nonadherent immature myeloid cell (TF1-GFP). Scale bar: $5\mu \mathrm{m}$; (b) intensity profile of the section marked with a white dashed line in (a); (c) space-scale representation of the modulus of the wavelet transform (WT) $|{\mathbf{T}}_{\mathbf{\Psi }}[I]|(b,a)$ of the section shown in (b); with a black dashed line we distinguish the wavelet transform modulus maxima (WTMM) profile. Note that we label the horizontal axis with the same variable $x$ to align the two signals (image section and WT), in that representation $b=x$; (d) $\partial \phi (\mathbf{b})/\partial x$ computed for each line of the original phase image with the method illustrated in (c), the gray coding spans the interval [$-2.7$ to 5.4] $\mathrm{rad}/\mu \mathrm{m}$; (e) color-coded optical path difference (OPD) $\mathrm{\Phi }=\phi \lambda /2\pi$ (in nm) computed from the arguments $A_{\mathrm{\Psi }}[I](\mathbf{b},a^{*})$ of the WT at the scale $a^{*}$ corresponding to a maximum of its modulus; and (f) DPM optical setup (see text).

2.2.

Cell Culture

3.1.

Diffraction Phase Microscopy Principle for Cell Imaging

DPM^23–25 allows a fast, nonintrusive, and high-sensitive measurement of the OPD produced by a transparent object embedded inside a homogeneous medium. The DPM optical setup sketched in Fig. 1(f) produces 2-D images with parallel fringe patterns [Fig. 1(a)] corresponding to a periodic modulation of the intensity:

3.2.

Diffraction Phase Microscopy Analysis of Model Spherical Cells

Let us first consider as a theoretical example, a spherical object of radius $R$ with a radial RI function varying from $n_0$ (the outer medium) to $n_C$ (at the center of the sphere):

Fig. 2

OPD and OPD derivative functions for spherical shell models with radial RI profiles: (a) monotonously increasing (or constant) RI profiles $\mathrm{\Delta }n(x,y=y_C,z=z_C)$ (from the border to the center) given by Eq. (5); (b) corresponding $\mathrm{\Phi }(x)$ (green lines) and $\partial \mathrm{\Phi }/\partial x$ (red lines) computed from the profiles in (a), using the first derivative of the Gaussian function as smoothing analysing wavelet (see text); (c) nonmonotonously increasing RI profiles $\mathrm{\Delta }n(x,y=y_C,z=z_C)$ described by Eq. (8); and (d) $\mathrm{\Phi }(x)$ (green lines) and $\partial \mathrm{\Phi }/\partial x$ (red lines) computed as before from the profiles in (c). $\alpha =0$ (respectively 0.25 and 0.5) is plotted with a green solid (respectively dashed-dotted and dotted) line. The underlying sphere shape is reported with a black dashed-dotted line. $x_C=0$, $y_C=0$, and $z_C=0$ correspond to the center of the sphere.

The OPD of this spherical object at position $\mathbf{x}$ with $x^2+y^2\le R^2$ is given by

To mimic an internal variation of the RI, we have constructed another structure with the same outer spherical shape, but containing an internal concentric spherical shell with a higher RI, the boundary of which also varyies smoothly with the radius $r$:

From this last model, we conclude that trusting the OPD isocontours to delineate regions of different RIs from a reconstructed phase image may be totally misleading even though we can still recover some information on the boundary (internal and external) properties with the computation of the local derivative of the OPD function.

3.3.

