Macrophage-based anti-cancer therapy: modelling different modes of tumour targeting

Quantification of spatial and phenotypic heterogeneity in an agent-based model of tumour-macrophage interactions

We introduce a new spatial statistic, the weighted pair correlation function (wPCF). The wPCF extends the existing pair correlation function (PCF) and cross-PCF to describe spatial relationships between points marked with combinations of discrete and continuous labels. We validate its use through application to a new agent-based model (ABM) which simulates interactions between macrophages and tumour cells. These interactions are influenced by the spatial positions of the cells and by macrophage phenotype, a continuous variable that ranges from anti-tumour to pro-tumour. By varying model parameters that regulate macrophage phenotype, we show that the ABM exhibits behaviours which resemble the 'three Es of cancer immunoediting': Equilibrium, Escape, and Elimination. We use the wPCF to analyse synthetic images generated by the ABM. We show that the wPCF generates a 'human readable' statistical summary of where macrophages with different phenotypes are located relative to both blood vessels and tumour cells. We also define a distinct 'PCF signature' that characterises each of the three Es of immunoediting, by combining wPCF measurements with the cross-PCF describing interactions between vessels and tumour cells. By applying dimension reduction techniques to this signature, we identify its key features and train a support vector machine classifier to distinguish between simulation outputs based on their PCF signature. This proof-of-concept study shows how multiple spatial statistics can be combined to analyse the complex spatial features that the ABM generates, and to partition them into interpretable groups. The intricate spatial features produced by the ABM are similar to those generated by state-of-the-art multiplex imaging techniques which distinguish the spatial distribution and intensity of multiple biomarkers in biological tissue regions. Applying methods such as the wPCF to multiplex imaging data would exploit the continuous variation in biomarker intensities and generate more detailed characterisation of the spatial and phenotypic heterogeneity in tissue samples.

Two-phase model of compressive stress induced on a surrounding hyperelastic medium by an expanding tumour

In vitro experiments in which tumour cells are seeded in a gelatinous medium, or hydrogel, show how mechanical interactions between tumour cells and the tissue in which they are embedded, together with local levels of an externally-supplied, diffusible nutrient (e.g., oxygen), affect the tumour's growth dynamics. In this article, we present a mathematical model that describes these in vitro experiments. We use the model to understand how tumour growth generates mechanical deformations in the hydrogel and how these deformations in turn influence the tumour's growth. The hydrogel is viewed as a nonlinear hyperelastic material and the tumour is modelled as a two-phase mixture, comprising a viscous tumour cell phase and an isotropic, inviscid interstitial fluid phase. Using a combination of numerical and analytical techniques, we show how the tumour's growth dynamics change as the mechanical properties of the hydrogel vary. When the hydrogel is soft, nutrient availability dominates the dynamics: the tumour evolves to a large equilibrium configuration where the proliferation rate of nutrient-rich cells on the tumour boundary balances the death rate of nutrient-starved cells in the central, necrotic core. As the hydrogel stiffness increases, mechanical resistance to growth increases and the tumour's equilibrium size decreases. Indeed, for small tumours embedded in stiff hydrogels, the inhibitory force experienced by the tumour cells may be so large that the tumour is eliminated. Analysis of the model identifies parameter regimes in which the presence of the hydrogel drives tumour elimination.

The impact of tumor associated macrophages on tumor biology under the lens of mathematical modelling: A review

In this article, we review the role of mathematical modelling to elucidate the impact of tumor-associated macrophages (TAMs) in tumor progression and therapy design. We first outline the biology of TAMs, and its current application in tumor therapies, and their experimental methods that provide insights into tumor cell-macrophage interactions. We then focus on the mechanistic mathematical models describing the role of macrophages as drug carriers, the impact of macrophage polarized activation on tumor growth, and the role of tumor microenvironment (TME) parameters on the tumor-macrophage interactions. This review aims to identify the synergies between biological and mathematical approaches that allow us to translate knowledge on fundamental TAMs biology in addressing current clinical challenges.

