Noether's theorem links the symmetries of a quantum system with its conserved quantities, and is a cornerstone of quantum mechanics. Here we prove a version of Noether's theorem for Markov processes. In quantum mechanics, an observable commutes with the Hamiltonian if and only if its expected value remains constant in time for every state. For Markov processes that no longer hold, but an observable commutes with the Hamiltonian if and only if both its expected value and standard deviation are constant in time for every state. © 2013 American Institute of Physics.

Purpose

Dendritic cell (DC) vaccines have recently emerged as an innovative therapeutic option for glioblastoma patients. To identify novel surrogates of anti-tumor immune responsiveness, we studied the dynamic expression of activation and inhibitory markers on peripheral blood lymphocyte (PBL) subsets in glioblastoma patients treated with DC vaccination at UCLA.

Experimental design

Pre-treatment and post-treatment PBL from 24 patients enrolled in two Phase I clinical trials of dendritic cell immunotherapy were stained and analyzed using flow cytometry. A univariate Cox proportional hazards model was utilized to investigate the association between continuous immune monitoring variables and survival. Finally, the immune monitoring variables were dichotomized and a recursive partitioning survival tree was built to obtain cut-off values predictive of survival.

Results

The change in regulatory T cell (CD3(+)CD4(+)CD25(+)CD127(low)) frequency in PBL was significantly associated with survival (p = 0.0228; hazard ratio = 3.623) after DC vaccination. Furthermore, the dynamic expression of the negative co-stimulatory molecule, CTLA-4, was also significantly associated with survival on CD3(+)CD4(+) T cells (p = 0.0191; hazard ratio = 2.840) and CD3(+)CD8(+) T cells (p = 0.0273; hazard ratio = 2.690), while that of activation markers (CD25, CD69) was not. Finally, a recursive partitioning tree algorithm was utilized to dichotomize the post/pre fold change immune monitoring variables. The resultant cut-off values from these immune monitoring variables could effectively segregate these patients into groups with significantly different overall survival curves.

Conclusions

Our results suggest that monitoring the change in regulatory T cell frequencies and dynamic expression of the negative co-stimulatory molecules on peripheral blood T cells, before and after DC vaccination, may predict survival. The cut-off point generated from these data can be utilized in future prospective immunotherapy trials to further evaluate its predictive validity.