Multiscale Method Based on the Continuous Two-Dimensional Wavelet Transform

Given that the OPD images are 2-D, computation of the derivatives must be performed along both directions $\mathbf{x}$ and $\mathbf{y}$ and ideally we must also include in the computation the possibility to smooth out the enhanced noise that could come from the derivative procedure. As originally noticed by Mallat and coworkers,^43,44 the 2-D WT^45–47 can be used to revisit Canny’s multiscale edge detector.⁴⁸ The principle of this analysis is to smooth the image by convolving it with a filter and then to compute the gradient of the smoothed image. Let us consider the two wavelets defined respectively as the partial derivatives with respect to $x$ and $y$ of a 2-D smoothing function $\psi (\mathbf{x})$:

In practice, at a given scale $a$, we first compute the 2-D fast Fourier transform (FFT) of ${\mathrm{\Psi }}_1$ and ${\mathrm{\Psi }}_2$ and we multiply these images by the FFT of $\mathrm{\Phi }$: $\widetilde{{\mathrm{\Psi }}_1}·\tilde{\mathrm{\Phi }}$ and $\widetilde{{\mathrm{\Psi }}_2}·\tilde{\mathrm{\Phi }}$ and from the inverse FFT of these products, we get the WTs $T_{{\mathrm{\Psi }}_1}[\mathrm{\Phi }]$ and $T_{{\mathrm{\Psi }}_2}[\mathrm{\Phi }]$. We then identify the so-called WTMM as the points $\mathbf{b}$, where the modulus ${\mathcal{M}}_{\mathrm{\Psi }}[\mathrm{\Phi }](\mathbf{b},a)$ is locally maximum for a given scale $a^{*}$. To increase the resolution of this local maxima detection, we transform the pixelated images into radial representations. To switch from Cartesian to cylindrical geometry, we interpolate the WT argument and modulus on 1440 radial axes crossing the center of the cell with an angular shift of $\delta \theta =4.4{10}^{-3}\mathrm{rad}$. On each of these rotating axes with $\theta$ varying from 0 to $2\pi$, we interpolate each pixelated image along the radial variable $r$ with a spatial resolution $\mathrm{d}r=1.8\mathrm{nm}$. This allows a very acute determination of both the local angle $\theta$ and the argument ${\mathcal{A}}_{\mathrm{\Psi }}$ of the WT vector. From the radial coordinates of the WTMMs, we reconstruct maxima chains as 2-D curves made of a sequence of neighboring points (distant of less than $2r\delta \theta$).

When the maxima chain is circular [Fig. 3(a)] and the WT vector ${\mathbf{T}}_{\mathrm{\Psi }}[\mathrm{\Phi }](\mathbf{b},a)$ is oriented outward [blue arrows in Fig. 3(a)], the argument of the WT is equal to the radial angle $\theta$, $\mathrm{\Delta }\theta ={\mathcal{A}}_{\mathrm{\Psi }}{[\mathrm{\Phi }]}_{\mathrm{WTMM}}-\theta =0$ [Fig. 3(b), blue circles]. If the WT vector is oriented inward [magenta arrows in [Fig. 3(a)], the argument is equal to $\pi$ [Fig. 3(b), magenta circles]. If instead we take an ellipsoidal-shaped maxima chain [Fig. 3(c)], the argument of the WT is no longer a constant function versus the angle $\theta$. Again we consider the two cases of outward WT vector [blue arrows in Fig. 3(c) and blue circles in Fig. 3(d)] and inward WT vector [magenta arrows in Fig. 3(c) and magenta circles in Fig. 3(d)]. It is important to note that when the WTMM chain deviates from a circular contour, the angle difference $\mathrm{\Delta }\theta$ may oscillate versus the radial angle $\theta$, with alternating increasing ($\theta <0.15\pi$) and decreasing ($0.15\pi <\theta <0.85\pi$) intervals in Fig. 3(d). The flatter the shape of the chain, the larger the slope of these curves (in absolute values). The positive slopes of $\mathrm{\Delta }\theta$ versus $\theta$ curves (which may reach vertical lines) correspond to highly curved chains (compared with a circle), whereas the negative slopes correspond to flatter chains (compared with a circle).

