Simple uniaxial and uniform biaxial deformation of nearly isotropic incompressible tissues

Recent advances in computational mechanics of the human knee joint

Computational mechanics has been advanced in every area of orthopedic biomechanics. The objective of this paper is to provide a general review of the computational models used in the analysis of the mechanical function of the knee joint in different loading and pathological conditions. Major review articles published in related areas are summarized first. The constitutive models for soft tissues of the knee are briefly discussed to facilitate understanding the joint modeling. A detailed review of the tibiofemoral joint models is presented thereafter. The geometry reconstruction procedures as well as some critical issues in finite element modeling are also discussed. Computational modeling can be a reliable and effective method for the study of mechanical behavior of the knee joint, if the model is constructed correctly. Single-phase material models have been used to predict the instantaneous load response for the healthy knees and repaired joints, such as total and partial meniscectomies, ACL and PCL reconstructions, and joint replacements. Recently, poromechanical models accounting for fluid pressurization in soft tissues have been proposed to study the viscoelastic response of the healthy and impaired knee joints. While the constitutive modeling has been considerably advanced at the tissue level, many challenges still exist in applying a good material model to three-dimensional joint simulations. A complete model validation at the joint level seems impossible presently, because only simple data can be obtained experimentally. Therefore, model validation may be concentrated on the constitutive laws using multiple mechanical tests of the tissues. Extensive model verifications at the joint level are still crucial for the accuracy of the modeling.

Review: Computer Methods in Membrane Biomechanics

The purpose of this paper is twofold: first, to review analytical, experimental, and numerical methods for studying the nonlinear, pseudoelastic behavior of membranes of interest in biomechanics, and second, to present illustrative examples from the literature for a variety of biomembranes (e.g., skin, pericardium, pleura, aneurysms, and cells) as well as elastomeric membranes used in balloon catheters and new cell stretching tests. Although a membrane approach affords great simplifications in comparison to the three-dimensional theory of nonlinear elasticity, associated problems are still challenging. Computer-based methods are essential, therefore, for performing the requisite experiments, analyzing data, and solving boundary and initial value problems. Emphasis is on stable equilibria although material instabilities and elastodynamics are discussed.

Mechanics of native bovine pericardium. II. A structure based model for the anisotropic mechanical behaviour of the tissue

A constitutive law to describe the mechanical behaviour of native bovine pericardial samples is proposed. The model is based upon an idealization of the tissue structure and in particular makes use of the preferential orientation of the collagen and elastin fibres in the plane of the tissue along a mechanically predominant direction. This non-axisymmetric layout of the tissue fibres has been shown in the past to cause a systematic variation of the tensile strength and the stiffness with the angle of orientation. The proposed constitutive relation provided a very good fit for results collected along two orthogonal principal directions in the past in uniaxial and biaxial tensile testing modes. It also allowed a better understanding of the interaction of the various load bearing elements and was able to describe some strange features of the tissue's anisotropic mechanical behaviour.

Viscoelastic properties of the visceral pleura and its contribution to lung impedance

The mechanical impedances of 10 dog lung lobes (ZL) and circular pleura samples 1.6 cm in diameter (Zpl) were measured with small-amplitude forced oscillations between 0.2 and 4.2 Hz. Two to four samples were ablated from each lobe after their in situ tension had been fixed at 5 cmH2O transpulmonary pressure with plastic rings. Lobe resistance was inversely proportional to frequency (f) and lobe elastance increased linearly with the logarithm of f by 23%/decade. The real part of Zpl (Rpl) decreased hyperbolically with f. Pleural elastance (Epl) showed only a 5%/decade increase with log f. The regional variability of Epl was large and the data allowed the lumping of Epl into only two groups. The variability of Epl was higher in the group of Epl values from the coastal surfaces than in the group of Epl values from any other surfaces (3105 +/- 2741 (SD) vs 2263 +/- 1152 cm H2O/L). The mean intraindividual variation of Epl corresponding to costal and to other surfaces was 38 +/- 25 and 32 +/- 23%, respectively. The hysteresivity index (Fredberg and Stamenovic, J. Appl. Physiol., 67: 2408-2419, 1989) of the pleura was significantly smaller than that of the lobes (0.025 vs 0.148). Extrapolation of Zpl to the entire lobe surface predicted pleural/lobar resistance and elastance ratios of 2.7 +/- 0.4 and 16.8 +/- 10.6%, respectively, at 0.2 Hz, and 0.4 +/- 0.4 and 15.5 +/- 10.6%, respectively, at 4.2 Hz. This suggests that for small deformations and medium lung volumes the pleural contribution to ZL is almost ideally elastic and only slightly frequency-dependent.

