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

Bistable dynamics of TAN-NK cells in tumor growth and control of radiotherapy-induced neutropenia in lung cancer treatment

Neutrophils play a crucial role in the innate immune response as a first line of defense in many diseases, including cancer. Tumor-associated neutrophils (TANs) can either promote or inhibit tumor growth in various steps of cancer progression via mutual interactions with cancer cells in a complex tumor microenvironment (TME). In this study, we developed and analyzed mathematical models to investigate the role of natural killer cells (NK cells) and the dynamic transition between N1 and N2 TAN phenotypes in killing cancer cells through key signaling networks and how adjuvant therapy with radiation can be used in combination to increase anti-tumor efficacy. We examined the complex immune-tumor dynamics among N1/N2 TANs, NK cells, and tumor cells, communicating through key extracellular mediators (Transforming growth factor (TGF-$ \beta $), Interferon gamma (IFN-$ \gamma $)) and intracellular regulation in the apoptosis signaling network. We developed several tumor prevention strategies to eradicate tumors, including combination (IFN-$ \gamma $, exogenous NK, TGF-$ \beta $ inhibitor) therapy and optimally-controlled ionizing radiation in a complex TME. Using this model, we investigated the fundamental mechanism of radiation-induced changes in the TME and the impact of internal and external immune composition on the tumor cell fate and their response to different treatment schedules.

Agent-based modeling for the tumor microenvironment (TME)

Cancer is a disease that arises from the uncontrolled growth of abnormal (tumor) cells in an organ and their subsequent spread into other parts of the body. If tumor cells spread to surrounding tissues or other organs, then the disease is life-threatening due to limited treatment options. This work applies an agent-based model to investigate the effect of intra-tumoral communication on tumor progression, plasticity, and invasion, with results suggesting that cell-cell and cell-extracellular matrix (ECM) interactions affect tumor cell behavior. Additionally, the model suggests that low initial healthy cell densities and ECM protein densities promote tumor progression, cell motility, and invasion. Furthermore, high ECM breakdown probabilities of tumor cells promote tumor invasion. Understanding the intra-tumoral communication under cellular stress can potentially lead to the design of successful treatment strategies for cancer.

Calibrating tumor growth and invasion parameters with spectral spatial analysis of cancer biopsy tissues

The reaction-diffusion equation is widely used in mathematical models of cancer. The calibration of model parameters based on limited clinical data is critical to using reaction-diffusion equation simulations for reliable predictions on a per-patient basis. Here, we focus on cell-level data as routinely available from tissue biopsies used for clinical cancer diagnosis. We analyze the spatial architecture in biopsy tissues stained with multiplex immunofluorescence. We derive a two-point correlation function and the corresponding spatial power spectral distribution. We show that this data-deduced power spectral distribution can fit the power spectrum of the solution of reaction-diffusion equations that can then identify patient-specific tumor growth and invasion rates. This approach allows the measurement of patient-specific critical tumor dynamical properties from routinely available biopsy material at a single snapshot in time.

Modeling of Mouse Experiments Suggests that Optimal Anti-Hormonal Treatment for Breast Cancer is Diet-Dependent

Estrogen receptor positive breast cancer is frequently treated with anti-hormonal treatment such as aromatase inhibitors (AI). Interestingly, a high body mass index has been shown to have a negative impact on AI efficacy, most likely due to disturbances in steroid metabolism and adipokine production. Here, we propose a mathematical model based on a system of ordinary differential equations to investigate the effect of high-fat diet on tumor growth. We inform the model with data from mouse experiments, where the animals are fed with high-fat or control (normal) diet. By incorporating AI treatment with drug resistance into the model and by solving optimal control problems we found differential responses for control and high-fat diet. To the best of our knowledge, this is the first attempt to model optimal anti-hormonal treatment for breast cancer in the presence of drug resistance. Our results underline the importance of considering high-fat diet and obesity as factors influencing clinical outcomes during anti-hormonal therapies in breast cancer patients.

Dynamics of a diffusive model for cancer stem cells with time delay in microRNA-differentiated cancer cell interactions and radiotherapy effects

Understand the dynamics of cancer stem cells (CSCs), prevent the non-recurrence of cancers and develop therapeutic strategies to destroy both cancer cells and CSCs remain a challenge topic. In this paper, we study both analytically and numerically the dynamics of CSCs under radiotherapy effects. The dynamical model takes into account the diffusion of cells, the de-differentiation (or plasticity) mechanism of differentiated cancer cells (DCs) and the time delay on the interaction between microRNAs molecules (microRNAs) with DCs. The stability of the model system is studied by using a Hopf bifurcation analysis. We mainly investigate on the critical time delay τ c , that represents the time for DCs to transform into CSCs after the interaction of microRNAs with DCs. Using the system parameters, we calculate the value of τ c for prostate, lung and breast cancers. To confirm the analytical predictions, the numerical simulations are performed and show the formation of spatiotemporal circular patterns. Such patterns have been found as promising diagnostic and therapeutic value in management of cancer and various diseases. The radiotherapy is applied in the particular case of prostate model. We calculate the optimum dose of radiation and determine the probability of avoiding local cancer recurrence after radiotherapy treatment. We find numerically a complete eradication of patterns when the radiotherapy is applied before a time t < τ c . This scenario induces microRNAs to act as suppressors as experimentally observed in prostate cancer. The results obtained in this paper will provide a better concept for the clinicians and oncologists to understand the complex dynamics of CSCs and to design more efficacious therapeutic strategies to prevent the non-recurrence of cancers.

A neural network computational structure for the fractional order breast cancer model

The current study provides the numerical performances of the fractional kind of breast cancer (FKBC) model, which are based on five different classes including cancer stem cells, healthy cells, tumor cells, excess estrogen, and immune cells. The motive to introduce the fractional order derivatives is to present more precise solutions as compared to integer order. A stochastic computing reliable scheme based on the Levenberg Marquardt backpropagation neural networks (LMBNNS) is proposed to solve three different cases of the fractional order values of the FKBC model. A designed dataset is constructed by using the Adam solver in order to reduce the mean square error by taking the data performances as 9% for both testing and validation, while 82% is used for training. The correctness of the solver is approved through the negligible absolute error and matching of the solutions for each model's case. To validates the accuracy, and consistency of the solver, the performances based on the error histogram, transition state, and regression for solving the FKBC model.

Modelling the Tumour Microenvironment, but What Exactly Do We Mean by "Model"?

The Oxford English Dictionary includes 17 definitions for the word "model" as a noun and another 11 as a verb. Therefore, context is necessary to understand the meaning of the word model. For instance, "model railways" refer to replicas of railways and trains at a smaller scale and a "model student" refers to an exemplary individual. In some cases, a specific context, like cancer research, may not be sufficient to provide one specific meaning for model. Even if the context is narrowed, specifically, to research related to the tumour microenvironment, "model" can be understood in a wide variety of ways, from an animal model to a mathematical expression. This paper presents a review of different "models" of the tumour microenvironment, as grouped by different definitions of the word into four categories: model organisms, in vitro models, mathematical models and computational models. Then, the frequencies of different meanings of the word "model" related to the tumour microenvironment are measured from numbers of entries in the MEDLINE database of the United States National Library of Medicine at the National Institutes of Health. The frequencies of the main components of the microenvironment and the organ-related cancers modelled are also assessed quantitatively with specific keywords. Whilst animal models, particularly xenografts and mouse models, are the most commonly used "models", the number of these entries has been slowly decreasing. Mathematical models, as well as prognostic and risk models, follow in frequency, and these have been growing in use.

A complex systems model of breast cancer etiology: The Paradigm II Model

BACKGROUND

Complex systems models of breast cancer have previously focused on prediction of prognosis and clinical events for individual women. There is a need for understanding breast cancer at the population level for public health decision-making, for identifying gaps in epidemiologic knowledge and for the education of the public as to the complexity of this most common of cancers.

METHODS AND FINDINGS

We developed an agent-based model of breast cancer for the women of the state of California using data from the U.S. Census, the California Health Interview Survey, the California Cancer Registry, the National Health and Nutrition Examination Survey and the literature. The model was implemented in the Julia programming language and R computing environment. The Paradigm II model development followed a transdisciplinary process with expertise from multiple relevant disciplinary experts from genetics to epidemiology and sociology with the goal of exploring both upstream determinants at the population level and pathophysiologic etiologic factors at the biologic level. The resulting model reproduces in a reasonable manner the overall age-specific incidence curve for the years 2008-2012 and incidence and relative risks due to specific risk factors such as BRCA1, polygenic risk, alcohol consumption, hormone therapy, breastfeeding, oral contraceptive use and scenarios for environmental toxin exposures.

