A Hybrid Discrete-Continuum Model for 3-D Skeletogenesis of the Vertebrate Limb. In: Sloot PMA, Chopard B, Hoekstra AG, editors. Cellular Automata. Berlin: Springer Berlin Heidelberg; 2004. pp. 543-52. [DOI: 10.1007/978-3-540-30479-1_56]

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.

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.

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.

Cell-oriented modeling of angiogenesis

Due to its significant involvement in various physiological and pathological conditions, angiogenesis (the development of new blood vessels from an existing vasculature) represents an important area of the actual biological research and a field in which mathematical modeling proved particularly useful in supporting the experimental work. In this paper, we focus on a specific modeling strategy, known as "cell-centered" approach. This type of mathematical models work at a "mesoscopic scale," assuming the cell as the natural level of abstraction for computational modeling of development. They treat cells phenomenologically, considering their essential behaviors to study how tissue structure and organization emerge from the collective dynamics of multiple cells. The main contributions of the cell-oriented approach to the study of the angiogenic process will be described. From one side, they have generated "basic science understanding" about the process of capillary assembly during development, growth, and pathology. On the other side, models were also developed supporting "applied biomedical research" for the purpose of identifying new therapeutic targets and clinically relevant approaches for either inhibiting or stimulating angiogenesis.

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.

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.

Distance between AER and ZPA is defined by feed-forward loop and is stabilized by their feedback loop in vertebrate limb bud

In the development of organs, multiple morphogen sources are often involved, and interact with each other. For example, the apical ectodermal ridge (AER) and the zone of polarizing activity (ZPA) are major morphogen sources in the limb bud formation of vertebrates. Fgf expression in the AER and Shh expression in the ZPA are maintained by their positive feedback regulation mediated by diffusible molecules, FGF and SHH. A recent experimental observation suggests that the FGF-signal regulates the Shh expression in a feed-forward manner with activation and repression regulatory pathways. We study the coupled dynamics of Shh expression in the ZPA and Fgf expression in the AER, and the relationship of the relative position between AER and ZPA. We first show that with the feed-forward regulation only, the peak of ZPA activity can be formed distant from the AER as observed experimentally. Then, we clarify that the robustness of the ZPA spatial pattern to changes in system parameters is enhanced by adding the feedback regulation between the AER and the ZPA. Furthermore, sensitivity analysis shows that there exists the optimal feedback strength where the robustness is the most improved.

A framework for three-dimensional simulation of morphogenesis

We present COMPUCELL3D, a software framework for three-dimensional simulation of morphogenesis in different organisms. COMPUCELL3D employs biologically relevant models for cell clustering, growth, and interaction with chemical fields. COMPUCELL3D uses design patterns for speed, efficient memory management, extensibility, and flexibility to allow an almost unlimited variety of simulations. We have verified COMPUCELL3D by building a model of growth and skeletal pattern formation in the avian (chicken) limb bud. Binaries and source code are available, along with documentation and input files for sample simulations, at http:// compucell.sourceforge.net.