Tamoxifen treatment failure in cancer and the nonlinear dynamics of TGFbeta

The force of cell-cell adhesion in determining the outcome in a nonlocal advection diffusion model of wound healing

A model of wound healing is presented to investigate the connection of the force of cell-cell adhesion to the sensing radius of cells in their spatial environment. The model consists of a partial differential equation with nonlocal advection and diffusion terms, describing the movement of cells in a spatial environment. The model is applied to biological wound healing experiments to understand incomplete wound closure. The analysis demonstrates that for each value of the force of adhesion parameter, there is a critical value of the sensing radius above which complete wound healing does not occur.

Dynamical density-functional-theory-based modeling of tissue dynamics: Application to tumor growth

We present a theoretical framework based on an extension of dynamical density-functional theory (DDFT) for describing the structure and dynamics of cells in living tissues and tumors. DDFT is a microscopic statistical mechanical theory for the time evolution of the density distribution of interacting many-particle systems. The theory accounts for cell-pair interactions, different cell types, phenotypes, and cell birth and death processes (including cell division), to provide a biophysically consistent description of processes bridging across the scales, including describing the tissue structure down to the level of the individual cells. Analysis of the model is presented for single-species and two-species cases, the latter aimed at describing competition between tumor and healthy cells. In suitable parameter regimes, model results are consistent with biological observations. Of particular note, divergent tumor growth behavior, mirroring metastatic and benign growth characteristics, are shown to be dependent on the cell-pair-interaction parameters.

Changing molecular profile of brain metastases compared with matched breast primary cancers and impact on clinical outcomes

Background:

Breast cancer commonly metastasises to the brain, but little is known about changes in the molecular profile of the brain secondaries and impact on clinical outcomes.

Methods:

Patients with samples from brain metastases and matched breast cancers were included. Immunohistochemical analysis for oestrogen receptor, progesterone receptor, p27kip1, cyclin D1, epidermal growth factor receptor, insulin like growth factor 1, insulin like growth factor 1 receptor, vascular endothelial growth factor A, transforming growth factor-β and HER2 receptor was performed. Borderline HER2 results were analysed by fluorescent in situ hybridisation. Levels of expression were compared, with review of effect on clinical outcomes.

Results:

A total of 41 patients were included. Of the patients, 20% had a change in oestrogen receptor or HER2 in their brain metastasis that could affect therapeutic decisions. There were statistically significant rises in brain metastases for p27kip1 (P=0.023) and cyclin D1 (P=0.030) and a fall in vascular endothelial growth factor A (P=0.012). Overall survival from the time of metastasis increased significantly with oestrogen receptor-positive (P=0.005) and progesterone receptor-positive (P=0.013) brain lesions and with a longer duration from diagnosis of the breast primary (P<0.001).

Conclusions:

In this cohort there were phenotypic differences in metastatic brain tumours compared with matched primary breast tumours. These could be relevant for aetiology, and have an impact on prognostication, current and future therapies.

The formation of tight tumor clusters affects the efficacy of cell cycle inhibitors: a hybrid model study

Cyclin-dependent kinases (CDKs) are vital in regulating cell cycle progression, and, thus, in highly proliferating tumor cells CDK inhibitors are gaining interest as potential anticancer agents. Clonogenic assay experiments are frequently used to determine drug efficacy against the survival and proliferation of cancer cells. While the anticancer mechanisms of drugs are usually described at the intracellular single-cell level, the experimental measurements are sampled from the entire cancer cell population. This approach may lead to discrepancies between the experimental observations and theoretical explanations of anticipated drug mechanisms. To determine how individual cell responses to drugs that inhibit CDKs affect the growth of cancer cell populations, we developed a spatially explicit hybrid agent-based model. In this model, each cell is equipped with internal cell cycle regulation mechanisms, but it is also able to interact physically with its neighbors. We model cell cycle progression, focusing on the G1 and G2/M cell cycle checkpoints, as well as on related essential components, such as CDK1, CDK2, cell size, and DNA damage. We present detailed studies of how the emergent properties (e.g., cluster formation) of an entire cell population depend on altered physical and physiological parameters. We analyze the effects of CDK1 and CKD2 inhibitors on population growth, time-dependent changes in cell cycle distributions, and the dynamic evolution of spatial cell patterns. We show that cell cycle inhibitors that cause cell arrest at different cell cycle phases are not necessarily synergistically super-additive. Finally, we demonstrate that the physical aspects of cell population growth, such as the formation of tight cell clusters versus dispersed colonies, alter the efficacy of cell cycle inhibitors, both in 2D and 3D simulations. This finding may have implications for interpreting the treatment efficacy results of in vitro experiments, in which treatment is applied before the cells can grow to produce clusters, especially because in vivo tumors, in contrast, form large masses before they are detected and treated.

