Complex Dynamics of Virus Spread from Low Infection Multiplicities: Implications for the Spread of Oncolytic Viruses

Choice of spatial discretisation influences the progression of viral infection within multicellular tissues

There has been an increasing recognition of the utility of models of the spatial dynamics of viral spread within tissues. Multicellular models, where cells are represented as discrete regions of space coupled to a virus density surface, are a popular approach to capture these dynamics. Conventionally, such models are simulated by discretising the viral surface and depending on the rate of viral diffusion and other considerations, a finer or coarser discretisation may be used. The impact that this choice may have on the behaviour of the system has not been studied. Here we demonstrate that under realistic parameter regimes - where viral diffusion is small enough to support the formation of familiar ring-shaped infection plaques - the choice of spatial discretisation of the viral surface can qualitatively change key model outcomes including the time scale of infection. Importantly, we show that the choice between implementing viral spread as a cell-scale process, or as a high-resolution converged PDE can generate distinct model outcomes, which raises important conceptual questions about the strength of assumptions underpinning the spatial structure of the model. We investigate the mechanisms driving these discretisation artefacts, the impacts they may have on model predictions, and provide guidance on the design and implementation of spatial and especially multicellular models of viral dynamics. We obtain our results using the simplest TIV construct for the viral dynamics, and therefore anticipate that the important effects we describe will also influence model predictions in more complex models of virus-cell-immune system interactions. This analysis will aid in the construction of models for robust and biologically realistic modelling and inference.

Mathematical Modeling of Oncolytic Virus Therapy Reveals Role of the Immune Response

Oncolytic adenoviruses (OAds) present a promising path for cancer treatment due to their selectivity in infecting and lysing tumor cells and their ability to stimulate the immune response. In this study, we use an ordinary differential equation (ODE) model of tumor growth inhibited by oncolytic virus activity to parameterize previous research on the effect of genetically re-engineered OAds in A549 lung cancer tumors in murine models. We find that the data are best fit by a model that accounts for an immune response, and that the immune response provides a mechanism for elimination of the tumor. We also find that parameter estimates for the most effective OAds share characteristics, most notably a high infection rate and low viral clearance rate, that might be potential reasons for these viruses' efficacy in delaying tumor growth. Further studies observing E1A and P19 recombined viruses in different tumor environments may further illuminate the extent of the effects of these genetic modifications.

Modelling the spatial dynamics of oncolytic virotherapy in the presence of virus-resistant tumour cells

Oncolytic virotherapy is a promising form of cancer treatment that uses native or genetically engineered viruses to target, infect and kill cancer cells. Unfortunately, this form of therapy is not effective in a substantial proportion of cancer patients, partly due to the occurrence of infection-resistant tumour cells. To shed new light on the mechanisms underlying therapeutic failure and to discover strategies that improve therapeutic efficacy we designed a cell-based model of viral infection. The model allows us to investigate the dynamics of infection-sensitive and infection-resistant cells in tumour tissue in presence of the virus. To reflect the importance of the spatial configuration of the tumour on the efficacy of virotherapy, we compare three variants of the model: two 2D models of a monolayer of tumour cells and a 3D model. In all model variants, we systematically investigate how the therapeutic outcome is affected by the properties of the virus (e.g. the rate of viral spread), the tumour (e.g. production rate of resistant cells, cost of resistance), the healthy stromal cells (e.g. degree of resistance to the virus) and the timing of treatment. We find that various therapeutic outcomes are possible when resistant cancer cells arise at low frequency in the tumour. These outcomes depend in an intricate but predictable way on the death rate of infected cells, where faster death leads to rapid virus clearance and cancer persistence. Our simulations reveal three different causes of therapy failure: rapid clearance of the virus, rapid selection of resistant cancer cells, and a low rate of viral spread due to the presence of infection-resistant healthy cells. Our models suggest that improved therapeutic efficacy can be achieved by sensitizing healthy stromal cells to infection, although this remedy has to be weighed against the toxicity induced in the healthy tissue.

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.

A combination therapy of oncolytic viruses and chimeric antigen receptor T cells: a mathematical model proof-of-concept

