In this paper, the dynamical behavior of a mathematical model of cancer including tumor cells, immune cells, and normal cells is investigated when a delay term is induced. Though the model was originally proposed by De Pillis et al. (Math. Comput. Model. 37:1221–1244, 2003), to make the model more realistic, we have added a delay term into the model, and it has incorporated novelty in our present work. The stability of existing equilibrium points in the delay-induced system is studied in detail. Global stability conditions of the tumor-free equilibrium point have been found. It is shown that due to this delay effect, the coexisting equilibrium point may lose its stability through a Hopf bifurcation. The implicit function theorem is applied to characterize a complex function in a neighborhood of delay terms. Additionally, the presence of Hopf bifurcation is demonstrated when the transversality conditions are satisfied. The length of delay for which the solutions preserve the stability of the limit cycle is estimated. Finally, through a series of numerical simulations, the theoretical results are formally examined.
15513 Background: E7389 is a synthetic macrocyclic ketone derivative of the marine sponge product halichondrin B. A unique tubulin depolymerizer, E7389 induces nonproductive tubulin aggregates and inhibits tubulin polymerization. E7389 has demonstrated activity in refractory breast cancer and non small cell lung cancer with response rates of 14.7% and 9.7%, respectively. E7389 inhibits the growth of prostate cancer cell lines, including those (DU145) with over-expression of beta-tubulin III, which may confer resistance to taxanes. Methods: Phase II Simon two-stage study explores the activity and safety of E7389 as monotherapy without concomitant steroids in patients with histologically proven adenocarcinoma of the prostate that has progressed despite maintained castration. Patients are stratified into two groups that are analyzed separately, including those who failed either no prior chemotherapy (except mitoxanthrone or estramustine) or no more than one prior regimen with a tubulin binding agent, such as docetaxel. E7389 1.4 mg/m2 is administered as a 2 to 5 minute bolus IV infusion on Days 1 and 8 of 21-day cycles. PSA measurements are obtained at the end of each cycle. The primary objective is to assess PSA response using Bubley criteria. Results: Thus far, 57 patients (37 taxane pretreated and 20 taxane naïve) have been treated. The median age is 71 (range 48–91) with 43% of patients over 75. A total of 160 treatment cycles have been given (median: 2; range: 2–7). Twelve study drug related serious adverse events have been reported in 9 patients: PE (2), melena (2), fever, neutropenia, febrile neutropenia, UTI, anemia, DVT, chest pain, and renal failure (1). Based on preliminary data, the taxane-treated group has 2 PSA responses in the first 21 patients, and the taxane-naïve group has 4 PSA responses in the first 14 patients, allowing both groups to progress to Stage 2 with further accrual. Conclusions: In patients with hormone refractory prostate cancer, there is some evidence of single agent activity for E7389 in taxane naïve and taxane pretreated patients with acceptable toxicity. Preliminary results of activity allow further recruitment to proceed. [Table: see text]
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