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

The anomalous diffusion of a tumor invading with different surrounding tissues

We simulated the invasion of a proliferating, diffusing tumor within different surrounding tissue conditions using a hybrid mathematical model. The in silico invasion of a tumor was addressed systematically for the first time within the framework of a generalized diffusion theory. Our results reveal that a tumor not only migrates using typical Fickian diffusion, but also migrates more generally using subdiffusion, superdiffusion, and even ballistic diffusion, with increasing mobility of the tumor cell when haptotaxis and chemotaxis toward the host tissue surrounding the proliferative tumor are involved. Five functional terms were included in the hybrid model and their effects on a tumor's invasion were investigated quantitatively: haptotaxis toward the extracellular matrix tissue that is degraded by matrix metalloproteinases; chemotaxis toward nutrients; cell-cell adhesion; the proliferation of the tumor; and the immune response toward the tumor. Haptotaxis and chemotaxis, which are initiated by extracellular matrix and nutrient supply (i.e., glucose) respectively, as well as cell-cell adhesions all drastically affect a tumor's diffusion mode when a tumor invades its surrounding host tissue and proliferates. We verified the in silico invasive behavior of a tumor by analyzing experimental data gathered from the in vitro culturing of different tumor cells and clinical imaging observations that used the same approach as was used to process the simulation data. The different migration modes of a tumor suggested by the simulations generally conform to the results observed in cell cultures and in clinical imaging. Our study not only discloses some migration modes of a tumor that proliferates and invades under different host tissues conditions, but also provides a heuristic method to characterize the invasion of a tumor in clinical medical imaging analysis.

Discrimination between glioblastoma multiforme and solitary metastasis using morphological features derived from the p:q tensor decomposition of diffusion tensor imaging

The management and treatment of high-grade glioblastoma multiforme (GBM) and solitary metastasis (MET) are very different and influence the prognosis and subsequent clinical outcomes. In the case of a solitary MET, diagnosis using conventional radiology can be equivocal. Currently, a definitive diagnosis is based on histopathological analysis on a biopsy sample. Here, we present a computerised decision support framework for discrimination between GBM and solitary MET using MRI, which includes: (i) a semi-automatic segmentation method based on diffusion tensor imaging; (ii) two-dimensional morphological feature extraction and selection; and (iii) a pattern recognition module for automated tumour classification. Ground truth was provided by histopathological analysis from pre-treatment stereotactic biopsy or at surgical resection. Our two-dimensional morphological analysis outperforms previous methods with high cross-validation accuracy of 97.9% and area under the receiver operating characteristic curve of 0.975 using a neural networks-based classifier.

Choindroitinase ABC I-mediated enhancement of oncolytic virus spread and anti tumor efficacy: a mathematical model

Oncolytic viruses are genetically engineered viruses that are designed to kill cancer cells while doing minimal damage to normal healthy tissue. After being injected into a tumor, they infect cancer cells, multiply inside them, and when a cancer cell is killed they move on to spread and infect other cancer cells. Chondroitinase ABC (Chase-ABC) is a bacterial enzyme that can remove a major glioma ECM component, chondroitin sulfate glycosoamino glycans from proteoglycans without any deleterious effects in vivo. It has been shown that Chase-ABC treatment is able to promote the spread of the viruses, increasing the efficacy of the viral treatment. In this paper we develop a mathematical model to investigate the effect of the Chase-ABC on the treatment of glioma by oncolytic viruses (OV). We show that the model's predictions agree with experimental results for a spherical glioma. We then use the model to test various treatment options in the heterogeneous microenvironment of the brain. The model predicts that separate injections of OV, one into the center of the tumor and another outside the tumor will result in better outcome than if the total injection is outside the tumor. In particular, the injection of the ECM-degrading enzyme (Chase-ABC) on the periphery of the main tumor core need to be administered in an optimal strategy in order to infect and eradicate the infiltrating glioma cells outside the tumor core in addition to proliferative cells in the bulk of tumor core. The model also predicts that the size of tumor satellites and distance between the primary tumor and multifocal/satellite lesions may be an important factor for the efficacy of the viral therapy with Chase treatment.

