Mathematical and statistical modeling and analysis of biological growth using images collected over time are important for understanding of normal and abnormal development. In computational anatomy, changes in the shape of a growing anatomical structure have been modeled by means of diffeomorphic transformations in the background coordinate space. Various image and landmark matching algorithms have been developed for inference of large transformations that perform image registration consistent with the material properties of brain anatomy under study. However, from a biological perspective, it is not material constants that regulate growth, it is the genetic control system. A pattern theoretic model called the Growth as Random Iterated Diffeomorphisims (GRID) introduced by Ulf Grenander (Brown University) constructs growth-induced transformations according to fundamental biological principles of growth. They are governed by an underlying genetic control that is expressed in terms of probability laws governing the spatial-temporal patterns of elementary cell decisions (e.g., cell division/death). This thesis addresses computational and stochastic aspects of the GRID model and develops its application to image analysis of growth. The first part of the thesis introduces the original GRID view of growth-induced deformation on a fine time scale as a composition of several, elementary, local deformations each resulting from a random cell decision, a highly localized event in space-time called a seed. A formalization of the proposed model using theory of stochastic processes is presented, namely, an approximation of the GRID model by the diffusion process and the Fokker-Planck equation describing the evolution of the probability density of seed trajectories in space-time. Its time-dependent and stationary numerical solutions reveal bimodal distribution of a random seed trajectory in space-time. The second part of the thesis considers the growth pattern on a coarse time scale which underlies visible shape changes seen in images. It is shown that such a "macroscopic" growth pattern is a solution to a deterministic integro-differential equation in the form of a diffeomorphic flow dependent on the GRID growth variables such as the probability density of cell decisions and the rate of contraction/expansion. Since the GRID variables are unobserved, they have to be estimated from image data. Using the GRID macroscopic growth equation such an estimation problem is formulated as an optimal control problem. The estimated GRID variables are optimal controls that force the image of an initial organism to be continuously transformed into the image of a grown organism. The GRID-based inference method is implemented for inference of growth properties of the Drosophila wing disc directly from confocal micrographs of Wingless gene expression patterns.

Mathematical and statistical modeling and analysis of biological growth using images collected over time are important for understanding of normal and abnormal development. In computational anatomy, changes in the shape of a growing anatomical structure have been modeled by means of diffeomorphic transformations in the background coordinate space. Various image and landmark matching algorithms have been developed for inference of large transformations that perform image registration consistent with the material properties of brain anatomy under study. However, from a biological perspective, it is not material constants that regulate growth, it is the genetic control system. A pattern theoretic model called the Growth as Random Iterated Diffeomorphisims (GRID) introduced by Ulf Grenander (Brown University) constructs growth-induced transformations according to fundamental biological principles of growth. They are governed by an underlying genetic control that is expressed in terms of probability laws governing the spatial-temporal patterns of elementary cell decisions (e.g., cell division/death). This thesis addresses computational and stochastic aspects of the GRID model and develops its application to image analysis of growth. The first part of the thesis introduces the original GRID view of growth-induced deformation on a fine time scale as a composition of several, elementary, local deformations each resulting from a random cell decision, a highly localized event in space-time called a seed. A formalization of the proposed model using theory of stochastic processes is presented, namely, an approximation of the GRID model by the diffusion process and the Fokker-Planck equation describing the evolution of the probability density of seed trajectories in space-time. Its time-dependent and stationary numerical solutions reveal bimodal distribution of a random seed trajectory in space-time. The second part of the thesis considers the growth pattern on a coarse time scale which underlies visible shape changes seen in images. It is shown that such a "macroscopic" growth pattern is a solution to a deterministic integro-differential equation in the form of a diffeomorphic flow dependent on the GRID growth variables such as the probability density of cell decisions and the rate of contraction/expansion. Since the GRID variables are unobserved, they have to be estimated from image data. Using the GRID macroscopic growth equation such an estimation problem is formulated as an optimal control problem. The estimated GRID variables are optimal controls that force the image of an initial organism to be continuously transformed into the image of a grown organism. The GRID-based inference method is implemented for inference of growth properties of the Drosophila wing disc directly from confocal micrographs of Wingless gene expression patterns.

