Towards whole-organ modelling of tumour growth

A bio-lattice deep learning framework for modeling discrete biological materials

Biological tissues dynamically adapt their mechanical properties at the microscale in response to stimuli, often governed by discrete interacting mechanisms that dictate the material's behavior at the macroscopic scale. An approach to model the discrete nature of these elemental units is the Lattice Spring Modeling (LSM). However, the interactions in biological matter can present a high degree of complexity and heterogeneity at the macroscale, posing a computational challenge in multiscale modeling. In this work, we propose a novel machine learning-based multiscale framework that integrates deep neural networks (DNNs), the finite element method (FEM), and a LSM-inspired microstructure description to investigate the behavior of discrete, spatially heterogeneous materials. We develop a versatile, assumption-free lattice framework for interacting discrete units, and derive a consistent multiscale connection with our FEM implementation. A single DNN is trained to learn the constitutive equations of various particle configurations and boundary conditions, enabling rapid response predictions of heterogeneous biological tissues. We demonstrate the effectiveness of our approach with extensive testing, starting with benchmark cases and progressively increasing the complexity of the microstructures. We explored materials ranging from soft to hard inclusions, then combined them to form a macroscopically homogeneous material, a gradient-varying polycrystalline solid, and fully randomized configurations. Our results show that the model accurately captures the material response across these spatially varying structures.

Image-guided patient-specific prediction of interstitial fluid flow and drug transport in solid tumors

Tumor fluid dynamics and drug delivery simulations in solid tumors are highly relevant topics in clinical oncology. The current study introduces a novel method combining computational fluid dynamics (CFD) modeling, quantitative magnetic resonance imaging (MRI; including dynamic contrast-enhanced (DCE) MRI and diffusion-weighted (DW) MRI), and a novel ex-vivo protocol to generate patient-specific models of solid tumors in four patients with peritoneal metastases. DCE-MRI data were analyzed using the extended Tofts model to estimate the spatial distribution of tumor capillary permeability using the Ktrans parameter. DW-MRI data analysis provided a 3D representation of drug diffusivity, and DW-MRI coupled to an ex-vivo measurement protocol informed the spatial heterogeneity of the hydraulic conductivity of tumor tissue. The patient-specific data were subsequently incorporated into a computational fluid dynamics (CFD) model to simulate individualized tumor perfusion and drug transport maps. The results on interstitial fluid flow demonstrated noticeable heterogeneity of interstitial fluid pressure and velocity within the tumor, along with heterogeneous drug penetration profiles among different tumors, even with a similar drug administration regimen.

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.

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.

A rigid body framework for multicellular modeling

Off-lattice models are a well-established approach in multicellular modeling, where cells are represented as points that are free to move in space. The representation of cells as point objects is useful in a wide range of settings, particularly when large populations are involved; however, a purely point-based representation is not naturally equipped to deal with objects that have length, such as cell boundaries or external membranes. Here we introduce an off-lattice modeling framework that exploits rigid body mechanics to represent objects using a collection of conjoined one-dimensional edges in a viscosity-dominated system. This framework can be used to represent cells as free moving polygons, to allow epithelial layers to smoothly interact with themselves, to model rod-shaped cells such as bacteria and to robustly represent membranes. We demonstrate that this approach offers solutions to the problems that limit the scope of current off-lattice multicellular models.

Multiscale modeling of tumor growth and angiogenesis: Evaluation of tumor-targeted therapy

The dynamics of tumor growth and associated events cover multiple time and spatial scales, generally including extracellular, cellular and intracellular modifications. The main goal of this study is to model the biological and physical behavior of tumor evolution in presence of normal healthy tissue, considering a variety of events involved in the process. These include hyper and hypoactivation of signaling pathways during tumor growth, vessels' growth, intratumoral vascularization and competition of cancer cells with healthy host tissue. The work addresses two distinctive phases in tumor development-the avascular and vascular phases-and in each stage two cases are considered-with and without normal healthy cells. The tumor growth rate increases considerably as closed vessel loops (anastomoses) form around the tumor cells resulting from tumor induced vascularization. When taking into account the host tissue around the tumor, the results show that competition between normal cells and cancer cells leads to the formation of a hypoxic tumor core within a relatively short period of time. Moreover, a dense intratumoral vascular network is formed throughout the entire lesion as a sign of a high malignancy grade, which is consistent with reported experimental data for several types of solid carcinomas. In comparison with other mathematical models of tumor development, in this work we introduce a multiscale simulation that models the cellular interactions and cell behavior as a consequence of the activation of oncogenes and deactivation of gene signaling pathways within each cell. Simulating a therapy that blocks relevant signaling pathways results in the prevention of further tumor growth and leads to an expressive decrease in its size (82% in the simulation).

Matrix adhesion and remodeling diversifies modes of cancer invasion across spatial scales

The metastasis of malignant epithelial tumors begins with the egress of transformed cells from the confines of their basement membrane (BM) to their surrounding collagen-rich stroma. Invasion can be morphologically diverse: when breast cancer cells are separately cultured within BM-like matrix, collagen I (Coll I), or a combination of both, they exhibit collective-, dispersed mesenchymal-, and a mixed collective-dispersed (multimodal)- invasion, respectively. In this paper, we asked how distinct these invasive modes are with respect to the cellular and microenvironmental cues that drive them. A rigorous computational exploration of invasion was performed within an experimentally motivated Cellular Potts-based modeling environment. The model comprised of adhesive interactions between cancer cells, BM- and Coll I-like extracellular matrix (ECM), and reaction-diffusion-based remodeling of ECM. The model outputs were parameters cognate to dispersed- and collective- invasion. A clustering analysis of the output distribution curated through a careful examination of subsumed phenotypes suggested at least four distinct invasive states: dispersed, papillary-collective, bulk-collective, and multimodal, in addition to an indolent/non-invasive state. Mapping input values to specific output clusters suggested that each of these invasive states are specified by distinct input signatures of proliferation, adhesion and ECM remodeling. In addition, specific input perturbations allowed transitions between the clusters and revealed the variation in the robustness between the invasive states. Our systems-level approach proffers quantitative insights into how the diversity in ECM microenvironments may steer invasion into diverse phenotypic modes during early dissemination of breast cancer and contributes to tumor heterogeneity.

