Lattice Boltzmann simulations of the time evolution of living multicellular systems

Embryonic tissues and multicellular aggregates of adult cells mimic the behavior of highly viscous liquids. The liquid analogy helps to understand morphogenetic phenomena, such as cell sorting and tissue fusion, observed in developmental biology and tissue engineering. Tissue fusion is vital in tissue printing, an emergent technique based on computer-controlled deposition of tissue fragments and biocompatible materials. Computer simulations proved useful in predicting post-printing shape changes of tissue constructs. The simulation methods available to date, however, are unable to describe the time evolution of living systems made of millions of cells. The Lattice Boltzmann (LB) approach allows the implementation of interaction forces between the constituents of the system and yields time evolution in terms of distribution functions. With tissue engineering applications in mind, we have developed a finite difference Lattice Boltzmann model of a multicellular system and applied it to simulate the sidewise fusion of two contiguous cylinders made of cohesive cells and embedded in a medium (hydrogel). We have identified a biologically relevant range of model parameters. The proposed LB model may be extended to describe the time evolution of more complex multicellular structures such as sheets or tubes produced by tissue printing.     

Osteoblasts infill irregular pores under curvature and porosity controls: a hypothesis-testing analysis of cell behaviours

The geometric control of bone tissue growth plays a significant role in bone remodelling, age-related bone loss, and tissue engineering. However, how exactly geometry influences the behaviour of bone-forming cells remains elusive. Geometry modulates cell populations collectively through the evolving space available to the cells, but it may also modulate the individual behaviours of cells. To factor out the collective influence of geometry and gain access to the geometric regulation of individual cell behaviours, we develop a mathematical model of the infilling of cortical bone pores and use it with available experimental data on cortical infilling rates. Testing different possible modes of geometric controls of individual cell behaviours consistent with the experimental data, we find that efficient smoothing of irregular pores only occurs when cell secretory rate is controlled by porosity rather than curvature. This porosity control suggests the convergence of a large scale of intercellular signalling to single bone-forming cells, consistent with that provided by the osteocyte network in response to mechanical stimulus. After validating the mathematical model with the histological record of a real cortical pore infilling, we explore the infilling of a population of randomly generated initial pore shapes. We find that amongst all the geometric regulations considered, the collective influence of curvature on cell crowding is a dominant factor for how fast cortical bone pores infill, and we suggest that the irregularity of cement lines thereby explains some of the variability in double labelling data as well as the overall speed of osteon infilling.     

A model of growth restraints to explain the development and evolution of tooth shapes in mammals

The problem investigated here is control of the development of tooth shape. Cells at the growing soft tissue interface between the ectoderm and mesoderm in a tooth anlage are observed to buckle and fold into a template for the shape of the tooth crown. The final shape is created by enamel secreted onto the folds. The pattern in which the folds develop is generally explained as a response to the pattern in which genes are locally expressed at the interface. This congruence leaves the problem of control unanswered because it does not explain how either pattern is controlled. Obviously, cells are subject to Newton's laws of motion so that mechanical forces and constraints must ultimately cause the movements of cells during tooth morphogenesis. A computer model is used to test the hypothesis that directional resistances to growth of the epithelial part of the interface could account for the shape into which the interface folds. The model starts with a single epithelial cell whose growth is constrained by 4 constant directional resistances (anterior, posterior, medial and lateral). The constraints force the growing epithelium to buckle and fold. By entering into the model different values for these constraints the modeled epithelium is induced to buckle and fold into the different shapes associated with the evolution of a human upper molar from that of a reptilian ancestor. The patterns and sizes of cusps and the sequences in which they develop are all correctly reproduced. The model predicts the changes in the 4 directional constraints necessary to develop and evolve from one tooth shape into another. I conclude more generally expressed genes that control directional resistances to growth, not locally expressed genes, may provide the information for the shape into which a tooth develops.     

Population dynamics in modeling tumor growth

A new approach based on local interaction between cancer and tissue cells is applied to the problem of the onset and growth of solid tumors in homogeneous tissues and effects associated with dramatic changes in tumor growth after crossing the boundary between different tissues. The characteristic sizes and growth rates of spherical tumors, the points of the beginning and the end of spherical growth, and further development of complex structures from the spherical ones (rough interface between the tumor and the host tissue, elongate outgrowths, dendritic structures, and metastases) are inferred assuming that the reproduction rate of a population of cancer cells is a nonmonotonous function of their local concentration and thus of the local curvature of the tumor surface. The growth behavior changes dramatically when the tumor crosses a boundary between two tissues.     

