Biological inferences from a mathematical model of comedo ductal carcinoma in situ of the breast

Gell: A GPU-powered 3D hybrid simulator for large-scale multicellular system

As a powerful but computationally intensive method, hybrid computational models study the dynamics of multicellular systems by evolving discrete cells in reacting and diffusing extracellular microenvironments. As the scale and complexity of studied biological systems continuously increase, the exploding computational cost starts to limit large-scale cell-based simulations. To facilitate the large-scale hybrid computational simulation and make it feasible on easily accessible computational devices, we develop Gell (GPU Cell), a fast and memory-efficient open-source GPU-based hybrid computational modeling platform for large-scale system modeling. We fully parallelize the simulations on GPU for high computational efficiency and propose a novel voxel sorting method to further accelerate the modeling of massive cell-cell mechanical interaction with negligible additional memory footprint. As a result, Gell efficiently handles simulations involving tens of millions of cells on a personal computer. We compare the performance of Gell with a state-of-the-art paralleled CPU-based simulator on a hanging droplet spheroid growth task and further demonstrate Gell with a ductal carcinoma in situ (DCIS) simulation. Gell affords ~150X acceleration over the paralleled CPU method with one-tenth of the memory requirement.

Bio-Mechanical Model of Osteosarcoma Tumor Microenvironment: A Porous Media Approach

Osteosarcoma is the most common malignant bone tumor in children and adolescents with a poor prognosis. To describe the progression of osteosarcoma, we expanded a system of data-driven ODE from a previous study into a system of Reaction-Diffusion-Advection (RDA) equations and coupled it with Biot equations of poroelasticity to form a bio-mechanical model. The RDA system includes the spatio-temporal information of the key components of the tumor microenvironment. The Biot equations are comprised of an equation for the solid phase, which governs the movement of the solid tumor, and an equation for the fluid phase, which relates to the motion of cells. The model predicts the total number of cells and cytokines of the tumor microenvironment and simulates the tumor's size growth. We simulated different scenarios using this model to investigate the impact of several biomedical settings on tumors' growth. The results indicate the importance of macrophages in tumors' growth. Particularly, we have observed a high co-localization of macrophages and cancer cells, and the concentration of tumor cells increases as the number of macrophages increases.

A PDE Model of Breast Tumor Progression in MMTV-PyMT Mice

The evolution of breast tumors greatly depends on the interaction network among different cell types, including immune cells and cancer cells in the tumor. This study takes advantage of newly collected rich spatio-temporal mouse data to develop a data-driven mathematical model of breast tumors that considers cells' location and key interactions in the tumor. The results show that cancer cells have a minor presence in the area with the most overall immune cells, and the number of activated immune cells in the tumor is depleted over time when there is no influx of immune cells. Interestingly, in the case of the influx of immune cells, the highest concentrations of both T cells and cancer cells are in the boundary of the tumor, as we use the Robin boundary condition to model the influx of immune cells. In other words, the influx of immune cells causes a dominant outward advection for cancer cells. We also investigate the effect of cells' diffusion and immune cells' influx rates in the dynamics of cells in the tumor micro-environment. Sensitivity analyses indicate that cancer cells and adipocytes' diffusion rates are the most sensitive parameters, followed by influx and diffusion rates of cytotoxic T cells, implying that targeting them is a possible treatment strategy for breast cancer.

The Multiple Dimensions of Networks in Cancer: A Perspective

This perspective article gathers the latest developments in mathematical and computational oncology tools that exploit network approaches for the mathematical modelling, analysis, and simulation of cancer development and therapy design. It instigates the community to explore new paths and synergies under the umbrella of the Special Issue “Networks in Cancer: From Symmetry Breaking to Targeted Therapy”. The focus of the perspective is to demonstrate how networks can model the physics, analyse the interactions, and predict the evolution of the multiple processes behind tumour-host encounters across multiple scales. From agent-based modelling and mechano-biology to machine learning and predictive modelling, the perspective motivates a methodology well suited to mathematical and computational oncology and suggests approaches that mark a viable path towards adoption in the clinic.

