A quantitative theory of solid tumor growth, metabolic rate and vascularization

The relationships between cellular, structural and dynamical properties of tumors have traditionally been studied separately. Here, we construct a quantitative, predictive theory of solid tumor growth, metabolic rate, vascularization and necrosis that integrates the relationships between these properties. To accomplish this, we develop a comprehensive theory that describes the interface and integration of the tumor vascular network and resource supply with the cardiovascular system of the host. Our theory enables a quantitative understanding of how cells, tissues, and vascular networks act together across multiple scales by building on recent theoretical advances in modeling both healthy vasculature and the detailed processes of angiogenesis and tumor growth. The theory explicitly relates tumor vascularization and growth to metabolic rate, and yields extensive predictions for tumor properties, including growth rates, metabolic rates, degree of necrosis, blood flow rates and vessel sizes. Besides these quantitative predictions, we explain how growth rates depend on capillary density and metabolic rate, and why similar tumors grow slower and occur less frequently in larger animals, shedding light on Peto's paradox. Various implications for potential therapeutic strategies and further research are discussed.     

Exercise delays allogeneic tumor growth and reduces intratumoral inflammation and vascularization.

This investigation determined whether daily strenuous exercise would alter the progression and regression of an allogeneic lymphoid tumor in mice. We also determined whether exercise would alter the cellular composition and vascularity of the tumor. Female BALB/c mice (age 6-8 wk) were randomly assigned to sedentary control (Con) or daily exercised groups (EXH). EXH mice ran on a treadmill at incremental speeds (20-40 m/min) for 3 h or until fatigue. Each mouse was subcutaneously injected with 20 x 10(6) EL-4 lymphoma cells immediately after the first exercise bout (day 1) and run daily. Tumor volume was measured daily with calipers. In some experiments, mice were euthanized on days 5-10, 12, and 14. Tumors were excised and stained with hematoxylin and eosin or for Factor VIII-associated antigen using immunohistochemistry and analyzed in a blinded fashion under a light microscope. There was no significant treatment main effect found for tumor volumes. Interestingly, a significant treatment x time interaction was found, such that there was a 2-day delay in peak tumor volume and a more rapid tumor regression in EXH. Tumors isolated from Con exhibited significantly higher numbers of apoptotic bodies, blood vessels, macrophages, and neutrophils when compared with EXH. Intratumoral lymphocytes were higher in Con early in tumor growth but higher in EXH at peak tumor size. These data indicate that daily strenuous exercise may influence tumor growth by affecting the microenvironment of the tumor, resulting in a delay in tumor growth and a more rapid regression.     

Tristetraprolin inhibits Ras-dependent tumor vascularization by inducing vascular endothelial growth factor mRNA degradation

Vascular endothelial growth factor (VEGF) is one of the most important regulators of physiological and pathological angiogenesis. Constitutive activation of the extracellular signal-regulated kinase (ERK) pathway and overexpression of VEGF are common denominators of tumors from different origins. We have established a new link between these two fundamental observations converging on VEGF mRNA stability. In this complex phenomenon, tristetraprolin (TTP), an adenylate and uridylate-rich element-associated protein that binds to VEGF mRNA 3′-untranslated region, plays a key role by inducing VEGF mRNA degradation, thus maintaining basal VEGF mRNA amounts in normal cells. ERKs activation results in the accumulation of TTP mRNA. However, ERKs reduce the VEGF mRNA-destabilizing effect of TTP, leading to an increase in VEGF expression that favors the angiogenic switch. Moreover, TTP decreases RasVal12-dependent VEGF expression and development of vascularized tumors in nude mice. As a consequence, TTP might represent a novel antiangiogenic and antitumor agent acting through its destabilizing activity on VEGF mRNA. Determination of TTP and ERKs status would provide useful information for the evaluation of the angiogenic potential in human tumors.     

Assessment of tumor vascularization: immunohistochemical and non-invasive methods

Growth of solid tumors beyond a certain mass is dependent on the vascular bed from pre-existing host vasculature. The process of angiogenesis is essential not only for primary tumor growth but also for metastasis. The number of microvessels within the invasive component of a primary tumor reflects the degree of tumor angiogenesis. At present the most widely used method to assess neovascularization is the quantitation of intratumoral microvessel density (IMD) by immunohistochemical methods in which specific markers for endothelial cells are employed. In this paper we analyze the different methods used to assess IMD, as well as their advantages and potential methodological pitfalls. Several studies have shown a close correlation between IMD, tumor growth and the occurrence of metastasis, suggesting that IMD is a prognostic indicator of clinical relevance. Furthermore, preliminary studies suggest that determination of angiogenesis may predict responsiveness to some forms of conventional anticancer therapy. Although the histological microvessel density technique is the current gold standard to characterize tumor angiogenesis, it may not be the ideal tool for clinical purposes because it needs to be performed on biopsy material and does not assess the functional pathways involved in the angiogenic activity of tumors. Non-invasive assessment of tumor vascularity is possible in vivo by means of Doppler sonography, dynamic contrast-enhanced magnetic resonance imaging (MRI) and positron emission tomography (PET). These methods may be preferable to histological assay because they are non-invasive, survey the entire tumor, reflect both anatomic and physiologic characteristics, and may be useful to monitor the activity of antiangiogenic therapies.     

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.     

A simplified mathematical model of tumor growth

A one-dimensional model of tumor tissue growth is presented in which the source of mitotic inhibitor is nonuniformly distributed within the tissue (in contrast to many earlier models). As a result, stable and unstable regimes of growth become significantly modified from the uniform-source case, indicating that the model, schematic though it is, is very sensitive to the type of source term assumed, and this has implications for experimental and theoretical comparisons in more realistic geometries.     

A multiscale model for avascular tumor growth

The desire to understand tumor complexity has given rise to mathematical models to describe the tumor microenvironment. We present a new mathematical model for avascular tumor growth and development that spans three distinct scales. At the cellular level, a lattice Monte Carlo model describes cellular dynamics (proliferation, adhesion, and viability). At the subcellular level, a Boolean network regulates the expression of proteins that control the cell cycle. At the extracellular level, reaction-diffusion equations describe the chemical dynamics (nutrient, waste, growth promoter, and inhibitor concentrations). Data from experiments with multicellular spheroids were used to determine the parameters of the simulations. Starting with a single tumor cell, this model produces an avascular tumor that quantitatively mimics experimental measurements in multicellular spheroids. Based on the simulations, we predict: 1), the microenvironmental conditions required for tumor cell survival; and 2), growth promoters and inhibitors have diffusion coefficients in the range between 10(-6) and 10(-7) cm2/h, corresponding to molecules of size 80-90 kDa. Using the same parameters, the model also accurately predicts spheroid growth curves under different external nutrient supply conditions.     

Autowaves in a model of invasive tumor growth

A mathematical model for invasive tumor growth is proposed, which takes into account cell division, death, and motility. The model includes local cell density and the distribution of nutrient (oxygen) concentration. Cancer cells die in the absence of nutrients; therefore, the distribution of oxygen in tissue substantially affects both the tumor proliferation rate and its structure. The model adequately describes the experimentally measured rate of tumor proliferation. The existence of autowave solutions is demonstrated, and their properties are investigated. The results are compared with the properties of the Kolmogorov-Petrovskii-Piskunov and Fisher equations. It is shown that the nutrient distribution influences the selection of speed and the convergence of the initial conditions to the automodel solution.     

A delay differential equation model for tumor growth

We present a competition model of tumor growth that includes the immune system response and a cycle-phase-specific drug. The model considers three populations: Immune system, population of tumor cells during interphase and population of tumor during mitosis. Delay differential equations are used to model the system to take into account the phases of the cell cycle. We analyze the stability of the system and prove a theorem based on the argument principle to determine the stability of a fixed point and show that the stability may depend on the delay. We show theoretically and through numerical simulations that periodic solutions may arise through Hopf Bifurcations.Send offprint requests to:Minaya Villasana     

Delay-induced model for tumor-immune interaction and control of malignant tumor growth

The paper deals with the qualitative analysis of the solutions of a system of delay differential equations describing the interaction between tumor and immune cells. The asymptotic stability of the possible steady states is showed and the occurrence of limit cycle of the system around the interior equilibrium is proved by the application of Hopf bifurcation theorem by using the delay as a bifurcation parameter. The length of the delay parameter for preserving stability of the system is also estimated, which gives the idea about the mode of action for controlling oscillations in malignant tumor cell growth. The theoretical and numerical outcomes have been supported through experimental results from literatures. This approach gives new insight of modeling tumor-immune interactions and provides significant control strategies to overcome the large oscillations in tumor cells.     

Activation of CD40 favors the growth and vascularization of Kaposi's sarcoma

Although CD40 is expressed by several tumor lines and is up-regulated in tumor vascular endothelium, its role in tumor biology is still unclear. In the present study, we investigated the role of CD40 in the growth and vascularization of Kaposi's sarcoma (KS). In vitro, stimulation of CD40 induced migration of KS cells and inhibited vincristine-induced apoptosis. Similarly, the CD40 engagement on endothelial cells resulted in cell contraction, migration, and prevention of serum withdrawal-apoptosis. To understand the biological relevance of CD40 in vivo, KS cells were engineered to express and release a soluble form of CD40 (KS-sCD40) able to disrupt CD40-CD154 interaction. SCID mice s.c. injected with KS-sCD40 cells developed tumors that were significantly smaller than those induced by control cells (KS-neo). In addition, KS-sCD40 tumors showed several areas of necrosis, diffuse presence of apoptotic cells, and poor vascularization. In contrast, KS-neo tumors showed few or absent areas of necrosis and apoptosis and intense vascularization. Moreover, anti-CD40 Abs stimulated neo-angiogenesis in a murine model in which s.c. implantation of Matrigel was used as a vehicle for the delivery of mediators. These observations provide demonstration that CD40 supports tumor cell survival, growth, and neo-vascularization of KS.     

Formulation and numerical simulations of a continuum model of avascular tumor growth

In this paper we present a continuum mathematical model for a multicellular spheroid that mimics the micro-environment within avascular tumor growth. The model consists of a coupled system of non-linear convection-diffusion-reaction equations. This system is solved using a previously developed conservative Galerkin characteristics method. In the model considered, there are three cell types: the proliferative cells, the quiescent non-dividing cells which stay in the G₀ phase of the cell cycle and the necrotic cells. The model includes viable cell diffusion, diffusion of cellular material and the removal of necrotic cells. We assume that the nutrients diffuse passively and are consumed by the proliferative and quiescent tumor cells depending on the availability of resources (oxygen, glucose, etc.). The numerical simulations are performed using different sets of parameters, including biologically realistic ones, to explore the effects of each of these model parameters on reaching the steady state. The present results, taken together with those reported earlier, indicate that the removal of necrotic cells and the diffusion of cellular material have significant effects on the steady state, reflecting growth saturation, the number of viable cells, and the spheroid size.     

A model of tumor growth based on cell cycle kinetics

This work describes a mathematical model of growth based on the kinetics of the cell cycle. A traditional model of the cell cycle has been used, with the addition of a resting (G₀) state from which cells could reenter the reproductive cycle. The model assumes that a growth regulatory substance regulates the transition of cells to and from the resting state. Other transitions between the phases of the cycle were modeled as a first order process. Cell loss is an important feature of growth kinetics, and has been represented by a general but tractable mathematical form. The resulting model forms a system of ordinary nonlinear differential equations. Analytic methods are employed first in the study of this system. Simplifying assumptions regarding cell loss give rise to special cases for which equilibrium solutions can be found. One special case, which assumes first order loss from all cell cycle phases at equal rates, is presented here. For small time values, approximations corresponding to exponential growth were developed. The equations describing an intrinsic growth rate were derived. Simulation methods were used to further characterize the behavior of this model. Parameter values were chosen based on animal tumor cell cycle kinetic data, resulting in a set of 45 model simulations. Several tumor treatment protocols were simulated which illustrated the importance of the intrinsic growth rate and cell loss concepts. Although the qualitative behavior regarding absolute and relative growth is reasonable, this model awaits data for model fitting, parameter estimation, or revision of the equations.     

A nonlinear structured population model of tumor growth with quiescence

A nonlinear structured cell population model of tumor growth is considered. The model distinguishes between two types of cells within the tumor: proliferating and quiescent. Within each class the behavior of individual cells depends on cell size, whereas the probabilities of becoming quiescent and returning to the proliferative cycle are in addition controlled by total tumor size. The asymptotic behavior of solutions of the full nonlinear model, as well as some linear special cases, is investigated using spectral theory of positive simigroup of operators. Supported in part by the National Science Foundation under Grant No. DMS-8722947     

A mathematical model of tumor growth and its response to single irradiation

Background

Mathematical modeling of biological processes is widely used to enhance quantitative understanding of bio-medical phenomena. This quantitative knowledge can be applied in both clinical and experimental settings. Recently, many investigators began studying mathematical models of tumor response to radiation therapy. We developed a simple mathematical model to simulate the growth of tumor volume and its response to a single fraction of high dose irradiation. The modelling study may provide clinicians important insights on radiation therapy strategies through identification of biological factors significantly influencing the treatment effectiveness.

