A mathematical model of breast cancer development, local treatment and recurrence

Modelling circulating tumour cells for personalised survival prediction in metastatic breast cancer

Ductal carcinoma is one of the most common cancers among women, and the main cause of death is the formation of metastases. The development of metastases is caused by cancer cells that migrate from the primary tumour site (the mammary duct) through the blood vessels and extravasating they initiate metastasis. Here, we propose a multi-compartment model which mimics the dynamics of tumoural cells in the mammary duct, in the circulatory system and in the bone. Through a branching process model, we describe the relation between the survival times and the four markers mainly involved in metastatic breast cancer (EPCAM, CD47, CD44 and MET). In particular, the model takes into account the gene expression profile of circulating tumour cells to predict personalised survival probability. We also include the administration of drugs as bisphosphonates, which reduce the formation of circulating tumour cells and their survival in the blood vessels, in order to analyse the dynamic changes induced by the therapy. We analyse the effects of circulating tumour cells on the progression of the disease providing a quantitative measure of the cell driver mutations needed for invading the bone tissue. Our model allows to design intervention scenarios that alter the patient-specific survival probability by modifying the populations of circulating tumour cells and it could be extended to other cancer metastasis dynamics.

Chemotherapy in heterogeneous cultures of cancer cells with interconversion

Recently, the interconversion between differentiated and stem-like cancer cells has been observed. Here, we model the in vitro growth of heterogeneous cell cultures in the presence of interconversion from differentiated cancer cells to cancer stem cells (CSCs), showing that, by targeting only CSC with cytotoxic agents, it is not always possible to eradicate cancer. We have determined the kinetic conditions under which cytotoxic agents in in vitro heterogeneous cultures of cancer cells eradicate cancer. In particular, we have shown that the chemotherapeutic elimination of in vitro cultures of heterogeneous cancer cells is effective only if it targets all cancer cell types, and if the induced death rates for the different subpopulations of cancer cell types are large enough. The quantitative results of the model are compared and validated with experimental data.

Evolution of pre-existing versus acquired resistance to platinum drugs and PARP inhibitors in BRCA-associated cancers

Platinum drugs and PARP inhibitors (“PARPis”) are considered to be effective in BRCA-associated cancers with impaired DNA repair. These agents cause stalled and collapsed replication forks and create double-strand breaks effectively in the absence of repair mechanisms, resulting in arrest of the cell cycle and induction of cell death. However, recent studies have shown failure of these chemotherapeutic agents due to emerging drug resistance. In this study, we developed a stochastic model of BRCA-associated cancer progression in which there are four cancer populations: those with (i) functional BRCA, (ii) dysfunctional BRCA, (iii) functional BRCA and a growth advantage, and (iv) dysfunctional BRCA and a growth advantage. These four cancer populations expand from one cancer cell with normal repair function until the total cell number reaches a detectable amount. We derived formulas for the probability and expected numbers of each population at the time of detection. Furthermore, we extended the model to consider the tumor dynamics during treatment. Results from the model were validated and showed good agreement with clinical and experimental evidence in BRCA-associated cancers. Based on the model, we investigated conditions in which drug resistance during the treatment course originated from either a pre-existing drug-resistant population or a de novo population, due to secondary mutations. Finally, we found that platinum drugs and PARPis were effective if (i) BRCA inactivation is present, (ii) the cancer was diagnosed early, and (iii) tumor growth is rapid. Our results indicate that different types of cancers have a preferential way of acquiring resistance to platinum drugs and PARPis according to their growth and mutational characteristics.

Simulating cancer growth with multiscale agent-based modeling

There have been many techniques developed in recent years to in silico model a variety of cancer behaviors. Agent-based modeling is a specific discrete-based hybrid modeling approach that allows simulating the role of diversity in cell populations as well as within each individual cell; it has therefore become a powerful modeling method widely used by computational cancer researchers. Many aspects of tumor morphology including phenotype-changing mutations, the adaptation to microenvironment, the process of angiogenesis, the influence of extracellular matrix, reactions to chemotherapy or surgical intervention, the effects of oxygen and nutrient availability, and metastasis and invasion of healthy tissues have been incorporated and investigated in agent-based models. In this review, we introduce some of the most recent agent-based models that have provided insight into the understanding of cancer growth and invasion, spanning multiple biological scales in time and space, and we further describe several experimentally testable hypotheses generated by those models. We also discuss some of the current challenges of multiscale agent-based cancer models.

Estimating dose painting effects in radiotherapy: a mathematical model

Tumor heterogeneity is widely considered to be a determinant factor in tumor progression and in particular in its recurrence after therapy. Unfortunately, current medical techniques are unable to deduce clinically relevant information about tumor heterogeneity by means of non-invasive methods. As a consequence, when radiotherapy is used as a treatment of choice, radiation dosimetries are prescribed under the assumption that the malignancy targeted is of a homogeneous nature. In this work we discuss the effects of different radiation dose distributions on heterogeneous tumors by means of an individual cell-based model. To that end, a case is considered where two tumor cell phenotypes are present, which we assume to strongly differ in their respective cell cycle duration and radiosensitivity properties. We show herein that, as a result of such differences, the spatial distribution of the corresponding phenotypes, whence the resulting tumor heterogeneity can be predicted as growth proceeds. In particular, we show that if we start from a situation where a majority of ordinary cancer cells (CCs) and a minority of cancer stem cells (CSCs) are randomly distributed, and we assume that the length of CSC cycle is significantly longer than that of CCs, then CSCs become concentrated at an inner region as tumor grows. As a consequence we obtain that if CSCs are assumed to be more resistant to radiation than CCs, heterogeneous dosimetries can be selected to enhance tumor control by boosting radiation in the region occupied by the more radioresistant tumor cell phenotype. It is also shown that, when compared with homogeneous dose distributions as those being currently delivered in clinical practice, such heterogeneous radiation dosimetries fare always better than their homogeneous counterparts. Finally, limitations to our assumptions and their resulting clinical implications will be discussed.

Risk-adapted targeted intraoperative radiotherapy versus whole-breast radiotherapy for breast cancer: 5-year results for local control and overall survival from the TARGIT-A randomised trial

BACKGROUND

The TARGIT-A trial compared risk-adapted radiotherapy using single-dose targeted intraoperative radiotherapy (TARGIT) versus fractionated external beam radiotherapy (EBRT) for breast cancer. We report 5-year results for local recurrence and the first analysis of overall survival.