Fig. 3

WT argument for a spheroid and an ellipsoid cell model: (a) circular chain model with outward (respectively inward) WT vectors [${\mathbf{T}}_{\mathrm{\Psi }}[\mathrm{\Phi }](\mathbf{r},a)]$; (b) $\mathrm{\Delta }\theta ={\mathcal{A}}_{\mathrm{\Psi }}{[\mathrm{\Phi }]}_{\mathrm{WTMM}}-\theta$ on the WTMM chain line in (a); (c) ellipsoidal chain model with outward and inward wavelet vectors; and (d) $\mathrm{\Delta }\theta ={\mathcal{A}}_{\mathrm{\Psi }}{[\mathrm{\Phi }]}_{\mathrm{WTMM}}-\theta$ on the WTMM chain line in (c). The outward (respectively inward) vectors in (a) and (c) correspond to blue (respectively magenta) circles in (b) and (d).

3.4.

Wavelet-Based Analysis of Model Spherical Cells with Noise

We illustrate the WTMM method for detecting the local maxima chains from noisy data, taking again the previous model of a spherical object with an internal spherical shell of higher RI [Eq. (8)], adding a white noise term to the RI function before computing the OPD image:

Fig. 4

WTMM chain line detection from the OPD of a spherical cell model with noise: (a) OPD $\mathrm{\Phi }(x)$ (green line) computed from model [Eq. (15)] for $\alpha =0.25$ and its WT modulus ${\mathcal{M}}_{\mathrm{\Psi }}[\mathrm{\Phi }](b,a)$ estimated for two scales $a=2$ (red) and 10 (magenta dashed line); (b) two-dimensional (2-D) color-coded image of ${\mathcal{M}}_{\mathrm{\Psi }}[\mathrm{\Phi }]$ for $a=2$, color coded from dark blue to red in the interval [0, 0.001]; (c) local maxima of ${\mathcal{M}}_{\mathrm{\Psi }}[\mathrm{\Phi }]$ for $a=2$; (d) 2-D color-coded representation of $\mathrm{\Phi }$ (in the interval [0, 133 nm]), computed from model [Eq. (15)]; (e) 2-D color-coded image of ${\mathcal{M}}_{\mathrm{\Psi }}[\mathrm{\Phi }]$ for $a=10$, color coded in the interval [0, 0.001]; and (f) local maxima of ${\mathcal{M}}_{\mathrm{\Psi }}[\mathrm{\Phi }]$ for $a=10$.

4.1.

Red Blood Cells

A typical healthy RBC has a maximum and minimum thicknesses of 2.84 and $1.44\mu \mathrm{m}$, respectively, and a diameter $d=7.5\pm 0.5\mu \mathrm{m}$. We recognize in Fig. 5(a) the characteristic OPD “donut” shape of a RBC,^26,55 with a central hole and cylindrical symmetry. This example is particularly interesting to test the performance of WTMM detection method, as shown in Figs. 5(b)–5(d). Figure 5(c) shows the corresponding sections of $\mathrm{\Phi }$ (green) and ${\mathcal{M}}_{\mathrm{\Psi }}[\mathrm{\Phi }]$ (red) along the $x$ direction, taken at the barycenter of the cell. The experimental green profile in Fig. 5(c) is in very good agreement with the biconcave shape (black line) predicted by Eq. (16). This method detects two WTMM chains, one exterior and one interior. Note that the exterior WTMM chain is color coded in hot (red to brown red) colors in Fig. 5(d) as an indicator of larger modulus values than the interior WTMM chain coded in cold (dark blue to blue) colors. The plot of the argument of the WT ${\mathcal{A}}_{\mathrm{\Psi }}[\mathrm{\Phi }]$ versus $\theta$ along each of the two WTMM chains shows a clear separation of the two chains [Fig. 5(e)]. The exterior chain corresponds roughly to the diagonal (red to brown red color) and the interior chain is globally shifted from the diagonal by $\pi$. This shift corresponds simply to the inward direction of the wavelet vector (as already illustrated in Fig. 3). The two plots of the evolution of $\mathrm{\Delta }\theta ={\mathcal{A}}_{\mathrm{\Psi }}[\mathrm{\Phi }]-\theta$ in Fig. 5(f) for each WTMM chain confirm the reversal of the direction of the wavelet vector from the outer to the inner WTMM chain. Indeed this vector gives the direction of the steepest descent of the WT modulus. More interestingly, we note that the fluctuations of $\mathrm{\Delta }\theta$ on the inner chain are much larger than on the outer chain, meaning a more irregular shape (loss of circularity) distribution of the internal part of this cell. Given the predicted minimal thickness of a RBC,^26,52–55 $h_{\min }\sim 1.44\mu \mathrm{m}$, we can use the averaged $\mathrm{\Phi }$ values in the hole of this RBC ${\mathrm{\Phi }}_{\min }=90\pm 5\mathrm{nm}=h_{\min }\mathrm{\Delta }n$ to estimate the RI drop: $\mathrm{\Delta }n=0.063\pm 0.004$, leading to the following estimate of the RBC RI: $n=1.333+0.063=1.396\pm 0.004$.

