A Microenvironment Based Model of Antimitotic Therapy of Gompertzian Tumor Growth

A model of tumor growth, based on two-compartment cell population dynamics, and an overall Gompertzian growth has been previously developed. The main feature of the model is an inter-compartmental transfer function that describes the net exchange between proliferating (P) and quiescent (Q) cells and yields Gompertzian growth for tumor cell population N = P + Q. Model parameters provide for cell reproduction and cell death. This model is further developed here and modified to simulate antimitotic therapy. Therapy decreases the reproduction-rate constant and increases the death-rate constant of proliferating cells with no direct effect on quiescent cells. The model results in a system of two ODE equations (in N and P/N) that has an analytical solution. Net tumor growth depends on support from the microenvironment. Indirectly, this is manifested in the transfer function, which depends on the proliferation ratio, P/N. Antimitotic therapy will change P/N, and the tumor responds by slowing the transfer rate from P to Q. While the cellular effects of therapy are modeled as dependent only on antimitotic activity of the drug, the tumor response also depends on the tumor age and any previous therapies—after therapy, it is not the same tumor. The strength of therapy is simulated by the parameter λ, which is the ratio of therapy induced net proliferation rate constant versus the original. A pharmacodynamic factor inversely proportional to tumor size is implemented. Various chemotherapy regimens are simulated and the outcomes of therapy administered at different time points in the life history of the tumor are explored. Our analysis shows: (1) for a constant total dose administered, a decreasing dose schedule is marginally superior to an increasing or constant scheme, with more pronounced benefit for faster growing tumors, (2) the minimum dose to stop tumor growth is age dependent, and (3) a dose-dense schedule is favored. Faster growing tumors respond better to dose density.     

Inferring the effect of therapy on tumors showing stochastic Gompertzian growth

The present work deals with a Gompertz-type diffusion process, which includes in the drift term a time-dependent function C(t) representing the effect of a therapy able to modify the dynamics of the underlying process. However, in experimental studies is not immediate to deduce the functional form of C(t) from a treatment protocol. So a statistical approach is proposed in order to estimate this function when a control group and one or more treated groups are observed. In order to validate the proposed strategy a simulation study for several interesting functional forms of C(t) has been carried out. Finally, an application to infer the net effect of cisplatin and doxorubicin+cyclophosphamide in actual murine models is presented.     

Observer-based techniques for the identification and analysis of avascular tumor growth

Cancer represents one of the most challenging issues for the biomedical research, due its large impact on the public health state. For this reason, many mathematical methods have been proposed to forecast the time evolution of cancer size and invasion. In this paper, we study how to apply the Gompertz’s model to describe the growth of an avascular tumor in a realistic setting. To this aim, we introduce mathematical techniques to discretize the model, an important requirement when discrete-time measurements are available. Additionally, we describe observed-based techniques, borrowed from the field of automation theory, as a tool to estimate the model unknown parameters. This identification approach is a promising alternative to traditional statistical methods, and it can be easily extended to other models of cancer growth as well as to the evaluation of not measurable variables, on the basis of the available measurements. We show an application of this method to the analysis of solid tumor growth and parameters estimation in presence of a chemotherapy agent.     

Stochastic and deterministic simulations of heterogeneous cell population dynamics

A Monte Carlo algorithm, which can accurately simulate the dynamics of entire heterogeneous cell populations, was developed. The algorithm takes into account the random nature of cell division as well as unequal partitioning of cellular material at cell division. Moreover, it is general in the sense that it can accommodate a variety of single-cell, deterministic reaction kinetics as well as various stochastic division and partitioning mechanisms. The validity of the algorithm was assessed through comparison of its results with those of the corresponding deterministic cell population balance model in cases where stochastic behavior is expected to be quantitatively negligible. Both algorithms were applied to study: (a) linear intracellular kinetics and (b) the expression dynamics of a genetic network with positive feedback architecture, such as the lac operon. The effects of stochastic division as well as those of different division and partitioning mechanisms were assessed in these systems, while the comparison of the stochastic model with a continuum model elucidated the significance of cell population heterogeneity even in cases where only the prediction of average properties is of primary interest.     