METHODS

TARGIT-A was a randomised, non-inferiority trial. Women aged 45 years and older with invasive ductal carcinoma were enrolled and randomly assigned in a 1:1 ratio to receive TARGIT or whole-breast EBRT, with blocks stratified by centre and by timing of delivery of targeted intraoperative radiotherapy: randomisation occurred either before lumpectomy (prepathology stratum, TARGIT concurrent with lumpectomy) or after lumpectomy (postpathology stratum, TARGIT given subsequently by reopening the wound). Patients in the TARGIT group received supplemental EBRT (excluding a boost) if unforeseen adverse features were detected on final pathology, thus radiotherapy was risk-adapted. The primary outcome was absolute difference in local recurrence in the conserved breast, with a prespecified non-inferiority margin of 2·5% at 5 years; prespecified analyses included outcomes as per timing of randomisation in relation to lumpectomy. Secondary outcomes included complications and mortality. This study is registered with ClinicalTrials.gov, number NCT00983684.

FINDINGS

Patients were enrolled at 33 centres in 11 countries, between March 24, 2000, and June 25, 2012. 1721 patients were randomised to TARGIT and 1730 to EBRT. Supplemental EBRT after TARGIT was necessary in 15·2% [239 of 1571] of patients who received TARGIT (21·6% prepathology, 3·6% postpathology). 3451 patients had a median follow-up of 2 years and 5 months (IQR 12-52 months), 2020 of 4 years, and 1222 of 5 years. The 5-year risk for local recurrence in the conserved breast was 3·3% (95% CI 2·1-5·1) for TARGIT versus 1·3% (0·7-2·5) for EBRT (p=0·042). TARGIT concurrently with lumpectomy (prepathology, n=2298) had much the same results as EBRT: 2·1% (1·1-4·2) versus 1·1% (0·5-2·5; p=0·31). With delayed TARGIT (postpathology, n=1153) the between-group difference was larger than 2·5% (TARGIT 5·4% [3·0-9·7] vs EBRT 1·7% [0·6-4·9]; p=0·069). Overall, breast cancer mortality was much the same between groups (2·6% [1·5-4·3] for TARGIT vs 1·9% [1·1-3·2] for EBRT; p=0·56) but there were significantly fewer non-breast-cancer deaths with TARGIT (1·4% [0·8-2·5] vs 3·5% [2·3-5·2]; p=0·0086), attributable to fewer deaths from cardiovascular causes and other cancers. Overall mortality was 3·9% (2·7-5·8) for TARGIT versus 5·3% (3·9-7·3) for EBRT (p=0·099). Wound-related complications were much the same between groups but grade 3 or 4 skin complications were significantly reduced with TARGIT (four of 1720 vs 13 of 1731, p=0·029).

INTERPRETATION

TARGIT concurrent with lumpectomy within a risk-adapted approach should be considered as an option for eligible patients with breast cancer carefully selected as per the TARGIT-A trial protocol, as an alternative to postoperative EBRT.

FUNDING

University College London Hospitals (UCLH)/UCL Comprehensive Biomedical Research Centre, UCLH Charities, National Institute for Health Research Health Technology Assessment programme, Ninewells Cancer Campaign, National Health and Medical Research Council, and German Federal Ministry of Education and Research.

Spatial invasion dynamics on random and unstructured meshes: implications for heterogeneous tumor populations

In this work we discuss a spatial evolutionary model for a heterogeneous cancer cell population. We consider the gain-of-function mutations that not only change the fitness potential of the mutant phenotypes against normal background cells but may also increase the relative motility of the mutant cells. The spatial modeling is implemented as a stochastic evolutionary system on a structured grid (a lattice, with random neighborhoods, which is not necessarily bi-directional) or on a two-dimensional unstructured mesh, i.e. a bi-directional graph with random numbers of neighbors. We present a computational approach to investigate the fixation probability of mutants in these spatial models. Additionally, we examine the effect of the migration potential on the spatial dynamics of mutants on unstructured meshes. Our results suggest that the probability of fixation is negatively correlated with the width of the distribution of the neighborhood size. Also, the fixation probability increases given a migration potential for mutants. We find that the fixation probability (of advantaged, disadvantaged and neutral mutants) on unstructured meshes is relatively smaller than the corresponding results on regular grids. More importantly, in the case of neutral mutants the introduction of a migration potential has a critical effect on the fixation probability and increases this by orders of magnitude. Further, we examine the effect of boundaries and as intuitively expected, the fixation probability is smaller on the boundary of regular grids when compared to its value in the bulk. Based on these computational results, we speculate on possible better therapeutic strategies that may delay tumor progression to some extent.

Dynamics of tissue topology during cancer invasion and metastasis

During tumor progression, cancer cells mix with other cell populations including epithelial and endothelial cells. Although potentially important clinically as well as for our understanding of basic tumor biology, the process of mixing is largely a mystery. Furthermore, there is no rigorous, analytical measure available for quantifying the mixing of compartments within a tumor. I present here a mathematical model of tissue repair and tumor growth based on collective cell migration that simulates a wide range of observed tumor behaviors with correct tissue compartmentalization and connectivity. The resulting dynamics are analyzed in light of the Euler characteristic number (χ), which describes key topological features such as fragmentation, looping and cavities. The analysis predicts a number of regimes in which the cancer cells can encapsulate normal tissue, form a co-interdigitating mass, or become fragmented and encapsulated by endothelial or epithelial structures. Key processes that affect the topological changes are the production of provisional matrix in the tumor, and the migration of endothelial or epithelial cells on this matrix. Furthermore, the simulations predict that topological changes during tumor invasion into blood vessels may contribute to metastasis. The topological analysis outlined here could be useful for tumor diagnosis or monitoring response to therapy and would only require high resolution, 3D image data to resolve and track the various cell compartments.

Symmetric vs. asymmetric stem cell divisions: an adaptation against cancer?