Fig. 5

WTMM chain detection from the OPD of a living red blood cell (RBC): (a) OPD phase image, color coded from dark blue ($\mathrm{\Phi }=0\mathrm{nm}$) to brown red ($\mathrm{\Phi }=163\mathrm{nm}$); (b) ${\mathcal{M}}_{\mathrm{\Psi }}[\mathrm{\Phi }](\mathbf{b},a)$ for $a=15$, color coded in the interval [0, 0.09]; (c) horizontal sections through the barycenter of the cell of the OPD $\mathrm{\Phi }$ (green line) and of ${\mathcal{M}}_{\mathrm{\Psi }}[\mathrm{\Phi }]$ (red line). The black line corresponds to the biconcave shape predicted by Eq. (16); (d) WTMM chains of the RBC cell shown in (a), color coded according to the value of ${\mathcal{M}}_{\mathrm{\Psi }}[\mathrm{\Phi }]$; (e) plot of the argument ${\mathcal{A}}_{\mathrm{\Psi }}{[\mathrm{\Phi }]}_{\mathrm{WTMM}}$ of the WTMM chains versus $\theta$ [same color coding as in (d)]; and (f) plot of $\mathrm{\Delta }\theta ={\mathcal{A}}_{\mathrm{\Psi }}{[\mathrm{\Phi }]}_{\mathrm{WTMM}}-\theta$ versus $\theta$.

4.2.

Primary Immature Blood Cells

We consider now spherical mononucleated immature blood cells (nonadherent). These CD34+ cells are sorted from the bone marrow or peripheral blood by the CD34 antigen; they are a mixture of hematopoietic stem and progenitor cells with various degrees of maturity. In healthy conditions, these cells remain mostly in the bone marrow. In chronic myeloid leukemia (CML), these immature cells can also be found in the blood. These cells have a rather high nuclear:cytoplasmic ratio ($\mathrm{N}:\mathrm{C}$) in the interphase.^56–58 This ratio indicates the maturity of the cell; for example for immature leukocytes, it may reach $4:1$.⁵⁹ If we assume that the nucleus is a concentric sphere (which can be applied to CD34+ cells) of the cell of radius $R_N$, a $4:1$ $\mathrm{N}:\mathrm{C}$ would give $R_N={(4/5)}^{1/3}·R_C\sim 0.93R_C$ ($R_C$ is the cell radius). If $R_C=4\mu \mathrm{m}$, this would give $R_N=3.7\mu \mathrm{m}$, leaving only a 300-nm distance in between the outer cytoplasmic and the inner nuclear membranes. Such a large nucleus should not be distinguishable from the outer membrane shell in our microscope device, since one fringe produced by the grating is too thick $\sim 400\mathrm{nm}$ (in the scale of the cell). If the $\mathrm{N}:\mathrm{C}$ ratio drops to $3:1$, the radius of the nucleus decreases only by 70 nm, which should also be undetectable with our optical setup. The impact of the nucleus should, therefore, only be visible on the amplitude of the OPD $\mathrm{\Phi }$ and/or its derivative. However, we should be able to detect internal structures of the nuclei on this type of cells. Figure 6 illustrates the WTMM boundary detection on a rounded nonadherent living CD34+ cell. The outer boundary of this spherical cell is detected straightforwardly by the maxima chain with maximum mean $⟨{\mathcal{M}}_{\mathrm{\Psi }}{[\mathrm{\Phi }]}_{\mathrm{WTMM}}⟩$. While the external contour of the cell is very circular [${\mathcal{A}}_{\mathrm{\Psi }}{[\mathrm{\Phi }]}_{\mathrm{WTMM}}$ versus $\theta$ is a straight line in Fig. 6(e)], with few fluctuations [Fig. 6(f)], the internal shape of $\mathrm{\Phi }$ [Fig. 6(a)] presents a single slightly off-centered small bump, leading to the conical shape of the $\mathrm{\Phi }$ profile in Fig. 6(c) (green line). If we fit the outer part of this $\mathrm{\Phi }$ profile by the prediction for a homogeneous sphere with radius $R_C=4.5\pm 0.08\mu \mathrm{m}$ [$\alpha =0$ in Eq. (7)]:

