Ernesto A. B. F. Lima

A hybrid ten-species phase-field model of tumor growth

The development of predictive computational models of tumor initiation, growth, and decline is faced with many formidable challenges. Phenomenological models which attempt to capture the complex interactions of multiple tissue and cellular species must cope with moving interfaces of heterogeneous media and the sprouting vascular structures due to angiogenesis and their evolution. They must be able to deliver predictions consistent with events that take place at cellular scales, and they must faithfully depict biological mechanisms and events that are known to be associated with various forms of cancer. In the present work, a ten-species vascular model for the tumor growth is presented which falls within the framework of phase-field (or diffuse-interface) models suggested by continuum mixture theory. This framework provides for the simultaneous treatment of interactions of multiple evolving species, such as tumor cells, necrotic cell cores, nutrients, and other cellular and tissue types that exist and interact in living tissue. We develop a hybrid model that couples the tumor growth with sprouting through angiogenesis. The model is able to represent the branching of new vessels through coupling a discrete model for which the angiogenesis is started upon pre-defined conditions on the nutrient deprivation in the continuum model. Such conditions are represented by hypoxic cells that release tumor growth factors that ultimately trigger vascular growth. We discuss the numerical approximation of the model using mixed finite elements. The results of numerical experiments are also presented and discussed.

Selection and validation of predictive models of radiation effects on tumor growth based on noninvasive imaging data

The use of mathematical and computational models for reliable predictions of tumor growth and decline in living organisms is one of the foremost challenges in modern predictive science, as it must cope with uncertainties in observational data, model selection, model parameters, and model inadequacy, all for very complex physical and biological systems. In this paper, large classes of parametric models of tumor growth in vascular tissue are discussed including models for radiation therapy. Observational data is obtained from MRI of a murine model of glioma and observed over a period of about three weeks, with X-ray radiation administered 14.5 days into the experimental program. Parametric models of tumor proliferation and decline are presented based on the balance laws of continuum mixture theory, particularly mass balance, and from accepted biological hypotheses on tumor growth. Among these are new model classes that include characterizations of effects of radiation and simple models of mechanical deformation of tumors. The Occam Plausibility Algorithm (OPAL) is implemented to provide a Bayesian statistical calibration of the model classes, 39 models in all, as well as the determination of the most plausible models in these classes relative to the observational data, and to assess model inadequacy through statistical validation processes. Discussions of the numerical analysis of finite element approximations of the system of stochastic, nonlinear partial differential equations characterizing the model classes, as well as the sampling algorithms for Monte Carlo and Markov chain Monte Carlo (MCMC) methods employed in solving the forward stochastic problem, and in computing posterior distributions of parameters and model plausibilities are provided. The results of the analyses described suggest that the general framework developed can provide a useful approach for predicting tumor growth and the effects of radiation.

A hybrid three-scale model of tumor growth

Cancer results from a complex interplay of different biological, chemical, and physical phenomena that span a wide range of time and length scales. Computational modeling may help to unfold the role of multiple evolving factors that exist and interact in the tumor microenvironment. Understanding these complex multiscale interactions is a crucial step towards predicting cancer growth and in developing effective therapies. We integrate different modeling approaches in a multiscale, avascular, hybrid tumor growth model encompassing tissue, cell, and sub-cell scales. At the tissue level, we consider the dispersion of nutrients and growth factors in the tumor microenvironment, which are modeled through reaction-diffusion equations. At the cell level, we use an agent based model (ABM) to describe normal and tumor cell dynamics, with normal cells kept in homeostasis and cancer cells differentiated apoptotic, hypoxic, and necrotic states. Cell movement is driven by the balance of a variety of forces according to Newtons second law, including those related to growth-induced stresses. Phenotypic transitions are defined by specific rule of behaviors that depend on microenvironment stimuli. We integrate in each cell/agent a branch of the epidermal growth factor receptor (EGFR) pathway. This pathway is modeled by a system of coupled nonlinear differential equations involving the mass laws of 20 molecules. The rates of change in the concentration of some key molecules trigger proliferation or migration advantage response. The bridge between cell and tissue scales is built through the reaction and source terms of the partial differential equations. Our hybrid model is built in a modular way, enabling the investigation of the role of different mechanisms at multiple scales on tumor progression. This strategy allows representating both the collective behavior due to cell assembly as well as microscopic intracellular phenomena described by signal transduction pathways. Here, we investigate the impact of some mechanisms associated with sustained proliferation on cancer progression. Specifically, we focus on the intracellular proliferation/migration-advantage-response driven by the EGFR pathway and on proliferation inhibition due to accumulation of growth-induced stresses. Simulations demonstrate that the model can adequately describe some complex mechanisms of tumor dynamics, including growth arrest in avascular tumors. Both the sub-cell model and growth-induced stresses give rise to heterogeneity in the tumor expansion and a rich variety of tumor behaviors.

