Mathematical Model Of Tumor Growth Research Articles

A validated mathematical model of tumor growth including tumor-host interaction, cell-mediated immune response and chemotherapy.

We consider a dynamical model of cancer growth including three interacting cell populations of tumor cells, healthy host cells and immune effector cells. The tumor-immune and the tumor-host interactions are characterized to reproduce experimental results. A thorough dynamical analysis of the model is carried out, showing its capability to explain theoretical and empirical knowledge about tumor development. A chemotherapy treatment reproducing different experiments is also introduced. We believe that this simple model can serve as a foundation for the development of more complicated and specific cancer models.

Abstract 5333: A mathematical model for the kinetics of metastasis formation based on SEER data

Abstract Metastasis is responsible for most cancer mortality, yet many questions remain about how best to describe the kinetics of metastasis formation mathematically. Because the initial step of metastasis is the detachment of cells from a tumor mass, the rate of formation of new metastases is assumed to depend on the size of the original tumor mass. A power-law dependence, rather than linear proportionality, has been assumed, since access to blood vessels and lymphatics becomes more limited as tumor mass increases. Under this assumption, the cumulative distribution of fraction of patients not showing detectable metastases at diagnosis should decrease monotonically as tumor size at detection increases. Using data from the NCI's SEER (Surveillance, Epidemiology, and End Results) 1973-2010 database, it is shown here that this relation in fact shows non-monotonic behavior at larger detection sizes. A mathematical model for tumor growth and metastasis kinetics is developed to show that this implies that proliferation rates and metastatic rates are not independently distributed across the patient population, but rather, show some correlation. The model is further applied to deduce quantitative information about the size-dependence of the rate of metastasis. Kaplan-Meier curves stratified by tumor size, also computed from SEER 1973-2010 data, provide an additional data source for determining model parameters. It is found that accounting for age dependence is necessary for fitting these curves. Two possible age-related effects: a lower lethal tumor burden threshold; and an increased coefficient of metastatic rate due to declining immunity; are considered in the model, and a comparison is made of their ability to describe the observed age-dependence. In sum, the model shows how SEER data can be used to deduce information about the kinetics of metastasis formation, and is expected to have application in optimization of the treatment of metastatic tumors as well as screening schedules. Citation Format: Ardith W. El-Kareh, Timothy W. Secomb. A mathematical model for the kinetics of metastasis formation based on SEER data. [abstract]. In: Proceedings of the 105th Annual Meeting of the American Association for Cancer Research; 2014 Apr 5-9; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2014;74(19 Suppl):Abstract nr 5333. doi:10.1158/1538-7445.AM2014-5333

Traveling wave solution of the Hele–Shaw model of tumor growth with nutrient

Several mathematical models of tumor growth are now commonly used to explain medical observations and predict cancer evolution based on images. These models incorporate mechanical laws for tissue compression combined with rules for nutrients availability which can differ depending on the situation under consideration, in vivo or in vitro. Numerical solutions exhibit, as expected from medical observations, a proliferative rim and a necrotic core. However, their precise profiles are rather complex, both in one and two dimensions. We study a simple free boundary model formed of a Hele–Shaw equation for the cell number density coupled to a diffusion equation for a nutrient. We can prove that a traveling wave solution exists with a healthy region separated from the progressing tumor by a sharp front (the free boundary) while the transition to the necrotic core is smoother. Remarkable is the pressure distribution which vanishes at the boundary of the proliferative rim with a vanishing derivative at the transition point to the necrotic core.

Mathematical Modeling of Tumor Growth and Treatment

Using mathematical models to simulate dynamic biological processes has a long history. Over the past couple of decades or so, quantitative approaches have also made their way into cancer research. An increasing number of mathematical, physical, computational and engineering techniques have been applied to various aspects of tumor growth, with the ultimate goal of understanding the response of the cancer population to clinical intervention. So-called in silico trials that predict patient-specific response to various dose schedules or treatment combinations and sequencing are on the way to becoming an invaluable tool to optimize patient care. Herein we describe fundamentals of mathematical modeling of tumor growth and tumor-host interactions, and summarize some of the seminal and most prominent approaches.