Traditionally, it has been held that a central characteristic of stem cells is their ability to divide asymmetrically. Recent advances in inducible genetic labeling provided ample evidence that symmetric stem cell divisions play an important role in adult mammalian homeostasis. It is well understood that the two types of cell divisions differ in terms of the stem cells' flexibility to expand when needed. On the contrary, the implications of symmetric and asymmetric divisions for mutation accumulation are still poorly understood. In this paper we study a stochastic model of a renewing tissue, and address the optimization problem of tissue architecture in the context of mutant production. Specifically, we study the process of tumor suppressor gene inactivation which usually takes place as a consequence of two “hits”, and which is one of the most common patterns in carcinogenesis. We compare and contrast symmetric and asymmetric (and mixed) stem cell divisions, and focus on the rate at which double-hit mutants are generated. It turns out that symmetrically-dividing cells generate such mutants at a rate which is significantly lower than that of asymmetrically-dividing cells. This result holds whether single-hit (intermediate) mutants are disadvantageous, neutral, or advantageous. It is also independent on whether the carcinogenic double-hit mutants are produced only among the stem cells or also among more specialized cells. We argue that symmetric stem cell divisions in mammals could be an adaptation which helps delay the onset of cancers. We further investigate the question of the optimal fraction of stem cells in the tissue, and quantify the contribution of non-stem cells in mutant production. Our work provides a hypothesis to explain the observation that in mammalian cells, symmetric patterns of stem cell division seem to be very common.

How changes in extracellular matrix mechanics and gene expression variability might combine to drive cancer progression

Changes in extracellular matrix (ECM) structure or mechanics can actively drive cancer progression; however, the underlying mechanism remains unknown. Here we explore whether this process could be mediated by changes in cell shape that lead to increases in genetic noise, given that both factors have been independently shown to alter gene expression and induce cell fate switching. We do this using a computer simulation model that explores the impact of physical changes in the tissue microenvironment under conditions in which physical deformation of cells increases gene expression variability among genetically identical cells. The model reveals that cancerous tissue growth can be driven by physical changes in the microenvironment: when increases in cell shape variability due to growth-dependent increases in cell packing density enhance gene expression variation, heterogeneous autonomous growth and further structural disorganization can result, thereby driving cancer progression via positive feedback. The model parameters that led to this prediction are consistent with experimental measurements of mammary tissues that spontaneously undergo cancer progression in transgenic C3(1)-SV40Tag female mice, which exhibit enhanced stiffness of mammary ducts, as well as progressive increases in variability of cell-cell relations and associated cell shape changes. These results demonstrate the potential for physical changes in the tissue microenvironment (e.g., altered ECM mechanics) to induce a cancerous phenotype or accelerate cancer progression in a clonal population through local changes in cell geometry and increased phenotypic variability, even in the absence of gene mutation.

A mathematical model of cancer stem cell driven tumor initiation: implications of niche size and loss of homeostatic regulatory mechanisms

Hierarchical organized tissue structures, with stem cell driven cell differentiation, are critical to the homeostatic maintenance of most tissues, and this underlying cellular architecture is potentially a critical player in the development of a many cancers. Here, we develop a mathematical model of mutation acquisition to investigate how deregulation of the mechanisms preserving stem cell homeostasis contributes to tumor initiation. A novel feature of the model is the inclusion of both extrinsic and intrinsic chemical signaling and interaction with the niche to control stem cell self-renewal. We use the model to simulate the effects of a variety of types and sequences of mutations and then compare and contrast all mutation pathways in order to determine which ones generate cancer cells fastest. The model predicts that the sequence in which mutations occur significantly affects the pace of tumorigenesis. In addition, tumor composition varies for different mutation pathways, so that some sequences generate tumors that are dominated by cancerous cells with all possible mutations, while others are primarily comprised of cells that more closely resemble normal cells with only one or two mutations. We are also able to show that, under certain circumstances, healthy stem cells diminish due to the displacement by mutated cells that have a competitive advantage in the niche. Finally, in the event that all homeostatic regulation is lost, exponential growth of the cancer population occurs in addition to the depletion of normal cells. This model helps to advance our understanding of how mutation acquisition affects mechanisms that influence cell-fate decisions and leads to the initiation of cancers.

Stochastic Tunneling of Two Mutations in a Population of Cancer Cells

Cancer initiation, progression, and the emergence of drug resistance are driven by specific genetic and/or epigenetic alterations such as point mutations, structural alterations, DNA methylation and histone modification changes. These alterations may confer advantageous, deleterious or neutral effects to mutated cells. Previous studies showed that cells harboring two particular alterations may arise in a fixed-size population even in the absence of an intermediate state in which cells harboring only the first alteration take over the population; this phenomenon is called stochastic tunneling. Here, we investigated a stochastic Moran model in which two alterations emerge in a cell population of fixed size. We developed a novel approach to comprehensively describe the evolutionary dynamics of stochastic tunneling of two mutations. We considered the scenarios of large mutation rates and various fitness values and validated the accuracy of the mathematical predictions with exact stochastic computer simulations. Our theory is applicable to situations in which two alterations are accumulated in a fixed-size population of binary dividing cells.

Population dynamics of cancer cells with cell state conversions

Cancer stem cell (CSC) theory suggests a cell-lineage structure in tumor cells in which CSCs are capable of giving rise to the other non-stem cancer cells (NSCCs) but not vice versa. However, an alternative scenario of bidirectional interconversions between CSCs and NSCCs was proposed very recently. Here we present a general population model of cancer cells by integrating conventional cell divisions with direct conversions between different cell states, namely, not only can CSCs differentiate into NSCCs by asymmetric cell division, NSCCs can also dedifferentiate into CSCs by cell state conversion. Our theoretical model is validated when applying the model to recent experimental data. It is also found that the transient increase in CSCs proportion initiated from the purified NSCCs subpopulation cannot be well predicted by the conventional CSC model where the conversion from NSCCs to CSCs is forbidden, implying that the cell state conversion is required especially for the transient dynamics. The theoretical analysis also gives the condition such that our general model can be equivalently reduced into a simple Markov chain with only cell state transitions keeping the same cell proportion dynamics.

Dealing with diversity in computational cancer modeling

This paper discusses the need for interconnecting computational cancer models from different sources and scales within clinically relevant scenarios to increase the accuracy of the models and speed up their clinical adaptation, validation, and eventual translation. We briefly review current interoperability efforts drawing upon our experiences with the development of in silico models for predictive oncology within a number of European Commission Virtual Physiological Human initiative projects on cancer. A clinically relevant scenario, addressing brain tumor modeling that illustrates the need for coupling models from different sources and levels of complexity, is described. General approaches to enabling interoperability using XML-based markup languages for biological modeling are reviewed, concluding with a discussion on efforts towards developing cancer-specific XML markup to couple multiple component models for predictive in silico oncology.