A Mathematical Model to Study the Effectiveness of Some of the Strategies Adopted in Curtailing the Spread of COVID-19

In this paper, we developed a model that suggests the use of robots in identifying COVID-19-positive patients and which studied the effectiveness of the government policy of prohibiting migration of individuals into their countries especially from those countries that were known to have COVID-19 epidemic. Two compartmental models consisting of two equations each were constructed. The models studied the use of robots for the identification of COVID-19-positive patients. The effect of migration ban strategy was also studied. Four biologically meaningful equilibrium points were found. Their local stability analysis was also carried out. Numerical simulations were carried out, and the most effective strategy to curtail the spread of the disease was shown.

Towards personalized computational oncology: from spatial models of tumour spheroids, to organoids, to tissues

A main goal of mathematical and computational oncology is to develop quantitative tools to determine the most effective therapies for each individual patient. This involves predicting the right drug to be administered at the right time and at the right dose. Such an approach is known as precision medicine. Mathematical modelling can play an invaluable role in the development of such therapeutic strategies, since it allows for relatively fast, efficient and inexpensive simulations of a large number of treatment schedules in order to find the most effective. This review is a survey of mathematical models that explicitly take into account the spatial architecture of three-dimensional tumours and address tumour development, progression and response to treatments. In particular, we discuss models of epithelial acini, multicellular spheroids, normal and tumour spheroids and organoids, and multi-component tissues. Our intent is to showcase how these in silico models can be applied to patient-specific data to assess which therapeutic strategies will be the most efficient. We also present the concept of virtual clinical trials that integrate standard-of-care patient data, medical imaging, organ-on-chip experiments and computational models to determine personalized medical treatment strategies.

Simultaneous Model Selection and Model Calibration for the Proliferation of Tumor and Normal Cells During In Vitro Chemotherapy Experiments

In vitro experiments were conducted in this work to analyze the proliferation of tumor (DU-145) and normal (macrophage RAW 264.7) cells under the influence of a chemotherapeutic drug (doxorubicin). Approximate Bayesian Computation (ABC) was used to select among four competing models to represent the number of cells and to estimate the model parameters, based on the experimental data. For one case, the selected model was validated in a replicated experiment, through the solution of a state estimation problem with a particle filter algorithm, thus demonstrating the robustness of the ABC procedure used in this work.

Spatial moment dynamics for collective cell movement incorporating a neighbour-dependent directional bias

The ability of cells to undergo collective movement plays a fundamental role in tissue repair, development and cancer. Interactions occurring at the level of individual cells may lead to the development of spatial structure which will affect the dynamics of migrating cells at a population level. Models that try to predict population-level behaviour often take a mean-field approach, which assumes that individuals interact with one another in proportion to their average density and ignores the presence of any small-scale spatial structure. In this work, we develop a lattice-free individual-based model (IBM) that uses random walk theory to model the stochastic interactions occurring at the scale of individual migrating cells. We incorporate a mechanism for local directional bias such that an individual's direction of movement is dependent on the degree of cell crowding in its neighbourhood. As an alternative to the mean-field approach, we also employ spatial moment theory to develop a population-level model which accounts for spatial structure and predicts how these individual-level interactions propagate to the scale of the whole population. The IBM is used to derive an equation for dynamics of the second spatial moment (the average density of pairs of cells) which incorporates the neighbour-dependent directional bias, and we solve this numerically for a spatially homogeneous case.

Proteolytic and non-proteolytic regulation of collective cell invasion: tuning by ECM density and organization

Cancer cells manoeuvre through extracellular matrices (ECMs) using different invasion modes, including single cell and collective cell invasion. These modes rely on MMP-driven ECM proteolysis to make space for cells to move. How cancer-associated alterations in ECM influence the mode of invasion remains unclear. Further, the sensitivity of the two invasion modes to MMP dynamics remains unexplored. In this paper, we address these open questions using a multiscale hybrid computational model combining ECM density-dependent MMP secretion, MMP diffusion, ECM degradation by MMP and active cell motility. Our results demonstrate that in randomly aligned matrices, collective cell invasion is more efficient than single cell invasion. Although increase in MMP secretion rate enhances invasiveness independent of cell-cell adhesion, sustenance of collective invasion in dense matrices requires high MMP secretion rates. However, matrix alignment can sustain both single cell and collective cell invasion even without ECM proteolysis. Similar to our in-silico observations, increase in ECM density and MMP inhibition reduced migration of MCF-7 cells embedded in sandwich gels. Together, our results indicate that apart from cell intrinsic factors (i.e., high cell-cell adhesion and MMP secretion rates), ECM density and organization represent two important extrinsic parameters that govern collective cell invasion and invasion plasticity.

