Large deformation of isotropic biological tissue

The requirements for the development of a general potential function describing the behavior of homogeneous, isotropic biological tissue are discussed and such a function is proposed. It is shown that this function yields both tensile and torsional results compatible with existing data.     

Linear viscoelastic properties of bovine brain tissue in shear

We report the results from a series of rheological tests of fresh bovine brain tissue. Using a standard Bohlin VOR shear rheometer, shear relaxation and oscillating strain sweep experiments were performed on disks of brain tissue 30 mm in diameter, with a thickness of 1.5–2 mm. The strain sweep experiment showed that the viscoelastic strain limit is of the order of 0.1% strain. Shear relaxation data do not indicate the presence of a long-term elastic modulus, indicating fluid-like behavior. A relaxation spectrum was calculated by inverting the experimental data and used to predict oscillatory response, which agreed well with measured data.     

The shear mechanical properties of diabetic and non-diabetic plantar soft tissue

Changes in the plantar soft tissue shear properties may contribute to ulceration in diabetic patients, however, little is known about these shear parameters. This study examines the elastic and viscoelastic shear behavior of both diabetic and non-diabetic plantar tissue. Previously compression tested plantar tissue specimens (n=54) at six relevant plantar locations (hallux, first, third, and fifth metatarsal heads, lateral midfoot, and calcaneus) from four cadaveric diabetic feet and five non-diabetic feet were utilized. Per in vivo data (i.e., combined deformation patterns of compression followed by shear), an initial static compressive strain (36-38%) was applied to the tissue followed by target shear strains of 50% and 85% of initial thickness. Triangle waves were used to quantify elastic parameters at both strain levels and a stress relaxation test (0.25 s ramp and 300 s hold) was used to quantify the viscoelastic parameters at the upper strain level. Several differences were found between test groups including a 52-62% increase in peak shear stress, a 63% increase in toe shear modulus, a 47% increase in final shear modulus, and a 67% increase in middle slope magnitude (sharper drop in relaxation) in the diabetic tissue. Beyond a 54% greater peak compressive stress in the third metatarsal compared to the lateral midfoot, there were no differences in shear properties between plantar locations. Notably, this study demonstrates that plantar soft tissue with diabetes is stiffer than healthy tissue, thereby compromising its ability to dissipate shear stresses borne by the foot that may increase ulceration risk.     

Large deformation of elastic tubes

In this paper a study is made of the large deformation of elastic tubes. Analysis is confined to the case of statical deformation. The energy of deformation is considered to be a function of the three strain invariants only. In order to determine its form for actual materials it is expanded in its Taylor's series. From the theoretical analysis of the inflation and extension of tubes, linear algebraic equations in the elastic constants are developed. Using these and appropriate measurements of diametral change and longitudinal stretch, obtained from experiments, elastic constants for the non-linear theory of elasticity are explicity determined with the aid of a computer program. The highly non-linear forms of deformation of tubes computed on the basis of the theory are compared with our experimental findings for a latex rubber tubing and for some bio-physical data taken from the literature.     

Scale-dependent mechanical properties of native and decellularized liver tissue

Decellularization, a technique used in liver regenerative medicine, is the removal of all the cellular components from a tissue or organ, leaving behind an intact structure of extracellular matrix. The biomechanical properties of this novel scaffold material are currently unknown and are important due to the mechanosensitivity of liver cells. Characterizing this material is important for bioengineering liver tissue from this decellularized scaffold as well as creating new 3-dimensional mimetic structures of liver extracellular matrix. This study set out to characterize the biomechanical properties of perfused liver tissue in its native and decellularized states on both a macro- and nano-scale. Poroviscoelastic finite element models were then used to extract the fluid and solid mechanical properties from the experimental data. Tissue-level spherical indentation-relaxation tests were performed on 5 native livers and 8 decellularized livers at two indentation rates and at multiple perfusion rates. Cellular-level spherical nanoindentation was performed on 2 native livers and 1 decellularized liver. Tissue-level results found native liver tissue to possess a long-term Young’s modulus of 10.5 kPa and decellularized tissue a modulus of 1.18 kPa. Cellular-level testing found native tissue to have a long-term Young’s modulus of 4.40 kPa and decellularized tissue to have a modulus of 0.91 kPa. These results are important for regenerative medicine and tissue engineering where cellular response is dependent on the mechanical properties of the engineered scaffold.     

Large deformation analysis of orthodontic appliances

The deformations of orthodontic appliances used for space closure are large so that any mathematical analysis will require a nonlinear approach. Existing incremental finite element and finite difference numerical methods suffer from excessive computational effort when analyzing these problems. An accurate segmental technique is proposed to handle these difficulties in an extremely efficient fashion. The segmental technique starts by assuming that an orthodontic appliance is composed of a number of smaller segments, the ends of which undergo small relative rotation. With an appropriate choice of local coordinate system the equilibrium equations for each segment are linearized and solved in a straightforward manner. The segments are then assembled using geometric and force compatibility relations similar to the transfer matrix method. Consequently, the original nonlinear boundary value problem is solved as a sequence of linear initial value problems which converge to the required boundary conditions. As only one segment need be considered at a time, the computations can be performed accurately and efficiently on a PC type computer. Although an iterative solution is used to match the boundary conditions, the time required to solve a given problem ranges from a few seconds to a couple of minutes depending on the initial geometric complexity. The accuracy of the segmental technique is verified by comparison with an exact solution for an initially curved cantilever beam with an end load. In addition, comparisons are made with existing experimental and numerical results as well as with a new set of experimental data. In all cases the segmental technique is in excellent agreement with the results of these other studies.     

