Growth factors and growth control of heterogeneous cell populations

Interactions between the immune system and cancer: a brief review of non-spatial mathematical models

We briefly review spatially homogeneous mechanistic mathematical models describing the interactions between a malignant tumor and the immune system. We begin with the simplest (single equation) models for tumor growth and proceed to consider greater immunological detail (and correspondingly more equations) in steps. This approach allows us to clarify the necessity for expanding the complexity of models in order to capture the biological mechanisms we wish to understand. We conclude by discussing some unsolved problems in the mathematical modeling of cancer-immune system interactions.

Cancer dissemination: a consequence of limited carrying capacity?

Assuming that there is feedback between an expanding cancer system and its organ-typical microenvironment, we argue here that such local tumor growth is initially guided by co-existence rather than competition with the surrounding tissue. We then present a novel concept that understands cancer dissemination as a biological mechanism to evade the specific carrying capacity limit of its host organ. This conceptual framework allows us to relate the tumor system's volumetric growth rate to the host organ's functionality-conveying composite infrastructure, and, intriguingly, already provides useful insights into several clinical findings.

Modeling positive regulatory feedbacks in cell–cell interactions

Our current understanding of molecular mechanisms of cellular regulation still does not support quantitative predictions of the overall growth kinetics of normal or malignant tissues. However, discernment of the role of growth-factor mediated cell-cell communication in tissue kinetics is possible by the use of simple mathematical models. Here we discuss the design and use of mathematical models in quantifying the contribution of autocrine and paracrine (i.e., humoral) interactions to the kinetics of tissue growth. We present models that include a humorally mediated regulatory feedback among cells built into phenomenological mathematical models of growth. Application of these models to data exemplifies the finite contributions of positive feedback in cell-cell interactions to the overall tissue growth. In addition, we propose a perturbation approach to allow separation of cell-cell interactions dependent on the perturbing agent (such as hormone antagonists in hormone-dependent tissues) from cell-cell interactions independent of it.

Positive feedback and angiogenesis in tumor growth control

In vivo tumor growth data from experiments performed in our laboratory suggest that basic fibroblast growth factor (bFGF) and vascular endothelial growth factor (VEGF) are angiogenic signals emerging from an up-regulated genetic message in the proliferating rim of a solid tumor in response to tumor-wide hypoxia. If these signals are generated in response to unfavorable environmental conditions, i.e. a decrease in oxygen tension, then the tumor may play an active role in manipulating its own environment. We have idealized this type of adaptive behavior in our mathematical model via a parameter which represents the carrying capacity of the host for the tumor. If that model parameter is held constant, then environmental control is limited to tumor shape and mitogenic signal processing. However, if we assume that the response of the local stroma to these signals is an increase in the host's ability to support an ever larger tumor, then our models describe a positive feedback control system. In this paper, we generalize our previous results to a model including a carrying capacity which depends on the size of the proliferating compartment in the tumor. Specific functional forms for the carrying capacity are discussed. Stability criteria of the system and steady state conditions for these candidate functions are analyzed. The dynamics needed to generate stable tumor growth, including countervailing negative feedback signals, are discussed in detail with respect to both their mathematical and biological properties.

A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competitive environment

A competition model describing tumor-normal cell interaction with the added effects of periodically pulsed chemotherapy is discussed. The model describes parameter conditions needed to prevent relapse following attempts to remove the tumor or tumor metastasis. The effects of resistant tumor subpopulations are also investigated and recurrence prevention strategies are explored.

Secretion rates and levels of vascular endothelial growth factor in clone A or HCT-8 human colon tumour cells as a function of oxygen concentration

