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 simulation of tumor growth

 We study solid tumor (carcinoma) growth in the nonlinear regime using boundary-integral simulations. The tumor core is nonnecrotic and no inhibitor chemical species are present. A new formulation of the classical models [18,24,8,3] is developed and it is demonstrated that tumor evolution is described by a reduced set of two dimensionless parameters and is qualitatively unaffected by the number of spatial dimensions. One parameter describes the relative rate of mitosis to the relaxation mechanisms (cell mobility and cell-to-cell adhesion). The other describes the balance between apoptosis (programmed cell-death) and mitosis. Both parameters also include the effect of vascularization. Our analysis and nonlinear simulations reveal that the two new dimensionless groups uniquely subdivide tumor growth into three regimes associated with increasing degrees of vascularization: low (diffusion dominated, e.g., in vitro), moderate and high vascularization, that correspond to the regimes observed in vivo. We demonstrate that critical conditions exist for which the tumor evolves to nontrivial dormant states or grows self-similarly (i.e., shape invariant) in the first two regimes. This leads to the possibility of shape control and of controlling the release of tumor angiogenic factors by restricting the tumor volume-to-surface-area ratio. Away from these critical conditions, evolution may be unstable leading to invasive fingering into the external tissues and to topological transitions such as tumor breakup and reconnection. Interestingly we find that for highly vascularized tumors, while they grow unbounded, their shape always stays compact and invasive fingering does not occur. This is in agreement with recent experimental observations [30] of in vivo tumor growth, and suggests that the invasive growth of highly-vascularized tumors is associated to vascular and elastic anisotropies, which are not included in the model studied here. Received: 1 May 2002 / Revised version: 26 August 2002 / Published online: 18 December 2002 Current address: Department of Biomedical Engineering, and Department of Mathematics, University of California at Irvine, Irvine CA 92697. e-mail: cristini@math.uci.edu; lowengrb@math.umn.edu. Key words or phrases: Tumor growth – Linear stability analysis – Self-similarity – Boundary-integral simulations     

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.     

Studies of the oxysterol inhibition of tumor cell growth

The oxysterols 3 beta-hydroxy-5 alpha-cholest-8-en-11-one, 3 beta-hydroxy-5 alpha-cholest-8-en-7-one, 3 beta-hydroxy-5 alpha-cholest-8(14)-en-7-one, 3 beta-hydroxy-4,4'-dimethylcholest-5-ene-7 one, 4,4'-dimethylcholest-5-ene-3 beta, 7 alpha-diol, 4,4'-dimethylcholest-5-ene-3 beta, 7 beta-diol, lanost-8-ene-3 beta, 25-diol, 25-hydroxylanost-8-en-3-one, 9 alpha, 11 alpha-epoxy-5 alpha-cholest-7-en-3 beta-ol, 3 beta-hydroxycholest-5 alpha-en-22-one, and 3 beta-hydroxycholest-5-en-22-one oxime were evaluated with respect to their ability to inhibit cell growth. All of the sterols were found to possess cytotoxicity when incubated with hepatoma (HTC) and lymphoma (RDM-4) cells in culture at 10-30 microM concentrations.     

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.     

Modeling growth of a heterogeneous tumor

It has long been recognized that the growth of tumor population depends on the initial age distribution of the cells in the tumor and the age-dependent cellular birth rate. Deterministic dual-cell models have been available for sometime; these models take into account the effects of the resultant cell heterogeneity. Nevertheless, these models ignore various variables significantly affecting the growth, such as those characterizing the cells' inherent properties and environmental factors. Uncertainties, or fluctuations, arise when the growth is simulated with the models. Stochastic analysis of these fluctuations is the focus of the current work.Two types of cells are visualized to proliferate separately and to transform mutually during the process. The master equations of the system have been formulated through probabilistic population balance around a particular state by considering all mutually exclusive events. The governing equations for the means, variances, and covariance of the random variables have been derived through the system-size expansion of these nonlinear master equations. The stochastic pathways of the two different types of cells have been numerically simulated by the algorithm derived from the master equation for two different physical situations, one without and, the other, with the chemotherapeutic treatment. The results of the current study illuminate the significance of stochastically modeling the responses of the tumor to a variety of medicinal treatments: The coefficient of variation of the malignant cells' population magnifies with time under chemotherapeutic regimens. Consequently, the impact of the uncertainties in the exact number of malignant cells as expressed by this coefficient of variation is highly unpredictable. For example, it becomes increasingly uncertain if or how fast these cells will reactivate to become a full-blown carcinogenic tumor after treatment.     

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.     

Inhibition of the MtTF4 tumor growth by dexamethasone

It is known that estradiol, but not progesterone or dihydrotestosterone, slows down the growth of the MtTF4 tumor. In the present paper, it is shown that: (1) this tumor contains glucocorticoid receptors, (2) its growth is also inhibited by treatment with dexamethasone (Dex), and (3) the growth rate of a cell line and several clones established from the tumor is negatively controlled by Dex 10(-7) M in culture medium containing 10% gelding serum. Unlike estradiol, Dex does not induce cell hypertrophy. This work suggests that the inhibition of the MtTF4 tumor growth by Dex may be due in part to a direct action on tumor cells and, taking into consideration previous reports, it allows us to forward the hypothesis that both Dex and estradiol inhibit MtTF4 tumor growth in two different ways.     

Methemoglobin contributes to the growth of human tumor cells

Methemoglobin (metHb) has been reported to be present in areas surrounding solid tumors. The effects of human metHb on the growth of one human hepatocellular carcinoma cell line and one human glioma cell line that simply replicate in Ham's nutrient mixture F12 (F12) were investigated. MetHb, depending on its concentration, stimulated or inhibited the in vitro growth of both cancer cell lines. The stimulatory or inhibitory effect was due to the release of hemin from metHb, which was recognized by its characteristic light absorption spectrum. The possibility of metHb or hemin acting initially through a 3', 5'-cyclic guanosine monophosphate- (cGMP-) or prostaglandin E2- (PGE2-) mediated pathway to enhance cell growth was excluded. Ferric iron derived from the catabolic degradation of hemin increased cell growth, whereas biliverdin (Bv) and its reduction product, bilirubin (Br), decreased cell growth. Hemoglobin oxidized to metHb in conditions found in tumors showing neovascularization and hemorrhage may contribute significantly to increased proliferation of cancerous cells.