Design of a planar resonant structure sensitive to out-of-plane forces 

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Overview of microfabricated tools for conducting me-chanical studies on individual cells

The author now surveys different types of microfabricated tools that have been reported for conducting mechanical studies on individual cells. For the sake of clarity and concise-ness, these microfabricated tools will now be referred as microelectromechanical systems (MEMS). By essence, the acronym MEMS is however a broad definition. Throughout the thesis, the terminology MEMS employed will refer to systems that encompass elec-trical, mechanical, but also optical or fluidic parts manufactured via microfabrication processes. In particular, focus will be given on MEMS with a high functional density, namely MEMS that try be as portable and autonomous as possible by maximizing the integration of actuation or/and sensing capabilities in a compact single-chip piece.
Therefore, devices such as micropipettes [10, 11, 12], microcantilevers used in atomic force microscopes (AFM) [13, 14, 15, 16, 17, 18], microplates [19, 20, 21, 22, 23] or mi-croindenters [24, 25, 26] will not be described in details in this chapter (further details can be found in several relevant reviews [7, 27, 28, 29, 30, 31, 32, 33] ). Notwithstand-ing the presence of microscopic components, actuators (e.g., positioning stages) and measurement means (e.g., microscopes, cameras, etc.) utilized in these systems are all distant (off-chip) from the extremity entering in contact with the cells. Besides, such tools are usually considered as experimental techniques or laboratory apparatus by the research community (see for instance classifications adopted in [2, 32]).
The choice to exclude micropipettes, AFM cantilevers, microplates or microindenters is representative of the difficulty to adopt a proper classification for introducing MEMS dedicated to cell mechanics. This classification is further complexified by the fact that a broad class of cell mechanical studies can be carried out. These studies may concern how cells move, deform and interact, as well as how cells sense, generate, and respond to mechanical forces. Accordingly, a large variety of MEMS can be found in the literature.
However, in many cell studies, stresses (i.e., forces) or/and strains (i.e., deformations) must be imposed or/and measured. Hereafter, Section 1.2.2 first lists several MEMS that encompass actuation means capable of imposing stresses/strains on cells. Then, Section 1.2.3 reports MEMS capable of measuring stresses/strains developed during cell responses.

MEMS encompassing actuation means

In this section, it is pointed out that, to avoid too many confusing subcategories, no particular distinction is made between MEMS applying a prescribed force or a prescribed displacement. Similarly, MEMS that provide stimulation globally (i.e., a stress/strain is provided to the entire cell structure) or locally (i.e., only a given cellular region is excited) are not dissociated. In addition, MEMS that target adherent cells or suspension cells (see Section 1.3.1) are not differentiated. Finally, auxiliary equipment (e.g., laser sources, peristaltic pumps, electric power supplies) are not considered.

Electrostatic comb drives

In [34, 35], interdigitated comb fingers exploiting electrostatic phenomena were used to carry out stress-strain experiments on individual collagen fibrils. A multidimensional approach based on a single linear electrostatic structure was also reported by Scuor et al. [36], who conceived a micro in-plane biaxial cell stretcher (see Fig. 1.2). The quadrants of a sliced circular plate were actuated in mutually-orthogonal directions, that is to say that the quadrants moved in horizontal and vertical directions simultaneously. The net force developed by such a comb drive actuator is given by F = N ce ǫ tce U 2 (1.1).
where Nce is the number of comb electrodes, ǫ is the permittivity constant of the di-electric medium, tce is the comb thickness, gce is the comb electrode gap and U is the driving voltage. Theoretically, Scuor et al. claimed that a nominal voltage of 100 V permitted such an electrostatic structure to generate actuation forces up to 60 N. In practice, only translation amplitudes of the plate were reported. In ambient conditions, a power supply of 100 V led to a maximum space between the quadrants of 3.4 m.

MEMS providing sensing capabilities

Applying a stress or a strain to a cell is a requirement for many studies. But once the cell has been mechanically stimulated, the possibility to evaluate the mechanical behavior of the cell is also often a sine qua non condition. In the following paragraphs, one hence enumerates different MEMS capable of extracting various mechanical properties of cells.

Deformable beam-based sensors

Cells can migrate in response to multiple situations (e.g., wound healing). During loco-motion, cells pull themselves and develop forces to move from one location to another. In precursory works, soft silicon substrates that wrinkled during cell movements con-stituted a first mean to detect these forces [87, 88, 89]. Distortions of the substrates were however highly chaotic and nonlinear. A quantitative evaluation of the cell forces generated was hence difficult.
To alleviate this difficulty, locomotion forces developed during cell migration were studied with cantilever-like configurations. For instance, Galbraith and Sheetz [90] re-ported a high functional density MEMS made up of 5904 horizontal microlevers (see Fig. 1.12). Each lever was ended by a pad. The area of the pads ranged from 4 to 25 m2. The whole set of pads constituted a sensitive surface where a chicken fibroblast was seeded. Centroid of the pads were monitored optically during locomotion of the cell. The forces that the cell exerted on the pads could be determined by calculating the product of the pad displacement and the stiffness of the levers (see image (d) in Fig. 1.12). Traction forces < 1 nN and up to 100 nN were measured for different regions of the cell.