Combined effects of dynamic tissue shear deformation and insulin-like growth factor I on chondrocyte biosynthesis in cartilage explants

Biophysical forces and biochemical factors play crucial roles in the maintenance of the integrity of articular cartilage. In this study, we explored the effect of dynamic tissue shear deformation and insulin-like growth factor I (IGF-I) on matrix synthesis by chondrocytes within native cartilage explants. Dynamic tissue shear in the range of 0.5-6% strain amplitude at 0.1 Hz was applied to cartilage explants cultured in serum-free medium. Dynamic tissue shear above 1.5% strain amplitude significantly stimulated protein and proteoglycan synthesis, by maximum values of 35 and 25%, respectively, over statically held control specimens. In the absence of tissue shear, IGF-I augmented protein and proteoglycan synthesis up to twofold at IGF-I concentrations in the range of 100-300 ng/ml. When tissue shear and IGF-I stimuli were combined, matrix biosynthesis levels were significantly higher than the maximal effect caused by either stimulus alone. However, there was no significant interaction between tissue shear and IGF-I as determined by two-way ANOVA. We then quantified the effect of dynamic tissue shear on the transport of IGF-I into and within cartilage explants. [125I]IGF-I was added to the medium, and the levels of intratissue [125I]IGF-I were directly measured as a function of time over 48 h in the presence and absence of continuous dynamic shear strain. Dynamic shear did not alter the rate of uptake of [125I]IGF-I into the explants, suggesting that convective diffusion of [125I]IGF-I is negligible under the shear strain conditions used. This is in marked contrast to the enhancement of transport reported in response to uniaxial dynamic compression. Taken together, these data suggest that (1) the stimulatory effect of tissue shear is via mechanotransduction pathways and not by facilitated transport of biochemical factors and (2) chondrocytes may possess complementary signal transduction pathways for biophysical and biochemical factors leading to changes in metabolic activity.     

Large strain behaviour of brain tissue in shear: some experimental data and differential constitutive model.

In this paper, some experimental measurements of the behaviour of bovine brain tissue under large shear strains in vitro are reported, and a constitutive model which is consistent with the data is developed. It was determined that brain tissue is not strain-time separable, showing slower relaxation at higher strains, and that the stresses in shear are not linear with increasing shear strain. The new constitutive model is a differential model, including both an "elastic" term, of the Mooney type and a nonlinear viscoelastic term. The latter allows for the change in relaxation behaviour with strain, by modifying an upper convected multimode Maxwell model with a damping function. The model shows good agreement with the experimental shear results and could be used to describe other types of data.     

Coupling bending and shear effects on liposome deformation

Cell membrane deformation induced by external mechanical stimuli has been studied extensively over the past three decades. The present study focuses on the coupling of in-plane shear H and out-of-plane bending B of liposome membrane and its influences on the deformation of a single vesicle subjected to (i) external compressive load via two parallel platens and (ii) contact forces caused by a rigid substrate. Our results show that the increase of membrane resultant stress in both loading configurations causes the liposome to become more rigid and the degree of vesicle deformation decreases when the in-plane shearing effect is dominant. A theoretical approach is developed to facilitate cell membrane characterization under different biomechanical stimuli.     

Nanoscale shear deformation mechanisms of opposing cartilage aggrecan macromolecules

The nanoscale shear deformation behavior of two opposing end-grafted aggrecan layers was studied in aqueous solutions using atomic force microscopy, and was observed to depend markedly on bath ionic strength, the presence of calcium ions, and the applied lateral displacement rate. These results provide molecular-level insights into the contribution of aggrecan deformation mechanisms to cartilage tissue-level material properties.     

Large deformation mechanics of the enucleated eyeball

Large deformation of enucleated pig eyeballs under rigid cylindrical indenters was studied analytically and experimentally. The analytic model for the eyeball consists of a fluid-filled spherical membrane composed of an incompressible, elastic material with an exponential strain energy function. The Rayleigh-Ritz technique provided an approximate solution via a potential energy formulation. Comparison with results from tests on eyeballs and a water-filled rubber (Mooney-Rivlin) shell shows good agreement at large deflection, where membrane action dominates. Due to the highly nonlinear stress-strain relations for the sclera, the load remains relatively small until the indenter displacement approaches 40-60 percent of the eyeball radius, and then the load increases rapidly. Depending on the indenter size, either a perforation or a rupture type of failure occurs.     

