A multiscale mathematical model of avascular tumor growth to investigate the therapeutic benefit of anti-invasive agents

Simulating the behaviour of glioblastoma multiforme based on patient MRI during treatments

Glioblastoma multiforme is a brain cancer that still shows poor prognosis for patients despite the active research for new treatments. In this work, the goal is to model and simulate the evolution of tumour associated angiogenesis and the therapeutic response to glioblastoma multiforme. Multiple phenomena are modelled in order to fit different biological pathways, such as the cellular cycle, apoptosis, hypoxia or angiogenesis. This leads to a nonlinear system with 4 equations and 4 unknowns: the density of tumour cells, the [Formula: see text] concentration, the density of endothelial cells and the vascular endothelial growth factor concentration. This system is solved numerically on a mesh fitting the geometry of the brain and the tumour of a patient based on a 2D slice of MRI. We show that our numerical scheme is positive, and we give the energy estimates on the discrete solution to ensure its existence. The numerical scheme uses nonlinear control volume finite elements in space and is implicit in time. Numerical simulations have been done using the different standard treatments: surgery, chemotherapy and radiotherapy, in order to conform to the behaviour of a tumour in response to treatments according to empirical clinical knowledge. We find that our theoretical model exhibits realistic behaviours.

Kinetic modeling and numerical simulations to predict patient-specific responses to radiotherapy

Recent research indicates that quiescent tumor cells are significantly less radiosensitive with a greater repair capacity than proliferative cells. In order to better predict patient-specific responses to radiotherapy, we develop a mathematical model with treatment terms to describe dynamical behaviors of tumor growth. The global stabilities of the tumor-free equilibrium, the tumor-present equilibrium and the corresponding sufficient criteria are obtained. In addition, we simulate volumetric imaging data from 12 head-and-neck cancer patients and estimate the patient-specific responses to radiotherapy. Results indicate that radiosensitivity of proliferative cells is a critical factor that determines a successful radiotherapy. By comparison with previous simulation results, we find that the model presented in this paper is more suitable to describe the radiotherapy procedure of head-and-neck cancer. Finally, we discuss the influences of different radiotherapy strategies on therapeutic effect. The results show that treatment strategies with large dose or short treatment cycle can obtain better treatment effect.

Scientific and regulatory evaluation of mechanistic in silico drug and disease models in drug development: Building model credibility

The value of in silico methods in drug development and evaluation has been demonstrated repeatedly and convincingly. While their benefits are now unanimously recognized, international standards for their evaluation, accepted by all stakeholders involved, are still to be established. In this white paper, we propose a risk‐informed evaluation framework for mechanistic model credibility evaluation. To properly frame the proposed verification and validation activities, concepts such as context of use, regulatory impact and risk‐based analysis are discussed. To ensure common understanding between all stakeholders, an overview is provided of relevant in silico terminology used throughout this paper. To illustrate the feasibility of the proposed approach, we have applied it to three real case examples in the context of drug development, using a credibility matrix currently being tested as a quick‐start tool by regulators. Altogether, this white paper provides a practical approach to model evaluation, applicable in both scientific and regulatory evaluation contexts.

Mathematical simulation and prediction of tumor volume using RBF artificial neural network at different circumstances in the tumor microenvironment

Uncontrolled proliferation of cells in a tissue caused by genetic mutations inside a cell is referred to as a tumor. A tumor which grows rapidly encounters a barrier when it grows to a certain size in presence of preexisting vasculature. This is the time when it has to find a way to go on the growth. The tumor starts to secrete tumor angiogenic factors (TAFs) and stimulate preexisting vessels to grow new sprouts. These new sprouts will find their way to the tumor in the extracellular matrix (ECM) by the gradient of TAF. As these new capillaries anastomose and reach tumor, fresh oxygen is available for the tumor and it will reinitiate the growth. Number of initial sprouts, distance of initial tumor cells from the vessel(s) and initial density of the tumor at the time of sprout formation are questions which are to be investigated. In the present study, the aim is to find the response of tumor cells and vessels to the reciprocal effects of each other in different circumstances in the tissue. Together with a mathematical formulation, a radial basis function (RBF) neural network is established to predict the number of tumor cells at different circumstances including size and distance of initial tumors from the parent vessel. A final formulation is given for the final number of tumor cells as a function of initial tumor size and distance between a parent vessel and a tumor. Results of this simulation demonstrate that, increasing the distance between a tumor and a parent vessel decreases the number of final tumor cells. Specially, this decrement becomes faster beyond a certain distance. Moreover, initial tumors in bigger domains must become much bigger before inducing angiogenesis which makes it harder for them to survive.