CONCLUSIONS

The Paradigm II model illustrates the role of multiple etiologic factors in breast cancer from domains of biology, behavior and the environment. The value of the model is in providing a virtual laboratory to evaluate a wide range of potential interventions into the social, environmental and behavioral determinants of breast cancer at the population level.

Mathematical Modelling of Cervical Precancerous Lesion Grade Risk Scores: Linear Regression Analysis of Cellular Protein Biomarkers and Human Papillomavirus E6/E7 RNA Staining Patterns

The current practice of determining histologic grade with a single molecular biomarker can facilitate differential diagnosis but cannot predict the risk of lesion progression. Cancer is caused by complex mechanisms, and no single biomarker can both make accurate diagnoses and predict progression risk. Modelling using multiple biomarkers can be used to derive scores for risk prediction. Mathematical models (MMs) may be capable of making predictions from biomarker data. Therefore, this study aimed to develop MM-based scores for predicting the risk of precancerous cervical lesion progression and identifying precancerous lesions in patients in northern Thailand by evaluating the expression of multiple biomarkers. The MMs (Models 1-5) were developed in the test sample set based on patient age range (five categories) and biomarker levels (cortactin, p16INK4A, and Ki-67 by immunohistochemistry [IHC], and HPV E6/E7 ribonucleic acid (RNA) by in situ hybridization [ISH]). The risk scores for the prediction of cervical lesion progression ("risk biomolecules") ranged from 2.56-2.60 in the normal and low-grade squamous intraepithelial lesion (LSIL) cases and from 3.54-3.62 in cases where precancerous lesions were predicted to progress. In Model 4, 23/86 (26.7%) normal and LSIL cases had biomolecule levels that suggested a risk of progression, while 5/86 (5.8%) cases were identified as precancerous lesions. Additionally, histologic grading with a single molecular biomarker did not identify 23 cases with risk, preventing close patient monitoring. These results suggest that biomarker level-based risk scores are useful for predicting the risk of cervical lesion progression and identifying precancerous lesion development. This multiple biomarker-based strategy may ultimately have utility for predicting cancer progression in other contexts.

Systems biology of autophagy in leishmanial infection and its diverse role in precision medicine

Autophagy is a contentious issue in leishmaniasis and is emerging as a promising therapeutic regimen. Published research on the impact of autophagic regulation on Leishmania survival is inconclusive, despite numerous pieces of evidence that Leishmania spp. triggers autophagy in a variety of cell types. The mechanistic approach is poorly understood in the Leishmania parasite as autophagy is significant in both Leishmania and the host. Herein, this review discusses the autophagy proteins that are being investigated as potential therapeutic targets, the connection between autophagy and lipid metabolism, and microRNAs that regulate autophagy and lipid metabolism. It also highlights the use of systems biology to develop novel autophagy-dependent therapeutics for leishmaniasis by utilizing artificial intelligence (AI), machine learning (ML), mathematical modeling, network analysis, and other computational methods. Additionally, we have shown many databases for autophagy and metabolism in Leishmania parasites that suggest potential therapeutic targets for intricate signaling in the autophagy system. In a nutshell, the detailed understanding of the dynamics of autophagy in conjunction with lipids and miRNAs unfolds larger dimensions for future research.

Mathematical modelling identifies conditions for maintaining and escaping feedback control in the intestinal epithelium

The intestinal epithelium is one of the fastest renewing tissues in mammals. It shows a hierarchical organisation, where intestinal stem cells at the base of crypts give rise to rapidly dividing transit amplifying cells that in turn renew the pool of short-lived differentiated cells. Upon injury and stem-cell loss, cells can also de-differentiate. Tissue homeostasis requires a tightly regulated balance of differentiation and stem cell proliferation, and failure can lead to tissue extinction or to unbounded growth and cancerous lesions. Here, we present a two-compartment mathematical model of intestinal epithelium population dynamics that includes a known feedback inhibition of stem cell differentiation by differentiated cells. The model shows that feedback regulation stabilises the number of differentiated cells as these become invariant to changes in their apoptosis rate. Stability of the system is largely independent of feedback strength and shape, but specific thresholds exist which if bypassed cause unbounded growth. When dedifferentiation is added to the model, we find that the system can recover faster after certain external perturbations. However, dedifferentiation makes the system more prone to losing homeostasis. Taken together, our mathematical model shows how a feedback-controlled hierarchical tissue can maintain homeostasis and can be robust to many external perturbations.

Modeling and Analysis of Breast Cancer with Adverse Reactions of Chemotherapy Treatment through Fractional Derivative

The abnormal growth of cells in the breast is called malignancy or breast cancer; it is a life-threatening and dangerous cancer in women around the world. In the treatment of cancer, the doctors apply different techniques to stop cancer cell development, remove cancer cells through surgery, or kill cancer cells. In chemotherapy treatment, powerful drugs are used to kill abnormal cells; however, it has adverse reactions on the patient heart which is called cardiotoxicity. In this paper, we formulate the dynamics of cancer in the breast with adverse reactions of chemotherapy treatment on the heart of a patient in the fractional framework to visualize its dynamical behaviour. We listed the fundamental results of the fractional calculus for the analysis of our model. The model is then analyzed for the basic properties, and the existence and uniqueness of the proposed breast cancer system are investigated through fixed point theory. Furthermore, the Adams-Bashforth numerical technique is presented for the solution of fractional-order system to illustrate the time series of breast cancer model. The dynamical behaviour of different stages of breast cancer is then highlighted numerically to show the effect of fractional-order ϑ and to visualize the role of input parameter on the dynamics of breast cancer.

Estimating latent, dynamic processes of breast cancer tumour growth and distant metastatic spread from mammography screening data

We propose a framework for jointly modelling tumour size at diagnosis and time to distant metastatic spread, from diagnosis, based on latent dynamic sub-models of growth of the primary tumour and of distant metastatic detection. The framework also includes a sub-model for screening sensitivity as a function of latent tumour size. Our approach connects post-diagnosis events to the natural history of cancer and, once refined, may prove useful for evaluating new interventions, such as personalised screening regimes. We evaluate our model-fitting procedure using Monte Carlo simulation, showing that the estimation algorithm can retrieve the correct model parameters, that key patterns in the data can be captured by the model even with misspecification of some structural assumptions, and that, still, with enough data it should be possible to detect strong misspecifications. Furthermore, we fit our model to observational data from an extension of a case-control study of post-menopausal breast cancer in Sweden, providing model-based estimates of the probability of being free from detected distant metastasis as a function of tumour size, mode of detection (of the primary tumour), and screening history. For women with screen-detected cancer and two previous negative screens, the probabilities of being free from detected distant metastases 5 years after detection and removal of the primary tumour are 0.97, 0.89 and 0.59 for tumours of diameter 5, 15 and 35 mm, respectively. We also study the probability of having latent/dormant metastases at detection of the primary tumour, estimating that 33% of patients in our study had such metastases.

Adenosine 2A receptor and TIM3 suppress cytolytic killing of tumor cells via cytoskeletal polarization

Tumors generate an immune-suppressive environment that prevents effective killing of tumor cells by CD8⁺ cytotoxic T cells (CTL). It remains largely unclear upon which cell type and at which stage of the anti-tumor response mediators of suppression act. We have combined an in vivo tumor model with a matching in vitro reconstruction of the tumor microenvironment based on tumor spheroids to identify suppressors of anti-tumor immunity that directly act on interaction between CTL and tumor cells and to determine mechanisms of action. An adenosine 2A receptor antagonist, as enhanced by blockade of TIM3, slowed tumor growth in vivo. Engagement of the adenosine 2A receptor and TIM3 reduced tumor cell killing in spheroids, impaired CTL cytoskeletal polarization ex vivo and in vitro and inhibited CTL infiltration into tumors and spheroids. With this role in CTL killing, blocking A_2AR and TIM3 may complement therapies that enhance T cell priming, e.g. anti-PD-1 and anti-CTLA-4.

