Recent developments in Computer-Integrated and Robot-Aided Surgery (in particular, the emergence of automatic surgical tools and robots (as well as advances in Virtual Reality techniques, call for closer examination of the mechanical properties of very soft tissues (such as brain, liver, kidney, etc.). Moreover, internal organs are very susceptible to trauma. In order to protect them properly against car crash and other impact consequences we need to be able to predict the organ deformation. Such prediction can be achieved by proper mathematical modelling followed by a computer simulation. The ultimate goal of our research into the biomechanics of these tissues is development of corresponding, realistic mathematical models. This paper contains experimental results of in vitro, uniaxial, unconfined compression of swine brain tissue obtained by the author in Mechanical Engineering Laboratory, Japan, and discusses liver and kidney in vivo compression experiments conducted in Highway Safety Research Institute and the Medical Centre of The University of Michigan. The stress-strain curves for investigated tissues are concave upward for all compression rates containing no linear portion from which a meaningful elastic modulus might be determined. The tissue response stiffened as the loading speed increased, indicating a strong stress (strain rate dependence. As the step in the direction towards realistic computer simulation of injuries and surgical procedures, this paper presents two mathematical representations of brain, liver and kidney tissue stiffness. Biphasic and single-phase models are discussed. The biphasic model is shown to be inappropriate due to its inability to account for strong stress-strain relationship. Agreement between the proposed single-phase models and experiment is good for compression levels reaching 30% and for loading velocities varying over five orders of magnitude. Presented mathematical models can find applications in computer and robot assisted surgery, e. g. the realistic simulation of surgical procedures (including virtual reality), control systems of surgical robots, and non-rigid registration, as well as ergonomic design for injury prevention.

Keywords: mathematical modelling, mechanical properties, Kidney, Liver