# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

2013 , Volume 18 , Issue 4

Special issue on cancer modeling, analysis and control

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2013, 18(4): i-iii doi: 10.3934/dcdsb.2013.18.4i +[Abstract](449) +[PDF](139.1KB)
Abstract:
This special issue of Discrete and Continuous Dynamical Systems Series B in Cancer Modeling, Analysis and Control originated as a result of the Casablanca International Workshop on Mathematical Biology, held in Casablanca, Morocco from June 20-24, 2011. This five-day workshop was aimed at: (1) bringing together US, international, and African experts in the field of mathematical biology to exchange ideas, (2) advancing the state of research in the field of mathematical modeling of infectious and in host diseases, and (3) providing a platform for the participants to explore opportunities for collaboration across disciplinary boundaries, as well as across countries and continents. The general themes of the workshop were the modeling and analysis of infectious diseases in Africa such as HIV, tuberculosis, influenza, malaria, cholera, and within-host diseases such as cancer. A particular focus of this workshop was on the analysis and control of cancer models.

2013, 18(4): 847-863 doi: 10.3934/dcdsb.2013.18.847 +[Abstract](336) +[PDF](439.8KB)
Abstract:
This paper deals with the derivation of macroscopic tissue models from the underlying description delivered by a class of equations modeling binary mixtures of multi-cellular systems by methods of the kinetic theory for active particles. Cellular interactions generate both modification of biological functions and proliferative-destructive events. The analysis refers to a suitable hyperbolic approximation to show how the macroscopic tissue behavior can be described from the underlying cellular description. The approach is specifically focused on the modeling of chemotaxis phenomena by the Keller--Segel approximation.
2013, 18(4): 865-889 doi: 10.3934/dcdsb.2013.18.865 +[Abstract](308) +[PDF](981.2KB)
Abstract:
We review the main types of mathematical models that have been designed to represent and predict the evolution of a cell population under the action of anti-cancer drugs that are in use in the clinic, with effects on healthy and cancer tissue growth, which from a cell functional point of view are classically divided between proliferation, death and differentiation''. We focus here on the choices of the drug targets in these models, aiming at showing that they must be linked in each case to a given therapeutic application. We recall some analytical results that have been obtained in using models of proliferation in cell populations with control in recent years. We present some simulations performed when no theoretical result is available and we state some open problems. In view of clinical applications, we propose possible ways to design optimal therapeutic strategies by using combinations of drugs, cytotoxic, cytostatic, or redifferentiating agents, depending on the type of cancer considered, acting on different targets at the level of cell populations.
2013, 18(4): 891-914 doi: 10.3934/dcdsb.2013.18.891 +[Abstract](398) +[PDF](1492.2KB)
Abstract:
This paper presents a mathematical model for immune response against cancer aimed at reproducing emerging phenomena arising from the interactions between tumor and immune cells. The model is stated in terms of integro-differential equations and describes the dynamics of tumor cells, characterized by heterogeneous antigenic expressions, antigen-presenting cells and T-cells. Asymptotic analysis and simulations, developed with an exploratory aim, are addressed to verify the consistency of the model outputs as well as to provide biological insights into the mechanisms that rule tumor-immune interactions. In particular, the present model seems able to mimic the recognition, learning and memory aspects of immune response and highlights how the immune system might act as an additional selective pressure leading, eventually, to the selection for the most resistant cancer clones.
2013, 18(4): 915-943 doi: 10.3934/dcdsb.2013.18.915 +[Abstract](335) +[PDF](933.1KB)
Abstract:
We present a mathematical model to study the effects of the regulatory T cells (T$_{\textrm{reg}}$) on Renal Cell Carcinoma (RCC) treatment with sunitinib. The drug sunitinib inhibits the natural self-regulation of the immune system, allowing the effector components of the immune system to function for longer periods of time. This mathematical model builds upon our non-linear ODE model by de Pillis et al. (2009) [13] to incorporate sunitinib treatment, regulatory T cell dynamics, and RCC-specific parameters. The model also elucidates the roles of certain RCC-specific parameters in determining key differences between in silico patients whose immune profiles allowed them to respond well to sunitinib treatment, and those whose profiles did not.
Simulations from our model are able to produce results that reflect clinical outcomes to sunitinib treatment such as: (1) sunitinib treatments following standard protocols led to improved tumor control (over no treatment) in about 40% of patients; (2) sunitinib treatments at double the standard dose led to a greater response rate in about 15% the patient population; (3) simulations of patient response indicated improved responses to sunitinib treatment when the patient's immune strength scaling and the immune system strength coefficients parameters were low, allowing for a slightly stronger natural immune response.
