A quantitative cellular automaton model of in vitro multicellular spheroid tumour growth

Phenotype-structured model of intra-clonal heterogeneity and drug resistance in multiple myeloma

Multiple myeloma (MM) is a genetically complex hematological cancer characterized by the abnormal proliferation of malignant plasma cells in the bone marrow. This disease progresses from a premalignant condition known as monoclonal gammopathy of unknown significance (MGUS) through sequential genetic alterations involving various genes. These genetic changes contribute to the uncontrolled growth of multiple clones of plasma cells. In this study, we present a phenotype-structured model that captures the intra-clonal heterogeneity and drug resistance in multiple myeloma (MM). The model accurately reproduces the branching evolutionary pattern observed in MM progression, aligning with a previously developed multiscale model. Numerical simulations reveal that higher mutation rates enhance tumor phenotype diversity, while access to growth factors accelerates tumor evolution and increases its final size. Interestingly, the model suggests that further increasing growth factor access primarily amplifies tumor size rather than altering clonal dynamics. Additionally, the model emphasizes that higher mutation frequencies and growth factor availability elevate the chances of drug resistance and relapse. It indicates that the timing of the treatment could trajectory of tumor evolution and clonal emergence in the case of branching evolutionary pattern. Given its low computational cost, our model is well-suited for quantitative studies on MM clonal heterogeneity and its interaction with chemotherapeutic treatments.

Evaluation of entropy and fractal dimension as biomarkers for tumor growth and treatment response using cellular automata

Cell-based models provide a helpful approach for simulating complex systems that exhibit adaptive, resilient qualities, such as cancer. Their focus on individual cell interactions makes them a particularly appropriate strategy to study cancer therapies' effects, which are often designed to disrupt single-cell dynamics. In this work, we propose them as viable methods for studying the time evolution of cancer imaging biomarkers (IBM). We propose a cellular automata model for tumor growth and three different therapies: chemotherapy, radiotherapy, and immunotherapy, following well-established modeling procedures documented in the literature. The model generates a sequence of tumor images, from which a time series of two biomarkers: entropy and fractal dimension, is obtained. Our model shows that the fractal dimension increased faster at the onset of cancer cell dissemination. At the same time, entropy was more responsive to changes induced in the tumor by the different therapy modalities. These observations suggest that the prognostic value of the proposed biomarkers could vary considerably with time. Thus, it is essential to assess their use at different stages of cancer and for different imaging modalities. Another observation derived from the results was that both biomarkers varied slowly when the applied therapy attacked cancer cells scattered along the automatons' area, leaving multiple independent clusters of cells at the end of the treatment. Thus, patterns of change of simulated biomarkers time series could reflect on essential qualities of the spatial action of a given cancer intervention.

Hybrid computational models of multicellular tumour growth considering glucose metabolism

Cancer cells metabolize glucose through metabolic pathways that differ from those used by healthy and differentiated cells. In particular, tumours have been shown to consume more glucose than their healthy counterparts and to use anaerobic metabolic pathways, even under aerobic conditions. Nevertheless, scientists have still not been able to explain why cancer cells evolved to present an altered metabolism and what evolutionary advantage this might provide them. Experimental and computational models have been increasingly used in recent years to understand some of these biological questions. Multicellular tumour spheroids are effective experimental models as they replicate the initial stages of avascular solid tumour growth. Furthermore, these experiments generate data which can be used to calibrate and validate computational studies that aim to simulate tumour growth. Hybrid models are of particular relevance in this field of research because they model cells as individual agents while also incorporating continuum representations of the substances present in the surrounding microenvironment that may participate in intracellular metabolic networks as concentration or density distributions. Henceforth, in this review, we explore the potential of computational modelling to reveal the role of metabolic reprogramming in tumour growth.

