Introduction

This special section is a collection of selected papers from two symposia we organized during 2014: one as a part of the Seventh World Congress of Biomechanics (WCB) in Boston and one as a part of Fifth European Conference on Computational Mechanics (ECCM V) in Barcelona. Both symposia covered different topics related to the multiscale mechanics of cells, tissues, and biomaterials. A variety of topics from different areas of research within the biomechanics, biophysics, and biomaterials community were presented during those symposia. This editorial aims at summarizing the current trends of research that we identified during those symposia and that we would like to briefly highlight here as a context within which the contributions of this special section could be discussed.

From morphogenesis, which initially forms the bodies of organisms, to regenerative medicine, which promises to improve the quality of life in our graying societies, mechanical forces and displacements play extremely important roles. The effects of mechanical forces and corresponding displacements travel back and forth between several spatial and time scales. Therefore, mechanics-based study of living organisms and the biomaterials that are used to replace them either temporarily or permanently is in the center of many research programs worldwide. In addition to traditional areas of research, new approaches are emerging not only as a result of our improved understanding of the important biological and physical aspects that play a role in sustaining life but also due to specific measurement techniques and computational approaches that open new avenues of research.

Physical phenomena involving material features with characteristic sizes spanning several orders of magnitude are frequently found in nature. During recent years, our improved capability in modeling those phenomena has advanced our comprehension of the intrinsic nature of things in many different areas of research. In particular, the mechanics of tissues, biomechanics, and biomaterials has greatly benefited from this approach, as evidenced by the increasing attention these areas of research have been receiving recently [1–3]. Living organisms are often characterized by complex hierarchical architectures with constituents varying in size between the nanometer range and the whole organ level. Three different length scales, namely, macro, micro, and nanoscales are usually identified with respect to the modeling approaches. At the macroscale, tissues and organs are often studied using the continuum mechanics theories that aim to capture the complex constitutive behavior of tissues and organs including the anisotropic, nonlinear, time-dependent, and adaptive aspects. At the microscale, cells and tissues are studied using continuum mechanics theories, polymer physics theories, and other classical mechanics approaches. At the nanoscale, proteins and other biomolecules are studied using Newtonian molecular dynamics techniques, ab initio molecular dynamics models, and Monte Carlo methods. There is also an active area of research that tries to bridge the different scales to create multiscale models that span and connect physical phenomena happening at several length and time scales.

Under the roof of multiple-scales investigation, three specific topics are highlighted in this editorial including the multiphysics and multidomain approaches to the study of living systems, relatively new experimental techniques that are used for assessment of the mechanical behavior at different scales, and relevant inverse problems in the context of the above-mentioned computational and experimental approaches.

Multiphysics and Multidomain Approaches

An exciting area of research recently receiving attention within the biomechanics community is development of multiphysics and multidomain models for explaining the behavior of cells and tissues. In living organisms, mechanical forces and displacements interact with a multitude of other phenomena including chemical, biochemical, and electrical phenomena.

An example of that is the relationship between prestress and mechanical deformation in cartilage and diffusion of molecules into the tissue. The chemical concentration gradients and fixed electrical charges cause swelling of the cartilage tissue that greatly contributes to the mechanical properties and the multiphasic behavior of cartilage. In this section, Arbabi et al. present a biphasic-solute model of cartilage to study the transport of neutral solutes across cartilage. The diffusion of solutes is linked to chemical concentration gradients that cause exodus or inflow of water molecules, thereby causing mechanical deformations that are taken into account in the model.

Several other multiphysics and multidomain studies can be found in this special section. Frotscher et al. use a combined experimental and computational approach to study the effects of several cardioactive drugs on the behavior of a tissue monolayer. Giverso et al. study the mechanically driven branching of bacterial colonies using computational models that also take into account the “chemotactic response of bacteria, the viscous interaction between the colony and the underlying agar, and the effects of the surface tension at the boundary.” Grillo et al. present a comprehensive two-phases constitutive formulation for cartilage with the purpose to generate benchmark tests for the validation of complex user-defined material models within Finite Element implementations.

Experimental Techniques at Different Scales and Relevant Modeling

The emergence of new experimental techniques has enabled measurement of certain phenomena and quantities that were previously very difficult to study. Two particularly important classes of experimental techniques are indentation experiments and full-field strain measurement techniques. As for nano-indentation, it is now possible to probe the mechanical properties of cells, tissues, and biomaterials at length scales ranging from a few nanometers to a few millimeters and larger using indentation-type experiments. The indentation experiments are performed using Atomic Force Microscopes (AFM) and dedicated nano-indentation setups at the nano- to microscales, while more traditional loading frames could be used to perform indentation tests at larger scales. For example, Taffetani et al. perform nano-indentation experiments using AFM to characterize the material properties of cartilage subject to harmonic loading; a frequency domain study is presented.

Full-field strain measurement techniques have enabled measurement of strain during mechanical testing of many different types of tissues and biomaterials. This rich set of information could be very useful for the study of mechanical behavior, deformation, and failure mechanisms and for determining the mechanical properties of tissues and biomaterials. Two specific measurement techniques are more frequently used in the literature, namely, Digital Image Correlation (DIC) [4] and Digital Volume Correlation (DVC) [5]. While DIC is capable of measuring full-field strains throughout the outer (i.e., visible) surface of test objects, DVC could measure the strains everywhere within the volume of the test objects. An interesting example of full-field strain measurement is contributed to this special section by Palanca et al. The study compares three different approaches for measurement of 3D strains in bone using DVC.

Coudrillier et al. study the effects of age and diabetes on scleral stiffness at the macroscale. Continuum mechanics approach and three dimensional imaging correlation technique are used to estimate tissue stiffness, and wide angle X-ray scattering technique is used to measure collagen organization.

Parameter Identification and Inverse Problems

In biomechanical studies of cells and tissues, inverse problems are often encountered [6–10]. For example, the use of above-mentioned experimental techniques is often associated with the solution of an inverse problem. This is particularly the case for indentation-type experiments where one tries to use the measured force–displacement curve to determine the mechanical properties of the cells or tissues at different scales [9]. The associated inverse problem can be then formulated as: what material properties will give rise to a theoretically (i.e., computationally or analytically) determined force–displacement curve as close as possible to the measured force–displacement curve? Similar inverse problems could be formulated for full-field strain measurement techniques such as DIC [10] and DVC: when the material is subjected to a given set of loading conditions, what material properties should be used in a computational model to obtain strain distributions as close as possible to the measured strain distribution?

In this section, Taffetani et al. use a manual fitting procedure allowing for storage and loss data of harmonic loading to estimate the material properties of a cartilage from indentation-type AFM experiments. Arbabi et al. use a similar approach for calculating the diffusion coefficient of the different zones of cartilage from concentration curves obtained using contrast agent enhanced micro Computer Tomography (micro-CT) images. Coudrillier et al. use inverse finite element models to determine the material properties of sclera by matching the DIC-measured and computationally determined displacement fields.

Conclusions

This special section presents a carefully selected collection of studies at the forefront of developments in computational mechanics of proteins, cells, tissues, and biomaterials. The types of tissues and cells, the scale of study, and the applied experimental and computational techniques cover a wide range of current approaches to the mechanics-based study of living organisms. However, a number of recurrent trends and approaches could be identified within the included studies, which are highlighted in this editorial.

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