Developing and studying the dynamical behavior of a nonlinear mathematical model for cancers with tumor by considering immune system role

We are constrained by widespread cancerous diseases to improve treatment methods which save patients and provide better living conditions during and after the treatment period. Because of the complexity of the treatment process, mathematical models need to be used in order to have a better understanding of the process. However, deriving an adequate complex model that can capture the disease pattern which could be confirmed by simulations and experiments has its own barriers. In this paper, a new mathematical model is developed concerning immune system effect on cancer. The model is introduced using nonlinear ordinary differential equations. Also, the qualitative behavior of the proposed system is studied in order to examine the extent of the model with respect to the nature of tumor evolution. Thus, number and status of equilibria points in line with the existence of limit cycles are obtained for sub-systems and the whole system. Meanwhile, possible bifurcations are mentioned, and the consequent evolutions are described. It is shown that the model conforms well to natural possibilities, cancer growth or remission. Thus, the model would be fit for further studies for prediction and contemplating treatment method, especially for immune stimulating drugs and immunotherapy.

A multi-scale agent-based model for avascular tumour growth

In this paper, we have developed a multi-scale, lattice-free, agent based model of avascular tumour growth in epithelial tissue. The model integrates different events to identify the underlying diversity within intracellular, cellular, and extracellular layer dynamics. The model considers every cell as an agent. A cellular agent may proliferate, spawns two identical daughter agents, or it may be transformed into other phenotypes during its life time depending on its internal proteins' activity as well as its external microenvironment. In this context, a simplified age-structured cell cycle model is adopted from the existing literature. The model considers that the intracellular events are regulated by p27 gene expression. In this model, p27 protein controls the overall tumour growth dynamics. Moreover, p27 is controlled by the external oxygen and nutrients that are modelled with the reaction-diffusion equations. The model also considers several biophysical forces which directly effect on the tumour growth dynamics. This modelling framework offers biologically realistic outcomes and also covers important criteria of the hallmarks of cancer which include oxygen and nutrient consumptions, micro-environmental heterogeneity, tumour cell proliferation by avoiding growth suppressor signals, replication of tumour cells at an abnormally faster rate, and resistance of apoptosis. The avascular tumour growth model is validated with immunohistochemistry and histopathology data. The outcome of the proposed model is very close to the range of the patient data, which concludes that the model is capable enough to mimic these complex biophysical phenomena.

A Model-Strengthened Imaging Biomarker for Survival Prediction in EGFR-Mutated Non-small-cell Lung Carcinoma Patients Treated with Tyrosine Kinase Inhibitors

Non-small-cell lung carcinoma is a frequent type of lung cancer with a bad prognosis. Depending on the stage and genomics, several therapeutical approaches are used. Tyrosine Kinase Inhibitors (TKI) may be successful for a time in the treatment of EGFR-mutated non-small cells lung carcinoma. Our objective is here to introduce a survival assessment as their efficacy in the long run is challenging to evaluate. The study includes 17 patients diagnosed with EGFR-mutated non-small cell lung cancer and exposed to an EGFR-targeting TKI with 3 computed tomography (CT) scans of the primary tumor (one before the TKI introduction and two after). An imaging biomarker based on evolution of texture heterogeneity between the first and the third exams is derived and computed from a mathematical model and patient data. Defining the overall survival as the time between the introduction of the TKI treatment and the patient death, we obtain a statistically significant correlation between the overall survival and our imaging marker ([Formula: see text]). Using the ROC curve, the patients are separated into two populations and the comparison of the survival curves is statistically significant ([Formula: see text]). The baseline exam seems to have a significant role in the prediction of response to TKI treatment. More precisely, our imaging biomarker defined using only the CT scan before the TKI introduction allows to determine a first classification of the population which is improved over time using the imaging marker as soon as more CT scans are available. This exploratory study leads us to think that it is possible to obtain a survival assessment using only few CT scans of the primary tumor.

Spatial mechanistic modeling for prediction of the growth of asymptomatic meningiomas

BACKGROUND AND OBJECTIVE

Mathematical modeling of tumor growth draws interest from the medical community as they have the potential to improve patients' care and the use of public health resources. The main objectives of this work are to model the growth of meningiomas - slow-growing benign tumors requiring extended imaging follow-up - and to predict tumor volume and shape at a later desired time using only two times examinations.

