A mathematical model for pattern formation of glioma cells outside the tumor spheroid core

A simple agent-based hybrid model to simulate the biophysics of glioblastoma multiforme cells and the concomitant evolution of the oxygen field

BACKGROUND AND OBJECTIVES

Glioblastoma multiforme (GBM) is one of the most aggressive cancers of the central nervous system. It is characterized by a high mitotic activity and an infiltrative ability of the glioma cells, neovascularization and necrosis. GBM evolution entails the continuous interplay between heterogeneous cell populations, chemotaxis, and physical cues through different scales. In this work, an agent-based hybrid model is proposed to simulate the coupling of the multiscale biological events involved in the GBM invasion, specifically the individual and collective migration of GBM cells and the concurrent evolution of the oxygen field and phenotypic plasticity. An asset of the formulation is that it is conceptually and computationally simple but allows to reproduce the complexity and the progression of the GBM micro-environment at cell and tissue scales simultaneously.

METHODS

The migration is reproduced as the result of the interaction between every single cell and its micro-environment. The behavior of each individual cell is formulated through genotypic variables whereas the cell micro-environment is modeled in terms of the oxygen concentration and the cell density surrounding each cell. The collective behavior is formulated at a cellular scale through a flocking model. The phenotypic plasticity of the cells is induced by the micro-environment conditions, considering five phenotypes.

RESULTS

The model has been contrasted by benchmark problems and experimental tests showing the ability to reproduce different scenarios of glioma cell migration. In all cases, the individual and collective cell migration and the coupled evolution of both the oxygen field and phenotypic plasticity have been properly simulated. This simple formulation allows to mimic the formation of relevant hallmarks of glioblastoma multiforme, such as the necrotic cores, and to reproduce experimental evidences related to the mitotic activity in pseudopalisades.

CONCLUSIONS

In the collective migration, the survival of the clusters prevails at the expense of cell mitosis, regardless of the size of the groups, which delays the formation of necrotic foci and reduces the rate of oxygen consumption.

Data driven modeling of pseudopalisade pattern formation

Pseudopalisading is an interesting phenomenon where cancer cells arrange themselves to form a dense garland-like pattern. Unlike the palisade structure, a similar type of pattern first observed in schwannomas by pathologist J.J. Verocay (Wippold et al. in AJNR Am J Neuroradiol 27(10):2037-2041, 2006), pseudopalisades are less organized and associated with a necrotic region at their core. These structures are mainly found in glioblastoma (GBM), a grade IV brain tumor, and provide a way to assess the aggressiveness of the tumor. Identification of the exact bio-mechanism responsible for the formation of pseudopalisades is a difficult task, mainly because pseudopalisades seem to be a consequence of complex nonlinear dynamics within the tumor. In this paper we propose a data-driven methodology to gain insight into the formation of different types of pseudopalisade structures. To this end, we start from a state of the art macroscopic model for the dynamics of GBM, that is coupled with the dynamics of extracellular pH, and formulate a terminal value optimal control problem. Thus, given a specific, observed pseudopalisade pattern, we determine the evolution of parameters (bio-mechanisms) that are responsible for its emergence. Random histological images exhibiting pseudopalisade-like structures are chosen to serve as target pattern. Having identified the optimal model parameters that generate the specified target pattern, we then formulate two different types of pattern counteracting ansatzes in order to determine possible ways to impair or obstruct the process of pseudopalisade formation. This provides the basis for designing active or live control of malignant GBM. Furthermore, we also provide a simple, yet insightful, mechanism to synthesize new pseudopalisade patterns by linearly combining the optimal model parameters responsible for generating different known target patterns. This particularly provides a hint that complex pseudopalisade patterns could be synthesized by a linear combination of parameters responsible for generating simple patterns. Going even further, we ask ourselves if complex therapy approaches can be conceived, such that some linear combination thereof is able to reverse or disrupt simple pseudopalisade patterns; this is investigated with the help of numerical simulations.

Role of senescent tumor cells in building a cytokine shield in the tumor microenvironment: mathematical modeling

Cellular senescence can induce dual effects (promotion or inhibition) on cancer progression. While immune cells naturally respond and migrate toward various chemotactic sources from the tumor mass, various factors including senescent tumor cells (STCs) in the tumor microenvironment may affect this chemotactic movement. In this work, we investigate the mutual interactions between the tumor cells and the immune cells that either inhibit or facilitate tumor growth by developing a mathematical model that consists of taxis-reaction-diffusion equations and receptor kinetics for the key players in the interaction network. We apply a mathematical model to a transwell Boyden chamber invasion assay used in the experiments to illustrate that STCs can play a pivotal role in negating immune attack through tight regulation of intra- and extra-cellular signaling molecules. In particular, we show that senescent tumor cells in cell cycle arrest can block intratumoral infiltration of CD8+ T cells by secreting a high level of CXCL12, which leads to significant reduction its receptors, CXCR4, on T cells, and thus impaired chemotaxis. The predictions of nonlinear responses to CXCL12 were in good agreement with experimental data. We tested several hypotheses on immune-tumor interactions under various biochemical conditions in the tumor microenvironment and developed new concepts for anti-tumor strategies targeting senescence induced immune impairment.

Macrophage polarization as a potential therapeutic target for atherosclerosis: a dynamic stochastic modelling study

We proposed a dynamic stochastic mathematical model to evaluate the role of macrophage polarization in plaque development. The dynamic process of macrophages from proliferation to death was simulated under different lipid microenvironments. The probability of macrophage phenotypic switching was described using a Bernoulli distribution where the stochastic variable was determined by the local lipid level. Moreover, the interactions between macrophages and microenvironmental factors vary with macrophage phenotype. We investigated the distribution of key microenvironmental factors, the dynamics of macrophage polarization and its influence on foam cell formation. M1 macrophages were found to predominate in advanced plaque corresponding to the exacerbated inflammation observed in mice experiments. The imbalance between the deposition of oxidized low-density lipoprotein and phagocytic effects of macrophages governed the formation of foam cells. Furthermore, we simulated targeted therapies by either directly inhibiting the polarization probability to M1 macrophages or indirectly regulating macrophage polarization due to high-density lipoprotein levels. Comparison of simulation results with experimental findings in both therapies indicated that the intervention and regulation of macrophage polarization could influence plaque microenvironment and subsequently induce plaque regression, especially in the early stage. The proposed modelling system can facilitate the evaluation of novel therapies targeting macrophage polarization.