Spatial Metrics of Tumour Vascular Organisation Predict Radiation Efficacy in a Computational Model

Intratumoural heterogeneity is known to contribute to poor therapeutic response. Variations in oxygen tension in particular have been correlated with changes in radiation response in vitro and at the clinical scale with overall survival. Heterogeneity at the microscopic scale in tumour blood vessel architecture has been described, and is one source of the underlying variations in oxygen tension. We seek to determine whether histologic scale measures of the erratic distribution of blood vessels within a tumour can be used to predict differing radiation response. Using a two-dimensional hybrid cellular automaton model of tumour growth, we evaluate the effect of vessel distribution on cell survival outcomes of simulated radiation therapy. Using the standard equations for the oxygen enhancement ratio for cell survival probability under differing oxygen tensions, we calculate average radiation effect over a range of different vessel densities and organisations. We go on to quantify the vessel distribution heterogeneity and measure spatial organization using Ripley’s L function, a measure designed to detect deviations from complete spatial randomness. We find that under differing regimes of vessel density the correlation coefficient between the measure of spatial organization and radiation effect changes sign. This provides not only a useful way to understand the differences seen in radiation effect for tissues based on vessel architecture, but also an alternate explanation for the vessel normalization hypothesis.

In this paper we use a mathematical model, called a hybrid cellular automaton, to study the effect of different vessel distributions on radiation therapy outcomes at the cellular level. We show that the correlation between radiation outcome and spatial organization of vessels changes signs between relatively low and high vessel density. Specifically, that for relatively low vessel density, radiation efficacy is decreased when vessels are more homogeneously distributed, and the opposite is true, that radiation efficacy is improved, when vessel organisation is normalised in high densities. This result suggests an alteration to the vessel normalization hypothesis which states that normalisation of vascular beds should improve radio- and chemo-therapeutic response, but has failed to be validated in clinical studies. In this alteration, we show that Ripley’s L function allows discrimination between vascular architectures in different density regimes in which the standard hypothesis holds and does not hold. Further, we find that this information can be used to augment quantitative histologic analysis of tumours to aid radiation dose personalisation.

Spatio-Temporal Dynamics of Hypoxia during Radiotherapy

Tumour hypoxia plays a pivotal role in cancer therapy for most therapeutic approaches from radiotherapy to immunotherapy. The detailed and accurate knowledge of the oxygen distribution in a tumour is necessary in order to determine the right treatment strategy. Still, due to the limited spatial and temporal resolution of imaging methods as well as lacking fundamental understanding of internal oxygenation dynamics in tumours, the precise oxygen distribution map is rarely available for treatment planing. We employ an agent-based in silico tumour spheroid model in order to study the complex, localized and fast oxygen dynamics in tumour micro-regions which are induced by radiotherapy. A lattice-free, 3D, agent-based approach for cell representation is coupled with a high-resolution diffusion solver that includes a tissue density-dependent diffusion coefficient. This allows us to assess the space- and time-resolved reoxygenation response of a small subvolume of tumour tissue in response to radiotherapy. In response to irradiation the tumour nodule exhibits characteristic reoxygenation and re-depletion dynamics which we resolve with high spatio-temporal resolution. The reoxygenation follows specific timings, which should be respected in treatment in order to maximise the use of the oxygen enhancement effects. Oxygen dynamics within the tumour create windows of opportunity for the use of adjuvant chemotherapeutica and hypoxia-activated drugs. Overall, we show that by using modelling it is possible to follow the oxygenation dynamics beyond common resolution limits and predict beneficial strategies for therapy and in vitro verification. Models of cell cycle and oxygen dynamics in tumours should in the future be combined with imaging techniques, to allow for a systematic experimental study of possible improved schedules and to ultimately extend the reach of oxygenation monitoring available in clinical treatment.

Estimation of Tumor Size Evolution Using Particle Filters

Cancer is characterized by the uncontrolled growth of cells with the ability of invading local organs and/or tissues and of spreading to other sites. Several kinds of mathematical models have been proposed in the literature, involving different levels of refinement, for the evolution of tumors and their interactions with chemotherapy drugs. In this article, we present the solution of a state estimation problem for tumor size evolution. A system of nonlinear ordinary differential equations is used as the state evolution model, which involves as state variables the numbers of tumor, normal and angiogenic cells, as well as the masses of the chemotherapy and anti-angiogenic drugs in the body. Measurements of the numbers of tumor and normal cells are considered available for the inverse analysis. Parameters appearing in the formulation of the state evolution model are treated as Gaussian random variables and their uncertainties are taken into account in the estimation of the state variables, by using an algorithm based on the auxiliary sampling importance resampling particle filter. Test cases are examined in the article dealing with a chemotherapy protocol for pancreatic cancer.

Multiscale modelling in immunology: a review

One of the greatest challenges in biomedicine is to get a unified view of observations made from the molecular up to the organism scale. Towards this goal, multiscale models have been highly instrumental in contexts such as the cardiovascular field, angiogenesis, neurosciences and tumour biology. More recently, such models are becoming an increasingly important resource to address immunological questions as well. Systematic mining of the literature in multiscale modelling led us to identify three main fields of immunological applications: host-virus interactions, inflammatory diseases and their treatment and development of multiscale simulation platforms for immunological research and for educational purposes. Here, we review the current developments in these directions, which illustrate that multiscale models can consistently integrate immunological data generated at several scales, and can be used to describe and optimize therapeutic treatments of complex immune diseases.

Strategies for efficient numerical implementation of hybrid multi-scale agent-based models to describe biological systems

Biologically related processes operate across multiple spatiotemporal scales. For computational modeling methodologies to mimic this biological complexity, individual scale models must be linked in ways that allow for dynamic exchange of information across scales. A powerful methodology is to combine a discrete modeling approach, agent-based models (ABMs), with continuum models to form hybrid models. Hybrid multi-scale ABMs have been used to simulate emergent responses of biological systems. Here, we review two aspects of hybrid multi-scale ABMs: linking individual scale models and efficiently solving the resulting model. We discuss the computational choices associated with aspects of linking individual scale models while simultaneously maintaining model tractability. We demonstrate implementations of existing numerical methods in the context of hybrid multi-scale ABMs. Using an example model describing Mycobacterium tuberculosis infection, we show relative computational speeds of various combinations of numerical methods. Efficient linking and solution of hybrid multi-scale ABMs is key to model portability, modularity, and their use in understanding biological phenomena at a systems level.