METHODS

We develop two variants of a 3D partial differential system of equations (PDE) which yield after a spatial integration systems of ordinary differential equations (ODE) that relate tumor volume with time. Estimation of models parameters is a crucial step to obtain a personalized model for a patient that can be used for descriptive or predictive purposes. As PDE and ODE systems share the same parameters, they are both estimated by fitting the ODE systems to the tumor volumes obtained from MRI examinations acquired at different times. A population approach allows to compensate for sparse sampling times and measurement uncertainties by constraining the variability of the parameters in the population.

RESULTS

Description capabilities of the models are investigated in 39 patients with benign asymptomatic meningiomas who had had at least three surveillance MRI examinations. The two models can fit to the data accurately and more realistically than a naive linear regression. Prediction performances are validated for 33 patients using a population approach. Mean relative errors in volume predictions are less than 10% with ODE systems versus 12.5% with the naive linear model using only two times examinations. Concerning the shape, the mean Sørensen-Dice coefficients are 85% with the PDE systems in a subset of 10 representative patients.

CONCLUSIONS

Our strategy - based on personalization of mathematical model - provides a good insight on meningioma growth and may help decide whether to extend the follow-up or to treat the tumor.

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.

Mathematical modeling of the proliferation gradient in multicellular tumor spheroids

MultiCellular Tumor Spheroids are 3D cell cultures that can accurately reproduce the behavior of solid tumors. It has been experimentally observed that large spheroids exhibit a decreasing gradient of proliferation from the periphery to the center of these multicellular 3D models: the proportion of proliferating cells is higher in the periphery while the non-proliferating quiescent cells increase in depth. In this paper, we propose to investigate the key mechanisms involved in the establishment of this gradient with a Partial Differential Equations model that mimics the experimental set-up of growing spheroids under different nutrients supply conditions. The model consists of mass balance equations on the two cell populations observed in the data: the proliferating cells and the quiescent cells. The spherical symmetry is used to rewrite the model in radial and relative coordinates. Thanks to a rigorous data postprocessing the model is then fit and compared quantitatively with the experimental quantification of the percentage of proliferating cells from EdU immunodetection on 2D spheroid cryosection images. The results of this calibration show that the proliferation gradient observed in spheroids can be quantitatively reproduced by our model.

Multi-Scale Modeling and Oxygen Impact on Tumor Temporal Evolution: Application on Rectal Cancer During Radiotherapy

We present a multi-scale approach of tumor modeling in order to predict its evolution during radiotherapy. Within this context we focus on three different scales of tumor modeling: microscopic (individual cells in a voxel), mesoscopic (population of cells in a voxel) and macroscopic (whole tumor), with transition interfaces between these three scales. At the cellular level, the description is based on phase transfer probabilities in the cellular cycle. At the mesoscopic scale we represent populations of cells according to different stages in a cell cycle. Finally, at the macroscopic scale, the tumor description is based on the use of FDG PET image voxels. These three scales exist naturally: biological data are collected at the macroscopic scale, but the pathological behavior of the tumor is based on an abnormal cell-cycle at the microscopic scale. On the other hand, the introduction of a mesoscopic scale is essential in order to reduce the gap between the two extreme, in terms of resolution, description levels. It also reduces the computational burden of simulating a large number of individual cells. As an application of the proposed multi-scale model, we simulate the effect of oxygen on tumor evolution during radiotherapy. Two consecutive FDG PET images of 17 rectal cancer patients undergoing radiotherapy are used to simulate the tumor evolution during treatment. The simulated results are compared with those obtained on a third FDG PET image acquired two weeks after the beginning of the treatment.

Tumor growth model of ductal carcinoma: from in situ phase to stroma invasion

This paper aims at modeling breast cancer transition from the in situ stage -when the tumor is confined to the duct- to the invasive phase. Such a transition occurs thanks to the degradation of the duct membrane under the action of specific enzymes so-called matrix metalloproteinases (MMPs). The model consists of advection-reaction equations that hold in the duct and in the surrounding tissue, in order to describe the proliferation and the necrosis of the cancer cells in each subdomain. The divergence of the velocity is given by the increase of the cell densities. Darcy law is imposed in order to close the system. The key-point of the modeling lies in the description of the transmission conditions across the duct. Nonlinear Kedem-Katchalsky transmission conditions across the membrane describe the discontinuity of the pressure as a linear function of the flux. These transmission conditions make it possible to describe the transition from the in situ stage to the invasive phase at the macroscopic level. More precisely, the membrane permeability increases with respect to the local concentration of MMPs. The cancer cells are no more confined to the duct and the tumor invades the surrounding tissue. The model is enriched by the description of nutrients concentration, tumor necrosis factors, and MMPs production. The mathematical model is implemented in a 3D C⁺⁺-code, which is based on well-adapted finite difference schemes on Cartesian grid. The membrane interface is described by a level-set, and the transmission conditions are precisely approached at the second order thanks to well-suited sharp stencils. Our continuous approach provides new significant insights in the macroscopic modeling of the breast cancer phase transition, due to the membrane degradation by MMP enzymes.

