Adhesion between cells, diffusion of growth factors, and elasticity of the AER produce the paddle shape of the chick limb

Why Animal Experiments Are Still Indispensable in Bone Research: A Statement by the European Calcified Tissue Society

Major achievements in bone research have always relied on animal models and in vitro systems derived from patient and animal material. However, the use of animals in research has drawn intense ethical debate and the complete abolition of animal experimentation is demanded by fractions of the population. This phenomenon is enhanced by the reproducibility crisis in science and the advance of in vitro and in silico techniques. 3D culture, organ-on-a-chip, and computer models have improved enormously over the last few years. Nevertheless, the overall complexity of bone tissue cross-talk and the systemic and local regulation of bone physiology can often only be addressed in entire vertebrates. Powerful genetic methods such as conditional mutagenesis, lineage tracing, and modeling of the diseases enhanced the understanding of the entire skeletal system. In this review endorsed by the European Calcified Tissue Society (ECTS), a working group of investigators from Europe and the US provides an overview of the strengths and limitations of experimental animal models, including rodents, fish, and large animals, as well the potential and shortcomings of in vitro and in silico technologies in skeletal research. We propose that the proper combination of the right animal model for a specific hypothesis and state-of-the-art in vitro and/or in silico technology is essential to solving remaining important questions in bone research. This is crucial for executing most efficiently the 3R principles to reduce, refine, and replace animal experimentation, for enhancing our knowledge of skeletal biology, and for the treatment of bone diseases that affect a large part of society. © 2023 The Authors. Journal of Bone and Mineral Research published by Wiley Periodicals LLC on behalf of American Society for Bone and Mineral Research (ASBMR).

Finite Element Models of Osteocytes and Their Load-Induced Activation

PURPOSE OF REVIEW

Osteocytes are the conductors of bone adaptation and remodelling. Buried inside the calcified matrix, they sense mechanical cues and signal osteoclasts in case of low activity, and osteoblasts when stresses are high. How do osteocytes detect mechanical stress? What physical signal do they perceive? Finite element analysis is a useful tool to address these questions as it allows calculating stresses, strains and fluid flow where they cannot be measured. The purpose of this review is to evaluate the capabilities and challenges of finite element models of bone, in particular the osteocytes and load-induced activation mechanisms.

RECENT FINDINGS

High-resolution imaging and increased computational power allow ever more detailed modelling of osteocytes, either in isolation or embedded within the mineralised matrix. Over the years, homogeneous models of bone and osteocytes got replaced by heterogeneous and microstructural models, including, e.g. the lacuno-canalicular network and the cytoskeleton. The lacuno-canalicular network induces strain amplifications and the osteocyte protrusions seem to be stimulated much more than the cell body, both by strain and fluid flow. More realistic cell geometries, like minute constrictions of the canaliculi, increase this effect. Microstructural osteocyte models describe the transduction of external stimuli to the nucleus. Supracellular multiscale models (e.g. of a tunnelling osteon) allow to study differential loading of osteocytes and to distinguish between strain and fluid flow as the pivotal stimulatory cue. In the future, the finite element models may be enhanced by including chemical transport and intercellular communication between osteocytes, osteoclasts and osteoblasts.

Mechanical Regulation of Limb Bud Formation

Early limb bud development has been of considerable interest for the study of embryological development and especially morphogenesis. The focus has long been on biochemical signalling and less on cell biomechanics and mechanobiology. However, their importance cannot be understated since tissue shape changes are ultimately controlled by active forces and bulk tissue rheological properties that in turn depend on cell-cell interactions as well as extracellular matrix composition. Moreover, the feedback between gene regulation and the biomechanical environment is still poorly understood. In recent years, novel experimental techniques and computational models have reinvigorated research on this biomechanical and mechanobiological side of embryological development. In this review, we consider three stages of early limb development, namely: outgrowth, elongation, and condensation. For each of these stages, we summarize basic biological regulation and examine the role of cellular and tissue mechanics in the morphogenetic process.

Quantification of cell behaviors and computational modeling show that cell directional behaviors drive zebrafish pectoral fin morphogenesis

MOTIVATION

Understanding the mechanisms by which the zebrafish pectoral fin develops is expected to produce insights on how vertebrate limbs grow from a 2D cell layer to a 3D structure. Two mechanisms have been proposed to drive limb morphogenesis in tetrapods: a growth-based morphogenesis with a higher proliferation rate at the distal tip of the limb bud than at the proximal side, and directed cell behaviors that include elongation, division and migration in a non-random manner. Based on quantitative experimental biological data at the level of individual cells in the whole developing organ, we test the conditions for the dynamics of pectoral fin early morphogenesis.

RESULTS

We found that during the development of the zebrafish pectoral fin, cells have a preferential elongation axis that gradually aligns along the proximodistal (PD) axis of the organ. Based on these quantitative observations, we build a center-based cell model enhanced with a polarity term and cell proliferation to simulate fin growth. Our simulations resulted in 3D fins similar in shape to the observed ones, suggesting that the existence of a preferential axis of cell polarization is essential to drive fin morphogenesis in zebrafish, as observed in the development of limbs in the mouse, but distal tip-based expansion is not.

