A suggestion for a new approach to the mathematical theory of imitative behavior

The theory of imitative behavior as developed hitherto by the author was based on the assumption that each individual has a natural preference for one of the two mutually exclusive behaviors. The endogenous fluctuations in the central nervous system then result in the individual’s exhibiting the two behaviors alternately with a relative frequency determined by the natural preference. Imitation shifts the natural preference towards one or the other of the two mutually exclusive behaviors. In the present approach it is suggested that the relative frequency of the two mutually exclusive behaviors exhibited alternately is determined by maximizing the “satisfaction function” of the individual, that is by hedonistic factors rather than by purely random fluctuations. Corresponding equations are developed. It is shown that in certain cases, even when the imitation effect is absent, a sort of “pseudoimitation” may occur. Another situation leads, in the case of two individuals only, to a complete “division of labor” between them, with respect to the two behaviors. Each one exhibits only one behavior. After that imitation is introduced explicitly by assuming that imitation by one individual or another increases the satisfaction function of the imitating individual. Results thus obtained show similarities to the results of the old theory.     

A new mathematical model for avascular tumour growth

The early development of solid tumours has been extensively studied, both experimentally via the multicellular spheroid assay, and theoretically using mathematical modelling. The vast majority of previous models apply specifically to multicell spheroids, which have a characteristic structure of a proliferating rim and a necrotic core, separated by a band of quiescent cells. Many previous models represent these as discrete layers, separated by moving boundaries. Here, the authors develop a new model, formulated in terms of continuum densities of proliferating, quiescent and necrotic cells, together with a generic nutrient/growth factor. The model is oriented towards an in vivo rather than in vitro setting, and crucially allows for nutrient supply from underlying tissue, which will arise in the two-dimensional setting of a tumour growing within an epithelium. In addition, the model involves a new representation of cell movement, which reflects contact inhibition of migration. Model solutions are able to reproduce the classic three layer structure familiar from multicellular spheroids, but also show that new behaviour can occur as a result of the nutrient supply from underlying tissue. The authors analyse these different solution types by approximate solution of the travelling wave equations, enabling a detailed classification of wave front solutions.     

Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching

We develop a thermodynamically consistent mixture model for avascular solid tumor growth which takes into account the effects of cell-to-cell adhesion, and taxis inducing chemical and molecular species. The mixture model is well-posed and the governing equations are of Cahn-Hilliard type. When there are only two phases, our asymptotic analysis shows that earlier single-phase models may be recovered as limiting cases of a two-phase model. To solve the governing equations, we develop a numerical algorithm based on an adaptive Cartesian block-structured mesh refinement scheme. A centered-difference approximation is used for the space discretization so that the scheme is second order accurate in space. An implicit discretization in time is used which results in nonlinear equations at implicit time levels. We further employ a gradient stable discretization scheme so that the nonlinear equations are solvable for very large time steps. To solve those equations we use a nonlinear multilevel/multigrid method which is of an optimal order O(N) where N is the number of grid points. Spherically symmetric and fully two dimensional nonlinear numerical simulations are performed. We investigate tumor evolution in nutrient-rich and nutrient-poor tissues. A number of important results have been uncovered. For example, we demonstrate that the tumor may suffer from taxis-driven fingering instabilities which are most dramatic when cell proliferation is low, as predicted by linear stability theory. This is also observed in experiments. This work shows that taxis may play a role in tumor invasion and that when nutrient plays the role of a chemoattractant, the diffusional instability is exacerbated by nutrient gradients. Accordingly, we believe this model is capable of describing complex invasive patterns observed in experiments.     

A new mathematical approach predicts individual cell growth behavior using bacterial population information

A theoretical methodology has been developed for studying the growth kinetics of bacterial cells. It utilizes the steady-state cell length distribution in a bacterial population to predict the dependency of growth and division rates on cell length and age. The mathematical model has been applied to the analysis of two bacterial populations, a wild-type strain of Bacillus subtilis, and a minicell-producing strain that carries the divIVB1 mutation. The results show that our model describes the wild-type population very well and that the assumptions typically used in traditional methods are unrealistic. In the case of the minicell-producing mutant we find evidence that the rate of cell division must be a function not only of cell size but also of cell age.     

A weakly nonlinear analysis of a model of avascular solid tumour growth

 In this paper we study a mathematical model that describes the growth of an avascular solid tumour. Our analysis concentrates on the stability of steady, radially-symmetric model solutions with respect to perturbations taken from the class of spherical harmonics. Using weakly nonlinear analysis, previous results are extended to show how the amplitudes of the asymmetric modes interact. Attention focuses on a special case for which the model equations simplify. Analysis of the simplified model equations leads to the identification of a two-parameter family of asymmetric steady solutions, the dimensions of whose stable and unstable manifolds depend on the system parameters. The asymmetric steady solutions limit the basin of attraction of the radially-symmetric steady state when it is linearly stable. On the basis of these numerical and analytical results we postulate the existence of fully nonlinear steady solutions which are stable with respect to time-dependent perturbations. Received: 25 October 1998 / Revised version: 20 June 1998     

A simple mathematical model based on the cancer stem cell hypothesis suggests kinetic commonalities in solid tumor growth

Background

The Cancer Stem Cell (CSC) hypothesis has gained credibility within the cancer research community. According to this hypothesis, a small subpopulation of cells within cancerous tissues exhibits stem-cell-like characteristics and is responsible for the maintenance and proliferation of cancer.