Mathematical investigation into the role of macrophage heterogeneity on the temporal and spatio-temporal dynamics of non-small cell lung cancers

Non Small Cell Lung Cancer (NSCLC) is the most common type of lung cancer, and represents the leading cause of cancer-related deaths worldwide. Experimental studies have shown that these solid cancers are heavily infiltrated with macrophages: anti-tumour M1 macrophages, pro-tumour M2 macrophages, and macrophage subtypes sharing both M1 and M2 properties. In this study we aim to investigate qualitatively the role of macrophages with different functional phenotypes (especially those with mixed phenotypes) on cancer dynamics and the success of different immunotherapies for cancer. To this end, we start with two time-evolving mathematical models for cancer-immune interactions that consider: (i) the effect of the two extreme phenotypes, M1 and M2 cells; (ii) the effect of M1 and M2 cells, as well as a macrophage sub-population with a mixed phenotype (throughout this theoretical study we call these cells "M12 cells"). We compare the dynamics of the two models using computational approaches, paying particular attention to the effect of different anti-cancer immunotherapies that focus on macrophages. Since data available for NSCLC and macrophage interactions are incomplete, we perform a global sensitivity analysis to see the influence of input parameters on model outcomes. Finally, we consider extensions of the previous two models to include also the spatial movement of cells, and investigate the role of macrophages with extreme phenotypes and with mixed phenotypes, on the invasion of cancer cells into the surrounding extracellular matrix (ECM). We use numerical simulations to investigate the macrophages phenotypes at the tumour center versus the invasive margin. Again, we examine the impact of immunotherapies for cancer on the spatial dynamics of cancers and immune cells, and observe a shift in the phenotype of macrophages distributed at the tumour center and invasive margin.

Tracing New Landscapes in the Arena of Nanoparticle-Based Cancer Immunotherapy

Over the past two decades, unique and comprehensive cancer treatment has ushered new hope in the holistic management of the disease. Cancer immunotherapy, which harnesses the immune system of the patient to attack the cancer cells in a targeted manner, scores over others by being less debilitating compared to the existing treatment strategies. Significant advancements in the knowledge of immune surveillance in the last few decades have led to the development of several types of immune therapy like monoclonal antibodies, cancer vaccines, immune checkpoint inhibitors, T-cell transfer therapy or adoptive cell therapy (ACT) and immune system modulators. Intensive research has established cancer immunotherapy to be a safe and effective method for improving survival and the quality of a patient’s life. However, numerous issues with respect to site-specific delivery, resistance to immunotherapy, and escape of cancer cells from immune responses, need to be addressed for expanding and utilizing this therapy as a regular mode in the clinical treatment. Development in the field of nanotechnology has augmented the therapeutic efficiency of treatment modalities of immunotherapy. Nanocarriers could be used as vehicles because of their advantages such as increased surface areas, targeted delivery, controlled surface and release chemistry, enhanced permeation and retention effect, etc. They could enhance the function of immune cells by incorporating immunomodulatory agents that influence the tumor microenvironment, thus enabling antitumor immunity. Robust validation of the combined effect of nanotechnology and immunotherapy techniques in the clinics has paved the way for a better treatment option for cancer than the already existing procedures such as chemotherapy and radiotherapy. In this review, we discuss the current applications of nanoparticles in the development of ‘smart’ cancer immunotherapeutic agents like ACT, cancer vaccines, monoclonal antibodies, their site-specific delivery, and modulation of other endogenous immune cells. We also highlight the immense possibilities of using nanotechnology to accomplish leveraging the coordinated and adaptive immune system of a patient to tackle the complexity of treating unique disease conditions and provide future prospects in the field of cancer immunotherapy.

