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    Mathematics of cell growth : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand
    (Massey University, 2014) Zaidi, Ali Ashher
    We present a model that describes growth, division and death of cells structured by size. Here, size can be interpreted as DNA content or physical size. The model is an extension of that studied by Hall and Wake [24] and incorporates the symmetric as well as the asymmetric division of cells. We first consider the case of symmetric cell division. This leads to an initial boundary value problem that involves a first-order linear PDE with a functional term. We study the separable solution to this problem which plays an important role in the long term behaviour of solutions. We also derive a solution to the problem for arbitrary initial cell distributions. The method employed exploits the hyperbolic character of the underlying differential operator, and the advanced nature of the functional argument to reduce the problem to a sequence of simple Cauchy problems. The existence of solutions for arbitrary initial distributions is established along with uniqueness. The asymptotic relationship with the separable solution is established, and because the solution is known explicitly, higher order terms in the asymptotics can be obtained. Adding variability to the growth rate of cells leads to a modified Fokker-Planck equation with a functional term. We find the steady size distribution solution to this equation. We also obtain a constructive existence and uniqueness theorem for this equation with an arbitrary initial size-distribution and with a no-flux condition. We then proceed to study the binary asymmetric division of cells. This leads to an initial boundary value problem that involves a first-order linear PDE with two functional terms. We find and prove the unimodality of the steady size distribution solution to this equation. The existence of higher eigenfunctions is also discussed. Adding stochasticity to the growth rate of cells yields a second-order functional differential equation with two non-local terms. These problems, being a particular kind of functional differential equations exhibit unusual characteristics. Although the associated boundary value problems are well-posed, the spectral problems that arise by separating the variables, cannot be easily shown to have a complete set of eigenfunctions or the usual orthogonality properties.
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    An initial-boundary value problem arising in cell population growth modelling : a thesis submitted in partial fulfillment of the requirement for the degree of Master of Science in Mathematics at Massey University, (Manawatu) Institute of Fundamental Sciences, Mathematics
    (Massey University, 2012) Almalki, Adel
    A partial differential equation modelling cell populations undergoing growth and division characterised by size is studied. This equation is a special case of the fragmentation equation studied by Michel et al. [5] with no dispersion term, and the problem is of the initial-boundary value type. Eigenfunction solutions are derived for separable solutions and related properties such as spectrum, uniqueness and unimodality are investigated. We show that the spectrum is continuous and that the decay of the eigenfunctions is exponential at a critical eigenvalue and algebraic otherwise. The existence of a fast decay general solution n(x; t) is then established. The problem can be solved analytically, and it is shown that the solution is unique and smooth. The solution properties are illustrated with some numerical simulations. Finally, the role of exponential decaying eigenfunction solutions is interpreted from the standpoint of the general solution. The asymptotic behaviour as t ! 1 of the general solution is examined. Slow decay eigenfunction solutions are briefly discussed, but their mathematical role remains to be explored.
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    Steady size distributions in cell populations : a thesis presented in partial fulfilment of the requirements for the degree of Doctor Philosophy in Mathematics at Massey University
    (Massey University, 1991) Hall, Alistair John
    In any population of cells, individual cells grow for some period of time and then divide into two or more parts, called daughters. To describe this process mathematically, we need to specify functions describing the growth rate, size at division, and proportions into which each cell divides. In this thesis, it is assumed that the growth rate of a cell can be determined precisely from its size, but that both its size at division and the proportions into which it divides may be described stochastically, by probability density functions whose parameters are dependent on cell size and age (or birth-size). Special cases are also considered where all cells with the same birth-size divide at the same size, or where all cells divide exactly in half. We consider a population of cells growing and dividing steadily, such that the total cell population is increasing, but the proportion of cells in any size class remains constant. In Chapter 1, equations are derived which need to be solved in order to deduce the shape of the steady size distribution (or steady size/age or size/birth-size distributions) from any given growth rate and probability distributions describing the division rate and division proportions. In the general case, a Fredholm-type integral equation is obtained, but if the probability of cell division depends on cell size only (i.e. not age or birth-size), and all cells divide into equal-sized daughters, then we obtain a functional differential equation. In two special cases, the resulting equations simplify considerably, and it is these cases which are explored further in this thesis. The first is where the probability of a cell dividing in any instant of time is a constant, independent of cell age or size. In Chapter 2, the functional differential equation resulting when cells divide into equal-sized daughters is solved for the special case where the growth rate is constant, and in an appendix the case where the growth rate is described by a power law is dealt with. The second case which simplifies is where the time-independent part of the growth rate of a cell is proportional to cell size. This case is particularly important, as it is a good first-order approximation to the real cell growth rate in some structured tissues, and in some bacteria. The special case in which this leads to a functional differential equation is discussed in Chapter 3, and the integral equation arising in the general case is dealt with in Chapter 4. Finally, the conditions under which the integral operator in Chapter 4 will be both square-integrable and non-factorable are discussed in Chapter 5. It is shown that if these conditions are satisfied then a unique, stable, steady size distribution will exist.