Mathematical modeling and machine learning for public health decision-making: the case of breast cancer in Benin

Breast cancer is the most common type of cancer in women. Its mortality rate is high due to late detection and cardiotoxic effects of chemotherapy. In this work, we used the Support Vector Machine (SVM) method to classify tumors and proposed a new mathematical model of the patient dynamics of the breast cancer population. Numerical simulations were performed to study the behavior of the solutions around the equilibrium point. The findings revealed that the equilibrium point is stable regardless of the initial conditions. Moreover, this study will help public health decision-making as the results can be used to minimize the number of cardiotoxic patients and increase the number of recovered patients after chemotherapy.

Atorvastatin-mediated rescue of cancer-related cognitive changes in combined anticancer therapies

Acute administration of trastuzumab (TZB) may induce various forms of cognitive impairment. These cancer-related cognitive changes (CRCC) are regulated by an adverse biological process involving cancer stem cells (CSCs) and IL-6. Recent studies have reported that atorvastatin (ATV) may change the dynamic of cognitive impairment in a combination (TZB+ATV) therapy. In this study, we investigate the mutual interactions between cancer stem cells and the tumor cells that facilitate cognitive impairment during long term TZB therapy by developing a mathematical model that involves IL-6 and the key apoptotic regulation. These include the densities of tumor cells and CSCs, and the concentrations of intracellular signaling molecules (NFκB, Bcl-2, BAX). We apply the mathematical model to a single or combination (ATV+TZB) therapy used in the experiments to demonstrate that the CSCs can enhance CRCC by secreting IL-6 and ATV may interfere the whole regulation. We show that the model can both reproduce the major experimental observation on onset and prevention of CRCC, and suggest several important predictions to guide future experiments with the goal of the development of new anti-tumor and anti-CRCC strategies. Moreover, using this model, we investigate the fundamental mechanism of onset of cognitive impairment in TZB-treated patients and the impact of alternating therapies on the anti-tumor efficacy and intracellular response to different treatment schedules.

A conventional drug, trastuzumab (TZB), was shown to be an effective weapon in killing cancer cells in brain. However, long term treatment of TZB increases the proportion of cancer stem cells (CSCs) in the tumour microenvironment (TME) and induces up-regulation of pro-tumoral molecules such as IL-6 in TME. These cancer cells then become more resistant to this chemotherapy through the IL-mediated up-regulation of NFκB and CSCs. More importantly, these changes in TME result in a serious side effect, cognitive impairment called cancer-related cognitive changes (CRCC). The detailed mechanism of CRCC is still poorly understood. However, cancer patients with chemotherapy-induced cognitive impairment can have long-term or delayed mental changes. In this study, we investigated the fundamental mechanism of CRCC in cancer patients based on experiments and a mathematical model that describes how tumor cells interact with CSCs in response to chemo drugs. In particular, we investigate how TZB-induced CSCs with modified IL-6 landscapes shape the cognitive functions in cancer patients. We showed that the combination treatment with another drug, atorvastatin (ATV), can abrogate the TZB-induced CRCC and enhance the survival probability of cancer patients by synergistic anti-tumor effect. We demonstrate that the cognitive functions and survival rates in cancer patients depend on the apoptotic signaling pathways via the critical communication and IL-6 landscapes of stimulated CTCs.

A Mathematical Model of Breast Tumor Progression Based on Immune Infiltration

Breast cancer is the most prominent type of cancer among women. Understanding the microenvironment of breast cancer and the interactions between cells and cytokines will lead to better treatment approaches for patients. In this study, we developed a data-driven mathematical model to investigate the dynamics of key cells and cytokines involved in breast cancer development. We used gene expression profiles of tumors to estimate the relative abundance of each immune cell and group patients based on their immune patterns. Dynamical results show the complex interplay between cells and molecules, and sensitivity analysis emphasizes the direct effects of macrophages and adipocytes on cancer cell growth. In addition, we observed the dual effect of IFN-γ on cancer proliferation, either through direct inhibition of cancer cells or by increasing the cytotoxicity of CD8+ T-cells.

Mathematical model of STAT signalling pathways in cancer development and optimal control approaches

In various diseases, the STAT family display various cellular controls over various challenges faced by the immune system and cell death programs. In this study, we investigate how an intracellular signalling network (STAT1, STAT3, Bcl-2 and BAX) regulates important cellular states, either anti-apoptosis or apoptosis of cancer cells. We adapt a mathematical framework to illustrate how the signalling network can generate a bi-stability condition so that it will induce either apoptosis or anti-apoptosis status of tumour cells. Then, we use this model to develop several anti-tumour strategies including IFN-β infusion. The roles of JAK-STATs signalling in regulation of the cell death program in cancer cells and tumour growth are poorly understood. The mathematical model unveils the structure and functions of the intracellular signalling and cellular outcomes of the anti-tumour drugs in the presence of IFN-β and JAK stimuli. We identify the best injection order of IFN-β and DDP among many possible combinations, which may suggest better infusion strategies of multiple anti-cancer agents at clinics. We finally use an optimal control theory in order to maximize anti-tumour efficacy and minimize administrative costs. In particular, we minimize tumour volume and maximize the apoptotic potential by minimizing the Bcl-2 concentration and maximizing the BAX level while minimizing total injection amount of both IFN-β and JAK2 inhibitors (DDP).

Mechanical Pressure Driving Proteoglycan Expression in Mammographic Density: a Self-perpetuating Cycle?

Regions of high mammographic density (MD) in the breast are characterised by a proteoglycan (PG)-rich fibrous stroma, where PGs mediate aligned collagen fibrils to control tissue stiffness and hence the response to mechanical forces. Literature is accumulating to support the notion that mechanical stiffness may drive PG synthesis in the breast contributing to MD. We review emerging patterns in MD and other biological settings, of a positive feedback cycle of force promoting PG synthesis, such as in articular cartilage, due to increased pressure on weight bearing joints. Furthermore, we present evidence to suggest a pro-tumorigenic effect of increased mechanical force on epithelial cells in contexts where PG-mediated, aligned collagen fibrous tissue abounds, with implications for breast cancer development attributable to high MD. Finally, we summarise means through which this positive feedback mechanism of PG synthesis may be intercepted to reduce mechanical force within tissues and thus reduce disease burden.

Optimal solution of the fractional order breast cancer competition model

In this article, a fractional order breast cancer competition model (F-BCCM) under the Caputo fractional derivative is analyzed. A new set of basis functions, namely the generalized shifted Legendre polynomials, is proposed to deal with the solutions of F-BCCM. The F-BCCM describes the dynamics involving a variety of cancer factors, such as the stem, tumor and healthy cells, as well as the effects of excess estrogen and the body's natural immune response on the cell populations. After combining the operational matrices with the Lagrange multipliers technique we obtain an optimization method for solving the F-BCCM whose convergence is investigated. Several examples show that a few number of basis functions lead to the satisfactory results. In fact, numerical experiments not only confirm the accuracy but also the practicability and computational efficiency of the devised technique.

Data Driven Mathematical Model of FOLFIRI Treatment for Colon Cancer

Many colon cancer patients show resistance to their treatments. Therefore, it is important to consider unique characteristic of each tumor to find the best treatment options for each patient. In this study, we develop a data driven mathematical model for interaction between the tumor microenvironment and FOLFIRI drug agents in colon cancer. Patients are divided into five distinct clusters based on their estimated immune cell fractions obtained from their primary tumors' gene expression data. We then analyze the effects of drugs on cancer cells and immune cells in each group, and we observe different responses to the FOLFIRI drugs between patients in different immune groups. For instance, patients in cluster 3 with the highest T-reg/T-helper ratio respond better to the FOLFIRI treatment, while patients in cluster 2 with the lowest T-reg/T-helper ratio resist the treatment. Moreover, we use ROC curve to validate the model using the tumor status of the patients at their follow up, and the model predicts well for the earlier follow up days.

Global Dynamics of a Breast Cancer Competition Model

In this paper, we present a system of five ordinary differential equations which consider population dynamics among cancer stem cells, tumor cells, and healthy cells. Additionally, we consider the effects of excess estrogen and the body's natural immune response on the aforementioned cell populations. Employing a variety of analytical methods, we study the global dynamics of the full system, along with various submodels. We find sufficient conditions on parameter values to ensure cancer persistence in the absence of immune cells, and cancer eradication when an immune response is included. We conclude with a discussion on the biological implications of the resulting global dynamics.