Combining chimeric antigen receptor T (CAR-T) cells with oncolytic viruses (OVs) has recently emerged as a promising treatment approach in preclinical studies that aim to alleviate some of the barriers faced by CAR-T cell therapy. In this study, we address by means of mathematical modeling the main question of whether a single dose or multiple sequential doses of CAR-T cells during the OVs therapy can have a synergetic effect on tumor reduction. To that end, we propose an ordinary differential equations-based model with virus-induced synergism to investigate potential effects of different regimes that could result in efficacious combination therapy against tumor cell populations. Model simulations show that, while the treatment with a single dose of CAR-T cells is inadequate to eliminate all tumor cells, combining the same dose with a single dose of OVs can successfully eliminate the tumor in the absence of virus-induced synergism. However, in the presence of virus-induced synergism, the same combination therapy fails to eliminate the tumor. Furthermore, it is shown that if the intensity of virus-induced synergy and/or virus oncolytic potency is high, then the induced CAR-T cell response can inhibit virus oncolysis. Additionally, the simulations show a more robust synergistic effect on tumor cell reduction when OVs and CAR-T cells are administered simultaneously compared to the combination treatment where CAR-T cells are administered first or after OV injection. Our findings suggest that the combination therapy of CAR-T cells and OVs seems unlikely to be effective if the virus-induced synergistic effects are included when genetically engineering oncolytic viral vectors.

Viral Infection Spreading and Mutation in Cell Culture

A new model of viral infection spreading in cell cultures is proposed taking into account virus mutation. This model represents a reaction-diffusion system of equations with time delay for the concentrations of uninfected cells, infected cells and viral load. Infection progression is characterized by the virus replication number Rv, which determines the total viral load. Analytical formulas for the speed of propagation and for the viral load are obtained and confirmed by numerical simulations. It is shown that virus mutation leads to the emergence of a new virus variant. Conditions of the coexistence of the two variants or competitive exclusion of one of them are found, and different stages of infection progression are identified.

The Influence of Immune Response on Spreading of Viral Infection

In this work we develop a model of viral infection in host tissues in order to study the influence of the immune response on the infection spreading speed and on the viral load characterizing, respectively, severity of symptoms and infection transmission rate. Dynamics of the interaction between viral infection and the immune response is studied with nonlocal reaction-diffusion equations for the concentrations of virus, interferon, immune cells and antibodies. Analytical results for infection spreading speed and viral load are completed by numerical simulations. At the first stage, progression of viral infection is confronted by the innate immune response mostly determined by the local interferon production. The modeling results show in this case that infection spreading speed does not depend on interferon concentration, while the total viral load decreases with the increase of its concentration. Next, we consider the influence of globally circulating interferon and show that, in contrast to local interferon diffusion, infection spreading speed decreases with increasing of global interferon level, and the total viral load also decreases. At the next stage, adaptive immune response mediated by antibodies and cytotoxic T cells (CTL) further influences infection progression. In this case, the infection propagation speed and the total viral load are decreased by the immune response. The humoral adaptive response (antibodies) increases the global interferon concentration through the viral load, while the cellular adaptive response (CTL) decreases it.

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.

Natural Killer Cells Recruitment in Oncolytic Virotherapy: A Mathematical Model

In this paper, we investigate how natural killer (NK) cell recruitment to the tumor microenvironment (TME) affects oncolytic virotherapy. NK cells play a major role against viral infections. They are, however, known to induce early viral clearance of oncolytic viruses, which hinders the overall efficacy of oncolytic virotherapy. Here, we formulate and analyze a simple mathematical model of the dynamics of the tumor, OV and NK cells using currently available preclinical information. The aim of this study is to characterize conditions under which the synergistic balance between OV-induced NK responses and required viral cytopathicity may or may not result in a successful treatment. In this study, we found that NK cell recruitment to the TME must take place neither too early nor too late in the course of OV infection so that treatment will be successful. NK cell responses are most influential at either early (partly because of rapid response of NK cells to viral infections or antigens) or later (partly because of antitumoral ability of NK cells) stages of oncolytic virotherapy. The model also predicts that: (a) an NK cell response augments oncolytic virotherapy only if viral cytopathicity is weak; (b) the recruitment of NK cells modulates tumor growth; and (c) the depletion of activated NK cells within the TME enhances the probability of tumor escape in oncolytic virotherapy. Taken together, our model results demonstrate that OV infection is crucial, not just to cytoreduce tumor burden, but also to induce the stronger NK cell response necessary to achieve complete or at least partial tumor remission. Furthermore, our modeling framework supports combination therapies involving NK cells and OV which are currently used in oncolytic immunovirotherapy to treat several cancer types.

Beyond the pair approximation: Modeling colonization population dynamics

The process of range expansion (colonization) is one of the basic types of biological dynamics, whereby a species grows and spreads outwards, occupying new territories. Spatial modeling of this process is naturally implemented as a stochastic cellular automaton, with individuals occupying nodes on a rectangular grid, births and deaths occurring probabilistically, and individuals only reproducing onto unoccupied neighboring spots. In this paper we derive several approximations that allow prediction of the expected range expansion dynamics, based on the reproduction and death rates. We derive several approximations, where the cellular automaton is described by a system of ordinary differential equations that preserves correlations among neighboring spots (up to a distance). This methodology allows us to develop accurate approximations of the population size and the expected spatial shape, at a fraction of the computational time required to simulate the original stochastic system. In addition, we provide simple formulas for the steady-state population densities for von Neumann and Moore neighborhoods. Finally, we derive concise approximations for the speed of range expansion in terms of the reproduction and death rates, for both types of neighborhoods. The methodology is generalizable to more complex scenarios, such as different interaction ranges and multiple-species systems.