Web-based workflow planning platform supporting the design and execution of complex multiscale cancer models

Significant Virtual Physiological Human efforts and projects have been concerned with cancer modeling, especially in the European Commission Seventh Framework research program, with the ambitious goal to approach personalized cancer simulation based on patient-specific data and thereby optimize therapy decisions in the clinical setting. However, building realistic in silico predictive models targeting the clinical practice requires interactive, synergetic approaches to integrate the currently fragmented efforts emanating from the systems biology and computational oncology communities all around the globe. To further this goal, we propose an intelligent graphical workflow planning system that exploits the multiscale and modular nature of cancer and allows building complex cancer models by intuitively linking/interchanging highly specialized models. The system adopts and extends current standardization efforts, key tools, and infrastructure in view of building a pool of reliable and reproducible models capable of improving current therapies and demonstrating the potential for clinical translation of these technologies.

Modeling the role of TGF-β in regulation of the Th17 phenotype in the LPS-driven immune system

Airway exposure levels of lipopolysaccharide (LPS) are known to determine type I versus type II helper T cell induced experimental asthma. While low doses of LPS derive Th2 inflammatory responses, high (and/or intermediate) LPS levels induce Th1- or Th17-dominant responses. The present paper develops a mathematical model of the phenotypic switches among three Th phenotypes (Th1, Th2, and Th17) in response to various LPS levels. In the present work, we simplify the complex network of the interactions between cells and regulatory molecules. The model describes the nonlinear cross-talks between the IL-4/Th2 activities and a key regulatory molecule, transforming growth factor β (TGF-β), in response to high, intermediate, and low levels of LPS. The model characterizes development of three phenotypes (Th1, Th2, and Th17) and predicts the onset of a new phenotype, Th17, under the tight control of TGF-β. Analysis of the model illustrates the mono-, bi-, and oneway-switches in the key regulatory parameter sets in the absence or presence of time delays. The model also predicts coexistence of those phenotypes and Th1- or Th2-dominant immune responses in a spatial domain under various biochemical and bio-mechanical conditions in the microenvironment.

On-lattice agent-based simulation of populations of cells within the open-source Chaste framework

Over the years, agent-based models have been developed that combine cell division and reinforced random walks of cells on a regular lattice, reaction-diffusion equations for nutrients and growth factors; and ordinary differential equations for the subcellular networks regulating the cell cycle. When linked to a vascular layer, this multiple scale model framework has been applied to tumour growth and therapy. Here, we report on the creation of an agent-based multi-scale environment amalgamating the characteristics of these models within a Virtual Physiological Human (VPH) Exemplar Project. This project enables reuse, integration, expansion and sharing of the model and relevant data. The agent-based and reaction-diffusion parts of the multi-scale model have been implemented and are available for download as part of the latest public release of Chaste (Cancer, Heart and Soft Tissue Environment; http://www.cs.ox.ac.uk/chaste/), part of the VPH Toolkit (http://toolkit.vph-noe.eu/). The environment functionalities are verified against the original models, in addition to extra validation of all aspects of the code. In this work, we present the details of the implementation of the agent-based environment, including the system description, the conceptual model, the development of the simulation model and the processes of verification and validation of the simulation results. We explore the potential use of the environment by presenting exemplar applications of the 'what if' scenarios that can easily be studied in the environment. These examples relate to tumour growth, cellular competition for resources and tumour responses to hypoxia (low oxygen levels). We conclude our work by summarizing the future steps for the expansion of the current system.

Visualizing time-related data in biology, a review

Time is of the essence in biology as in so much else. For example, monitoring disease progression or the timing of developmental defects is important for the processes of drug discovery and therapy trials. Furthermore, an understanding of the basic dynamics of biological phenomena that are often strictly time regulated (e.g. circadian rhythms) is needed to make accurate inferences about the evolution of biological processes. Recent advances in technologies have enabled us to measure timing effects more accurately and in more detail. This has driven related advances in visualization and analysis tools that try to effectively exploit this data. Beyond timeline plots, notable attempts at more involved temporal interpretation have been made in recent years, but awareness of the available resources is still limited within the scientific community. Here, we review some advances in biological visualization of time-driven processes and consider how they aid data analysis and interpretation.

Validity of the Cauchy-Born rule applied to discrete cellular-scale models of biological tissues

The development of new models of biological tissues that consider cells in a discrete manner is becoming increasingly popular as an alternative to continuum methods based on partial differential equations, although formal relationships between the discrete and continuum frameworks remain to be established. For crystal mechanics, the discrete-to-continuum bridge is often made by assuming that local atom displacements can be mapped homogeneously from the mesoscale deformation gradient, an assumption known as the Cauchy-Born rule (CBR). Although the CBR does not hold exactly for noncrystalline materials, it may still be used as a first-order approximation for analytic calculations of effective stresses or strain energies. In this work, our goal is to investigate numerically the applicability of the CBR to two-dimensional cellular-scale models by assessing the mechanical behavior of model biological tissues, including crystalline (honeycomb) and noncrystalline reference states. The numerical procedure involves applying an affine deformation to the boundary cells and computing the quasistatic position of internal cells. The position of internal cells is then compared with the prediction of the CBR and an average deviation is calculated in the strain domain. For center-based cell models, we show that the CBR holds exactly when the deformation gradient is relatively small and the reference stress-free configuration is defined by a honeycomb lattice. We show further that the CBR may be used approximately when the reference state is perturbed from the honeycomb configuration. By contrast, for vertex-based cell models, a similar analysis reveals that the CBR does not provide a good representation of the tissue mechanics, even when the reference configuration is defined by a honeycomb lattice. The paper concludes with a discussion of the implications of these results for concurrent discrete and continuous modeling, adaptation of atom-to-continuum techniques to biological tissues, and model classification.