Stress-strain characteristics of normal and emphysematous hamster lung strips

A simple mathematical model of the one dimensional, stress-strain behavior of hamster lung tissue based on strain energy considerations was tested in degassed, uniaxially stretched strips obtained from normal and emphysematous hamster lungs cycled in saline. The relationship between Eulerian stress (sigma) and extension ratio (lambda) was found to take the form sigma = (lambda 2-1/lambda) x f(lambda) where the function f(lambda) was experimentally determined. Stress in six normal and five emphysematous strips was calculated by dividing the tension at each stretch increment by the strip cross-sectional area. Plotting sigma lambda/(lambda 2-1) versus a function of the form e eta lambda yielded a linear expression for f(lambda), me eta lambda + b, where n = 2. The complete stress-strain behavior of hamster lung strip tissue could then be expressed as a simple function of lambda over a range of lambda = 1.0-2.0: sigma = (lambda 2-1/lambda)(me2 lambda+b) The values of the constants m and b depend solely upon the mechanical properties of the elastic and collagen fiber networks in these atelectatic, saline cycled lung strips. The slope m = 0.151, and the intercept b = 0.416 in normal strips (r = 0.98). In emphysematous strips m = 0.016 and b = -0.199 (r = 0.82). Given the smaller m found for emphysematous strips, less strain energy accumulated with increasing stretch and did not even begin in these strips until lambda = 1.3. Further, the fit of the equation to the data was not as good for emphysematous as for normal strips. We conclude that the above equation adequately describes the stress-strain properties of normal hamster lung strips tissue but is not as good in emphysematous strips where the disease is patchy.

Experimental determination of the force required for insertion of a thermoseed into deep brain tissues

Our laboratories are developing a new technique for delivering localized hyperthermia to deep-seated brain tumors. In this technique, a spherical thermoseed is stereotactically navigated through the brain and tumour tissues via the noncontact application of an external magnetic force. The force required to produce motion of a 3 mm diameter sphere through in vitro brain tissues was measured to be 0.07 +/- 0.03 N. This result was obtained from a series of experiments performed on whole brain specimens extracted from adult canines. Data were also taken with a 3 mm x 3 mm cylinder and a 5 mm sphere. An experimental procedure simulating physiological conditions was developed prior to testing. Evaluations of systematic effects included determinations of the calibration uncertainties, tests of the dependence of the measured force on temperature, and studies of the effects of method of storage of the tissue specimens. The results obtained are compared with (and confirmed by) two different series of experiments performed in vivo on adult canines and with another series of experiments using brain phantom gelatin.

The bovine pericardial xenograft: II. Effect of tethering or pressurization during fixation on the tensile viscoelastic properties of bovine pericardium

Our previous article suggested that control of the extensibility of aldehyde-fixed pericardium could be achieved by controlling shrinkage during fixation. Therefore, to prevent shrinkage, we have used sandpaper-lined plexiglass plates to clamp circular samples of bovine pericardium during fixation in glutaraldehyde, tethering them at their original dimensions. As well, we have applied transmural pressures of 50 or 100 mm Hg during fixation using a hydraulic column of glutaraldehyde solution. Strips cut at 0 degree, 30 degrees, 60 degrees, and 90 degrees to the base-to-apex cardiac direction have been examined for cyclic stress-strain response, stress relaxation, plastic deformation, and fracture behavior. Under physiological stresses, tethered and pressure-fixed materials were both nearly isotropic. Tethering during fixation produced a material with extensibility nearly identical to that of fresh tissue. Plastic deformation during cyclic loading was reduced below that seen in simple fixation while stress relaxation was unchanged. Pressure-fixation produced reduced extensibility similar to that produced in porcine aortic valve leaflets. Plastic deformation and stress relaxation were both markedly reduced. Pressure-fixation reduced the strain at fracture, but fracture behavior was otherwise unaffected. Tethering and pressure-fixation offer attractive means to control the mechanical behavior of bovine xenograft materials.

