Ab initio phenomenological simulation of the growth of large tumor cell populations

In a previous paper we have introduced a phenomenological model of cell metabolism and of the cell cycle to simulate the behavior of large tumor cell populations (Chignola and Milotti 2005 Phys. Biol. 2 8). Here we describe a refined and extended version of the model that includes some of the complex interactions between cells and their surrounding environment. The present version takes into consideration several additional energy-consuming biochemical pathways such as protein and DNA synthesis, the tuning of extracellular pH and of the cell membrane potential. The control of the cell cycle, which was previously modeled by means of ad hoc thresholds, has been directly addressed here by considering checkpoints from proteins that act as targets for phosphorylation on multiple sites. As simulated cells grow, they can now modify the chemical composition of the surrounding environment which in turn acts as a feedback mechanism to tune cell metabolism and hence cell proliferation: in this way we obtain growth curves that match quite well those observed in vitro with human leukemia cell lines. The model is strongly constrained and returns results that can be directly compared with actual experiments, because it uses parameter values in narrow ranges estimated from experimental data, and in perspective we hope to utilize it to develop in silico studies of the growth of very large tumor cell populations (10(6) cells or more) and to support experimental research. In particular, the program is used here to make predictions on the behavior of cells grown in a glucose-poor medium: these predictions are confirmed by experimental observation.     

Recent progress in modelling and simulation of three-dimensional tumor growth and treatment

This paper illustrates how to apply methods of systems analysis, control theory and simulation to the field of biology and medicine. For this purpose normal and abnormal cell growth has been modelled at different levels. It was possible to simulate three-dimensional tumor growth and different kinds of treatment. The paper shows how tumor treatment may be optimized in the long run using computer simulation experiments as a powerful new tool prior to clinical therapy.     

Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching

We develop a thermodynamically consistent mixture model for avascular solid tumor growth which takes into account the effects of cell-to-cell adhesion, and taxis inducing chemical and molecular species. The mixture model is well-posed and the governing equations are of Cahn-Hilliard type. When there are only two phases, our asymptotic analysis shows that earlier single-phase models may be recovered as limiting cases of a two-phase model. To solve the governing equations, we develop a numerical algorithm based on an adaptive Cartesian block-structured mesh refinement scheme. A centered-difference approximation is used for the space discretization so that the scheme is second order accurate in space. An implicit discretization in time is used which results in nonlinear equations at implicit time levels. We further employ a gradient stable discretization scheme so that the nonlinear equations are solvable for very large time steps. To solve those equations we use a nonlinear multilevel/multigrid method which is of an optimal order O(N) where N is the number of grid points. Spherically symmetric and fully two dimensional nonlinear numerical simulations are performed. We investigate tumor evolution in nutrient-rich and nutrient-poor tissues. A number of important results have been uncovered. For example, we demonstrate that the tumor may suffer from taxis-driven fingering instabilities which are most dramatic when cell proliferation is low, as predicted by linear stability theory. This is also observed in experiments. This work shows that taxis may play a role in tumor invasion and that when nutrient plays the role of a chemoattractant, the diffusional instability is exacerbated by nutrient gradients. Accordingly, we believe this model is capable of describing complex invasive patterns observed in experiments.     

Nonlinear simulation of tumor necrosis,neo-vascularization and tissue invasion via an adaptive finite-element/level-set method

We present a multi-scale computer simulator of cancer progression at the tumoral level, from avascular stage growth, through the transition from avascular to vascular growth (neo-vascularization), and into the later stages of growth and invasion of normal tissue. We use continuum scale reaction-diffusion equations for the growth component of the model, and a combined continuum-discrete model for the angiogenesis component. We use the level set method for describing complex topological changes observed during growth such as tumor splitting and reconnection, and capture of healthy tissue inside the tumor. We use an adaptive, unstructured finite element mesh that allows for finely resolving important regions of the computational domain such as the necrotic rim, the tumor interface and around the capillary sprouts. We present full nonlinear, two-dimensional simulations, showing the potential of our virtual cancer simulator. We use microphysical parameters characterizing malignant glioma cells, obtained from recent in vitro experiments from our lab and from clinical data, and provide insight into the mechanisms leading to infiltration of the brain by the cancer cells. The results indicate that diffusional instability of tumor mass growth and the complex interplay with the developing neo-vasculature may be powerful mechanisms for tissue invasion.     

Modeling tumor growth.

The meaning and limitations of certain mathematical models of tumor growth are discussed, and some new derivations of the existing models are given. A theoretical justification for Gompertz's law of growth for tumors is presented. An age-dependent Von Bertalanffy's equation and diffusion models are introduced, and existence and uniqueness problems are addressed.     

Nonlinear stochastic modeling of aphid population growth

This paper develops a stochastic population size model for the black-margined pecan aphid. Prajneshu [Prajneshu, A nonlinear statistical model for aphid population growth. J. Indian Soc. Agric. Statist. 51 (1998), p. 73] proposes a novel nonlinear deterministic model for aphid abundance. The per capita death rate in his model is proportional to the cumulative population size, and the solution is a symmetric analytical function. This paper fits Prajneshu's deterministic model to data. An analogous stochastic model, in which both the current and the cumulative aphid counts are state variables, is then proposed. The bivariate solution of the model, with parameter values suggested by the data, is obtained by solving a large system of Kolmogorov equations. Differential equations are derived for the first and second order cumulants, and moment closure approximations are obtained for the means and variances by solving the set of only five equations. These approximations, which are simple for ecologists to calculate, are shown to give accurate predictions of the two endpoints of applied interest, namely (1) the peak aphid count and (2) the final cumulative aphid count.     

