A Coupled Mathematical Model of Cell Migration,Vessel Cooption and Tumour Microenvironment during the Initiation of Micrometastases

We propose a coupled mathematical model for the detailed quantitative analyses of initial microtumour and micrometastases formation by including cancer cell migration, host vessel cooption and changes in microenvironment. Migrating cells are included as a new phenotype to describe the migration behaviour of malignant tumour cells. Migration probability of a migrating cell is assumed to be influenced by local chemical microenvironment. Pre-existing vessel cooption and remodelling are introduced according to the local haemodynamical microenvironment, such as interstitial pressure and vessel wall permeability. After the tumour cells and tumour vessels distribution are updated, the chemical substances are coupled calculated with the haemodynamical environment. The simulation results clearly reproduce the tumour cells migrate and proliferate along the pre-existing vessels at the very early stage of growth, which are consistent with many published experimental observations. In addition, the model demonstrates the interactions of tumour cells with the pre-existing vessels, which are believed to be essential for initial adhesion, proliferation, invasion, and micrometastases establishment. Quantitative analysis of tumour expansion in longitudinal and transverse directions shows that the cooption and migration along host vessels will be inhibited once angiogenesis phase occurs. The influences of the ability of cell migration and the inclusion of vessel cooption on the formation of micrometastases are discussed.     

A Mathematical Model Coupling Tumor Growth and Angiogenesis

We present a mathematical model for vascular tumor growth. We use phase fields to model cellular growth and reaction-diffusion equations for the dynamics of angiogenic factors and nutrients. The model naturally predicts the shift from avascular to vascular growth at realistic scales. Our computations indicate that the negative regulation of the Delta-like ligand 4 signaling pathway slows down tumor growth by producing a larger density of non-functional capillaries. Our results show good quantitative agreement with experiments.     

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.     

A Multiscale Mathematical Model of Tumour Invasive Growth

Known as one of the hallmarks of cancer (Hanahan and Weinberg in Cell 100:57–70, 2000) cancer cell invasion of human body tissue is a complicated spatio-temporal multiscale process which enables a localised solid tumour to transform into a systemic, metastatic and fatal disease. This process explores and takes advantage of the reciprocal relation that solid tumours establish with the extracellular matrix (ECM) components and other multiple distinct cell types from the surrounding microenvironment. Through the secretion of various proteolytic enzymes such as matrix metalloproteinases or the urokinase plasminogen activator (uPA), the cancer cell population alters the configuration of the surrounding ECM composition and overcomes the physical barriers to ultimately achieve local cancer spread into the surrounding tissue. The active interplay between the tissue-scale tumour dynamics and the molecular mechanics of the involved proteolytic enzymes at the cell scale underlines the biologically multiscale character of invasion and raises the challenge of modelling this process with an appropriate multiscale approach. In this paper, we present a new two-scale moving boundary model of cancer invasion that explores the tissue-scale tumour dynamics in conjunction with the molecular dynamics of the urokinase plasminogen activation system. Building on the multiscale moving boundary method proposed in Trucu et al. (Multiscale Model Simul 11(1):309–335, 2013), the modelling that we propose here allows us to study the changes in tissue-scale tumour morphology caused by the cell-scale uPA microdynamics occurring along the invasive edge of the tumour. Our computational simulation results demonstrate a range of heterogeneous dynamics which are qualitatively similar to the invasive growth patterns observed in a number of different types of cancer, such as the tumour infiltrative growth patterns discussed in Ito et al. (J Gastroenterol 47:1279–1289, 2012).     