2013, 18(4): 945-967 doi: 10.3934/dcdsb.2013.18.945 +[Abstract](342) +[PDF](759.8KB)
Abstract:
Due to its dependence on androgens, metastatic prostate cancer is typically treated with continuous androgen ablation. However, such therapy eventually fails due to the emergence of castration-resistance cells. It has been hypothesized that intermittent androgen ablation can delay the onset of this resistance. In this paper, we present a biochemically-motivated ordinary differential equation model of prostate cancer response to anti-androgen therapy, with the aim of predicting optimal treatment protocols based on individual patient characteristics. Conditions under which intermittent scheduling is preferable over continuous therapy are derived analytically for a variety of castration-resistant cell phenotypes. The model predicts that while a cure is not possible for androgen-independent castration-resistant cells, continuous therapy results in longer disease-free survival periods. However, for androgen-repressed castration-resistant cells, intermittent therapy can significantly delay the emergence of resistance, and in some cases induce tumor regression. Numerical simulations of the model lead to two interesting cases, where even though continuous therapy may be non-viable, an optimally chosen intermittent schedule leads to tumor regression, and where a sub-optimally chosen intermittent schedule can initially appear to result in a cure, it eventually leads to resistance emergence. These results demonstrate the model's potential impact in a clinical setting.
2013, 18(4): 969-1015 doi: 10.3934/dcdsb.2013.18.969 +[Abstract](413) +[PDF](5181.5KB)
Abstract:
Glioblastoma is the most aggressive type of brain cancer with the median survival time of one year. A particular microRNA, miR-451, and its counterpart, AMPK complex are known to play a key role in controlling a balance between rapid proliferation and aggressive invasion in response to metabolic stress in the microenvironment. The present paper develops a hybrid model of glioblastoma that identifies a key mechanism behind the molecular switches between proliferative phase and migratory phase in response to metabolic stress and biophysical interaction between cells. We focus on the core miR-451-AMPK control system and show how up- or down-regulation of components in these pathways affects cell proliferation and migration. The model predicts the larger window of bistable systems when there exists a time delay in the inhibitory pathway from CAB39/LKB1/STRAD/AMPK to miR-451. Delayed down-regulation of miR-451 along this pathway would let glioma cells stay longer in the proliferative stage despite relatively low glucose levels, making it a possible therapeutic target. Analysis of the model predicts the existence of a limit cycle with two time delays. We then study a hybrid model for the biomechanical interaction between invasive and proliferative cells, in which all cells are modeled individually, and show how biophysical properties of cells and core miR-451-AMPK control system affect the growth/invasion patterns of glioma spheroids in response to various glucose levels in the microenvironment. The model predicts that cell migration not only depends on glucose availability but also on mechanical constraints between cells. The model suggests that adhesion strength between cells plays an important role in cell shedding from the main core and the disruption of cell-cell adhesion is a pre-requisite for glioma cell invasion. The model also suggests that injection of glucose after surgery will increase visibility of individual migratory cells and the second surgery may eradicate the remaining cancer cells, preventing regrowth of the invisible migratory glioma cells.
2013, 18(4): 1017-1030 doi: 10.3934/dcdsb.2013.18.1017 +[Abstract](286) +[PDF](1799.5KB)
Abstract:
Aggressive cancers develop immune suppression mechanisms, allowing them to evade specific immune responses. Patients with active melanoma are polarized towards a T helper (Th) 2-type immune phenotype, which subverts effective anticancer Th1-type cellular immunity. The pro-inflammatory factor, interleukin (IL)-12, can potentially restore Th1 responses in such patients, but still shows limited clinical efficacy and substantial side effects. We developed a model for the Th1/Th2 imbalance in melanoma patients and its regulation via IL-12 treatment. The model focuses on the interactions between the two Th cell types as mediated by their respective key cytokines, interferon (IFN)-$\gamma$ and IL-10. Theoretical and numerical analysis showed a landscape consisting of a single, globally attracting steady state, which is stable under large ranges of relevant parameter values. Our results suggest that in melanoma, the cellular arm of the immune system cannot reverse tumor immunotolerance naturally, and that immunotherapy may be the only way to overturn tumor dominance. We have shown that given a toxicity threshold for IFN$\gamma$, the maximal allowable IL-12 concentration to yield a Th1-polarized state can be estimated. Moreover, our analysis pinpoints the IL-10 secretion rate as a significant factor influencing the Th1:Th2 balance, suggesting its use as a personal immunomarker for prognosis.
