Non-linear model of cancer growth and metastasis: a limiting nutrient as a major determinant of tumor shape and diffusion

A New Mathematical Model for Controlling Tumor Growth Based on Microenvironment Acidity and Oxygen Concentration

Hypoxia and the pH level of the tumor microenvironment have a great impact on the treatment of tumors. Here, the tumor growth is controlled by regulating the oxygen concentration and the acidity of the tumor microenvironment by introducing a two-dimensional multiscale cellular automata model of avascular tumor growth. The spatiotemporal evolution of tumor growth and metabolic variations is modeled based on biological assumptions, physical structure, states of cells, and transition rules. Each cell is allocated to one of the following states: proliferating cancer, nonproliferating cancer, necrotic, and normal cells. According to the response of the microenvironmental conditions, each cell consumes/produces metabolic factors and updates its state based on some stochastic rules. The input parameters are compatible with cancer biology using experimental data. The effect of neighborhoods during mitosis and simulating spatial heterogeneity is studied by considering multicellular layer structure of tumor. A simple Darwinist mutation is considered by introducing a critical parameter (Nmm) that affects division probability of the proliferative tumor cells based on the microenvironmental conditions and cancer hallmarks. The results show that Nmm regulation has a significant influence on the dynamics of tumor growth, the growth fraction, necrotic fraction, and the concentration levels of the metabolic factors. The model not only is able to simulate the in vivo tumor growth quantitatively and qualitatively but also can simulate the concentration of metabolic factors, oxygen, and acidity graphically. The results show the spatial heterogeneity effects on the proliferation of cancer cells and the rest of the system. By increasing Nmm, tumor shrinkage and significant increasing in the oxygen concentration and the pH value of the tumor microenvironment are observed. The results demonstrate the model's ability, providing an essential tool for simulating different tumor evolution scenarios of a patient and reliable prediction of spatiotemporal progression of tumors for utilizing in personalized therapy.

Competition and niche construction in a model of cancer metastasis

Niche construction theory states that not only does the environment act on populations to generate Darwinian selection, but organisms reciprocally modify the environment and the sources of natural selection. Cancer cells participate in niche construction as they alter their microenvironments and create pre-metastatic niches; in fact, metastasis is a product of niche construction. Here, we present a mathematical model of niche construction and metastasis. Our model contains producers, which pay a cost to contribute to niche construction that benefits all tumor cells, and cheaters, which reap the benefits without paying the cost. We derive expressions for the conditions necessary for metastasis, showing that the establishment of a mutant lineage that promotes metastasis depends on niche construction specificity and strength of interclonal competition. We identify a tension between the arrival and invasion of metastasis-promoting mutants, where tumors composed only of cheaters remain small but are susceptible to invasion whereas larger tumors containing producers may be unable to facilitate metastasis depending on the level of niche construction specificity. Our results indicate that even if metastatic subclones arise through mutation, metastasis may be hindered by interclonal competition, providing a potential explanation for recent surprising findings that most metastases are derived from early mutants in primary tumors.

A Mathematical Model for the Onset of Avascular Tumor Growth in Response to the Loss of P53 Function

We present a mathematical model for the formation of an avascular tumor based on the loss by gene mutation of the tumor suppressor function of p53. The wild type p53 protein regulates apoptosis, cell expression of growth factor and matrix metalloproteinase, which are regulatory functions that many mutant p53 proteins do not possess. The focus is on a description of cell movement as the transport of cell population density rather than as the movement of individual cells. In contrast to earlier works on solid tumor growth, a model is proposed for the initiation of tumor growth. The central idea, taken from the mathematical theory of dynamical systems, is to view the loss of p53 function in a few cells as a small instability in a rest state for an appropriate system of differential equations describing cell movement. This instability is shown (numerically) to lead to a second, spatially inhomogeneous, solution that can be thought of as a solid tumor whose growth is nutrient diffusion limited. In this formulation, one is led to a system of nine partial differential equations. We show computationally that there can be tumor states that coexist with benign states and that are highly unstable in the sense that a slight increase in tumor size results in the tumor occupying the sample region while a slight decrease in tumor size results in its ultimate disappearance.

Development of an in silico stochastic 4D model of tumor growth with angiogenesis

PURPOSE

A stochastic computer model of tumour growth with spatial and temporal components that includes tumour angiogenesis was developed. In the current work it was used to simulate head and neck tumour growth. The model also provides the foundation for a 4D cellular radiotherapy simulation tool.

