The dynamics of tumour-vasculature interaction suggests low-dose, time-dense anti-angiogenic schedulings

Angiogenesis and vessel co-option in a mathematical model of diffusive tumor growth: The role of chemotaxis

This work considers the propagation of a tumor from the stage of a small avascular sphere in a host tissue and the progressive onset of a tumor neovasculature stimulated by a pro-angiogenic factor secreted by hypoxic cells. The way new vessels are formed involves cell sprouting from pre-existing vessels and following a trail via a chemotactic mechanism (CM). Namely, it is first proposed a detailed general family of models of the CM, based on a statistical mechanics approach. The key hypothesis is that the CM is composed by two components: i) the well-known bias induced by the angiogenic factor gradient; ii) the presence of stochastic changes of the velocity direction, thus giving rise to a diffusive component. Then, some further assumptions and simplifications are applied in order to derive a specific model to be used in the simulations. The tumor progression is favored by its acidic aggression towards the healthy cells. The model includes the evolution of many biological and chemical species. Numerical simulations show the onset of a traveling wave eventually replacing the host tissue with a fully vascularized tumor. The results of simulations agree with experimental measures of the vasculature density in tumors, even in the case of particularly hypoxic tumors.

Chronic hepatitis B virus and liver fibrosis: A mathematical model

Hepatitis B virus (HBV) infection is a liver disorder that can result in cirrhosis, liver failure and hepatocellular carcinoma. HBV infection remains a major global health problem, as it affects more 350 million people chronically and kills roughly 600,000 people annually. Drugs currently used against HBV include IFN-α that decreases viremia, inflammation and the growth of liver fibrosis, and adefovir that decreases the viral load. Each of these drugs can have severe side-effects. In the present paper, we consider the treatment of chronic HBV by a combination of IFN-α and adefovir, and raise the following question: What should be the optimal ratio between IFN-α and adefovir in order to achieve the best 'efficacy' under constraints on the total amount of the drugs; here the efficacy is measured by the reduction of the levels of inflammation and of fibrosis? We develop a mathematical model of HBV pathogenesis by a system of partial differential equations (PDEs) and use the model to simulate a 'synergy map' which addresses the above question.

Application of mathematical models to metronomic chemotherapy: What can be inferred from minimal parameterized models?

Metronomic chemotherapy refers to the frequent administration of chemotherapy at relatively low, minimally toxic doses without prolonged treatment interruptions. Different from conventional or maximum-tolerated-dose chemotherapy which aims at an eradication of all malignant cells, in a metronomic dosing the goal often lies in the long-term management of the disease when eradication proves elusive. Mathematical modeling and subsequent analysis (theoretical as well as numerical) have become an increasingly more valuable tool (in silico) both for determining conditions under which specific treatment strategies should be preferred and for numerically optimizing treatment regimens. While elaborate, computationally-driven patient specific schemes that would optimize the timing and drug dose levels are still a part of the future, such procedures may become instrumental in making chemotherapy effective in situations where it currently fails. Ideally, mathematical modeling and analysis will develop into an additional decision making tool in the complicated process that is the determination of efficient chemotherapy regimens. In this article, we review some of the results that have been obtained about metronomic chemotherapy from mathematical models and what they infer about the structure of optimal treatment regimens.

Modeling and predicting optimal treatment scheduling between the antiangiogenic drug sunitinib and irinotecan in preclinical settings

We present a system of nonlinear ordinary differential equations used to quantify the complex dynamics of the interactions between tumor growth, vasculature generation, and antiangiogenic treatment. The primary dataset consists of longitudinal tumor size measurements (1,371 total observations) in 105 colorectal tumor‐bearing mice. Mice received single or combination administration of sunitinib, an antiangiogenic agent, and/or irinotecan, a cytotoxic agent. Depending on the dataset, parameter estimation was performed either using a mixed‐effect approach or by nonlinear least squares. Through a log‐likelihood ratio test, we conclude that there is a potential synergistic interaction between sunitinib when administered in combination with irinotecan in preclinical settings. Model simulations were then compared to data from a follow‐up preclinical experiment. We conclude that the model has predictive value in identifying the therapeutic window in which the timing between the administrations of these two drugs is most effective.

Maximum tolerated dose versus metronomic scheduling in the treatment of metastatic cancers

Although optimal control theory has been used for the theoretical study of anti-cancerous drugs scheduling optimization, with the aim of reducing the primary tumor volume, the effect on metastases is often ignored. Here, we use a previously published model for metastatic development to define an optimal control problem at the scale of the entire organism of the patient. In silico study of the impact of different scheduling strategies for anti-angiogenic and cytotoxic agents (either in monotherapy or in combination) is performed to compare a low-dose, continuous, metronomic administration scheme with a more classical maximum tolerated dose schedule. Simulation results reveal differences between primary tumor reduction and control of metastases but overall suggest use of the metronomic protocol.

Fine-tuning anti-tumor immunotherapies via stochastic simulations

BACKGROUND

Anti-tumor therapies aim at reducing to zero the number of tumor cells in a host within their end or, at least, aim at leaving the patient with a sufficiently small number of tumor cells so that the residual tumor can be eradicated by the immune system. Besides severe side-effects, a key problem of such therapies is finding a suitable scheduling of their administration to the patients. In this paper we study the effect of varying therapy-related parameters on the final outcome of the interplay between a tumor and the immune system.