The LDL-HDL profile determines the risk of atherosclerosis: a mathematical model

Atherosclerosis, the leading death in the United State, is a disease in which a plaque builds up inside the arteries. As the plaque continues to grow, the shear force of the blood flow through the decreasing cross section of the lumen increases. This force may eventually cause rupture of the plaque, resulting in the formation of thrombus, and possibly heart attack. It has long been recognized that the formation of a plaque relates to the cholesterol concentration in the blood. For example, individuals with LDL above 190 mg/dL and HDL below 40 mg/dL are at high risk, while individuals with LDL below 100 mg/dL and HDL above 50 mg/dL are at no risk. In this paper, we developed a mathematical model of the formation of a plaque, which includes the following key variables: LDL and HDL, free radicals and oxidized LDL, MMP and TIMP, cytockines: MCP-1, IFN-γ, IL-12 and PDGF, and cells: macrophages, foam cells, T cells and smooth muscle cells. The model is given by a system of partial differential equations with in evolving plaque. Simulations of the model show how the combination of the concentrations of LDL and HDL in the blood determine whether a plaque will grow or disappear. More precisely, we create a map, showing the risk of plaque development for any pair of values (LDL,HDL).

On a multiphase multicomponent model of biofilm growth

Biofilms are formed when free-floating bacteria attach to a surface and secrete polysaccharide to form an extracellular polymeric matrix (EPS). A general model of biofilm growth needs to include the bacteria, the EPS, and the solvent within the biofilm region Ω(t), and the solvent in the surrounding region D(t). The interface between the two regions, Γ(t), is a free boundary. In this paper, we consider a mathematical model, which consists of a Stokes equation for the EPS with bacteria attached to it, and a Stokes equation for the solvent in Ω(t) and a different one for the solvent in D(t). The volume fraction of the EPS is another unknown satisfying a reaction-diffusion equation. The entire system is coupled nonlinearly within Ω(t), and across the free surface Γ(t). We prove the existence and uniqueness of solution, with a smooth surface Γ(t), for a small time interval.

Mathematical modelling of spatio-temporal glioma evolution

Background

Gliomas are the most common types of brain cancer, well known for their aggressive proliferation and the invasive behavior leading to a high mortality rate. Several mathematical models have been developed for identifying the interactions between glioma cells and tissue microenvironment, which play an important role in the mechanism of the tumor formation and progression.

Methods

Building and expanding on existing approaches, this paper develops a continuous three-dimensional model of avascular glioma spatio-temporal evolution. The proposed spherical model incorporates the interactions between the populations of four different glioma cell phenotypes (proliferative, hypoxic, hypoglychemic and necrotic) and their tissue microenvironment, in order to investigate how they affect tumor growth and invasion in an isotropic and homogeneous medium. The model includes two key variables involved in the proliferation and invasion processes of cancer cells; i.e. the extracellular matrix and the matrix-degradative enzymes concentrations inside the tumor and its surroundings. Additionally, the proposed model focuses on innovative features, such as the separate and independent impact of two vital nutrients, namely oxygen and glucose, in tumor growth, leading to the formation of cell populations with different metabolic profiles. The model implementation takes under consideration the variations of particular factors, such as the local cell proliferation rate, the variable conversion rates of cells from one category to another and the nutrient-dependent thresholds of conversion. All model variables (cell densities, ingredients concentrations) are continuous and described by reaction-diffusion equations.

Results

Several simulations were performed using combinations of growth and invasion rates, for different evolution times. The model results were evaluated by medical experts and validated on experimental glioma models available in the literature, revealing high agreement between simulated and experimental results.

Conclusions

Based on the experimental validation, as well as the evaluation by clinical experts, the proposed model may provide an essential tool for the patient-specific simulation of different tumor evolution scenarios and reliable prognosis of glioma spatio-temporal progression.

The role of CD200-CD200R in tumor immune evasion

CD200 is a cell membrane protein that interacts with CD200 receptor (CD200R) of myeloid lineage cells. During tumor initiation and progression, CD200-positive tumor cells can interact with M1 and M2 macrophages through CD200-CD200R-compex, and downregulate IL-10 and IL-12 productions secreted primarily by M2 and M1 macrophages, respectively. In the tumor microenvironment, IL-10 inhibits the activation of cytotoxic T lymphocytes (CTL), while IL-12 enhances CTL activation. In this paper, we used a system approach to determine the combined effect of CD200-CD200R interaction on tumor proliferation by developing a mathematical model. We demonstrate that blocking CD200 on tumor cells may have opposite effects on tumor proliferation depending on the "affinity" of the macrophages to form the CD200-CD200R-complex with tumor cells. Our results help understanding the complexities of tumor microenvironment.