A mathematical investigation into the uptake kinetics of nanoparticles in vitro

Nanoparticles have the potential to increase the efficacy of anticancer drugs whilst reducing off-target side effects. However, there remain uncertainties regarding the cellular uptake kinetics of nanoparticles which could have implications for nanoparticle design and delivery. Polymersomes are nanoparticle candidates for cancer therapy which encapsulate chemotherapy drugs. Here we develop a mathematical model to simulate the uptake of polymersomes via endocytosis, a process by which polymersomes bind to the cell surface before becoming internalised by the cell where they then break down, releasing their contents which could include chemotherapy drugs. We focus on two in vitro configurations relevant to the testing and development of cancer therapies: a well-mixed culture model and a tumour spheroid setup. Our mathematical model of the well-mixed culture model comprises a set of coupled ordinary differential equations for the unbound and bound polymersomes and associated binding dynamics. Using a singular perturbation analysis we identify an optimal number of ligands on the polymersome surface which maximises internalised polymersomes and thus intracellular chemotherapy drug concentration. In our mathematical model of the spheroid, a multiphase system of partial differential equations is developed to describe the spatial and temporal distribution of bound and unbound polymersomes via advection and diffusion, alongside oxygen, tumour growth, cell proliferation and viability. Consistent with experimental observations, the model predicts the evolution of oxygen gradients leading to a necrotic core. We investigate the impact of two different internalisation functions on spheroid growth, a constant and a bond dependent function. It was found that the constant function yields faster uptake and therefore chemotherapy delivery. We also show how various parameters, such as spheroid permeability, lead to travelling wave or steady-state solutions.

Directionality of Macrophages Movement in Tumour Invasion: A Multiscale Moving-Boundary Approach

Invasion of the surrounding tissue is one of the recognised hallmarks of cancer (Hanahan and Weinberg in Cell 100: 57–70, 2000. 10.1016/S0092-8674(00)81683-9), which is accomplished through a complex heterotypic multiscale dynamics involving tissue-scale random and directed movement of the population of both cancer cells and other accompanying cells (including here, the family of tumour-associated macrophages) as well as the emerging cell-scale activity of both the matrix-degrading enzymes and the rearrangement of the cell-scale constituents of the extracellular matrix (ECM) fibres. The involved processes include not only the presence of cell proliferation and cell adhesion (to other cells and to the extracellular matrix), but also the secretion of matrix-degrading enzymes. This is as a result of cancer cells as well as macrophages, which are one of the most abundant types of immune cells in the tumour micro-environment. In large tumours, these tumour-associated macrophages (TAMs) have a tumour-promoting phenotype, contributing to tumour proliferation and spread. In this paper, we extend a previous multiscale moving-boundary mathematical model for cancer invasion, by considering also the multiscale effects of TAMs, with special focus on the influence that their directional movement exerts on the overall tumour progression. Numerical investigation of this new model shows the importance of the interactions between pro-tumour TAMs and the fibrous ECM, highlighting the impact of the fibres on the spatial structure of solid tumour.

Identifying and characterising the impact of excitability in a mathematical model of tumour-immune interactions

We study a five-compartment mathematical model originally proposed by Kuznetsov et al. (1994) to investigate the effect of nonlinear interactions between tumour and immune cells in the tumour microenvironment, whereby immune cells may induce tumour cell death, and tumour cells may inactivate immune cells. Exploiting a separation of timescales in the model, we use the method of matched asymptotics to derive a new two-dimensional, long-timescale, approximation of the full model, which differs from the quasi-steady-state approximation introduced by Kuznetsov et al. (1994), but is validated against numerical solutions of the full model. Through a phase-plane analysis, we show that our reduced model is excitable, a feature not traditionally associated with tumour-immune dynamics. Through a systematic parameter sensitivity analysis, we demonstrate that excitability generates complex bifurcating dynamics in the model. These are consistent with a variety of clinically observed phenomena, and suggest that excitability may underpin tumour-immune interactions. The model exhibits the three stages of immunoediting - elimination, equilibrium, and escape, via stable steady states with different tumour cell concentrations. Such heterogeneity in tumour cell numbers can stem from variability in initial conditions and/or model parameters that control the properties of the immune system and its response to the tumour. We identify different biophysical parameter targets that could be manipulated with immunotherapy in order to control tumour size, and we find that preferred strategies may differ between patients depending on the strength of their immune systems, as determined by patient-specific values of associated model parameters.