AVAILABILITYAND IMPLEMENTATION

Upon publication, biological data will be available at http://bioemergences.eu/modelingFin, and source code at https://github.com/guijoe/MaSoFin.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Biofabrication of spheroids fusion-based tumor models: computational simulation of glucose effects

In vitro tumor models consisting of cell spheroids are increasingly used for mechanistic studies and pharmacological testing. However, unless vascularized, the availability of nutrients such as glucose to deeper layers of multicellular aggregates is limited. In addition, recent developments in cells-only biofabrication (e.g. 'scaffold-free bioprinting'), allow the creation of more complex spheroid-based structures, further exposing the cells to nutrient deprivation within these constructs. To explore the impact of glucose availability on such tumor-like structures, we used the CompuCell3D (CC3D) platform for modeling of tumor spheroids. By monitoring the types of cells, fusing pairs geometry and the distance between spheroids centers of mass, we made novel heuristic observations on how binary- and multi-spheroid fusions are impacted by glucose availability. At limiting glucose concentrations mimicking hypoglycemia we noted an abrupt collapse of the tumor spheroids, unexpectedly amplified by the contact with normal cell spheroids. At higher glucose concentrations, we found an increased intermixing of cancerous cells, strong anti-phase oscillations between proliferating and quiescent tumor cells and a structural instability of fusing tumor spheroids, leading to their re-fragmentation. In a model of tumor microenvironment composed of normal cell spheroids fusing around a tumoral one, the competition for glucose lead to either the tumor's disappearance, or to its steady expansion. Moreover, the invasion of this microenvironment by individual tumor cells was also strongly depended on the available glucose. In conclusion, we demonstrate the value of computational simulations for anticipating the properties of biofabricated tumor models, and in generating testable hypotheses regarding the relationship between cancer, nutrition and diabetes.

Self-sustained planar intercalations due to mechanosignaling feedbacks lead to robust axis extension during morphogenesis

Tissue elongation is a necessary process in metazoans to implement their body plans that is not fully understood. Here we propose a mechanism based on the interplay between cellular mechanics and primordia patterning that results in self-sustained planar intercalations. Thus, we show that a location-dependent modulation of the mechanical properties of cells leads to robust axis extension. To illustrate the plausibility of this mechanism, we test it against different patterning models by means of computer simulations of tissues where we implemented mechano-signaling feedbacks. Our results suggest that robust elongation relies on a trade-off between cellular and tissue strains that is orchestrated through the cleavage orientation. In the particular context of axis extension in Turing-patterned tissues, we report that different directional cell activities cooperate synergetically to achieve elongation. Altogether, our findings help to understand how the axis extension phenomenon emerges from the dynamics of individual cells.

Multiscale modeling of vertebrate limb development

We review the current state of mathematical modeling of cartilage pattern formation in vertebrate limbs. We place emphasis on several reaction-diffusion type models that have been proposed in the last few years. These models are grounded in more detailed knowledge of the relevant regulatory processes than previous ones but generally refer to different molecular aspects of these processes. Considering these models in light of comparative phylogenomics permits framing of hypotheses on the evolutionary order of appearance of the respective mechanisms and their roles in the fin-to-limb transition. This article is categorized under: Analytical and Computational Methods > Computational Methods Models of Systems Properties and Processes > Mechanistic Models Developmental Biology > Developmental Processes in Health and Disease Analytical and Computational Methods > Analytical Methods.

Mathematical modeling of chondrogenic pattern formation during limb development: Recent advances in continuous models

The phenomenon of chondrogenic pattern formation in the vertebrate limb is one of the best studied examples of organogenesis. Many different models, mathematical as well as conceptual, have been proposed for it in the last fifty years or so. In this review, we give a brief overview of the fundamental biological background, then describe in detail several models which aim to describe qualitatively and quantitatively the corresponding biological phenomena. We concentrate on several new models that have been proposed in recent years, taking into account recent experimental progress. The major mathematical tools in these approaches are ordinary and partial differential equations. Moreover, we discuss models with non-local flux terms used to account for cell-cell adhesion forces and a structured population model with diffusion. We also include a detailed list of gene products and potential morphogens which have been identified to play a role in the process of limb formation and its growth.

Medical Systems Biology

The aim of this volume is to encourage the use of systems-level methodologies to contribute to the improvement of human-health . We intend to motivate biomedical researchers to complement their current theoretical and empirical practice with up-to-date systems biology conceptual approaches. Our perspective is based on the deep understanding of the key biomolecular regulatory mechanisms that underlie health, as well as the emergence and progression of human-disease . We strongly believe that the contemporary systems biology perspective opens the door to the effective development of novel methodologies to the improvement of prevention . This requires a deeper and integrative understanding of the involved underlying systems-level mechanisms. In order to explain our proposal in a simple way, in this chapter we privilege the conceptual exposition of our chosen framework over formal considerations. The formal exposition of our proposal will be expanded and discussed later in the next chapters.

Simulation of Morphogen and Tissue Dynamics

Morphogenesis, the process by which an adult organism emerges from a single cell, has fascinated humans for a long time. Modeling this process can provide novel insights into development and the principles that orchestrate the developmental processes. This chapter focuses on the mathematical description and numerical simulation of developmental processes. In particular, we discuss the mathematical representation of morphogen and tissue dynamics on static and growing domains, as well as the corresponding tissue mechanics. In addition, we give an overview of numerical methods that are routinely used to solve the resulting systems of partial differential equations. These include the finite element method and the Lattice Boltzmann method for the discretization as well as the arbitrary Lagrangian-Eulerian method and the Diffuse-Domain method to numerically treat deforming domains.