Fig. 6

WTMM method analysis of the OPD of a living domed CD34+ cell: (a) OPD phase image, color coded from dark blue ($\mathrm{\Phi }=0\mathrm{nm}$) to brown red ($\mathrm{\Phi }=320\mathrm{nm}$); (b) ${\mathcal{M}}_{\mathrm{\Psi }}[\mathrm{\Phi }](\mathbf{b},a)$ for $a=15$, color coded in the interval [0, 0.16]; (c) horizontal sections through the barycenter of the cell of the OPD $\mathrm{\Phi }$ (green line) and of ${\mathcal{M}}_{\mathrm{\Psi }}[\mathrm{\Phi }]$ (red line). The black line corresponds to the OPD profile for a homogeneous sphere with radius $4.5\mu \mathrm{m}$ and index $n=1.365$; (d) WTMM chains of the CD34+ cell shown in (a), color coded according to the value of ${\mathcal{M}}_{\mathrm{\Psi }}[\mathrm{\Phi }]$; (e) plot of the argument ${\mathcal{A}}_{\mathrm{\Psi }}{[\mathrm{\Phi }]}_{\mathrm{WTMM}}$ of the WTMM chains versus $\theta$ [same color coding as in (d)]; and (f) plot of $\mathrm{\Delta }\theta ={\mathcal{A}}_{\mathrm{\Psi }}{[\mathrm{\Phi }]}_{\mathrm{WTMM}}-\theta$ versus $\theta$.

The small off-centered dome of the CD34+ cell shown in Fig. 6 corresponds to a higher density zone of the nucleus which is mainly detectable by the WTMM method on its border oriented toward the center of the cell image (computed from the center of mass of the projected shadow of the cell image onto the $\mathbf{x}$ plane, where its gradient is stronger). The outer contour of this small dome is shrouded in the nuclear-extracellular borders and is hardly detectable due to a limited number of fringes per micrometer.

It is interesting to compare this WTMM analysis of a domed CD34+ cell with a flatter CD34+ cell from the same bone marrow sample (Fig. 7). This new cell has an average radius of $5.53\pm 0.18\mu \mathrm{m}$ and is only 23% larger than the previous CD34+ cell (Fig. 6). Its OPD topography is drastically different, since its internal nucleus is more inhomogeneous with a larger set of WTMM chains. One may wonder if this roughening of the internal core of the cell could be accounted for by its larger size, which would facilitate the local maxima detection. Actually, the threshold for local maxima detection is chosen to be small enough ($1\times {10}^{-3}$) to collect all the local events given the wavelet scale $a$. The occurrence of these WTMM chains is really reflecting a modification of the internal structure of the cell. Another way to confirm this transformation is to compare the OPVs: in that latter case, $\mathrm{OPV}=13.3\mu {\mathrm{m}}^3$, which, surprisingly, is very close to the OPV of the previous CD34+ cell. According to Eq. (20), if we divide the OPV by the volume of the ideal sphere with the same radius $R_C=5.53\mu \mathrm{m}$ ($V_S=708\mu {\mathrm{m}}^3$), we get an average RI drop: $\mathrm{\Delta }{n}_C=0.0188\pm 0.005$, leading to a much smaller averaged RI of the cell: $n_C=1.352\pm 0.005$. Since the OPV remains invariant, we know that the total material of the cell is not changed. This means that its apparent surface and its internal structure have changed, possibly due to a local condensation of the intranuclear material. Since we did not stain the nucleus to avoid any alteration of the cell with an external agent that could also modify the interferometric measure, we cannot conclusivelydetermine this possibility. Further experiments could be needed to confirm this hypothesis, in particular by following the transformation of a living cell during a whole division cycle.