Editorial: Targeting Neuregulin1 and HER Receptor Tyrosine Kinases for Therapy of Breast Cancer and Heart Failure

The Neuregulin1 proteins and HER receptor tyrosine kinases are pivotal for function and disease development of multiple organ systems, including breast cancer and the heart [1, 2]. The HER2 receptor is mutated or overexpressed in about 20-30% of metastatic breast cancers [3]. Therefore, it is a major drug target of breast cancer therapy. Trastuzumab, a monoclonal antibody that directly binds and blocks the HER2 receptor, is among the first drugs approved by the US Food and Drug Administration for targeted cancer therapy, which proven to be effective for improving the overall survival of breast cancer patients [4]. However, Trastuzumab can cause severe heart failure in patients, especially when used in combination with the chemotherapy drug doxorubicin [4]. Clinical studies show that the New York Heart Association class III/IV heart failure is significantly increased in breast cancer patients treated concurrently with Trastuzumab and doxorubicin compared to doxorubicin alone (16 vs. 4 %), suggesting the HER2 receptor is necessary for protecting the heart from doxorubicin-induced cardiotoxicity [4]. This clinical finding led to a decade of vigorous research on the HER signaling in heart failure, as well as the identification of Neuregulin1 as a cardio-protective factor [5-10]. Clinical trials are ongoing to test the therapeutic effects of using recombinant Neuregulin1 for the treatment of heart failure [11]. With the success of inhibiting the HER2 signals in cancer and using Neuregulin1 for heart failure, major clinical challenges remain which are: how to inhibit the HER signaling for cancer therapy while sparing the heart, and whether Neuregulin1 proteins can be safely used in patients without exacerbating the existing cancer burden or increasing the risk of cancer. In this issue, scientists from both the oncology and cardiology fields are getting together to review the current knowledge of the Neuregulin1-HER signaling in breast cancer and the heart. In the paper “Neuregulin Signaling in Pieces – Evolution of the Gene Family”, Mark Marchionni utilizes gene structure and protein sequence searches and alignments, as well as construction of phylogenetic tree to illustrate how Neuregulins and HER receptor tyrosine kinases evolve during evolution to form a complex signaling network that is currently known. In the paper “Heregulin in Breast Cancer: Old Story, New Paradigm”, Ruth Lupu et al. review Neuregulins’ biological role in the development, progression and maintenance of breast cancer. In the paper “Breast Cancer Biomarkers: Risk Assessment, Diagnosis, Prognosis, Prediction of Treatment Efficacy and Toxicity, and Recurrence”, Adedayo Onitilo et al. review the biomarker development for assisting breast cancer diagnosis, prediction of prognosis, therapeutic response and toxicity. In the paper “Cardiovascular Effects of Neuregulin1-1/ErbB Signaling: Role in Vascular Signaling and Angiogenesis”, Kerry Russell et al. review the role of Neuregulin-HER signaling in endothelial cell angiogenesis, vascular smooth muscle cell proliferation and migration, and maintenance of vascular structure and function. This paper also reviews the cross-talk between cardiac endothelial cells and cardiomyocytes, Neuregulin1 and VEGF signaling, as well as Neuregulin1 and inflammatory cytokines. In the paper “The Developing Role of Neuregulin1 in Cardiac Regenerative Stem Cell Therapy”, James Morgan et al. review the findings of using Neuregulin1 proteins to direct stem cell differentiation into the cardiac lineage in cell culture and discuss the potential use of Neuregulin1 to improve the therapeutic efficacy of stem cell therapy for heart failure. In the paper “Anti-HER2 Cancer Therapy and Cardiotoxicity”, Xinhua Yan et al. review the current anti-HER2 therapies, the preclinical and clinical data on cardiotoxicity induced by anti-HER2 therapy and the mechanisms of how HER2 signaling protects the heart from stress. In the paper “Tumor Dormancy and the Angiogenic Switch: Possible Implications of Bone Marrow-Derived Cells”, Nava Almog et al. review the molecular and cellular regulators, including growth stimulatory signals for angiogenic switch in dormant tumors. In the paper “Mathematical Modeling of Tumor Growth and Treatment”, Heiko Enderling et al. discuss how to use mathematic modeling to simulate the dynamic biological process of tumor growth, tumor-host interactions and to predict treatment response.