Computer modeling in developmental biology: growing today, essential tomorrow

D'Arcy Thompson was a true pioneer, applying mathematical concepts and analyses to the question of morphogenesis over 100 years ago. The centenary of his famous book, On Growth and Form, is therefore a great occasion on which to review the types of computer modeling now being pursued to understand the development of organs and organisms. Here, I present some of the latest modeling projects in the field, covering a wide range of developmental biology concepts, from molecular patterning to tissue morphogenesis. Rather than classifying them according to scientific question, or scale of problem, I focus instead on the different ways that modeling contributes to the scientific process and discuss the likely future of modeling in developmental biology.

A node-based version of the cellular Potts model

The cellular Potts model (CPM) is a lattice-based Monte Carlo method that uses an energetic formalism to describe the phenomenological mechanisms underlying the biophysical problem of interest. We here propose a CPM-derived framework that relies on a node-based representation of cell-scale elements. This feature has relevant consequences on the overall simulation environment. First, our model can be implemented on any given domain, provided a proper discretization (which can be regular or irregular, fixed or time evolving). Then, it allowed an explicit representation of cell membranes, whose displacements realistically result in cell movement. Finally, our node-based approach can be easily interfaced with continuous mechanics or fluid dynamics models. The proposed computational environment is here applied to some simple biological phenomena, such as cell sorting and chemotactic migration, also in order to achieve an analysis of the performance of the underlying algorithm. This work is finally equipped with a critical comparison between the advantages and disadvantages of our model with respect to the traditional CPM and to some similar vertex-based approaches.

A Cellular Potts Model of single cell migration in presence of durotaxis

Cell migration is a fundamental biological phenomenon during which cells sense their surroundings and respond to different types of signals. In presence of durotaxis, cells preferentially crawl from soft to stiff substrates by reorganizing their cytoskeleton from an isotropic to an anisotropic distribution of actin filaments. In the present paper, we propose a Cellular Potts Model to simulate single cell migration over flat substrates with variable stiffness. We have tested five configurations: (i) a substrate including a soft and a stiff region, (ii) a soft substrate including two parallel stiff stripes, (iii) a substrate made of successive stripes with increasing stiffness to create a gradient and (iv) a stiff substrate with four embedded soft squares. For each simulation, we have evaluated the morphology of the cell, the distance covered, the spreading area and the migration speed. We have then compared the numerical results to specific experimental observations showing a consistent agreement.

Emergent Stratification in Solid Tumors Selects for Reduced Cohesion of Tumor Cells: A Multi-Cell, Virtual-Tissue Model of Tumor Evolution Using CompuCell3D

Tumor cells and structure both evolve due to heritable variation of cell behaviors and selection over periods of weeks to years (somatic evolution). Micro-environmental factors exert selection pressures on tumor-cell behaviors, which influence both the rate and direction of evolution of specific behaviors, especially the development of tumor-cell aggression and resistance to chemotherapies. In this paper, we present, step-by-step, the development of a multi-cell, virtual-tissue model of tumor somatic evolution, simulated using the open-source CompuCell3D modeling environment. Our model includes essential cell behaviors, microenvironmental components and their interactions. Our model provides a platform for exploring selection pressures leading to the evolution of tumor-cell aggression, showing that emergent stratification into regions with different cell survival rates drives the evolution of less cohesive cells with lower levels of cadherins and higher levels of integrins. Such reduced cohesivity is a key hallmark in the progression of many types of solid tumors.

Lattice-based model of ductal carcinoma in situ suggests rules for breast cancer progression to an invasive state

Ductal carcinoma in situ (DCIS) is a heterogeneous group of non-invasive lesions of the breast that result from abnormal proliferation of mammary epithelial cells. Pathologists characterize DCIS by four tissue morphologies (micropapillary, cribriform, solid, and comedo), but the underlying mechanisms that distinguish the development and progression of these morphologies are not well understood. Here we explored the conditions leading to the emergence of the different morphologies of DCIS using a two-dimensional multi-cell lattice-based model that incorporates cell proliferation, apoptosis, necrosis, adhesion, and contractility. We found that the relative rates of cell proliferation and apoptosis governed which of the four morphologies emerged. High proliferation and low apoptosis favored the emergence of solid and comedo morphologies. In contrast, low proliferation and high apoptosis led to the micropapillary morphology, whereas high proliferation and high apoptosis led to the cribriform morphology. The natural progression between morphologies cannot be investigated in vivo since lesions are usually surgically removed upon detection; however, our model suggests probable transitions between these morphologies during breast cancer progression. Importantly, cribriform and comedo appear to be the ultimate morphologies of DCIS. Motivated by previous experimental studies demonstrating that tumor cells behave differently depending on where they are located within the mammary duct in vivo or in engineered tissues, we examined the effects of tissue geometry on the progression of DCIS. In agreement with our previous experimental work, we found that cells are more likely to invade from the end of ducts and that this preferential invasion is regulated by cell adhesion and contractility. This model provides additional insight into tumor cell behavior and allows the exploration of phenotypic transitions not easily monitored in vivo.

