Hybrid mathematical model of glioma progression

A dynamical model of combination therapy applied to glioma

Glioma is a human brain tumor that is very difficult to treat at an advanced stage. Studies of glioma biomarkers have shown that some markers are released into the bloodstream, so data from these markers indicate a decrease in the concentration of blood glucose and serum glucose in patients with glioma; these suggest an association between glucose and glioma. This decrease mechanism in glucose concentration can be described by the coupled ordinary differential equations of the early-stage glioma growth and interactions between glioma cells, immune cells, and glucose concentration. In this paper, we propose developing a new mathematical model to explain how glioma cells evolve and survive combination therapy between chemotherapy and oncolytic virotherapy, as an alternative to glioma treatment. In this study, three therapies were applied for analysis, that is, (1) chemotherapy, (2) virotherapy, and (3) a combination of chemotherapy and virotherapy. Virotherapy uses specialist viruses that only attack tumor cells. Based on the simulation results of the therapy carried out, we conclude that combination therapy can reduce the glioma cells significantly compared to the other two therapies. The simulation results of this combination therapy can be an alternative to glioma therapy.

Coupling solid and fluid stresses with brain tumour growth and white matter tract deformations in a neuroimaging-informed model

Brain tumours are among the deadliest types of cancer, since they display a strong ability to invade the surrounding tissues and an extensive resistance to common therapeutic treatments. It is therefore important to reproduce the heterogeneity of brain microstructure through mathematical and computational models, that can provide powerful instruments to investigate cancer progression. However, only a few models include a proper mechanical and constitutive description of brain tissue, which instead may be relevant to predict the progression of the pathology and to analyse the reorganization of healthy tissues occurring during tumour growth and, possibly, after surgical resection. Motivated by the need to enrich the description of brain cancer growth through mechanics, in this paper we present a mathematical multiphase model that explicitly includes brain hyperelasticity. We find that our mechanical description allows to evaluate the impact of the growing tumour mass on the surrounding healthy tissue, quantifying the displacements, deformations, and stresses induced by its proliferation. At the same time, the knowledge of the mechanical variables may be used to model the stress-induced inhibition of growth, as well as to properly modify the preferential directions of white matter tracts as a consequence of deformations caused by the tumour. Finally, the simulations of our model are implemented in a personalized framework, which allows to incorporate the realistic brain geometry, the patient-specific diffusion and permeability tensors reconstructed from imaging data and to modify them as a consequence of the mechanical deformation due to cancer growth.

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.

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.

Cognitive impairment and morphological changes after radiation therapy in brain tumors: A review

Life expectancy of patients treated for brain tumors has lengthened due to the therapeutic improvements. Cognitive impairment has been described following brain radiotherapy, but the mechanisms leading to this adverse event remain mostly unknown. Technical evolutions aim at enhancing the therapeutic ratio. Sparing of the healthy tissues has been improved using various approaches; however, few dose constraints have been established regarding brain structures associated with cognitive functions. The aims of this literature review are to report the main brain areas involved in cognitive adverse effects induced by radiotherapy as described in literature, to better understand brain radiosensitivity and to describe potential future improvements.

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.

MATHEMATICAL MODELING OF THE INTERPLAY BETWEEN STRESS AND ANISOTROPIC GROWTH OF AVASCULAR TUMORS

In this work, we propose a new mathematical framework for the study of the mutual interplay between anisotropic growth and stresses of an avascular tumor surrounded by an external medium. The mechanical response of the tumor is dictated by anisotropic growth, and reduces to that of an elastic, isotropic, and incompressible material when the latter is not taking place. Both proliferation and death of tumor cells are in turn assumed to depend on the stresses. We perform a parametric analysis in terms of key parameters representing growth anisotropy and the influence of stresses on tumor growth in order to determine how these effects affect tumor progression. We observe that tumor progression is enhanced when anisotropic growth is considered, and that mechanical stresses play a major role in limiting tumor growth.

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.

Non-standard radiotherapy fractionations delay the time to malignant transformation of low-grade gliomas

Grade II gliomas are slowly growing primary brain tumors that affect mostly young patients. Cytotoxic therapies (radiotherapy and/or chemotherapy) are used initially only for patients having a bad prognosis. These therapies are planned following the “maximum dose in minimum time” principle, i. e. the same schedule used for high-grade brain tumors in spite of their very different behavior. These tumors transform after a variable time into high-grade gliomas, which significantly decreases the patient’s life expectancy. In this paper we study mathematical models describing the growth of grade II gliomas in response to radiotherapy. We find that protracted metronomic fractionations, i.e. therapeutical schedules enlarging the time interval between low-dose radiotherapy fractions, may lead to a better tumor control without an increase in toxicity. Other non-standard fractionations such as protracted or hypoprotracted schemes may also be beneficial. The potential survival improvement depends on the tumor’s proliferation rate and can be even of the order of years. A conservative metronomic scheme, still being a suboptimal treatment, delays the time to malignant progression by at least one year when compared to the standard scheme.