Modeling of Nanotherapy Response as a Function of the Tumor Microenvironment: Focus on Liver Metastasis

The tumor microenvironment (TME) presents a challenging barrier for effective nanotherapy-mediated drug delivery to solid tumors. In particular for tumors less vascularized than the surrounding normal tissue, as in liver metastases, the structure of the organ itself conjures with cancer-specific behavior to impair drug transport and uptake by cancer cells. Cells and elements in the TME of hypovascularized tumors play a key role in the process of delivery and retention of anti-cancer therapeutics by nanocarriers. This brief review describes the drug transport challenges and how they are being addressed with advanced in vitro 3D tissue models as well as with in silico mathematical modeling. This modeling complements network-oriented techniques, which seek to interpret intra-cellular relevant pathways and signal transduction within cells and with their surrounding microenvironment. With a concerted effort integrating experimental observations with computational analyses spanning from the molecular- to the tissue-scale, the goal of effective nanotherapy customized to patient tumor-specific conditions may be finally realized.

Long term survival and local control outcomes from single dose targeted intraoperative radiotherapy during lumpectomy (TARGIT-IORT) for early breast cancer: TARGIT-A randomised clinical trial

OBJECTIVE

To determine whether risk adapted intraoperative radiotherapy, delivered as a single dose during lumpectomy, can effectively replace postoperative whole breast external beam radiotherapy for early breast cancer.

DESIGN

Prospective, open label, randomised controlled clinical trial.

SETTING

32 centres in 10 countries in the United Kingdom, Europe, Australia, the United States, and Canada.

PARTICIPANTS

2298 women aged 45 years and older with invasive ductal carcinoma up to 3.5 cm in size, cN0-N1, eligible for breast conservation and randomised before lumpectomy (1:1 ratio, blocks stratified by centre) to either risk adapted targeted intraoperative radiotherapy (TARGIT-IORT) or external beam radiotherapy (EBRT).

INTERVENTIONS

Random allocation was to the EBRT arm, which consisted of a standard daily fractionated course (three to six weeks) of whole breast radiotherapy, or the TARGIT-IORT arm. TARGIT-IORT was given immediately after lumpectomy under the same anaesthetic and was the only radiotherapy for most patients (around 80%). TARGIT-IORT was supplemented by EBRT when postoperative histopathology found unsuspected higher risk factors (around 20% of patients).

MAIN OUTCOME MEASURES

Non-inferiority with a margin of 2.5% for the absolute difference between the five year local recurrence rates of the two arms, and long term survival outcomes.

RESULTS

Between 24 March 2000 and 25 June 2012, 1140 patients were randomised to TARGIT-IORT and 1158 to EBRT. TARGIT-IORT was non-inferior to EBRT: the local recurrence risk at five year complete follow-up was 2.11% for TARGIT-IORT compared with 0.95% for EBRT (difference 1.16%, 90% confidence interval 0.32 to 1.99). In the first five years, 13 additional local recurrences were reported (24/1140 v 11/1158) but 14 fewer deaths (42/1140 v 56/1158) for TARGIT-IORT compared with EBRT. With long term follow-up (median 8.6 years, maximum 18.90 years, interquartile range 7.0-10.6) no statistically significant difference was found for local recurrence-free survival (hazard ratio 1.13, 95% confidence interval 0.91 to 1.41, P=0.28), mastectomy-free survival (0.96, 0.78 to 1.19, P=0.74), distant disease-free survival (0.88, 0.69 to 1.12, P=0.30), overall survival (0.82, 0.63 to 1.05, P=0.13), and breast cancer mortality (1.12, 0.78 to 1.60, P=0.54). Mortality from other causes was significantly lower (0.59, 0.40 to 0.86, P=0.005).

CONCLUSION

For patients with early breast cancer who met our trial selection criteria, risk adapted immediate single dose TARGIT-IORT during lumpectomy was an effective alternative to EBRT, with comparable long term efficacy for cancer control and lower non-breast cancer mortality. TARGIT-IORT should be discussed with eligible patients when breast conserving surgery is planned.

TRIAL REGISTRATION

ISRCTN34086741, NCT00983684.

Effect of Delayed Targeted Intraoperative Radiotherapy vs Whole-Breast Radiotherapy on Local Recurrence and Survival: Long-term Results From the TARGIT-A Randomized Clinical Trial in Early Breast Cancer

Importance

Conventional adjuvant radiotherapy for breast cancer given daily for several weeks is onerous and expensive. Some patients may be obliged to choose a mastectomy instead, and some may forgo radiotherapy altogether. We proposed a clinical trial to test whether radiotherapy could be safely limited to the tumor bed.

Objective

To determine whether delayed second-procedure targeted intraoperative radiotherapy (TARGIT-IORT) is noninferior to whole-breast external beam radiotherapy (EBRT) in terms of local control.

Design, Setting, and Participants

In this prospective, randomized (1:1 ratio) noninferiority trial, 1153 patients aged 45 years or older with invasive ductal breast carcinoma smaller than 3.5 cm treated with breast conservation were enrolled from 28 centers in 9 countries. Data were locked in on July 3, 2019.

Interventions

The TARGIT-A trial was started in March 2000; patients were randomized after needle biopsy to receive TARGIT-IORT immediately after lumpectomy under the same anesthetic vs EBRT and results have been shown to be noninferior. A parallel study, described in this article, was initiated in 2004; patients who had their cancer excised were randomly allocated using separate randomization tables to receive EBRT or delayed TARGIT-IORT given as a second procedure by reopening the lumpectomy wound.

Main Outcomes and Measures

A noninferiority margin for local recurrence rate of 2.5% at 5 years, and long-term survival outcomes.

Results

Overall, 581 women (mean [SD] age, 63 [7] years) were randomized to delayed TARGIT-IORT and 572 patients (mean [SD] age, 63 [8] years) were randomized to EBRT. Sixty patients (5%) had tumors larger than 2 cm, or had positive nodes and only 32 (2.7%) were younger than 50 years. Delayed TARGIT-IORT was not noninferior to EBRT. The local recurrence rates at 5-year complete follow-up were: delayed TARGIT-IORT vs EBRT (23/581 [3.96%] vs 6/572 [1.05%], respectively; difference, 2.91%; upper 90% CI, 4.4%). With long-term follow-up (median [IQR], 9.0 [7.5-10.5] years), there was no statistically significant difference in local recurrence-free survival (HR, 0.75; 95% CI, 0.57-1.003; P = .052), mastectomy-free survival (HR, 0.88; 95% CI, 0.65-1.18; P = .38), distant disease-free survival (HR, 1.00; 95% CI, 0.72-1.39; P = .98), or overall survival (HR, 0.96; 95% CI, 0.68-1.35; P = .80).

Conclusions and Relevance

These long-term data show that despite an increase in the number of local recurrences with delayed TARGIT-IORT, there was no statistically significant decrease in mastectomy-free survival, distant disease-free survival, or overall survival.

Trial Registration

ISRCTN34086741, ClinicalTrials.gov Identifier: NCT00983684.

Crosstalk between HER2 and PD-1/PD-L1 in Breast Cancer: From Clinical Applications to Mathematical Models

Breast cancer is one of the major causes of mortality in women worldwide. The most aggressive breast cancer subtypes are human epidermal growth factor receptor-positive (HER2+) and triple-negative breast cancers. Therapies targeting HER2 receptors have significantly improved HER2+ breast cancer patient outcomes. However, several recent studies have pointed out the deficiency of existing treatment protocols in combatting disease relapse and improving response rates to treatment. Overriding the inherent actions of the immune system to detect and annihilate cancer via the immune checkpoint pathways is one of the important hallmarks of cancer. Thus, restoration of these pathways by various means of immunomodulation has shown beneficial effects in the management of various types of cancers, including breast. We herein review the recent progress in the management of HER2+ breast cancer via HER2-targeted therapies, and its association with the programmed death receptor-1 (PD-1)/programmed death ligand-1 (PD-L1) axis. In order to link research in the areas of medicine and mathematics and point out specific opportunities for providing efficient theoretical analysis related to HER2+ breast cancer management, we also review mathematical models pertaining to the dynamics of HER2+ breast cancer and immune checkpoint inhibitors.

