The Impact of Time Delay in a Tumor Model

We consider a free boundary tumor growth model with a time delay in cell proliferation and study how time delay affects the stability and the size of the tumor. The model consists of a coupled system of an elliptic equation, a parabolic equation and an ordinary differential equation to describe the cell location under the presence of time delay, with the tumor boundary as a free boundary. A parameter $\mu$ in the model is proportional to the “aggressiveness” of the tumor. It is proved that there exists a unique classical radially symmetric stationary solution $(\sigma_*,p_*,R_*)$ which is stable for any $\mu\u3e0$ with respect to all radially symmetric perturbations [S. Xu, Q. Zhou, and M. Bai, Qualitative analysis of a time-delayed free boundary problem for tumor growth under the action of external inhibitors]. However, under non-radially symmetric perturbations, we prove that there exists a critical number $\mu_*$ such that if $\mu\mu_*$ then the stationary solution is unstable. It is actually unrealistic to expect the problem to be stable for large tumor aggressiveness parameter, therefore our result is more reasonable. Furthermore, it is also proved by the authors that adding the time delay in the model would result in a larger tumor, and if $\mu$ is larger, then the time delay would have a greater impact on the size of the tumor

Synthesis and Modification of [2+3] Imine Cage Based on 2,7,14-Triaminotriptycene

This thesis is focused on the synthesis and modification of [2+3] organic imine cages based on 2,7,14-triaminotriptycene. Many porous cage compounds have been synthesized based on triptycene precursors because of its rigid 3D skeleton and a D3h symmetry. However, modifying the cage compounds with functional groups which may empower the cages is still challenging. In general, post-synthetic modification in pre-synthesized cages and modification of molecular precursors are two main strategies to introduce functional groups into cage compounds. By modifying aldehyde building blocks, ten different side-chains were introduced to the [2+3] terphenyl imine cages. Solubility in organic solvent was improved for all these new cages compared with the unfunctionalized one. Among these cages, the perfluorobutyl chain modified cage exhibits specific surface area up to SABET = 588 m2/g and high Henry selectivity SSF6/N2 = 107 for sulphur hexafluoride over nitrogen, which shows a potential application in the separation of sulphur hexafluoride. Besides the aldehyde building blocks, a new amine building block was constructed by triaminotriptycene and pyridine dicarbonyl dichloride. This new amino building block was used to react with a series of salicylaldehydes with different distance between the aldehyde groups, and the results prove topologies of cage can be affected by the length of the aldehyde precursors. In the third part, through a post-synthetic modification strategy, a carbamate cage was synthesized after two steps based on a [2+3] imine cage (formed by triaminotriptycene and bisphenyl salicylaldehyde). This carbamate cage exhibits high pH stability from pH =-1 to 14 which is proved by NMR spectra, SEM and gas sorption experiments. This is one of the most stable cage compounds reported so fa