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.     

Modeling of self-organized avascular tumor growth with a hybrid cellular automaton

Pattern formation in multicellular spheroids is addressed with a hybrid lattice-gas cellular automaton model. Multicellular spheroids serve as experimental model system for the study of avascular tumor growth. Typically, multicellular spheroids consist of a necrotic core surrounded by rings of quiescent and proliferating tumor cells, respectively. Furthermore, after an initial exponential growth phase further spheroid growth is significantly slowed down even if further nutrient is supplied. The cellular automaton model explicitly takes into account mitosis, apoptosis and necrosis as well as nutrient consumption and a diffusible signal that is emitted by cells becoming necrotic. All cells follow identical interaction rules. The necrotic signal induces a chemotactic migration of tumor cells towards maximal signal concentrations. Starting from a small number of tumor cells automaton simulations exhibit the self-organized formation of a layered structure consisting of a necrotic core, a ring of quiescent tumor cells and a thin outer ring of proliferating tumor cells.     

Modeling and computer simulations of tumor growth and tumor response to radiotherapy

A model of tumor growth and tumor response to radiation is introduced in which each tumor cell is taken into account individually. Each cell is assigned a set of radiobiological parameters, and the status of each cell is checked in discrete intervals. Tumor proliferation is governed by the cell cycle times of tumor cells, the growth fraction, the apoptotic capacity of the tumor, and the degree of tumor angiogenesis. The response of tumor cells to radiation is determined by the radiosensitivities and the oxygenation status. Computer simulation is performed on a 3D rigid cubic lattice, starting out from a single tumor cell. Random processes are simulated by Monte Carlo methods. Short cell cycle time, high growth fraction, and tumor angiogenesis all increase tumor proliferation rates. Accelerated time-dose patterns result in lower total doses needed for tumor control, but the extent of dose reduction depends on the kinetics and the radiosensitivities of tumor cells. Tumor angiogenesis alters fully oxygenated and hypoxic fractions within the tumor and subsequently affects the radiation response. It is demonstrated for selected radiobiological parameters that the simulation tools are suitable to quantitatively assess the total doses needed for tumor control. Using the simulation tools, it is feasible to simulate time-dependent effects during fractionated radiotherapy and to compare different time-dose patterns in terms of their tumor control.     

Modeling of the metabolic energy dissipation for restricted tumor growth

Energy dissipation mostly represents unwanted outcome but in the biochemical processes it may alter the biochemical pathways. However, it is rarely considered in the literature although energy dissipation and its alteration due to the changes in cell microenvironment may improve methods for guiding chemical and biochemical processes in the desired directions. Deeper insight into the changes of metabolic activity of tumor cells exposed to osmotic stress or irradiation may offer the possibility of tumor growth reduction. In this work effects of the osmotic stress and irradiation on the thermodynamical affinity of tumor cells and their damping effects on metabolic energy dissipation were investigated and modeled. Although many various models were applied to consider the tumor restrictive growth they have not considered the metabolic energy dissipation. In this work a pseudo rheological model in the form of “the metabolic spring-pot element” is formulated to describe theoretically the metabolic susceptibility of tumor spheroid. This analog model relates the thermodynamical affinity of cell growth with the volume expansion of tumor spheroid under isotropic loading conditions. Spheroid relaxation induces anomalous nature of the metabolic energy dissipation which causes the damping effects on cell growth. The proposed model can be used for determining the metabolic energy “structure” in the context of restrictive cell growth as well as for predicting optimal doses for cancer curing in order to tailor the clinical treatment for each person and each type of cancer.     