Simulating the behaviour of glioblastoma multiforme based on patient MRI during treatments

Glioblastoma multiforme is a brain cancer that still shows poor prognosis for patients despite the active research for new treatments. In this work, the goal is to model and simulate the evolution of tumour associated angiogenesis and the therapeutic response to glioblastoma multiforme. Multiple phenomena are modelled in order to fit different biological pathways, such as the cellular cycle, apoptosis, hypoxia or angiogenesis. This leads to a nonlinear system with 4 equations and 4 unknowns: the density of tumour cells, the [Formula: see text] concentration, the density of endothelial cells and the vascular endothelial growth factor concentration. This system is solved numerically on a mesh fitting the geometry of the brain and the tumour of a patient based on a 2D slice of MRI. We show that our numerical scheme is positive, and we give the energy estimates on the discrete solution to ensure its existence. The numerical scheme uses nonlinear control volume finite elements in space and is implicit in time. Numerical simulations have been done using the different standard treatments: surgery, chemotherapy and radiotherapy, in order to conform to the behaviour of a tumour in response to treatments according to empirical clinical knowledge. We find that our theoretical model exhibits realistic behaviours.

Optimal regulation of tumour-associated neutrophils in cancer progression

In a tumour microenvironment, tumour-associated neutrophils could display two opposing differential phenotypes: anti-tumour (N1) and pro-tumour (N2) effector cells. Converting N2 to N1 neutrophils provides innovative therapies for cancer treatment. In this study, a mathematical model for N1-N2 dynamics describing the cancer survival and immune inhibition in response to TGF-β and IFN-β is considered. The effects of exogenous intervention of TGF-β inhibitor and IFN-β are examined in order to enhance N1 recruitment to combat tumour progression. Our approach employs optimal control theory to determine drug infusion protocols that could minimize tumour volume with least administration cost possible. Four optimal control scenarios corresponding to different therapeutic strategies are explored, namely, TGF-β inhibitor control only, IFN-β control only, concomitant TGF-β inhibitor and IFN-β controls, and alternating TGF-β inhibitor and IFN-β controls. For each scheme, different initial conditions are varied to depict different pathophysiological condition of a cancer patient, leading to adaptive treatment schedule. TGF-β inhibitor and IFN-β drug dosages, total drug amount, infusion times and relative cost of drug administrations are obtained under various circumstances. The control strategies achieved could guide in designing individualized therapeutic protocols.

Spatiotemporal dynamics of a glioma immune interaction model

We report a mathematical model which depicts the spatiotemporal dynamics of glioma cells, macrophages, cytotoxic-T-lymphocytes, immuno-suppressive cytokine TGF-β and immuno-stimulatory cytokine IFN-γ through a system of five coupled reaction-diffusion equations. We performed local stability analysis of the biologically based mathematical model for the growth of glioma cell population and their environment. The presented stability analysis of the model system demonstrates that the temporally stable positive interior steady state remains stable under the small inhomogeneous spatiotemporal perturbations. The irregular spatiotemporal dynamics of gliomas, macrophages and cytotoxic T-lymphocytes are discussed extensively and some numerical simulations are presented. Performed some numerical simulations in both one and two dimensional spaces. The occurrence of heterogeneous pattern formation of the system has both biological and mathematical implications and the concepts of glioma cell progression and invasion are considered. Simulation of the model shows that by increasing the value of time, the glioma cell population, macrophages and cytotoxic-T-lymphocytes spread throughout the domain.

Atorvastatin-mediated rescue of cancer-related cognitive changes in combined anticancer therapies

Acute administration of trastuzumab (TZB) may induce various forms of cognitive impairment. These cancer-related cognitive changes (CRCC) are regulated by an adverse biological process involving cancer stem cells (CSCs) and IL-6. Recent studies have reported that atorvastatin (ATV) may change the dynamic of cognitive impairment in a combination (TZB+ATV) therapy. In this study, we investigate the mutual interactions between cancer stem cells and the tumor cells that facilitate cognitive impairment during long term TZB therapy by developing a mathematical model that involves IL-6 and the key apoptotic regulation. These include the densities of tumor cells and CSCs, and the concentrations of intracellular signaling molecules (NFκB, Bcl-2, BAX). We apply the mathematical model to a single or combination (ATV+TZB) therapy used in the experiments to demonstrate that the CSCs can enhance CRCC by secreting IL-6 and ATV may interfere the whole regulation. We show that the model can both reproduce the major experimental observation on onset and prevention of CRCC, and suggest several important predictions to guide future experiments with the goal of the development of new anti-tumor and anti-CRCC strategies. Moreover, using this model, we investigate the fundamental mechanism of onset of cognitive impairment in TZB-treated patients and the impact of alternating therapies on the anti-tumor efficacy and intracellular response to different treatment schedules.

A conventional drug, trastuzumab (TZB), was shown to be an effective weapon in killing cancer cells in brain. However, long term treatment of TZB increases the proportion of cancer stem cells (CSCs) in the tumour microenvironment (TME) and induces up-regulation of pro-tumoral molecules such as IL-6 in TME. These cancer cells then become more resistant to this chemotherapy through the IL-mediated up-regulation of NFκB and CSCs. More importantly, these changes in TME result in a serious side effect, cognitive impairment called cancer-related cognitive changes (CRCC). The detailed mechanism of CRCC is still poorly understood. However, cancer patients with chemotherapy-induced cognitive impairment can have long-term or delayed mental changes. In this study, we investigated the fundamental mechanism of CRCC in cancer patients based on experiments and a mathematical model that describes how tumor cells interact with CSCs in response to chemo drugs. In particular, we investigate how TZB-induced CSCs with modified IL-6 landscapes shape the cognitive functions in cancer patients. We showed that the combination treatment with another drug, atorvastatin (ATV), can abrogate the TZB-induced CRCC and enhance the survival probability of cancer patients by synergistic anti-tumor effect. We demonstrate that the cognitive functions and survival rates in cancer patients depend on the apoptotic signaling pathways via the critical communication and IL-6 landscapes of stimulated CTCs.

Glioblastoma radiomics to predict survival: Diffusion characteristics of surrounding nonenhancing tissue to select patients for extensive resection

BACKGROUND AND PURPOSE

Glioblastoma (GBM) is an aggressive primary CNS neoplasm with poor overall survival (OS) despite standard of care. On MRI, GBM is usually characterized by an enhancing portion (CET) (surgery target) and a nonenhancing surrounding (NET). Extent of resection is a long debated issue in GBM, with recent evidence suggesting that both CET and NET should be resected in <65 years old patients, regardless of other risk factors (i.e., molecular biomarkers). Our aim was to test a radiomic model for patient survival stratification in <65 years old patients, by analyzing MRI features of NET, to aid tumor resection.

METHODS

Sixty-eight <65 years old GBM patients, with extensive CET resection, were selected. Resection was evaluated by manually segmenting CET on volumetric T1-weighted MRI pre and postsurgery (within 72 h). All patients underwent the same treatment protocol including chemoradiation. NET radiomic features were extracted with a custom version of Pyradiomics. Feature selection was performed with principal component analysis (PCA) and its effect on survival tested with Cox regression model. Twelve months OS discrimination was tested by t-test followed by logistic regression. Statistical significance was set at p<0.05. The most relevant features were identified from the component matrix.