Modelling the Role of Angiogenesis and Vasculogenesis in Solid Tumour Growth

Recent experimental evidence suggests that vasculogenesis may play an important role in tumour vascularisation. While angiogenesis involves the proliferation and migration of endothelial cells (ECs) in pre-existing vessels, vasculogenesis involves the mobilisation of bone-marrow-derived endothelial progenitor cells (EPCs) into the bloodstream. Once blood-borne, EPCs home in on the tumour site, where subsequently they may differentiate into ECs and form vascular structures. In this paper, we develop a mathematical model, formulated as a system of nonlinear ordinary differential equations (ODEs), which describes vascular tumour growth with both angiogenesis and vasculogenesis contributing to vessel formation. Submodels describing exclusively angiogenic and exclusively vasculogenic tumours are shown to exhibit similar growth dynamics. In each case, there are three possible scenarios: the tumour remains in an avascular steady state, the tumour evolves to a vascular equilibrium, or unbounded vascular growth occurs. Analysis of the full model reveals that these three behaviours persist when angiogenesis and vasculogenesis act simultaneously. However, when both vascularisation mechanisms are active, the tumour growth rate may increase, causing the tumour to evolve to a larger equilibrium size or to expand uncontrollably. Alternatively, the growth rate may be left unaffected, which occurs if either vascularisation process alone is able to keep pace with the demands of the growing tumour. To clarify further the effects of vasculogenesis, the full model is also used to compare possible treatment strategies, including chemotherapy and antiangiogenic therapies aimed at suppressing vascularisation. This investigation highlights how, dependent on model parameter values, targeting both ECs and EPCs may be necessary in order to effectively reduce tumour vasculature and inhibit tumour growth.     

A Functional Requirement for Astroglia in Promoting Blood Vessel Development in the Early Postnatal Brain

Astroglia are a major cell type in the brain and play a key role in many aspects of brain development and function. In the adult brain, astrocytes are known to intimately ensheath blood vessels and actively coordinate local neural activity and blood flow. During development of the neural retina, blood vessel growth follows a meshwork of astrocytic processes. Several genes have also been implicated in retinal astrocytes for regulating vessel development. This suggests a role of astrocytes in promoting angiogenesis throughout the central nervous system. To determine the roles that astrocytes may play during brain angiogenesis, we employ genetic approaches to inhibit astrogliogenesis during perinatal corticogenesis and examine its effects on brain vessel development. We find that conditional deletion from glial progenitors of orc3, a gene required for DNA replication, dramatically reduces glial progenitor cell number in the subventricular zone and astrocytes in the early postnatal cerebral cortex. This, in turn, results in severe reductions in both the density and branching frequency of cortical blood vessels. Consistent with a delayed growth but not regression of vessels, we find neither significant net decreases in vessel density between different stages after normalizing for cortical expansion nor obvious apoptosis of endothelial cells in these mutants. Furthermore, concomitant with loss of astroglial interactions, we find increased endothelial cell proliferation, enlarged vessel luminal size as well as enhanced cytoskeletal gene expression in pericytes, which suggests compensatory changes in vascular cells. Lastly, we find that blood vessel morphology in mutant cortices recovers substantially at later stages, following astrogliosis. These results thus implicate a functional requirement for astroglia in promoting blood vessel growth during brain development.     

A Computational Model Predicting Disruption of Blood Vessel Development

Vascular development is a complex process regulated by dynamic biological networks that vary in topology and state across different tissues and developmental stages. Signals regulating de novo blood vessel formation (vasculogenesis) and remodeling (angiogenesis) come from a variety of biological pathways linked to endothelial cell (EC) behavior, extracellular matrix (ECM) remodeling and the local generation of chemokines and growth factors. Simulating these interactions at a systems level requires sufficient biological detail about the relevant molecular pathways and associated cellular behaviors, and tractable computational models that offset mathematical and biological complexity. Here, we describe a novel multicellular agent-based model of vasculogenesis using the CompuCell3D (http://www.compucell3d.org/) modeling environment supplemented with semi-automatic knowledgebase creation. The model incorporates vascular endothelial growth factor signals, pro- and anti-angiogenic inflammatory chemokine signals, and the plasminogen activating system of enzymes and proteases linked to ECM interactions, to simulate nascent EC organization, growth and remodeling. The model was shown to recapitulate stereotypical capillary plexus formation and structural emergence of non-coded cellular behaviors, such as a heterologous bridging phenomenon linking endothelial tip cells together during formation of polygonal endothelial cords. Molecular targets in the computational model were mapped to signatures of vascular disruption derived from in vitro chemical profiling using the EPA''s ToxCast high-throughput screening (HTS) dataset. Simulating the HTS data with the cell-agent based model of vascular development predicted adverse effects of a reference anti-angiogenic thalidomide analog, 5HPP-33, on in vitro angiogenesis with respect to both concentration-response and morphological consequences. These findings support the utility of cell agent-based models for simulating a morphogenetic series of events and for the first time demonstrate the applicability of these models for predictive toxicology.     

