Mathematical Modelling of the Interaction Between Cancer Cells and an Oncolytic Virus: Insights into the Effects of Treatment Protocols

Oscillations in a Spatial Oncolytic Virus Model

Virotherapy treatment is a new and promising target therapy that selectively attacks cancer cells without harming normal cells. Mathematical models of oncolytic viruses have shown predator-prey like oscillatory patterns as result of an underlying Hopf bifurcation. In a spatial context, these oscillations can lead to different spatio-temporal phenomena such as hollow-ring patterns, target patterns, and dispersed patterns. In this paper we continue the systematic analysis of these spatial oscillations and discuss their relevance in the clinical context. We consider a bifurcation analysis of a spatially explicit reaction-diffusion model to find the above mentioned spatio-temporal virus infection patterns. The desired pattern for tumor eradication is the hollow ring pattern and we find exact conditions for its occurrence. Moreover, we derive the minimal speed of travelling invasion waves for the cancer and for the oncolytic virus. Our numerical simulations in 2-D reveal complex spatial interactions of the virus infection and a new phenomenon of a periodic peak splitting. An effect that we cannot explain with our current methods.

Optimal strategies of oncolytic virus-bortezomib therapy via the apoptotic, necroptotic, and oncolysis signaling network

Bortezomib and oncolytic virotherapy are two emerging targeted cancer therapies. Bortezomib, a proteasome inhibitor, disrupts protein degradation in cells, leading to the accumulation of unfolded proteins that induce apoptosis. On the other hand, virotherapy uses genetically modified oncolytic viruses (OVs) to infect cancer cells, trigger cell lysis, and activate anti-tumor response. Despite progress in cancer treatment, identifying administration protocols for therapeutic agents remains a significant concern, aiming to strike a balance between efficacy, minimizing toxicity, and administrative costs. In this work, optimal control theory was employed to design a cost-effective and efficient co-administration protocols for bortezomib and OVs that could significantly diminish the population of cancer cells via the cell death program with the NF$ \kappa $B-BAX-RIP1 signaling network. Both linear and quadratic control strategies were explored to obtain practical treatment approaches by adapting necroptosis protocols to efficient cell death programs. Our findings demonstrated that a combination therapy commencing with the administration of OVs followed by bortezomib infusions yields an effective tumor-killing outcome. These results could provide valuable guidance for the development of clinical administration protocols in cancer treatment.

Mathematical modeling predicts pathways to successful implementation of combination TRAIL-producing oncolytic virus and PAC-1 to treat granulosa cell tumors of the ovary

The development of new cancer therapies requires multiple rounds of validation from in vitro and in vivo experiments before they can be considered for clinical trials. Mathematical models assist in this preclinical phase by combining experimental data with human parameters to provide guidance about potential therapeutic regimens to bring forward into trials. However, granulosa cell tumors of the ovary lack a relevant mouse model, complexifying preclinical drug development for this rare tumor. To bridge this gap, we established a mathematical model as a framework to explore the potential of using a tumor necrosis factor-related apoptosis-inducing ligand (TRAIL)-producing oncolytic vaccinia virus in combination with the chemotherapeutic agent first procaspase activating compound (PAC-1). We have previously shown that TRAIL and PAC-1 act synergistically on granulosa tumor cells. In line with our previous results, our current model predicts that, although it is unable to stop the tumor from growing in its current form, combination oral PAC-1 with oncolytic virus (OV) provides the best result compared to monotherapies. Encouragingly, our results suggest that increases to the OV infection rate can lead to the success of this combination therapy within a year. The model developed here can continue to be improved as more data become available, allowing for regimen-tailoring via virtual clinical trials, ultimately shepherding effective regimens into trials.

On the treatment of melanoma: A mathematical model of oncolytic virotherapy

We develop and analyze a mathematical model of oncolytic virotherapy in the treatment of melanoma. We begin with a special, local case of the model, in which we consider the dynamics of the tumour cells in the presence of an oncolytic virus at the primary tumour site. We then consider the more general regional model, in which we incorporate a linear network of lymph nodes through which the tumour cells and the oncolytic virus may spread. The modelling also considers the impact of hypoxia on the disease dynamics. The modelling takes into account both the effects of hypoxia on tumour growth and spreading, as well as the impact of hypoxia on oncolytic virotherapy as a treatment modality. We find that oxygen-rich environments are favourable for the use of adenoviruses as oncolytic agents, potentially suggesting the use of complementary external oxygenation as a key aspect of treatment. Furthermore, the delicate balance between a virus' infection capabilities and its oncolytic capabilities should be considered when engineering an oncolytic virus. If the virus is too potent at killing tumour cells while not being sufficiently effective at infecting them, the infected tumour cells are destroyed faster than they are able to infect additional tumour cells, leading less favourable clinical results. Numerical simulations are performed in order to support the analytic results and to further investigate the impact of various parameters on the outcomes of treatment. Our modelling provides further evidence indicating the importance of three key factors in treatment outcomes: tumour microenvironment oxygen concentration, viral infection rates, and viral oncolysis rates. The numerical results also provide some estimates on these key model parameters which may be useful in the engineering of oncolytic adenoviruses.