Cellular potts modeling of tumor growth, tumor invasion, and tumor evolution

Despite a growing wealth of available molecular data, the growth of tumors, invasion of tumors into healthy tissue, and response of tumors to therapies are still poorly understood. Although genetic mutations are in general the first step in the development of a cancer, for the mutated cell to persist in a tissue, it must compete against the other, healthy or diseased cells, for example by becoming more motile, adhesive, or multiplying faster. Thus, the cellular phenotype determines the success of a cancer cell in competition with its neighbors, irrespective of the genetic mutations or physiological alterations that gave rise to the altered phenotype. What phenotypes can make a cell "successful" in an environment of healthy and cancerous cells, and how? A widely used tool for getting more insight into that question is cell-based modeling. Cell-based models constitute a class of computational, agent-based models that mimic biophysical and molecular interactions between cells. One of the most widely used cell-based modeling formalisms is the cellular Potts model (CPM), a lattice-based, multi particle cell-based modeling approach. The CPM has become a popular and accessible method for modeling mechanisms of multicellular processes including cell sorting, gastrulation, or angiogenesis. The CPM accounts for biophysical cellular properties, including cell proliferation, cell motility, and cell adhesion, which play a key role in cancer. Multiscale models are constructed by extending the agents with intracellular processes including metabolism, growth, and signaling. Here we review the use of the CPM for modeling tumor growth, tumor invasion, and tumor progression. We argue that the accessibility and flexibility of the CPM, and its accurate, yet coarse-grained and computationally efficient representation of cell and tissue biophysics, make the CPM the method of choice for modeling cellular processes in tumor development.

Acid-mediated tumour cell invasion: a discrete modelling approach using the extended Potts model

Acidic extracellular pH has been shown to play a crucial part in the invasive and metastatic cascade of some tumours. In this study, we examine the effect of extracellular acidity on tumour invasion focusing, in particular, on cellular adhesion, proteolytic enzyme activity and cellular proliferation. Our numerical simulations using a cellular Potts model show that, under acidic extracellular pH, changes in cell-matrix adhesion strength has a comparable effect on tumour invasiveness as the increase in proteolytic enzyme activity. We also show that tumour cells cultured under physiological pH tend to be large and the tumours develop a "diffuse" morphology compared to those cultured at acidic pH, which display protruding "fingers" at the advancing front. A key model prediction is the observation that the main effect on invasion from culturing cells at low extracellular pH stems from changes in the intercellular and cell-matrix adhesion strengths and proteolytic enzyme secretion rate. However, we show that the effects of proteolysis needs to be significant as low to moderate changes only has nominal effects on cell invasiveness. We find that the low pH e effects on cell size and proliferation rate have much lower influence on cell invasiveness.

Hybrid models of tumor growth

Cancer is a complex, multiscale process in which genetic mutations occurring at a subcellular level manifest themselves as functional changes at the cellular and tissue scale. The multiscale nature of cancer requires mathematical modeling approaches that can handle multiple intracellular and extracellular factors acting on different time and space scales. Hybrid models provide a way to integrate both discrete and continuous variables that are used to represent individual cells and concentration or density fields, respectively. Each discrete cell can also be equipped with submodels that drive cell behavior in response to microenvironmental cues. Moreover, the individual cells can interact with one another to form and act as an integrated tissue. Hybrid models form part of a larger class of individual-based models that can naturally connect with tumor cell biology and allow for the integration of multiple interacting variables both intrinsically and extrinsically and are therefore perfectly suited to a systems biology approach to tumor growth.