Modeling the effects of vasculature evolution on early brain tumor growth

Mathematical modeling of both tumor growth and angiogenesis have been active areas of research for the past several decades. Such models can be classified into one of two categories: those that analyze the remodeling of the vasculature while ignoring changes in the tumor mass, and those that predict tumor expansion in the presence of a non-evolving vasculature. However, it is well accepted that vasculature remodeling and tumor growth strongly depend on one another. For this reason, we have developed a two-dimensional hybrid cellular automaton model of early brain tumor growth that couples the remodeling of the microvasculature with the evolution of the tumor mass. A system of reaction-diffusion equations has been developed to track the concentration of vascular endothelial growth factor (VEGF), Ang-1, Ang-2, their receptors and their complexes in space and time. The properties of the vasculature and hence of each cell are determined by the relative concentrations of these key angiogenic factors. The model exhibits an angiogenic switch consistent with experimental observations on the upregulation of angiogenesis. Particularly, we show that if the pathways that produce and respond to VEGF and the angiopoietins are properly functioning, angiogenesis is initiated and a tumor can grow to a macroscopic size. However, if the VEGF pathway is inhibited, angiogenesis does not occur and tumor growth is thwarted beyond 1-2mm in size. Furthermore, we show that tumor expansion can occur in well-vascularized environments even when angiogenesis is inhibited, suggesting that anti-angiogenic therapies may not be sufficient to eliminate a population of actively dividing malignant cells.     

Modeling tumor cell shedding

Cell shedding is an important step in the development of tumor invasion and metastasis. It influences growth saturation, latency, and tumor surface roughness. In spite of careful experiments carried out using multicellular tumor spheroids (MTS), the effects of the shedding process are not yet completely understood. Using a simulational model, we study how the nature and intensity of cell shedding may influence tumor morphology and examine the dependence of the total number of shed cells with the relevant parameters, finding the ranges that maximize cell detachment. These ranges correspond to intermediate values of the adhesion, for which we observe the emergence of a rough tumor surface. They are also likely to maximize the probability of generating invasion and metastases. Using numerical values taken from experiments, we find that the shedding-induced reduction in the growth rate is not intense enough to lead to latency, except when cell adhesion is assumed to be very weak. This suggests that the presence of inhibitors is a necessary condition for the observed MTS growth saturation.     

Immunotherapy with SLPI over-expressing mammary tumor cells decreases tumor growth

We have demonstrated previously that the inoculation of murine mammary tumor cells genetically modified to express high levels of secretory leukocyte protease inhibitor (2C1) do not develop tumors in immunocompetent mice and these cells are more prone to apoptosis than control cells. The aim of the present study was to evaluate the role of the adaptive immune response in the lack of tumor growth of 2C1 cells and the possibility of using these cells for immunotherapy. The s.c. administration of mock transfected F3II cells induces tumor in BALB/c and Nude mice. However, the inoculation of 2C1 cells develops tumor in Nude but not in BALB/c mice. The inoculation of mock transfected F3II cells to 2C1 immunized BALB/c mice by repeated administration of 2C1 cells (once a week for 3 weeks) developed significantly smaller tumors than those observed in non-immunized mice. Remarkably, survival of tumor-bearing immunized mice was higher than non-immunized animals. Herein, we demonstrate that an immunotherapy with SLPI over-expressing non-irradiated tumor cells which do not develop tumor in immunocompetent mice, partially restrain the tumor growth induced by F3II cells and increase the survival of the mice.     

Modeling phytoplankton growth rates

Mathematical models of planktonic ecosystems use a variety ofdifferent formulations to relate phytoplankton growth ratesto environmental conditions. Does the formulation influencethe model result? We have modified the model of Fasham, Ducklowarid McKelvie (J. Mar. Res., 34,591–639, 1990) to testhow its results would respond to changes in algal growth rateformulations. The original model uses a Monod relationship betweennutrient concentration and relative growth rate, and a multiplicativerule to combine light and nutrient effects. Use of a Droop formulationfor algal growth rate or a threshold (Blackman's law) mechanismto combine light and nutrient limitation produced significantchanges in simulation results. One important effect was to increasezooplankton population and, as a result, the regenerated production.While there are aesthetic reasons to prefer these alternateformulations, a more accurate formulation will require morelaboratory work on algal physiology. Such laboratory work shouldbe encouraged as an adjunct to modeling work.     

Modeling bacterial growth responses

Summary The main steps of modeling bacterial growth responses are summarized and a new model for growth curves is shown. Its advantages are analyzed from some theoretical and practical points of view. The new model fits better and has more advantageous statistical properties than the Gompertz curve.     

