Exploiting the synergy between carboplatin and ABT-737 in the treatment of ovarian carcinomas

Platinum drug-resistance in ovarian cancers is a major factor contributing to chemotherapeutic resistance of recurrent disease. Members of the Bcl-2 family such as the anti-apoptotic protein Bcl-XL have been shown to play a role in this resistance. Consequently, concurrent inhibition of Bcl-XL in combination with standard chemotherapy may improve treatment outcomes for ovarian cancer patients. Here, we develop a mathematical model to investigate the potential of combination therapy with ABT-737, a small molecule inhibitor of Bcl-XL, and carboplatin, a platinum-based drug, on a simulated tumor xenograft. The model is calibrated against in vivo\ud experimental data, wherein tumor xenografts were established in mice and treated with ABT-737 and carboplatin on a fixed periodic schedule, alone or in combination, and tumor sizes recorded regularly. We show that the validated model can be used to predict the minimum drug load that will achieve a predetermined level of tumor growth inhibition, thereby maximizing the synergy between the two drugs. Our simulations suggest that the time of infusion of each carboplatin dose is a critical parameter, with an 8-hour infusion of carboplatin administered each week combined with a daily bolus dose of ABT-737 predicted to minimize residual disease. We also investigate the potential of ABT-737 co-therapy with carboplatin to prevent or delay the onset of carboplatin-resistance under two scenarios. When resistance is acquired as a result of aberrant DNA-damage repair in cells treated with carboplatin, the model is used to identify drug delivery schedules that induce tumor remission with even low doses of combination therapy. When resistance is intrinsic, due to a pre-existing cohort of resistant cells, tumor remission is no longer feasible, but our model can be used to identify dosing strategies that extend disease-free survival periods. These results underscore the potential of our model to accelerate the development of novel therapeutics such as ABT-737, by predicting optimal treatment strategies when these drugs are given in combination with currently approved cancer medications

A mathematical model of systemic inhibition of angiogenesis in metastatic development

We present a mathematical model describing the time development of a population of tumors subject to mutual angiogenic inhibitory signaling. Based on biophysical derivations, it describes organism-scale population dynamics under the influence of three processes: birth (dissemination of secondary tumors), growth and inhibition (through angiogenesis). The resulting model is a nonlinear partial differential transport equation with nonlocal boundary condition. The nonlinearity stands in the velocity through a nonlocal quantity of the model (the total metastatic volume). The asymptotic behavior of the model is numerically investigated and reveals interesting dynamics ranging from convergence to a steady state to bounded non-periodic or periodic behaviors, possibly with complex repeated patterns. Numerical simulations are performed with the intent to theoretically study the relative impact of potentiation or impairment of each process of the birth/growth/inhibition balance. Biological insights on possible implications for the phenomenon of "cancer without disease" are also discussed

Hypoxic Cell Waves around Necrotic Cores in Glioblastoma: A Biomathematical Model and its Therapeutic Implications

Glioblastoma is a rapidly evolving high-grade astrocytoma that is distinguished pathologically from lower grade gliomas by the presence of necrosis and microvascular hiperplasia. Necrotic areas are typically surrounded by hypercellular regions known as "pseudopalisades" originated by local tumor vessel occlusions that induce collective cellular migration events. This leads to the formation of waves of tumor cells actively migrating away from central hypoxia. We present a mathematical model that incorporates the interplay among two tumor cell phenotypes, a necrotic core and the oxygen distribution. Our simulations reveal the formation of a traveling wave of tumor cells that reproduces the observed histologic patterns of pseudopalisades. Additional simulations of the model equations show that preventing the collapse of tumor microvessels leads to slower glioma invasion, a fact that might be exploited for therapeutic purposes.Comment: 29 pages, 9 figure

Lytic cycle: A defining process in oncolytic virotherapy

The viral lytic cycle is an important process in oncolytic virotherapy. Most mathematical models for oncolytic virotherapy do not incorporate this process. In this article, we propose a mathematical model with the viral lytic cycle based on the basic mathematical model for oncolytic virotherapy. The viral lytic cycle is characterized by two parameters, the time period of the viral lytic cycle and the viral burst size. The time period of the viral lytic cycle is modeled as a delay parameter. The model is a nonlinear system of delay differential equations. The model reveals a striking feature that the critical value of the period of the viral lytic cycle is determined by the viral burst size. There are two threshold values for the burst size. Below the first threshold, the system has an unstable trivial equilibrium and a globally stable virus free equilibrium for any nonnegative delay, while the system has a third positive equilibrium when the burst size is greater than the first threshold. When the burst size is above the second threshold, there is a functional relation between the bifurcation value of the delay parameter for the period of the viral lytic cycle and the burst size. If the burst size is greater than the second threshold, the positive equilibrium is stable when the period of the viral lytic cycle is smaller than the bifurcation value, while the system has orbitally stable periodic solutions when the period of the lytic cycle is longer than the bifurcation value. However, this bifurcation value becomes smaller when the burst size becomes bigger. The viral lytic cycle may explain the oscillation phenomena observed in many studies. An important clinic implication is that the burst size should be carefully modified according to its effect on the lytic cycle when a type of a virus is modified for virotherapy, so that the period of the viral lytic cycle is in a suitable range which can break away the stability of the positive equilibria or periodic solutions. (C) 2012 Elsevier Inc. All rights reserved

Stability of Hahnfeldt Angiogenesis Models with Time Lags

Mathematical models of angiogenesis, pioneered by P. Hahnfeldt, are under study. To enrich the dynamics of three models, we introduced biologically motivated time-varying delays. All models under study belong to a special class of nonlinear nonautonomous systems with delays. Explicit conditions for the existence of positive global solutions and the equilibria solutions were obtained. Based on a notion of an M-matrix, new results are presented for the global stability of the system and were used to prove local stability of one model. For a local stability of a second model, the recent result for a Lienard-type second-order differential equation with delays was used. It was shown that models with delays produce a complex and nontrivial dynamics. Some open problems are presented for further studies