Optomechanical microrheology of single adherent cancer cells

There is a close relationship between the mechanical properties of cells and their physiological function. Non-invasive measurements of the physical properties of cells, especially of adherent cells, are challenging to perform. Through a non-contact optical interferometric technique, we measure and combine the phase, amplitude, and frequency of vibrating silicon pedestal micromechanical resonant sensors to quantify the “loss tangent” of individual adherent human colon cancer cells (HT-29). The loss tangent, a dimensionless ratio of viscoelastic energy loss and energy storage — a measure of the viscoelasticity of soft materials, obtained through an optical path length model, was found to be 1.88 ± 0.08 for live cells and 4.32 ± 0.13 for fixed cells, revealing significant changes (p < 0.001) in mechanical properties associated with estimated nanoscale cell membrane fluctuations of 3.86 ± 0.2 nm for live cells and 2.87 ± 0.1 nm for fixed cells. By combining these values with the corresponding two-degree-of-freedom Kelvin-Voigt model, we obtain the elastic stiffness and viscous loss associated with each individual cell rather than estimations from a population. The technique is unique as it decouples the heterogeneity of individual cells in our population and further refines the viscoelastic solution space.


INTRODUCTION
The mechanical response of soft materials is generally reported as the frequency-dependent viscoelastic behavior determined using rheological techniques. However, performing rheology on living cells is particularly challenging due to invasiveness of applied forces and choosing appropriate time-dependent loading conditions to probe viscous material properties. 1 Recently, it has been shown that the mechanical properties of cells are directly correlated with biological processes, such as in cancer [2][3][4] and blood diseases. 5,6 For example, normal cells are known to be stiffer than their cancerous counterparts. In practice, these mechanical biomarkers have the potential to contribute to early detection and diagnostic techniques.
In the past few years, there have been a variety of experimental techniques and models to measure and describe the rheological behavior of cells and tissue under different physiological conditions. These quantitative experimental techniques can be divided into two main categories: passive and active. Passive techniques involve the measurement of thermal fluctuations of embedded particles. 7 Approaches for active microrheology generally involve the application of a local force and the measurement of material response. Some of these prominent techniques for active microrheology measurements include atomic force microscopy, 8 cell poking, 9 shear flow cytometry, 10 microplates, [11][12][13] optical tweezers, [14][15][16] optical stretchers, 17,18 magnetic tweezers, 19,20 and optomechanical techniques. 21 These techniques probe the behavior of cells at different length scales and timescales and employ different stress-strain magnitudes and behaviors.
In this paper, we present a technique for measuring the viscoelastic properties of individual adherent cells using a microresonator sensor. This technique simply combines the simultaneous measurement of the resonant frequency shift, 22 amplitude change, 23 and mechanically driven optical path phase 21 within a cell using a microresonator to more uniquely determine the viscoelastic properties of a single cell without requiring averaging over a population of cells. This overcomes limitations of our previous techniques that depended on additional measurements of the average cell height, reference measurements, and other ensemble average cell group measurements. 23 Here, we measure the loss tangent of adherent colon cancer (HT-29) cells, a descriptor of the relative viscoelastic properties of a cell. Using this technique, we are also able to quantify the nanoscale membrane fluctuations (height oscillations) of individual live HT-29 cells and their stiffer fixed counterparts. Figure 1(a) depicts an increase in the membrane fluctuation amplitude of the live cells relative to fixed cells for individual cells at the resonant frequency of each respective sensor. Qualitatively, this indicates that the live cells are more compliant (softer). To quantitatively illustrate this trend, we fit a normal distribution to our membrane fluctuation data and compare as shown in Fig. 1(b). Live values of membrane fluctuations show a narrower distribution with a mean and standard deviation of 3.86 6 0.2 nm compared to fixed cells with 2.87 6 0.1 nm. A paired t-test value (p < 0.001) further confirmed the statistically high significance of the observed fluctuation decrease after fixation. These experimental findings agree well with previous studies on the increase in cell stiffness and viscosity after fixation. 3 Other cell vibration and membrane fluctuation measurements on red blood cells [24][25][26] show significantly higher amplitude values; however, this difference in the amplitude can be attributed to the reasons of integrininduced tension 27 in adherent cancer cells. 26 In addition, resonant sensors in other methods were more compliant (0.01-0.05 N/m) as compared to our resonant sensors ($19.4 N/m), thus supporting larger deflections of the sensor and the cell membrane. 28 The amplitude ratio is another key variable in determining the individual cell loss tangent. Figure 1 shows a plot of the observed amplitude ratio (DA) of live and fixed cells, where there is an increase in the amplitude ratio for the fixed cells over live cells, matching well with previous studies. 23 A higher amplitude ratio for fixed cells implies that a relative increase in the refractive index (and viscoelasticity) exceeds the relative change in the actual cell-to-sensor amplitude ratio. We can further exemplify the amplitude change for each cell by plotting the live amplitude ratio against the fixed amplitude ratio [ Fig. 1(d)] and comparing with a slope of unity (dotted line). Figure 1(e) shows a comparative difference between the loss tangent of live and fixed cells at resonance ($32.0 kHz and $31.8 kHz, respectively), where values above unity indicate a more dissipative, viscous-like state and values below unity indicate an elastic-like state shown graphically in Fig. 1(f). Our observed data show that all cells tend toward a more viscous state with a bias for fixed cells being more viscous than live cells. Fixed cells show an increase in the loss tangent over live cells, indicating that fixed cells exhibit more energy dissipated (viscous) [tanðdÞ > 1] than energy stored (elastic) within the resonant frequency regime. The average loss tangent values for live and fixed cells are 1.876 6 0.075 and 4.316 6 0.126, respectively.