Mathematical modelling reveals cellular dynamics within tumour spheroids

Tumour spheroids are widely used as an in vitro assay for characterising the dynamics and response to treatment of different cancer cell lines. Their popularity is largely due to the reproducible manner in which spheroids grow: the diffusion of nutrients and oxygen from the surrounding culture medium, and their consumption by tumour cells, causes proliferation to be localised at the spheroid boundary. As the spheroid grows, cells at the spheroid centre may become hypoxic and die, forming a necrotic core. The pressure created by the localisation of tumour cell proliferation and death generates an cellular flow of tumour cells from the spheroid rim towards its core. Experiments by Dorie et al. showed that this flow causes inert microspheres to infiltrate into tumour spheroids via advection from the spheroid surface, by adding microbeads to the surface of tumour spheroids and observing the distribution over time. We use an off-lattice hybrid agent-based model to re-assess these experiments and establish the extent to which the spatio-temporal data generated by microspheres can be used to infer kinetic parameters associated with the tumour spheroids that they infiltrate. Variation in these parameters, such as the rate of tumour cell proliferation or sensitivity to hypoxia, can produce spheroids with similar bulk growth dynamics but differing internal compositions (the proportion of the tumour which is proliferating, hypoxic/quiescent and necrotic/nutrient-deficient). We use this model to show that the types of experiment conducted by Dorie et al. could be used to infer spheroid composition and parameters associated with tumour cell lines such as their sensitivity to hypoxia or average rate of proliferation, and note that these observations cannot be conducted within previous continuum models of microbead infiltration into tumour spheroids as they rely on resolving the trajectories of individual microbeads.

Tumour spheroids are an experimental assay used to characterise the dynamics and response to treatment of different cancer cell lines. Previous experiments have demonstrated that the localisation of tumour cell proliferation to the spheroid edge (due to the gradient formed by nutrient diffusing from the surrounding medium) causes cells to be pushed from the proliferative rim towards the nutrient-deficient necrotic core. This movement allows inert particles to infiltrate tumour spheroids. We use a hybrid agent-based model to reproduce this data. We show further how data from individual microbead trajectories can be used to infer the composition of simulated tumour spheroids, and to estimate model parameters pertaining to tumour cell proliferation rates and their responses to hypoxia. Since these measurements are possible using modern imaging techniques, this could motivate new experiments in which spheroid composition could be inferred by observing passive infiltration of inert particles.

Spatiotemporal dynamics of immunogenic tumors

A mathematical model, consisting of a system of two coupled reaction-diffusion partial differential equations describing the interaction between solid tumor and immune system (termed as effector cells), is proposed here. The main focus is on the analysis of the spatiotemporal dynamics of tumor cells and immune cells. The resulting system is analyzed and numerical simulations are presented. Different types of spatial patterns with respect to different initial conditions, and time are observed. Their analysis and mechanism of spatiotemporal pattern formation in immunogenic tumor are studied. Spatiotemporal perturbation around non-spatial steady state beyond the linear regime is obtained based on the analysis of higher-order perturbation terms.

Mesenchymal stem cells used as carrier cells of oncolytic adenovirus results in enhanced oncolytic virotherapy

Mesenchymal stem cells (MSCs) loaded with oncolytic viruses are presently being investigated as a new modality of advanced/metastatic tumors treatment and enhancement of virotherapy. MSCs can, however, either promote or suppress tumor growth. To address the critical question of how MSCs loaded with oncolytic viruses affect virotherapy outcomes and tumor growth patterns in a tumor microenvironment, we developed and analyzed an integrated mathematical-experimental model. We used the model to describe both the growth dynamics in our experiments of firefly luciferase-expressing Hep3B tumor xenografts and the effects of the immune response during the MSCs-based virotherapy. We further employed it to explore the conceptual clinical feasibility, particularly, in evaluating the relative significance of potential immune promotive/suppressive mechanisms induced by MSCs loaded with oncolytic viruses. We were able to delineate conditions which may significantly contribute to the success or failure of MSC-based virotherapy as well as generate new hypotheses. In fact, one of the most impactful outcomes shown by this investigation, not inferred from the experiments alone, was the initially counter-intuitive fact that using tumor-promoting MSCs as carriers is not only helpful but necessary in achieving tumor control. Considering the fact that it is still currently a controversial debate whether MSCs exert a pro- or anti-tumor action, mathematical models such as this one help to quantitatively predict the consequences of using MSCs for delivering virotherapeutic agents in vivo. Taken together, our results show that MSC-mediated systemic delivery of oncolytic viruses is a promising strategy for achieving synergistic anti-tumor efficacy with improved safety profiles.

