Mathematical models of cell colonization of uniformly growing domains.

During the development of vertebrate embryos, cell migrations occur on an underlying tissue domain in response to some factor, such as nutrient. Over the time scale of days in which this cell migration occurs, the underlying tissue is itself growing. Consequently cell migration and colonization is strongly affected by the tissue domain growth. Numerical solutions for a mathematical model of chemotactic migration with no domain growth can lead to travelling waves of cells with constant velocity; the addition of domain growth can lead to travelling waves with nonconstant velocity. These observations suggest a mathematical approximation to the full system equations, allowing the method of characteristics to be applied to a simplified chemotactic migration model. The evolution of the leading front of the migrating cell wave is analysed. Linear, exponential and logistic uniform domain growths are considered. Successful colonization of a growing domain depends on the competition between cell migration velocity and the velocity and form of the domain growth, as well as the initial penetration distance of the cells. In some instances the cells will never successfully colonize the growing domain. These models provide an insight into cell migration during embryonic growth, and its dependence upon the form and timing of the domain growth.

Tumour dynamics and necrosis: surface tension and stability.

A model is developed for the motion of cells within a multicell spherical tumour. The model allows either for the intercellular forces to be in compression and cells to be compacted to a fixed number density, or for the cell number density to fall and cells to become isolated from each other. The model develops necrotic regions naturally due to force balances rather than being directly attributable to a critical oxygen concentration. These necrotic regions may result in a gradual reduction in local cell density rather than jump to a completely dead region. Numerical and analytical analysis of the spherically symmetric model shows that the long time behaviour of the spheroid depends on any surface tension effects created by cells on the outer surface. For small surface tension the spheroid grows linearly in time developing a large necrotic region, while for larger surface tension the growth can be halted. The linear stability to spherically symmetric perturbations of all the possible resulting steady states is revealed.

Age-related changes in the kinetics of water transport in normal human lenses.

Magnetic resonance microscopy (MRM) has been used to study the kinetics of water transport in human eye lenses. Fresh lenses obtained from the Queensland Eye Bank were incubated at 34.5 degrees C in artificial aqueous humour (AAH) containing nutrients and metabolites similar to those that are present in vivo. MR images were acquired over approximately a 20 hr period following replacement of H(2)O based AAH with deuterium oxide (D(2)O) based AAH. NMR signal intensity from the lenses decreased with time corresponding to a decrease in concentration of H(2)O within the lenses. A statistically significant correlation (P<0.001) was found between the rate of NMR signal loss from the lens nuclei and increasing age of the lenses. The results show that as lenses age, there is a reduction in the rate at which water and presumably also water soluble low molecular weight metabolites, can enter the cells of the lens nucleus via the epithelium and cortex. A decrease in the rate of transport of water, nutrients and anti-oxidants (e.g. glutathione) would be expected to lead to progressive oxidative damage to lenses with age, and may ultimately contribute to presbyopia and senile nuclear cataract.

Modelling the action of proteinase and inhibitor in tissue invasion.

Tissue invasion processes have been receiving considerable attention in recent years. The successful invasion and subsequent implantation of a mammalian trophoblast into maternal uterine tissue is under the control of a self-regulating mechanism, as yet not fully determined, which if operating incorrectly can lead to the excessive placental invasion associated with choriocarcinoma or insufficient placental invasion as occurs in the condition of pre-eclampsia. It is hoped that developing an understanding of the mechanisms governing the invasion of trophoblast cells into uterine tissue may lead to an understanding of the control of invasive mechanisms such as those observed in the development of invasive cancers. We have developed here a number of mathematical models in order to gain a better understanding of the role of proteinase and inhibitors in the context of trophoblast invasion. The models, which may be extended to other tissue invasion mechanisms, are based upon a series of proposed experiments involving blastocysts invading reconstituted basement membrane (Matrigel) allowing the quantification of the proteinase-Matrigel and proteinase-inhibitor interactions.

A continuum model for surface contraction waves.

The periodic travelling waves which appear on some animal eggs after fertilization are considered here. These are thought to be caused by a calcium initiated calcium release on the surface, causing calcium waves. A continuum model is developed where the cell is treated as a small viscous droplet with a surface contamination. When a periodic source of surfactant acts at one pole and propagates down the cell surface to the opposite pole, the drop responds by forming constriction rings which move from pole to pole.

Diffusion of a contaminant in fractured porous media.

A theoretical model is developed for diffusion from a finite source of buried contaminant through a porous medium containing a fracture that reaches to the atmosphere. Important quantities are shown to be the burial depth, source width and fracture width, the soil porosity, the diffusion coefficients in the soil and the (air-filled) fracture, and the decay constant of a radioactive contaminant. The flux across the soil-air interface is calculated and shown to be greatly increased by the presence of the fracture. The flux is found to increase with fracture width and source width, to decrease with increasing soil porosity and decay constant, and to decrease with increasing ratio of soil diffusion coefficient to air diffusion coefficient.

A continuum model for a red blood cell transformation: sphere to crenated sphere.

The evolution of crenations on the spherical red blood cell, which is part of the reversible disc-sphere transformation of these cells, is considered here. A continuum model is developed where the cell is treated as a small viscous droplet encapsulated by a viscoelastic solid membrane. When a small amount of material is deposited into the membrane, the drop responds by increasing its surface area as manifested by the appearance of ripples. A preferred number of crenations form and they are sustained for long time periods. The inception of crenations is due to a dynamic instability in the governing set of nonlinear equations, and depends on the rheological properties of the droplets' interior and membrane.

Transport of radon through cracks in a concrete slab.

A model involving the use of line sources is developed to describe the transport of radon through the cracks or gaps which appear in concrete slabs used in building foundations. The strength of these sources is determined from the results of the diffusion model proposed by Landman in a previous work. Once the strength of the source is known, additional transport mechanisms can be treated in a simple manner. Pressure differences across the slab and in the underlying soil are discussed. The rate of exhalation through a portion of the cracked slab is determined and compared to the rate of exhalation from the same surface area of bare soil. In typical cases, their ratios vary from 0.25 to 0.50. Therefore, these transport mechanisms account for a larger portion of the levels of radon found in many houses than do previous models.

Diffusion of radon through cracks in a concrete slab.

A mathematical model is developed to describe diffusion of radon through cracks or gaps in concrete slabs which are used in building foundations. As radon approaches the soil surface from underlying soil, it encounters a concrete slab. The radon will diffuse toward any air-filled cracks. The rate of exhalation through a portion of a cracked slab is determined and compared to the rate of exhalation from the same surface area of bare soil. In a typical case, this ratio is approx. 0.25. This is about a 20-fold increase to the ratio found when the concrete slab has no cracks. Therefore crack pathways are potentially a major source of indoor radon.