METHODS

The model, developed in Matlab, contains cell positions randomised in 3D space without overlap. Blood vessels are represented by strings of blood vessel units which branch outwards to achieve the desired tumour relative vascular volume. Hypoxic cells have an increased cell cycle time and become quiescent at oxygen tensions less than 1 mmHg. Necrotic cells are resorbed. A hierarchy of stem cells, transit cells and differentiated cells is considered along with differentiated cell loss. Model parameters include the relative vascular volume (2-10%), blood oxygenation (20-100 mmHg), distance from vessels to the onset of necrosis (80-300 μm) and probability for stem cells to undergo symmetric division (2%). Simulations were performed to observe the effects of hypoxia on tumour growth rate for head and neck cancers. Simulations were run on a supercomputer with eligible parts running in parallel on 12 cores.

RESULTS

Using biologically plausible model parameters for head and neck cancers, the tumour volume doubling time varied from 45 ± 5 days (n = 3) for well oxygenated tumours to 87 ± 5 days (n = 3) for severely hypoxic tumours.

CONCLUSIONS

The main achievements of the current model were randomised cell positions and the connected vasculature structure between the cells. These developments will also be beneficial when irradiating the simulated tumours using Monte Carlo track structure methods.

Oscillations in growth of multicellular tumour spheroids: a revisited quantitative analysis

OBJECTIVES

Multicellular tumour spheroids (MTS) provide an important tool for study of the microscopic properties of solid tumours and their responses to therapy. Thus, observation of large-scale volume oscillations in MTS, reported several years ago by two independent groups (1,2), in our opinion represent a remarkable discovery, particularly if this could promote careful investigation of the possible occurrence of volume oscillations of tumours 'in vivo'.

MATERIALS AND METHODS

Because of high background noise, quantitative analysis of properties of observed oscillations has not been possible in previous studies. Such an analysis can be now performed, thanks to a recently proposed approach, based on formalism of phenomenological universalities (PUN).

RESULTS

Results have provided unambiguous confirmation of the existence of MTS volume oscillations, and quantitative evaluation of their properties, for two tumour cell lines. Proof is based not only on quality of fitting of the experimental datasets, but also on determination of well-defined values of frequency and amplitude of the oscillations for each line investigated, which would not be consistent with random fluctuation.

CONCLUSIONS

Biological mechanisms, which can be directly responsible for observed oscillations, are proposed, which relates also to recent work on related topics. Further investigations, both at experimental and at modelling levels, are also suggested. Finally, from a methodological point of view, results obtained represent further confirmation of applicability and usefulness of the PUN approach.

Hematopoietic- and neurologic-expressed sequence 1 expression in the murine GL261 and high-grade human gliomas

The hematopoietic- and neurologic-expressed sequence 1 (Hn1) gene encodes a highly conserved protein that is expressed in developing and regenerating tissues. In this study, Hn1 expression was evaluated in human and murine malignant gliomas. Hn1 mRNA and protein were detected in the murine GL261 glioma cell line and in GL261 brain tumors in vivo. HN1 is also expressed in human U118MG and U87MG cell lines. Evaluation of human brain tumors using an anti-Hn1 polyclonal antibody detected strong immunoreactivity in high-grade (WHO III and IV) malignant gliomas. The rate of GL261 cell proliferation in vitro was unaltered by Hn1 depletion using an anti-Hn1 siRNA. However, tumors established from Hn1-depleted GL261 cells formed significantly smaller volumes than those established from control-treated cells. These data suggest a role for Hn1 in the biology of malignant brain tumors.

Beyond the oncogene paradigm: understanding complexity in cancerogenesis

In the past decades, an enormous amount of precious information has been collected about molecular and genetic characteristics of cancer. This knowledge is mainly based on a reductionistic approach, meanwhile cancer is widely recognized to be a 'system biology disease'. The behavior of complex physiological processes cannot be understood simply by knowing how the parts work in isolation. There is not solely a matter how to integrate all available knowledge in such a way that we can still deal with complexity, but we must be aware that a deeply transformation of the currently accepted oncologic paradigm is urgently needed. We have to think in terms of biological networks: understanding of complex functions may in fact be impossible without taking into consideration influences (rules and constraints) outside of the genome. Systems Biology involves connecting experimental unsupervised multivariate data to mathematical and computational approach than can simulate biologic systems for hypothesis testing or that can account for what it is not known from high-throughput data sets. Metabolomics could establish the requested link between genotype and phenotype, providing informations that ensure an integrated understanding of pathogenic mechanisms and metabolic phenotypes and provide a screening tool for new targeted drug.

The biocompatibility of quantum dot probes used for the targeted imaging of hepatocellular carcinoma metastasis

Semiconductor quantum dots (QDs) have several photo-physical advantages over organic dyes making them good markers in biomedical application. We used CdSe/ZnS QDs with maximum emission wavelength of 590nm (QD590) linked to alpha-fetoprotein (AFP) monoclonal antibody (Ab) to detect AFP in cytoplasm of human hepatocellular carcinoma (HCC) cell line HCCLM6. For the in vivo studies, we used QD-AFP-Ab probes for targeted imaging of human HCC xenograft growing in nude mice by injecting them into the tail vein. In addition, the cytotoxicity in vitro, the acute toxicity in vivo, the hemodynamics and tissue distribution of these probes were also investigated. The results in vitro and in vivo indicate that our QD-based probes have good stability, specificity and biocompatibility for ultrasensitive fluorescence imaging of molecular targets in our liver cancer model system.

