Modeling and analysis of a virus that replicates selectively in tumor cells

Agent-Based and Continuum Models for Spatial Dynamics of Infection by Oncolytic Viruses

The use of oncolytic viruses as cancer treatment has received considerable attention in recent years, however the spatial dynamics of this viral infection is still poorly understood. We present here a stochastic agent-based model describing infected and uninfected cells for solid tumours, which interact with viruses in the absence of an immune response. Two kinds of movement, namely undirected random and pressure-driven movements, are considered: the continuum limit of the models is derived and a systematic comparison between the systems of partial differential equations and the individual-based model, in one and two dimensions, is carried out. In the case of undirected movement, a good agreement between agent-based simulations and the numerical and well-known analytical results for the continuum model is possible. For pressure-driven motion, instead, we observe a wide parameter range in which the infection of the agents remains confined to the center of the tumour, even though the continuum model shows traveling waves of infection; outcomes appear to be more sensitive to stochasticity and uninfected regions appear harder to invade, giving rise to irregular, unpredictable growth patterns. Our results show that the presence of spatial constraints in tumours' microenvironments limiting free expansion has a very significant impact on virotherapy. Outcomes for these tumours suggest a notable increase in variability. All these aspects can have important effects when designing individually tailored therapies where virotherapy is included.

Oncolytic virus treatment of human breast cancer cells: Modelling therapy efficacy

Oncolytic viruses are a promising new treatment for cancer, whereby viruses are engineered to selectively destroy cancer cells. Mathematical modelling of the dynamics of the virus-tumour system can be modelled to provide insight into the system outcomes under different treatment protocols. In this study key metrics of treatment efficacy were identified and the mathematical model used to develop a decision framework to assess different treatment protocols. The optimal treatment outcome is the interplay between the virus application protocol and the uncertainty about the tumour characteristics. The uncertainty in the model parameters decreases as more data is available for their inference - however to obtain more data more time is required and the tumour then grows in size. Thus, there is an inherent tension whether it is better to wait to know the characteristics of the tumour system better or immediately initiating treatment. It is shown that, for small tumours, parameter inference with limited data does not constrain the choice of treatment protocol and rather only influences longer term decisions.

Oncolytic viral kinetics mechanistic modeling of Talimogene Laherparepvec (T-VEC) a first-in-class oncolytic viral therapy in patients with advanced melanoma

Talimogene Laherparepvec (T-VEC) is a first-in-class oncolytic virotherapy approved for the treatment of unresectable melanoma recurrent after initial surgery. Biodistribution data from a phase II study was used to develop a viral kinetic mechanistic model describing the interaction between cytokines such as granulocyte-macrophage colony-stimulating factor (GM-CSF), the immune system, and T-VEC treatment. Our analysis found that (1) the viral infection rate has a great influence on T-VEC treatment efficacy; (2) an increase in T-VEC dose of 10² plaque-forming units/ml 21 days and beyond after the initial dose of T-VEC resulted in an ~12% increase in response; and (3) at the systemic level, the ratio of resting innate immune cells to the death rate of innate immune impact T-VEC treatment efficacy. This analysis clarifies under which condition the immune system either assists in eliminating tumor cells or inhibits T-VEC treatment efficacy, which is critical to both efficiently design future oncolytic agents and understand cancer development.

Slow-Fast Model and Therapy Optimization for Oncolytic Treatment of Tumors

The present work studies models of oncolytic virotherapy without space variable in which virus replication occurs at a faster time scale than tumor growth. We address the questions of the modeling of virus injection in this slow-fast system and of the optimal timing for different treatment strategies. To this aim, we first derive the asymptotic of a three-species slow-fast model and obtain a two-species dynamical system, where the variables are tumor cells and infected tumor cells. We fully characterize the behavior of this system depending on the various biological parameters. In the second part, we address the modeling of virus injection and its expression in the two-species system, where the amount of virus does not appear explicitly. We prove that the injection can be described by an instantaneous jump in the phase plane, where a certain amount of tumors cells are transformed instantly into infected tumor cells. This description allows discussing qualitatively the timing of different injections in the frame of successive treatment strategies. This work is illustrated by numerical simulations. The timing and amount of injected virus may have counterintuitive optimal values; nevertheless, the understanding is clear from the phase space analysis.

Bifurcation analysis of the cancer virotherapy system with time delay and diffusion

In this paper, we propose a cancer virotherapy model with virus lytic cycle and diffusion term. Spatiotemporal dynamic properties of the cancer virotherapy system are studied. First, by analyzing the roots distribution of the characteristic equation and transcendental equation, the conditions for the local stability of the constant equilibria of system are given. Second, we select delay as the bifurcation parameter, the existence conditions of Hopf bifurcation are given. By using the center manifold theory and normal form method of partial functional differential equation, the detailed formulae for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are given. Finally, some numerical simulations are given.

Bistability in a model of tumor-immune system interactions with an oncolytic viral therapy

A mathematical model of tumor-immune system interactions with an oncolytic virus therapy for which the immune system plays a twofold role against cancer cells is derived. The immune cells can kill cancer cells but can also eliminate viruses from the therapy. In addition, immune cells can either be stimulated to proliferate or be impaired to reduce their growth by tumor cells. It is shown that if the tumor killing rate by immune cells is above a critical value, the tumor can be eradicated for all sizes, where the critical killing rate depends on whether the immune system is immunosuppressive or proliferative. For a reduced tumor killing rate with an immunosuppressive immune system, that bistability exists in a large parameter space follows from our numerical bifurcation study. Depending on the tumor size, the tumor can either be eradicated or be reduced to a size less than its carrying capacity. However, reducing the viral killing rate by immune cells always increases the effectiveness of the viral therapy. This reduction may be achieved by manipulating certain genes of viruses via genetic engineering or by chemical modification of viral coat proteins to avoid detection by the immune cells.