Tumor growth and calcification in evolving microenvironmental geometries

In this paper, we apply the diffuse domain framework developed in Chen and Lowengrub (Tumor growth in complex, evolving microenvironmental geometries: A diffuse domain approach, J. Theor. Biol. 361 (2014) 14-30) to study the effects of a deformable basement membrane (BM) on the growth of a tumor in a confined, ductal geometry, such as ductal carcinoma in situ (DCIS). We use a continuum model of tumor microcalcification and investigate the tumor extent beyond the microcalcification. In order to solve the governing equations efficiently, we develop a stable nonlinear multigrid finite difference method. Two dimensional simulations are performed where the adhesion between tumor cells and the basement membrane is varied. Additional simulations considering the variation of duct radius and membrane stiffness are also conducted. The results demonstrate that enhanced membrane deformability promotes tumor growth and tumor calcification. When the duct radius is small, the cell-BM adhesion is weak or when the membrane is slightly deformed, the mammographic and pathologic tumor extents are linearly correlated, as predicted by Macklin et al. (J. Theor. Biol. 301 (2012) 122-140) using an agent-based model that does not account for the deformability of the basement membrane and the active forces that the membrane imparts on the tumor cells. Interestingly, we predict that when the duct radius is large, there is strong cell-BM adhesion or the membrane is highly deformed, the extents of the mammographic and pathologic tumors are instead quadratically correlated. The simulations can help surgeons to measure DCIS surgical margins while removing less non-cancerous tissue, and can improve targeting of intra- and post-operative radiotherapy.

An agent-based model of triple-negative breast cancer: the interplay between chemokine receptor CCR5 expression, cancer stem cells, and hypoxia

Background

Triple-negative breast cancer lacks estrogen, progesterone, and HER2 receptors and is thus not possible to treat with targeted therapies for these receptors. Therefore, a greater understanding of triple-negative breast cancer is necessary for the treatment of this cancer type. In previous work from our laboratory, we found that chemokine ligand-receptor CCL5-CCR5 axis is important for the metastasis of human triple-negative breast cancer cell MDA-MB-231 to the lymph nodes and lungs, in a mouse xenograft model. We collected relevant experimental data from our and other laboratories for numbers of cancer stem cells, numbers of CCR5+ cells, and cell migration rates for different breast cancer cell lines and different experimental conditions.

Results

Using these experimental data we developed an in silico agent-based model of triple-negative breast cancer that considers surface receptor CCR5-high and CCR5-low cells and breast cancer stem cells, to predict the tumor growth rate and spatio-temporal distribution of cells in primary tumors. We find that high cancer stem cell percentages greatly increase tumor growth. We find that anti-stem cell treatment decreases tumor growth but may not lead to dormancy unless all stem cells get eliminated. We further find that hypoxia increases overall tumor growth and treatment with a CCR5 inhibitor maraviroc slightly decreases overall tumor growth. We also characterize 3D shapes of solid and invasive tumors using several shape metrics.

Conclusions

Breast cancer stem cells and CCR5+ cells affect the overall growth and morphology of breast tumors. In silico drug treatments demonstrate limited efficacy of incomplete inhibition of cancer stem cells after which tumor growth recurs, and CCR5 inhibition causes only a slight reduction in tumor growth.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-017-0445-x) contains supplementary material, which is available to authorized users.

Environmental factors in breast cancer invasion: a mathematical modelling review

This review presents a brief overview of breast cancer, focussing on its heterogeneity and the role of mathematical modelling and simulation in teasing apart the underlying biophysical processes. Following a brief overview of the main known pathophysiological features of ductal carcinoma, attention is paid to differential equation-based models (both deterministic and stochastic), agent-based modelling, multi-scale modelling, lattice-based models and image-driven modelling. A number of vignettes are presented where these modelling approaches have elucidated novel aspects of breast cancer dynamics, and we conclude by offering some perspectives on the role mathematical modelling can play in understanding breast cancer development, invasion and treatment therapies.