Are Tumor Cell Lineages Solely Shaped by Mechanical Forces?

This paper investigates cell proliferation dynamics in small tumor cell aggregates using an individual-based model (IBM). The simulation model is designed to study the morphology of the cell population and of the cell lineages as well as the impact of the orientation of the division plane on this morphology. Our IBM model is based on the hypothesis that cells are incompressible objects that grow in size and divide once a threshold size is reached, and that newly born cell adhere to the existing cell cluster. We performed comparisons between the simulation model and experimental data by using several statistical indicators. The results suggest that the emergence of particular morphologies can be explained by simple mechanical interactions.

Near-criticality underlies the behavior of early tumor growth

The controlling factors that underlie the growth of tumors have often been hard to identify because of the presence in this system of a large number of intracellular biochemical parameters. Here, we propose a simplifying framework to identify the key physical parameters that govern the early growth of tumors. We model growth by means of branching processes where cells of different types can divide and differentiate. First, using this process that has only one controlling parameter, we study a one cell type model and compute the probability for tumor survival and the time of tumor extinction. Second, we show that when cell death and cell division are perfectly balanced, stochastic effects dominate the growth dynamics and the system exhibits a near-critical behavior that resembles a second-order phase transition. We show, in this near-critical regime, that the time interval before tumor extinction is power-law distributed. Finally, we apply this branching formalism to infer, from experimental growth data, the number of different cell types present in the observed tumor.

Systems Medicine: The Future of Medical Genomics, Healthcare, and Wellness

Recent advances in genomics have led to the rapid and relatively inexpensive collection of patient molecular data including multiple types of omics data. The integration of these data with clinical measurements has the potential to impact on our understanding of the molecular basis of disease and on disease management. Systems medicine is an approach to understanding disease through an integration of large patient datasets. It offers the possibility for personalized strategies for healthcare through the development of a new taxonomy of disease. Advanced computing will be an important component in effectively implementing systems medicine. In this chapter we describe three computational challenges associated with systems medicine: disease subtype discovery using integrated datasets, obtaining a mechanistic understanding of disease, and the development of an informatics platform for the mining, analysis, and visualization of data emerging from translational medicine studies.

Proposal of a hybrid approach for tumor progression and tumor-induced angiogenesis

One of the main challenges in cancer modelling is to improve the knowledge of tumor progression in areas related to tumor growth, tumor-induced angiogenesis and targeted therapies efficacy. For this purpose, incorporate the expertise from applied mathematicians, biologists and physicians is highly desirable. Despite the existence of a very wide range of models, involving many stages in cancer progression, few models have been proposed to take into account all relevant processes in tumor progression, in particular the effect of systemic treatments and angiogenesis. Composite biological experiments, both in vitro and in vivo, in addition with mathematical modelling can provide a better understanding of theses aspects. In this work we proposed that a rational experimental design associated with mathematical modelling could provide new insights into cancer progression. To accomplish this task, we reviewed mathematical models and cancer biology literature, describing in detail the basic principles of mathematical modelling. We also analyze how experimental data regarding tumor cells proliferation and angiogenesis in vitro may fit with mathematical modelling in order to reconstruct in vivo tumor evolution. Additionally, we explained the mathematical methodology in a comprehensible way in order to facilitate its future use by the scientific community.

Predictive Modeling of Drug Response in Non-Hodgkin's Lymphoma

We combine mathematical modeling with experiments in living mice to quantify the relative roles of intrinsic cellular vs. tissue-scale physiological contributors to chemotherapy drug resistance, which are difficult to understand solely through experimentation. Experiments in cell culture and in mice with drug-sensitive (Eµ-myc/Arf-/-) and drug-resistant (Eµ-myc/p53-/-) lymphoma cell lines were conducted to calibrate and validate a mechanistic mathematical model. Inputs to inform the model include tumor drug transport characteristics, such as blood volume fraction, average geometric mean blood vessel radius, drug diffusion penetration distance, and drug response in cell culture. Model results show that the drug response in mice, represented by the fraction of dead tumor volume, can be reliably predicted from these inputs. Hence, a proof-of-principle for predictive quantification of lymphoma drug therapy was established based on both cellular and tissue-scale physiological contributions. We further demonstrate that, if the in vitro cytotoxic response of a specific cancer cell line under chemotherapy is known, the model is then able to predict the treatment efficacy in vivo. Lastly, tissue blood volume fraction was determined to be the most sensitive model parameter and a primary contributor to drug resistance.