Mathematical modeling of tumor-immune cell interactions

The anti-tumor activity of the immune system is increasingly recognized as critical for the mounting of a prolonged and effective response to cancer growth and invasion, and for preventing recurrence following resection or treatment. As the knowledge of tumor-immune cell interactions has advanced, experimental investigation has been complemented by mathematical modeling with the goal to quantify and predict these interactions. This succinct review offers an overview of recent tumor-immune continuum modeling approaches, highlighting spatial models. The focus is on work published in the past decade, incorporating one or more immune cell types and evaluating immune cell effects on tumor progression. Due to their relevance to cancer, the following immune cells and their combinations are described: macrophages, Cytotoxic T Lymphocytes, Natural Killer cells, dendritic cells, T regulatory cells, and CD4+ T helper cells. Although important insight has been gained from a mathematical modeling perspective, the development of models incorporating patient-specific data remains an important goal yet to be realized for potential clinical benefit.

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.

Nanomaterials for modulating innate immune cells in cancer immunotherapy

Cancer immunotherapy has been intensively investigated in both preclinical and clinical studies. Whereas chemotherapies use cytotoxic drugs to kill tumor cells, cancer immunotherapy is based on the ability of the immune system to fight cancer. Tumors are intimately associated with the immune system: they can suppress the immune response and/or control immune cells to support tumor growth. Immunotherapy has yielded promising results in clinical practice, but some patients show limited responses. This may reflect the complexities of the relationship between a tumor and the immune system. In an effort to improve the current immunotherapies, researchers have exploited nanomaterials in creating new strategies to cure tumors via modulation of the immune system in tumor tissues. Although extensive studies have examined the use of immune checkpoint-based immunotherapy, rather less work has focused on manipulating the innate immune cells. This review examines the recent approaches and challenges in the use of nanomaterials to modulate innate immune cells.

Near-Infrared Light-Excited Upconverting Persistent Nanophosphors in Vivo for Imaging-Guided Cell Therapy

Optical imaging for biological applications is in need of more sensitive tool. Persistent luminescent nanophosphors enable highly sensitive in vivo optical detection and almost completely avoid tissue autofluorescence. Nevertheless, the actual persistent luminescent nanophosphors necessitate ex vivo activation before systemic operation, which severely restricted the use of long-term imaging in vivo. Hence, we introduced a novel generation of optical nanophosphors, based on (Zn₂SiO₄:Mn):Y³⁺, Yb³⁺, Tm³⁺ upconverting persistent luminescent nanophosphors; these nanophosphors can be excited in vivo through living tissues by highly penetrating near-infrared light. We can trace labeled tumor therapeutic macrophages in vivo after endocytosing these nanophosphors in vitro and follow macrophages biodistribution by a simple whole animal optical detection. These nanophosphors will open novel potentials for cell therapy research and for a variety of applications in diagnosis in vivo.

Mathematical modeling of tumor-associated macrophage interactions with the cancer microenvironment

Background

Immuno-oncotherapy has emerged as a promising means to target cancer. In particular, therapeutic manipulation of tumor-associated macrophages holds promise due to their various and sometimes opposing roles in tumor progression. It is established that M1-type macrophages suppress tumor progression while M2-types support it. Recently, Tie2-expressing macrophages (TEM) have been identified as a distinct sub-population influencing tumor angiogenesis and vascular remodeling as well as monocyte differentiation.

Methods

This study develops a modeling framework to evaluate macrophage interactions with the tumor microenvironment, enabling assessment of how these interactions may affect tumor progression. M1, M2, and Tie2 expressing variants are integrated into a model of tumor growth representing a metastatic lesion in a highly vascularized organ, such as the liver. Behaviors simulated include M1 release of nitric oxide (NO), M2 release of growth-promoting factors, and TEM facilitation of angiogenesis via Angiopoietin-2 and promotion of monocyte differentiation into M2 via IL-10.

