A. A. Stein

Dynamics of the intraocular fluid: Mathematical model and its main consequences

The problem of interaction between the intraocular pressure and the hydrodynamic processes in the eye is investigated. A lumped-parameter model taking into account a wide range of physically permissible mechanisms of the inflow and outflow of intraocular fluid is proposed. In the cases of stationary and nonstationary flow regimes, the consequences of various assumptions are analyzed. The flow in the presence of an external load is considered. It is shown that the physical interpretation of measurements used in ophthalmology strongly depends on the hypotheses adopted, the choice between which needs further investigation.

Investigation of the properties of the two-component eyeball model and opportunities to use it for practically estimating the mechanical characteristics of the human eye

On the basis of the two-component eyeball model proposed in [1], the loadings of the cornea by a rod and wide stamps with flat, convex and concave bases are investigated and compared. It is shown that for all loading types studied the measured geometric deformation characteristic is, over the physiological range of parameters, determined by the pressure under load and depends on the elastic properties of the eye shell only slightly. The problem of eyeball deformation at a constant volume, which corresponds to the clinical procedure of measuring the intraocular pressure, is considered. The pressure dependence on the applied load is studied. It is shown that for the rod and convex stamp the inclination of the straight line which approximates this dependence over the practically important range of loads, as well as the difference between the pressures under load and before loading, depend on the cornea stiffness weakly but are significantly affected by the stiffness of the scleral segment. On the other hand, for the flat and concave stamps these characteristics substantially depend on both elastic constants. A method of measurements and calculations, which makes it possible to increase the informativeness and reliability of the data obtained in clinics by static tonometry, is proposed.

Estimating the elastic properties of the eye from differential tonometry by the Schiøtz tonometer: Analysis of the measurement procedure on the basis of a two-component mathematical model

The procedure of differential tonometry by the Schiøtz tonometer is modeled using a two-component model of the eyeball in which the cornea is represented by a momentless deformable and linearly elastic surface and the sclera region by an elastic element that responds to changes in intraocular pressure via changes in volume. Numerical calculations showed that the dependence of intraocular pressure on the weight of a cornea- deforming rod (a plunger) is almost linear. This makes it possible to take the slope of its plot (pressure difference coefficient) as a characteristic of the eye. The coefficient is studied as a function of the elastic characteristics of the eye and the pressure in the unloaded eye. An analysis based on the dimensional theory and confirmed by calculations made it possible to conclude that in the first approximation the pressure-difference coefficient depends on the elastic constants through their ratios to the intraocular pressure. A comparison was made with the standard method of processing the data of differential tonometry.

The Effect of Spatial Inhomogeneity of the Cornea on the Deformation Properties of the Eyeball and the Results of Maklakoff Applanation Tonometry

A two-component model of the eyeball that represents the cornea as a momentless, linearly elastic deformable surface and the scleral region, as an elastic element that responds to intraocular pressure changes by volume changes, has been used to analyze the effect of spatial inhomogeneity in the distribution of effective corneal stiffness on the mechanical properties of the eye. The effective stiffness of the cornea characterized both the elastic properties and the thickness of the cornea within the framework of the model. Various axisymmetric forms of the effective stiffness distribution characterized by monotonic increase along the arc between a point on the corneal surface and the apex of the cornea were studied. The considered distributions simulated both natural inhomogeneity and apical region weakening due to surgical interventions. Numerical simulation yielded the dependences of deformation parameters on intraocular pressure changes. These parameters characterized the deformation properties of both the cornea (apex displacement) and the eyeball as a whole (intraocular volume change). In the case of moderate inhomogeneity, the dependences were only slightly different from those for a homogeneous cornea with an effective stiffness equal to the mean value for the corresponding inhomogeneous distribution. A noticeable increase in the integral response of the cornea and the eyeball as a whole to changes in pressure was observed if the effective stiffness amplitude was very high (two or more times higher than the mean value). The effect of inhomogeneity on the results of tonometric measurements with a Maklakoff tonometer (flat stamp) was studied. The tonometric difference, that is, the difference between the tonometric pressure (in the loaded eye) and the true pressure (before loading), mainly depended on the average stiffness of the cornea in this case as well, with a substantial increase observed at very high stiffness amplitudes only. Apical weakening of the cornea led to an increase (although not very pronounced) of the tonometric difference.

A Mathematical Model of Spatial Self-Organization in a Mechanically Active Cellular Medium

A general continual model of a medium composed of mechanically active cells is proposed. The medium is considered to be formed by three phases: cells, extracellular fluid, and an additional phase that is responsible for active interaction forces between cells and, for instance, may correspond to a system of protrusions that provide the development of active contractile forces. The deformation of the medium, which is identified with the deformation of the cell phase, consists of two components: elastic deformation of individual cells and cell rearrangements. The elastic deformation is associated with stresses in the cell phase. The spherical component of the stress tensor describes the nonlinear resistance of the cellular medium, which leads to the impossibility of its excessive compression. The constitutive equation for pressure in the cell phase is taken in the form of a nonlinear dependence on the volume cell density. The rearrangement of cells is considered as a flow controlled by stresses in the cell phase, active stresses, and fluid pressure. The tensor of active stresses is assumed to be spherical and nonlocally dependent on the cell density. Assuming that the process of biological tissue deformation is slow, we obtained a reduced model that neglects the elastic deformation of cells, compared to the inelastic deformation. A linear stability analysis of a spatially uniform steady-state solution was performed. The hydrostatic pressure of fluid is present among the parameters that are responsible for the loss of stability of the steady-state solution: an increase in it has a destabilizing effect owing to the action of the component of the interphase interaction force that is determined by the fluid pressure. The model we obtained can be used to describe the process of cavity formation in an initially homogeneous cell spheroid. The role of local and nonlocal mechanisms of active stress generation in the formation of cavity is investigated.