Modeling of oncolytic viruses in a heterogeneous cell population to predict spread into non-cancerous cells

New cancer treatment modalities that limit patient discomfort need to be developed. One possible new therapy is the use of oncolytic (cancer-killing) viruses. It is only recently that our ability to manipulate viral genomes has allowed us to consider deliberately infecting cancer patients with viruses. One key consideration is to ensure that the virus exclusively targets cancer cells and does not harm nearby non-cancerous cells. Here, we use a mathematical model of viral infection to determine the characteristics a virus would need to have in order to eradicate a tumor, but leave non-cancerous cells untouched. We conclude that the virus must differ in its ability to infect the two different cell types, with the infection rate of non-cancerous cells needing to be less than one hundredth of the infection rate of cancer cells. Differences in viral production rate or infectious cell death rate alone are not sufficient to protect non-cancerous 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.

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.

Spatial cumulant models enable spatially informed treatment strategies and analysis of local interactions in cancer systems

Theoretical and applied cancer studies that use individual-based models (IBMs) have been limited by the lack of a mathematical formulation that enables rigorous analysis of these models. However, spatial cumulant models (SCMs), which have arisen from theoretical ecology, describe population dynamics generated by a specific family of IBMs, namely spatio-temporal point processes (STPPs). SCMs are spatially resolved population models formulated by a system of differential equations that approximate the dynamics of two STPP-generated summary statistics: first-order spatial cumulants (densities), and second-order spatial cumulants (spatial covariances). We exemplify how SCMs can be used in mathematical oncology by modelling theoretical cancer cell populations comprising interacting growth factor-producing and non-producing cells. To formulate model equations, we use computational tools that enable the generation of STPPs, SCMs and mean-field population models (MFPMs) from user-defined model descriptions (Cornell et al. Nat Commun 10:4716, 2019). To calculate and compare STPP, SCM and MFPM-generated summary statistics, we develop an application-agnostic computational pipeline. Our results demonstrate that SCMs can capture STPP-generated population density dynamics, even when MFPMs fail to do so. From both MFPM and SCM equations, we derive treatment-induced death rates required to achieve non-growing cell populations. When testing these treatment strategies in STPP-generated cell populations, our results demonstrate that SCM-informed strategies outperform MFPM-informed strategies in terms of inhibiting population growths. We thus demonstrate that SCMs provide a new framework in which to study cell-cell interactions, and can be used to describe and perturb STPP-generated cell population dynamics. We, therefore, argue that SCMs can be used to increase IBMs' applicability in cancer research.

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.

Dynamic analysis of an age structure model for oncolytic virus therapy

Cancer is recognized as one of the serious diseases threatening human health. Oncolytic therapy is a safe and effective new cancer treatment method. Considering the limited ability of uninfected tumor cells to infect and the age of infected tumor cells have a significant effect on oncolytic therapy, an age-structured model of oncolytic therapy involving Holling-Ⅱ functional response is proposed to investigate the theoretical significance of oncolytic therapy. First, the existence and uniqueness of the solution is obtained. Furthermore, the stability of the system is confirmed. Then, the local stability and global stability of infection-free homeostasis are studied. The uniform persistence and local stability of the infected state are studied. The global stability of the infected state is proved by constructing the Lyapunov function. Finally, the theoretical results are verified by numerical simulation. The results show that when the tumor cells are at the appropriate age, injection of the right amount of oncolytic virus can achieve the purpose of tumor treatment.