An in silico model to study the impact of carbonic anhydrase IX expression on tumour growth and anti-PD-1 therapy

Immune checkpoint inhibitors (ICIs) are revolutionary cancer treatments. However, the mechanisms behind their effectiveness are not yet fully understood. Here, we aimed to investigate the role of the pH-regulatory enzyme carbonic anhydrase IX (CAIX) in ICI success. Consequently, we developed an in silico model of the tumour microenvironment. The hybrid model consists of an agent-based model of tumour-immune cell interactions, coupled with a set of diffusion-reaction equations describing substances in the environment. It is calibrated with data from the literature, enabling the study of its qualitative behaviour. In our model, CAIX-expressing tumours acidified their neighbourhood, thereby reducing immune infiltration by 90% (p < 0.001) and resulting in a 25% increase in tumour burden (p < 0.001). Moreover, suppression of CAIX improved the response to anti-PD-1 (23% tumour reduction in CAIX knockouts and 6% in CAIX-expressing tumours, p < 0.001), independently of initial PD-L1 expression. Our simulations suggest that patients with CAIX-expressing tumours could respond favourably to combining ICIs with CAIX suppression, even in the absence of pre-treatment PD-L1 expression. Furthermore, when calibrated with tumour-type-specific data, our model could serve as a high-throughput tool for testing the effectiveness of such a combinatorial approach.

Mathematical modelling reveals cellular dynamics within tumour spheroids

Tumour spheroids are widely used as an in vitro assay for characterising the dynamics and response to treatment of different cancer cell lines. Their popularity is largely due to the reproducible manner in which spheroids grow: the diffusion of nutrients and oxygen from the surrounding culture medium, and their consumption by tumour cells, causes proliferation to be localised at the spheroid boundary. As the spheroid grows, cells at the spheroid centre may become hypoxic and die, forming a necrotic core. The pressure created by the localisation of tumour cell proliferation and death generates an cellular flow of tumour cells from the spheroid rim towards its core. Experiments by Dorie et al. showed that this flow causes inert microspheres to infiltrate into tumour spheroids via advection from the spheroid surface, by adding microbeads to the surface of tumour spheroids and observing the distribution over time. We use an off-lattice hybrid agent-based model to re-assess these experiments and establish the extent to which the spatio-temporal data generated by microspheres can be used to infer kinetic parameters associated with the tumour spheroids that they infiltrate. Variation in these parameters, such as the rate of tumour cell proliferation or sensitivity to hypoxia, can produce spheroids with similar bulk growth dynamics but differing internal compositions (the proportion of the tumour which is proliferating, hypoxic/quiescent and necrotic/nutrient-deficient). We use this model to show that the types of experiment conducted by Dorie et al. could be used to infer spheroid composition and parameters associated with tumour cell lines such as their sensitivity to hypoxia or average rate of proliferation, and note that these observations cannot be conducted within previous continuum models of microbead infiltration into tumour spheroids as they rely on resolving the trajectories of individual microbeads.

Tumour spheroids are an experimental assay used to characterise the dynamics and response to treatment of different cancer cell lines. Previous experiments have demonstrated that the localisation of tumour cell proliferation to the spheroid edge (due to the gradient formed by nutrient diffusing from the surrounding medium) causes cells to be pushed from the proliferative rim towards the nutrient-deficient necrotic core. This movement allows inert particles to infiltrate tumour spheroids. We use a hybrid agent-based model to reproduce this data. We show further how data from individual microbead trajectories can be used to infer the composition of simulated tumour spheroids, and to estimate model parameters pertaining to tumour cell proliferation rates and their responses to hypoxia. Since these measurements are possible using modern imaging techniques, this could motivate new experiments in which spheroid composition could be inferred by observing passive infiltration of inert particles.

MpFEW RHIZOIDS1 miRNA-Mediated Lateral Inhibition Controls Rhizoid Cell Patterning in Marchantia polymorpha

Lateral inhibition patterns differentiated cell types among equivalent cells during development in bacteria, metazoans, and plants. Tip-growing rhizoid cells develop among flat epidermal cells in the epidermis of the early-diverging land plant Marchantia polymorpha. We show that the majority of rhizoid cells develop individually, but some develop in linear, one-dimensional groups (chains) of between 2 and 7 rhizoid cells in wild-type plants. The distribution of rhizoid cells can be accounted for within a simple cellular automata model of lateral inhibition. The model predicted that in the absence of lateral inhibition, two-dimensional rhizoid cell groups (clusters) form. These can be larger than those formed with lateral inhibition. M. polymorpha rhizoid differentiation is positively regulated by the ROOT HAIR DEFECTIVE SIX-LIKE1 (MpRSL1) basic-helix-loop-helix (bHLH) transcription factor, which is directly repressed by the FEW RHIZOIDS1 (MpFRH1) microRNA (miRNA). To test if MpFRH1 miRNA acts during lateral inhibition, we generated loss-of-function (lof) mutants without the MpFRH1 miRNA. Two-dimensional clusters of rhizoids develop in Mpfrh1^lof mutants as predicted by the model for plants that lack lateral inhibition. Furthermore, two-dimensional clusters of up to 9 rhizoid cells developed in the Mpfrh1^lof mutants compared to a maximum number of 7 observed in wild-type groups. The higher steady-state levels of MpRSL1 mRNA in Mpfrh1^lof mutants indicate that MpFRH1-mediated lateral inhibition involves the repression of MpRSL1 activity. Together, the modeling and genetic data indicate that MpFRH1 miRNA mediates lateral inhibition by repressing MpRSL1 during pattern formation in the M. polymorpha epidermis.