Evolution and morphology of microenvironment-enhanced malignancy of three-dimensional invasive solid tumors

The emergence of invasive and metastatic behavior in malignant tumors can often lead to fatal outcomes for patients. The collective malignant tumor behavior resulting from the complex tumor-host interactions and the interactions between the tumor cells is currently poorly understood. In this paper, we employ a cellular automaton (CA) model to investigate microenvironment-enhanced malignant behaviors and morphologies of in vitro avascular invasive solid tumors in three dimensions. Our CA model incorporates a variety of microscopic-scale tumor-host interactions, including the degradation of the extracellular matrix by the malignant cells, nutrient-driven cell migration, pressure buildup due to the deformation of the microenvironment by the growing tumor, and its effect on the local tumor-host interface stability. Moreover, the effects of cell-cell adhesion on tumor growth are explicitly taken into account. Specifically, we find that while strong cell-cell adhesion can suppress the invasive behavior of the tumors growing in soft microenvironments, cancer malignancy can be significantly enhanced by harsh microenvironmental conditions, such as exposure to high pressure levels. We infer from the simulation results a qualitative phase diagram that characterizes the expected malignant behavior of invasive solid tumors in terms of two competing malignancy effects: the rigidity of the microenvironment and cell-cell adhesion. This diagram exhibits phase transitions between noninvasive and invasive behaviors. We also discuss the implications of our results for the diagnosis, prognosis, and treatment of malignant tumors.

Determining duration of HER2-targeted therapy using stem cell extinction models

Introduction

Trastuzumab dramatically improves survival in breast cancer patients whose tumor overexpresses HER2. A subpopulation of cells in human breast tumors has been identified with characteristics of cancer stem cells. These breast cancer stem-like cells (BCSCs) rely on HER2 signaling for self-renewal, suggesting that HER2-targeted therapy targets BCSCs even when the bulk of the tumor does not overexpress HER2. In order to guide clinical trials examining HER2-targeted therapy in the adjuvant setting, we propose a mathematical model to examine BCSC population dynamics and predict optimal duration of therapy.

Methods

Varying the susceptibility of BCSCs to HER2-targeted therapy, we quantify the average time to extinction of BCSCs. We expand our model using stochastic simulation to include the partially differentiated tumor cells (TCs) that represent bulk tumor population and examine effects of plasticity on required duration of therapy.

Results

Lower susceptibility of BCSCs and increased rates of dedifferentiation entail longer extinction times, indicating a need for prolonged administration of HER2-targeted therapy. We predict that even when therapy does not appreciably reduce tumor size in the advanced cancer setting, it will eventually eradicate the tumor in the adjuvant setting as long as there is at least a modest effect on BCSCs.

Conclusions

We anticipate that our results will inform clinical trials of targeted therapies in planning the duration of therapy needed to eradicate BCSCs. Our predictions also address safety, as longer duration of therapy entails a greater potential impact on normal stem cells that may also be susceptible to stem cell-targeted therapies.

Acute and fractionated irradiation differentially modulate glioma stem cell division kinetics

Glioblastoma multiforme (GBM) is one of the most aggressive human malignancies with a poor patient prognosis. Ionizing radiation either alone or adjuvant after surgery is part of standard treatment for GBM but remains primarily noncurative. The mechanisms underlying tumor radioresistance are manifold and, in part, accredited to a special subpopulation of tumorigenic cells. The so-called glioma stem cells (GSC) are bestowed with the exclusive ability to self-renew and repopulate the tumor and have been reported to be less sensitive to radiation-induced damage through preferential activation of DNA damage checkpoint responses and increased capacity for DNA damage repair. During each fraction of radiation, non-stem cancer cells (CC) die and GSCs become enriched and potentially increase in number, which may lead to accelerated repopulation. We propose a cellular Potts model that simulates the kinetics of GSCs and CCs in glioblastoma growth and radiation response. We parameterize and validate this model with experimental data of the U87-MG human glioblastoma cell line. Simulations are conducted to estimate GSC symmetric and asymmetric division rates and explore potential mechanisms for increased GSC fractions after irradiation. Simulations reveal that in addition to their higher radioresistance, a shift from asymmetric to symmetric division or a fast cycle of GSCs following fractionated radiation treatment is required to yield results that match experimental observations. We hypothesize a constitutive activation of stem cell division kinetics signaling pathways during fractionated treatment, which contributes to the frequently observed accelerated repopulation after therapeutic irradiation.

A hybrid cellular automaton model of solid tumor growth and bioreductive drug transport

Bioreductive drugs are a class of hypoxia selective drugs that are designed to eradicate the hypoxic fraction of solid tumors. Their activity depends upon a number of biological and pharmacological factors and we used a mathematical modeling approach to explore the dynamics of tumor growth, infusion, and penetration of the bioreductive drug Tirapazamine (TPZ). An in-silico model is implemented to calculate the tumor mass considering oxygen and glucose as key microenvironmental parameters. The next stage of the model integrated extra cellular matrix (ECM), cell-cell adhesion, and cell movement parameters as growth constraints. The tumor microenvironments strongly influenced tumor morphology and growth rates. Once the growth model was established, a hybrid model was developed to study drug dynamics inside the hypoxic regions of tumors. The model used 10, 50 and 100 \mu {\rm M} as TPZ initial concentrations and determined TPZ pharmacokinetic (PK) (transport) and pharmacodynamics (cytotoxicity) properties inside hypoxic regions of solid tumor. The model results showed that diminished drug transport is a reason for TPZ failure and recommend the optimization of the drug transport properties in the emerging TPZ generations. The modeling approach used in this study is novel and can be a step to explore the behavioral dynamics of TPZ.

Open biomedical pluralism: formalising knowledge about breast cancer phenotypes

We demonstrate a heterogeneity of representation types for breast cancer phenotypes and stress that the characterisation of a tumour phenotype often includes parameters that go beyond the representation of a corresponding empirically observed tumour, thus reflecting significant functional features of the phenotypes as well as epistemic interests that drive the modes of representation. Accordingly, the represented features of cancer phenotypes function as epistemic vehicles aiding various classifications, explanations, and predictions. In order to clarify how the plurality of epistemic motivations can be integrated on a formal level, we give a distinction between six categories of human agents as individuals and groups focused around particular epistemic interests. We analyse the corresponding impact of these groups and individuals on representation types, mapping and reasoning scenarios. Respecting the plurality of representations, related formalisms, expressivities and aims, as they are found across diverse scientific communities, we argue for a pluralistic ontology integration. Moreover, we discuss and illustrate to what extent such a pluralistic integration is supported by the distributed ontology language DOL, a meta-language for heterogeneous ontology representation that is currently under standardisation as ISO WD 17347 within the OntoIOp (Ontology Integration and Interoperability) activity of ISO/TC 37/SC 3. We particularly illustrate how DOL supports representations of parthood on various levels of logical expressivity, mapping of terms, merging of ontologies, as well as non-monotonic extensions based on circumscription allowing a transparent formal modelling of the normal/abnormal distinction in phenotypes.

