How to test very soft biological tissues in extension?

Mechanical properties of very soft tissues, such as brain, liver and kidney, until recently have largely escaped the attention of researchers because these tissues do not bear mechanical loads. However, 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, require closer examination of the mechanical properties of very soft tissues and, ultimately, the construction of corresponding, realistic mathematical models. A body of knowledge about mechanical properties of very soft tissues, assembled in recent years, has been almost exclusively based on the results of compression, indentation and impact tests. There are no results of tensile tests available. This state of affairs, in the author's opinion, is caused by the lack of analytical solution relating a measured quantity - machine head displacement - to strain in simple extension experiments of cylindrical samples with low aspect ratio. In the paper this important solution is presented. The theoretical solution obtained is valid for isotropic, incompressible materials for moderate deformations (<30%) when it can be assumed that planes initially perpendicular to the direction of applied extension remain plane. Two astonishing results are obtained: (i) deformed shape of a cylindrical sample subjected to uniaxial extension is independent on the form of constitutive law, (ii) vertical extension in the plane of symmetry lambda(z) is proportional to the total change of height for strains as large as 30%. The importance and relevance of these results to testing procedures in Biomechanics is highlighted.     

Dynamic and quasi-static compressive response of porcine muscle

Research and application activities in impact biomechanics require dynamic response of biological tissues under high-rate loading. However, experimental difficulties have limited the characterization of soft tissues under such loading conditions. In this paper, we identify these technical challenges in dynamic compression experiments using a split Hopkinson pressure bar (SHPB) and present the remedies to overcome them. In order to subject the specimens to valid dynamic testing conditions, in addition to developing new pulse-shaping techniques and incorporating highly sensitive load-measuring transducers, annular thin-disc specimens radically different from regular solid specimens were used to minimize radial inertia effects that may overshadow the intrinsic material properties. By using this modified SHPB, the compressive stress-strain behavior of soft porcine muscle tissue was obtained along and perpendicular to the muscle fiber direction from quasi-static to dynamic strain rates. The results show that the non-linear compressive stress-strain responses in both directions are strongly strain-rate sensitive.     

Constitutive modelling of brain tissue: Experiment and theory

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.). The ultimate goal of our research into the biomechanics of these tissues is the development of corresponding, realistic mathematical models. This paper contains experimental results of in vitro, uniaxial, unconfined compression of swine brain tissue and discusses a single-phase, non-linear, viscoelastic tissue model. The experimental results obtained for three loading velocities, ranging over five orders of magnitude, are presented. The applied strain rates have been much lower than those applied in previous studies, focused on injury modelling. The stress-strain curves 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. The use of the single-phase model is recommended for applications in registration, surgical operation planning and training systems as well as a control system of an image-guided surgical robot. The material constants for the brain tissue are evaluated. Agreement between the proposed theoretical model and experiment is good for compression levels reaching 30% and for loading velocities varying over five orders of magnitude.     

The nonlinear material properties of liver tissue determined from no-slip uniaxial compression experiments

The mechanical response of soft tissue is commonly characterized from unconfined uniaxial compression experiments on cylindrical samples. However, friction between the sample and the compression platens is inevitable and hard to quantify. One alternative is to adhere the sample to the platens, which leads to a known no-slip boundary condition, but the resulting nonuniform state of stress in the sample makes it difficult to determine its material parameters. This paper presents an approach to extract the nonlinear material properties of soft tissue (such as liver) directly from no-slip experiments using a set of computationally determined correction factors. We assume that liver tissue is an isotropic, incompressible hyperelastic material characterized by the exponential form of strain energy function. The proposed approach is applied to data from experiments on bovine liver tissue. Results show that the apparent material properties, i.e., those determined from no-slip experiments ignoring the no-slip conditions, can differ from the true material properties by as much as 50% for the exponential material model. The proposed correction approach allows one to determine the true material parameters directly from no-slip experiments and can be easily extended to other forms of hyperelastic material models.     

A transversely isotropic biphasic model for unconfined compression of growth plate and chondroepiphysis.

Using the biphasic theory for hydrated soft tissues (Mow et al., 1980) and a transversely isotropic elastic model for the solid matrix, an analytical solution is presented for the unconfined compression of cylindrical disks of growth plate tissues compressed between two rigid platens with a frictionless interface. The axisymmetric case where the plane of transverse isotropy is perpendicular to the cylindrical axis is studied, and the stress-relaxation response to imposed step and ramp displacements is solved. This solution is then used to analyze experimental data from unconfined compression stress-relaxation tests performed on specimens from bovine distal ulnar growth plate and chondroepiphysis to determine the biphasic material parameters. The transversely isotropic biphasic model provides an excellent agreement between theory and experimental results, better than was previously achieved with an isotropic model, and can explain the observed experimental behavior in unconfined compression of these tissues.     