Tumor growth in complex, evolving microenvironmental geometries: A diffuse domain approach

We develop a mathematical model of tumor growth in complex, dynamic microenvironments with active, deformable membranes. Using a diffuse domain approach, the complex domain is captured implicitly using an auxiliary function and the governing equations are appropriately modified, extended and solved in a larger, regular domain. The diffuse domain method enables us to develop an efficient numerical implementation that does not depend on the space dimension or the microenvironmental geometry. We model homotypic cell–cell adhesion and heterotypic cell–basement membrane (BM) adhesion with the latter being implemented via a membrane energy that models cell–BM interactions. We incorporate simple models of elastic forces and the degradation of the BM and ECM by tumor-secreted matrix degrading enzymes. We investigate tumor progression and BM response as a function of cell–BM adhesion and the stiffness of the BM. We find tumor sizes tend to be positively correlated with cell–BM adhesion since increasing cell–BM adhesion results in thinner, more elongated tumors. Prior to invasion of the tumor into the stroma, we find a negative correlation between tumor size and BM stiffness as the elastic restoring forces tend to inhibit tumor growth. In order to model tumor invasion of the stroma, we find it necessary to downregulate cell–BM adhesiveness, which is consistent with experimental observations. A stiff BM promotes invasiveness because at early stages the opening in the BM created by MDE degradation from tumor cells tends to be narrower when the BM is stiffer. This requires invading cells to squeeze through the narrow opening and thus promotes fragmentation that then leads to enhanced growth and invasion. In three dimensions, the opening in the BM was found to increase in size even when the BM is stiff because of pressure induced by growing tumor clusters. A larger opening in the BM can increase the potential for further invasiveness by increasing the possibility that additional tumor cells could invade the stroma.

WE‐E‐17A‐08: Prediction of Response to Neoadjuvant Chemotherapy Using a Mechanically Coupled Reaction‐Diffusion Model

Purpose:To develop a clinically‐relevant patient‐specific modeling framework for oncology that is amenable to readily available clinical imaging data and yet retains the most salient features of response prediction. We use a mechanically coupled mathematical model of tumor growth that is initialized and constrained by MRI data early in the course of therapy, to guide the determination of model parameters and predict the response of breast cancers to neoadjuvant chemotherapy (NAC).Methods:We adopt a patient‐scale spatiotemporal tumor growth modeling framework and apply patient‐specific predictive modeling, constrained by quantitative imaging data, to a group of 26 patients exhibiting a varying degree of response to NAC. Dynamic contrast enhanced MRI, diffusion weighted MRI, and anatomical T1‐weighted MRI volumes were acquired prior to beginning NAC, after one cycle of NAC, and at the conclusion of NAC. Tumor response is parameterized using data from before and after the first cycle of therapy, and the model is driven forward in time to predict tumor burden at the conclusion of therapy. Model reconstructed parameters and predictions are retrospectively assessed for prognostic value in predicting patients that eventually respond or do not respond to NAC.Results:Using our mechanics‐coupled modeling approach, we are able to discriminate, after the first cycle of therapy, breast cancer patients that would eventually achieve a complete pathological response and those who would not, with an area under the receiver operator characteristic curve of 0.81, sensitivity of 90%, and specificity of 56%.Conclusion:We show the potential for model‐predictions at the conclusion of therapy for use as a prognostic indicator of response to therapy. This work provides considerable promise for predictive modeling centered on integrating quantitative in vivo imaging data with biomechanical models of tumor growth.National Institutes of Health NCI 1U01CA142565, NCI U01CA174706, NCI R25CA092043, NCI 1P50 098131, NCI P30CA68485, NCI R01CA138599, NINDS R01NS049251. The Vanderbilt initiative in Surgery and Engineering Pilot Award Program and the Whitaker Foundation.