Modeling phytoplankton growth rates

Equation 14 on p. 68 should read: Equation 20 on p. 69 should read:      

Resistance to tumor growth in guinea pigs immunized with lyophilized tumor cells

Summary Living BCG, killed Mycobacterium tuberculosis cells, or BCG cell walls (CW) augmented the immunogenicity of lyophilized syngeneic ascites hepatoma (line 10) of strain-2 guinea pigs. Effective vaccine contained living BCG and lyophilized line-10 cells, or mycobacterial cells or CW attached to oil droplets and lyophilized line-10 cells. Protection against the challenge tumor was evident 14 or 21 days after one administration of either vaccine.     

Copper and zinc metabolism with solid tumor growth

The variation in copper and zinc metabolism with tumor growth appears to relate directly to progression or regression of the disease. Historically, elevations in serum copper have been used as clinical indicators in hematological neoplasms since the early 1960s. More recently, we have monitored breast, colo-rectal, and lung cancer patients for a six-month period through courses of cytotoxic chemotherapy to determine copper and zinc changes with tumor growth. Groups were divided into responders and nonresponders blind to their serum copper and zinc levels. Trends in elevated serum copper with active disease have shown similar trends in decreasing values with effective therapy, but normalization was at a slower rate. Serum zinc levels in the same patients were markedly below normal and did not increase in the study period. The clinical significance or elevated serum copper and depressed serum zinc is discussed and the potential relationship between the two elements is explored. A solid tumor-bearing rat model, mammary adenocarcinoma R 3230 AC, has detailed more of the changes in copper and zinc metabolism with solid tumor growth. Serum copper and zinc varied with tumor mass, as in clinical studies. Liver values of the two essential metals did not change significantly, but liver-related copper-containing enzymes showed marked variations. Ceruloplasmin in serum increased with increasing tumor mass, as would be expected with the increased serum copper levels. Cytochrome c oxidase activity in liver homogenates from tumor-bearing animals was significantly depressed.     

Modeling plant growth and development

Computational plant models or 'virtual plants' are increasingly seen as a useful tool for comprehending complex relationships between gene function, plant physiology, plant development, and the resulting plant form. The theory of L-systems, which was introduced by Lindemayer in 1968, has led to a well-established methodology for simulating the branching architecture of plants. Many current architectural models provide insights into the mechanisms of plant development by incorporating physiological processes, such as the transport and allocation of carbon. Other models aim at elucidating the geometry of plant organs, including flower petals and apical meristems, and are beginning to address the relationship between patterns of gene expression and the resulting plant form.     

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     

Modeling tumor invasion and metastasis in Drosophila

Conservation of major signaling pathways between humans and flies has made Drosophila a useful model organism for cancer research. Our understanding of the mechanisms regulating cell growth, differentiation and development has been considerably advanced by studies in Drosophila. Several recent high profile studies have examined the processes constraining the metastatic growth of tumor cells in fruit fly models. Cell invasion can be studied in the context of an in vivo setting in flies, enabling the genetic requirements of the microenvironment of tumor cells undergoing metastasis to be analyzed. This Perspective discusses the strengths and limitations of Drosophila models of cancer invasion and the unique tools that have enabled these studies. It also highlights several recent reports that together make a strong case for Drosophila as a system with the potential for both testing novel concepts in tumor progression and cell invasion, and for uncovering players in metastasis.     

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.     

Reduction of tumor growth following treatment with a glutamine antimetabolite

Assessment of arterial-venous differences across transplanted methylcholanthrene-induced sarcomas in rats revealed significant decreases in plasma concentrations of glutamine, serine and glucose. Treatment with the glutamine antimetabolite, acivicin, significantly reduced tumor weights by 65% at the conclusion of the experiment 34 days after tumor induction. These results suggest that glutamine is an essential metabolic substrate for tumor growth and that blockade of glutamine utilization can inhibit the growth of these transplantable sarcomas.