Breast cancer is a complex disease that affects women worldwide. One heterogeneous group of lesions, ductal carcinoma in situ (DCIS), often begins as a nonmalignant disease but can readily progress if left untreated. The progression of this disease is not well understood because DCIS is typically removed upon detection. Therefore, computational models might help predict whether DCIS will remain nonmalignant or progress towards invasive ductal carcinoma. Here we used a multi-cell lattice-based model to explore the relative effects of cell proliferation, death, division axis, adhesion and contractility on the development and progression of DCIS. We also examined the emergence and progression of DCIS in physiologically relevant geometries of the mammary duct. Our model suggests several plausible progressions between morphologies of DCIS, and predicts that some regions of a duct are preferential for tumor cell invasion.

Bayesian inference of whole-organ deformation dynamics from limited space-time point data

To understand the morphogenetic mechanisms of organ development and regeneration, it is essential to clarify the inter-hierarchical relationship between microscopic, molecular/cellular activities and organ-level tissue deformation dynamics. While the former have been studied for several decades, the latter - macroscopic geometrical information about physical tissue deformation - is often missing, especially for many vertebrates. This is mainly because live recording of detailed cell behaviors in whole tissues during vertebrate organogenesis is technically difficult. In this study, we have developed a novel method that combines snapshot lineage tracing with Bayesian statistical estimation to construct whole-organ deformation maps from landmark data on limited numbers of space-time points. Following the validation of the method using artificially generated data sets, we applied it to the analysis of tissue deformation dynamics in chick limb development. A quantitative tissue deformation map for St.23-St.24 has been constructed, and its precision has been proven by evaluating its predictive performance. Geometrical analyses of the map have revealed a spatially heterogeneous volume growth pattern that is consistent with the expression pattern of a major morphogen and anisotropic tissue deformation along an axis. Thus, our method enables deformation dynamics analysis in organogenesis using practical lineage marking techniques.

A computational model predicting disruption of blood vessel development

Vascular development is a complex process regulated by dynamic biological networks that vary in topology and state across different tissues and developmental stages. Signals regulating de novo blood vessel formation (vasculogenesis) and remodeling (angiogenesis) come from a variety of biological pathways linked to endothelial cell (EC) behavior, extracellular matrix (ECM) remodeling and the local generation of chemokines and growth factors. Simulating these interactions at a systems level requires sufficient biological detail about the relevant molecular pathways and associated cellular behaviors, and tractable computational models that offset mathematical and biological complexity. Here, we describe a novel multicellular agent-based model of vasculogenesis using the CompuCell3D (http://www.compucell3d.org/) modeling environment supplemented with semi-automatic knowledgebase creation. The model incorporates vascular endothelial growth factor signals, pro- and anti-angiogenic inflammatory chemokine signals, and the plasminogen activating system of enzymes and proteases linked to ECM interactions, to simulate nascent EC organization, growth and remodeling. The model was shown to recapitulate stereotypical capillary plexus formation and structural emergence of non-coded cellular behaviors, such as a heterologous bridging phenomenon linking endothelial tip cells together during formation of polygonal endothelial cords. Molecular targets in the computational model were mapped to signatures of vascular disruption derived from in vitro chemical profiling using the EPA's ToxCast high-throughput screening (HTS) dataset. Simulating the HTS data with the cell-agent based model of vascular development predicted adverse effects of a reference anti-angiogenic thalidomide analog, 5HPP-33, on in vitro angiogenesis with respect to both concentration-response and morphological consequences. These findings support the utility of cell agent-based models for simulating a morphogenetic series of events and for the first time demonstrate the applicability of these models for predictive toxicology.

A mathematical model of tumour self-seeding reveals secondary metastatic deposits as drivers of primary tumour growth

Two models of circulating tumour cell (CTC) dynamics have been proposed to explain the phenomenon of tumour 'self-seeding', whereby CTCs repopulate the primary tumour and accelerate growth: primary seeding, where cells from a primary tumour shed into the vasculature and return back to the primary themselves; and secondary seeding, where cells from the primary first metastasize into a secondary tissue and form microscopic secondary deposits, which then shed cells into the vasculature returning to the primary. These two models are difficult to distinguish experimentally, yet the differences between them is of great importance to both our understanding of the metastatic process and also for designing methods of intervention. Therefore, we developed a mathematical model to test the relative likelihood of these two phenomena in the subset of tumours whose shed CTCs first encounter the lung capillary bed, and show that secondary seeding is several orders of magnitude more likely than primary seeding. We suggest how this difference could affect tumour evolution, progression and therapy, and propose several possible methods of experimental validation.

Mathematical modeling of vertebrate limb development

In this paper, we review the major mathematical and computational models of vertebrate limb development and their roles in accounting for different aspects of this process. The main aspects of limb development that have been modeled include outgrowth and shaping of the limb bud, establishment of molecular gradients within the bud, and formation of the skeleton. These processes occur interdependently during development, although (as described in this review), there are various interpretations of the biological relationships among them. A wide range of mathematical and computational methods have been used to study these processes, including ordinary and partial differential equation systems, cellular automata and discrete, stochastic models, finite difference methods, finite element methods, the immersed boundary method, and various combinations of the above. Multiscale mathematical modeling and associated computational simulation have become integrated into the study of limb morphogenesis and pattern formation to an extent with few parallels in the field of developmental biology. These methods have contributed to the design and analysis of experiments employing microsurgical and genetic manipulations, evaluation of hypotheses for limb bud outgrowth, interpretation of the effects of natural mutations, and the formulation of scenarios for the origination and evolution of the limb skeleton.