Calibration of Multi-Parameter Models of Avascular Tumor Growth Using Time Resolved Microscopy Data

Two of the central challenges of using mathematical models for predicting the spatiotemporal development of tumors is the lack of appropriate data to calibrate the parameters of the model, and quantitative characterization of the uncertainties in both the experimental data and the modeling process itself. We present a sequence of experiments, with increasing complexity, designed to systematically calibrate the rates of apoptosis, proliferation, and necrosis, as well as mobility, within a phase-field tumor growth model. The in vitro experiments characterize the proliferation and death of human liver carcinoma cells under different initial cell concentrations, nutrient availabilities, and treatment conditions. A Bayesian framework is employed to quantify the uncertainties in model parameters. The average difference between the calibration and the data, across all time points is between 11.54% and 14.04% for the apoptosis experiments, 7.33% and 23.30% for the proliferation experiments, and 8.12% and 31.55% for the necrosis experiments. The results indicate the proposed experiment-computational approach is generalizable and appropriate for step-by-step calibration of multi-parameter models, yielding accurate estimations of model parameters related to rates of proliferation, apoptosis, and necrosis.

A Influencia da Diferenciação Fenotípica na Dinâmica do Crescimento Tumoral

O crescimento tumoral e resultado de mecanismos nao lineares complexos que ocorrem em diversas escalas de tempo e espaco. A modelagem computacional pode ajudar na compreensao de tais mecanismos, assim como auxiliar no desenvolvimento de terapias efetivas. Neste trabalho, desenvolvemos um modelo hibrido avascular que integra tres escalas espaciais (tecidual, celular e molecular) com o objetivo de estudar a influencia da resposta regulatoria intra-celular nos processos de migracao e proliferacao. Essas repostas resultam de reacoes bioquimicas de sinalizacao molecular iniciadas por estimulos extracelulares. Experimentos computacionais sao realizados para demonstrar a importância da diferenciacao fenotipica na dinâmica do crescimento tumoral.

Calibração de um Modelo de Crescimento Tumoral Avascular

Neste trabalho utilizamos um modelo matematico simples para representar o crescimento de tumores avasculares. Atraves de dados da literatura realizamos a calibracao do modelo proposto tendo como base a abordagem Bayesiana. Nesta abordagem, as distribuicoes a posteriori dos parâmetros sao obtidas utilizando duas tecnicas: amostragem via algoritmo de Metropolis-Hastings, um metodo de Monte Carlo via cadeias de Markov, e amostragem via grade fixa. Ambas conduziram a resultados satisfatorios para os experimentos realizados e sao por natureza mais informativas que os metodos tradicionais de regressao.

Mathematical models of tumor cell proliferation: A review of the literature

ABSTRACT Introduction: A defining hallmark of cancer is aberrant cell proliferation. Efforts to understand the generative properties of cancer cells span all biological scales: from genetic deviations and alterations of metabolic pathways to physical stresses due to overcrowding, as well as the effects of therapeutics and the immune system. While these factors have long been studied in the laboratory, mathematical and computational techniques are being increasingly applied to help understand and forecast tumor growth and treatment response. Advantages of mathematical modeling of proliferation include the ability to simulate and predict the spatiotemporal development of tumors across multiple experimental scales. Central to proliferation modeling is the incorporation of available biological data and validation with experimental data. Areas covered: We present an overview of past and current mathematical strategies directed at understanding tumor cell proliferation. We identify areas for mathematical development as motivated by available experimental and clinical evidence, with a particular emphasis on emerging, non-invasive imaging technologies. Expert commentary: The data required to legitimize mathematical models are often difficult or (currently) impossible to obtain. We suggest areas for further investigation to establish mathematical models that more effectively utilize available data to make informed predictions on tumor cell proliferation.