Mathematical Models of Cancer Cell Plasticity

Quantitative modelling is increasingly important in cancer research, helping to integrate myriad diverse experimental data into coherent pictures of the disease and able to discriminate between competing hypotheses or suggest specific experimental lines of enquiry and new approaches to therapy. Here, we review a diverse set of mathematical models of cancer cell plasticity (a process in which, through genetic and epigenetic changes, cancer cells survive in hostile environments and migrate to more favourable environments, respectively), tumour growth, and invasion. Quantitative models can help to elucidate the complex biological mechanisms of cancer cell plasticity. In this review, we discuss models of plasticity, tumour progression, and metastasis under three broadly conceived mathematical modelling techniques: discrete, continuum, and hybrid, each with advantages and disadvantages. An emerging theme from the predictions of many of these models is that cell escape from the tumour microenvironment (TME) is encouraged by a combination of physiological stress locally (e.g., hypoxia), external stresses (e.g., the presence of immune cells), and interactions with the extracellular matrix. We also discuss the value of mathematical modelling for understanding cancer more generally.

Computational modeling of therapy on pancreatic cancer in its early stages

More than eighty percent of pancreatic cancer involves ductal adenocarcinoma with an abundant desmoplastic extracellular matrix surrounding the solid tumor entity. This aberrant tumor microenvironment facilitates a strong resistance of pancreatic cancer to medication. Although various therapeutic strategies have been reported to be effective in mice with pancreatic cancer, they still need to be tested quantitatively in wider animal-based experiments before being applied as therapies. To aid the design of experiments, we develop a cell-based mathematical model to describe cancer progression under therapy with a specific application to pancreatic cancer. The displacement of cells is simulated by solving a large system of stochastic differential equations with the Euler-Maruyama method. We consider treatment with the PEGylated drug PEGPH20 that breaks down hyaluronan in desmoplastic stroma followed by administration of the chemotherapy drug gemcitabine to inhibit the proliferation of cancer cells. Modeling the effects of PEGPH20 + gemcitabine concentrations is based on Green's fundamental solutions of the reaction-diffusion equation. Moreover, Monte Carlo simulations are performed to quantitatively investigate uncertainties in the input parameters as well as predictions for the likelihood of success of cancer therapy. Our simplified model is able to simulate cancer progression and evaluate treatments to inhibit the progression of cancer.

Optimal modality selection in external beam radiotherapy

The goal in external beam radiotherapy (EBRT) for cancer is to maximize damage to the tumour while limiting toxic effects on the organs-at-risk. EBRT can be delivered via different modalities such as photons, protons and neutrons. The choice of an optimal modality depends on the anatomy of the irradiated area and the relative physical and biological properties of the modalities under consideration. There is no single universally dominant modality. We present the first-ever mathematical formulation of the optimal modality selection problem. We show that this problem can be tackled by solving the Karush-Kuhn-Tucker conditions of optimality, which reduce to an analytically tractable quartic equation. We perform numerical experiments to gain insights into the effect of biological and physical properties on the choice of an optimal modality or combination of modalities.

Mathematical Models of Cancer: When to Predict Novel Therapies, and When Not to

The number of publications on mathematical modeling of cancer is growing at an exponential rate, according to PubMed records, provided by the US National Library of Medicine and the National Institutes of Health. Seminal papers have initiated and promoted mathematical modeling of cancer and have helped define the field of mathematical oncology (Norton and Simon in J Natl Cancer Inst 58:1735-1741, 1977; Norton in Can Res 48:7067-7071, 1988; Hahnfeldt et al. in Can Res 59:4770-4775, 1999; Anderson et al. in Comput Math Methods Med 2:129-154, 2000. https://doi.org/10.1080/10273660008833042 ; Michor et al. in Nature 435:1267-1270, 2005. https://doi.org/10.1038/nature03669 ; Anderson et al. in Cell 127:905-915, 2006. https://doi.org/10.1016/j.cell.2006.09.042 ; Benzekry et al. in PLoS Comput Biol 10:e1003800, 2014. https://doi.org/10.1371/journal.pcbi.1003800 ). Following the introduction of undergraduate and graduate programs in mathematical biology, we have begun to see curricula developing with specific and exclusive focus on mathematical oncology. In 2018, 218 articles on mathematical modeling of cancer were published in various journals, including not only traditional modeling journals like the Bulletin of Mathematical Biology and the Journal of Theoretical Biology, but also publications in renowned science, biology, and cancer journals with tremendous impact in the cancer field (Cell, Cancer Research, Clinical Cancer Research, Cancer Discovery, Scientific Reports, PNAS, PLoS Biology, Nature Communications, eLife, etc). This shows the breadth of cancer models that are being developed for multiple purposes. While some models are phenomenological in nature following a bottom-up approach, other models are more top-down data-driven. Here, we discuss the emerging trend in mathematical oncology publications to predict novel, optimal, sometimes even patient-specific treatments, and propose a convention when to use a model to predict novel treatments and, probably more importantly, when not to.

Engineered 3D Model of Cancer Stem Cell Enrichment and Chemoresistance

Intraperitoneal dissemination of ovarian cancers is preceded by the development of chemoresistant tumors with malignant ascites. Despite the high levels of chemoresistance and relapse observed in ovarian cancers, there are no in vitro models to understand the development of chemoresistance in situ. Method: We describe a highly integrated approach to establish an in vitro model of chemoresistance and stemness in ovarian cancer, using the 3D hanging drop spheroid platform. The model was established by serially passaging non-adherent spheroids. At each passage, the effectiveness of the model was evaluated via measures of proliferation, response to treatment with cisplatin and a novel ALDH1A inhibitor. Concomitantly, the expression and tumor initiating capacity of cancer stem-like cells (CSCs) was analyzed. RNA-seq was used to establish gene signatures associated with the evolution of tumorigenicity, and chemoresistance. Lastly, a mathematical model was developed to predict the emergence of CSCs during serial passaging of ovarian cancer spheroids. Results: Our serial passage model demonstrated increased cellular proliferation, enriched CSCs, and emergence of a platinum resistant phenotype. In vivo tumor xenograft assays indicated that later passage spheroids were significantly more tumorigenic with higher CSCs, compared to early passage spheroids. RNA-seq revealed several gene signatures supporting the emergence of CSCs, chemoresistance, and malignant phenotypes, with links to poor clinical prognosis. Our mathematical model predicted the emergence of CSC populations within serially passaged spheroids, concurring with experimentally observed data. Conclusion: Our integrated approach illustrates the utility of the serial passage spheroid model for examining the emergence and development of chemoresistance in ovarian cancer in a controllable and reproducible format.

Modeling dynamics for oncogenesis encompassing mutations and genetic instability

Tumorigenesis has been described as a multistep process, where each step is associated with a genetic alteration, in the direction to progressively transform a normal cell and its descendants into a malignant tumour. Into this work, we propose a mathematical model for cancer onset and development, considering three populations: normal, premalignant and cancer cells. The model takes into account three hallmarks of cancer: self-sufficiency on growth signals, insensibility to anti-growth signals and evading apoptosis. By using a nonlinear expression to describe the mutation from premalignant to cancer cells, the model includes genetic instability as an enabling characteristic of tumour progression. Mathematical analysis was performed in detail. Results indicate that apoptosis and tissue repair system are the first barriers against tumour progression. One of these mechanisms must be corrupted for cancer to develop from a single mutant cell. The results also show that the presence of aggressive cancer cells opens way to survival of less adapted premalignant cells. Numerical simulations were performed with parameter values based on experimental data of breast cancer, and the necessary time taken for cancer to reach a detectable size from a single mutant cell was estimated with respect to some parameters. We find that the rates of apoptosis and mutations have a large influence on the pace of tumour progression and on the time it takes to become clinically detectable.

Role of tumor-associated neutrophils in regulation of tumor growth in lung cancer development: A mathematical model

Neutrophils display rapid and potent innate immune responses in various diseases. Tumor-associated neutrophils (TANs) however either induce or overcome immunosuppressive functions of the tumor microenvironment through complex tumor-stroma crosstalk. We developed a mathematical model to address the question of how phenotypic alterations between tumor suppressive N1 TANS, and tumor promoting N2 TANs affect nonlinear tumor growth in a complex tumor microenvironment. The model provides a visual display of the complex behavior of populations of TANs and tumors in response to various TGF-β and IFN-β stimuli. In addition, the effect of anti-tumor drug administration is incorporated in the model in an effort to achieve optimal anti-tumor efficacy. The simulation results from the mathematical model were in good agreement with experimental data. We found that the N2-to-N1 ratio (N21R) index is positively correlated with aggressive tumor growth, suggesting that this may be a good prognostic factor. We also found that the antitumor efficacy increases when the relative ratio (Dap) of delayed apoptotic cell death of N1 and N2 TANs is either very small or relatively large, providing a basis for therapeutically targeting prometastatic N2 TANs.