Dynamic modeling of genes controlling cancer stem cell proliferation

The growing evidence that cancer originates from stem cells (SC) holds a great promise to eliminate this disease by designing specific drug therapies for removing cancer SC. Translation of this knowledge into predictive tests for the clinic is hampered due to the lack of methods to discriminate cancer SC from non-cancer SC. Here, we address this issue by describing a conceptual strategy for identifying the genetic origins of cancer SC. The strategy incorporates a high-dimensional group of differential equations that characterizes the proliferation, differentiation, and reprogramming of cancer SC in a dynamic cellular and molecular system. The deployment of robust mathematical models will help uncover and explain many still unknown aspects of cell behavior, tissue function, and network organization related to the formation and division of cancer SC. The statistical method developed allows biologically meaningful hypotheses about the genetic control mechanisms of carcinogenesis and metastasis to be tested in a quantitative manner.

Integrated intravital microscopy and mathematical modeling to optimize nanotherapeutics delivery to tumors

Inefficient vascularization hinders the optimal transport of cell nutrients, oxygen, and drugs to cancer cells in solid tumors. Gradients of these substances maintain a heterogeneous cell-scale microenvironment through which drugs and their carriers must travel, significantly limiting optimal drug exposure. In this study, we integrate intravital microscopy with a mathematical model of cancer to evaluate the behavior of nanoparticle-based drug delivery systems designed to circumvent biophysical barriers. We simulate the effect of doxorubicin delivered via porous 1000 x 400 nm plateloid silicon particles to a solid tumor characterized by a realistic vasculature, and vary the parameters to determine how much drug per particle and how many particles need to be released within the vasculature in order to achieve remission of the tumor. We envision that this work will contribute to the development of quantitative measures of nanoparticle design and drug loading in order to optimize cancer treatment via nanotherapeutics.

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.

Exploiting clinical trial data drastically narrows the window of possible solutions to the problem of clinical adaptation of a multiscale cancer model

The development of computational models for simulating tumor growth and response to treatment has gained significant momentum during the last few decades. At the dawn of the era of personalized medicine, providing insight into complex mechanisms involved in cancer and contributing to patient-specific therapy optimization constitute particularly inspiring pursuits. The in silico oncology community is facing the great challenge of effectively translating simulation models into clinical practice, which presupposes a thorough sensitivity analysis, adaptation and validation process based on real clinical data. In this paper, the behavior of a clinically-oriented, multiscale model of solid tumor response to chemotherapy is investigated, using the paradigm of nephroblastoma response to preoperative chemotherapy in the context of the SIOP/GPOH clinical trial. A sorting of the model's parameters according to the magnitude of their effect on the output has unveiled the relative importance of the corresponding biological mechanisms; major impact on the result of therapy is credited to the oxygenation and nutrient availability status of the tumor and the balance between the symmetric and asymmetric modes of stem cell division. The effect of a number of parameter combinations on the extent of chemotherapy-induced tumor shrinkage and on the tumor's growth rate are discussed. A real clinical case of nephroblastoma has served as a proof of principle study case, demonstrating the basics of an ongoing clinical adaptation and validation process. By using clinical data in conjunction with plausible values of model parameters, an excellent fit of the model to the available medical data of the selected nephroblastoma case has been achieved, in terms of both volume reduction and histological constitution of the tumor. In this context, the exploitation of multiscale clinical data drastically narrows the window of possible solutions to the clinical adaptation problem.

Physical oncology: a bench-to-bedside quantitative and predictive approach

Cancer models relating basic science to clinical care in oncology may fail to address the nuances of tumor behavior and therapy, as in the case, discussed herein, of the complex multiscale dynamics leading to the often-observed enhanced invasiveness, paradoxically induced by the very antiangiogenic therapy designed to destroy the tumor. Studies would benefit from approaches that quantitatively link the multiple physical and temporal scales from molecule to tissue in order to offer outcome predictions for individual patients. Physical oncology is an approach that applies fundamental principles from the physical and biological sciences to explain certain cancer behaviors as observable characteristics arising from the underlying physical and biochemical events. For example, the transport of oxygen molecules through tissue affects phenotypic characteristics such as cell proliferation, apoptosis, and adhesion, which in turn underlie the patient-scale tumor growth and invasiveness. Our review of physical oncology illustrates how tumor behavior and treatment response may be a quantifiable function of marginally stable molecular and/or cellular conditions modulated by inhomogeneity. By incorporating patient-specific genomic, proteomic, metabolomic, and cellular data into multiscale physical models, physical oncology could complement current clinical practice through enhanced understanding of cancer behavior, thus potentially improving patient survival.

Quantitative modeling of tumor dynamics and radiotherapy

Cancer is a complex disease, necessitating research on many different levels; at the subcellular level to identify genes, proteins and signaling pathways associated with the disease; at the cellular level to identify, for example, cell-cell adhesion and communication mechanisms; at the tissue level to investigate disruption of homeostasis and interaction with the tissue of origin or settlement of metastasis; and finally at the systems level to explore its global impact, e.g. through the mechanism of cachexia. Mathematical models have been proposed to identify key mechanisms that underlie dynamics and events at every scale of interest, and increasing effort is now being paid to multi-scale models that bridge the different scales. With more biological data becoming available and with increased interdisciplinary efforts, theoretical models are rendering suitable tools to predict the origin and course of the disease. The ultimate aims of cancer models, however, are to enlighten our concept of the carcinogenesis process and to assist in the designing of treatment protocols that can reduce mortality and improve patient quality of life. Conventional treatment of cancer is surgery combined with radiotherapy or chemotherapy for localized tumors or systemic treatment of advanced cancers, respectively. Although radiation is widely used as treatment, most scheduling is based on empirical knowledge and less on the predictions of sophisticated growth dynamical models of treatment response. Part of the failure to translate modeling research to the clinic may stem from language barriers, exacerbated by often esoteric model renderings with inaccessible parameterization. Here we discuss some ideas for combining tractable dynamical tumor growth models with radiation response models using biologically accessible parameters to provide a more intuitive and exploitable framework for understanding the complexity of radiotherapy treatment and failure.