Bayesian Calibration,Validation and Uncertainty Quantification for Predictive Modelling of Tumour Growth: A Tutorial

In this work, we present a pedagogical tumour growth example, in which we apply calibration and validation techniques to an uncertain, Gompertzian model of tumour spheroid growth. The key contribution of this article is the discussion and application of these methods (that are not commonly employed in the field of cancer modelling) in the context of a simple model, whose deterministic analogue is widely known within the community. In the course of the example, we calibrate the model against experimental data that are subject to measurement errors, and then validate the resulting uncertain model predictions. We then analyse the sensitivity of the model predictions to the underlying measurement model. Finally, we propose an elementary learning approach for tuning a threshold parameter in the validation procedure in order to maximize predictive accuracy of our validated model.     

A Theory and a Model to Understand Glioblastoma Development Both in the Bulk and in the Microinfiltrated Brain Parenchyma

The prognosis of patients affected by glioblastoma remains dismal despite many efforts have been devoted worldwide in research and therapeutic strategies. Reasons of our failure include the fact that the patient harboring a glioblastoma always has two problems inside the brain, the bulk tumor and the parenchyma microinfiltrated; the other reason is that the tumor is able to grow dynamically adapting to the mutated conditions of its growth microenvironment. This paper tries to give an interpretation to the dynamic process of the tumor growth, from the beginning to the end of its natural history, dividing it in three phases, one pre-hypoxia and two post-hypoxia, and these are then correlated with the types of cancer stem cells (CSCs) involved. Furthermore, the paper proposes an original animal model to follow glioblastoma development in only one generation of mice, both in the bulk and in the brain parenchyma.     

Diosgenin as a Novel Alternative Therapy for Inhibition of Growth,Invasion, and Angiogenesis Abilities of Different Glioblastoma Cell Lines

Neurochemical Research - Fenugreek (Trigonella foenum-graecum) seeds and roots of wild yam (Dioscorea villosa) possess nutritional and medicinal properties and have been used for centuries in...     

A Mathematical Model of the Metabolic and Perfusion Effects on Cortical Spreading Depression

Cortical spreading depression (CSD) is a slow-moving ionic and metabolic disturbance that propagates in cortical brain tissue. In addition to massive cellular depolarizations, CSD also involves significant changes in perfusion and metabolism—aspects of CSD that had not been modeled and are important to traumatic brain injury, subarachnoid hemorrhage, stroke, and migraine. In this study, we develop a mathematical model for CSD where we focus on modeling the features essential to understanding the implications of neurovascular coupling during CSD. In our model, the sodium-potassium–ATPase, mainly responsible for ionic homeostasis and active during CSD, operates at a rate that is dependent on the supply of oxygen. The supply of oxygen is determined by modeling blood flow through a lumped vascular tree with an effective local vessel radius that is controlled by the extracellular potassium concentration. We show that during CSD, the metabolic demands of the cortex exceed the physiological limits placed on oxygen delivery, regardless of vascular constriction or dilation. However, vasoconstriction and vasodilation play important roles in the propagation of CSD and its recovery. Our model replicates the qualitative and quantitative behavior of CSD—vasoconstriction, oxygen depletion, extracellular potassium elevation, prolonged depolarization—found in experimental studies. We predict faster, longer duration CSD in vivo than in vitro due to the contribution of the vasculature. Our results also help explain some of the variability of CSD between species and even within the same animal. These results have clinical and translational implications, as they allow for more precise in vitro, in vivo, and in silico exploration of a phenomenon broadly relevant to neurological disease.     