RESULTS
While a large loss tangent identifies a more viscous (or fluid-like) behavior, it does not automatically mean that the stiffness of the cell is low, rather it means that viscosity dominates over elasticity within our regime of vibration. Our derived tensile storage and loss moduli solution pair is used to obtain the apparent elasticity (EÞ and viscosity (lÞ for each cell as shown in Fig. 2, where E 0 $ E and E 00 ¼ lx (see supplementary material). Figure 2 compiles the result of our rheological investigation for the 14 different cells. The loss tangent, tan d ð Þ ; does not assume a particular cell-sensor geometry. Thus, it is independent of the sensor material property, and the details of contact mechanics provide a quantitative measure of the overall rheological behavior of the sample. Figure 2(a-i-ii) show that there is no correlation between the loss tangent and the apparent mass for (i) live and (ii) fixed cells (R 2 ¼ 0.232 and 0.083, respectively) to further demonstrate that cell viscosity (c 2 ) also affects the dynamics of the system, in addition to cell elastic stiffness (k 2 ). For consistency, we measured the fixed cell mass values (frequency shift) and compared with the same cells before fixation.
The computed apparent elasticity (storage modulus) and viscosity are shown in Figs. 2(b) and 2(c). Live cells showed a lower elasticity of 259 6 44 Pa, whereas fixed cells showed a higher elasticity of 712 6 40 Pa. Our reported elasticity range for live cells (164-310 Pa) and higher fixed cell values agree well with other methods. [29][30][31] The cell-to-cell variability in viscoelastic moduli elucidates the phenotypic differences in each cell and shows a lack of synchrony in their individual growth cycles despite being the same cell type and probed at the same measurement time. In a previous similar study, 23 the maximum likelihood of elasticity and viscosity values obtained from an ensemble cell group amplitude and frequency measurements was reported to be 100 Pa and 3.1 mPa s. This work uses an additional optical phase shift measurement ðD/) and membrane fluctuation, A c , of each cell to further derive a more accurate cell-by-cell viscoelasticity solution which is consistent with the order of magnitude in our previous work. 23 The apparent viscosity values showed a similar trend to elasticity with lower viscosity values for live cells at 2.

CONCLUSION
In this study, we derived an expression for the loss tangent measurement of each cell by using a MEMS resonant sensor to capture three observables, frequency shift, amplitude ratio, and phase shift, allowing us to uniquely determine more accurately the viscoelastic moduli and therefore quantify the ratio of energy dissipation upon deformation of each probed cell. The developed model resolves the cell-to-cell heterogeneity in the viscoelastic properties of a sample population, thereby elucidating the phenotypic difference in the growth cycles of individual cells. We found that fixed cells show a substantially larger loss tangent than their live counterparts, revealing that fixed cells are more dissipative and elastic at our resonant frequency. By constraining our governing dynamic equations with an additional optical phase shift and membrane fluctuations of individual cells, we are able to obtain a tighter viscoelastic modulus solution space from our resonant sensors. Our technique can open opportunities to allow for continuous real-time measurements of changes in mass, stiffness, and viscoelasticity over the growth period of single cells, allowing them to serve as a diagnostic tool to identify phenotypes by their mechanical signature and response to mechanical stimulation.