RESULTS

Five PCA components (PC1-5) explained 90% of the variance. PC5 resulted significant in the Cox model (p = 0.002; exp(B) = 0.686), at t-test (p = 0.002) and logistic regression analysis (p = 0.006). Apparent diffusion coefficient (ADC)-based features were the most significant for patient survival stratification.

CONCLUSIONS

ADC radiomic features on NET predict survival after standard therapy and could be used to improve patient selection for more extensive surgery.

In vitro biomimetic models for glioblastoma-a promising tool for drug response studies

The poor response of glioblastoma to current treatment protocols is a consequence of its intrinsic drug resistance. Resistance to chemotherapy is primarily associated with considerable cellular heterogeneity, and plasticity of glioblastoma cells, alterations in gene expression, presence of specific tumor microenvironment conditions and blood-brain barrier. In an attempt to successfully overcome chemoresistance and better understand the biological behavior of glioblastoma, numerous tri-dimensional (3D) biomimetic models were developed in the past decade. These novel advanced models are able to better recapitulate the spatial organization of glioblastoma in a real time, therefore providing more realistic and reliable evidence to the response of glioblastoma to therapy. Moreover, these models enable the fine-tuning of different tumor microenvironment conditions and facilitate studies on the effects of the tumor microenvironment on glioblastoma chemoresistance. This review outlines current knowledge on the essence of glioblastoma chemoresistance and describes the progress achieved by 3D biomimetic models. Moreover, comprehensive literature assessment regarding the influence of 3D culturing and microenvironment mimicking on glioblastoma gene expression and biological behavior is also provided. The contribution of the blood-brain barrier as well as the blood-tumor barrier to glioblastoma chemoresistance is also reviewed from the perspective of 3D biomimetic models. Finally, the role of mathematical models in predicting 3D glioblastoma behavior and drug response is elaborated. In the future, technological innovations along with mathematical simulations should create reliable 3D biomimetic systems for glioblastoma research that should facilitate the identification and possibly application in preclinical drug testing and precision medicine.

Role of neutrophil extracellular traps in regulation of lung cancer invasion and metastasis: Structural insights from a computational model

Lung cancer is one of the leading causes of cancer-related deaths worldwide and is characterized by hijacking immune system for active growth and aggressive metastasis. Neutrophils, which in their original form should establish immune activities to the tumor as a first line of defense, are undermined by tumor cells to promote tumor invasion in several ways. In this study, we investigate the mutual interactions between the tumor cells and the neutrophils that facilitate tumor invasion by developing a mathematical model that involves taxis-reaction-diffusion equations for the critical components in the interaction. These include the densities of tumor and neutrophils, and the concentrations of signaling molecules and structure such as neutrophil extracellular traps (NETs). We apply the mathematical model to a Boyden invasion assay used in the experiments to demonstrate that the tumor-associated neutrophils can enhance tumor cell invasion by secreting the neutrophil elastase. We show that the model can both reproduce the major experimental observation on NET-mediated cancer invasion and make several important predictions to guide future experiments with the goal of the development of new anti-tumor strategies. Moreover, using this model, we investigate the fundamental mechanism of NET-mediated invasion of cancer cells and the impact of internal and external heterogeneity on the migration patterning of tumour cells and their response to different treatment schedules.

An Integrative and Modular Framework to Recapitulate Emergent Behavior in Cell Migration

Cell migration has been a subject of study in a broad variety of biological systems, from morphogenetic events during development to cancer progression. In this work, we describe single-cell movement in a modular framework from which we simulate the collective behavior of glioblastoma cells, the most prevalent and malignant primary brain tumor. We used the U87 cell line, which can be grown as a monolayer or spatially closely packed and organized in 3D structures called spheroids. Our integrative model considers the most relevant mechanisms involved in cell migration: chemotaxis of attractant factor, mechanical interactions and random movement. The effect of each mechanism is integrated into the overall probability of the cells to move in a particular direction, in an automaton-like approach. Our simulations fit and reproduced the emergent behavior of the spheroids in a set of migration assays where single-cell trajectories were tracked. We also predicted the effect of migration inhibition on the colonies from simple experimental characterization of single treated cell tracks. The development of tools that allow complementing molecular knowledge in migratory cell behavior is relevant for understanding essential cellular processes, both physiological (such as organ formation, tissue regeneration among others) and pathological perspectives. Overall, this is a versatile tool that has been proven to predict individual and collective behavior in U87 cells, but that can be applied to a broad variety of scenarios.

Cell – extracellular matrix interaction in glioma growth. In silico model

The study aims to investigate the role of viscoelastic interactions between cells and extracellular matrix (ECM) in avascular tumor growth. Computer simulations of glioma multicellular tumor spheroid (MTS) growth are being carried out for various conditions. The calculations are based on a continuous model, which simulates oxygen transport into MTS; transitions between three cell phenotypes, cell transport, conditioned by hydrostatic forces in cell–ECM composite system, cell motility and cell adhesion. Visco-elastic cell aggregation and elastic ECM scaffold represent two compressible constituents of the composite. Cell–ECM interactions form a Transition Layer on the spheroid surface, where mechanical characteristics of tumor undergo rapid transition. This layer facilitates tumor progression to a great extent. The study demonstrates strong effects of ECM stiffness, mechanical deformations of the matrix and cell–cell adhesion on tumor progression. The simulations show in particular that at certain, rather high degrees of matrix stiffness a formation of distant multicellular clusters takes place, while at further increase of ECM stiffness subtumors do not form. The model also illustrates to what extent mere mechanical properties of cell–ECM system may contribute into variations of glioma invasion scenarios.

A Flux-Limited Model for Glioma Patterning with Hypoxia-Induced Angiogenesis

We propose a model for glioma patterns in a microlocal tumor environment under the influence of acidity, angiogenesis, and tissue anisotropy. The bottom-up model deduction eventually leads to a system of reaction–diffusion–taxis equations for glioma and endothelial cell population densities, of which the former infers flux limitation both in the self-diffusion and taxis terms. The model extends a recently introduced (Kumar, Li and Surulescu, 2020) description of glioma pseudopalisade formation with the aim of studying the effect of hypoxia-induced tumor vascularization on the establishment and maintenance of these histological patterns which are typical for high-grade brain cancer. Numerical simulations of the population level dynamics are performed to investigate several model scenarios containing this and further effects.

Mathematical models for cell migration: a non-local perspective

We provide a review of recent advancements in non-local continuous models for migration, mainly from the perspective of its involvement in embryonal development and cancer invasion. Particular emphasis is placed on spatial non-locality occurring in advection terms, used to characterize a cell's motility bias according to its interactions with other cellular and acellular components in its vicinity (e.g. cell-cell and cell-tissue adhesions, non-local chemotaxis), but we also briefly address spatially non-local source terms. Following a short introduction and description of applications, we give a systematic classification of available PDE models with respect to the type of featured non-localities and review some of the mathematical challenges arising from such models, with a focus on analytical aspects. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.

