Modelling the formation of necrotic regions in avascular tumours

Computational model of the cancer necrotic core formation in a tumor-on-a-chip device

The mechanisms underlying the formation of necrotic regions within avascular tumors are complex and poorly understood. In this paper, we investigate the formation of a necrotic core in a 3D tumor cell culture within a microfluidic device, considering oxygen, nutrients, and the microenvironment acidification by means of a computational-mathematical model. Our objective is to simulate cell processes, including proliferation and death inside a microfluidic device, according to the microenvironmental conditions. We employed approximation utilizing finite element models taking into account glucose, oxygen, and hydrogen ions diffusion, consumption and production, as well as cell proliferation, migration and death, addressing how tumor cells evolve under different conditions. The resulting mathematical model was examined under different scenarios, being capable of reproducing cell death and proliferation under different cell concentrations, and the formation of a necrotic core, in good agreement with experimental data reported in the literature. This approach not only advances our fundamental understanding of necrotic core formation but also provides a robust computational platform to study personalized therapeutic strategies, offering an important tool in cancer research and treatment design.

Echo Chambers: Emotional Contagion and Group Polarization on Facebook

Recent findings showed that users on Facebook tend to select information that adhere to their system of beliefs and to form polarized groups - i.e., echo chambers. Such a tendency dominates information cascades and might affect public debates on social relevant issues. In this work we explore the structural evolution of communities of interest by accounting for users emotions and engagement. Focusing on the Facebook pages reporting on scientific and conspiracy content, we characterize the evolution of the size of the two communities by fitting daily resolution data with three growth models - i.e. the Gompertz model, the Logistic model, and the Log-logistic model. Although all the models appropriately describe the data structure, the Logistic one shows the best fit. Then, we explore the interplay between emotional state and engagement of users in the group dynamics. Our findings show that communities' emotional behavior is affected by the users' involvement inside the echo chamber. Indeed, to an higher involvement corresponds a more negative approach. Moreover, we observe that, on average, more active users show a faster shift towards the negativity than less active ones.

Oxygen-Driven Tumour Growth Model: A Pathology-Relevant Mathematical Approach

Xenografts -as simplified animal models of cancer- differ substantially in vasculature and stromal architecture when compared to clinical tumours. This makes mathematical model-based predictions of clinical outcome challenging. Our objective is to further understand differences in tumour progression and physiology between animal models and the clinic.

To achieve that, we propose a mathematical model based upon tumour pathophysiology, where oxygen -as a surrogate for endocrine delivery- is our main focus. The Oxygen-Driven Model (ODM), using oxygen diffusion equations, describes tumour growth, hypoxia and necrosis. The ODM describes two key physiological parameters. Apparent oxygen uptake rate (kR′) represents the amount of oxygen cells seem to need to proliferate. The more oxygen they appear to need, the more the oxygen transport. kR′ gathers variability from the vasculature, stroma and tumour morphology. Proliferating rate (k_p) deals with cell line specific factors to promote growth. The K_H,K_N describe the switch of hypoxia and necrosis. Retrospectively, using archived data, we looked at longitudinal tumour volume datasets for 38 xenografted cell lines and 5 patient-derived xenograft-like models.

Exploration of the parameter space allows us to distinguish 2 groups of parameters. Group 1 of cell lines shows a spread in values of kR′ and lower k_p, indicating that tumours are poorly perfused and slow growing. Group 2 share the value of the oxygen uptake rate (kR′) and vary greatly in k_p, which we interpret as having similar oxygen transport, but more tumour intrinsic variability in growth.

However, the ODM has some limitations when tested in explant-like animal models, whose complex tumour-stromal morphology may not be captured in the current version of the model. Incorporation of stroma in the ODM will help explain these discrepancies. We have provided an example. The ODM is a very simple -and versatile- model suitable for the design of preclinical experiments, which can be modified and enhanced whilst maintaining confidence in its predictions.