Regulation of cell proliferation and migration in glioblastoma: new therapeutic approach

Glioblastoma is the most aggressive brain cancer with the poor survival rate. A microRNA, miR-451, and its downstream molecules, CAB39/LKB1/STRAD/AMPK, are known to play a critical role in regulating a biochemical balance between rapid proliferation and invasion in the presence of metabolic stress in microenvironment. We develop a novel multi-scale mathematical model where cell migration and proliferation are controlled through a core intracellular control system (miR-451-AMPK complex) in response to glucose availability and physical constraints in the microenvironment. Tumor cells are modeled individually and proliferation and migration of those cells are regulated by the intracellular dynamics and reaction-diffusion equations of concentrations of glucose, chemoattractant, extracellular matrix, and MMPs. The model predicts that invasion patterns and rapid growth of tumor cells after conventional surgery depend on biophysical properties of cells, dynamics of the core control system, and microenvironment as well as glucose injection methods. We developed a new type of therapeutic approach: effective injection of chemoattractant to bring invasive cells back to the surgical site after initial surgery, followed by glucose injection at the same location. The model suggests that a good combination of chemoattractant and glucose injection at appropriate time frames may lead to an effective therapeutic strategy of eradicating tumor cells.

A hybrid model of tumor-stromal interactions in breast cancer

Ductal carcinoma in situ (DCIS) is an early stage noninvasive breast cancer that originates in the epithelial lining of the milk ducts, but it can evolve into comedo DCIS and ultimately, into the most common type of breast cancer, invasive ductal carcinoma. Understanding the progression and how to effectively intervene in it presents a major scientific challenge. The extracellular matrix (ECM) surrounding a duct contains several types of cells and several types of growth factors that are known to individually affect tumor growth, but at present the complex biochemical and mechanical interactions of these stromal cells and growth factors with tumor cells is poorly understood. Here we develop a mathematical model that incorporates the cross-talk between stromal and tumor cells, which can predict how perturbations of the local biochemical and mechanical state influence tumor evolution. We focus on the EGF and TGF-β signaling pathways and show how up- or down-regulation of components in these pathways affects cell growth and proliferation. We then study a hybrid model for the interaction of cells with the tumor microenvironment (TME), in which epithelial cells (ECs) are modeled individually while the ECM is treated as a continuum, and show how these interactions affect the early development of tumors. Finally, we incorporate breakdown of the epithelium into the model and predict the early stages of tumor invasion into the stroma. Our results shed light on the interactions between growth factors, mechanical properties of the ECM, and feedback signaling loops between stromal and tumor cells, and suggest how epigenetic changes in transformed cells affect tumor progression.

Invasion from a cell aggregate--the roles of active cell motion and mechanical equilibrium

Cell invasion from an aggregate into a surrounding extracellular matrix (ECM) is an important process during development disease, e.g., vascular network assembly or tumor progression. To describe the behavior emerging from autonomous cell motility, cell-cell adhesion and contact guidance by ECM filaments, we propose a suitably modified cellular Potts model. We consider an active cell motility process in which internal polarity is governed by a positive feedback from cell displacements, a mechanism that can result in highly persistent motion when constrained by an oriented ECM structure. The model allows us to explore the interplay between haptotaxis, matrix degradation and active cell movement. We show that for certain conditions the cells are able to both invade the ECM and follow the ECM tracks. Furthermore, we argue that enforcing mechanical equilibrium within a bulk cell mass is of key importance in multicellular simulations.

miR451 and AMPK mutual antagonism in glioma cell migration and proliferation: a mathematical model

Glioblastoma multiforme (GBM) is the most common and the most aggressive type of brain cancer; the median survival time from the time of diagnosis is approximately one year. GBM is characterized by the hallmarks of rapid proliferation and aggressive invasion. miR-451 is known to play a key role in glioblastoma by modulating the balance of active proliferation and invasion in response to metabolic stress in the microenvironment. The present paper develops a mathematical model of GBM evolution which focuses on the relative balance of growth and invasion. In the present work we represent the miR-451/AMPK pathway by a simple model and show how the effects of glucose on cells need to be “refined” by taking into account the recent history of glucose variations. The simulations show how variations in glucose significantly affect the level of miR-451 and, in turn, cell migration. The model predicts that oscillations in the levels of glucose increase the growth of the primary tumor. The model also suggests that drugs which upregulate miR-451, or block other components of the CAB39/AMPK pathway, will slow down glioma cell migration. The model provides an explanation for the growth-invasion cycling patterns of glioma cells in response to high/low glucose uptake in microenvironment in vitro, and suggests new targets for drugs, associated with miR-451 upregulation.