Agent-based computational modeling of glioblastoma predicts that stromal density is central to oncolytic virus efficacy

Oncolytic viruses (OVs) are emerging cancer immunotherapy. Despite notable successes in the treatment of some tumors, OV therapy for central nervous system cancers has failed to show efficacy. We used an ex vivo tumor model developed from human glioblastoma tissue to evaluate the infiltration of herpes simplex OV rQNestin (oHSV-1) into glioblastoma tumors. We next leveraged our data to develop a computational, model of glioblastoma dynamics that accounts for cellular interactions within the tumor. Using our computational model, we found that low stromal density was highly predictive of oHSV-1 therapeutic success, suggesting that the efficacy of oHSV-1 in glioblastoma may be determined by stromal-to-tumor cell regional density. We validated these findings in heterogenous patient samples from brain metastatic adenocarcinoma. Our integrated modeling strategy can be applied to suggest mechanisms of therapeutic responses for central nervous system cancers and to facilitate the successful translation of OVs into the clinic.

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.

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.

Iterative community-driven development of a SARS-CoV-2 tissue simulator

The 2019 novel coronavirus, SARS-CoV-2, is a pathogen of critical significance to international public health. Knowledge of the interplay between molecular-scale virus-receptor interactions, single-cell viral replication, intracellular-scale viral transport, and emergent tissue-scale viral propagation is limited. Moreover, little is known about immune system-virus-tissue interactions and how these can result in low-level (asymptomatic) infections in some cases and acute respiratory distress syndrome (ARDS) in others, particularly with respect to presentation in different age groups or pre-existing inflammatory risk factors. Given the nonlinear interactions within and among each of these processes, multiscale simulation models can shed light on the emergent dynamics that lead to divergent outcomes, identify actionable "choke points" for pharmacologic interventions, screen potential therapies, and identify potential biomarkers that differentiate patient outcomes. Given the complexity of the problem and the acute need for an actionable model to guide therapy discovery and optimization, we introduce and iteratively refine a prototype of a multiscale model of SARS-CoV-2 dynamics in lung tissue. The first prototype model was built and shared internationally as open source code and an online interactive model in under 12 hours, and community domain expertise is driving regular refinements. In a sustained community effort, this consortium is integrating data and expertise across virology, immunology, mathematical biology, quantitative systems physiology, cloud and high performance computing, and other domains to accelerate our response to this critical threat to international health. More broadly, this effort is creating a reusable, modular framework for studying viral replication and immune response in tissues, which can also potentially be adapted to related problems in immunology and immunotherapy.

Optimal solution of the fractional order breast cancer competition model

In this article, a fractional order breast cancer competition model (F-BCCM) under the Caputo fractional derivative is analyzed. A new set of basis functions, namely the generalized shifted Legendre polynomials, is proposed to deal with the solutions of F-BCCM. The F-BCCM describes the dynamics involving a variety of cancer factors, such as the stem, tumor and healthy cells, as well as the effects of excess estrogen and the body's natural immune response on the cell populations. After combining the operational matrices with the Lagrange multipliers technique we obtain an optimization method for solving the F-BCCM whose convergence is investigated. Several examples show that a few number of basis functions lead to the satisfactory results. In fact, numerical experiments not only confirm the accuracy but also the practicability and computational efficiency of the devised technique.

Oncolytic virus therapy benefits from control theory

Oncolytic virus therapy aims to eradicate tumours using viruses which only infect and destroy targeted tumour cells. It is urgent to improve understanding and outcomes of this promising cancer treatment because oncolytic virus therapy could provide sensible solutions for many patients with cancer. Recently, mathematical modelling of oncolytic virus therapy was used to study different treatment protocols for treating breast cancer cells with genetically engineered adenoviruses. Indeed, it is currently challenging to elucidate the number, the schedule, and the dosage of viral injections to achieve tumour regression at a desired level and within a desired time frame. Here, we apply control theory to this model to advance the analysis of oncolytic virus therapy. The control analysis of the model suggests that at least three viral injections are required to control and reduce the tumour from any initial size to a therapeutic target. In addition, we present an impulsive control strategy with an integral action and a state feedback control which achieves tumour regression for different schedule of injections. When oncolytic virus therapy is evaluated in silico using this feedback control of the tumour, the controller automatically tunes the dose of viral injections to improve tumour regression and to provide some robustness to uncertainty in biological rates. Feedback control shows the potential to deliver efficient and personalized dose of viral injections to achieve tumour regression better than the ones obtained by former protocols. The control strategy has been evaluated in silico with parameters that represent five nude mice from a previous experimental work. Together, our findings suggest theoretical and practical benefits by applying control theory to oncolytic virus therapy.