Front instabilities and invasiveness of simulated avascular tumors

We study the interface morphology of a 2D simulation of an avascular tumor composed of identical cells growing in an homogeneous healthy tissue matrix (TM), in order to understand the origin of the morphological changes often observed during real tumor growth. We use the Glazier-Graner-Hogeweg model, which treats tumor cells as extended, deformable objects, to study the effects of two parameters: a dimensionless diffusion-limitation parameter defined as the ratio of the tumor consumption rate to the substrate transport rate, and the tumor-TM surface tension. We model TM as a nondiffusing field, neglecting the TM pressure and haptotactic repulsion acting on a real growing tumor; thus, our model is appropriate for studying tumors with highly motile cells, e.g., gliomas. We show that the diffusion-limitation parameter determines whether the growing tumor develops a smooth (noninvasive) or fingered (invasive) interface, and that the sensitivity of tumor morphology to tumor-TM surface tension increases with the size of the dimensionless diffusion-limitation parameter. For large diffusion-limitation parameters, we find a transition (missed in previous work) between dendritic structures, produced when tumor-TM surface tension is high, and seaweed-like structures, produced when tumor-TM surface tension is low. This observation leads to a direct analogy between the mathematics and dynamics of tumors and those observed in nonbiological directional solidification. Our results are also consistent with the biological observation that hypoxia promotes invasive growth of tumor cells by inducing higher levels of receptors for scatter factors that weaken cell-cell adhesion and increase cell motility. These findings suggest that tumor morphology may have value in predicting the efficiency of antiangiogenic therapy in individual patients.

A mathematical model quantifies proliferation and motility effects of TGF-β on cancer cells

Transforming growth factor (TGF)-β is known to have properties of both a tumour suppressor and a tumour promoter. While it inhibits cell proliferation, it also increases cell motility and decreases cell-cell adhesion. Coupling mathematical modelling and experiments, we investigate the growth and motility of oncogene-expressing human mammary epithelial cells under exposure to TGF-β. We use a version of the well-known Fisher-Kolmogorov equation, and prescribe a procedure for its parametrisation. We quantify the simultaneous effects of TGF-β to increase the tendency of individual cells and cell clusters to move randomly and to decrease overall population growth. We demonstrate that in experiments with TGF-β treated cells in vitro, TGF-β increases cell motility by a factor of 2 and decreases cell proliferation by a factor of 1/2 in comparison with untreated cells.

Growth based morphogenesis of vertebrate limb bud

Many genes and their regulatory relationships are involved in developmental phenomena. However, by chemical information alone, we cannot fully understand changing organ morphologies through tissue growth because deformation and growth of the organ are essentially mechanical processes. Here, we develop a mathematical model to describe the change of organ morphologies through cell proliferation. Our basic idea is that the proper specification of localized volume source (e.g., cell proliferation) is able to guide organ morphogenesis, and that the specification is given by chemical gradients. We call this idea "growth-based morphogenesis." We find that this morphogenetic mechanism works if the tissue is elastic for small deformation and plastic for large deformation. To illustrate our concept, we study the development of vertebrate limb buds, in which a limb bud protrudes from a flat lateral plate and extends distally in a self-organized manner. We show how the proportion of limb bud shape depends on different parameters and also show the conditions needed for normal morphogenesis, which can explain abnormal morphology of some mutants. We believe that the ideas shown in the present paper are useful for the morphogenesis of other organs.

