Emergence of fractal geometry on the surface of human cervical epithelial cells during progression towards cancer

Primary cell cultures of human cervical epithelial cells were prepared directly from human cervical tissues collected from the transformation zone of the cervix. The cell isolation was performed by a two-stage enzymatic digestion using dispase to remove the epithelium and then trypsin to disperse the individual epithelial cells [22]. All tissues were obtained from the Cooperative Human Tissue Network (informed consent was obtained from patients according to their published guidelines [23]). All experiments were performed in accordance with relevant guidelines and regulations. All experimental protocols used in this work were exempted from the regular Institution Review Board (IRB) review by the IRB committee of Clarkson University. Each tissue was digested for 16 h at 4 °C in dispase. Then, the layer of epithelial cells was removed from the underlying connective tissue by gentle scraping. The sheet of epithelial cells was cut into ∼1 mm² pieces and digested in 0.25% trypsin at 37 °C for 10 min. Trypsin was neutralized by adding 10% fetal bovine serum. The cells were collected by low speed centrifugation. Cultures consisting of ≥95% epithelial cells were maintained in keratinocyte serum-free medium (KSFM, Invitrogen, Carlsbad, CA) which prevents outgrowth of fibroblasts and other stromal cells. Six cancer lines were derived in this way from six cancer patients, and six normal cell stains from six healthy individuals.

Six immortalized (pre-malignant) cell lines were prepared separately in two steps: transfection of normal cervical cells with the complete HPV-16 genome, and subsequent immortalization of the transfected cells. HPV-16 genome was introduced into cultured cervical cells by transfection with plasmid DNA containing the complete HPV-16 genome in combination with the neomycin resistance gene [24]. Subsequently, the medium was changed and cells grew for 24 h before cultures were split 1:3. After 24 h, transfected cells were selected by growth for 2 days in KSFM containing 200 ug ml⁻¹ G418 and used immediately. Only immortalized cells survived after 60-150 population doublings (PDs). Normal cervical cells were used between 20 to 40 PDs, and cancer cell lines were used at 40 to 290 PDs. The slightly higher number of PDs for (pre)cancer cell lines avoids potential confusion because any normal cells, which may contaminate the premalignant and cancer culture dishes, would die out by that number of PDs. Normal HCX-160, 265, 277, 278, 369, 372 strains, precancerous CX-16-2, 16-4, 16-11, 16-12, 16-14, 16-15 and cancerous CXT-2, 3, 5, 6,7, 8 cell lines were analyzed in this work.

To increase lateral resolution of the AFM images, and to mimic the processing of cells in the screening liquid cytology tests, cells were fixed and dried before the imaging with AFM. To avoid drying artifacts, freeze-drying was used. Specifically, all cells were cultured in 60 mm tissue culture dishes. The cells were analysed when the cells reached <50% confluency. The cells were then washed twice with phosphate buffered solution (PBS), and then treated with 4 ml of Karnovsky's fixative overnight at 4 °C. After that treatment, the cells were washed twice with 4 ml of PBS at an interval of 2 h to remove excess Karnovsky's fixative and kept overnight at 4 °C in water. Finally, the cells were washed with 5 ml of DI water twice before freeze-drying. (The cell samples thus prepared can be preserved for several weeks with DI water at 4 °C before freeze-drying them.) After fixing, water was removed by freeze-drying (using Labconco Lyph-Lock 12 freeze dryer). After freeze-drying, cells were preserved in a dessicator. The cells were imaged under AFM directly in culture dishes after not more than 30 min after removing from the dessicator. The dried samples can be preserved at least for several weeks in a desiccator.

Nanoscope™ Dimension 3100 (Bruker/Veeco, Inc., Santa Barbara, CA) atomic force microscope controlled by Nanoscope V controller were used in the present study. AFM cantilever holders for operation in air were employed. To collect the maps of cell adhesion, the HarmoniX mode of AFM operation was utilized. Bruker/Veeco cantilevers for imaging in air were used. HarmoniX standard cantilevers were used.