Compression testing of very soft biological tissues using semi-confined configuration--a word of caution

We analyse semi-confined (i.e. using no-slip boundary conditions) compression experiment of very soft tissue sample using finite element method. We show that the assumption that the planes perpendicular to the direction of the applied force remain plane during the experiments is not satisfied for compression levels lower than previously stated in Miller [2005. Method for testing very soft biological tissues in compression. Journal of Biomechanics 38, 153-158]. Therefore, we recommend that the parameters for constitutive models of very soft tissues be determined by fitting a solution of the finite element models of the experimental set-up to the measurements obtained using semi-confined compression experiments.     

Analysis of effects of friction on the deformation behavior of soft tissues in unconfined compression tests

Frictionless specimen/platen contact in unconfined compression tests has traditionally been assumed in determining material properties of soft tissues via an analytical solution. In the present study, the suitability of this assumption was examined using a finite element method. The effect of the specimen/platen friction on the mechanical characteristics of soft tissues in unconfined compression was analyzed based on the published experimental data of three different materials (pigskin, pig brain, and human calcaneal fat). The soft tissues were considered to be nonlinear and viscoelastic; the friction coefficient at the contact interface between the specimens and platens was assumed to vary from 0.0 to 0.5. Our numerical simulations show that the tissue specimens are, due to the specimen/platen friction, not compressed in a uniform stress/strain state, as has been traditionally assumed in analytical analysis. The stress of the specimens obtained with the specimen/platen friction can be greater than those with the frictionless specimen/platen contact by more than 50%, even in well-controlled test conditions.     

Inexpensive, semi-automated system for measuring mechanical properties of soft tissues.

Stiffness and strength are important properties of many tissues, but standard material-testing equipment is expensive, often ill-suited for testing soft tissues, and rarely accessible to biologists. We describe a system built around a microcomputer and an electronic balance which is particularly well-suited for measuring stress and strain in small samples of soft tissue. We use a discarded floppy disk drive as a linear actuator to strain the sample, while an electronic balance measures the tension (used to calculate stress). We give an algorithm for a program to drive a microcomputer which controls the floppy disk drive via its parallel port and records the balance measurements via its serial port. We used this system to obtain stress-strain curves from a sample of latex rubber and a sample of soft insect cuticle. Three tests of the rubber sample gave nearly identical results, with smooth, J-shaped stress-strain curves. The stress-strain curves gave a modulus elasticity value of 1.72 Mpa over the steep, straight region, well within the range for natural latex rubber. We also tested a sample of abdominal cuticle from a caterpillar (Manduca sexta). The caterpillar cuticle had a J-shaped stress-strain curve with a modulus of elasticity of 2.11 Mpa over the steep part of the curve. J. Exp. Zool. 284:374-378, 1999.     

Lubricated squeezing flow: a useful method for measuring the viscoelastic properties of soft tissues

Conducting experiments on very soft biological tissues can be difficult. Traditionally, unconfined compression and shear have been used. Here, an improved method of compression testing, lubricated squeezing flow is described. This gives a uniform compression along the squeezing axis and almost uniform equi-biaxial elongation at right angles to the squeezing axis, with minimal shear deformation due to the constant lubrication of the sample surfaces during testing. Sample results for porcine liver obtained using this method are described here.     

Biomechanical properties of abdominal organs in vivo and postmortem under compression loads

Accurate knowledge of biomechanical characteristics of tissues is essential for developing realistic computer-based surgical simulators incorporating haptic feedback, as well as for the design of surgical robots and tools. As simulation technologies continue to be capable of modeling more complex behavior, an in vivo tissue property database is needed. Most past and current biomechanical research is focused on soft and hard anatomical structures that are subject to physiological loading, testing the organs in situ. Internal organs are different in that respect since they are not subject to extensive loads as part of their regular physiological function. However, during surgery, a different set of loading conditions are imposed on these organs as a result of the interaction with the surgical tools. Following previous research studying the kinematics and dynamics of tool/tissue interaction in real surgical procedures, the focus of the current study was to obtain the structural biomechanical properties (engineering stress-strain and stress relaxation) of seven abdominal organs, including bladder, gallbladder, large and small intestines, liver, spleen, and stomach, using a porcine animal model. The organs were tested in vivo, in situ, and ex corpus (the latter two conditions being postmortem) under cyclical and step strain compressions using a motorized endoscopic grasper and a universal-testing machine. The tissues were tested with the same loading conditions commonly applied by surgeons during minimally invasive surgical procedures. Phenomenological models were developed for the various organs, testing conditions, and experimental devices. A property database-unique to the literature-has been created that contains the average elastic and relaxation model parameters measured for these tissues in vivo and postmortem. The results quantitatively indicate the significant differences between tissue properties measured in vivo and postmortem. A quantitative understanding of how the unconditioned tissue properties and model parameters are influenced by time postmortem and loading condition has been obtained. The results provide the material property foundations for developing science-based haptic surgical simulators, as well as surgical tools for manual and robotic systems.     