Computational and mathematical models of chondrogenesis in vertebrate limbs

The production of cartilage (chondrogenic patterning) in the limb is one of the best-studied examples of the emergence of form in developmental biology. At the core of the theoretical study is an effort to understand the mechanism that establishes the characteristic distribution of cartilage in the embryonic limb, which defines the future sites and shapes of bones that will be present in the mature limb. This review article gives an overview of the history and current state of a rich literature of mathematical and computational models that seek to contribute to this problem. We describe models for the mechanisms of limb growth and shaping via interaction with various chemical fields, as well as models addressing the intrinsic self-organization capabilities of the embryonic mesenchymal tissue, such as reaction-diffusion and mechanochemical models. We discuss the contributions of these models to the current understanding of chondrogenesis in vertebrate limbs, as well as their relation to the varied conceptual models that have been proposed by experimentalists.

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

INTRODUCTION

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

AREAS COVERED

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

EXPERT OPINION

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

Adhesion failures determine the pattern of choroidal neovascularization in the eye: a computer simulation study

Choroidal neovascularization (CNV) of the macular area of the retina is the major cause of severe vision loss in adults. In CNV, after choriocapillaries initially penetrate Bruch's membrane (BrM), invading vessels may regress or expand (CNV initiation). Next, during Early and Late CNV, the expanding vasculature usually spreads in one of three distinct patterns: in a layer between BrM and the retinal pigment epithelium (sub-RPE or Type 1 CNV), in a layer between the RPE and the photoreceptors (sub-retinal or Type 2 CNV) or in both loci simultaneously (combined pattern or Type 3 CNV). While most studies hypothesize that CNV primarily results from growth-factor effects or holes in BrM, our three-dimensional simulations of multi-cell model of the normal and pathological maculae recapitulate the three growth patterns, under the hypothesis that CNV results from combinations of impairment of: 1) RPE-RPE epithelial junctional adhesion, 2) Adhesion of the RPE basement membrane complex to BrM (RPE-BrM adhesion), and 3) Adhesion of the RPE to the photoreceptor outer segments (RPE-POS adhesion). Our key findings are that when an endothelial tip cell penetrates BrM: 1) RPE with normal epithelial junctions, basal attachment to BrM and apical attachment to POS resists CNV. 2) Small holes in BrM do not, by themselves, initiate CNV. 3) RPE with normal epithelial junctions and normal apical RPE-POS adhesion, but weak adhesion to BrM (e.g. due to lipid accumulation in BrM) results in Early sub-RPE CNV. 4) Normal adhesion of RBaM to BrM, but reduced apical RPE-POS or epithelial RPE-RPE adhesion (e.g. due to inflammation) results in Early sub-retinal CNV. 5) Simultaneous reduction in RPE-RPE epithelial binding and RPE-BrM adhesion results in either sub-RPE or sub-retinal CNV which often progresses to combined pattern CNV. These findings suggest that defects in adhesion dominate CNV initiation and progression.

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

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

Investigation of bone resorption within a cortical basic multicellular unit using a lattice-based computational model

In this paper we develop a lattice-based computational model focused on bone resorption by osteoclasts in a single cortical basic multicellular unit (BMU). Our model takes into account the interaction of osteoclasts with the bone matrix, the interaction of osteoclasts with each other, the generation of osteoclasts from a growing blood vessel, and the renewal of osteoclast nuclei by cell fusion. All these features are shown to strongly influence the geometrical properties of the developing resorption cavity including its size, shape and progression rate, and are also shown to influence the distribution, resorption pattern and trajectories of individual osteoclasts within the BMU. We demonstrate that for certain parameter combinations, resorption cavity shapes can be recovered from the computational model that closely resemble resorption cavity shapes observed from microCT imaging of human cortical bone.

Multi-scale modeling of tissues using CompuCell3D

The study of how cells interact to produce tissue development, homeostasis, or diseases was, until recently, almost purely experimental. Now, multi-cell computer simulation methods, ranging from relatively simple cellular automata to complex immersed-boundary and finite-element mechanistic models, allow in silico study of multi-cell phenomena at the tissue scale based on biologically observed cell behaviors and interactions such as movement, adhesion, growth, death, mitosis, secretion of chemicals, chemotaxis, etc. This tutorial introduces the lattice-based Glazier-Graner-Hogeweg (GGH) Monte Carlo multi-cell modeling and the open-source GGH-based CompuCell3D simulation environment that allows rapid and intuitive modeling and simulation of cellular and multi-cellular behaviors in the context of tissue formation and subsequent dynamics. We also present a walkthrough of four biological models and their associated simulations that demonstrate the capabilities of the GGH and CompuCell3D.

Computer simulations of cell sorting due to differential adhesion

The actions of cell adhesion molecules, in particular, cadherins during embryonic development and morphogenesis more generally, regulate many aspects of cellular interactions, regulation and signaling. Often, a gradient of cadherin expression levels drives collective and relative cell motions generating macroscopic cell sorting. Computer simulations of cell sorting have focused on the interactions of cells with only a few discrete adhesion levels between cells, ignoring biologically observed continuous variations in expression levels and possible nonlinearities in molecular binding. In this paper, we present three models relating the surface density of cadherins to the net intercellular adhesion and interfacial tension for both discrete and continuous levels of cadherin expression. We then use then the Glazier-Graner-Hogeweg (GGH) model to investigate how variations in the distribution of the number of cadherins per cell and in the choice of binding model affect cell sorting. We find that an aggregate with a continuous variation in the level of a single type of cadherin molecule sorts more slowly than one with two levels. The rate of sorting increases strongly with the interfacial tension, which depends both on the maximum difference in number of cadherins per cell and on the binding model. Our approach helps connect signaling at the molecular level to tissue-level morphogenesis.