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.

A mathematical model of pre-diagnostic glioma growth

Due to their location, the malignant gliomas of the brain in humans are very difficult to treat in advanced stages. Blood-based biomarkers for glioma are needed for more accurate evaluation of treatment response as well as early diagnosis. However, biomarker research in primary brain tumors is challenging given their relative rarity and genetic diversity. It is further complicated by variations in the permeability of the blood brain barrier that affects the amount of marker released into the bloodstream. Inspired by recent temporal data indicating a possible decrease in serum glucose levels in patients with gliomas yet to be diagnosed, we present an ordinary differential equation model to capture early stage glioma growth. The model contains glioma-glucose-immune interactions and poses a potential mechanism by which this glucose drop can be explained. We present numerical simulations, parameter sensitivity analysis, linear stability analysis and a numerical experiment whereby we show how a dormant glioma can become malignant.

Towards the Personalized Treatment of Glioblastoma: Integrating Patient-Specific Clinical Data in a Continuous Mechanical Model

Glioblastoma multiforme (GBM) is the most aggressive and malignant among brain tumors. In addition to uncontrolled proliferation and genetic instability, GBM is characterized by a diffuse infiltration, developing long protrusions that penetrate deeply along the fibers of the white matter. These features, combined with the underestimation of the invading GBM area by available imaging techniques, make a definitive treatment of GBM particularly difficult. A multidisciplinary approach combining mathematical, clinical and radiological data has the potential to foster our understanding of GBM evolution in every single patient throughout his/her oncological history, in order to target therapeutic weapons in a patient-specific manner. In this work, we propose a continuous mechanical model and we perform numerical simulations of GBM invasion combining the main mechano-biological characteristics of GBM with the micro-structural information extracted from radiological images, i.e. by elaborating patient-specific Diffusion Tensor Imaging (DTI) data. The numerical simulations highlight the influence of the different biological parameters on tumor progression and they demonstrate the fundamental importance of including anisotropic and heterogeneous patient-specific DTI data in order to obtain a more accurate prediction of GBM evolution. The results of the proposed mathematical model have the potential to provide a relevant benefit for clinicians involved in the treatment of this particularly aggressive disease and, more importantly, they might drive progress towards improving tumor control and patient’s prognosis.

Extreme protraction for low-grade gliomas: theoretical proof of concept of a novel therapeutical strategy

Grade II gliomas are slowly growing primary brain tumours that affect mostly young patients and become fatal after a variable time period. Current clinical handling includes surgery as first-line treatment. Cytotoxic therapies (radiotherapy RT or chemotherapy QT) are used initially only for patients having a bad prognosis. Therapies are administered following the 'maximum dose in minimum time' principle, which is the same schedule used for high-grade brain tumours. Using mathematical models describing the growth of these tumours in response to radiotherapy, we find that an extreme protraction therapeutical strategy, i.e. enlarging substantially the time interval between RT fractions, may lead to better tumour control. Explicit formulas are found providing the optimal spacing between doses in a very good agreement with the simulations of the full 3D mathematical model approximating the tumour spatiotemporal dynamics. This idea, although breaking the well-established paradigm, has biological meaning since, in these slowly growing tumours, it may be more favourable to treat the tumour as the tumour cells leave the quiescent compartment and move into the cell cycle.

Computer implementation of a new therapeutic model for GBM tumor

Modeling the tumor behavior in the host organ as function of time and radiation dose has been a major study in the previous decades. Here the effort in estimation of cancerous and normal cell proliferation and growth in glioblastoma multiform (GBM) tumor is presented. This paper introduces a new mathematical model in the form of differential equation of tumor growth. The model contains dose delivery amount in the treatment scheme as an input term. It also can be utilized to optimize the treatment process in order to increase the patient survival period. Gene expression programming (GEP) as a new concept is used for estimating this model. The LQ model has also been applied to GEP as an initial value, causing acceleration and improvement of the algorithm estimation. The model shows the number of the tumor and normal brain cells during the treatment process using the status of normal and cancerous cells in the initiation of treatment, the timing and amount of dose delivery to the patient, and a coefficient that describes the brain condition. A critical level is defined for normal cell when the patient's death occurs. In the end the model has been verified by clinical data obtained from previous accepted formulae and some of our experimental resources. The proposed model helps to predict tumor growth during treatment process in which further treatment processes can be controlled.