Lattice-based model of ductal carcinoma in situ suggests rules for breast cancer progression to an invasive state

Ductal carcinoma in situ (DCIS) is a heterogeneous group of non-invasive lesions of the breast that result from abnormal proliferation of mammary epithelial cells. Pathologists characterize DCIS by four tissue morphologies (micropapillary, cribriform, solid, and comedo), but the underlying mechanisms that distinguish the development and progression of these morphologies are not well understood. Here we explored the conditions leading to the emergence of the different morphologies of DCIS using a two-dimensional multi-cell lattice-based model that incorporates cell proliferation, apoptosis, necrosis, adhesion, and contractility. We found that the relative rates of cell proliferation and apoptosis governed which of the four morphologies emerged. High proliferation and low apoptosis favored the emergence of solid and comedo morphologies. In contrast, low proliferation and high apoptosis led to the micropapillary morphology, whereas high proliferation and high apoptosis led to the cribriform morphology. The natural progression between morphologies cannot be investigated in vivo since lesions are usually surgically removed upon detection; however, our model suggests probable transitions between these morphologies during breast cancer progression. Importantly, cribriform and comedo appear to be the ultimate morphologies of DCIS. Motivated by previous experimental studies demonstrating that tumor cells behave differently depending on where they are located within the mammary duct in vivo or in engineered tissues, we examined the effects of tissue geometry on the progression of DCIS. In agreement with our previous experimental work, we found that cells are more likely to invade from the end of ducts and that this preferential invasion is regulated by cell adhesion and contractility. This model provides additional insight into tumor cell behavior and allows the exploration of phenotypic transitions not easily monitored in vivo.

Breast cancer is a complex disease that affects women worldwide. One heterogeneous group of lesions, ductal carcinoma in situ (DCIS), often begins as a nonmalignant disease but can readily progress if left untreated. The progression of this disease is not well understood because DCIS is typically removed upon detection. Therefore, computational models might help predict whether DCIS will remain nonmalignant or progress towards invasive ductal carcinoma. Here we used a multi-cell lattice-based model to explore the relative effects of cell proliferation, death, division axis, adhesion and contractility on the development and progression of DCIS. We also examined the emergence and progression of DCIS in physiologically relevant geometries of the mammary duct. Our model suggests several plausible progressions between morphologies of DCIS, and predicts that some regions of a duct are preferential for tumor cell invasion.

Mathematical modeling for novel cancer drug discovery and development

INTRODUCTION

Mathematical modeling enables: the in silico classification of cancers, the prediction of disease outcomes, optimization of therapy, identification of promising drug targets and prediction of resistance to anticancer drugs. In silico pre-screened drug targets can be validated by a small number of carefully selected experiments.

AREAS COVERED

This review discusses the basics of mathematical modeling in cancer drug discovery and development. The topics include in silico discovery of novel molecular drug targets, optimization of immunotherapies, personalized medicine and guiding preclinical and clinical trials. Breast cancer has been used to demonstrate the applications of mathematical modeling in cancer diagnostics, the identification of high-risk population, cancer screening strategies, prediction of tumor growth and guiding cancer treatment.

EXPERT OPINION

Mathematical models are the key components of the toolkit used in the fight against cancer. The combinatorial complexity of new drugs discovery is enormous, making systematic drug discovery, by experimentation, alone difficult if not impossible. The biggest challenges include seamless integration of growing data, information and knowledge, and making them available for a multiplicity of analyses. Mathematical models are essential for bringing cancer drug discovery into the era of Omics, Big Data and personalized medicine.

Modeling the mechanics of cancer: effect of changes in cellular and extra-cellular mechanical properties

Malignant transformation, though primarily driven by genetic mutations in cells, is also accompanied by specific changes in cellular and extra-cellular mechanical properties such as stiffness and adhesivity. As the transformed cells grow into tumors, they interact with their surroundings via physical contacts and the application of forces. These forces can lead to changes in the mechanical regulation of cell fate based on the mechanical properties of the cells and their surrounding environment. A comprehensive understanding of cancer progression requires the study of how specific changes in mechanical properties influences collective cell behavior during tumor growth and metastasis. Here we review some key results from computational models describing the effect of changes in cellular and extra-cellular mechanical properties and identify mechanistic pathways for cancer progression that can be targeted for the prediction, treatment, and prevention of cancer.

