# American Institute of Mathematical Sciences

ISSN:
1551-0018

eISSN:
1547-1063

All Issues

## Mathematical Biosciences & Engineering

2013 , Volume 10 , Issue 1

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2013, 10(1): i-ii doi: 10.3934/mbe.2013.10.1i +[Abstract](2559) +[PDF](116.6KB)
Abstract:
There is an increasing awareness that to properly understand how tumors originate and grow, and then how to develop effective cures, it must be taken into account the dynamics of tumors and its great complexity. Tumors are characterized not only by the coexistence of multiple scales, both temporal and spatial, but also by multiple and quite different kinds of interactions, from chemical to mechanical. This makes the study of tumors remarkably complicated. A genuine explosion of data concerning the multi-faceted aspects of this family of phenomena collectively called cancers, is now becoming available. At the same time, it is becoming quite evident that traditional tools from biostatistics and bioinformatics cannot manage these data. Mathematics, theoretical biophysics and computer sciences are needed to qualitatively and quantitatively interpret experimental and clinical results, in order to make realistic predictions.

2013, 10(1): 1-17 doi: 10.3934/mbe.2013.10.1 +[Abstract](3101) +[PDF](538.4KB)
Abstract:
Cell proliferation is controlled by many complex regulatory networks. Our purpose is to analyse, through mathematical modeling, the effects of growth factors on the dynamics of the division cycle in cell populations.
Our work is based on an age-structured PDE model of the cell division cycle within a population of cells in a common tissue. Cell proliferation is at its first stages exponential and is thus characterised by its growth exponent, the first eigenvalue of the linear system we consider here, a growth exponent that we will explicitly evaluate from biological data. Moreover, this study relies on recent and innovative imaging data (fluorescence microscopy) that make us able to experimentally determine the parameters of the model and to validate numerical results. This model has allowed us to study the degree of simultaneity of phase transitions within a proliferating cell population and to analyse the role of an increased growth factor concentration in this process.
This study thus aims at helping biologists to elicit the impact of growth factor concentration on cell cycle regulation, at making more precise the dynamics of key mechanisms controlling the division cycle in proliferating cell populations, and eventually at establishing theoretical bases for optimised combined anticancer treatments.
2013, 10(1): 19-35 doi: 10.3934/mbe.2013.10.19 +[Abstract](2255) +[PDF](501.3KB)
Abstract:
In the paper we consider the model of tumour angiogenesis process proposed by Bodnar&Foryś (2009). The model combines ideas of Hahnfeldt et al. (1999) and Agur et al. (2004) describing the dynamics of tumour, angiogenic proteins and effective vessels density. Presented analysis is focused on the dependance of the model dynamics on delays introduced to the system. These delays reflect time lags in the proliferation/death term and the vessel formation/regression response to stimuli. It occurs that the dynamics strongly depends on the model parameters and the behaviour independent of the delays magnitude as well as multiple stability switches with increasing delay can be obtained.
2013, 10(1): 37-57 doi: 10.3934/mbe.2013.10.37 +[Abstract](2618) +[PDF](2344.7KB)
Abstract:
A tumor is kinetically characterized by the presence of multiple spatio-temporal scales in which its cells interplay with, for instance, endothelial cells or Immune system effectors, exchanging various chemical signals. By its nature, tumor growth is an ideal object of hybrid modeling where discrete stochastic processes model low-numbers entities, and mean-field equations model abundant chemical signals. Thus, we follow this approach to model tumor cells, effector cells and Interleukin-2, in order to capture the Immune surveillance effect.
We here present a hybrid model with a generic delay kernel accounting that, due to many complex phenomena such as chemical transportation and cellular differentiation, the tumor-induced recruitment of effectors exhibits a lag period. This model is a Stochastic Hybrid Automata and its semantics is a Piecewise Deterministic Markov process where a two-dimensional stochastic process is interlinked to a multi-dimensional mean-field system. We instantiate the model with two well-known weak and strong delay kernels and perform simulations by using an algorithm to generate trajectories of this process.
Via simulations and parametric sensitivity analysis techniques we $(i)$ relate tumor mass growth with the two kernels, we $(ii)$ measure the strength of the Immune surveillance in terms of probability distribution of the eradication times, and $(iii)$ we prove, in the oscillatory regime, the existence of a stochastic bifurcation resulting in delay-induced tumor eradication.
2013, 10(1): 59-73 doi: 10.3934/mbe.2013.10.59 +[Abstract](1881) +[PDF](435.3KB)
Abstract:
In a previous paper a mathematical model was developed for the dynamics of activation and clonal expansion of T cells during the immune response to a single type of antigen challenge, constructed phenomenologically in the macroscopic framework of a thermodynamic theory of continuum mechanics for reacting and proliferating fluid mixtures. The present contribution deals with approximate smooth solutions, called asymptotic waves, of the system of PDEs describing the introduced model, obtained using a suitable perturbative method. In particular, in the one-dimensional case, after deriving the expression of the velocity along the characteristic rays and the equation of the wave front, the transport equation for the first perturbation term of the asymptotic solution is obtained. Finally, it is shown that this transport equation can be reduced to an equation similar to Burgers equation.