Population dynamics in tumor growth modeling

A new approach based on local interaction between cancer and tissue cells was applied to the problem of the onset and growth of solid tumors in homogeneous tissues and effects associated with dramatic changes in tumor growth after crossing the boundary between different tissues. The characteristic sizes and growth rates of spherical tumors, the points of the beginning and the end of spherical growth, and the further development of complex structures from the spherical ones (rough interface between the tumor and the host tissue, elongate outgrowths, dendritic structures, and metastases) were inferred assuming that the reproduction rate of a population of cancer cells is a nonmonotone function of their local concentration and thus of the local curvature of the tumor surface.     

Physical Model of the Dynamic Instability in an Expanding Cell Culture

Collective cell migration is of great significance in many biological processes. The goal of this work is to give a physical model for the dynamics of cell migration during the wound healing response. Experiments demonstrate that an initially uniform cell-culture monolayer expands in a nonuniform manner, developing fingerlike shapes. These fingerlike shapes of the cell culture front are composed of columns of cells that move collectively. We propose a physical model to explain this phenomenon, based on the notion of dynamic instability. In this model, we treat the first layers of cells at the front of the moving cell culture as a continuous one-dimensional membrane (contour), with the usual elasticity of a membrane: curvature and surface-tension. This membrane is active, due to the forces of cellular motility of the cells, and we propose that this motility is related to the local curvature of the culture interface; larger convex curvature correlates with a stronger cellular motility force. This shape-force relation gives rise to a dynamic instability, which we then compare to the patterns observed in the wound healing experiments.     

A 3-D model of tumor progression based on complex automata driven by particle dynamics

The dynamics of a growing tumor involving mechanical remodeling of healthy tissue and vasculature is neglected in most of the existing tumor models. This is due to the lack of efficient computational framework allowing for simulation of mechanical interactions. Meanwhile, just these interactions trigger critical changes in tumor growth dynamics and are responsible for its volumetric and directional progression. We describe here a novel 3-D model of tumor growth, which combines particle dynamics with cellular automata concept. The particles represent both tissue cells and fragments of the vascular network. They interact with their closest neighbors via semi-harmonic central forces simulating mechanical resistance of the cell walls. The particle dynamics is governed by both the Newtonian laws of motion and the cellular automata rules. These rules can represent cell life-cycle and other biological interactions involving smaller spatio-temporal scales. We show that our complex automata, particle based model can reproduce realistic 3-D dynamics of the entire system consisting of the tumor, normal tissue cells, blood vessels and blood flow. It can explain phenomena such as the inward cell motion in avascular tumor, stabilization of tumor growth by the external pressure, tumor vascularization due to the process of angiogenesis, trapping of healthy cells by invading tumor, and influence of external (boundary) conditions on the direction of tumor progression. We conclude that the particle model can serve as a general framework for designing advanced multiscale models of tumor dynamics and it is very competitive to the modeling approaches presented before.     

A validated model of GAG deposition,cell distribution,and growth of tissue engineered cartilage cultured in a rotating bioreactor

In this work a new phenomenological model of growth of cartilage tissue cultured in a rotating bioreactor is developed. It represents an advancement of a previously derived model of deposition of glycosaminoglycan (GAG) in engineered cartilage by (i) introduction of physiological mechanisms of proteoglycan accumulation in the extracellular matrix (ECM) as well as by correlating (ii) local cell densities and (iii) tissue growth to the ECM composition. In particular, previously established predictions and correlations of local oxygen concentrations and GAG synthesis rates are extended to distinguish cell secreted proteoglycan monomers free to diffuse in cell surroundings and outside from the engineered construct, from large aggrecan molecules, which are constrained within the ECM and practically immovable. The model includes kinetics of aggregation, that is, transformation of mobile GAG species into immobile aggregates as well as maintenance of the normal ECM composition after the physiological GAG concentration is reached by incorporation of a product inhibition term. The model also includes mechanisms of the temporal evolution of cell density distributions and tissue growth under in vitro conditions. After a short initial proliferation phase the total cell number in the construct remains constant, but the local cell distribution is leveled out by GAG accumulation and repulsion due to negative molecular charges. Furthermore, strong repulsive forces result in expansion of the local tissue elements observed macroscopically as tissue growth (i.e., construct enlargement). The model is validated by comparison with experimental data of (i) GAG distribution and leakage, (ii) spatial‐temporal distributions of cells, and (iii) tissue growth reported in previous works. Validation of the model predictive capability—against a selection of measured data that were not used to construct the model—suggests that the model successfully describes the interplay of several simultaneous processes carried out during in vitro cartilage tissue regeneration and indicates that this approach could also be attractive for application in other tissue engineering systems. Biotechnol. Bioeng. 2010. 105: 842–853. © 2009 Wiley Periodicals, Inc.     