Cellular automata coupled with steady-state nutrient solution permit simulation of large-scale growth of tumours

We model complete growth of an avascular tumour by employing cellular automata for the growth of cells and steady-state equation to solve for nutrient concentrations. Our modelling and computer simulation results show that, in the case of a brain tumour, oxygen distribution in the tumour volume may be sufficiently described by a time-independent steady-state equation without losing the characteristics of a time-dependent diffusion equation. This makes the solution of oxygen concentration in the tumour volume computationally more efficient, thus enabling simulation of tumour growth on a large scale. We solve this steady-state equation using a central difference method. We take into account the composition of cells and intercellular adhesion in addition to processes involved in cell cycle--proliferation, quiescence, apoptosis, and necrosis--in the tumour model. More importantly, we consider cell mutation that gives rise to different phenotypes and therefore a tumour with heterogeneous population of cells. A new phenotype is probabilistically chosen and has the ability to survive at lower levels of nutrient concentration and reproduce faster. We show that heterogeneity of cells that compose a tumour leads to its irregular growth and that avascular growth is not supported for tumours of diameter above 18 mm. We compare results from our growth simulation with existing experimental data on Ehrlich ascites carcinoma and tumour spheroid cultures and show that our results are in good agreement with the experimental findings.

Optimization of vascular-targeting drugs in a computational model of tumor growth

A biophysical tool is introduced that seeks to provide a theoretical basis for helping drug design teams assess the most promising drug targets and design optimal treatment strategies. The tool is grounded in a previously validated computational model of the feedback that occurs between a growing tumor and the evolving vasculature. In this paper, the model is particularly used to explore the therapeutic effectiveness of two drugs that target the tumor vasculature: angiogenesis inhibitors (AIs) and vascular disrupting agents (VDAs). Using sensitivity analyses, the impact of VDA dosing parameters is explored, as is the effects of administering a VDA with an AI. Further, a stochastic optimization scheme is utilized to identify an optimal dosing schedule for treatment with an AI and a chemotherapeutic. The treatment regimen identified can successfully halt simulated tumor growth, even after the cessation of therapy.

On the evolution of morphogenetic models: mechano-chemical interactions and an integrated view of cell differentiation, growth, pattern formation and morphogenesis

In the 1950s, embryology was conceptualized as four relatively independent problems: cell differentiation, growth, pattern formation and morphogenesis. The mechanisms underlying the first three traditionally have been viewed as being chemical in nature, whereas those underlying morphogenesis have usually been discussed in terms of mechanics. Often, morphogenesis and its mechanical processes have been regarded as subordinate to chemical ones. However, a growing body of evidence indicates that the biomechanics of cells and tissues affect in striking ways those phenomena often thought of as mainly under the control of cell-cell signalling. This accumulation of data has led to a revival of the mechano-transduction concept in particular, and of complexity in general, causing us now to consider whether we should retain the traditional conceptualization of development. The researchers' semantic preferences for the terms 'patterning', 'pattern formation' or 'morphogenesis' can be used to describe three main 'schools of thought' which emerged in the late 1970s. In the 'molecular school', the term patterning is deeply tied to the positional information concept. In the 'chemical school', the term 'pattern formation' regularly implies reaction-diffusion models. In the 'mechanical school', the term 'morphogenesis' is more frequently used in relation to mechanical instabilities. Major differences among these three schools pertain to the concept of self-organization, and models can be classified as morphostatic or morphodynamic. Various examples illustrate the distorted picture that arises from the distinction among differentiation, growth, pattern formation and morphogenesis, based on the idea that the underlying mechanisms are respectively chemical or mechanical. Emerging quantitative approaches integrate the concepts and methods of complex sciences and emphasize the interplay between hierarchical levels of organization via mechano-chemical interactions. They draw upon recent improvements in mathematical and numerical morphogenetic models and upon considerable progress in collecting new quantitative data. This review highlights a variety of such models, which exhibit important advances, such as hybrid, stochastic and multiscale simulations.

Role for stromal heterogeneity in prostate tumorigenesis

Prostate cancer develops through a stochastic mechanism whereby precancerous lesions on occasion progress to multifocal adenocarcinoma. Analysis of human benign and cancer prostate tissues revealed heterogeneous loss of TGF-β signaling in the cancer-associated stromal fibroblastic cell compartment. To test the hypothesis that prostate cancer progression is dependent on the heterogeneous TGF-β responsive microenvironment, a tissue recombination experiment was designed in which the ratio of TGF-β responsive and nonresponsive stromal cells was varied. Although 100% TGF-β responsive stromal cells supported benign prostate growth and 100% TGF-β nonresponsive stromal cells resulted in precancerous lesions, only the mixture of TGF-β responsive and nonresponsive stromal cells resulted in adenocarcinoma. A computational model was used to resolve a mechanism of tumorigenic progression in which proliferation and invasion occur in two independent steps mediated by distinct stromally derived paracrine signals produced by TGF-β nonresponsive and responsive stromal cells. Complex spatial relationships of stromal and epithelial cells were incorporated into the model on the basis of experimental data. Informed by incorporation of experimentally derived spatial parameters for complex stromal-epithelial relationships, the computational model indicated ranges for the relative production of paracrine factors by each cell type and provided bounds for the diffusive range of the molecules. Because SDF-1 satisfied model predictions for an invasion-promoting paracrine factor, a more focused computational model was subsequently used to investigate whether SDF-1 was the invasion signal. Simulations replicating SDF-1 expression data revealed the requirement for cooperative SDF-1 expression, a prediction supported biologically by heterotypic stromal interleukin-1β signaling between fibroblastic cell populations. The cancer stromal field effect supports a functional role for the unaltered fibroblasts as a cooperative mediator of cancer progression.