FEI Phenom SEM was used in this study. For the SEM imaging, the cells in a culture dish were washed twice with PBS, and then treated with 4 ml of Karnovsky's fixative overnight at 4 °C. After that treatment, the cells were washed twice with 4 ml of PBS at an interval of 2 h to remove excess Karnovsky's fixative and kept overnight at 4 °C in water. Finally, the cells were washed with 5 ml of DI water twice before freeze-drying. Freeze-drying was in a Labconco Lyph-Lock 12 freeze dryer. Osmium salt and/or gold coating were used to enhance contrast.

Fractal analysis of AFM images/maps was done with the help of the Fourier analysis. This procedure is equivalent to the standard self-correlation function analysis [25]. Specifically, 2D Fourier magnitude $F\left( u,v \right)$ of the AFM images can be found as follows:

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} F\left( u,v \right)=\sum \limits_{x=0}^{{{N}_{x}}-1} \sum \limits_{y=0}^{{{N}_{y}}-1} z\left( x,y \right){\rm exp} \left[ -i2\pi \right. \\ \quad \quad \quad \ \;\ \;\ \;\;\times \left. \left( ux/{{N}_{x}}+vy/{{N}_{y}} \right) \right]/{{N}_{x}}{{N}_{y}}, \\ \end{array}\end{eqnarray} \tag{ 1 }$$

where z(x,y) is the value of the image at point/pixel (x,y), N_x, N_y are the number of pixels in the x, y directions. The magnitude was then written in polar coordinates and averaged over the polar angles:

$$\begin{eqnarray}&&A(Q)=1/\pi \int \limits_{0}^{\pi } F(Q{\rm cos} \varphi ,Q{\rm sin} \varphi )d\varphi .\end{eqnarray} \tag{ 2 }$$

A is a function of reciprocal space Q (inverse lateral size of the geometrical features on the AFM image, L). Linear behavior of A(Q) in the log-log scale (or $A\left( Q \right)\tilde{\ }{{Q}^{b}})$ is a defining feature of fractals. An important parameter of fractals is the fractal dimensionality α, which can be defined as α = 2 − b. Such a definition of the fractal dimension gives α = 2 for flat and α = 3 for infinitely rough surfaces. To calculate the Fourier spectra of the AFM images, Scanning Probe Image Processor (SPIP) software (Image Metrology A/S, Denmark) was utilized. The fractal dimensionality was found by fitting the obtained spectra with $A\left( Q \right)\tilde{\ }{{Q}^{b}}$ function by using Origin 9.0 software (OriginLabs, Inc.).

It should be noted that, strictly speaking, it is impossible to obtain the exact fractal behavior from the maps observed for any realistic surface. According to the classical definition of fractal, the power dependence of the magnitude which defines the fractal should be observed in the entire geometrical (infinite) scale of Q. However, this is impractical because of natural limitations [26] due to the finite size of data and finite digitalization/ pixelization of any image. It implies that the fractal geometry cannot be even considered for the size of geometrical features L (=Q⁻¹) that are either greater than the size of the recorded image or smaller than the size of each pixel. For example, if the analyzed AFM images of 5 × 5 μm² are recorded with 256 × 256 pixels, the fractal behavior can be analyzed for L ranging between ∼5 μm and 20 nm (∼5 μm/256). So, for fractal behavior discussed in this work, we define it within these limits. One can argue that this might be insufficient to declare fractal behavior. To make our statement rigorous, we just say that we analyzed the fractal behavior (scaling self-affinity) within a limited geometrical scale.

We study the fractal geometry of images/maps obtained by means of atomic force microscopy (AFM). Following methods described in [18], we use the maps of physical adhesion between the AFM probe and cell surface to study the fractal geometry. As was demonstrated in [18], the maps of adhesion gave the highest spatial resolution over the cell surface. Secondly, fractal properties of the maps of adhesion are very robust with respect to the variation in the AFM scanning parameters, see below. Finally, the analysis of the AFM height images did not show any statistically significant segregation between normal and cancer cells.

For comparison, figure 1 demonstrates both AFM adhesion and the scanning electron microscopy (SEM) images of cells. SEM is a more traditional method of studying the cell surface compared to AFM. Therefore, it is instructional to compare SEM and AFM techniques to understand the uniqueness and advantages of AFM. AFM demonstrates a higher 3D resolution compared to SEM. Secondly, as we demonstrate later (figure 6), the maps of adhesion allow detection of the local surface curvature of the order of the molecular scale, ∼1 nm. This is impractical to attain with SEM of the cell surface due to low electron contrast. Therefore, hereafter we utilize AFM working in the adhesion mode to analyze the change of the cell surface during progression towards cancer. It is worth mentioning here that the analysis of the AFM height images did not show any statistically significant change in the fractal behavior because of presumably poor resolution, see figure 6 for detail.