Systems biology of the microvasculature

The vascular network carries blood throughout the body, delivering oxygen to tissues and providing a pathway for communication between distant organs. The network is hierarchical and structured, but also dynamic, especially at the smaller scales. Remodeling of the microvasculature occurs in response to local changes in oxygen, gene expression, cell-cell communication, and chemical and mechanical stimuli from the microenvironment. These local changes occur as a result of physiological processes such as growth and exercise, as well as acute and chronic diseases including stroke, cancer, and diabetes, and pharmacological intervention. While the vasculature is an important therapeutic target in many diseases, drugs designed to inhibit vascular growth have achieved only limited success, and no drug has yet been approved to promote therapeutic vascular remodeling. This highlights the challenges involved in identifying appropriate therapeutic targets in a system as complex as the vasculature. Systems biology approaches provide a means to bridge current understanding of the vascular system, from detailed signaling dynamics measured in vitro and pre-clinical animal models of vascular disease, to a more complete picture of vascular regulation in vivo. This will translate to an improved ability to identify multi-component biomarkers for diagnosis, prognosis, and monitoring of therapy that are easy to measure in vivo, as well as better drug targets for specific disease states. In this review, we summarize systems biology approaches that have advanced our understanding of vascular function and dysfunction in vivo, with a focus on computational modeling.

Mathematical modelling of tumour volume dynamics in response to stereotactic ablative radiotherapy for non-small cell lung cancer

This paper reports a modelling study of tumour volume dynamics in response to stereotactic ablative radiotherapy (SABR). The main objective was to develop a model that is adequate to describe tumour volume change measured during SABR, and at the same time is not excessively complex as lacking support from clinical data. To this end, various modelling options were explored, and a rigorous statistical method, the Akaike information criterion, was used to help determine a trade-off between model accuracy and complexity. The models were calibrated to the data from 11 non-small cell lung cancer patients treated with SABR. The results showed that it is feasible to model the tumour volume dynamics during SABR, opening up the potential for using such models in a clinical environment in the future.

A methodological framework for drug development in rare diseases

INTRODUCTION

Developing orphan drugs is challenging because of their severity and the requisite for effective drugs. The small number of patients does not allow conducting adequately powered randomized controlled trials (RCTs). There is a need to develop high quality, ethically investigated, and appropriately authorized medicines, without subjecting patients to unnecessary trials.

AIMS AND OBJECTIVES

The main aim is to develop generalizable framework for choosing the best-performing drug/endpoint/design combinations in orphan drug development using an in silico modeling and trial simulation approach. The two main objectives were (i) to provide a global strategy for each disease to identify the most relevant drugs to be evaluated in specific patients during phase III RCTs, (ii) and select the best design for each drug to be used in future RCTs.

METHODOLOGICAL APPROACH

In silico phase III RCT simulation will be used to find the optimal trial design and was carried out in two steps: (i) statistical analysis of available clinical databases and (ii) integrative modeling that combines mathematical models for diseases with pharmacokinetic-pharmacodynamics models for the selected drug candidates.

CONCLUSION

There is a need to speed up the process of orphan drug development, develop new methods for translational research and personalized medicine, and contribute to European Medicines Agency guidelines. The approach presented here offers many perspectives in clinical trial conception.

Modeling of human tumor xenografts and dose rationale in oncology

Xenograft models are commonly used in oncology drug development. Although there are discussions about their ability to generate meaningful data for the translation from animal to humans, it appears that better data quality and better design of the preclinical experiments, together with appropriate data analysis approaches could make these data more informative for clinical development. An approach based on mathematical modeling is necessary to derive experiment-independent parameters which can be linked with clinically relevant endpoints. Moreover, the inclusion of biomarkers as predictors of efficacy is a key step towards a more general mechanism-based strategy.

Computational modeling of 3D tumor growth and angiogenesis for chemotherapy evaluation

Solid tumors develop abnormally at spatial and temporal scales, giving rise to biophysical barriers that impact anti-tumor chemotherapy. This may increase the expenditure and time for conventional drug pharmacokinetic and pharmacodynamic studies. In order to facilitate drug discovery, we propose a mathematical model that couples three-dimensional tumor growth and angiogenesis to simulate tumor progression for chemotherapy evaluation. This application-oriented model incorporates complex dynamical processes including cell- and vascular-mediated interstitial pressure, mass transport, angiogenesis, cell proliferation, and vessel maturation to model tumor progression through multiple stages including tumor initiation, avascular growth, and transition from avascular to vascular growth. Compared to pure mechanistic models, the proposed empirical methods are not only easy to conduct but can provide realistic predictions and calculations. A series of computational simulations were conducted to demonstrate the advantages of the proposed comprehensive model. The computational simulation results suggest that solid tumor geometry is related to the interstitial pressure, such that tumors with high interstitial pressure are more likely to develop dendritic structures than those with low interstitial pressure.