Modelo Hı́brido para o Crescimento Tumoral do Carcinoma Avascular

crescimento tumoral e resultado de uma serie de complexos fenomenos que ocorrem em multiplas escalas de tempo e espaco. Na escala sub-celular ocorre uma variedade de cascatas de reacoes moleculares que regulam as atividades celulares. Na escala da celula, destacam-se os mecanismos de agregacao, adesao e sinalizacao entre celulas e entre os componentes do microambiente. Fenomenos tipicos dos meios continuos ocorrem na escala do tecido, tais como difusao de nutrientes, fatores de crescimento, etc. Eventos que ocorrem em uma escala interferem com os que ocorrem em outras e vice-versa, de modo que o entendimento destes mecanismos e fundamental para a compreensao da doenca e para o desenvolvimento de terapias. Neste trabalho, desenvolvemos um modelo hibrido que representa fenomenos que ocorrem nas escalas tecidual e celular. A escala celular e descrita atraves de um modelo baseado em agentes (MBA), que possibilita tratar cada celula individualmente e descrever seu comportamento no microambiente. Na escala do tecido representamos a dispersao de nutrientes no meio atraves de uma equacao diferencial parcial de difusao-reacao. Sem perda de generalidade, este modelo e usado para descrever o crescimento tumoral do carcinoma avascular e experimentos computacionais sao realizados para demonstrar o potencial uso da metodologia desenvolvida.

Hierarchical Models of Tumor Growth

We propose in this work a simple framework to build a hierarchical family of tumor growth models by selecting a subset of the most important parameters of our base model with respect to the evolution of the tumor volume. The importance of each parameter is identified through a model-free sensitivity analysis technique, the elementary effects (EE), due to its simplicity and low computational cost. This model framework encompasses the essential hypotheses and the limited set of important parameters acquired from the sensitivity analysis. In this way, we are able to create a family of models described by at least the same essential conditions and parameters but with different complexities regarding the number of parameters used. Numerical experiments are conducted to show the reasoning behind the hierarchical developed family of tumor growth modes. The modeling framework in this manner provides a powerful way for studying a model itself or either its simplification or extension. The framework can also be tailored to form the basis for future models, incorporating new processes and phenomena.

Toxicity Management in CAR T cell therapy for B-ALL: Mathematical modelling as a new avenue for improvement.

Advances in genetic engineering have made it possible to reprogram individual immune cells to express receptors that recognise markers on tumour cell surfaces. The process of re-engineering T cell lymphocytes to express Chimeric Antigen Receptors (CARs), and then re-infusing the CAR-modified T cells into patients to treat various cancers is referred to as CAR T cell therapy. This therapy is being explored in clinical trials - most prominently for B Cell Acute Lymphoblastic Leukaemia (B-ALL), a common B cell malignancy, for which CAR T cell therapy has led to remission in up to 90% of patients. Despite this extraordinary response rate, however, potentially fatal inflammatory side effects occur in up to 10% of patients who have positive responses. Further, approximately 50% of patients who initially respond to the therapy relapse. Significant improvement is thus necessary before the therapy can be made widely available for use in the clinic. To inform future development, we develop a mathematical model to explore interactions between CAR T cells, inflammatory toxicity, and individual patients’ tumour burdens in silico. This paper outlines the underlying system of coupled ordinary differential equations designed based on well-known immunological principles and widely accepted views on the mechanism of toxicity development in CAR T cell therapy for B-ALL - and reports in silico outcomes in relationship to standard and recently conjectured predictors of toxicity in a heterogeneous, randomly generated patient population. Our initial results and analyses are consistent with and connect immunological mechanisms to the clinically observed, counterintuitive hypothesis that initial tumour burden is a stronger predictor of toxicity than is the dose of CAR T cells administered to patients. We outline how the mechanism of action in CAR T cell therapy can give rise to such non-standard trends in toxicity development, and demonstrate the utility of mathematical modelling in understanding the relationship between predictors of toxicity, mechanism of action, and patient outcomes.