A nonlinear viscoelastic finite element model of polyethylene

A nonlinear viscoelastic finite element model of ultra-high molecular weight polyethylene (UHMWPE) was developed in this study. Eight cylindrical specimens were machined from ram extruded UHMWPE bar stock (GUR 1020) and tested under constant compression at 7% strain for 100 sec. The stress strain data during the initial ramp up to 7% strain was utilized to model the "instantaneous" stress-strain response using a Mooney-Rivlin material model. The viscoelastic behavior was modeled using the time-dependent relaxation in stress seen after the initial maximum stress was achieved using a stored energy formulation. A cylindrical model of similar dimensions was created using a finite element analysis software program. The cylinder was made up of hexahedral elements, which were given the material properties utilizing the "instantaneous" stress-strain curve and the energy-relaxation curve obtained from the experimental data. The cylinder was compressed between two flat rigid bodies that simulated the fixtures of the testing machine. Experimental stress-relaxation, creep and dynamic testing data were then used to validate the model. The mean error for predicted versus experimental data for stress relaxation at different strain levels was 4.2%. The mean error for the creep test was 7% and for dynamic test was 5.4%. Finally, dynamic loading in a hip arthroplasty was modeled and validated experimentally with an error of 8%. This study establishes a working finite element material model of UHMWPE that can be utilized to simulate a variety of postoperative arthroplasty conditions.     

High strain rate compressive properties of bovine muscle tissue determined using a split Hopkinson bar apparatus

The polymeric split Hopkinson pressure bar (PSHPB) apparatus is introduced as a means for measuring the high strain rate (1,000-2,500 s(-1)) compressive properties of soft tissues. Issues related to specimen design are discussed, and protocols are presented for specimen preparation. Proposed specimen geometries were validated using high-speed photography. Stress-strain data were obtained for high strain rate compression of bovine muscle tissue to strains as high as 80%. The stress-strain curves were found to be strain rate-sensitive and concave upward, as is typical of soft tissues. Rigor had a significant impact on the material properties between 5 and 24 h post mortem, while at longer times, properties returned essentially to their pre-rigor values. This study presents some of the first published high rate properties of muscle tissue, data that are urgently for advanced modeling of the human body and for evaluation of safety systems for the human body.     

Non-ideal effects in indentation testing of soft tissues

Indentation has several advantages as a loading mode for determining constitutive behavior of soft, biological tissues. However, indentation induces a complex, spatially heterogeneous deformation field that creates analytical challenges for the calculation of constitutive parameters. As a result, investigators commonly assume small indentation depths and large sample thicknesses to simplify analysis and then restrict indentation depth and sample geometry to satisfy these assumptions. These restrictions limit experimental resolution in some fields, such as brain biomechanics. However, recent experimental evidence suggests that conventionally applied limits are in fact excessively conservative. We conducted a parametric study of indentation loading with various indenter geometries, surface interface conditions, sample compressibility, sample geometry and indentation depth to quantitatively describe the deviation from previous treatments that results from violation of the assumptions of small indentation depth and large sample thickness. We found that the classical solution was surprisingly robust to violation of the assumption of small strain but highly sensitive to violation of the assumption of large sample thickness, particularly if the indenter was cylindrical. The ramifications of these findings for design of indentation experiments are discussed and correction factors are presented to allow future investigators to account for these effects without recreating our finite element models.     

Effect of non-uniform thickness of samples in stress relaxation tests under unconfined compression of samples of articular discs

A precise information of the biomechanical properties of soft tissues is required to develop a suitable simulation model, with which the distribution of stress and strain in the complex structures can be estimated. Many soft tissues have been mechanically characterized by stress relaxation tests under unconfined or confined compression. In general, full-thickness samples are extracted to reduce the damage in the tissue as much as possible. However, it is not guaranteed that these samples have a uniform thickness or, in other words, planar parallel faces. In particular, in the articular disc of the temporomandibular joint, many studies can be found testing full-thickness samples for which that thickness is known to be non-uniform, while making the assumption of uniaxial stress state to extract the mechanical properties from those tests. That inaccuracy may have a strong influence in some cases and needs a profound revision. The main goal of this work is to quantify the error committed in that assumption and the influence of the variation of thickness on that error in a particular test: stress relaxation tests under unconfined compression. Based on this error and defining an allowable tolerance, a criterion is established to reject samples depending on their aspect ratio.     