Cell Geometry Guides the Dynamic Targeting of Apoplastic GPI-Linked Lipid Transfer Protein to Cell Wall Elements and Cell Borders in Arabidopsis thaliana

During cellular morphogenesis, changes in cell shape and cell junction topology are fundamental to normal tissue and organ development. Here we show that apoplastic Glycophosphatidylinositol (GPI)-anchored Lipid Transfer Protein (LTPG) is excluded from cell junctions and flat wall regions, and passively accumulates around their borders in the epidermal cells of Arabidopsis thaliana. Beginning with intense accumulation beneath highly curved cell junction borders, this enrichment is gradually lost as cells become more bulbous during their differentiation. In fully mature epidermal cells, YFP-LTPG often shows a fibrous cellulose microfibril-like pattern within the bulging outer faces. Physical contact between a flat glass surface and bulbous cell surface induces rapid and reversible evacuation from contact sites and accumulation to the curved wall regions surrounding the contact borders. Thus, LTPG distribution is dynamic, responding to changes in cell shape and wall curvature during cell growth and differentiation. We hypothesize that this geometry-based mechanism guides wax-carrying LTPG to functional sites, where it may act to “seal” the vulnerable border surrounding cell-cell junctions and assist in cell wall fortification and cuticular wax deposition.     

A Computational Framework for 3D Mechanical Modeling of Plant Morphogenesis with Cellular Resolution

The link between genetic regulation and the definition of form and size during morphogenesis remains largely an open question in both plant and animal biology. This is partially due to the complexity of the process, involving extensive molecular networks, multiple feedbacks between different scales of organization and physical forces operating at multiple levels. Here we present a conceptual and modeling framework aimed at generating an integrated understanding of morphogenesis in plants. This framework is based on the biophysical properties of plant cells, which are under high internal turgor pressure, and are prevented from bursting because of the presence of a rigid cell wall. To control cell growth, the underlying molecular networks must interfere locally with the elastic and/or plastic extensibility of this cell wall. We present a model in the form of a three dimensional (3D) virtual tissue, where growth depends on the local modulation of wall mechanical properties and turgor pressure. The model shows how forces generated by turgor-pressure can act both cell autonomously and non-cell autonomously to drive growth in different directions. We use simulations to explore lateral organ formation at the shoot apical meristem. Although different scenarios lead to similar shape changes, they are not equivalent and lead to different, testable predictions regarding the mechanical and geometrical properties of the growing lateral organs. Using flower development as an example, we further show how a limited number of gene activities can explain the complex shape changes that accompany organ outgrowth.     

Kalman smoothing improves the estimation of joint kinematics and kinetics in marker-based human gait analysis

We developed a Kalman smoothing algorithm to improve estimates of joint kinematics from measured marker trajectories during motion analysis. Kalman smoothing estimates are based on complete marker trajectories. This is an improvement over other techniques, such as the global optimisation method (GOM), Kalman filtering, and local marker estimation (LME), where the estimate at each time instant is only based on part of the marker trajectories. We applied GOM, Kalman filtering, LME, and Kalman smoothing to marker trajectories from both simulated and experimental gait motion, to estimate the joint kinematics of a ten segment biomechanical model, with 21 degrees of freedom. Three simulated marker trajectories were studied: without errors, with instrumental errors, and with soft tissue artefacts (STA). Two modelling errors were studied: increased thigh length and hip centre dislocation. We calculated estimation errors from the known joint kinematics in the simulation study. Compared with other techniques, Kalman smoothing reduced the estimation errors for the joint positions, by more than 50% for the simulated marker trajectories without errors and with instrumental errors. Compared with GOM, Kalman smoothing reduced the estimation errors for the joint moments by more than 35%. Compared with Kalman filtering and LME, Kalman smoothing reduced the estimation errors for the joint accelerations by at least 50%. Our simulation results show that the use of Kalman smoothing substantially improves the estimates of joint kinematics and kinetics compared with previously proposed techniques (GOM, Kalman filtering, and LME) for both simulated, with and without modelling errors, and experimentally measured gait motion.     