A systems biology view of blood vessel growth and remodelling

Blood travels throughout the body in an extensive network of vessels – arteries, veins and capillaries. This vascular network is not static, but instead dynamically remodels in response to stimuli from cells in the nearby tissue. In particular, the smallest vessels – arterioles, venules and capillaries – can be extended, expanded or pruned, in response to exercise, ischaemic events, pharmacological interventions, or other physiological and pathophysiological events. In this review, we describe the multi-step morphogenic process of angiogenesis – the sprouting of new blood vessels – and the stability of vascular networks in vivo. In particular, we review the known interactions between endothelial cells and the various blood cells and plasma components they convey. We describe progress that has been made in applying computational modelling, quantitative biology and high-throughput experimentation to the angiogenesis process.

In vivo mathematical modeling of tumor growth from imaging data: soon to come in the future?

The future challenges in oncology imaging are to assess the response to treatment even earlier. As an addition to functional imaging, mathematical modeling based on the imaging is an alternative, cross-disciplinary area of development. Modeling was developed in oncology not only in order to understand and predict tumor growth, but also to anticipate the effects of targeted and untargeted therapies. A very wide range of these models exist, involving many stages in the progression of tumors. Few models, however, have been proposed to reproduce in vivo tumor growth because of the complexity of the mechanisms involved. Morphological imaging combined with "spatial" models appears to perform well although functioning imaging could still provide further information on metabolism and the micro-architecture. The combination of imaging and modeling can resolve complex problems and describe many facets of tumor growth or response to treatment. It is now possible to consider its clinical use in the medium term. This review describes the basic principles of mathematical modeling and describes the advantages, limitations and future prospects for this in vivo approach based on imaging data.

Advances in the formation, use and understanding of multi-cellular spheroids

INTRODUCTION

Developing in vitro models for studying cell biology and cell physiology is of great importance to the fields of biotechnology, cancer research, drug discovery, toxicity testing, as well as the emerging fields of tissue engineering and regenerative medicine. Traditional two-dimensional (2D) methods of mammalian cell culture have several limitations and it is increasingly recognized that cells grown in a three-dimensional (3D) environment more closely represent normal cellular function due to the increased cell-to-cell interactions, and by mimicking the in vivo architecture of natural organs and tissues.

AREAS COVERED

In this review, we discuss the methods to form 3D multi-cellular spheroids, the advantages and limitations of these methods, and assays used to characterize the function of spheroids. The use of spheroids has led to many advances in basic cell sciences, including understanding cancer cell interactions, creating models for drug discovery and cancer metastasis, and they are being investigated as basic units for engineering tissue constructs. As so, this review will focus on contributions made to each of these fields using spheroid models.

EXPERT OPINION

Multi-cellular spheroids are rich in biological content and mimic better the in vivo environment than 2D cell culture. New technologies to form and analyze spheroids are rapidly increasing their adoption and expanding their applications.

Mathematical and statistical modeling in cancer systems biology

Cancer is a major health problem with high mortality rates. In the post-genome era, investigators have access to massive amounts of rapidly accumulating high-throughput data in publicly available databases, some of which are exclusively devoted to housing Cancer data. However, data interpretation efforts have not kept pace with data collection, and gained knowledge is not necessarily translating into better diagnoses and treatments. A fundamental problem is to integrate and interpret data to further our understanding in Cancer Systems Biology. Viewing cancer as a network provides insights into the complex mechanisms underlying the disease. Mathematical and statistical models provide an avenue for cancer network modeling. In this article, we review two widely used modeling paradigms: deterministic metabolic models and statistical graphical models. The strength of these approaches lies in their flexibility and predictive power. Once a model has been validated, it can be used to make predictions and generate hypotheses. We describe a number of diverse applications to Cancer Biology, including, the system-wide effects of drug-treatments, disease prognosis, tumor classification, forecasting treatment outcomes, and survival predictions.