Modelling of the heart and pericardium at end-diastole

Herein we present a refined version of Vito's two-sphere static model of the heart with pericardium and discuss its possible applications. The improvements we make on Vito's model are: (i) Vito assumed that the elastic materials which constitute the model 'heart' and 'pericardium' are isotropic; we relax this assumption to that of transverse-isotropy. (ii) Our analysis, which does not assume the existence of stored-energy functions, links the model directly to empirical stress-strain relations of suitable biaxial uniform-extension tests; two such stress-strain relations (one for the pericardium, one for the myocardium, both of which may be described by the same equation except for difference in the values of response parameters) now define the model completely, so we avoid altogether the difficult task of determining full-fledged constitutive equations for the pericardium and myocardium. As for applications, we contend that the concentric spheres in static equilibrium can be taken as a model of the left ventricle and pericardium at end-diastole. We show that the model when equipped with suitable stress-strain relations does give good fit to the pressure-volume data which Spotnitz et al. (1966, Circulation Res., 18, 49-66) obtained from excised canine left ventricles and to the pericardium data which Pegram et al. (1975, Circulation Res., 9, 707-714) obtained from closed chest, anaesthetized dogs. Three different empirical formulae were tried in the data-fitting as the equation that describes the requisite stress-strain relations. The 'exponential law' gave the best results.

Quantification of the mechanical properties of noncontracting canine myocardium under simultaneous biaxial loading

Detailed understanding of cardiac mechanics depends upon accurate and complete characterization of the three-dimensional properties of both normal and diseased myocardial tissue. This, however, can only be obtained by performing multiaxial tests on cardiac tissue. In this study we subjected thin sheets of passive canine left ventricular myocardium to various combinations of simultaneous biaxial stretching. During each stretch the ratio of the orthogonal strains was kept constant and the corresponding stresses remained proportional. We fitted the biaxial stress-strain data both with exponential strain-energy functions with quadratic powers of strains as well as with an alternative function with nonintegral powers of strains. We used our recently developed nonparametric method to assess the reliability of the coefficients for each of these functions. The quadratic strain-energy functions resulted in wide intra- and interspecimen variability in the coefficients. Moreover, both their absolute and relative values demonstrated marked load history dependence such that interpretation of the direction of anisotropy was difficult. Fitting the data with the alternative nonintegral strain-energy function seemed to alleviate these problems. This alternative strain-energy function may provide more self-consistent results than the more commonly used quadratic strain-energy functions.

Mechanical behavior of excised canine visceral pleura

A computer-controlled optical electromechanical biaxial test system was employed to study the mechanical response of excised sheets of canine visceral pleura. Three classes of tests were performed: uniform biaxial stretching tests and tests in which the specimen was cyclically stretched along one axis while either the load or dimension was maintained at a prescribed level in the orthogonal direction. The tests were defined completely within the software. Strain was inferred from tracking four particles affixed to the central region of the specimen surface. The visceral pleura was found to behave similarly to other biological soft tissues and required preconditioning to yield repeatable responses. In addition, the visceral pleura appeared to possess in-plane transverse isotropic material symmetry and to exhibit strong in-plane mechanical coupling at lower loads. The data presented herein is sufficient for determination of certain three-dimensional constitutive laws which are essential for further biomechanical analyses of the visceral pleura's role in lung response.