Long-term results of targeted intraoperative radiotherapy (Targit) boost during breast-conserving surgery

PURPOSE

We have previously shown that delivering targeted radiotherapy to the tumour bed intraoperatively is feasible and desirable. In this study, we report on the feasibility, safety, and long-term efficacy of TARGeted Intraoperative radioTherapy (Targit), using the Intrabeam system.

METHODS AND MATERIALS

A total of 300 cancers in 299 unselected patients underwent breast-conserving surgery and Targit as a boost to the tumor bed. After lumpectomy, a single dose of 20 Gy was delivered intraoperatively. Postoperative external beam whole-breast radiotherapy excluded the usual boost. We also performed a novel individualized case control (ICC) analysis that computed the expected recurrences for the cohort by estimating the risk of recurrence for each patient using their characteristics and follow-up period.

RESULTS

The treatment was well tolerated. The median follow up was 60.5 months (range, 10-122 months). Eight patients have had ipsilateral recurrence: 5-year Kaplan Meier estimate for ipsilateral recurrence is 1.73% (SE 0.77), which compares well with that seen in the boosted patients in the European Organization for Research and Treatment of Cancer study (4.3%) and the UK STAndardisation of breast RadioTherapy study (2.8%). In a novel ICC analysis of 242 of the patients, we estimated that there should be 11.4 recurrences; in this group, only 6 recurrences were observed.

CONCLUSIONS

Lumpectomy and Targit boost combined with external beam radiotherapy results in a low local recurrence rate in a standard risk patient population. Accurate localization and the immediacy of the treatment that has a favorable effect on tumour microenvironment may contribute to this effect. These long-term data establish the long-term safety and efficacy of the Targit technique and generate the hypothesis that Targit boost might be superior to an external beam boost in its efficacy and justifies a randomized trial.

Mathematical modeling of cancer cell invasion of tissue: biological insight from mathematical analysis and computational simulation

The ability of cancer cells to break out of tissue compartments and invade locally gives solid tumours a defining deadly characteristic. One of the first steps of invasion is the remodelling of the surrounding tissue or extracellular matrix (ECM) and a major part of this process is the over-expression of proteolytic enzymes, such as the urokinase-type plasminogen activator (uPA) and matrix metalloproteinases (MMPs), by the cancer cells to break down ECM proteins. Degradation of the matrix enables the cancer cells to migrate through the tissue and subsequently to spread to secondary sites in the body, a process known as metastasis. In this paper we undertake an analysis of a mathematical model of cancer cell invasion of tissue, or ECM, which focuses on the role of the urokinase plasminogen activation system. The model consists of a system of five reaction-diffusion-taxis partial differential equations describing the interactions between cancer cells, uPA, uPA inhibitors, plasmin and the host tissue. Cancer cells react chemotactically and haptotactically to the spatio-temporal effects of the uPA system. The results obtained from computational simulations carried out on the model equations produce dynamic heterogeneous spatio-temporal solutions and using linear stability analysis we show that this is caused by a taxis-driven instability of a spatially homogeneous steady-state. Finally we consider the biological implications of the model results, draw parallels with clinical samples and laboratory based models of cancer cell invasion using three-dimensional invasion assay, and go on to discuss future development of the model.

Targeted intraoperative radiotherapy versus whole breast radiotherapy for breast cancer (TARGIT-A trial): an international, prospective, randomised, non-inferiority phase 3 trial

BACKGROUND

After breast-conserving surgery, 90% of local recurrences occur within the index quadrant despite the presence of multicentric cancers elsewhere in the breast. Thus, restriction of radiation therapy to the tumour bed during surgery might be adequate for selected patients. We compared targeted intraoperative radiotherapy with the conventional policy of whole breast external beam radiotherapy.

METHODS

Having safely piloted the new technique of single-dose targeted intraoperative radiotherapy with Intrabeam, we launched the TARGIT-A trial on March 24, 2000. In this prospective, randomised, non-inferiority trial, women aged 45 years or older with invasive ductal breast carcinoma undergoing breast-conserving surgery were enrolled from 28 centres in nine countries. Patients were randomly assigned in a 1:1 ratio to receive targeted intraoperative radiotherapy or whole breast external beam radiotherapy, with blocks stratified by centre and by timing of delivery of targeted intraoperative radiotherapy. Neither patients nor investigators or their teams were masked to treatment assignment. Postoperative discovery of predefined factors (eg, lobular carcinoma) could trigger addition of external beam radiotherapy to targeted intraoperative radiotherapy (in an expected 15% of patients). The primary outcome was local recurrence in the conserved breast. The predefined non-inferiority margin was an absolute difference of 2.5% in the primary endpoint. All randomised patients were included in the intention-to-treat analysis. This trial is registered with ClinicalTrials.gov, number NCT00983684.

FINDINGS

1113 patients were randomly allocated to targeted intraoperative radiotherapy and 1119 were allocated to external beam radiotherapy. Of 996 patients who received the allocated treatment in the targeted intraoperative radiotherapy group, 854 (86%) received targeted intraoperative radiotherapy only and 142 (14%) received targeted intraoperative radiotherapy plus external beam radiotherapy. 1025 (92%) patients in the external beam radiotherapy group received the allocated treatment. At 4 years, there were six local recurrences in the intraoperative radiotherapy group and five in the external beam radiotherapy group. The Kaplan-Meier estimate of local recurrence in the conserved breast at 4 years was 1.20% (95% CI 0.53-2.71) in the targeted intraoperative radiotherapy and 0.95% (0.39-2.31) in the external beam radiotherapy group (difference between groups 0.25%, -1.04 to 1.54; p=0.41). The frequency of any complications and major toxicity was similar in the two groups (for major toxicity, targeted intraoperative radiotherapy, 37 [3.3%] of 1113 vs external beam radiotherapy, 44 [3.9%] of 1119; p=0.44). Radiotherapy toxicity (Radiation Therapy Oncology Group grade 3) was lower in the targeted intraoperative radiotherapy group (six patients [0.5%]) than in the external beam radiotherapy group (23 patients [2.1%]; p=0.002).

INTERPRETATION

For selected patients with early breast cancer, a single dose of radiotherapy delivered at the time of surgery by use of targeted intraoperative radiotherapy should be considered as an alternative to external beam radiotherapy delivered over several weeks.

FUNDING

University College London Hospitals (UCLH)/UCL Comprehensive Biomedical Research Centre, UCLH Charities, National Institute for Health Research Health Technology Assessment programme, Ninewells Cancer Campaign, National Health and Medical Research Council, and German Federal Ministry of Education and Research (BMBF).