Integrating intracellular dynamics using CompuCell3D and Bionetsolver: applications to multiscale modelling of cancer cell growth and invasion

In this paper we present a multiscale, individual-based simulation environment that integrates CompuCell3D for lattice-based modelling on the cellular level and Bionetsolver for intracellular modelling. CompuCell3D or CC3D provides an implementation of the lattice-based Cellular Potts Model or CPM (also known as the Glazier-Graner-Hogeweg or GGH model) and a Monte Carlo method based on the metropolis algorithm for system evolution. The integration of CC3D for cellular systems with Bionetsolver for subcellular systems enables us to develop a multiscale mathematical model and to study the evolution of cell behaviour due to the dynamics inside of the cells, capturing aspects of cell behaviour and interaction that is not possible using continuum approaches. We then apply this multiscale modelling technique to a model of cancer growth and invasion, based on a previously published model of Ramis-Conde et al. (2008) where individual cell behaviour is driven by a molecular network describing the dynamics of E-cadherin and β-catenin. In this model, which we refer to as the centre-based model, an alternative individual-based modelling technique was used, namely, a lattice-free approach. In many respects, the GGH or CPM methodology and the approach of the centre-based model have the same overall goal, that is to mimic behaviours and interactions of biological cells. Although the mathematical foundations and computational implementations of the two approaches are very different, the results of the presented simulations are compatible with each other, suggesting that by using individual-based approaches we can formulate a natural way of describing complex multi-cell, multiscale models. The ability to easily reproduce results of one modelling approach using an alternative approach is also essential from a model cross-validation standpoint and also helps to identify any modelling artefacts specific to a given computational approach.

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.

Unraveling the complex regulatory relationships between metabolism and signal transduction in cancer

Cancer cells exhibit an altered metabolic phenotype, known as the Warburg effect, which is characterized by high rates of glucose uptake and glycolysis, even under aerobic conditions. The Warburg effect appears to be an intrinsic component of most cancers and there is evidence linking cancer progression to mutations, translocations, and alternative splicing of genes that directly code for or have downstream effects on key metabolic enzymes. Many of the same signaling pathways are routinely dysregulated in cancer and a number of important oncogenic signaling pathways play important regulatory roles in central carbon metabolism. Unraveling the complex regulatory relationship between cancer metabolism and signaling requires the application of systems biology approaches. Here we discuss computational approaches for modeling protein signal transduction and metabolism as well as how the regulatory relationship between these two important cellular processes can be combined into hybrid models.

Bayesian enrichment strategies for randomized discontinuation trials

We propose optimal choice of the design parameters for random discontinuation designs (RDD) using a Bayesian decision-theoretic approach. We consider applications of RDDs to oncology phase II studies evaluating activity of cytostatic agents. The design consists of two stages. The preliminary open-label stage treats all patients with the new agent and identifies a possibly sensitive subpopulation. The subsequent second stage randomizes, treats, follows, and compares outcomes among patients in the identified subgroup, with randomization to either the new or a control treatment. Several tuning parameters characterize the design: the number of patients in the trial, the duration of the preliminary stage, and the duration of follow-up after randomization. We define a probability model for tumor growth, specify a suitable utility function, and develop a computational procedure for selecting the optimal tuning parameters.

Cancer stem cells in solid tumors: is 'evading apoptosis' a hallmark of cancer?

Conventional wisdom has long held that once a cancer cell has developed it will inevitably progress to clinical disease. Updating this paradigm, it has more recently become apparent that the tumor interacts with its microenvironment and that some environmental bottlenecks, such as the angiogenic switch, must be overcome for the tumor to progress. In parallel, attraction has been drawn to the concept that there is a minority population of cells - the cancer stem cells - bestowed with the exclusive ability to self-renew and regenerate the tumor. With therapeutic targeting issues at stake, much attention has shifted to the identification of cancer stem cells, the thinking being that the remaining non-stem population, already fated to die, will play a negligible role in tumor development. In fact, the newly appreciated importance of intercellular interactions in cancer development also extends in a unique and unexpected way to interactions between the stem and non-stem compartments of the tumor. Here we discuss recent findings drawn from a hybrid mathematical-cellular automaton model that simulates growth of a heterogeneous solid tumor comprised of cancer stem cells and non-stem cancer cells. The model shows how the introduction of cell fate heterogeneity paradoxically influences the tumor growth dynamic in response to apoptosis, to reveal yet another bottleneck to tumor progression potentially exploitable for disease control.

Coupled mathematical model of tumorigenesis and angiogenesis in vascular tumours

OBJECTIVES

Mathematical models are useful for studying vascular and avascular tumours, because these allow for more logical experimental design and provide valuable insights into the underlying mechanisms of their growth and development. The processes of avascular tumour growth and the development of capillary networks through tumour-induced angiogenesis have already been extensively investigated, albeit separately. Despite the clinical significance of vascular tumours, few studies have combined these approaches to develop a single comprehensive growth and development model.

