Liquid biopsies have the utility for detecting minimal residual disease in several cancer types. Here, we investigate if liquid biopsy tracking on-treatment informs on tumour phenotypes by longitudinally quantifying circulating Epstein-barr virus (EBV) DNA copy number in 673 nasopharyngeal carcinoma patients undergoing radical induction chemotherapy (IC) and chemo-radiotherapy (CRT). We observe significant inter-patient heterogeneity in viral copy number clearance that is classifiable into eight distinct patterns based on clearance kinetics and bounce occurrence, including a substantial proportion of complete responders (≈30%) to only one IC cycle. Using a supervised statistical clustering of disease relapse risks, we further bin these eight subgroups into four prognostic phenotypes (early responders, intermediate responders, late responders, and treatment resistant) that are correlated with efficacy of chemotherapy intensity. Taken together, we show that real-time monitoring of liquid biopsy response adds prognostic information, and has the potential utility for risk-adapted treatment de-intensification/intensification in nasopharyngeal carcinoma.
We study two mathematical models for the growth of tumors with time delays in cell proliferation, one for nonnecrotic tumors in the presence of inhibitors, and the other for necrotic tumors. Mathematical formulations of these models are retarded differential equations. By using a comparison method, we make rigorous analysis of these models. The results show that dynamical behavior of solutions of these models are similar to that of solutions for corresponding nonretarded problems, and the tumor will tend to a dormant state as time goes to infinity in favorable conditions (nutrient sufficient, inhibitor less), while it will finally disappear in unfavorable conditions (nutrient insufficient, inhibitor much).
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