Measurement scheme: Combining frequency, amplitude, and phase shift
As shown in Fig. 3(a), the technique to obtain the loss tangent of individual cells involves measuring the frequency (f Þ, amplitude ðAÞ, and optical phase shift ðD/Þ in four distinct schemes. First, the response of a dry (unloaded) sensor and an unloaded wet sensor (in-media) is measured to determine the baseline resonant frequency and amplitude. Second, a live cell is loaded on the sensor, and then, the frequency and amplitude shifts are measured, along with optical path length (OPL) variations between laser paths inside and outside the cell. The laser path length will change due to height oscillations of the cell during vibration, resulting in a phase shift. 3,23,32,33 Figures 3(b)-3(c) show SEM images of our MEMS resonant sensor structure that is electromagnetically actuated to produce vertical motion in the first resonance mode. The details of the resonant sensors and the experimental setup used in this study are described elsewhere. 33 The viscoelastic effect of HT-29 cells on the resonant frequency and amplitude of the sensor were previously quantified to generate a large space of potential solutions. 23

Optical path length (OPL) model
The current study takes a step further by focusing on combining the complementary optical path phase-shift and membrane fluctuation measurements to improve the stiffness and viscous dissipation solution space. As was previously established, a laser Doppler vibrometer (LDV) can measure a time-derivative of the OPL and therefore can determine the membrane oscillation of the cell. 21,34 Figure 4 where n cell represents the refractive index of the cell, n GM represents the refractive index of the media, A c represents the amplitude of the cell height oscillation with respect to the cell initial height also denoted as the cell membrane fluctuation (amplitude), A s represents the amplitude of the sensor oscillation, h represents the phase of the cell height oscillation with respect to the sensor, x represents the oscillating resonant frequency, and D/ is our measured maximum phase shift of the OPL. P sensor t ð Þ and h t ð Þ are the instantaneous sensor position and cell height.

Membrane fluctuation determination: Analytical estimation and simulation
We can uniquely compute the individual cell membrane fluctuation at resonance, A c for one cell per sensor, which overcomes the initial bulk estimate limitation. This is calculated via Eq. (2) by estimating A s ¼ 180 pm (from measured velocity and oscillating frequency) for a 35 nN (1 Vrms) excitatory input in media, n GM ¼ 1.35 and n cell ¼ 1.38 and 1.55 for live and fixed cells, respectively. 21 To better understand how the changes in cell membrane fluctuations occur, we modeled our sensor-cell system as a two-degree-of-freedom (2-DOF) suspended mass model where the cell mass (m 2 ) is considered a Kelvin-Voigt viscoelastic solid with elastic stiffness (k 2 ) and the viscous coefficient (c 2 ) connected to the sensor and the sensor mass (m 1 ) is connected to the fixed substrate by a second Kelvin-Voigt spring-damper (k 1 , c 1 ) (see supplementary material). The model assumes an oscillatory force F(t) applied to the sensor mass. Figure 4(b) illustrates the effect of both high and low viscoelastic cell mass (m 2 ) loaded sensors on the membrane fluctuations by simulating the time-dependent transient and steady state dynamic responses at a representative resonant frequency (xÞ of our sensors. In these simulations, the ratio c 2 k 2 is held constant and the stiffness/dissipation pair (k 2 , c 2 ) is scaled for cases of high and low viscoelasticities. These simulations depend on a few required assumptions that the cell membrane is homogeneous and smooth as well as the average cell-surface oscillation is calculated in the steady state.

Data analysis and cell culture
We combined interferometric techniques to measure the viscoelasticity of both live and fixed individual HT-29 cells. Cells were cultured and fixed on resonant sensors as described in previous work 33 (see supplementary material). To fully investigate the mechanical response of live cells compared to fixed cells, we started with the data collection of the necessary established methods, including amplitude ratios, 23 phase shift, 21 and frequency shift, 33 to ultimately convey a meaningful rheological translation. Ethics approval is not required for this study.
We model the 2-DOF system behavior as described fully in the supplementary material. The experimental observables based on the LDV measurements are related to mechanical quantities as given in Eqs. (2)-(3). Knowing the mechanical amplitude ratio and the phase shift between the cell and sensor oscillations and calibrating sensor mass, stiffness, and dissipation allow us to obtain cell viscoelastic properties at the probed resonant frequencies.
Estimation of viscoelastic coefficients (k 2 , c 2 ) The mechanical phase difference (h) is related to cell viscoelastic properties that we extract by considering the full 2-DOF suspended mass system. To estimate the viscoelastic coefficients (k 2 , c 2 ), our microresonator is modeled as a 2-DOF Kevin-Voigt model as described in the following equation in simplified matrix form: Note that the physical quantities of interest are found by taking the imaginary components, e.g., F t ð Þ ¼ Fsin xt ¼ I m Fe jxt f g , where A s ¼ A R s þ jA I s and A c ¼ A R s þ jA I c are the vector amplitudes of the sensor and cell states x sensor and x cell ; respectively, and Fe jxt is the input force. Equation (4) is further decomposed as the instantaneous sensor and cell responses in the following equations: x sensor t ð Þ ¼ A s j jsin xt þ h s ð Þ ; (5a)