A mathematical model to study the effects of drugs administration on tumor growth dynamics

A mathematical model for describing the cancer growth dynamics in response to anticancer agents administration in xenograft models is discussed. The model consists of a system of ordinary differential equations involving five parameters (three for describing the untreated growth and two for describing the drug action). Tumor growth in untreated animals is modelled by an exponential growth followed by a linear growth. In treated animals, tumor growth rate is decreased by an additional factor proportional to both drug concentration and proliferating cells. The mathematical analysis conducted in this paper highlights several interesting properties of this tumor growth model. It suggests also effective strategies to design in vivo experiments in animals with potential saving of time and resources. For example, the drug concentration threshold for the tumor eradication, the delay between drug administration and tumor regression, and a time index that measures the efficacy of a treatment are derived and discussed. The model has already been employed in several drug discovery projects. Its application on a data set coming from one of these projects is discussed in this paper.     

A stochastic model in tumor growth

A stochastic model of solid tumor growth based on deterministic Gompertz law is presented. Tumor cells evolution is described by a one-dimensional diffusion process limited by two absorbing boundaries representing healing threshold and patient death (carrying capacity), respectively. Via a numerical approach the first exit time problem is analysed for the process inside the region restricted by the boundaries. The proposed model is also implemented to simulate the effects of a time-dependent therapy. Finally, some numerical results are obtained for the specific case of a parathyroid tumor.     

Modeling and testing treated tumor growth using cubic smoothing splines

Human tumor xenograft models are often used in preclinical study to evaluate the therapeutic efficacy of a certain compound or a combination of certain compounds. In a typical human tumor xenograft model, human carcinoma cells are implanted to subjects such as severe combined immunodeficient (SCID) mice. Treatment with test compounds is initiated after tumor nodule has appeared, and continued for a certain time period. Tumor volumes are measured over the duration of the experiment. It is well known that untreated tumor growth may follow certain patterns, which can be described by certain mathematical models. However, the growth patterns of the treated tumors with multiple treatment episodes are quite complex, and the usage of parametric models is limited. We propose using cubic smoothing splines to describe tumor growth for each treatment group and for each subject, respectively. The proposed smoothing splines are quite flexible in modeling different growth patterns. In addition, using this procedure, we can obtain tumor growth and growth rate over time for each treatment group and for each subject, and examine whether tumor growth follows certain growth pattern. To examine the overall treatment effect and group differences, the scaled chi-squared test statistics based on the fitted group-level growth curves are proposed. A case study is provided to illustrate the application of this method, and simulations are carried out to examine the performances of the scaled chi-squared tests.     

Angiopoietin decoy secreted at tumor site impairs tumor growth and metastases by inducing local inflammation and altering neoangiogenesis

The extracellular domain of the receptor tyrosine kinase Tie2/TEK (exTEK) has been used as an angiopoietin decoy to study the role of angiopoietins in the tumor–host interactions, using a syngeneic model of experimental metastases and subcutaneous tumor. Soluble exTEK secreted by transfected tumor cells inhibited HUVECs from forming tubes in Matrigel. ExTEK-transfected C26 colon carcinoma and TS/A mammary tumor cells displayed reduced growth rate when injected subcutaneously, and reduced ability to form experimental metastases when injected intravenously. Immunohistochemical analysis of tumors and metastases showed increased leukocytes infiltration and signs of inflammation in exTEK-secreting compared to parental tumor, as well as impairment in neo-vessel growth and organization. However, while neoangiogenesis eventually rescued in the subcutis, it failed to organize in the experimental metastases of exTEK-secreting tumor, contributing to the hampering of metastatic growth and to increased mice survival. The reactive infiltrate of C26TEK contained a different percentage of leukocytes and was responsible for the tumor inhibition. In fact, leukopenia induced by -irradiation of recipient mice or injection into interferon gamma (IFN-) gene knockout (GKO) mice resulted in reduced mouse survival and an increased number of lung metastases. On the other hand, interleukin (IL)-12 treatment prolonged the survival of mice bearing subcutaneous C26TEK but not of those bearing lung metastases, suggesting that IL-12 could exert further antiangiogenic effects at the site where the tumor can restore neoangiogenesis. These results show in vivo that reduced angiopoietin availability at the tumor site induces a local inflammatory response and impairment of neoangiogenesis which act synergistically to limit tumor growth and metastasis.Abbreviations AEC amino-ethylcarbazole - ELISA enzyme-linked immunosorbent assay - HRP horseradish peroxidase - HUVEC human umbilical vascular endothelial cell - i.v. intravenous - s.c. subcutaneous - TBS Tris-HCl buffered solution     

How regulatory CD25(+)CD4(+) T cells impinge on tumor immunobiology? On the existence of two alternative dynamical classes of tumors