Population Dynamics with Threshold Effects Give Rise to a Diverse Family of Allee Effects

The Allee effect describes populations that deviate from logistic growth models and arises in applications including ecology and cell biology. A common justification for incorporating Allee effects into population models is that the population in question has altered growth mechanisms at some critical density, often referred to as a threshold effect. Despite the ubiquitous nature of threshold effects arising in various biological applications, the explicit link between local threshold effects and global Allee effects has not been considered. In this work, we examine a continuum population model that incorporates threshold effects in the local growth mechanisms. We show that this model gives rise to a diverse family of Allee effects, and we provide a comprehensive analysis of which choices of local growth mechanisms give rise to specific Allee effects. Calibrating this model to a recent set of experimental data describing the growth of a population of cancer cells provides an interpretation of the threshold population density and growth mechanisms associated with the population.

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.

Recent advances in oncolytic virus-based cancer therapy

Administration of oncolytic viruses (OVs) is an emerging anticancer strategy that exploits the lytic nature of viral replication to enhance the killing of malignant cells. OVs can be used as tools to directly induce cancer cell death and to trigger local and/or systemic immune responses to metastatic cancer in vivo. The effectiveness of OV therapy was initially highlighted by the clinical use of the genetically modified herpes virus, talimogene laherparepvec, for melanoma therapy. A number of OVs are now being evaluated as potential treatments for cancer in clinical trials. In spite of being engineered to specifically target tumor cells, the safety and off-target effects of OV therapy are a concern. The potential safety concerns of OVs are highlighted by current clinical trial criteria, which exclude individuals harbouring other viral infections and people who are immunocompromised. Despite the potential for adverse effects, clinical trials to date revealed relatively minimal adverse immune-related effects, such as fever. With advances in our understanding of virus replication cycles, several novel OVs have emerged. Reverse genetic systems have facilitated the insertion of anticancer genes into a range of OVs to further enhance their tumor-killing capacity. In this review, we highlight the recent advances in OV therapy for a range of human cancers in in vitro and in in vivo animal studies. We further discuss the future of OVs as a therapeutic strategy for a range of life-threatening cancers.

Developing a Minimally Structured Mathematical Model of Cancer Treatment with Oncolytic Viruses and Dendritic Cell Injections

Mathematical models of biological systems must strike a balance between being sufficiently complex to capture important biological features, while being simple enough that they remain tractable through analysis or simulation. In this work, we rigorously explore how to balance these competing interests when modeling murine melanoma treatment with oncolytic viruses and dendritic cell injections. Previously, we developed a system of six ordinary differential equations containing fourteen parameters that well describes experimental data on the efficacy of these treatments. Here, we explore whether this previously developed model is the minimal model needed to accurately describe the data. Using a variety of techniques, including sensitivity analyses and a parameter sloppiness analysis, we find that our model can be reduced by one variable and three parameters and still give excellent fits to the data. We also argue that our model is not too simple to capture the dynamics of the data, and that the original and minimal models make similar predictions about the efficacy and robustness of protocols not considered in experiments. Reducing the model to its minimal form allows us to increase the tractability of the system in the face of parametric uncertainty.

Modelling heterogeneity in viral-tumour dynamics: The effects of gene-attenuation on viral characteristics

The use of viruses as a cancer treatment is becoming increasingly more robust; however, there is still a long way to go before a completely successful treatment is formulated. One major challenge in the field is to select which virus, out of a burgeoning number of oncolytic viruses and engineered derivatives, can maximise both treatment spread and anticancer cytotoxicity. To assist in solving this problem, an in-depth understanding of the virus-tumour interaction is crucial. In this article, we present a novel integro-differential system with distributed delays embodying the dynamics of an oncolytic adenovirus with a fixed population of tumour cells in vitro, allowing for heterogeneity to exist in the virus and cell populations. The parameters of the model are optimised in a hierarchical manner, the purpose of which is not to obtain a perfect representation of the data. Instead, we place our parameter values in the correct region of the parameter space. Due to the sparse nature of the data it is not possible to obtain the parameter values with any certainty, but rather we demonstrate the suitability of the model. Using our model we quantify how modifications to the viral genome alter the viral characteristics, specifically how the attenuation of the E1B 19 and E1B 55 gene affect the system performance, and identify the dominant processes altered by the mutations. From our analysis, we conclude that the deletion of the E1B 55 gene significantly reduces the replication rate of the virus in comparison to the deletion of the E1B 19 gene. We also found that the deletion of both the E1B 19 and E1B 55 genes resulted in a long delay in the average replication start time of the virus. This leads us to propose the use of E1B 19 gene-attenuated adenovirus for cancer therapy, as opposed to E1B 55 gene-attenuated adenoviruses.