Effect of cellular de-differentiation on the dynamics and evolution of tissue and tumor cells in mathematical models with feedback regulation

Tissues are maintained by adult stem cells that self-renew and also differentiate into functioning tissue cells. Homeostasis is achieved by a set of complex mechanisms that involve regulatory feedback loops. Similarly, tumors are believed to be maintained by a minority population of cancer stem cells, while the bulk of the tumor is made up of more differentiated cells, and there is indication that some of the feedback loops that operate in tissues continue to be functional in tumors. Mathematical models of such tissue hierarchies, including feedback loops, have been analyzed in a variety of different contexts. Apart from stem cells giving rise to differentiated cells, it has also been observed that more differentiated cells can de-differentiate into stem cells, both in healthy tissue and tumors, aspects of which have also been investigated mathematically. This paper analyses the effect of de-differentiation on the basic and evolutionary dynamics of cells in the context of tissue hierarchy models that include negative feedback regulation of the cell populations. The models predict that in the presence of de-differentiation, the fixation probability of a neutral mutant is lower than in its absence. Therefore, if de-differentiation occurs, a mutant with identical parameters compared to the wild-type cell population behaves like a disadvantageous mutant. Similarly, the process of de-differentiation is found to lower the fixation probability of an advantageous mutant. These results indicate that the presence of de-differentiation can lower the rates of tumor initiation and progression in the context of the models considered here.

Revealing age-related changes of adult hippocampal neurogenesis using mathematical models

New neurons are continuously generated in the dentate gyrus of the adult hippocampus. This continuous supply of newborn neurons is important to modulate cognitive functions. Yet the number of newborn neurons declines with age. Increasing Wnt activity upon loss of dickkopf 1 can counteract both the decline of newborn neurons and the age-related cognitive decline. However, the precise cellular changes underlying the age-related decline or its rescue are fundamentally not understood. The present study combines a mathematical model and experimental data to address features controlling neural stem cell (NSC) dynamics. We show that available experimental data fit a model in which quiescent NSCs may either become activated to divide or may undergo depletion events, such as astrocytic transformation and apoptosis. Additionally, we demonstrate that old NSCs remain quiescent longer and have a higher probability of becoming re-activated than depleted. Finally, our model explains that high NSC-Wnt activity leads to longer time in quiescence while enhancing the probability of activation. Altogether, our study shows that modulation of the quiescent state is crucial to regulate the pool of stem cells throughout the life of an animal.

Summary: New deterministic and stochastic mathematical models are proposed to investigate adult neurogenesis in young, old and perturbed hippocampus, and quantified using population-level and clonal experimental data.

The role of Allee effect in modelling post resection recurrence of glioblastoma

Resection of the bulk of a tumour often cannot eliminate all cancer cells, due to their infiltration into the surrounding healthy tissue. This may lead to recurrence of the tumour at a later time. We use a reaction-diffusion equation based model of tumour growth to investigate how the invasion front is delayed by resection, and how this depends on the density and behaviour of the remaining cancer cells. We show that the delay time is highly sensitive to qualitative details of the proliferation dynamics of the cancer cell population. The typically assumed logistic type proliferation leads to unrealistic results, predicting immediate recurrence. We find that in glioblastoma cell cultures the cell proliferation rate is an increasing function of the density at small cell densities. Our analysis suggests that cooperative behaviour of cancer cells, analogous to the Allee effect in ecology, can play a critical role in determining the time until tumour recurrence.

Mathematical models of propagating fronts have been used to represent a wide variety of biological phenomena from action potentials in neural cells to invasive species in ecology and epidemic spreading. Here we show that when such models are used to predict the effects of external perturbations the results can be very sensitive to certain details of the local dynamics. For example, the post resection recurrence of tumour growth depends strongly on the density dependence of the proliferation of cancer cells. This suggests that targeting the cooperative behaviour of cancer cells could be an efficient strategy for delaying the recurrence of diffuse aggressive brain tumours.

Cell dynamics in tumour environment after treatments

Most cancer treatments cause necrotic cell deaths in the tumour microenvironment. Necrotic cells send signals to immune cells to start the wound healing process in the tissue. Therefore, we assume after stopping treatments there is a wound that needs to be healed. We develop a simple computational model to investigate cell dynamics during the wound healing process after treatments. The model predicts that the involvement of high-fitness cancer cells in the wound healing leads to fast relapse, and cancer cells outside of the wound can cause a slow recurrence of the tumour. Therefore, the absence of relapse after treatments may imply a slow-developing tumour that might not reach an observable size in the patients' lifetime. Additionally, the model indicates that the location of remaining cancer cells after treatments is an important factor in the recurrence time. The fastest recurrence happens when high-fitness cancer cells remain inside of the wound. However, the longest time to recurrence corresponds to cancer cells located outside of the wound. Note that this model is the first attempt to study cell dynamics in the wound healing process after cancer treatments, and it has some limitations that might influence the results. Experiments are needed to validate the results.

Determining the control networks regulating stem cell lineages in colonic crypts

The question of stem cell control is at the center of our understanding of tissue functioning, both in healthy and cancerous conditions. It is well accepted that cellular fate decisions (such as divisions, differentiation, apoptosis) are orchestrated by a network of regulatory signals emitted by different cell populations in the lineage and the surrounding tissue. The exact regulatory network that governs stem cell lineages in a given tissue is usually unknown. Here we propose an algorithm to identify a set of candidate control networks that are compatible with (a) measured means and variances of cell populations in different compartments, (b) qualitative information on cell population dynamics, such as the existence of local controls and oscillatory reaction of the system to population size perturbations, and (c) statistics of correlations between cell numbers in different compartments. Using the example of human colon crypts, where lineages are comprised of stem cells, transit amplifying cells, and differentiated cells, we start with a theoretically known set of 32 smallest control networks compatible with tissue stability. Utilizing near-equilibrium stochastic calculus of stem cells developed earlier, we apply a series of tests, where we compare the networks' expected behavior with the observations. This allows us to exclude most of the networks, until only three, very similar, candidate networks remain, which are most compatible with the measurements. This work demonstrates how theoretical analysis of control networks combined with only static biological data can shed light onto the inner workings of stem cell lineages, in the absence of direct experimental assessment of regulatory signaling mechanisms. The resulting candidate networks are dominated by negative control loops and possess the following properties: (1) stem cell division decisions are negatively controlled by the stem cell population, (2) stem cell differentiation decisions are negatively controlled by the transit amplifying cell population.

Significance of Inactivated Genes in Leukemia: Pathogenesis and Prognosis

Epigenetic and genetic alterations are two mechanisms participating in leukemia, which can inactivate genes involved in leukemia pathogenesis or progression. The purpose of this review was to introduce various inactivated genes and evaluate their possible role in leukemia pathogenesis and prognosis. By searching the mesh words "Gene, Silencing AND Leukemia" in PubMed website, relevant English articles dealt with human subjects as of 2000 were included in this study. Gene inactivation in leukemia is largely mediated by promoter's hypermethylation of gene involving in cellular functions such as cell cycle, apoptosis, and gene transcription. Inactivated genes, such as ASPP1, TP53, IKZF1 and P15, may correlate with poor prognosis in acute lymphoid leukemia (ALL), chronic lymphoid leukemia (CLL), chronic myelogenous leukemia (CML) and acute myeloid leukemia (AML), respectively. Gene inactivation may play a considerable role in leukemia pathogenesis and prognosis, which can be considered as complementary diagnostic tests to differentiate different leukemia types, determine leukemia prognosis, and also detect response to therapy. In general, this review showed some genes inactivated only in leukemia (with differences between B-ALL, T-ALL, CLL, AML and CML). These differences could be of interest as an additional tool to better categorize leukemia types. Furthermore; based on inactivated genes, a diverse classification of Leukemias could represent a powerful method to address a targeted therapy of the patients, in order to minimize side effects of conventional therapies and to enhance new drug strategies.