Fig. 7

WTMM method analysis of the OPD of a living flattened shape CD34+ cell: (a) OPD phase image, color coded from dark blue ($\mathrm{\Phi }=0\mathrm{nm}$) to brown red ($\mathrm{\Phi }=230\mathrm{nm}$); (b) ${\mathcal{M}}_{\mathrm{\Psi }}[\mathrm{\Phi }](\mathbf{b},a)$ for $a=15$, color coded in the interval [0, 0.092]; (c) horizontal sections through the barycenter of the cell of the OPD $\mathrm{\Phi }$ (green line) and of ${\mathcal{M}}_{\mathrm{\Psi }}[\mathrm{\Phi }]$ (red line). The black line corresponds to the OPD profile for a homogeneous sphere with radius $5.53\mu \mathrm{m}$ and index $n=1.35$; (d) WTMM chains of the CD34+ cell shown in (a), color coded according to the value of ${\mathcal{M}}_{\mathrm{\Psi }}[\mathrm{\Phi }]$; (e) plot of the argument ${\mathcal{A}}_{\mathrm{\Psi }}{[\mathrm{\Phi }]}_{\mathrm{WTMM}}$ of the WTMM chains versus $\theta$ [same color coding as in (d)]; and (f) plot of $\mathrm{\Delta }\theta ={\mathcal{A}}_{\mathrm{\Psi }}{[\mathrm{\Phi }]}_{\mathrm{WTMM}}-\theta$ versus $\theta$.

4.3.

Cell Models for Hematopoietic Stem Cells

Primary CD34+ cells are much more difficult to maintain alive than laboratory cell lines. We used the TF1 cell line as a model of immature CD34+ cells, because it displays clonogenic ability similar to human bone marrow CD34+ cells and is able to differentiate into myeloid lineages.⁶¹ As compared with a wild-type or GFP-transduced TF1 cells, BCR-ABL-transduced (CML oncogene) TF1 cells (TF1-BCR-ABL) increase their transcriptional levels of BCR-ABL and ABL.³⁴ These cells could bring information on the impact of BCR-ABL oncogene transduction on immature cells. Figure 8 shows the results of the WTMM analysis of the OPD of a nontransformed (control) TF1-GFP cell. Immediately, we notice that the size of this cell ($R_C=8.22\pm 0.2\mu \mathrm{m}$) is definitely larger than that of the CD34+ primary cells. This cell looks rather homogeneous in its composition because we do not detect so many WTMM chains. The parametrization of the OPD section [Fig. 8(c)] and the computation of the $\mathrm{OPV}=76\mu \mathrm{m}$ lead to a mean RI, $n_C=1.363\pm 0.004$, which is not very far from the one estimated for the first domed CD34+ cell (Fig. 6). It seems that even if this cell has significantly increased its size compared with primary cells, its optical properties are not very different. In the sampling of these control and transformed cells, we have observed very drastic changes, as illustrated in Fig. 9, for the TF1-GFP-BCR-ABL cell line whose morphology is dramatically different from the TF1-GFP cell shown in Fig. 8. This type of transformation occurs in less than 10% of the cells transduced by the BCR-ABL oncogene, but since it is accompanied by a drastic reorganization of the cytoskeleton, we think it is important to show how the QPM-WTMM method can interpret and quantify this transformation. BCR-ABL has previously been demonstrated to bind actin filaments (F-actin),⁶² one of the major force transducers in cellular adhesion and motility,⁶³ and to induce its redistribution into punctate, juxtanuclear aggregates,⁶⁴ implying a reorganization of the whole cytoskeleton. In Fig. 9, we immediately notice that the cell radius has increased by a factor $\sim 3/2$. The number of chains detected by the WTMM method has also increased by a factor $\sim 5$ (we count only the chains with a length larger than 100 nm, as smaller chain detection may be spoiled by background noise). The mean RI of this TF1-GFP-BCR-ABL cell, computed from the $\mathrm{OPV}/V_S$ ratio [Eq. (20)], is not distinguishable from the mean RI of the previous TF1-GFP cell. When comparing Figs. 8(b) and 8(c) and 9(b) and 9(c), we realize that the difference between the 2-D OPD derivatives of control and cancer cells is higher in the inner cell structures (${\mathcal{M}}_{\mathrm{\Psi }}[\mathrm{\Phi }]=0.011\pm 0.009$ for TF1-GFP cell and $0.035\pm 0.02$ for TF1-GFP-BCR-ABL cell) than along the outer contour (${\mathcal{M}}_{\mathrm{\Psi }}[\mathrm{\Phi }]=0.061\pm 0.047$ for TF1-GFP cell and $0.07\pm 0.0085$ for TF1-GFP-BCR-ABL cell). This internal reorganization of the TF1-GFP-BCR-ABL is also visible on the higher dispersion of the WT arguments [Figs. 8(e) and 8(f) and 9(e) and 9(f)]. If this preliminary discussion on these two cells does not allow us to make general conclusions on the transformation of TF1 cells upon BCR-ABL oncogen transduction, it simply illustrates the fact that the internal structure of these cells may appear very different through QPM analysis.