Results

The results show that M2 presence leads to larger tumor growth regardless of TEM effects, implying that immunotherapeutic strategies that lead to TEM ablation may fail to restrain growth when the M2 represents a sizeable population. As TEM pro-tumor effects are less pronounced and on a longer time scale than M1-driven tumor inhibition, a more nuanced approach to influence monocyte differentiation taking into account the tumor state (e.g., under chemotherapy) may be desirable.

Conclusions

The results highlight the dynamic interaction of macrophages within a growing tumor, and, further, establish the initial feasibility of a mathematical framework that could longer term help to optimize cancer immunotherapy.

Mathematical Models for Immunology: Current State of the Art and Future Research Directions

The advances in genetics and biochemistry that have taken place over the last 10 years led to significant advances in experimental and clinical immunology. In turn, this has led to the development of new mathematical models to investigate qualitatively and quantitatively various open questions in immunology. In this study we present a review of some research areas in mathematical immunology that evolved over the last 10 years. To this end, we take a step-by-step approach in discussing a range of models derived to study the dynamics of both the innate and immune responses at the molecular, cellular and tissue scales. To emphasise the use of mathematics in modelling in this area, we also review some of the mathematical tools used to investigate these models. Finally, we discuss some future trends in both experimental immunology and mathematical immunology for the upcoming years.

Mathematical Based Calculation of Drug Penetration Depth in Solid Tumors

Cancer is a class of diseases characterized by out-of-control cells' growth which affect cells and make them damaged. Many treatment options for cancer exist. Chemotherapy as an important treatment option is the use of drugs to treat cancer. The anticancer drug travels to the tumor and then diffuses in it through capillaries. The diffusion of drugs in the solid tumor is limited by penetration depth which is different in case of different drugs and cancers. The computation of this depth is important as it helps physicians to investigate about treatment of infected tissue. Although many efforts have been made on studying and measuring drug penetration depth, less works have been done on computing this length from a mathematical point of view. In this paper, first we propose phase lagging model for diffusion of drug in the tumor. Then, using this model on one side and considering the classic diffusion on the other side, we compute the drug penetration depth in the solid tumor. This computed value of drug penetration depth is corroborated by comparison with the values measured by experiments.

Mathematical model of macrophage-facilitated breast cancer cells invasion

Mortality from breast cancer stems from its tendency to invade into surrounding tissues and organs. Experiments have shown that this metastatic process is facilitated by macrophages in a short-ranged chemical signalling loop. Macrophages secrete epidermal growth factor, EGF, and respond to the colony stimulating factor 1, CSF-1. Tumor cells secrete CSF-1 and respond to EGF. In this way, the cells coordinate aggregation and cooperative migration. Here we investigate this process in a model for in vitro interactions using two distinct but related mathematical approaches. In the first, we analyze and simulate a set of partial differential equations to determine conditions for aggregation. In the second, we use a cell-based discrete 3D simulation to follow the fates and motion of individual cells during aggregation. Linear stability analysis of the PDE model reveals that decreasing the chemical secretion, chemotaxis coefficients or density of cells or increasing the chemical degradation in the model could eliminate the spontaneous aggregation of cells. Simulations with the discrete model show that the ratio between tumor cells and macrophages in aggregates increases when the EGF secretion parameter is increased. The results also show how CSF-1/CSF-1R autocrine signalling in tumor cells affects the ratio between the two cell types. Comparing the continuum results with simulations of a discrete cell-based model, we find good qualitative agreement.

The critical role of the tumor microenvironment in shaping natural killer cell-mediated anti-tumor immunity

Considerable evidence has been gathered over the last 10 years showing that the tumor microenvironment (TME) is not simply a passive recipient of immune cells, but an active participant in the establishment of immunosuppressive conditions. It is now well documented that hypoxia, within the TME, affects the functions of immune effectors including natural killer (NK) cells by multiple overlapping mechanisms. Indeed, each cell in the TME, irrespective of its transformation status, has the capacity to adapt to the hostile TME and produce immune modulatory signals or mediators affecting the function of immune cells either directly or through the stimulation of other cells present in the tumor site. This observation has led to intense research efforts focused mainly on tumor-derived factors. Notably, it has become increasingly clear that tumor cells secrete a number of environmental factors such as cytokines, growth factors, exosomes, and microRNAs impacting the immune cell response. Moreover, tumor cells in hostile microenvironments may activate their own intrinsic resistance mechanisms, such as autophagy, to escape the effective immune response. Such adaptive mechanisms may also include the ability of tumor cells to modify their metabolism and release several metabolites to impair the function of immune cells. In this review, we summarize the different mechanisms involved in the TME that affect the anti-tumor immune function of NK cells.