Collective invasion of glioma cells through OCT1 signalling and interaction with reactive astrocytes after surgery

Glioblastoma multiforme (GBM) is the most aggressive form of brain cancer with a short median survival time. GBM is characterized by the hallmarks of aggressive proliferation and cellular infiltration of normal brain tissue. miR-451 and its downstream molecules are known to play a pivotal role in regulation of the balance of proliferation and aggressive invasion in response to metabolic stress in the tumour microenvironment (TME). Surgery-induced transition in reactive astrocyte populations can play a significant role in tumour dynamics. In this work, we develop a multi-scale mathematical model of miR-451-LKB1-AMPK-OCT1-mTOR pathway signalling and individual cell dynamics of the tumour and reactive astrocytes after surgery. We show how the effects of fluctuating glucose on tumour cells need to be reprogrammed by taking into account the recent history of glucose variations and an AMPK/miR-451 reciprocal feedback loop. The model shows how variations in glucose availability significantly affect the activity of signalling molecules and, in turn, lead to critical cell migration. The model also predicts that microsurgery of a primary tumour induces phenotypical changes in reactive astrocytes and stem cell-like astrocytes promoting tumour cell proliferation and migration by Cxcl5. Finally, we investigated a new anti-tumour strategy by Cxcl5-targeting drugs. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.

Overcoming Drug Resistance to BRAF Inhibitor

One of the most frequently found mutations in human melanomas is in the B-raf gene, making its protein BRAF a key target for therapy. However, in patients treated with BRAF inhibitor (BRAFi), although the response is very good at first, relapse occurs within 6 months, on the average. In order to overcome this drug resistance to BRAFi, various combinations of BRAFi with other drugs have been explored, and some are being applied clinically, such as a combination of BRAF and MEK inhibitors. Experimental data for melanoma in mice show that under continuous treatment with BRAFi, the pro-cancer MDSCs and chemokine CCL2 initially decrease but eventually increase to above their original level, while the anticancer T cells continuously decrease. In this paper, we develop a mathematical model that explains these experimental results. The model is used to explore the efficacy of combinations of BRAFi with anti-CCL2, anti-PD-1 and anti-CTLA-4, with the aim of eliminating or reducing drug resistance to BRAFi.

Continuous Dynamic Modeling of Regulated Cell Adhesion: Sorting, Intercalation, and Involution

Cell-cell adhesion is essential for tissue growth and multicellular pattern formation and crucial for the cellular dynamics during embryogenesis and cancer progression. Understanding the dynamical gene regulation of cell adhesion molecules (CAMs) responsible for the emerging spatial tissue behaviors is a current challenge because of the complexity of these nonlinear interactions and feedback loops at different levels of abstraction-from genetic regulation to whole-organism shape formation. To extend our understanding of cell and tissue behaviors due to the regulation of adhesion molecules, here we present a novel, to our knowledge, model for the spatial dynamics of cellular patterning, growth, and shape formation due to the differential expression of CAMs and their regulation. Capturing the dynamic interplay between genetic regulation, CAM expression, and differential cell adhesion, the proposed continuous model can explain the complex and emergent spatial behaviors of cell populations that change their adhesion properties dynamically because of inter- and intracellular genetic regulation. This approach can demonstrate the mechanisms responsible for classical cell-sorting behaviors, cell intercalation in proliferating populations, and the involution of germ layer cells induced by a diffusing morphogen during gastrulation. The model makes predictions on the physical parameters controlling the amplitude and wavelength of a cellular intercalation interface, as well as the crucial role of N-cadherin regulation for the involution and migration of cells beyond the gradient of the morphogen Nodal during zebrafish gastrulation. Integrating the emergent spatial tissue behaviors with the regulation of genes responsible for essential cellular properties such as adhesion will pave the way toward understanding the genetic regulation of large-scale complex patterns and shapes formation in developmental, regenerative, and cancer biology.

Strategies in regulating glioblastoma signaling pathways and anti-invasion therapy

Glioblastoma multiforme is one of the most invasive type of glial tumors, which rapidly grows and commonly spreads into nearby brain tissue. It is a devastating brain cancer that often results in death within approximately 12 to 15 months after diagnosis. In this work, optimal control theory was applied to regulate intracellular signaling pathways of miR-451–AMPK–mTOR–cell cycle dynamics via glucose and drug intravenous administration infusions. Glucose level is controlled to activate miR-451 in the up-stream pathway of the model. A potential drug blocking the inhibitory pathway of mTOR by AMPK complex is incorporated to explore regulation of the down-stream pathway to the cell cycle. Both miR-451 and mTOR levels are up-regulated inducing cell proliferation and reducing invasion in the neighboring tissues. Concomitant and alternating glucose and drug infusions are explored under various circumstances to predict best clinical outcomes with least administration costs.

Spatiotemporal dynamics of different growth-diffusion systems on a percolation lattice

To investigate the proliferation and invasion of a tumor within an inhomogeneous matrix, we studied the spatiotemporal dynamics of two types of growth-diffusion systems (GDSs) with logistic or Allee growth occurring on a two-dimensional square site percolation lattice via numerical computation and finite-size scaling approaches. A critical percolation threshold exists in the two systems, but becomes obscure with an increasing Allee effect in Allee growth. The two systems evidently differ in their short-time spatiotemporal patterns: The tumor number density in the logistic model grows and spreads continuously and subdiffusively or weakly superdiffusively while that in the Allee model does so discretely and strongly superdiffusively. This difference is attributed to a lack of cooperation between sites for growth and diffusion in the logistic model as compared to its Allee counterpart. The Allee growth pattern is characterized by a rougher border and more inhomogeneous interior than its logistic counterpart. Judging from their growth-diffusion feature in combination with a clinical image analysis, we conclude that Allee growth is more suitable for modeling the proliferation and invasion of an early-stage malignant tumor than is logistic growth. A phase diagram that correlates a tumor's growth and diffusion on a percolation lattice with a site occupation fraction and Allee effect was established to reveal the sensitivity on proliferation and spreading of a tumor towards the above parameters. The Allee effect was also found to induce diverse dynamic features on its short-time growth and diffusion in the GDS, which brings in an opposite trend toward a tumor's growth and diffusion.

How to schedule VEGF and PD-1 inhibitors in combination cancer therapy?

BACKGROUND

One of the questions in the design of cancer clinical trials with combination of two drugs is in which order to administer the drugs. This is an important question, especially in the case where one agent may interfere with the effectiveness of the other agent.

RESULTS

In the present paper we develop a mathematical model to address this scheduling question in a specific case where one of the drugs is anti-VEGF, which is known to affect the perfusion of other drugs. As a second drug we take anti-PD-1. Both drugs are known to increase the activation of anticancer T cells. Our simulations show that in the case where anti-VEGF reduces the perfusion, a non-overlapping schedule is significantly more effective than a simultaneous injection of the two drugs, and it is somewhat more beneficial to inject anti-PD-1 first.