Glioma follow white matter tracts: a multiscale DTI-based model

Gliomas are a class of rarely curable tumors arising from abnormal glia cells in the human brain. The understanding of glioma spreading patterns is essential for both radiological therapy as well as surgical treatment. Diffusion tensor imaging (DTI) allows to infer the white matter fibre structure of the brain in a noninvasive way. Painter and Hillen (J Theor Biol 323:25-39, 2013) used a kinetic partial differential equation to include DTI data into a class of anisotropic diffusion models for glioma spread. Here we extend this model to explicitly include adhesion mechanisms between glioma cells and the extracellular matrix components which are associated to white matter tracts. The mathematical modelling follows the multiscale approach proposed by Kelkel and Surulescu (Math Models Methods Appl Sci 23(3), 2012). We use scaling arguments to deduce a macroscopic advection-diffusion model for this process. The tumor diffusion tensor and the tumor drift velocity depend on both, the directions of the white matter tracts as well as the binding dynamics of the adhesion molecules. The advanced computational platform DUNE enables us to accurately solve our macroscopic model. It turns out that the inclusion of cell binding dynamics on the microlevel is an important factor to explain finger-like spread of glioma.

Oedema-based model for diffuse low-grade gliomas: application to clinical cases under radiotherapy

OBJECTIVES

Diffuse low-grade gliomas are characterized by slow growth. Despite appropriate treatment, they change inexorably into more aggressive forms, jeopardizing the patient's life. Optimizing treatments, for example with the use of mathematical modelling, could help to prevent tumour regrowth and anaplastic transformation. Here, we present a model of the effect of radiotherapy on such tumours. Our objective is to explain observed delay of tumour regrowth following radiotherapy and to predict its duration.

MATERIALS AND METHODS

We have used a migration-proliferation model complemented by an equation describing appearance and draining of oedema. The model has been applied to clinical data of tumour radius over time, for a population of 28 patients.

RESULTS

We were able to show that draining of oedema accounts for regrowth delay after radiotherapy and have been able to fit the clinical data in a robust way. The model predicts strong correlation between high proliferation coefficient and low progression-free gain of lifetime, due to radiotherapy among the patients, in agreement with clinical studies. We argue that, with reasonable assumptions, it is possible to predict (precision ~20%) regrowth delay after radiotherapy and the gain of lifetime due to radiotherapy.

CONCLUSIONS

Our oedema-based model provides an early estimation of individual duration of tumour response to radiotherapy and thus, opens the door to the possibility of personalized medicine.

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.

Optimal control oriented to therapy for a free-boundary tumor growth model

This paper is devoted to present and solve some optimal control problems, oriented to therapy, for a particular model of tumor growth. In the considered systems, the state is given by one or several functions that provide information on the cell population and also the tumor shape evolution and the control is a time dependent function associated to the therapy strategy (in practice, a cytotoxic drug). We first present and analyze the model (based on PDEs) and the related optimal control problems. The solutions are expected to provide the best therapy strategies for a given set of constraints (here, the cost or objective function is a measure of the number of cells at a given final time T). We also recall some mathematical techniques for solving the related optimization problems and we illustrate the behavior of the methods and the validity of the models with several numerical experiments. In view of the results, we are able to design appropriate strategies that, at least to some extent, are confirmed by real data. Finally, we present some conclusions and indications on future work.

Dynamic density functional theory of solid tumor growth: Preliminary models

Cancer is a disease that can be seen as a complex system whose dynamics and growth result from nonlinear processes coupled across wide ranges of spatio-temporal scales. The current mathematical modeling literature addresses issues at various scales but the development of theoretical methodologies capable of bridging gaps across scales needs further study. We present a new theoretical framework based on Dynamic Density Functional Theory (DDFT) extended, for the first time, to the dynamics of living tissues by accounting for cell density correlations, different cell types, phenotypes and cell birth/death processes, in order to provide a biophysically consistent description of processes across the scales. We present an application of this approach to tumor growth.

Improving the time-machine: estimating date of birth of grade II gliomas

OBJECTIVES

Here we present a model aiming to provide an estimate of time from tumour genesis, for grade II gliomas. The model is based on a differential equation describing the diffusion-proliferation process. We have applied our model to situations where tumour diameter was shown to increase linearly with time, with characteristic diametric velocity.