Evolving homeostatic tissue using genetic algorithms

Multicellular organisms maintain form and function through a multitude of homeostatic mechanisms. The details of these mechanisms are in many cases unknown, and so are their evolutionary origin and their link to development. In order to illuminate these issues we have investigated the evolution of structural homeostasis in the simplest of cases, a tissue formed by a mono-layer of cells. To this end, we made use of a 3-dimensional hybrid cellular automaton, an individual-based model in which the behaviour of each cell depends on its local environment. Using an evolutionary algorithm (EA) we evolved cell signalling networks, both with a fixed and an incremental fitness evaluation, which give rise to and maintain a mono-layer tissue structure. Analysis of the solutions provided by the EA shows that the two evaluation methods gives rise to different types of solutions to the problem of homeostasis. The fixed method leads to almost optimal solutions, where the tissue relies on a high rate of cell turnover, while the solutions from the incremental scheme behave in a more conservative manner, only dividing when necessary. In order to test the robustness of the solutions we subjected them to environmental stress, by wounding the tissue, and to genetic stress, by introducing mutations. The results show that the robustness very much depends on the mechanism responsible for maintaining homeostasis. The two evolved cell types analysed present contrasting mechanisms by which tissue homeostasis can be maintained. This compares well to different tissue types found in multicellular organisms. For example the epithelial cells lining the colon in humans are shed at a considerable rate, while in other tissue types, which are not as exposed, the conservative type of homeostatic mechanism is normally found. These results will hopefully shed light on how multicellular organisms have evolved homeostatic mechanisms and what might occur when these mechanisms fail, as in the case of cancer.

Budding behaviors: Growth of the limb as a model of morphogenesis

Questions regarding morphogenesis have played second fiddle to those pertaining to pattern formation among the limb development set for some time. A recent series of publications has reinvigorated the search for mechanisms by which the limb bud arises, elongates and acquires its peculiar shape. While there are stage-specific variations, the theme that resonates across these studies is that mesoderm and cartilage cells in the limb bud exhibit polarity that drives directional movement and oriented division. Noncanonical Wnt signalling is important for these cell behaviors at all stages of limb development. While the emerging morphogenetic mechanisms underlying limb bud outgrowth are partly analogous to those of other developing structures, insights from the limb have the potential to reveal intriguing new mechanisms by which three dimensional mesoderm changes shape.

The Hemimelic extra toes mouse mutant: Historical perspective on unraveling mechanisms of dysmorphogenesis

Hemimelic extra toes (Hx) arose spontaneously as a dominant mutation in B10.D2/nSnJ mice in 1967. It specifically affects the appendicular skeleton, causing variable foreshortening of the tibia (radius) and preaxial polydactylism. Early anatomical studies revealed anterior overgrowth of the autopod, with decreased apoptosis and increased mitosis in the anterior apical ectodermal ridge and underlying mesenchyme; overextension of apoptosis in the central zeugopod accounted for hemimelia. The Hx mutant phenotype was coarsely mapped to mouse chromosome (Chr) 5 and closely linked to engrailed-2 (En2) and Sonic hedgehog (Shh). This region is syntenic to human Chr 7q36 that harbors several dominant mutations affecting the hand. High-resolution genome mapping identified the Hx mutation as a G --> A base pair transition within Intron 5 of the murine Lmbr1 locus. The critical effect is on a multifunctional conserved regulatory element that acts as a limb-specific, long-distance cis-acting enhancer of Shh expression. As such, the Hx mutant phenotype results from ectopic Shh signals at the anterior margin of the limb bud that directly or indirectly alter FGF4 signaling from the apical ectodermal ridge. Given significant advances in understanding of embryonic development in general and limb development in particular, this review article reveals how research that once attracted interest of teratologists has advanced across the decades to pinpoint a critical molecular lesion and reveal a potential mechanism of a specific malformation that is found commonly in experimental teratology.

The role of spatially controlled cell proliferation in limb bud morphogenesis

Although the vertebrate limb bud has been studied for decades as a model system for spatial pattern formation and cell specification, the cellular basis of its distally oriented elongation has been a relatively neglected topic by comparison. The conventional view is that a gradient of isotropic proliferation exists along the limb, with high proliferation rates at the distal tip and lower rates towards the body, and that this gradient is the driving force behind outgrowth. Here we test this hypothesis by combining quantitative empirical data sets with computer modelling to assess the potential role of spatially controlled proliferation rates in the process of directional limb bud outgrowth. In particular, we generate two new empirical data sets for the mouse hind limb--a numerical description of shape change and a quantitative 3D map of cell cycle times--and combine these with a new 3D finite element model of tissue growth. By developing a parameter optimization approach (which explores spatial patterns of tissue growth) our computer simulations reveal that the observed distribution of proliferation rates plays no significant role in controlling the distally extending limb shape, and suggests that directional cell activities are likely to be the driving force behind limb bud outgrowth. This theoretical prediction prompted us to search for evidence of directional cell orientations in the limb bud mesenchyme, and we thus discovered a striking highly branched and extended cell shape composed of dynamically extending and retracting filopodia, a distally oriented bias in Golgi position, and also a bias in the orientation of cell division. We therefore provide both theoretical and empirical evidence that limb bud elongation is achieved by directional cell activities, rather than a PD gradient of proliferation rates.