A device for testing mechanical properties of biological materials--the "Biodyne".

An electromechanical servo-controlled device has been developed. This device can be used to test the mechanical behavior of a wide variety of biological soft tissues. Control and execution of material testing procedures such as stress-strain, vibration, relaxation, creep etc. can be performed by manual operation of the device or by interfacing it with a laboratory type minicomputer. Experiments on excitable tissues such as muscle can also be executed. The design details and system performance are discussed.     

Effects of friction on the unconfined compressive response of articular cartilage: a finite element analysis

A finite element analysis is used to study a previously unresolved issue of the effects of platen-specimen friction on the response of the unconfined compression test; effects of platen permeability are also determined. The finite element formulation is based on the linear KLM biphasic model for articular cartilage and other hydrated soft tissues. A Galerkin weighted residual method is applied to both the solid phase and the fluid phase, and the continuity equation for the intrinsically incompressible binary mixture is introduced via a penalty method. The solid phase displacements and fluid phase velocities are interpolated for each element in terms of unknown nodal values, producing a system of first order differential equations which are solved using a standard numerical finite difference technique. An axisymmetric element of quadrilateral cross-section is developed and applied to the mechanical test problem of a cylindrical specimen of soft tissue in unconfined compression. These studies show that interfacial friction plays a major role in the unconfined compression response of articular cartilage specimens with small thickness to diameter ratios.     

Subject-specific non-linear biomechanical model of needle insertion into brain

The previous models for predicting the forces acting on a needle during insertion into very soft organs (such as, e.g. brain) relied on oversimplifying assumptions of linear elasticity and specific experimentally derived functions for determining needle-tissue interactions. In this contribution, we propose a more general approach in which the needle forces are determined directly from the equations of continuum mechanics using fully non-linear finite element procedures that account for large deformations (geometric non-linearity) and non-linear stress-strain relationship (material non-linearity) of soft tissues. We applied these procedures to model needle insertion into a swine brain using the constitutive properties determined from the experiments on tissue samples obtained from the same brain (i.e. the subject-specific constitutive properties were used). We focused on the insertion phase preceding puncture of the brain meninges and obtained a very accurate prediction of the needle force. This demonstrates the utility of non-linear finite element procedures in patient-specific modelling of needle insertion into soft organs such as, e.g. brain.     

Technical Note: Modelling Soft Tissue Using Biphasic Theory - A Word of Caution

In recent years the biphasic approach initiated by Mow and coworkers, has been very popular in modelling soft, hydrated, cartilage tissues as well as other soft tissues, such as the brain. This work points out that due to the inherent inability of biphasic models in their present form to account for stress-strain rate dependence resulting from the viscoelasticity of the solid phase, the applicability of these models is limited to the loading conditions producing large relative velocities of phases.     

Modelling Soft Tissue Using Biphasic Theory — A Word of Caution

In recent years the biphasic approach initiated by Mow and coworkers, has been very popular in modelling soft, hydrated, cartilage tissues as well as other soft tissues, such as the brain. This work points out that due to the inherent inability of biphasic models in their present form to account for stress-strain rate dependence resulting from the viscoelasticity of the solid phase, the applicability of these models is limited to the loading conditions producing large relative velocities of phases.     

Constitutive model for brain tissue under finite compression

While advances in computational models of mechanical phenomena have made it possible to simulate dynamically complex problems in biomechanics, accurate material models for soft tissues, particularly brain tissue, have proven to be very challenging. Most studies in the literature on material properties of brain tissue are performed in shear loading and very few tackle the behavior of brain in compression. In this study, a viscoelastic constitutive model of bovine brain tissue under finite step-and-hold uniaxial compression with 10 s(-1) ramp rate and 20 s hold time has been developed. The assumption of quasi-linear viscoelasticity (QLV) was validated for strain levels of up to 35%. A generalized Rivlin model was used for the isochoric part of the deformation and it was shown that at least three terms (C(10), C(01) and C(11)) are needed to accurately capture the material behavior. Furthermore, for the volumetric deformation, a two parameter Ogden model was used and the extent of material incompressibility was studied. The hyperelastic material parameters were determined through extracting and fitting to two isochronous curves (0.06 s and 14 s) approximating the instantaneous and steady-state elastic responses. Viscoelastic relaxation was characterized at five decay rates (100, 10, 1, 0.1, 0 s(-1)) and the results in compression and their extrapolation to tension were compared against previous models.