Aiming to get a better insight on the impact of regulatory CD25(+)CD4(+) T cells in tumor immunobiology, a simple mathematical model was formulated and studied. This model is an extension of a previous model for the dynamics of autoreactive regulatory cells and effector cells that interact upon their co-localized activation at the antigen presenting cells (APCs). It assumes that tumor growth stimulates the activation and migration to the adjacent lymph node of fresh APCs loaded with tumor antigens. These APCs stimulate the growth of both effector and regulatory T cells, which may then migrate to the tumor site and induce tumor cell destruction. Our results predict the existence of two alternative dynamic modes of unbounded tumor growth. In the first mode, the tumor induces the expansion of effector T cells that outcompete regulatory T cells, but nevertheless fail to control the tumor. In the second mode, the tumor induces a balanced expansion of both effector and regulatory T cells, which prevents the tumor from being destroyed by the immune cells. Tumors characterized by a high specific growth rate, low immunogenicity, and that are relatively resistant to T cell destructive functions, will grow in the first mode; conversely, tumors that have a slow specific growth rate, that are immunogenic, and/or that are more sensitive to destruction by T cells will grow in the second mode. Overall, this result provides a simple explanation to the fact that the development of some tumors expands regulatory T cells while others do not, predicting how some key dynamical properties of the tumor determine either one or the other type of behavior.     

A mechanical model of tumor encapsulation and transcapsular spread

In this paper, a mathematical modeling framework is presented which describes the growth, encapsulation, and transcapsular spread of solid tumors. The model is based on the physical forces and cellular interactions involved in tumorigenesis and is used to test and compare the active (foreign body hypothesis) and passive (expansive growth hypothesis) hypotheses of capsule formation, such investigations being ideally suited to our mechanical model. The model simulations lead us to predict that, although an active response can successfully control tumor growth via the deposition of large amounts of collagen, this alone is insufficient for capsule formation. In contrast, a solely passive responsive is capable of producing an encapsulated tumor with minimal accumulation of connective tissue within the tumor. When both responses are active, a denser capsule forms and there is a significant increase in connective tissue within the tumor. Using a modified version of the model, in which tumor cells are assumed to produce degradative proteases at a rate which depends on the pressure they experience, it is also possible to show that transcapsular spread or invasion of the tumor may be due to the production by the tumor cells of proteases and their subsequent action.     

THE CIRCADIAN CLOCK GENE PER1 SUPPRESSES CANCER CELL PROLIFERATION AND TUMOR GROWTH AT SPECIFIC TIMES OF DAY

Cell cycle progression is tightly regulated. The expressions of cell cycle regulators, the products of which either promote or inhibit cell proliferation, oscillate during each cell cycle. Cellular proliferation and the expression of cell cycle regulators are also controlled by the circadian clock. Disruption of the circadian clock may thereby lead to deregulated cell proliferation. Mammalian Per2 is a core clock gene, the product of which suppresses cancer cell proliferation and tumor growth in vivo and in vitro. Because Per1, another key clock gene, is mutated in human breast cancers, and because its clock functions are similar and complementary to those of Per2, we have studied its role in modulating breast cancer cell proliferation and tumor growth. We find that breast cancer growth rate is gated by the circadian clock with two daily peaks and troughs, and that they are coupled to the daily expression patterns of clock-controlled genes that regulate cell proliferation. Down-regulation of the expression of tumor Per1 increases cancer cell growth in vitro and tumor growth in vivo by enhancing the circadian amplitude of the two daily tumor growth peaks. The data of the study suggest Per1 has tumor-suppressor function that diminishes cancer proliferation and tumor growth, but only at specific times of day. (Author correspondence: williamhrushesky@gmail.com).     

Phenomenological modeling of tumor diameter growth based on a mixed effects model

Over the last few years, taking advantage of the linear kinetics of the tumor growth during the steady-state phase, tumor diameter-based rather than tumor volume-based models have been developed for the phenomenological modeling of tumor growth. In this study, we propose a new tumor diameter growth model characterizing early, late and steady-state treatment effects. Model parameters consist of growth rhythms, growth delays and time constants and are meaningful for biologists. Biological experiments provide in vivo longitudinal data. The latter are analyzed using a mixed effects model based on the new diameter growth function, to take into account inter-mouse variability and treatment factors. The relevance of the tumor growth mixed model is firstly assessed by analyzing the effects of three therapeutic strategies for cancer treatment (radiotherapy, concomitant radiochemotherapy and photodynamic therapy) administered on mice. Then, effects of the radiochemotherapy treatment duration are estimated within the mixed model. The results highlight the model suitability for analyzing therapeutic efficiency, comparing treatment responses and optimizing, when used in combination with optimal experiment design, anti-cancer treatment modalities.