Electric breakdown of lipid vesicles

Theoretical expression for free energy F of spherical lipid vesicle containing through pore in the presence of diffusional potential difference is derived. It is assumed that the pore radius is small in comparison with vesicle size. According to estimation the variation of elastic energy of vesicle membrane with pore radius is small. Therefore electrical breakdown becomes reversible for reasonable region of r values. Conditions of equilibrium and dynamic modes of breakdown are analyzed. Random oscillation mode of intravesicular label discharge is shown for some region of vesicle parameters.     

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.     

Body forces and pressures in elastic models of the myocardium.

Tension strands are introduced to represent active myocardial fibers. They create one body force proportional to the divergence of the tension-direction vector, and a second equal to the tension divided by the radius of curvature. Explicit solutions to isotropic linearly elastic tensor equations with these body forces are found for the radially-symmetric, linearly-isotropic, elastic spherical heart with arbitrary radial body force. They confirm experiments showing supraluminal intramural pressures. Such pressures may affect coronary perfusion. A tension strand model which is a reasonable compromise between actual myofibrillar geometry and analytical simplicity is the iso-oblique, terminating, nonintersecting model. The body force from that or any other axially symmetric body force can be the forcing term for equations in which the heart is modeled as a thin, ellipsoidal, elastic membrane.     

Elastic instability in growing yeast colonies

The differential adhesion between cells is believed to be the major driving force behind the formation of tissues. The idea is that an aggregate of cells minimizes the overall adhesive energy between cell surfaces. We demonstrate in a model experimental system that there exist conditions where a slowly growing tissue does not minimize this adhesive energy. A mathematical model demonstrates that the instability of a spherical shape is caused by the competition between elastic and surface energies.     

Modeling electrical power absorption and thermally-induced biological tissue damage

This work develops a model for thermally induced damage from high current flow through biological tissue. Using the first law of thermodynamics, the balance of energy produced by the current and the energy absorbed by the tissue are investigated. The tissue damage is correlated with an evolution law that is activated upon exceeding a temperature threshold. As an example, the Fung material model is used. For certain parameter choices, the Fung material law has the ability to absorb relatively significant amounts of energy, due to its inherent exponential response character, thus, to some extent, mitigating possible tissue damage. Numerical examples are provided to illustrate the model’s behavior.     

Stability analysis of micropipette aspiration of neutrophils

During micropipette aspiration, neutrophil leukocytes exhibit a liquid-drop behavior, i.e., if a neutrophil is aspirated by a pressure larger than a certain threshold pressure, it flows continuously into the pipette. The point of the largest aspiration pressure at which the neutrophil can still be held in a stable equilibrium is called the critical point of aspiration. Here, we present a theoretical analysis of the equilibrium behavior and stability of a neutrophil during micropipette aspiration with the aim to rigorously characterize the critical point. We take the energy minimization approach, in which the critical point is well defined as the point of the stability breakdown. We use the basic liquid-drop model of neutrophil rheology extended by considering also the neutrophil elastic area expansivity. Our analysis predicts that the behavior at large pipette radii or small elastic area expansivity is close to the one predicted by the basic liquid-drop model, where the critical point is attained slightly before the projection length reaches the pipette radius. The effect of elastic area expansivity is qualitatively different at smaller pipette radii, where our analysis predicts that the critical point is attained at the projection lengths that may significantly exceed the pipette radius.     

Role of the membrane cortex in neutrophil deformation in small pipets.

The simplest model for a neutrophil in its "passive" state views the cell as consisting of a liquid-like cytoplasmic region surrounded by a membrane. The cell surface is in a state of isotropic contraction, which causes the cell to assume a spherical shape. This contraction is characterized by the cortical tension. The cortical tension shows a weak area dilation dependence, and it determines the elastic properties of the cell for small curvature deformations. At high curvature deformations in small pipets (with internal radii less than 1 micron), the measured critical suction pressure for cell flow into the pipet is larger than its estimate from the law of Laplace. A model is proposed where the region consisting of the cytoplasm membrane and the underlying cortex (having a finite thickness) is introduced at the cell surface. The mechanical properties of this region are characterized by the apparent cortical tension (defined as a free contraction energy per unit area) and the apparent bending modulus (introduced as a bending free energy per unit area) of its middle plane. The model predicts that for small curvature deformations (in pipets having radii larger than 1.2 microns) the role of the cortical thickness and the resistance for bending of the membrane-cortex complex is negligible. For high curvature deformations, they lead to elevated suction pressures above the values predicted from the law of Laplace. The existence of elevated suction pressures for pipets with radii from 1 micron down to 0.24 micron is found experimentally. The measured excess suction pressures cannot be explained only by the modified law of Laplace (for a cortex with finite thickness and negligible bending resistance), because it predicts unacceptable high cortical thicknesses (from 0.3 to 0.7 micron). It is concluded that the membrane-cortex complex has an apparent bending modulus from 1 x 10(-18) to 2 x 10(-18) J for a cortex with a thickness from 0.1 micron down to values much smaller than the radius of the smallest pipet (0.24 micron) used in this study.     