A numerical simulation of plant growth

A simulation model has been formulated to calculate the growth of plants as a function of the environmental factors: light intensity, temperature and water. The environmental factors are derived from conventional meteorological variables. A plant is defined in terms of its physiological requirements for light, heat and water. Observed plant responses are used to develop functional relationships between the rate of biomass accumulation and the environmental factors. Simulations of corn growth based upon numerical integrations of observed meteorological variables are compared with measured crop yields.Presented at the Seventh International Biometeorological Congress, 17–23 August 1975, College Park, Maryland, USA.     

Organ-specific pancreatic tumor growth properties and tumor immunity

We established a model of orthotopic injection of a syngeneic pancreatic tumor cell line in C57BL/6 mice and evaluated the effects of organ site on induction of immunity to a tumor-specific antigen, MUC1. Mice were challenged with a syngeneic pancreatic adenocarcinoma cell line that expressed MUC1 (Panc02-MUC1) by orthotopic injection into the pancreas, or by subcutaneous injection. Tumor cells injected into the pancreas grew much faster than those injected subcutaneously. Mice challenged subcutaneously with Panc02-MUC1 rejected tumors or developed slowly growing tumors that were negative for MUC1 expression. In contrast, mice challenged orthotopically into the pancreas developed progressive tumors that were positive for MUC1 expression. Sera from mice that rejected Panc02-MUC1 (tumor-immune mice) showed no detectable IgG1 and IgM titers against the MUC1 tandem-repeat peptide, whereas mice with progressive tumor growth had significant titers of IgG1 and IgM specific for MUC1. This suggests that the humoral immune response was ineffective in mediating tumor rejection. The results show that the growth properties and immunological rejection of pancreatic tumors is affected by the organ site at which the tumor grows. Received: 25 April 1998 / Accepted: 7 October 1998     

Polypeptide growth factors in embryogenesis and tumor growth

Polypeptide growth factors, which belong to different families (epidermal growth factors, insulin-like growth factors, fibroblast growth factors, transforming growth factors-beta, and some others), were characterized regarding their specific role in embryogenesis and tumor growth. Differences and parallels of the functioning of growth factors in these processes have been noted. Potential significance of the described characteristics of growth factors for directed modulation of embryogenesis and tumor growth is discussed.     

Nonlinear age-dependent models for prediction of population growth

In this paper, some new algorithms are proposed to estimate parameter functions in nonlinear age-dependent population models by practical data. These algorithms together with a numerical method are applied to compute the human population using the data provided by the United Nations Demographic Yearbook.     

DNA polymerases during tumor growth

DNA polymerase activity was studied as a function of stage of tumor growth and correlated with DNA synthesis measured by 3H-TdR uptake. Considerable variations in DNA synthesis activity occur at different growth stages and following host death. DNA alpha-polymerase activity did vary with growth stage in the ascites tumor. However, it did not have a clear correlation with DNA synthesis or with tumor growth. No striking fall in DNA polymerase enzyme levels occurred as the ascites tumor reached stationary phase in contrast to reports in some cell culture systems. A decrease occurred with advanced tumor stage and after host death. DNA beta-polymerase activity did not change with tumor growth stage.     

The universal dynamics of tumor growth

Scaling techniques were used to analyze the fractal nature of colonies of 15 cell lines growing in vitro as well as of 16 types of tumor developing in vivo. All cell colonies were found to exhibit exactly the same growth dynamics-which correspond to the molecular beam epitaxy (MBE) universality class. MBE dynamics are characterized by 1), a linear growth rate, 2), the constraint of cell proliferation to the colony/tumor border, and 3), surface diffusion of cells at the growing edge. These characteristics were experimentally verified in the studied colonies. That these should show MBE dynamics is in strong contrast with the currently established concept of tumor growth: the kinetics of this type of proliferation rules out exponential or Gompertzian growth. Rather, a clear linear growth regime is followed. The importance of new cell movements-cell diffusion at the tumor border-lies in the fact that tumor growth must be conceived as a competition for space between the tumor and the host, and not for nutrients or other factors. Strong experimental evidence is presented for 16 types of tumor, the growth of which cell surface diffusion may be the main mechanism responsible in vivo. These results explain most of the clinical and biological features of colonies and tumors, offer new theoretical frameworks, and challenge the wisdom of some current clinical strategies.     

Fungal polysaccharopeptide inhibits tumor angiogenesis and tumor growth in mice

Angiogenesis is crucial to tumor growth and metastasis, and interruption of this process is a prime avenue for therapeutic intervention of tumor proliferation. The present study has made use of the S180 tumor-bearing mouse model to investigate the polysaccharopeptide, PSP, isolated from the edible mushroom Coriolus versicolor, a herbal medicine known for its anti-angiogenesis properties. Quantitative analysis of microcorrosion casting of the tumor tissue showed more angiogenic features such as dense sinusoids and hot spots, in control (untreated) than in PSP-treated animals. Immunostaining of tumor tissues with antibody against the endothelial cell marker (Factor VIII) demonstrated a positive correlation in that both the vascular density and tumor weight were lower in mice treated with PSP. Morphometric analysis of corrosion casts revealed that, even though the total amount of new vessel production was reduced, the basic tumor type-specific vascular architecture was retained. However, the expression of vascular endothelial cell growth factor (VEGF) in these tumors was suppressed. In conclusion, anti-angiogenesis should be one of the pathways through which PSP mediated its anti-tumor activity.