Mathematical Modeling of Tumor Growth, Drug-Resistance, Toxicity, and Optimal Therapy Design

The combination of mathematical modeling and optimal control techniques holds great potential for quantitatively describing tumor progression and optimal treatment planning. Hereby, we use a Gompertz-type growth law and a pharmacokinetic-pharmacodynamic approach for modeling the effects of drugs on tumor progression in tumor bearing mice, and we combine these in order to design optimal therapeutic patterns. Specifically, we describe colon cancer progression in both untreated mice as well as mice treated with widely used anticancer agents. We also present a pharmacokinetic model to describe the kinetics of drugs in the body as well as detailed toxicity models to describe the severity of side effects. Finally, we propose a promising methodology by which cancer progression in mice with drug resistance can be controlled. By using optimal control, we demonstrate that the optimal planning of the frequency and magnitude of treatment interruptions is key to the control of cancer progression in subjects with resistance and should be further investigated in an experimental setting, which is currently underway.

Non-dimensional system for analysis equilibrium point mathematical model of tumor growth

In this paper we presented analysis equilibrium point of mathematical model tumor growth use non-dimensional technique. This technique is useful to simply equation that complicated as a mathematical model that present in this paper. This mathematical model describes the effect of tumor infiltrating lymphocytes (TIL) and interleukin-2 (IL-2) on the dynamics of tumor cells. The steps of this technique are presented. With this technique we can be to determine more accurately the equilibria of our system which form nonlinear dynamics of system ordinary differential equation that coupled.

Predictive Modeling of In Vivo Response to Gemcitabine in Pancreatic Cancer

A clear contradiction exists between cytotoxic in-vitro studies demonstrating effectiveness of Gemcitabine to curtail pancreatic cancer and in-vivo studies failing to show Gemcitabine as an effective treatment. The outcome of chemotherapy in metastatic stages, where surgery is no longer viable, shows a 5-year survival <5%. It is apparent that in-vitro experiments, no matter how well designed, may fail to adequately represent the complex in-vivo microenvironmental and phenotypic characteristics of the cancer, including cell proliferation and apoptosis. We evaluate in-vitro cytotoxic data as an indicator of in-vivo treatment success using a mathematical model of tumor growth based on a dimensionless formulation describing tumor biology. Inputs to the model are obtained under optimal drug exposure conditions in-vitro. The model incorporates heterogeneous cell proliferation and death caused by spatial diffusion gradients of oxygen/nutrients due to inefficient vascularization and abundant stroma, and thus is able to simulate the effect of the microenvironment as a barrier to effective nutrient and drug delivery. Analysis of the mathematical model indicates the pancreatic tumors to be mostly resistant to Gemcitabine treatment in-vivo. The model results are confirmed with experiments in live mice, which indicate uninhibited tumor proliferation and metastasis with Gemcitabine treatment. By extracting mathematical model parameter values for proliferation and death from monolayer in-vitro cytotoxicity experiments with pancreatic cancer cells, and simulating the effects of spatial diffusion, we use the model to predict the drug response in-vivo, beyond what would have been expected from sole consideration of the cancer intrinsic resistance. We conclude that this integrated experimental/computational approach may enhance understanding of pancreatic cancer behavior and its response to various chemotherapies, and, further, that such an approach could predict resistance based on pharmacokinetic measurements with the goal to maximize effective treatment strategies.