Cell shape dynamics: from waves to migration

We observe and quantify wave-like characteristics of amoeboid migration. Using the amoeba Dictyostelium discoideum, a model system for the study of chemotaxis, we demonstrate that cell shape changes in a wave-like manner. Cells have regions of high boundary curvature that propagate from the leading edge toward the back, usually along alternating sides of the cell. Curvature waves are easily seen in cells that do not adhere to a surface, such as cells that are electrostatically repelled from surfaces or cells that extend over the edge of micro-fabricated cliffs. Without surface contact, curvature waves travel from the leading edge to the back of a cell at -35 μm/min. Non-adherent myosin II null cells do not exhibit these curvature waves. At the leading edge of adherent cells, curvature waves are associated with protrusive activity. Like regions of high curvature, protrusive activity travels along the boundary in a wave-like manner. Upon contact with a surface, the protrusions stop moving relative to the surface, and the boundary shape thus reflects the history of protrusive motion. The wave-like character of protrusions provides a plausible mechanism for the zig-zagging of pseudopods and for the ability of cells both to swim in viscous fluids and to navigate complex three dimensional topography.     

Generation of diverse biological forms through combinatorial interactions between tissue polarity and growth

A major problem in biology is to understand how complex tissue shapes may arise through growth. In many cases this process involves preferential growth along particular orientations raising the question of how these orientations are specified. One view is that orientations are specified through stresses in the tissue (axiality-based system). Another possibility is that orientations can be specified independently of stresses through molecular signalling (polarity-based system). The axiality-based system has recently been explored through computational modelling. Here we develop and apply a polarity-based system which we call the Growing Polarised Tissue (GPT) framework. Tissue is treated as a continuous material within which regionally expressed factors under genetic control may interact and propagate. Polarity is established by signals that propagate through the tissue and is anchored in regions termed tissue polarity organisers that are also under genetic control. Rates of growth parallel or perpendicular to the local polarity may then be specified through a regulatory network. The resulting growth depends on how specified growth patterns interact within the constraints of mechanically connected tissue. This constraint leads to the emergence of features such as curvature that were not directly specified by the regulatory networks. Resultant growth feeds back to influence spatial arrangements and local orientations of tissue, allowing complex shapes to emerge from simple rules. Moreover, asymmetries may emerge through interactions between polarity fields. We illustrate the value of the GPT-framework for understanding morphogenesis by applying it to a growing Snapdragon flower and indicate how the underlying hypotheses may be tested by computational simulation. We propose that combinatorial intractions between orientations and rates of growth, which are a key feature of polarity-based systems, have been exploited during evolution to generate a range of observed biological shapes.     

The interplay between tissue growth and scaffold degradation in engineered tissue constructs

In vitro tissue engineering is emerging as a potential tool to meet the high demand for replacement tissue, caused by the increased incidence of tissue degeneration and damage. A key challenge in this field is ensuring that the mechanical properties of the engineered tissue are appropriate for the in vivo environment. Achieving this goal will require detailed understanding of the interplay between cell proliferation, extracellular matrix (ECM) deposition and scaffold degradation. In this paper, we use a mathematical model (based upon a multiphase continuum framework) to investigate the interplay between tissue growth and scaffold degradation during tissue construct evolution in vitro. Our model accommodates a cell population and culture medium, modelled as viscous fluids, together with a porous scaffold and ECM deposited by the cells, represented as rigid porous materials. We focus on tissue growth within a perfusion bioreactor system, and investigate how the predicted tissue composition is altered under the influence of (1) differential interactions between cells and the supporting scaffold and their associated ECM, (2) scaffold degradation, and (3) mechanotransduction-regulated cell proliferation and ECM deposition. Numerical simulation of the model equations reveals that scaffold heterogeneity typical of that obtained from $\mu $ CT scans of tissue engineering scaffolds can lead to significant variation in the flow-induced mechanical stimuli experienced by cells seeded in the scaffold. This leads to strong heterogeneity in the deposition of ECM. Furthermore, preferential adherence of cells to the ECM in favour of the artificial scaffold appears to have no significant influence on the eventual construct composition; adherence of cells to these supporting structures does, however, lead to cell and ECM distributions which mimic and exaggerate the heterogeneity of the underlying scaffold. Such phenomena have important ramifications for the mechanical integrity of engineered tissue constructs and their suitability for implantation in vivo.     

Three-dimensional osseo-ligamentous model of the thorax representing initiation of scoliosis by asymmetric growth

A biomechanical model of the human thorax was constructed to investigate how asymmetric growth of the thorax might initiate spinal lateral curvature and axial rotation as seen in scoliosis deformities. Geometric data specifying nodal points of the model were taken from stereo-radiographs of an adolescent subject. An initially symmetrical geometry was created by 'mirroring' measurements of a hemi-thorax and spine. Published data provided cross-sectional measurements of the ribs, material properties of tissues and global flexibilities of the intervertebral motion segments. The ribs, sternum, intervertebral motion segments and intercostal ligaments were represented by elastic elements. Model deformations were calculated by the direct stiffness finite element method, with growth represented by an initial strain term in the constitutive law. Non-linear behavior was accommodated by running the model recursively, with updated node locations at each step. Both stress relaxation and stress modulation of growth in the component tissues were simulated. Thoracic growth of 20% with asymmetric growth of the ribs was simulated to give rib length asymmetries of 11%. similar to that observed in a previous study of patients with idiopathic scoliosis. This resulted in the model having a small thoracic scoliosis curvature convex toward the side of the longer ribs. Variations of the model which permitted free motion at the costo-vertebral joints or produced changes in the curvature of the posterior parts of the ribs resulted in axial rotation of the vertebrae similar to that observed clinically. The model supports the idea that growth asymmetry could initiate a small scoliosis during adolescence.     