MATERIALS AND METHODS

We develop a continuum-based mathematical model of vascular tumour growth. In the model, angiogenesis is initiated through the release of angiogenic growth factors (AGFs) by cells in the hypoxic regions of the tumour. The nutrient concentration within the tumour reflects the influence of capillary growth and invasion induced by AGF.

RESULTS AND CONCLUSIONS

Parametric and sensitivity studies were performed to evaluate the influence of different model parameters on tumour growth and to identify the parameters with the most influence, which include the rates of proliferation, apoptosis and necrosis, as well as the diffusion of sprout tips and the size of the region affected by angiogenesis. An optimization was performed for values of the model parameters that resulted in the best agreement with published experimental data. The resulting model solution matched the experimental data with a high degree of correlation (r = 0.85).

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

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

Phenomenological modeling of tumor diameter growth based on a mixed effects model

Over the last few years, taking advantage of the linear kinetics of the tumor growth during the steady-state phase, tumor diameter-based rather than tumor volume-based models have been developed for the phenomenological modeling of tumor growth. In this study, we propose a new tumor diameter growth model characterizing early, late and steady-state treatment effects. Model parameters consist of growth rhythms, growth delays and time constants and are meaningful for biologists. Biological experiments provide in vivo longitudinal data. The latter are analyzed using a mixed effects model based on the new diameter growth function, to take into account inter-mouse variability and treatment factors. The relevance of the tumor growth mixed model is firstly assessed by analyzing the effects of three therapeutic strategies for cancer treatment (radiotherapy, concomitant radiochemotherapy and photodynamic therapy) administered on mice. Then, effects of the radiochemotherapy treatment duration are estimated within the mixed model. The results highlight the model suitability for analyzing therapeutic efficiency, comparing treatment responses and optimizing, when used in combination with optimal experiment design, anti-cancer treatment modalities.

Homeostatic competition drives tumor growth and metastasis nucleation

We propose a mechanism for tumor growth emphasizing the role of homeostatic regulation and tissue stability. We show that competition between surface and bulk effects leads to the existence of a critical size that must be overcome by metastases to reach macroscopic sizes. This property can qualitatively explain the observed size distributions of metastases, while size-independent growth rates cannot account for clinical and experimental data. In addition, it potentially explains the observed preferential growth of metastases on tissue surfaces and membranes such as the pleural and peritoneal layers, suggests a mechanism underlying the seed and soil hypothesis introduced by Stephen Paget in 1889, and yields realistic values for metastatic inefficiency. We propose a number of key experiments to test these concepts. The homeostatic pressure as introduced in this work could constitute a quantitative, experimentally accessible measure for the metastatic potential of early malignant growths.

Nanovehicular intracellular delivery systems

This article provides an overview of principles and barriers relevant to intracellular drug and gene transport, accumulation and retention (collectively called as drug delivery) by means of nanovehicles (NV). The aim is to deliver a cargo to a particular intracellular site, if possible, to exert a local action. Some of the principles discussed in this article apply to noncolloidal drugs that are not permeable to the plasma membrane or to the blood-brain barrier. NV are defined as a wide range of nanosized particles leading to colloidal objects which are capable of entering cells and tissues and delivering a cargo intracelullarly. Different localization and targeting means are discussed. Limited discussion on pharmacokinetics and pharmacodynamics is also presented. NVs are contrasted to micro-delivery and current nanotechnologies which are already in commercial use. Newer developments in NV technologies are outlined and future applications are stressed. We also briefly review the existing modeling tools and approaches to quantitatively describe the behavior of targeted NV within the vascular and tumor compartments, an area of particular importance. While we list "elementary" phenomena related to different level of complexity of delivery to cancer, we also stress importance of multi-scale modeling and bottom-up systems biology approach.

Modelling methodology in physiopathology

Diseases are complex systems. Modelling them, i.e. systems physiopathology, is a quite demanding, complicated, multidimensional, multiscale process. As such, in order to achieve the goal of the model and further to optimise a rather-time and resource-consuming process, a relevant and easy to practice methodology is required. It includes guidance for validation. Also, the model development should be managed as a complicated process, along a strategy which has been elaborated in the beginning. It should be flexible enough to meet every case. A model is a representation of the available knowledge. All available knowledge does not have the same level of evidence and, further, there is a large variability of the values of all parameters (e.g. affinity constant or ionic current) across the literature. In addition, in a complex biological system there are always values lacking for a few or sometimes many parameters. All these three aspects are sources of uncertainty on the range of validity of the models and raise unsolved problems for designing a relevant model. Tools and techniques for integrating the parameter range of experimental values, level of evidence and missing data are needed.