Computation of the conditions for anti-angiogenesis and gene therapy synergistic effects: Sensitivity analysis and robustness of target solutions

Both anti-angiogenesis and gene therapy involve complex processes depending on non-point parameters belonging to a space of values. To successfully overcome the challenges involved in their therapeutic approaches, there is a need to analyze the sensitivity of these parameters. In this paper, a new mathematical model that combines immune system stimulations, inflammatory processes associated with tumor development, and gene therapy aimed at enhancing the efficacy of both treatments are explored. Using the global sensitivity methods of Sobol and Morris, the most important parameters are estimated. Estimation of the sensitivity variance revealed a strong interdependence between the parameters. Also, determinations of the conditions for effective therapy lead to a target of reducing the cancer cell numbers by at least 50%. This opened the way for delimiting the parameter spaces making it possible to reach the treatment target in addition to enhancing the estimation of the minimum time of remission. The combination of therapies and sensitivity analysis have demonstrated the robustness of therapy success.

Quantitative imaging and dynamics of tumor therapy with viruses

Cancer therapy remains challenging due to the myriad presentations of the disease and the vast genetic diversity of tumors that continuously evolve and often become resistant to therapy. Viruses can be engineered to specifically infect, replicate, and kill tumor cells (tumor virotherapy). Moreover, the viruses can be "armed" with therapeutic genes to enhance their oncolytic effect. Using viruses to treat cancer is exciting and novel and in principle can be used for a broad variety of tumors. However, the approach is distinctly different from other cancer therapies since success depends on establishment of an infection within the tumor and ongoing propagation of the oncolytic virus within the tumor itself. Therefore, the target itself amplifies the therapy. This introduces complex dynamics especially when the immune system is taken into consideration as well as the physical and other biological barriers to virus growth. Understanding these dynamics not only requires mathematical and computational models but also approaches for the noninvasive monitoring of the virus and tumor populations. In this perspective, we discuss strategies and current results to achieve this important goal of understanding these dynamics in pursuit of optimization of oncolytic virotherapy.

The role of viral infectivity in oncolytic virotherapy outcomes: A mathematical study

A model capturing the dynamics between virus and tumour cells in the context of oncolytic virotherapy is presented and analysed. The ability of the virus to be internalised by uninfected cells is described by an infectivity parameter, which is inferred from available experimental data. The parameter is also able to describe the effects of changes in the tumour environment that affect viral uptake from tumour cells. Results show that when a virus is inoculated inside a growing tumour, strategies for enhancing infectivity do not lead to a complete eradication of the tumour. Within typical times of experiments and treatments, we observe the onset of oscillations, which always prevent a full destruction of the tumour mass. These findings are in good agreement with available laboratory results. Further analysis shows why a fully successful therapy cannot exist for the proposed model and that care must be taken when designing and engineering viral vectors with enhanced features. In particular, bifurcation analysis reveals that creating longer lasting virus particles or using strategies for reducing infected cell lifespan can cause unexpected and unwanted surges in the overall tumour load over time. Our findings suggest that virotherapy alone seems unlikely to be effective in clinical settings unless adjuvant strategies are included.

Mathematical modeling of PDGF-driven glioma reveals the dynamics of immune cells infiltrating into tumors

Background: Tumor-infiltrated immune cells compose a significant component of many cancers. They have been observed to have contradictory impacts on tumors. Although the primary reasons for these observations remain elusive, it is important to understand how immune cells infiltrating into tumors is regulated. Recently our group conducted a series of experimental studies, which showed that muIDH1 gliomas have a significant global reduction of immune cells and suggested that the longer survival time of mice with CIMP gliomas may be due to the IDH mutation and its effect on reducing of the tumor-infiltrated immune cells. However, to comprehend how IDH1 mutants regulate infiltration of immune cells into gliomas and how they affect the aggressiveness of gliomas, it is necessary to integrate our experimental data into a dynamical system to acquire a much deeper understanding of subtle regulation of immune cell infiltration. Methods: The method is integration of mathematical modeling and experiments. According to mass conservation laws and assumption that immune cells migrate into the tumor site along a chemotactic gradient field, a mathematical model is formulated. Parameters are estimated from our experiments. Numerical methods are developed to solve the problem. Numerical predictions are compared with experimental results. Results: Our analysis shows that the net rate of increase of immune cells infiltrated into the tumor is approximately proportional to the 4/5 power of the chemoattractant production rate, and it is an increasing function of time while the percentage of immune cells infiltrated into the tumor is a decreasing function of time. Our model predicts that wtIDH1 mice will survive longer if the immune cells are blocked by reducing chemotactic coefficient. For more aggressive gliomas, our model shows that there is little difference in their survivals between wtIDH1 and muIDH1 tumors, and the percentage of immune cells infiltrated into the tumor is much lower. These predictions are verified by our experimental results. In addition, wtIDH1 and muIDH1 can be quantitatively distinguished by their chemoattractant production rates, and the chemotactic coefficient determines possibilities of immune cells migration along chemoattractant gradient fields. Conclusions: The chemoattractant gradient field produced by tumor cells may facilitate immune cells migration to the tumor cite. The chemoattractant production rate may be utilized to classify wtIDH1 and muIDH1 tumors. The dynamics of immune cells infiltrating into tumors is largely determined by tumor cell chemoattractant production rate and chemotactic coefficient.