Fig. 8

WTMM method analysis of the OPD of a living hematopoietic model cell TF1-GFP: (a) OPD phase image, color coded from dark blue ($\mathrm{\Phi }=0\mathrm{nm}$) to brown red ($\mathrm{\Phi }=507\mathrm{nm}$); (b) ${\mathcal{M}}_{\psi }[\mathrm{\Phi }](\mathbf{b},a)$ for $a=15$, color coded in the interval [0, 0.2]; (c) horizontal sections through the barycenter of the cell of the OPD $\mathrm{\Phi }$ (green line) and of ${\mathcal{M}}_{\mathrm{\Psi }}[\mathrm{\Phi }]$ (red line). The black line corresponds to the OPD profile for a homogeneous sphere with radius $8.22\mu \mathrm{m}$ and index $n=1.363$; (d) WTMM chains of the TF1-GFP cell shown in (a), color coded according to the value of ${\mathcal{M}}_{\mathrm{\Psi }}[\mathrm{\Phi }]$; (e) plot of the argument ${\mathcal{A}}_{\mathrm{\Psi }}{[\mathrm{\Phi }]}_{\mathrm{WTMM}}$ of the WTMM chains versus $\theta$ [same color coding as in (d)]; and (f) plot of $\mathrm{\Delta }\theta ={\mathcal{A}}_{\mathrm{\Psi }}{[\mathrm{\Phi }]}_{\mathrm{WTMM}}-\theta$ versus $\theta$.

Fig. 9

WTMM method analysis of the OPD of a living TF1-BCR-ABL cell: (a) OPD phase image, color coded from dark blue ($\mathrm{\Phi }=0\mathrm{nm}$) to brown red ($\mathrm{\Phi }=1050\mathrm{nm}$); (b) ${\mathcal{M}}_{\mathrm{\Psi }}[\mathrm{\Phi }](\mathbf{b},a)$ for $a=15$, color coded in the interval [0, 0.2]; (c) horizontal sections through the barycenter of the cell of the OPD $\mathrm{\Phi }$ (green line) and of ${\mathcal{M}}_{\mathrm{\Psi }}[\mathrm{\Phi }]$ (red line). The black line corresponds to the OPD profile for a homogeneous sphere with radius $13.77\mu \mathrm{m}$ and index $n=1.36$; (d) WTMM chains of the TF1-GFP-BCR-ABL cell shown in (a), color coded according to the value of ${\mathcal{M}}_{\mathrm{\Psi }}[\mathrm{\Phi }]$; (e) plot of the argument ${\mathcal{A}}_{\mathrm{\Psi }}{[\mathrm{\Phi }]}_{\mathrm{WTMM}}$ of the WTMM chains versus $\theta$ [same color coding as in (d)]; and (f) plot of $\mathrm{\Delta }\theta ={\mathcal{A}}_{\mathrm{\Psi }}{[\mathrm{\Phi }]}_{\mathrm{WTMM}}-\theta$ versus $\theta$.