Models for estimating empirical Gompertz mortality: With an application to evolution of the Gompertzian slope

Using data from the human mortality database (HMD), and five different modeling approaches, we estimate Gompertz mortality parameters for 7,704 life tables. To gauge model fit, we predict life expectancy at age 40 from these parameters, and compare predicted to empirical values. Across a diversity of human populations, and both sexes, the overall best way to estimate Gompertz parameters is weighted least squares, although Poisson regression performs better in 996 cases for males and 1,027 cases for females, out of 3,852 populations per sex. We recommend against using unweighted least squares unless death counts (to use as weights or to allow Poisson estimation) are unavailable. We also recommend fitting to logged death rates. Over time in human populations, the Gompertz slope parameter has increased, indicating a more severe increase in mortality rates as age goes up. However, it is well-known that the two parameters of the Gompertz model are very tightly (and negatively) correlated. When the slope goes up, the level goes down, and, overall, mortality rates are decreasing over time. An analysis of Gompertz parameters for all of the HMD countries shows a distinct pattern for males in the formerly socialist economies of Europe.     

A mathematical model to study the effects of drugs administration on tumor growth dynamics

A mathematical model for describing the cancer growth dynamics in response to anticancer agents administration in xenograft models is discussed. The model consists of a system of ordinary differential equations involving five parameters (three for describing the untreated growth and two for describing the drug action). Tumor growth in untreated animals is modelled by an exponential growth followed by a linear growth. In treated animals, tumor growth rate is decreased by an additional factor proportional to both drug concentration and proliferating cells. The mathematical analysis conducted in this paper highlights several interesting properties of this tumor growth model. It suggests also effective strategies to design in vivo experiments in animals with potential saving of time and resources. For example, the drug concentration threshold for the tumor eradication, the delay between drug administration and tumor regression, and a time index that measures the efficacy of a treatment are derived and discussed. The model has already been employed in several drug discovery projects. Its application on a data set coming from one of these projects is discussed in this paper.     

Determinants of intra-annual population dynamics in a tropical soil arthropod

Studies on inter-annual dynamics of populations from temperate regions have shown that density dependence and climate effects are relatively common, albeit weak. Yet, for short-lived organisms, intra-annual variation may be at least as important. Furthermore, tropical species commonly experience temperatures close to their upper thermal limit and thus may be more likely to experience heat stress. Here, we used the soil mite Rostrozetes ovulum to investigate the drivers of intra-annual population dynamics in an Amazonian rain forest. We sampled 3,600 soil cores from 20 transects during 13 months, obtaining 180 mite counts. Next, we built a dynamic N-mixture model accounting for different detection probabilities between soil types. In a Bayesian framework, we used this model to estimate (a) the strength of density dependence and (b) per capita growth rates, which were then tested against environmental variables. We found that the intra-annual population dynamics of R. ovulum were weakly density dependent. Further, per capita growth rates increased with resource supply (litterfall) and decreased with maximum temperature over much of the observed thermal range, although these effects explained relatively little variance. Yet, the seasonal correlation between these factors created a trade-off, so that realized population growth was highest when neither resource supply nor thermal suitability was optimal. Overall, our results suggest that the mechanisms shaping soil animal population dynamics may be surprisingly similar across latitudes. Our model offers a starting point for analyses of soil animal counts when extraction from soil samples is imperfect.     

Parameter estimation in a Gompertzian stochastic model for tumor growth

The problem of estimating parameters in the drift coefficient when a diffusion process is observed continuously requires some specific assumptions. In this paper, we consider a stochastic version of the Gompertzian model that describes in vivo tumor growth and its sensitivity to treatment with antiangiogenic drugs. An explicit likelihood function is obtained, and we discuss some properties of the maximum likelihood estimator for the intrinsic growth rate of the stochastic Gompertzian model. Furthermore, we show some simulation results on the behavior of the corresponding discrete estimator. Finally, an application is given to illustrate the estimate of the model parameters using real data.     