Basic ingredients for mathematical modeling of tumor growth in vitro: Cooperative effects and search for space

Based on the literature data from HT-29 cell monolayers, we develop a model for its growth, analogous to an epidemic model, mixing local and global interactions. First, we propose and solve a deterministic equation for the progress of these colonies. Thus, we add a stochastic (local) interaction and simulate the evolution of an Eden-like aggregate by using dynamical Monte Carlo methods. The growth curves of both deterministic and stochastic models are in excellent agreement with the experimental observations. The waiting times distributions, generated via our stochastic model, allowed us to analyze the role of mesoscopic events. We obtain log-normal distributions in the initial stages of the growth and Gaussians at long times. We interpret these outcomes in the light of cellular division events: in the early stages, the phenomena are dependent each other in a multiplicative geometric-based process, and they are independent at long times. We conclude that the main ingredients for a good minimalist model of tumor growth, at mesoscopic level, are intrinsic cooperative mechanisms and competitive search for space.

3D numerical simulation of avascular tumour growth: effect of hypoxic micro-environment in host tissue

A three-dimensional (3D) mathematical model of tumour growth at the avascular phase and vessel remodelling in host tissues is proposed with emphasis on the study of the interactions of tumour growth and hypoxic micro-environment in host tissues. The hybrid based model includes the continuum part, such as the distributions of oxygen and vascular endothelial growth factors (VEGFs), and the discrete part of tumour cells (TCs) and blood vessel networks. The simulation shows the dynamic process of avascular tumour growth from a few initial cells to an equilibrium state with varied vessel networks. After a phase of rapidly increasing numbers of the TCs, more and more host vessels collapse due to the stress caused by the growing tumour. In addition, the consumption of oxygen expands with the enlarged tumour region. The study also discusses the effects of certain factors on tumour growth, including the density and configuration of preexisting vessel networks and the blood oxygen content. The model enables us to examine the relationship between early tumour growth and hypoxic micro-environment in host tissues, which can be useful for further applications, such as tumour metastasis and the initialization of tumour angiogenesis.

Coevolution of Solid Stress and Interstitial Fluid Pressure in Tumors During Progression: Implications for Vascular Collapse

The stress harbored by the solid phase of tumors is known as solid stress. Solid stress can be either applied externally by the surrounding normal tissue or induced by the tumor itself due to its growth. Fluid pressure is the isotropic stress exerted by the fluid phase. We recently showed that growth-induced solid stress is on the order of 1.3 to 13.0 kPa (10-100 mmHg)--high enough to cause compression of fragile blood vessels, resulting in poor perfusion and hypoxia. However, the evolution of growth-induced stress with tumor progression and its effect on cancer cell proliferation in vivo is not understood. To this end, we developed a mathematical model for tumor growth that takes into account all three types of stresses: growth-induced stress, externally applied stress, and fluid pressure. First, we conducted in vivo experiments and found that growth-induced stress is related to tumor volume through a biexponential relationship. Then, we incorporated this information into our mathematical model and showed that due to the evolution of growth-induced stress, total solid stress levels are higher in the tumor interior and lower in the periphery. Elevated compressive solid stress in the interior of the tumor is sufficient to cause the collapse of blood vessels and results in a lower growth rate of cancer cells compared with the periphery, independently from that caused by the lack of nutrients due to vessel collapse. Furthermore, solid stress in the periphery of the tumor causes blood vessels in the surrounding normal tissue to deform to elliptical shapes. We present histologic sections of human cancers that show such vessel deformations. Finally, we found that fluid pressure increases with tumor growth due to increased vascular permeability and lymphatic impairment, and is governed by the microvascular pressure. Crucially, fluid pressure does not cause vessel compression of tumor vessels.

SU-E-J-199: An Image-Based Model of Glioblastoma Growth for Treatment Response Assessment