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.

A checkpoint-oriented cell cycle simulation model

Modeling and in silico simulations are of major conceptual and applicative interest in studying the cell cycle and proliferation in eukaryotic cells. In this paper, we present a cell cycle checkpoint-oriented simulator that uses agent-based simulation modeling to reproduce the dynamics of a cancer cell population in exponential growth. Our in silico simulations were successfully validated by experimental in vitro supporting data obtained with HCT116 colon cancer cells. We demonstrated that this model can simulate cell confluence and the associated elongation of the G1 phase. Using nocodazole to synchronize cancer cells at mitosis, we confirmed the model predictivity and provided evidence of an additional and unexpected effect of nocodazole on the overall cell cycle progression. We anticipate that this cell cycle simulator will be a potential source of new insights and research perspectives.

Dynamic Load Balancing Strategy for Parallel Tumor Growth Simulations

In this paper, we propose a parallel cellular automaton tumor growth model that includes load balancing of cells distribution among computational threads with the introduction of adjusting parameters. The obtained results show a fair reduction in execution time and improved speedup compared with the sequential tumor growth simulation program currently referenced in tumoral biology. The dynamic data structures of the model can be extended to address additional tumor growth characteristics such as angiogenesis and nutrient intake dependencies.

Dynamical density-functional-theory-based modeling of tissue dynamics: Application to tumor growth

We present a theoretical framework based on an extension of dynamical density-functional theory (DDFT) for describing the structure and dynamics of cells in living tissues and tumors. DDFT is a microscopic statistical mechanical theory for the time evolution of the density distribution of interacting many-particle systems. The theory accounts for cell-pair interactions, different cell types, phenotypes, and cell birth and death processes (including cell division), to provide a biophysically consistent description of processes bridging across the scales, including describing the tissue structure down to the level of the individual cells. Analysis of the model is presented for single-species and two-species cases, the latter aimed at describing competition between tumor and healthy cells. In suitable parameter regimes, model results are consistent with biological observations. Of particular note, divergent tumor growth behavior, mirroring metastatic and benign growth characteristics, are shown to be dependent on the cell-pair-interaction parameters.

A general framework dedicated to computational morphogenesis Part I - Constitutive equations

In order to understand living organisms, considerable experimental efforts and resources have been devoted to correlate genes and their expressions with cell, tissue, organ and whole organisms' phenotypes. This data driven approach to knowledge discovery has led to many breakthrough in our understanding of healthy and diseased states, and is paving the way to improve the diagnosis and treatment of diseases. Complementary to this data-driven approach, computational models of biological systems based on first principles have been developed in order to deepen our understanding of the multi-scale dynamics that drives normal and pathological biological functions. In this paper we describe the biological, physical and mathematical concepts that led to the design of a Computational Morphogenesis (CM) platform baptized Generic Modeling and Simulating Platform (GMSP). Its role is to generate realistic 3D multi-scale biological tissues from virtual stem cells and the intended target applications include in virtuo studies of normal and abnormal tissue (re)generation as well as the development of complex diseases such as carcinogenesis. At all space-scales of interest, biological agents interact with each other via biochemical, bioelectrical, and mechanical fields that operate in concert during embryogenesis, growth and adult life. The spatio-temporal dependencies of these fields can be modeled by physics-based constitutive equations that we propose to examine in relation to the landmark biological events that occur during embryogenesis.