Dissecting cancer through mathematics: from the cell to the animal model

This Timeline article charts progress in mathematical modelling of cancer over the past 50 years, highlighting the different theoretical approaches that have been used to dissect the disease and the insights that have arisen. Although most of this research was conducted with little involvement from experimentalists or clinicians, there are signs that the tide is turning and that increasing numbers of those involved in cancer research and mathematical modellers are recognizing that by working together they might more rapidly advance our understanding of cancer and improve its treatment.

From ABC genes to regulatory networks, epigenetic landscapes and flower morphogenesis: making biological sense of theoretical approaches

The ABC model postulates that expression combinations of three classes of genes (A, B and C) specify the four floral organs at early stages of flower development. This classic model provides a solid framework to study flower development and has been the foundation for multiple studies in different plant species, as well as for new evolutionary hypotheses. Nevertheless, it has been shown that in spite of being necessary, these three gene classes are not sufficient for flower organ specification. Rather, flower organ specification depends on complex interactions of several genes, and probably other non-genetic factors. Being useful to study systems of complex interactions, mathematical and computational models have enlightened the origin of the A, B and C stereotyped and robust expression patterns and the process of early flower morphogenesis. Here, we present a brief introduction to basic modeling concepts and techniques and review the results that these models have rendered for the particular case of the Arabidopsis thaliana flower organ specification. One of the main results is the uncovering of a robust functional module that is sufficient to recover the gene configurations characterizing flower organ primordia. Another key result is that the temporal sequence with which such gene configurations are attained may be recovered only by modeling the aforementioned functional module as a noisy or stochastic system. Finally, modeling approaches enable testable predictions regarding the role of non-genetic factors (noise, mechano-elastic forces, etc.) in development. These predictions, along with some perspectives for future work, are also reviewed and discussed.

Introduction of hypermatrix and operator notation into a discrete mathematics simulation model of malignant tumour response to therapeutic schemes in vivo. Some operator properties

The tremendous rate of accumulation of experimental and clinical knowledge pertaining to cancer dictates the development of a theoretical framework for the meaningful integration of such knowledge at all levels of biocomplexity. In this context our research group has developed and partly validated a number of spatiotemporal simulation models of in vivo tumour growth and in particular tumour response to several therapeutic schemes. Most of the modeling modules have been based on discrete mathematics and therefore have been formulated in terms of rather complex algorithms (e.g. in pseudocode and actual computer code). However, such lengthy algorithmic descriptions, although sufficient from the mathematical point of view, may render it difficult for an interested reader to readily identify the sequence of the very basic simulation operations that lie at the heart of the entire model. In order to both alleviate this problem and at the same time provide a bridge to symbolic mathematics, we propose the introduction of the notion of hypermatrix in conjunction with that of a discrete operator into the already developed models. Using a radiotherapy response simulation example we demonstrate how the entire model can be considered as the sequential application of a number of discrete operators to a hypermatrix corresponding to the dynamics of the anatomic area of interest. Subsequently, we investigate the operators’ commutativity and outline the “summarize and jump” strategy aiming at efficiently and realistically address multilevel biological problems such as cancer. In order to clarify the actual effect of the composite discrete operator we present further simulation results which are in agreement with the outcome of the clinical study RTOG 83–02, thus strengthening the reliability of the model developed.

3D multi-cell simulation of tumor growth and angiogenesis

We present a 3D multi-cell simulation of a generic simplification of vascular tumor growth which can be easily extended and adapted to describe more specific vascular tumor types and host tissues. Initially, tumor cells proliferate as they take up the oxygen which the pre-existing vasculature supplies. The tumor grows exponentially. When the oxygen level drops below a threshold, the tumor cells become hypoxic and start secreting pro-angiogenic factors. At this stage, the tumor reaches a maximum diameter characteristic of an avascular tumor spheroid. The endothelial cells in the pre-existing vasculature respond to the pro-angiogenic factors both by chemotaxing towards higher concentrations of pro-angiogenic factors and by forming new blood vessels via angiogenesis. The tumor-induced vasculature increases the growth rate of the resulting vascularized solid tumor compared to an avascular tumor, allowing the tumor to grow beyond the spheroid in these linear-growth phases. First, in the linear-spherical phase of growth, the tumor remains spherical while its volume increases. Second, in the linear-cylindrical phase of growth the tumor elongates into a cylinder. Finally, in the linear-sheet phase of growth, tumor growth accelerates as the tumor changes from cylindrical to paddle-shaped. Substantial periods during which the tumor grows slowly or not at all separate the exponential from the linear-spherical and the linear-spherical from the linear-cylindrical growth phases. In contrast to other simulations in which avascular tumors remain spherical, our simulated avascular tumors form cylinders following the blood vessels, leading to a different distribution of hypoxic cells within the tumor. Our simulations cover time periods which are long enough to produce a range of biologically reasonable complex morphologies, allowing us to study how tumor-induced angiogenesis affects the growth rate, size and morphology of simulated tumors.

The influence of P-glycoprotein expression and its inhibitors on the distribution of doxorubicin in breast tumors

Background

Anti-cancer drugs access solid tumors via blood vessels, and must penetrate tumor tissue to reach all cancer cells. Previous studies have demonstrated steep gradients of decreasing doxorubicin fluorescence with increasing distance from blood vessels, such that many tumor cells are not exposed to drug. Studies using multilayered cell cultures show that increased P-glycoprotein (PgP) is associated with better penetration of doxorubicin, while PgP inhibitors decrease drug penetration in tumor tissue. Here we evaluate the effect of PgP expression on doxorubicin distribution in vivo.