Stochastic Gompertz model of tumour cell growth

In this communication, based upon the deterministic Gompertz law of cell growth, a stochastic model in tumour growth is proposed. This model takes account of both cell fission and mortality too. The corresponding density function of the size of the tumour cells obeys a functional Fokker--Planck equation which can be solved analytically. It is found that the density function exhibits an interesting "multi-peak" structure generated by cell fission as time evolves. Within this framework the action of therapy is also examined by simply incorporating a therapy term into the deterministic cell growth term.     

Correlating single cell motility with population growth dynamics for flagellated bacteria

Many bacteria used for biotechnological applications are naturally motile. Their "bio-nanopropeller" driven movement allows searching for better environments in a process called chemotaxis. Since bacteria are extremely small in size compared to the bulk fluid volumes in bioreactors, single cell motility is not considered to influence bioreactor operations. However, with increasing interest in localized fluid flow inside reactors, it is important to ask whether individual motility characteristics of bacteria are important in bioreactor operations. The first step in this direction is to try to correlate single cell measurements with population data of motile bacteria in a bioreactor. Thus, we observed the motility behavior of individual bacterial cells, using video microscopy with 33 ms time resolution, as a function of population growth dynamics of batch cultures in shake flasks. While observing the motility behavior of the most intensively studied bacteria, Escherichia coli, we find that overall bacterial motility decreases with progression of the growth curve. Remarkably, this is due to a decrease in a specific motility behavior called "running". Our results not only have direct implications on biofilm formations, but also provide a new direction in bioprocess design research highlighting the role of individual bacterial cell motility as an important parameter.     

Epidermal growth factor (EGF) induces apoptosis in a transfected cell line expressing EGF receptor on its membrane

Interaction between epidermal growth factor (EGF) and EGF receptor (EGFR) promotes cell growth in most cell lines, but in a number of cell lines, EGF paradoxically inhibits proliferation. In the present study, we established a cell line expressing full-length human EGFR on membrane with a GFP fluorescence reporter at the C-terminal and studied the effects of EGF on cell proliferation in the transfected cell line. Our results suggested that low concentrations of EGF promoted proliferation, while high concentrations of EGF induced loss of adhesion, cell cycle arrest, apoptosis, and inhibition of proliferation. The effects of EGF on cell proliferation correlated well with the expression levels of EGFR. High concentrations of EGF induced both EGFR expression and apoptosis in a dose-dependent manner. Our study reported, for the first time, a relationship between the effects of EGF on cell proliferation and levels of EGFR expression in one cell line expressing different levels of EGFR caused by different concentrations of EGF treatment. The study should provide considerable insight into the effects of EGF on cell proliferation and tumor cell metastasis.     

An approximation method for estimating cell loss from a growing tumor

A simple self-consistent calculational scheme is developed for estimating cell loss for a growing tumor (or other population) when the growth fraction can be estimated at regular intervals. This is applied to published data for a particular much-studied Ehrlich ascites tumor. The loss rate is found to be substantially higher than that estimated by previous, less precise means.     

Individual‐based and stochastic modeling of cell population dynamics considering substrate dependency

In this article, we propose an individual‐based and stochastic modeling approach that is capable of describing the bacterial cell population dynamics during a batch culture. All stochastic nature inherent in intracellular molecular level reactions and cell division processes were considered in a single model framework by embedding a sub‐model describing individual cell's growth kinetics in a discrete event simulation algorithm. The resultant unique feature of the model is that the effects of the stochasticities on the cell population dynamics can be investigated for different substrate‐dependent cell growth kinetics. When Monod kinetics was used as the sub‐model, the stochasticities only slightly affected the cell mass increase and substrate consumption profiles during the batch culture although they were still important in describing the changes of cell population distributions. When Andrews substrate inhibition kinetics was used, however, it was revealed that the overall cell population dynamics could be seriously influenced by the stochasticities. Under a critical initial substrate level, the cell population could proliferate against the substrate inhibition only when the stochasticities were considered. Biotechnol. Bioeng. 2009;103: 891–899. © 2009 Wiley Periodicals, Inc.     