Stability of Control Networks in Autonomous Homeostatic Regulation of Stem Cell Lineages

Design principles of biological networks have been studied extensively in the context of protein-protein interaction networks, metabolic networks, and regulatory (transcriptional) networks. Here we consider regulation networks that occur on larger scales, namely the cell-to-cell signaling networks that connect groups of cells in multicellular organisms. These are the feedback loops that orchestrate the complex dynamics of cell fate decisions and are necessary for the maintenance of homeostasis in stem cell lineages. We focus on "minimal" networks that are those that have the smallest possible numbers of controls. For such minimal networks, the number of controls must be equal to the number of compartments, and the reducibility/irreducibility of the network (whether or not it can be split into smaller independent sub-networks) is defined by a matrix comprised of the cell number increments induced by each of the controlled processes in each of the compartments. Using the formalism of digraphs, we show that in two-compartment lineages, reducible systems must contain two 1-cycles, and irreducible systems one 1-cycle and one 2-cycle; stability follows from the signs of the controls and does not require magnitude restrictions. In three-compartment systems, irreducible digraphs have a tree structure or have one 3-cycle and at least two more shorter cycles, at least one of which is a 1-cycle. With further work and proper biological validation, our results may serve as a first step toward an understanding of ways in which these networks become dysregulated in cancer.

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.

An international randomised controlled trial to compare TARGeted Intraoperative radioTherapy (TARGIT) with conventional postoperative radiotherapy after breast-conserving surgery for women with early-stage breast cancer (the TARGIT-A trial)

BACKGROUND

Based on our laboratory work and clinical trials we hypothesised that radiotherapy after lumpectomy for breast cancer could be restricted to the tumour bed. In collaboration with the industry we developed a new radiotherapy device and a new surgical operation for delivering single-dose radiation to the tumour bed - the tissues at highest risk of local recurrence. We named it TARGeted Intraoperative radioTherapy (TARGIT). From 1998 we confirmed its feasibility and safety in pilot studies.

OBJECTIVE

To compare TARGIT within a risk-adapted approach with whole-breast external beam radiotherapy (EBRT) over several weeks.

DESIGN

The TARGeted Intraoperative radioTherapy Alone (TARGIT-A) trial was a pragmatic, prospective, international, multicentre, non-inferiority, non-blinded, randomised (1 : 1 ratio) clinical trial. Originally, randomisation occurred before initial lumpectomy (prepathology) and, if allocated TARGIT, the patient received it during the lumpectomy. Subsequently, the postpathology stratum was added in which randomisation occurred after initial lumpectomy, allowing potentially easier logistics and a more stringent case selection, but which needed a reoperation to reopen the wound to give TARGIT as a delayed procedure. The risk-adapted approach meant that, in the experimental arm, if pre-specified unsuspected adverse factors were found postoperatively after receiving TARGIT, EBRT was recommended. Pragmatically, this reflected how TARGIT would be practised in the real world.

SETTING

Thirty-three centres in 11 countries.

PARTICIPANTS

Women who were aged ≥ 45 years with unifocal invasive ductal carcinoma preferably ≤ 3.5 cm in size.

INTERVENTIONS

TARGIT within a risk-adapted approach and whole-breast EBRT.

MAIN OUTCOME MEASURES

The primary outcome measure was absolute difference in local recurrence, with a non-inferiority margin of 2.5%. Secondary outcome measures included toxicity and breast cancer-specific and non-breast-cancer mortality.

RESULTS

In total, 3451 patients were recruited between March 2000 and June 2012. The following values are 5-year Kaplan-Meier rates for TARGIT compared with EBRT. There was no statistically significant difference in local recurrence between TARGIT and EBRT. TARGIT was non-inferior to EBRT overall [TARGIT 3.3%, 95% confidence interval (CI) 2.1% to 5.1% vs. EBRT 1.3%, 95% CI 0.7% to 2.5%; p = 0.04; P_{non-inferiority} = 0.00000012] and in the prepathology stratum (n = 2298) when TARGIT was given concurrently with lumpectomy (TARGIT 2.1%, 95% CI 1.1% to 4.2% vs. EBRT 1.1%, 95% CI 0.5% to 2.5%; p = 0.31; P_{non-inferiority} = 0.0000000013). With delayed TARGIT postpathology (n = 1153), the between-group difference was larger than 2.5% and non-inferiority was not established for this stratum (TARGIT 5.4%, 95% CI 3.0% to 9.7% vs. EBRT 1.7%, 95% CI 0.6% to 4.9%; p = 0.069; P_{non-inferiority} = 0.06640]. The local recurrence-free survival was 93.9% (95% CI 90.9% to 95.9%) when TARGIT was given with lumpectomy compared with 92.5% (95% CI 89.7% to 94.6%) for EBRT (p = 0.35). In a planned subgroup analysis, progesterone receptor (PgR) status was found to be the only predictor of outcome: hormone-responsive patients (PgR positive) had similar 5-year local recurrence with TARGIT during lumpectomy (1.4%, 95% CI 0.5% to 3.9%) as with EBRT (1.2%, 95% CI 0.5% to 2.9%; p = 0.77). Grade 3 or 4 radiotherapy toxicity was significantly reduced with TARGIT. Overall, breast cancer mortality was much the same between groups (TARGIT 2.6%, 95% CI 1.5% to 4.3% vs. EBRT 1.9%, 95% CI 1.1% to 3.2%; p = 0.56) but there were significantly fewer non-breast-cancer deaths with TARGIT (1.4%, 95% CI 0.8% to 2.5% vs. 3.5%, 95% CI 2.3% to 5.2%; p = 0.0086), attributable to fewer deaths from cardiovascular causes and other cancers, leading to a trend in reduced overall mortality in the TARGIT arm (3.9%, 95% CI 2.7% to 5.8% vs. 5.3%, 95% CI 3.9% to 7.3%; p = 0.099]. Health economic analyses suggest that TARGIT was statistically significantly less costly than EBRT, produced similar quality-adjusted life-years, had a positive incremental net monetary benefit that was borderline statistically significantly different from zero and had a probability of > 90% of being cost-effective. There appears to be little uncertainty in the point estimates, based on deterministic and probabilistic sensitivity analyses. If TARGIT were given instead of EBRT in suitable patients, it might potentially reduce costs to the health-care providers in the UK by £8-9.1 million each year. This does not include environmental, patient and societal costs.

LIMITATIONS

The number of local recurrences is small but the number of events for local recurrence-free survival is not as small (TARGIT 57 vs. EBRT 59); occurrence of so few events (< 3.5%) also implies that both treatments are effective and any difference is unlikely to be large. Not all 3451 patients were followed up for 5 years; however, more than the number of patients required to answer the main trial question (n = 585) were followed up for > 5 years.

CONCLUSIONS

For patients with breast cancer (women who are aged ≥ 45 years with hormone-sensitive invasive ductal carcinoma that is up to 3.5 cm in size), TARGIT concurrent with lumpectomy within a risk-adapted approach is as effective as, safer than and less expensive than postoperative EBRT.

FUTURE WORK

The analyses will be repeated with longer follow-up. Although this may not change the primary result, the larger number of events may confirm the effect on overall mortality and allow more detailed subgroup analyses. The TARGeted Intraoperative radioTherapy Boost (TARGIT-B) trial is testing whether or not a tumour bed boost given intraoperatively (TARGIT) boost is superior to a tumour bed boost given as part of postoperative EBRT.

TRIAL REGISTRATION

Current Controlled Trials ISRCTN34086741 and ClinicalTrials.gov NCT00983684.

FUNDING

University College London Hospitals (UCLH)/University College London (UCL) Comprehensive Biomedical Research Centre, UCLH Charities, Ninewells Cancer Campaign, National Health and Medical Research Council and German Federal Ministry of Education and Research (BMBF). From September 2009 this project was funded by the NIHR Health Technology Assessment programme and will be published in full in Health Technology Assessment; Vol. 20, No. 73. See the NIHR Journals Library website for further project information.

Near Equilibrium Calculus of Stem Cells in Application to the Airway Epithelium Lineage

Homeostatic maintenance of tissues is orchestrated by well tuned networks of cellular signaling. Such networks regulate, in a stochastic manner, fates of all cells within the respective lineages. Processes such as symmetric and asymmetric divisions, differentiation, de-differentiation, and death have to be controlled in a dynamic fashion, such that the cell population is maintained at a stable equilibrium, has a sufficiently low level of stochastic variation, and is capable of responding efficiently to external damage. Cellular lineages in real tissues may consist of a number of different cell types, connected by hierarchical relationships, albeit not necessarily linear, and engaged in a number of different processes. Here we develop a general mathematical methodology for near equilibrium studies of arbitrarily complex hierarchical cell populations, under regulation by a control network. This methodology allows us to (1) determine stability properties of the network, (2) calculate the stochastic variance, and (3) predict how different control mechanisms affect stability and robustness of the system. We demonstrate the versatility of this tool by using the example of the airway epithelium lineage. Recent research shows that airway epithelium stem cells divide mostly asymmetrically, while the so-called secretory cells divide predominantly symmetrically. It further provides quantitative data on the recovery dynamics of the airway epithelium, which can include secretory cell de-differentiation. Using our new methodology, we demonstrate that while a number of regulatory networks can be compatible with the observed recovery behavior, the observed division patterns of cells are the most optimal from the viewpoint of homeostatic lineage stability and minimizing the variation of the cell population size. This not only explains the observed yet poorly understood features of airway tissue architecture, but also helps to deduce the information on the still largely hypothetical regulatory mechanisms governing tissue turnover, and lends insight into how different control loops influence the stability and variance properties of cell populations.