2013, 18(4): 1031-1051 doi: 10.3934/dcdsb.2013.18.1031 +[Abstract](288) +[PDF](1147.5KB)
Abstract:
An optimal control problem for combination of cancer chemotherapy with immunotherapy in form of a boost to the immune system is considered as a multi-input optimal control problem. The objective to be minimized is chosen as a weighted average of (i) the number of cancer cells at the terminal time, (ii) a measure for the immunocompetent cell densities at the terminal point (included as a negative term), the overall amounts of (iii) cytotoxic agents and (iv) immune boost given as a measure for the side effects of treatment and (v) a small penalty on the free terminal time that limits the overall therapy horizon. This last term is essential in obtaining a mathematically well-posed problem formulation. Both analytical and numerical results about the structures of optimal controls will be presented that give some insights into the structure of optimal protocols, i.e., the dose rates and sequencing of drugs in these combination treatments.
2013, 18(4): 1053-1076 doi: 10.3934/dcdsb.2013.18.1053 +[Abstract](311) +[PDF](1076.8KB)
Abstract:
B cell chronic lymphocytic leukemia (B-CLL) is known to have substantial clinical heterogeneity. There is no cure, but treatments allow for disease management. However, the wide range of clinical courses experienced by B-CLL patients makes prognosis and hence treatment a significant challenge. In an attempt to study disease progression across different patients via a unified yet flexible approach, we present a mathematical model of B-CLL with immune response, that can capture both rapid and slow disease progression. This model includes four different cell populations in the peripheral blood of humans: B-CLL cells, NK cells, cytotoxic T cells and helper T cells. We analyze existing data in the medical literature, determine ranges of values for parameters of the model, and compare our model outcomes to clinical patient data. The goal of this work is to provide a tool that may shed light on factors affecting the course of disease progression in patients. This modeling tool can serve as a foundation upon which future treatments can be based.
2013, 18(4): 1077-1108 doi: 10.3934/dcdsb.2013.18.1077 +[Abstract](335) +[PDF](1445.4KB)
Abstract:
Healthy vasculature exhibits a hierarchical branching structure in which, on average, vessel radius and length change systematically with branching order. In contrast, tumor vasculature exhibits less hierarchy and more variability in its branching patterns. Although differences in vasculature have been highlighted in the literature, there has been very little quantification of these differences. Fractal analysis is a natural tool for comparing tumor and healthy vasculature, especially because it has already been used extensively to model healthy tissue. In this paper, we provide a fractal analysis of existing vascular data, and we present a new mathematical framework for predicting tumor growth trajectories by coupling: (1) the fractal geometric properties of tumor vascular networks, (2) metabolic properties of tumor cells and host vascular systems, and (3) spatial gradients in resources and metabolic states within the tumor. First, we provide a new analysis for how the mean and variation of scaling exponents for ratios of vessel radii and lengths in tumors differ from healthy tissue. Next, we use these characteristic exponents to predict metabolic rates for tumors. Finally, by combining this analysis with general growth equations based on energetics, we derive universal growth curves that enable us to compare tumor and ontogenetic growth. We also extend these growth equations to include necrotic, quiescent, and proliferative cell states and to predict novel growth dynamics that arise when tumors are treated with drugs. Taken together, this mathematical framework will help to anticipate and understand growth trajectories across tumor types and drug treatments.
2013, 18(4): 1109-1154 doi: 10.3934/dcdsb.2013.18.1109 +[Abstract](390) +[PDF](1126.6KB)
Abstract:
This work presents a continuous model for three early stage events in angiogenesis: initiation, sprout extension, and vessel maturation. We carefully examine the regulating mechanisms of vascular endothelial growth factor (VEGF) and angiopoietins (Ang1 and Ang2) on the proliferation, migration and maturation of endothelial cells through their endothelium-specific receptor tyrosine kinase VEGFR2 and Tie2, respectively. We also consider the effect of platelet-derived growth factor-B (PDGF-B) on the proliferation and migration of pericytes. For growth factors, we present a mathematical model integrating molecular reactions on blood vessels with tissue-level diffusion. For capillary extension, we develop a visco-elastic model to couple tip cell protrusion, endothelium elasticity, and stalk cell proliferation. Our model reproduces corneal angiogenesis experiments and several anti-angiogenesis therapy results. This model also demonstrates that (1) the competition between Ang1 and Ang2 is the angiogenic switch; (2) the maturation process modulated by pericytes and angiopoietins is crucial to vessel normalization and can explain the resistance to anti-VEGF therapy; (3) combined anti-pericyte and anti-VEGF therapy enhances blood vessel regression over anti-VEGF therapy alone.

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