Backward Hopf bifurcation in a mathematical model for oncolytic virotherapy with the infection delay and innate immune effects

In this paper, we consider a system of delay differential equations that models the oncolytic virotherapy on solid tumours with the delay of viral infection in the presence of the innate immune response. We conduct qualitative and numerical analysis, and provide possible medical implications for our results. The system has four equilibrium solutions. Fixed point analysis indicates that increasing the burst size and infection rate of the viruses has positive contribution to the therapy. However, increasing the immune killing infection rate, the immune stimulation rate, or the immune killing virus rate may lead the treatment failed. The viral infection time delay induces backward Hopf bifurcations, which means that the therapy may fail before time delay increases passing through a Hopf bifurcation. The parameter analysis also shows how saddle-node and Hopf bifurcations occur as viral burst size and other parameters vary, which yields further insights into the dynamics of the virotherapy.

Analysis of a Multiple Delays Model for Treatment of Cancer with Oncolytic Virotherapy

Despite advanced discoveries in cancerology, conventional treatments by surgery, chemotherapy, or radiotherapy remain ineffective in some situations. Oncolytic virotherapy, i.e., the involvement of replicative viruses targeting specific tumor cells, opens new perspectives for better management of this disease. Certain viruses naturally have a preferential tropism for the tumor cells; others are genetically modifiable to present such properties, as the lytic cycle virus, which is a process that represents a vital role in oncolytic virotherapy. In the present paper, we present a mathematical model for the dynamics of oncolytic virotherapy that incorporates multiple time delays representing the multiple time periods of a lytic cycle. We compute the basic reproductive ratio R ₀, and we show that there exist a disease-free equilibrium point (DFE) and an endemic equilibrium point (DEE). By formulating suitable Lyapunov function, we prove that the disease-free equilibrium (DFE) is globally asymptotically stable if R ₀ < 1 and unstable otherwise. We also demonstrate that under additional conditions, the endemic equilibrium is stable. Also, a Hopf bifurcation analysis of our dynamic system is used to understand how solutions and their stability change as system parameters change in the case of a positive delay. To illustrate the effectiveness of our theoretical results, we give numerical simulations for several scenarios.

Spatial Model for Oncolytic Virotherapy with Lytic Cycle Delay

We formulate a mathematical model of functional partial differential equations for oncolytic virotherapy which incorporates virus diffusivity, tumor cell diffusion, and the viral lytic cycle based on a basic oncolytic virus dynamics model. We conduct a detailed analysis for the dynamics of the model and carry out numerical simulations to demonstrate our analytic results. Particularly, we establish the positive invariant domain for the [Formula: see text] limit set of the system and show that the model has three spatially homogenous equilibriums solutions. We prove that the spatially uniform virus-free steady state is globally asymptotically stable for any viral lytic period delay and diffusion coefficients of tumor cells and viruses when the viral burst size is smaller than a critical value. We obtain the conditions, for example the ratio of virus diffusion coefficient to that of tumor cells is greater than a value and the viral lytic cycle, is greater than a critical value, under which the spatially uniform positive steady state is locally asymptotically stable. We also obtain conditions under which the system undergoes Hopf bifurcations, and stable periodic solutions occur. We point out medical implications of our results which are difficult to obtain from models without combining diffusive properties of viruses and tumor cells with viral lytic cycles.

In vitro and in silico multidimensional modeling of oncolytic tumor virotherapy dynamics

Tumor therapy with replication competent viruses is an exciting approach to cancer eradication where viruses are engineered to specifically infect, replicate, spread and kill tumor cells. The outcome of tumor virotherapy is complex due to the variable interactions between the cancer cell and virus populations as well as the immune response. Oncolytic viruses are highly efficient in killing tumor cells in vitro, especially in a 2D monolayer of tumor cells, their efficiency is significantly lower in a 3D environment, both in vitro and in vivo. This indicates that the spatial dimension may have a major influence on the dynamics of virus spread. We study the dynamic behavior of a spatially explicit computational model of tumor and virus interactions using a combination of in vitro 2D and 3D experimental studies to inform the models. We determine the number of nearest neighbor tumor cells in 2D (median = 6) and 3D tumor spheroids (median = 16) and how this influences virus spread and the outcome of therapy. The parameter range leading to tumor eradication is small and even harder to achieve in 3D. The lower efficiency in 3D exists despite the presence of many more adjacent cells in the 3D environment that results in a shorter time to reach equilibrium. The mean field mathematical models generally used to describe tumor virotherapy appear to provide an overoptimistic view of the outcomes of therapy. Three dimensional space provides a significant barrier to efficient and complete virus spread within tumors and needs to be explicitly taken into account for virus optimization to achieve the desired outcome of therapy.

Mathematical modelling of a hypoxia-regulated oncolytic virus delivered by tumour-associated macrophages

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A continuum model of macrophages releasing an oncolytic virus within a tumour spheroid.

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Predictive modelling of this treatment given in combination with radiotherapy.

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Investigation into how radiotherapy and oncolytic virotherapy should be scheduled.

Tumour hypoxia has long presented a challenge for cancer therapy: Poor vascularisation in hypoxic regions hinders both the delivery of chemotherapeutic agents and the response to radiotherapy, and hypoxic cancer cells that survive treatment can trigger tumour regrowth after treatment has ended. Tumour-associated macrophages are attractive vehicles for drug delivery because they localise in hypoxic areas of the tumour. In this paper, we derive a mathematical model for the infiltration of an in vitro tumour spheroid by macrophages that have been engineered to release an oncolytic adenovirus under hypoxic conditions. We use this model to predict the efficacy of treatment schedules in which radiotherapy and the engineered macrophages are given in combination. Our work suggests that engineered macrophages should be introduced immediately after radiotherapy for maximum treatment efficacy. Our model provides a framework that may guide future experiments to determine how multiple rounds of radiotherapy and macrophage virotherapy should be coordinated to maximise therapeutic responses.