The role of spatially controlled cell proliferation in limb bud morphogenesis

Although the vertebrate limb bud has been studied for decades as a model system for spatial pattern formation and cell specification, the cellular basis of its distally oriented elongation has been a relatively neglected topic by comparison. The conventional view is that a gradient of isotropic proliferation exists along the limb, with high proliferation rates at the distal tip and lower rates towards the body, and that this gradient is the driving force behind outgrowth. Here we test this hypothesis by combining quantitative empirical data sets with computer modelling to assess the potential role of spatially controlled proliferation rates in the process of directional limb bud outgrowth. In particular, we generate two new empirical data sets for the mouse hind limb--a numerical description of shape change and a quantitative 3D map of cell cycle times--and combine these with a new 3D finite element model of tissue growth. By developing a parameter optimization approach (which explores spatial patterns of tissue growth) our computer simulations reveal that the observed distribution of proliferation rates plays no significant role in controlling the distally extending limb shape, and suggests that directional cell activities are likely to be the driving force behind limb bud outgrowth. This theoretical prediction prompted us to search for evidence of directional cell orientations in the limb bud mesenchyme, and we thus discovered a striking highly branched and extended cell shape composed of dynamically extending and retracting filopodia, a distally oriented bias in Golgi position, and also a bias in the orientation of cell division. We therefore provide both theoretical and empirical evidence that limb bud elongation is achieved by directional cell activities, rather than a PD gradient of proliferation rates.

Front instabilities and invasiveness of simulated 3D avascular tumors

We use the Glazier-Graner-Hogeweg model to simulate three-dimensional (3D), single-phenotype, avascular tumors growing in an homogeneous tissue matrix (TM) supplying a single limiting nutrient. We study the effects of two parameters on tumor morphology: a diffusion-limitation parameter defined as the ratio of the tumor-substrate consumption rate to the substrate-transport rate, and the tumor-TM surface tension. This initial model omits necrosis and oxidative/hypoxic metabolism effects, which can further influence tumor morphology, but our simplified model still shows significant parameter dependencies. The diffusion-limitation parameter determines whether the growing solid tumor develops a smooth (noninvasive) or fingered (invasive) interface, as in our earlier two-dimensional (2D) simulations. The sensitivity of 3D tumor morphology to tumor-TM surface tension increases with the size of the diffusion-limitation parameter, as in 2D. The 3D results are unexpectedly close to those in 2D. Our results therefore may justify using simpler 2D simulations of tumor growth, instead of more realistic but more computationally expensive 3D simulations. While geometrical artifacts mean that 2D sections of connected 3D tumors may be disconnected, the morphologies of 3D simulated tumors nevertheless correlate with the morphologies of their 2D sections, especially for low-surface-tension tumors, allowing the use of 2D sections to partially reconstruct medically-important 3D-tumor structures.

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.

Front instabilities and invasiveness of simulated avascular tumors

We study the interface morphology of a 2D simulation of an avascular tumor composed of identical cells growing in an homogeneous healthy tissue matrix (TM), in order to understand the origin of the morphological changes often observed during real tumor growth. We use the Glazier-Graner-Hogeweg model, which treats tumor cells as extended, deformable objects, to study the effects of two parameters: a dimensionless diffusion-limitation parameter defined as the ratio of the tumor consumption rate to the substrate transport rate, and the tumor-TM surface tension. We model TM as a nondiffusing field, neglecting the TM pressure and haptotactic repulsion acting on a real growing tumor; thus, our model is appropriate for studying tumors with highly motile cells, e.g., gliomas. We show that the diffusion-limitation parameter determines whether the growing tumor develops a smooth (noninvasive) or fingered (invasive) interface, and that the sensitivity of tumor morphology to tumor-TM surface tension increases with the size of the dimensionless diffusion-limitation parameter. For large diffusion-limitation parameters, we find a transition (missed in previous work) between dendritic structures, produced when tumor-TM surface tension is high, and seaweed-like structures, produced when tumor-TM surface tension is low. This observation leads to a direct analogy between the mathematics and dynamics of tumors and those observed in nonbiological directional solidification. Our results are also consistent with the biological observation that hypoxia promotes invasive growth of tumor cells by inducing higher levels of receptors for scatter factors that weaken cell-cell adhesion and increase cell motility. These findings suggest that tumor morphology may have value in predicting the efficiency of antiangiogenic therapy in individual patients.

Multicell simulations of development and disease using the CompuCell3D simulation environment

Mathematical modeling and computer simulation have become crucial to biological fields from genomics to ecology. However, multicell, tissue-level simulations of development and disease have lagged behind other areas because they are mathematically more complex and lack easy-to-use software tools that allow building and running in silico experiments without requiring in-depth knowledge of programming. This tutorial introduces Glazier-Graner-Hogeweg (GGH) multicell simulations and CompuCell3D, a simulation framework that allows users to build, test, and run GGH simulations.