Mechanistic modeling of the effects of myoferlin on tumor cell invasion

Myoferlin (MYOF) is a member of the evolutionarily conserved ferlin family of proteins, noted for their role in a variety of membrane processes, including endocytosis, repair, and vesicular transport. Notably, ferlins are implicated in Caenorhabditis elegans sperm motility (Fer-1), mammalian skeletal muscle development and repair (MYOF and dysferlin), and presynaptic transmission in the auditory system (otoferlin). In this paper, we demonstrate that MYOF plays a previously unrecognized role in cancer cell invasion, using a combination of mathematical modeling and in vitro experiments. Using a real-time impedance-based invasion assay (xCELLigence), we have shown that lentiviral-based knockdown of MYOF significantly reduced invasion of MDA-MB-231 breast cancer cells in Matrigel bioassays. Based on these experimental data, we developed a partial differential equation model of MYOF effects on cancer cell invasion, which we used to generate mechanistic hypotheses. The mathematical model predictions revealed that matrix metalloproteinases (MMPs) may play a key role in modulating this invasive property, which was supported by experimental data using quantitative RT-PCR screens. These results suggest that MYOF may be a promising target for biomarkers or drug target for metastatic cancer diagnosis and therapy, perhaps mediated through MMPs.

Combined experimental and mathematical approach for development of microfabrication-based cancer migration assay

Migration of cancer cells is a key determinant of metastasis, which is correlated with poor prognosis in patients. Evidence shows that cancer cell motility is regulated by stromal cell interactions. To quantify the role of homotypic and heterotypic cell-cell interaction in migration, a two-dimensional migration assay has been developed by microfabrication techniques. Two breast cancer cell lines, MDA-MB-231 and MDA-MB-453, were used to develop micropatterns of cancer cells (cell islands) that revealed distinct migration profiles in this assay. Although the individual migration rates of these cells showed only a sevenfold difference, MDA-MB-453 islands migrated significantly lower than MDA-MB-231 islands, indicating differential regulation of migration in isolated cells vs. islands. Island size had the greatest impact on migration, primarily for MDA-MB-231 cells. Migration of MDA-MB-231 islands was decreased by interaction with homotypic cells, and significantly more by heterotypic non-cancer-associated fibroblasts. In addition, a mathematical model of island migration in multi-cellular population has been developed using Stefan-Maxwell's equation. The model showed qualitative agreement with experimental results and predicted a biphasic relation between cell densities and island sizes. The combined experimental and mathematical model can be used to quantitatively study the impact of cell-cell interactions on migration.

Cancer stem cells in solid tumors: is 'evading apoptosis' a hallmark of cancer?

Conventional wisdom has long held that once a cancer cell has developed it will inevitably progress to clinical disease. Updating this paradigm, it has more recently become apparent that the tumor interacts with its microenvironment and that some environmental bottlenecks, such as the angiogenic switch, must be overcome for the tumor to progress. In parallel, attraction has been drawn to the concept that there is a minority population of cells - the cancer stem cells - bestowed with the exclusive ability to self-renew and regenerate the tumor. With therapeutic targeting issues at stake, much attention has shifted to the identification of cancer stem cells, the thinking being that the remaining non-stem population, already fated to die, will play a negligible role in tumor development. In fact, the newly appreciated importance of intercellular interactions in cancer development also extends in a unique and unexpected way to interactions between the stem and non-stem compartments of the tumor. Here we discuss recent findings drawn from a hybrid mathematical-cellular automaton model that simulates growth of a heterogeneous solid tumor comprised of cancer stem cells and non-stem cancer cells. The model shows how the introduction of cell fate heterogeneity paradoxically influences the tumor growth dynamic in response to apoptosis, to reveal yet another bottleneck to tumor progression potentially exploitable for disease control.

Tumor cells proliferation and migration under the influence of their microenvironment

It is well known that tumor microenvironment affects tumor growth and metastasis: Tumor cells may proliferate at different rates and migrate in different patterns depending on the microenvironment in which they are embedded. There is a huge literature that deals with mathematical models of tumor growth and proliferation, in both the avascular and vascular phases. In particular, a review of the literature of avascular tumor growth (up to 2006) can be found in Lolas (G. Lolas, Lecture Notes in Mathematics, Springer Berlin / Heidelberg, 1872, 77 (2006)). In this article we report on some of our recent work. We consider two aspects, proliferation and of migration, and describe mathematical models based on in vitro experiments. Simulations of the models are in agreement with experimental results. The models can be used to generate hypotheses regarding the development of drugs which will confine tumor growth.