Cell viscoelasticity is linked to fluctuations in cell biomass distributions

The viscoelastic properties of mammalian cells can vary with biological state, such as during the epithelial-to-mesenchymal (EMT) transition in cancer, and therefore may serve as a useful physical biomarker. To characterize stiffness, conventional techniques use cell contact or invasive probes and as a result are low throughput, labor intensive, and limited by probe placement. Here, we show that measurements of biomass fluctuations in cells using quantitative phase imaging (QPI) provides a probe-free, contact-free method for quantifying changes in cell viscoelasticity. In particular, QPI measurements reveal a characteristic underdamped response of changes in cell biomass distributions versus time. The effective stiffness and viscosity values extracted from these oscillations in cell biomass distributions correlate with effective cell stiffness and viscosity measured by atomic force microscopy (AFM). This result is consistent for multiple cell lines with varying degrees of cytoskeleton disruption and during the EMT. Overall, our study demonstrates that QPI can reproducibly quantify cell viscoelasticity.


Summary:
We provide a derivation for how quantitative phase rheology (QPR) generates effective cell stiffness and viscosity using the autovariance of quantitative phase imaging (QPI) data. To do this, we establish a basic definition of the autovariance function and a basic equation that extracts stiffness and viscosity from dry mass, or non-aqueous biomass, displacement. We combine this definition and equation with QPI data to extract stiffness and viscosity regimes from the autovariance function.

Autocovariance
To measure the similarity of quantitative phase data over time we used an unbiased estimate of autocovariance 1 of the phase-shift signal, which is an autocorrelation of the mean subtracted data. We normalized the temporal autocovariance to the number of data points used in each autocovariance window, referenced to the end of the time shift window (t0), and defined as: Where x and y are positions after removing rigid translational motion of a cell cluster, t0 is the time, ϕ is phase-shift, N is the number of data points used to calculate the signal, w is the number of images, Δt is time between measurements, and τ is time lag. The autocovariance was then averaged over a cell or cell cluster area as: 00 , all , in 1 ( , ) ( , , , ) where A is the area of a cell or cell cluster in imaging pixels. We also took the average of the autocovariance through time for all times corresponding to cells in interphase of the cell cycle, where n is the number of different end time points.

Two-parameter
We treat the cellular structures imaged by quantitative phase as particles immersed in a Maxwell liquid (Fig. 1A). Therefore, these structures feel the effect of a spring damper system in series described as the following system of equations: where k is the long term effective spring constant of the cell felt by a particle, μ is the effective damping coefficient from the viscous forces of the cell felt by a particle, f(t) is the applied impulse force, X1 is the elastic displacement, X2 is the viscous displacement, Xtot is the total displacement of the biomass, and m is the average biomass of particles in the system. We observe long timescales that are much greater than the average relaxation times of a cell ( Fig.   S3), so the long timescale effects dominate and the active force can be considered as applied nearly instantaneously. Rearranging equation (S4) in terms of only the elastic displacement X1 or only the viscous displacement X2 yields the following differential equations: Assuming that the total displacement Xtot contributes to the majority of biomass rearrangement and oscillation, we integrate equation (S8) over time, add it to equation (S7), and rearrange this equation of a damper spring system in series to obtain an inhomogeneous ordinary differential equation (ODE) for the total displacement: where c1 could be seen as a buildup of stress from past deformation or a memory function.
where Xtot,0 is the initial displacement and Xtot,rest is the long term resting displacement of our system. Because the relaxation timescale (Fig. S3) is over an order of magnitude lower than the period of measurement, the active force can be modeled as an instantaneous displacement represented as a delta function, δ, at some time, tj, not equal to zero. Solution of this spring damper system without this active force yields:   where T is the period of observation. We then establish the relationship between biomass and displacement of biomass by: (S15) where M is biomass as a function of position x and time t, and v is velocity as a function of x and t. We assume that the main contribution to the partial derivative of biomass with time is due to growth and since our measurement occurs over a short time interval, growth is negligible, therefore: . (S16) We further assume that the cell velocity, v, and biomass, M, fields are isotropic with no dependence on direction. Averaging over θ in polar coordinates yields: Assuming that this change in biomass over radial distance is small compared to the total biomass over a radial distance r we obtain: Averaging over a radial distance and assuming that velocity, v, radial position, r, and biomass, M, do not correlate over the radial distance because the system is isotropic we obtain: (S19) Assuming this system is ergodic, the local spatial average of biomass is equal to the temporal average biomass, which is constant with respect to time, and therefore the average biomass over radial distance term is only a function of r, which we call κ(r): where γ(r) is the local spatial average of the biomass, which is constant over time and is therefore only a function of radial position: Since κ is independent of time, we can integrate and obtain: ,, Thus, the ratio of biomass over initial biomass is equivalent to the displacement over initial displacement: , , , The biomass for a particular area is directly proportional to the phase-shift 4-7 : where ϕ is phase-shift, and α is the specific refractive index, which is determined experimentally.
Therefore, phase-shift data, ϕ, obtained via QPI can be used to obtain information about the displacement of cell biomass over time.

Predicted autocovariance of cell biomass distributions
Using biomass as a tracer for displacement and translating this equation into autocovariance space yields: where a and b are described in terms of coefficients as: