Optimal control of tumor size used to maximize survival time when cells are resistant to chemotherapy.

The high failure rates encountered in the chemotherapy of some cancers suggest that drug resistance is a common phenomenon. In the current study, the tumor burden during therapy is used to slow the growth of the drug-resistant cells, thereby maximizing the survival time of the host. Three types of tumor growth model are investigated--Gompertz, logistic, and exponential. For each model, feedback controls are constructed that specify the optimal tumor mass as a function of the size of the resistant subpopulation. For exponential and logistic tumor growth, the tumor burden during therapy is shown to have little impact upon survival time. When the tumor is in Gompertz growth, therapies maintaining a large tumor burden double and sometimes triple the survival time under aggressive therapies. Aggressive therapies aim for a rapid reduction in the sensitive cell subpopulation. These conclusions are not dependent upon the values of the model constants that determine the mass of resistant cells. Since treatments maintaining a high tumor burden are optimal for Gompertz tumor growth and close to optimal for exponential and logistic tumor growth, it may no longer be necessary to know the growth characteristics of a tumor to schedule anticancer drugs.     

Mechanistic characterization of oscillatory patterns in unperturbed tumor growth dynamics: The interplay between cancer cells and components of tumor microenvironment

Mathematical modeling of unperturbed and perturbed tumor growth dynamics (TGD) in preclinical experiments provides an opportunity to establish translational frameworks. The most commonly used unperturbed tumor growth models (i.e. linear, exponential, Gompertz and Simeoni) describe a monotonic increase and although they capture the mean trend of the data reasonably well, systematic model misspecifications can be identified. This represents an opportunity to investigate possible underlying mechanisms controlling tumor growth dynamics through a mathematical framework. The overall goal of this work is to develop a data-driven semi-mechanistic model describing non-monotonic tumor growth in untreated mice. For this purpose, longitudinal tumor volume profiles from different tumor types and cell lines were pooled together and analyzed using the population approach. After characterizing the oscillatory patterns (oscillator half-periods between 8–11 days) and confirming that they were systematically observed across the different preclinical experiments available (p<10^?9), a tumor growth model was built including the interplay between resources (i.e. oxygen or nutrients), angiogenesis and cancer cells. The new structure, in addition to improving the model diagnostic compared to the previously used tumor growth models (i.e. AIC reduction of 71.48 and absence of autocorrelation in the residuals (p>0.05)), allows the evaluation of the different oncologic treatments in a mechanistic way. Drug effects can potentially, be included in relevant processes taking place during tumor growth. In brief, the new model, in addition to describing non-monotonic tumor growth and the interaction between biological factors of the tumor microenvironment, can be used to explore different drug scenarios in monotherapy or combination during preclinical drug development.     

Model-Based Tumor Growth Dynamics and Therapy Response in a Mouse Model of De Novo Carcinogenesis

Tumorigenesis is a complex, multistep process that depends on numerous alterations within the cell and contribution from the surrounding stroma. The ability to model macroscopic tumor evolution with high fidelity may contribute to better predictive tools for designing tumor therapy in the clinic. However, attempts to model tumor growth have mainly been developed and validated using data from xenograft mouse models, which fail to capture important aspects of tumorigenesis including tumor-initiating events and interactions with the immune system. In the present study, we investigate tumor growth and therapy dynamics in a mouse model of de novo carcinogenesis that closely recapitulates tumor initiation, progression and maintenance in vivo. We show that the rate of tumor growth and the effects of therapy are highly variable and mouse specific using a Gompertz model to describe tumor growth and a two-compartment pharmacokinetic/ pharmacodynamic model to describe the effects of therapy in mice treated with 5-FU. We show that inter-mouse growth variability is considerably larger than intra-mouse variability and that there is a correlation between tumor growth and drug kill rates. Our results show that in vivo tumor growth and regression in a double transgenic mouse model are highly variable both within and between subjects and that mathematical models can be used to capture the overall characteristics of this variability. In order for these models to become useful tools in the design of optimal therapy strategies and ultimately in clinical practice, a subject-specific modelling strategy is necessary, rather than approaches that are based on the average behavior of a given subject population which could provide erroneous results.     

Influence of interstitial fluid dynamics on growth and therapy of angiogenic tumor. Analysis by mathematical model

We have developed a spatially distributed mathematical model of angiogenic tumor growth in tissue with account of interstitial fluid dynamics and bevacizumab monotherapy. In this model the process of neovascularization is initiated by tumor cells in a state of metabolic stress, vascular endothelial growth factor (VEGF) being its main mediator. The model takes into consideration the convection flows arising in dense tissue due to active proliferation and migration of tumor cells as well as interstitial fluid inflow from blood vascular system, its outflow through lymphatic system and redistribution in the area of tumor growth. The work considers the diffusive approximation of interstitial fluid dynamics in tumor and normal tissue. Numerical study of the model showed that in absence of therapy a peritumoral edema is formed due to the increase of interstitial fluid inflow from angiogenic capillaries. In the case of rapid interstitial fluid outflow through lymphatic system and its fast transport from necrotic zone to normal tissue the regimes of full growth stop are observed in case of low-invasive tumor. Under bevacizumab monotherapy the peritumoral edema vanishes and low-invasive tumor may not only decelerate its growth, but also start shrinking for a large range of parameters.     

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.     

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.     

Endocrine characterization of a human ovarian carcinoma (BG-1) established in nude mice

The steroid receptor-positive human ovarian cancer (BG-1) was evaluated to determine its usefulness as a tumor model. This tumor grows in intact male and female nude mice without hormone supplements. Moreover, its growth was significantly accelerated in ovariectomized mice, and the increased growth rate could be reversed by estradiol administration. Evaluation of tumor growth following endocrine therapy revealed that, while antiandrogens did not affect the tumor growth, both an aromatase inhibitor and a luteinizing hormone-releasing hormone agonist significantly impaired growth of this human ovarian tumor. Estradiol was also shown to up-regulate both estrogen and progesterone receptors in tumors grown in ovariectomized mice. Therefore, the BG-1 human ovarian carcinoma grows without hormonal supplements and yet responds to specific forms of endocrine therapy. Moreover, the steroid receptors present in this tumor respond to exogenous steroids. In conclusion, this tumor may serve as an ideal model for the study of hormonal regulation of ovarian tumor growth.     

Predictive Modeling of Neuroblastoma Growth Dynamics in Xenograft Model After Bevacizumab Anti-VEGF Therapy

Neuroblastoma is the leading cause of cancer death in young children. Although treatment for neuroblastoma has improved, the 5-year survival rate of patients still remains less than half. Recent studies have indicated that bevacizumab, an anti-VEGF drug used in treatment of several other cancer types, may be effective for treating neuroblastoma as well. However, its effect on neuroblastoma has not been well characterized. While traditional experiments are costly and time-consuming, mathematical models are capable of simulating complex systems quickly and inexpensively. In this study, we present a model of vascular tumor growth of neuroblastoma IMR-32 that is complex enough to replicate experimental data across a range of tumor cell properties measured in a suite of in vitro and in vivo experiments. The model provides quantitative insight into tumor vasculature, predicting a linear relationship between vasculature and tumor volume. The tumor growth model was coupled with known pharmacokinetics and pharmacodynamics of the VEGF blocker bevacizumab to study its effect on neuroblastoma growth dynamics. The results of our model suggest that total administered bevacizumab concentration per week, as opposed to dosage regimen, is the major determining factor in tumor suppression. Our model also establishes an exponentially decreasing relationship between administered bevacizumab concentration and tumor growth rate.     

Does tumor growth follow a "universal law"?

A general model for the ontogenetic growth of living organisms has been recently proposed. Here we investigate the extension of this model to the growth of solid malignant tumors. A variety of in vitro and in vivo data are analysed and compared with the prediction of a "universal" law, relating properly rescaled tumor masses and tumor growth times. The results support the notion that tumor growth follows such a universal law. Several important implications of this finding are discussed, including its relevance for tumor metastasis and recurrence, cell turnover rates, angiogenesis and invasion.     

Emergent behaviors from a cellular automaton model for invasive tumor growth in heterogeneous microenvironments

Understanding tumor invasion and metastasis is of crucial importance for both fundamental cancer research and clinical practice. In vitro experiments have established that the invasive growth of malignant tumors is characterized by the dendritic invasive branches composed of chains of tumor cells emanating from the primary tumor mass. The preponderance of previous tumor simulations focused on non-invasive (or proliferative) growth. The formation of the invasive cell chains and their interactions with the primary tumor mass and host microenvironment are not well understood. Here, we present a novel cellular automaton (CA) model that enables one to efficiently simulate invasive tumor growth in a heterogeneous host microenvironment. By taking into account a variety of microscopic-scale tumor-host interactions, including the short-range mechanical interactions between tumor cells and tumor stroma, degradation of the extracellular matrix by the invasive cells and oxygen/nutrient gradient driven cell motions, our CA model predicts a rich spectrum of growth dynamics and emergent behaviors of invasive tumors. Besides robustly reproducing the salient features of dendritic invasive growth, such as least-resistance paths of cells and intrabranch homotype attraction, we also predict nontrivial coupling between the growth dynamics of the primary tumor mass and the invasive cells. In addition, we show that the properties of the host microenvironment can significantly affect tumor morphology and growth dynamics, emphasizing the importance of understanding the tumor-host interaction. The capability of our CA model suggests that sophisticated in silico tools could eventually be utilized in clinical situations to predict neoplastic progression and propose individualized optimal treatment strategies.     

荧光标记技术在肿瘤模型研究中的应用

目的利用绿色荧光小鼠和红色荧光蛋白标记肿瘤细胞,建立荧光标记的小鼠肿瘤模型,并建立活体荧光成像和荧光显微镜成像在整体和细胞水平直接观察肿瘤的技术。方法将小鼠B16黑色素瘤细胞接种到绿色荧光蛋白转基因小鼠皮下,建立GFP小鼠肿瘤模型。以红色荧光蛋白作为标记基因导入小鼠黑色素瘤细胞B16细胞,建立稳定表达红色荧光蛋白的细胞株。将表达红色荧光蛋白B16细胞接种到绿色荧光转基因小鼠皮下,建立双荧光小鼠肿瘤模型。用荧光显微镜和活体荧光成像系统检测小鼠肿瘤的发生发展。结果分别建立了GFP小鼠肿瘤模型和双色荧光小鼠肿瘤模型。利用活体荧光影像仪可以观察双色荧光小鼠模型中受体绿色荧光组织和红色荧光移植肿瘤相互融合。利用荧光显微镜,可以观察到肿瘤内绿色荧光标记的来源于受体小鼠的血管和免疫细胞。经香菇多糖刺激的GFP小鼠肿瘤模型的移植瘤组织中,来源于受体小鼠绿色荧光标记的免疫细胞明显多于经生理盐水刺激的对照小鼠。结论利用绿色荧光小鼠和红色荧光RFP标记肿瘤细胞建立荧光标记的小鼠肿瘤模型,采用活体荧光成像仪和荧光显微镜可在整体和细胞水平直接观察肿瘤的生长以及肿瘤与宿主的相互作用。     

Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis

We present a mathematical model of the cytotoxic T lymphocyte response to the growth of an immunogenic tumor. The model exhibits a number of phenomena that are seenin vivo, including immunostimulation of tumor growth, “sneaking through” of the tumor, and formation of a tumor “dormant state”. The model is used to describe the kinetics of growth and regression of the B-lymphoma BCL₁ in the spleen of mice. By comparing the model with experimental data, numerical estimates of parameters describing processes that cannot be measuredin vivo are derived. Local and global bifurcations are calculated for realistic values of the parameters. For a large set of parameters we predict that the course of tumor growth and its clinical manifestation have a recurrent profile with a 3- to 4-month cycle, similar to patterns seen in certain leukemias.     

A Multilayer Grow-or-Go Model for GBM: Effects of Invasive Cells and Anti-Angiogenesis on Growth

The recent use of anti-angiogenesis (AA) drugs for the treatment of glioblastoma multiforme (GBM) has uncovered unusual tumor responses. Here, we derive a new mathematical model that takes into account the ability of proliferative cells to become invasive under hypoxic conditions; model simulations generate the multilayer structure of GBM, namely proliferation, brain invasion, and necrosis. The model is able to replicate and justify the clinical observation of rebound growth when AA therapy is discontinued in some patients. The model is interrogated to derive fundamental insights int cancer biology and on the clinical and biological effects of AA drugs. Invasive cells promote tumor growth, which in the long run exceeds the effects of angiogenesis alone. Furthermore, AA drugs increase the fraction of invasive cells in the tumor, which explain progression by fluid-attenuated inversion recovery (FLAIR) signal and the rebound tumor growth when AA is discontinued.     

A mathematical algorithm that computes breast cancer sizes and doubling times detected by screening

This paper presents a mathematical algorithm that computes the sizes and growth rates of breast cancer detected in a hypothetical population that is screened for the disease. The algorithm works by simulating the outcomes of the hypothetical population twice, first without screening and then with screening. The simulation without screening relies on an underlying model of the natural history of the disease. The simulation with screening uses this natural history model to track the growth of breast tumors backwards in the time starting from the time they would have been detected without screening. The method of tracking tumor growth backward in time is different from methods that track tumor growth forward in time by starting from an estimated time of tumor onset. The screening algorithm combines the natural history model, the method tracking of tumor growth backward in time, the age group, the interval between screening exams, and the detection threshold of the screening exam to compute the joint distribution of tumor size and growth rate among screen-detected and interval patients. The algorithm also computes the sensitivity and leadtime distribution. It allows for arbitrary age groups, detection thresholds and screening intervals and may contribute to the design of future screening trials.     