A Heat-Shock Protein Axis Regulates VEGFR2 Proteolysis,Blood Vessel Development and Repair

Vascular endothelial growth factor A (VEGF-A) binds to the VEGFR2 receptor tyrosine kinase, regulating endothelial function, vascular physiology and angiogenesis. However, the mechanism underlying VEGFR2 turnover and degradation in this response is unclear. Here, we tested a role for heat-shock proteins in regulating the presentation of VEGFR2 to a degradative pathway. Pharmacological inhibition of HSP90 stimulated VEGFR2 degradation in primary endothelial cells and blocked VEGF-A-stimulated intracellular signaling via VEGFR2. HSP90 inhibition stimulated the formation of a VEGFR2-HSP70 complex. Clathrin-mediated VEGFR2 endocytosis is required for this HSP-linked degradative pathway for targeting VEGFR2 to the endosome-lysosome system. HSP90 perturbation selectively inhibited VEGF-A-stimulated human endothelial cell migration in vitro. A mouse femoral artery model showed that HSP90 inhibition also blocked blood vessel repair in vivo consistent with decreased endothelial regeneration. Depletion of either HSP70 or HSP90 caused defects in blood vessel formation in a transgenic zebrafish model. We conclude that perturbation of the HSP70-HSP90 heat-shock protein axis stimulates degradation of endothelial VEGFR2 and modulates VEGF-A-stimulated intracellular signaling, endothelial cell migration, blood vessel development and repair.     

Computational Modelling of Cancer Development and Growth: Modelling at Multiple Scales and Multiscale Modelling

In this paper, we present two mathematical models related to different aspects and scales of cancer growth. The first model is a stochastic spatiotemporal model of both a synthetic gene regulatory network (the example of a three-gene repressilator is given) and an actual gene regulatory network, the NF-\(\upkappa \)B pathway. The second model is a force-based individual-based model of the development of a solid avascular tumour with specific application to tumour cords, i.e. a mass of cancer cells growing around a central blood vessel. In each case, we compare our computational simulation results with experimental data. In the final discussion section, we outline how to take the work forward through the development of a multiscale model focussed at the cell level. This would incorporate key intracellular signalling pathways associated with cancer within each cell (e.g. p53–Mdm2, NF-\(\upkappa \)B) and through the use of high-performance computing be capable of simulating up to \(10^9\) cells, i.e. the tissue scale. In this way, mathematical models at multiple scales would be combined to formulate a multiscale computational model.     

Effects of Anti-Angiogenesis on Glioblastoma Growth and Migration: Model to Clinical Predictions

Glioblastoma multiforme (GBM) causes significant neurological morbidity and short survival times. Brain invasion by GBM is associated with poor prognosis. Recent clinical trials of bevacizumab in newly-diagnosed GBM found no beneficial effects on overall survival times; however, the baseline health-related quality of life and performance status were maintained longer in the bevacizumab group and the glucocorticoid requirement was lower. Here, we construct a clinical-scale model of GBM whose predictions uncover a new pattern of recurrence in 11/70 bevacizumab-treated patients. The findings support an exception to the Folkman hypothesis: GBM grows in the absence of angiogenesis by a cycle of proliferation and brain invasion that expands necrosis. Furthermore, necrosis is positively correlated with brain invasion in 26 newly-diagnosed GBM. The unintuitive results explain the unusual clinical effects of bevacizumab and suggest new hypotheses on the dynamic clinical effects of migration by active transport, a mechanism of hypoxia-driven brain invasion.     

Cell Growth and Division: III. Conditions for Balanced Exponential Growth in a Mathematical Model

In a previous paper, we proposed a model in which the volume growth rate and probability of division of a cell were assumed to be determined by the cell's age and volume. Some further mathematical implications of the model are here explored. In particular we seek properties of the growth and division functions which are required for the balanced exponential growth of a cell population. Integral equations are derived which relate the distribution of birth volumes in successive generations and in which the existence of balanced exponential growth can be treated as an eigenvalue problem. The special case in which all cells divide at the same age is treated in some detail and conditions are derived for the existence of a balanced exponential solution and for its stability or instability. The special case of growth rate proportional to cell volume is seen to have neutral stability. More generally when the division probability depends on age only and growth rate is proportional to cell volume, there is no possibility of balanced exponential growth. Some comparisons are made with experimental results. It is noted that the model permits the appearance of differentiated cells. A generalization of the model is formulated in which cells may be described by many state variables instead of just age and volume.     