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 Oncolytic Virotherapy

Mathematical modeling in biology has a long history as it allows the analysis and simulation of complex dynamic biological systems at little cost. A mathematical model trained on experimental or clinical data can be used to generate and evaluate hypotheses, to ask "what if" questions, and to perform in silico experiments to guide future experimentation and validation. Such models may help identify and provide insights into the mechanisms that drive changes in dynamic systems. While a mathematical model may never replace actual experiments, it can synergize with experiments to save time and resources by identifying experimental conditions that are unlikely to yield favorable outcomes, and by using optimization principles to identify experiments that are most likely to be successful. Over the past decade, numerous models have also been developed for oncolytic virotherapy, ranging from merely theoretic frameworks to fully integrated studies that utilize experimental data to generate actionable hypotheses. Here we describe how to develop such models for specific oncolytic virotherapy experimental setups, and which questions can and cannot be answered using integrated mathematical oncology.

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.

Effects of spatiotemporal HSV-2 lesion dynamics and antiviral treatment on the risk of HIV-1 acquisition

Patients with Herpes Simplex Virus-2 (HSV-2) infection face a significantly higher risk of contracting HIV-1. This is thought to be due to herpetic lesions serving as entry points for HIV-1 and tissue-resident CD4+ T cell counts increasing during HSV-2 lesional events. We have created a stochastic and spatial mathematical model describing the dynamics of HSV-2 infection and immune response in the genital mucosa. Using our model, we first study the dynamics of a developing HSV-2 lesion. We then use our model to quantify the risk of infection with HIV-1 following sexual exposure in HSV-2 positive women. Untreated, we find that HSV-2 infected women are up to 8.6 times more likely to acquire HIV-1 than healthy patients. However, when including the effects of the HSV-2 antiviral drug, pritelivir, the risk of HIV-1 infection is predicted to decrease by up to 35%, depending on drug dosage. We estimate the relative importance of decreased tissue damage versus decreased CD4+ cell presence in determining the effectiveness of pritelivir in reducing HIV-1 infection. Our results suggest that clinical trials should be performed to evaluate the effectiveness of pritelivir or similar agents in preventing HIV-1 infection in HSV-2 positive women.

The risk of contracting HIV-1 is significantly higher in people who have genital HSV-2 infections. Here, we put forward a new mathematical model to describe HSV-2 infection and the process of HIV-1 infection in the genital mucosa surrounding HSV-2 lesions. We determine how the characteristics of HSV-2 infection affect the risk of HIV-1 infection, and determine whether reducing the severity of HSV-2 symptoms with antiviral drugs can be expected to decrease the risk of HIV-1 infection. We find that the risk of HIV-1 infection is dependent on three factors: the amount of HIV-1 the patient is exposed to, the severity of HSV-2 lesions, and the number of CD4+ T immune cells in the genital mucosa. Our model predicts that antiviral drugs targeting HSV-2 can cause a therapeutic decrease in lesion severity and CD4+ T cell count in the genital mucosa. This furthermore causes a significant decrease in the risk of HIV-1 infection but the dose of HSV-2 antiviral drug must be sufficiently high. Our results support further development and testing of new HSV-2 antiviral drugs to help decrease the world-wide burden of HIV-1.

The importance of imaging strategies for pre-clinical and clinical in vivo distribution of oncolytic viruses

Oncolytic viruses (OVs) are an emergent and unique therapy for cancer patients. Similar to chemo- and radiation therapy, OV can lyse (kill) cancer cell directly. In general, the advantages of OVs over other treatments are primarily: a higher safety profile (as shown by less adverse effects), ability to replicate, transgene(s) delivery, and stimulation of a host’s immune system against cancer. The latter has prompted successful use of OVs with other immunotherapeutic strategies in a synergistic manner. In spite of extended testing in pre-clinical and clinical setting, using biologically derived therapeutics like virus always raises potential concerns about safety (replication at non-intended locations) and bio-availability of the product. Recent advent in in vivo imaging techniques dramatically improves the convenience of use, quality of pictures, and amount of information acquired. Easy assessing of safety/localization of the biotherapeutics like OVs became a new potential weapon in the physician’s arsenal to improve treatment outcome. Given that OVs are typically replicating, in vivo imaging can also track virus replication and persistence as well as precisely mapping tumor tissues presence. This review discusses the importance of imaging in vivo in evaluating OV efficacy, as well as currently available tools and techniques.

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.