Glioma growth modeling based on the effect of vital nutrients and metabolic products

Glioma brain tumors exhibit considerably aggressive behavior leading to high mortality rates. Mathematical modeling of tumor growth aims to explore the interactions between glioma cells and tissue microenvironment, which affect tumor evolution. Leveraging this concept, we present a three-dimensional model of glioma spatio-temporal evolution based on existing continuum approaches, yet incorporating novel factors of the phenomenon. The proposed model involves the interactions between different tumor cell phenotypes and their microenvironment, investigating how tumor growth is affected by complex biological exchanges. It focuses on the separate and combined effect of vital nutrients and cellular wastes on tumor expansion, leading to the formation of cell populations with different metabolic, proliferative, and diffusive profiles. Several simulations were performed on a virtual and a real glioma, using combinations of proliferation and diffusion rates for different evolution times. The model results were validated on a glioma model available in the literature and a real case of tumor progression. The experimental observations indicate that our model estimates quite satisfactorily the expansion of each region and the overall tumor growth. Based on the individual results, the proposed model may provide an important research tool for patient-specific simulation of different tumor evolution scenarios and reliable estimation of glioma evolution. Graphical Abstract Outline of the mathematical model functionality and application to glioma growth with indicative results.

A mechanically coupled reaction-diffusion model that incorporates intra-tumoural heterogeneity to predict in vivo glioma growth

While gliomas have been extensively modelled with a reaction-diffusion (RD) type equation it is most likely an oversimplification. In this study, three mathematical models of glioma growth are developed and systematically investigated to establish a framework for accurate prediction of changes in tumour volume as well as intra-tumoural heterogeneity. Tumour cell movement was described by coupling movement to tissue stress, leading to a mechanically coupled (MC) RD model. Intra-tumour heterogeneity was described by including a voxel-specific carrying capacity (CC) to the RD model. The MC and CC models were also combined in a third model. To evaluate these models, rats (n = 14) with C6 gliomas were imaged with diffusion-weighted magnetic resonance imaging over 10 days to estimate tumour cellularity. Model parameters were estimated from the first three imaging time points and then used to predict tumour growth at the remaining time points which were then directly compared to experimental data. The results in this work demonstrate that mechanical-biological effects are a necessary component of brain tissue tumour modelling efforts. The results are suggestive that a variable tissue carrying capacity is a needed model component to capture tumour heterogeneity. Lastly, the results advocate the need for additional effort towards capturing tumour-to-tissue infiltration.

A hybrid discrete-continuous model of in vitro spheroid tumor growth and drug response

Anti-cancer therapy efficacy in solid tumors mainly depends on drug transportation through the vasculature system and the extracellular matrix, on diffusion gradients and clonal heterogeneity within the tumor mass, as well as on the responses of the individual tumor cells to drugs and their interactions with each other and their local microenvironment. In this work, we develop a mathematical predictive model for tumor growth and drug response based on 3D spheroids experiments that possess several in vivo features of tumors and are considered better for drug screening. The model takes into account the diffusion gradients of both oxygen and drug through the tumor volume, describes the tumor population at cell level and assumes a simple underlying cellular dose-response curve that is translated to a cell death probability. The model shows that although the endpoint tumor regression can be well approximated, the effects of the drug on cell fate necessitate a more sophisticated model to explain the temporal evolution of tumor regression and more quantitative information regarding the number and topology of dead and living cells, which is highly important for in vivo clinical relevant predictions. The model is built in a way that can be constrained by experimentally derived set of parameters and is capable of accommodating cell heterogeneity, sub-cellular regulatory mechanisms and drug-induced signaling cascades, as well as additional mechanisms of adapted resistance.

A Glycolysis-Based In Silico Model for the Solid Tumor Growth

Cancer-tumor growth is a complex process depending on several biological factors, such as the chemical microenvironment of the tumor, the cellular metabolic profile, and its proliferation rate. Several mathematical models have been developed for identifying the interactions between tumor cells and tissue microenvironment, since they play an important role in tumor formation and progression. Toward this direction we propose a new continuum model of avascular glioma-tumor growth, which incorporates a new factor, namely, the glycolytic potential of cancer cells, to express the interactions of three different tumor-cell populations (proliferative, hypoxic, and necrotic) with their tissue microenvironment. The glycolytic potential engages three vital nutrients, i.e., oxygen, glucose, and lactate, which provide cells with the necessary energy for their survival and proliferation. Extensive simulations are performed for different evolution times and various proliferation rates, in order to investigate how the tumor growth is affected. According to medical experts, the experimental observations indicate that the model predicts quite satisfactorily the overall tumor growth as well as the expansion of each region separately. Following extensive evaluation, the proposed model may provide an essential tool for patient-specific tumor simulation and reliable prediction of glioma spatiotemporal expansion.