Methods

Mice bearing tumor sublines with either high or low expression of PgP were treated with doxorubicin, with or without pre-treatment with the PgP inhibitors verapamil or PSC 833. The distribution of doxorubicin in relation to tumor blood vessels was quantified using immunofluorescence.

Results

Our results indicate greater uptake of doxorubicin by cells near blood vessels in wild type as compared to PgP-overexpressing tumors, and pre-treatment with verapamil or PSC 833 increased uptake in PgP-overexpressing tumors. However, there were steeper gradients of decreasing doxorubicin fluorescence in wild-type tumors compared to PgP overexpressing tumors, and treatment of PgP overexpressing tumors with PgP inhibitors led to steeper gradients and greater heterogeneity in the distribution of doxorubicin.

Conclusion

PgP inhibitors increase uptake of doxorubicin in cells close to blood vessels, have little effect on drug uptake into cells at intermediate distances, and might have a paradoxical effect to decrease doxorubicin uptake into distal cells. This effect probably contributes to the limited success of PgP inhibitors in clinical trials.

An integrative computational model for intestinal tissue renewal

OBJECTIVES

The luminal surface of the gut is lined with a monolayer of epithelial cells that acts as a nutrient absorptive engine and protective barrier. To maintain its integrity and functionality, the epithelium is renewed every few days. Theoretical models are powerful tools that can be used to test hypotheses concerning the regulation of this renewal process, to investigate how its dysfunction can lead to loss of homeostasis and neoplasia, and to identify potential therapeutic interventions. Here we propose a new multiscale model for crypt dynamics that links phenomena occurring at the subcellular, cellular and tissue levels of organisation.

METHODS

At the subcellular level, deterministic models characterise molecular networks, such as cell-cycle control and Wnt signalling. The output of these models determines the behaviour of each epithelial cell in response to intra-, inter- and extracellular cues. The modular nature of the model enables us to easily modify individual assumptions and analyse their effects on the system as a whole.

RESULTS

We perform virtual microdissection and labelling-index experiments, evaluate the impact of various model extensions, obtain new insight into clonal expansion in the crypt, and compare our predictions with recent mitochondrial DNA mutation data.

CONCLUSIONS

We demonstrate that relaxing the assumption that stem-cell positions are fixed enables clonal expansion and niche succession to occur. We also predict that the presence of extracellular factors near the base of the crypt alone suffices to explain the observed spatial variation in nuclear beta-catenin levels along the crypt axis.

In silico biology of bone modelling and remodelling: regeneration

Bone regeneration is the process whereby bone is able to (scarlessly) repair itself from trauma, such as fractures or implant placement. Despite extensive experimental research, many of the mechanisms involved still remain to be elucidated. Over the last decade, many mathematical models have been established to investigate the regeneration process in silico. The first models considered only the influence of the mechanical environment as a regulator of the healing process. These models were followed by the development of bioregulatory models where mechanics was neglected and regeneration was regulated only by biological stimuli such as growth factors. The most recent mathematical models couple the influences of both biological and mechanical stimuli. Examples are given to illustrate the added value of mathematical regeneration research, specifically in the in silico design of treatment strategies for non-unions. Drawbacks of the current continuum-type models, together with possible solutions in extending the models towards other time and length scales are discussed. Finally, the demands for dedicated and more quantitative experimental research are presented.

Mathematical and computational approaches can complement experimental studies of host-pathogen interactions

In addition to traditional and novel experimental approaches to study host-pathogen interactions, mathematical and computer modelling have recently been applied to address open questions in this area. These modelling tools not only offer an additional avenue for exploring disease dynamics at multiple biological scales, but also complement and extend knowledge gained via experimental tools. In this review, we outline four examples where modelling has complemented current experimental techniques in a way that can or has already pushed our knowledge of host-pathogen dynamics forward. Two of the modelling approaches presented go hand in hand with articles in this issue exploring fluorescence resonance energy transfer and two-photon intravital microscopy. Two others explore virtual or 'in silico' deletion and depletion as well as a new method to understand and guide studies in genetic epidemiology. In each of these examples, the complementary nature of modelling and experiment is discussed. We further note that multi-scale modelling may allow us to integrate information across length (molecular, cellular, tissue, organism, population) and time (e.g. seconds to lifetimes). In sum, when combined, these compatible approaches offer new opportunities for understanding host-pathogen interactions.

A computational framework for modelling solid tumour growth

The biology of cancer is a complex interplay of many underlying processes, taking place at different scales both in space and time. A variety of theoretical models have been developed, which enable one to study certain components of the cancerous growth process. However, most previous approaches only focus on specific aspects of tumour development, largely ignoring the influence of the evolving tumour environment. In this paper, we present an integrative framework to simulate tumour growth, including those model components that are considered to be of major importance. We start by addressing issues at the tissue level, where the phenomena are modelled as continuum partial differential equations. We extend this model with relevant components at the cellular or even sub-cellular level in a vertical fashion. We present an implementation of this framework, covering the major processes and treat the mechanical deformation due to growth, the biochemical response to hypoxia, blood flow, oxygenation and the explicit development of a vascular system in a coupled way. The results demonstrate the feasibility of the approach and its applicability to in silico studies of the influence of different treatment strategies (like the usage of novel anti-cancer drugs) for more effective therapy design.