Predicting prognosis using molecular profiling in estrogen receptor-positive breast cancer treated with tamoxifen

Background

Estrogen receptor positive (ER+) breast cancers (BC) are heterogeneous with regard to their clinical behavior and response to therapies. The ER is currently the best predictor of response to the anti-estrogen agent tamoxifen, yet up to 30–40% of ER+BC will relapse despite tamoxifen treatment. New prognostic biomarkers and further biological understanding of tamoxifen resistance are required. We used gene expression profiling to develop an outcome-based predictor using a training set of 255 ER+ BC samples from women treated with adjuvant tamoxifen monotherapy. We used clusters of highly correlated genes to develop our predictor to facilitate both signature stability and biological interpretation. Independent validation was performed using 362 tamoxifen-treated ER+ BC samples obtained from multiple institutions and treated with tamoxifen only in the adjuvant and metastatic settings.

Results

We developed a gene classifier consisting of 181 genes belonging to 13 biological clusters. In the independent set of adjuvantly-treated samples, it was able to define two distinct prognostic groups (HR 2.01 95%CI: 1.29–3.13; p = 0.002). Six of the 13 gene clusters represented pathways involved in cell cycle and proliferation. In 112 metastatic breast cancer patients treated with tamoxifen, one of the classifier components suggesting a cellular inflammatory mechanism was significantly predictive of response.

Conclusion

We have developed a gene classifier that can predict clinical outcome in tamoxifen-treated ER+ BC patients. Whilst our study emphasizes the important role of proliferation genes in prognosis, our approach proposes other genes and pathways that may elucidate further mechanisms that influence clinical outcome and prediction of response to tamoxifen.

A continuum approach to modelling cell-cell adhesion

Cells adhere to each other through the binding of cell adhesion molecules at the cell surface. This process, known as cell-cell adhesion, is fundamental in many areas of biology, including early embryo development, tissue homeostasis and tumour growth. In this paper we develop a new continuous mathematical model of this phenomenon by considering the movement of cells in response to the adhesive forces generated through binding. We demonstrate that our model predicts the aggregation behaviour of a disassociated adhesive cell population. Further, when the model is extended to represent the interactions between multiple populations, we demonstrate that it is capable of replicating the different types of cell sorting behaviour observed experimentally. The resulting pattern formation is a direct consequence of the relative strengths of self-population and cross-population adhesive bonds in the model. While cell sorting behaviour has been captured previously with discrete approaches, it has not, until now, been observed with a fully continuous model.

Using cell potential energy to model the dynamics of adhesive biological cells

Developing a continuous mathematical model of a physical phenomenon which is based on a discrete model of the same system is not straightforward. Yet such a process is useful in illustrating the link between the individual behavior of the elements comprising a system and its macroscopic behavior. Collections of biological cells can exhibit phenomena such as pattern formation, aggregation, and invasion, and mathematics has proven useful in elucidating the underlying dynamics of these phenomena. The continuous models formulated are frequently of reaction-diffusion form, and central to their application is a knowledge of the diffusion coefficient of a collection of the elements comprising the system. Cohen and Murray [J. Math Biol. 12, 237 (1981)] developed a means of deriving this quantity which has since been largely neglected by model developers, and which is based on a knowledge of the potential energy associated with the mutual interaction between the cells. In this work, we begin by deriving the energy of interaction of biological cells modeled as adhesive, deformable spheres. In so doing, we are able to quantify the equilibrium density of a biological cell aggregate, and also obtain a quantitative estimate of the diffusion coefficient of a collection of cells modeled in this way. In so doing, we are able to use experimental data from single-cell studies of the adhesiveness and cell membrane elasticity of a biological cell to derive the diffusion coefficient of a cell mass composed of a collection of identical cells. This allows us to better inform the parameter values used in reaction-diffusion models of biological systems. We go on to apply this technique to a particular situation: modeling the dynamics of a collection of biological cells which experience strong cell-cell adhesion. In so doing, we derive a nonlinear fourth-order partial differential equation to model this system. We conclude by discussing the practical utility of this work in illuminating the link between the microscopic behavior of individual biological cells and the macroscopic behavior of the aggregate to which they give rise, and also by giving some insights into how the modeling of cell-cell adhesion may be treated mathematically.