In Vivo Estimation of Oncolytic Virus Populations within Tumors

The use of replication-competent viruses as oncolytic agents is rapidly expanding, with several oncolytic viruses approved for cancer therapy. As responses to therapy are highly variable, understanding the dynamics of therapy is critical for optimal application of virotherapy in practice. Although mathematical models have been developed to understand the dynamics of tumor virotherapy, a scarcity of in vivo data has made difficult parametrization of these models. To tackle this problem, we studied the in vitro and in vivo spread of two oncolytic measles viruses that induce expression of the sodium iodide symporter (NIS) in cells. NIS expression enabled infected cells to concentrate radioactive isotopes that could be reproducibly and quantitatively imaged using SPECT/CT. We observed a strong linear relationship in vitro between infectious virus particles, viral N and NIS gene expression, and radioactive isotope uptake. In vivo radioisotope uptake was highly correlated with viral N and NIS gene expression. Similar expression patterns between viral N and NIS gene expression in vitro and in vivo implied that the oncolytic virus behaved similarly in both scenarios. Significant titers of viable virus were consistently isolated from tumors explanted from mice that had been injected with oncolytic measle viruses. We observed a weaker but positive in vivo relationship between radioisotope uptake and the viable virus titer recovered from tumors; this was likely due to anisotropies in the viral distribution in vivo These data suggest that methods that enable quantitation of in vivo anisotropies are required for continuing development of oncolytic virotherapy.Significance: These findings address a fundamental gap in our knowledge of oncolytic virotherapy by presenting technology that gives insight into the behavior of oncolytic viruses in vivo Cancer Res; 78(20); 5992-6000. ©2018 AACR.

The Role of the Innate Immune System in Oncolytic Virotherapy

The complexity of the immune responses is a major challenge in current virotherapy. This study incorporates the innate immune response into our basic model for virotherapy and investigates how the innate immunity affects the outcome of virotherapy. The viral therapeutic dynamics is largely determined by the viral burst size, relative innate immune killing rate, and relative innate immunity decay rate. The innate immunity may complicate virotherapy in the way of creating more equilibria when the viral burst size is not too big, while the dynamics is similar to the system without innate immunity when the viral burst size is big.

Growth of adult spinal cord in knifefish: Development and parametrization of a distributed model

The study of indeterminate-growing organisms such as teleost fish presents a unique opportunity for improving our understanding of central nervous tissue growth during adulthood. Integrating the existing experimental data associated with this process into a theoretical framework through mathematical or computational modeling provides further research avenues through sensitivity analysis and optimization. While this type of approach has been used extensively in investigations of tumor growth, wound healing, and bone regeneration, the development of nervous tissue has been rarely studied within a modeling framework. To address this gap, the present work introduces a distributed model of spinal cord growth in the knifefish Apteronotus leptorhynchus, an established teleostean model of adult growth in the central nervous system. The proposed model incorporates two mechanisms, cell proliferation by active stem/progenitor cells and cell drift due to population pressure, both of which are subject to global constraints. A coupled reaction-diffusion equation approach was adopted to represent the densities of actively-proliferating and non-proliferating cells along the longitudinal axis of the spinal cord. Computer simulations using this model yielded biologically-feasible growth trajectories. Subsequent comparisons with whole-organism growth curves allowed the estimation of previously-unknown parameters, such as relative growth rates.

Oncolytic potency and reduced virus tumor-specificity in oncolytic virotherapy. A mathematical modelling approach

In the present paper, we address by means of mathematical modeling the following main question: How can oncolytic virus infection of some normal cells in the vicinity of tumor cells enhance oncolytic virotherapy? We formulate a mathematical model describing the interactions between the oncolytic virus, the tumor cells, the normal cells, and the antitumoral and antiviral immune responses. The model consists of a system of delay differential equations with one (discrete) delay. We derive the model's basic reproductive number within tumor and normal cell populations and use their ratio as a metric for virus tumor-specificity. Numerical simulations are performed for different values of the basic reproduction numbers and their ratios to investigate potential trade-offs between tumor reduction and normal cells losses. A fundamental feature unravelled by the model simulations is its great sensitivity to parameters that account for most variation in the early or late stages of oncolytic virotherapy. From a clinical point of view, our findings indicate that designing an oncolytic virus that is not 100% tumor-specific can increase virus particles, which in turn, can further infect tumor cells. Moreover, our findings indicate that when infected tissues can be regenerated, oncolytic viral infection of normal cells could improve cancer treatment.

Fighting Cancer with Mathematics and Viruses

After decades of research, oncolytic virotherapy has recently advanced to clinical application, and currently a multitude of novel agents and combination treatments are being evaluated for cancer therapy. Oncolytic agents preferentially replicate in tumor cells, inducing tumor cell lysis and complex antitumor effects, such as innate and adaptive immune responses and the destruction of tumor vasculature. With the availability of different vector platforms and the potential of both genetic engineering and combination regimens to enhance particular aspects of safety and efficacy, the identification of optimal treatments for patient subpopulations or even individual patients becomes a top priority. Mathematical modeling can provide support in this arena by making use of experimental and clinical data to generate hypotheses about the mechanisms underlying complex biology and, ultimately, predict optimal treatment protocols. Increasingly complex models can be applied to account for therapeutically relevant parameters such as components of the immune system. In this review, we describe current developments in oncolytic virotherapy and mathematical modeling to discuss the benefit of integrating different modeling approaches into biological and clinical experimentation. Conclusively, we propose a mutual combination of these research fields to increase the value of the preclinical development and the therapeutic efficacy of the resulting treatments.