Growth factors and growth control of heterogeneous cell populations

In an earlier work a model of the autocrine and paracrine pathways of tumor growth control was developed (Michelson and Leith. 1991. Autocrine and paracrine growth factors in tumor growth.Bull. math. Biol. 53, 639–656). The target population, a generic tumor, was modeled as a single, homogeneous population using the standard Verhulst equation of logistic growth. Mitogenic signals were represented by modifications to the Malthusian growth parameter and adaptational signals were represented by modifications to the carrying capacity. Three growth scenarios were described: (1) normal tissue wound healing, (2) unperturbed tumor growth, and (3) tumor growth in a radiation damaged environment, a phenomenon termed the Tumor Bed Effect (TBE). In this paper, we extend those results to include a “triad” of growth factor controls (autocrine, paracrine and endocrine) and heterogeneity of the target population. The heterogeneous factors in the model represent either intrinsic, epigenetic or environmental differences in both normally differentiating tissues and tumors. Three types of growth are modeled: (1) normal tissue differentiation or wound healing, assuming no communication between differentiated and undifferentiated cell compartments; (2) normal wound healing with feedback inhibition, due to signalling from the differentiated compartment; and (3) the development of hypoxia in a spherical tumor. The signal processing within the triad is discussed for each model and biologically reasonable constraints are defined for limits on growth control.     

Simulated brain tumor growth dynamics using a three-dimensional cellular automaton

We have developed a novel and versatile three-dimensional cellular automaton model of brain tumor growth. We show that macroscopic tumor behavior can be realistically modeled using microscopic parameters. Using only four parameters, this model simulates Gompertzian growth for a tumor growing over nearly three orders of magnitude in radius. It also predicts the composition and dynamics of the tumor at selected time points in agreement with medical literature. We also demonstrate the flexibility of the model by showing the emergence, and eventual dominance, of a second tumor clone with a different genotype. The model incorporates several important and novel features, both in the rules governing the model and in the underlying structure of the model. Among these are a new definition of how to model proliferative and non-proliferative cells, an isotropic lattice, and an adaptive grid lattice.     

人宫颈癌细胞Hela移植模型肿瘤生长的荧光分析和卡尺测量的比较

目的建立稳定表达绿色荧光蛋白的人宫颈癌细胞系,建立移植瘤模型并比较移植模型肿瘤生长的荧光分析和卡尺测量的优缺点。方法以Lipofectamine 2000介导chickenβ-actin-GFP-NEO转染人宫颈癌细胞Hela,经梯度浓度G418筛选获得稳定表达绿色荧光蛋白的细胞克隆并扩大培养。BALB/cA-nu裸鼠皮下接种1×10^6个发光细胞使其成瘤,利用活体荧光成像系统和游标卡尺观察肿瘤的生长情况。结果获得了稳定表达GFP的人宫颈癌细胞株,将其接种到裸鼠体内可成瘤。活体荧光成像观察发现,1至3周随着肿瘤体积逐渐增大,平均荧光光子数逐渐增加;4周时随着肿瘤出现明显坏死,平均荧光光子数呈现下降趋势,而游标卡尺测量结果显示肿瘤在4至5周仍然不断的增大。结论绿色荧光蛋白能够在人宫颈癌细胞Hela中长期稳定表达,用绿色荧光蛋白标记的人宫颈癌细胞Hela建立的裸鼠肿瘤模型可以为人宫颈癌研究提供理想的实验材料,应用小动物活体成像系统能够客观定量评价活的肿瘤细胞在动物体内的生长情况,而不是肿瘤体积的变化。     

Variation of cell kinetic parameters in relation to the growth rate of the Ehrlich ascites tumor.

A multicompartmental model of the cell cycle and proliferation kinetics was used to analyse the time-course behavior of the cell cycle time, the growth fraction, and the cell loss rate during Ehrlich ascites tumor growth. The growth rate of Ehrlich ascites tumor cells as the tumor aged was significantly influenced by change in the cell cycle time.     

VARIATION OF CELL KINETIC PARAMETERS IN RELATION TO THE GROWTH RATE OF THE EHRLICH ASCITES TUMOR

A multicompartmental model of the cell cycle and proliferation kinetics was used to analyse the time-course behavior of the cell cycle time, the growth fraction, and the cell loss rate during Ehrlich ascites tumor growth. The growth rate of Ehrlich ascites tumor cells as the tumor aged was significantly influenced by change in the cell cycle time.