Methodologies/Principal Findings

We present a simple compartmental pseudo-chemical mathematical model for tumor growth, based on the CSC hypothesis, and derived using a “chemical reaction” approach. We defined three cell subpopulations: CSCs, transit progenitor cells, and differentiated cells. Each event related to cell division, differentiation, or death is then modeled as a chemical reaction. The resulting set of ordinary differential equations was numerically integrated to describe the time evolution of each cell subpopulation and the overall tumor growth. The parameter space was explored to identify combinations of parameter values that produce biologically feasible and consistent scenarios.

Conclusions/Significance

Certain kinetic relationships apparently must be satisfied to sustain solid tumor growth and to maintain an approximate constant fraction of CSCs in the tumor lower than 0.01 (as experimentally observed): (a) the rate of symmetrical and asymmetrical CSC renewal must be in the same order of magnitude; (b) the intrinsic rate of renewal and differentiation of progenitor cells must be half an order of magnitude higher than the corresponding intrinsic rates for cancer stem cells; (c) the rates of apoptosis of the CSC, transit amplifying progenitor (P) cells, and terminally differentiated (D) cells must be progressively higher by approximately one order of magnitude. Simulation results were consistent with reports that have suggested that encouraging CSC differentiation could be an effective therapeutic strategy for fighting cancer in addition to selective killing or inhibition of symmetric division of CSCs.     

A mathematical programming approach for gene selection and tissue classification

MOTIVATION: Extracting useful information from expression levels of thousands of genes generated with microarray technology needs a variety of analytical techniques. Mathematical programming approaches for classification analysis outperform parametric methods when the data depart from assumptions underlying these methods. Therefore, a mathematical programming approach is developed for gene selection and tissue classification using gene expression profiles. RESULTS: A new mixed integer programming model is formulated for this purpose. The mixed integer programming model simultaneously selects genes and constructs a classification model to classify two groups of tissue samples as accurately as possible. Very encouraging results were obtained with two data sets from the literature as examples. These results show that the mathematical programming approach can rival or outperform traditional classification methods.     

A new nonlinear model of age-dependent population growth

A new nonlinear age-structured population model is presented. Within its framework the occurence of time-persistent age distributions is possible, even if the population sizes are nonstationary. The age distribution as well as the moments of the generation index can be determined analytically. The proposed model is a nonlinear generalization of Lotka's theory of stable populations.     

Gene selection for tumor classification using a novel bio-inspired multi-objective approach

Identifying the informative genes has always been a major step in microarray data analysis. The complexity of various cancer datasets makes this issue still challenging. In this paper, a novel Bio-inspired Multi-objective algorithm is proposed for gene selection in microarray data classification specifically in the binary domain of feature selection. The presented method extends the traditional Bat Algorithm with refined formulations, effective multi-objective operators, and novel local search strategies employing social learning concepts in designing random walks. A hybrid model using the Fisher criterion is then applied to three widely-used microarray cancer datasets to explore significant biomarkers which reveal the effectiveness of the proposed method for genomic analysis. Experimental results unveil new combinations of informative biomarkers have association with other studies.     

Investigation of solid tumor progression with account of proliferation/migration dichotomy via Darwinian mathematical model

Journal of Mathematical Biology - A new continuous spatially-distributed model of solid tumor growth and progression is presented. The model explicitly accounts for mutations/epimutations of tumor...     

A quantitative theory of solid tumor growth, metabolic rate and vascularization

The relationships between cellular, structural and dynamical properties of tumors have traditionally been studied separately. Here, we construct a quantitative, predictive theory of solid tumor growth, metabolic rate, vascularization and necrosis that integrates the relationships between these properties. To accomplish this, we develop a comprehensive theory that describes the interface and integration of the tumor vascular network and resource supply with the cardiovascular system of the host. Our theory enables a quantitative understanding of how cells, tissues, and vascular networks act together across multiple scales by building on recent theoretical advances in modeling both healthy vasculature and the detailed processes of angiogenesis and tumor growth. The theory explicitly relates tumor vascularization and growth to metabolic rate, and yields extensive predictions for tumor properties, including growth rates, metabolic rates, degree of necrosis, blood flow rates and vessel sizes. Besides these quantitative predictions, we explain how growth rates depend on capillary density and metabolic rate, and why similar tumors grow slower and occur less frequently in larger animals, shedding light on Peto's paradox. Various implications for potential therapeutic strategies and further research are discussed.     

A mathematical model of tumor growth and its response to single irradiation

Background

Mathematical modeling of biological processes is widely used to enhance quantitative understanding of bio-medical phenomena. This quantitative knowledge can be applied in both clinical and experimental settings. Recently, many investigators began studying mathematical models of tumor response to radiation therapy. We developed a simple mathematical model to simulate the growth of tumor volume and its response to a single fraction of high dose irradiation. The modelling study may provide clinicians important insights on radiation therapy strategies through identification of biological factors significantly influencing the treatment effectiveness.