Mathematical and Computational Modeling for Tumor Virotherapy with Mediated Immunity

We propose a new mathematical modeling framework based on partial differential equations to study tumor virotherapy with mediated immunity. The model incorporates both innate and adaptive immune responses and represents the complex interaction among tumor cells, oncolytic viruses, and immune systems on a domain with a moving boundary. Using carefully designed computational methods, we conduct extensive numerical simulation to the model. The results allow us to examine tumor development under a wide range of settings and provide insight into several important aspects of the virotherapy, including the dependence of the efficacy on a few key parameters and the delay in the adaptive immunity. Our findings also suggest possible ways to improve the virotherapy for tumor treatment.

The role of TNF-α inhibitor in glioma virotherapy: A mathematical model

Virotherapy, using herpes simplex virus, represents a promising therapy of glioma. But the innate immune response, which includes TNF-α produced by macrophages, reduces the effectiveness of the treatment. Hence treatment with TNF-α inhibitor may increase the effectiveness of the virotherapy. In the present paper we develop a mathematical model that includes continuous infusion of the virus in combination with TNF-α inhibitor. We study the efficacy of the treatment under different combinations of the two drugs for different scenarios of the burst size of newly formed virus emerging from dying infected cancer cells. The model may serve as a first step toward developing an optimal strategy for the treatment of glioma by the combination of TNF-α inhibitor and oncolytic virus injection.

A generic viral dynamic model to systematically characterize the interaction between oncolytic virus kinetics and tumor growth

Oncolytic viruses (OV) represent an encouraging new therapeutic concept for treatment of human cancers. OVs specifically replicate in tumor cells and initiate cell lysis whilst tumor cells act as endogenous bioreactors for virus amplification. This complex bidirectional interaction between tumor and oncolytic virus hampers the establishment of a straight dose-concentration-effect relation. We aimed to develop a generic mathematical pharmacokinetic/pharmacodynamics (PK/PD) model to characterize the relationship between tumor cell growth and kinetics of different OVs. U87 glioblastoma cell growth and titer of Newcastle disease virus (NDV), reovirus (RV) and parvovirus (PV) were systematically determined in vitro. PK/PD analyses were performed using non-linear mixed effects modeling. A viral dynamic model (VDM) with a common structure for the three different OVs was developed which simultaneously described tumor growth and virus replication. Virus specific parameters enabled a comparison of the kinetics and tumor killing efficacy of each OV. The long-term interactions of tumor cells with NDV and RV were simulated to predict tumor reoccurrence. Various treatment scenarios (single and multiple dosing with same OV, co-infection with different OVs and combination with hypothetical cytotoxic compounds) were simulated and ranked for efficacy using a newly developed treatment rating score. The developed VDM serves as flexible tool for the systematic cross-characterization of tumor-virus relationships and supports preselection of the most promising treatment regimens for follow-up in vivo analyses.

Dynamical properties and tumor clearance conditions for a nine-dimensional model of bladder cancer immunotherapy

Understanding the global interaction dynamics between tumor and the immune system plays a key role in the advancement of cancer therapy. Bunimovich-Mendrazitsky et al. (2015) developed a mathematical model for the study of the immune system response to combined therapy for bladder cancer with Bacillus Calmette-Guérin (BCG) and interleukin-2 (IL-2) . We utilized a mathematical approach for bladder cancer treatment model for derivation of ultimate upper and lower bounds and proving dissipativity property in the sense of Levinson. Furthermore, tumor clearance conditions for BCG treatment of bladder cancer are presented. Our method is based on localization of compact invariant sets and may be exploited for a prediction of the cells populations dynamics involved into the model.

Resources, mortality, and disease ecology: Importance of positive feedbacks between host growth rate and pathogen dynamics

Resource theory and metabolic scaling theory suggest that the dynamics of a pathogen within a host should strongly depend upon the rate of host cell metabolism. Once an infection occurs, key ecological interactions occur on or within the host organism that determine whether the pathogen dies out, persists as a chronic infection, or grows to densities that lead to host death. We hypothesize that, in general, conditions favoring rapid host growth rates should amplify the replication and proliferation of both fungal and viral pathogens. If a host population experiences an increase in mortality, to persist it must have a higher growth rate, per host, often reflecting greater resource availability per capita. We hypothesize that this could indirectly foster the pathogen, which also benefits from increased within-host resource turnover. We first bring together in a short review a number of key prior studies which illustrate resource effects on viral and fungal pathogen dynamics. We then report new results from a semi-continuous cell culture experiment with SHIV, demonstrating that higher mortality rates indeed can promote viral proliferation. We develop a simple model that illustrates dynamical consequences of these resource effects, including interesting effects such as alternative stable states and oscillatory dynamics. Our paper contributes to a growing body of literature at the interface of ecology and infectious disease epidemiology, emphasizing that host abundances alone do not drive community dynamics: the physiological state and resource content of infected hosts also strongly influence host-pathogen interactions.

Modeling oncolytic virotherapy: is complete tumor-tropism too much of a good thing?

The specific targeting of tumor cells by replication-competent oncolytic viruses is considered indispensable for realizing the potential of oncolytic virotherapy. Yet off-target infections by oncolytic viruses may increase virus production, further reducing tumor load. This ability may be critical when tumor-cell scarcity or the onset of an adaptive immune response constrain viral anti-tumoral efficacy. Here we develop a mathematical framework for assessing whether oncolytic viruses with reduced tumor-specificity can more effectively eliminate tumors while keeping losses to normal cell populations low. We find viruses that infect some normal cells can potentially balance the competing goals of tumor elimination and minimizing the effects on normal cell populations. Particularly when infected tissues can be regenerated, moderating rather than completely eliminating the ability of oncolytic viruses to infect and lyse normal cells could improve cancer treatment, with potentially fewer side-effects than conventional treatments such as chemotherapy.