A matter of timing: identifying significant multi-dose radiotherapy improvements by numerical simulation and genetic algorithm search

Multi-dose radiotherapy protocols (fraction dose and timing) currently used in the clinic are the product of human selection based on habit, received wisdom, physician experience and intra-day patient timetabling. However, due to combinatorial considerations, the potential treatment protocol space for a given total dose or treatment length is enormous, even for relatively coarse search; well beyond the capacity of traditional in-vitro methods. In constrast, high fidelity numerical simulation of tumor development is well suited to the challenge. Building on our previous single-dose numerical simulation model of EMT6/Ro spheroids, a multi-dose irradiation response module is added and calibrated to the effective dose arising from 18 independent multi-dose treatment programs available in the experimental literature. With the developed model a constrained, non-linear, search for better performing cadidate protocols is conducted within the vicinity of two benchmarks by genetic algorithm (GA) techniques. After evaluating less than 0.01% of the potential benchmark protocol space, candidate protocols were identified by the GA which conferred an average of 9.4% (max benefit 16.5%) and 7.1% (13.3%) improvement (reduction) on tumour cell count compared to the two benchmarks, respectively. Noticing that a convergent phenomenon of the top performing protocols was their temporal synchronicity, a further series of numerical experiments was conducted with periodic time-gap protocols (10 h to 23 h), leading to the discovery that the performance of the GA search candidates could be replicated by 17-18 h periodic candidates. Further dynamic irradiation-response cell-phase analysis revealed that such periodicity cohered with latent EMT6/Ro cell-phase temporal patterning. Taken together, this study provides powerful evidence towards the hypothesis that even simple inter-fraction timing variations for a given fractional dose program may present a facile, and highly cost-effecitive means of significantly improving clinical efficacy.

Modeling intrinsic heterogeneity and growth of cancer cells

Intratumoral heterogeneity has been found to be a major cause of drug resistance. Cell-to-cell variation increases as a result of cancer-related alterations, which are acquired by stochastic events and further induced by environmental signals. However, most cellular mechanisms include natural fluctuations that are closely regulated, and thus lead to asynchronization of the cells, which causes intrinsic heterogeneity in a given population. Here, we derive two novel mathematical models, a stochastic agent-based model and an integro-differential equation model, each of which describes the growth of cancer cells as a dynamic transition between proliferative and quiescent states. These models are designed to predict variations in growth as a function of the intrinsic heterogeneity emerging from the durations of the cell-cycle and apoptosis, and also include cellular density dependencies. By examining the role all parameters play in the evolution of intrinsic tumor heterogeneity, and the sensitivity of the population growth to parameter values, we show that the cell-cycle length has the most significant effect on the growth dynamics. In addition, we demonstrate that the agent-based model can be approximated well by the more computationally efficient integro-differential equations when the number of cells is large. This essential step in cancer growth modeling will allow us to revisit the mechanisms of multidrug resistance by examining spatiotemporal differences of cell growth while administering a drug among the different sub-populations in a single tumor, as well as the evolution of those mechanisms as a function of the resistance level.

Biphasic modulation of cancer stem cell-driven solid tumour dynamics in response to reactivated replicative senescence

OBJECTIVES

Cell senescence is a physiological programme of irreversible mitotic arrest that is triggered after a variety of intracellular and extracellular events. Its purpose is to protect tissue integrity by disabling mitosis in stressed or damaged cells. The senescence program serves as a tumour suppressor, and cancer cells are believed to bypass senescence to advance to malignancy. Recent studies have shown that senescence can be reactivated in cancer cells through a number of external perturbations, including oncogene activation, tumour suppressor gene withdrawal and irradiation.

MATERIALS AND METHODS

We have developed an agent-based model of solid tumour growth whose input population composition is based on the cancer stem-cell hypothesis. It is used to show how cancer stem cells can drive tumour progression, while non-stem cancer cells (CCs) interfere with this by impeding cancer stem-cell dynamics.