Zoom In Zoom Out Reset image size

Analysing the adhesion maps of individual cells (as described in the Method section), we find that some pre-malignant and cancerous cells demonstrate simple fractal behaviour. The rest of the cells can only be approximated at best as multi-fractal [26, 27]. This is explained in figure 2 in which the magnitude of the adhesion map is plotted versus the reciprocal space. A straight line in the log-log scale of figure 2 is a definitive characteristic of fractal. However, one can see from figure 2 that the magnitude can be approximated as a straight line in the log-log scale on the whole range of scales Q (reciprocal of the range L = 5 μm–20 nm) for only some immortal and malignant cells. For other pre-malignant, most of the malignant, and all normal cells, the magnitude A(Q) plotted in log-log scale can be well-approximated as a broken straight line with the break point at Q ∼ 5 μm⁻¹ (L ∼ 200 nm). Such an object, multifractal is described with two fractal dimensions.

Zoom In Zoom Out Reset image size

Here we introduce a parameter, which we suggest calling 'multi-fractality', which describes the deviation of the observed geometry from a simple fractal. It is equal to the difference between two fractal dimensions below and above a break point. Figure 2 demonstrates that the break point for the presented data can be taken at L ∼ 200 nm (Q ∼ 5 μm⁻¹). In other words, the multi-fractality is defined as the change of the fractal dimension when defined above and below Q = 5 μm⁻¹. The multi-fractality is zero for the ideal (simple) fractal.

It should be noted that the value of magnitude A for the larger geometrical scale is rather noisy. This noise comes from a relatively small number of points available in the AFM maps to calculate A(Q) at those scales. Comparatively, the scales below 200 nm (or Q > 5 μm⁻¹) give relatively low noise to the magnitude A. The fractal dimension of the large scale does not carry any signal statistically discriminating normal, pre-malignant, and malignant cells. Therefore, to exclude this noise, we will use the average fractal dimension for the larger geometrical scale when calculating multi-fractality.

3.3.1. Different scanning/peak force

The same parts of cancer cell were scanned with different scanning (peak) force. Figure S1 shows the multi-fractality, which was calculated from maps collected using peak forces of 20–105 nN. (Note that the starting peak force of 20 nN is the force when HarmoniX force curve can be resolved.) One can see in figure 3 that the multi-fractality is practically independent of the scanning force. The standard deviation of calculated multi-fractality is just 0.01. It should be noted that there may be scanning artifacts when approaching the saturation (maximum) scanning force. Such artifacts could appear due to nonlinearity of the AFM probe motion or inelastic interaction with the cell surface. Therefore, all measurements reported in this work were done in the linear regime with forces 30–70 nN (marked by the dashed lines in figure 3). This broad range can easily be located by any AFM user.

Zoom In Zoom Out Reset image size

3.3.2. Simulation of dependence on AFM probe radius

To check the influence of the AFM probe radius on the multi-fractality of the adhesion maps, we model the adhesion for different probe radii. The adhesion force can be estimated as:

$$\begin{eqnarray}&&{{F}_{Ad}}=\frac{{{R}_{t}}{{R}_{f}}}{{{R}_{t}}+{{R}_{f}}}2\pi W\left( d \right),\end{eqnarray} \tag{ 3 }$$

where R_t is the probe radius, R_f is the radius of curvature of the surface at the contact point, W(d) is the adhesion energy per unit area between a flat contact of two materials of the probe and surface.

Equation (3) is derived based on the assumption that the adhesion force is simply defined by the surface geometry if we exclude the viscoelastic response of the surface. The amount of viscoelastic response indeed seems to be negligible. This conclusion can be made because the viscoelastic response depends on the load force, whereas we do not observe such dependence (see, figure 3). Next, we assume that the chemical interaction of the cell surface and probe does not depend on the location on the surface. This is justified by the fact that W(d) is defined by mostly the van der Walls force (we do not observe specific protein-probe interactions which are revealed in a specific stretching behavior of the retraction AFM force curves [28]) and the capillary force. Van der Waals force remains virtually the same over the cell surface. While adsorbed water definitely plays an important role in the observed adhesion, the multi-fractality did not change noticeably when imaging was done in summer time (relative humidity ∼60%) and winter time (∼30%). This assumption gives a possibility to estimate the change of the multi-fractality for different probe radii.