CONCLUSION

The method and results of the paper can be extended to other combinations, and they could play an important role in the design of clinical trials with combination therapy, where scheduling strategies may significantly affect the outcome.

Integrating in vitro experiments with in silico approaches for Glioblastoma invasion: the role of cell-to-cell adhesion heterogeneity

Glioblastoma cells adopt migration strategies to invade into the brain parenchyma ranging from individual to collective mechanisms, whose role and dynamics are not yet fully understood. In this work, we explore Glioblastoma heterogeneity and recapitulate its invasive patterns both in vitro, by utilizing primary cells along with the U87MG cell line, and in silico, by adopting discrete, individual cell-based mathematics. Glioblastoma cells are cultured three-dimensionally in an ECM-like substrate. The primary Glioblastoma spheroids adopt a novel cohesive pattern, mimicking perivascular invasion in the brain, while the U87MG adopt a typical, starburst invasive pattern under the same experimental setup. Mathematically, we focus on the role of the intrinsic heterogeneity with respect to cell-to-cell adhesion. Our proposed mathematical approach mimics the invasive morphologies observed in vitro and predicts the dynamics of tumour expansion. The role of the proliferation and migration is also explored showing that their effect on tumour morphology is different per cell type. The proposed model suggests that allowing cell-to-cell adhesive heterogeneity within the tumour population is sufficient for variable invasive morphologies to emerge which remain originally undetectable by conventional imaging, indicating that exploration in pathological samples is needed to improve our understanding and reveal potential patient-specific therapeutic targets.

Modeling three-dimensional invasive solid tumor growth in heterogeneous microenvironment under chemotherapy

A systematic understanding of the evolution and growth dynamics of invasive solid tumors in response to different chemotherapy strategies is crucial for the development of individually optimized oncotherapy. Here, we develop a hybrid three-dimensional (3D) computational model that integrates pharmacokinetic model, continuum diffusion-reaction model and discrete cell automaton model to investigate 3D invasive solid tumor growth in heterogeneous microenvironment under chemotherapy. Specifically, we consider the effects of heterogeneous environment on drug diffusion, tumor growth, invasion and the drug-tumor interaction on individual cell level. We employ the hybrid model to investigate the evolution and growth dynamics of avascular invasive solid tumors under different chemotherapy strategies. Our simulations indicate that constant dosing is generally more effective in suppressing primary tumor growth than periodic dosing, due to the resulting continuous high drug concentration. In highly heterogeneous microenvironment, the malignancy of the tumor is significantly enhanced, leading to inefficiency of chemotherapies. The effects of geometrically-confined microenvironment and non-uniform drug dosing are also investigated. Our computational model, when supplemented with sufficient clinical data, could eventually lead to the development of efficient in silico tools for prognosis and treatment strategy optimization.

Role of extracellular matrix and microenvironment in regulation of tumor growth and LAR-mediated invasion in glioblastoma

The cellular dispersion and therapeutic control of glioblastoma, the most aggressive type of primary brain cancer, depends critically on the migration patterns after surgery and intracellular responses of the individual cancer cells in response to external biochemical cues in the microenvironment. Recent studies have shown that miR-451 regulates downstream molecules including AMPK/CAB39/MARK and mTOR to determine the balance between rapid proliferation and invasion in response to metabolic stress in the harsh tumor microenvironment. Surgical removal of the main tumor is inevitably followed by recurrence of the tumor due to inaccessibility of dispersed tumor cells in normal brain tissue. In order to address this complex process of cell proliferation and invasion and its response to conventional treatment, we propose a mathematical model that analyzes the intracellular dynamics of the miR-451-AMPK- mTOR-cell cycle signaling pathway within a cell. The model identifies a key mechanism underlying the molecular switches between proliferative phase and migratory phase in response to metabolic stress in response to fluctuating glucose levels. We show how up- or down-regulation of components in these pathways affects the key cellular decision to infiltrate or proliferate in a complex microenvironment in the absence and presence of time delays and stochastic noise. Glycosylated chondroitin sulfate proteoglycans (CSPGs), a major component of the extracellular matrix (ECM) in the brain, contribute to the physical structure of the local brain microenvironment but also induce or inhibit glioma invasion by regulating the dynamics of the CSPG receptor LAR as well as the spatiotemporal activation status of resident astrocytes and tumor-associated microglia. Using a multi-scale mathematical model, we investigate a CSPG-induced switch between invasive and non-invasive tumors through the coordination of ECM-cell adhesion and dynamic changes in stromal cells. We show that the CSPG-rich microenvironment is associated with non-invasive tumor lesions through LAR-CSGAG binding while the absence of glycosylated CSPGs induce the critical glioma invasion. We illustrate how high molecular weight CSPGs can regulate the exodus of local reactive astrocytes from the main tumor lesion, leading to encapsulation of non-invasive tumor and inhibition of tumor invasion. These different CSPG conditions also change the spatial profiles of ramified and activated microglia. The complex distribution of CSPGs in the tumor microenvironment can determine the nonlinear invasion behaviors of glioma cells, which suggests the need for careful therapeutic strategies.

The biology and mathematical modelling of glioma invasion: a review

Adult gliomas are aggressive brain tumours associated with low patient survival rates and limited life expectancy. The most important hallmark of this type of tumour is its invasive behaviour, characterized by a markedly phenotypic plasticity, infiltrative tumour morphologies and the ability of malignant progression from low- to high-grade tumour types. Indeed, the widespread infiltration of healthy brain tissue by glioma cells is largely responsible for poor prognosis and the difficulty of finding curative therapies. Meanwhile, mathematical models have been established to analyse potential mechanisms of glioma invasion. In this review, we start with a brief introduction to current biological knowledge about glioma invasion, and then critically review and highlight future challenges for mathematical models of glioma invasion.