Automated reconstruction algorithm for identification of 3D architectures of cribriform ductal carcinoma in situ

Ductal carcinoma in situ (DCIS) is a pre-invasive carcinoma of the breast that exhibits several distinct morphologies but the link between morphology and patient outcome is not clear. We hypothesize that different mechanisms of growth may still result in similar 2D morphologies, which may look different in 3D. To elucidate the connection between growth and 3D morphology, we reconstruct the 3D architecture of cribriform DCIS from resected patient material. We produce a fully automated algorithm that aligns, segments, and reconstructs 3D architectures from microscopy images of 2D serial sections from human specimens. The alignment algorithm is based on normalized cross correlation, the segmentation algorithm uses histogram equilization, Otsu's thresholding, and morphology techniques to segment the duct and cribra. The reconstruction method combines these images in 3D. We show that two distinct 3D architectures are indeed found in samples whose 2D histological sections are similarly identified as cribriform DCIS. These differences in architecture support the hypothesis that luminal spaces may form due to different mechanisms, either isolated cell death or merging fronds, leading to the different architectures. We find that out of 15 samples, 6 were found to have 'bubble-like' cribra, 6 were found to have 'tube-like' criba and 3 were 'unknown.' We propose that the 3D architectures found, 'bubbles' and 'tubes', account for some of the heterogeneity of the disease and may be prognostic indicators of different patient outcomes.

Patient-calibrated agent-based modelling of ductal carcinoma in situ (DCIS): from microscopic measurements to macroscopic predictions of clinical progression

Ductal carcinoma in situ (DCIS)--a significant precursor to invasive breast cancer--is typically diagnosed as microcalcifications in mammograms. However, the effective use of mammograms and other patient data to plan treatment has been restricted by our limited understanding of DCIS growth and calcification. We develop a mechanistic, agent-based cell model and apply it to DCIS. Cell motion is determined by a balance of biomechanical forces. We use potential functions to model interactions with the basement membrane and amongst cells of unequal size and phenotype. Each cell's phenotype is determined by genomic/proteomic- and microenvironment-dependent stochastic processes. Detailed "sub-models" describe cell volume changes during proliferation and necrosis; we are the first to account for cell calcification. We introduce the first patient-specific calibration method to fully constrain the model based upon clinically-accessible histopathology data. After simulating 45 days of solid-type DCIS with comedonecrosis, the model predicts: necrotic cell lysis acts as a biomechanical stress relief and is responsible for the linear DCIS growth observed in mammography; the rate of DCIS advance varies with the duct radius; the tumour grows 7-10mm per year--consistent with mammographic data; and the mammographic and (post-operative) pathologic sizes are linearly correlated--in quantitative agreement with the clinical literature. Patient histopathology matches the predicted DCIS microstructure: an outer proliferative rim surrounds a stratified necrotic core with nuclear debris on its outer edge and calcification in the centre. This work illustrates that computational modelling can provide new insight on the biophysical underpinnings of cancer. It may 1-day be possible to augment a patient's mammography and other imaging with rigorously-calibrated models that help select optimal surgical margins based upon the patient's histopathologic data.

Normal morphogenesis of epithelial tissues and progression of epithelial tumors

Epithelial cells organize into various tissue architectures that largely maintain their structure throughout the life of an organism. For decades, the morphogenesis of epithelial tissues has fascinated scientists at the interface of cell, developmental, and molecular biology. Systems biology offers ways to combine knowledge from these disciplines by building integrative models that are quantitative and predictive. Can such models be useful for gaining a deeper understanding of epithelial morphogenesis? Here, we take inventory of some recurring themes in epithelial morphogenesis that systems approaches could strive to capture. Predictive understanding of morphogenesis at the systems level would prove especially valuable for diseases such as cancer, where epithelial tissue architecture is profoundly disrupted.

The role of the microenvironment in tumor growth and invasion

Mathematical modeling and computational analysis are essential for understanding the dynamics of the complex gene networks that control normal development and homeostasis, and can help to understand how circumvention of that control leads to abnormal outcomes such as cancer. Our objectives here are to discuss the different mechanisms by which the local biochemical and mechanical microenvironment, which is comprised of various signaling molecules, cell types and the extracellular matrix (ECM), affects the progression of potentially-cancerous cells, and to present new results on two aspects of these effects. We first deal with the major processes involved in the progression from a normal cell to a cancerous cell at a level accessible to a general scientific readership, and we then outline a number of mathematical and computational issues that arise in cancer modeling. In Section 2 we present results from a model that deals with the effects of the mechanical properties of the environment on tumor growth, and in Section 3 we report results from a model of the signaling pathways and the tumor microenvironment (TME), and how their interactions affect the development of breast cancer. The results emphasize anew the complexities of the interactions within the TME and their effect on tumor growth, and show that tumor progression is not solely determined by the presence of a clone of mutated immortal cells, but rather that it can be 'community-controlled'.