W(d) was estimated on a flat cell area (when R_f tends to infinity) as ∼1.8 N m⁻¹. With the known W(d) and R_t (10 nm), the distribution of R_f was calculated for a chosen reference image, which had the multi-fractality of −0.13. Based on the observed distribution R_f and known W(d), one can now calculate the adhesion map for different AFM probe radii R_t. All modeled images were processed though SPIP software, and the multi-fractality was plotted as a function of a tip radius multiplier coefficient R_t/10 nm, figure 4. The multi-fractality of the initial image is marked with a grey horizontal line. One can see that multi-fractality of the modeled images is virtually constant for a wide range of the probe radii.

Zoom In Zoom Out Reset image size

3.3.3. On possible contamination of AFM probe

3.3.4. Stability of multi-fractality with respect to image digital resolution.

Here we check how the multi-fractality of the adhesion maps is stable with respect to the imaging taking with different digital (pixel) resolution. We measured the same cell with different resolution (which can be changed through the AFM software). The reason for imaging with a resolution lesser than maximum is the imaging time. It takes almost twenty minutes to record the image with the resolution described in this work. Imaging with the maximum resolution may take almost 10 times longer, which makes it less interesting from a practical point of view. Thus, testing the stability of the multi-fractality, we can also find the minimal resolution that allows reliable separation of normal and cancer cells.

The adhesion maps of the same place on a cancer cell were recorded with resolution of 3.9, 7.8 and 19 nm/pixel (this resolution was used to obtain the reported results). For the two former maps, the image size was 2 μm, whereas it was 10 μm for the 19 nm/pixel map. To find the multi-fractality, the images were processed through SPIP software using the described procedure. Figure 5 shows the spectral magnitudes of the Fourier spectra in log-log scale as a function of reciprocal space Q for all three resolutions. One can see that within the Q range of 5–25 1/μm (marked by vertical lines), which were used for the calculation, the curve slopes are virtually the same. The maximum reciprocal length Q at which cancer and normal cells can be distinguished (Fourier spectra slopes are different) is about 15 1/μm, which corresponds to a resolution of about 33 nm/pixel. Scanning with such resolution will be faster than the scanning used in this work. While it may be useful for the future clinical trials, the resolution used in this work was higher for the sake of broader research interest.

Zoom In Zoom Out Reset image size

3.3.5. Size of geometrical features contributing to the maps of adhesion

Here we demonstrate that difference in multi-fractality can be seen in the maps of adhesion (but not in the height images) because the adhesion maps are effectively probing local geometry, the curvature of the cell surface at each pixel. The most abundant curvature is of the order of 1 nm, see the insert of figure 6(b). The radii are obtained by processing the adhesion maps through equation (3). One can derive the radius of curvature of the surface features at each point of the image,

$$\begin{eqnarray}&&{{R}_{f}}={{F}_{Ad}}{{R}_{t}}/\left( 2\pi {{R}_{t}}W\left( d \right)-{{F}_{Ad}} \right).\end{eqnarray} \tag{ 4 }$$

For comparison, the same figure shows the histogram of the radii derived from the height information recorded in parallel, figure 6(a). One can see that it is impossible to record the radii information in the height mode that would be comparable to the radii derived from the adhesion maps. The most abundant (mean) radius resolved in the adhesion map is 1 nm versus 200–300 nm in the height mode.