Microenvironmental influence on microtumour infiltration patterns: 3D-mathematical modelling supported by in vitro studies

Mathematical modelling approaches have become increasingly abundant in cancer research. Tumour infiltration extent and its spatial organization depend both on the tumour type and stage and on the bio-physicochemical characteristics of the microenvironment. This sets a complex scenario that often requires a multidisciplinary and individually adjusted approach. The ultimate goal of this work is to present an experimental/numerical combined method for the development of a three-dimensional mathematical model with the ability to reproduce the growth and infiltration patterns of a given avascular microtumour in response to different microenvironmental conditions. The model is based on a diffusion-convection reaction equation that considers logistic proliferation, volumetric growth, a rim of proliferative cells at the tumour surface, and invasion with diffusive and convective components. The parameter values of the model were fitted to experimental results while radial velocity and diffusion coefficients were made spatially variable in a case-specific way through the introduction of a shape function and a diffusion-limited-aggregation (DLA)-derived fractal matrix, respectively, according to the infiltration pattern observed. The in vitro model consists of multicellular tumour spheroids (MTSs) of an epithelial mammary tumour cell line (LM3) immersed in a collagen I gel matrix with a standard culture medium ("naive" matrix) or a conditioned medium from adipocytes or preadipocytes ("conditioned" matrix). It was experimentally determined that both adipocyte and preadipocyte conditioned media had the ability to change the MTS infiltration pattern from collective and laminar to an individual and atomized one. Numerical simulations were able to adequately reproduce qualitatively and quantitatively both kinds of infiltration patterns, which were determined by area quantification, analysis of fractal dimensions and lacunarity, and Bland-Altman analysis. These results suggest that the combined approach presented here could be established as a new framework with interesting potential applications at both the basic and clinical levels in the oncology area.

Chronic hepatitis B virus and liver fibrosis: A mathematical model

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

Glioma growth modeling based on the effect of vital nutrients and metabolic products

Glioma brain tumors exhibit considerably aggressive behavior leading to high mortality rates. Mathematical modeling of tumor growth aims to explore the interactions between glioma cells and tissue microenvironment, which affect tumor evolution. Leveraging this concept, we present a three-dimensional model of glioma spatio-temporal evolution based on existing continuum approaches, yet incorporating novel factors of the phenomenon. The proposed model involves the interactions between different tumor cell phenotypes and their microenvironment, investigating how tumor growth is affected by complex biological exchanges. It focuses on the separate and combined effect of vital nutrients and cellular wastes on tumor expansion, leading to the formation of cell populations with different metabolic, proliferative, and diffusive profiles. Several simulations were performed on a virtual and a real glioma, using combinations of proliferation and diffusion rates for different evolution times. The model results were validated on a glioma model available in the literature and a real case of tumor progression. The experimental observations indicate that our model estimates quite satisfactorily the expansion of each region and the overall tumor growth. Based on the individual results, the proposed model may provide an important research tool for patient-specific simulation of different tumor evolution scenarios and reliable estimation of glioma evolution. Graphical Abstract Outline of the mathematical model functionality and application to glioma growth with indicative results.

A simple and accurate rule-based modeling framework for simulation of autocrine/paracrine stimulation of glioblastoma cell motility and proliferation by L1CAM in 2-D culture

Background

Glioblastoma multiforme (GBM) is a devastating brain cancer for which there is no known cure. Its malignancy is due to rapid cell division along with high motility and invasiveness of cells into the brain tissue. Simple 2-dimensional laboratory assays (e.g., a scratch assay) commonly are used to measure the effects of various experimental perturbations, such as treatment with chemical inhibitors. Several mathematical models have been developed to aid the understanding of the motile behavior and proliferation of GBM cells. However, many are mathematically complicated, look at multiple interdependent phenomena, and/or use modeling software not freely available to the research community. These attributes make the adoption of models and simulations of even simple 2-dimensional cell behavior an uncommon practice by cancer cell biologists.

Results

Herein, we developed an accurate, yet simple, rule-based modeling framework to describe the in vitro behavior of GBM cells that are stimulated by the L1CAM protein using freely available NetLogo software. In our model L1CAM is released by cells to act through two cell surface receptors and a point of signaling convergence to increase cell motility and proliferation. A simple graphical interface is provided so that changes can be made easily to several parameters controlling cell behavior, and behavior of the cells is viewed both pictorially and with dedicated graphs. We fully describe the hierarchical rule-based modeling framework, show simulation results under several settings, describe the accuracy compared to experimental data, and discuss the potential usefulness for predicting future experimental outcomes and for use as a teaching tool for cell biology students.

Conclusions

It is concluded that this simple modeling framework and its simulations accurately reflect much of the GBM cell motility behavior observed experimentally in vitro in the laboratory. Our framework can be modified easily to suit the needs of investigators interested in other similar intrinsic or extrinsic stimuli that influence cancer or other cell behavior. This modeling framework of a commonly used experimental motility assay (scratch assay) should be useful to both researchers of cell motility and students in a cell biology teaching laboratory.

Electronic supplementary material

The online version of this article (10.1186/s12918-017-0516-z) contains supplementary material, which is available to authorized users.

Combination therapy for melanoma with BRAF/MEK inhibitor and immune checkpoint inhibitor: a mathematical model

BACKGROUND

The B-raf gene is mutated in up to 66% of human malignant melanomas, and its protein product, BRAF kinase, is a key part of RAS-RAF-MEK-ERK (MAPK) pathway of cancer cell proliferation. BRAF-targeted therapy induces significant responses in the majority of patients, and the combination BRAF/MEK inhibitor enhances clinical efficacy, but the response to BRAF inhibitor and to BRAF/MEK inhibitor is short lived. On the other hand, treatment of melanoma with an immune checkpoint inhibitor, such as anti-PD-1, has lower response rate but the response is much more durable, lasting for years. For this reason, it was suggested that combination of BRAF/MEK and PD-1 inhibitors will significantly improve overall survival time.

RESULTS

This paper develops a mathematical model to address the question of the correlation between BRAF/MEK inhibitor and PD-1 inhibitor in melanoma therapy. The model includes dendritic and cancer cells, CD 4+ and CD 8+ T cells, MDSC cells, interleukins IL-12, IL-2, IL-6, IL-10 and TGF- β, PD-1 and PD-L1, and the two drugs: BRAF/MEK inhibitor (with concentration γ B ) and PD-1 inhibitor (with concentration γ A ). The model is represented by a system of partial differential equations, and is used to develop an efficacy map for the combined concentrations (γ B ,γ A ). It is shown that the two drugs are positively correlated if γ B and γ A are at low doses, that is, the growth of the tumor volume decreases if either γ B or γ A is increased. On the other hand, the two drugs are antagonistic at some high doses, that is, there are zones of (γ B ,γ A ) where an increase in one of the two drugs will increase the tumor volume growth, rather than decrease it.

CONCLUSIONS

It will be important to identify, by animal experiments or by early clinical trials, the zones of (γ B ,γ A ) where antagonism occurs, in order to avoid these zones in more advanced clinical trials.

The role of myosin II in glioma invasion: A mathematical model

Gliomas are malignant tumors that are commonly observed in primary brain cancer. Glioma cells migrate through a dense network of normal cells in microenvironment and spread long distances within brain. In this paper we present a two-dimensional multiscale model in which a glioma cell is surrounded by normal cells and its migration is controlled by cell-mechanical components in the microenvironment via the regulation of myosin II in response to chemoattractants. Our simulation results show that the myosin II plays a key role in the deformation of the cell nucleus as the glioma cell passes through the narrow intercellular space smaller than its nuclear diameter. We also demonstrate that the coordination of biochemical and mechanical components within the cell enables a glioma cell to take the mode of amoeboid migration. This study sheds lights on the understanding of glioma infiltration through the narrow intercellular spaces and may provide a potential approach for the development of anti-invasion strategies via the injection of chemoattractants for localization.