Molecular and in situ hybridization studies have shown, in a number of cell types, that under hypoxic conditions, vascular endothelial growth factor (VEGF) mRNA expression is up-regulated and VEGF protein is concomitantly increased. To establish a quantitative relationship between VEGF protein levels and oxygenation, we exposed exponentially growing clone A or HCT-8 human colon tumour cells in vitro (22 h at 37 degrees C) to oxygen concentrations from 21% (air mixture) to 0.01%. Protein levels in cells and medium were then assayed using an enzyme-linked immunoabsorbent assay (ELISA). Intracellular levels of VEGF in clone A or HCT-8 cells exposed to either air (21% O2) or the 0.01% O2 mixture respectively increased from about 73 to 1270, and 1.5 to 1180 pg/10(6) cells (about 17- and 80-fold increases). The shapes of the response curves (log of the intracellular VEGF concentrations v. log oxygen concentration) for both cell types were sigmoidal. However, intracellular VEGF levels in HCT-8 cells were always less than that of clone A cells until levels of about 0.3 to 0.1% O2 were reached. Levels of VEGF in the supernatant were also increased after the 22 h hypoxic exposures. Because cell proliferation and clonogenicity were also measured, it was possible to estimate the secretion rates of VEGF for both cell lines as a function of oxygen percentage. For clone A cells, the secretion rate (pg/10(6) cells/h) in 21% O2 was 62.5. This rate increased to 428.8 pg/10(6) cells/h at 0.01% O2, a 7-fold increase. For HCT-8 cells, levels in the medium at 21% O2 were too low to be measured by ELISA. However, between 10% and 0.01% O2, secretion rates increased from 5.0 to 376.0 pg/10(6) cells/h, a 75-fold increase. Therefore, at very low O2 levels, VEGF secretion rates were similar in the two cell lines. We propose that the different VEGF responses of clone A and HCT-8 colon tumour cells to hypoxic stress in vitro are related to the in vivo observation that the respective hypoxic percentages of solid neoplasms originating from these cell lines are markedly different (i.e. about 3 versus 80%) at equivalent volumes of 750 mm3.

Comparison of basic fibroblast growth factor levels in clone A human colon cancer cells in vitro with levels in xenografted tumours

We measured levels of basic fibroblast growth factor (FGF-2) in human colon cancer cells (clone A) in vitro and in xenografted solid tumours using a commercial enzyme-linked immunoassay. In Vitro, levels in unfed plateau phase or exponentially growing cells were low, averaging respectively about 2 and 8 pg 10(-6) cells. However, when solid tumours (average volumes 787 mm3) were cut into halves and either enzymatically disaggregated to obtain a cellular fraction or extracted in toto, levels were much higher. In the cellular fraction, values averaged 110 pg 10(-6) cells, while in whole tumour extracts, average values were 24 pg mg-1 tumour tissue. These results indicate that growth factor levels in solid neoplasms may differ markedly from those predicted from in vitro measurements. We hypothesise that the apparent increase in FGF-2 levels in vivo results primarily from the presence of a significant fraction of host cells (in particular, macrophages, which may contain high levels of FGF-2) within xenografted clone A neoplasms.

Levels of selected growth factors in viable and necrotic regions of xenografted HCT-8 human colon tumours

Xenografted tumours were produced in nude mice by injection of HCT-8 human colon tumour cells. At average volumes of about 750 mm3, animals were injected with fast green vital dye, and 20 min later, tumours were excised and dissected into viable (stained) and necrotic portions (unstained). Viable and necrotic regions were then examined for cell yields, colony forming efficiencies, and levels of basic fibroblast growth factor (FGF-2), transforming growth factors-beta 1 and -alpha (TGF-beta 1, TGF-alpha), platelet derived growth factor (PDGF), and vascular endothelial growth factor (VEGF) using enzyme-linked immunoassay (ELISA) procedures. Levels in the viable and necrotic regions were compared to levels in unseparated tumours. The average extent of necrosis in HCT-8 tumours of this size was 64%. The data for cell yields, colony forming efficiencies FGF-2, VEGF, TGF-beta 1 and TGF-alpha indicated that values determined in the unseparated tumours could be understood on the basis of the weighted average between viable and necrotic tissue, with the higher values occurring in the viable tissue. Low levels of FGF-2 and VEGF were found in the necrotic portions of the tumour while no measurable levels of TGF-beta 1 and TGF-alpha could be determined. PDGF levels were, however, equivalent in both the viable and necrotic regions indicating that necrotic tissue could be an important reservoir for this growth factor.

Interlocking triads of growth control in tumors

A pair of growth control triads are used to describe coincident tumor growth and liver regeneration after partial hepatectomy. The models are extensions of previous growth control models which describe tumor growth in an unperturbed host (Michelson and Leith, 1991, Bull. math. Biol. 53, 639-656; idem, 1992, Proceedings of the Third International Conference on Communications and Control, Vol. 2, pp. 481-490; idem, 1992 Bull. math. Biol. 55, 993-1011; idem, J. theor. Biol. 169, 327-338). The linkage between the two triads depends upon systemic signals carried by soluble factors, and mathematical descriptors based upon biological first principals are proposed. The sources of the growth factors, their targets and the processing of their signals are investigated. Analyses of equilibrium in the constant coefficients case and simulated growth curves for the dynamic system are presented, and the effects of growth factor-induced mitogenesis and angiogenesis are discussed in particular. A case is made for early and late responses in the coupled control system. The biology of the signal processing paradigm is placed within a new theoretical context and discussed with regard to tumor adaptation, liver differentiation and the development of a tumor hypoxic fraction.