RESULTS

This work generalizes our previous study on hybrid models of such an interplay where interleukins are modeled as a continuous variable, and the tumor and the immune system as a discrete-state continuous-time stochastic process. The hybrid model we use is obtained by modifying the corresponding deterministic model, originally proposed by Kirschner and Panetta. We consider Adoptive Cellular Immunotherapies and Interleukin-based therapies, as well as their combination. By asymptotic and transitory analyses of the corresponding deterministic model we find conditions guaranteeing tumor eradication, and we tune the parameters of the hybrid model accordingly. We then perform stochastic simulations of the hybrid model under various therapeutic settings: constant, piece-wise constant or impulsive infusion and daily or weekly delivery schedules.

CONCLUSIONS

Results suggest that, in some cases, the delivery schedule may deeply impact on the therapy-induced tumor eradication time. Indeed, our model suggests that Interleukin-based therapies may not be effective for every patient, and that the piece-wise constant is the most effective delivery to stimulate the immune-response. For Adoptive Cellular Immunotherapies a metronomic delivery seems more effective, as it happens for other anti-angiogenesis therapies and chemotherapies, and the impulsive delivery seems more effective than the piece-wise constant. The expected synergistic effects have been observed when the therapies are combined.

Anti-VEGF and beyond: shaping a new generation of anti-angiogenic therapies for cancer

The anti-angiogenic class of drugs is one of the few where representatives have gained international approval for clinical use in oncology during the past decade. Most of the biological and clinical activity of the currently available generation of anti-angiogenic drugs targets vascular endothelial growth factor (VEGF) and its related pathways. However, the clinical benefits associated with the use of these drugs have, so far, been limited. There is, therefore, an unmet need for biomarkers that can be used to identify patients who are most likely to benefit therapeutically and also to predict the best schedule and dosage for these drugs. Here, we discuss some of the emerging new combination strategies involving the approved anti-angiogenic drugs, some of the emerging targets associated with neoplastic angiogenesis and some novel agents used as a paradigm of the next generation of anti-angiogenic drugs.

Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics

An optimal control problem for cancer chemotherapy is considered that includes immunological activity. In the objective a weighted average of several quantities that describe the effectiveness of treatment is minimized. These terms include (i) the number of cancer cells at the terminal time, (ii) a measure for the immunocompetent cell densities at the terminal point (included as a negative term), (iii) the overall amount of cytotoxic agents given as a measure for the side effects of treatment and (iv) a small penalty on the terminal time that limits the overall therapy horizon which is assumed to be free. This last term is essential in obtaining a well-posed problem formulation. Employing a Gompertzian growth model for the cancer cells, for various scenarios optimal controls and corresponding responses of the system are calculated. Solutions initially follow a full dose treatment, but then at one point switch to a singular regimen that only applies partial dosages. This structure is consistent with protocols that apply an initial burst to reduce the tumor volume and then maintain a small volume through lower dosages. Optimal controls end with either a prolonged period of no dose treatment or, in a small number of scenarios, this no dose interval is still followed by one more short burst of full dose treatment.

New approach to modeling of antiangiogenic treatment on the basis of Hahnfeldt et al. model

In the paper we propose a new methodology in modeling of antiangiogenic treatment on the basis of well recognized model formulated by Hahnfeldt et al. in 1999. On the basis of the Hahnfeldt et al. model, with the usage of the optimal control theory, some protocols of antiangiogenic treatment were proposed. However, in our opinion the formulation of that model is valid only for the antivascular treatment, that is treatment that is focused on destroying endothelial cells. Therefore, we propose a modification of the original model which is valid in the case of the antiangiogenic treatment, that is treatment which is focused on blocking angiogenic signaling. We analyze basic mathematical properties of the proposed model and present some numerical simulations.

Resistance to antitumor chemotherapy due to bounded-noise-induced transitions

Tumor angiogenesis is a landmark of solid tumor development, but it is also directly relevant to chemotherapy. Indeed, the density and quality of neovessels may influence the effectiveness of therapies based on blood-born agents. In this paper, first we define a deterministic model of antiproliferative chemotherapy in which the drug efficacy is a unimodal function of vessel density, and then we show that under constant continuous infusion therapy the tumor-vessel system may be multistable. However, the actual drug concentration profiles are affected by bounded even if possibly large fluctuations. Through numerical simulations, we show that the tumor volume may undergo transitions to the higher equilibrium value induced by the bounded noise. In case of periodically delivered boli-based chemotherapy, we model the fluctuations due to time variability of both the drug clearance rate and the distribution volume, as well as those due to irregularities in drug delivery. We observed noise-induced transitions also in case of periodic delivering. By applying a time dense scheduling with constant average delivered drug (metronomic scheduling), we observed an easier suppression of the transitions. Finally, we propose to interpret the above phenomena as an unexpected non-genetic kind of resistance to chemotherapy.

Chemotherapy of vascularised tumours: role of vessel density and the effect of vascular "pruning"

In this work we propose to model chemotherapy taking into account the mutual interaction between tumour growth and the development of tumour vasculature. By adopting a simple model for this interaction, and assuming that the efficacy of a drug can be modulated by the vessel density, we study the constant continuous therapy, the periodic bolus-based therapy, and combined therapy in which a chemotherapic drug is associated with an anti-angiogenic agent. The model allows to represent the vessel-disrupting activity of some standard chemotherapic drugs, and shows, in the case of constant continuous drug administration, the possibility of multiple stable equilibria. The multistability suggests an explanation for some sudden losses of control observed during therapy, and for the beneficial effect of vascular "pruning" exerted by anti-angiogenic agents in combined therapy. Moreover, in case of periodic therapies in which the drug amount administered per unit time is constant ("metronomic" delivery), the model predicts a response, as a function of the bolus frequency, significantly influenced by the extent of the anti-angiogenic activity of the chemotherapic drug and by the dependence of the drug efficacy on the vessel density.