Suzanne L. Weekes

A Multicompartment Mathematical Model of Cancer Stem Cell-Driven Tumor Growth Dynamics

Tumors are appreciated to be an intrinsically heterogeneous population of cells with varying proliferation capacities and tumorigenic potentials. As a central tenet of the so-called cancer stem cell hypothesis, most cancer cells have only a limited lifespan, and thus cannot initiate or reinitiate tumors. Longevity and clonogenicity are properties unique to the subpopulation of cancer stem cells. To understand the implications of the population structure suggested by this hypothesis—a hierarchy consisting of cancer stem cells and progeny non-stem cancer cells which experience a reduction in their remaining proliferation capacity per division—we set out to develop a mathematical model for the development of the aggregate population. We show that overall tumor progression rate during the exponential growth phase is identical to the growth rate of the cancer stem cell compartment. Tumors with identical stem cell proportions, however, can have different growth rates, dependent on the proliferation kinetics of all participating cell populations. Analysis of the model revealed that the proliferation potential of non-stem cancer cells is likely to be small to reproduce biologic observations. Furthermore, a single compartment of non-stem cancer cell population may adequately represent population growth dynamics only when the compartment proliferation rate is scaled with the generational hierarchy depth.

A dispersive effective equation for wave propagation through spatio-temporal laminates

We consider the problem of wave propagation through one-dimensional spatio-temporal or dynamic laminates when the wavelength of the disturbance is large relative to the scale of the microstructure. Dynamic materials are heterogeneous formations assembled from materials which are distributed on a microscale in space and in time. Using the techniques of Floquet analysis and asymptotic expansions, we reveal the dispersive nature of the effective medium. The effects are supported by direct numerical simulation of the heterogeneous problem. These results are compared with the exact solution of the effective equation.

Dispersion effects in dynamic laminates

We consider the problem of wave propagation through one-dimensional dynamic laminates when the wavelength of the disturbance is large relative to the scale of the microstructure. Dynamic materials are heterogeneous formations assembled from materials which are distributed on a microscale in space and in time. Using the techniques of Floquet analysis and asymptotic expansions, we uncover the dispersive effects of the effective medium.

Energy Accumulation in a Functionally Graded Spatial-Temporal Checkerboard

This letter extends the analysis of wave propagation in transmission lines with LC-parameters varying in space and time and the related effect of energy accumulation emerging from the concept of dynamic materials. We consider a practically important scenario of functionally graded checkerboard in space and time, i.e., the assembly combined of two dielectrics with material property transition zones applied instead of sharp interfaces. It is shown that the energy accumulation in traveling waves is preserved for certain ranges of material and geometric parameters.