To test the statistical behaviour of multi-fractality, we analysed a representative number of cells from several human subjects. Figure 7(a) shows histograms of distribution of multi-fractality calculated for 540 cervical epithelial cells collected from 6 healthy individuals (normal cells), immortalized cells (derived from normal cells of 6 healthy individuals), and 6 cancer patients (cancer cells). One can see a strong correlation between multi-fractality and the cancer progression stage. Positive multi-fractality of normal cells decreases through the stage of immortalization, and changes to negative for cancer cells. One can see that no normal cells reached the level zero of 'ideal fractal' whereas immortalized and cancer cells do. At the same time, one can see that the majority of the cancer cells deviate from the ideal fractal behaviour, and the deviation is opposite to that of the normal cells. Figure 7(b) demonstrates a box-graph of multi-fractality for the normal, immortal, and cancer cells presented in figure 7(a). One can see a clear statistically significant trend (p < 0.0001), the decrease of multi-fractality with the development towards cancer.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

It is known that the risk of cancer typically increases with age. Similarly, malignant phenotype of immortal (precancerous) cells increases with the number of cell divisions (the number of passages in vitro) [29]. Cancerous cells become more aggressive with cancer progression. Therefore, we analyze the change of multi-fractality for a different number of passages (more precisely, population doublings) of cells. Figure 8 shows the dependence of multi-fractality on the number of population doublings (PD) of cells starting from their extraction from tissues. Average values and one-standard deviations are shown. Some cells with close PDs are grouped to make the advance in PD and the total number of cells in each point more uniform. Zero line corresponding to an ideal fractal is also shown. One can see that the multi-fractality crosses zero mostly for immortal cells with high PD and cancer cells with low PD.

Zoom In Zoom Out Reset image size

Here we studied the dependence of multi-fractality as the function of the peak force. The scan force is to be chosen by the AFM operator. Therefore, independence of our results of this parameter is important to check. We do see such impendence in a broad range of scan forces. It was checked as follows. For example, while working in HarmoniX and using standard HarmoniX AFM cantilevers, one can get reliable imaging starting from the peak force of ∼20 nN and until the force reaches ∼105 nN. A range of 30–70 nN is the force needed for a robust imaging for the time needed to get several scans of the cell surface. We observed variation of multi-fractality within the range of 0.02 when changing the peak force within the range of long-time robust imaging (30–70 nN). A typical dependence of multi-fractality on the peak force is shown in figure 3. One can see that the variation of multi-fractality is negligible compared to the standard deviation shown in figure 8.

Variation of the radius of curvature of commercially available AFM probes is typically within 5–20 nm. If we assume no specific chemical interactions between the AFM probe and the cell surface (it can be seen/verified by the lack of specific force signatures [28], which was not observed here), the adhesion force could be simulated as a contact of two spherical surfaces [30], as described by equation (4). Using this approximation, we can recover the distribution of effective radii at each pixel of the cell surface. Changing the radius of the AFM probe, one can now recover a new map of adhesion collected with the new probe. Performing fractal analysis of those new surfaces, we found that the multi-fractality changes within 1% when an initial probe diameter of 10 nm changes within the range of 4–18 nm. Thus, we expect no substantial variation of the multi-fractality for a variety of commercial AFM probes. It should be noted that the probe radius can be verified by inverse imaging of a calibrated grid, or tip-check samples [31].

As one can judge from the weak dependence of the probe radius, small contaminations on the AFM probe that do not change the probe radius substantially are not dangerous for robust measurements of multi-fractality. In the present measurements, no statistically detectable change of multi-fractality was observed when imaging at least 30 cells. (The probes used can still be cleaned with water plasma cleaners and reused.) All large contaminations are easily detectable by a sudden change of the image. The imaging protocol used here allows avoiding such artifacts. Specifically, we image larger than needed areas of 10 × 10 um², and analyze the areas of smaller size that do not have clear artifacts (see the Method section for more detail).

This is easily controlled through the AFM software. We observed no dependence of multi-fractality on the speed of scanning until visible deterioration of the recorded image.

This one is also software controlled. It could be advantageous to find the minimum pixel resolution suitable for robust recovery of the fractal dimension because this would correspond to the faster scanning, image recording. For example, we estimated that for reliable calculation of the multi-fractality value on the cell surface, we need a resolution of at least 20 nm/pixel.

Although humidity can be controlled during the AFM scanning, the role of humidity can be evaluated. As was shown and tested in [18], the fractal dimension of cells, and consequently, multi-fractality, is rather stable up to the relative humidity of 60–70%. The imaging here was done below that level.