Mathematical model for radial expansion and conflation of intratumoral infectious centers predicts curative oncolytic virotherapy parameters

Simple, inductive mathematical models of oncolytic virotherapy are needed to guide protocol design and improve treatment outcomes. Analysis of plasmacytomas regressing after a single intravenous dose of oncolytic vesicular stomatitis virus in myeloma animal models revealed that intratumoral virus spread was spatially constrained, occurring almost exclusively through radial expansion of randomly distributed infectious centers. From these experimental observations we developed a simple model to calculate the probability of survival for any cell within a treated tumor. The model predicted that small changes to the density of initially infected cells or to the average maximum radius of infected centers would have a major impact on treatment outcome, and this was confirmed experimentally. The new model provides a useful and flexible tool for virotherapy protocol optimization.

A mathematical model for cell cycle-specific cancer virotherapy

Oncolytic viruses preferentially infect and replicate in cancerous cells, leading to elimination of tumour populations, while sparing most healthy cells. Here, we study the cell cycle-specific activity of viruses such as vesicular stomatitis virus (VSV). In spite of its capacity as a robust cytolytic agent, VSV cannot effectively attack certain tumour cell types during the quiescent, or resting, phase of the cell cycle. In an effort to understand the interplay between the time course of the cell cycle and the specificity of VSV, we develop a mathematical model for cycle-specific virus therapeutics. We incorporate the minimum biologically required time spent in the non-quiescent cell cycle phases using systems of differential equations with incorporated time delays. Through analysis and simulation of the model, we describe how varying the minimum cycling time and the parameters that govern viral dynamics affect the stability of the cancer-free equilibrium, which represents therapeutic success.

Dynamics of melanoma tumor therapy with vesicular stomatitis virus: explaining the variability in outcomes using mathematical modeling

Tumor selective, replication competent viruses are being tested for cancer gene therapy. This approach introduces a new therapeutic paradigm due to potential replication of the therapeutic agent and induction of a tumor-specific immune response. However, the experimental outcomes are quite variable, even when studies utilize highly inbred strains of mice and the same cell line and virus. Recognizing that virotherapy is an exercise in population dynamics, we utilize mathematical modeling to understand the variable outcomes observed when B16ova malignant melanoma tumors are treated with vesicular stomatitis virus in syngeneic, fully immunocompetent mice. We show how variability in the initial tumor size and the actual amount of virus delivered to the tumor have critical roles on the outcome of therapy. Virotherapy works best when tumors are small, and a robust innate immune response can lead to superior tumor control. Strategies that reduce tumor burden without suppressing the immune response and methods that maximize the amount of virus delivered to the tumor should optimize tumor control in this model system.

Modeling of tumor growth undergoing virotherapy

Tumor growth models subject to virotherapy treatment are analyzed and compared in this paper. Tumor growth conditions are obtained for each model type based on the virus infection rate and immune suppressive drug delivery. Equilibrium conditions resulted into quadratic functions for which the tumor radius remained constant during virotherapy. An irrigation tumor model for virotherapy treatment was also proposed. This model consists of irrigation layers distributed radially along the tumor and attached to a common blood circulation compartment. The irrigation model has similar dynamic and steady state characteristics to the diffusion model, which has been supported by experimental results. The irrigation model considers the immune system cell generation and consumption outside the tumor boundary but inside the blood circulation compartment. These characteristics provide a great potential for advanced cancer treatment applications because therapy dose delivery and immune system measurements can be made at the blood compartment level of the irrigation model.

The replicability of oncolytic virus: defining conditions in tumor virotherapy

The replicability of an oncolytic virus is measured by its burst size. The burst size is the number of new viruses coming out from a lysis of an infected tumor cell. Some clinical evidences show that the burst size of an oncolytic virus is a defining parameter for the success of virotherapy. This article analyzes a basic mathematical model that includes burst size for oncolytic virotherapy. The analysis of the model shows that there are two threshold values of the burst size: below the first threshold, the tumor always grows to its maximum (carrying capacity) size; while passing this threshold, there is a locally stable positive equilibrium solution appearing through transcritical bifurcation; while at or above the second threshold, there exits one or three families of periodic solutions arising from Hopf bifurcations. The study suggests that the tumor load can drop to a undetectable level either during the oscillation or when the burst size is large enough.

The parvoviral capsid controls an intracellular phase of infection essential for efficient killing of stepwise-transformed human fibroblasts

Members of the rodent subgroup of the genus Parvovirus exhibit lytic replication and spread in many human tumor cells and are therefore attractive candidates for oncolytic virotherapy. However, the significant variation in tumor tropism observed for these viruses remains largely unexplained. We report here that LuIII kills BJ-ELR 'stepwise-transformed' human fibroblasts efficiently, while MVM does not. Using viral chimeras, we mapped this property to the LuIII capsid gene, VP2, which is necessary and sufficient to confer the killer phenotype on MVM. LuIII VP2 facilitates a post-entry, pre-DNA-amplification step early in the life cycle, suggesting the existence of an intracellular moiety whose efficient interaction with the incoming capsid shell is critical to infection. Thus targeting of human cancers of different tissue-type origins will require use of parvoviruses with capsids that effectively make this critical interaction.

Dynamics of multiple myeloma tumor therapy with a recombinant measles virus

Replication-competent viruses are being tested as tumor therapy agents. The fundamental premise of this therapy is the selective infection of the tumor cell population with the amplification of the virus. Spread of the virus in the tumor ultimately should lead to eradication of the cancer. Tumor virotherapy is unlike any other form of cancer therapy as the outcome depends on the dynamics that emerge from the interaction between the virus and tumor cell populations both of which change in time. We explore these interactions using a model that captures the salient biological features of this system in combination with in vivo data. Our results show that various therapeutic outcomes are possible ranging from tumor eradication to oscillatory behavior. Data from in vivo studies support these conclusions and validate our modeling approach. Such realistic models can be used to understand experimental observations, explore alternative therapeutic scenarios and develop techniques to optimize therapy.