Current advances in mathematical modeling of anti-cancer drug penetration into tumor tissues

Delivery of anti-cancer drugs to tumor tissues, including their interstitial transport and cellular uptake, is a complex process involving various biochemical, mechanical, and biophysical factors. Mathematical modeling provides a means through which to understand this complexity better, as well as to examine interactions between contributing components in a systematic way via computational simulations and quantitative analyses. In this review, we present the current state of mathematical modeling approaches that address phenomena related to drug delivery. We describe how various types of models were used to predict spatio-temporal distributions of drugs within the tumor tissue, to simulate different ways to overcome barriers to drug transport, or to optimize treatment schedules. Finally, we discuss how integration of mathematical modeling with experimental or clinical data can provide better tools to understand the drug delivery process, in particular to examine the specific tissue- or compound-related factors that limit drug penetration through tumors. Such tools will be important in designing new chemotherapy targets and optimal treatment strategies, as well as in developing non-invasive diagnosis to monitor treatment response and detect tumor recurrence.

Opportunities and challenges for use of tumor spheroids as models to test drug delivery and efficacy

Multicellular spheroids are three dimensional in vitro microscale tissue analogs. The current article examines the suitability of spheroids as an in vitro platform for testing drug delivery systems. Spheroids model critical physiologic parameters present in vivo, including complex multicellular architecture, barriers to mass transport, and extracellular matrix deposition. Relative to two-dimensional cultures, spheroids also provide better target cells for drug testing and are appropriate in vitro models for studies of drug penetration. Key challenges associated with creation of uniformly sized spheroids, spheroids with small number of cells and co-culture spheroids are emphasized in the article. Moreover, the assay techniques required for the characterization of drug delivery and efficacy in spheroids and the challenges associated with such studies are discussed. Examples for the use of spheroids in drug delivery and testing are also emphasized. By addressing these challenges with possible solutions, multicellular spheroids are becoming an increasingly useful in vitro tool for drug screening and delivery to pathological tissues and organs.

Mathematical modeling predicts synergistic antitumor effects of combining a macrophage-based, hypoxia-targeted gene therapy with chemotherapy

Tumor hypoxia is associated with low rates of cell proliferation and poor drug delivery, limiting the efficacy of many conventional therapies such as chemotherapy. Because many macrophages accumulate in hypoxic regions of tumors, one way to target tumor cells in these regions could be to use genetically engineered macrophages that express therapeutic genes when exposed to hypoxia. Systemic delivery of such therapeutic macrophages may also be enhanced by preloading them with nanomagnets and applying a magnetic field to the tumor site. Here, we use a new mathematical model to compare the effects of conventional cyclophosphamide therapy with those induced when macrophages are used to deliver hypoxia-inducible cytochrome P450 to locally activate cyclophosphamide. Our mathematical model describes the spatiotemporal dynamics of vascular tumor growth and treats cells as distinct entities. Model simulations predict that combining conventional and macrophage-based therapies would be synergistic, producing greater antitumor effects than the additive effects of each form of therapy. We find that timing is crucial in this combined approach with efficacy being greatest when the macrophage-based, hypoxia-targeted therapy is administered shortly before or concurrently with chemotherapy. Last, we show that therapy with genetically engineered macrophages is markedly enhanced by using the magnetic approach described above, and that this enhancement depends mainly on the strength of the applied field, rather than its direction. This insight may be important in the treatment of nonsuperficial tumors, where generating a specific orientation of a magnetic field may prove difficult. In conclusion, we demonstrate that mathematical modeling can be used to design and maximize the efficacy of combined therapeutic approaches in cancer.