Predicting the efficacy of radiotherapy in individual glioblastoma patients in vivo: a mathematical modeling approach

Glioblastoma multiforme (GBM) is the most malignant form of primary brain tumors known as gliomas. They proliferate and invade extensively and yield short life expectancies despite aggressive treatment. Response to treatment is usually measured in terms of the survival of groups of patients treated similarly, but this statistical approach misses the subgroups that may have responded to or may have been injured by treatment. Such statistics offer scant reassurance to individual patients who have suffered through these treatments. Furthermore, current imaging-based treatment response metrics in individual patients ignore patient-specific differences in tumor growth kinetics, which have been shown to vary widely across patients even within the same histological diagnosis and, unfortunately, these metrics have shown only minimal success in predicting patient outcome. We consider nine newly diagnosed GBM patients receiving diagnostic biopsy followed by standard-of-care external beam radiation therapy (XRT). We present and apply a patient-specific, biologically based mathematical model for glioma growth that quantifies response to XRT in individual patients in vivo. The mathematical model uses net rates of proliferation and migration of malignant tumor cells to characterize the tumor's growth and invasion along with the linear-quadratic model for the response to radiation therapy. Using only routinely available pre-treatment MRIs to inform the patient-specific bio-mathematical model simulations, we find that radiation response in these patients, quantified by both clinical and model-generated measures, could have been predicted prior to treatment with high accuracy. Specifically, we find that the net proliferation rate is correlated with the radiation response parameter (r = 0.89, p = 0.0007), resulting in a predictive relationship that is tested with a leave-one-out cross-validation technique. This relationship predicts the tumor size post-therapy to within inter-observer tumor volume uncertainty. The results of this study suggest that a mathematical model can create a virtual in silico tumor with the same growth kinetics as a particular patient and can not only predict treatment response in individual patients in vivo but also provide a basis for evaluation of response in each patient to any given therapy.

Perspectives on tissue interactions in development and disease

From the morphogenetic movements of the three germ layers during development to the reactive stromal microenvironment in cancer, tissue interactions are vital to maintaining healthy organ morphologic architecture and function. The stromal compartment is thought to be complicit in tumor progression and, as such, represents an opportune target for disease therapies. However, recent developments in our understanding of the diversity of the stromal compartment and the lack of appropriate models to study its relevance in human disease have limited our further understanding of the role of tissue interactions in tumor progression. The failure any model to fully recapitulate the complexities of systemic biology continue to create a higher imperative for incorporating various perspectives into a broader understanding for the ultimate goal of designing interventional therapies. Understanding this potential, this review examines the biological models used to study stromal-epithelial interactions and includes an attempt to incorporate behavioral terminology to define and mathematically model ecological relationships in stromal-epithelial interactions. In addition, the current attempt to incorporate these diverse ecological perspectives into in silico mathematical models through cross-disciplinary coordination is reviewed, which will provide a fresh perspective on defining cell group behavior and tissue ecology in disease and hopefully lead to the generation of new hypotheses to be empirically validated.

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.

Differential destruction of stem cells: implications for targeted cancer stem cell therapy

Cancer stem cells represent a novel therapeutic target. The major challenge in targeting leukemic stem cells (LSC) is finding therapies that largely spare normal hematopoietic stem cells (HSC) while eradicating quiescent LSCs. We present a mathematical model to predict how selective a therapy must be to ensure that enough HSCs survive when LSCs have been eradicated. Stem cell population size is modeled as a birth-death process. This permits comparison of LSC and HSC eradication times under therapy and calculation of the number of HSCs at the time of LSC eradication for varied initial population sizes and stem cell death rates. We further investigate the effects of LSC quiescence and resistance mutations on our predictions. From a clinical point of view, our models suggest criteria by which cancer stem cell therapy safety can be assessed. We anticipate that in conjunction with experimental observation of cancer stem cell killing rates, our results will be useful in screening targeted therapies for both hematologic and solid tumor malignancies.

Evolutionary theory of cancer

As Theodosius Dobzhansky famously noted in 1973, "Nothing in biology makes sense except in the light of evolution," and cancer is no exception to this rule. Our understanding of cancer initiation, progression, treatment, and resistance has advanced considerably by regarding cancer as the product of evolutionary processes. Here we review the literature of mathematical models of cancer evolution and provide a synthesis and discussion of the field.

Early breast cancer

Adoption of urbanised lifestyles together with changes in reproductive behaviour might partly underlie the continued rise in worldwide incidence of breast cancer. Widespread mammographic screening and effective systemic therapies have led to a stage shift at presentation and mortality reductions in the past two decades. Loco-regional control of the disease seems to affect long-term survival, and attention to surgical margins together with improved radiotherapy techniques could further contribute to mortality gains. Developments in oncoplastic surgery and partial-breast reconstruction have improved cosmetic outcomes after breast-conservation surgery. Optimum approaches for delivering chest-wall radiotherapy in the context of immediate breast reconstruction present special challenges. Accurate methods for intraoperative assessment of sentinel lymph nodes remain a clinical priority. Clinical trials are investigating combinatorial therapies that use novel agents targeting growth factor receptors, signal transduction pathways, and tumour angiogenesis. Gene-expression profiling offers the potential to provide accurate prognostic and predictive information, with selection of best possible therapy for individuals and avoidance of overtreatment and undertreatment of patients with conventional chemotherapy. Short-term presurgical studies in the neoadjuvant setting allow monitoring of proliferative indices, and changes in gene-expression patterns can be predictive of response to therapies and long-term outcome.