Modeling of cancer virotherapy with recombinant measles viruses

The Edmonston vaccine strain of measles virus has potent and selective activity against a wide range of tumors. Tumor cells infected by this virus or genetically modified strains express viral proteins that allow them to fuse with neighboring cells to form syncytia that ultimately die. Moreover, infected cells may produce new virus particles that proceed to infect additional tumor cells. We present a model of tumor and virus interactions based on established biology and with proper accounting of the free virus population. The range of model parameters is estimated by fitting to available experimental data. The stability of equilibrium states corresponding to complete tumor eradication, therapy failure and partial tumor reduction is discussed. We use numerical simulations to explore conditions for which the model predicts successful therapy and tumor eradication. The model exhibits damped, as well as stable oscillations in a range of parameter values. These oscillatory states are organized by a Hopf bifurcation.

Systemic armed oncolytic and immunologic therapy for cancer with JX-594, a targeted poxvirus expressing GM-CSF

Targeted oncolytic viruses and immunostimulatory therapeutics are being developed as novel cancer treatment platforms. These approaches can be combined through the expression of immunostimulatory cytokines from targeted viruses, including adenoviruses and herpesviruses. Although intratumoral injection of such viruses has been associated with tumor growth inhibition, eradication of distant metastases was not reported. The major limitations for this approach to date have been (1) inefficient intravenous virus delivery to tumors and (2) the lack of predictive, immunocompetent preclinical models. To overcome these hurdles, we developed JX-594, a targeted, thymidine kinase(-) vaccinia virus expressing human GM-CSF (hGM-CSF), for intravenous (i.v.) delivery. We evaluated two immunocompetent liver tumor models: a rabbit model with reproducible, time-dependent metastases to the lungs and a carcinogen-induced rat liver cancer model. Intravenous JX-594 was well tolerated and had highly significant efficacy, including complete responses, against intrahepatic primary tumors in both models. In addition, whereas lung metastases developed in all control rabbits, none of the i.v. JX-594-treated rabbits developed detectable metastases. Tumor-specific virus replication and gene expression, systemically detectable levels of hGM-CSF, and tumor-infiltrating CTLs were also demonstrated. JX-594 holds promise as an i.v.-delivered, targeted virotherapeutic. These two tumor models hold promise for the optimization of this approach.

Glioma virotherapy: effects of innate immune suppression and increased viral replication capacity

Oncolytic viruses are genetically altered replication-competent viruses that infect, and reproduce in, cancer cells but do not harm normal cells. On lysis of the infected cells, the newly formed viruses burst out and infect other tumor cells. Experiments with injecting mutant herpes simplex virus 1 (hrR3) into glioma implanted in brains of rats show lack of efficacy in eradicating the cancer. This failure is attributed to interference by the immune system. Initial pretreatment with immunosuppressive agent cyclophosphamide reduces the percentage of immune cells. We introduce a mathematical model and use it to determine how different protocols of cyclophosphamide treatment and how increased burst size of the mutated virus will affect the growth of the cancer. One of our conclusions is that the diameter of the cancer will decrease from 4 mm to eventually 1 mm if the burst size of the virus is triple that which is currently available. The effect of repeated cyclophosphamide treatment is to maintain a low density of uninfected cells in the tumor, thus reducing the probability of migration of tumor cells to other locations in the brain.

Oncolytic activity of vesicular stomatitis virus in primary adult T-cell leukemia

Treatments for hematological malignancies have improved considerably over the past decade, but the growing therapeutic arsenal has not benefited adult T-cell leukemia (ATL) patients. Oncolytic viruses such as vesicular stomatitis virus (VSV) have recently emerged as a potential treatment of solid tumors and leukemias in vitro and in vivo. In the current study, we investigated the ability of VSV to lyse primary human T-lymphotropic virus type 1 (HTLV-1)-infected T-lymphocytes from patients with ATL. Ex vivo primary ATL cells were permissive for VSV and underwent rapid oncolysis in a time-dependent manner. Importantly, VSV infection showed neither viral replication nor oncolysis in HTLV-1-infected, nonleukemic cells from patients with HTLV-1-associated myelopathy/tropical spastic paraparesis (HAM/TSP), and in naive CD4(+) T-lymphocytes from normal individuals or in ex vivo cell samples from patients with chronic lymphocytic leukemia (CLL). Interestingly, activation of primary CD4(+) T-lymphocytes with anti-CD3/CD28 monoclonal antibody, and specifically with anti-CD3, was sufficient to induce limited viral replication and oncolysis. However, at a similar level of T-cell activation, VSV replication was increased fourfold in ATL cells compared to activated CD4(+) T-lymphocytes, emphasizing the concept that VSV targets genetic defects unique to tumor cells to facilitate its replication. In conclusion, our findings provide the first essential information for the development of a VSV-based treatment for ATL.

Mathematical modeling of tumor therapy with oncolytic viruses: regimes with complete tumor elimination within the framework of deterministic models

Background

Oncolytic viruses that specifically target tumor cells are promising anti-cancer therapeutic agents. The interaction between an oncolytic virus and tumor cells is amenable to mathematical modeling using adaptations of techniques employed previously for modeling other types of virus-cell interaction.

Results

A complete parametric analysis of dynamic regimes of a conceptual model of anti-tumor virus therapy is presented. The role and limitations of mass-action kinetics are discussed. A functional response, which is a function of the ratio of uninfected to infected tumor cells, is proposed to describe the spread of the virus infection in the tumor. One of the main mathematical features of ratio-dependent models is that the origin is a complicated equilibrium point whose characteristics determine the main properties of the model. It is shown that, in a certain area of parameter values, the trajectories of the model form a family of homoclinics to the origin (so-called elliptic sector). Biologically, this means that both infected and uninfected tumor cells can be eliminated with time, and complete recovery is possible as a result of the virus therapy within the framework of deterministic models.