Tumour-bearing animal models of cancer are needed to discover new drugs to treat cancer. We aim in this article to capture—through mathematics- some underlying phenomena of tumour growth in animals. We propose a set of equations that, despite being very simple, describe tumour growth, hypoxia and necrosis. Cells under low oxygen levels change into a stress state called “hypoxia”, which will ultimately lead to tissue death, also known as “necrosis” and “apoptosis”. Hypoxic cells undergo a variety of changes which impact tumour growth, development, metastasis and -most importantly- response to therapy. Hence, oxygen distribution is important. We simulate oxygen profiles to locate hypoxic and necrotic tumour regions. Finally, this mathematical model allows us to compare and classify animal models from a grounded and physiological perspective, rather than a more convenient and empirical one. This will help us understand how well (or poorly) animal tumours mimic tumours in patients. The simplicity of our mathematical model allows us to obtain more information out of the same animal sets without any further experiments, hopefully saving time, money and animal usage.

Highly effective antiangiogenesis via magnetic mesoporous silica-based siRNA vehicle targeting the VEGF gene for orthotopic ovarian cancer therapy

Therapeutic antiangiogenesis strategies have demonstrated significant antitumor efficacy in ovarian cancer. Recently, RNA interference (RNAi) has come to be regarded as a promising technology for treatment of disease, especially cancer. In this study, vascular endothelial growth factor (VEGF)-small interfering RNA (siRNA) was encapsulated into a magnetic mesoporous silica nanoparticle (M-MSN)-based, polyethylenimine (PEI)-capped, polyethylene glycol (PEG)-grafted, fusogenic peptide (KALA)-functionalized siRNA delivery system, termed M-MSN_VEGF siRNA@PEI-PEG-KALA, which showed significant effectiveness with regard to VEGF gene silencing in vitro and in vivo. The prepared siRNA delivery system readily exhibited cellular internalization and ease of endosomal escape, resulting in excellent RNAi efficacy without associated cytotoxicity in SKOV3 cells. In in vivo experiments, notable retardation of tumor growth was observed in orthotopic ovarian tumor-bearing mice, which was attributed to significant inhibition of angiogenesis by systemic administration of this nanocarrier. No obvious toxic drug responses were detected in major organs. Further, the magnetic core of M-MSN_VEGF siRNA@PEI-PEG-KALA proved capable of probing the site and size of the ovarian cancer in mice on magnetic resonance imaging. Collectively, the results demonstrate that an M-MSN-based delivery system has potential to serve as a carrier of siRNA therapeutics in ovarian cancer.

GPI/AMF inhibition blocks the development of the metastatic phenotype of mature multi-cellular tumor spheroids

Epithelial-mesenchymal transition (EMT) and cellular invasiveness are two pivotal processes for the development of metastatic tumor phenotypes. The metastatic profile of non-metastatic MCF-7 cells growing as multi-cellular tumor microspheroids (MCTSs) was analyzed by determining the contents of the EMT, invasive and migratory proteins, as well as their migration and invasiveness potential and capacity to secrete active cytokines such as the glucose phosphate isomerase/AMF (GPI/AMF). As for the control, the same analysis was also performed in MCF-7 and MDA-MB-231 (highly metastatic, MDA) monolayer cells, and in stage IIIB and IV human metastatic breast biopsies. The proliferative cell layers (PRL) of mature MCF-7 MCTSs, MDA monolayer cells and metastatic biopsies exhibited increased cellular contents (2-15 times) of EMT (β-catenin, SNAIL), migratory (vimentin, cytokeratin, and fibronectin) and invasive (MMP-1, VEGF) proteins versus MCF-7 monolayer cells, quiescent cell layers of mature MCF-7 MCTS and non-metastatic breast biopsies. The increase in metastatic proteins correlated with substantially elevated cellular abilities for migration (18-times) and invasiveness (13-times) and with the higher level (6-times) of the cytokine GPI/AMF in the extracellular medium of PRL, as compared to MCF-7 monolayer cells. Interestingly, the addition of the GPI/AMF inhibitors erythrose-4-phosphate or 6-phosphogluconate at micromolar doses significantly decreased its extracellular activity (>80%), with a concomitant diminution in the metastatic protein content and migratory tumor cell capacity, and with no inhibitory effect on tumor lactate production or toxicity on 3T3 mouse fibroblasts. The present findings provide new insights into the discovery of metabolic inhibitors to be used as complementary therapy against metastatic and aggressive tumors.