Increased expression of centrosomal alpha, gamma-tubulin in atypical ductal hyperplasia and carcinoma of the breast

Centrosomal abnormalities have been found in various cancer types. We sought to determine whether centrosomal dysfunctions occur in the atypical ductal hyperplasia (ADH)-carcinoma sequence of breast cancer. As alpha and gamma-tubulins are the structural components of centrosomes, we performed real time quantitative polymerase chain reaction (qPCR), in situ hybridization (ISH) and immunnohistochemistry (IHC) to determine the DNA copy levels, messenger RNA (mRNA) expression, and protein expression of alpha and gamma-tubulins respectively. Gamma-tubulin staining was used for the localization and quantification of centrosomes. We found that alpha-tubulin or gamma-tubulin mRNA was increasingly expressed from normal breast tissue (NBT) to ADH, ductal carcinoma in situ (DCIS), and infiltrative ductal carcinoma (IDC), respectively, with the highest expressions being found in DCIS. The expression profiles of alpha, gamma-tubulin proteins were concordant with that of mRNA, except that the highest expression was found in IDC. Similarly, DNA copies of alpha, gamma-tubulins showed a rising tendency, with the highest level for gamma-tubulin attained in IDC and that for alpha-tubulin was found in DCIS. However, there was no significant difference of alpha, gamma-tubulin DNA copy levels, mRNA expression, and protein expression between DCIS and IDC. Our results demonstrate that centrosomal aberrations may play key roles in the early stage of breast tumorogenesis. The malignant transformation sequence is probably attributable to the amplification of centrosomal DNA leading to mRNA and protein over-expression of these centrosomal proteins. Furthermore, determination of alpha, gamma-tubulins using combined qPCR with ISH may be useful in assisting the diagnosis of premalignant lesions of the breast.

Numerical Simulation of Bone Regeneration in a Bone Chamber

While mathematical models are able to capture essential aspects of biological processes like fracture healing and distraction osteogenesis, their predictive capacity in peri-implant osteogenesis remains uninvestigated. We tested the hypothesis that a mechano-regulatory model has the potential to predict bone regeneration around implants. In an in vivo bone chamber set-up allowing for controlled implant loading (up to 90 μ m axial displacement), bone tissue formation was simulated and compared qualitatively and quantitatively with histology. Furthermore, the model was applied to simulate excessive loading conditions. Corresponding to literature data, implant displacement magnitudes larger than 90 μ m predicted the formation of fibrous tissue encapsulation of the implant. In contradiction to findings in orthopedic implant osseointegration, implant displacement frequencies higher than 1 Hz did not favor the formation of peri-implant bone in the chamber. Additional bone chamber experiments are needed to test these numerical predictions.

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.

A mathematical model for brain tumor response to radiation therapy

Gliomas are highly invasive primary brain tumors, accounting for nearly 50% of all brain tumors (Alvord and Shaw in The pathology of the aging human nervous system. Lea & Febiger, Philadelphia, pp 210-281, 1991). Their aggressive growth leads to short life expectancies, as well as a fairly algorithmic approach to treatment: diagnostic magnetic resonance image (MRI) followed by biopsy or surgical resection with accompanying second MRI, external beam radiation therapy concurrent with and followed by chemotherapy, with MRIs conducted at various times during treatment as prescribed by the physician. Swanson et al. (Harpold et al. in J Neuropathol Exp Neurol 66:1-9, 2007) have shown that the defining and essential characteristics of gliomas in terms of net rates of proliferation (rho) and invasion (D) can be determined from serial MRIs of individual patients. We present an extension to Swanson's reaction-diffusion model to include the effects of radiation therapy using the classic linear-quadratic radiobiological model (Hall in Radiobiology for the radiologist. Lippincott, Philadelphia, pp 478-480, 1994) for radiation efficacy, along with an investigation of response to various therapy schedules and dose distributions on a virtual tumor (Swanson et al. in AACR annual meeting, Los Angeles, 2007).

Cancer: shift of the paradigm

Cancer is usually considered to be a by-product of design limitations of a multicellular organism and its intrinsic fallibility. However, recent data prompt a revision of some established notions about carcinogenesis and form a new paradigm of carcinogenesis as a highly conserved biological phenomenon - a programmed death of an organism. This altruistic program, which is unleashed when mutagenesis surpasses a certain critical threshold, gives a population the important benefit acting as a guardian of the gene pool against the spread of certain mutant genes. A growing body of evidence supports this point of view: (i) epigenetic changes leading to cancer arise early, simultaneously in many cells and look like deterministic regulation; (ii) concept of cancer stem cell suggests a view of carcinogenesis not as vague transformation but as well known differentiation; (iii) tumor/host relations usually perceived as antagonistic are, in reality, synergistic; (iv) death of an individual from cancer is predetermined and results apparently from a specific activity (killer function) of cancer cell and (v) evolutionary conservation indicates that cancer comes with a general advantage that explains its evolutionary success. A holistic approach to carcinogenesis suggests new avenues of research and new therapeutic strategy.

Mathematical models of cancer stem cells

Human cancers are thought to be sustained in their growth by a pathologic counterpart of normal adult stem cells: cancer stem cells. This concept was first developed in human myeloid leukemias and is today being extended to solid tumors such as breast and brain cancers. A quantitative understanding of cancer stem cells requires a mathematical framework to describe the dynamics of cancer initiation and progression, the response to treatment, and the evolution of resistance. In this review, I use chronic myeloid leukemia as an example to discuss how mathematical and computational techniques have been used to gain insights into the biology of cancer stem cells.

Integrative mathematical oncology

Cancer research attracts broad resources and scientists from many disciplines, and has produced some impressive advances in the treatment and understanding of this disease. However, a comprehensive mechanistic view of the cancer process remains elusive. To achieve this it seems clear that one must assemble a physically integrated team of interdisciplinary scientists that includes mathematicians, to develop mathematical models of tumorigenesis as a complex process. Examining these models and validating their findings by experimental and clinical observations seems to be one way to reconcile molecular reductionist with quantitative holistic approaches and produce an integrative mathematical oncology view of cancer progression.

A hybrid cellular automaton model of clonal evolution in cancer: the emergence of the glycolytic phenotype

We present a cellular automaton model of clonal evolution in cancer aimed at investigating the emergence of the glycolytic phenotype. In the model each cell is equipped with a micro-environment response network that determines the behaviour or phenotype of the cell based on the local environment. The response network is modelled using a feed-forward neural network, which is subject to mutations when the cells divide. This implies that cells might react differently to the environment and when space and nutrients are limited only the fittest cells will survive. With this model we have investigated the impact of the environment on the growth dynamics of the tumour. In particular, we have analysed the influence of the tissue oxygen concentration and extra-cellular matrix density on the dynamics of the model. We found that the environment influences both the growth and the evolutionary dynamics of the tumour. For low oxygen concentration we observe tumours with a fingered morphology, while increasing the matrix density gives rise to more compact tumours with wider fingers. The distribution of phenotypes in the tumour is also affected, and we observe that the glycolytic phenotype is most likely to emerge in a poorly oxygenated tissue with a high matrix density. Our results suggest that it is the combined effect of the oxygen concentration and matrix density that creates an environment where the glycolytic phenotype has a growth advantage and consequently is most likely to appear.