Constraints imposed by the cell environment

Living cells are obviously entities of exquisite fragility. For successful mechanical studies conducted on cells, a first priority is to preserve their integrity so that they can survive after experiments. Ideally, cells should be conserved in specific medium during exper-iments. Such cell medium allow the continuous delivery of vital nutrients in order to maintain cells alive.
But major challenges arise as soon as MEMS must face such a liquid environment. For instance, in the presence of aqueous solutions, capillary meniscus arise at the air-water interface when soft force sensors (e.g., [97, 98, 103]) are immersed and removed. Soft structures must then withstand large capillary forces. Besides, capillary forces can engender measurement artifacts.
MEMS exploiting electrostatic comb drives (see Fig. 1.2) experience significantly re-duced performances when immersed in cell medium. Due to the hydrophobic nature of the silicon-water interface, air trapping between the comb drive teeth and the MEMS ground plane may arise. Furthermore, the enhanced electrical conductivity of liquids usually reduces their initial stroke. Similarly, electrothermal beams cope with challenging phenomena if they are plunged in a liquid environment. For instance, electrothermal beams cannot be supplied with continuous power for underwater operation due to electrolysis. For the MEMS shown in Fig. 1.4, alternating voltages were used in electrolytic solutions. But the initial travel range of 9 m measured in air was restricted to 4 m in liquids. An additional feature of electrothermal actuators relates to the high temperature that they can reach during operation. Since cells are particularly sensitive to temperature fluctuations, high tem-peratures may potentially cause irreversible damages. Special precautions should hence be taken accordingly.
The latter remark may be extended to all types of contact-based MEMS, that is to say all MEMS for whom an extremity directly touches the cells. For instance, sharp tips (e.g., such of those used in conventional AFM) may cause damages to external lipid cell biomembranes. Moreover, contamination problems may arise once the tool has touched a cell. Therefore, the tips should be properly cleaned before each new experiment. This additional laborious step may further prevent repetitive analysis.
MEMS that exploit non-contact based techniques may alleviate such problems. Op-tical gradients, electric fields and magnetic fields have indeed the potential to carry out cell stimulation without direct physical interaction with the cells. However, electric fields can directly affect cells under test [118]. Although no direct contact occurs dur-ing cell stimulation (see Fig. 1.8), electric fields cause power dissipation in the form of Joules heating in a conductive medium. Therefore, and as in the case of electrothermal actuators, the usage of electric fields requires to monitor changes in temperature that can affect cell phenotypes.
The wavelength of some highly concentrated laser beams may also be hazardous for cells [119, 120, 121, 122]. Comparatively, magnetic fields (see Fig. 1.6) are nowadays considered safe for cells. In effect, magnetic fields do not significantly disturb the cell re-sponse upon short times of exposure. Unfortunately, restrictions of magnetic setups are related the microbeads that must be locally attached to the cell membrane. Magnetic forces applied strongly depend on the beads size whereas it may be difficult to avoid size variations from bead to bead in experimental conditions. Likewise, material prop-erties of the beads used (e.g., magnetic moment) cannot be easily controlled. Moreover, the adhesion procedure of the beads remains an unpredictable process and formation of bead aggregates may appear. Finally, since bead immersion is unpredictable, the force distribution around adhesion sites can actually be highly heterogeneous.
Taking into account the above discussion, it is clear that using MEMS in cell media remains challenging. Besides, several results of cell mechanical studies reported in the literature have been obtained in air. However, living cells naturally evolve in a liquid environment and conducting experiments with cells in a dry environment may signif-icantly impact their mechanical properties. Therefore, it may be reasonably presume that results obtained in air could have been partly biased and could have possibly led to some misinterpretations.