An approach to quantification of biaxial tissue stress-strain data

Delineation of the mechanical properties of biologic tissues is one of the cornerstones of biomechanics. Abundant data from uniaxial tests exist but these cannot be extrapolated to describe three-dimensional properties of tissue. Biaxial stress-strain studies have been performed using skin, blood vessels and pericardium. Quantitative description of tissue properties in these studies has employed either polynomial or exponential strain-energy functions. Interpretation of these data, however, is difficult because of wide variability of the estimated coefficients of these functions. This variability has been attributed to experimental noise, numerical instabilities in the algorithms, or to strain-history dependence. No systematic method has been proposed to evaluate the variability. This paper describes a statistically based approach to assessing the sources of and accounting for variability of coefficients in describing biaxial stress-strain data. Our data are from canine pericardium subjected to various combinations of simultaneous biaxial stretching. We first determine a suitable strain-energy function with the least number of free parameters that will fit the data reasonably. We then perform residual analysis to see if standard statistical methods can be used to assess the variability. If not, we use a nonparametric method called bootstrapping that is suitable for assessing the uncertainty in the coefficients. Using a five-parameter exponential strain-energy function, pericardial tissue is found to be strain-history dependent and anisotropic. These findings cannot be attributed to either experimental noise or instability in the numerical algorithms.

Analysis of the pressure-volume relationship of excised lungs

The pressure-volume relationship of excised lungs is explicitly defined in the form of a mathematical model. In the model, lung volume (V) is given by the function V = VmaxF(Ptp,T*)H(Ptp). Vmax is maximum lung volume. F, which describes the recruitment of air-filled units, is a function of transpulmonary pressure (Ptp) and surface tension (T*), whereas H, which is also a function of transpulmonary pressure, describes the expansion of recruited units against tissue forces. F is shown to be the integral of the normalized distribution function of the lung units and remains constant so long as the number of air-filled units does not change. H, on the other hand, is shown to be the product of the elastic properties of the tissues and is responsible for the characteristic non-linear sigmoid shape of lung deflation curves. Results obtained with the model are consistent with the hypothesis that tissue elasticity, tissue hysteresis, area dependent surface tension, and recruitment share responsibility for the characteristic hysteresis of excised lungs.

Mechanical suitability of glycerol-preserved human dura mater for construction of prosthetic cardiac valves

We have examined the tensile viscoelastic properties of fresh and glycerol-preserved human dura mater, and correlated the results with structural information from the scanning electron microscope. The interwoven laminar structure of dura produces rather high flexural stiffness, while the crossed-fibrillar laminae produce planar mechanical isotropy. Glycerol storage shifts the stress-strain curve to lower strain, reduces stress relaxation and creep, and lowers the ultimate tensile strength and strain at fracture. These changes may be due to glyceraldehyde crosslinking, or to increased interfibrillar friction. The latter hypothesis suggests that glycerol storage may reduce the fatigue lifetime of the tissue.

Constitutive equations for fibrous connective tissues

A general multiaxial theory for the constitutive relations in fibrous connective tissues is developed on the basis of microstructural and thermodynamic considerations. It is compatible with existing general material theories. In elastic tissues, the theory considers the strain-energy function to be the sum of strain-energies of the tissue's components. The stresses are derived from this strain-energy function. Viscoelastic constitutive relations are obtained in an analogous manner. Few examples are developed in detail. The results of the present strain-energy based theory are identical with those of the author's previous structural models (Lanir, 1979a, b) which are based on detailed equilibrium analysis. It turns out, however, that the analytical work involved in solving boundary value problems is considerably shorter if the present theory is used. The advantages of structural theories in avoiding ambiguity in material characterization and in offering an insight into the function, structure and mechanics of tissue components are discussed.

The elastic behavior of the urinary bladder for large deformations

The purpose of this paper is to investigate the theoretical basis for the pressure-distension behavior of the urinary bladder. A finite strain theory is developed for hollow spherical structures and it is shown that the Treloar model is a good prototype only for rubber balloons. The pressure-extension ratio relationship is inverted to lead a general form of strain energy function, and fitted by an empirical relation involving one exponential. The following form of strain energy function is derived: W(lambda, lambda, lambda -2) = C1 (P(1), a) + P(1)C2 (a, lambda)ea(lambda -1). Where C1(P(1), a) is a constant (N m-2), P(1) is the initial pressure, a is the rate of pressure increase and C2 (a, lambda) a third degree polynomial relation. P(1) and a are experimentally determined through volumetric pressure-distension data. It is verified that this type of energy function is also valid for uniaxial loading experiments by testing strips coming from the same bladder for which P(1) and a were computed. There is a good agreement between the experimental points and the theoretical stress-strain relation. Finally, the strain energy function is plotted as a function of the first strain invariant and appears to be of an exponential nature.