Physico-chemical model of a protocell

An immersed boundary framework for modelling the growth of individual cells: an application to the early tumour development

A biomechanical approach in modelling the growth and division of a single fully deformable cell by using an immersed boundary method with distributed sources is presented, and its application to model the early tumour development is discussed. This mathematical technique couples a continuous description of a viscous incompressible cytoplasm with the dynamics of separate elastic cells, containing their own point nuclei, elastic plasma membranes with membrane receptors, and individually regulated cell processes. This model enables one to focus on the biomechanical properties of individual cells and on communication between cells and their microenvironment, simultaneously allowing for the formation of clusters or sheets of cells that act together as one complex tissue. Several examples of early tumours growing in various geometrical configurations and with distinct conditions of their initiation and progression are also presented to show the strength of our approach in modelling different topologies of the growing tissues in distinct biochemical conditions of the surrounding media.

Modelling the growth of solid tumours and incorporating a method for their classification using nonlinear elasticity theory

Medically, tumours are classified into two important classes--benign and malignant. Generally speaking, the two classes display different behaviour with regard to their rate and manner of growth and subsequent possible spread. In this paper, we formulate a new approach to tumour growth using results and techniques from nonlinear elasticity theory. A mathematical model is given for the growth of a solid tumour using membrane and thick-shell theory. A central feature of the model is the characterisation of the material composition of the model through the use of a strain-energy function, thus permitting a mathematical description of the degree of differentiation of the tumour explicitly in the model. Conditions are given in terms of the strain-energy function for the processes of invasion and metastasis occurring in a tumour, being interpreted as the bifurcation modes of the spherical shell which the tumour is essentially modelled as. Our results are compared with actual experimental results and with the general behaviour shown by benign and malignant tumours. Finally, we use these results in conjunction with aspects of surface morphogenesis of tumours (in particular, the Gaussian and mean curvatures of the surface of a solid tumour) in an attempt to produce a mathematical formulation and description of the important medical processes of staging and grading cancers. We hope that this approach may form the basis of a practical application.

A mathematical model for the growth and classification of a solid tumor: a new approach via nonlinear elasticity theory using strain-energy functions

Medically, tumors are classified into two important classes--benign and malignant. Generally speaking, the two classes display different behaviour with regard to their rate and manner of growth and subsequent possible spread. In this paper, we formulate a new approach to tumor growth using results and techniques from nonlinear elasticity theory. A mathematical model is given for the growth of a solid tumor using membrane and thick-shell theory. A central feature of the model is the characterization of the material composition of the tumor through the use of a strain energy function, thus permitting a mathematical description of the degree of differentiation of the tumor explicitly in the model. Conditions are given in terms of the strain energy function for the processes of invasion and metastasis occurring in a tumor, being interpreted as the bifurcation modes of the spherical shell, which the tumor is essentially modeled as. Our results are compared with actual medical experimental results and with the general behavior shown by benign and malignant tumors. Finally, we use these results in conjunction with aspects of surface morphogenesis of tumors (in particular, the Gaussian and mean curvatures of the surface of a solid tumor) in an attempt to produce a mathematical formulation and description of the important medical processes of staging and grading cancers. We hope that this approach may form the basis of a practical application.

The "protocell": a mathematical model of self-maintenance

The concepts of self-generation, autonomous boundary and self-maintenance are explained briefly. The "protocell" is presented as a model of self-maintenance which is based on simple physical mechanisms of diffusion and reaction. The time evolution of the surface of the protocell is taken into account explicitly in the form of a Stefan condition giving rise to a non-linear feedback of the surface motion to the reaction and diffusion processes inside the protocell. The spatio-temporal dynamics are investigated, particularly in the neighbourhood of the stationary states, showing a self-maintaining behaviour under a certain range of nutritional conditions. Under another set of conditions we find an instability leading to a division process so that the population of protocells becomes self-maintaining instead of the single individual. The presented formulation of the protocell model is crucially improved compared with a previous version which required boundary conditions at infinity. The previous version was not strictly self-maintaining since dynamics outside the cell were essential for its behaviour.