The Role of the Tumor Microenvironment in Glioblastoma: A Mathematical Model

Glioblastoma multiforme is one of the deadliest human cancers and is characterized by tumor cells that hijack immune system cells in a deadly symbiotic relationship. Microglia and glioma infiltrating macrophages, which in principle should mount an immune response to the tumor, are subverted by tumor cells to facilitate growth in several ways. In this study, we seek to understand the interactions between the tumor cells and the microglia that enhance tumor growth, and for this purpose, we develop a mathematical and computational model that involves reaction-diffusion equations for the important components in the interaction. These include the densities of tumor and microglial cells, and the concentrations of growth factors and other signaling molecules. We apply this model to a transwell assay used in the laboratory to demonstrate that microglia can stimulate tumor cell invasion by secreting the growth factor TGF- β. We show that the model can both replicate the major components of the experimental findings and make new predictions to guide future experiments aimed at the development of new therapeutic approaches. Sensitivity analysis is used to identify the most important parameters as an aid to future experimental work. This study is the first step in a program that involves development of detailed 3-D models of the mechanical and biochemical interactions between a glioblastoma and the tumor microenvironment.

A quantitative study of shape descriptors from glioblastoma multiforme phenotypes for predicting survival outcome

OBJECTIVE

Predicting the survival outcome of patients with glioblastoma multiforme (GBM) is of key importance to clinicians for selecting the optimal course of treatment. The goal of this study was to evaluate the usefulness of geometric shape features, extracted from MR images, as a potential non-invasive way to characterize GBM tumours and predict the overall survival times of patients with GBM.

METHODS

The data of 40 patients with GBM were obtained from the Cancer Genome Atlas and Cancer Imaging Archive. The T₁ weighted post-contrast and fluid-attenuated inversion-recovery volumes of patients were co-registered and segmented into delineate regions corresponding to three GBM phenotypes: necrosis, active tumour and oedema/invasion. A set of two-dimensional shape features were then extracted slicewise from each phenotype region and combined over slices to describe the three-dimensional shape of these phenotypes. Thereafter, a Kruskal-Wallis test was employed to identify shape features with significantly different distributions across phenotypes. Moreover, a Kaplan-Meier analysis was performed to find features strongly associated with GBM survival. Finally, a multivariate analysis based on the random forest model was used for predicting the survival group of patients with GBM.

RESULTS

Our analysis using the Kruskal-Wallis test showed that all but one shape feature had statistically significant differences across phenotypes, with p-value < 0.05, following Holm-Bonferroni correction, justifying the analysis of GBM tumour shapes on a per-phenotype basis. Furthermore, the survival analysis based on the Kaplan-Meier estimator identified three features derived from necrotic regions (i.e. Eccentricity, Extent and Solidity) that were significantly correlated with overall survival (corrected p-value < 0.05; hazard ratios between 1.68 and 1.87). In the multivariate analysis, features from necrotic regions gave the highest accuracy in predicting the survival group of patients, with a mean area under the receiver-operating characteristic curve (AUC) of 63.85%. Combining the features of all three phenotypes increased the mean AUC to 66.99%, suggesting that shape features from different phenotypes can be used in a synergic manner to predict GBM survival.

CONCLUSION

Results show that shape features, in particular those extracted from necrotic regions, can be used effectively to characterize GBM tumours and predict the overall survival of patients with GBM. Advances in knowledge: Simple volumetric features have been largely used to characterize the different phenotypes of a GBM tumour (i.e. active tumour, oedema and necrosis). This study extends previous work by considering a wide range of shape features, extracted in different phenotypes, for the prediction of survival in patients with GBM.

Tumor proliferation and diffusion on percolation clusters

We study in silico the influence of host tissue inhomogeneity on tumor cell proliferation and diffusion by simulating the mobility of a tumor on percolation clusters with different homogeneities of surrounding tissues. The proliferation and diffusion of a tumor in an inhomogeneous tissue could be characterized in the framework of the percolation theory, which displays similar thresholds (0.54, 0.44, and 0.37, respectively) for tumor proliferation and diffusion in three kinds of lattices with 4, 6, and 8 connecting near neighbors. Our study reveals the existence of a critical transition concerning the survival and diffusion of tumor cells with leaping metastatic diffusion movement in the host tissues. Tumor cells usually flow in the direction of greater pressure variation during their diffusing and infiltrating to a further location in the host tissue. Some specific sites suitable for tumor invasion were observed on the percolation cluster and around these specific sites a tumor can develop into scattered tumors linked by some advantage tunnels that facilitate tumor invasion. We also investigate the manner that tissue inhomogeneity surrounding a tumor may influence the velocity of tumor diffusion and invasion. Our simulation suggested that invasion of a tumor is controlled by the homogeneity of the tumor microenvironment, which is basically consistent with the experimental report by Riching et al. as well as our clinical observation of medical imaging. Both simulation and clinical observation proved that tumor diffusion and invasion into the surrounding host tissue is positively correlated with the homogeneity of the tissue.

Optimal control strategies of eradicating invisible glioblastoma cells after conventional surgery

Glioblastoma, the most aggressive type of brain cancer, has median survival time of 1 year after diagnosis. It is characterized by alternating modes of rapid proliferation and aggressive invasion in response to metabolic stress in the microenvironment. A particular microRNA, miR-451, and its downstream signalling molecules, AMPK complex, are known to be key determinants in switching cell fate. These components form a core control system determining a balance between cell growth and migration which is regulated by fluctuating glucose levels in the microenvironment. An important factor from the treatment point of view is that low levels of glucose affect metabolism and activate cell migration through the miR-451-AMPK control system, creating 'invisible' migratory cells and making them inaccessible by conventional surgery. In this work, we apply optimal control theory to deal with the problem of maintaining upregulated miR-451 levels that prevent cell infiltration to surrounding brain tissue and thus induce localization of these cancer cells at the surgical site. The model also considers the effect of a drug that blocks inhibitive pathways of miR-451 from AMPK complex. Glucose infusion control and drug infusion control are chosen to represent dose rates of glucose and drug intravenous administrations, respectively. The characteristics of optimal control lead us to investigate the structure of optimal intravenous infusion regimen under various circumstances and predict best clinical outcomes with minimum expense possible.