Current trends in mathematical modeling of tumor-microenvironment interactions: a survey of tools and applications

In its simplest description, a tumor is comprised of an expanding population of transformed cells supported by a surrounding microenvironment termed the tumor stroma. The tumor microenvironment has a very complex composition, including multiple types of stromal cells, a dense network of various extracellular matrix (ECM) fibers interpenetrated by the interstitial fluid and gradients of several chemical species that either are dissolved in the fluid or are bound to the ECM structure. In order to study experimentally such complex interactions between multiple players, cancer is dissected and considered at different scales of complexity, such as protein interactions, biochemical pathways, cellular functions or whole organism studies. However, the integration of information acquired from these studies into a common description is as difficult as the disease itself. Computational models of cancer can provide cancer researchers with invaluable tools that are capable of integrating the complexity into organizing principles as well as suggesting testable hypotheses. We will focus in this Minireview on mathematical models in which the whole cell is a main modeling unit. We will present a current stage of such cell-focused mathematical modeling incorporating different stromal components and their interactions with growing tumors, and discuss what modeling approaches can be undertaken to complement the in vivo and in vitro experimentation.

A 2D mechanistic model of breast ductal carcinoma in situ (DCIS) morphology and progression

Ductal carcinoma in situ (DCIS) of the breast is a non-invasive tumor in which cells proliferate abnormally, but remain confined within a duct. Although four distinguishable DCIS morphologies are recognized, the mechanisms that generate these different morphological classes remain unclear, and consequently the prognostic strength of DCIS classification is not strong. To improve the understanding of the relation between morphology and time course, we have developed a 2D in silico particle model of the growth of DCIS within a single breast duct. This model considers mechanical effects such as cellular adhesion and intra-ductal pressure, and biological features including proliferation, apoptosis, necrosis, and cell polarity. Using this model, we find that different regions of parameter space generate distinct morphological subtypes of DCIS, so elucidating the relation between morphology and time course. Furthermore, we find that tumors with similar architectures may in fact be produced through different mechanisms, and we propose future work to further disentangle the mechanisms involved in DCIS progression.

Magnetic resonance imaging of the natural history of in situ mammary neoplasia in transgenic mice: a pilot study

Introduction

Because of the small size of in situ mammary cancers in mouse models, high-resolution imaging techniques are required to effectively observe how lesions develop, grow and progress over time. The purpose of this study was to use magnetic resonance (MR) imaging to track in vivo the transition from in situ neoplasia to invasive cancer in a transgenic mouse model of human cancer.

Methods

MR images of 12 female C3(1) SV40 Tag mice that develop mammary intraepithelial neoplasia (MIN) were obtained. MIN is believed to be similar to human ductal carcinoma in situ (DCIS) and is considered a precursor of invasive tumors. Images were serially obtained from 10-21 weeks of age at 2-3 week intervals. MIN lesions were identified based on their morphology on MR images. Lesions were followed over time and several lesion features were measured including volume, growth rate and morphology. For those MIN lesions that progressed to invasive cancer the progression time was measured.

Results

Overall, 21 MIN lesions were initially detected at an average initial volume of 0.3 ± 0.2 mm³ with an average growth rate of -0.15 ± 0.66 week^-1. Even though all mice were inbred to express the SV40 Tag transgene in the mammary epithelium and expected to develop invasive carcinoma, the individual MIN lesions took vastly different progression paths: (i) 9 lesions progressed to invasive tumors with an average progression time of 4.6 ± 1.9 weeks; (ii) 2 lesions regressed, i.e., were not detected on future images; and (iii) 5 were stable for over 8 weeks, and were demonstrated by a statistical model to represent indolent disease.

Conclusions

To our knowledge, the results reported here are the first measurements of the timescale and characteristics of progression from in situ neoplasia to invasive carcinoma and provide image-based evidence that DCIS may be a non-obligate precursor lesion with highly variable outcomes. In addition, this study represents a first step towards developing methods of image acquisition for identifying radiological characteristics that might predict which in situ neoplasias will become invasive cancers and which are unlikely to progress.