RESULTS

Here we show that intratumoural competition between the two cell types may arise to modulate tumour progression and ultimately cancer presentation risk. Model simulations reveal that reactivation of the replicative senescence programme in CCs initially increases total tumour burden, as attrition from cell death is partially averted, but evolves to provide tumour control in the long-term through increasing constraints on stem-cell compartment kinetics.

CONCLUSIONS

Reactivation of replicative senescence can prolong CC competition with cancer stem cells, thereby ultimately inhibiting malignant progression regardless of tumour size.

A High-Performance Cellular Automaton Model of Tumor Growth with Dynamically Growing Domains

Tumor growth from a single transformed cancer cell up to a clinically apparent mass spans many spatial and temporal orders of magnitude. Implementation of cellular automata simulations of such tumor growth can be straightforward but computing performance often counterbalances simplicity. Computationally convenient simulation times can be achieved by choosing appropriate data structures, memory and cell handling as well as domain setup. We propose a cellular automaton model of tumor growth with a domain that expands dynamically as the tumor population increases. We discuss memory access, data structures and implementation techniques that yield high-performance multi-scale Monte Carlo simulations of tumor growth. We discuss tumor properties that favor the proposed high-performance design and present simulation results of the tumor growth model. We estimate to which parameters the model is the most sensitive, and show that tumor volume depends on a number of parameters in a non-monotonic manner.

A numerical model of EMT6/Ro spheroid dynamics under irradiation: calibration and estimation of the underlying irradiation-induced cell survival probability

We present extensions to our quasi-2D cellular automata spheroid model that add a cellular kinetics module together with an irradiation and repair module. Significantly, our approach is not based on the Linear Quadratic (LQ) model, instead, we propose a simple two-parameter, algorithmic model which captures the essential biological features of irradiation-induced cell death, repair and associated cell cycle delays. This approach allows us to estimate directly the underlying irradiation-induced cell survival probability. We present the calibration of this extended model both with and without the application of single irradiation doses to the commonly studied (in vitro) EMT6/Ro (mammary carcinoma) cell line. A comparison of the estimated underlying cell survival probability with the in vitro survival probability data confirms the expected differences in the measures.

A mathematical model for describing the metastasis of cancer in bone tissue

Metastasis is the rapid proliferation of cancer cells (secondary tumour) at a specific place, generally leading to death. This occurs at anatomical parts providing the necessary environment for vascularity, oxygen and food to hide their actions and trigger the rapid growth of cancer. Prostate and breast cancers, for example, use bone marrow for their proliferation. Bone-supporting cancer cells thus adapt to the environment, mimicking the behaviour of genetic and molecular bone cells. Evidence of this has been given in Cecchini et al. (2005, EAU Update Ser. 3:214-226), providing arguments such as how cancer cell growth is so active during bone reabsorption. This paper simulates metastasis activation in bone marrow. A mathematical model has been developed involving the activation of molecules from bone tissue cells, which are necessary for cancer to proliferate. Here, we simulate two forms of secondary tumour growth depending on the type of metastasis: osteosclerosis and osteolysis.

Phenotypic transition maps of 3D breast acini obtained by imaging-guided agent-based modeling

We introduce an agent-based model of epithelial cell morphogenesis to explore the complex interplay between apoptosis, proliferation, and polarization. By varying the activity levels of these mechanisms we derived phenotypic transition maps of normal and aberrant morphogenesis. These maps identify homeostatic ranges and morphologic stability conditions. The agent-based model was parameterized and validated using novel high-content image analysis of mammary acini morphogenesis in vitro with focus on time-dependent cell densities, proliferation and death rates, as well as acini morphologies. Model simulations reveal apoptosis being necessary and sufficient for initiating lumen formation, but cell polarization being the pivotal mechanism for maintaining physiological epithelium morphology and acini sphericity. Furthermore, simulations highlight that acinus growth arrest in normal acini can be achieved by controlling the fraction of proliferating cells. Interestingly, our simulations reveal a synergism between polarization and apoptosis in enhancing growth arrest. After validating the model with experimental data from a normal human breast line (MCF10A), the system was challenged to predict the growth of MCF10A where AKT-1 was overexpressed, leading to reduced apoptosis. As previously reported, this led to non growth-arrested acini, with very large sizes and partially filled lumen. However, surprisingly, image analysis revealed a much lower nuclear density than observed for normal acini. The growth kinetics indicates that these acini grew faster than the cells comprising it. The in silico model could not replicate this behavior, contradicting the classic paradigm that ductal carcinoma in situ is only the result of high proliferation and low apoptosis. Our simulations suggest that overexpression of AKT-1 must also perturb cell-cell and cell-ECM communication, reminding us that extracellular context can dictate cellular behavior.