MATERIALS AND METHODS

We have performed numerical simulations to analyse data, on patients with grade II gliomas and to extract information concerning time of tumour biological onset, as well as radiology and distribution of model parameters.

RESULTS AND CONCLUSIONS

We show that the estimate of tumour onset obtained from extrapolation using a constant velocity assumption, always underestimates biological tumour age, and that the correction one should add to this estimate is given roughly by 20/v (year), where v is the diametric velocity of expansion of the tumour (expressed in mm/year). Within the assumptions of the model, we have identified two types of tumour: the first corresponds to very slowly growing tumours that appear during adolescence, and the second type corresponds to slowly growing tumours that appear later, during early adulthood. That all these tumours become detectable around a mean patient age of 30 years could be interesting for formulation of strategies for early detection of tumours.

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.

Identification of intrinsic in vitro cellular mechanisms for glioma invasion

Invasion of malignant glioma is a highly complex phenomenon involving molecular and cellular processes at various spatio-temporal scales, whose precise interplay is still not fully understood. In order to identify the intrinsic cellular mechanisms of glioma invasion, we study an in vitro culture of glioma cells. By means of a computational approach, based on a cellular automaton model, we compare simulation results to the experimental data and deduce cellular mechanisms from microscopic and macroscopic observables (experimental data). For the first time, it is shown that the migration/proliferation dichotomy plays a central role in the invasion of glioma cells. Interestingly, we conclude that a diverging invasive zone is a consequence of this dichotomy. Additionally, we observe that radial persistence of glioma cells in the vicinity of dense areas accelerates the invasion process. We argue that this persistence results from a cell-cell repulsion mechanism. If glioma cell behavior is regulated through a migration/proliferation dichotomy and a self-repellent mechanism, our simulations faithfully reproduce all the experimental observations.

Mathematical model of the role of intercellular signalling in intercellular cooperation during tumorigenesis

OBJECTIVES

Intercellular cooperation has been hypothesized to enhance cell proliferation during cancer metastasis through autocrine signalling cascades and mathematical models can provide valuable insights into underlying mechanisms of metastatic tumorigenesis. Here, we present a model that incorporates signal-stimulated cell proliferation, and investigate influences of diffusion-driven heterogeneity in signal concentration on proliferation dynamics.

MATERIALS AND METHODS

Our model incorporates signal production through both autocrine and paracrine pathways, and signal diffusion and loss for a metastasizing cell population at a host site. We use the signalling pathway of IL-6 for illustration where this signalling species forms an intermediate complex with its receptor IL-6R. This in turn forms a heterodimeric complex with transmembrane protein gp130, ultimately resulting in production of downstream signals. Cell population dynamics are taken to follow a modified logistic equation for which the rate term is dependent on local IL-6 concentration.

RESULTS AND CONCLUSIONS

Our spatiotemporal model agrees closely with experimental results. The model is also able to predict two phenomena typical of metastatic tumorigenesis - host tissue preference and long periods of proliferation dormancy. It confirms that diffusivity of the signalling species in a host tissue plays a significant role during the process. Our results show that the proliferation-apoptosis balance is tipped in favour of the former for host sites that have relatively smaller signal diffusivities.

Coupled mathematical model of tumorigenesis and angiogenesis in vascular tumours

OBJECTIVES

Mathematical models are useful for studying vascular and avascular tumours, because these allow for more logical experimental design and provide valuable insights into the underlying mechanisms of their growth and development. The processes of avascular tumour growth and the development of capillary networks through tumour-induced angiogenesis have already been extensively investigated, albeit separately. Despite the clinical significance of vascular tumours, few studies have combined these approaches to develop a single comprehensive growth and development model.

MATERIALS AND METHODS

We develop a continuum-based mathematical model of vascular tumour growth. In the model, angiogenesis is initiated through the release of angiogenic growth factors (AGFs) by cells in the hypoxic regions of the tumour. The nutrient concentration within the tumour reflects the influence of capillary growth and invasion induced by AGF.

RESULTS AND CONCLUSIONS

Parametric and sensitivity studies were performed to evaluate the influence of different model parameters on tumour growth and to identify the parameters with the most influence, which include the rates of proliferation, apoptosis and necrosis, as well as the diffusion of sprout tips and the size of the region affected by angiogenesis. An optimization was performed for values of the model parameters that resulted in the best agreement with published experimental data. The resulting model solution matched the experimental data with a high degree of correlation (r = 0.85).