https://link.springer.com/content/pdf/10.1016/j.bulm.2003.11.002.pdf

A history of the study of solid tumour growth: the contribution of mathematical modelling.

Not long ago it was thought that the most important characteristics of the mechanics of soft tissues were their complex mechanical properties: they often exhibit nonlinear, anisotropic, nearly inco...

A constrained mixture model for growth and remodeling of soft tissues

Since its coming of age in the mid 1960s, continuum biomechanics has contributed much to our understanding of human health as well as to disease, injury, and their treatment. Nevertheless, biomechanics has yet to reach its full potential as a consistent contributor to the improvement of health–care delivery. Because of the inherent complexities of the microstructure and biomechanical behaviour of biological cells and tissues, there is a need for new theoretical frameworks to guide the design and interpretation of new classes of experiments. Because of continued advances in experimental technology, and the associated rapid increase in information on molecular and cellular contributions to behaviour at tissue and organ levels, there is a pressing need for mathematical models to synthesize and predict observations across multiple length– and time–scales. And because of the complex geometries and loading conditions, there is a need for new computational approaches to solve the boundaryand initial–value problems of clinical, industrial, and academic importance. Clearly, much remains to be done. The purpose of this paper is twofold: to review a few of the many achievements in the biomechanics of soft tissues and the tools that allowed them, but, more importantly, to identify some of the open problems that merit increased attention from those in applied mechanics, biomechanics, mathematics and mechanobiology.

Review Paper: Continuum biomechanics of soft biological tissues

https://imechanica.org/files/gel_in_equilibrium_2008_05_06%20submit.pdf

Inhomogeneous swelling of a gel in equilibrium with a solvent and mechanical load

In this paper the theory of mixtures is used to develop a two-phase model of an avascular tumour, which comprises a solid, cellular, phase and a liquid phase. Mass and momentum balances which are used to derive the governing equations are supplemented by constitutive laws that distinguish the two phases and enable the stresses within the tumour to be calculated. Novel features of the model include the dependence of the cell proliferation rate on the cellular stress and the incorporation of mass exchange between the two phases. A combination of numerical and analytical techniques is used to investigate the sensitivity of equilibrium tumour configurations to changes in the model parameters. Variation of parameters such as the maximum cell proliferation rate and the rate of natural cell death yield results which are consistent with analyses performed on simpler tumour growth models and indicate that the two-phase formulation is a natural extension of the earlier models. New predictions relate to the impact of mechanical effects on the tumour's equilibrium size which decreases under increasing stress and/or external loading. In particular, as a parameter which measures the reduction in cell proliferation due to cell stress is increased a critical value is reached, above which the tumour is eliminated.

Modelling solid tumour growth using the theory of mixtures.

Kinematics partial stress and total stress balance laws and the entropy inequality constitutive theory steady state problems diffusing singular surface wave propagation in solids infused with fluids epilogue some results from differential geometry status of Darcy's law within the context of mixture theory a brief discussion of a mixture of immiscible fluids.

Mechanics of Mixtures

Rotor-bearing systems supported on squeeze film dampers (SFDs) show large amplitude vibratory motions when traversing critical speeds. At these operating conditions air is drawn into the damper thin film clearance generating a bubbly mixture with the lubricant and producing SFD forces not readily predictable with currently available analysis. A continuum model is proposed for describing the motion of a bubbly fluid in an open ended SFD operating with circular centered journal orbits. Computed predictions for peak-peak dynamic pressures and fluid film forces agree reasonably well with experimental measurements conducted on a SFD test rig operating with a controlled air in oil mixture. The bubbly flow model provides an initial procedure towards the reliable design of SFDs in actual operating conditions.

Analysis of Squeeze Film Dampers Operating With Bubbly Lubricants

A mixture theory for heat-induced alterations in hydration and mechanical properties in soft tissues

En The increased use of heating in many medical specialties is driven by the availability of new devices, not an understanding of the underlying physics. Whereas most prior studies have quantified material response characteristics before and after thermal damage, the goal of this work is to describe changing response characteristics during a particular class of thermal damage tests—biaxial isometric tests on collagenous biomembranes. Evolution equations are derived/postulated that provide a good fit to data but more generally provide direction for future work.

/pdf/a-model-for-the-kinetics-of-thermal-damage-of-biological-2x044i7bf5.pdf

Luoyi Tao

Papers

Mechanics of Mixtures

Analysis of Squeeze Film Dampers Operating With Bubbly Lubricants

A mixture theory for heat-induced alterations in hydration and mechanical properties in soft tissues

A model for the kinetics of thermal damage of biological membranes