2013, 10(1): 75-101 doi: 10.3934/mbe.2013.10.75 +[Abstract](3156) +[PDF](1172.6KB)
Abstract:
The basement membrane (BM) and extracellular matrix (ECM) play critical roles in developmental and cancer biology, and are of great interest in biomathematics. We introduce a model of mechanical cell-BM-ECM interactions that extends current (visco)elastic models (e.g. [8,16]), and connects to recent agent-based cell models (e.g. [2,3,20,26]). We model the BM as a linked series of Hookean springs, each with time-varying length, thickness, and spring constant. Each BM spring node exchanges adhesive and repulsive forces with the cell agents using potential functions. We model elastic BM-ECM interactions with analogous ECM springs. We introduce a new model of plastic BM and ECM reorganization in response to prolonged strains, and new constitutive relations that incorporate molecular-scale effects of plasticity into the spring constants. We find that varying the balance of BM and ECM elasticity alters the node spacing along cell boundaries, yielding a nonuniform BM thickness. Uneven node spacing generates stresses that are relieved by plasticity over long times. We find that elasto-viscoplastic cell shape response is critical to relieving uneven stresses in the BM. Our modeling advances and results highlight the importance of rigorously modeling of cell-BM-ECM interactions in clinically important conditions with significant membrane deformations and time-varying membrane properties, such as aneurysms and progression from in situ to invasive carcinoma.
2013, 10(1): 103-120 doi: 10.3934/mbe.2013.10.103 +[Abstract](2144) +[PDF](5062.7KB)
Abstract:
We started offering an introduction to very basic aspects of molecular biology, for the reader coming from computer sciences, information technology, mathematics. Similarly we offered a minimum of information about pathways and networks in graph theory, for a reader coming from the bio-medical sector. At the crossover about the two different types of expertise, we offered some definition about Systems Biology. The core of the article deals with a Molecular Interaction Map (MIM), a network of biochemical interactions involved in a small signaling-network sub-region relevant in breast cancer. We explored robustness/sensitivity to random perturbations. It turns out that our MIM is a non-isomorphic directed graph. For non physiological directions of propagation of the signal the network is quite resistant to perturbations. The opposite happens for biologically significant directions of signal propagation. In these cases we can have no signal attenuation, and even signal amplification. Signal propagation along a given pathway is highly unidirectional, with the exception of signal-feedbacks, that again have a specific biological role and significance. In conclusion, even a relatively small network like our present MIM reveals the preponderance of specific biological functions over unspecific isomorphic behaviors. This is perhaps the consequence of hundreds of millions of years of biological evolution.
2013, 10(1): 121-150 doi: 10.3934/mbe.2013.10.121 +[Abstract](2705) +[PDF](5049.8KB)
Abstract:
Experimental evidence suggests that a tumor's environment may be critical to designing successful therapeutic protocols: Modeling interactions between a tumor and its environment could improve our understanding of tumor growth and inform approaches to treatment. This paper describes an efficient, flexible, hybrid cellular automaton-based implementation of numerical solutions to multiple time-scale reaction-diffusion equations, applied to a model of tumor proliferation. The growth and maintenance of cells in our simulation depend on the rate of cellular energy (ATP) metabolized from nearby nutrients such as glucose and oxygen. Nutrient consumption rates are functions of local pH as well as local concentrations of oxygen and other fuels. The diffusion of these nutrients is modeled using a novel variation of random-walk techniques. Furthermore, we detail the effects of three boundary update rules on simulations, describing their effects on computational efficiency and biological realism. Qualitative and quantitative results from simulations provide insight on how tumor growth is affected by various environmental changes such as micro-vessel density or lower pH, both of high interest in current cancer research.
2013, 10(1): 151-165 doi: 10.3934/mbe.2013.10.151 +[Abstract](2386) +[PDF](353.7KB)
Abstract:
In this work an optimization problem for a leukemia treatment model based on the Gompertzian law of cell growth is considered. The quantities of the leukemic and of the healthy cells at the end of the therapy are chosen as the criterion of the treatment quality. In the case where the number of healthy cells at the end of the therapy is higher than a chosen desired number, an analytical solution of the optimization problem for a wide class of therapy processes is given. If this is not the case, a control strategy called alternative is suggested.
2013, 10(1): 167-184 doi: 10.3934/mbe.2013.10.167 +[Abstract](2662) +[PDF](1037.1KB)
Abstract:
The vascular endothelial growth factor (VEGF) is known as one of the main promoter of angiogenesis - the process of blood vessel formation. Angiogenesis has been recognized as a key stage for cancer development and metastasis. In this paper, we propose a structural model of the main molecular pathways involved in the endothelial cells response to VEGF stimuli. The model, built on qualitative information from knowledge databases, is composed of 38 ordinary differential equations with 78 parameters and focuses on the signalling driving endothelial cell proliferation, migration and resistance to apoptosis. Following a VEGF stimulus, the model predicts an increase of proliferation and migration capability, and a decrease in the apoptosis activity. Model simulations and sensitivity analysis highlight the emergence of robustness and redundancy properties of the pathway. If further calibrated and validated, this model could serve as tool to analyse and formulate new hypothesis on th e VEGF signalling cascade and its role in cancer development and treatment.
2013, 10(1): 185-198 doi: 10.3934/mbe.2013.10.185 +[Abstract](2037) +[PDF](362.8KB)
Abstract:
This paper is devoted to a nonlinear system of partial differential equations modeling the effect of an anti-angiogenic therapy based on an agent that binds to the tumor angiogenic factors. The main feature of the model under consideration is a nonlinear flux production of tumor angiogenic factors at the boundary of the tumor. It is proved the global existence for the nonlinear system and the effect in the large time behavior of the system for high doses of the therapeutic agent.
2013, 10(1): 199-219 doi: 10.3934/mbe.2013.10.199 +[Abstract](1958) +[PDF](464.9KB)
Abstract:
In this article we show how dichotomic classes, binary variables naturally derived from a new mathematical model of the genetic code, can be used in order to characterize different parts of the genome. In particular, we analyze and compare different parts of whole chromosome 1 of Arabidopsis thaliana: genes, exons, introns, coding sequences (CDS), intergenes, untranslated regions (UTR) and regulatory sequences. In order to accomplish the task we encode each sequence in the 3 possible reading frames according to the definitions of the dichotomic classes (parity, Rumer and hidden). Then, we perform a statistical analysis on the binary sequences. Interestingly, the results show that coding and non-coding sequences have different patterns and proportions of dichotomic classes. This suggests that the frame is important only for coding sequences and that dichotomic classes can be useful to recognize them. Moreover, such patterns seem to be more enhanced in CDS than in exons. Also, we derive an independence test in order to assess whether the percentages observed could be considered as an expression of independent random processes. The results confirm that only genes, exons and CDS seem to possess a dependence structure that distinguishes them from i.i.d sequences. Such informational content is independent from the global proportion of nucleotides of a sequence. The present work confirms that the recent mathematical model of the genetic code is a new paradigm for understanding the management and the organization of genetic information and is an innovative tool for investigating informational aspects of error detection/correction mechanisms acting at the level of DNA replication.
2013, 10(1): 221-234 doi: 10.3934/mbe.2013.10.221 +[Abstract](2029) +[PDF](526.8KB)
Abstract:
Dosage and frequency of treatment schedules are important for successful chemotherapy. However, in this work we argue that cell-kill response and tumoral growth should not be seen as separate and therefore are essential in a mathematical cancer model. This paper presents a mathematical model for sequencing of cancer chemotherapy and surgery. Our purpose is to investigate treatments for large human tumours considering a suitable cell-kill dynamics. We use some biological and pharmacological data in a numerical approach, where drug administration occurs in cycles (periodic infusion) and surgery is performed instantaneously. Moreover, we also present an analysis of stability for a chemotherapeutic model with continuous drug administration. According to Norton & Simon [22], our results indicate that chemotherapy is less efficient in treating tumours that have reached a plateau level of growing and that a combination with surgical treatment can provide better outcomes.
2013, 10(1): 235-261 doi: 10.3934/mbe.2013.10.235 +[Abstract](3184) +[PDF](1467.3KB)
Abstract:
Cell migration on and through extracellular matrix is fundamental in a wide variety of physiological and pathological phenomena, and is exploited in scaffold-based tissue engineering. Migration is regulated by a number of extracellular matrix- or cell-derived biophysical parameters, such as matrix fiber orientation, pore size, and elasticity, or cell deformation, proteolysis, and adhesion. We here present an extended Cellular Potts Model (CPM) able to qualitatively and quantitatively describe cell migration efficiencies and phenotypes both on two-dimensional substrates and within three-dimensional matrices, close to experimental evidence. As distinct features of our approach, cells are modeled as compartmentalized discrete objects, differentiated into nucleus and cytosolic region, while the extracellular matrix is composed of a fibrous mesh and a homogeneous fluid. Our model provides a strong correlation of the directionality of migration with the topological extracellular matrix distribution and a biphasic dependence of migration on the matrix structure, density, adhesion, and stiffness, and, moreover, simulates that cell locomotion in highly constrained fibrillar obstacles requires the deformation of the cell's nucleus and/or the activity of cell-derived proteolysis. In conclusion, we here propose a mathematical modeling approach that serves to characterize cell migration as a biological phenomenon in healthy and diseased tissues and in engineering applications.
2013, 10(1): 263-278 doi: 10.3934/mbe.2013.10.263 +[Abstract](2418) +[PDF](347.9KB)
Abstract:
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.

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