Simulation of single-species bacterial-biofilm growth using the Glazier-Graner-Hogeweg model and the CompuCell3D modeling environment

The CompuCell3D modeling environment provides a convenient platform for biofilm simulations using the Glazier-Graner-Hogeweg (GGH) model, a cell-oriented framework designed to simulate growth and pattern formation due to biological cells' behaviors. We show how to develop such a simulation, based on the hybrid (continuum-discrete) model of Picioreanu, van Loosdrecht, and Heijnen (PLH), simulate the growth of a single-species bacterial biofilm, and study the roles of cell-cell and cell-field interactions in determining biofilm morphology. In our simulations, which generalize the PLH model by treating cells as spatially extended, deformable bodies, differential adhesion between cells, and their competition for a substrate (nutrient), suffice to produce a fingering instability that generates the finger shapes of biofilms. Our results agree with most features of the PLH model, although our inclusion of cell adhesion, which is difficult to implement using other modeling approaches, results in slightly different patterns. Our simulations thus provide the groundwork for simulations of medically and industrially important multispecies biofilms.

Coordinated action of N-CAM, N-cadherin, EphA4, and ephrinB2 translates genetic prepatterns into structure during somitogenesis in chick

During gastrulation in vertebrates, mesenchymal cells at the anterior end of the presomitic mesoderm (PSM) periodically compact, transiently epithelialize and detach from the posterior PSM to form somites. In the prevailing clock-and-wavefront model of somitogenesis, periodic gene expression, particularly of Notch and Wnt, interacts with an FGF8-based thresholding mechanism to determine cell fates. However, this model does not explain how cell determination and subsequent differentiation translates into somite morphology. In this paper, we use computer simulations of chick somitogenesis to show that experimentally-observed temporal and spatial patterns of adhesive N-CAM and N-cadherin and repulsive EphA4-ephrinB2 pairs suffice to reproduce the complex dynamic morphological changes of somitogenesis in wild-type and N-cadherin (-/-) chick, including intersomitic separation, boundary-shape evolution and sorting of misdifferentiated cells across compartment boundaries. Since different models of determination yield the same, experimentally-observed, distribution of adhesion and repulsion molecules, the patterning is independent of the details of this mechanism.

Regular mosaic pattern development: a study of the interplay between lateral inhibition, apoptosis and differential adhesion

Background

A significant body of literature is devoted to modeling developmental mechanisms that create patterns within groups of initially equivalent embryonic cells. Although it is clear that these mechanisms do not function in isolation, the timing of and interactions between these mechanisms during embryogenesis is not well known. In this work, a computational approach was taken to understand how lateral inhibition, differential adhesion and programmed cell death can interact to create a mosaic pattern of biologically realistic primary and secondary cells, such as that formed by sensory (primary) and supporting (secondary) cells of the developing chick inner ear epithelium.

Results

Four different models that interlaced cellular patterning mechanisms in a variety of ways were examined and their output compared to the mosaic of sensory and supporting cells that develops in the chick inner ear sensory epithelium. The results show that: 1) no single patterning mechanism can create a 2-dimensional mosaic pattern of the regularity seen in the chick inner ear; 2) cell death was essential to generate the most regular mosaics, even through extensive cell death has not been reported for the developing basilar papilla; 3) a model that includes an iterative loop of lateral inhibition, programmed cell death and cell rearrangements driven by differential adhesion created mosaics of primary and secondary cells that are more regular than the basilar papilla; 4) this same model was much more robust to changes in homo- and heterotypic cell-cell adhesive differences than models that considered either fewer patterning mechanisms or single rather than iterative use of each mechanism.

Conclusion

Patterning the embryo requires collaboration between multiple mechanisms that operate iteratively. Interlacing these mechanisms into feedback loops not only refines the output patterns, but also increases the robustness of patterning to varying initial cell states.

Simulating invasion with cellular automata: connecting cell-scale and population-scale properties

Interpretive and predictive tools are needed to assist in the understanding of cell invasion processes. Cell invasion involves cell motility and proliferation, and is central to many biological processes including developmental morphogenesis and tumor invasion. Experimental data can be collected across a wide range of scales, from the population scale to the individual cell scale. Standard continuum or discrete models used in isolation are insufficient to capture this wide range of data. We develop a discrete cellular automata model of invasion with experimentally motivated rules. The cellular automata algorithm is applied to a narrow two-dimensional lattice and simulations reveal the formation of invasion waves moving with constant speed. The simulation results are averaged in one dimension-these data are used to identify the time history of the leading edge to characterize the population-scale wave speed. This allows the relationship between the population-scale wave speed and the cell-scale parameters to be determined. This relationship is analogous to well-known continuum results for Fisher's equation. The cellular automata algorithm also produces individual cell trajectories within the invasion wave that are analogous to cell trajectories obtained with new experimental techniques. Our approach allows both the cell-scale and population-scale properties of invasion to be predicted in a way that is consistent with multiscale experimental data. Furthermore we suggest that the cellular automata algorithm can be used in conjunction with individual data to overcome limitations associated with identifying cell motility mechanisms using continuum models alone.

Differential adhesion in morphogenesis: a modern view

The spreading of one embryonic tissue over another, the sorting out of their cells when intermixed and the formation of intertissue boundaries respected by the motile border cells all have counterparts in the behavior of immiscible liquids. The 'differential adhesion hypothesis' (DAH) explains these liquid-like tissue behaviors as consequences of the generation of tissue surface and interfacial tensions arising from the adhesion energies between motile cells. The experimental verification of the DAH, the recent computational models simulating adhesion-mediated morphogenesis, and the evidence concerning the role of differential adhesion in a number of morphodynamic events, including teleost epiboly, the specification of boundaries between rhombomeres in the developing vertebrate hindbrain, epithelial-mesenchymal transitions in embryos, and malignant invasion are reviewed here.