The Initiation, Growth, and Characteristics of a Tissue 2

The rate and extent of initiation of callus from potato tuberdiscs depends on the concentrations of auxin and kinetin inthe medium on which they are grown. NAA is the most effectiveauxin, initiating callus at a concentration (0. 01 mg/1) anorder of magnitude lower than for IAA or 2,4-D. There is a week'slag before initiation begins with IAA or 2,4-D. In combinationwith each auxin, kinetin is inhibitory to initiation of callusand its growth on the explant. High-intensity light and lowtemperature are also inhibitory. In isolated callus subcultured so as to prevent dilution ofits accumulated auxin, the only effect of varying exogenousauxin levels is as a progressive inhibition by NAA. If thisdilution is permitted, however, 2,4-D and IAA have an optimumgrowth promoting activity at 1 mg/1, whereas the effect of NAAincreases up to 10 mg/1. The growth of the callus is affectedby agar concentration (1 per cent optimum), and is halted bypH values below 5. The callus grows on various carbon sourcesbut is dependent upon one or more components of N. Z. Amine;it also requires a number of micronutrients. A suspension culture from the callus exhibits the usual growthcurve. The phenolic content follows a pattern different fromthat of growth, protein, and RNA content, and phenolics arerapidly synthesized as active growth ceases. In contrast tothe callus tissue, the suspension culture grows at a wide rangeof pH values and buffers the medium. At low temperatures in the light, potato discs produce greencallus with a chlorophyll content corresponding to that of thediscs from which they grew. The isolated callus tissue doesnot require kinetin and produces and excretes its own cytokinin(s).The amount synthesized varies over the growth cycle.     

The Role of a Single Angiogenesis Inhibitor in the Treatment of Recurrent Glioblastoma Multiforme: A Meta-Analysis and Systematic Review

Background

Currently, the standard treatment for newly diagnosed glioblastoma multiforme (GBM) is maximal safe surgical resection followed by radiation therapy with concurrent and adjuvant temozolomide. However, disease recurs in almost all patients, and the optimal salvage treatment for recurrent GBM remains unclear. We conducted a systematic review and meta-analysis of published clinical trials to assess the efficacy and toxicities of angiogenesis inhibitors alone as salvage treatment in these patients.

Methods

Trials published between 1994 and 2015 were identified by an electronic search of public databases (MEDLINE, EMBASE, Cochrane library). Demographic data, treatment regimens, objective response rate (ORR), median progression-free survival (PFS), median overall survival (OS), 6-months PFS rate, 1-year OS and grade 3/4 toxicities were extracted. We also compared the main outcomes of interest between bevacizumab and other angiogenesis inhibitors. All analyses were performed using Comprehensive Meta Analysis software (Version 2.0).

Results

A total of 842 patients were included for analysis: 343 patients were treated with bevacizumab, 386 with other angiogenesis inhibitors and 81 with thalidomide. The pooled ORR, 6-months PFS, and 1-year OS for recurrent GBM patients receiving angiogenesis inhibitors was 20.1%, 19.5% and 29.3%, respectively. The use of single agent bevacizumab in recurrent GBM significantly improved ORR and 6-months PFS when compared to other angiogenesis inhibitors [relative risk (RR) 2.93, 95% CI 1.38–6.21; p = 0.025; and RR 2.36 95% CI 1.46–3.82; p<0.001, respectively], while no significant difference in 1-year OS was found between the two groups (p = 0.07). when compared to thalidomide, bevacizumab treatment in recurrent GBM significantly improved ORR (RR 6.8, 95%CI: 2.64–17.6, p<0.001), but not for 6-months PFS (p = 0.07) and 1-year OS (p = 0.31). As for grade 3/4 toxicities, the common toxicity was hypertension with pooled incidence of 12.1%, while high-grade thromboembolic events (2.2%), hemorrhage (5.1%) and GI perforation (2.8%) associated with angiogenesis inhibitors were relatively low.

Conclusions

In comparison with other angiogenesis inhibitors and thalidomide, the use of single agent bevacizumab as salvage treatment for recurrent GBM patients improve ORR and 6-months PFS, but not for 1-year OS.