Tissue stability is the basic property of healthy organs, and yet the mechanisms governing the stable, long-term maintenance of cell numbers in tissues are poorly understood. While more and more signaling pathways are being discovered, for the most part it remains unknown how they are being put together by different cell types into complex, nonlinear, hierarchical control networks that, on the one hand, reliably maintain constant cell numbers, and on the other hand, quickly adjust to oversee the robust response to tissue damage. Theoretical approaches can fill the gap by being able to reconstruct the underlying control network, based on the observations about the aspects of cellular dynamics. We argue that while many hypothetical networks may be capable of basic cell lineage maintenance, some are much more efficient from the viewpoint of variance minimization. Thus, we developed a new methodology that can test various control networks for stability, variance, and robustness. In the example of the airway epithelium that we highlight, it turns out that the evolutionary selected, actual architecture coincides with the mathematically optimal solution that minimizes the fluctuations of cell numbers at homeostasis.

Mathematical models of breast and ovarian cancers

Women constitute the majority of the aging United States (US) population, and this has substantial implications on cancer population patterns and management practices. Breast cancer is the most common women's malignancy, while ovarian cancer is the most fatal gynecological malignancy in the US. In this review, we focus on these subsets of women's cancers, seen more commonly in postmenopausal and elderly women. In order to systematically investigate the complexity of cancer progression and response to treatment in breast and ovarian malignancies, we assert that integrated mathematical modeling frameworks viewed from a systems biology perspective are needed. Such integrated frameworks could offer innovative contributions to the clinical women's cancers community, as answers to clinical questions cannot always be reached with contemporary clinical and experimental tools. Here, we recapitulate clinically known data regarding the progression and treatment of the breast and ovarian cancers. We compare and contrast the two malignancies whenever possible in order to emphasize areas where substantial contributions could be made by clinically inspired and validated mathematical modeling. We show how current paradigms in the mathematical oncology community focusing on the two malignancies do not make comprehensive use of, nor substantially reflect existing clinical data, and we highlight the modeling areas in most critical need of clinical data integration. We emphasize that the primary goal of any mathematical study of women's cancers should be to address clinically relevant questions. WIREs Syst Biol Med 2016, 8:337-362. doi: 10.1002/wsbm.1343 For further resources related to this article, please visit the WIREs website.

In silico study of the impact of cancer stem cell dynamics and radiobiological hypoxia on tumour response to hyperfractionated radiotherapy

OBJECTIVES

Advanced head and neck carcinomas (HNCs) are aggressive tumours, mainly due to hypoxia and a cancer stem cell (CSC) subpopulation. The aim of this study was to simulate tumour growth and behaviour during radiotherapy of three HNC groups (governed by different growth kinetics, hypoxia levels and CSC division pattern) to determine correlation between resistance factors and responses to hyperfractionated radiotherapy.

METHODS

An in silico HNC model was developed based on biologically realistic input parameters. During radiotherapy simulation, three parameters were studied: growth kinetics, hypoxia and probability of CSC symmetrical division. Both independent and combined effects on tumour response to hyperfractionated radiotherapy were assessed.

RESULTS

Oxic and very mildly hypoxic HNCs were revealed to be controlled by hyperfractionated radiotherapy, irrespective of growth kinetics and CSC division pattern. Moderately hypoxic tumours had different responses to radiotherapy: while slowly proliferating HNCs were still controllable, tumours with higher cell turnover were more resistant. In rapidly proliferating tumours, the number of fractions needed for tumour control increased exponentially with the probability of CSC symmetrical division, whereas in moderately growing HNC, this behaviour was linear. Severely hypoxic tumours could not be controlled by radiotherapy alone. Tumours with CSCs in a severely hypoxic niche required adjuvant therapies to be eradicated.

CONCLUSIONS

Growth kinetics strongly influence tumour responses to treatment. Slowly growing tumours showed linear dependence between dose and hypoxia/CSC, whereas rapidly growing tumours followed exponential behaviour.

Serum uPAR as Biomarker in Breast Cancer Recurrence: A Mathematical Model

There are currently over 2.5 million breast cancer survivors in the United States and, according to the American Cancer Society, 10 to 20 percent of these women will develop recurrent breast cancer. Early detection of recurrence can avoid unnecessary radical treatment. However, self-examination or mammography screening may not discover a recurring cancer if the number of surviving cancer cells is small, while biopsy is too invasive and cannot be frequently repeated. It is therefore important to identify non-invasive biomarkers that can detect early recurrence. The present paper develops a mathematical model of cancer recurrence. The model, based on a system of partial differential equations, focuses on tissue biomarkers that include the plasminogen system. Among them, only uPAR is known to have significant correlation to its concentration in serum and could therefore be a good candidate for serum biomarker. The model includes uPAR and other associated cytokines and cells. It is assumed that the residual cancer cells that survived primary cancer therapy are concentrated in the same location within a region with a very small diameter. Model simulations establish a quantitative relation between the diameter of the growing cancer and the total uPAR mass in the cancer. This relation is used to identify uPAR as a potential serum biomarker for breast cancer recurrence.

The Role of Symmetric Stem Cell Divisions in Tissue Homeostasis

Successful maintenance of cellular lineages critically depends on the fate decision dynamics of stem cells (SCs) upon division. There are three possible strategies with respect to SC fate decision symmetry: (a) asymmetric mode, when each and every SC division produces one SC and one non-SC progeny; (b) symmetric mode, when 50% of all divisions produce two SCs and another 50%-two non-SC progeny; (c) mixed mode, when both the asymmetric and two types of symmetric SC divisions co-exist and are partitioned so that long-term net balance of the lineage output stays constant. Theoretically, either of these strategies can achieve lineage homeostasis. However, it remains unclear which strategy(s) are more advantageous and under what specific circumstances, and what minimal control mechanisms are required to operate them. Here we used stochastic modeling to analyze and quantify the ability of different types of divisions to maintain long-term lineage homeostasis, in the context of different control networks. Using the example of a two-component lineage, consisting of SCs and one type of non-SC progeny, we show that its tight homeostatic control is not necessarily associated with purely asymmetric divisions. Through stochastic analysis and simulations we show that asymmetric divisions can either stabilize or destabilize the lineage system, depending on the underlying control network. We further apply our computational model to biological observations in the context of a two-component lineage of mouse epidermis, where autonomous lineage control has been proposed and notable regional differences, in terms of symmetric division ratio, have been noted-higher in thickened epidermis of the paw skin as compared to ear and tail skin. By using our model we propose a possible explanation for the regional differences in epidermal lineage control strategies. We demonstrate how symmetric divisions can work to stabilize paw epidermis lineage, which experiences high level of micro-injuries and a lack of hair follicles as a back-up source of SCs.

Analysis of stochastic stem cell models with control

Understanding the dynamics of stem cell lineages is of central importance both for healthy and cancerous tissues. We study stochastic population dynamics of stem cells and differentiated cells, where cell decisions, such as proliferation vs. differentiation decisions, or division and death decisions, are under regulation from surrounding cells. The goal is to understand how different types of control mechanisms affect the means and variances of cell numbers. We use the assumption of weak dependencies of the regulatory functions (the controls) on the cell populations near the equilibrium to formulate moment equations. We then study three different methods of closure, showing that they all lead to the same results for the highest order terms in the expressions for the moments. We derive simple explicit expressions for the means and the variances of stem cell and differentiated cell numbers. It turns out that the variance is expressed as an algebraic function of partial derivatives of the controls with respect to the population sizes at the equilibrium. We demonstrate that these findings are consistent with the results previously obtained in the context of particular systems, and also present two novel examples with negative and positive control of division and differentiation decisions. This methodology is formulated without any specific assumptions on the functional form of the controls, and thus can be used for any biological system.