Modeling cancer: integration of "omics" information in dynamic systems

The last 10 years have seen the rise of many technologies that produce an unprecedented amount of genome-scale data from many organisms. Although the research community has been successful in exploring these data, many challenges still persist. One of them is the effective integration of such data sets directly into approaches based on mathematical modeling of biological systems. Applications in cancer are a good example. The bridge between information and modeling in cancer can be achieved by two major types of complementary strategies. First, there is a bottom-up approach, in which data generates information about structure and relationship between components of a given system. In addition, there is a top-down approach, where cybernetic and systems-theoretical knowledge are used to create models that describe mechanisms and dynamics of the system. These approaches can also be linked to yield multi-scale models combining detailed mechanism and wide biological scope. Here we give an overall picture of this field and discuss possible strategies to approach the major challenges ahead.

Elucidating the interactions between the adhesive and transcriptional functions of -catenin in normal and cancerous cells

Wnt signalling is involved in a wide range of physiological and pathological processes. The presence of an extracellular Wnt stimulus induces cytoplasmic stabilisation and nuclear translocation of beta-catenin, a protein that also plays an essential role in cadherin-mediated adhesion. Two main hypotheses have been proposed concerning the balance between beta-catenin's adhesive and transcriptional functions: either beta-catenin's fate is determined by competition between its binding partners, or Wnt induces folding of beta-catenin into a conformation allocated preferentially to transcription. The experimental data supporting each hypotheses remain inconclusive. In this paper we present a new mathematical model of the Wnt pathway that incorporates beta-catenin's dual function. We use this model to carry out a series of in silico experiments and compare the behaviour of systems governed by each hypothesis. Our analytical results and model simulations provide further insight into the current understanding of Wnt signalling and, in particular, reveal differences in the response of the two modes of interaction between adhesion and signalling in certain in silico settings. We also exploit our model to investigate the impact of the mutations most commonly observed in human colorectal cancer. Simulations show that the amount of functional APC required to maintain a normal phenotype increases with increasing strength of the Wnt signal, a result which illustrates that the environment can substantially influence both tumour initiation and phenotype.

Dynamics of tumor growth and combination of anti-angiogenic and cytotoxic therapies

Tumors cannot grow beyond a certain size (about 1-2 mm in diameter) through simple diffusion of oxygen and other essential nutrients into the tumor. Angiogenesis, the formation of blood vessels from pre-existing vessels, is a crucial and observed step, through which a tumor obtains its own blood supply. Thus, strategies that interfere with the development of this tumor vasculature, known as anti-angiogenic therapy, represent a novel approach to controlling tumor growth. Several pre-clinical studies have suggested that currently available angiogenesis inhibitors are unlikely to yield significant sustained improvements in tumor control on their own, but rather will need to be used in combination with conventional treatments to achieve maximal benefit. Optimal sequencing of anti-angiogenic treatment and radiotherapy or chemotherapy is essential to the success of these combined treatment strategies. Hence, a major challenge to mathematical modeling and computer simulations is to find appropriate dosages, schedules and sequencing of combination therapies to control or eliminate tumor growth. Here, we present a mathematical model that incorporates tumor cells and the vascular network, as well as their interplay. We can then include the effects of two different treatments, conventional cytotoxic therapy and anti-angiogenic therapy. The results are compared with available experimental and clinical data.

Mathematical modelling of liver regeneration after intoxication with CCl4

Liver regeneration is a complex process, having evolved to protect animals from the consequences of liver loss caused by food toxins. In this study, we established a mathematical spatial-temporal model of the liver lobule regenerating after CCl(4) intoxication. The aim of modelling the regeneration process by matching experimental observations with those from a mathematical model is to gain a better understanding of the process and to recognize which parameters are relevant for specific phenomena. In order to set up a realistic minimal model, we first reconstructed a schematised liver lobule after determination of: (i) the mean number of hepatocytes between the central vein and the periphery of the lobule, (ii) the mean size of the hepatocytes and (iii) the mean number of hepatocyte columns in the inner, midzonal and peripheral ring of the lobule. In a next step, we determined the time course of cell death and BrdU incorporation after intoxication of male Sprague Dawley rats with CCl(4), thereby differentiating between inner, midzonal and peripheral hepatocytes. These parameters were used to construct a model. The basic unit of this model is the individual cell. The detailed behaviour of the cells is studied, controlled by the model parameters: (1) probability of cell division at defined positions of the lobule at a given time, (2) "coordinated cell orientation", i.e., the ability of the cells to align during the regeneration process into columns towards the central vein of a liver lobule, (3) cell cycle duration, (4) the migration activity and (5) the polarity of the hepatocytes resulting in polar cell-cell adhesion between them. In a schematised lobule, the model shows that CCl(4) initially induced cell death of a pericentral ring of hepatocytes, followed by a wave of proliferation that starts in the surviving hepatocytes next to the inner ring of dead cells and continues to the peripheral hepatocytes, finally restoring the characteristic micro-architecture of the lobule in a 7-day process. This model was used to systematically analyze the influence of parameters 1-5. Interestingly, coordinated cell orientation and cell polarity were identified to be the most critical parameters. Elimination led to destruction of the characteristic micro-architecture of the lobule and to a high degree of disorder characterized by hexagonal cell structures. Our model suggests that the ability of hepatocytes to realign after cell division by a process of coordinated cell orientation (model parameter 2) in combination with cell polarity (model parameter 5) may be at least as critical as hepatocyte proliferation (model parameter 1) itself.

Toward a multiscale model of antigen presentation in immunity

A functioning immune system and the process of antigen presentation in particular encompass events that occur at multiple length and time scales. Despite a wealth of information in the biological literature regarding each of these scales, no single representation synthesizing this information into a model of the overall immune response as it depends on antigen presentation is available. In this article, we outline an approach for integrating information over relevant biological and temporal scales to generate such a representation for major histocompatibility complex class II-mediated antigen presentation. In addition, we begin to address how such models can be used to answer questions about mechanisms of infection and new strategies for treatment and vaccines.