Dynamics of metastasis suppressor gene inactivation

For most cancer cell types, the acquisition of metastatic ability leads to clinically incurable disease. Twelve metastasis suppressor genes (MSGs) have been identified that reduce the metastatic propensity of cancer cells. If these genes are inactivated in both alleles, metastatic ability is promoted. Here, we develop a mathematical model of the dynamics of MSG inactivation and calculate the expected number of metastases formed by a tumor. We analyse the effects of increased mutation rates and different fitness values of cells with one or two inactivated alleles on the ability of a tumor to form metastases. We find that mutations that are negatively selected in the main tumor are unlikely to be responsible for the majority of metastases produced by a tumor. Most metastases-causing mutations will be present in all (or most) cells in the main tumor.

Stochastic dynamics of metastasis formation

Tumor metastasis accounts for the majority of deaths in cancer patients. The metastatic behavior of cancer cells is promoted by mutations in many genes, including activation of oncogenes such as RAS and MYC. Here, we develop a mathematical framework to analyse the dynamics of mutations enabling cells to metastasize. We consider situations in which one mutation is necessary to confer metastatic ability to the cell. We study different population sizes of the main tumor and different somatic fitness values of metastatic cells. We compare mutations that are positively selected in the main tumor with those that are neutral or negatively selected, but faster at forming metastases. We study whether metastatic potential is the property of all (or the majority of) cells in the main tumor or only the property of a small subset. Our theory shows how to calculate the expected number of metastases that are formed by a tumor.

Retrograde regulation due to mitochondrial dysfunction may be an important mechanism for carcinogenesis

Mitochondrial dysfunction has crucial importance in carcinogenesis. Due to several reasons, it may lead to insufficiency in the electron transport chain, which activates a series of cytosolic proteins. These proteins are transported to the nucleus and promote the activation of genes leading to intracellular diverse metabolic, regulatory, signalization and stress-related pathways. Retrograde regulation is the general term for mitochondrial signaling, and is broadly defined as cellular responses to alterations in functional state of mitochondria. This signaling pathway is triggered by mitochondrial dysfunction. The retrograde response is not a simple On-Off switch, but rather it responds in a continuous manner to the changing metabolic needs of the cell. Communication between mitochondria and the nucleus is important for a variety of cellular processes such as carbohydrate and nitrogen metabolism, cell cycle and proliferation, and cell growth and morphogenesis. As a result of retrograde regulation, the cell, actually a component of the multicellular organism, transforms to a unicellular lifestyle and initiates a developing course, independent of the systemic structure. This transformed cell runs metabolic regulations effectively in order to utilize all energy depots, mainly the adipose tissue of the multicellular organism. The most important one is the active utilization of glyoxylate cycle, through which the malign cells supply glucose from fats. Continuously acting glycolysis and gluconeogenesis, fatty acid oxidation and de novo lipogenesis constitute futile cycles. This in turn causes cachexia by maintaining the organism in constant negative energy balance. Mitochondria-to-nucleus stress signaling activates some of the genes implicated in tumor progression and tumor cell metastasis. Retrograde regulation also renders the cell more resistant to apoptosis. It is becoming clearer which genes control the retrograde response in human cells. Most probably, MYC is one of the transcription factors necessary for this response.

Effects of anatomical constraints on tumor growth

Competition for available nutrients and the presence of anatomical barriers are major determinants of tumor growth in vivo. We extend a model recently proposed to simulate the growth of neoplasms in real tissues to include geometrical constraints mimicking pressure effects on the tumor surface induced by the presence of rigid or semirigid structures. Different tissues have different diffusivities for nutrients and cells. Despite the simplicity of the approach, based on a few inherently local mechanisms, the numerical results agree qualitatively with clinical data (computed tomography scans of neoplasms) for the larynx and the oral cavity.

Analysis of a "phase transition" from tumor growth to latency

A mathematical model, based on the local interaction simulation approach, is developed in order to allow simulations of the spatiotemporal evolution of neoplasies. The model consists of a set of rules, which govern the interaction of cancerous cells among themselves and in competition with other cell populations for the acquisition of essential nutrients. As a result of small variations in the basic parameters, it leads to four different outcomes: indefinite growth, metastasis, latency, and complete regression. In the present contribution a detailed analysis of the dormant phase is carried on and the critical parameters for the transition to other phases are computed. Interesting chaotic behaviors can also be observed, with different attractors in the parameters space. Interest in the latency phase has been aroused by therapeutical strategies aiming to reduce a growing tumor to dormancy. The effect of such strategies may be simulated with our approach.