Nonlinear modelling of cancer: bridging the gap between cells and tumours

Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.

A quantitative theoretical model for the development of malignancy in ductal carcinoma in situ

Mathematical models and clinical observations have demonstrated that microenvironmental hypoxia and acidosis are important selection factors during the later stages of the somatic evolution of breast cancer. The consequent promotion of constitutive upregulation of glycolysis and resistance to acid-induced cellular toxicity is hypothesized to be critical for the ability of cancer cells to invade host tissue. In this work we developed a 3D fixed lattice cellular automata model to study the role of these two phenotypes in determining morphology and the potential for invasion of ductal carcinoma in situ (DCIS), which in this work is defined as the erosion of a healthy epithelial cell layer and direct contact with the basement membrane. The model was conceived as a 40-cell wide epithelial duct surrounded by blood vessels and composed of a basement membrane and one internal layer of epithelial cells. Our results show that an increment in the order of 8-fold in glucose metabolism and an increase in acid resistance corresponding to pH thresholds of approximately 6.8 and 6.45 for quiescence and death, respectively, are required for the tumor to breach through the layer of healthy epithelial cells and reach the basement membrane as a first step for invasion. Our model also suggests correlations between classic morphologies and different values of hyperglycolytic and acid-resistant phenotypes, indicating that immunohistochemistry studies targeting these genes may improve the predictive power of morphological analyses of biopsies.

Comedo-ductal carcinoma in situ: A paradoxical role for programmed cell death

Comedo-DCIS is a histologic subtype of preinvasive breast neoplasia that is characterized by prominent apoptotic cell death and has greater malignant potential than other DCIS subtypes. We investigated the mechanisms of apoptosis in comedo-DCIS and its role in conversion of comedo-DCIS to invasive cancer. Clinical comedo-DCIS excisions and the MCF10DCIS.com human breast cancer model which produces lesions resembling comedo-DCIS were analyzed. Apoptotic luminal and myoepithelial cells were identified by TUNEL and reactivity to cleaved PARP antibody and cell death assessed by Western blotting, Mitocapture and immunohistochemical assays. MCF10DCIS.com cells undergo spontaneous apoptosis in vitro, both in monolayers and multicellular spheroids; it is associated with increased mitochondrial membrane permeability, increase in Bax/Bcl-2 ratio and occurs via caspase-9-dependent p53-independent pathway. This suggests that apoptosis is stromal-independent and that the cells are programmed to undergo apoptosis. Immunostaining with cleaved PARP antibody showed that myoepithelial apoptosis occurs before lesions progress to comedo-DCIS in both clinical comedo-DCIS and in vivo MCF10DCIS.com lesions. Intense staining for MMP-2, MMP-3, MMP-9 and MMP-11 was observed in the stroma and epithelia of solid DCIS lesions prior to conversion to comedo-DCIS in clinical and MCF10DCIS.com lesions. Gelatin zymography showed higher MMP-2 levels in lysates and conditioned media of MCF10DCIS. com cells undergoing apoptosis. These data suggest that signals arising from the outside (microenvironmental) and inside (internal genetic alterations) of the duct act in concert to trigger apoptosis of myoepithelial and luminal epithelial cells. Our findings implicate spontaneous apoptosis in both the etiology and progression of comedo-DCIS. It is possible that spontaneous apoptosis facilitates elimination of cells thus permitting expansion and malignant transformation of cancer cells that are resistant to spontaneous apoptosis.

Computational model of ductal carcinoma in situ: the effects of contact inhibition on pattern formation

The computational model presented in this paper focuses on modeling ductal carcinoma in situ (DCIS), which is the most commonly detected preinvasive form of breast cancer. The model aims to understand the biological mechanisms and resultant growth dynamics of DCIS. The cellular automaton model based on observed phenotypic characteristics of DCIS emphasize the important role of contact inhibition on lesion pattern formation. Computer simulations resembled the cribriform, micropapillary, solid, and comedo patterns of DCIS. The model has led to insights about the progression of the preinvasive disease such as possible explanations for coexisting micropapillary and cribriform patterns commonly found through histological analyses.