Nonlinear simulation of tumor necrosis,neo-vascularization and tissue invasion via an adaptive finite-element/level-set method

We present a multi-scale computer simulator of cancer progression at the tumoral level, from avascular stage growth, through the transition from avascular to vascular growth (neo-vascularization), and into the later stages of growth and invasion of normal tissue. We use continuum scale reaction-diffusion equations for the growth component of the model, and a combined continuum-discrete model for the angiogenesis component. We use the level set method for describing complex topological changes observed during growth such as tumor splitting and reconnection, and capture of healthy tissue inside the tumor. We use an adaptive, unstructured finite element mesh that allows for finely resolving important regions of the computational domain such as the necrotic rim, the tumor interface and around the capillary sprouts. We present full nonlinear, two-dimensional simulations, showing the potential of our virtual cancer simulator. We use microphysical parameters characterizing malignant glioma cells, obtained from recent in vitro experiments from our lab and from clinical data, and provide insight into the mechanisms leading to infiltration of the brain by the cancer cells. The results indicate that diffusional instability of tumor mass growth and the complex interplay with the developing neo-vasculature may be powerful mechanisms for tissue invasion.     

The biological regulation of sea urchin larval skeletogenesis – From genes to biomineralized tissue

Biomineralization is the process in which soft organic tissues use minerals to produce shells, skeletons and teeth for various functions such as protection and physical support. The ability of the cells to control the time and place of crystal nucleation as well as crystal orientation and stiffness is far beyond the state-of-the art of human technologies. Thus, understanding the biological control of biomineralization will promote our understanding of embryo development as well as provide novel approaches for material engineering. Sea urchin larval skeletogenesis offers an excellent platform for functional analyses of both the molecular control system and mineral uptake and deposition. Here we describe the current understanding of the genetic, molecular and cellular processes that underlie sea urchin larval skeletogenesis. We portray the regulatory genes that define the specification of the skeletogenic cells and drive the various morphogenetic processes that occur in the skeletogenic lineage, including: epithelial to mesenchymal transition, cell migration, spicule cavity formation and mineral deposition into the spicule cavity. We describe recent characterizations of the size, motion and mineral concentration of the calcium-bearing vesicles in the skeletogenic cells. We review the distinct specification states within the skeletogenic lineage that drive localized skeletal growth at the tips of the spicules. Finally, we discuss the surprising similarity between the regulatory network and cellular processes that drive sea urchin skeletogenesis and those that control vertebrate vascularization. Overall, we illustrate the novel insights on the biological regulation and evolution of biomineralization, gained from studies of the sea urchin larval skeletogenesis.     

异质性干细胞增殖过程中的熵变化

干细胞在多细胞生物体内广泛存在,其增殖过程在生命体的生长、发育、衰老、组织修复过程中起着重要作用。正常组织中的细胞增殖过程受到严格的控制,干细胞的异常增殖与恶性肿瘤、肥胖症、再生障碍性贫血等疾病有密切关系。生命体内异质性细胞的增殖过程是复杂的动力系统行为,干细胞异常增殖过程伴随细胞的可塑性变化和细胞间相互作用的再平衡过程,如何对这一过程进行定量描述是重要的研究课题。本文构建包含细胞的增殖分化指标和异常增殖性指标异质性的干细胞增殖模型,通过所建立的模型研究由于微环境变化引起的细胞异常增殖过程的熵变化,建立不同增殖条件下的系统熵变化与宏观动力学和系统参数之间的关系。结果表明,在细胞微环境变化引起异常增殖和恢复的过程中,系统的熵与细胞数量之间存在对应关系,而与微环境变化的路径无关。 此外,熵对细胞数量的依赖关系在异常增殖和恢复阶段表现出不同的行为,显示了生物过程的微观不可逆性。本文从物理学的角度对细胞异常增殖过程中熵变化与细胞数量变化的动力学给出定量刻画,为定量描述异质性干细胞增殖过程给出新的研究思路。