Ion induced deformation of soft tissue

In this paper the effects of changing the ion concentration in and around a sample of soft tissue are investigated. The triphasic theory developed by Laiet al. (1990,Biomechanics of Diarthrodial Joints, Vol. 1, Berlin, Springer-Verlag) is reduced to two coupled partial differential equations involving fluid ion concentration and tissue solid deformation. These equations are given in general form for Cartesian, cylindrical and spherical geometries. After solving the two equations quantities such as fluid velocity, fluid pressure, chemical potentials and chemical expansion stress may be easily calculated. In the Cartesian geometry comparison is made with the experimental and theoretical work of Myerset al. (1984,ASME J. biomech. Engng,106, 151–158). This dealt with changing the ion concentration of a salt shower on a strip of bovine articular cartilage. Results were obtained in both free swelling and isometric tension states, using an empirical formula to acount for ion induced deformation. The present theory predicts lower ion concentrations inside the tissue than this earlier work. A spherical sample of tissue subjected to a change in salt bath ion concentration is also considered. Numerical results are obtained for both hypertonic and hypotonic bathing solutions. Of particular interest is the finding that tissue may contract internally before reaching a final swollen equilibrium state or swell internally before finally contracting. By considering the relative magnitude, and also variation throughout the time course of terms in the governing equations, an even simpler system is deduced. As well as being linear the concentration equation in the new system is uncoupled. Results obtained from the linear system compare well with those from the spherical section. Thus, biological swelling situations may be modelled by a simple system of equations with the possibility, of approximate analytic solutions in certain cases.     

Propagation of shear waves in muscle tissue

A mathematical model of the propagation of acoustic shear waves in muscle tissue is considered. Muscle is modeled as an incompressible transversely isotropic viscoelastic continuum with quasi-one-dimensional active tension. There are two types of shear waves in an infinite medium. Waves of the second type (transverse) propagate without decay even when myofibril viscosity is taken into account. A problem of standing transverse waves in a rectangular layer was investigated numerically. The values of the problem parameters are found for which one can easily estimate the active tension (or muscle tone) from the characteristics of standing waves. This value is informative for diagnostics of the muscle state.     

Measurement of surface deformation of soft tissue

A method is described for measuring the surface shape and deformations of soft tissue in three dimensions. The method uses close range stereophotogrammetry to record the three-dimensional locations of miniature optical targets applied to the tissue surface. This has been applied to study of human lumbar intervertebral disc. Measurements of the strain along surface annular fibers have been made under varying loads. In this case the maximum expected errors are about 0.15 mm, which corresponds to a strain of less than 1%. Preliminary findings have differed from predictions made in published mathematical models for the disc in that they show very little strain of the annulus in compression loading, but confirm axial torsional loading as liable to produce mechanical disruption of the disc annulus.     

Oscillatory deformation of human erythrocytes in sinusoidally modulated shear flow

The red cell deformation under oscillatory shear stress was studied. Shear stress was sinusoidally modulated between 8 and 32 dyn/cm2, thus, the extent of cellular deformation altered sinusoidally. At a low modulation frequency (less than 1.8 Hz), intact red cells perfectly responded to the shear stress applied on cells, and they could deform as much as the deformation in stationary shear flow. Above 2 Hz, the cellular deformation could not follow changes in shear stress along up-phase in the shear stress cycle. As decreasing the intracellular hemoglobin concentration, the cellular response to oscillatory shear stress became better. Treatment of cells with low concentrations of diamide impaired the response of intact cells to oscillatory shear stress, but unaffected the response of partially hemolyzed cells. These data suggest that the cellular response to oscillatory shear stress is determined by the cytoskeletal structure and the intracellular viscosity.     

Mechanical asymmetry during articulation of tibial and femoral cartilages: Local and overall compressive and shear deformation and properties

During knee movement, femoral cartilage articulates against cartilage from the tibial plateau, and the resulting mechanical behavior is yet to be fully characterized. The objectives of this study were to determine (1) the overall and depth-varying axial and shear strains and (2) the associated moduli, of femoral and tibial cartilages during the compression and shearing of apposing tibial and femoral samples. Osteochondral blocks from human femoral condyles (FCs) characterized as normal and donor-matched lateral tibial plateau (TP) were apposed, compressed 13%, and subjected to relative lateral motion. When surfaces began to slide, axial (?E_zz) and shear (E_xz) strains and compressive (E) and shear (G) moduli, overall and as a function of depth, were determined for femoral and tibial cartilages. Tibial ?E_zz was ～2-fold greater than FC ?E_zz near the surface (0.38 versus 0.22) and overall (0.16 versus 0.07). Near the surface, E_xz of TP was 8-fold higher than that of FC (0.41 versus 0.05), while overall E_xz was 4-fold higher (0.09 versus 0.02). For TP and FC, ?E_zz and E_xz were greatest near the surface and decreased monotonically with depth. E for FC was 1.7-fold greater than TP, both near the surface (0.40 versus 0.24 MPa) and overall (0.76 versus 0.47 MPa). Similarly, G was 7-fold greater for FC (0.22 MPa) than TP near the surface (0.03 MPa) and 3-fold higher for FC (0.38 MPa) than TP (0.13 MPa) overall. These results indicate that tibial cartilage deforms and strains more axially and in shear than the apposing femoral cartilage during tibial–femoral articulation, reflecting their respective moduli.     

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.