Hybrid models of tumor growth

Cancer is a complex, multiscale process in which genetic mutations occurring at a subcellular level manifest themselves as functional changes at the cellular and tissue scale. The multiscale nature of cancer requires mathematical modeling approaches that can handle multiple intracellular and extracellular factors acting on different time and space scales. Hybrid models provide a way to integrate both discrete and continuous variables that are used to represent individual cells and concentration or density fields, respectively. Each discrete cell can also be equipped with submodels that drive cell behavior in response to microenvironmental cues. Moreover, the individual cells can interact with one another to form and act as an integrated tissue. Hybrid models form part of a larger class of individual-based models that can naturally connect with tumor cell biology and allow for the integration of multiple interacting variables both intrinsically and extrinsically and are therefore perfectly suited to a systems biology approach to tumor growth.

Cellular automata modelling of biomolecular networks dynamics

The modelling of biological systems dynamics is traditionally performed by ordinary differential equations (ODEs). When dealing with intracellular networks of genes, proteins and metabolites, however, this approach is hindered by network complexity and the lack of experimental kinetic parameters. This opened the field for other modelling techniques, such as cellular automata (CA) and agent-based modelling (ABM). This article reviews this emerging field of studies on network dynamics in molecular biology. The basics of the CA technique are discussed along with an extensive list of related software and websites. The application of CA to networks of biochemical reactions is exemplified in detail by the case studies of the mitogen-activated protein kinase (MAPK) signalling pathway, the FAS-ligand (FASL)-induced and Bcl-2-related apoptosis. The potential of the CA method to model basic pathways patterns, to identify ways to control pathway dynamics and to help in generating strategies to fight with cancer is demonstrated. The different line of CA applications presented includes the search for the best-performing network motifs, an analysis of importance for effective intracellular signalling and pathway cross-talk.

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

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

Rule-Based Simulation of Multi-Cellular Biological Systems-A Review of Modeling Techniques

Emergent behaviors of multi-cellular biological systems (MCBS) result from the behaviors of each individual cells and their interactions with other cells and with the environment. Modeling MCBS requires incorporating these complex interactions among the individual cells and the environment. Modeling approaches for MCBS can be grouped into two categories: continuum models and cell-based models. Continuum models usually take the form of partial differential equations, and the model equations provide insight into the relationship among the components in the system. Cell-based models simulate each individual cell behavior and interactions among them enabling the observation of the emergent system behavior. This review focuses on the cell-based models of MCBS, and especially, the technical aspect of the rule-based simulation method for MCBS is reviewed. How to implement the cell behaviors and the interactions with other cells and with the environment into the computational domain is discussed. The cell behaviors reviewed in this paper are division, migration, apoptosis/necrosis, and differentiation. The environmental factors such as extracellular matrix, chemicals, microvasculature, and forces are also discussed. Application examples of these cell behaviors and interactions are presented.

Rule-Based Simulation of Multi-Cellular Biological Systems-A Review of Modeling Techniques

Emergent behaviors of multi-cellular biological systems (MCBS) result from the behaviors of each individual cells and their interactions with other cells and with the environment. Modeling MCBS requires incorporating these complex interactions among the individual cells and the environment. Modeling approaches for MCBS can be grouped into two categories: continuum models and cell-based models. Continuum models usually take the form of partial differential equations, and the model equations provide insight into the relationship among the components in the system. Cell-based models simulate each individual cell behavior and interactions among them enabling the observation of the emergent system behavior. This review focuses on the cell-based models of MCBS, and especially, the technical aspect of the rule-based simulation method for MCBS is reviewed. How to implement the cell behaviors and the interactions with other cells and with the environment into the computational domain is discussed. The cell behaviors reviewed in this paper are division, migration, apoptosis/necrosis, and differentiation. The environmental factors such as extracellular matrix, chemicals, microvasculature, and forces are also discussed. Application examples of these cell behaviors and interactions are presented.