A principle of fractal-stochastic dualism and Gompertzian dynamics of growth and self-organization

The emergence of Gompertzian dynamics at the macroscopic, tissue level during growth and self-organization is determined by the existence of fractal-stochastic dualism at the microscopic level of supramolecular, cellular system. On one hand, Gompertzian dynamics results from the complex coupling of at least two antagonistic, stochastic processes at the molecular cellular level. It is shown that the Gompertz function is a probability function, its derivative is a probability density function, and the Gompertzian distribution of probability is of non-Gaussian type. On the other hand, the Gompertz function is a contraction mapping and defines fractal dynamics in time-space; a prerequisite condition for the coupling of processes. Furthermore, the Gompertz function is a solution of the operator differential equation with the Morse-like anharmonic potential. This relationship indicates that distribution of intrasystemic forces is both non-linear and asymmetric. The anharmonic potential is a measure of the intrasystemic interactions. It attains a point of the minimum (U(0), t(0)) along with a change of both complexity and connectivity during growth and self-organization. It can also be modified by certain factors, such as retinoids.     

Cell adhesion molecules, signal transduction and cell growth

Signals from dynamic cellular interactions between the extracellular matrix and neighboring cells ultimately input into the cellular decision-making process. These interactions form the basis of anchorage-dependent growth. Recent advances have provided the mechanistic details behind the ability of integrins, and other cell adhesion molecules (CAMs), to regulate both early signal transduction events initiated by soluble factors and downstream events more proximally involved in cell cycle progression. These actions appear to depend on the ability of CAMs to initiate the formation of organized structures that permit the efficient flow of information.     

A proapoptotic peptide conjugated to penetratin selectively inhibits tumor cell growth

The peptide KLA (acetyl-(KLAKLAK)₂-NH₂), which is rather non toxic for eukaryotic cell lines, becomes active when coupled to the cell penetrating peptide, penetratin (Pen), by a disulfide bridge. Remarkably, the conjugate KLA–Pen is cytotoxic, at low micromolar concentrations, against a panel of seven human tumor cell lines of various tissue origins, including cells resistant to conventional chemotherapy agents but not to normal human cell lines. Live microscopy on cells possessing fluorescent labeled mitochondria shows that in tumor cells, KLA–Pen had a strong impact on mitochondria tubular organization instantly resulting in their aggregation, while the unconjugated KLA and pen peptides had no effect. But, mitochondria in various normal cells were not affected by KLA–Pen. The interaction with membrane models of KLA–Pen, KLA and penetratin were studied using dynamic light scattering, calorimetry, plasmon resonance, circular dichroism and ATR-FTIR to unveil the mode of action of the conjugate. To understand the selectivity of the conjugate towards tumor cell lines and its action on mitochondria, lipid model systems composed of zwitterionic lipids were used as mimics of normal cell membranes and anionic lipids as mimics of tumor cell and mitochondria membrane. A very distinct mode of interaction with the two model systems was observed. KLA–Pen may exert its deleterious and selective action on cancer cells by the formation of pores with an oblique membrane orientation and establishment of important hydrophobic interactions. These results suggest that KLA–Pen could be a lead compound for the design of cancer therapeutics.     

Estrogen receptor beta in breast cancer—Diagnostic and therapeutic implications

More than 10 years have passed since the discovery of the second estrogen receptor, estrogen receptor β (ERβ). It is now evident that ERα is not the only ER in breast cancer cells; in fact, ERβ is expressed in the majority of breast cancers although at lower levels than in the normal breast. In addition, ERβ is expressed in breast cancer infiltrating lymphocytes, fibroblasts and endothelial cells, all known to influence tumor growth. By overexpressing or knocking-out ERβ in breast cancer cell lines, several researchers have investigated its function with respect to proliferation and tumor growth. It appears that ERβ is anti-proliferative, in many ways antagonising the function of ERα. Furthermore, phytoestrogens have a binding-preference for ERβ and several epidemiological studies indicate a breast cancer preventing effect of this class of compounds. Tamoxifen is one of the standard, adjuvant treatments for ERα positive breast cancer, classically thought to mediate its effect through ERα. However, in several recent studies, ERβ has been described as a potential marker for tamoxifen response. In summary, experimental, epidemiological as well as diagnostic studies point towards ERβ as an important factor in breast cancer, opening up the possibility for novel ERβ-selective therapies in the treatment of breast cancer.