Biomedical subjects

G C Cruywagen

Publications and source records attributed to G C Cruywagen.

6 recordsLinked to original sources

PubMed1996

Competition in a spatially heterogeneous environment: modelling the risk of spread of a genetically engineered population.

In recent years regulations have been developed to address the risks of releasing genetically engineered organisms into the natural environment. These risks are generally considered to be proportional to the exposure multiplied by the hazard. Exposure is, in part, determined by the spatial spread of the organisms, a component of risk suited to mathematical analysis. In this paper we exampine a mathematical model describing the spread of organisms introduced into a hetereogeneous environment, focusing on the risk of spread and plausibility of containment strategies. Two competing populations are assumed, one the natural species and the other an engineered species or strain, both of which move randomly in a spatially heterogenous environment consisting of alternating favourable and unfavourable patches. The classical Lotka-Volterra competition model with diffusion is used. Analyses of the possible spread and invasion of engineered organisms are thus reduced to finding periodic travelling wave solutions to the model equations. We focus on whether a very small number of engineered organisms can spatially invade a natural population. Initially we investigate the problem for spatially periodic diffusion coefficients and demonstrate that, under the right circumstances and a large enough unfavourable patch, invasion does not succeed. However, if spatially periodic carrying capacities are assumed along with spatially varying diffusion rates, the situation is far more complex. In this case containment of the engineered species is no longer only a simple function of the unfavourable patch length. By using perturbation solutions to the nonuniform steady states, approximate invasion conditions are obtained.

DNA, Recombinant↗

PubMed1996

A sex-structured delayed recruitment model.

A delay-difference model for a sex-structured population with delayed recruitment is presented. Constant-effort harvesting is introduced for examining the model's sensitivity to harvesting. Linear analyses about the steady states are performed under various parameter choices. The effects of the delay period, the survival parameter, and the harvesting effort on population stability are examined. It is shown that differences in the male and female delays to recruitment can give rise to very different stability diagrams. Our analyses indicate, for example, that a population with equal male and female delays to recruitment is the most robust to recruitment failures. Possible forms that the male and female recruitment functions could take are suggested and evaluated. Finally, the very encouraging result is obtained that maximum sustainable yield is attained at a stable steady-state population level.

Animals↗

PubMed1996

A mathematical model of glioma growth: the effect of extent of surgical resection.

We have developed a mathematical model based on proliferation and infiltration of neoplastic cells that allows predictions to be made concerning the life expectancies following various extents of surgical resection of gliomas of all grades of malignancy. The key model parameters are the growth rate and the diffusion rate. These rates were initially derived from analysis of a case of recurrent anaplastic astrocytoma treated by chemotherapies. Numerical simulations allow us to estimate what would have happened to that patient if various extents of surgical resection, rather than chemotherapies, had been used. In each case, the shell of the infiltrating tumour that remains after 'gross total removal' or even a maximal excision continues to grow and regenerates the tumour mass remarkably rapidly. By developing a model that allows the growth and diffusion rates to define the distribution of cells at the time of diagnosis, and then varying these rates by about 50%, we created a hypothetical tumour patient population whose survival times show good agreement with the results recently reported by Kreth for treatments of glioblastomas. Tenfold decreases in the rates of growth and diffusion mimic the results reported by many other investigators with more slowly growing gliomas. Thus, the model quantitatively supports the ideas that (i) gliomas infiltrate so diffusely that they cannot be cured by resection alone, surgical or radiological, no matter how extensive that may be; (ii) the more extensive the resection, regardless of the degree of malignancy of the glioma, the greater the life expectancy; and (iii) measurements of the two rates, growth and diffusion, may be able to predict survival rates better than the current histological estimates of the type and grade of gliomas.

Cell Division↗

PubMed1995

A mathematical model of glioma growth: the effect of chemotherapy on spatio-temporal growth.

During the past two decades computerized tomography (CT) and magnetic resonance imaging (MRI) have permitted the detection of tumours at much earlier stages in their development than was previously possible. In spite of this earlier diagnosis the effects of earlier and more extensive treatments have been difficult to document. This failure has led to an increasing awareness of the importance of infiltration of glioma cells into surrounding grossly normal brain tissue such that recurrence still occurs. In this paper a simple mathematical model for the proliferation and infiltration of such tumours is introduced, based in part on quantitative image analysis of histological sections of a human brain glioma and especially on cross-sectional area/volume measurements of serial CT images while the patient was undergoing chemotherapy. The model parameters were estimated using optimization techniques to give the best fit of the simulated tumour area to the CT scan data. Numerical solution of the model on a two-dimensional domain, which took into account the geometry of the brain and its natural barriers to diffusion, was used to determine the effect of chemotherapy on the spatio-temporal growth of the tumour.

Antineoplastic Combined Chemotherapy Protocols↗

PubMed1994

Travelling waves in a tissue interaction model for skin pattern formation.

Tissue interaction plays a major role in many morphogenetic processes, particularly those associated with skin organ primordia. We examine travelling wave solutions in a tissue interaction model for skin pattern formation which is firmly based on the known biology. From a phase space analysis we conjecture the existence of travelling waves with specific wave speeds. Subsequently, analytical approximations to the wave profiles are derived using perturbation methods. We then show numerically that such travelling wave solutions do exist and that they are in good agreement with out analytical results. Finally, the biological implications of our analysis are discussed.

Animals↗

PubMed1992

Sequential pattern formation in a model for skin morphogenesis.

During morphogenesis regular patterns often develop behind a frontier of pattern formation which travels across the prospective tissue. Here the authors consider the propagating patterns exhibited in a two-dimensional domain by a tissue interaction mechanochemical model for skin pattern formation. It is shown that the model can exhibit travelling waves of complex spatial pattern formation. Two alternative mechanisms that can produce such sequential patterning are presented. In particular, it is demonstrated that the specification of a simple quasi-one-dimensional pattern is all that is required to determine a complex two-dimensional pattern. Finally, the model solutions are related to actual pattern propagation during chick feather primordia initiation.

Animals↗