Conclusion

Our model, in contrast to the previously published models of oncolytic virus-tumor interaction, exhibits all possible outcomes of oncolytic virus infection, i.e., no effect on the tumor, stabilization or reduction of the tumor load, and complete elimination of the tumor. The parameter values that result in tumor elimination, which is, obviously, the desired outcome, are compatible with some of the available experimental data.

Reviewers

This article was reviewed by Mikhail Blagosklonny, David Krakauer, Erik Van Nimwegen, and Ned Wingreen.

Mathematical modeling of cancer radiovirotherapy

Cancer virotherapy represents a dynamical system that requires mathematical modeling for complete understanding of the outcomes. The combination of virotherapy with radiation (radiovirotherapy) has been recently shown to successfully eliminate tumors when virotherapy alone failed. However, it introduces a new level of complexity. We have developed a mathematical model, based on population dynamics, that captures the essential elements of radiovirotherapy. The existence of corresponding equilibrium points related to complete cure, partial cure, and therapy failure is proved and discussed. The parameters of the model were estimated by fitting to experimental data. By using simulations we analyzed the influence of parameters that describe the interaction between virus and tumor cell on the outcome of the therapy. Furthermore, we evaluated relevant therapeutic scenarios for radiovirotherapy, and offered elements for optimization.

The competitive dynamics between tumor cells, a replication-competent virus and an immune response

Replication-competent viruses have been used as an alternative therapeutic approach for cancer treatment. However, new clinical data revealed an innate immune response to virus that may mitigate the effects of treatment. Recently, Wein, Wu and Kirn have established a model which describes the interaction between tumor cells, a replication-competent virus and an immune response (Cancer Research 63 (2003):1317-1324). The purpose of this paper is to extend their model from the viewpoints of mathematics and biology and then prove global existence and uniqueness of solution to this new model, to study the dynamics of this novel therapy for cancers, and to explore a explicit threshold of the intensity of the immune response for controlling the tumor. We also study a time-delayed version of the model. We analytically prove that there exists a critical value tau0 of the time-delay tau such that the system has a periodic solution if tau > tau0. Numerical simulations are given to verify the analytical results. Furthermore, we numerically study the spatio-temporal dynamics of the model. The effects of the diffusivity of the immune response on the tumor growth are also discussed.

Mathematical modelling of the use of macrophages as vehicles for drug delivery to hypoxic tumour sites

Poor drug delivery and low rates of cell proliferation are two factors associated with hypoxia that diminish the efficacy of many chemotherapeutic drugs. Since macrophages are known to migrate specifically towards, and localize within, hypoxic tumour regions, a promising resolution to these problems involves genetically engineering macrophages to perform such anti-tumour functions as inducing cell lysis and inhibiting angiogenesis. In this paper we outline a modelling approach to characterize macrophage infiltration into early avascular solid tumours, and extensions to study the interaction of these cells with macrophages already present within the tumour. We investigate the role of chemotaxis and chemokine production, and the efficacy of macrophages as vehicles for drug delivery to hypoxic tumour sites. The model is based upon a growing avascular tumour spheroid, in which volume is filled by tumour cells, macrophages and extracellular material, and tumour cell proliferation and death is regulated by nutrient diffusion. Crucially, macrophages occupy volume, and hence contribute to the volume balance and hence the size of the tumour. We also include oxygen-dependent production of macrophage chemokines, which can lead to accumulations in the hypoxic region of the tumour. We find that the macrophage chemotactic sensitivity is a key determinant of macrophage infiltration and tumour size. Although increased infiltration should be beneficial from the point of view of macrophage-based therapies, such infiltration in fact leads to increased tumour sizes. Finally, we include terms representing the induced death of tumour cells by hypoxic engineered macrophages. We demonstrate that reductions in tumour size can be achieved, but predict that a combination of therapies would be required for complete eradication. We also highlight some counter-intuitive predictions-for example, absolute and relative measures of tumour burden lead to different conclusions about prognosis. In summary, this paper illustrates how mathematical models may be used to investigate promising macrophage-based therapies.

Novel immunocompetent murine tumor models for the assessment of replication-competent oncolytic adenovirus efficacy

Oncolytic replication-selective adenoviruses constitute a rapidly expanding experimental approach to the treatment of cancer. However, due to the lack of an immunocompetent and replication-competent efficacy model, the role of the host immune response and viral E3 immunoregulatory genes remained unknown. We screened nine murine carcinoma lines for adenovirus (Ad5) uptake, gene expression, replication, and cytopathic effects. In seven of these murine cell lines the infectability and cytopathic effects were similar to those seen with human carcinoma lines. Surprisingly, productive viral replication was demonstrated in several lines; replication varied from levels similar to those for some human carcinoma lines (e.g., CMT-64) to very low levels. Seven of these lines were grown as subcutaneous xenografts in immunocompetent mice and were subsequently injected directly with Ad5, saline, or a replication-deficient control adenovirus particle to assess intratumoral viral gene expression, replication, and antitumoral effects. E1A, coat protein expression, and cytopathic effects were documented in five xenografts; Ad5 replication was demonstrated in CMT-64 and JC xenografts. Ad5 demonstrated significant efficacy compared to saline and nonreplicating control Ad particles in both replication-permissive xenografts (CMT-64, JC) and poorly permissive tumors (CMT-93); efficacy against CMT-93 tumors was significantly greater in immunocompetent mice compared to athymic mice. These murine tumor xenograft models have potential for elucidating viral and host immune mechanisms involved in oncolytic adenovirus antitumoral effects.