Figure 7 shows the statistical behaviour of the multi-fractality for the cells of study. One can see from figure 7(b) that positive multi-fractality of normal cells drops through the stage of immortalization, and changes to negative for cancer cells. One can see from figure 7(a) that no normal cells crossed zero multi-fractality. At the same time, many of the immortal and some of the cancerous cells have multi-fractality within the vicinity of zero. It should be noted that cells, which we call cancerous, were extracted directly from cervical tumours. Tumours can contain multiple cell phenotypes. By preparation, we excluded somatic and normal epithelial cells. However, it is plausible to expect that a population of precancerous cells is still there. It might be plausible to speculate that there is a double peak in the cancer histogram of figure 7(a) that might correspond to malignant and precancerous cell populations.

Figure 7(b) shows the averaged trend of the change of multi-fractality during progression towards cancer. Just by simple interpolation (drawing a straight line) between the mean values of multi-fractality, one can see that the ideal fractal (zero multi-fractality) is reached between the stages of immortalization and malignancy. To test this hypothesis in a more quantitative way, we analyzed multi-fractality not only as a function of phenotype but also its dependence on the population doubling (PD) of cells studied from the moment of their extraction from tissue. It was possible because we prepared the cell lines and strains.

A careful analysis of various AFM scanning parameters was done. One can see from the results that the errors in definition of multi-fractality due to various scanning variations are within single percents. This excludes coincidental correlation between the multi-fractality and cancer development reported here.

Figure 8 demonstrates the cell multi-fractality (the same as presented in figure 7) ordered by the different PDs. One can see from figure 8 that zero multi-fractality is indeed reached at the point when immortal cells turn into cancerous, specifically at the transition between immortal cells of high PD and cancerous cells with low PD. Since risk of cancer development typically increases with age, the number of PD (cell age in vitro) correlates with the cancer progression in precancerous (immortal) cells [29]. The dysplastic phenotype was shown to increase with the number of passages [32]. Cancerous cells become more aggressive with cancer progression. Therefore, this result supports the conclusion that zero multi-fractality is reached at the moment precancerous cells (immortal cells with high PD) just turn into cancerous (of low PDs). However, the deviation from simple fractal increases with the increase of PD.

Before interpreting the above results, it is useful to analyze bio-physical and -chemical reasons of the observed changes in multi-fractality. As was suggested in the literature [10, 12], generally speaking, imbalance of various biochemical reactions, which is typically associated with cancer, could result in chaos and the appearance of fractal geometry of cancer. However, a specific path is yet to be found. Since the AFM measures the physical interaction between the cell surface and AFM probe, one has to expect physical changes of the surfaces of cells during progression towards cancer. Such surface difference was reported when measuring forces and adhesion on viable epithelial cervical cells [33–36]. It was shown that the difference could be attributed to the variations in pericellular coating, or cellular brush. The cellular brush, which is measured by AFM, consists of microvilli, membrane 'wrinkles' (microridges), glycocalyx, etc. To amplify, we are speaking not about chemical changes in the brush but rather geometrical alteration of the cell surface.

Because AFM measures the physical adhesion of the AFM probe to the cell surface, the difference reported here should come from the difference in the organization of the pericellular brush at the nanometer level. This should then come from the difference in intrinsic membrane roughness (cortical layer of cytoskeleton), microvilli, and/or glycocalyx brush on the cell surface. The reported results allow estimation that the difference should be seen at the scales less than 200 nm (and most probably down to 1 nm, see figure 6 showing the main contribution to the adhesion maps coming from the local curvatures of the order of 1 nm). It will be a task for the future to find the specific pathway influencing the changes in the pericellular brush at that scale.

As we mentioned in the introduction, it was found that the fractal patterns are typically formed under far-from-equilibrium conditions, or emerge from chaos. Thus, the results reported here could be interpreted in favour of chaotic or far-from-equilibrium imbalance during transformation of precancerous cells towards cancer. However, the results vote against cancer itself as a chaotic or far-from-equilibrium imbalance in cells, because the multi-fractality substantially deviates from zero with cancer progression (increasing with the increase of the population doublings of malignant cells). This result will be useful for the development of models describing cancer development and progression. It may also bring a new means of attack on cancer, for example, by searching the points of instability [5] that influence chaotic development the most. Biologists could start identifying those instabilities in the pathways connected to formation of the pericellular coating. Besides the fundamental interest, the obtained results could be used to measure the degree of possible progression towards cancer.