A comparison of mechanical models for the viscoelastic response of human breast carcinomas
The mechanical response of a living cell is notoriously complicated. The complex, heterogeneous characteristics of cellular structure introduce difficulties that simple linear models of viscoelasticity cannot overcome, particularly at moderate indentation depths. Herein, a nano-scale stress-relaxation analysis performed with an Atomic Force Microscope reveals that isolated human breast cells do not exhibit simple exponential relaxation capable of being modeled by the Standard Linear Solid (SLS) model. Therefore, this work proposes the application of a progression of more sophisticated models that may extract the mechanical parameters from the entire relaxation response, improving upon existing physical techniques to probe isolated cells. The first model under consideration is the Generalized Maxwell (GM) model that distributes the response of the cell across multiple time scales in an attempt to replicate the interaction of subcellular components. The second is a fractional model that operates without a priori assumptions of the cell's internal structure and describes the fractional time-derivative dependence of the response. The results show an exceptional increase in conformance to the experimental data compared to that predicted by the SLS model. Both models excel at mapping the relaxation behavior of the cells that occurs within a few seconds of the initial force. This area is generally ignored with an SLS fit and therefore not included in most cell differentiation studies. The results of the GM model show a significant change in the mechanical properties of the first relaxation mode, which validates the necessity of the early behavior's inclusion. The FZ model preserves the distinctions highlighted in the SLS model, but also incorporates the disparity in the early-relaxation times seen in the GM model as a change in the composite relaxation time.