Methods

We made several key assumptions of the model. Tumor volume is composed of proliferating (or dividing) cancer cells and non-dividing (or dead) cells. Tumor growth rate (or tumor volume doubling time) is proportional to the ratio of the volumes of tumor vasculature and the tumor. The vascular volume grows slower than the tumor by introducing the vascular growth retardation factor, θ. Upon irradiation, the proliferating cells gradually die over a fixed time period after irradiation. Dead cells are cleared away with cell clearance time. The model was applied to simulate pre-treatment growth and post-treatment radiation response of rat rhabdomyosarcoma tumors and metastatic brain tumors of five patients who were treated with Gamma Knife stereotactic radiosurgery (GKSRS).

Results

By selecting appropriate model parameters, we showed the temporal variation of the tumors for both the rat experiment and the clinical GKSRS cases could be easily replicated by the simple model. Additionally, the application of our model to the GKSRS cases showed that the α-value, which is an indicator of radiation sensitivity in the LQ model, and the value of θ could be predictors of the post-treatment volume change.

Conclusions

The proposed model was successful in representing both the animal experimental data and the clinically observed tumor volume changes. We showed that the model can be used to find the potential biological parameters, which may be able to predict the treatment outcome. However, there is a large statistical uncertainty of the result due to the small sample size. Therefore, a future clinical study with a larger number of patients is needed to confirm the finding.

     

A new mathematical continuum theory of axoplasmic transport.

The hypothesis is advanced that successive waves of apparent contraction-relaxation (due perhaps to filament sliding) propagate along the filamentous proteins that pierce axoplasm oriented parallel to the axon length. A mathematical continuum model is developed to characterize the flow that could result in the viscous fluid bathing the moving filamentous proteins. This flow is complicated and oscillatory in time and space, but, on the average, it yields a bi-directional drift of fluid. It would transport various substances riding in the fluid, soluble and particulate, at various distinct speeds both up and down the axon. The model can thus account qualitatively for many observed features of axoplasmic transport.     

A mathematical model for storage and recall functions in plants

In plantlets of Bidens pilosa L., under severely limiting environmental conditions the growth of the buds at the axil of the cotyledons (cotyledonary buds) is asymmetric (i.e. one of the buds starts growing before the other one), this asymmetry being oriented by the pricking of one of the cotyledons (i.e. pricking one cotyledon increases the probability that the bud at the axil of the other cotyledon be the first to start to grow). As long as the plant apex (i.e. the terminal bud) is present, the growth of the cotyledonary buds is inhibited (apical dominance), but the souvenir of the asymmetric message caused by sub-optimal environmental conditions and the orientation given by the cotyledon pricking is always present in the plant and can be revealed by removing the apex. Depending on the conditions for removing the plant apex and/or on the application of a variety of symmetrical treatments (e.g. thermal treatment, symmetrical pricking treatments, etc.) the stored asymmetry will either take effect (the bud at the axil of the non-pricked cotyledon will be the first to start to grow more often than the other one) or not (both buds will have equal chance to be the first to start to grow). This has been termed 'recalling' the stored asymmetry. By combining several successive symmetrical treatments, it is possible to reversibly switch on and off the recall function several times. This recall of the stored plant-asymmetry is analogous to the evocation function of a memory system. In this paper, we will present first a discrete logical version of the observed interaction structure between the main components of the bud growth system, then a continuous differential version, taking into account the main features of the observed experimental reality and trying to explain this phenomenology. The interaction structure of both the discrete and the continuous models presents similar positive and negative feedback circuits, necessary condition for observing multistationarity and stability.     

The threshold bias model: a mathematical model for the nomothetic approach of suicide

Background

Comparative and predictive analyses of suicide data from different countries are difficult to perform due to varying approaches and the lack of comparative parameters.

Methodology/Principal Findings

A simple model (the Threshold Bias Model) was tested for comparative and predictive analyses of suicide rates by age. The model comprises of a six parameter distribution that was applied to the USA suicide rates by age for the years 2001 and 2002. Posteriorly, linear extrapolations are performed of the parameter values previously obtained for these years in order to estimate the values corresponding to the year 2003. The calculated distributions agreed reasonably well with the aggregate data. The model was also used to determine the age above which suicide rates become statistically observable in USA, Brazil and Sri Lanka.

Conclusions/Significance

The Threshold Bias Model has considerable potential applications in demographic studies of suicide. Moreover, since the model can be used to predict the evolution of suicide rates based on information extracted from past data, it will be of great interest to suicidologists and other researchers in the field of mental health.     

Kinetic study of a heterogeneous tumor cell population using a mathematical model

A mathematical model of a heterogenous tumor as a system of interrelating cell populations is described, including a pool of quiescent cells, cell-to-cell variability in maturation rates, and cell migration from growth area to necrotic one. Computer simulation results are given, model labeled mitoses and labeled index curves for the Lewis carcinoma are compared with experimental data.