Current trends in mathematical modeling of tumor-microenvironment interactions: a survey of tools and applications

In its simplest description, a tumor is comprised of an expanding population of transformed cells supported by a surrounding microenvironment termed the tumor stroma. The tumor microenvironment has a very complex composition, including multiple types of stromal cells, a dense network of various extracellular matrix (ECM) fibers interpenetrated by the interstitial fluid and gradients of several chemical species that either are dissolved in the fluid or are bound to the ECM structure. In order to study experimentally such complex interactions between multiple players, cancer is dissected and considered at different scales of complexity, such as protein interactions, biochemical pathways, cellular functions or whole organism studies. However, the integration of information acquired from these studies into a common description is as difficult as the disease itself. Computational models of cancer can provide cancer researchers with invaluable tools that are capable of integrating the complexity into organizing principles as well as suggesting testable hypotheses. We will focus in this Minireview on mathematical models in which the whole cell is a main modeling unit. We will present a current stage of such cell-focused mathematical modeling incorporating different stromal components and their interactions with growing tumors, and discuss what modeling approaches can be undertaken to complement the in vivo and in vitro experimentation.

Oscillatory dynamics in a model of vascular tumour growth--implications for chemotherapy

BACKGROUND

Investigations of solid tumours suggest that vessel occlusion may occur when increased pressure from the tumour mass is exerted on the vessel walls. Since immature vessels are frequently found in tumours and may be particularly sensitive, such occlusion may impair tumour blood flow and have a negative impact on therapeutic outcome. In order to study the effects that occlusion may have on tumour growth patterns and therapeutic response, in this paper we develop and investigate a continuum model of vascular tumour growth.

RESULTS

By analysing a spatially uniform submodel, we identify regions of parameter space in which the combination of tumour cell proliferation and vessel occlusion give rise to sustained temporal oscillations in the tumour cell population and in the vessel density. Alternatively, if the vessels are assumed to be less prone to collapse, stable steady state solutions are observed. When spatial effects are considered, the pattern of tumour invasion depends on the dynamics of the spatially uniform submodel. If the submodel predicts a stable steady state, then steady travelling waves are observed in the full model, and the system evolves to the same stable steady state behind the invading front. When the submodel yields oscillatory behaviour, the full model produces periodic travelling waves. The stability of the waves (which can be predicted by approximating the system as one of lambda-omega type) dictates whether the waves develop into regular or irregular spatio-temporal oscillations. Simulations of chemotherapy reveal that treatment outcome depends crucially on the underlying tumour growth dynamics. In particular, if the dynamics are oscillatory, then therapeutic efficacy is difficult to assess since the fluctuations in the size of the tumour cell population are enhanced, compared to untreated controls.

CONCLUSIONS

We have developed a mathematical model of vascular tumour growth formulated as a system of partial differential equations (PDEs). Employing a combination of numerical and analytical techniques, we demonstrate how the spatio-temporal dynamics of the untreated tumour may influence its response to chemotherapy.

REVIEWERS

This manuscript was reviewed by Professor Zvia Agur and Professor Marek Kimmel.

Interactions between the immune system and cancer: a brief review of non-spatial mathematical models

We briefly review spatially homogeneous mechanistic mathematical models describing the interactions between a malignant tumor and the immune system. We begin with the simplest (single equation) models for tumor growth and proceed to consider greater immunological detail (and correspondingly more equations) in steps. This approach allows us to clarify the necessity for expanding the complexity of models in order to capture the biological mechanisms we wish to understand. We conclude by discussing some unsolved problems in the mathematical modeling of cancer-immune system interactions.

Tie2-expressing monocytes (TEMs): novel targets and vehicles of anticancer therapy?

There is a growing interest in understanding the complex interactions between bone marrow-derived myeloid-lineage cells and angiogenesis in tumors. Such interest has been revived recently by the observation that tumor-infiltrating myeloid cells convey proangiogenic programs that can counteract the activity of antiangiogenic drugs in mouse tumor models. Among myeloid cells, Tie2-expressing monocytes (TEMs) appear to have nonredundant function in promoting tumor angiogenesis and growth in mouse models. The identification and functional characterization of TEMs in mice and humans may provide novel molecular targets for anticancer therapy. Moreover, TEMs may be exploited to deliver antitumor drugs specifically to the tumor microenvironment.