A membrane model from implicit elasticity theory: application to visceral pleura

A Fungean solid is derived for membranous materials as a body defined by isotropic response functions whose mathematical structure is that of a Hookean solid where the elastic constants are replaced by functions of state derived from an implicit, thermodynamic, internal energy function. The theory utilizes Biot’s (Lond Edinb Dublin Philos Mag J Sci 27:468–489, 1939) definitions for stress and strain that, in one-dimension, are the stress/strain measures adopted by Fung (Am J Physiol 28:1532–1544, 1967) when he postulated what is now known as Fung’s law. Our Fungean membrane model is parameterized against a biaxial data set acquired from a porcine pleural membrane subjected to three, sequential, proportional, planar extensions. These data support an isotropic/deviatoric split in the stress and strain-rate hypothesized by our theory. These data also demonstrate that the material response is highly nonlinear but, otherwise, mechanically isotropic. These data are described reasonably well by our otherwise simple, four-parameter, material model.     

A mathematical model of the dynamics of odontogenic cyst growth

OBJECTIVE: To formulate a mathematical model of odontogenic cyst growth and establish the dynamics of cyst enlargement and role of osmotic pressure forces throughout its growth. STUDY DESIGN: The model assumed a spherical cyst with a semipermeable lining of living cells and a core consisting of degraded cellular material, including generic osmotic material, fed by the continuous death of epithelial cells in the lining. The lining cells were assumed to have both elastic and viscous properties, reflecting the action of physical stresses by the surrounding cyst capsule, composed of fibroblasts and collagen fibers. The model couples the cyst radius and osmotic pressure differences resulting in a system of 2 nonlinear ordinary differential equations. RESULTS: The model predicts that in all parameter regimens the long-time behavior of the cyst is the same and that linear radial expansion results. CONCLUSION: In the early and intermediate stages of cystic growth, osmotic pressure differences play an important role; however, in very large cysts, this role becomes negligible, and cell birth in the lining dominates growth.     

Affinity of red blood cell membrane for particle surfaces measured by the extent of particle encapsulation.

An experimental technique and a simple analysis are presented that can be used to quantitate the affinity of red blood cell membrane for surfaces of small beads or microsomal particles up to 3 micrometers Diam. The technique is demonstrated with an example of dextran-mediated adhesion of small spherical red cell fragments to normal red blood cells. Cells and particles are positioned for contact by manipulation with glass micropipets. The mechanical equilibrium of the adhesive contact is represented by the variational expression that the decrease in interfacial free energy due to a virtual increase in contact area is balanced by the increase in elastic energy of the membrane due to virtual deformation. The surface affinity is the reduction in free energy per unit area of the interface associated with the formation of adhesive contact. From numerical computations of equilibrium configurations, the surface affinity is derived as a function of the fractional extent of particle encapsulation. The range of surface affinities for which the results are applicable is increased over previous techniques to several times the value of the elastic shear modulus. It is shown that bending rigidity of the membrane has little effect on the analytical results for particles 1--3 micrometers Diam and that results are essentially the same for both cup- and disk-shaped red cells. A simple analytical model is shown to give a good approximation for surface affinity (normalized by the elastic shear modulus) as a function of the fractional extent of particle encapsulation. The model predicts that a particle would be almost completely vacuolized for surface affinities greater than or equal to 10 times the elastic shear modulus. Based on an elastic shear modulus of 6.6 x 10(-3) dyn/cm, the range for the red cell-particle surface affinity as measured by this technique is from approximately 7 x 10(-4) to 7 x 10(-2) erg/cm2. Also, an approximate relation is derived for the level of surface affinity necessary to produce particle vacuolization by a phospholipid bilayer surface which possesses bending rigidity and a fixed tension.     

Spherical indentation of soft matter beyond the Hertzian regime: numerical and experimental validation of hyperelastic models

The lack of practicable nonlinear elastic contact models frequently compels the inappropriate use of Hertzian models in analyzing indentation data and likely contributes to inconsistencies associated with the results of biological atomic force microscopy measurements. We derived and validated with the aid of the finite element method force-indentation relations based on a number of hyperelastic strain energy functions. The models were applied to existing data from indentation, using microspheres as indenters, of synthetic rubber-like gels, native mouse cartilage tissue, and engineered cartilage. For the biological tissues, the Fung and single-term Ogden models achieved the best fits of the data while all tested hyperelastic models produced good fits for the synthetic gels. The Hertz model proved to be acceptable for the synthetic gels at small deformations (strain < 0.05 for the samples tested), but not for the biological tissues. Although this finding supports the generally accepted view that many soft materials can be assumed to be linear elastic at small deformations, the nonlinear models facilitate analysis of intrinsically nonlinear tissues and large-strain indentation behavior.     