We repeated this analysis on two large sets of TF1-GFP (294) and TF1-GFP-BCR-ABL (216) cells, and we computed the statistical distributions of the mean radius of the outer chain, the OPV, the mean RI drop [computed from Eq. (20)], the angle difference $\mathrm{\Delta }\theta$, the number of chains per cells, and the chain length (Fig. 10). The cell radius distribution is clearly shifted and spread to larger values with $\overline{R_C}=7.2\pm 1.2\mu \mathrm{m}$ for the TF1-GFP cells and $\overline{R_C}=8.2\pm 2.1\mu \mathrm{m}$ for the TF1-GFP-BCR-ABL cells. The OPV follows the same trend with $\overline{\mathrm{OPV}}=56\pm 3\mu {\mathrm{m}}^3$ for the TF1-GFP cells and $\overline{\mathrm{OPV}}=89\pm 5\mu {\mathrm{m}}^3$ for the TF1-GFP-BCR-ABL cells (we use the error of the mean for these quantities). Again, we note that the OPV values are much more dispersed for the transformed cells. The oncogene transduction seems to increase the variability of the cell structural properties. One more surprising result is that the mean RI drop (inner to outer media) of these cells is slightly decreasing from $0.0351\pm 0.0075$ to $0.0331\pm 0.019$, suggesting that the apparent swelling of the TF1-GFP-BCR-ABL cells is not followed by an adapted increase of the intracellular concentration of proteins to keep the mean RI invariant. The distribution of angle differences $\mathrm{\Delta }\theta$ follows a power law distribution $P(\mathrm{\Delta }\theta )\propto {|\mathrm{\Delta }\theta |}^{-\alpha }$, with $\alpha =1$. The fact that the shape of this distribution does not change when switching from control to oncogene-transduced TF1 cell means that the statistics of angular orientation of the maxima chains are not immediately impacted by the cell transformation. In contrast, the number of chains per cell is affected by oncogene transduction. The median value of the two distributions in Fig. 10(e) increases from 21 to 30 chains per cell (considering only the chains of length larger than 100 nm). Again, we observe that the distribution of these chain numbers for TF1-GFP-BCR-ABL cells is more spread out than for control TF1-GFP cells. The distribution of length of these chains (above 100 nm) follows a smoothly decreasing (logarithmic decrease) function for chains smaller than $5\mu \mathrm{m}$, which drops rapidly to zero for larger chains. The peak popping up around $45\mu \mathrm{m}$ corresponds to the outer chain length, and the slight flattening and shifting to higher values of this peak for transduced TF1-GFP-BCR-ABL cells mean that the circumference of these cells increases (as their radius) and is more variable than for nontransformed cells. This observation corroborates our previous remarks on the cell radius distribution.

Fig. 10

Statistical analysis of cell structural parameters from the WTMM analysis of the OPD of control and cancer cell lines: (a) distribution of mean cell radius, estimated from the outermost contour WTMM chain; (b) distributions of OPVs computed from Eq. (19); (c) distributions of RI drops computed from Eq. (20); (d) distribution of $\mathrm{\Delta }\theta$ values; (e) distribution of the numbers of chains per cell; $\overline{N_C}=21\pm 7$ for TF1-GFP cells and $\overline{N_C}=30\pm 9$ for TF1-GFP-BCR-ABL cells; and (f) distribution of chain lengths (${\log }_{10}$ of the length in $\mu \mathrm{m}$). The green (respectively red) lines are obtained from a set of 294 (respectively 216) TF1-GFP (respectively TF1-GFP-BCR-ABL) cells.