Stochastic model explains formation of cell arrays on H/O-diamond patterns

Cell migration plays an important role in many biological systems. A relatively simple stochastic model is developed and used to describe cell behavior on chemically patterned substrates. The model is based on three parameters: the speed of cell movement (own and external), the probability of cell adhesion, and the probability of cell division on the substrate. The model is calibrated and validated by experimental data obtained on hydrogen- and oxygen-terminated patterns on diamond. Thereby, the simulations reveal that: (1) the difference in the cell movement speed on these surfaces (about 1.5×) is the key factor behind the formation of cell arrays on the patterns, (2) this difference is provided by the presence of fetal bovine serum (validated by experiments), and (3) the directional cell flow promotes the array formation. The model also predicts that the array formation requires mean distance of cell travel at least 10% of intended stripe width. The model is generally applicable for biosensors using diverse cells, materials, and structures.

Hybrid models of cell and tissue dynamics in tumor growth

Hybrid models of tumor growth, in which some regions are described at the cell level and others at the continuum level, provide a flexible description that allows alterations of cell-level properties and detailed descriptions of the interaction with the tumor environment, yet retain the computational advantages of continuum models where appropriate. We review aspects of the general approach and discuss applications to breast cancer and glioblastoma.

A Glycolysis-Based In Silico Model for the Solid Tumor Growth

Cancer-tumor growth is a complex process depending on several biological factors, such as the chemical microenvironment of the tumor, the cellular metabolic profile, and its proliferation rate. Several mathematical models have been developed for identifying the interactions between tumor cells and tissue microenvironment, since they play an important role in tumor formation and progression. Toward this direction we propose a new continuum model of avascular glioma-tumor growth, which incorporates a new factor, namely, the glycolytic potential of cancer cells, to express the interactions of three different tumor-cell populations (proliferative, hypoxic, and necrotic) with their tissue microenvironment. The glycolytic potential engages three vital nutrients, i.e., oxygen, glucose, and lactate, which provide cells with the necessary energy for their survival and proliferation. Extensive simulations are performed for different evolution times and various proliferation rates, in order to investigate how the tumor growth is affected. According to medical experts, the experimental observations indicate that the model predicts quite satisfactorily the overall tumor growth as well as the expansion of each region separately. Following extensive evaluation, the proposed model may provide an essential tool for patient-specific tumor simulation and reliable prediction of glioma spatiotemporal expansion.

Mathematically modeling the biological properties of gliomas: A review

Although mathematical modeling is a mainstay for industrial and many scientific studies, such approaches have found little application in neurosurgery. However, the fusion of biological studies and applied mathematics is rapidly changing this environment, especially for cancer research. This review focuses on the exciting potential for mathematical models to provide new avenues for studying the growth of gliomas to practical use. In vitro studies are often used to simulate the effects of specific model parameters that would be difficult in a larger-scale model. With regard to glioma invasive properties, metabolic and vascular attributes can be modeled to gain insight into the infiltrative mechanisms that are attributable to the tumor's aggressive behavior. Morphologically, gliomas show different characteristics that may allow their growth stage and invasive properties to be predicted, and models continue to offer insight about how these attributes are manifested visually. Recent studies have attempted to predict the efficacy of certain treatment modalities and exactly how they should be administered relative to each other. Imaging is also a crucial component in simulating clinically relevant tumors and their influence on the surrounding anatomical structures in the brain.

Morphometric model for discrimination between glioblastoma multiforme and solitary metastasis using three-dimensional shape analysis

PURPOSE

Glioblastoma multiforme (GBM) and brain metastasis (MET) are the most common intra-axial brain neoplasms in adults and often pose a diagnostic dilemma using standard clinical MRI. These tumor types require different oncological and surgical management, which subsequently influence prognosis and clinical outcome.

METHODS

Here, we hypothesize that GBM and MET possess different three-dimensional (3D) morphological attributes based on their physical characteristics. A 3D morphological analysis was applied on the tumor surface defined by our diffusion tensor imaging (DTI) segmentation technique. It segments the DTI data into clusters representing different isotropic and anisotropic water diffusion characteristics, from which a distinct surface boundary between healthy and pathological tissue was identified. Morphometric features of shape index and curvedness were then computed for each tumor surface and used to build a morphometric model of GBM and MET pathology with the goal of developing a tumor classification method based on shape characteristics.

RESULTS

Our 3D morphometric method was applied on 48 untreated brain tumor patients. Cross-validation resulted in a 95.8% accuracy classification with only two shape features needed and that can be objectively derived from quantitative imaging methods.

CONCLUSION

The proposed 3D morphometric analysis framework can be applied to distinguish GBMs from solitary METs. Magn Reson Med 75:2505-2516, 2016. © 2015 Wiley Periodicals, Inc.

Strategies of eradicating glioma cells: a multi-scale mathematical model with MiR-451-AMPK-mTOR control

The cellular dispersion and therapeutic control of glioblastoma, the most aggressive type of primary brain cancer, depends critically on the migration patterns after surgery and intracellular responses of the individual cancer cells in response to external biochemical and biomechanical cues in the microenvironment. Recent studies have shown that a particular microRNA, miR-451, regulates downstream molecules including AMPK and mTOR to determine the balance between rapid proliferation and invasion in response to metabolic stress in the harsh tumor microenvironment. Surgical removal of main tumor is inevitably followed by recurrence of the tumor due to inaccessibility of dispersed tumor cells in normal brain tissue. In order to address this multi-scale nature of glioblastoma proliferation and invasion and its response to conventional treatment, we propose a hybrid model of glioblastoma that analyses spatio-temporal dynamics at the cellular level, linking individual tumor cells with the macroscopic behaviour of cell organization and the microenvironment, and with the intracellular dynamics of miR-451-AMPK-mTOR signaling within a tumour cell. The model identifies a key mechanism underlying the molecular switches between proliferative phase and migratory phase in response to metabolic stress and biophysical interaction between cells in response to fluctuating glucose levels in the presence of blood vessels (BVs). The model predicts that cell migration, therefore efficacy of the treatment, not only depends on oxygen and glucose availability but also on the relative balance between random motility and strength of chemoattractants. Effective control of growing cells near BV sites in addition to relocalization of invisible migratory cells back to the resection site was suggested as a way of eradicating these migratory cells.

Mathematical modeling of Interleukin-35 promoting tumor growth and angiogenesis

Interleukin-35 (IL-35), a cytokine from the Interleukin-12 cytokine family, has been considered as an anti-inflammatory cytokine which promotes tumor progression and tumor immune evasion. It has also been demonstrated that IL-35 is secreted by regulatory T cells. Recent mouse experiments have shown that IL-35 produced by cancer cells promotes tumor growth via enhancing myeloid cell accumulation and angiogenesis, and reducing the infiltration of activated CD8[Formula: see text] T cells into tumor microenvironment. In the present paper we develop a mathematical model based on these experimental results. We include in the model an anti-IL-35 drug as treatment. The extended model (with drug) is used to design protocols of anti-IL-35 injections for treatment of cancer. We find that with a fixed total amount of drug, continuous injection has better efficacy than intermittent injections in reducing the tumor load while the treatment is ongoing. We also find that the percentage of tumor reduction under anti-IL-35 treatment improves when the production of IL-35 by cancer is increased.