Purpose: To develop a mathematical model of tumor growth based on MRI imaging to predict tumor growth, with potential applications in treatment response assessment to differentiate responding and non‐responding tumor types. Method: The model is composed of 3 terms describing tumor proliferation, diffusion in various tissue types, and radiation treatment effect. The proliferation terms describes the rate at which the cell replicates while the diffusion term describes tumor invasion into adjacent tissue. The radiotherapy effect term models cell response to radiation through a linear quadratic radiobiological model. Tissue‐specific evolution taking into account the differential diffusion in white and gray matter is accomplished by segmenting T1Pre, T1POST and FLAIR images using a support vector machine (SVM) algorithm. Patient specific model parameters of perfusion and diffusion are deduced from 3 serial apparent diffusion coefficients (ADC) imaging maps that are evolved under the guidance of a partial differential equation to deduce tumor evolution at later times. Results: The model tumor was applied on 19 glioblastoma patients and initialized from their ADC maps acquired before, after and 1 month after treatment completion. Voxel wise maps of the perfusion, diffusion and tumor spreading speed were created to assess treatment efficiency. The proliferation was measured in region enhancing in the pre‐treatment imaging and ranged between −0.0189 and 0.00820 mm/day while the diffusion coefficients ranged between 0.0317 and 0.4721 per day. Conclusion: We have developed a tumor growth model and applied it on clinical patients to estimate response to treatment. The model may find also applications follow‐up method for radiation treatments as well as in designing treatment margins that take into account tumor diffusion speed and direction.

Serial Diffusion MRI to Monitor and Model Treatment Response of the Targeted Nanotherapy CRLX101

Targeted nanotherapies are being developed to improve tumor drug delivery and enhance therapeutic response. Techniques that can predict response will facilitate clinical translation and may help define optimal treatment strategies. We evaluated the efficacy of diffusion-weighted magnetic resonance imaging to monitor early response to CRLX101 (a cyclodextrin-based polymer particle containing the DNA topoisomerase I inhibitor camptothecin) nanotherapy (formerly IT-101), and explored its potential as a therapeutic response predictor using a mechanistic model of tumor cell proliferation. Diffusion MRI was serially conducted following CRLX101 administration in a mouse lymphoma model. Apparent diffusion coefficients (ADCs) extracted from the data were used as treatment response biomarkers. Animals treated with irinotecan (CPT-11) and saline were imaged for comparison. ADC data were also input into a mathematical model of tumor growth. Histological analysis using cleaved-caspase 3, terminal deoxynucleotidyl transferase-mediated dUTP nick end labeling, Ki-67, and hematoxylin and eosin (H&E) were conducted on tumor samples for correlation with imaging results. CRLX101-treated tumors at day 2, 4, and 7 posttreatment exhibited changes in mean ADC = 16 ± 9%, 24 ± 10%, 49 ± 17%, and size (TV) = -5 ± 3%, -30 ± 4%, and -45 ± 13%, respectively. Both parameters were statistically greater than controls [p(ADC) ≤ 0.02, and p(TV) ≤ 0.01 at day 4 and 7], and noticeably greater than CPT-11-treated tumors (ADC = 5 ± 5%, 14 ± 7%, and 18 ± 6%; TV = -15 ± 5%, -22 ± 13%, and -26 ± 8%). Model-derived parameters for cell proliferation obtained using ADC data distinguished CRLX101-treated tumors from controls (P = 0.02). Temporal changes in ADC specified early CRLX101 treatment response and could be used to model image-derived cell proliferation rates following treatment. Comparisons of targeted and nontargeted treatments highlight the utility of noninvasive imaging and modeling to evaluate, monitor, and predict responses to targeted nanotherapeutics.

In vivo mathematical modeling of tumor growth from imaging data: Soon to come in the future?

The future challenges in oncology imaging are to assess the response to treatment even earlier. As an addition to functional imaging, mathematical modeling based on the imaging is an alternative, cross-disciplinary area of development. Modeling was developed in oncology not only in order to understand and predict tumor growth, but also to anticipate the effects of targeted and untargeted therapies. A very wide range of these models exist, involving many stages in the progression of tumors. Few models, however, have been proposed to reproduce in vivo tumor growth because of the complexity of the mechanisms involved. Morphological imaging combined with "spatial" models appears to perform well although functioning imaging could still provide further information on metabolism and the micro-architecture. The combination of imaging and modeling can resolve complex problems and describe many facets of tumor growth or response to treatment. It is now possible to consider its clinical use in the medium term. This review describes the basic principles of mathematical modeling and describes the advantages, limitations and future prospects for this in vivo approach based on imaging data.

