Keywords: kinetics, shapes, curve fitting, geometry
In this talk we consider dimension reduction for complex biological shapes. We focus on two motivating applications that drove the work. The first application is a study of viral kinetics in the human colon. Here interest lies in determining the kinetics of a microbicide lubricant and its effective coverage for HIV prevention after receptive anal intercourse. The second is the study of diffusion tensor imaging (DTI) of the brain in humans. The corpus callosum is a central white matter structure that serves as the primary mode of connection between the left and right hemispheres of the brain. The corpus callosum is well imaged by DTI resulting in a complex multidimensional shape.
In both applications, central interest lies in a form of geometric dimension reduction for the purpose of subsequent statistical analyses. In the case of modeling lubricant in the colon, we focus on curve and tube fitting while in the case of the corpus callosum, we focus on manifolds. In each case, the image is reduced to a structure more amenable to analysis. Our approach relies heavily on the principal curves and surfaces methodology developed in Hastie and Stuetzle (JASA 1989).