The trajectory of a stiff rod in a curved potential energy trough. An initial result for short nucleosomal rods

The equilibrium trajectory of the axis of a rod subject to an externally imposed curved potential energy trough tends to conform to the shape of the curved trough, but also tends to be straight because of elastic resistance to bending. The actual path of the axis is a balance between the two extremes. We consider a potential energy trough centered along a circular arc of radius R. For a rod of small length compared to R, we show that the axis at equilibrium forms an arc of a circle of radius greater than R. The value of the radius of the axial path depends on the relative values of the Hooke's Law bending constant for the rod and the depth and width of the trough. Motivation for the calculation is provided by nucleosomal DNA, which conforms to the surface of a roughly cylindrical histone core at physiological ionic strength, but is observed to unwind into a partially extended conformation at very low ionic strength. We suggest that the rigidity to bending of short DNA segments becomes sufficiently great at low ionic strength to overcome attractive interactions with the histone surface. Alternately, of course, if during the cell cycle mutually attractive forces between DNA and histone core are weakened at constant ionic strength, the same type of unfolding would be expected to occur as the strength of the DNA-histone contacts drops below the level required to overcome elastic resistance to bending of the DNA rod.     

On the influence of wall properties and Poiseuille flow in peristalsis

The present study extends the two-dimensional analysis of peristaltic motion by Fung and Yih to include an elastic or viscoelastic wall, and a Poiseuille flow. This fluid-solid interaction problem is investigated by considering equations of motion of both the fluid and the deformable boundaries. The wall characteristics appear in their equations of motion, which are solved to represent boundary conditions of fluid motion. The influence of Poiseuille flow on pure peristalisis is also investigated.

The phenomenon of the ‘mean flow reversal’ is found to exist both at the center and at the boundaries of the channel. When the walls of the channel are elastic, pure peristalsis involves flow reversal only at the center. This position may shift drastically to the boundaries, if viscous damping forces are considered.     

Loss of stability-a new model for stress relaxation in plant cell walls

This study addresses the mechanism of wall stress relaxation in growing plant cells. The current viscoelastic model of cell wall relaxation, which dates from the work of Preston, Cleland, Lockhart, and others in the 1960s, has serious shortcomings. It has been shown however that the theory of loss of stability (LOS) can be applied to materials in tension, leading to the conclusion that the relaxation of stresses in the walls of any pressure vessel is rigorously modeled using LOS. We propose that LOS also provides a more appropriate and versatile model of stress relaxation in growing plant cells. We argue that when treated as a manifestation of LOS, the regulation of cell turgor has a rigorous and demonstrable basis in the geometrical and physical properties of the cell wall and the cell's ability to import water. Thus plant cell growth can be regarded as an inherently self-limiting process, tunable by biochemical or structural means. Lastly, despite the current limitations of our model, we apply direct measurement of elastic modulus, wall thickness and cell radius obtained from cylindrical Chara corallina cells to generate an initial calculation of critical pressures in a hypothetical spherical cell with the same material properties.     

A description of arterial wall mechanics using limiting chain extensibility constitutive models

Certain aspects of the mechanical response of arterial walls can be described using nonlinear elasticity theory. Uniaxial tests on vascular walls reveal nonlinear stress-strain behavior, with higher extensibility in the low stretch range and progressively lower extensibility with increasing stretch. This phenomenon is well known in the framework of rubber-like materials where it is called a strain-hardening or strain-stiffening effect. Constitutive models of incompressible hyperelasticity that take this into account include power-law models and limiting chain extensibility models. Our purpose in this paper is to bring to the attention of the biomechanics community some essential features of one such model of the latter type due to Gent. This model is compared with isotropic versions of biomechanical constitutive models by Takamizawa-Hayashi and Fung; the latter is a limiting version of a power-law material. Two particular problems are considered for which experimental data on arterial wall deformations are available. The first concerns small oscillations superposed on a large static stretch of a vertical string of arterial tissue. It is shown that the exponential model of Fung and the Gent model match well with the experimental data. The second problem is the extension of an internally pressurized circular cylindrical tube. It is shown that an inversion phenomenon observed experimentally for the human iliac artery can be described within a membrane theory by the Gent model whereas this cannot be described using the exponential model. The foregoing considerations are carried out for isotropic elastic materials in the absence of residual stress. Extensions to include anisotropy are also indicated.