The study examined interactions between morogo plants, arbuscular mycorrhizal fungi (AMF) and Fusarium species. Morogo refers to traditional leafy vegetables that, together with maize porridge, are dominant staple foods in rural areas of the Limpopo Province such as the Dikgale Demographic Surveillance Site (DDSS). Morogo plants grow either as weeds (often among maize), occur naturally in the field or are cultivated as subsistence crops by rural communities. Botanical species of morogo plants consumed in the DDSS were determined. Colonisation of morogo plant roots by AMF and Fusarium species composition in the immediate soil environment were investigated in four of eight DDSS subsistence communities, Isolated AMF were shown to belong to the genera Acaulospora and Glomus. Twelve Fusarium species were isolated from soil among which Fusariurn verticilliodes and Fusarium proliferaturn occurred predominantly. Greenhouse pot trials were conducted to examine the effect of AMF on morogo plant growth (cowpea; Mgna unguiculata) and Fusarium proliferatum levels in soil, Interaction between plants and AMF, as well as tripartite interactions of cowpea plants, AMF and Fusarium proliferatum were investigated. Non-inoculated cowpea plants served as controls for the following inoculations of cowpea in pots: (i) Fusarium proliferatum; (ii) commercial AMF from Mycoroot (PTY) Ltd. (a mixture of selected indigenous Glomus spp referred to commercial AMF for the purpose of this study); (iii) indigenous AMF obtained from DDSS soil (referred to iocal AMF for the purpose of this study); (iv) commercial AMF plus Fusarium proliferatum; (v) local AMF plus Fusariurn proliferatum. Results showed reduced root colonization by local as well as commercial AMF when Fusarium proliferatum were present. Local AMF significantly enhanced cowpea growth while commercial AMF apparently reduced the level of Fusarium proliferatum in the rhizosphere and surrounding soil. Results suggest that AMF may have potential as biological growth enhancers and bioprotective agents against Fusarium proliferatum. / Thesis (M. Environmental Science (Water Science))--North-West University, Potchefstroom Campus, 2007.

The study examined interactions between morogo plants, arbuscular mycorrhizal fungi (AMF) and Fusarium species. Morogo refers to traditional leafy vegetables that, together with maize porridge, are dominant staple foods in rural areas of the Limpopo Province such as the Dikgale Demographic Surveillance Site (DDSS). Morogo plants grow either as weeds (often among maize), occur naturally in the field or are cultivated as subsistence crops by rural communities. Botanical species of morogo plants consumed in the DDSS were determined. Colonisation of morogo plant roots by AMF and Fusarium species composition in the immediate soil environment were investigated in four of eight DDSS subsistence communities, Isolated AMF were shown to belong to the genera Acaulospora and Glomus. Twelve Fusarium species were isolated from soil among which Fusariurn verticilliodes and Fusarium proliferaturn occurred predominantly. Greenhouse pot trials were conducted to examine the effect of AMF on morogo plant growth (cowpea; Mgna unguiculata) and Fusarium proliferatum levels in soil, Interaction between plants and AMF, as well as tripartite interactions of cowpea plants, AMF and Fusarium proliferatum were investigated. Non-inoculated cowpea plants served as controls for the following inoculations of cowpea in pots: (i) Fusarium proliferatum; (ii) commercial AMF from Mycoroot (PTY) Ltd. (a mixture of selected indigenous Glomus spp referred to commercial AMF for the purpose of this study); (iii) indigenous AMF obtained from DDSS soil (referred to iocal AMF for the purpose of this study); (iv) commercial AMF plus Fusarium proliferatum; (v) local AMF plus Fusariurn proliferatum. Results showed reduced root colonization by local as well as commercial AMF when Fusarium proliferatum were present. Local AMF significantly enhanced cowpea growth while commercial AMF apparently reduced the level of Fusarium proliferatum in the rhizosphere and surrounding soil. Results suggest that AMF may have potential as biological growth enhancers and bioprotective agents against Fusarium proliferatum. / Thesis (M. Environmental Science (Water Science))--North-West University, Potchefstroom Campus, 2007.