Multiphase modelling of desmoplastic tumour growth

It is well-known that the microenvironment of solid tumours is a significant component of the processes of tumour growth and invasion. Interactions between tumour cells and stromal components play a crucial role in tumour progression as well as suppression. We describe a mathematical model of tumour growth within a host tissue which takes into account both cell-extracellular matrix interactions and tissue compression effects. This multiphase model consisting of three coupled partial differential equations captures the dynamics of tumour progression, particularly of a desmoplastic tumour (i.e. a tumour rich in fibrous connective tissue). The model is analysed in terms of stability in a spatially homogenous case. Computer simulations agree with the biological picture of the disease and may help to understand the process leading to the pathology.

Abstract A2: The importance of metabolic heterogeneity and its consequences on tumor invasion, metastatic growth, and treatment

Abstract Heterogeneity of both the tumor microenvironment and cellular phenotype plays a key role in guiding tumor behavior. Metabolism acts as a central integrator to mediate cellular response to the changing environment, linking the effects of hypoxia, acidosis, proliferation and necrosis. Using a hybrid multi-scale mathematical model of tumor growth in a vascularized tissue, we investigate the selection pressures exerted by the tumor microenvironment as the cancer progresses. A key focus is the metabolism of both normal and tumor cells, and how small changes in phenotype can lead to the development of more metabolically aggressive groups of cells. Development of a glycolytic phenotype due to microenvironmental selection pressures leads to acid-mediated invasion. Results from the model suggest that these aggressive cells have a dramatic advantage as a metastatic seed, compared to less aggressive seeds from the same primary. In addition, there are significant differences in metastatic growth rate depending on the soil, suggesting that prevention or treatment of metastases in different organs may require different strategies. Application of pH buffering therapy has been shown in mouse models to slow down or prevent both primary tumor and metastatic growth. This treatment has been simulated in the model, and the results show that the effect of the treatment is highly sensitive to the timing of application. Additional results suggest that diffusible cytotoxic treatments such as chemotherapy may increase the metabolic aggressiveness of a tumor post-treatment due to the altered selection pressure caused by the drug. Chemotherapy removes the metabolic stratification of the tumor and allows more aggressive cells to grow towards blood vessels and normal tissue. A third treatment we consider is the use of anti-angiogenic therapy, which foments the development of aggressive phenotypes due to degradation of the tumor microenvironment. The model highlights the role of metabolic heterogeneity and microenvironmental selection in driving tumor invasion and metastatic development. Citation Format: Mark Robertson-Tessi, Robert J. Gillies, Robert A. Gatenby, Alexander R.A. Anderson. The importance of metabolic heterogeneity and its consequences on tumor invasion, metastatic growth, and treatment. [abstract]. In: Proceedings of the AACR Special Conference on Tumor Invasion and Metastasis; Jan 20-23, 2013; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2013;73(3 Suppl):Abstract nr A2.

On a mathematical model of tumor growth based on cancer stem cells

We consider a simple mathematical model of tumor growth based on cancer stem cells. The model consists of four hyperbolic equations of first order to describe the evolution of different subpopulations of cells: cancer stem cells, progenitor cells, differentiated cells and dead cells. A fifth equation is introduced to model the evolution of the moving boundary. The system includes non-local terms of integral type in the coefficients. Under some restrictions in the parameters we show that there exists a unique homogeneous steady state which is stable.

Mathematical Models of Tumor Growth and Angiogenesis

In this paper we present several mathematical models of tumor growth and angiogenesis expressed by systems of ODE’s encoding the most essential observations and assumptions about the complex hierarchical interactive processes of tumor neo-vascularization (angiogenesis). The simplest modeling option presented merely captures the three independent variables mentioned earlier-tumor size N, total vessel volume V and the amount of protein P. We modify this model assuming that the protein is additionally consumed by growing vessels and obtain a model with protein consumption. Next models with time-delays are introduced. To make our models more realistic, two more compartments representing more complex vascularity and protein effects are introduced. Hence five dimensional models are obtained.