Type of mechanical properties probed: relevance of the cell elas-tic modulus

MEMS reported in Section 1.2 have permitted to investigate different biophysical prop-erties of cells. Examples of these biophysical properties are summarized in Table 1.2. In essence, all these properties permit to gain insights in the mechanics of cells. Accord-ingly, they all deserved to be thoroughly investigated.
Nevertheless, the notion of cell deformability appears today as an increasingly im-portant physical marker for future biomedical applications. Cell are indeed constantly stabilized by their internal scaffolding, the cytoskeleton network. As seen in Section 1.1, the cytoskeleton is a complex biopolymer network that may undergo structural alter-ations leading to changes in cell rigidity. Deformation characteristics of cells may hence provide relevant information about the biological and structural function of cells.
The notion of cell deformability is however general and must be clarified. Basically, studying cell deformability requires application of an external force, and a correspond-ing quantification of the cellular deformation in response. Intuitively, a stiffer cell is less deformable. In the literature, quantitative measures of cell deformability are however reported in different manners. Cell deformability can indeed be expressed in terms of elasticity (i.e., elastic modulus), viscoelasticity or stiffness. Although all of these param-eters provide information about the resistance of a cell to deformation, they describe distinctly different properties. For the sake of clarity, some essential distinctions be-tween these parameters are first reminded.
A cell exhibits an elastic behavior if it deforms under stress (i.e., an external force) and returns to its orginal shape when the stress is removed (see left image in Fig. 1.17). The relationship between stress and strain (force-deformation) is linear, and the de-formation energy is returned completely. Elasticity is often referred to as the elastic modulus or Young’s modulus. Given the large values typical for many common materi-als, the Young’s modulus is usually quoted in MPa or GPa. For instance, the Young’s modulus of steel, bone, polystyrene or soft silicon rubber are about 200 GPa, 17 GPa, 3MPa and 2 MPa, respectively. Comparatively, most living cells range from 1 kPa to 100 kPa (see Fig. 1.18).

Table of contents :

Introduction
1 Microfabricated tools for conducting mechanical studies on individual cells: state-of-the art 
1.1 The cell architecture: some fundamentals
1.2 Overview of microfabricated tools for conducting mechanical studies on individual cells
1.2.1 Preliminary remarks
1.2.2 MEMS encompassing actuation means
1.2.2.1 Electrostatic comb drives
1.2.2.2 Electrothermal beams
1.2.2.3 Electro-active polymers
1.2.2.4 Magnetic fields
1.2.2.5 Electric fields
1.2.2.6 Optical gradients
1.2.2.7 Fluid flows
1.2.3 MEMS providing sensing capabilities
1.2.3.1 Deformable beam-based sensors
1.2.3.2 Piezoresistive strain gauges
1.2.3.3 Capacitive sensors
1.3 Discussions
1.3.1 Nature and number of cells targeted
1.3.2 Constraints imposed by the cell environment
1.3.3 Type of mechanical properties probed: relevance of the cell elastic modulus
1.4 Summary and conclusions
2 Sensing forces in cell studies with beam resonators: theoretical back- ground 
2.1 Vibration of a CC beam subjected to an axial force: exact solution
2.1.1 Effects of an axial force on the fundamental frequency
2.1.2 Effects of an axial force on the first mode shape
2.2 Vibrations of a CC beam subjected to an axial force: approximate solution via energy methods
2.3 Vibration of a CC beam in fluids
2.3.1 Presence of a fluid: impact on the resonance frequency and oscillation amplitude
2.3.2 Energy losses: notion of quality factor (Q factor)
2.3.3 Vibration of a beam in air and in water: numerical application .
2.4 Vibrations of a CC beam in fluids: parametric analysis
2.4.1 Varying the beam geometry: influence on the mass added by a fluid
2.4.2 Varying the beam geometry: influence on the resonance frequency and Q factor
2.4.3 Varying the beam geometry: influence on the force sensitivity
2.5 Conclusion
3 Design of a planar resonant structure sensitive to out-of-plane forces 
3.1 Overall description and key features of the structure
3.2 Theoretical analysis
3.2.1 Preliminary remarks
3.2.2 Static behavior: large deflection of the planar structure
3.2.3 Dynamic analysis: effects of a static predeflection on the oscillation of the outer beams
3.3 Discussion about the dimensions of the planar structure
3.4 Static and dynamic behavior of the structure: numerical application
3.4.1 Theoretical results: static deflection
3.4.2 Theoretical results: variations of resonance frequency
3.5 Conclusion
4 Experimental validation and first investigations conducted on biologi- cal samples 
4.1 Experimental arrangement
4.1.1 Overview
4.1.2 Implementation of an optical fiber displacement probe
4.2 Comparison between theory and experiments
4.2.1 Evaluation of static deflections
4.2.2 Evaluation of dynamic performances
4.2.2.1 Resonance frequency
4.2.2.2 Vibrations at the central beam
4.2.2.3 Quality factor
4.2.2.4 Frequency variations induced by large displacements
4.2.2.5 Frequency variations induced by small forces
4.3 Measuring the elastic properties of supersoft materials
4.3.1 Calibration of the prototype with gel samples
4.3.2 Direct extraction of the Young’s modulus of a lobster egg
4.4 Conclusion
Conclusions and suggestions for future research 94
Appendix
Abbreviations and Notations
List of publications
Bibliography

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