Quiescence as a mechanism for cyclical hypoxia and acidosis

Tumour tissue characteristically experiences fluctuations in substrate supply. This unstable microenvironment drives constitutive metabolic changes within cellular populations and, ultimately, leads to a more aggressive phenotype. Previously, variations in substrate levels were assumed to occur through oscillations in the haemodynamics of nearby and distant blood vessels. In this paper we examine an alternative hypothesis, that cycles of metabolite concentrations are also driven by cycles of cellular quiescence and proliferation. Using a mathematical modelling approach, we show that the interdependence between cell cycle and the microenvironment will induce typical cycles with the period of order hours in tumour acidity and oxygenation. As a corollary, this means that the standard assumption of metabolites entering diffusive equilibrium around the tumour is not valid; instead temporal dynamics must be considered.

An immersed boundary framework for modelling the growth of individual cells: an application to the early tumour development

A biomechanical approach in modelling the growth and division of a single fully deformable cell by using an immersed boundary method with distributed sources is presented, and its application to model the early tumour development is discussed. This mathematical technique couples a continuous description of a viscous incompressible cytoplasm with the dynamics of separate elastic cells, containing their own point nuclei, elastic plasma membranes with membrane receptors, and individually regulated cell processes. This model enables one to focus on the biomechanical properties of individual cells and on communication between cells and their microenvironment, simultaneously allowing for the formation of clusters or sheets of cells that act together as one complex tissue. Several examples of early tumours growing in various geometrical configurations and with distinct conditions of their initiation and progression are also presented to show the strength of our approach in modelling different topologies of the growing tissues in distinct biochemical conditions of the surrounding media.

Cellular automaton simulation examining progenitor hierarchy structure effects on mammary ductal carcinoma in situ

A computer simulation is used to model ductal carcinoma in situ, a form of non-invasive breast cancer. The simulation uses known histological morphology, cell types, and stochastic cell proliferation to evolve tumorous growth within a duct. The ductal simulation is based on a hybrid cellular automaton design using genetic rules to determine each cell's behavior. The genetic rules are a mutable abstraction that demonstrate genetic heterogeneity in a population. Our goal was to examine the role (if any) that recently discovered mammary stem cell hierarchies play in genetic heterogeneity, DCIS initiation and aggressiveness. Results show that simpler progenitor hierarchies result in greater genetic heterogeneity and evolve DCIS significantly faster. However, the more complex progenitor hierarchy structure was able to sustain the rapid reproduction of a cancer cell population for longer periods of time.

Modelling aspects of cancer dynamics: a review

Cancer is a complex disease in which a variety of factors interact over a wide range of spatial and temporal scales with huge datasets relating to the different scales available. However, these data do not always reveal the mechanisms underpinning the observed phenomena. In this paper, we explain why mathematics is a powerful tool for interpreting such data by presenting case studies that illustrate the types of insight that realistic theoretical models of solid tumour growth may yield. These range from discriminating between competing hypotheses for the formation of collagenous capsules associated with benign tumours to predicting the most likely stimulus for protease production in early breast cancer. We will also illustrate the benefits that may result when experimentalists and theoreticians collaborate by considering a novel anti-cancer therapy.

A cellular automata model of tumor-immune system interactions

We present a hybrid cellular automata-partial differential equation model of moderate complexity to describe the interactions between a growing tumor next to a nutrient source and the immune system of the host organism. The model allows both temporal and two-dimensional spatial evolution of the system under investigation and is comprised of biological cell metabolism rules derived from both the experimental and mathematical modeling literature. We present numerical simulations that display behaviors which are qualitatively similar to those exhibited in tumor-immune system interaction experiments. These include spherical tumor growth, stable and unstable oscillatory tumor growth, satellitosis and tumor infiltration by immune cells. Finally, the relationship between these different growth regimes and key system parameters is discussed.

The role of acidity in solid tumour growth and invasion

Acidic pH is a common characteristic of human tumours. It has a significant impact on tumour progression and response to therapies. In this paper, we develop a simple model of three-dimensional tumour growth to examine the role of acidosis in the interaction between normal and tumour cell populations. Both vascular and avascular tumour dynamics are investigated, and a number of different behaviours are observed. Whilst an avascular tumour always proceeds to a benign steady state, a vascular tumour may display either benign or invasive dynamics, depending on the value of a critical parameter. Analysis of the model allows us to assess novel therapies directed towards changing the level of acidity within the tumour.