Methods

We made several key assumptions of the model. Tumor volume is composed of proliferating (or dividing) cancer cells and non-dividing (or dead) cells. Tumor growth rate (or tumor volume doubling time) is proportional to the ratio of the volumes of tumor vasculature and the tumor. The vascular volume grows slower than the tumor by introducing the vascular growth retardation factor, θ. Upon irradiation, the proliferating cells gradually die over a fixed time period after irradiation. Dead cells are cleared away with cell clearance time. The model was applied to simulate pre-treatment growth and post-treatment radiation response of rat rhabdomyosarcoma tumors and metastatic brain tumors of five patients who were treated with Gamma Knife stereotactic radiosurgery (GKSRS).

Results

By selecting appropriate model parameters, we showed the temporal variation of the tumors for both the rat experiment and the clinical GKSRS cases could be easily replicated by the simple model. Additionally, the application of our model to the GKSRS cases showed that the α-value, which is an indicator of radiation sensitivity in the LQ model, and the value of θ could be predictors of the post-treatment volume change.

Conclusions

The proposed model was successful in representing both the animal experimental data and the clinically observed tumor volume changes. We showed that the model can be used to find the potential biological parameters, which may be able to predict the treatment outcome. However, there is a large statistical uncertainty of the result due to the small sample size. Therefore, a future clinical study with a larger number of patients is needed to confirm the finding.

     

A single-cell-based model of tumor growth in vitro: monolayers and spheroids

To what extent the growth dynamics of tumors is controlled by nutrients, biomechanical forces and other factors at different stages and in different environments is still largely unknown. Here we present a biophysical model to study the spatio-temporal growth dynamics of two-dimensional tumor monolayers and three-dimensional tumor spheroids as a complementary tool to in vitro experiments. Within our model each cell is represented as an individual object and parametrized by cell-biophysical and cell-kinetic parameters that can all be experimentally determined. Hence our modeling strategy allows us to study which mechanisms on the microscopic level of individual cells may affect the macroscopic properties of a growing tumor. We find the qualitative growth kinetics and patterns at early growth stages to be remarkably robust. Quantitative comparisons between computer simulations using our model and published experimental observations on monolayer cultures suggest a biomechanically-mediated form of growth inhibition during the experimentally observed transition from exponential to sub-exponential growth at sufficiently large tumor sizes. Our simulations show that the same transition during the growth of avascular tumor spheroids can be explained largely by the same mechanism. Glucose (or oxygen) depletion seems to determine mainly the size of the necrotic core but not the size of the tumor. We explore the consequences of the suggested biomechanical form of contact inhibition, in order to permit an experimental test of our model. Based on our findings we propose a phenomenological growth law in early expansion phases in which specific biological small-scale processes are subsumed in a small number of effective parameters.     

Apatinib-loaded nanoparticles inhibit tumor growth and angiogenesis in a model of melanoma

Anti-angiogenic drugs are an effective therapeutic method for the treatment of melanomas. Apatinib is a small-molecule tyrosine kinase inhibitor, which has potent inhibitory activity on tumor angiogenesis. Due to the low water solubility and stability of Apatinib, we aimed to design and develop poly (lactic-co-glycolic acid) (PLGA) and Poloxamer 407 nanoparticles to encapsulate Apatinib (Apa/p NPs) to improve the efficacy of application in melanoma treatment. The size and morphology of the nanoparticles were characterized by dynamic light scattering (DLS) and transmission electron microscopy (TEM). In vitro proliferation assays were used to assess the capacity of Apa/p NPs to suppress the growth of B16 cells. Furthermore, we constructed melanoma models using C57BL/6 mice, and preliminary evaluation of the effect and mechanism of Apa/p NPs on tumor inhibition was performed in vivo. The results showed that the size of Apa/p NPs averaged 136 ± 0.27 nm and the nanoparticles were evenly dispersed. Moreover, Apa/p NPs significantly inhibited the growth of B16 cells and melanoma tumors, compared with the naked drug treatment and control groups. The protein levels of VEGFR-2, phosphorylated (p)-VEGFR-2 and p-ERK1/2 in tumor tissues were inhibited by Apa/p NP treatment, as detected by Western blot. The results of this study suggested that Apa/p NPs could inhibit the growth of melanoma tumors by inhibiting the phosphorylation and expression of VEGFR-2 and downstream ERK1/2, providing a theoretical basis for the clinical application of Apatinib in the treatment of melanoma.