Interplay between distribution of live cells and growth dynamics of solid tumours

Experiments show that simple diffusion of nutrients and waste molecules is not sufficient to explain the typical multilayered structure of solid tumours, where an outer rim of proliferating cells surrounds a layer of quiescent but viable cells and a central necrotic region. These experiments challenge models of tumour growth based exclusively on diffusion. Here we propose a model of tumour growth that incorporates the volume dynamics and the distribution of cells within the viable cell rim. The model is suggested by in silico experiments and is validated using in vitro data. The results correlate with in vivo data as well, and the model can be used to support experimental and clinical oncology.

Modelling acidosis and the cell cycle in multicellular tumour spheroids

A partial differential equation model is developed to understand the effect that nutrient and acidosis have on the distribution of proliferating and quiescent cells and dead cell material (necrotic and apoptotic) within a multicellular tumour spheroid. The rates of cell quiescence and necrosis depend upon the local nutrient and acid concentrations and quiescent cells are assumed to consume less nutrient and produce less acid than proliferating cells. Analysis of the differences in nutrient consumption and acid production by quiescent and proliferating cells shows low nutrient levels do not necessarily lead to increased acid concentration via anaerobic metabolism. Rather, it is the balance between proliferating and quiescent cells within the tumour which is important; decreased nutrient levels lead to more quiescent cells, which produce less acid than proliferating cells. We examine this effect via a sensitivity analysis which also includes a quantification of the effect that nutrient and acid concentrations have on the rates of cell quiescence and necrosis.

Bright solitary waves in malignant gliomas

We put forward a nonlinear wave model describing the fundamental dynamical features of an aggressive type of brain tumors. Our model accounts for the invasion of normal tissue by a proliferating and propagating rim of active glioma cancer cells in the tumor boundary and the subsequent formation of a necrotic core. By resorting to numerical simulations, phase space analysis, and exact solutions we prove that bright solitary tumor waves develop in such systems. Possible implications of our model as a tool to extract relevant patient specific tumor parameters in combination with standard preoperative clinical imaging are also discussed.

Formulation and numerical simulations of a continuum model of avascular tumor growth

In this paper we present a continuum mathematical model for a multicellular spheroid that mimics the micro-environment within avascular tumor growth. The model consists of a coupled system of non-linear convection-diffusion-reaction equations. This system is solved using a previously developed conservative Galerkin characteristics method. In the model considered, there are three cell types: the proliferative cells, the quiescent non-dividing cells which stay in the G(0) phase of the cell cycle and the necrotic cells. The model includes viable cell diffusion, diffusion of cellular material and the removal of necrotic cells. We assume that the nutrients diffuse passively and are consumed by the proliferative and quiescent tumor cells depending on the availability of resources (oxygen, glucose, etc.). The numerical simulations are performed using different sets of parameters, including biologically realistic ones, to explore the effects of each of these model parameters on reaching the steady state. The present results, taken together with those reported earlier, indicate that the removal of necrotic cells and the diffusion of cellular material have significant effects on the steady state, reflecting growth saturation, the number of viable cells, and the spheroid size.

Nonlinear modelling of cancer: bridging the gap between cells and tumours

Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.

Cell killing and resistance in pre-operative breast cancer chemotherapy

Background

Despite the recent development of technologies giving detailed images of tumours in vivo, direct or indirect ways to measure how many cells are actually killed by a treatment or are resistant to it are still beyond our reach.

Methods

We designed a simple model of tumour progression during treatment, based on descriptions of the key phenomena of proliferation, quiescence, cell killing and resistance, and giving as output the macroscopically measurable tumour volume and growth fraction. The model was applied to a database of the time course of volumes of breast cancer in patients undergoing pre-operative chemotherapy, for which the initial estimate of proliferating cells by the measure of the percentage of Ki67-positive cells was available.

Results

The analysis recognises different patterns of response to treatment. In one subgroup of patients the fitting implied drug resistance. In another subgroup there was a shift to higher sensitivity during the therapy. In the subgroup of patients where killing of cycling cells had the highest score, the drugs showed variable efficacy against quiescent cells.

Conclusion

The approach was feasible, providing items of information not otherwise available. Additional data, particularly sequential Ki67 measures, could be added to the system, potentially reducing uncertainty in estimates of parameter values.