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We present a novel method of 3-dimensional surface fitting of a droplet using ellipsoids such that the droplet is a combination of segments of two to four distinct ellipsoids. Further, this fitting method has been used to develop an analytical model estimating the volume of a droplet resting over compliant as well as non-compliant substrate. Here, we have used Glass and Poly(methyl methacrylate) (PMMA) substrates as rigid, and, Polydimethylsiloxane (PDMS) free-hanging thin membranes (with thickness ranging from 20 − 40 𝜇𝑚) as compliant substrates. The analytical model considers the base length, width, height, and contact angles of the droplet captured from the experiment, and estimates the droplet volume. The proposed analytical model could predict the volume correctly for droplets resting over compliant as well as non-compliant substrates with a maximum deviation of 16.6 % for the volume range of 5 − 70 𝜇𝐿. Further, the predictions from the proposed analytical model are compared with the spherical cap-based model for droplets placed over compliant as well as non-compliant substrates. The spherical cap-based model failed to estimate the droplet volume over a compliant substrate with an error of > 50 %. Whereas, the proposed ellipsoid-based model could predict the droplet volume over compliant substrate correctly with a maximum error of 16.6 %. Also, the proposed analytical model predicts the volume of droplets even at high contact angle hysteresis (> 50𝑜) where the droplet has high radial asymmetry. Further, the study also illustrates how Artificial Neural Networks (ANNs) can be used to forecast droplet width and contact angle hysteresis (CAH). The droplet width predicted from ANN could be used to eliminate the requirement of measuring droplet width from the top view experimental image. The volume of the droplet can thus be predicted from its side profile alone when utilized in conjunction with the theoretical model. Further, we developed an ANN model which predicts the CAH of the droplet by considering the length scales of the droplet. The developed ANN models performed a very good prediction with an R-value of > 0.98.

This paper presents a novel approach for the stiffness characterization of thin membranes using the droplet motion characteristics over the membrane. The droplet motion over an inclined free hanging thin compliant membrane is investigated with the help of numerical model. Effect of substrate compliance over the displacement of the droplet has been studied. Further, the wall deformation of the substrate with the droplet motion has also been investigated. The numerical study has highlighted that the distance moved by a droplet over a membrane reduces with reduction in flexural rigidity of the membrane. Also, the dependency of motion of droplet with contact angle hysteresis (CAH) is presented. It was observed that the CAH increases with decrease in flexural rigidity of the membrane. These characteristics of the droplet motion over inclined free-hanging thin complaint membrane have been utilized to predict the Young's modulus of PDMS membranes using a novel Artificial Neural Network (ANN) model. The ANN model is developed for the same purpose. The input parameters for the ANN model are the droplet's displacement over a fixed time interval (𝑥), the PDMS membrane's deflection (𝑌), the angle of inclination (𝜃), the thickness of the membrane (𝑡), and the droplet's volume (𝑉). The output parameter of the ANN is the Young’s Modulus of the PDMS membrane. The created ANN network was found to have good accuracy (𝑅𝑜𝑣𝑒𝑟𝑎𝑙𝑙 = 0.992) in predicting the Young's modulus of the PDMS membrane. The proposed approach is a simple, low-cost method for prediction of Young’s modulus of PDMS membranes.

Mechanical property of biological cells can act as an indicator for the health state of a human being. Mathematical modelling of cells help to understand and predict the cell deformation patterns that might provide insightful finding for cell mechanics. Many models have been developed that tries to explain the cell mechanics at the cellular level. We propose an analytical model that considers the poroelastic nature of cells to understand their deformation behavior. The validity of the model is tested by comparing the predicted cell deformations against the experimental observations reported by Raj et al. (2017). Also, a computational study is performed, where we employ an in house Python code along with MS Excel GRG solver which incorporates the cell deformation predictions from developed poroelastic model and predicts the Young’s modulus value of the cells. The predictions using GRG based approach showed a good match with the experimental results with a maximum error of 12.09% in the case of MDA MB-231 cells. Further, we present an artificial neural network model to predict the Young’s modulus and viscosity of cells based on the experimentally measured input parameters such as entry time, transit velocity, initial cell diameter and extension ratio from the cell migration process through a micro-constriction channel. It was found that the neural network with architecture of 4-5-1 was best suited for the MCF-10A cells while the 4-8-1 architecture was giving better results for the MDA MB-231 cells. The developed ANN model is further tested for Young’s Modulus prediction of HeLa cells with completely new set of data. The predictions from ANN based model for HeLa cells matched well the experimental prediction within 4.5 % of error.

Mechanical properties are vital biomarkers for the health state of biological cells and indirectly human health. Mathematical and computational models relating the mechanical properties of cells with their deformation are valuable tools for understanding and predicting cellular behavior. Numerous models and techniques have been developed to measure the stiffness and viscosity of biological cells. Recent experimental investigations demonstrated that biological cells are poroelastic materials of solid networks bathed with cytosol liquid in the pores (Moeendarbary et al. (2013), Nature Materials). However, a mathematical model relating the deformation of poroelastic cell material with Young's modulus of Solid networks has not been reported yet to the best of our knowledge. This paper presents a combined mathematical and computational approach to the mechanophenotyping of biological cells. First, an analytical model is presented that considers the poroelastic nature of cells and relates Young's modulus of solid network with cell deformation. The developed model has been validated by predicting its Young’s Modulus based on the experimental data on deformation characteristics of cells squeezing through constriction microchannel. Model’s predicted Young’s Modulus for three different cell lines; HeLa, MCF-10A, and MDAMB-231 are \(153.64\pm 60.3 \text{k}\text{P}\text{a}, 97.84 \pm 41.7 \text{k}\text{P}\text{a}, \text{a}\text{n}\text{d} 67.9 \pm 48.8 \text{k}\text{P}\text{a}\) respectively, which matches well with the measurements reported using conventional techniques in literature. Furthermore, two artificial neural network (ANN) models are developed to predict Young's modulus and viscosity of cells based on measured deformation parameters for HeLa, MCF 10A, and MDA MB-231 cells. The neural network with an architecture of 4-8-9-1 is found to be best suited for Young's modulus predictions with \(R\sim0.974\). In contrast, the 4-7-8-1 architecture could provide better results for viscosity predictions of the given cell lines with \(R\sim0.999\). Further, a linear Support Vector Machine (SVM) model is also presented to classify the three given cell lines based on their initial diameter and elongation behavior in the constriction microchannel. To the best of our knowledge, this is the first study to present a poroelasticity-based mathematical model for biological cells predicting cell stiffness based on their deformation characteristics. Additionally, it is the first one to explore the classification of different cell lines based on their elongation ratio, derived from the analysis of static images within a constriction channel, eliminating the need for time-based studies. This combined analytical and computational approach can prove to be very useful for the direct estimation of mechanical properties of cells based on their squeezing behavior through constriction microchannel.